III European Conference on Computational Mechanics: Solids, Structures and Coupled Problems in Engineering: Book of Abstracts 1402049943, 9781402049941

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III EUROPEAN CONFERENCE ON COMPUTATIONAL MECHANICS

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering: Book of Abstracts

Edited by

C. A. MOTA SOARES J. A. C. MARTINS H. C. RODRIGUES JORGE A. C. AMBRÓSIO C. A. B. PINA C. M. MOTA SOARES E. B. R. PEREIRA and

J. FOLGADO Instituto Superior Técnico, Lisbon, Portugal

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN-10 1-4020-4994-3 (HB) ISBN-13 978-1-4020-4994-1 (HB)

Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. www.springer.com

Printed on acid-free paper

All Rights Reserved © 2006 Springer No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed in the Netherlands.

Table of Contents Preface Plenary Lectures M. Bendsoe Computational Challenges for Multi-Physics Topology Optimization ...................1 J.A. Cottrell, A. Reali, Y. Bazilev, T.J.R. Hughes Computational Geometry and the Analysis of Solids and Structures......................2 E. Oñate, S.R. Idelsohn, M.A. Celigueta, R. Rossi Advances in the Particle Finite Element Method for Fluid-Structure Interaction Problems................................................................................................3 W. Schiehlen, R. Seifried Elastic and Plastic Impacts in Multibody Dynamics ...............................................4 A. Suleman, P. Moniz Active Aeroelastic Aircraft Structures.....................................................................5 Keynote Lectures O. Allix Multiscale Strategy for Solving Industrial Problem ................................................6 F. Armero, C. Zambrana Numerical Integration of the Nonlinear Dynamics of Elastoplastic Solids .............7 T. Belytschko, S. Wang Computational Methods for Dynamic Crack Propagation.......................................8 P. Bergan, K. Bakken, K.C. Thienel Analysis and Design of Sandwich Structures Made of Steel and Lightweight Concrete ..................................................................................................................9 R. Borja Multiscale Modeling of Pore Collapse Instability in High-Porosity Solids...........10 R. de Borst, M.A. Abellan, J. Réthoré Instabilities and Discontinuities in Two-Phase Media ..........................................11 C. Bottaso Towards Maneuvering Aeroelasticity - Progress in the Simulation of Large Fluid-Structure Interaction Problems......................................................12 v

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K. K. Choi, L. Du A Design Optimization Formulation for Problems with Random and Fuzzy Input Variables Using Performance Measure Approach ............................13 I. Doltsinis Strength of Porous Ceramics - Mechanical Testing and Numerical Modelling ....14 E. A. Dowel, K.C. Hall, J.P. Thomas, R.E. Kielb,M.A. Spiker, C.M. Denegri, Jr. Reduced Order Models in Unsteady Aerodynamic Models, Aeroelasticity and Molecular Dynamics.......................................................................................15 G. S. Dulikravich, H.R.B. Orlande, B.H. Dennis Inverse Engineering...............................................................................................16 J. Fish, W. Chen Multiscale Approaches for Bridging Discrete and Continuum Scales ..................17 P. L. George Adapative Mesh Generation in 3 Dimensions by Means of a Delaunay Based Method - Applications to Mechanical Problems.........................................18 C. Hellmich, K. Hofstetter, C. Kober Computational Micromechanics of Biological Materials: Bone and Wood ..........19 G. Holzapfel,C.T. Gasser, D. Kiousis Mechanobiology: Computation and Clinical Application .....................................20 B. Karihaloo, Q.Z. Xiao Recent Developments of Hybrid Crack Element: Determination of its Complete Displacement Field and Combination with XFEM...............................21 P. Ladevèze, P. Enjalbert, G. Puel, T. Romeuf Structural Model Validation and the Lack-of-Knowledge Theory........................22 J.V. Lemos Modeling of Historical Masonry with Discrete Elements .....................................23 G.-R. Liu, B.B.T. Kee A Regularized Strong-form Meshfree Method for Adaptive Analysis..................24 W. K. Liu Multiresolution Analysis for Material Design.......................................................25 R. Lackner, R. Blab, J. Eberhardsteiner, H. A. Mang Characterization and Multiscale Modeling of Asphalt – Recent Developments in Upscaling of Viscous and Strength Properties ..........................26 J.F. Dëu, W. Larbi, R. Ohayon Dissipative Interface Modeling for Vibroacoustic Problems – A New Symmetric Formulation.........................................................................................27

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M. Papadrakakis, N.D. Lagaros, M. Fragiadakis Seismic Design Procedures in the Framework of Evolutionary Based Structural Optimization .........................................................................................28 L.F. González, L.H. Damp, J. Periaux, K. Srinivas High-fidelity Multi-Criteria Aero-Structural Optimisation Using Hierarchical Parallel Evolutionary Algorithms .....................................................29 E. Ramm, C. Förster, M. Neumann, W.A. Wall Interaction of Shells and Membranes with Incompressible Flows ........................30 J. N. Reddy, R.A. Arciniega Nonlinear Analysis of Composite and FGM Shells Using Tensor-Based Shell Finite Elements.............................................................................................31 R. Rolfes, G. Ernst, D. Hartung, J. Teßmer Strength of Textile Composites – A Voxel Based Continuum Damage Mechanics Approach .............................................................................................32 B. Schrefler, F. Pesavento, D. Gawin, M. Wyrzykowski Concrete at Early Ages and Beyond: Numerical Model and Validation ...............33 G. Schueller Uncertainty & Reliability Analysis of Structural Dynamical Systems..................34 Z. Waszczyszyn, L. ZiemiaĔski Neural Networks: New Results and Prospects of Applications in Structural Engineering ...........................................................................................................35 T. Ekevid, P. Kettil, H. Lane, N. E. Wiberg Computational Railway Dynamics........................................................................36 P. Wriggers and M. Hain Micro-Meso-Macro Modelling of Composite Materials .......................................37 A Computational Methods Organizers: Pereira, E.

M. Baitsch, T. Sikiwat, D. Hartmann (ID-1667) An Object-Oriented Approach to High Order Finite Element Analysis of Three-Dimensional Continua ............................................................. 38 E. Boerner, D. Mueller-Hoeppe, S. Loehnert, P. Wriggers A Finite Element Formulation Based on the Theory of a Cosserat Point - Extension to Ogden Material ..................................................................... 39

(ID-1984)

J. Eom, B. Lee A Macro Tetrahedral Element with Vertex Rotational D.O.F.s ................ 40

(ID-2456)

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S. Fialko (ID-1659) Application of Aggregation Multilevel Iterative Solver to Problems of Structural Mechanics......................................................................... 41 A. Hansen, F. Rochinha Convergence Analysis of a Domain Decomposition Method with Augmented Lagrangian Formulation..................................................................... 42

(ID-2357)

Y. Kholodov, A. Kholodov, N. Kovshov, S. Simakov, D. Severov, A. Bordonos, A. Bapayev (ID-2279) Computational Models on Graphs for Nonlinear Hyperbolic and Parabolic System of Equations .............................................................................. 43 Ü. Lepik Haar Wavelet Method for Solving Integral Equations and Evolution Equations .............................................................................................. 44

(ID-1478)

H. Ostad-Hossein, S. Mohammadi A New Approach for Elimination of Dissipation and Dispersion Errors in Particle Methods..................................................................................... 45

(ID-2368)

Z. Pawlak, J. Rakowski Solution of Stability Problem of Infinite Plate Strips................................ 46

(ID-2155)

R. Robalo, M. Coimbra, A. Rodrigues Modeling Time-Dependent Partial Equations with Moving Boundaries by the Moving Finite Element Method............................................... 47

(ID-1838)

D. Rypl (ID-2072)

Discretization of Three-Dimensional Aggregate Particles ........................ 48

K. Sato Bending of an Elliptical Plate on Elastic Foundation and Under the Combined Action of Lateral Load and In-Plane Force.................................... 49

(ID-1733)

M. Silva, E. Pereira A Modal Analysis Approach Using an Hybrid-Mixed Formulation to Solve 2D Elastodynamic Problems ................................................................... 50

(ID-2560)

L. Veiga, J. Niiranen, R. Stenberg A New Finite Element Method for Kirchhoff Plates................................. 51

(ID-1970)

E. Zieniuk, A. Boltuc Non-Element Method for Solving 2D Boundary Problems Defined on Connected Polygonal Domains Described by Navier Equation.......... 52

(ID-2406)

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B Computational Solid Mechanics Organizers: Martins, J.

S. Aitalyev, R. Baimakhan, Z. Masanov, N. Kurmanbekkyzy, G. Ylyasova (ID-2497) Computational Methods of Anisotropic Massif Mechanics Under Different Types of External Actions ..................................................................... 53 N. Alexandrova, P. Real Effect of Plastic Anisotropy on the Size of Elastic-Plastic Boundary in a Rotating Disk Problem................................................................... 54

(ID-1019)

E. Artioli, F. Auricchio, L. Veiga Numerical Testing on Return Map Algorithms for Von-Mises Plasticity with Nonlinear Hardening ..................................................................... 55

(ID-2068)

M. Asik (ID-2250)

A Model for the Analysis of Plates on a Layered Elastic Medium ........... 56

S. Benke Modeling of Solid State Transformations Using a Phase Field Model with Transformation Plasticity ................................................................... 57

(ID-1817)

S. Bosiakov, M. Zhuravkov Computer Modeling of Three-Dimensional Wave Movements in Anisotropic Elastic Environments ......................................................................... 58

(ID-1574)

R. Cardoso, V. Silva, H. Varum Visco-Elastic Regularization and Strain Softening ................................... 59

(ID-2411)

J. Cela, A. Piriz, M. Serna Numerical Simulations of the Rayleigh-Taylor Instability in Accelerated Solids ................................................................................................. 60

(ID-1108)

B. Diouf, F. Houdaigui, S. Poortmans, B. Verlinden, A. Habraken A Phenomenological Model to Simulate Mechanical Tests on Ultrafine-Grained Aluminum Produced by ECAE................................................ 61

(ID-2439)

E. Emmrich, O. Weckner The Peridynamic Equation of Motion in Non-Local Elasticity Theory ................................................................................................................... 62

(ID-1797)

A. Eraslan, E. Arslan Numerical Solution of Partially Plastic Curved Beam Problem................ 63

(ID-1232)

J. Fernandes, A. Alvim, R. Rocha A Proposal of Strain-Gage Rosette for Measurement Residual Stress Around a Circular Hole in a Plate with Circular Hole ................................ 64 (ID-1610)

J. Gómez, J. Royo, F. Martínez, E. Liarte, M. Jiménez (ID-1871) Prediction of Dynamic Stiffness of Filled Rubber Mounts ....................... 65

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P. Grammenoudis, C. Tsakmakis (ID-1546) Size Effects in Finite Deformation Micropolar Plasticity ......................... 66 M. Grekov (ID-1888) An Application of a Boundary Perturbation Method to Some Problems of Elasticity............................................................................................ 67 E. Grosu, I. Harari Spatial Stabilization of Semidiscrete Elastodynamics............................... 68

(ID-1005)

M. Grymer, M. Ekh, K. Runesson, T. Svedberg (ID-1931) Modeling the Grain Size Effect Using Gradient Hardening and Damage in Crystal (Visco) Plasticity .................................................................... 69 T. Hata Stress-Focusing Effect Following Dynamically Transforming Strains in a Spherical Zirconia Inclusion............................................................... 70

(ID-1208)

E. Heikkola, S. Mönkölä, A. Pennanen, T. Rossi Controllability Method for the Solution of Linear Elastic Wave Equations............................................................................................................... 71

(ID-1469)

C. Husson, J. Richeton, S. Ahzi Development of a Flow Stress Model From Metals Using the Strain Rate/ Temperature Superposition Principle ................................................ 72

(ID-1590)

T. Kato, M. Kawahara (ID-1827) Analysis of Elastic Body Using Kalman Filter Finite Element Method .................................................................................................................. 73 A. Lew, A. Eick Discontinuous Galerkin Methods for Nonlinear Elasticity ....................... 74

(ID-2100)

S. Marguet, P. Rozycki, L. Gornet A Rate Dependent Constitutive Model for Carbon-Fibre / EpoxyMatrix Woven Fabrics Submitted to Dynamic Loadings ...................................... 75

(ID-1870)

V. Matveyenko Solution of Viscoelasticity Problems Using Special Forms of Elastic Solutions .................................................................................................... 76

(ID-1798)

M. Mazdziarz Finite Elements Method Analysis of Influence of Contact Phenomena on Structure-Subsoil Interaction ........................................................ 77

(ID-1832)

M. Mizuno, Y. Sanomura Simulation of Inelastic Deformation of Polyethylene in Multiaxial State of Stress by Viscoelastic Constitutive Equtaion ........................................... 78

(ID-1286)

I. Pavlov Modeling of Granular Media by the 2D Discrete Lattice.......................... 79

(ID-1982)

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M. Rabahallah, B. Bacroix, S. Bouvier, T. Balan (ID-2510) Crystal Plasticity Based Identification of Anisotropic Strain Rate Potentials for Sheet Metal Forming Simulation .................................................... 80 G. Romano, M. Diaco, R. Barretta A Geometric Approach to the Algorithmic Tangent Stiffness .................. 81

(ID-2044)

V. Sadovskii, O. Sadovskaya Parallel Computation of 3D Problems of the Dynamics of ElasticPlastic Granular Material Under Small Strains ..................................................... 82

(ID-1277)

Z. Uthman, H. Askes A Hyperelastodynamic Ale Formulation Based on Spatial and Material Forces...................................................................................................... 83

(ID-1328)

Y. Zhuk Monoharmonic Approach to Investigation of Heat Generation in the Viscoplastic Solids Under Harmonic Loading ................................................ 84

(ID-1073)

C Coupled Problems Organizers: Rodrigues, H.

S. Bargmann, P. Steinmann (ID-1021) A Continuous Galerkin Finite Element Method for Thermoelasticity Without Energy Dissipation ...................................................... 85 J. Blaszczuk, Z. Domanski The Model Coupling Liquid Bridge Between Ellipsoidal Grains ............. 86

(ID-2171)

M. Brehm, C. Bucher Reliability of Wavelet Packet System Identification................................. 87

(ID-1652)

L. Ecsi, P. Elesztos An Attempt to Simulate More Precisely the Behavior of a Solid Body Using New Energy Conservation Equation ................................................. 88

(ID-1112)

J. Gawinecki Mathematical Aspects of the Initial-Boundary Value Problems in Nonlinear Thermoelasticity of Simple and Non-Simple Materials ....................... 89

(ID-1361)

M. Guerich, S. Chaabane Effect of Parameter Uncertainties on a Vibro-Acoustic Design................ 90

(ID-2610)

G. Khodabakhshi, V. Nassehi, L. Shojai, R. Wakeman A Numerical Method for Solid-Liquid Interaction ................................... 91

(ID-2533)

A. Kiselev, O. Nekhaeva, A. Privalsky Computational Simulation of Irreversible Deforming and Fracture of Damageable Solids and Structures .................................................................... 92

(ID-1121)

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M. Kropac, J. Murin (ID-2469) Solution of the Coupled Light-Mechanical Problems ............................... 93 I. Kurzhöfer, J. Schröder, H. Romanowski Simulation of the Ferroelectric Hysteresis Using a Hybrid Finite Element Formulation ............................................................................................. 94

(ID-1322)

R. Lanrivain, L. Silva, T. Coupez A Two-Phase Numerical Modelling of the Liquid Solid Transition in Polymer Processing ........................................................................................... 95

(ID-1338)

C. Leppert, D. Dinkler A Two-Phase Model for Granular Flows Applied to Avalanches............. 96

(ID-2596)

L. Margetts, I. Smith, J. Leng Parallel 3D Finite Element Analysis of Coupled Problems....................... 97

(ID-2218)

P. Porta, C. Vega Numerical Simulation of Rubber Curing Process with Application to Bladders Manufacture ....................................................................................... 98

(ID-1360)

E. Stupak, R. Kacianauskas, A. Dementjev, A. Jovaisa Coupled Finite Element Analysis of Composite Laser Rods Thermal Characteristics Under Longitudinal Diode Pumping .............................. 99

(ID-1475)

D Computational Structural Mechanics Organizers: Ambrósio, J.

T. Akis, T. Tokdemir, C. Yilmaz (ID-2480) On the Modeling of Nonplanar Shear Walls in Shear Wall-Frame Building Structures.............................................................................................. 100 F. Alamo, H. Weber, H. Espinoza Directional Drillstring Dynamics ............................................................ 101

(ID-2624)

M. Alinia, A. Rahai, S. Kazemi Mvm Energy Method for Buckling Analysis of Tapered Plates ............. 102

(ID-2651)

A. Alvarenga, R. Silveira Considerations on Advanced Analysis of Steel Portal Frames ............... 103

(ID-2119)

N. Azevedo, J. Lemos, J. Almeida A Discrete Element Model for the Fracture Analysis of Reinforced Concrete............................................................................................ 104

(ID-2569)

A. Baptista Analytical Criteria for the Evaluation of the Internal Forces at the Elastic and Plastic Limit States of Lozenge and Triangular Cross-Sections ....... 105

(ID-2700)

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M. Benachour, A. Hadjoui, M. Benguediab, N. Benachour, F. Hadjoui (ID-2252) Comparative Study of Aluminum Alloy Plate 2024/7050 Under the Effect of Internal Damping............................................................................ 106 A. Correia, F. Virtuoso Nonlinear Analysis of Space Frames ...................................................... 107

(ID-2648)

A. Davaran, A. Kashefi, S. Amiri (ID-1504) Investigation of Shear Wall Behavior with Composite Boundary Elements .............................................................................................................. 108 M. Fard, S. Amiri, A. Kashefi An Investigation on Dynamic Behavior of Shear Walls on Flexible Foundation............................................................................................. 109

(ID-1096)

T. Fiedler, A. Öchsner, J. Grácio Influence of the Morphology of Adhesive Joining on the Mechanical Properties of Periodic Metal Hollow-Sphere-Structures.................. 110

(ID-1084)

H. Figueiredo, R. Calçada, R. Delgado Dynamic Behaviour of a Composite Twin Girder Bridge in a High Speed Interoperable Line..................................................................................... 111

(ID-2620)

A. Foces, J. Garrido, A. Moreno A Finite Element Model for Beam to Column Bolted Connections with Semi Rigid Behaviour ................................................................................. 112

(ID-1702)

G. Garcea, A. Madeo Rational Strain Measures - the Implicit Corotational Method................. 113

(ID-2353)

V. Kulbach, J. Idnurm (ID-1662) Discrete and Continuous Analysis of Different Cable Structures ........... 114 Q. Li, V. Iu Three-Dimensional Vibration Analysis of Crystal Plates Via Ritz Method ................................................................................................................ 115

(ID-1072)

G. Lykidis, K. Spiliopoulos A 3D Solid Finite Element for Reinforced Concrete Analysis Allowing Slippage of Reinforcement .................................................................. 116

(ID-2233)

M. Lyly, J. Niiranen, R. Stenberg Some New Results on Mitc Plate Elements ............................................ 117

(ID-2000)

J. Neto, A. Assan (ID-1341) Nonlinear Analysis of Reinforced Concrete Beams Considering the Slip Between Steel and Concrete................................................................... 118 J. Rijn Novel Semi-Analytical Methodology to Determine Model Parameters for a Simple Finite Element Bolt Model........................................... 119

(ID-2276)

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W. Rust, J. Overberg (ID-1803) Accounting for Fuselage Instabilities in the Coarse Model of an Aircraft Fuselage by Means of a Material Law ................................................... 120 J. Sosa, D. Owen, E. Neto, N. Petrinic (ID-1530) Modelling of Reinforced Materials by a Subcycling Algorithm ............. 121 K. Spiliopoulos, T. Patsios (ID-2329) Limit Analysis of Cable-Tied Structures................................................. 122 R. Steenbergen, J. Blaauwendraad Smart Super Elements in Slender Structures Subjected to Wind ............ 123

(ID-1516)

B. Trogrlic, A. Mihanovic, Z. Nikolic Nonlinear Analysis of Space R/C Frames with Non-Uniform Torsion ................................................................................................................ 124

(ID-2121)

G. Turvey, P. Wang An Fe Analysis of the Stresses in Pultruded GRP Single-Bolt Tension Joints and Their Implications for Joint Design ...................................... 125

(ID-1921)

J. Veiga, A. Henriques, J. Delgado An Efficient Evaluation of Structural Safety Applying Perturbation Techniques...................................................................................... 126

(ID-2688)

G. Yaoqing, L. Ke Semi-Analytical Analysis of Super Tall Building Bundled-Tube Structures............................................................................................................. 127

(ID-1214)

N. Zivaljic, A. Mihanovic, B. Trogrlic Large Displacements in Nonlinear Numerical Analyses for Cable Structures............................................................................................................. 128

(ID-1892)

E Industrial Applications Organizers: Pina, C.

V. Abadjiev, D. Petrova, E. Abadjieva (ID-1325) Mathematical Modeling for Synthesis and Design of NonOrthogonal Worm Gears with a Straight-Line Tooth Contact............................. 129 N. Ahmed, A. Mitrofanov, V. Silberschmidt, V. Babitsky Computational Modeling of Ultrasonically Assisted Turning................. 130

(ID-1675)

G. Anjos, R. Cunha, P. Kuklik, B. Miroslav Numerical Evaluation of Bored Piles in Tropical Soils by Means of the Geotechnical Engineering “Geo4” Fine Software..................................... 131

(ID-1694)

B. Atahanovich, B. Gayratovich Kinematics and Force Interaction of Screw Shaft with Variable Screw Course....................................................................................................... 132

(ID-1176)

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J. Barglik, B. Ulrych (ID-2343) Optimal Construction of the Thermo-Elastic Actuator ........................... 133 M. Barros, C. Oliveira (ID-2490) Determination of Moment-Curvature Diagrams and MomentDeflection Curves in Reinforced Concrete Beams .............................................. 134 P. Berke, T. Massart Numerical Simulation of the Nanoindentation Experiment: Sensitivity Analysis of the Experimental Parameters.......................................... 135

(ID-2274)

M. Chuda-Kowalska, A. Garstecki, Z. Pozorski Numerical Evaluation of Wrinkling Stress in Sandwich Panels ............. 136

(ID-1599)

I. Conde, M. Jiménez, J. Bielsa, E. Liarte, M. Laspalas Application of Fea As a Predictive Tool in the Corrugated Paperboard Industry ............................................................................................ 137

(ID-1881)

J. Dib, F. Bilteryst, J. Batoz, I. Lewon Implementation of 3D Homogenization Techniques for the Thermo-Elastic FEM Analysis of Brazed Plate-Fin Heat Exchangers ................ 138

(ID-2631)

R. Hoffmann Hierarchical Treecode for Optimized Collision Checking in Dem Simulations – Application on Electrophotographic Toner Simulations .............. 139

(ID-1003)

J. Kazanecki, Z. Pater, J. Bartnicki 3D FEM Analysis of Basic Process Parameters in Rotary Piercing Mill ...................................................................................................................... 140

(ID-2559)

M. Khoshravan, J. Shahimehr Numerical Evaluation of the Influence of Stiffener Rings on the Critical Buckling Pressure of the Vessels............................................................ 141

(ID-2239)

G. Kiziltas Topology Optimization and Fabrication of Multi-Material Dielectrics for Antenna Performance Improvements .......................................... 142

(ID-1754)

P. Klinge Reduction Method Independent Substructure Synthesis ......................... 143

(ID-1882)

M. Lancini, A. Magalini, D. Vetturi (ID-1321) Discrete Models for the Simulation of Rubber Components Dynamics............................................................................................................. 144 A. Liao, H. Zhang, C. Wu Contact Analysis of Impeller-Shaft Assembly and Reasonably Designing the Amount Interference of Turbocompressors.................................. 145

(ID-1102)

Y. Sheng, C. Lawrence, B. Briscoe Study of the Stress Induced Granular Consolidation Process by 3D DEM Simulation............................................................................................ 146

(ID-1483)

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J. Shiau, C. Smith (ID-2577) Numerical Analysis of Passive Earth Pressures with Interfaces ............. 147 W. Wang, J. Wang, W. Shen, Z. Xu, W. Fan (ID-1273) Three-Dimensional Finite Element Analysis of a Multi-Propped Deep Excavation in Shanghai Soft Deposit......................................................... 148 M. Zhuravkov, S. Bosiakov, S. Pronckevich The Computer Analysis of the Temperature Fields Arising in Bearing Node at Rotation of a Rotor ................................................................... 149 (ID-1579)

MS.01 Acoustics Structural Interactions Organizers:Tadeu, A.

V. Decouvreur, P. Ladevèze, P. Bouillard (ID-2023) Updating 3D Acoustic Models with the Constitutive Law Error Method. a Two Step Approach for Absorbing Material Characterization........... 150 J. García-Andujar, L. Fritz, J. López-Díez Analysis of Fluid-Structure Coupling by Statistical Energy Analysis............................................................................................................... 151 (ID-2109)

K. Ito, J. Toivanen (ID-1608) Efficient Iterative Solution of Time-Harmonic Scattering by Objects in Layered Fluid ..................................................................................... 152 B. Neuhierl, E. Rank Computational Aeroacoustics by Coupling the Finite-Element and the Lattice-Boltzmann-Method............................................................................ 153

(ID-1186)

A. Panteghini, F. Genna, E. Piana Analysis of a Perforated Panel for the Correction of Low Frequency Resonances in Domestic Rooms........................................................ 154

(ID-1132)

A. Pereira, A. Tadeu Sound Insulation Provided by a Multi-Layer System Containing a Heterogeneity: a BEM Approach ........................................................................ 155

(ID-1252)

B. Pluymers, W. Desmet, D. Vandepitte, P. Sas An Efficient Wave Based Method for Steady-State VibroAcoustic Transmission Calculations ................................................................... 156

(ID-1864)

P. Santos, A. Tadeu Modeling Sound Radiation by Structures Caused by a Ground Impact Load: a BEM Approach........................................................................... 157

(ID-2270)

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MS.02 Smart Structures and Materials Organizers: Suleman, A., Benjeddou, A.

I. Arias, S. Serebrinsky, M. Ortiz (ID-1763) Cohesive Model of Electromechanical Fatigue for Ferroelectric Materials and Structures ...................................................................................... 158 H. Atzrodt, S. Herold, D. Mayer (ID-1500) Simulation of Active Systems in a NVH Full Car Model ....................... 159 F. Auricchio, L. Petrini, A. Reali (ID-1911) Toward an Exhaustive Modeling of the Macroscopic Behaviour of Shape Memory Alloys......................................................................................... 160 A. Benjeddou, J. Ranger Vibration Damping Using Resonant Shunted Shear-Mode Piezoceramics...................................................................................................... 161

(ID-1207)

I. Figueiredo, G. Stadler Optimal Control of Piezoelectric Anisotropic Plates .............................. 162

(ID-2619)

W. Gambin, A. Zarzycki Residual Internal Forces in Stiffened Thermal-Bimorph Actuator After Forming Process......................................................................................... 163 (ID-1559)

A. Gomes, A. Suleman (ID-1523) Spectral Level Set Methodology in the Optimal Design of Adaptive Aeroelastic Structures .......................................................................... 164 D. Marinova, D. Lukarski, G. Stavroulakis, E. Zacharenakis Nondestructive Identification of Defects for Smart Plates in Bending Using Genetic Algorithms .................................................................... 165

(ID-2195)

J. Pereira, M. Pacheco, P. Pacheco, R. Aguiar, M. Savi (ID-2097) Modeling Shape Memory Alloy Plane Truss Structures Using the Finite Element Method ........................................................................................ 166 G. Pirge, N. Kiliç, O. Uçan, S. Altintas Evaluation of Nimnga Magnetic Shape Memory Alloys Using Cellular Neural Networks.................................................................................... 167

(ID-2575)

C. Ramos, R. Oliveira, R. Campilho, A. Marques Modelling of Fibre Bragg Grating Sensor Plates .................................... 168

(ID-2645)

M. Trindade, A. Benjeddou (ID-1309) Refined Finite Element Model for Vibration Analysis of Sandwich Beams with Shear Piezoelectric Actuators and Sensors ..................... 169 S. Ueda, H. Kondo Thermoelectromechanical Response of a Parallel Crack in a Functionally Graded Piezoelectric Strip.............................................................. 170

(ID-1724)

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Y. Uetsuji, E. Nakamachi (ID-1833) Multi-Scale Finite Element Modeling of Piezoelectric Materials by a Crystallographic Homogenization Method.................................................. 171 H. Uyanik, Z. Mecitoglu Vibration Control of a Laminated Composite Plate Subjected to Blast Loading ...................................................................................................... 172

(ID-1490)

C. Vasques, R. Moreira, J. Rodrigues Experimental Identification of GHM and ADF Parameters for Viscoelastic Damping Modeling ......................................................................... 173

(ID-1960)

L. Wang, R. Melnik Numerical Aspects of Modelling Thermo-Mechanical Wave Propagation with Phase Transformations ............................................................ 174

(ID-1438)

A. Zabihollah, R. Ganesan, R. Sedaghati Analysis and Design Optimization of Smart Laminated Composite Beams Using Layerwise Theory.......................................................................... 175

(ID-2191)

MS.03 Asphalt Mechanics and Pavement Engineering Organizers: Lackner, R., Blab, R.

H. Benedetto, B. Delaporte, C. Sauzéat, M. Neifar (ID-2475) A Thermo-Viscoplastic Model for Bituminous Materials....................... 176 J. Croll From Asphalt to the Arctis: New Insights Into ThermoMechanical Ratchetting Processes....................................................................... 177

(ID-1149)

A. Holanda, E. Junior, T. Araujo, L. Melo, F. Junior, J. Soares An Object-Oriented System for Finite Element Analysis of Pavements............................................................................................................ 178

(ID-2557)

B. Lenhof, P. Kettil, K. Runesson, N. Wiberg On the Treatment of Convective Terms in Coupled HydroMechanics for Porous Media Subjected to Dynamic Loading ............................ 179

(ID-1804)

A. Molenaar Asphalt Mechanics, a Key Tool for Improved Pavement Performance Predictions...................................................................................... 180

(ID-1187)

MS.04 Biomechanical Simulations Organizers: Eriksson, A.

P. Blanco, I. Larrabide, S. Urquiza, R. Feijóo (ID-1858) Sensitivity of Blood Flow Patterns to the Constitutive Law of the Fluid .................................................................................................................... 181

Table of Contents

xix

A. Completo, F. Fonseca, J. Simoes (ID-2035) The Influence of Stem Design on Strains and Micromotion in Revision Total Knee Arthroplasty: Finite Element Analysis .............................. 182 A. Eriksson Optimization of Targeted Movements .................................................... 183

(ID-1192)

B. Heidari, D. Fitzpatrick, D. Mccormack, K. Synnott (ID-1697) Persistence of Axial Rotation in Idiopathic Scoliosis Due to the Structural Changes of the Intervertebral Disc ..................................................... 184 H. Iwase, R. Himeno Numerical Simulation of Hemodynamcs in a Cerebral Artery ............... 185

(ID-1571)

A. John, M. Mazdziarz, J. Rojek, J. Telega, P. Maldyk Analysis of Some Contact Problems in Human Joints After Arthroplasty......................................................................................................... 186

(ID-1851)

P. Khayyer, A. Zolghadrasli, F. Daneshmand, A. Najafi Cardiovascular Disease Diagnosis Before Birth by Means of Chaotic Analysis on the Heart Rate Signal.......................................................... 187 (ID-1343)

I. Larrabide, R. Feijóo, E. Taroco, A. Novotny (ID-1845) Configurational Derivative As a Tool for Image Segmentation.............. 188 G. Link, M. Kaltenbacher, R. Lerch Numerical Simulations to Analyze and Optimize the Human Substitute Voice .................................................................................................. 189

(ID-1670)

C. Nabais, R. Guedes, J. Simoes The Thaw Time of Frozen Cancellous Bone for Mechanical Testing................................................................................................................. 190

(ID-1843)

T. Olsson, J. Martins Modeling of Passive Behavior of Soft Tissues Including Viscosity and Damage......................................................................................................... 191

(ID-2414)

M. Racila, J. Crolet Nano and Macro Structure of Cortical Bone: Numerical Investigations....................................................................................................... 192

(ID-2288)

S. Rues, H. Schindler, K. Schweizerhof, J. Lenz Calculation of Muscle and Joint Forces in the Masticatory System........ 193

(ID-2403)

V. Vondrák, J. Rasmussen, M. Damsgaard, Z. Dostál The Algorithms of Mathematical Programming in Muscle Recruitment and Muscle Wrapping Problems ..................................................... 194

(ID-2177)

C. Yu, Y. Wang, Y. Liu, X. Sun Three-Dimensional Numerical Simulation of Airflow and Vibration Analysis for Upper Airway of Humans............................................... 195

(ID-2011)

xx

Table of Contents

MS.05 Computational Biomechanics Organizers: Rodrigues, H.

D. Balzani, J. Schröder, D. Gross (ID-1317) Computer Simulation of Anisotropic Damage and Residual Stresses in Atherosclerotic Arteries..................................................................... 196 J. Folgado, R. Andrade, P. Fernandes Computational Study on Stability and Bone Remodeling for a Hip Replacement Using a "Minimal Invasive" Femoral Stem ................................... 197

(ID-2493)

A. Fritsch, L. Dormieux, C. Hellmich Porous Polycrystals Built Up by Uniformly and Axisymmetrically Oriented Needles: Homogenization of Elastic Properties ................................... 198

(ID-1549)

U. Görke, H. Günther, M. Wimmer A Poroviscoelastic Overlay Model for Finite Element Analyses of Articular Cartilage at Large Strains..................................................................... 199

(ID-1924)

R. Guimaraes, S. Taylor, G. Blunn A Finite Element Study of Strain Distribution in an Instrumented Knee Prosthesis for Full Force Measurement in Vivo......................................... 200 (ID-2341)

I. Larrabide, R. Feijóo, E. Taroco, A. Novotny (ID-1850) Topological Derivative Applied to Image Enhancement ........................ 201 J. Mcgarry, A. Pathak, L. Valdevit, A. Evans, P. Mchugh, R. Mcmeeking Determination of Contractile Forces Generated by Actin Fibre Networks ............................................................................................................. 202

(ID-2219)

R. Ruben, P. Fernandes, J. Folgado, H. Rodrigues Hip Prosthesis Design Using a Multi-Criteria Formulation .................... 203

(ID-2476)

S. Simakov, A. Kholodov, Y. Kholodov, A. Nadolskiy, A. Shushlebin Global Dynamical Model of the Cardiovascular System ........................ 204

(ID-1464)

S. Simakov, A. Kholodov, Y. Kholodov, A. Nadolskiy, A. Shushlebin Computational Study of the Vibrating Disturbances to the Lung Function............................................................................................................... 205

(ID-1467)

B. Simon, P. Rigby, T. Newberg, R. Park, S. Williams Abaqus-Based, Coupled Porohyperelastic Transport Finite Element Models for Soft Hydrated Biological Structures ................................... 206

(ID-1736)

J. Soares, J. Moore, K. Rajagopal Theoretical Modeling of Cyclically Loaded, Biodegradable Cylinders ............................................................................................................. 207

(ID-1979)

K. Subramani, M. Oliveira, J. Simoes Validation of a Non-Linear Wear Model for UHMWPE ........................ 208

(ID-1699)

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xxi

L. Zach, S. Konvickova, P. Ruzicka (ID-1782) FEA of Human Knee Joint Replacement Using Real Bone Models ....... 209 MS.06 Boundary Elements Organizers: Leitão, V.

G. Dziatkiewicz, P. Fedelinski (ID-1318) Subregion Boundary Element Method for Piezoelectric Structures........ 210 M. Guminiak, R. Sygulski Vibrations of System of Plates Immersed in Fluid by Bem .................... 211

(ID-1260)

S. Gupta, G. Degrande, H. Chebli, D. Clouteau, M. Hussein, H. Hunt A Coupled Periodic FE-BE Model for Ground-Borne Vibrations From Underground Railways .............................................................................. 212

(ID-2365)

A. Iban, J. Garcia-Teran, I. Rico On the Application of the Bem to Rubber-Elastic Materials................... 213

(ID-2163)

J. Rungamornrat Modeling of Darcy's Flow in Generally Anisotropic Porous Media Containing Discontinuity Surface by SGBEM-FEM Coupling........................... 214

(ID-1568)

E. Sapountzakis, V. Protonotariou A Displacement Solution to Transverse Shear Loading of Beams by BEM ............................................................................................................... 215

(ID-1002)

M. Schanz, T. Rüberg Non-Conforming Coupled Time Domain Boundary Element Analysis............................................................................................................... 216

(ID-2103)

MS.08 Advanced Composites Organizers: Reddy, J. N., Mota Soares, C.M., Benjeddou, A.

F. Ashida, S. Sakata, K. Matsumoto (ID-1749) Control of Thermal Stress in a Piezoelectric Composite Disk by a Stepwise Applied Electric Potential Distribution ................................................ 217 J. Baucom, M. Qidwai, J. Thomas Mitigation of Free-Edge Effects by Meso-Scale Structuring .................. 218

(ID-2372)

J. Bekuit, D. Oguamanam, O. Damisa A Quasi-2D Finite Element Formulation for Static and Dynamic Analysis of Sandwich Beams .............................................................................. 219

(ID-2151)

J. Belinha, L. Dinis A Numerical Comparison of Distinct Meshless Methods for the Analysis of Composite Laminates....................................................................... 220

(ID-1521)

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Table of Contents

A. Benjeddou, S. Vijayakumar, I. Tawfiq (ID-1178) A New Damage Identification and Quantification Indicator for Piezoelectric Advanced Composites ................................................................... 221 A. Blom, S. Setoodeh, J. Hol, Z. Gürdal Design of Variable-Stiffness Conical Shells for Maximum Fundamental Frequency ...................................................................................... 222

(ID-1941)

P. Camanho, J. Mayugo, P. Maimí, C. Dávila A Micromechanics Based Damage Model for the Strength Prediction of Composite Laminates .................................................................... 223

(ID-1661)

J. Cardoso, N. Benedito, A. Valido Finite Element Analysis of Geometrically Nonlinear Thin-Walled Composite Laminated Beams.............................................................................. 224

(ID-2555)

J. Cardoso, A. Valido Design Sensitivity Analysis of Composite Thin-Walled Profiles Including Torsion and Shear Warping................................................................. 225

(ID-2626)

A. Carpentier, J. Barrau, L. Michel, S. Grihon Buckling Optimisation of Composite Panels Via Lay-Up Tables........... 226

(ID-2530)

J. Cognard, R. Créac'Hcadec Numerical Approach for the Design of Adhesively-Bonded Assemblies .......................................................................................................... 227

(ID-1335)

G. Dvorak, Y. Bahei-El-Din Enhancement of Blast Resistance of Sandwich Plates ............................ 228

(ID-2689)

C. Friebel, I. Doghri, V. Legat Mechanics and Acoustics of Viscoelastic Composites by a MicroMacro Mean-Field Approach .............................................................................. 229

(ID-1337)

P. Fuschi, A. Pisano Numerical Evaluation of Upper and Lower Bounds to the Collapse Limit Load for Composite Laminates................................................... 230

(ID-2346)

K. Kalnins, J. Auzins, R. Rikards Material Degradation Assessment for Stiffened Composite Shells Using Metamodelling Approach ......................................................................... 231

(ID-2049)

M. Karama, K. Afaq, S. Mistou A New Model for the Behaviour of the Multi-Layer Material Interfaces ............................................................................................................. 232

(ID-1981)

L. Kärger, J. Baaran, J. Teßmer Fe-Tool Codac for an Efficient Simulation of Low-Velocity Impacts on Composite Sandwich Structures ....................................................... 233

(ID-1101)

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xxiii

J. Kato, A. Lipka, E. Ramm (ID-2603) Preliminary Investigation for Optimization of Fiber Reinforced Cementitious Composite Structures .................................................................... 234 P. Kere, M. Lyly On Post-Buckling Analysis and Experimental Correlation of Cylindrical Composite Shells .............................................................................. 235

(ID-1801)

M. Laspalas, S. Maynar, C. Crespo, M. Jiménez, B. García Application of Failure Criteria to Short Fiber Reinforced Composites and Experimental Validation ........................................................... 236

(ID-1891)

C. Lopes, Z. Gürdal, P. Camanho Tow-Placed, Variable-Stiffness Composite Panels: Damage Tolerance Improvements Over Traditional Straight-Fibre Laminates................. 237

(ID-2153)

F. Melo, R. Moreira, J. Rodrigues A Mixed-Formulation Four-Node Rectangular Element in the Modeling of Laminate Composite Beam Stuctures............................................. 238

(ID-2052)

T. Messager, P. Chauchot, B. Bigourdan Optimal Design of Stiffened Composite Underwater Hulls .................... 239

(ID-1333)

J. Moita, C. Soares, C. Soares Higher Order Model for Analysis of Magneto-Electro-Elastic Plates ................................................................................................................... 240

(ID-1115)

F. Moleiro, C. Soares, C. Soares, J. Reddy Mixed Least-Squares Finite Element Model for the Static Analysis of Laminated Composite Plates ............................................................ 241

(ID-1508)

M. Nader, H. Garssen, H. Irschik Applications of Distributed Piezoelectric Electrode Patches for Active Noise and Vibration Control.................................................................... 242

(ID-1289)

E. Ng, A. Suleman Computational Elastoplatic Modeling of Multi-Phase FiberReinforced Composites ....................................................................................... 243

(ID-1554)

K. Niessen, J. Moreno, R. Gadow Evolution and Analysis of Stresses in Thixoforged Metal Matrix Composites .......................................................................................................... 244

(ID-2236)

M. Pietrzakowski Piezoelectric Control of Composite Plate Vibration: Effect of Electric Field Distribution ................................................................................... 245

(ID-1222)

H. Santos, C. Soares, C. Soares, J. Reddy A Finite Element Model for the Analysis of 3D Axisymmetric Laminated Shells with Embedded Piezoelectric Sensors and Actuators ............. 246

(ID-1082)

xxiv Table of Contents

J. Santos, H. Lopes, M. Vaz, C. Soares, C. Soares, M. Freitas (ID-1239) Damage Localization in Laminated Composite Plates Using Double Pulse-Electronic Holographic Interferometry ......................................... 247 E. Sapountzakis Shear Deformation Effect in Nonlinear Analysis of Spatial Composite Beams in Variable Axial Loading by BEM ...................................... 248 (ID-1139)

C. Schuecker, H. Pettermann (ID-1759) Constitutive Ply Damage Modeling, FEM Implementation, and Analyses of Laminated Structures....................................................................... 249 W. Wagner, C. Balzani Simulation of Delamination in Stringer Stiffened FiberReinforced Composite Shells .............................................................................. 250

(ID-1204)

MS.10 Computational Fracture Mechanics Organizers: Leung, A.

A. Andreev (ID-1632) Development of Methods of Numerical Solution of Singular Integro-Differential Equations for Solid Mechanics Problems ........................... 251 I. Arias, J. Knap, V. Chalivendra, S. Hong, M. Ortiz, A. Rosakis Validation of Large Scale Simulations of Dynamic Fracture.................. 252

(ID-2247)

E. Giner, A. Vercher, O. González, J. Tarancón, F. Fuenmayor Crack Growth in Fretting-Fatigue Problems Using the Extended Finite Element Method ........................................................................................ 253

(ID-2196)

S. Jox, P. Dumstorff, G. Meschke Aspects of Crack Propagation and Hygro-Mechanical Coupling Using X-FEM ...................................................................................................... 254

(ID-2536)

S. Klishin Destruction of Rocks by Directional Hydraulic Fracturing on the Basis of Models of Plasticity with Internal Variables ......................................... 255

(ID-1287)

N. Krukova, I. Lavit The Finite-Element Method in Linear Fracture Mechanics Problems.............................................................................................................. 256 (ID-1638)

R. Larsson, M. Fagerström (ID-1471) Finite Deformation Fracture Modelling of a Thermo-Mechanical Cohesive Zone..................................................................................................... 257 N. Myagkov, T. Shumikhin Critical Behavior and Energy Dependence of Mass Distribution in High-Velocity Impact Fragmentation.................................................................. 258

(ID-1145)

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xxv

S. Reese, P. Wriggers (ID-2419) One 3D Adaptive Fragmentation Procedure for the Explicit Simulation of Brittle Material Cracking .............................................................. 259 G. Tsamasphyros, T. Papathanassiou Finite Element Analysis of Cracked Plates with Circular Stress Raisers Used for S.I.F. Reduction ....................................................................... 260

(ID-2441)

N. Vlasov, I. Fedik Simulation of Materials Damage in the Field of Internal Stresses .......... 261

(ID-2230)

MS.11 Computational Mathematics Organizers: Stenberg, R., Figueiredo, I.

F. Bourquin (ID-2063) Computational Methods for the Fast Boundary Stabilization of Flexible Plates ..................................................................................................... 262 L. Costa, I. Figueiredo, P. Oliveira Sensor and Actuator Capabilities of a Laminated Piezoelectric Plate Model ......................................................................................................... 263

(ID-1835)

A. Loula, M. Correa Numerical Analysis of Stabilized Finite Element Methods for Darcy Flow .......................................................................................................... 264 (ID-2187)

C. Lovadina, R. Stenberg (ID-1223) An Error Estimator for the Reissner-Mindlin Plate Problem .................. 265 N. Troyani, E. Gutiérrez A Convergence Study of the Numerical Solution of Two BiDirectionally Coupled Partial Differential Equations in Thermoelectricity ........ 266

(ID-1069)

MS.12 Computational Methods for Anisotropic Material Behaviour at Large Strains Organizers: Sansour, C.

M. Böl, S. Reese (ID-1440) Numerical Simulations of Rubber-Like Materials Under Changing Directions ............................................................................................................ 267 A. Bucher, U. Görke, R. Kreißig About an Efficient and Consistent Numerical Strategy for the Solution of the Initial-Boundary Value Problem................................................. 268

(ID-1920)

H. Hein Vibrations of Composite Beams with Multiple Delaminations............... 269

(ID-1482)

xxvi Table of Contents

I. Karsaj, C. Sansour, J. Soric (ID-2382) Computational Aspects of Anisotropic Finite Strain Plasticity Based on the Multiplicative Decomposition........................................................ 270 A. Menzel Adaptation of Biological Tissues - a Fibre Reorientation Model for Orthotropic Multiplicative Growth ................................................................ 271

(ID-1883)

J. Schröder, P. Neff Polyconvex Anisotropic Hyperelastic Energies ...................................... 272

(ID-1589)

J. Wang, V. Levkovitch, B. Svendsen Micromechanically Motivated Phenomenological Modeling of Induced Flow Anisotropy and Its Application to Sheet Forming Processes........ 273

(ID-1486)

MS.13 Computational Modelling of Masonry Structures Organizers: Lourenço, P.

A. Barbieri, A. Cecchi, A. Tommaso (ID-1199) 3D Homogenization Procedure for Load Bearing Masonry Columns .............................................................................................................. 274 C. Calderini, S. Lagomarsino Non Linear Modelling of Masonry Structures Under Cyclic Loads ....... 275

(ID-1809)

S. Casolo, C. Sanjust Macroscale Modelling of Structured Materials with Damage by a Specific Rigid Element Model ............................................................................ 276

(ID-1862)

A. Cecchi Plate Micromechanical Models for 3D Periodic Brickworks.................. 277

(ID-1027)

A. Cecchi, G. Milani, A. Tralli Limit Analysis of Out-Of-Plane Loaded Running Bond Masonry Walls Under Mindlin-Reissner Plate Hypotheses ............................................... 278

(ID-1140)

M. Malena, A. Bilotta, A. Lanzo Nonlinear Analysis of Brittle Materials .................................................. 279

(ID-1762)

T. Massart, R. Peerlings, M. Geers Damage Localisation in Computational Homogenisation of Masonry and Its Incorporation in a Two-Scale Computational Framework........ 280

(ID-2179)

G. Milani, P. Lourenço, A. Tralli 3D Homogenized Limit Analysis of Masonry Buildings Subjected to Horizontal Loads ............................................................................................. 281

(ID-1131)

F. Peña, P. Lourenço, J. Lemos Modelling the Dynamic Behaviour of Masonry Walls As Rigid Blocks.................................................................................................................. 282

(ID-1256)

Table of Contents xxvii

G. Zingone, G. Canio, L. Cavaleri (ID-2514) On the Improvement of Monumental Structure Safety: a Case Study.................................................................................................................... 283 A. Zucchini, P. Lourenço Homogenization of Masonry Using a Micro-Mechanical Model: Compressive Behaviour....................................................................................... 284

(ID-1056)

MS.14 Computational Stochastic Failure Mechanics Organizers: Gutierrez, M. A.

R. Jimenez-Rodriguez, L. Lacoma (ID-1756) Uncertainty Characterization and Settlement Analyses: the Importance of Distribution Types........................................................................ 285 M. Vorechovsky, R. Chudoba, J. Jerábek Adaptive Probabilistic Modeling of Localization, Failure and Size Effect of Quasi-Brittle Materials ......................................................................... 286

(ID-1885)

MS.15 Computational Stochastic Structural and Uncertainty Analysis Organizers: Schueller, G.

D. Alvarez (ID-1625) On the Use of Infinite Random Sets for Bounding the Probability of Failure in the Case of Parameter Uncertainty.................................................. 287 J. Colliat, M. Krosche, M. Krosche, M. Hautefeuille, A. Ibrahimbegovic, R. Niekamp, H. Matthies (ID-2470) Stochastic Analysis of Coupled Nonlinear Thermo-Mechanical Problems: SFEM Model ...................................................................................... 288 J. Crempien-Laborie (ID-1064) Response of a Single Degree of Freedom Elastic Perfectly Plastic System Under Non-Stationary Gaussian Seismic Excitation .............................. 289 D. Degrauwe, G. Roeck, G. Lombaert Fuzzy Frequency Response Function of a Composite Floor Subject to Uncertainty by Application of the GĮd Algorithm............................. 290

(ID-1999)

J. Fernandes, M. Pinho, A. Alvim, A. Pithon Comparative Numerical Evaluation of Angra I Auxiliary Feedwater System Reliability by the Method of Suplementary Variables .......... 291

(ID-1611)

M. Galffy, M. Baitsch, A. Wellmann-Jelic, D. Hartmann Lifetime Estimation of Vertical Bridge Tie Rods Exposed to Wind-Induced Vibrations .................................................................................... 292

(ID-1088)

xxviii Table of Contents

G. Giunta, E. Carrera (ID-1544) Stochastic Static Analyses of FE Models by Means of Newton's Series Expansions................................................................................................ 293 M. Grigoriu A Galerkin Solution for Stochastic Algebraic Equations ........................ 294

(ID-1262)

R. Iwankiewicz, M. Vasta Approximate Method for Probability Density of the Response of a Linear Oscillator to a Non-Poisson Impulse Process........................................ 295

(ID-1906)

H. Kang, Y. Lee, J. Huh, B. Kwak Comparative Study of RBDO Algorithms Based on Form and FAMM................................................................................................................. 296

(ID-2625)

M. Mailhé, S. Chaabane, F. Léné, G. Duvaut, S. Grihon Probabilistic Analysis and Optimization of a Fully Composite Cylinder............................................................................................................... 297

(ID-2589)

A. Martowicz, L. Pieczonka, T. Uhl Assessment of Dynamic Behaviour of Spot Welds with Uncertain Parameters Using Genetic Algorithms Application............................................. 298

(ID-1316)

M. Munck, D. Moens, W. Desmet, D. Vandepitte Optimisation Algorithms for Non-Deterministic Dynamic Finite Element Analysis of Imprecisely Defined Structures.......................................... 299

(ID-1926)

M. Pellissetti, H. Pradlwarter, G. Schuëller Relative Importance of Uncertain Parameters in Aerospace Applications ........................................................................................................ 300

(ID-2699)

V. Potapov Stability of Elastic and Viscoelastic Systems Under Stochastic Non-Gaussian Excitation..................................................................................... 301

(ID-1175)

J. Santos, B. Mace Modelling Uncertainty in Mechanical Joint Parameters Using Component Modal and Fuzzy Approaches.......................................................... 302

(ID-2348)

E. Sibilio, J. Beck, M. Muto, M. Ciampoli Bayesian Model Updating Approach for Ground-Motion Attenuation Relations .......................................................................................... 303

(ID-2650)

C. Verhoosel, M. Gutiérrez, S. Hulshoff Iterative Solution of the Random Eigenvalue Problem ........................... 304

(ID-1205)

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xxix

MS.16 Contact Mechanics Organizers: Martins, J.

J. Bielsa, R. Rodríguez, L. Vila, M. Jiménez (ID-2076) Parametrized Finite Element Analysis of Tribological Instabilities on Polymer-Metal Sliding Contacts .................................................................... 305 D. Boso, P. Litewka, B. Schrefler, P. Wriggers Thermo-Electro-Mechanical Coupling in Beam-to-Beam Contact ......... 306

(ID-2104)

M. Campo, J. Fernandez, K. Kuttler, M. Shillor (ID-1133) Numerical Analysis of a Dynamic Frictional Viscoelastic Contact Problem with Damage ......................................................................................... 307 A. Chernov, M. Maischak, E. Stephan Hp-Mortar Boundary Element Method and FE/BE Coupling for Multibody Contact Problems with Friction ......................................................... 308

(ID-1991)

G. Chevallier, D. Nizerhy Friction Induced Vibrations in a Clutch System. Consequences on the Apparent Friction Torque. ............................................................................. 309

(ID-2328)

A. Chudzikiewicz, A. Myslinski Thermoelastic Wheel - Rail Contact Problem with Temperature Dependent Friction Coefficient ........................................................................... 310

(ID-1704)

M. Cocou, M. Raous, M. Schryve Analysis of a Dynamic Contact Problem with Adhesion and Friction in Viscoelasticity.................................................................................... 311

(ID-2078)

G. Drossopoulos, G. Stavroulakis, C. Massalas Influence of the FRP Strengthening, the Shape and the Movement of Abutments on the Collapse of Arch Stone Bridges......................................... 312

(ID-2194)

R. Dzonou, M. Marques, L. Paoli Sweeping Process for Vibro-Impact Problem with a General Inertia Operator ................................................................................................... 313

(ID-1020)

A. Eddhahak, L. Chevalier, S. Cloupet On a Simplified Method for Wear Simulation in Rolling Contact Problems.............................................................................................................. 314

(ID-1251)

G. Festa, J. Vilotte Spectral Element Simulations of Rupture Dynamics Along Planar and Kinked Frictional Faults .............................................................................. 315

(ID-1312)

M. Foerg, T. Geier, L. Neumann, H. Ulbrich R-Factor Strategies for the Augmented Lagrangian Approach in Multi-Body Contact Mechanics........................................................................... 316

(ID-1480)

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L. Fourment, S. Guerdoux (ID-1217) A Simple Smoothing Procedure of 3D Surfaces for Accurate Contact Analysis: Application to Metal Forming Problems................................ 317 D. Gabriel, J. Plesek, F. Vales, M. Okrouhlík Symmetry Preserving Algorithm for a Dynamic Contact-Impact Problem ............................................................................................................... 318

(ID-2524)

H. Georgiadis, D. Anagnostou Problems of Concentrated Loads in Microstructured Solids Characterized by Dipolar Gradient Elasticity...................................................... 319 (ID-1342)

F. Gutzeit, M. Wangenheim, M. Kröger (ID-1925) An Experimentally Validated Model for Unsteady Rolling .................... 320 S. Hartmann, S. Brunssen, E. Ramm, B. Wohlmuth A Primal-Dual Active Set Strategy for Unilateral Non-Linear Dynamic Contact Problems of Thin-Walled Structures ...................................... 321

(ID-2576)

Y. Kanno, J. Martins Arc-Length Method for Frictional Contact with a Criterion of Maximum Dissipation of Energy ........................................................................ 322 (ID-1573)

H. Khenous, P. Laborde, Y. Renard (ID-2170) A Energy Conserving Approximation for Elastodynamic Contact Problems.............................................................................................................. 323 A. Konyukhov, K. Schweizerhof, P. Vielsack On Models of Contact Surfaces Including Anisotropy for Friction and Adhesion and Their Experimental Validations............................................. 324

(ID-2320)

R. Krause Fast and Robust Solution Methods for Dynamic Contact Problems ....... 325

(ID-2147)

M. Marques, L. Paoli A Velocity-Based Time-Stepping Method for Frictional Dynamics....... 326

(ID-1865)

T. Meyer, A. Gal, M. Klüppel (ID-1785) Mechanical Modeling of Friction and Adhesion of Elastomers at Rough Interfaces.................................................................................................. 327 C. Miranda, M. Neto, G. Fainer A 9M Drop Test Simulation of a Dual Purpose Cask for Nuclear Research Reactors Spent Fuel Elements.............................................................. 328 (ID-1347)

U. Nackenhorst, M. Ziefle (ID-1745) A Discontinuous Galerkin Approach for the Numerical Treatment........ 329 P. Neittaanmäki, A. Kravchuk, I. Goryacheva An Iterative Method with BEM Discretization for the Friction Contact Problems ................................................................................................ 330

(ID-1297)

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xxxi

J. Nettingsmeier, P. Wriggers (ID-1299) Frictional Contact of Elastomer Materials on Rough Rigid Surfaces ............................................................................................................... 331 M. Oliveira, J. Alves, L. Menezes Optimizing the Description of Forming Tools with Bézier Surfaces in the Numerical Simulation of the Deep Drawing Process.................. 332

(ID-1856)

E. Pratt, A. Léger Exploring the Dynamics of a Simple System Involving Coulomb Friction ................................................................................................................ 333

(ID-2367)

F. Rauter, J. Pombo, J. Ambrósio, M. Pereira Multibody Modeling of Pantographs for Catenary-Pantograph Interaction............................................................................................................ 334

(ID-1653)

M. Renouf, V. Acary Comparison and Coupling of Algorithms for Collisions, Contact and Friction in Rigid Multibody Simulations...................................................... 335

(ID-2436)

M. Renouf, A. Saulot, Y. Berthier Third-Body Flow During Wheel-Rail Interaction .................................. 336

(ID-2440)

J. Sá, S. Grégoire, P. Moreau, D. Lochegnies (ID-2352) Modelling Thermal Contact Resistance on Glass Forming Processes with Special Interface Finite Elements................................................ 337 J. Shiau Numerical Investigation of Shakedown Residual Stresses Under Moving Surface Loads ........................................................................................ 338

(ID-1734)

N. Strömberg Frictional Contact/Impact Between a Hyperelastic Body and Moving Rigid Obstacles ...................................................................................... 339

(ID-1578)

J. Svígler Incorrect Contact of Screw Surfaces and Its Consequences.................... 340

(ID-1485)

K. Uenishi (ID-1104) Three-Dimensional Rupture Instability of a DisplacementSoftening Interface Under Nonuniform Loading................................................. 341 K. Willner, D. Görke Contact of Rough Surfaces - a Comparison of Numerical and Experimental Results........................................................................................... 342

(ID-1288)

xxxii Table of Contents

MS.17 Continnum Models for Nano-Structures Organizers: Kompis, V.

M. Arroyo, T. Belytschko (ID-1543) Continuum Mechanics Modelling and Simulation of Carbon Nanotubes............................................................................................................ 343 L. Bochkareva, M. Kireitseu, G. Tomlinson Comparison of Computational Efficiency of Modeling Approaches to Prediction of Damping Behavior................................................. 344

(ID-1144)

P. Dluzewski, M. Mazdziarz, G. Jurczak, P. Traczykowski, S. Nagao, R. Nowak, K. Kurzydlowski (ID-1849) A Hybrid Atomistic-–Continuum Finite Element Modelling of Nanoindentation Test on Copper......................................................................... 345 V. Kompis, M. Stiavnický, M. Kaukic Continuum Models for Composites Reinforced by Micro/Nano Fibers................................................................................................................... 346

(ID-1741)

A. Kushima, Y. Umeno, T. Kitamura First Principles Evaluation of Ideal Strength of Cu Nanowire ................ 347

(ID-2339)

S. Lisina, A. Potapov Variation Descriptions of Nano-Structured Media.................................. 348

(ID-1668)

T. Messager, P. Cartraud (ID-1234) Homogenization of Single-Walled Carbon Nanotubes ........................... 349 M. Rabia Phonon Scattering by Perturbed Multichannel Waveguides ................... 350

(ID-1010)

H. Wu, X. Wang (ID-1086) An Atomistic-Information-Based Continuum Inhomogeneous Material Model for Metal Nanorod ..................................................................... 351 MS.18 Coupling Problems Organizers: Schrefler, B.

F. Duda, L. Guimarães, A. Souza, J. Barbosa (ID-2349) On the Modeling of Deformation-Diffusion-Damage Coupling in Elastic Solids ....................................................................................................... 352 J. Gatica, V. Pita, N. Brum Frost Growth on Cold Flat Plate: a Correlation for the Diffusion Resistance Factors ............................................................................................... 353 (ID-1698)

Table of Contents xxxiii

D. Khalmanova, F. Costanzo (ID-1532) Discontinuous Space-Time Galerkin Finite Element Method in Linear Dynamic Fully Coupled Thermoelastic Problems with Strain and Heat Flux Discontinuities .................................................................................... 354 M. Landervik, R. Larsson Pore Gas Interaction in Polymeric Foams with Respect to Energy Absorption........................................................................................................... 355

(ID-1463)

F. Meftah, H. Sabeur A Thermo-Hydro–Damage Model for the Dehydration Creep of Concrete Subjected to High Temperature............................................................ 356

(ID-2290)

D. Néron, L. Pierre, B. Schrefler A Time-Space Framework Suitable for the LATIN Computational Strategy for Multiphysics Problems .................................................................... 357

(ID-2483)

W. Oliveira, M. Savi, P. Pacheco, L. Souza Finite Element Analysis of the Thermomechanical Coupling in Quenching of Steel Cylinders Using a Constitutive Model................................. 358

(ID-2176)

B. Pichler, C. Hellmich, H. Mang A Combined Fracture-Micromechanics Model for Tensile StrainSoftening in Brittle Materials .............................................................................. 359 (ID-1550)

Ö. Sen, D. Turhan (ID-1491) Transient Dynamic Response of Thermoelastic Cylindrical Layered Media..................................................................................................... 360 MS.19 Damage Organizers: Alfaiate, J.

H. Askes, M. Gutiérrez, A. Rodriguez-Ferran (ID-2083) Novel Nonlocal Continuum Formulations. Part 1: Gradient Elasticity Based on Nonlocal Displacements and Nonlocal Strains .................... 361 T. Bennett, S. Kulasegaram On the Use of a Damage Model Based on Non-Local Displacements in the Element-Free Galerkin Method......................................... 362

(ID-1514)

T. Domingues, J. Alfaiate Modelling of Reinforced Concrete Beams Strengthened with PreStressed CFRP..................................................................................................... 363

(ID-2350)

T. Fiedler, L. Cunda, A. Öchsner, G. Creus, J. Grácio Numerical and Experimental Studies of Damage in Porous Materials.............................................................................................................. 364

(ID-1203)

xxxiv Table of Contents

R. Frizzell, C. Mccarthy, D. Cronin, M. Mccarthy, R. O'Higgins (ID-2369) The Development of a Continuum Damage Model for Fibre Metal Laminate Structures............................................................................................. 365 M. Hassanzadeh, G. Fagerlund Residual Strength of the Frost-Damaged Reinforced Concrete Beams .................................................................................................................. 366

(ID-1872)

M. Konrad, R. Chudoba, B. Kang Numerical and Experimental Evaluation of Damage Parameters for Textile Reinforced Concrete Under Cyclic Loading..................................... 367

(ID-2057)

P. Neto, J. Alfaiate, J. Vinagre Modeling the Behavior of Reinforced Concrete Beams Strengthened with FRP........................................................................................ 368

(ID-2380)

S. Oliveira, N. Gaspar, P. Dinis Cracking Analysis in Concrete Dams Using Isotropic Damage Models. Objectivity of Numerical Solutions ....................................................... 369

(ID-2649)

R. Pedersen, A. Simone, B. Sluys Continuous-Discontinuous Modelling of Dynamic Failure of Concrete Using a Viscoelastic Viscoplastic Damage Model............................... 370

(ID-1962)

J. Pituba On the Formulation of Damage Constitutive Models for Bimodular Anisotropic Media ............................................................................. 371

(ID-2089)

L. Rosa, E. Carvalho, B. Danziger Soil-Structure Interaction - Case History Analysis Involving Structural Damage............................................................................................... 372

(ID-2540)

Y. Sanomura, K. Saitoh Evolution Equation of Creep Damage Under Stress Variation ............... 373

(ID-1737)

A. Sichaib, G. Mounajed, C. Laborderie, H. Boussa, H. Quoc (ID-1746) Concrete Damage Model Adaptation for Cyclic Loading....................... 374 C. Silva, L. Castro Hybrid and Mixed Finite Element Formulations for Softening Materials.............................................................................................................. 375

(ID-1107)

D. Sornin, K. Saanouni Theoretical and Computational Aspects of an Elastoplastic Damage Gradient Non Local Model ................................................................... 376

(ID-1326)

J. Wu, J. Li On a New Framework for Anisotropic Damage Model .......................... 377

(ID-1616)

Table of Contents xxxv

MS.20 Design Optimization Under Uncertainty Organizers: Choi, K. K. (USA), Kwak, B. M., Gorsich, D.

N. Banichuk, S. Ivanova, E. Makeev, A. Sinitsin (ID-2572) Shell Optimization Under Constraint on Damage Accumulation ........... 378 J. Casaca, A. Gomes (ID-1249) Design of Acceptance-Sampling Plans Under Bayesian Risk................. 379 E. Cherkaev, A. Cherkaev Optimal Design for the Worst Case Scenario.......................................... 380

(ID-2192)

K. Choi, I. Lee, D. Gorsich Dimension Reduction Method for Reliability-Based Robust Design Optimization............................................................................................ 381

(ID-2522)

R. Dippolito, S. Donders, L. Hermans, M. Hack, J. Peer, N. Tzannetakis A Fatigue Life Reliability-Based Design Optimization of a Slat Track Using Mesh Morphing .............................................................................. 382

(ID-1231)

B. Kwak, J. Chang, J. Kim A New Approach of Robust Design Based on the Concept of Allowable Load Set ............................................................................................. 383

(ID-2571)

T. Lee, J. Jung, D. Jung A Sampling Technique Enhancing Accuracy and Efficiency of Metamodel-Based RBDO: Constraint Boundary Sampling ................................ 384

(ID-2593)

D. Moens, D. Vandepitte Interval Sensitivity Analysis of Dynamic Response Envelopes for Uncertain Mechanical Structures......................................................................... 385

(ID-2139)

C. Poloni, P. Geremia, A. Clarich Multi-Objective Robust Design Optimization of an Engine Crankshaft ........................................................................................................... 386

(ID-1904)

Y. Ryu Development and Application of a New Metropolis GA for the Structural Design Optimization ........................................................................... 387

(ID-2570)

B. Youn, Z. Xi, L. Wells, D. Lamb Stochastic Response Surface Using the Enhanced DimensionReduction (EDR) Method for Reliability-Based Robust Design......................... 388

(ID-2539)

MS.21 Differential Quadrature, Generalized Methods and Related Discrete Element Analysis Methods Organizers: Chen, C.

R. Balevicius, R. Kacianauskas (ID-1499) DEM Analysis of Granular Flow in Pyramidal Hoppers ........................ 389

xxxvi Table of Contents

C. Chen (ID-2085) DQEM and DQFDM for Computational Mechanics Problems .............. 390 C. Cinquini, M. Bruggi, P. Venini An Innovative Truly-Mixed Method for Cohesive-Crack Propagation Problems.......................................................................................... 391

(ID-1191)

D. Rosillo, F. Pérez Compactly Supported Fundamental Functions for Spline-Based Differential Quadrature ....................................................................................... 392

(ID-1876)

F. Tornabene, E. Viola Differential Quadrature Solution for Parabolic Structural Shell Elements .............................................................................................................. 393

(ID-2347)

MS.23 Enriched and Enhanced Finite Element Technology Organizers: Areias, P.

E. Budyn, L. Henry, T. Hoc (ID-2167) Multiple Crack Growth Failure in Cortical Bone Under Tension by the Extended Finite Element Method ............................................................. 394 F. Cirak Subdivision Shells................................................................................... 395

(ID-1710)

C. Foster, R. Borja Capturing Slip Weakening and Variable Frictional Response in Localizing Geomaterials Using an Enhanced Strain Finite Element................... 396

(ID-2040)

L. Hazard, P. Bouillard, J. Sener A Partition of Unity Finite Element Method Applied to the Study of Viscoelastic Sandwich Structures ................................................................... 397

(ID-1972)

A. Kölke, A. Legay An Enriched Space-Time Finite Element Method for FluidStructure Interaction - Part Ii: Thin Flexible Structures ...................................... 398

(ID-2318)

A. Legay, A. Kölke An Enriched Space-Time Finite Element Method for FluidStructure Interaction- Part I: Prescribed Structural Displacement....................... 399

(ID-2316)

I. Moldovan, J. Freitas Hybrid-Trefftz Finite Element Models for Bounded and Unbounded Elastodynamic Problems.................................................................. 400

(ID-2246)

P. Rozycki, E. Bechet, N. Möes Explicit Dynamic with X-FEM to Handle Complex Geometries............ 401

(ID-2010)

L. Stankovíc, J. Mosler Prediction of Macroscopic Material Failure Based on Microscopic Cohesive Laws .................................................................................................... 402

(ID-1255)

Table of Contents xxxvii

MS.24 Error Analysis and Adaptivity Organizers: Wiberg, N. E., Moitinho de Almeida, J., Diez, P.

L. Chamoin, P. Ladevèze (ID-2111) Strict, Sharp and Practical Bounds of Computed Outputs of Interest for Evolution Problems........................................................................... 403 É. Florentin, P. Ladevèze, J. Bellec Error Bounds on Outputs of Interest for Linear Stochastic Problems.............................................................................................................. 404

(ID-2299)

M. Jasinski, G. Zboinski An Hp-Adaptive Analysis of Some Linear Free Vibration Problems.............................................................................................................. 405

(ID-1313)

H. Lane, P. Kettil, N. Wiberg Moving Mesh Adaptivity Applied to Railway Dynamics ....................... 406

(ID-2373)

J. Mosler, M. Ortiz Finite Strain R-Adaption Based on a Fully Variational Framework ....... 407

(ID-1254)

E. Rank, V. Nübel, A. Düster Extension Processes, Adaptivity and Remeshing for Elasto-Plastic Problems.............................................................................................................. 408

(ID-2351)

J. Ródenas, J. Albelda, C. Corral, J. Mas Efficient Implementation of Domain Decomposition Methods Using a Hierarchical H-Adaptive Finite Element Program ................................. 409

(ID-2107)

W. Wall, T. Erhart, E. Ramm Adaptive Remeshing in Transient Impact Processes with Large Deformations and Nonlinear Material Behavior ................................................. 410

(ID-2615)

MS.25 Evolutionary Methods for Design Organizers: Periaux, J.

B. Bochenek, P. Forys (ID-1826) Particle Swarms in Engineering Design Problems .................................. 411 E. Campana, G. Fasano, D. Peri, A. Pinto Particle Swarm Optimization: Efficient Globally Convergent Modifications....................................................................................................... 412

(ID-2025)

G. Dulikravich, I. Egorov, N. Jelisavcic Evolutionary Optimization of Chemistry of Bulk Metallic Glasses........ 413

(ID-2228)

T. Naskar Introduction of Control Points in Splines for Synthesis of Optimized Cam Motion Programme ................................................................... 414

(ID-1576)

xxxviii Table of Contents

G. Oliveira, S. Saramago, P. Oliveira (ID-1837) On the Use of Differential Evolution in the Trajectory Modeling of Parallel Architecture Robot............................................................................. 415 M. Teixeira, M. Brandão Evolutionary Topologic Optimization Using the Finite Element Method ................................................................................................................ 416

(ID-2327)

MS.27 Fluid-Structure Interactions Organizers: Idelsohn S.

E. Aulisa, S. Manservisi, P. Seshaiyer (ID-1566) A Multilevel Domain Decomposition Methodology for Solving Coupled Problems in Fluid-Structure-Thermal Interaction ................................. 417 A. Boer, M. Schoot, H. Bijl Moving Mesh Algorithm for Unstructured Grids Based on Interpolation with Radial Basis Functions........................................................... 418

(ID-1781)

F. Daneshmand, S. Niroomandi Vibrational Analysis of Fluid-Structure Systems Using Natural Neighbour Galerkin Method................................................................................ 419

(ID-2310)

A. Kraker, D. Rixen, R. Ostayen Fluid-Structure Interaction in FEM Journal Bearing Simulations........... 420

(ID-1923)

A. Kupzok, R. Wüchner, K. Bletzinger Numerical Simulation of Wind-Structure Interaction for Thin Shells and Membranes......................................................................................... 421

(ID-1109)

J. Li, H. Chen, J. Chen Reliability Analysis of Prestressed Egg-Shaped Digester ....................... 422

(ID-1119)

P. Matyushin, V. Gushchin The Vortex Structures in the Sphere Wakes in the Wide Range of the Reynolds and Froude Numbers ..................................................................... 423

(ID-1303)

R. Niesner, M. Haupt, P. Horst Transient Analysis Methods for Hypersonic Applications with Thermo-Mechanical Fluid-Structure Interaction ................................................. 424

(ID-1965)

S. Sarkar, H. Bijl Stall Induced Vibration & Flutter in a Symmetric Airfoil....................... 425

(ID-1520)

R. Unger, M. Haupt, P. Horst (ID-1946) Application of Lagrange Multipliers for Computational Aeroelasticity....................................................................................................... 426

Table of Contents xxxix

MS.28 Fracture and Fatigue Mechanics Organizers: Karihaloo, B.

J. Assis, V. Monine, S. Filippov, S. Iglesias (ID-1717) Computer Simulation of Diffraction Technique Applied for Measurements of Surface Stress Gradients ......................................................... 427 K. Enakoutsa, J. Leblond, G. Perrin Numerical Assessment of a Micromorphic Model of Ductile Rupture ................................................................................................................ 428

(ID-2408)

H. Hosseini-Toudeshky, B. Mohammadi, S. Bakhshandeh Fatigue Crack Trajectory Analysis of Single-Side Repaired Thin Aluminum Panels with Various Composite Patch Lay-Up Configurations ........ 429

(ID-1555)

S. Jog, R. Baddam Prediction of the Crack Initiation Life of Turbine Blade ........................ 430

(ID-1120)

M. Khoshravan, A. Hamidi (ID-2240) Numerical Analysis of the Influence of Location of the Stopping Holes and Their Diameter in the Crack Growth of Ductile Metals ..................... 431 M. Kopecky Computed Analysis to Determine Service Life Criteria of Special Elements and Applications .................................................................................. 432 (ID-1110)

T. Luther, C. Könke (ID-1796) Analysis of Crack Initiation and Propagation in Polycrystalline Meso and Microstructures of Metal Materials..................................................... 433 E. Szymczyk, A. Derewonko, J. Jachimowicz Analysis of Displacement and Stress Distributions in Riveted Joints.................................................................................................................... 434

(ID-1669)

MS.29 Genetic Algorithms Organizers: António, C.

C. António (ID-2124) The Synergetic Effects of Hybrid Crossover Operators in Structural Optimisation ....................................................................................... 435 G. Bugeda, J. Ródenas, E. Pahl, E. Oñate An Adaptive Mesh Generation Strategy for the Solution of Structural Shape Optimization Problems Using Evolutionary Methods ............. 436

(ID-2428)

J. Dias, R. Corrêa Multiobjective Optimization of Multibody Systems with Genetic Algorithms........................................................................................................... 437

(ID-2472)

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L. Iuspa, V. Minutolo, E. Ruocco (ID-2064) Buckling Optimization of Grid Structures Via Genetic Algorithms ....... 438 D. Kim, Y. Kim Multiscale Multiresolution Genetic Algorithm Using Diverse Population Groups............................................................................................... 439

(ID-1740)

P. Montrull, O. Querin, C. Gomez An Adaptive Correction Function for Structural Optimization with Genetic Algorithms ............................................................................................. 440

(ID-2450)

L. Sousa, C. Castro, C. António Optimization of the Topology of Masonry Units From the Thermal Point of View Using a Genetic Algorithm ............................................ 441

(ID-2292)

M. Victoria, P. Marti Topology Optimization of Bidimensional Continuum Structures by Genetic Algorithms and Stress Iso-Lines ....................................................... 442

(ID-2452)

MS.31 Impact and Control Organizers: Holnicki-Szulc, J.

I. Ario, P. Pawlowski, J. Holnicki-Szulc (ID-1476) Dynamic Analysis of Folding Patterns for Multi-Folding Structures............................................................................................................. 443 J. Cardoso, P. Moita, A. Valido Mechanical Systems Design and Control Optimization with Varying Time Domain......................................................................................... 444

(ID-2627)

S. Pashah, E. Jacquelin, J. Lainé, M. Massenzio Modelling for the Determination of the Interaction Force of Impacted Structures ............................................................................................. 445

(ID-1750)

MS.33 Intelligent Computing in Solid and Structural Mechanics Organizers: Burczynski, T.

W. Beluch (ID-1812) Evolutionary Identification and Optimization of Composite Structures............................................................................................................. 446 A. Dlugosz Parallel Evolutionary Optimization of Heat Radiators by Using MSC MARC/MENTAT Software....................................................................... 447

(ID-1853)

G. Dziatkiewicz, P. Fedelinski Evolutionary Algorithm and Boundary Element Method for Solving Inverse Problems of Piezoelectricity...................................................... 448

(ID-1930)

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xli

W. Kus Evolutionary Optimization of Preform and Die Shape in Forging Using Computational Grid .................................................................................. 449

(ID-1813)

W. Kus, T. Burczynski Optimization of Mechanical Structures Using Serial and Parallel Artificial Immune Systems.................................................................................. 450

(ID-1918)

P. Nazarko, L. Ziemianski Experiments of Damage Detection in Strips Based on Soft Computing Methods and Wave Propagation....................................................... 451

(ID-2581)

P. Orantek The Optimization and Identification Problems of Structures with Fuzzy Parameters ................................................................................................ 452

(ID-1811)

A. Poteralski Topology Optimization of the 3-D Structures for Various Criteria Using Evolutionary Algorithm ............................................................................ 453

(ID-1818)

A. Skrobol, T. Burczynski Computational Intelligence System in Non-Destructive Identification of Internal Defects......................................................................... 454

(ID-1816)

M. Szczepanik Optimization of Topology and Stiffeners Locations in 2-D Structures Using Evolutionary Methods.............................................................. 455

(ID-1825)

N. Wang, K. Tai A Structural Optimization Problem Formulation for Design of Compliant Gripper Using a Genetic Algorithm................................................... 456

(ID-2638)

MS.34 Intelligent Optimization Organizers: Sousa, J. C.

R. Almeida, S. Vieira, J. Sousa, U. Kaymak (ID-2061) The Prediction of Bankruptcy Using Weighted Fuzzy Classifiers.......... 457 P. Pinto, T. Runkler, J. Sousa Optimization of a Logistic Process by Ant Colonies, Wasp Swarms and Genetic Post-Optimization.............................................................. 458

(ID-1128)

S. Sakata, F. Ashida, M. Zako Kriging-Based Estimation with Noisy Data ............................................ 459

(ID-1735)

C. Silva, J. Faria, D. Naso Distributed Optimization Using ACO for Concrete Delivery ................. 460

(ID-1894)

F. Viana, G. Kotinda, V. Steffen Tuning a Vibrating Blade Dynamic Vibration Absorber by Using Ant Colony Optimization and Finite Element Modeling..................................... 461 (ID-1100)

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MS.35 Interrelation of Numerical and Asymptotical Approaches in Solid and Structural Mechanics Organizers: Manevitch, L., Lamarque, C. H.

C. Lamarque, E. Gourdon (ID-2579) Energy Pumping of Systems Connected to a Nonlinear Energy Sink Device ......................................................................................................... 462 V. Lyakh, V. Meleshko Stress Analysis of Curved Elastic Bar and Elastic Wedge Under Bending Load; Infinite Systems and Asymptotic ................................................ 463

(ID-1712)

A. Potapov, I. Miloserdova, I. Potapov Nonlinear Oscillations in Discretely Continual System .......................... 464

(ID-1672)

F. Romeo, G. Rega Propagation Properties of Bi-Coupled Nonlinear Oscillatory Chains.................................................................................................................. 465

(ID-1519)

MS.36 Inverse Engineering Organizers: Dulikravich, G. S., Orlande, H.

S. Abboudi, E. Artioukhine (ID-2202) Numerical Analysis of the Estimation of Three Boundary Conditions in Two Dimensional Inverse Heat Conduction Problem................... 466 J. Auzins, S. Ruchevskis, R. Rikards, A. Chate Metamodeling for the Identification of Composite Material Properties............................................................................................................. 467

(ID-2037)

T. Baranger, S. Andrieux An Energy Approach for a Cauchy Problem in Elasticity....................... 468

(ID-2062)

R. Fedele, G. Maier, L. Marazza Flat-Jack Tests and Parameter Identification for Diagnostic Analysis of Dams ................................................................................................ 469 (ID-1955)

H. Gualous, D. Zibret, E. Artioukhine (ID-2693) Convective Boiling in Mini-Channels: Flow Visualization and Inverse Thermal Characterization ....................................................................... 470 E. Katamine, H. Azegami, M. Hirai Solution of Shape Identification Problem on Thermoelastic Solids........ 471

(ID-1716)

L. Lage, A. Cuco, F. Folly, F. Soeiro, A. Neto Stochastic and Hybrid Methods for the Solution of an Inverse Mass Transfer Problem........................................................................................ 472

(ID-2437)

P. Masson, S. Rouquette, T. Loulou, E. Artioukhine Inverse Heat Conduction Problem: Estimation of a Source Term for an Electron Beam Welding; Theoretical and Experimental........................... 473

(ID-1346)

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xliii

H. Nakamura, M. Kawahara (ID-1829) The Parameter Identification of Elastic Modulus at Futatsuishi Site....................................................................................................................... 474 K. Ogura, M. Kawahara Parameter Identification of the Attenuation Using First Order Adjoint Method ................................................................................................... 475

(ID-1819)

K. Sassi, S. Andrieux Parameters Identification of a Nonlinear Viscoelastic Model Via an Energy Error Functional ................................................................................. 476

(ID-1292)

F. Soeiro, A. Neto Inverse Radiative Transfer Problems in Two-Layer Participating Media................................................................................................................... 477

(ID-2435)

N. Thomson, H. Orlande Computation of Sensitivity Coefficients and Estimation of Thermophysical Properties with the Line Heat Source Method.......................... 478

(ID-1026)

H. Velho, S. Sambatti, L. Chiwiacowsky Combining a Parallel Genetic Algorithm with Variational Approach for Assessing Structural Damage........................................................ 479

(ID-1558)

MS.37 Large Scale Shape and Topology Optimization Organizers: Pedersen, P., Bendsoe, M., Sigmund O.

M. Abdalla, S. Setoodeh, Z. Gurdal (ID-1621) Design of Variable-Stiffness Composite Panels for Maximum Buckling Load ..................................................................................................... 480 G. Cheng, L. Liu, J. Yan Optimum Structure with Homogeneous Optimum Truss-Like Material ............................................................................................................... 481

(ID-1722)

P. Clausen, C. Pedersen Non-Parametric Large Scale Structural Optimization............................. 482

(ID-2583)

A. Diaz, R. Mukherjee (ID-1708) Optimal Joint Placement and Modal Disparity in Control of Flexible Structures............................................................................................... 483 A. Gersborg-Hansen Topology Optimization of 3D Stokes Flow Problems ............................ 484

(ID-1157)

J. Jensen Efficient Optimization of Dynamic Systems Using Pade Approximants ...................................................................................................... 485

(ID-1800)

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M. Kim, G. Jang, Y. Kim (ID-1743) Ground Structure Based Joint Stiffness Controlling Method for Joint Compliant Mechanism Design.................................................................... 486 S. Lambert, E. Pagnacco, L. Khalij, A. Hami Topology Optimization of Structures Subject to Random Excitations with Fatigue Life Constraints ........................................................... 487

(ID-2099)

E. Lemaire, P. Duysinx, V. Rochus, J. Golinval Improvement of Pull-In Voltage of Electromechanical Microbeams Using Topology Optimization ........................................................ 488

(ID-2065)

E. Lund Large Scale Optimization of Compression Loaded Composite Structures............................................................................................................. 489

(ID-1879)

C. Narváez, A. Tovar, D. Garzon Topology Synthesis of Compliant Mechanisms Using the Hybrid Cellular Automaton Method with an Efficient Mass Control STR ..................... 490

(ID-2215)

N. Olhoff, J. Du Topological Design for Minimum Sound Radiation From Structures Subjected to Forced Vibration............................................................ 491

(ID-1792)

P. Pedersen Aspects of 3D Shape and Topology Optimization with Multiple Load Cases .......................................................................................................... 492

(ID-1250)

N. Pedersen On Shape, Material and Orientational Design of Plates in Relation to Dynamics......................................................................................................... 493

(ID-1910)

N. Perchikov, M. Fuchs Optimal Layouts of Stiffeners for Plates in Bending - Topology Optimization Approach ....................................................................................... 494

(ID-1678)

U. Schramm How Topology Optimization Changed the Design Process .................... 495

(ID-2156)

Y. Sia, O. Querin (ID-1141) Structural Shape Optimisation by Using Multi-Direction Boundary Points Movement Method................................................................... 496 O. Sigmund On Topology Optimization with Manufacturing Constraints.................. 497

(ID-1895)

K. Svanberg, M. Werme (ID-1769) Sequential Integer Programming Methods for Stress-Constrained Shape and Topology Optimization ...................................................................... 498

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xlv

A. Tovar, W. Quevedo, N. Patel, J. Renaud (ID-2185) Topology Optimization with Stress and Displacement Constraints Using the Hybrid Cellular Automaton Method ................................................... 499 G. Yoon, J. Jensen, O. Sigmund Topological Design of Acoustic-Structure Interaction Structures with the Mixed Finite Element Method............................................................... 500

(ID-2008)

MS.38 Composite Molding-Numerical Simulations and Applications Organizers: Dimitrovova, Z., Bassir, D.

N. Abdelkader, Y. Chevalier, S. Aguib, N. Chikh (ID-2643) Cinematique Influence on the Vibrations of Stratified Plates ................. 501 D. Bassir, W. Zhang, S. Guessasma Optimization of Resign Transfer Molding Process by a Virtual Manufacturing and a Genetic Algorithms ........................................................... 502

(ID-1604)

Z. Dimitrovova Post-Processing Techniques Suitability for Mesolevel Free Boundary Flows .................................................................................................. 503

(ID-2335)

M. Nossek, M. Sauer, K. Thoma Adaptive Simulation of Cohesive Interface Debonding for Crash and Impact Analyses............................................................................................ 504 (ID-1942)

S. Shigehisa, Y. Miyata, H. Ochiai (ID-2454) Failure Analysis of Cement-Treated Soil by FEM Implemented with Particle Discretization ................................................................................. 505 A. Zoheir, T. Nabil, R. Ayad, A. Samir, B. Malek An Homogenisation Procedure for Cardboard and Stitched Sandwiches Using Respectively Analytical and Numerical Simulation ............. 506

(ID-2551)

MS.39 Material Models for Composites at Different Length Scales Organizers: Rolfes, R.

D. Ballhause, M. König, B. Kröplin (ID-1489) Modelling of Woven Fabrics with the Discrete Element Method........... 507 M. Böl, S. Reese (ID-1441) A New Approach for the Simulation of Damage Effects in Rubber-Like Materials Using Chain Statistics .................................................... 508 L. Gornet, S. Marguet, G. Marckmann Numerical Modelling of Nomex® Honeycomb Cores : Failure and Effective Elastic Properties........................................................................... 509

(ID-2066)

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G. Haasemann, M. Kästner, V. Ulbricht (ID-2024) Multi-Scale Modelling and Simulation of Textile Reinforced Materials.............................................................................................................. 510 L. Kaczmarczyk, Z. Waszczyszyn Enforcing Boundary Conditions in Micro-Macro Transition for Second Order Continuum .................................................................................... 511

(ID-2562)

M. Klüppel, J. Meier, M. Ramspeck Micro-Structure Based Modeling of Elastomer Materials ...................... 512

(ID-1511)

F. Lipperman, M. Ryvkin, M. Fuchs Analysis and Effective Properties of Honeycombs with NonSymmetric Unit Cells .......................................................................................... 513

(ID-1802)

M. Luxner, J. Stampfl, A. Woesz, P. Fratzl, H. Pettermann Influence of Defects and Perturbations on the Performance of 3D Open Cell Structures ........................................................................................... 514

(ID-1752)

A. Muhr Mechanics of Elastometer-Shim Laminates............................................ 515

(ID-2012)

J. Skocek, J. Zeman, M. Šejnoha Application of the Mori-Tanaka Method to Analysis of Woven Composites with Imperfections ........................................................................... 516

(ID-1636)

S. Smaoui, B. Abelwahed, D. Irini, D. Hélène An Homogenization Iterative Process for Nonlinear Materials Applied to Compacted Clays............................................................................... 517 (ID-2241)

M. Timmel, M. Kaliske, S. Kolling, R. Mueller (ID-1938) A Micromechanical Approach for the Simulation of Rubberlike Materials with Damage........................................................................................ 518 W. Wajda, H. Paul Influence of Grains Misorientation on Material Hardening on Example of Aluminum Bicrystals Deformed in Channel Die ............................. 519 (ID-2544)

MS.40 Meshless Methods Organizers: Alves, C.

C. Alves, P. Antunes (ID-2532) The Method of Fundamental Solutions Applied to the Calculation of Eigensolutions for Simply Connected Plates .................................................. 520 C. Alves, S. Valtchev Comparison Between Meshfree and Boundary Element Methods Applied to BVPS in Domains with Corners ........................................................ 521

(ID-2558)

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C. Alves, N. Martins (ID-2618) The Method of Fundamental Solutions Applied to a Heat Conduction Inverse Problem ............................................................................... 522 M. Arroyo, M. Ortiz (ID-1540) Local Maximum-Entropy Approximation Schemes................................ 523 J. Belinha, L. Dinis Analysis of 2D Problems Resorting to a New Meshless Method............ 524

(ID-1935)

M. Bitaraf, S. Mohammadi (ID-1887) Solving the Chloride Diffusion Equation in Concrete Structures for Prediction of Initiation Time of Corrosion .................................................... 525 S. Chantasiriwan Solution of the Stationary Three-Dimensional Navier-Stokes Equations by Using Radial Basis Functions........................................................ 526

(ID-1732)

T. Most, C. Bucher An Enhanced Moving Least Squares Interpolation for the Element-Free Galerkin Method ........................................................................... 527

(ID-1233)

H. Netuzhylov Enforcement of Boundary Conditions in Meshfree Methods Using Interpolating Moving Least Squares.................................................................... 528

(ID-1969)

V. Rosca, V. Leitão A Simple and Less-Costly Integration of Meshless Galerkin Weak Form .................................................................................................................... 529

(ID-2574)

C. Tiago, P. Pimenta Geometrically Exact Analysis of Shells by a Meshless Approach.......... 530

(ID-2159)

O. Valencia, F. Gómez-Escalonilla, F. Urbinati, J. López-Díez Weight Functions Analysis in Elastostatic Problems for Meshless Element Free Galerkin Method ........................................................................... 531

(ID-2398)

MS.41 Metal Forming Organizers: Cesar Sá, J., Pietrzyk, M.

F. Abbassi, O. Pantale, A. Zghal, R. Rakotomalala (ID-1575) Prediction of Sheet Metal Formability (FLD) by Using Diverse Method ................................................................................................................ 532 J. Alves, M. Oliveira, L. Menezes Modeling Drawbeads in Deep Drawing Simulations .............................. 533

(ID-1848)

S. Benke, G. Laschet On a 2-Phase Finite Element Model for the Coherent Mushy Zone in Casting Applications ....................................................................................... 534

(ID-1501)

xlviii Table of Contents

H. Christophe, A. Said, D. Loic (ID-1588) Modeling of Ductile Behavior of Metals Under a Wide Range of Loading Rates: Semi-Empirical Approach.......................................................... 535 S. Ding, B. Daniel, P. Meehan A New Relaxation Method for Roll Forming Problems.......................... 536

(ID-1362)

M. Glowacki, M. Hojny Computer Modelling of Deformation of Steel Samples with Mushy Zone......................................................................................................... 537

(ID-1753)

B. Haddag, F. Abed-Meraim, T. Balan Prediction of Strain Localisation in Forming Process Using Advanced Elastic-Plastic Behaviour Models Coupled with Damage.................. 538

(ID-2055)

R. Jorge, A. Roque, M. Parente, A. Fernandes, R. Valente Symulation of Hydroforming on Tailor-Welded Tubular Blanks Using Solid-Shell Finite Elements ...................................................................... 539

(ID-1988)

T. Lelotte, L. Duchêne, A. Habraken Fast Method to Predict an Earing Profile Based on Lankford's Coefficients and Yield Locus .............................................................................. 540

(ID-2402)

L. Madej, A. Zmudzki, P. Hodgson, M. Pietrzyk Possibilities of Application of the Multi Scale Strain Localization Café ..................................................................................................................... 541

(ID-2231)

K. Nabil, K. Yahia Thermal Modeling of D. C. Continuous Casting Process of a AlMg Alloy ............................................................................................................. 542

(ID-1915)

J. Ponthot, R. Boman, L. Papeleux, Q. Bui A 3D Arbitrary Lagrangian Eulerian Formulation for the Numerical Simulation of Forming Processes. ..................................................... 543

(ID-2053)

M. Poursina, H. Ebrahimi, J. Parvizian A Study for the Constitutive Equations of 1.4021 and 1.4841 Stainless Steels in Hot Deformation .................................................................... 544

(ID-2503)

W. Rasp, A. Yusupov Modelling of Spread and Side-Form Function in Hot Rolling by Different Upper-Bound Approaches ................................................................... 545

(ID-2243)

J. Sá, C. Zheng Non Local Models and Length Scale Effects on Metal Forming Processes ............................................................................................................. 546

(ID-2336)

M. Salmanitehrani, M. Poursina Analysis of Thermal Cracking of an Industrial Duct Using Finite Element Simulation ............................................................................................. 547

(ID-2500)

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xlix

M. Schwarze, S. Reese (ID-2393) Analysis of Forming Processes with Efficient Finite Element Procedures ........................................................................................................... 548 P. Teixeira, F. Pires, A. Santos, J. Sá (ID-2413) Numerical Investigation of Fracture Onset in Sheet Metal Forming ............................................................................................................... 549 MS.42 Modeling in Mechanobiology Organizers: Lekszycki, T.

A. Andreykiv, F. Keulen (ID-2446) Effect of Surface Geometry and Local Mechanical Environment on Periimplant Tissue Differentiation:A Finite Element Study........................... 550 C. Bonifasi-Lista, E. Cherkaev Identification of Bone Structure From Effective Measurements............. 551

(ID-2190)

M. Cioffi, F. Galbusera, M. Raimondi, F. Boschetti, G. Dubini Computational Modeling of Mechanical Environment Within Tissue Engineered Cartilage................................................................................ 552

(ID-1868)

H. Kim, P. Clement, J. Cunningham Stress-Based Optimum and Bone Architecture....................................... 553

(ID-2340)

P. Kowalczyk Parameterized Orthotropic Cellular Microstructures As Mechanical Models of Cancellous Bone ............................................................. 554

(ID-1151)

M. Nowak Modeling of Cancellous Bone Surface Adaptation Based on the 3Dimensional Trabeculae Topology Evolution.................................................... 555

(ID-1004)

J. Pierre, C. Oddou Theoretical Analysis of the Remodeling Processes in Bony Tissue Engineered Implants............................................................................................ 556

(ID-2294)

G. Sciarra, T. Lekszycki Bone Remodeling Description Based on Micro Mechanical/Biological Effects ............................................................................ 557

(ID-1768)

MS.43 Modelling of Functionally graded materials and structures Organizers: Batra, R., Ferreira, A.

L. Aebi, J. Vollmann, J. Dual (ID-1497) Two-Dimensional Elastodynamic Wave Propagation in Graded Structures............................................................................................................. 558

l

Table of Contents

H. Argeso, A. Eraslan (ID-1184) A Computational Study on Functionally Graded Rotating Solid Shafts: Analysis of Preliminary Results .............................................................. 559 R. Batra, J. Xiao, D. Gilhooley, M. Mccarthy, J. Gillespie (ID-2013) Static Analysis of Thick Functionally Graded Plates by Using a Higher-Order Shear and Normal Deformable Plate Theory ................................ 560 M. Bocciarelli, G. Bolzon, G. Maier Three-Point-Bending and Indentation Tests for the Calibration of Functionally Graded Material Models by Inverse Analysis ................................ 561

(ID-1747)

A. Ferreira, G. Fasshauer, C. Roque, R. Jorge Static Deformations and Natural Frequencies of Functionally Graded Plates by a Hybrid Meshless Method...................................................... 562

(ID-1339)

S. Hamza-Cherif, A. Houmat, A. Hadjoui Graded Fourier P-Element Calculation of Steady State Heat Conduction in Functionally Graded Materials .................................................... 563

(ID-1111)

T. Nguyen, K. Sab, G. Bonnet A Reissner-Mindlin Plate Model for Functionally Graded Materials.............................................................................................................. 564

(ID-2254)

W. Szymezyk A Review of the Chosen Problems of FEM Modeling of Surface Coatings............................................................................................................... 565

(ID-1823)

MS.44 Multibody Dynamics Organizers: Ambrósio, J.

C. Bottasso, D. Dopico, L. Trainelli (ID-1764) On the Optimal Scaling of Index Three DAEs in Multibody Dynamics............................................................................................................. 566 S. Breun, R. Callies Redundant Optimal Control of Manipulators Along Specified Paths .................................................................................................................... 567

(ID-1874)

M. Ebbesen, M. Hansen, T. Andersen Optimal Tool Point Control of Hydraulically Actuated Flexible Multibody System with an Operator-In-The-Loop.............................................. 568

(ID-2433)

D. Franitza, T. Lichtneckert Slim Elastic Structures with Transversal Isotropic Material Properties Under Finite Deformations................................................................. 569

(ID-2281)

M. Hajzman, P. Polach Modelling of Flexible Rods Falling in Fluid with Possible Contacts............................................................................................................... 570

(ID-1912)

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li

H. Irschik, M. Nader, C. Zehetner (ID-1934) Tracking of Displacements in Smart Elastic Beams Subjected to Rigid Body Motions ............................................................................................ 571 E. Lens, A. Cardona (ID-2509) Dynamic Analysis of Constrained Nonlinear Multibody Systems with Intermittent Contact..................................................................................... 572 D. Leshchenko, L. Akulenko, S. Suksova Evolution of Rotation of a Triaxial Satellite Under the Action of Gravitational and Light Pressure Torques ........................................................... 573 (ID-1085)

S. Leyendecker, P. Betsch, P. Steinmann (ID-2309) Mechanical Integrators for Nonlinear Flexible Multibody Dynamics............................................................................................................. 574 K. Lipinski Some Really Simple But Useful Model of Substitutable Elasticity Modelled As Elasticity in Six Subsequent Joints ................................................ 575

(ID-1757)

K. Lipinski Optimal Dampers Localization for a Body Under Double Load and the Body Behaviour for Some Intermediate Loads....................................... 576

(ID-1758)

P. Polach, M. Hajzman Design of Characteristics of Air Pressure Controlled Hydraulic Shock Absorbers in an Intercity Bus ................................................................... 577

(ID-1860)

R. Santos, V. Steffen, S. Saramago Robot Path Planning in a Constrained Workspace by Using Optimal Control Techniques ............................................................................... 578

(ID-1628)

G. Schanzer, R. Callies Optimal Control of Multi-Link Manipulators with Rivalling Actuators ............................................................................................................. 579

(ID-1771)

L. Sousa, P. Verissimo, J. Ambrósio Development of Validated Generic Road Vehicles for Crashworthiness Through Optimization Procedures ........................................... 580

(ID-2363)

M. Szczotka, I. Adamiec-Wójcik, S. Wojciech Developing Mathematical and Computer Models for Car Dynamics Using Joint Co-Ordinates and Homogenous Transformations ........... 581

(ID-1577)

MS.45 Multiphysics Modelling in Geomechanics Organizers: Borja, R. I., Montáns, F.J., Tamagnini, C.

J. Andrade, R. Borja (ID-1563) Finite Element Simulation of Deformation Bands in Saturated Granular Media with Inhomogeneous Porosities at the Meso-Scale ................... 582

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M. Caminero, F. Montáns (ID-1700) Computational Framework for Multilayer Plasticity Based on Critical State Soil Mechanics .............................................................................. 583 L. Hu, T. Hueckel Creep of Geomaterials Due to Coupled Damage and Spontaneous Mineral Dissolution............................................................................................. 584

(ID-2222)

R. Kohler, G. Hofstetter Validation of an Extended Cap Model for Partially Saturated Soils ....... 585

(ID-1257)

A. Mesgouez, G. Lefeuve-Mesgouez, A. Chambarel Partially Saturated Porous Medium Vibration Induced by an Impulsional Load................................................................................................. 586

(ID-1964)

M. Murad, C. Moyne Macroscopic Behavior of Smectitic Clays Derived From Nanostructure ...................................................................................................... 587

(ID-1103)

P. Rehermann, R. Borja, D. Pollard Mechanical Modeling of Multi-Layer Sedimentary Rock Folding ......... 588

(ID-1567)

D. Souza, A. Coutinho, J. Alves Fast Numerical Simulation of Porous Media Flows................................ 589

(ID-1536)

MS.46 Multiscale Mechanics of Biological Materials and Other Natural Composites Organizers: Hellmich, C.

F. Genna (ID-1129) A Micromechanically-Based Interface Model for the Periodontal Ligament.............................................................................................................. 590 Q. Grimal, K. Raum, P. Laugier Computation of Cortical Bone Macroscopic Properties From Microscopic Elastic Data..................................................................................... 591

(ID-1300)

R. Grytz, G. Meschke Computational Homogenization in Multi-Scale Shell Analysis at Large Strains........................................................................................................ 592

(ID-1238)

D. Katti, P. Ghosh, K. Katti Mineral Proximity Influences Protein Unfolding: a Molecular Dynamics Study .................................................................................................. 593

(ID-1998)

K. Katti, R. Bhowmik, D. Katti, D. Verma Modeling the Role of Interfaces on Mechanical Response in Composite Bone Biomaterials ............................................................................. 594

(ID-2018)

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P. Laugier (ID-2022) Finite-Difference Computations of Ultrasound Wave Propagation in Bone ................................................................................................................ 595 F. Moravec, M. Muller Microstructural Model of the Viscoelastic Behaviour of Biological Tissues ............................................................................................... 596

(ID-1815)

K. Okumura Scaling Views on Strength of Soft/Hard Composites ............................ 597

(ID-2501)

MS.47 Multiscale Method for Structural Non-Linear and Dynamic Problems Organizers: Allix, O., Rey, C.

A. Amini, D. Dureisseix, P. Cartraud, N. Buannic (ID-1310) A Micro-Macro Strategy for Ship Structural Analysis with FETIDP Method .......................................................................................................... 598 E. Baranger, O. Allix, L. Blanchard On the Use of Fourier Expansions for the Simulation of Elastic Composite Pipes with Defects............................................................................. 599

(ID-2550)

B. Bourel, A. Combescure, L. Valentin A Multigrid Approach for Non-Linear Structural Analysis in Explicit Dynamics ............................................................................................... 600

(ID-2647)

V. Carvelli, C. Corazza, C. Poggi Mechanical Behaviour of Textile Structures: Two-Scales Approach ............................................................................................................. 601

(ID-1336)

P. Cresta, O. Allix, C. Rey, S. Guinard On Multilevel Strategies for Nonlinear Computations with Domain Decomposition: Application to Post-Buckling ...................................... 602

(ID-2138)

M. Kuprys, R. Barauskas Textile Fabric Simulator: Collisions Handling at the Level of Yarns ................................................................................................................... 603

(ID-1839)

J. Pebrel, P. Gosselet, C. Rey A Computational Strategy for Contact Simulation.................................. 604

(ID-2345)

C. Rickelt, S. Reese (ID-2361) A Simulation Strategy for Life Time Calculations of Large, Partially Damaged Structures .............................................................................. 605 Y. Shiihara, O. Kuwazuru, N. Yoshikawa Finite Element Method in First-Principles Calculation........................... 606

(ID-2355)

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M. Spiridonakos, S. Fassois (ID-1840) Modelling and Simulation of Earthquake Ground Motion Via Functional Series TARMA Models with Wavelet Basis Functions .................... 607 MS.48 Neural Networks and Soft Computing in Solid and Structural Mechanics Organizers: Waszczyszyn, Z.

A. Borowiec, L. Ziemianski (ID-2582) Identification of Damage in Multispan Beams Using ParameterDependent Frequency Changes and Neural Networks ........................................ 608 A. Kucerova, M. Leps, J. Zeman Microplane Model Parameters Estimation Using Neural Networks ....... 609

(ID-1992)

K. Kuzniar (ID-2592) Anns and Linguistic Variables in the Analysis of Mine Induced Rockbursts Transmission to the High Building................................................... 610 W. Lu, L. Duan, F. Mora-Camino, R. Faye Differential Flatness of Aircraft Flight Dynamics and Neural Inversion.............................................................................................................. 611

(ID-1591)

MS.49 Nonlinear Dynamics of Moving Structures Organizers: Zahariev, E., Mayo Nunez, J.

A. Ayestarán, C. Graciano (ID-2635) Finite Element Analysis of an Energy Absorbing Crush Zone Using Expanded Metal ........................................................................................ 612 L. Desouza Design of Satellite Control System Using the Optimal Nonlinear Theory ................................................................................................................. 613

(ID-1017)

J. Dias, F. Antunes, M. Pereira Design for Crashworthiness of Train Structures with Simplified Multibody Models ............................................................................................... 614

(ID-2499)

J. Gerstmayr, M. Matikainen Analysis of Stress and Strain in the Absolute Nodal Coordinate Formulation with Nonlinear Material Behavior .................................................. 615

(ID-2612)

S. Hornstein, O. Gottlieb Nonlinear Multimode Dynamics of a Moving Microbeam for Noncontacting Atomic Force Microscopy........................................................... 616

(ID-1172)

Y. Lin, P. Nikravesh Model Reduction with Mean-Axes in Deformable Multibody Dynamics............................................................................................................. 617

(ID-2105)

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lv

J. Mayo (ID-1841) Impacts with Friction in Flexible Multibody Dynamics ......................... 618 J. Orden, R. Ortega A Conservative Augmented Lagrangian Algorithm for the Dynamics of Constrained Mechanical Systems................................................... 619

(ID-1968)

M. Piovan, R. Sampaio Non Linear Model for Coupled Axial/Torsional/Flexural Vibrations of Drill-Strings................................................................................... 620

(ID-1561)

A. Poulimenos, M. Spiridonakos, S. Fassois Identification of Time-Varying Structures Under Unobservable Excitation: an Overview and Experimental Comparison..................................... 621

(ID-1810)

I. Romero Variational Integrators for the Rigid Body Dynamics............................. 622

(ID-1961)

E. Zahariev (ID-1548) Planning and Optimization of Maneuver Strategy of Large Flexible Space Structures .................................................................................... 623 R. Zander, H. Ulbrich Free Plain Motion of Flexible Beams in MBS - a Comparison of Models................................................................................................................. 624

(ID-1515)

MS.50 Non-Linear Vibration of Structures Organizers: Ribeiro, P.

R. Arquier, B. Cochelin (ID-2051) Numerical Computation of Non Linear Modes of Elastic Structures............................................................................................................. 625 S. Bellizzi, R. Bouc Nonlinear Modes: Amplitude-Phase Formulation and Bifurcation Analysis............................................................................................................... 626

(ID-2455)

M. Haterbouch, P. Ribeiro, R. Benamar Multi-Modal Non-Linear Free Vibration of Thin Isotropic Circular Plates ..................................................................................................... 627

(ID-1603)

M. Kireitseu, G. Tomlinson Computational Approaches to Prediction of Damping Behavior of Nanoparticle-Reinforced Coatings and Foamy Structures .................................. 628

(ID-1143)

A. Kocsis, G. Károlyi Buckling Under Conservative and Nonconservative Load ..................... 629

(ID-1531)

N. Krancevic, M. Stegic, N. Vrankovic Bifurcation of Periodic Solutions in the Two-Degree-of-Freedom System with Clearances....................................................................................... 630

(ID-1218)

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R. Lewandowski (ID-1873) Perturbation Method for Strongly Non-Linear Free Vibrations of Beams .................................................................................................................. 631 P. Ribeiro, B. Cochelin, S. Bellizzi Vibrations of Shallow and Deep Shells by the P-Version Finite Element Method .................................................................................................. 632

(ID-2129)

N. Srinil, G. Rega Resonant Non-Linear Dynamic Responses of Horizontal Cables Via Kinematically Non-Condensed/Condensed Modeling.................................. 633

(ID-1627)

J. Thomsen Computing Effective Properties of Nonlinear Structures Exposed to Strong High-Frequency Loading at Multiple Frequencies .............................. 634

(ID-2552)

J. Zapomel Implementation of a Vapour Cavitation Into Computational Models of Rotors Supported by Long Journal Bearings...................................... 635

(ID-1105)

MS.51 Optimization and Robust Design for Industrial-sized Problems Organizers: Bletzinger, K. U., Duddeck, F., Meyer M.

F. Duddeck (ID-1470) Multi-Criteria Optimizations and Robustness Estimations for Crashworthiness, Structural Dynamics, and Acoustics ....................................... 636 K. Grossenbacher, F. Duddeck, M. Ganser, P. Hora Process Robustness in Sheet Metal Forming by an Integrated Engineering Strategy ........................................................................................... 637

(ID-2030)

M. Hansen, T. Andersen, J. Hansen, O. Mouritsen Topology Optimization of Robots Using Mapping Techniques.............. 638

(ID-2606)

H. Müllerschön, M. Hove, B. Mlekusch Optimization Strategies for Highly Non-Linear FE-Applications As Crashworthiness Applications........................................................................ 639

(ID-2388)

M. Ohsaki Local and Global Searches of Approximate Optimal Designs of Regular Frames.................................................................................................... 640

(ID-1174)

J. Reger, T. Schneider, C. Ehlert Optimisation of Car Body Parts Regarding Equivalent Radiation Power Using a Genetic Algorithm and Morphing ............................................... 641

(ID-1596)

K. Sedlaczek, P. Eberhard Grid-Based Topology Optimization of Rigid Body Mechanisms Using Different Problem Formulations ............................................................... 642

(ID-1498)

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lvii

H. Wenzel (ID-1903) Combining Optimization and Robust Engineering Methods in the Engineering Product Design Process................................................................... 643 MS.56 Shape and Topological Sensitivity Analysis: Theory and Applications Organizers: Feijóo, R.., Taroco, E.

G. Allaire, F. Jouve, F. Gournay, A. Toader (ID-2691) Combining Topological and Shape Derivatives in Structural Optimization........................................................................................................ 644 F. Barthold, K. Wiechmann A Comparison of Displacement and Mixed Finite Element Formulations for Variational Design Sensitivity Analysis .................................. 645

(ID-2003)

M. Bonnet (ID-2677) Inverse Acoustic Scattering by Small-Obstacle Expansion of Misfit Function .................................................................................................... 646 J. Faria, R. Feijóo, A. Novotny, E. Taroco, C. Padra Second Order Topological Sensitivity Analysis...................................... 647

(ID-2029)

M. Langelaar, F. Keulen (ID-1905) Sensitivity Analysis of Shape Memory Alloy Shells .............................. 648 L. Miegroet, T. Jacobs, E. Lemaire, P. Duysinx (ID-2026) Stress Constrained Optimization Using X-FEM and Level Set Description .......................................................................................................... 649 A. Myslinski Level Set Method for Optimization of Contact Problems ....................... 650

(ID-2067)

H. Pham, C. Bucher An Iterative Procedure for Model Updating Based on Selective Sensitivity............................................................................................................ 651

(ID-1083)

MS.57 Shell and Spatial Structures Organizers: Ramm, E.

M. Birsan (ID-1462) Extension, Bending and Torsion of Cylindrical Cosserat Shells Made From a Porous Elastic Material ................................................................. 652 N. Dung, G. Wells A Study of Discontinuous Galerkin Methods for Thin Bending Problems.............................................................................................................. 653

(ID-2449)

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M. Frenzel, M. Bischoff, W. Wall (ID-2611) Discrete Strain Gap (Dsg) Solid Finite Elements at Large Deformations for Non-Linear Analysis of Shells and Solids .............................. 654 E. Gal, M. Zelkha, R. Levy A Simple Co-Rotational Geometrically Non-Linear Membrane Finite Element Wrinkling Analysis ..................................................................... 655

(ID-1629)

B. Izzuddin Co-Rotational System Definitions for Large Displacement Triangular and Quadrilateral Shell Elements ...................................................... 656

(ID-1609)

P. Khosravi, R. Ganesan, R. Sedaghati Non-Linear Analysis of Composite Plates and Shells Using a New Shell Element ...................................................................................................... 657

(ID-1614)

J. Mäkinen, H. Marjamäki Total and Updated Lagrangian Geometrically Exact Beam Elements .............................................................................................................. 658

(ID-1683)

E. Maunder, B. Izzuddin Large Displacement Analysis of Plates Using Hybrid Equilibrium Elements .............................................................................................................. 659 (ID-1541)

H. Noguchi, F. Fujii, Y. Ishihara (ID-2271) Buckling Modes of Large-Scale Shell Structures Automatically Detected From Linearized Stiffness by Iterative Solvers .................................... 660 I. Oliveira, E. Campello, P. Pimenta (ID-2415) Finite Element Analysis of the Wrinkling of Orthotropic Membranes .......................................................................................................... 661 R. Schlebusch, B. Zastrau Theory and Numerics of a Surface-Related Shell Formulation............... 662

(ID-2284)

M. Spalatelu-Lazar, F. Léné, N. Turbé Modelling and Optimization of Sails ...................................................... 663

(ID-1989)

M. Tanaka, H. Noguchi Instability Analysis of Thin-Walled Structures Using Incompressible Hyperelastic Shell Elements....................................................... 664

(ID-2253)

MS.58 Simulation of Non-Gaussian Stochastic Processes and Fields with Applications to Structural Engineering Problems Organizers: Papadrakakis, M., Stefanou, G.

J. Cruz, M. Gutiérrez, L. Koene (ID-1206) Stochastic Simulation of Pitting Corrosion ............................................. 665

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lix

V. Denoël, H. Degée (ID-1227) Non Gaussian Response of Bridges Subjected to Turbulent Wind Effect of the Non Linearity of Aerodynamic Coefficients .................................. 666 M. Grigoriu Simulation of Non-Gaussian Stochastic Processes and Fields with Applications to Structural Engineering Problems ............................................... 667

(ID-1261)

N. Lagaros, G. Stefanou, M. Papadrakakis A Novel Approach for the Efficient Simulation of Highly Skewed Non-Gaussian Stochastic Fields .......................................................................... 668

(ID-1932)

M. Schevenels, G. Lombaert, G. Degrande, D. Degrauwe, B. Schoors The Wave Propagation in a Vertically Inhomogeneous Soil with a Random Dynamic Shear Modulus....................................................................... 669

(ID-1854)

G. Stefanou, M. Papadrakakis On the Karhunen-Loeve Expansion and Spectral Representation Methods for the Simulation of Gaussian Stochastic Fields ................................. 670

(ID-1940)

MS.59 Soft Tissue Organizers: Jorge, R.

J. Barbosa, R. Jorge, M. Parente, A. Fernandes, T. Mascarenhas, B. Patricio (ID-1452) Study of Mechanical Properties of Human Skin ..................................... 671 R. Basto, C. Costa, W. Pereira, M. Krüger, H. Orlande, H. Fonseca Thermophysical Properties of Different Samples of TissueMimicking Materials for Ultrasound Hyperthermia Phantoms ........................... 672

(ID-2160)

A. Cavicchi, L. Gambarotta, R. Massabò Different Computational Approaches in the Modeling of Wrinkling of Biological Membranes................................................................... 673

(ID-2426)

F. Gentil, R. Jorge, A. Ferreira, M. Parente, M. Moreira, E. Almeida Dynamic Study of the Middle Ear........................................................... 674

(ID-1496)

S. Kuzukami, N. Yoshikawa, O. Kuwazuru Image-Base Inverse Problems to Identify Three-Dimensional Displacement Field.............................................................................................. 675

(ID-1425)

P. Martins, R. Jorge, A. Ferreira, F. Gentil Experimental Study of the Middle Ear Biological Support Structures............................................................................................................. 676

(ID-1527)

T. Miyashita, H. Yamauchi, M. Inui, H. Yamakawa A Real-Time FEM Simulation for Cutting Operation Using Haptic Device.................................................................................................................. 677

(ID-2338)

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Table of Contents

M. Parente, R. Jorge, A. Fernandes, T. Mascarenhas, J. Martins (ID-1473) The Biomechanical Behavior of the Pelvic Floor Muscles During a Vaginal Delivery............................................................................................... 678 MS.60 Stability and Non-Linear Behaviour of Thin-Walled Members and Structures Organizers: Camotim, D.

S. Ádány, N. Silvestre, B. Schafer, D. Camotim (ID-1443) Buckling Analysis of Unbranched Thin-Walled Members: Generalised Beam Theory and Constrained Finite Strip Method ........................ 679 A. Andrade, D. Camotim, P. Costa Lateral-Torsional Buckling Analysis of Singly Symmetric WebTapered I-Beams Using Finite Elements and Finite Differences ........................ 680

(ID-1974)

J. André, A. Baptista Stability of Telescopic Props for Temporary Structures ......................... 681

(ID-1545)

C. Basaglia, D. Camotim, N. Silvestre Formulation of a GBT-Based Finite Element to Analyse the Global Buckling Behaviour of Plane/Spatial Thin-Walled Frames..................... 682

(ID-1943)

R. Bebiano, N. Silvestre, D. Camotim GBT-Based Finite Element to Analyse the Buckling Behaviour of Thin-Walled Members Subjected to Non-Uniform Bending............................... 683

(ID-2039)

N. Boissonnade, H. Degée A Non-Linear 3-D Beam Finite Element for the Study of Steel Frames with Tapered Members. .......................................................................... 684 (ID-1248)

R. Castro, J. Silva, P. Vellasco, S. Andrade, L. Lima, L. Neves (ID-1726) Non-Linear Dynamical Response of Steel Portal Frames with Semi-Rigid Connections...................................................................................... 685 N. Chen, C. Soares Buckling Analysis of Stiffened Composite Panels.................................. 686

(ID-2002)

H. Degée, N. Boissonnade (ID-1228) Interactive Buckling of Thin-Walled Rectangular Hollow Sections - Comparison Between Modified Beam Models and .......................................... 687 R. Degenhardt, A. Kling, K. Rohwer Design and Analysis of Composite Panels.............................................. 688

(ID-1914)

P. Dinis, D. Camotim On the Use of Shell Fea to Assess the Local Buckling & PostBuckling Behaviour of Cold-Formed Steel Thin-Walled Members.................... 689

(ID-1828)

N. Fallah A Finite Volume Method for Plate Buckling Analysis ........................... 690

(ID-1353)

Table of Contents

lxi

A. Formisano, G. Matteis, M. Maruzzelli, F. Mazzolani (ID-2517) Design of Slender Steel Shear Panels: a Numerical Study...................... 691 R. Goncalves, P. Grognec, D. Camotim (ID-1953) Plastic Bifurcation Fea of Thin-Walled Members: Thin Shell Elements Vs. GBT-Based Beam Elements.......................................................... 692 R. Gonçalves, M. Ritto-Corrêa, D. Camotim A Large Displacement and Finite Rotation Thin-Walled Beam Finite Element Formulation................................................................................. 693

(ID-2017)

R. Goñi, E. Bayo A New Method to Assess the Rotation Capacity of Structural Hollow Sections Based in Multibody Theory ..................................................... 694

(ID-1474)

M. Haßler, K. Schweizerhof On the Stability Analysis of Thin Walled Shell Structures Containing Gas Or Fluid ..................................................................................... 695

(ID-2376)

M. Khedmati, M. Zareei Sensitivity Analysis on Ultimate Strength of Stiffened Aluminum Plates Under Combined Inplane Compression and Lateral Presuure .................. 696

(ID-2326)

H. Ovesy, S. Ghannadpour Large Deflection Behavior of Functionally Graded Plates Under Pressure Loads, Using Finite Strip Method......................................................... 697

(ID-1243)

H. Ovesy, H. Hosseini-Toudeshky, M. Kharazi (ID-1744) Buckling Analysis of Laminates with Multiple Through-TheWidth Delaminations by Using Spring Simulated Model ................................... 698 P. Real, N. Lopes, L. Silva, C. Rebelo Numerical Validation of the Eurocode 3 Design Rules for LateralTorsional Buckling of I-Beams ........................................................................... 699

(ID-2126)

C. Rebelo, L. Silva, P. Real, N. Lopes Statistical Evaluation of the Eurocode 3 Design Rules for LateralTorsional Buckling of I-Beams ........................................................................... 700

(ID-2416)

M. Ritto-Corrêa, D. Camotim On the Interpolation of Rotations and Rigid-Body Motions in Nonlinear Beam Finite Elements......................................................................... 701

(ID-2050)

K. Rzeszut, A. Garstecki Stability of Beams and Columns Made of Thin-Walled ColdFormed Sections Accounting for Imperfections.................................................. 702

(ID-1944)

W. Schneider, M. Gettel Equivalent Geometric Imperfections for Steel Shell Structures Subject to Combined Loading ............................................................................. 703

(ID-2004)

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Table of Contents

I. Shufrin, M. Eisenberger (ID-2404) Shear Buckling of Thin Plates with Constant In-Plane Stresses ............. 704 N. Silva, N. Silvestre, D. Camotim GBT Formulation to Analyse the Buckling Behaviour of Frp Composite Branched Thin-Walled Members ...................................................... 705

(ID-2282)

N. Silvestre, N. Silva On the Influence of Material Couplings on the Buckling Behaviour of FRP Thin-Walled Columns – a GBT-Based Approach ................. 706

(ID-2272)

C. Soares, R. Luís Ultimate Strength of Plate Assemblies with Localized Imperfection Subjected to Compressive Loads ................................................... 707

(ID-1948)

R. Vieira, F. Virtuoso, E. Pereira Higher Order Analysis of a Thin-Walled Beam...................................... 708

(ID-2370)

MS.61 Structural and Multidisciplinary Optimization Organizers: Herskovits, J.

M. Abolbashari, M. Majdi (ID-2387) Structural Optimization Using Optimizer Program................................. 709 D. Babkin, E. Kligman, V. Matveyenko, N. Yurlova Optimization of Dissipative Characteristics of Structures on the Basis of Problems on Natural Vibrations of Viscoelastic Solids......................... 710 (ID-1807)

K. Bhagate, P. Pawar, A. Singh, J. Ye (ID-1477) Accuracy of Design Sensitivity Analysis for Optimization of Structures for Small Strain Theory by Finite Element Method ........................... 711 D. Bojczuk Method of Optimal Reinforcement of Structures Based on Topological Derivative........................................................................................ 712

(ID-1601)

R. Botez, A. Dinu, I. Cotoi Optimisation of Unsteady Aerodynamic Forces for Aircraft Aeroservoelastic Studies ..................................................................................... 713

(ID-1606)

R. Cadete, J. Dias, M. Pereira A Multidisciplinary Design Optimization Framework Applied to Mechanical Systems ............................................................................................ 714

(ID-2371)

A. Canelas, P. Mappa, J. Herskovits, J. Telles Shape Optimization Using the Boundary Elements and a Sand Interior Point Algorithm for Constrained Optimization ...................................... 715

(ID-2448)

P. Coelho, P. Fernandes, J. Cardoso, J. Guedes, H. Rodrigues A Three-Dimensional Hierarchical Model for Topology Optimization of Structures .................................................................................. 716

(ID-2481)

Table of Contents

lxiii

A. Csébfalvi, G. Csébfalvi (ID-2019) A New Hybrid Meta-Heuristic Method for Optimal Design of Space Trusses with Elastic-Plastic Collapse Constraints..................................... 717 B. Desmorat The Concept of Homogeneous Thermodynamical Potentials for Nonlinear Structural Rigidity Optimization ........................................................ 718

(ID-2042)

L. Desouza Satellite Attitude Control System Parameters Optimization ................... 719

(ID-1097)

V. Dubeux, J. Herskovits, S. Mazorche A Limited Memory Quasi-Newton Preconditioner for Large Scale Optimization........................................................................................................ 720 (ID-2600)

M. Ebbesen, M. Hansen, N. Pedersen (ID-2001) Design Optimization of Conveyor Systems ............................................ 721 J. Herskovits, E. Goulart, M. Aroztegui Sparse Quasi-Newton Matrices for Large Size Optimization with FAIPA, the Feasible Arc Interior Point Algorithm.............................................. 722

(ID-1977)

S. Holopainen Sensitivity and Sizing of Nonlinear Structures Made of Anisotropic Rubber-Like Material ...................................................................... 723

(ID-1241)

L. Johansen, E. Lund Optimization of Laminated Composite Structures Using Delamination Criteria and Adaptive Models ....................................................... 724

(ID-1896)

J. Kock, C. Strauss, C. Pohl, P. Wyk, P. Botes Yeast Biomechanics ................................................................................ 725

(ID-1057)

J. Kruzelecki, M. Król Optimization of Postbuckling Path for Cylindrical Shells Under External Pressure................................................................................................. 726

(ID-2123)

R. López, A. Tovar, C. Narváez Structural Analysis in Continuum Media Using Cellular Automata ....... 727

(ID-1742)

K. Mikhail, I. Egorov, S. Yuri, F. Konstantin Multidisciplinary Optimization of Complex Technical Systems ............ 728

(ID-2211)

J. Roche Adaptive Shape Optimization Method .................................................... 729

(ID-1235)

M. Secanell, B. Carnes, A. Suleman, N. Djilali A Pem Fuel Cell Cathode Model for Gradient-Based Optimization ....... 730

(ID-1553)

I. Shevtsov, V. Markine, C. Esveld, M. Markina Optimisation of a Railway Wheel Profile ............................................... 731

(ID-1774)

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Table of Contents

J. Stegmann, M. Stolpe (ID-1861) Discrete Material Optimization of Laminated Composites - Simp Vs. Global Optimization...................................................................................... 732 K. Svanberg, M. Werme First and Second Order Sensitivities of Functions with Respect to Binary Variables and Their Application in Topology Optimization ................... 733

(ID-1867)

A. Vincenti, P. Vannucci Optimal Design of Smart Composite Laminates by the Polar Method and the Genetic Algorithm Bianca ......................................................... 734

(ID-2184)

C. Wang, B. Duan, F. Zheng, H. Cao On Optimization Platform for Coupled Structural-Electromagnetic Performances of Large Reflector Antennas......................................................... 735

(ID-1570)

H. Wang, J. Croll, N. Yamamoto, S. Yamada Optimising Buckling Capacities for Composite Shells ........................... 736

(ID-2511)

B. Wilczynski, Z. Mróz Multiaxial Plastic Hardening Models Used in Shape Optimization with Respect to Fatigue Life................................................................................ 737

(ID-2269)

A. Wit, A. Lipka, E. Ramm, F. Keulen Multi-Level Optimization of Material and Structural Layout ................. 738

(ID-2588)

MS.62 Structural Dynamics Organizers: Azevedo, J.

R. Almeida, J. Silva (ID-1727) A Stochastic Modelling of the Dynamical Response of Highway Bridge Decks Under Traffic Loads ..................................................................... 739 S. Amiri, M. Barghian, M. Fard, J. Safadoust Comparisation of Two Dimensional Nonlinear Analysis of Integral Abutment Bridge and Simply Supported Bridge ................................... 740

(ID-1095)

J. Chang, I. Huang, W. Hou, P. Chang Dynamic Stability Analysis of Truss Structures Under Nonconservative Constant and Pulsating Follower Forces ................................. 741

(ID-2597)

A. Davaran, M. Fard, S. Amiri, A. Kashefi Comparison of Concrete Tall Building Behavior Using an Intermittent Shear Walls Form in One Frame ..................................................... 742

(ID-1602)

S. François, G. Degrande An Iterative Coupled Boundary-Finite Element Method for the Dynamic Response of Structures......................................................................... 743

(ID-2666)

Table of Contents

lxv

S. Fransen, D. Rixen (ID-2090) Pendulum Mode Control in the Dynamic Analysis of Lift-Off of Launchers ............................................................................................................ 744 M. Gao, J. Pan, J. Xiong Dynamic Response of Long Span Cable-Stayed Bridge Subjected to Earthquake and Moving Train......................................................................... 745

(ID-2291)

M. Gutiérrez, H. Askes Parametrisation of the Newmark Time Integrator for Non-Linear Solid Dynamics ................................................................................................... 746

(ID-1201)

S. Krenk Global Formulation of Conservative Time Integration by the Increment of the Geometric Stiffness .................................................................. 747

(ID-2477)

D. Makovicka, D. Makovicka Response Analysis of Building Loaded by Groundborne Transient Vibration.............................................................................................................. 748

(ID-1220)

H. Marjamäki, J. Mäkinen Different Approaches in Modelling Boom Lifting Movement................ 749

(ID-1528)

B. Möller, W. Graf, A. Hoffmann, J. Sickert, F. Steinigen (ID-2232) Textile Reinforced Concrete Structures Under Uncertain Dynamic Loading Processes ............................................................................................... 750 M. Ursu Shock Response Spectrum Analysis for Measured Earthquake Data ..................................................................................................................... 751

(ID-1973)

MS.64 System Identification and Finite Element Updating Organizers: Cunha, A.

A. Frigerio, E. Bom, G. Mazzà (ID-1444) Histride: an Integrated Software for Dynamic Structural Identification ....................................................................................................... 752 J. Lee, J. Kim, Y. Kim Damage Detection by the Topology Design Formulation Using Modal Parameters................................................................................................ 753

(ID-1739)

F. Magalhães, B. Costa, A. Cunha, E. Caetano Experimental Validation of the Finite Element Modelling of Pinhão Bridge ...................................................................................................... 754

(ID-2661)

S. Oliveira, P. Mendes Development of a Cabril Dam Finite Element Model for Dynamic Analysis Using Ambient Vibration Tests Results ............................................... 755

(ID-2077)

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Table of Contents

D. Rebolho, L. Souza, E. Belo, F. Marques (ID-2054) Application of Eera Method for Identification of Modal Parameters of a Simulated Aircraft ..................................................................... 756 E. Reynders, G. Roeck Reference-Based Combined Deterministic-Stochastic Subspace Identification for Experimental and Operational Model Analysis....................... 757

(ID-2508)

MS.65 Temperature and time dependent effects in steel and concrete structures Organizers: Silva, L. S., Júlio, E.

D. Costa, V. Silva, E. Júlio (ID-1253) Numerical Modelling of Time-Dependent Behaviour of High Strength Concrete Beams .................................................................................... 758 G. Ranzi, M. Bradford, P. Ansourian Behaviour of Composite Steel-Concrete Beams with Longitudinal and Transverse Partial Interaction in Fire............................................................ 759

(ID-1721)

G. Ranzi, P. Ansourian, L. Dezi, S. Zhang Partial Interaction Analysis of Multi-Layered Composite Beams Accounting for Time Effects ............................................................................... 760

(ID-1731)

A. Santiago, L. Silva, P. Real Numerical Behaviour of Steel Sub-Frame System in Fire ...................... 761

(ID-1814)

MS. 66 Vehicle Dynamics Organizers: Schiehlen, W.

J. Borges, M. Leal, R. Filho, J. Rezende (ID-2556) Structure Design and Dynamic Analysis of Vehicle Using Metamodeling and Optimization Techniques...................................................... 762 F. Braghin, E. Sabbioni, F. Cheli Race Driver Model: Identification of the Driver’s Inputs ....................... 763

(ID-1449)

F. Braghin, S. Melzi, F. Cheli Race Driver Model: Trajectory Planning ................................................ 764

(ID-1450)

F. Cheli, E. Giangiulio, E. Sabbioni, A. Concas A Simplified Abs Numerical Model for Actively Controlled Vehicle Dynamic Simulations: Validation with Experimental Data ................... 765

(ID-2096)

F. Cheli, G. Diana, F. Ripamonti, G. Tomasini, G. Zanetti Aerodynamic Sensitivity Analysis of the New Emuv250 Train to Cross Wind by Wind Tunnel Tests and CFD Analysis ....................................... 766

(ID-2391)

Table of Contents

lxvii

K. Dufva, K. Kerkkänen, L. Maqueda, A. Shabana (ID-1169) Three-Dimensional Large Deformation Finite Elment Analysis of Belt Drives........................................................................................................... 767 A. Freitas, R. Silva, J. Dias Multibody and Finite Element Models for the Design of Motorcyclist’s Roadside Protections ................................................................... 768

(ID-2397)

J. Landre, L. Saturnino, M. Becker, L. Patrício, C. Barcellos An Integrated Educational Tool for Vehicle Dynamical Response Studies ................................................................................................................. 769

(ID-1162)

J. Massat, A. Bobillot, J. Laine Robust Methods for Detecting Defects in Overhead Contact Line Based on Simulation Results ............................................................................... 770

(ID-1479)

B. Mavroudakis, P. Eberhard Mode Decoupling Vehicle Suspension System Applied to Race Car ....................................................................................................................... 771

(ID-1751)

J. Meijaard, A. Schwab Linearized Equations for an Extended Bicycle Model............................ 772

(ID-1715)

L. Patrício, M. Becker, J. Landre, C. Barcellos A New Vehicle 3D Model with 7 Degrees of Freedom for Vehicle Dynamical Response Studies .............................................................................. 773

(ID-1163)

J. Pombo, J. Ambrósio A Hertzian Contact Formulation for the Wheel-Rail Contact Problem in Railway Dynamics ............................................................................ 774 (ID-2639)

R. Portal, J. Dias (ID-2396) Multibody Models for Vehicle Accident Reconstruction........................ 775 G. Rill (ID-2242)

First Order Tire Dynamics ...................................................................... 776

A. Schwab, J. Meijaard, J. Kooijman Experimental Validation of a Model of an Uncontrolled Bicycle ........... 777

(ID-1624)

G. Tomasini, A. Collina, E. Leo, F. Resta, F. Cheli Numerical-Experimental Methodology for Runnability Analysis and Wind-Bridge-Vehicle Interaction Study ....................................................... 778

(ID-2392)

P. Verissimo, J. Ambrósio Improved Bushing Models for Vehicle Dynamics .................................. 779

(ID-2362)

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MS.68 Behaviour of Structures Submitted to Extreme Event Organizers: Simões. L.C.

A. Dias Simulation of Large Deformations on Timber Joints Using 3D FEM Models........................................................................................................ 780

(ID-1967)

A. Freire, J. Negrão Nonlinear Dynamics of Flexible Partially Collapsed Structures............. 781

(ID-1787)

K. Jármai, J. Rodrigues Optimal Steel Frame Design for Fire Resistance .................................... 782

(ID-1116)

M. Larcher, L. Stempniewski Simulation of Shock Wave Loaded Concrete with Discrete Cracks ....... 783

(ID-1869)

A. Lopes, A. Cunha, L. Simões (ID-2443) CFD Based Evaluation of the Lock-In Phenomenon of a Bridge Under Wind Load................................................................................................ 784 A. Lopes, D. Gomes, L. Simões CFD Based Aerodynamic Study to Discrete Optimization of Bridge Cross Sections.......................................................................................... 785

(ID-2444)

K. Menchel, P. Bouillard, T. Massart Progressive Collapse Simulation in RC Structures ................................. 786

(ID-1986)

V. Terzi, M. Alexoudi, K. Pitilakis, T. Hatzigogos Vulnerability Assessment for Pipelines Under Permanent Ground Deformatioon. Comparison Between Analytical and Empirical ......................... 787

(ID-1775)

R. Vicente, H. Rodrigues, H. Varum Seismic Performance and Strengthening of Traditional Masonry Buildings in the City Centre of Coimbra............................................................. 788

(ID-1359)

Preface This book contains the edited version of the Abstracts of Plenary and Keynote Lectures and Papers, and a companion CD-ROM with the full-length papers, presented at the III European Conference on Computational Mechanics: Solids, Structures and Coupled Problems in Engineering (ECCM-2006), held in the National Laboratory of Civil Engineering, Lisbon, Portugal 5th - 8th June 2006. The book reflects the state-of-art of Computation Mechanics in Solids, Structures and Coupled Problems in Engineering and it includes contributions by the world most active researchers in this field. ECCM-2006 is a continuation of the very successful Conferences held in Munich, Germany (1999) and Cracow, Poland (2001) and it is organized by the European Committee of Computational Solid and Structural Mechanics (ECCSM) of the European Community on Computational Methods in Applied Science (ECCOMAS) in collaboration with the Portuguese Association of Theoretical, Applied and Computational Mechanics (APMTAC), the Technical University of Lisbon and the National Laboratory of Civil Engineering. ECCM-2006 is attended by about 1000 participants from 70 countries. More than 1300 Abstracts were submitted to ECCM-2006. Altogether, 6 plenary lectures, 35 keynote lectures and 800 papers are presented in 58 organized symposia. A companion book, containing the edited version of the majority of Plenary and Keynote Lectures, entitled Computational Mechanics: Solids, Structures and Coupled Problems in Engineering is also published by Springer 2006. The Proceedings of the Conference could not be possible without the sponsorship and financial support of: European Community on Computational Methods in Applied Science (ECCOMAS); European Committee of Computational Solids and Structural Mechanics (ECCSM); International Association of Computational Mechanics (IACM); Portuguese Association of Theoretical, Applied and Computational Mechanics (APMTAC); Foundation of Science and Technology (Portugal); Calouste Gulbenkian Foundation (Portugal); Technical University of Lisbon (Portugal); National Laboratory of Civil Engineering (Portugal); Instituto Superior Técnico (Portugal). The Editors are grateful to all authors and to the reviewers that helped ensuring the scientific quality, allowing for this book to be published before ECCM2006. We acknowledge the support of Mr. Pedro Pinto of the Technical University of Lisbon, in the editing of the book. The Editors are also grateful to all Members of Executive, Organizing, Advisory and Scientific Committees and to the organizers of the Symposia, whose work made possible the success of ECCM-2006. We are grateful to Andrea Marques and Ana Catarina Amador for their effort and valuable assistance in the conference and preparation of this book. Technical University of Lisbon, Portugal, June 2006 Carlos A. Mota Soares, João A.C. Martins Helder C. Rodrigues, Jorge A.C. Ambrósio, Carlos A.B. Pina Cristóvão M. Mota Soares, Eduardo B.R. Pereira, João Folgado lxix

lxxi

Executive Committee E. Oñate (President of IACM), Polytechnic University of Catalunya, Spain A. Mang (President of ECCOMAS), Technical University of Vienna, Austria C. A. Mota Soares (Co-Chairperson), Technical University of Lisbon, Portugal M. Papadrakakis (Co-Chairperson), National Technical University of Athens, Greece B. Schrefler, University of Padua, Italy J. Teixeira de Freitas, Technical University of Lisbon, Portugal

Organizing Committee E. Arantes e Oliveira (Honorary Co-Chairperson), Technical Univ. of Lisbon, Portugal E. Stein (Honorary Co-Chairperson), University of Hannover, Germany J. Ambrósio, Technical University of Lisbon, Portugal M. Bernadou, University of Leonard de Vinci, France T. Burczynski, Silesian University of Technology, Poland P. Diez, Polytechnic University of Catalunya, Spain M. Kleiber, IPPT PAN, Poland O. Mahrenholz, Technische Universitat Hamburg, Germany J. Martins, Technical University of Lisbon, Portugal C.M. Mota Soares, Technical University of Lisbon, Portugal P. Neittaanmaki, University of Jyväskylä, Finland E. Pereira, Technical University of Lisbon, Portugal J. Periaux, Dassault Aviation, France C. Pina, National Laboratory of Civil Engineering, Portugal P.Steinmann, University of Kaiserslautern, Germany E. Ramm, University of Stuttgart, Germany H.C. Rodrigues, Technical University of Lisbon, Portugal N. E. Wiberg, Chalmers University of Techonology, Sweden O. C. Zienkiewicz, University of Wales, Swansea, United Kingdom

Management Committee Carlos Pina, National Laboratory of Civil Engineering, Portugal Helder Rodrigues, Technical University of Lisbon, Portugal João Martins, Technical University of Lisbon, Portugal

Organizing Institution Portuguese Association of Theoretical, Applied and Computational Mechanics LNEC, Av. do Brasil 101, 1700-066 Lisboa, Portugal

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Advisory Committee F. Auricchio, Associazione Italiana di Meccanica Teorica e Applicata, Italy N. Bicanic, Association for Computer Methods in Engineering (ACME), UK M.H. Boduroglu, Turkish Committee on Computational Mechanics, Turkey T. Burczynski, Polish Association for Computational Mechanics (PACM), Poland M. Casteleiro, Sociedad Española de Métodos Numéricos en Ingenieria (SEMNI), Spain M. Cerrolaza, Sociedad Venezolana de Métodos Numéricos en Ingenieria, Venezuela P. Chauchot, Computational Structural Mechanics Association (CSMA), France C. K. Choi, Korean Association on Computational Mechanics (KACM), Korea J. Crempien, Sociedade Chilena de Mecánica Computacional (SCMC), Chile R. Delgado, Portuguese Association of Theoretical, Applied and Computational Mechanics (APMTAC), Portugal A. Eriksson, The Nordic Association for Computational Mechanics (NOACM), Denmark, Estonia, Finland, Iceland, Latvia, Lithuania, Norway, Sweden J. Fish, U. S. Association for Computational Mechanics (USACM), USA I. Harari, The Israel Association of Computational Methods in Mechanics, Israel I. Herrera, Sociedad Mexicana de Métodos Numéricos en Ingenieria, México S.R. Idelsohn, Asociación Argentina de Mecánica Computational (AMCA), Argentina W. Kanok-Nukulchai, Thailand Society for Computational Mechanics (TSCM), Thailand T. Kant, India Association of Computation Mechanics, India M. Kleiber, The Central European Association for Computational Mechanics (CEACM), Austria, Croatia, Hungary, Poland, Slovenia, The Czech Republic V. Kompis, Slovakia Association for Computational Mechanics, Slovakia G. R. Liu, Singapore Association for Computational Mechanics (SACM), Singapore P.R. Lyra, Brazilian Association for Computational Mechanics (ABMEC), Brazil J. Miller, Irish Society for Scientifc and Engineering Computation (ISSEC), Ireland H. Ohtsubo, Japan Society of Computational Engineering Science (JSCES), Japan M. Papadrakakis, The Greek Association of Computational Mechanics (GRACM), Greece E. Ramm, German Association of Computational Mechanics (GACM), Germany B. D. Reddy, South Africa Association for Theorical and Applied Mechanics (SAAM), South Africa S. Valliapan, Australian Association of Computational Mechanics, Australia T. Yabe, Japan Association for Computational Mechanics (JACM), Japan M. W. Yuan, Chinese Association of Computational Mechanics, China

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Scientific Committee S. Adali (South Africa) J. Alfaiate (Portugal) O. Allix (France) C. Alves (Portugal) C. António (Portugal) F. Amero (USA) J. Azevedo (Portugal) N. Banichuk (Russia) A.L. Baptista (Portugal) K.J. Bathe (USA) J. L. Batoz (France) P. Beckers (Belgium) T. Belytschko (USA) M. Bendsoe (Denmark) A. Benjeddou (France) P. Bergan (Norway) R. Borja (USA) R. de Borst (Netherland) F. Branco (Portugal) C.L. Bottasso (USA) E. Carrera (Italy) D. Camotim (Portugal) M. Casteleiro (Spain) J. Cesar Sá (Portugal) C. Chen (Taiwan) K.K. Choi, (USA) C. Cinquini (Italy) S. Cowin (USA) A. Cunha (Portugal) M. Doblaré (Spain) E.H .Dowel (USA) G. Dulikravich (USA) I. Doltsinis (Germany) R. Feijoo (Brazil) I. Figueiredo (Portugal) J. Fish (USA) L. Gaul (Germany) P.L. George (France) M. Geradin (Italy) E. Van der Giessen (Netherlands) M. Goicolea (Spain) C. Hellmich (Austria) J. Herskovits (Brazil) J. Holnicki-Szulc (Poland) G. Holzapfel (Austria) T.J.R. Hughes (USA) S. R. Idelsohn (Argentina)

R. Jorge (Portugal) J.J. Judice (Portugal) B. Karihaloo (UK) N. Kikuchi (USA) M. Kojic (USA) P. Ladevèze (France) V. Leitão (Portugal) T. Lekszycki (Poland) J. Vieira de Lemos (Portugal) A. Leung (China) W. K. Liu (USA) P. Lourenço (Portugal) B. Mace (UK) P. Martins (Portugal) R. Melchers (Australia) J. Moitinho de Almeida (Portugal) C. Navarro (Spain) P. Nikravesh (USA) J.T. Oden (USA) R. Ohayon (France) N. Olhoff (Denmark) X. Oliver (Spain) H. Orlande (Brasil) P. Pedersen (Denmark) H. Pina (Portugal) P. Prendergast (Ireland) A. Preumont (Belgium) J. N. Reddy (USA) P. Ribeiro (Portugal) R. Rolfes (Germany) W. Schiehlen (Germany) G.I. Schueller (Austria) O. Sigmund (Denmark) L. Simões (Portugal) J.C. Sousa (Portugal) R. Stenberg (Finland) A. Suleman (Portugal) A. Tadeu (Portugal) A. Torres Marques (Portgual) Z. Waszcyszyn (Poland) P. Wriggers (Germany) M. Zmindak (Slovak Republic) T.I. Zohdi (USA) Liu Gui-Rong (Singapore) M. Pietrzyk (Poland)

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International Correspondents M. Bercovier, The Hebrew University, Jerusalem, Israel S. Botello, University of Guanajuato, Mexico M. Cerrolaza, Central University of Venezuela, Venezuela C. K. Choi, Korea Advanced Inst. of Sc. & Techn., Korea R. Correa, Universidad de Chile, Chile S. M. Desphande, Indian Inst. of Science, India P. R. B. Devloo, Universidade Estadual de Campinas, Brazil A. Gaona, Universidad Nat. de Asuncion, Paraguay I. Herrera, UNAM, Mexico J. Herskovits, Federal University of Rio de Janeiro, Brazil J. Hurtado, Universidad Manizales, Colombia W. Kanok-Nukulchai, Asian Institute of Technology, Thailand M. Kawahara, Chuo University, Japan T. Kawai, University of Science in Tokyo, Japan A. M. Kharitonov, Russian Academy of Sciences, Russia T. Kobayashi, Japan E. B. Las Casas, Brasilian Association for Comp. Mech., Brazil C. H. Lee, University of Aeronautics, China A. Leung, City University of Hong-Kong, China C. A. Lin, National Tsing Hua University, Taiwan L. Quiroz, Universidad de Concepcion, Chile C. V. Ramakrishnan, Indian Institute of Technology, India R. Sampaio, PUC Rio, Rio de Janeiro, Brazil L. Súarez, Universidad de Puerto Rico, Puerto Rico K.Y. Sze, The University of Hong Kong, China S. Valliappan, University of New South Wales, Australia K. William, University of Colorado USA J. Yagawa, University of Tokyo, Japan Y.B. Yang, National University of Taiwan, Taiwan M. Yuan, University of Beijing, China W.X. Zhong, Dalian University of Technology, China F. G. Zhuang, CNSA Chinese Aerodynamics, China M. Xiao, NUAA, China

Sponsors European Community on Computational Methods in Applied Science International Association of Computational Mechanics Portuguese Association of Theoretical, Applied and Computational Mechanics Foundation of Science and Technology (Portugal); Calouste Gulbenkian Foundation (Portugal); Technical University of Lisbon (Portugal); National Laboratory of Civil Engineering (Portugal); Instituto Superior Técnico (Portugal)

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Computational Challenges for Multi-Physics Topology Optimization Martin P. Bendsøe Department of Mathematics, Technical University of Denmark Matematiktorvet B303, DK-2800 Kgs. Lyngby, Denmark [email protected] ABSTRACT Topology design involves working with multiple fields. Of primary interest is the density distribution of material that makes a certain objective function minimal while other objectives are satisfied as constraints in a mathematical programming statement. The evaluation of the objective and constraint functions will involve state variables that are fields that relate the design variables to physical behavior. The state fields are scalar or vector fields and multiple fields representing various physical responses may be involved; these fields will typically be coupled in multi-physics applications. Computational procedures for topology design (and for design optimization as a whole) thus encompass discretization schemes for design and state fields together with algorithms for optimization and for analysis. The prevailing computational approach to structural design and topology design in particular is to view the optimization procedure as a problem in the design variables only. This means that analysis is treated as a function call that provides information on function values and derivatives as a function of design. In optimization terms this is a nested format. An alternative to the nested format is to treat design and state fields on equal terms and formulate one unified optimization problem that involves also state equations as constraints. This is the typical approach in the areas of Mathematical Programs with Equilibrium Constraints (MPECs) and PDE-constrained optimization. Finally, in some cases it turns out to be advantageous to treat the design variables as functions of the states, for example where an explicit calculation of the optimal design for a fixed state is possible. This then leads to a variational statement for the optimal state field in itself. Computational challenges are thus by the nature of the problem two-fold and successful implementations rely on both efficient analysis (and the associated sensitivity analysis) and on the efficiency of optimization algorithms. The discretization of the analysis and design fields play here a significant role for stability and for obtaining relevant results, and it is typical that the optimization will utilize a poor model to give results of little physical meaning. For the design field, the computational model will typically also include a relaxation of an integer valued field and suitable ways to handle this is a crucial issue especially when extending topology design from continuum structural applications to multi-physics settings. The different approaches and the associated computational issues involved in their resolution will be illustrated by considering some recent work on design of multi-physics devices and the design of articulated mechanisms.

References [1] M.P. Bendsøe & O. Sigmund, Topology Optimization - Theory, Methods, and Applications. Springer-Verlag, Berlin-Heidelberg, 2003 (revised printing, 2004). [2] K.K. Choi & N.-H. Kim, Structural Sensitivity Analysis and Optimization, Vol 1 & 2. SpringerVerlag, New York, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Computational Geometry and the Analysis of Solids and Structures J.A. Cottrell , A. Reali†, Y. Bazilevs , T.J.R. Hughes Institute for Computational Engineering and Sciences The University of Texas at Austin 201 East 24th Street 1 University Station C0200 Austin, TX 78712, USA {jac3, bazily, hughes}@ices.utexas.edu †European

School for Advanced Studies in Reduction of Seismic Risk [email protected]

ABSTRACT The concept of Isogeometric Analysis is described. Basis functions generated from NURBS (NonUniform Rational B-Splines) are employed to construct exact geometric models. For purposes of analysis, the basis is refined and/or its order elevated without changing the geometry or its parameterization. Analogues of finite-element, h-refinement and p- refinement schemes are presented, and a new, more efficient, higher-order concept, k-refinement, is introduced. Refinements are easily implemented and exact geometry is maintained at all levels without the necessity of subsequent communication with a CAD (Computer Aided Design) description. In the context of structural mechanics, it is established that the basis functions are complete with respect to affine transformations, meaning that all rigid body motions and constant strain states are exactly represented. Standard patch tests are likewise satisfied. Numerical examples exhibit optimal rates of convergence for linear elasticity problems and convergence to thin elastic shell solutions. The concept of k-refinement is explored and shown to produce more accurate and robust results than standard finite elements for problems of structural vibrations. Through the use of nonlinear parameterizations, optical branches of frequency spectra are eliminated for k-refined meshes. Optical branches have been identified as contributors to Gibbs phenomena in wave propagation problems and the cause of rapid degradation of higher modes in p-method finite elements. A geometrically exact model of the NASA Aluminum Testbed Cylinder is constructed and frequencies and mode shapes are computed and shown to compare favorably with experimental results. It is argued that Isogeometric Analysis is a powerful generalization of standard, polynomial-based, finite element analysis.

References [1] T.J.R. Hughes, J.A. Cottrell, Y. Bazilevs. Isogeometric Analysis: CAD, finite elements, NURBS, edxact geometry, and mesh refinement. Computer Methods in Applied Mechanics and Engineering, 194:4135–4195, 2005. [2] J.A. Cottrell, A. Reali, Y. Bazilevs, T.J.R. Hughes. Isogeometric Analysis of structural vibration. Computer Methods in Applied Mechanics and Engineering, in press, 2005. Also as an ICES report http://www.ices.utexas.edu.research/reports/2005/0527.pdf.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Advances in the Particle Finite Element Method for Fluid-Structure Interaction Problems Eugenio Oñate, Sergio R. Idelsohn, Miguel A. Celigueta and Ricardo Rossi International Center for Numerical Methods in Engineering (CIMNE) Universidad Politécnica de Cataluña Campus Norte UPC, 08034 Barcelona, Spain [email protected]

ABSTRACT There is an increasing interest in the development of robust and efficient numerical methods for analysis of engineering problems involving the interaction of fluids and structures accounting for large motions of the fluid free surface and the existence of fully or partially submerged bodies. Examples of this kind are common in ship hydrodynamics, off-shore structures, spillways in dams, free surface channel flows, liquid containers, stirring reactors, mould filling processes, etc. We present a general formulation for analysis of fluid-structure interaction problems using the particle finite element method (PFEM). The key feature of the PFEM is the use of a Lagrangian description to model the motion of nodes (particles) in both the fluid and the structure domains. Nodes are thus viewed as particles which can freely move and even separate from the main analysis domain representing, for instance, the effect of water drops. A mesh connects the nodes defining the discretized domain where the governing equations, expressed in an integral from, are solved as in the standard FEM. The necessary stabilization for dealing with the incompressibility condition in the fluid is introduced via the finite calculus (FIC) method. A fractional step scheme for the transient coupled fluid-structure solution is described. Examples of application of the PFEM method to solve a number of fluid-structure interaction problems involving large motions of the free surface and splashing of waves are presented.

References [1] E. Oñate, S.R. Idelsohn, F. Del Pin, R. Aubry, The particle finite element method. An overview, Int. J. Comput. Meth., Vol. 1 (2), 267-307, 2004. [2] S.R. Idelsohn, E. Oñate, F. Del Pin, The particle finite element method: a powerful tool to solve incompressible flows with free-surfaces and breaking waves, Int. J. for Num. Meth. in Engrg. Vol. 61, 964-989, 2004. [3] www.cimne.com/pfem

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Elastic and Plastic Impacts in Multibody Dynamics Werner Schiehlen, Robert Seifried Institute of Engineering and Computational Mechanics Paffenwaldring 9, 70569 Stuttgart, Germany [email protected] [email protected]

ABSTRACT Many mechanical systems are subject to impacts modeled as unilateral constraints using the multibody system approach and the coefficient of restitution found from measurements. A multi-scale method is presented for the computation of the coefficient of restitution considering elastic wave propagation and plastic deformation which may occur simultaneously. Different models are presented for the impact period: a continuum model and a modal model with elastostatic Hertzian contact resulting in a boundary approach, a linear modal model with precomputed and concurrently computed finite elements in the contact region, and a completely nonlinear finite element model. In engineering, impacts are usually emerging from repeated processes with repeated collisions occurring on a previously deformed contact area. Therefore, the efficient modal models are extended for the evaluation of repeated impacts. The influence of the initial velocity prior to the impact as well as the shape and the yield stress of the bodies involved are investigated. For the experiments a special test bench is designed using Laser-DopplerVibrometers for velocity and displacement measurements on a fast time scale.

References [1] C. Glocker, On frictionless impact models in rigid-body systems. Philosophical Transactions of the Royal Society of London, A359, 2385–2404, 2001. [2] W. Goldsmith, Impact: The Theory and Physical Behaviour of Colliding Solids, Edward Arnold Ltd, London, 1960. [3] B. Hu, W. Schiehlen, Multi-time scale simulation for impact systems: from wave propagation to rigid body motion. Archive of Applied Mechanics, 72, 885–897, 2003. [4] W. Schiehlen, R. Seifried, Three approaches for elastodynamic contact in multibody systems. Multibody System Dynamics, 12, 1–16, 2004. [5] W. Schiehlen, R, Seifried, P. Eberhard, Elastoplastic phenomena in multibody impact dynamics. Computer Methods in Applied Mechanics and Engineering, in press, [doi:10.1016/j.cma.2005.08.011]. [6] R. Seifried, W. Schiehlen, P. Eberhard, Numerical and experimental evaluation of the coefficient of restitution for repeated impacts. International Journal of Impact Engineering, 32, 508–524, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Active Aeroelastic Aircraft Structures A. Suleman and P. A. Moniz Instituto de Engenharia Mecânica, Instituto Superior Técnico (IDMEC-IST), Lisbon, Portugal. Av. Rovisco Pais 1, 1049-001 Lisboa PORTUGAL [email protected]

ABSTRACT Aeroelasticity results in problems such as structural divergence, aileron reversal, and flutter stability due to insufficient torsional stiffness of the wings and “aeroelastic weight penalty” became a widely used expression by engineers in aircraft design. Aeroelastic solutions generally involve increasing the structure stiffness or mass balance (passive solutions), which typically involve increase of weight and cost while decreasing performance. In the seventies, composite materials with highly anisotropic directional stiffness properties enabled the introduction of aeroelastic tailoring methods where the composition of thickness and orientation of the individual material layers could be tailored to minimize the added structural weight necessary to minimize the detrimental effects due to aeroelastic behavior. This technology paved the way for looking at aeroelasticity from a different perspective. The new paradigm consists of looking at the structural deformations, caused by aerodynamic forces, to be used intentionally and in a beneficial way in order to improve aerodynamic performance and help to create the required control forces. In the last two decades, a new actuation concept for structural control has emerged. This concept uses the multifunctional materials properties to control the structural stiffness and shape of composite materials. Several studies have been performed to demonstrate applications of adaptive structures in aircraft, helicopters and submarines. This paper presents the research and development of novel active aeroelastic control strategies, aimed at improved aircraft performance (structural weight, better control effectiveness) by controlling structural deformations to modulate the desired aerodynamic deformations. The proposed research was carried out in the framework of the European research project 3AS (Active Aeroelastic Aircraft Structures). To this end, the following research issues to enable active aeroelastic aircraft structures have been addressed: • Demonstrate the application of piezoelectric actuators and sensors to dynamic aeroelastic control of a structure and design an experimental and computational setup that allows the quantification of the performance in flutter suppression, buffeting vibration reduction, and attenuation of other external mechanical vibrations. • Develop an airborne flight test platform (RPV - Remotely Piloted Vehicle) to demonstrate the proposed concepts both in the wind tunnel and in actual flight conditions; • Develop a methodology to design and integrate the proposed adaptive structures technology in real aircraft while reducing the weight of a wing for a given flight envelope.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Multiscale Strategy for Solving Industrial Problems O. Allix L.M.T., E.N.S. de Cachan/ C.N.R.S./Univ. P. et M. Curie 61, avenue du Président Wilson, 94235 Cachan, France [email protected]

ABSTRACT Multi-scale strategies are now mature for most of linear spatial problem. They can also be applied as solver in the case of non-linear problem but it is our opinion that more robust and efficient methods can be designed in that case. Two examples will serve as illustration. The case of crack propagation [1] and the case of post-buckling analyses of aeronautical structures [2]. In [3] Ladevèze proposed a multi-scale strategy for solving non-linear problem. This method offers several possibilities to adapt it in order that the main difficulties of the problem are specifically treated. One of its appealing features is to split the macro part from the micro one at interface level only. This allow to work on the most adapted splitting of the interfacial quantities. The classical choice which allows to incorporate homogenization at the macro-level automatically is the one where the interfacial macro extractor is associated with the linear part. In the case of crack propagation it appears more adapted to introduce a discontinuous scheme at the macro level, the use of the PUM method [4] for crack allows improving the micro description of the solution. In the case of the post-buckling analysis of large aeronautical, the non-linear scheme used, allows to iterate where it is needed only, that is in the sub-structures prone to the highest degree of nonlinearity. As non-linearities are often localized (i. e; corresponds to a local buckling giving rise to large displacement of the whole structure) the computational effort can be reduced drastically in comparison with a classical [2].

References [1] P.-A. Guidault, O. Allix, L. Champaney , J.-P. Navarro “A micro-macro approach for crack propagation with local enrichment”, to appear in CST [2] Ph. Cresta, O. Allix, C Rey, S Guinard "Nonlinear localization strategies for domain decomposition methods: application to post-buckling analyses" to apppear in CMAME [3] P. Ladevèze, O. Loiseau, D. Dureisseix, “A micro-macro and parallel computational strategy for highly heterogeneous structures”, IJNME, 52 (1–2) 121–138,2001. [4] J. Melenk, I. Babuska, “The partition of unity finite element method: Basic theory and applications”, Computer Methods in Applied Mechanics and Engineering 139, 289–314, 1996.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Numerical Integration of the Nonlinear Dynamics of Elastoplastic Solids Francisco Armero & Christian Zambrana Structural Engineering, Mechanics and Materials University of California at Berkeley Berkeley, CA 94720 [email protected]

ABSTRACT Classical numerical schemes for the numerical integration of the equations of solid and structural dynamics in time show serious limitations when applied to the finite deformation range. These includes the classical Newmark and HHT type schemes. The unconditional stability property of some of these schemes is lost in the nonlinear range, with the simulations showing an unbounded growth of energy. Furthermore, the law of conservation of angular momentum is, in general, not inherited by the schemes in this nonlinear range either. To handle these difficulties, the formulation of the so-called energymomentum schemes has received a great deal of attention lately. As illustrated in [1], these difficulties are also present in the physically dissipative problem of finite strain plasticity. The schemes do not lead to dissipative discrete dynamical systems. We present in this contribution a new class of integration schemes for multiplicative finite strain plasticity that exhibit the non-negative energy dissipation characteristic of these physical systems and that preserve the conservation laws of linear and angular momenta in time. The exact physical dissipation is obtained, hence recovering previously developed energy-momentum schemes with the exact energy conservation for elastic problems in elastic steps of the elastoplastic simulations. Furthermore, the schemes allow the consideration of extensions showing a controllable high-frequency (numerical) energy dissipation to handle the numerical schemes of typical problems of interest. The new schemes rely on an alternative integration of the plastic evolution equations, or return mapping algorithm, that define the discrete in time evolution of the plastic internal variables and the exact enforcement of the yield constraint on the stresses driving the motion. We built on the developments presented in [1] and incorporate an additional set of new considerations for the volumetric contributions, thus arriving to algorithms that also preserve exactly the isochoric character of the plastic flow in some common models of finite strain plasticity (e.g. J2 -flow theory based on von Mises yield criteria). Moreover, we discuss the implementation of the new time-stepping schemes in the context of new formulations of assumed strain and mixed finite elements for the handling of the well-known volumetric locking in these cases, while obtaining the aforementioned conservation and dissipation properties in time. Several numerical simulations are presented illustrating the performance of the new schemes.

References [1] F. Armero, Energy-Dissipative Momentum-Conserving Time-Stepping Algorithms for Finite Strain Multiplicative Plasticity, Computer Methods in Applied Mechanics and Engineering, in press.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Computational Methods for Dynamic Crack Propagation Ted Belytschko, Song, Wang Northwestern University, Department of Mechanical Engineering 2145 North Sheridan Road, Evanston, IL 60208-3111 [email protected]

ABSTRACT Several methods for dynamic computational fracture mechanics are reviewed, and their performance in some benchmark problems is reviewed. The methods considered include element deletion, interelement unzipping models and the extended finite element method. Element deletion consists of simply deleting the element when a material criterion is met. The inter-element crack models are of the type proposed by Xu and Needleman, and by Ortiz and Pandolfi. The extended finite element method permits arbitrary crack propagation within the mesh. Advancement of the crack by several criteria is studied. Among the results that are compared are the crack paths, the speed of the crack and the energy dissipated by the fracture process.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Analysis and Design of Sandwich Structures Made of Steel and Lightweight Concrete Pål G. Bergan*, Kåre Bakken*, Karl-Christian Thienel† *

Det Norske Veritas Research NO-1322 Høvik Norway [email protected] [email protected]

†

Universität der Bundeswehr DE-85577 Neubiberg Germany [email protected]

ABSTRACT Reinforced concrete has been used in ship building for more than 150 years; particularly during war periods with shortage of steel. Unfortunately, this inexpensive material has proven not to be commercially viable, primarily because of added weight. A new concept for design of ships and marine structures has been developed using sandwich plates composed of steel skins with lightweight concrete as core material. New types of light-weight concretes have been developed and tested for this purpose. Very large structures can be built without increasing the core thickness considerably; scalability is achieved by way of a cellular structural concept. A Panmax type bulk carrier ship has been designed and analyzed by way of linear finite elements. The study shows great promise for this new sandwich concept in that it appears that it saves 30 to 40 percent of the steel and comes out with about the same weight compared with a standard steel ship design. It also seems clear that the sandwich concept may offer important advantages when it comes to safety, design life, and operational performance. The current type of concretes is very light with wet, demoulding density of less than 1000 kg/m3. Its use in sandwich panels has been extensively tested in the laboratory. Some testing has focused on strength of the light-weight concrete in compression and shear, whereas other tests have dealt with bond and pull-out strength in the steel-concrete interaction. The laboratory program has included static and fatigue testing of series of sandwich beams; it has to a large extent focused on interaction effects and the failure of the core material, particularly in shear failure. The testing shows how cracks initiate and develop. The cracks in the test specimen have a highly jagged appearance, indicating extensive capability of shear force transfer across cracks (shear locking). The tests also show that delamination primarily occurs as a secondary phenomenon, normally a short distance away from the contact plane between steel and concrete. The most remarkable finding is that the beams exhibit extensive ductility and ability to absorb energy far beyond initial cracking. A key to this behaviour is that the surface skins are locked by end plates at the end of the beams. The study reveals that there is a need for developing good mathematical models for this unconventional concrete. There is also a need for further developing nonlinear finite elements models that can simulate the complex behaviour and interaction effects of steel-concrete sandwich plates during all stages of damage development. The practical potential of this technology is very extensive.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Multiscale Modeling of Pore Collapse Instability in High-Porosity Solids Ronaldo I. Borja* *

Department of Civil and Environmental Engineering Stanford University, Stanford, CA 94305-4020, USA [email protected]

ABSTRACT High-porosity solids include elastomeric foams and other cellular materials, Aeolian sands, poorly cemented coquina, diatomite, and chalk. A majority of these materials are known to exhibit several regions of behavior in simple uniaxial or conventional triaxial compression: a nearly linearly elastic behavior at small strain, plastic behavior at larger strain, a plateau region in which strain increases at nearly constant stress, and, finally, a densification region characterized by pore collapse. In this paper we address the problem of pore collapse instability as a local bifurcation from a homogeneous solution driven by a singular constitutive tangent operator. We identify different eigenmodes (emodes) of bifurcation and propose an approach for constitutive branching useful for multiscale modeling of the pore collapse/densification process.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Instabilities and Discontinuities in Two-Phase Media René de Borst 1, 2, Marie-Angèle Abellan 3, Julien Réthoré 1 1

2

Faculty of Aerospace Engineering, Delft University of Technology P.O. Box 5058, 2600 GB Delft, The Netherlands [email protected]

LaMCoS – UMR CNRS 5514, INSA de Lyon, 69621 Villeurbanne, France 3

LTDS-ENISE – UMR CNRS 5513, Saint-Etienne, France

ABSTRACT Within the framework of the generalised theory of heterogeneous media, the complete set of equations is derived for a three-dimensional fluid-saturated porous medium. Subsequently, dispersion analyses are carried out for an infinite one-dimensional continuum, that has been deforming homogeneously prior to the application of the perturbation. A dispersive wave is obtained, but the internal length scale associated with it vanishes in the short wave--length limit, at least for the assumptions made regarding the constitutive behaviour of the solid and of the fluid. This result leads to the conclusion that, upon the introduction of softening, localisation in a zero width will occur and no regularisation will be present. This conclusion is corroborated by the results of numerical analyses of wave propagation in a finite one-dimensional bar [1]. The result has severe implications for finite element analyses of damaging multiphase media, since they will be mesh-dependent. To avoid mesh dependence, either the constitutive model for the solid must be equipped with a nonvanishing internal length scale, or any damage that occurs must be modelled in a strictly discrete manner. The second part of the contribution therefore focuses on the proper modelling of discontinuities in fluid-saturated porous media. A two-field finite element formulation has been set up with the displacements and the fluid pressure as the fundamental unknowns [2]. At discontinuities in the body, e.g. cracks, the displacement and the fluid pressure fields are allowed to be discontinuous. Numerically, the discontinuities in the displacements and the fluid pressure are incorporated using the partition-of-unity property of finite element shape functions. The tractions at the interface are related to the displacement jumps using a cohesive zone model, where the behaviour in the direction normal to the interface can be different from that in the tangential direction. Regarding the pore fluid flow, it has been assumed that the normal flux to the interface is proportional to the jump in pore pressures at both sides of the discontinuity, which can be conceived as the discrete analogon of Darcy’s relation for fluid flow in porous media. The paper concludes with some examples of finite element analyses of fluid-saturated porous media with discontinuities.

References [1] M.-A. Abellan, R, de Borst, Wave propagation and localisation in a softening two-phase medium. Computer Methods in Applied Mechanics and Engineering, doi: 10.1016/j.cma.2005.05.056. [2] R. de Borst, J.J.C. Remmers, A. Needleman, M.-A. Abellan, Discrete vs smeared crack models for concrete fracture: Bridging the gap. International Journal for Numerical and Analytical Methods in Geomechanics, 28, 583-607, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Towards Maneuvering Aeroelasticity — Progress in the Simulation of Large Fluid-Structure Interaction Problems Carlo L. Bottasso Politecnico di Milano Via La Masa 34, 20156 Milano, Italy [email protected]

ABSTRACT Advanced multidisciplinary software tools enable the development of sophisticated models of fixed and rotary wing vehicles. Such comprehensive vehicle models account for the interactions between the structural and aerodynamic fields, often coupled with hydraulic and electromechanical subsystem models and with control laws (aero-servo-elasticity). With these models it is now possible to predict with a growing level of confidence a variety of phenomena that are of crucial importance in the design phase, including loads, performance, stability, vibratory response, handling qualities and flight mechanics characteristics of the vehicle. The quick pace of the evolution of comprehensive vehicle models must be accompanied by a similar growth in the range of problems that can be addressed by simulation. In particular, it is clear that quite often the limiting factors that constrain the design are found in the maneuvering regime at the boundaries of the flight envelope. For example, a design engineer might be interested in flying a minimum time turn with a virtual model of a helicopter while not exceeding a maximum allowable load factor, in order to assess the vibratory characteristics of the machine in this extreme turn. Current aeroelastic simulation tools are not directly equipped to solve this class of problems, since in fact they all compute the dynamic response of the model under the action of assigned control inputs. Unfortunately, control inputs that will fly a given maneuver are in general not available, and must be computed. In this paper we describe a methodology for maneuvering aeroelasticity that is applicable to arbitrarily complex vehicle models. The approach is based on model-based virtual pilots that, on the basis of a formal maneuver description, first plan the path of the vehicle throughout the maneuver and then track it by driving the vehicle model along it, as proposed in Reference [1]. Both planning and tracking pilots are based on adaptive reduced models of the system and have the ability to learn, and hence improve their driving performance, as they steer the vehicle. We illustrate the use of this emerging technology with the help of relevant examples in the area of rotorcraft technology.

References [1] C.L. Bottasso, C.-S. Chang, A. Croce, D. Leonello, L. Riviello, Adaptive planning and tracking of trajectories for the simulation of maneuvers with multibody models. Computer Methods in Applied Mechanics and Engineering, in press, available online 10 October 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A Design Optimization Formulation for Problems with Random and Fuzzy Input Variables Using Performance Measure Approach K.K. Choi and Liu Du Department of Mechanical and Industrial Engineering The University of Iowa, Iowa City, IA 52242, U.S.A. [email protected] [email protected]

ABSTRACT To obtain reliable designs, aleatory and epistemic uncertainties are considered recently in the structural analysis and design optimization. The reliability based de-sign optimization (RBDO) method [1] is used when the amount of input data is sufficient enough to create accurate statistical distribution. On the other hand, when the sufficient input data are not available due to limitations in time, human, and facility resources, the optimum design may not be reliable if RBDO method is used. To deal with the situation that input uncertainties have insufficient information, a possibility (or fuzzy set) method can be used for structural analysis and possibility based design optimization (PBDO) [2]. However, in many industry design problems, we may have to deal with design problems that involve with the mixed input statistical random and fuzzy variables simultaneously. For these problems, RBDO may yield unreliable optimum designs because of insufficient data. On the other hand, treating the random variables as fuzzy variables and invoking PBDO to solve the mixed design variable problem may yield too conservative designs with higher optimum costs. This paper proposes a new mixed variable design optimization (MVDO) problem based on the performance measure approach (PMA) [1]. To evaluate the possibilistic constraint in MVDO, a sub-optimization problem for inverse analysis is carried out using a hyper-cylinder domain. To solve this sub-problem efficiently and effectively, a new numerical algorithm, maximum failure search (MFS) method, is proposed in this paper by combining the enhanced hybrid mean value (HMV+) method [3] for the inverse reliability analysis in RBDO and the maximal possibility search (MPS) method [2] for the inverse possibility analysis in PBDO. Some mathematical examples are used to demonstrate the efficiency and effectiveness of the proposed numerical MFS method. Some physical design examples are used to compare the proposed MVDO results with RBDO and PBDO results.

References [1] Youn BD, Choi KK, Park YH (2003) Hybrid analysis method for reliability-based de-sign optimization. Journal of Mechanical Design, ASME 125(2): 221-232 [2] Du L, Choi, KK, Youn BD. An Inverse Possibility Analysis Method For Possibility-Based Design Optimization. AIAA Journal, to be appear. [3] Youn BD, Choi KK, Du L (2005) Enriched Performance Measure Approach for Reliability-Based Design Optimization. AIAA Journal 43(4): 874-884

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Strength of Porous Ceramics – Mechanical Testing and Numerical Modelling Ioannis Doltsinis Faculty of Aerospace Engineering and Geodesy, University of Stuttgart Pfaffenwaldring 27, D-70569 Stuttgart [email protected]

ABSTRACT The lecture addresses modelling of the damage of porous ceramics on the microstructural level, and the failure of structural components. The research has been occasioned by the industrial interest in the strength of ceramic filter supports used in nanofiltration, where internal pressure is of importance. Porous ceramics subjected to fluid pressure in the pores are prone to brittle microfracturing which may progress to brittle or quasi-brittle rupture of the component. Apart from physical and numerical modelling on the microscale, subtle laboratory testing of components has been of paramount importance; brittleness suggests statistical evaluation. The study focuses on mechanical issues, but conceptual thoughts on stress-enhanced corrosion are included [4]. The phenomenon is significant to aging which determines the life-time of parts exposed to chemically aggressive media. A brief discussion on the formalism for the fracturing continuum [1] is followed by the modelling of cracking by separation of grain boundaries. The associated algorithm simulates progressive damage on artificial microstructures generated in the computer for given material characteristics [2], and determines rupture statistics in dependence of various parameters. Laboratory measurements on the structural parts of interest (circular cylinders with longitudinal channels) refer to the diametral compression (Brazilian) test which adequately replaces the condition of channel pressure [3], as justified by finite element stress analysis. The impact of the two loading cases on Weibull statistics is formally explained, effects of basic material and doping on component rupture are investigated, the damage tolerance confirmed, corrosion discussed. Synthesis incorporates material strength statistics in the finite element model and estimates critical locations in the structural part under internal pressure. There is quantitative agreement with pressure levels actually registered at rupture. The account refers to cooperative research performed at the Universities of Stuttgart and Caen [4].

References [1] I. Doltsinis, Issues in modelling distributed fracturing in brittle solids with microstructure, in: S.R. Idelsohn et al. (Eds.), Computational Mechanics - New Trends and Applications. (CDROM) CIMNE, Barcelona, 1998. [2] I. Doltsinis and R. Dattke, Modelling the damage of ceramics under pore pressure, Comput. Meths. Appl. Mech. Engng., 191, 29-46, 2001. [3] F. Osterstock, I. Doltsinis, O. Vansse O, The Brazilian reliability test and micromechanical modelling for channelled cylinders of multiphase porous ceramics, in: High-Performance Ceramics II, Trans. Tech. Publications, 2004. [4] I. Doltsinis and F. Osterstock, Modelling and experimentation on the strength of porous ceramics, Arch. Comput. Meth. Engng., 12, 303-336, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Reduced Order Models in Unsteady Aerodynamic Models, Aeroelasticity and Molecular Dynamics Earl H. Dowell*, Kenneth C. Hall*, Jeffrey P. Thomas*, Robert E. Kielb*, Meredith A. Spiker*, Charles M. Denegri, Jr. † *

†

Duke University, Durham, North Carolina, United States [email protected]

U.S. Air Force SEEK EAGLE Office, Eglin Air Force Base, Florida, United States

ABSTRACT The state of reduced order modeling of unsteady aerodynamic flows for the efficient calculation of fluid-structure interaction (aeroelasticity) is discussed. Reduced order modeling is a set of conceptually novel and computationally efficient techniques for computing unsteady flow about airfoils, wings, and turbomachinery cascades. Starting with either a time domain or frequency domain computational fluid dynamics (CFD) analysis of unsteady aerodynamic flows, a large, sparse eigenvalue problem is solved. Then, using just a few of the resulting aerodynamic eigenmodes, a Reduced Order Model (ROM) of the unsteady flow is constructed. The aerodynamic ROM can then be combined with a similar ROM for the structure to provide a Reduced Order Aeroelastic Model that reduces computational model sized and cost by several orders of magnitude. Moreover, the aerodynamic and aeroelastic eigenvalue and eigenmode information provides important insights into the physics of unsteady flows and fluidstructure interaction. The method is particularly well suited for use in the active control of aeroelastic (fluid-structural) and unsteady aerodynamic phenomena as well as in standard aeroelastic analysis. As an alternative to the use of aerodynamic eigenmodes, Proper Orthogonal Decomposition (POD) has also been explored. POD is an attractive alternative because of the greater simplicity of calculating POD modes rather than fluid eigenmodes per se. Moreover once the POD modes have been used to construct a Reduced Order Model, this ROM may be used to find a good approximation to the dominant aerodynamic eigenmodes. After the Hopf Bifurcation (flutter) condition is determined for the fluid-structural system, a novel High Dimensional Harmonic Balance (HDHB) solution method for the fluid (and structural) model(s) proves to be a very efficient technique for determining limit cycle oscillations in fluid-structural systems. In this approach one exploits the knowledge of the aeroelastic eigenmode determined from the aeroelastic ROM. Several examples will be discussed including the limit cycle oscillations (LCO) of the F-16 aircraft and the limit cycle oscillations (LCO) of the Von Karman vortex street behind a cylinder in a cross-flow. The latter is a prototypical example of self-excited fluid oscillations that occur for bluff bodies including wings at high angles of attack. Correlation of theoretical calculations with experiment will also be shown.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Inverse Engineering George S. Dulikravich1, Helcio R. B. Orlande2 and Brian H. Dennis3 1

2

3

Florida International University, Department of Mechanical & Materials Eng. 10555 West Flagler Street, Room EC 3474, Miami, Florida 33174, U.S.A. [email protected]

Federal University of Rio de Janeiro, Department of Mechanical Eng., COPPE, Cid. Universitaria, Cx. Postal 68503, Rio de Janeiro, RJ, 21941-972, Brazil [email protected]

Department of Mechanical and Aerospace Eng., University of Texas at Arlington, Arlington, Texas 78712, U.S.A. [email protected]

ABSTRACT Inverse problems are rapidly becoming a multi-disciplinary field with many practical engineering applications. The objective of this lecture is to present several such multi-disciplinary concepts and applications. In some examples, sophisticated regularization formulations were used. In other examples, different optimization algorithms were used as tools to solve de facto inverse problems. Due to the mathematical complexity of these multi-disciplinary and often multi-scale inverse problems, the most widely acceptable formulations eventually result in a need for minimization of a certain norm or a simultaneous extremization of several such norms. These single-objective and multiobjective minimization problems are then solved using appropriate robust evolutionary optimization algorithms. Specifically, we focus here on inverse problems of determining spatial distribution of a heat source for specified thermal boundary conditions, finding simultaneously thermal and stress/deformation boundary conditions on inaccessible boundaries, and determining chemical compositions of steel alloys for specified multiple properties.

References [1] M.N. Özisik, HRB Orlande, Inverse Heat Transfer: Fundamentals and Applications, Taylor & Francis, New York, NY, USA, 2000. [2] P.M.P. Silva, H.R.B. Orlande, M.J. Colaco, P.S. Shiakolas, G.S. Dulikravich, Estimation of Spatially and Time Dependent Source Term in a Two-Region Problem. In Proceedings of the 5th International Conference on Inverse Problems in Engineering: Theory and Practic (Lesnic, D. ed.) July 11-15, 2005, Cambridge, United Kingdom [3] B.H. Dennis, G.S. Dulikravich, Z.-X. Han, Determination of Temperatures and Heat Fluxes on Surfaces and Interfaces of Multi-domain Three-Dimensional Electronic Components. ASME Journal of Electronic Packaging, Vol. 126, No. 4, December 2004, pp. 457-464. [4] B.H. Dennis, G.S. Dulikravich, S. Yoshimura, A Finite Element Formulation for the Determination of Unknown Boundary Conditions for Three-Dimensional Steady Thermoelastic Problems. ASME J. of Heat Transfer, Vol. 126, February 2004, pp. 110-118. [5] I.N. Egorov-Yegorov, G.S. Dulikravich, Inverse Design of Alloys for Specified Stress, Temperature and Time-to-Rupture by Using Stochastic Optimization. International Symposium on Inverse Problems, Design and Optimization – IPDO (Eds: Colaco, M., Orlande, H., Dulikravich, G.), Rio de Janeiro, Brazil, March 17-19, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Multiscale Approaches for Bridging Discrete and Continuum Scales Jacob Fish and Wen Chen Rensselaer Polytechnic Institute [email protected], [email protected]

ABSTRACT In this talk, I will present information-passing and concurrent discrete-to-continuum scale bridging approaches. In the concurrent approach both, the discrete and continuum scales are simultaneously resolved, whereas in the information-passing schemes, the discrete scale is modeled and its gross response is infused into the continuum scale. For the information-passing multiscale methods to be valid both the temporal and spatial scales should be separable. Among the information-passing bridging techniques, I will present the Generalized Mathematical Homogenization (GMH) theory [1,2] and the Multiscale Enrichment based on the Partition of Unity (MEPU) method [3]. The GMH constructs an equivalent continuum description directly from molecular dynamics (MD) equations. The MEPU approach gives rise to the enriched quasicontinuum formulation, capable of dealing with heterogeneous inter-atomic potentials, nonperiodic fields and high velocity impact applications. The second part of the talk will focus on multiscale systems, whose response depend inherently on physics at multiple scales, such as turbulence, crack propagation, friction, and problems involving nano-like devices. For these types of problems, multiple scales have to be simultaneously resolved in different portions of the problem domain. Among the concurrent bridging techniques, attention will be restricted to multilevel-like methods [4,5]. A space-time multilevel method for bridging discrete scales with either coarse grained discrete or continuum scales will be presented. The method consists of the wave-form relaxation scheme aimed at capturing the high frequency response of the atomistic vibrations and the coarse scale solution (explicit or implicit) intended to resolve the coarse scale features (in both space and time domains) of the discrete medium

References [1] J. Fish and C. Schwob, “Towards Constitutive Model Based on Atomistics,” International Journal of Multiscale Computational Engineering, Vol. 1 pp. 43-56, (2003). [2] W. Chen and J. Fish, “A Generalized Space-Time Mathematical Homogenization Theory for Bridging Atomistic and Continuum Scales,” to appear in Int. J. Num. Meth. Eng., (2005). [3] J. Fish and Z. Yuan, “Multiscale Enrichment based on Partition of Unity,” International Journal for Numerical Methods in Engineering, (2004), in print. [4] J. Fish and W. Chen, “Discrete-to-Continuum Bridging Based on Multigrid Principles,” Comp. Meth. Appl. Mech. Engng., Vol. 193, pp. 1693-1711, (2004). [5] H. Waisman and J. Fish, “Space-Time multigrid method for bridging discrete scales,” submitted to International Journal for Numerical Methods in Engineering (2005)

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Adaptive Mesh Generation in 3 Dimensions by Means of a Delaunay Based Method. Applications to Mechanical Problems. Paul Louis George ∗ INRIA,

Domaine de Voluceau, F 78153 Le Chesnay Cedex [email protected]

ABSTRACT Mesh adaptation is recognized as a powerful tool to compute accurate solution while minimizing the required resources (CPU time and memory space), therefore avoiding using parallel computing. It is also a way to accurately capture the physical behavior of the PDE problem in hand and, in some cases, this is the only way to access to a reasonable solution. Mesh adaptation in two dimensions can be considered as mature and is used in various problems. Right now, in three dimensions, the question is much more tedious and only a limited number of works can be reported. Mesh adaptation can be considered in two different ways. The first makes use of local modification of the current mesh so as to adapt it. Modification tools include well known operators such as point relocation, collapse (mesh coarsening), edge flips, point addition (mesh enrichment). Relatively easy to implement, such methods proved to give nice results in a number of cases but are not so flexible in specific when anisotropic features are desired. The second is based on the full generation of a new (adapted) mesh based on the current one and metric data provided at the nodes of this mesh. The generation method is then a variant (widely different in various aspects) of the well know mesh generation method. The aim being not only to mesh at the best a given domain but to match the given metric, which is much more demanding. We are concerned with this second approach and we propose a Delaunay based mesh generation method capable to complete adapted meshes. The mesh generation aspect is driven by metric data (element size and directional specification) which are defined by means of error estimates. Isotropic and anisotropic meshes can be produced. Concrete application examples will demonstrate the flexibility of the proposed method, show the low cost of the approach which compares well with the first approach.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

19

Computational Micromechanics of Biological Materials: Bone and Wood Christian Hellmich∗ , Karin Hofstetter∗ , Cornelia Kober† ∗ Vienna

University of Technology (TU Wien), Institute for Mechanics of Materials and Structures Karlsplatz 13/202, A-1040 Wien (Vienna), Austria [email protected], [email protected]

† Osnabr¨ uck

University of Applied Science, Faculty of Engineering and Computer Science Albrechtsstraße 30, D-49009 Osnabr¨uck, Germany [email protected] ABSTRACT

Despite complex hierarchical organization of bone and wood, it was recently possible to identify a few elementary components at the micro and nanolevel of these material classes for the explanation of the diversity of macroscopic (poro-)elastic properties of different bones and woods [2, 3]. The mechanical properties (i.e. elasticity) of these elementary components are (up to experimental scattering) the same across a variety of different bones and woods, respectively; they are ’universal’, i.e., independent of tissue-type, species, and anatomical location. The mechanical interaction between these elementary components (mechanical morphology) and the dosages of these components in different tissues determine the macroscopic material properties. Having in mind that, as regards bone, these dosages are dependent on complex biochemical control cycles (defining the metabolism of the organism), the purely mechanical theory can be linked to biology, biochemistry, and, on the applied side, to clinical practice. Drug-driven or genetically driven changes in metabolisms lead to changes in the dosages of elementary components. The effects of these metabolic changes on the mechanical behavior of skeletal (sub)systems under well-defined loading conditions (e.g., downfall of elderly persons with osteoporosis) can then be studied by feeding structural models (e.g., Finite Element models [4]) of whole bones with the aforementioned macroscopic material properties - the output of the micromechanical models. This is probably highly relevant for patient-specific non-invasive bone disease diagnosis and therapy. As regards wood, our nano-to-macro approach is expected to support optimization of technological processes, such as drying.

References [1] L. Dormieux and F.-J. Ulm, editors. CISM Vol.480 – Applied Micromechanics of Porous Media. Springer, Wien - New York, 2005. [2] Ch. Hellmich. Microelasticity of bone, pages 289 – 331. In Dormieux and Ulm [1], 2005. [3] K. Hofstetter, C. Hellmich, and J. Eberhardsteiner. Development and experimental validation of a continuum micromechanics model for the elasticity of wood. European Journal of Mechanics A Solids, 24:1030 – 1053, 2005. [4] C. Kober, B. Erdmann, Ch. Hellmich, R. Sader, and H.-F. Zeilhofer. Validation of interdependency between inner structure visualization and structural mechanics simulation. International Congress Series, 1281:1373, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Mechanobiology: Computation and Clinical Application Gerhard A. Holzapfel∗† , Christian T. Gasser∗ , Dimitris Kiousis∗ ∗ Royal

Institute of Technology (KTH), School of Engineering Sciences Osquars backe 1, 100 44 Stockholm, Sweden {gh,tg,dk}@hallf.kth.se

† Graz

University of Technology, Computational Biomechanics Schiesstattgasse 14-B, 8010 Graz, Austria [email protected] ABSTRACT

Some reasons for the present worldwide procession of biomechanics are the exciting new developments in biology and the rapidly expanding field of mechanobiology, which aims to understand how cells respond to changes in their mechanical environment. Mechanobiology also studies the mechanical factors that may be important in, e.g., triggering the onset of atherosclerosis or aneurysms. Because of the inherent geometric, structural and material complexities of biological tissues [1], and the spatially non-uniform and time-varying boundary conditions, these type of problems demand computational methods [2], sufficient computational resources and graphics capability to display three-dimensional results. Computational models offer the potential to simulate multifield coupled processes encountered in the micro-heterogeneous biological tissues, and to realistically predict physiological functional interactions. Computational mechanobiology of biological tissue is increasing our ability to address multidisciplinary problems of academic, industrial and clinical importance. This presentation deals with the 3D modeling of balloon angioplasty, in particular of the interaction of balloon, stent and an atherosclerotic carotid artery. The artery is modeled as a heterogenous structure composed of three layers and a plaque. Distinctive attention is paid to the 3D contact of the artery with a stent and a balloon catheter, inflated with the goal to enlarge the area of the vessel. We use a smooth contact representation which is shown to be superior with respect to facet elements discretizing the interacting bodies. In addition, the dissection of the plaque, typically occurring during balloon inflation, is modeled by means of strong discontinuities and the application of the theory of cohesive zones using the partition of unity finite element method [3]. Contact pressure between the balloon-stent structure and the arterial wall leads to non-physiological stress concentration that can trigger adverse biological responses of the cells culminating in in-stent restenosis. Indeed, it has been shown that the design of a stent is a major risk factor for restenosis. A strategy of designing novel stents is shown with the goal to minimize vascular injury and to optimize long-term success [1]. Acknowledgement. Financial support for this research was partly provided by the Austrian Science Foundation under START-Award Y74-TEC, and KTH.

References [1] G. A. Holzapfel and R. W. Ogden, editors. Mechanics of Biological Tissue. Springer, Heidelberg, 2005. [2] G. A. Holzapfel. Computational biomechanics of soft biological tissue. In E. Stein, R. de Borst, and T. J. R. Hughes, editors, Encyclopedia of Computational Mechanics. Volume 2 Solids and Structures, pages 605– 635, Chichester, 2004. John Wiley & Sons. [3] T. C. Gasser and G. A. Holzapfel. Physical and numerical modeling of dissection propagation in arteries caused by balloon angioplasty. In M. H. Hamza, editor, Proceedings of the 3rd IASTED International Conference on Biomechanics, pages 229–233. ACTA Press, Anaheim, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Recent Developments of Hybrid Crack Element: Determination of Its Complete Displacement Field and Combination with XFEM B.L. Karihaloo, Q.Z. Xiao School of Engineering, Cardiff University, Queen’s Buildings, The Parade, Newport Road, Cardiff CF24 3AA, UK [email protected] (BL Karihaloo), [email protected] (QZ Xiao)

ABSTRACT The hybrid crack element (HCE) [1] is one of the most accurate and convenient finite elements (FEs) for the direct calculation of the stress intensity factor (SIF) and coefficients of the higher order terms of the Williams expansion [2, 3]. It represents a crack by only one super-element which is connected compatibly with the surrounding elements. It is very efficient for analysing bodies with many cracks [4]. The HCE is formulated from a simplified variational functional using truncated asymptotic crack tip displacement and stress expansions and interelement boundary displacements compatible with the surrounding regular elements. In the implementation, a general FE mesh can be used by forming the HCE from elements surrounding the crack tip [5]. The HCE can thus be included in any commercial package as conveniently as normal hybrid stress elements. However, the exclusion of the rigid body modes in the truncated asymptotic displacements creates jumps between these displacements and element boundary displacements. In this study, the rigid body modes are recovered by minimising these jumps via a least squares method. If the HCE only is used, the part of the crack inside the HCE need not conform to the mesh. However, crack faces away from the crack tip (outside the HCE) need to conform to the mesh. This disadvantage can be avoided by combining the HCE with the extended FEM (XFEM) [6]. The XFEM enriches the standard local FE approximations with a displacement discontinuity across a crack, and the asymptotic solution at the crack tip, with the use of the partition of unity (PU). It avoids using meshes conforming with the discontinuity and also adaptive remeshing as the discontinuity grows as is the case with the FEM. XFEM offers great flexibility in the modelling of the fracture process. However, the accuracy of the displacements and/or stresses in a few layers of elements surrounding the crack tip is low. The combined method using both HCE and XFEM inherits the flexibility of the XFEM and the high accuracy of the HCE. Typical static and propagating crack problems will be presented to demonstrate the efficiency and accuracy of this method.

References [1] P. Tong, T.H.H. Pian, S.J. Lasry, A hybrid element approach to crack problems in plane elasticity. Int J Numer Meth Engng, 7, 297-308, 1973. [2] B.L. Karihaloo, Q.Z. Xiao, Accurate determination of the coefficients of elastic crack tip asymptotic field by a hybrid crack element with p-adaptivity. Engng Fract Mech, 68, 1609-30, 2001. [3] Q.Z. Xiao, B.L. Karihaloo, X.Y. Liu, Direct determination of SIF and higher order terms of mixed mode cracks by a hybrid crack element. Int J Fract, 125, 207-25, 2004. [4] D. Zeng, N. Katsube, J.M. Zhang, W. Soboyejo, Hybrid crack-tip element and its applications. Finite Elem Anal Des, 38, 319–35, 2002. [5] B.L. Karihaloo, Q.Z. Xiao, Implementation of HCE on a general FE mesh for interacting multiple cracks. Proc ECCOMAS 2004. Jyväskylä, Finland, 24 - 28 July, 2004. CD-ROM. [6] N. Moës, J. Dolbow, T. Belytschko, A finite element method for crack growth without remeshing. Int J Numer Meth Eng, 46, 131-150, 1999.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Structural Model Validation and the Lack-of-Knowledge Theory Pierre Ladevèze *,†, Paul Enjalbert *, Guilllaume Puel •, Thierry Romeuf ռ * LMT-Cachan (ENS Cachan/CNRS/Paris 6 University) 61 avenue du Président Wilson, F-94235 Cachan Cedex, France {pierre.ladeveze,paul.enjalbert}@lmt.ens-cachan.fr †

EADS Foundation Chair Advanced Computational Structural Mechanics • LMMSMat (École Centrale Paris/CNRS) Grande Voie des Vignes, F-92295 Châtenay-Malabry Cedex, France [email protected] ռ EADS Space Transportation Route de Verneuil BP96, F-78133 Les Mureaux Cedex, France [email protected]

ABSTRACT Today, the validation of complex structural models - i.e. the assessment of their quality compared to an experimental reference - remains a major issue. Most advanced approaches rely on the updating of deterministic dynamic parameters (stiffness, mass, damping) based on free or forced vibration tests. Uncertainties and probabilistic models can also be taken into account. In these works, the model validation is performed in a restricted sense. The true validation problem should be addressed through the comparison between the model - whether deterministic or not - used classically and the complete reality: such an issue raises philosophical questions. Here, we introduce a tentative answer through the Lack-Of-Knowledge (LOK) Theory, whose aim is to ”model the unknown”. In a certain way, this can be interpreted as an extension of what design engineers do when they introduce safety factors. Of course, the theory takes into account all the sources of uncertainties, including modeling errors, through the concept of basic LOKs. So far, two types of basic LOKs have been introduced: stiffness and excitation. For example, the structure being considered as an assembly of substructures, the basic stiffness LOKs are defined on the substructure level: each LOK is a scalar internal variable which quantifies the substructure’s LOK state in terms of structural stiffness; in mathematical terms, this variable is bounded by two stochastic bounds which follow probabilistic laws. Finally, a set of basic LOKs is added to the classical model to constitute the true model. This leads to an envelope of the actual responses; in particular, we can derive for the whole structure the effective LOK of a quantity of interest D, resulting in an interval with stochastic bounds. Another major question is the reduction of the LOKs using additional experimental information; the starting point could be an overestimated initial LOK level coming from experience. The paper focuses on the basic ideas of the Lack-Of-Knowledge Theory and on its first applications. Academic examples as well as industrial cases will be presented.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Modeling of Historical Masonry with Discrete Elements J. V. Lemos Laboratório Nacional de Engenharia Civil (LNEC) Av. do Brasil, 101, 1700-066 Lisboa, Portugal [email protected]

ABSTRACT Masonry is an inherently discontinuous material, formed by various components (stones, bricks, mortar), and its mechanical behavior reflects this internal structure. Engineering modeling of masonry structures is often based on continuum representations, using appropriate constitutive models, which provide an adequate solution for many practical cases. However, discontinuum models, which attempt to represent more closely the masonry components, are today applied with increasing frequency. They are preferred in research studies intended at understanding the mechanical behavior of masonry structures, but they are now also applied in engineering analysis, as the progress in computational resources is making them more accessible. Discontinuum representations may be achieved with various formulations of finite element methods. The present paper focuses on discrete elements models, a designation that covers a variety of representations of a structure as a system of blocks (rigid or deformable) or particles. Simplified contact formulations are generally used, to allow the analysis of very large systems, and explicit large displacement algorithms are employed. The paper discusses, in particular, two types of discrete element model, in relation to the specific requirements of masonry analysis. The first type is block models (rigid or deformable), which have proved very effective in the seismic analysis of historical stone masonry structures. Several applications are reviewed and key issues arising in this field are addressed, namely the variability of response observed in rigid block dynamics and the simplifications required in the analysis of large or complex structures. The second type of discrete element models examined is circular particle models, which represent each masonry block as a set of disks linked by contacts with tensile and shear bonds. Particles may also be used to simulate mortar or infill, and contacts between the various components are assigned appropriate constitutive behavior. An example of analysis of a multi-leaf wall is presented. The potential uses of these models in the study of the fundamental mechanics of masonry, in particular irregular masonry constructions, are discussed.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A Regularized Strong-form Meshfree Method for Adaptive Analysis G.R. Liu, Bernard B.T. Kee Centre for ACES, Department of Mechanical Engineering, National University of Singapore, 9 Engineering Drive 1, Singapore 117576 [email protected]; [email protected]

ABSTRACT This talk presents an adaptive meshfree method which is based on strongform formulation and does not use any mesh predefined through node connectivity. In this present formulation, a radial point collocation procedure is used to discretize the system governing equations. Techniques are presented to stabilize the solution to obtain stable and accurate results. Adaptive scheme adopted in this work uses an error indicator based on residuals. Simple and practical refinement procedures are also presented for additional node insertion at each adaptive step. Numerical examples are presented to demonstrate that the proposed adaptive meshfree method can obtain efficiently stable solutions of desired accuracy.

References [1] G. R. Liu and Y. T. Gu, An introduction to meshfree methods and their programming, Springer, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Multiresolution Analysis for Material Design Wing Kam Liu Department of Mechanical Engineering, Northwestern University, Evanston, IL, 60208 [email protected]

ABSTRACT The relationship between material microstructure and properties is the key to optimization and design of lightweight, strong, tough materials. Material properties are inherently a function of the microscale interactions at each distinct scale of deformation in a material. Currently, we rely on empirical data to define the structure-property link in the material design chain. A model is proposed here in which a material is physically and mathematically decomposed to each individual scale of interest. Material deformation can subsequently be resolved to each of these scales. Constitutive behavior at each scale can be determined by analytically or computationally examining the micromechanics at each scale. The proposed multiresolution technique is capable of linking overall material properties to the underlying microstructure via the micromechanics at each scale of interest. The small scale deformation phenomena which have a profound impact on macroscale properties are captured. The technique is general enough to be used in any material which exhibits different constitutive behavior at each scale. It can be implemented in a general finite element framework. This is illustrated for a polycrystalline material, a granular material, an alloy containing particles at two scales. A potential use for a bio-inspired self healing composite is also discussed. The theory can then be applied computationally in a finite element framework to determine the overall material properties in terms of the constitutive behavior at each scale, without resorting to empiricism.

References [1] Cahal McVeigh, Franck Vernerey, Wing Kam Liu and L. Cate Brinson, Multiresolution Analysis for Material Design, To appear in the special issue of Computer Methods in Applied Mechanics and Engineering, in memory of Professor J H Argyris. [2] S. Li, W.K. Liu, Meshfree Particle Methods, Springer (2004) [3] S. Hao, W.K. Liu, C.T. Chang, Computer Implementation of Damage Models by Finite Element and Meshfree Methods, Computational Methods for Applied Mechanics and Engineering 187, (2000) 401-440 [4] S. Hao, W.K. Liu, B. Moran, F. Vernerey, G.B. Olson, Multi-scale constitutive model and computational framework for the design of ultra- high strength, high toughness steels, Computer Methods in Applied Mechanics and Engineering 193 n17-20 (2004) 1865-1908 [5] F. Vernerey, W.K. Liu, B. Moran, Continuum theory for multi- scale micromorphic materials, Northwestern University Report, Illinois (2005) [6] H. Kadowaki, W.K. Liu, Bridging Multiscale method for localization problems, Computer methods in applied mechanics and engineering, 193 (2004) 3267-3302 [7] L.C. Brinson, One Dimensional Constitutive Behavior of Shape Memory Alloys: Thermomechanical derivation with non-constant material functions, Journal of Intelligent Material Systems and Structures 4(2) (1993) 229-242 [8] W.K. Liu, E.G. Karpov, S. Zhang, H.S. Park, An introduction to computational nanomechanics and materials, Computer Methods in Applied Mechanics and Engineering, 193 n17-20 May (2004) 1529-1578

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Characterization and Multiscale Modeling of Asphalt - Recent Developments in Upscaling of Viscous and Strength Properties Roman Lackner*, Ronald Blab†, Josef Eberhardsteiner*, and Herbert A. Mang* *

Institute for Mechanics of Materials and Structures, Vienna University of Technology (TU Wien) Karlsplatz 13/202, 1040 Vienna, Austria {Roman.Lackner,Josef.Eberhardsteiner,Herbert.Mang}@tuwien.ac.at †

Institute for Road Construction and Maintenance, Vienna University of Technology (TU Wien) Gußhausstraße 28/233, 1040 Vienna, Austria [email protected]

ABSTRACT The assessment and prediction of the performance of multi-composed materials, such as e.g. asphalt, requires suitable procedures for identification of their mechanical properties. In case of asphalt used for trafficked pavements, these properties vary with the underlying mix design (volume fractions and used constituents) and additives (e.g., polymers). In the past, the mix design and the allowance of additives were optimized, aiming at (a) a low viscosity at high temperatures (T > 135 °C) for the construction and compaction process of high-quality asphalt layers, (b) a significantly higher viscosity at medium temperature in order to minimize the development of permanent deformations (rutting), and (c) sufficient relaxation behavior at sub-zero temperatures, avoiding low-temperature cracking (see failure modes in Figure (left)). Motivated by the large variety of asphalt mixtures resulting from this optimization process and the necessity of predicting the future performance of pavements, a multiscale model for asphalt is currently developed at TU Wien. It relates macroscopic properties to finer-scale information (such as volume fractions, morphology, and the behavior of material phases) by introducing, in addition to the so-called macroscale (i.e., the scale at which prediction analyses are performed), four finer scales of observation, ranging down to the so-called bitumen-scale (see Figure (right)).

Figure: (left) failure modes in asphalt pavements and (right) multiscale model for asphalt In this lecture, recent results on upscaling information from finer observation scales towards the macroscale are presented. For this purpose, both analytical methods (continuum micromechanics) and numerical schemes (limit analysis) are employed. In addition to the theoretical work, the presented multiscale model for asphalt requires a significant amount of experimental work, covering both the identification of properties of material phases at the different observation scales and the validation of the employed upscaling schemes. Test results as well as advanced test methods employed at the Christian Doppler Laboratory for “Performance-based optimization of flexible pavements” at TU Wien for the characterization of asphalt at different scales are presented.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Dissipative Interface Modeling for Vibroacoustic Problems - A New Symmetric Formulation ¨ Walid Larbi, Roger Ohayon Jean-Franc¸ois Deu, Structural Mechanics and Coupled Systems Laboratory Conservatoire National des Arts et M´etiers Chair of Mechanics - case 353 292 rue Saint-Martin, 75141 Paris Cedex 03, France [email protected]

ABSTRACT This work concerns the variational formulation and the numerical computation of vibroacoustic interior problems with interface damping. The coupled system consists of an elastic structure (described by a displacement field) containing an inviscid, compressible and barotropic fluid (described by a pressure field), gravity effects being neglected. Within the context of noise reduction techniques, we propose to investigate the effect of introducing a thin layer of damping material at the fluid-structure interface. The originality of this work lies in the introduction of an additional unknown field at the fluid-structure interface, namely the normal fluid displacement field [1, 2]. With this new scalar unknown, various interface damping models can be introduced in the variational formulation. Moreover, the associated finite element matrix system can be solved in frequency and time domains. Here, a Kelvin-Voigt rheological model is used to take into account the interface damping. For a given material, the damping parameters can be found from the experimental acoustic impedance in a particular frequency range [3]. Following the procedure developed in [4], the proposed variational formulation is written in a symmetric form through the introduction of a displacement potential of the fluid. Numerical examples are presented in order to validate and analyze the new formulation.

References [1] J.-F. De¨u, W. Larbi and R. Ohayon, Structural-acoustic vibration and transient problems with interface damping. Third M.I.T. Conf. on Computational Fluid and Solid Mechanics, 14-17 June, 2005, Cambridge, USA. [2] W. Larbi, J.-F. De¨u and R. Ohayon, A new finite element formulation for internal acoustic problems with dissipative walls. International Journal for Numerical Methods in Engineering, accepted for publication, 2006. [3] V. Kehr-Candille and R. Ohayon, Elastoacoustic damped vibrations – Finite element and modal reduction methods. P. Ladev`eze, O.C. Zienkiewicz eds. New Advances in Computational Structural Mechanics, Elsevier, Amsterdam, 321–334, 1992. [4] H.J.-P. Morand and R. Ohayon, Fluid-Structure Interaction. Wiley, New York, 1995.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Seismic Design Procedures in the Framework of Evolutionary Based Structural Optimization Manolis Papadrakakis, Nikolaos D. Lagaros, Michalis Fragiadakis

Institute of Structural Analysis & Seismic Research, National Technical University of Athens 9 Iroon Polytechniou Str., Zografou Campus, 15780 Athens, Greece {mpapadra, nlagaros, mfrag}@central.ntua.gr

ABSTRACT Since the early seventies structural optimization has been the subject of intensive research and several different approaches have been advocated for the optimal design of structures in terms of optimization methods or problem formulation. Most of the attention of the engineering community has been directed towards the optimum design of structures under static loading conditions with the assumption of linear elastic structural behaviour. For a large number of real-life structural problems assuming linear response and ignoring the dynamic characteristics of the seismic action during the design phase may lead to structural configurations highly vulnerable to future earthquakes. Furthermore, seismic design codes suggest that under severe earthquake events the structures should be designed to deform inelastically due to the large intensity inertia loads imposed. The objective of this work is to evaluate various design procedures adopted by seismic codes and their influence on the performance of real-scale structures under an objective framework provided by structural optimization. Several studies have appeared in the literature where seismic design procedures based on non-linear response (e.g. [1,2]) are presented and compared. However, this task can be accomplished in a complete and elaborate manner only in the framework of structural optimization, where the designs obtained with different procedures can be directly evaluated by comparing the value of the objective function of the optimization problem and the seismic performance of the optimum solution achieved. In this work evolutionary methods are implemented [3-5] to address the optimization problem and replace the conventional trial and trial and adjustmentbased procedures.

References [1] Han SW, Wen YK, Method for reliability-based seismic design: Equivalent nonlinear systems, II: Calibration of code parameters. ASCE Journal of Structural Engineering, 123(3): 256-270, 1997. [2] Bazzuro P, Cornell CA, Shome N, Carballo JE. Three proposals for characterizing MDOF nonlinear response. ASCE Journal of Structural Engineering 1998; 124(11): 1281-1289. [3] Foley CM. Optimized performance-based design for buildings. In Burns S.A. (Ed.) Recent Advances in Optimal Structural Design, ASCE Publications, 2002: 169-240. [4] M. Fragiadakis, N.D. Lagaros and M. Papadrakakis, Performance based earthquake engineering using structural optimization tools, International Journal of Reliability and Safety, 2005. [5] M. Fragiadakis, N.D. Lagaros and M. Papadrakakis Performance-based optimum design of steel structures considering life cycle cost, Structural and Multidisciplinary Optimization, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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High-Fidelity Multi-Criteria Aero-Structural Optimisation using Hierarchical Parallel Evolutionary Algorithms L. F. González*, L. H. Damp*, J. Périaux** and K. Srinivas* *

**

The University of Sydney, Sydney, NSW 2006, Australia {gonzalez, lloyd.damp, ragh}@aeromech.usyd.edu.au

INRIA Sophia Antipolis, OPALE project associate and CIMNE/UPC Barcelona [email protected]

ABSTRACT An emerging technique for the solution of Multi-criteria and Multidisciplinary Optimisation problems are Evolutionary Algorithms (EAs). This paper describes a parallel multi-criteria (multi-objective) evolutionary algorithms (PEAs) for aero-structural problems. The foundations of the algorithm are based upon traditional evolution strategies and incorporate the concepts of a multi-objective optimisation, hierarchical topology, asynchronous evaluation and parallel computing. The algorithm works as a black-box optimiser and has been coupled to several aerodynamic and aircraft conceptual design solvers. The paper describes the features of the method and the application of the method for aero- structural design problems. The coupling of the algorithm with a higher order panel method, a commercial FEA solver and an automated aerodynamic and structural mesh generation program is described. In this automated process the optimiser defines design variables for the automatic mesh generation program which defines and produces the external geometry of the wing (Root chord, tip chord, sweep, etc) and the internal geometry (number of spars, number of ribs skin thickness, etc). This geometry is then analysed by the aerodynamic solver and maps the pressure forces into the structural model. Single or multiple objectives can be defined by the optimiser to find non-dominated –Pareto optimal solutions or the Nash equilibrium. The parallel computing capabilities in combination with hierarchical levels of aerodynamic and FEA solvers are exploited to reduce computational expense. Comparisons will be presented between a traditional analytical approach -weakly couple, and a high fidelity –strongly coupled- approach for the aero-structural analysis of a high aspect ratio aircraft/UAV wing.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Interaction of Shells and Membranes with Incompressible Flows Ekkehard Ramm∗, Christiane F¨orster∗, Malte Neumann∗, Wolfgang A. Wall† ∗

Institute of Structural Mechanics, University of Stuttgart Pfaffenwaldring7, 70569 Stuttgart, Germany {ramm,foerster,neumann}@statik.uni-stuttgart.de

†Chair of Computational Mechanics, Technical University of Munich

Boltzmannstraße 15, 85747 Garching, Germany [email protected]

ABSTRACT For the dynamic behavior of lightweight structures like thin shells and membranes exposed to fluid flow the interaction between the two fields is often essential. Computational fluid-structure interaction provides a tool to predict this interaction and complement or eventually replace expensive wind tunnel experiments. Partitioned analyses techniques enjoy great popularity for the numerical simulation of these interactions. This is due to their computational superiority over simultaneous, i.e. fully coupled monolithic approaches, as they allow the independent use of suitable discretization methods and modular analysis software. We use, for the fluid, GLS stabilized finite elements on a moving domain based on the incompressible instationary Navier-Stokes equations, where the formulation guarantees geometric conservation on the deforming domain. The structure is discretized by nonlinear, three-dimensional shell elements. Commonly used sequential staggered coupling schemes may exhibit instabilities due to the so-called artificial added mass effect. As best remedy to this problem subiterations should be invoked to guarantee kinematic and dynamic continuity across the fluid-structure interface. Since iterative coupling algorithms are computationally very costly, their convergence rate is very decisive for their usability. To ensure and accelerate the convergence of this iteration the updates of the interface position are relaxed. The time dependent, ’optimal’ relaxation parameter is determined automatically without any user-input via exploiting a gradient method or applying an Aitken iteration scheme. A variety of numerical examples will show the capabilities of the presented methods.

References [1] Ramm, E., Wall, W. A.: Shell Structures - A Sensitive Interrelation between Physics and Numerics. International Journal for Numerical Methods in Engineering 60, 381-427, 2004. [2] Neumann, M., Tiyyagura, S.R., Wall, W.A., Ramm, E.: Robustness and Efficiency Aspects for Computational Fluid Structure Interaction. Proc. of the Second Russian-German Advanced Research Workshop on Computational Science and High Performance Computing, Stuttgart, Germany, March 14-16, 2005. In: Notes on Numerical Fluid Mechanics and Multidisciplinary Design (NNFM), Springer, 2005. ¨ [3] Forster , Ch., Wall, W.A., Ramm, E.: On the Geometric Conservation Law in Transient Flow Calculations on Deforming Domains. Int. J. Num. Meth. Fluids, 2005 (accepted).

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Nonlinear Analysis of Composite and FGM Shells using Tensor-Based Shell Finite Elements J. N. Reddy* and R. A. Arciniega Advanced Computational Mechanics Laboratory Texas A & M University, College Station, TX 77843-3123 *[email protected]

ABSTRACT In this paper, a finite element model for the nonlinear analysis of laminated shell structures and through-thickness functionally graded shells is presented. A tensor-based finite element formulation is presented to describe the deformation and constitutive laws of a shell in a natural and simple way by using curvilinear coordinates. In addition, a family of high-order elements with Lagrangian interpolations is used to avoid membrane and shear locking; no mixed interpolations are employed. A first-order shell theory with seven parameters is derived with exact nonlinear deformations and under the framework of the Lagrangian description. This approach takes into account thickness changes and, therefore, 3D constitutive equations are utilized. Numerical comparisons of the present results with those found in the literature for typical benchmark problems involving isotropic and laminated composite plates and shells as well as functionally graded plates and shells are found to be excellent and show the validity of the developed finite element model. Moreover, the simplicity of this approach makes it attractive for applications in contact mechanics and damage propagation in shells. Acknowledgement. The research results reported herein were obtained while the authors were supported by the Structural Dynamics Program of the Army Research Office (ARO) through Grant . 45508EG.

References [1] R.A. Arciniega, On a tensor-based finite element model for the analysis of shell structures, PhD dissertation, Dept. of Mechanical Engineering, Texas A&M University, December 2005. [2] J.N. Reddy, An Introduction to Nonlinear Finite Element Analysis, Oxford University Press, Cambridge, UK, 2004. [3] J.N. Reddy, Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, 2nd edition, CRC Press, Boca Raton, Florida, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Strength of Textile Composites – A Voxel Based Continuum Damage Mechanics Approach Raimund Rolfes*, Gerald Ernst*, Daniel Hartung†, Jan Teßmer† *

†

Institute for Structural Analysis, University of Hannover Appelstraße 9a, 30167 Hannover, Germany {r.rolfes, g.ernst}@isd.uni-hannover.de

Institute of Composite Structures and Adaptive Systems, DLR Braunschweig Lilienthalplatz 7, 38108 Braunschweig, Germany {jan.tessmer, daniel.hartung}@dlr.de

ABSTRACT Industrialised infusion processes enable a cost-effective possibility to produce textile composite structures compared to pre-impregnated composite systems (Prepregs). Particularly with regards to high performance structures one has to be familiar with the material behaviour and the failure characteristic to apply fibre reinforced composites profitable. In this publication a brief overview of the current research activities to characterise the mechanical behaviour, failure and strength of textile composites compared with prepreg systems is presented. Unlike pre-impregnated and filament winding composites, textiles are different in their mechanical behaviour due to various fibre architectures of the preforms (braided, woven, stitched, tufted, etc.). Therefore, a finite element analysis of a representative volume element (unit cell) on a micromechanical level is a promising possibility to analyse the mechanical behaviour and to predict the material failure. Currently different approaches are used to account for many material characteristics. The first results of an approach, which considers a continuum damage model to predict the first micromechanical material failure, will be presented. A lot of standards have been established to determine the material properties of Prepregs over the last years. Particularly textile composites require an adapted test setup to account for the characteristic material behaviour and to validate different failure criteria. The standards used to qualify a preimpregnated material and a short description of the requirements for textile composites are presented. While many failure theories were developed during the last years, there are some weaknesses even by the most popular failure theories. The results of the World Wide Failure Exercise have shown that there is a demand for an accurate failure prediction also for prepreg composites. Especially the comparatively complex failure of textile composites requires an advanced failure theory. The fracture plane concept originally proposed by Hashin is a promising method to describe the failure behaviour of prepreg composites. A three dimensional failure model was developed based on a fracture plane by Juhasz, who considers the characteristic material behaviour of orthogonally reinforced composites. This criterion was implemented in a three dimensional finite element to account for the three dimensional stress state in each layer of a lamina by Kuhlmann and Rolfes [1].

References [1] G. Kuhlmann and R. Rolfes, A hierarchic 3d finite element for laminated composites. International Journal for Numerical Methods in Engineering, 61, 96–116, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Concrete at Early Ages and Beyond: Numerical Model and Validation Bernhard A. Schrefler*, Francesco Pesavento*, Dariusz Gawin†, Mateusz Wyrzykowski† *

†

Dept. of Structural and Transportation Engineering, University of Padova, Italy Via F. Marzolo 9, 34131 Padova - Italy [email protected], [email protected]

Dept. of Building Physics and Building Materials, Technical University of Lodz, Poland Al. Politechniki 6, 93-590 Lódz, Poland [email protected]

ABSTRACT This work deals with a new mathematical/numerical model for the analysis of the behaviour of concrete considered as multiphase viscous porous material from early ages to long term periods. This is a solidification-type model where all changes of material properties are expressed as functions of hydration degree, and not maturity nor equivalent hydration period as in maturity-type models. A mechanistic approach has been used to obtain the governing equations, starting from micro-scale, by means of modified averaging theory, also called hybrid mixture theory, [1-2]. Constitutive laws are directly introduced at macroscopic level. An evolution equation for the internal variable, hydration degree, describes hydration rate as a function of chemical affinity, considering additionally to the existing models, an effect of the relative humidity on the process. The model takes into account full coupling between hygral, thermal and chemical phenomena, as well as changes of concrete properties caused by hydration process, i.e. porosity, density, permeability, and strength properties. Phase changes and chemical phenomena, as well as the related heat and mass sources are considered. Some examples showing possibilities of the model for analysis of autogenous self-heating and selfdesiccation phenomena, as well as autogenous shrinkage are presented and discussed. Creep processes are modelled considering concrete as viscous-elastic material with aging caused by solidification of non-aging constituent, i.e. solidification theory for the so-called basic creep [1-2]. A Kelvin-type chain has been chosen for the definition of the compliance function, which corresponds to an expansion of that function in a Dirichlet’s series. Shrinkage is defined using the effective stress principle, as usual in the mechanics of porous materials, and it is coupled to the creep model. In such a way it is possible to have creep strains even if the concrete structure is not externally loaded. Capillary shrinkage is, in fact, characterized from capillary tensions which can be seen as a sort of internal load for the microstructure of the material. A series of numerical computations compared to the experimental results are presented as validation of the model described above.

References [1] D. Gawin, F. Pesavento, B.A. Schrefler: Hygro-thermo-chemo-mechanical modelling of concrete at early ages and beyond. Part I: Hydration and hygro-thermal phenomena. Int. J. Num. Meth. Engng, in print. [2] D. Gawin, F. Pesavento, B.A. Schrefler: Hygro-thermo-chemo-mechanical modelling of concrete at early ages and beyond. Part II: Shrinkage and creep of concrete. Int. J. Num. Meth. Engng, in print.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Uncertainty & Reliability Analysis of Structural Dynamical Systems Gerhart I. Schuëller Chair of Engineering Mechanics, Leopold-Franzens University, Innsbruck, Austria, EU. [email protected]

ABSTRACT In this paper various methods to analyze structural systems under stochastic dynamic loading are qualitatively compared. While the Karhunen-Loève expansion proved to be advantageous when uncertainty estimation is required, (advanced) Monte Carlo simulation procedures are recommended for reliability estimates. The computational efficiency of the methods play an important role w.r.t. practical applications. A further quantitative benchmark study is recommend.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Neural Networks: New Results and Prospects of Applications in Structural Engineering Zenon Waszczyszyn*†, Leonard Ziemianski† *

Cracow University of Technology, Institute of Computer Methods in Civil Engineering Warszawska 24, 31-155 Krakow, Poland [email protected] †

Rzeszów University of Technology, Chair of Structural Mechanics W. Pola 2, 35-959 Rzeszów, Poland [email protected]

ABSTRACT NN is a new computational tool for data processing and this tool can be characterized as a “data dependent and model free” approach. Other features of NNs correspond to their applicability in the analysis of nonlinear direct and inverse problems. NNs can also be used in hybrid systems as a complementary part to conventional computational methods, especially to FEM. The applications of the feed-forward, multilayer, error back-propagation NN, called for short BPNN (back-propagation NN), are discussed focusing on two fields: 1) BPNN as a new independent computational tool, 2) hybrid FEM/BPNN systems. All the considered applications are based on data taken from tests on laboratory models or measurements performed on natural scale buildings. BPNN applications are illustrated on four selected problems. The first one is related to soil-structure interaction caused by paraseismic excitations. The mappings of displacement response spectra DRSg o DRSb were performed, where: DRSg  spectrum computed on the ground level at a monitored building, DRSb  neurally predicted spectrum inside the building on the basement level. It was proved that the application of Kalman filtering for the training of a BPNN leads to much more exact approximation than the application of Rprop learning method. The second problem is related to the identification of placement of an additional mass fastened to a steel plate. The dynamic response corresponding to natural eigenfrequencies of the plate with the mass was used as the BPNN input. Satisfactory results for the parametric identification of the mass location were obtained due to addition of the Gaussian white noise to perturb a small number of measured dynamic responses and due to the application of cascade architecture of BPNNs. The third problem deals with the application of hybrid FEM/BPNN Monte Carlo method in the reliability analysis of steel cylindrical panels. The results of laboratory tests were explored to update the FE model which was then used to compute the patterns for the BPNN training and testing. The trained network was explored to rapid computation of Monte Carlo simulations of the panel ultimate load. It was shown that the hybrid approach enables us to predict the probability of reliable curve very efficiently. The fourth problem is related to the hybrid approach of the FE model updating. It is discussed on the example of a simple plane frame tested on laboratory models. In the formulated hybrid system a FE program was used for generating the training and testing sets of patterns. The trained BPNN was then explored in the inverse analysis for calibrating of control parameters values which well corresponded to the measured eigenfrequencies of the frame laboratory models. Using BPNNs in the hybrid updating of FE models makes it possible to eliminate the optimization procedure in the corresponding inverse analysis.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Computational Railway Dynamics Torbjörn Ekevid∗ , Per Kettil† , Håkan Lane† , Nils-Erik Wiberg† ∗ School

of Technology and Design Växjö University 351 95 Växjö, Sweden [email protected]

† Department of Applied Mechanics Chalmers University of Technology 412 96 Göteborg, Sweden {per.kettil; hakan.lane; nils-erik.wiberg}@sem.chalmers.se

ABSTRACT High-quality and efficient means of transport is of high priority in the modern society. Railway traffic is environment friendly and economically very competitive for both freight and personal transports at mid-range distances. Although the railway technology has been improved substantially during the last decades, generated vibrations still impose annoyances to the surroundings environment and leads to deterioration of the track structure. To understand the physical phenomena and propose countermeasures/improvements, simulation tools to perform computations of the entire dynamical system including subground, track structure and the railway vehicle has been developed. In particular, large effort has been devoted to the special wave propagation problem related to high-speed trains running at soft ground materials. As the speed of the train approaches and exceeds the natural (Rayleigh) wave propagation velocity of the ground material, shock waves similar to sonic boom originates from the onrushing train. The problem area contains several computational challenges since it impose techniques to handle non-reflecting boundaries[1], time integration of large-scale problems, non-linear material response etc. Efficient solvers based on a combination of multigrid[2], error estimations and adaptive refinement has been developed to reach acceptable execution times. The solution time is substantially reduced compared to conventional implicit solvers based on factorization. Moreover, indefinite system from the Lagrange multiplier[3] approach to handle constraint equations imposes additional preconditioning to guarantee convergence and reducing the number of iterations. In the paper a number of numerical examples from railway applications are presented. Results from computation where the train is represented as a collection of moving loads as well as a multi body system with complete train-track interaction are demonstrated.

References [1] T. Ekevid, N.-E. Wiberg, Wave propagation related to high-speed train - a scaled boundary FEapproach for unbounded domains, Comput. Methods Appl. Mech. Engrg. 191, 3947–3964, 2002. [2] U. Trottenberg, C. Oosterlee, A. Schüller, Multigrid, Academic Press, London, 2001. [3] Z.-H. Zhong, Finite element procedures for contact-impact problems, Oxford Univ. Press, Oxford, 1993.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Micro-Meso-Macro Modelling of Composite Materials P. Wriggers, M. Hain Institute of Mechanics and Computational Mechanics University of Hannover Appelstr. 9a, D-30167 Hannover, Germany [email protected]

ABSTRACT Multi-scale models can be extremely helpful in the understanding of complex materials used in engineering practice. In the presentation the basic theoretical strategy is developed. Possible finite element methods to solve such problems are explained in detail and discussed. These are based on homogenization techniques but also on true multi-scale solutions. The developed methodology is then applied to a specific engineering material which is concrete. This construction material has to be investigated on three different scales, the hardened cement paste, the mortar and finally the concrete. Here a successive two-stage approach is followed in which first the multi-scale model of the cement paste and mortar is applied. The resulting homogenization can then used in a multi-scale mortar-concrete model. The model for the hardened cement paste is based on a three–dimensional computer–tomography at the micrometer length scale. For this a finite element model is developed with different constitutive equations for the three parts unhydrated residual clinker, pores and hydrated products. The volume fraction of the hydrated products is approximately 84 Vol.%. For this part, a visco–plastic material model of Perzyna-type including damage is applied. The other two parts are described with a linear– elastic material model. The constitutive equations at the micro–scale contains inelastic parameters, which cannot be obtained through experimental testings. Therefore, one has to solve an inverse problem which yields the identification of these properties. For computational efficiency and robustness, a combination of the stochastic genetic algorithm and the deterministic Levenberg-Marquardt method is used. In order to speed-up the computation time significantly, a client-server based system is used. Hence, all calculations are distributed automatically within a network environment. The resulting constitutive parameters on the micro-scale are then used in the homogenized constitutive model for the mortar. But also in the multi-scale model for the mortar. Both results are compared with each other but also with experimental data. A further interesting application occurs when the micro-structure of the cement paste is filled partly with water and a freezing process takes place. Due to frost, the moisture inside the microstructure freezes. A constitutive model for ice is applied to the water filled parts of the microstructure is then developed. The expansion of the ice leads to damage in the micro–structure which yields an inelastic material behavior on the macro–scale. If such a calculation is performed for different moistures and temperatures, a correlation between moisture, temperature and the inelastic material behavior is obtained. Numerical examples show, that the developed approach reproduces the material behavior realistically.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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An Object-Oriented Approach to High Order Finite Element Analysis of Three-Dimensional Continua M. Baitsch∗, T. Sikiwat, D. Hartmann∗ ∗ Department of Civil Engineering Ruhr-University of Bochum, Germany [email protected] [email protected]

ABSTRACT It has been shown recently that the p-version of the finite element method is well suited for the analysis of thin walled three-dimensional continua [2]. However, designing and implementing software for the p-version of FEM is a challenge because of the increased complexity compared to the h-version. In this paper, we present an object-oriented finite element system implemented in Java. It is pointed out how the object-oriented paradigm, suitable design patterns and thorough unit testing can help to develop and maintain a complex engineering application. The basic idea of the software design is to separate generally applicable mathematical concepts, like basis functions and geometrical mappings from concrete element formulations that contain the physics of the actual problem. In the mathematical package, there are interfaces representing the concepts of basis functions (forming an Ansatz space) and functions defined on R1 , R2 and R3 . Several concrete classes implement these basic interfaces and realize for instance Lagrangian basis functions, hierarchical basis functions, functions constructed by a linear combination of basis functions or functions constructed by the blending function method. The use of NURBS curves hereby allows for the representation of complex geometries like that of the shell structure shown below. The power of this approach lies in the fact that on the element level only the interface types are used. Thus, an element formulation solely contains the physics and is not limited to a certain type of Ansatz space. Also, new Ansatz spaces can be easily incorporated lateron. Generally Java is considered as being slow for numerically intense applications. On the other side there are many advantages that make Java attractive also for simulation software [1]. In this paper, a hybrid approach is presented where the overall program is implemented in Java but numerically intense linear algebra operations are delegated to native code. In the full paper, it is shown that the software can be easily applied to problems involving 75 000 DOFs and more.

Analysis of a Shell Structure: System and Deformation

References [1] R.F. Boisvert, J. Moreira, M. Philippsen, and R. Pozo. Numerical computing in Java. Computing in Science and Engineering, 3(2):18–24, 2001. [2] A. D¨uster, H. Br¨oker, and E. Rank. The p-version of the finite element method for three-dimensional curved thin walled structures. International Journal for Numerical Methods in Engineering, 52:673–703, 2001.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A finite element formulation based on the theory of a Cosserat point – Extension to Ogden material Eiris F.I. Boerner , Dana S. Mueller-Hoeppe , Stefan Loehnert , Peter Wriggers Institute for Mechanics and Computational Mechanics University of Hannover, Appelstrasse 9a, 30167 Hannover, Germany [email protected] Department of Mechanical Engineering Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USA ABSTRACT The theory of Cosserat points is the basis of a finite element formulation for a solid three-dimensional continuum, which was presented by [1]. Previous investigations [2] have revealed, that this formulation is free of showing undesired locking or hourglassing-phenomena. It additionally shows excellent behaviour for large deformations in any type of incompressible material and for sensitive structures such as plates or shells. Within the theory of Cosserat points, the position vectors X and x of an 8-node-brick element are described through director vectors Di and di .

X=

7
i=0

N i (θ1 , θ2 , θ3 )Di

,

x=

7


N i (θ1 , θ2 , θ3 )di

i=0 3 4

N 0 = 1, N 1 = θ1 , N 2 = θ2 , N 3 = θ , N = θ1 θ2 , N 5 = θ1 θ3 , N 6 = θ2 θ3 , N 7 = θ1 θ2 θ3 The special choice of shape functions N i allows to split the deformation as well as resulting stresses into homogeneous and inhomogeneous parts respectively. The stresses due to the inhomogeneous part of the deformation are obtained by incorporating analytical solutions to the deformation modes bending, torsion and higher-order hourglassing for a rectangular parallelepiped shaped reference element, see [1] and [2]. This work shows approaches on how to overcome the difficulty of initially distorted element geometries that differ strongly from the shape of a rectangular parallelepiped. The formulation initially was restricted to a Neo-Hookean material. This work will present the extension to a general elastic Ogden material as well as to metal plasticity for large deformations with isotropic hardening. It will also give insight to the properties of the Cosserat point element and its behaviour for rubber-like materials.

References [1] Nadler, B.; Rubin, M.B. (2003), A new 3-D finite element for nonlinear elasticity using the theory of a Cosserat point, Solids & Structures, 40, 4585-4614. [2] Loehnert, S.; Boerner, E.F.I.; Rubin, M.B.; Wriggers, P. (2005), Response of a nonlinear elastic general Cosserat brick element in simulations typically exhibiting locking and hourglassing, Computational Mechanics, Vol. 36:266-288.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A macro tetrahedral element with vertex rotational D.O.F.s Jaesung Eom*, Byungchai Lee† *

Department of mechanical engineering, Korea advanced institute of science and technology, 373-1 Guseong-dong, Yuseong-gu, Daejon, 305-701, Republic of Korea [email protected]

†

Department of mechanical engineering, Korea advanced institute of science and technology, 373-1 Guseong-dong, Yuseong-gu, Daejon, 305-701, Republic of Korea [email protected]

ABSTRACT In this study, a macro tetrahedral element with vertex rotations is presented with the systematic macro element fabrication and the inclusion of proper higher-order deformation modes. The individual element test (IET) proved sub-tetrahedral elements are basically combined for the macro element. During the construction of the macro element, a higher-order stiffness is emerged by moving the influence of removed virtual node stiffness into the deformation modes which are related to the vertex rotations. The fundamental decomposition into a basic stiffness and a higher-order stiffness allows scaling the performance of the macro element. To fulfill the accuracy and stability requirement of the element, the transformation between hierarchical rotations and nodal freedoms is employed to separate rotational freedoms into the constant strain related rotation and the higher-order behavior related one. Each parameter in the element stiffness is tuned through beam-type higher order patch test. The numerical performance of the proposed element is compared with other solid elements with vertex rotation in the benchmark test problem.

References [1] K.Y. Sze and Y. S. Pan, Hybrid stress tetrahedral elements with Allman’s rotational D.O.F.s, International journal for numerical methods in engineering, 48, 1055-1070, 2000. [2] Carlos A. Felippa, A study of optimal membrane triangles with drilling freedoms. Computer methods applied mechanicultibody System Dynamics, 1, 149-188, 1997. [3] P. G. Bergan and L. Hanssen, A new approach for deriving ‘good’ finite elements, Proc. MAFELAP II Conf., Academic Press, Brunel University, 483-497, 1976. [4] Jaesung Eom and Byungchai Lee, A study of a macro membrane triangle with drilling freedoms, Proc. WCCM VI Conf., Tsinghua University Press & Springer-Verlag, 23, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Application of aggregation multilevel iterative solver to problems of structural mechanics Sergiy Yu. Fialko* *

National University of Architecture and Constructions in Kiev, Ukraine & Software Company SCAD Soft 4, I. Klimenko str., Office 20 Kiev, Ukraine, 03037 [email protected]

ABSTRACT The application of aggregation multilevel iterative solver AMIS [2, 3] to the structural analysis of large-scale finite element problems is discussed. It is well-known that such problems usually are poorly conditioned and this leads to slow convergence of iterative methods [1]. The preconditioned conjugate gradient method with aggregation multilevel preconditioning is applied to solution the both: linear static problems and natural vibration ones. The effectiveness of several solution stages is considered: the creation of aggregation model, the application of sparse matrix technique to solution of the coarsest level problem, the smoothing algorithms and so on. The set of local rigid links are imposed to the given finite element model to decrease the number of degrees of freedom. The nodeby-node or element-by-element approaches are applied to obtain the reduced model on coarsest level. We try to keep as large number of equation on coarsest level as the computer resources permits it to do. The block sparse multifrontal solver [4] is applied to solve the coarsest level problem. The proper reordering method among multilevel reordering and multiple minimum degrees ones is chosen to reduce fill-inns. The several modifications of incomplete Cholesky factorization approaches are applied to smooth rapidly oscillating residuals after prolongation. The numerous numerical examples, taken from computational practice of SCAD Soft, illustrate the robustness and efficiency of proposed method.

References [1] A. Perelmuter, S. Fialko. Problems of computational mechanics relate to finite-element analysis of structural constructions. International Journal for Computational Civil and Structural Engineering, 1(2) 72-86, 2005. [2] S. Fialko. Application of iterative solvers in finite element analysis of structural mechanics. Linear statics and natural vibrations. Proceedings of 8-th international conference "Modern building materials, structures and techniques". May 19–22, 2004, Vilnius, Lithuania. P. 721 – 725. [3] S. Fialko. Aggregation multilevel iterative sSolver for analysis of large-scale finite element problems of structural mechanics: linear statics and natural vibrations. R.Wyrzykowski et al. (Eds.): PPAM 2001, LNCS 2328, pp. 663 –670, 2002. Springer-Verlag Berlin Heidelberg 2002.

[4] S. Fialko. A block sparse direct multifrontal solver in SCAD software. Proceedings of the CMM-2005 – Computer Methods in Mechanics. June 21-24, 2005, Czestochowa, Poland. 73 – 74.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Convergence Analysis of a Domain Decomposition Method with Augmented Lagrangian Formulation Alexandre Santos Hansen*, Fernando Alves Rochinha† *

Petrobras/ENGENHARIA/IEEPT/EEPTM/EDI Rua Almirante Barroso, 81, 11º.andar, Rio de Janeiro, RJ, 21031-004, Brasil [email protected] †

Dept. de Engenharia Mecânica – COPPE - UFRJ Cx. 68503, 21945-970, Rio de Janeiro, RJ, Brasil [email protected]

ABSTRACT Through the combination of an augmented Lagrangian formulation with a preconditioned inexact Uzawa algorithm, we construct a domain decomposition based method for finite element approximation of linear second-order elliptic partial differential equations. With this approach, the proposed method shares the main features of Lagrange multipliers based domain decomposition methods, i.e. number of iterations bounded by the local element size (H/h) using a simple coarse space and direct application to decompositions with irregular sub domain geometry, with the advantage that inexact solvers at sub domain level are allowed at sub domain level. An analysis of the method applied to the Poisson equation is presented to justify the preconditioners choices and to derive bounds for the convergence factor in function of the local element size.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Computational Models on Graphs for Nonlinear Hyperbolic and Parabolic System of Equations Yaroslav A. Kholodov1, Alexander S. Kholodov2, Nikolai V. Kovshov3, Sergey S. Simakov4, Dmitri S. Severov5, Alexey K. Bordonos6, Azilkhan Bapayev7 1-4,6,7 Moscow Institute of Physics and Technology 9, Institutski Line, Dolgoprudny City, Moscow Region, 141700, Russia 1 [email protected], 2 [email protected], 3 [email protected], 4 [email protected], 6 [email protected], 7 [email protected] 5 Open Technologies company Bld. 1, 30, Obrucheva str., Moscow, 117997, Russia 5 [email protected]

ABSTRACT For G GgraphG edge G each G with length X k we consider 1D nonlinear hyperbolic system of equations vGt  Fxk (v )G f , v {v1 ,..., vI } , t t 0 , 0 d xk d X k , k 1,..., K (1) with initial conditions v (0, xk ) v 0 ( xkG) , k 1,..., K and the next boundary conditions: for graph enters (l 0 1,... L0 , xk 0G) M li0 (t , v (t , 0)) 0 , i 1,..., rk0 d I (2) , for graph exits (l * 1,...L* , xk X k ) (3) and for graph branchpoints l 1,..., L M li* (t , v G(t , X i 1,..., rk* d I G k )) G0 , \ lm (t , wl , vl1 ,..., vlM l ) 0 m 1,..., M l (4). Here K is the number of graph edges, L0 - enters, L* exits, L - branchpoints, M l - incoming and outgoing graph edges for the l th branchpoint, G G G vl1 ,..., vlM l - required vectors in theG ends G of edges adjoining to branchpoin l , wl - required vector for the branchpoint l . The matrix wF / wv A {aij } i, j 1,..., I is Jacobi matrix and we can apply the identity A : 1/: , where / {Oi } is the diagonal matrix of the matrix A eigenvalues, : is the nonsingular matrix whose rows are linearly independent left-hand eigenvectors of the matrix A ( Det: z 0 ) and : 1 is the matrix inverse to : . G To enclose boundary conditions (2)-(4) we can use compatibility conditions Zi dv dti 0 , G G G ( dv / dti wv / wt  Oi wv / wxk ), i 1,..., I along the characteristics of the system (1) dx Oi dt directed inside integration domain. These compatibility conditions can be used to analyze correctness of the problem definition for system (1) through the all graph segments. The main idea of this approach is that the solution of the global 3D problem for the whole graph can be split out on the set of the 1D problem for the single graph elements (edges). Then on the each step of numerical integration we join these 1D problems by the additional equation systems in the graph nodes to get the common solution for the original problem. By this way we guarantee the global coupling for the all original problem variables. The same computational model can be applied for nonlinear parabolic equations. The numerical results are presented for the problem solution on the different graph systems: The global numerical models of blood circulation in the human body. The model of global regional electrical power systems. The model of bar structures and frames behavior under the different impacts. The model of the intensive information flows in the computer networks. The model of heavy traffic in the big cities. The model of flood water and pollution propagation in the large river systems.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

44

Haar wavelet method for solving integral equations and evolution equations Ülo Lepik Institute of Applied Mathematics, University of Tartu Liivi 2, 50409 Tartu, Estonia [email protected]

ABSTRACT An efficient numerical method for solving nonlinear integral equations and evolution equations based on the Haar wavelets approach is proposed. The method is tested for Fredholm and Volterra integral equations and for the Burgers and sine-Gordon equations. Results obtained by computer simulation are compared with other available solutions. These calculations show that the accuracy of the Haar wavelet approach is quite high even in the case of a small number of grid points.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

45

A new approach for elimination of dissipation and dispersion errors in particle methods Hassan Ostad-Hossein*, and Soheil Mohammadi† *School of Civil Engineering, University of Tehran,Tehran, Iran [email protected] † School Civil Engineering, University of Tehran,Tehran, Iran [email protected]

ABSTRACT Numerical errors may be introduced in some numerical methods of solving differential equations because of their nature. Discretizing a continuum medium would result in changing the wave velocity and inducing numerical errors into the solution. Some methods using strong formulations are based on the Taylor expansion. Therefore, using only a finite number of Taylor series terms for particle simulations introduces truncation errors. Truncation of the Taylor expansion is also the reason for developing two other types of error. The first, called dispersion error appears in the form of extra vibration in high frequency modes that can result in solution instability in some problems. Another type of error is dissipation and may cause decrease in wave amplitude. Particle methods such as SPH [1] and CSPM [2], are also involved with truncation errors. A number of methods have already been proposed for removing dispersion from particle methods such as adding artificial stress. However these methods become energy dissipative resulting in wave amplitude decays after several time steps. In this paper further investigation is performed to study the roots of dispersion and dissipation errors in particle methods. A new procedure is proposed for eliminating dispersion and stabilizing the solution, based on the CSPM particle method and the Newmark time integration scheme. The results are compared with other existing methods.

REFERENCES [1] Chen JK, Beraun JE, Jih CJ, An improvement for tensile instability in smoothed particle hydrodynamics, Comput. Mech. 23 (1999a) 279-287 [2] Chen JK, Beraun JE, Jih CJ, Completeness of corrective smoothed particle method for linear elastodynamics, Comput. Mech. 24 (1999b) 273-285 [3]Chen JK, Beraun JE, Jih CJ, A corrective smoothed particle method for transient elastoplastic dynamics, Comput. Mech. 27 (2001) 177-187 [4] M.B. Liu, G.R. Liu, Z. Zong, Constructing smoothing functions in smoothed particle hydrodynamics with applications, Journal of Computational and applied Mathematics, 155 (2003) 263-284 [5] Gray J.P., Monaghan J.J., Swift R.P., SPH elastic dynamics, Comput. Methods Appl. Mech. Engrg 190 (2001) 6641-6662

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

46

Solution of stability problem of infinite plate strips Zdzisław Pawlak*, Jerzy Rakowski† *

Institute of Structural Engineering, Poznan University of Technology ul. Piotrowo 5, 60-965 Poznań, Poland [email protected]

†

Institute of Structural Engineering, Poznan University of Technology ul. Piotrowo 5, 60-965 Poznań, Poland jerzy.rakowski@ put.poznan.pl

ABSTRACT The aim of the paper is to solve a stability problem of infinite plate strips using the finite strip method (FSM). Contrary to well-known solutions for 2D continuous systems the authors present an idea for solving a stability problem of infinite discrete plates. A continuous plate strip simply supported on its opposite edges is divided into a regular mesh of identical finite strips. According to the finite strip procedure the field of loading and displacement functions are expressed in the form of harmonic series [1]. Stiffness and geometrical matrices for a four-degree-of-freedom finite strip are determined. The unknowns are deflections and transverse slope amplitudes along the nodal lines. Equilibrium conditions are derived from the FSM formulation. An infinite set of linear equations being the equilibrium conditions for each nodal line is expressed in the form of two second-order difference equations [2]. For a regular discrete system these equations written in the recurrent form are equivalent to the FSM matrix formulation. The exact solution of these difference equations is found in an analytical form. The displacement function fulfils the boundary conditions of the analysed plate strip and is given as a discrete expression for an arbitrary nodal load [3]. The solution of eigenvalue problem of the difference equations enables one to determine the critical forces of the structure. The main advantage of the presented approach is the analytical form of the solution obtained in the discrete domain. This enables a detailed parametrical analysis and investigations of the influence of geometrical and physical properties on the critical force value.

References [1] Y. C. Loo, A. R. Cusens, The Finite Strip Method in Bridge Engineering, New York, Viewpoint Publications,1978. [2] J. Rakowski, A critical analysis of quadratic beam finite elements. International Journal for Numerical Methods in Engineering, 1991, vol. 31, pp. 949-966 [3] Z. Pawlak, J. Rakowski, Static problem of plate strip by finite strip method using difference equation method. In: Proceedings of the 15th International Conference on Computer Methods in Mechanics CMM-2003, Gliwice/Wisła, 3-6 June 2003, pp. 279-280.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

47

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III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

48

Discretization of Three-Dimensional Aggregate Particles Daniel Rypl Department of Mechanics Faculty of Civil Engineering, Czech Technical University in Prague Th´akurova 7, 166 29, Prague, Czech Republic [email protected] ABSTRACT The design of concrete with specified properties became of increasing importance with the wide use of high-performance concretes (HPCs), such as pumpable concrete or self compacting concrete (SCC). Many concrete properties, starting from the mechanical properties as the compressive strength and modulus of elasticity, over the rheological properties influencing the workability of fresh concrete, up to physical properties as diffusivity and thermal and electric conductivity, for example, can be assessed by appropriate computational model representing concrete as a multiscale random composite material with realistically described aggregates. However, incorporation of three dimensional aggregate particles into computational code requires their proper discretization. This is not straightforward due to rather difficult mathematical characterization of aggregate particles of random shape. Modern technologies as computer tomography (CT) or magnetic resonance tomography (MRT) offer a powerful nondestructive technique for digital representation of opaque solid objects. This voxel based representation can be then discretized using for example marching cubes algorithm [1]. The resolution of the resulting triangulation, however, is strongly dependent on the resolution of the digital representation which might be either too coarse (without important features being captured) or too fine (with unimportant features captured by excessive number of elements). In the present work, the digital representation is first used to derive a smooth representation of aggregate particle using the expansion into spherical harmonic functions [2]. Although this representation is not universal it is suitable for almost all aggregates used in structural concrete. The significant advantage of this approach is that resolution of the smooth representation can be flexibly controlled by the number of terms in the expansion. In the next phase, the surface of aggregate particle is subjected to discretization using the advancing front technique. Although the representation of the surface is parameterized (by two spherical angles), the actual triangulation is performed directly on the surface in the real space [3] and not in 2D parametric space with subsequent mapping to the real space. The advantage of this procedure consists in the fact that the anisotropic meshing of the parametric space as well as the demanding calculations related to the reparameterization or the inverse mapping are avoided.

References [1] W.E. Lorensen, H.E. Cline, Marching cubes: A high resolution 3D surface construction algorithm. Computer Graphics, 21, 163-169, 1987. [2] E.J. Garboczi, Three-dimensional mathematical analysis of particle shape using x-ray tomography and spherical harmonics: Application to aggregate used in concrete. Cement and Concrete Research, 32, 1621-1638, 2002. [3] D. Rypl, Sequential and parallel generation of unstructured 3D meshes. PhD. Thesis, CTU Reports, 2 (3), 1998.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

49

Bending of an Elliptical Plate on Elastic Foundation and under the Combined Action of Lateral Load and In-Plane Force Kenzo Sato Akita University Department of Engineerings and Information Sciences, Faculty of Education and Human Studies, 010-8502 Akita, Japan [email protected] ABSTRACT The plane plates resting on elastic foundations are of practical importance in many engineering fields as seen in plate-subgrade structure, floating plate structure, composite material and so on. In some cases, the plates are subjected to large temperature differences from which considerable in-plane forces in the plates result. As structural elements in the wide fields of engineering, various types of elliptical plates may be used in order to avoid the high stress concentration and improve the usability of the instrument and the beauty of the architecture. In recent years, the author has been studied the vibration, buckling and bending problems of elliptical plates subjected to the combined action of lateral load and in-plane force[1]-[5]. In the reference [2] has been discussed also the influence of elastic foundation on the vibration and buckling of a clamped elliptical plate. From the viewpoint of the usefulness of an elliptical plate as structural element and the importance of the analytical solution in the mathematical theory of elasticity, it is the aim of this report to develop the exact theoretical analysis on the bending of an elliptical plate resting on a Winkler-type elastic foundation and subjected to the combined action of uniform lateral load and in-plane force. Here is considered the case that the plate-edge conditions are clamped and simply supported. Based on the classical small-deflection theory, the theoretical analysis is rigorously made in the elliptical coordinate system and the deflection surface due to bending of the plate is obtained in the form of an infinite Mathieu function series. The influences of elastic foundation and in-plane force on the bending of the elliptical plate are calculated by digital computer, and the new results obtained here will be presented in tables and figures.

References [1] K. Sato, Free Flexural Vibrations of a Simply Supported Elliptical Plate Subjected to an In-Plane Force. Theoretical and Applied Mechanics, 50, 165–181, 2001. [2] K. Sato, Vibration and Buckling of a Clamped Elliptical Plate on Elastic Foundation and under Uniform In-Plane Force. Theoretical and Applied Mechanics, 51, 49–62, 2002. [3] K. Sato, Bending of a Clamped Elliptical Plate under the Combined Action of Uniform Lateral Load and In-Plane Force. Theoretical and Applied Mechanics, 53, 37–47, 2004. [4] K. Sato, Bending of a Simply Supported Elliptical Plate under the Combined Action of Uniform Lateral Load and In-Plane Force. Theoretical and Applied Mechanics, 54, 31–44, 2005. [5] K. Sato, Bending of an Elastically Restrained Elliptical Plate under the Combined Action of Lateral Load and In-Plane Force. JSME International Journal, Ser.A, 49, 130–137, 2006.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

50

A modal analysis approach using an Hybrid-Mixed formulation to solve 2D elastodynamic problems M. Vicente da Silva∗ , Eduardo Pereira† ∗ Universidade

Nova de Lisboa - FCT/UNL, Civil Engineering Department Quinta da Torre, 2829-516 Monte da Caparica [email protected]

† Instituto

Superior T´ecnico, Civil Engineering and Architecture Department Av. Rovisco Pais no 1, 1049-001 Lisboa [email protected] ABSTRACT

The purpose of this work is to present a Hybrid-Mixed finite element formulation to solve elastodynamic plate problems in frequency domain. This Hybrid-Mixed model is derived [2] establishing, non-locally, the dynamic equilibrium, compatibility and constitutive relations based on the Galerking weighted residual method. Two different approximations fields are used in the element domain, namely the stresses and the displacements fields. A third approximation independent from the previous ones is also required for the displacement field on the static boundary of the elements. Once obtained the governing system, the stresses and the boundary displacements degrees of freedom are eliminated, thus resulting a new condensed system, were only the domain displacements degrees of freedom (qV ) intervine: (K − ω 2 M )qV = QV (1) The condensed system assume a form analogous to the one obtained with the conventional FEM[1], however with a different and richer physical meaning. In a first stage natural frequencies and shape modes are identified. Then, modal analysis technique is performed to uncouple the governing system equations, and to assess the relevance of each mode to a specific action. Less relevant modes are eliminated, thus reducing substantially the computational costs without significant lost of accuracy. The Hybrid-Mixed model is implemented using 4-node serendipian standard master elements to define the finite element shape, and non-nodal, Legendre polynomials as approximation functions. This functions are suited for this purpose because they allow to introduce in the finite element code closed form solutions to compute the coefficients of the structural matrices, avoiding time consuming numerical integration. Good results can be reach using macro finite elements since h-refinements are easy to performe simply by increasing the maximum degree of the polynomials in the approximation functions. The efficiency and performance of the presented method is validated with the aid of numerical examples. Free vibrations natural frequencies are determined and transient response in undamped and viscous damped structures are studied.

References [1] R. W. Clough and J. Penzien. Dynamics of Structures. McGraw-Hill, New York, 2nd edition, copyright 1993. [2] E. M. B. R. Pereira and J. A. T. Freitas. A mixed-hybrid finite element model based on orthogonal functions. International Journal for Numerical Methods in Engineering, 39:1295–1312, 1996.

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A new finite element method for Kirchhoff plates Lourenc¸o Beir˜ao da Veiga ∗, Jarkko Niiranen†, and Rolf Stenberg† ∗ Dipartimento di Matematica ”Federigo Enriques”

via Saldini 50, 20133 Milano, Italy [email protected] †Institute of Mathematics, Helsinki University of Technology

P.O. Box 1100, 02015 TKK, Finland jarkko.niiranen@tkk.fi, rolf.stenberg@tkk.fi

ABSTRACT Based on the ideas from [1] and [2] we present a new finite element method for the Kirchhoff plate bending model [3]. The method uses C 0 basis functions for the deflection and the rotation, i.e. the same approach as used for the Reissner–Mindlin model. To account for the effective shear force at the free boundary a stabilization term is added. We prove optimal a-priori and a-posteriori error estimates. The corresponding results from benchmark problems are also reported.

References [1] P. Destuynder and T. Nevers, Une modification du mod`ele de Mindlin pour les plaques minces en flexion pr´esentant un bord libre. RAIRO Mod´el. Math. Anal. Num´er., 22, 217–242, 1988. [2] R. Stenberg, A new finite element formulation for the plate bending problem. P. G. Ciarlet, L. Trabucho, and M. Via˜no eds. Asymptotic Methods for Elastic Structures, Proceedings of the International Conference, Lisbon, October 4–8, 1993, Walter de Gruyter & Co., Berlin – NewYork, 209–221, 1995. [3] L. Beir˜ao da Veiga and J. Niiranen and R. Stenberg, A family of C 0 finite elements for Kirchhoff plates. Helsinki University of Technology, Institute of Mathematics, Research Reports, A 483, January, 2006 (http://www.math.tkk.fi/reports).

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III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A geometric approach to the algorithmic tangent stiffness G. Romano∗, M. Diaco†, R. Barretta† ∗ Dipartimento di Scienza delle Costruzioni (DiSCo)

Universit`a di Napoli Federico II, via Claudio 21- 80125 Napoli, Italy [email protected] †Dipartimento di Scienza delle Costruzioni (DiSCo) Universit`a di Napoli Federico II, via Claudio 21- 80125 Napoli, Italy [email protected] - [email protected]

ABSTRACT The elastoplastic tangent stiffness is the linear operator which provides the stress rate corresponding to a prescribed strain rate. As such, it plays a central role in the computational aspects of elastoplastic problems. According to the usual approach to the nonlinear evolutive analysis of elastoplastic models, a finite time-step is considered and the evolution law describing the constitutive behavior is reformulated as a finite step flow rule. The algorithmic tangent stiffness was first introduced by Simo and Taylor in [1]. They showed that the adoption of the algorithmic tangent stiffness leads to a significant improvement of the asympthotic convergence rate. The expression of the algorithmic tangent stiffness provided in [1], and in all the subsequent references to their contribution, was based on an explicit formulation of the elastoplastic constitutive law in terms of a plastic scalar multiplier. The geometrical analysis developed in the present paper is based on a formulation of the constitutive problem in terms of the nonlinear projector, in complementary elastic energy, on the convex elastic domain. The algorithmic tangent stiffness is evaluated as the composition between the derivative of the nonlinear projector and the elastic stiffness. The key point consists in the evaluation of the derivative of the nonlinear projector. A direct geometric argument, based on hypersurface theory, shows that the derivative can be expressed as the difference between the linear projector on the hyperplane tangent at the trial stress point and the shape operator of the parallel hypersurface passing thru the trial stress-state, multiplied by the distance between the trial stress and the projected stress, evaluated in the complementary elastic norm. The composition of the linear projector with the elastic stiffness is in fact the rate elastoplastic tangent stiffness. Since the analytic expression of the parallel hypersurface thru the trial stress point is available only in special cases, an effective procedure consists in substituting it with the level set of the yield function passing thru the trial stress point. In this way, the exact expression of the algorithmic stiffness is got when the level sets of the yield function are homothetic hypersurfaces, as in von Mises plasticity criterion, and a simple useful approximation is obtained in the general case. Indeed the proposed procedure greatly simplifies the computations while preserving the benefit of an improved convergence rate, since it takes effectively into account the curvature of the yield hypersurface, thus leading to a reduced tangent stiffness, in comparison with the rate tangent stiffness.

References [1] J.C. Simo, R.L. Taylor, Consistent tangent operators for rate-independent elastoplasticity, Comp. Meth. Appl. Mech. Engrg. , 48, 101–118, 1985.

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Parallel Computation of 3D Problems of the Dynamics of Elastic-Plastic Granular Material under Small Strains Vladimir M. Sadovskii, Oxana V. Sadovskaya∗ Institute of Computational Modelling SB RAS Akademgorodok, Krasnoyarsk, Russia, 660036 ∗[email protected] ABSTRACT A process of waves propagation in elastic-plastic granular material under small strains is described on the basis of a new rheological model taking into account different resistance of the material with respect to tension and compression [1]. This model leads to the system of nonlinear partial differential equations of non-classical type which has an evident structure for numerical realization. Parallel decomposition of the used shock-capturing numerical method [2] is founded on the space-variable decomposition procedure. The proposed algorithm is accomplished as a program system for supercomputers with parallel architecture by means of SPMD (Single Program – Multiple Data) technology in Fortran-90 with the use of MPI (Massage Passing Interface) library. Various variants of 1D, 2D and 3D division of the spatial computational domain between the computational nodes are used. The program system allows to simulate a propagation of elastic-plastic waves generated by mechanical impacts in a body, aggregated of an arbitrary number of heterogeneous blocks with curvilinear boundaries. The exact one-dimensional solutions with plane shock waves are used for testing. Some problems of the waves refraction on the interior surfaces between blocks of granular materials with different mechanical properties are solved.

Stress field in different time moments

By means of the numerical experiments it is shown that the plane fronts of two waves, bending due to inhomogeneous loosening, can be reflected with the formation of transverse cumulative splash. The curved fronts of shock waves for different time moments and the cumulative splash (a typical zone of the compressive stresses, moving bottom-up) are represented in the Figure.

This work was supported by the grant of the Russian Foundation for Basic Research no. 04-01-00267, the Complex Program of the Presidium of RAS no. 14 “Fundamental Problems of Informatics and Informational Technologies” and the Russian Science Support Foundation.

References [1] V.M. Sadovskii, O.V. Sadovskaya, Parallel Computation of Elastic-Plastic Waves Propagation in Granular Material. Proc. of the 7th Intern. Conf. on Mathematical and Numerical Aspects of Wave Propagation. Brown University, Providence, 223–225, 2005. [2] O.V. Sadovskaya, Shock-Capturing Method as Applied to the Analysis of Elastoplastic Waves in a Granular Material. Comput. Math. & Mathematical Phys. 44, 1818–1828, 2004.

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A hyperelastodynamics ALE formulation based on spatial and material forces. Z. Uthman, H. Askes University of Sheffield, Department of Civil and Structural Engineering. Mappin Street, Sheffield S1 3JD, United Kingdom [email protected] [email protected]

ABSTRACT This contribution aims at providing the formulation and implementation details of arbitrary Lagrangian Eulerian hyperelastodynamic problem classes. This ALE formulation is based on the dual balance of momentum in terms of spatial forces (the well-known Newtonian forces) as well as material forces (also known as configurational forces). The balance of spatial momentum results in the usual equation of motion, whereas the balance of the material momentum indicates deficiencies in the nodal positions, hence providing an objective criterion to optimize the finite element mesh. The main difference with traditional ALE approaches is that the combination of the Lagrangian and Eulerian description is no longer arbitrary, in other words the mesh motion is no longer user defined but completely embedded within the formulation. The present work aims at developing spatial and material variational equations based on the Hamiltonian principle. These equations will be discretised to obtain the weak form of the momentum and continuity equations. The discretized ALE Hamiltonian equations of the spatial motion problem introduces the balance of the discretised spatial momentum and the discretised spatial continuity equation while the corresponding material motion problem defines the balance of the discretised material (or configurational) momentum and the discretised material continuity equation. We will deal with two systems of partial differential equations: the scalar continuity equation and the vector momentum equation. The momentum and continuity equations will then be linearised. The time integration of both the spatial and the material equations is performed with the Newmark scheme. A monolithic solution strategy solving the spatial and the material momentum equations simultaneously has been carried out while updating of the spatial and the material densities was attained through solving the spatial and material continuity equations (mass conservation). The solution defines the optimal spatial and material configuration in the context of energy minimization.

References [1] T. Belytschko, W.K. Liu and B. Moran. Nonlinear Finite Elements for Continue and Structures. John Wiley & Sons Ltd, 2001. [2] E. Kuhl and P. Steinmann, A hyperelastodynamic ALE formulation based on referential, spatial and material settings of continuum mechanics: Acta Mechanica, 174, 201–222, 2005. [3] E. Kuhl, H. Askes and P. Steinmann, An ALE formulation based on spatial and material settings of continuum mechanics. Part 1: Generic hyperelastic formulation. Comput. Methods Appl. Mech. Engrg, 193, 4207-4222, 2004.

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Monoharmonic Approach to Investigation of Heat Generation in The Viscoplastic Solids under Harmonic Loading Yaroslav A. Zhuk Timoshenko Institute of Mechanics 3, Nesterov str, Kiev 03057, Ukraine [email protected]

ABSTRACT Intensive forced vibration of structural elements can be accompanied with significant temperature rise (heating) caused by internal dissipation of mechanical energy. Therefore, the thermomechanical coupling can cause the degradation in material properties, high strains and stresses and results in fracture of various engineering structures. In studying the nonstationary, in particular resonant, processes, it is important to take into account inelastic deformation and heat emission [1]. Indeed, the thermoviscoplastic behavior of the material and the coupling of the mechanical and thermal fields may affect substantially the characteristics of vibration dampers for engineering structures. An approach to the solution of coupled problems of vibrations and dissipative heating of viscoplastic bodies is developed. Two problem statements are elaborated: “exact” and “ approximate” ones. The former statement consists of universal balance equations of thermodynamics and constitutive equations derived from the general thermodynamic theory of viscoplastic bodies with internal state variables. For this aim the theory version that corresponds to the Bodner-Partom generalized flow theory is used. The approximate or monoharmonic problem statement for the case of steady-state vibrations under the harmonic loading is obtained by the application of modified harmonic linearizing technique to the initial system of equations. Thereby the subsystem of mechanical equations is formulated in the complex-value form and material properties are described by means of amplitudedependent complex-value moduli. Numerical technique for the solution of the problems of vibration and dissipative heating of spatial and thin-wall elements of structures are developed. They are based on the iterative linearization procedures in association with FEM. The capabilities of the monoharmonic approach are studied by the example of the resonant and quasistatic vibrations and dissipative heating of a viscoplastic disk excited by a harmonic load [2].Additional attention is paid to the case of inhomogeneous (layered) solids. Physical and mechanical behavior of such systems is very complex because of the heterogeneity of the stress-strain state. Modeling such a behavior is additionally complicated by the necessity of allowing for the coupling of the mechanical and thermal fields when cyclic loading is intensive and for dynamic effects, in particular, when intensive vibrations occur in the quasistatic or resonant domain. Comparing the solutions obtained in the frame of the exact and approximate statements demonstrates high accuracy of the monoharmonic model for the problems investigated (axisymmetric vibrations and heating of layered or homogeneous viscoplastic disk and rectangle).

References [1] I. Senchenkov, Y Zhuk and V. Karnaukhov, Modeling the thermomechanical behavior of physically nonlinear materials under monoharmonic loading. International Applied Mechanics, 40, 943-969, 2004.

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A continuous Galerkin finite element method for thermoelasticity without energy dissipation Swantje Bargmann∗ , Paul Steinmann∗ ∗ Chair

of Applied Mechanics, University of Kaiserslautern P.O. Box 3049, 67653 Kaiserslautern, Germany {bargmann,ps}@rhrk.uni-kl.de ABSTRACT

The classical thermal theory based on Fourier’s law leads to a diffusive regime. Contrary to that Green and Naghdi [3] developed a theory of thermoelasticity without energy dissipation whose temperature evolution equation is hyperbolic. Among others, the introduction of a new internal variable, i.e. the thermal displacement α with α˙ = T , leads to the theory without energy dissipation, or also called theory of type II, which does not involve energy dissipation. It incorporates thermal wave propagation in a very consistent way and is capable of modeling the second sound phenomenon. The governing equations of the dynamic, linear theory of isotropic and homogeneous thermoelasticity without energy dissipation are the temperature equation a ρcbT˙ = ρr + ρ κ∆α − ρbT0 3wKI : ε˙ (1) b and the balance of linear momentum ˙ = divσ + b, [ρv] (2) where the entropy flux p = − ρκ b ∇α is determined by the same potential which determines the mechanical stresses σ. This contribution concentrates on numerical aspects of the Green-Naghdi theory of type II. In order to perpetuate the consistency of their theory to the numerical setting we resort to a Galerkin finite element method in space and in time. As the theory itself does not admit energy dissipation, conserving time integration schemes that inherit the underlying conservation principles are of great interest. Customary implicit time-stepping schemes fail to conserve major invariants, for example the total energy. The coupled dynamic system of equations is discretized in time within the framework of finite element methods using a continuous Galerkin (cG) method. In general, the cG-method has proven to qualify well for hyperbolic problems. This fact also holds true for hyperbolic heat conduction and linear thermoelastostatics [1, 2]. The cG(k)- method approximates trial functions piecewise and continuously with polynomials of degree k and test functions piecewise and discontinuously across the element boundaries with polynomials of degree k − 1. The coupled system is solved monolithically. A numerical example is investigated in order to evaluate the performance of the proposed method.

References [1] S. Bargmann and P. Steinmann, Finite element approaches to non-classical heat conduction in solids. Comp. Model. Eng. Sci. 9(2), 133–150, 2005. [2] S. Bargmann and P. Steinmann Theoretical and computational aspects of non-classical thermoelasticity. submitted [3] A.E. Green and P.M. Naghdi A re-examination of the basic postulates of thermomechanics. Proc. R. Soc. Lond. 432, 171–194, 1991.

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The Model Coupling Liquid Bridge Between Ellipsoidal Grains Jolanta Báaszczuk*, Zbigniew DomaĔski† Institute of Mathematics and Computer Science, Czestochowa University of Technology Dabrowskiego 73, 42-200 Czestochowa, Poland * [email protected] † [email protected]

ABSTRACT Granular materials are the subject of scientific studies due to their unusual physical properties which differ significantly from solid and liquid states of matter. In this paper we are interested in the role played by humidity on static and dynamic properties of systems consisting of non-spherical grains. The addition of a liquid to the material adds an attractive force to the system and then, increased its stability. Quantitative description of wetting thermodynamics is sensitive not only to the contact angle between solid and the liquid but also to the to the shape of grains and thus we analyze ellipsoidal grains and we assume that liquid spreads uniformly over the whole grain’s surface. We consider a model grain’s surface consisting of asperities of equal size uniformly distributed over the grain’s surface. We also suppose that each asperity may be either totally filled with liquid or stay empty. Thus, in our approach we consider two regimes of the inter-grain adhesive force versus volume of the wetting layer. For very small amount of liquid, the capillary force comes from the fluid accumulated around a small number of asperities at which two neighbouring grains are in contact. If the fluid wets the surface of the grains then all asperities are filled and inter-grain adhesive energy is determined mainly by the macroscopic curvature of the grain, and the surface roughness does not play a crucial role. In this case, the distribution of values of inter-grain energies is determined only by macroscopic quantities, i.e. the geometry of grains and material characteristics. Using toroidal approximation for the shape of liquid bridges and some simple probabilistic arguments we analyze the influence of amount of liquid on mutual grain – grain orientation. We found two energetically favorable orientations: one with mutually parallel and second with perpendicular axes of contacted grains.

References [1] J. Báaszczuk, Z. DomaĔski, Liquid – induced glassy behaviour of dense non-spherical grain ensembles, 16th International Conference on Computer Methods in Mechanics, CMM 2005, Publishing House of Czestochowa University of Technology, 347-348, 2005. [2] J. Báaszczuk, Z. DomaĔski, Toroidal approximation for capillary bridges between ellipsoidal grains, Scientific Research of the Institute of Mathematics and Computer Science 1(4) 2005, Publishing House of Czestochowa University of Technology, 13-17, 2005. [3] J. Báaszczuk, Dynamics and statics of packing granular materials. PhD thesis, Czestochowa University of Technology, 2005.

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Reliability of Wavelet Packet System Identification Maik Brehm ∗, Christian Bucher† ∗Institute of Structural Mechanics, Bauhaus-University Weimar

Marienstraße 15, 99423 Weimar, Germany [email protected] † Institute of Structural Mechanics, Bauhaus-University Weimar

Marienstraße 15, 99423 Weimar, Germany [email protected] ABSTRACT In recent years various methods have been developed to identify structural system parameters. One approach for dynamical system identification is the application of wavelets. [1] uses Daubechies wavelets within a wavelet transformation algorithm and [2] investigates the biorthogonal wavelet transformation. The basic challenge of such wavelet-based methods is the determination of an optimal set of approximation and detail coefficients to set up the linear equation system for the identification of the unknown parameters. A major improvement compared to the simple wavelet transformation is achieved by using wavelet packets. [3] presents an automatic best basis search to determine a suitable set of coefficients. However, this algorithm is not unique and identifies only one of the best sets of coefficients. Due to the frequency decomposition of the wavelet analysis all these methods can handle a high noise level. Of course, the quality of the solution still depends on the noise level. The current practice of verifying the results are based on the known parameters or those obtained by other methods. Both ways are suboptimal to identify parameters of a genuine structure. This study investigates the possibilities to increase the reliability of the identification of the system parameters by means of wavelet transformation and the wavelet packet algorithms without the knowledge of the correct solution. Condition numbers and optimization methods are used to detect the best set of coefficients to set up an optimal equation system. Furthermore, the presented methods are discussed regarding their challenges and feasibilities for practical system identification. The developed algorithms have been implemented in the SLang Software package, which is available at the Bauhaus-University Weimar for research activities.

References [1] V. Zabel, Applications of Wavelet Analysis in System Identification. PhD Thesis, BauhausUniversity Weimar, 2003. [2] M. Brehm et al. , Applications of Biorthogonal Wavelets in System Identification. P. Neittaanm¨aki, T. Rossi, K. Majava, and O. Pironneau (eds.) Proceedings of the 4th European Congress on Computational Methods in Applied Sciences and Engineering, Jyv¨askyl¨a, 24-28 July, 2004. [3] M. Brehm et al. , Applications of Wavelet Packets in System Identification . 76. Annual Conference of the International Association of Applied Mathematics and Mechanics (GAMM), University of Luxembourg, March 28 - April 1, 2005.

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An attempt to simulate more precisely the behavior of a solid body using new energy conservation equation for fully coupled thermal structural analysis L. Écsi*, P. ÉlesztĘs† †*

Dept. of Strength of Mater, Faculty of Mech. Eng, Slovak University of Technology in Bratislava Námestie slobody 17, 812 31 Bratislava, Slovak Republic * [email protected], † [email protected]

ABSTRACT In this paper a numerical study of solid bar behavior under various mechanical loads is presented using fully coupled thermal structural analysis with large strain / large deformation formulation. The analysis is based on a new energy conservation equation [2], which the authors believe represents the most complete formulation of the first principle of thermodynamics in contemporary computational mechanics. In the numerical analysis was used the finite element method (FEM), utilizing the updated Lagrange method and the extended NoIHKH material model [1],[3], modified for large strain / large deformation cyclic plasticity of metals. In the stress update calculation [4] was used the Jaumann objective rate in the form of the Green-Naghdi objective rate, formulated with the aid of the rotated Cauchy stress tensor. The rotation tensor was expressed with the Rodriguez formula. The case was implemented into a finite element code using “proper” linearization, which means that no simplifications were used in a gradient or an element volume expression in the current configuration. The presented results represent some of the first outcomes of the fully coupled thermal structural analysis utilizing the new energy conservation equation with large strain/ large deformation formulation, which the authors consider to be positive. The new energy conservation equation can still be improved by introducing a heat source in it to take into account the amount of energy dissipated into heat during plastic deformation, but to propose a mathematical formula for the heat source is rather an experimental problem than a mathematical one. If the new energy conservation equation proves to be experimentally correct, in the future more complex problems will be able to be solved, mainly in the area of fast/ultra fast thermoelasticity or thermoplasticity.

References [1] L. Écsi, Extended NoIHKH model usage for cyclic plasticity of metals, In proceedings of the 7th. International Conference on Applied Mechanics (CD), Hrotovice, Czech Republic, 29.March1.April, 2005. [2] L. Écsi, Numerical behavior of a solid body under various mechanical loads using finite element method with new energy balance equation for fully coupled thermal structural analysis, In proceedings of the 6th. International Congress on Thermal Stresses, Vienna, Austria, 26-29. May, 2005, Vol. II, pp. 543-546. [3] J. Lemaitre, Handbook of Materials Behavior Models 1, Academic Press, NY, 2000 [4] E.A. Souza, D. Peric, D.R.J. Owen, Computational Plasticity, Small and Large Strain Finite Element Analysis of Elastic and Inelastic Solids, Swansea, to be published

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Mathematical aspects of the initial-boundary value problems in nonlinear thermoelasticity of simple and non-simple materials J.A. Gawinecki Institute of Mathematics and Cryptology, Faculty of Cybernetics Military University of Technology Str. Kaliskiego 2, 00-908 Warsaw, Poland [email protected] ABSTRACT In this lecture we report on diffrent coupled thermoelastic systems. They are models for the description of elastic heat conductive media. We will consider a coupled second-order hyperbolic system describing the thermoelasticity of simple materials and a coupled parabolichyperbolic systems of thermoelasticity of non-simple materials. It is known that the classical thermoelasticity theory (hyperbolic-parabolic) leads to a parabolic diffrential equation for the temperature distribution in rigid heat conductors. This implies that thermal perturbations are felt instantaneously in every part of the body. Although, at first sight, this outcome seems to contradict the physical intuition, it can be justified by resorting to the fact that molecular motion, which places a crucial part in transport phenomena, is very rapid except at extremely low temperatures. Hencen finite velocity of propagation for thermal perturbations is usually non-observable unless experiments are performed in some neighborhood of absolute zero as in the case of liquid helium. In fact, thermal waves, commonly known as second sound, are detected in some metals cooled approximately down to 20◦ K. In our lecture we consider the theory of thermoelasticity by considering the temperature-ratedependence and assigning an appropriate constitutive function to the entropy flux. Such a theory leads to a hyperbolic differential equation for thermal perturbations different from the equation describing propagation of thermal perturbation in classical thermoelasticity which is parabolic one. We consider also the nonlinear hyperbolic-parabolic system of coupled partial differential equation of fourth order describing the thermoelasticity of non-simple materials. We are interested in both the lineralized and the non-linear systems looking for the description of the asymptotic behavior of smooth solutions, for smoothing effects of the systems and specifically for the non-linear systems for the global existence in time of solutions. For hyperbolic systems it is well known that in many cases locally existing smooth solutions tend to develop singularities in finite time. The basic problem for the system in question here is whatever and in which way the added damping by heat conduction or viscosity will assure the global existence of solutions. From this point of view we investigate the global existence in time for large data in nonlinear thermoviscoelasticity.

References 1. 2.

J. A. Gawinecki, Global solutions to initial value problems in nonlinear hyperbolic thermoelasticity. Dissertationes Math. 344, 1–51, Warsaw Poland. J. A. Gawinecki. Global existence of solution for non-small data to non-linear spherically symmetric thermoviscoelasticity, Math. Met. Appl. Sci. 26, 970–936, 2003.

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Effect of Parameter Uncertainties on a Vibro-Acoustic Design M. Guerich, S. Ben Chaabane Pôle Universitaire Léonard de Vinci ESILV – DER MS 92916 Paris la Défense Cedex {Francemohamed.guerich, samir.benchaabane}@devinci.fr

ABSTRACT The prediction of vibro-acoustic behaviour of structures is of the greatest interest for the design of less noisy machines. In the low frequency (LF) range, using finite element method (FEM) and/or boundary element method (BEM) in general insure such predictions. In the case where the FEM is used, in general this corresponds to an internal vibro-acoustic coupling problem, the vibro-acoustic model is constituted of structural and acoustical models. The structural model is performed using finite element of plates and/or shells and the acoustical model is constituted of volumic finite elements. When an internal vibro-acoustic coupling problem is studied, the modal method is in general used. It consists of the computation of structural and acoustical modes, which will be used as a basis of the displacement for the structure and of the pressure for the fluid. These modes depend on mechanical properties of the structure and the cavity and on geometric dimensions. Moreover, to predict the vibro-acoustic behaviour of the coupled system, appropriate caseload and cinematic boundary conditions must be considered. Thus, the vibro-acoustic model involves in general, a big number of parameters. These parameters can be given with some uncertainties.The purpose of this paper is to measure the effect of these uncertainties on the vibro-acoustic behaviour of the whole system. With such study the reliability of the vibro-acoustic system can be evaluated. To illustrate these effects, we have considered a simple case of vibro-acoustic response of a coupled plate-cavity to a harmonic mechanical force in a LF range.Some of the design parameters are considered as uncertain. These uncertainties are represented by Gaussian distributions of the parameters.A large sample of design points is defined and the responses are evaluated in these design points. The distribution of the response is then extracted and its coefficient of variation (COV) is compared to those of the parameters. Failure criteria (pressure level on the cavity and/or cinematic energy of the structure) can be defined to estimate the failure probability of the design.

References [1] Morand J.P., Ohayon R., Fluid structure interaction, Masson, Paris, 1992. [2] Lesueur C., Rayonnement acoustique des structures (in French), Eyrolles, Paris, 1988. [3] Guerich M., Hamdi M.A, A Numerical Method for Vibro-Acoustic Problems with Incompatible Finite Element Meshes Using B-Spline Functions, JASA, 105 (3), 1999, pp 1682-1694. [4] Melchers R. E., Structural Reliability Analysis And Prediction, Jhon Wiley & Sons Chichester, 1999 [5] ", S. Bouabdallah, S. Ben Chaabane, S. Missoum, Analyse fiabiliste du procédé de mise en forme des tôles minces, 7ème COLLOQUE NATIONAL EN CALCUL DES STRUCTURES, Giens 2005.

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A numerical method for solid-liquid interaction Goodarz Khodabakhshi, Vahid Nassehi, Leila Shojai, Richard J.Wakeman Chemical Engineering Department Loughborough University, Loughborough Leicestershire, LE11 3TU, United Kingdom {g.khodabakhshi, v.nassehi, l.shojai, r.j.wakeman}@lboro.ac.uk

ABSTRACT This paper deals with the mathematical modelling of coupled fluid flow and solid deformation problems. A novel mathematical technique for linking of the two sets of governing equations in a single model has been proposed. Results obtained by this technique using a range of power-law index for fluid flow simulation and elasticity modulus for the solid displacement are presented and discussed . Changing the rheological behaviour of the fluid has a significant effect on the deformation of the solid. These results are found to be self consistent and as expected from a theoretical point of view.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Computational Simulation of Irreversible Deforming and Fracture of Damageable Solids and Structures Alexey B. Kiselev, Olga V. Nekhaeva, Anton V. Privalsky Mechanics and Mathematics Faculty of Moscow M.V. Lomonosov State University Leninskie Gory, MSU, Main Bldg, Moscow 119992, Russia [email protected]

ABSTRACT Thermomechanical processes, which proceed in deformable solids under intensive dynamic loading, consist of mechanical, thermal and structural ones, which correlate themselves. The structural processes involve the formation, motion and interaction of defects in metallic crystals, phase transitions, the breaking of bonds between molecules in polymers, the accumulation of microstructural damages (pores, cracks), etc. Irreversible deformations, zones of adiabatic shear and microfractures are caused by these processes. Dynamic fracture is a complicated multistage process including an appearance, evolution and confluence of microdefects and a formation of embryonic microcracks, pores, their grow up to the break-up of a bodies with division into separate parts. The present paper include new results in the next scopes: 1) development the thermodynamically correct mathematical models of damageable thermoelastoviscoplastic medium (microfracture); 2) development the methods for determination of “nonstandart” constants of medium models, connected with microfracture of material; 3) numerical simulation of destruction (fragmentation) of constructions (macrofracture); 4) numerical investigation of some problems for damageable solids and structures (dynamical deforming and fracture of thick-walled cylindrical and spherical shells under explosion; dynamical deforming and fracture of thick-walled two-layer shell, filled with liquid, under impact and high velocity penetration; the problems of dynamic deforming and destruction of an oil-holding layer in gydraulic fracturing). Some of previously obtained results in consideration domains are published in the papers [1-5]. Russian Foundation for Basic Research (grants No. 06-01-00185 and No. 05-08-01435) and ISTC (grant No. 2992) are acknowledged for financial support.

References [1] A.B. Kiselev, Mathematical modeling of dynamical deformation and combined microfracture of a thermoelastoviscoplastic medium, Moscow Univ. Mech. Bull., 53, No. 6, 32-40, 1998. [2] A.B. Kiselev, A.A. Lukyanov and M. Thiercelin, Numerical simulation of dynamic propagation of curvilinear cracks of hydraulic fracturing, Moscow Univ. Mech. Bull., 59, No. 1, 36-41, 2004. [3] A.B. Kiselev and O.V. Nekhaeva, Numerial modeling of dynamical deformation and fracture of a thick-walled spherical shell, Moscow Univ. Mech. Bull., 59, No. 5, 53-58, 2004. [4] A.B. Kiselev and O.V. Nekhaeva, Numerial simulation of dynamical deformation and fracture of a thick-walled cylindrical shell, Moscow Univ. Mech. Bull., 60, No. 2, 33-37, 2005. [5] A.B. Kiselev and O.V. Nekhaeva, Mathematical modelling of dynamic processes of irreversible deforming, micro- and macrofracture of solids and structures, 11th Int. Conference on Fracture (Turin, Italy, March 20-25, 2005), Abstract Book, Turin, CCI, 228 (Proc. on CD-ROM, 6 p), 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Solution of the coupled light-mechanical problems Miroslav Kropáþ*, Justín Murín† * Visteon-Autopal, s.r.o. Nový Jiþín Lužická 14, 74101 Nový Jiþín, Czech Republik [email protected] †

Department of Mechanics, Faculty of Electrical Engineering and Information Technology, Slovak University of Technology Ilkoviþova 3, 812 19 Bratislava, Slovak Republik [email protected]

ABSTRACT

Dynamic optical systems (e.g. automotive headlights) provide the changes of their optical parameters, on the one hand by the motion of the whole lighting unite or on the other hand by the rigid motion of the individual elements of the optical system [1]. Our principle of reflector area controlled deformation is a new point of view in a required light beam modification of headlights during the car drive. It is necessary to solve coupled light-mechanical task to examine the influence of reflector area deformation on the quality of beam pattern by numerical methods. Light-mechanical task of weak coupled field is solved by the sequential method. In the first process step the deformation analyses of the reflector is made by the FEM method [2], following by the light analyses. In the second step the beam pattern of the optical system is under evaluation. Searching algorithm has been developed for the lighting field solving on the base of the backward ray tracing algorithm, which analyzes optical system and seeks out optical system elements orienting light rays into the examining parts of the illuminated half-space. Light performance of optical system is qualified as the beam pattern on the plane in the specific distance out of optical midpoint that represents detector. For the coupling between active optical system elements and detector nodes, the finite elements of light beam have been developed as well as the alternative direct computation. For the documentation of above-mentioned process we solved a bilinear optical system consist of a light source, a reflector and a detector. The response of the optical system has been sought (the curve of the illuminance spreading on the detector) on the controlled deformation of the reflector curve. The numerical simulation outcomes has been verified with the experiment measurements of the optical parameters of recall optical system.

References [1] M. Kropáþ, J. Murín, Design and analysis of the automotive headlights. ýasopis pre elektrotechniku a energetiku. 11, 89-92, 2005. [2] ANSYS 8.1 - Theory manual, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Simulation of the ferroelectric hysteresis using a hybrid finite element formulation I. Kurzh¨ofer∗ , J. Schr¨oder∗ , H. Romanowski∗ ∗ Institute

of Mechanics, Department of Civil Engineering, University of Duisburg-Essen Universit¨atsstr. 15, 45117 Essen, Germany [email protected] ABSTRACT

Ferroelectric materials can be found in a wide range of applications in smart materials, e.g. vibration reducing sensors or fuel injection systems. A special characteristic feature of these materials is the appearance of a spontaneous polarization in a certain temperature range. This polarization can be reversed by an applied electric field of sufficient magnitude. The resulting nonlinear material behavior is expressed by characteristic dielectric and butterfly hysteresis loops. These effects are correlated to the structure of the crystal and especially to the axis of the spontaneous polarization. In the present work we present an electric hybrid element formulation where stresses and electric field are derived from constitutive relations. Therefore, we consider the electric displacement as additional degree of freedom, as presented in [1]. Furthermore, the finite element problem is condensed in a manner that the formulation is suitable for conventional boundary problems where the electric field is computed by the negative derivative of the basic field variable φ, used in the finite element approximation. The anisotropic material behavior is modeled within a coordinate-invariant formulation for an assumed transversely isotropic material, see [2] and [3]. The anisotropic response of the material is governed by isotropic tensor functions which depend on a finite set of invariants. The nonlinearities of ferroelectric materials are considered by the regard of internal variables which are governed by evolution equations. The key assumption of this procedure is the split of the strains and the electric displacement into reversible and remanent part. A switching surface is defined and a general mapping algorithm is applied to evaluate the remanent values at the actual timestep. The resulting hysteresis loops for a ferroelectric ceramic which are governed by one part of the Helmholtz free energy are discussed by considering a simple numerical example.

References [1] K. Ghandi & N.W. Hagood, A hybrid finite element model for phase transitions in nonlinear electro-mechanically coupled material. Smart Structures and Materials, Proceedings of SPIE, 3039, 97–112, 1997. [2] J. Schr¨oder & D. Gross, Invariant formulation of the electromechanical enthalpy function of transversely isotropic piezoelectric materials. Archive of Applied Mechanics, 73, 533–552, 2004. [3] H. Romanowski & J. Schr¨oder, Coordinate invariant modeling of the ferroelectric hysteresis within a thermodynamically consistent framework. A mesoscopic approach. Trends in Applications of Mathematics to Mechanics, Wang Y. & Hutter K. (eds.), Shaker Verlag, 419–428, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A Two-Phase Numerical Modelling of the Liquid Solid Transition in Polymer Processing R. Lanrivain†, L. Silva, T. Coupez CEMEF, Ecole des Mines de Paris 1 rue Claude Daunesse, 06904 Sophia Antipolis, FRANCE †[email protected] ABSTRACT In this paper, we present a two-phase model. It allows the numerical computation, during processing, of a polymer’s mechanical behavior at the liquid state, during the transition, and at the solid state. This work’s context is an industrial project aiming the accurate modelling of the injection molding process [1], [2]. It focuses on the 3D numerical simulation of the different stages encountered in this process : filling, packing, cooling and ejection. During processing, anisotropy of the stress state build-up affects its mechanical, optical or dimensional properties, and induces warpage once the part is ejected. In order to predict residual deformations in the mould, and after ejection, the whole process is simulated with a two-phase thermo-mechanical model, which includes dynamics of phase change. Thus, interactions between polymer in flow and solidified one are implicitly taken into account in the coupled mechanical formulation. Based on the two-phase flow theory [3], a diffuse interface model is derived. A phase field quantifies the presence of each phase at each point of the computational domain. This parameter is obtained by solving an evolution equation of the hyperbolic type, that can be function of temperature or strain. Different behavior laws can be associated to the liquid (like viscous or viscoplastic) and to the solid (like elastic or hyperelastic). So each phase is described by its specifical kinematic variables (velocity for the fluid, displacement for the solidified part). Furthermore, the system of conservation equations (mass and momentum of each phase) is closed by momentum and mass coupling relations (for example, friction type). The kinematics variables for both liquid and solid phase (velocity, liquid pressure, displacement, and solid pressure) are calculated from a single strongly coupled system (monolythic approach) by using the mixed finite elements method within an eulerian framework. Due to the large number of unknowns, parallel computation is obviously required. Validation tests of the model’s implementation are shown, using literature benchmark examples. Results obtained in 3D complex industrial parts underline the robustness and the efficiency of our model.

References [1] T. Coupez,C. Gruau, C. Pequet and J. Bruchon, Metric Map and Anisotropic Mesh Adaptation for Static and Moving Surfaces. The Sixth WCCM, Beijing, China, 2004. [2] L. Silva, R. Valette, and T. Coupez. Viscoelastic compressible modelling of 3d filling and postfilling of complex industrial parts. In V. Brucato, editor, The 6th Int. ESAFORM Conf. Mat. Form., 19-22. University of Salerno, 2003. [3] D.A. Drew, Mathematical modelling of two phase flow. Annu. Rev. Fluid Mech. 15, 261–291, 1983.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A two-phase model for granular flows applied to avalanches Caroline Leppert, Dieter Dinkler Institut f¨ur Statik Technische Univerist¨at Braunschweig, Germany [email protected] ABSTRACT Understanding of granular materials under rapid motion is of importance to many phenomena in nature and industrial applications. During landslides and avalanches as well as in mixing and storing processes as in silos granular materials undergo phase transitions from solid to fluid state of matter. At rest, the material is characterised by dilatant elasto-plastic behaviour that is rate independent. For low densities and high shear-rates fluid properties dominate. A continuum mechanical approach for dense granular flow including both aspects is presented. Therefore, frictional stresses, representing the rateindependent aspects, extend the viscous stresses using the generalised visco-plastic constitutive model with Coulomb friction as proposed by Chen and Ling [1]. The differentiation between solid and fluid state of matter results from a comparison between current shear stress and plastic yield criterion. The model is able to represent many experimental results for moderately fast granular flows introducing an artificial viscosity that includes internal friction and depends on the current strain rate. To simulate the flow of granular material that travels large distances the material is described by the Navier-Stokes-equations. This requires the formulation of all balance equations for momentum and mass and the constitutive equations using velocity variables. The space-time finite element method is applied to discretise the weak form of balance and constitutive equations. This allows the monolithic coupling between the flowing material and any possible surrounding structure [3]. The method is extended by the level set technique [4] to model the motion of the free surface between granular material and air in order to represent landslides and the complete discharge of silos

References [1] C. Chen and C. Ling, Granular-flow rheology: role of shear-rate number in transition regime, J. Engng. Mech., Vol. 122, 469–480, 1996 [2] S.B. Savage, Analyses of slow high-concentration flows of granular materials. J. Fluid Mech., Vol. 377, 1–26, 1998 [3] B. Hubner ¨ , E. Walhorn and D. Dinkler, A monolithic approach for fluid-structure interaction with space-time finite elements Comp. Meth. Appl. Mech. Eng., Vol. 193/23, 2069–2086, 2004 [4] S. Osher and R. Fedkiw, Level set methods: An overview and some recent results. J. Comp. Phys., Vol. 79, 463–502, 2001

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Parallel 3D Finite Element Analysis of Coupled Problems Lee Margetts *, Ian M. Smith† and Joanna M. Leng* * Manchester Computing, University of Manchester Kilburn Building, Oxford Road, Manchester M13 9PL [email protected] [email protected] †

School of Mechanical, Aeronautical and Civil Engineering, University of Manchester, P.O. Box 88, Manchester M60 1QD [email protected]

ABSTRACT Steady advances in computer power have enabled researchers to consider tackling increasingly complex problems. In the academic community, current focus is on multiscale modelling and multiphysics. The aim is for simulation to be more realistically representative of real world processes. This paper considers the simulation of coupled problems involving more than one physical process, multiphysics. In particular, the authors present some ideas and experiences regarding the use of the finite element method and parallel computers to solve 3D coupled problems. In the literature, two main approaches have been used to solve coupled problems. These are sometimes referred to as (i) fully coupled modelling and (ii) un-coupled multi-physics. Both methods have their advantages and disadvantages. In the paper, the authors discuss some of the issues that should be considered when selecting a particular strategy, to ensure computational efficiency. Particular attention is given to an example from the field of magnetohydrodynamics: three dimensional steady state flow in a perfectly insulated rectangular duct. The magnetohydrodynamics example involves solving a system in which both magnetic and hydrodynamic forces influence the behaviour of the fluid. Visualisation of the problem, using streamlines to represent fluid flow (Figure 1) shows that a three dimensional representation is essential to capture the full complexity of the flow. A fully coupled solution strategy is presented in which the full system is represented by a single “stiffness” matrix and solved by a single computer program. A parallel implementation of an element-by-element variant of BiCGStab(l) is used to solve the equations, demonstrating the efficient use of up to 128 processors.

slow

fast

Figure 1 Velocity Streamlines from Three Different Viewpoints Along a Rectangular Duct

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Numerical Simulation of Rubber Curing Process with Application to Bladders Manufacture Paulo Porta † ∗, Carlos Vega† † DARMEX S.A.C.I.F.I. Luis Mar´ıa Drago 1555 – B1852LGS Burzaco – Argentina [email protected] [email protected] ∗ Corresponding author

ABSTRACT A large number of polymer products are formed into their final shape by polymerization in situ. In particular, mold curing process is the final step in many rubber products manufacturing and determines both the quality of the resulting product as well as productions costs. During this process, important changes in the mechanical properties –e.g. viscosity and modulus– take place, changes which are generally hard to be experimentally characterised. In view of this, a mathematical model is proposed for rubber vulcanisation molding, its strategical value being two fold: from production standpoint, its ability to predict optimal production parameters –the optimal curing time being the most important– and from quality assessment perspective, its capacity of predicting molded part properties. Following the literature –see, e.g. [1]– the mathematical model is built from general mass–energy conservation principles. A series of plausible hypothesis are made in order to simplify the model, which results in a combination of i) the unsteady Fourier’s heat conduction equation –(1)– with a distributed internal heat source, resulting from the reaction –(2), ii) the reaction rate equation –(3)– and iii) closure (γ) constitutive equations, for the kinetic constants of the process, namely KC and tinc : ∂Θ ∂t

= div (k · gradΘ) +

q  = Hr dC dt

=

q ρcp

in Ω for t > 0

dC dt

(γ) KC (1

(1) (2)

− C)γ

for t > tinc

(3)

An algorithm is proposed to solve this coupled system. A coupled ODE–implicit in time finite element approximation is proposed and implemented under ALBERTA ([2]). After calibration of the computational model, the complete temperature and cross-links concentration is obtained for the typical bladder geometry. Performance results as well as a short discussion on error estimator behaviour are also presented.

References [1] In-Su Han and Chang-Bock Chung and Hyoeng-Gwan Jeong and Sung-Ju Kang and Seung-Jai Kim, Optimal cure steps for product quality in a tire curing process. Journal of Applied Polymer Science, 74, 2063–2071, 1999. [2] Schmidt, A. and Siebert, K., Design of adaptive finite element software, Springer, Berlin, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Coupled Finite Element Analysis of Composite Laser Rods Thermal Characteristics under Longitudinal Diode Pumping E. Stupak*, R. Kaþianauskas*, A. S. Dementjev†, A. Jovaiša† *Department of Strength of Materials, Vilnius Gediminas Technical University, Saulơtekio av. 11, LT10223, Vilnius-40, Lithuania [email protected], [email protected] †

Nonlinear Optics and Spectroscopy Laboratory, Institute of Physics, Savanoriǐ av. 231, LT02300, Vilnius, Lithuania [email protected], [email protected]

ABSTRACT Diode laser pumped solid-state lasers are covering a wide range of applications. The longitudinal pumping geometry of laser rod is particularly favorable as it provides a high degree of spatial overlap between pump and lasing modes. However, high pump-power densities are required to achieve sufficient inversion in the laser material. This produces high thermal loading in laser crystals, which, in turn, leads to undesirable thermal effects, such as temperature-dependent index change, temperature-dependent stresses, and end-surface deformation. All those effects are qualified as thermal lensing in laser material. Yttrium aluminum garnet (YAG) crystals doped with Nd, Er, Yb and other ions are currently the most popular laser crystals for diode laser pumped solid-state lasers, especially in end-pumped configurations, that is why laser rods with YAG host material are chosen as the objects of modelling. A multiphysics approach and finite-element method are used to the numerical simulation of thermal lensing in such rods. Thermo-mechanical behavior is considered by applying the ANSYS software, while particular pre-processor for generation of the heat source as well as postprocessor for evaluation of the optical path difference (OPD) is developed. The difference between standard crystal blocks and composite structures with undoped end caps is investigated. The results show that OPD for one pass through the crystal in all cases is practically the same. Thus, the composite rod geometry does not reduce significantly thermal lensing, however temperature gradients are considerable smaller. The obtained results also show that the use of the composite AE practically does not decrease the thermally induced aberrations of spherical type. Therefore, the quality of the probe or laser beam will change mainly in the same way in both the conventional and the composite AE pumped with Gaussian beams.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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On the Modeling of Nonplanar Shear Walls in Shear Wall - Frame Building Structures Tolga Akis1, Turgut Tokdemir2, Cetin Yilmaz3 1

Department of Civil Engineering Atılım University, Ankara, øncek 06836, Turkey [email protected] 2 Department of Engineering Sciences Middle East Technical University, Ankara 06531, Turkey [email protected] 3 Department of Civil Engineering Middle East Technical University, Ankara 06531, Turkey [email protected]

ABSTRACT The objective of this study is to model the non-planar shear walls of asymmetric shear wall-frame building structures in elastic range. The modeling work is based on open and closed sections shear wall assemblies for which two different three-dimensional models are developed and verified in comparison to common shear wall modeling techniques. Two-dimensional modeling of symmetric building structures having planar shear walls may be a practical method. However, especially for an asymmetric shear wall-frame building system that contains non-planar shear wall assemblies, the structural system should be modeled in three dimensions. In addition, the three dimensional behavior of the shear wall assemblies should also be taken into consideration. The proposed models are based on conventional wide column analogy, in which a planar shear wall is replaced by an idealized frame structure consisting of a column and rigid beams located at floor levels. For open section shear walls, the connections of the rigid beams are released against torsion in the model. For modeling closed section shear walls, in addition to this the torsional stiffness of the wide columns are adjusted by using a series of equations. In the modeling studies, the rigid diaphragm floor assumption is also taken into consideration. As an example, plan view of an open section U shaped shear wall assembly and the corresponding model is given in Fig. 1. The main goals of the models that are given in this study are to reduce the required time and capacity for the threedimensional analysis of shear wall-frame building systems. In the verification studies, two single shear wall assemblies (an open and a closed section) and a six storey shear wall-frame building system are considered. Static lateral load analysis and dynamic analysis are performed on these structures where the proposed models are used. The results of these analyses are compared with the results obtained by using common shear wall modeling techniques and it is observed that both results agree well with each other.

Figure 1. U-Shaped Shear Wall Section and Proposed Model

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Directional Drillstring Dynamics Fredy Coral Alamo∗ , Hans Ingo Weber∗ and Harry Saavedra Espinoza† ∗ Pontif´ıcia Universidade Cat´olica do Rio de Janeiro Rua Marquˆes de S˜ao Vicente 225, 22453-900, Rio de Janeiro - RJ - Brazil {fjcoral,hans}@mec.puc-rio.br † Universidad Nacional de Ingenieria

Av. Tupac Amaru S/N, Lima - Peru [email protected]

ABSTRACT A rotating rod under load may execute vibration in different ways: transversal (bending), longitudinal (axial), torsional, or a combination of any of those. In this article, the dynamics of a rotating directional drilling system, constrained to rotate in a borehole, is investigated using the finite element formulation. To study the behavior of the system, a rotating 3D Cosserat rod element is used, this is a newly non linear element developed in this work. The equations of coupled bending, axial and torsional motion of the rotating elastic rod element is derived using the Cosserat rod theory. In general, for slender structures, the shear deformation can be neglected, consequently, to model the drillstring the Bernoulli hypothesis is considered and the shear deformations are neglected. The finite rod element developed has 12 degrees of freedom and takes into account all the geometric nonlinearities. Explicit expressions for the element mass, gyroscopic, stiffness, and non linear terms are derived using Hamiltons principle. Moreover, the finite element discretization is employed and numerical solutions are obtained for the nonlinear drillstring dynamics. Overall, the Cosserat model provides an accurate way of modelling long slender rods.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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MVM Energy Method for Buckling Analysis of Tapered Plates M. M. Alinia1, A. R. Rahai2, and S. Kazemi1 1,2 Amirkabir University of Technology 424 Hafez Ave. Tehran 15875-4413, Iran {m.alinia, a_rahai}@aut.ac.ir

ABSTRACT Tapered plates are being increasingly used in modern engineering structures. The increasing use is due to the distributed flexural stiffness that helps reduce the weight of structural elements and improve the utilization of the material. The flexural stiffness, vibrational, and buckling capacities of these plates may be significantly increased by appropriate tapering. On the other hand, provision of openings in perforated plates which provide access for inspection, services, and maintenance, can greatly enhance the applicability of these members in many structures such as platforms, naval and aeronautical structures. Beside these advantages, openings result in a reduction of the buckling capacity, which should be taken into account. n this paper a new exact solution procedure using energy method based on modified vibrational mode shapes is formulated for the buckling analysis of simply supported rectangular plates of abruptly varying stiffness subjected to uniform edge stresses. It is shown that the vibrational mode shapes of a tapered plate is in fact a linear combination of various mode shapes of intact plates. This phenomenon is used to estimate the vibrational mode shapes of taper plates, and is then incorporated to evaluate the buckling loads.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Considerations on Advanced Analysis of Steel Portal Frames Arthur R. Alvarenga and Ricardo A. M. Silveira Department. of Civil Engineering of School of Mines – Federal University of Ouro Preto – UFOP Campus Universitário, Morro do Cruzeiro, s/n, 35400-000 Ouro Preto – MG – Brazil [email protected], [email protected]

ABSTRACT This paper presents a little study about the necessary steps to qualify a second- order inelastic analysis as advanced one. Plastic-zone approach is applied to steel plane frames (portals) and the numerical formulation is based on finite element model of a Bernoulli-Euler beam-column member using the called “slice technique”. This element is set on a Lagrangian updated co-rotational system. The nonlinear problem is solved using Newton-Raphson iterative strategy and a new axial force iterative integration is shown. This process was implemented on a computer program PPLANAV* and the minimum requirements of advanced analysis (initial geometrical imperfections and residual stress) are automatically generated. Two examples show good agreement with other researcher’s answers, but there’s a great elapsed computing time.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A Discrete Element Model For The Fracture Analysis of Reinforced Concrete N. Monteiro Azevedo*, J. V. de Lemos† ,J. R. de Almeida+ * UNL-FCT 2829-516 Monte de Caparica [email protected] † LNEC 1700-066 Lisboa [email protected] +

UNL-FCT 2829-516 Monte de Caparica [email protected]

ABSTRACT The Discrete Element Method was initially applied to the analysis of discontinuous media, e.g. in rock mechanics and soil mechanics. Recently the DEM has been used in fracture studies of non-homogeneous continuous media such as concrete and rock. A 2D circular rigid discrete element formulation based on the DEM that has been further developed to model concrete is adopted [1]. To simulate the concrete at the meso-level, random assemblies of particles based on a given sieve analysis have to be generated. The DEM model micro-properties also have to be previously calibrated through uniaxial tension and compression tests. The formulation of a 1D rigid discrete element that interacts with the discrete rigid particles through contact interfaces is presented. The DEM enhanced model with reinforcement capabilities is evaluated in a three point bending [2] and in a four point bending [3] tests experiments of reinforced concrete beams without stirrups. Under flexure loading conditions the model is shown to predict the expected final crack pattern, the crack propagation and the load displacement diagram. Under shear loading conditions the model is shown to predict the size effect behaviour that occurs on beams without stirrups failing under diagonal shear and also the expected crack propagation and final crack patterns.

References [1] N. Monteiro Azevedo, A rigid particle discrete element model for the fracture analysis of plain and reinforced concrete, PhD thesis, Heriot-Watt University, Scotland, 2003. [2] C. Bosco, A. Carpinteri and P. Debenardi, Minimum reinforcement in high strength concrete, Journal of Structural Engineering, 116, 427-437, 1990. [3] Z.P. Bazant and M. T. Kazemi, Size effect on Diagonal shear failure of Beams Without Stirrups, ACI-Structural Journal, 88, 268-276, 1991.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Analytical Criteria for the Evaluation of the Internal Forces at the Elastic and Plastic Limit States of Lozenge and Triangular Cross-Sections António M. Baptista Laboratório Nacional de Engenharia Civil (LNEC) Av. Brasil 101 - 1700-066 Lisboa - Portugal [email protected]

ABSTRACT The elastic-plastic methods for the design of steel structures have been introduced in some national codes, in Europe and America, for almost half a century. This innovation has resulted from the recognition of these design methods for a better estimation of the ultimate resistance of some types of steel structures. These methods are often based on some hypotheses, such as the formation of plastic hinges in the most stressed cross-sections. The development of these plastic hinges is affected by the interaction between the axial force and the bending moment acting on the cross-section. The interaction criteria between these internal forces depend on the cross-section shape. Therefore, specific analytical criteria are required for each type of cross-sections, and for each combination of bending moments over the two main axis of inertia of the cross section (in the case of bi-axial bending). However, these criteria are not usually available in the design codes or text books for most of the cross-section shapes, especially for those that are less common in steel construction. Even in the case of the most usual cross-sections, the analytical criteria are often defined by means of simplified expressions. In the meanwhile, the use of new shapes is becoming more frequent, due to bold innovative structural solutions conceived by modern architects. This paper presents a set of analytical criteria for the evaluation of the internal forces (axial force and bending moment) in lozenge and triangular steel cross-sections, at their elastic and plastic limit states. They are written in a non-dimensional form, which makes them independent of the cross-sections dimensions and of their width-to-height ratio. These criteria are based on the hypothesis of a full yielding of the cross-sections at their plastic limit state. The material is supposed to present elastic-plastic behaviour without hardening. The effects of eventual shear or torsion deformations on the cross section are supposed to be negligible. These analytical criteria put in evidence the different behaviour of symmetrical and non-symmetrical cross-sections, regarding their axis of bending, at their elastic and plastic limit states. They constitute a basis for the elaboration of more complex analytical criteria, for hollow lozenge and triangular cross-sections, or for rectangular full or hollow cross-sections submitted to axial forces and bi-axial bending. A worked example shows an application of these criteria to the evaluation of the elastic and plastic limit states in a square section submitted to an axial force and bi-axial bending. Another worked example presents the expressions for the evaluation of the maximum eccentricities of an axial force in a triangular section, at its elastic and plastic limit states.

References [1] Ch. Massonnet, M.Save, Calcul Plastique des Constructions. Vol 1 – Structures dépendant d’un Paramètre. Ed. B. Nelissen, 3rd Edition, 1976.

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Comparative Study of Aluminum Alloy Plate 2024/7050 Under the Effect of Internal Damping M. Benachour *, A. Hadjoui*, M. Benguediab**, N. Benachour***, F. Hadjoui* *Automatic Laboratory of Tlemcen / Faculty of Engineering Sciences / UABB - Tlemcen BP 230 – Tlemcen 13000 – Algeria {mbenachour_99, hadjoui_ab, hadjoui_fethi}@yahoo.fr ** University of Sidi Bel Abbes / Faculty of Engineering Sciences [email protected] *** Faculty of Sciences / Physical Department / UABB - Tlemcen [email protected]

ABSTRACT In the present work we determine the influence of structural damping, of an aluminum alloy on the dynamic behavior of a plate with various boundary conditions. Structural damping (loss factor energy) is a very significant parameter by its influence on the dynamic behavior of the mechanical structures. The material of the studied plate is the aluminum alloy 2024 T3 and 7050 T7351. The behavior vibratory of the plate is studied by the finite element method and a comparative study is presented for theses materials. We present also the influence of the loss factor energy on the variation of the inertance relating to each node and each degree of freedom. In parallel, we study the influence of the geometrical parameters of the structure in the frequency domain. In addition, we identify the excited mode of vibration where a comparative example for steel is presented in order to show the excited mode of vibration. The effect of the internal damping is significant in resonance peak, where the reduction of the vibratory amplitude (inertance) is significant. The increase of the geometrical parameters (thickness, ratio length/width), decreases the vibratory amplitude, and on the other hand shift the peaks of resonance towards the high frequencies. The boundary conditions have a great influence on the rotational inertances in a resonance peak.

References [1] M. I. Friswell, The direct updating of damping and stiffness matrices. AIAA Journal, 36, n° 3, 491-493, March 1998. [2] J. C SNOWDON, Vibration and shock in damped mechanical. John Wiley and Sons Inc, 1968. [3] R. D. Mindlin, H. Deresiewics and Schacknow, Flexural vibrations of rectangular plates. Int. J. Eng. Sc., 07, 99-113, 1969. [4] C.F BEARDS, Structural vibration analysis: modeling, analysis and damping for vibrating structures. Ellis Horwood limited, England 1983. [5] S. SRINIVAD, C. V. RAO and A. K. RAO, An exact analysis for vibration of simply supported homogenous and laminated thick rectangular plates. J. of Sound and Vibration, 12, 187-199, 1970. [6] A. HADJOUI, M. BENACHOUR, Application of the matrix connection to the study of the thick plates. Fourth Mechanical Congress, 13 – 16 April 1999, Mohammadia, Morocco. [7] R. MALY Joseph, A. BENDER Kirsten and C. PENDLETON Scott, Complex stiffness measurement of vibration damped structural elements. International Modal Analysis Conference, IMAC-XVIII, San Antonio, Texas, February 2000.

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Nonlinear Analysis of Space Frames António A. Correia*, Francisco B. E. Virtuoso† *

ICIST and Instituto Superior Técnico, DECivil Av. Rovisco Pais, 1049-001 Lisboa, Portugal [email protected]

†

ICIST and Instituto Superior Técnico, DECivil Av. Rovisco Pais, 1049-001 Lisboa, Portugal [email protected]

ABSTRACT This work presents a method developed and implemented to consider material and geometrical nonlinearities in the behaviour of space frames. A geometrically nonlinear formulation is developed in which the compatibility and equilibrium relations are established in the structure’s deformed configuration using a co-rotational description of the movement. The use of total lagrangian, updated lagrangian and co-rotational descriptions of the movement is discussed. An exact method is presented to consider three-dimensional finite rotations, which cannot be added like vectors as assumed in geometrically linear analysis. The finite rotations are considered by using Euler’s finite rotation formula. That formulation allows obtaining the correct relationships between the rotational degrees of freedom considering large displacements and rotations. The material nonlinear behaviour considered in this work is due to the use of general nonlinear axial stress-axial strain relationships. Based on these assumptions for the material behaviour, and by using an approximation for the internal forces field throughout the frame element, a material nonlinear formulation for three-dimensional structures is presented. The use of an approximation for the internal forces field instead of the usual approximation for the internal displacements field is discussed. An unusual method of integration over the cross-section is presented, where the integrations are carried out over its perimeter, thus allowing any geometrical polygonal shape for it. In this work, despite large displacements and rotations being allowed, only small deformations are considered. This allows the material and geometrical nonlinearities to be treated in a completely independent way. The incremental and iterative strategies used are discussed. The capabilities of the formulations presented here are exemplified by the analysis of a few benchmark problems.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Investigation of shear wall behavior with composite boundary elements Ali Davaran1, Armin Kashefi and Shahin Nayeri Amiri Department of Civil Engineering, Tabriz University Tabriz, Iran [email protected] {davaran, sh.nayeri}@tabrizu.ac.ir

ABSTRACT Composite construction in steel and concrete offers significant advantages for use as the primary lateral resistance system in building structures subjected to seismic loading. While composite beam and joist floor system have been common for over a half a century, through the use of over the past decade a substantial amount of researches have been conducted world wide on a wide range of composite lateral resistance systems. The appropriate behavior of composite structures and the economical considerations are resulted in offering a new detail of composite shear wall and evaluation of the behavior of this system. The non-linear static analysis has been used in this research. In the detail which is offered in this research steel plates are used to reinforce the boundary elements of concrete shear wall and these steel plates are connected with shear studs to concrete. The research results show that the behavior of this detail with lateral loading is appropriate and ductility of this system is favorable, but the economical features of this detail must be verified when it is necessary to reduce the size of the boundary elements and it is suggested for further researches

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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An investigation on dynamic behavior of shear walls on flexible foundation Dr. Mikail Yousefzadeh Fard1, Shahin Nayeri Amiri2and Armin Kashefi Department of Civil Engineering, University of Islamic Azad Tabriz, Iran {mikail,sh.nayeri}@tabrizu.ac.ir Department of Civil Engineering, Tabriz University Tabriz Iran [email protected]

ABSTRACT

Nowadays, shear walls are used as efficient structural systems for resisting against external lateral loads such as wind and earthquake loads. Regarding to this fact that the pervious studies on the behavior of shear wall systems were performed without considering the effect of foundation and interaction between soil and structure, it seems necessary to conduct some research in the evaluation of shear walls dynamic behavior rested on flexible foundation and with considering the effect of soilstructure interaction. The linear modeling of a shear wall rested on flexible foundation is conducted which is accompanied with the modeling of soil by Winkler `s springs under foundation. The dynamic behavior of shear wall system is assessed. The results obtained show that depending on types of soils under foundation, the values of displacements are more than displacements when the flexibility of foundation is ignored. Also the internal forces at the base of shear walls and the periods of the first three modes of the structure have significant differences between two cases. In one case the interaction of soil-structure is ignored and on the other one the interaction is considered.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Influence of the morphology of adhesive joining on the mechanical properties of periodic metal hollow-sphere-structures ∗,†, J. Gr´acio∗, † ¨ T. Fiedler∗,†, A. Ochsner ∗ Centre for Mechanical Technology and Automation, University of Aveiro, Aveiro, Portugal † Department of Mechanical Engineering, University of Aveiro, Portugal

tfi[email protected], [email protected], [email protected]

ABSTRACT Hollow-sphere-structures (HSS) originate a new group of cellular materials characterised by a high reproducibility of geometry and therefore mechanical properties. Well-known advantages of cellular materials are a high ability of energy adsorption, good damping, excellent heat insulation and a high specific stiffness. Combination of these properties opens a wide field of potential applications, e.g. automotive, aviation or space-industry. Essential Limiting factors for the utilisation of cellular materials are inconstant material parameters and relatively high production costs. Both factors can be reduced by the application of hollow-sphere-structures. A new powder metallurgy based manufacturing route enables the economic production of metallic hollow spheres of defined geometry. Different joining technologies such as sintering, soldering and adhering can be applied to assemble hollow spheres to interdependent structures. Adhering provides the most economic way of joining and allows for further cost reduction and therefore the expansion of the field of potential applications. Another important advantage it the possible utilisation of the mechanical behaviour and morphology of the adhesive layer as a further design parameter for the optimisation of the structure’s mechanical properties for specific applications. The influence of the morphology and mechanical properties of the adhesive layer is discussed in the scope of this article. Finite element (FE) analysis is performed for periodic structures and the results are compared for different geometries. Two different approaches for adhering are considered: hollow spheres completely embedded in a polymer matrix (syntactic foam) and alternatively, concentration of the adhesive layer in the contact points of the spheres (Partial-HSS). In contrast to earlier approaches (e.g. [1, 2]), the geometry is discretised based on regular hexahedron elements. This approach is much more time-consuming, but important in order to achieve more accurate simulation of nonlinear-behaviour (e.g. plasticity).

References [1] W.S. Sanders and L.J. Gibson, Mechanics of BCC and FCC hollow-sphere foams, Materials Science and Engineering A, 352, 150–161, 2003. [2] S. Gasser, F. Paun, A. Cayzeele and Y. Br´echet, Uniaxial tensile elastic properties of a regular stacking of brazed hollow spheres, Scripta Materialia, 48, 1617–1623, 2003.

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Dynamic behaviour of a composite twin girder bridge in a high speed interoperable line Helder Figueiredo*, Rui Calçada†, Raimundo Delgado† *

Faculdade de Engenharia da Universidade do Porto Rua Dr. Roberto Frias, s/n [email protected]

†

Faculdade de Engenharia da Universidade do Porto Rua Dr. Roberto Frias, s/n [email protected], [email protected]

ABSTRACT A high-speed railway system is presently under implementation in Portugal, which will allow the connection of the country with a large European network. At the moment, many countries have already developed their own structural solutions for dealing with the effects of high-speed trains in bridges and a great number of structures have been in operation for several years. These solutions were initially designed for specific train types, thus being checked for the dynamic effects of only a small part of the actual European high speed rail traffic. Recent advances in the understanding of the behaviour of high speed railway bridges have been introduced in the EN 1990-Annex A2 and EN 1991-2, reflecting the work undertaken by the ERRI committee D214. In the case of interoperable lines where the high speed TSI is applicable, this being the case of the future Portuguese high speed network, additional checks for dynamic analysis using High Speed Load Model should be performed. One of the solutions that has proven to be very competitive in France is the composite twin girder bridge. This type of deck is used in continuous schemes, with spans lengths ranging from 40m up to 65m. In this paper the dynamic behaviour of this type of bridge is assessed using as reference a 333m long composite twin girder deck located on the French TGV Nord line. The bridge is continuous over its entire length, comprising 7 intermediate spans of 40m and 2 end spans of 28m and 25m. Dynamic analyses of the bridge were performed for both the European high speed trains and the HSLM load schemes, using various types of FE models of increasing level of detail. The response of the bridge is checked in terms of structural safety (amplification levels and fatigue requirements), deck acceleration and passenger comfort.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A Finite Element Model for Beam to Column Bolted Connections with Semi Rigid Behaviour Foces A.*, Garrido, JA.†, Moreno A.†† *

ETS de Ingenieros Industriales, Universidad de Valladolid Paseo del Cauce, s/n, 47011 Valladolid [email protected]

†

ETS de Ingenieros Industriales, Universidad de Valladolid Paseo del Cauce, s/n, 47011 Valladolid [email protected] ††

Escuela Politécnica Superior. Universidade da Coruña. Campus de Esteiro, C/ Mendizábal, s/n, 15403 Ferrol [email protected]

ABSTRACT Nowadays it is recognized that connections and members of steel frameworks have a semi rigid and nonlinear behaviour. One of the main concerns is how to incorporate the connection characteristics into an analysis. In the present study, beam-to-column bolted end plate connections, widely used because of the economy and simplicity of fabrication and assembly, are investigated for predicting their rotational behaviour (moment-rotation curve) that can be used in the frame analysis. In order to predict the rotational behaviour of this type of connection, a three-dimensional finite element model has been developed by the COSMOS/M® code. The proposed model takes into consideration the interaction between the various components that are comprised in the connection. Thus, the modelling domain includes the beam, end plate, bolts and nuts and the column. The main novelty now presented consists of accounting for the flexibility of the column components in the analysis. The column is modeled using solid elements. Other authors employ shell elements or use a rigid surface representing the column flange. Besides, the analysis incorporates the effects of material nonlinearity, for the plates and bolts, using the elastic-perfecty plastic stress-strain relationship. The results obtained from the finite element analysis are evaluated and verified by comparing the numerically predicted results with those of the corresponding tests carried out. The numerical results are also compared with a simplified theoretical model based on yield line analysis and the stub-tee analogy. The 3D finite element model presented in this study can be used to generalize the rotational behaviour of the end plate connection through more extensive parametric studies in order to take into consideration the connection flexibility and its effect on the performance of steel frames. So, extension of the methodology now presented may involve an improvement in analysis and design procedures of the proposed type of connection according to modern design codes.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Rational strain measures - The implicit corotational method G. Garcea, A. Madeo Dipartimento di Strutture, Universit´a della Calabria, 87030 Arcavacata di Rende (Cosenza), Italy [email protected], [email protected] ABSTRACT The nonlinear analysis of slender structures undergoing large displacements–small strains requires a proper description of the kinematic strain– displacement relationship to define a rational strain measure. In particular the strain measure must be unaffected by finite rigid body motions, that is it must be objective. Rational strain measures for structural models are not usually available or are too complex to be used in a FEM context, while the technical models which are usually adopted do not satisfy the requirements for objectivity. A simple method, called implicit corotational, to obtain rational strain measures is proposed. The starting point is the corotational idea [1] which is now, following Biot [2], applied at the continuum level. The body is thought of as subdivided in finite parts, each one with a corotational frame that follows the rigid motion of the part. In the corotational frame a linearized kinematic and a linear strain measure may be used in this way. The strain measures become then more accurate with a reduction in the part size. Furthermore it becomes exact by an appropriate limit process. The final results is a rational strain measure. As the displacement field in the corotational reference is infinitesimal, linearized kinematical models for beams, plates and shells can usefully be employed. Since linearized models are always available even for complex structural models (plates or shells) it is easy to obtain the corresponding rational strain measures using this approach. The correctness of the strain measures obtained for 3D beams and thin plates is clear when they are compared with those available in literature [3, 4]. Numerical analysis are performed using Koiter’s asymptotic approach [5]. In this context, which is particularly sensitive to the exactness of the strain measures , the accuracy of the results can be seen.

References [1] B. Nour-Omid, C.C.Rankin, Finite rotation analysis and consistent linearization using projectors. Computer Methods in Applied Mechanics and Engineering, 93, 353-384, 1991. [2] Maurice A.Biot, Mechanics of Incremental Deformations. J. Wiley & Sons, New-York, 1995. [3] Stuart S.Antman, Nonlinear Problems of Elasticity. Springer-Verlag, New-York, 1995. [4] J.C. Simo, L.Vu-Quoc, A three-dimensional finite-strain rod model. Part II: Computational aspects. Computer Methods in Applied Mechanics and Engineering, 58, 79-116, 1986. [5] W.T.Koiter, On the stability of elastic equilibrium. Thesis, Delft, 1945. English transl. NASA TTF10, 883 (1967) and AFFDL\TR70-25 (1970).

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Discrete and continuous analysis of different cable structures V. Kulbach, J. Idnurm Department of Structural Design Tallinn University of Technology, Ehitajate tee 5, Tallinn, Estonia [email protected] Department of Transportation Tallinn University of Technology, Ehitajate tee 5, Tallinn, Estonia [email protected]

ABSTRACT Continuous and discrete analysis of different cable structures is under investigation in our report. In cases of distributed loads the non-linear conditions of equilibrium and equations of deformation compatibility are taken as initial equations [1]. To eliminate the horizontal displacements, the equations of deformation compatibility were used in integrated form; corresponding integrals
(du/dx)dx were replaced by respective displacements of supporting structures under action of cable forces. Determination of deflections and inner forces for certain cable structures under action of distributed vertical loads may be carried out by means of exact analysis. A girder-stiffened structure has also an exact solution but in the form of complicated transcendental equations. A simpler, compact solution may be found with a proper approximation of the deflection function in the form of trigonometric dependences. Continuous analysis may be also applied to spatial cable structures in the form of hypar-networks with elliptical contour beam [2]. Suitable approximation of the deflection function with use of Galyorkin procedures brings us to values of the network’s deflection and cable forces very near to exact ones. In discrete analysis of girder-stiffened cable structures, the condition of equilibruim is to be composed for every node and the equation of deformations compatibility for every section of the cable [3]. For every joint on stiffening girder were used equation which consider girder’s node’s deformations, internal force in the hangers and load, which is balanced by the stiffening girder. The load may be by distributed load or concentrated force, and may be applied on any point of girder. Using this equations, and moment equilibrium conditions for girder’s supports, were calculated vertical displacements of cable nodes and displacements of girder nodes. Using iteration can be found such value of cable force, which gives actual displacement for every node. After found cable force, all required parameters can be calculated. Examples of analysis of hypar-network and the bridge for a 6100 m strait crossing are presented in the full text of our paper.

References [1] V. Kulbach, Investigation of prestressed cable structures at Tallinn Technical University, Proc. Estonian Acad. Sci., Eng., 8/2, 68 – 83, 2002. [2] I. Tärno, Effects of contour ellipticity upon structural behaviour of hyparform suspended roofs, Publ. of Royal Institute of Technology, Stockholm, 1998. [3] V. Kulbach, S. Idnurm, J. Idnurm, Discrete and continuous modeling of suspension bridges , Proc. Estonian Acad. Sci., Eng., 8/2, 121 – 133, 2002.

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Three-dimensional Vibration Analysis of Crystal Plates via Ritz Method Qian LI∗ and VaiPan IU† ∗ Department of Civil and Environmental Engineering, University of Macau

Macao SAR, PRC [email protected] †Department of Civil and Environmental Engineering, University of Macau

Macao SAR, PRC [email protected] ABSTRACT In this paper, three-dimensional vibration of rectangular Y-cut crystal plate has been investigated. The three displacement components of plate are expanded in series of Chebyshev polynomial multiplied by the boundary function R which makes expansions satisfy the essential boundary conditions along the edges. Chebyshev polynomial series are chosen as admissible functions for its two distinct advantages: One is that it is a set of complete and orthogonal series in the interval [−1, 1]; the other is that it includes constant and proportional terms. The constant term can easily express the whole rigid displacement of the body. The proportional term can easily reflect the shear force effect along the thickness of a finite plate. The maximum energy function of a plate is expressed in terms of Chebyshev polynomial series. The eigenvalue matrix for natural vibration frequencies is obtained by Ritz method and then solved by computer program. Example of an infinite plate excited by thickness-shear deformation parallel to one edge is solved and verified by exact solutions. Other examples of four clamped edges and four simply supported edges rectangular Y-cut crystal plates are carried out. The trial plates are of three different edge lengths and two different thicknesses. The first twenty frequencies of natural free vibration are compared with those from a finite element method. It also shows that the results from present method and finite element method have a good agreement. Besides, for the advantage of the constant and proportional terms in Chebyshev polynomial series, convergence study demonstrates the rapid rate and high efficiency. The frequencies monotonically decrease and approach certain values with the increase in the number of terms of admissible functions. Three terms are sufficient for the requirement of expansion in the thickness direction. Finally, the free vibration of clamped square Y-cut crystal is investigated. Due to the three-dimensional expansions, any relative displacements in any points of the plate body of modes can be determined very easily by back substitution of the eigenvalues. The deflected shapes of first eight modes show the flexural and thickness extensional modes explicitly.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A 3D Solid Finite Element for Reinforced Concrete Analysis Allowing Slippage of Reinforcement Georgios Ch. Lykidis, Konstantinos V. Spiliopoulos Institute of Structural Analysis and Aseismic Research Department of Civil Engineering, National Technical University of Athens, Zografou Campus, Athens, 157 73, Greece [email protected], [email protected]

ABSTRACT In order to evaluate the safety levels of the design of reinforced concrete structures it is essential to be able to predict their response under any type and level of loading. To this end the finite element method of analysis may be used. For such an analysis to be realistic, one must take into account all aspects of the nonlinear behaviour of reinforced concrete. A simple smeared crack material model for concrete behaviour [1], which is based on experiments and uses the uniaxial compressive concrete strength as the only prerequisite has been used recently by the authors to analyze structures loaded statically and dynamically ([2]). A numerical method that treats crack opening and closure in a unified way and presents no numerical instability has been presented. Steel bars are taken into account using an embedded reinforcement formulation [3] and assuming perfect bond with surrounding concrete. The latter assumption is avoided in the present paper with an additional degree of freedom. A realistic model [4] is used to describe the interface behaviour along a reinforcing bar. Comparative analyses of the model with and without bar slipping are performed for static loading cases. The analyses show that the whole procedure manages to give stable and realistic results. This enhanced, therefore, oneparameter concrete model may be used in the analysis of reinforced concrete structures more effectively.

References [1] M. D. Kotsovos, M. N. Pavlovic, Structural concrete, finite element analysis for limit state design, Thomas Telford, London, 1995. [2] K.V. Spiliopoulos, G. Ch. Lykidis, An efficient three dimensional solid finite element dynamic analysis of reinforced concrete structures, Earthquake Engineering & Structural Dynamics, 35, 137-157, 2006. [3] A. E. Elwi, T. M. Hrudey, “Finite Element Model for Curved Embedded Reinforcement”, ASCE, Journal of Engineering Mechanics, 115, 740-754, 1989. [4] R. Eligehausen, E. P. Popov, & V. V. Bertero, Local bond stress - slip relationships of deformed bars under generalized excitations, Rep. UCB/EERC 83-23, Earthquake Engng, Res. Ctr., University of California, Berkeley, 1983.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Some new results on MITC plate elements Mikko Lyly∗, Jarkko Niiranen†, and Rolf Stenberg † ∗

CSC – Scientific Computing Ltd. P.O. Box 405, 02101 Espoo, Finland mikko.lyly@csc.fi †Institute of Mathematics, Helsinki University of Technology

P.O. Box 1100, 02015 TKK, Finland jarkko.niiranen@tkk.fi, rolf.stenberg@tkk.fi

ABSTRACT The approximation of the deflection for the MITC plate elements [1, 2] is shown to be superconvergent with respect to a special interpolation operator [3]. This property holds in the H 1 -norm and the interpolation operator is closely related to the reduction operator used in the MITC methods. A part of the superconvergence result is, roughly speaking, that the vertex values obtained with the MITC methods are superconvergent. This may be an explanation why these methods have become so popular. By utilizing the superconvergence property a postprocessing method has been introduced [3] in order to improve the accuracy of the approximation for the deflection. The new approximation is a piecewise polynomial of one degree higher than the original one and it is constructed element by element which implies low computational costs. We show various computational results illustrating the superconvergence properties of the original approximation and confirming the improved accuracy of the postprocessed approximation. In the numerical tests both uniform and non-uniform meshes are used and cases with different kinds of boundary conditions are studied.

References [1] K.-J. Bathe, F. Brezzi and M. Fortin, Mixed-interpolated elements for Reissner-Mindlin plates. Int. J. Num. Meths. Eng., 28, 1787–1801, 1989. [2] F. Brezzi, M. Fortin and R. Stenberg, Error analysis of mixed-interpolated elements for ReissnerMindlin plates. Math. Mod. Meth. Appl. Sci., 1, 125–151, 1991. [3] M. Lyly, J. Niiranen and R. Stenberg. Superconvergence and postprocessing of MITC plate elements. Helsinki University of Technology, Institute of Mathematics, Research Reports A 474, January, 2005 (http://math.tkk.fi/reports).

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Nonlinear Analysis of Reinforced Concrete Beams Considering the Slip Between Steel and Concrete. Joaquim Marins Neto*, Aloisio Ernesto Assan† *

Department of Structures, Faculty of Civil Engineering, State University of Campinas Av. Albert Einstein, 951, C.E.P.: 13084-971, P.O.Box 6021,Campinas, SP, Brazil [email protected]

†

Department of Structures, Faculty of Civil Engineering, State University of Campinas Av. Albert Einstein, 951, C.E.P.: 13084-971, P.O.Box 6021,Campinas, SP, Brazil [email protected]

ABSTRACT In this work aspects of interaction between steel and concrete, for reinforced concrete structures, with particular interest in the mechanism of slip that occurs in the steel-concrete interface are presented. A nonlinear computational model which considers the bond-slip behavior between reinforcing steel and concrete for the nonlinear analysis of beams subjected to bending is developed. The finite element method is used to predict the behavior of reinforced concrete structures based on the properties of the concrete, the reinforcing steel, and the relationship of the steel-concrete interface. The concept about equivalent uniaxial stress-strain model proposed by [1] is used to describe the nonlinear behavior of reinforced concrete which incorporates tensile cracking at a limiting stress and the strain-softening phenomenon beyond the maximum compressive strength from an incremental load procedure with an iterative approach to obtain an equilibrium position of the structure for each increment. The bond is modeled with interface element (bond-zone element) connecting the steel and concrete elements. The interface element presented by [2] has its stiffness based on the stages of relationship between the local bond stress and the relative slip of the bar, for incremental load process. Several numerical examples comparing results of bending beams are presented.

References [1] D. DARWIN; D. A. W. PECKNOLD, Inelastic model for cyclic biaxial loading of reinforced concrete. Civil Engineering Studies, University of Illinois, Illinois, USA, july, 1974. [2] A. K. DE GROOT; G. M. A. KUSTERS; T. MONNIER, Numerical modeling of bond-slip behavior. Heron, Conc. Mech., Vol. 26, 1981.

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Novel semi-analytical methodology to determine model parameters for simple finite element bolt model Joost C.F.N. van Rijn National Aerospace Laboratory NLR P.O. Box 153, 8300 AD Emmeloord, The Netherlands [email protected]

ABSTRACT Bolted joints are commonly used in aircraft structures to join the various parts. These joints are an important factor in the sizing of the construction, for composite to composite as well as for composite to metal joints, as a larger thickness is normally required in the joint area. Consequently joints have an important bearing on the weight of the structure and the material costs. Moreover the production costs of joining are significant. Aircraft structures are mostly modeled using a shell representation. Bolted joints are incorporated in these models using a simple model that represents the bolt by a beam element that connects to one node in each adherent finite element model. At these particular nodes a point force is exerted on the adherent models that will cause a certain, mesh dependent, deformation. Experimentally determined bolt stiffnesses are available from various sources. A finite element model of a bolted joint should provide the same bolt stiffness; otherwise the distribution of bolt loads will be erroneous especially in a multiple load path situation. The bolt stiffness can be changed by adaptation of the beam area or the moments of inertia. It is necessary to account for the deformation of the adherent mesh in order to obtain the right bolt stiffness. The novel semi-analytical methodology to determine the model parameters for a simple finite element bolt model, presented in this paper, provides for a better correspondence between computed and experimentally determined behavior of a bolted joint. It enhances the finite element representation of a bolt as it explicitly accounts for the local deformation in the finite element representation of the adherents. The influence of variation in mesh density and boundary conditions for the shell representation of the adherents is established through a simple finite element analysis. The influence of variations in material stiffness and adherent thickness for both metal and composite adherents is obtained with the presented methodology. A mathematical foundation is provided for the simple finite element bolt model that also enables a sensitivity analysis. The use of the methodology will be demonstrated as it was used to determined the bolt parameters for a total of 40 different configurations within a detailed model of a flap section The presented methodology was developed within the context of the Sixth Framework EU project Bond Assisted Single Step Assembly (BASSA). The financial contribution of the EU is gratefully acknowledged.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Accounting for Fuselage Instabilities in the Coarse Model of an Aircraft Fuselage by means of a Material Law Wilhelm Rust*, Josef Overberg† *

Fachhochschule Hannover – University of Applied Sciences Ricklinger Stadtweg 120, 30459 Hannover, Germany [email protected] †

CAD-FEM GmbH Schmiedestr. 31, 31303 Burgdorf, Germany [email protected]

ABSTRACT In the typical dimensioning and evaluation process of an aircraft fuselage nominal stresses were obtained from a linear elastic fuselage or section model (barrel) discretized with finite elements in the coarsest possible way. This model does not account for nonlinearities. However, the stresses are compared with allowable ones obtained from either tests or finite element analyses of cutouts (panels) with fine meshes accounting for contact, material and geometric nonlinearities including stability problems. The results are force-displacement curves ending up in the ultimate load. Divided by related areas nominal allowable stresses are obtained. If in one coarse element the stress exceeds the allowable one the total fuselage section is considered as failing. This method does not account neither for changes of the force fluxes due to reaching the local ultimate load nor for the fact that the onset of buckling locally weakens the structure before failing. Therefore, a method is presented here which transfers the local nonlinear behavior analyzed for the panels with fine meshes to the coarse global model. For that purpose all nonlinear effects on the local force-displacement behavior are introduced to the global model by an elasto-plastic material law. It is shown how the stress-strain curves for shear and compression loading (of different shape) are calculated, how they interact in the yield condition, how the flow rule is chosen and how the hardening rule accounts for different ratios of shear and compression. It is shown that the assumptions work well for the described purpose and that the global system can reach a higher ultimate load compared with the standard method. Possible Extensions and Limits of the method are outlined.

References 1. P. Linde, J. Pleitner, W. Rust, Virtual Testing of Aircraft Fuselage Stiffened Panels. Proceedings of 24th Int. Congr. of the Aeronautical Sciences ICAS 2004 2. W. Rust, P. Linde, Ultimate Load Analyses of Aircraft Fuselage Structures within the Virtual Test Rig, Proceedings of IASS/IACM 2005, 5th International Conference on Computation of Shell & Spatial Structures, Salzburg 2005 3. P. Linde, A. Schulz, W. Rust, Influence of modelling and solution methods on the postbuckling behaviour of stiffened aircraft fuselage panels, to appear in Composite Structures

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Modelling of reinforced materials by a subcycling algorithm J.L. Curiel Sosa∗, D.R.J. Owen†, E.A. de Souza Neto†, N. Petrinic ∗ ∗ Department of Engineering Science,University of Oxford

Parks Road, Oxford OX1 3PJ, UK [email protected] †Civil and Computational Engineering Centre, University of Wales Swansea

Singleton Park, Swansea SA2 8PP, UK [email protected]

ABSTRACT The high nonlinearity associated to the interface and constituents in reinforced materials -e.g. reinforced concrete- has motivated the development of this subcycling algorithm. The interface modelling and the complex material model used to represent the continuum implies a small critical time step when solving a spatial discretised finite element mesh with an explicit time integrator conditionally stable. Making two subcycles -one for the continuum and the other one for the reinforcement- the smallest critical time step does not rule the other sub cycle. The interface is modeled by transmission conditions including empirically-based bond stress-slip relationship. A set of pullout tests of reinforcing bar embedded in a surrounding continuum to demonstrate the efficiency of the scheme is presented and, then, validated against experimental results from the literature. The attractiveness of this scheme lies in the computational efficiency implied by running reinforcement and continuum at two different velocities of execution and solve the problem of nonlinearity created in the interface of very distinct materials.

References [1] J. L. Curiel Sosa, E. A. de Souza Neto, and D. R. J. Owen. A combined implicit-explicit algorithm in time for non-linear finite element analysis. Commun. Numer. Meth. Engng, 22:63–75, 2006. [2] W. J. T. Daniel. A study of the stability of subcycling algorithms in structural dynamics. Comput. Methods Appl. Mech. Engrg., 156:1–13, 1998. [3] I. W. Farmer. Stress distribution along a resin grouted rock anchor. Int. J. Rock Mech. Min. Sci. Geomech., 12:347–351, 1975. [4] M. O. Neal and T. Belytschko. Explicit-explicit subcycling with non-integer time step ratios for structural dynamic systems. Computer and Structures, 31(6):871–880, 1989.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Limit Analysis of Cable-Tied Structures Konstantinos V. Spiliopoulos, and Theodoros N. Patsios Institute of Structural Analysis & Aseismic Research National Technical University of Athens 9 Iroon Politehniou, Zografou Campus, 157-73, Athens, Greece [email protected]

ABSTRACT Cables are frequently used to strengthen existing framed structures. They are also the main members that support bridge-decks in cable-stayed bridges. This type of structures may be idealised to consist of beam-type members under pure bending and cabled members under pure tension. In order to get an estimate of the strength of such structures a step by step elasto-plastic analysis must be used. This procedure, however, is time-consuming as it has to follow every single plasticization and any deplasticization that may occur up to collapse. Limit analyses, based on the upper or lower bound theorems of plasticity provide a better alternative. In the present work, a limit analysis procedure, based on the upper bound theorem and leading to a linear programming problem, is followed. Plastic rotation in the form of a plastic hinge at the end of a beam-type member marks the plasticization of the corresponding section when its ultimate moment is exceeded, whereas a plastic extension occurs at one point inside the cabled member when its ultimate axial force is exceeded. The whole approach is formulated within the mesh description of statics which is a generalisation of the force method. This method is known to be the computationally most effective one for linear programming structural problems. The process consists of three distinct parts. The first part deals exclusively with the cabled members of the structure. The influence of their force on the rest of the structure is taken into account by satisfying equilibrium along the shortest path between its two ends. In the second part the indeterminacy of the rest of the structure that now consists of beam-type members is catered for using an existing algorithm, also based on a shortest path technique between two points of a connected planar graph. Cantilevers that follow the shortest path of each load to the ground are used in the third part to satisfy equilibrium with the applied loads. The whole process renders a fully automatic and computationally efficient numerical method to find the limit load of the above-mentioned structures. Two examples of strengthened frames, as well as an example of a cable-stayed bridge are analysed.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Smart Super Elements in Slender Structures Subjected to Wind Raphaël D.J.M. Steenbergen*, Johan Blaauwendraad† *

TU Delft, Faculty of Civil Engineering and Geosciences P.O. Box 5048, NL-2600 GA, Delft, The Netherlands. [email protected]

† Emeritus professor Structural Mechanics, TU Delft, Faculty of Civil Engineering and Geosciences [email protected]

ABSTRACT Structural analysis of tall buildings of asymmetric plan and irregular geometry subjected to wind load eventuates in complicated calculus. This is among others the case if parts of the building or stability elements stop at a lower height than the rest. FEM programs are at disposal; however the modeling takes a lot of time and a quick and deeper understanding of the force flow is not provided. In this paper this want is supplied by developing a closed-form super element method for two frequently occurring building types. For two types of a tall building of irregular geometry, an insight-providing closed-form analysis method of combining super elements is presented. The main-structure is subdivided in only two super elements. The super elements are based on closed-form solutions describing the force flow in the stability elements. Within an element no change of floor plan, wall and shaft geometry occurs. A node between elements is only chosen where the properties of the building change. The in-plane stiffnesses of the floors are included and act as distributed coupling springs between the stability elements. For each super element a set of simultaneous differential equations is derived and closed-form solutions are obtained; see [1]. For each super element the stiffness matrix is composed from the homogeneous solution and the load vector is composed from both the particular and the homogeneous solution. Foundation stiffness is accounted for. At each change of geometry (node) a marked disturbance in the moment and shear force diagram is found, attenuating along a number of storeys depending on the ratio of the characteristic length and the length of the building. Closed-form expressions for the influence lengths of these disturbances are obtained. Including the rotational stiffness of the foundation may result in substantial disturbances in the stress state at the base of the building. No disturbance occurs if the ratio of the rotational stiffnesses of wall and shaft equals the ratio of the base moments of wall and shaft for an ideal rigid foundation. Results have been presented in [1]. Because of the use of a very small number of super elements with closed-form solutions, the method contributes to the understanding of the behaviour of the considered tall buildings with a discrete change along the height. In a preliminary design stage a fast analysis can be made without spending much time in modelling. It is shown that the modelling and calculating time of the present method is reduced significantly in comparison with complete finite element analysis and accurate results are obtained.

References [1] R.D.J.M. Steenbergen, Static Analysis of Asymmetric Buildings. Delft University of Technology, Delft, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Nonlinear analysis of space R/C frames with non-uniform torsion Boris Trogrlic, Ante Mihanovic and Zeljana Nikolic University of Split, Faculty of Civil Engineering and Architecture Matice hrvatske 15, 21000 Split, Croatia [email protected] [email protected] [email protected]

ABSTRACT This paper presents a numerical model of stability and load-bearing capacity of space reinforced concrete (R/C) frame structures taking into account the material and geometric nonlinearity. The developed model describes the behavior of space frames with composite cross sections under a monotonically increasing load, from zero up to the ultimate load, i.e. collapse of the structure. The collapse of the structure occurs due to exceeding the limit load and/or loss of stability of space beams or whole structure. The fibre decomposition procedure is developed to solve material and geometrical nonlinear behaviour of composite cross-section in three-dimensional frames. The filaments in the fibre decomposition model of the cross-section, which describe uniaxial behaviour of materials, are extended over corresponding finite element and create a separate prismatic body discretised by brick finite elements. After mapping of boundary forces on prismatic body, i.e. ‘comparative body’, the capture of non-uniform torsion is applied. The main attention in this approach is concentrated on the evaluation of the torsional stiffness, which are strongly nonlinear. Three integration levels exist: the first along the beam-column finite element, the second over the fibre decomposed cross-section and the third over a prismatic comparative body. Behaviour of the space frames shall be more realistically described in this way, especially flexural, lateral and torsional stability effects. The global procedure includes an incremental-direct iteration step approach. The incremental step model of gravitational load level is applied. Geometrical nonlinearity is assumed by Total Lagrange small displacement formulation. The perfect bond-slip effect between concrete and rebars as well as smeared crack model is assumed. Two examples are studied to verify the accuracy of the program and demonstrate its application in practical engineering.

References [1] B.A. Izzuddin, A.A.F.M. Siyam and D.L. Smith, An efficient beam–column formulation for 3D reinforced concrete frames, Computers and Structures, 80, Issues 7-8, 659-676, 2002. [2] H.-G. Kwak and S.-P. Kim, Nonlinear analysis of RC beams based on moment–curvature relation, Computers and Structures, 80, Issues 7-8, 615-628, 2002.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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An FE Analysis of the Stresses in Pultruded GRP Single-Bolt Tension Joints and Their Implications for Joint Design Geoffrey J. Turvey*, Pu Wang† *

Lancaster University Engineering Department, Bailrigg, Lancaster, LA1 4YR, UK [email protected] † Schlumberger Stonehouse Technology Centre, Stroudwater Industrial Estate, Stonehouse, Gloucestershire, GL10 3SX, UK [email protected]

ABSTRACT FE (Finite Element) analysis is used to determine stresses on critical planes and around the hole edge in a two-dimensional model of a single-bolt tension joint in pultruded GRP (Glass Reinforced Plastic) plate material. The analysis takes account of bolt – hole clearance and friction at the contact surfaces between the bolt shank and the hole. It is shown that even when the hole clearance is nominally zero (~0.2mm) critical stress distributions, normalized with respect to the far field stress are not invariant but change as the tension increases. Friction between the bolt shank and the hole and the small hole clearance are the principal factors which cause the zone of contact (defined by the angle it subtends at the centre of the bolt) to increase with increasing tension, and produce significant changes in the stress distributions at critical locations. These observations cast doubt on the validity of the simplified method of design, given in the EUROCOMP code [1], for bolted tension joints in pultruded GRP plate material, because the method relies on normalized critical stress distributions remaining unchanged as the tension load applied to the joint increases.

References [1] J.L. Clarke ed, Structural design of polymer composites – EUROCOMP design code and handbook, E. & F.N. Spon, London, 1996

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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An Efficient Evaluation of Structural Safety applying Perturbation Techniques José M.G.C.Veiga*, António A. R. Henriques†, Jorge M. Delgado* *

†

Escola Superior de Tecnologia e Gestão - IPVC Viana do Castelo - Portugal [email protected] and [email protected]

Laboratório da Tecnologia do Betão e do Comportamento Estrutural Faculdade de Engenharia da Universidade do Porto - Portugal [email protected]

ABSTRACT The application of probabilistic techniques on structural safety evaluation has suffered a great expansion in the last years. However, one of the main problems in the introduction of these techniques is the long computational time consuming required, particularly when simulation methods as Monte-Carlo method are used, even when sampling reduction techniques are adopted [2]. In this paper is presented an efficient structural reliability method that couples perturbation techniques with the finite element method [1]. This method allows, in one only structural analysis, to evaluate the mean value and the standard deviation of the structural response, by defining previously the probability distribution of problem basic random variables. Consequently a much faster analysis is performed, when compared with the most frequent used methods based on reliability techniques. Considering a structural system, with n structural elements, submitted to a load defined by F·Φ = F·[Φ1, Φ2, …, Φn]; where F is the load intensity and [Φ1, Φ2, …, Φn] is the load distribution vector along the structure. According to the finite element method, the system equilibrium is defined by the following equation: K(u)·U = F·Φ ; where K(u) is the tangent stiffness matrix of the structure, defined as a function of the nodal displacements U and F·Φ is the nodal forces vector (it includes dead loads, live loads, wind, etc.). By applying perturbation techniques to this equation it is possible to quantify the mean structural response and its dispersion, in terms of displacements or forces. Finally comparative examples between the results obtained with this technique and other probabilistic methods are presented, allowing to appraise the potentialities of the proposed method.

References [1] J. Eibl and B. Schmidt-Hurtienne, General outline of a new safety format. New developments in non-linear analysis method, CEB Bulletin d’Information, 229, 33-48, 1991. [2] A. Haldar and S. Mahadevan, Probability, Reliability and Statistical Methods in Engineering Design. John Wiley & Sons, New York, 2000. [3] E. Altus, E. Totry and S. Givli, Optimized functional perturbation method and morphology based effective properties of randomly heterogeneous beams. Int. J. Solids Struct., 42, 2345-2359, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Semi-Analytical Analysis of Super Tall Building Bundled-Tube Structures Gong Yaoqing *, Li Ke *

Civil Engineering School of Henan Polytechnic University, Henan Jiaozuo, 454000, China [email protected]

ABSTRACT A new semi-analytical method is developed for the analysis of interactions between the subgrade and the foundation and the superstructure of the super tall building bundled-tube structure by threedimensional model which is a combination of stiffened-thin-wall tubes on semi-infinite elastic body. The subgrade is idealized as a semi-infinite elastic body, and the rigidities of the elastic body pertinent to various deformations of the foundation have been expressed as analytical equations [1], with which the reactions between the foundation and subgrade can be quantified expediently. The foundation is taken as a part of the superstructure. In fact, the foundation is the extension of the superstructure toward the underground. The only difference is the size, since in most cases the foundation must be large enough to make the soil stable. The superstructure and its foundation of the super tall building bundled-tube structure are simplified equivalently and continuously to a combination of stiffened-thin-wall tubes on semiinfinite elastic subgrade. Then discretization is made by some nodal lines, the unknown functions defined on the lines are used as primary unknowns, and interpolating functions are implemented between the lines. So the displacement field of the computing model can be expressed by the unknown functions. After using the principle of minimum potential energy, the governing equations will then be obtained, which is actually a group of ordinary differential equations. Therefore, analysis of a tall building structure will be changed into the solution of the boundary problem of a group of ordinary differential equations that can be solved by the precise and powerful Ordinary Differential Equation Solver—COLSYS [2] , a kind of computational software. The interactions between the subgrade and the foundation and the superstructure of a super tall building bundled-tube structure due to static loadings are analyzed by the method based on the model. The numerical results show that the analytical model is reasonable and feasible. Therefore, a practicable method for the global analysis of the super tall building bundled-tube structure is obtained, and some valuable conclusions are acquired through analyzing the computing results as well.

References [1] Gong Yaoqing, Tall Building Structures on Elastic Subgrade and Research of Semi-Analytical Method [D]. Beijing: Tsinghua University, 1999 (in Chinese). [2] Yuan Si, Introduction of a common compute program for boundary problems of ordinary differential equation—COLSYS [J]. Computational Mechanics and Application, 2, 104-105, 1990 (in Chinese).

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Large Displacements in Nonlinear Numerical Analyses for Cable Structures Nikolina Zivaljic *, Ante Mihanovic †, Boris Trogrlic † *

Faculty of Civil Engineering and Architecture, University of Split Matice hrvatske 15, 21 000 Split, Croatia Nikolina.Zivaljic@ gradst.hr

†

Faculty of Civil Engineering and Architecture, University of Split Matice hrvatske 15, 21 000 Split, Croatia [email protected] Boris.Trogrlic@ gradst.hr

ABSTRACT Method for defining appropriate form of prestressed, tensile cable structures and for calculating stress and displacements for such structures is presented. The developed numerical model is taking into account the material and geometric nonlinearity. The described model represents a practical way of implementing the large displacements theory in the analysis of finding appropriate form of prestressed cable structures. The behavior of the structure under an increasing load, from zero up to final is described. The load usually applied in two phases. The first phase can be prestressing. In the second phase, the structure is computed taking into account the dead and the live gravity load. An approach to solving the problem of large displacements in the theory of structures is presented, based on an incremental approach of the Total Lagrange formulation with the small displacements. The model is based on the assumption that the FE are linear and small enough and thus tracking of large translational displacements can be approximated by a simple geometrical model. The resulting force, i.e. stress, inside FE are expressed within the large displacements, based on the successive approach of small displacements of each increment, using a singular quasi-tangent stiffness matrix. A solution of renewable of the internal forces and stress and their influence is presented. The renewal of the large translational displacements is based on their vector from increment to increment. The renewal of the geometry configuration is influenced to the renewal of basic stiffness, geometrical stiffness and large displacement stiffness. Spatial discretization of the system is on two-node line elements. The fiber discretization of the cross section is on triangular elements where mechanical properties of each fiber are presented by the V  H diagram. The numerical nonlinear material model is based upon nonlinear material properties defined in the form of a uniaxial V  H diagram. The developed model was tested on few practical examples. They are compared by the research computations of the other authors.

References [1] Dvornik J, Lazarevic D. Fractals and formfinding – magic with real numbers. International journal for Engineering Modelling 16, 1-11, 2003. [2] Tabarrok B, Qin Z. Nonlinear Analysis of tension structures, Computers and Structures; 45, 973984, 1992.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Mathematical Modeling for Synthesis and Design of Non-orthogonal Worm Gears with a Straight-line Tooth Contact V. Abadjiev*, D. Petrova†, E. Abadjieva† *

Institute of Mechanics, Bulgarian Academy of Science Acad. G. Bonchev Str., block 4, Sofia 1113, Bulgaria [email protected]

ABSTRACT The development of the different kinds of transmissions in recent decades has pushed the processes related with the construction of electric transmissions when a preliminary given law of motions transformation is realized by applying a suitable electronic control. Created in this period field of mechatronics covers in particular the most of the mentioned mechanisms and its scientific achievements may be applied to them. Independently of the intensive development of this new part of the techniques, the practice shows that when constructing transmissions oriented to transform big powers, the power mechanical transmission have not found their alternatives. This circumstance motivates the researchers in their trials to create new types of mechanical gears and/or to improve the existing ones in order to find out their new exploitation qualities [1]. The presented research aims at defining of an adequate mathematical model and constructing of a computer program for insuring the process of constructive and technological synthesis and of the design of non-orthogonal worm gears of type Wildhaber. This class of hyperbolic gears is synthesized according to the second Olivier’s principle. The mentioned technological characteristic and the specific geometry of the active tooth surfaces of the gear pair allow this type of mechanical gear sets to be applied alike as power transmission as a kinematic one. The mathematical modeling for synthesis of worm gears of type Wildhaber is based on an approach upon “the region of mesh” [2]. Together with the gear pair design the worked out algorithm realizes a control of the quality of meshing in the whole mesh region or in its definite parts. This control consists of: defining of an optimal configuration of the region of mesh, and of suitable dimensions of the active tooth surfaces; a registration and a limitation of the singular points on the meshed tooth surfaces; a choice of a calculated variant of a gear pair with an optimum situation of the contact lines in the of mesh region from a point of view of hydrodynamic lubrication and related with it hydrodynamic loading capacity, etc. The computer program is directed to studying real models in the design process. Patterns having concrete practical applications are illustrated in the paper.

References [1] Litvin F. Gearing Geometry and Applied Theory. PTR Prentice Hall, A Paramount Communication Company, Englewood Eliffs, New Jarsy, 1994 [2] Abadjiev V. Mathematical Modelling for Synthesis of Spatial Gears. Journal of Process Mechanical Engineering. Proc Inst Mech Engrs, Vol. 216, Part E, 31-46, 2002

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Computational modeling of ultrasonically assisted turning Naseer Ahmed*, Alexander V. Mitrofanov†, Vadim V. Silberschmidt††, Vladimir I. Babitsky††† Wolfson School of Mechanical and Manufacturing Engineering, Loughborough University, UK *[email protected] † [email protected] †† [email protected] ††† [email protected]

ABSTRACT Ultrasonically assisted turning (UAT) is an advanced machining technique, where high frequency vibration (frequency f | 20 kHz, amplitude a | 15 Pm) is superimposed on the movement of the cutting tool. Compared to conventional turning (CT), this technique allows significant improvements in processing intractable materials, such as high-strength aerospace alloys, composites and ceramics. Superimposed ultrasonic vibration yields a noticeable decrease in cutting forces, as well as a superior surface finish [1]. The paper presents a three-dimensional thermomechanically-coupled finite element (FE) model of both UAT and CT that was recently developed as an extension of the initial 2D model [2]. The current model enables studies of various 3D effects in turning, such as oblique chip formation, as well as the influence of the tool geometry on process parameters, e.g. cutting forces and stresses generated in the workpiece material. The model allows transient, coupled thermomechanical simulations for elasto-plastic materials with strain-rate sensitivity. Chip shapes and forces acting on the cutting tool are analyzed. Stress, strain and temperature distributions in the cutting zone are studied. The effects of cutting parameters (such as the feed rate) and influence of friction on both UAT and CT are investigated. Numerical results are validated by the experimental tests performed at our in-house UAT prototype.

References [1] Babitsky, V.I., A. Kalashnikov, A. Meadows, and A. Wijesundara, Ultrasonically assisted turning of aviation materials. Journal of Materials Processing Technology, 132, 157-167, 2003. [2] Mitrofanov, A.V., V.I. Babitsky, and V.V. Silberschmidt, Thermomechanical finite element simulations of ultrasonically assisted turning. Computational Material Sciences, 32, 463-471, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Numerical evaluation of bored piles in tropical soils by means of the geotechnical engineering “GEO4” Fine Software Anjos, G.J.M.*, Cunha, R.P.†, Kuklik, P.‡, Miroslav, B.‡ *

Federal University of Pará, Belém, Brazil [email protected] †

University of Brasília, Brasília, Brazil [email protected]

‡

Czech Technical University, Prague, Czech Republic [email protected]; [email protected]

ABSTRACT This paper presents the back-analyses of field loading tests carried out with bored pile and drilled shaft founded in a tropical soil executed in the University of Brasília experimental research site. For this, a numerical simulation was carried via existing commercial application software denominated GEO4. This software computes the load-displacement curve of the pile’s head plus, distribution of normal and shear forces along the pile’s shaft. The Shear behavior of pile-soil interface is described using the elastic-plastic material model with Mohr-Coulomb yield condition. The complete response of any foundations is represented by determination of shaft and toe resistance plus settlement analyses. Hence, this paper focused in the determination of components of resistance (angle friction and cohesion) and settlement (Young Modulus) for this type of foundation, and explains and presents, in details, the software GEO4 from Fine Inc. Ltd. for foundation design. In relation to the cohesion, it was verified the important effect that this parameter have in the determination of shaft resistance. Moreover, few research topics nowadays deal with the determination of this particular parameter for bored piles. The assessment of geotechnical parameter is a vital component of geotechnical design and some formulation are also presented for this evaluation.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Kinematics and force interaction of screw shaft with variable screw course Bahadirov Gayrat Atahanovich*, Bahadirov Kudrat Gayratovich1 *700143,Uzbekistan, Tashkent, Akademgorodok, F.Khodjaev str. 31, Institute of mechanics and seismic stability of structures Uzbek Academy of sciences [email protected] 1

700100, Uzbekistan, Tashkent, Shahjahan str. 5, Institute of textile and light industry [email protected]

ABSTRACT The screw shafts with spiral blade are using in practice are not satisfied to modern demands in industry. For effective smoothing of sheet materials’ creases are necessary uniformly and simultaneously to smooth creases by screw blades from middle site toward ends of the shaft. For achieving of this goal the new construction of screw smoothing blade shaft are developed with so property like blade divergence from middle site toward ends of the shaft with variable (increasing) course. In this construction the next step are compiled by previous via multiplying it on the coefficient which taking into account a delay (sliding) of screw blade in direction of rotation axis of smoothing shaft. The laws of rotation movement of blades’ sides and contact surfaces of screw blade of the shaft with processed of sheet material are obtained. The equation of a curve line of a side of the screw blade is investigated and received too. The smoothing speed of creases and the blade’s speed of a rather sheet material is determined, also are determined it normal, tangent and complete acceleration. Radius of curvature of a side of the screw blade of shaft is determined. The mathematical interrelation of step changes of screw blade, velocity of contact site, and radius of curvature are constructed by depending on angle of shaft’s turn. At mechanical processing of sheet materials by taking into account technological demands directions and volume of acted forces are selected. The force interaction of the variable course blade and processed sheet material with taking into account resistance forces of processed material and variety of screw course, friction forces which arising in contact point of a blade’s surfaces and sides, axial forces of reaction of a support of the shaft, angle of ascent and friction of the screw blade are considered. The torque moment necessary for rotation of one blade and whole shaft is obtained in this work. The driving rate of dislocation of a contact point of a sheet material with the blade of the screw shaft are presented as relation of dislocation of a blade’s screw shaft around of an axis to the appropriate moving of a contact point of the blade on an rotation axis. The numerical solutions’ of equations are compiled by universal software MATHCAD for mathematical calculations with different variations of system’s parameters. Graphics allowing consider kinematical process occurring between blades and processed material are developed on the base of the obtained results. Obtained results used at designing of screw shaft with variable step.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Optimal construction of the thermo-elastic actuator J. Barglik1, B. Ulrych2 1 Silesian University of Technology Krasiēskiego 8, 40-019 Katowice, Poland [email protected] 2

University of West Bohemia Universitní 26, 306 14 Plzeė, Czech Republic [email protected]

ABSTRACT Electro-mechanical actuators seem to be one of devices which produce mechanical forces as a result of eddy-currents influence. The most optimal of them are electromagnetic actuators constructed as ferromagnetic devices and thermo-elastic actuators constructed as bimetallic or mono-metallic are shown in Fig. 1.

Figure 1: Arrangement of the thermo-elastic actuator and distribution of electromagnetic field: 1– dilatation element, 2 – field coil, 3 – sleeve, 4 – ring, 5 – frame, 6 – body The ferromagnetic cylinder 1,that plays the role of dilatation element, is clamped to non-ferromagnetic ring 4. The ring is pressed in a sleeve 3, which is fixed to frame 5 of the tool machine. A cylindrical coil 2 is replaced between the ring 4 and dilatation element 1. The coil supplied with harmonic current of effective value I ext and frequency f, generates time variable electromagnetic field. Due to consequent induction heating the temperature of dilatation element 1 rapidly rises. As a result of temperature rise element 1 is pushed to the body 6 with a contact force FC. This kind of the actuator is characterized by a relatively big dispersion of magnetic field. The paper deals with analysis of possibilities of construction optimization of the device in order to minimize losses of electromagnetic energy and consequently to increase total efficiency and to match electromagnetic compatibility requirements. The aim is to find the construction of the actuator making possible to use practically all the electromagnetic energy for induction heating of the dilatation element. The task formulated as a weakly- coupled electromagnetic – temperature - heat stress problem is solved by means of FEMbased numerical method. The results for illustrative example are presented and discussed.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Determination of Moment-Curvature Diagrams and Moment-Deflection Curves in Reinforced Concrete Beams M. H. F. M. Barros*, C. Oliveira† * Civil Engineering Department, FCTUC Polo II Pinhal de Marrocos, 3030 Coimbra Portugal [email protected] †

Civil Engineering Department, FCTUC Polo II Pinhal de Marrocos, 3030 Coimbra Portugal [email protected]

ABSTRACT The purpose of this paper is to establish an automated process in order to swiftly calculate momentcurvature and moment-deflection diagrams. Comparisons can, this way, be established between several compressed concrete behaviour theories. The model is implemented into a mathematical manipulation program. The consideration of concrete tensile stress-strain relations in structural analysis if often neglected, which leads to the consideration of a far different stiffness in analysis. Considering the tensile stress-strain model referred in Bazant et al[1], together with different compressive stress-strain models, a good comparison of the predicted theoretical behaviour of the several models is obtained

References [1] Deformation of Progressively Cracking Reinforced Beams – ZdenƟk P. Bazant and Byung H. OH

[2] – Eurocode 2: Design of Concrete Structures – CEN – European Committee for Standartization

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

135

Numerical simulation of the nanoindentation experiment: sensitivity analysis of the experimental parameters P. Berke∗ and T. J. Massart† Universit´e Libre de Bruxelles Structural and Material Computational Mechanics Department CP 194/5 avenue Roosevelt 50, B-1050 BRUSSELS (Belgium) ∗ [email protected] † [email protected]

ABSTRACT The use of certain metallic materials in micro-mechanical systems applications is promising for chirurgical applications because of their bio-compatibility and interesting mechanical and wear properties compared to the widely used silicon. The reliability of miniaturized components, the building blocks of such systems depends largely upon the reliability of the techniques applied to characterize the materials, in relation with numerical simulations. Nanoindentation is the method adapted to investigate the local mechanical properties of materials at the nanoscale. The inter-disciplinary nature of such an experiment makes the interpretation of the results difficult. The goal of the research is the use of a relatively simple but flexible computational tool for the simulation of the nanoindentation experiment in order to better understand the physics and the mechanics involved. A finite element code therefore has been developed and used to solve the simulation problem with all the non-linearities involved (finite deformations, plasticity, contact evolution), including isotropic plastic behavior with hardening and an accurate computational contact mechanics feature using the augmented Lagrangian scheme. These tools allow to investigate the most significant sources of dispersion in nanoindentation experiments and their influence. This analysis helps to fix the range of given experimental parameters for which the sensitivity of experimental results is important. The numerical simulation tool allows a parametric study to quantify the effects of some of these experimental conditions, such as the tip geometry and the surface roughness.

References [1] P.Wriggers, Computational Contact Mechanics. John Wiley and Sons Ltd, ISBN 0-471-49680-4, 2002 [2] T.Belytschko, W.Kam Liu, B.Moran, Nonlinear Finite Elements for Continua and Structures. John Wiley and Sons Ltd, ISBN 0-471-98773-5, 2000 [3] J.P.Ponthot, Unified stress update algorithms for the numerical simulation of large deformation elasto-plastic and elasto-viscoplastic processes, Int.J.Plas., 18, 91-126, 2002

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Numerical evaluation of wrinkling stress in sandwich panels Monika Chuda-Kowalska, Andrzej Garstecki and Zbigniew Pozorski Poznan University of Technology ul. Piotrowo 5, 60-965 Poznan [email protected]

ABSTRACT Sandwich panels have been for years used as structural and cladding elements. Characteristic feature of these structures is that important role is played by temperature actions, creep of the core and local instability of thin faces of the panel [2]. Proper estimation of these phenomena has become a challenging issue because of strong tendency to optimize technical parameters and costs. The aim of the study is numerical analysis of bending of three-layered panels. In the classical approach linear constitutive equations for faces and core materials are assumed. Moreover, identical tension and compression elasticity modules of the core are introduced [1]. For the plate loaded by axial force P and uniform transversal load q, the following differential equilibrium equation is used: Bw IV + Pw′′ + cw = − q , where w denotes deformation form of compressed face and c is a stiffness coefficient of the core treated as a Winkler foundation. The term B represents the bending stiffness of the plate. In practice, wrinkling stress depends on many more factors, neglected in analytical solutions, though observed in experiments [3]. In this paper we use numerical methods and hence we can allow for the loss of face adhesion and anisotropy of the core. The analysis is carried out for various dimensions (span and depth of the panel and face thickness) and for various material parameters. By the way of the parametric analysis, the sensitivity of structural response to variations of dimensional and material parameters will be studied. The range of applicability of classical theoretical models will be discussed basing on numerical examples. The study presented in the paper was inspired by one of the biggest in the world producers of sandwich panels, with the aim to increase safety and economy.

References [1] K. Stamm, H. Witte, Sandwichkonstruktionen. Berechnung, Fertigung, Ausführung (in German). Springer-Verlag, Wien, Austria, 1974. [2] D. W. Sleight, J. T. Wang, Buckling analysis of debonded sandwich panel under compression. NASA Technical Memorandum 4701, Langley Research Center, Hampton, Virginia, 1995. [3] O. T. Thomsen, Y. Frostig, Localized bending effects in sandwich beams: photoelastic investigations versus high-order sandwich theory results. Composite Structures, 37, 97-108, 1997.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

137

Application of FEA as a Predictive Tool in the Corrugated Paperboard Industry 

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III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

138

Implementation of 3D homogenization techniques for the thermo-elastic FEM analysis of brazed plate-fin heat exchangers J. Dib†,*, F. Bilteryst* , J.L. Batoz* , I. Lewon† †

Nordon-Cryogenie 25 bis, rue du Fort, BP 87 88194 Golbey, France [email protected] * ERMeP Institut Supérieur d’Ingénierie de la Conception (GIP-InSIC) 27 rue d’Hellieule, 88100 Saint-Dié-des-Vosges, France [email protected]

ABSTRACT The present study results from a research collaboration between Laboratory ERMeP (GIP-InSIC) and the company Nordon-Cryogenie (Vosges, France), one of the major world manufacturers of heat exchangers for cryogenic processes. A general description of a multi-stream brazed aluminium plate-fin heat exchanger is presented in Fig 1. The problem for Nordon Cryogénie is to guarantee a thermal performance as well as the safety of the heat exchanger by ensuring the structural integrity of each stream subjected to pressure and temperature gradients. The current research program consists in the development of a dedicated FEM solver for the global thermo-elastic analysis of an exchanger. Since a complete FEM model would lead to several millions of structural elements (solid, shell) we propose a simplified 3D FEM model based on homogeneization techniques to obtain the equivalent (effective) stress-strain relations and equivalent thermal load vectors of the corrugated fins brazed with the plate layers (Fig. 1). Two approaches are considered (kinematical and mechanical models) while considering periodicity. The implementation of these two methods is performed using the free finite element software Code_Aster.

Dib J., Bilteryst F., Batoz J.-L., Lewon I. : Application des techniques d’homogénéisation pour l’analyse tridimensionnelle d’échangeurs de chaleur à plaques et ondes. Actes du 7ème Colloque National en Calcul des Structures, Giens 2005, Vol.1, 417-422, Lavoisier, France, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

139

Hierarchical treecode for optimized collision checking in DEM simulations – application on electrophotographic toner simulations Rainer Hoffmann* *

Océ Printing Systems Siemensallee 2, 85581 Poing, Germany [email protected]

ABSTRACT Discrete element modelling (DEM) according to Cundall and Strack [1] of a large set of particles with long-ranging forces like gravitation or electrostatics has the disadvantage of a computation time dependence on the number of particles of O(N²). This can be overcome by the usage of a hierarchical tree code [2] which groups particles which are far away to virtual pseudoparticles. This reduces the number of force calculations so that the computation time dependence can be reduced to O(N log N). The criterion which determines a possible grouping of particles is the theta parameter which is a measure for the reciprocal distance of the particle of interest to the neighboring particles. The paper here shows that the tree algorithm can be also used for an efficient collision checking routine since particles that the algorithm determines to be far enough from the particle of interest can certainly be excluded for a collision checking. It is shown, however, that for a particle set with a uniform radius distribution an upper limit for the theta parameter exists. When this upper limit is exceeded collisions will be suppressed or artificial collisions will occur so that the simulation result is severely falsified. This limit can be overcome by extending the theta parameter to include the radius of the particle of interest. This allows the theta parameter to be increased significantly over the previously found limit, thus reducing the computation time by a further 30 % without introducing much additional error. The prerequisite for such a high theta parameter is that the simulated particle set is rather densely packed (packing density > 10%) so that the total behaviour is dominated by collisions not by the long-ranging forces. The algorithm is applied to the simulation of electrostatically charged toner particles used for the electrophotographic print process. To provide a test case for the simulation a simple transfer experiment is chosen: A roller covered thickly with charged toner is positioned next to a second roller with a thin air gap between them. An external voltage is applied to the rollers causing the particles to jump to the second roller. The results of this experiment can be easily measured and the comparison with the simulation shows an error below 5%.

References [1] P. Cundall and O. Strack. A discrete numerical model for granular assemblies. Geotechnique, 29, 47-65, 1979 [2] Joshua Barnes and Piet Hut. A hierarchical o(n log n) force calculation algorithm. Nature, 324, 1986

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

140

3D FEM analysis of basic process parameters in rotary piercing mill *

†

Jan Kazanecki , Zbigniew Pater , Jarosław Bartnicki *

†

AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Cracow, Poland [email protected] †

Department of Computer Modelling and Technology of Metal Forming, The Lublin University of Technology, 36,Nadbystrzycka, 20-618 Lublin, Poland [email protected] [email protected]

ABSTRACT In this paper the 3D FEM analysis of basic process parameters in rotary piercing mill is presented. In this process the material is formed by means of two skew rolls, two Diescher’s discs and a plug. The material is dragged by the rolls, it moves axially forward and rotates. The FEM analyze of the rotary piercing process was made under the conditions of 3D state of strain with taking into consideration the thermal phenomena. The calculations were made by means of commercial software MSC.SuperFORM2004 with application of different rolls’ skew angles, different plug designs and working positions. All the mentioned above variants were calculated at a different area reduction ratio. In the result, the progression of shapes, temperature and distributions of stress and strain in various sections of the analyzed tubes were characterized. The numerical results of calculations were justified during the stand test with the use of 100Cr6 steel. The comparisons of the numerical and experimental tests confirm good agreement between the obtained results.

References [1] J. Kazanecki: The seamless tube manufacturing, AGH Kraków 2003, p. 1-622 [2] Z.Malinowski, J. Kazanecki, S. Urba ski: Thermal – mechanical model of tube elongation process in Diescher’s mill, Journal of Material Processing Technology 60, 1996, p. 513-516

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

141

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Spectral element simulations of rupture dynamics along planar and kinked frictional faults Gaetano Festa1 and Jean-Pierre Vilotte1 1 Institut

de Physique du Globe de Paris 4, Place Jussieu, 75252 Paris Cedex 05, France [email protected],[email protected]

ABSTRACT Understanding earthquake source dynamics is a major topic in seismology. Earthquake faulting is mainly controlled by non-regular friction, describing the dissipation within the fault interface, by the geometrical complexity of the fault and by the off-fault damage interactions. Recent observations during the Denali and Izmit earthquakes have shed evidence for supershear propagation in relation with the fault geometry. Numerical simulations of earthquake rupturing can bridge the gap between laboratory experiments and observations of large earthquakes. The simulations are expected to capture the different space and time scales involved in the nucleation phase, the rutpure front propagation and the short wave radiation, owing to the fault heterogeneities and geometrical complexities. Non-smooth spectral element method allows for efficient simulations of dynamic rupture along planar and non-planar faults. We present here recent numerical results on sub- and supershear propagation along bending and kinked faults. In particular, we focus the attention on the subshear-to-supershear transition and on the interaction between fault geometry and high-frequency radiation emitted by the fault. This work is a part of the SPICE European Project

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r-Factor Strategies for the Augmented Lagrangian Approach in Multi-Body Contact Mechanics Martin Foerg∗ , Thomas Geier∗ , Lutz Neumann∗ , Heinz Ulbrich∗ ∗ Institute

for Applied Mechanics Technical University of Munich Boltzmannstr. 15 85748 Munich, Garching [email protected] ABSTRACT

Multi-body system theory including unilateral constraints is now well established by means of setvalued force laws in connection with measure equations of motion. The crucial point consists in the numerical solution of such equations, especially when dealing with large systems as they often appear in practical problems of industrial relevance. Therefore the improvement of numerical algorithms is a focus of ongoing research. In the meantime there are different approaches and algorithms in order to formulate and compute unilateral constrained mechanical systems. Besides (N)LCP-formulations the Augmented Lagrangian approach [1] becomes more and more popular in contact mechanics. Within this approach the equations of motion are augmented by projection equations representing the physical constraints. The overall set of non-smooth, nonlinear equations can be solved by a root-finding algorithm, e.g. a fixed-point iteration scheme. The projection equations depend on a non-negative auxiliar parameter r. Though this parameter r is arbitrary from the mathematical point of view, it plays a crucial role in view of the rootfinding method. In particular, the problem of finding an optimal r-factor turns out to be a constrained optimization problem: on the one hand the parameter r may be bounded to ensure the convergence of the algorithm, on the other hand an appropriate choice improves the rate of convergence. In the present paper two different r-factor strategies are presented considering a fixed-point iteration scheme in order to find the root of the Augmented Lagrangian. The first strategy proposes one global r-factor for all constraint equations. The second strategy considers local r-factors, i.e. a different parameter for each constraint. In both cases the conditions for convergence are given and an optimal choice of r is proposed. The paper discusses the treatment of planar and spatial contacts as well as systems that are statically indeterminate, where a unique solution for the constraint forces does not exist. The presented r-factor strategies are applied to several non-smooth systems including a push belt CVT [2]. This large industrial problem is characterized by a hybrid multi-body model with a large number of unilateral and bilateral constraints.

References [1] A. Klabring, Mathematical Programming and Augmented Lagrangian Methods for Frictional Contact Problems. Proceedings Contact Mechanics International Symposium, October 7-9, EPFL, Lausanne, Switzerland, 409–422, 1992. [2] T. Geier, M. Foerg, R. Zander, H. Ulbrich, F. Pfeiffer, A. Brandsma, A. Van der Velde Modeling of contacts in a push belt CVT. Second International Conference on Nonsmooth/Nonconvex Mechanics with Applications in Engineering, Thessaloniki, Greece, 2006 (to appear)

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A simple smoothing procedure of 3D surfaces for accurate contact analysis: application to metal forming problems. L. Fourment*, S. Guerdoux* *

CEMEF, Ecole des Mines de Paris, UMR CNRS n°7635 rue Claude Daunesse, B.P. 207, 06 904 Sophia Antipolis Cedex, France [email protected]

ABSTRACT In most finite element software, it is very convenient to discretise complex obstacles with segments in 2D and facets in 3D. It results in C0 continuity of the obstacle and consequent discontinuity of its normal, which is at the origin of convergence difficulties of algorithms, of numerical oscillations with exaggerated stresses or unjustified loss of contact. Several methods have been proposed in literature to smooth the contact obstacle. They are based on Bezier surfaces [1], Gregory patches [2], or similar methods, which are not easy to implement into an existing finite element code. In [3] a more simple method was proposed for concave parts of the obstacle. For a point M sliding along the obstacle surface, there is a discontinuity of the normal between two segments (or two facets). If we allow a small penetration of M inside the obstacle, as in the following figure (left), and if the normal is defined by the direction MP where P is the orthogonal projection of M onto the obstacle, then the normal varies continuously when M slides on the obstacle surface; the obtained smoothing is proportional to the authorized penetration. In order to avoid this numerical penetration, the discretised contact obstacle is first shifted in the opposite direction, as shown in the following figure (right), with the same amplitude. This way, when M “penetrates” the shifted surface, it actually moves along the actual contact surface, except in the smoothing area.

Symmetrically, this method can be applied to convex parts of the surface. The focus of this paper is on how to combine the smoothing of concave and convex parts. It actually requires shifting the obstacle discretisation in two opposite directions and carrying out a contact analysis with each. It so provides two possible normals, which combination gives an almost continuously varying normal on the contact surface. The method is applied to some metal forming problems with large deformations, such as machining, forging and friction stir welding.

References [1] L. Krstulovi-Opara, P. Wriggers, J. Korelec, A C1-continuous formulation for 3D finite deformation frictional contact, Comput. Mech. 29 n° 1, (2002) 27-42. [2] M. A. Puso, T. A. Laursen, A 3D contact smoothing method using Gregory patches, Int.J.for Numerical Methods in Engineering 54 n° 8, (2002) 1161-1194. [3] L. Fourment, J. L. Chenot, K. Mocellin, Numerical formulations and algorithms for solving contact problems in metal forming simulation, Int J Numer Methods Eng 46 n° 9, (1999) 14351462.

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Symmetry preserving algorithm for a dynamic contact-impact problem Duˇsan Gabriel∗ , Jiˇr´ı Pleˇsek∗ , Frantiˇsek Valeˇs† , and Miloslav Okrouhl´ık∗ ∗ Institute of Thermomechanics Academy of Sciences, Praha, Czech Republic [email protected], [email protected], [email protected] † Institute of Thermomechanics Academy of Sciences, Plzen, Czech Republic [email protected]

ABSTRACT In the finite element method, the contact constraints can be introduced either before or after the finite element discretization has been performed, leading to the so-called pre-discretization or postdiscretization techniques [1]. In the paper [2] we focused on the pre-discretization approach, showing this technique to lead naturally to the use of surface integration points as contactors. It was shown that the proposed method preserved the symmetry of the algorithmic approximation with respect to contact boundaries. On the outcome there was nothing like a master or slave definition of contact surface. In this work the expected symmetry of algorithm was tested. The observation or loss of symmetry is clearly manifested in numerical step-by-step procedures, for instance, in the direct integration methods for the equations of motion. Three-dimensional face to face impact of two cylinders was considered as in the Taylor test. Comparisons of the proposed algorithm with the analytical solution and the finite element code MARC were made. The analytical solution of this problem using the Laplace transform is quite complicated [3]. The distribution of displacements and stresses are expressed in the form of an infinite series of improper integrals which are evaluated numerically. A good agreement between the numerical and analytical solution derived for the summation of the first 150 terms of the series was observed. Next, the results were compared to the output of the finite element code MARC, in which the implemented contact algorithm [4] was based on the node-to-segment procedure. The numerical results of the proposed method and MARC s computation, with the central difference scheme, were almost the same. However, a lack of symmetry in all MARC s kinematic and stress quantities was observed if the implicit Newmark integration method had been used. It should be emphasized that the symmetry is perfectly preserved in the proposed algorithm.

References [1] N. Kikuchi, J.T. Oden, Contact problems in elasticity: A study of variational inequalities and finite element methods. SIAM, Philadelphia, 1988. [2] D. Gabriel, J. Plesek, M. Ulbin, Symmetry preserving algorithm for large displacement frictionless contact by the pre-discretization penalty method. Int. J. Num. Met. Engng, 61, 2615–2638, 2004. [3] F. Vales, S. Mor·avka, R. Brepta, J. Cerv, Wave propagation in a thick cylindrical bar due to longitudinal impact. JSME Int. J., Ser.A, 39(1), 60–70, 1996. [4] MARC Analysis Research Corporation. Volume A, Theory and User Information, version K7.3, 1998.

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Problems of Concentrated Loads in Microstructured Solids Characterized by Dipolar Gradient Elasticity H.G. Georgiadis * , D.S. Anagnostou Mechanics Division, National Technical University of Athens, GR-15773, Greece (* Corresponding author) [email protected]

ABSTRACT This work studies the response of bodies governed by dipolar gradient elasticity to concentrated loads. The use of the theory of gradient elasticity is intended here to model material microstructure and incorporate size effects into stress analysis in a manner that the classical theory cannot afford. A simple but yet rigorous version of the generalized elasticity theories of Toupin [1] and Mindlin [2] is employed that involves an isotropic linear response and only one material constant (the so-called gradient coefficient) additional to the standard Lame constants. This theory, which can be viewed as a first-step extension of the classical elasticity theory, assumes a strain-energy density function, which besides its dependence upon the standard strain terms, depends also on strain gradients [3]. Twodimensional configurations in the form of either a half-space (Flamant-Boussinesq type problem) or a full-space (Kelvin type problem) are treated and the concentrated loads are taken as line forces. The problems enjoy important applications in various areas, e.g., in Contact Mechanics and Tribology. Also, the Flamant-Boussinesq and Kelvin solutions serve as pertinent Green’s functions in a multitude of problems analyzed by the Boundary Element Method. Our main concern here is to determine possible deviations from the predictions of classical linear elastostatics when a more refined theory is employed to attack the problems. Of special importance is the behavior of the new solutions near to the point of application of the loads where pathological singularities exist in the classical solutions. The solution method is based on integral transforms and is exact. The present results show departure from the ones of the classical elasticity solutions. Indeed, bounded displacements are predicted even at the points of application of the loads. Such a behavior of the displacement fields seems to be more natural than the singular behavior present in the classical solutions. Acknowledgment: This paper is a partial result of the Project PYTHAGORAS II / EPEAEK II (Operational Programme for Educational and Vocational Training II) [Title of the individual program: “Micro-mechanics of contacts and diffusion of humidity in granular geomaterials”]. This Project is co-funded by the European Social Fund (75%) of the European Union and by National Resources (25%) of the Greek Ministry of Education.

References [1] R.A. Toupin, Elastic materials with couple-stresses. Arch. Rational Mech. Anal., 11, 385-414, 1962. [2] R.D. Mindlin, Micro-structure in linear elasticity. Arch. Rational Mech. Anal., 16, 51-78, 1964. [3] H.G. Georgiadis, I. Vardoulakis and E.G. Velgaki. Dispersive Rayleigh-wave propagation in microstructured solids characterized by dipolar gradient elasticity, J. Elasticity, 74, 17-45, 2004.

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An Experimentally Validated Model for Unsteady Rolling F. Gutzeit∗ , M. Wangenheim∗ , and M. Kr¨oger∗ ∗ Institute of Dynamics and Vibrations Appelstr. 11, 30167 Hannover, Germany [email protected]

ABSTRACT The description of the tyre road contact is important for brake or drive stability systems working in modern vehicles. Steady models loose accuracy, if slip or normal force contain higher frequencies. This is the case during an ABS brake process. At the Institute of Dynamics and Vibrations, a mobile test rig has been built up to investigate unsteady rolling. A small solid rubber wheel, which is applied for wear experiments in the tyre industry, serves as the specimen. Any given time characteristics of slip and normal force can be realized by the control modules. The forces and moments, as well as the temperatures of the wheel and the road are recorded. Based on the experiments, a numerically efficient model for the unsteady rolling contact was developed, see [1]. The mechanical model describes the dependency on the slip ν(t) for constant normal force FN . The numerical efficiency is achieved by applying a modal condensed formulation of the wheel structure. Furthermore, the contact area is discretized by point contact elements representing the local contact behavior. The model will be revised in order to be able to include time depending normal forces FN (t). The HurtyCraig-Bampton reduction is applied to access the contact nodes during the simulation. The method allows to condense modally the remaining nodes. The model divides the contact zone into a sticking and a sliding area. Within the sticking area, the displacements of the contact nodes are forced according to the kinematic conditions of the wheel. Within the sliding area, the local friction force is calculated for every node. After deformations and external forces have been calculated, appropriate sticking and sliding conditions are proved to compute the division of the contact patch for the next time step. In many respects, rubber is a highly nonlinear material. The strong temperature dependency must be considered in many applications. By means of an infrared camera, thermal images of the contact zone are taken. An analytical approach is adapted to the case of a solid rubber wheel and finally compared to the experimental data. The temperature dependence on the rolling contact behavior is discussed. The results of the modified model are shown and compared to experimental data recorded by the friction robot. The influence of the global slip on the sticking to sliding transition is shown.

References [1] Gutzeit, F., Sextro W., Kr¨oger M.: Unsteady Rolling Contact of Rubber Wheels, 4th Contact Mechanics International Symposium (CMIS), Hannover, Germany, 2005, in press.

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A primal-dual active set strategy for unilateral non-linear dynamic contact problems of thin-walled structures Stefan Hartmann ∗, Stephan Brunssen†, Ekkehard Ramm∗ , Barbara Wohlmuth† ∗ Institute of Structural Mechanics University of Stuttgart, Pfaffenwaldring 7, D-70550 Stuttgart, Germany hartmann/[email protected]

† Institut f¨ur Angewandte Analysis und Numerische Simulation Universit¨at Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, Germany brunssen/[email protected]

ABSTRACT The efficient modeling of 3D contact problems is still a challenge in non-linear implicit structural analysis. Most of the existing contact algorithms use penalty methods to satisfy the contact constraints, which necessitates a user defined penalty parameter. As it is well known, the choice of this additional parameter is somehow arbitrary, problem dependent and influences the accuracy of the analysis. We use a primal-dual active set strategy [1], based on dual Lagrange multipliers [4] to handle the nonlinearity of the contact conditions. This allows us to enforce the contact constraints in a weak, integral sense without any additional parameter. Due to the biorthogonality condition of the basis functions, the Lagrange multipliers can be locally eliminated. We perform a static condensation to get a reduced system for the displacements. The Lagrange multipliers, representing the contact pressure, can be easily recovered from the displacements in a variationally consistent way. For our application to thin-walled structures we adapt a three-dimensional non-linear shell formulation, including the thickness stretch of the shell to contact problems. A reparametrization of the geometric description of the shell body gives us a surface oriented shell element, which allows to apply the contact conditions directly to nodes lying on the contact surface. The discretization in time is done with the implicit Generalized Energy-Momentum Method [2]. To conserve the total energy within our contact framework, we follow an approach from Laursen and Love [3], who introduce a discrete contact velocity to update the velocity field in a post processing step. Various examples show the good performance of the primal-dual active set strategy applied to the implicit dynamic analysis of thin-walled structures.

References [1] M. Hinterm¨uller, K. Ito, K. Kunisch, The primal-dual active set strategy as a semismooth Newton method. SIAM J. Optim., 13: 865–888, 2003. [2] D. Kuhl, E. Ramm, Generalized Energy-Momentum Method for non-linear adaptive shell dynamics. Computer Methods in Applied Mechanics and Engineering, 178: 343–366, 1999. [3] T.A. Laursen, G.R. Love, Improved implicit integrators for transient impact problems - geometric admissibility within the conserving framework. International Journal for Numerical Methods in Engineering, 53: 245–274, 2002. [4] B. Wohlmuth, A mortar finite element method using dual spaces for the Lagrange multiplier. SIAM J. Numer. Anal., 38: 989–1012, 2000.

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Arc-Length Method for Frictional Contact with a Criterion of Maximum Dissipation of Energy Yoshihiro Kanno∗ , Jo˜ao A.C. Martins† ∗ Department

of Urban and Environmental Engineering, Kyoto University Sakyo, Kyoto 606-8501, Japan [email protected]

† Departamento

de Engenharia Civil and ICIST, Instituto Superior T´ecnico Avenida Rovisco Pais, 1049-001 Lisboa, Portugal [email protected]

ABSTRACT In this paper, we propose an arc-length equilibrium path-following method for quasi-static frictional contact problems incorporating a criterion of maximum dissipation of energy, which is applicable to cases in which there exist critical points along the equilibrium path. The Coulomb friction law and the unilateral contact condition are considered. It is well known that the frictional contact problems may have limit points and successive stable and/or unstable bifurcation points [1], even if small rotations and small strains are assumed. This implies that the corresponding incremental problem does not have unique solution in general. Moreover, this problem often has bifurcated paths such that most sliding contact nodes become stuck and the loading parameter decreases, which are referred to as trivial unloading paths. Our aim is to propose a path-following method that can automatically avoid tracing trivial unloading paths, which seem to be uninteresting from the practical point of view. To this end, we attempt to follow the path with the maximum dissipation of energy when the corresponding incremental problem has some solutions. At each loading stage, the incremental displacements and the reactions are obtained by solving a mathematical program with complementarity constraints (MPEC). Algorithms that do not have any criterion to select among multiple solutions may compute trivial unloading solutions. A regularization scheme of the MPEC is also proposed. In contrast with the fact that the original MPEC fails to satisfy any standard constraint qualification, it is shown that the regularized problem satisfies the linear independence constraint qualification at a feasible solution. This implies that, in the inner iteration of the arc-length method, we can solve the proposed regularized problem by using the conventional nonlinear programming approach. It has been shown in the numerical examples that the proposed method can automatically avoid tracing trivial unloading paths, even when there exist successive bifurcation points due to friction and/or some limit points.

References [1] J.A.C. Martins, A. Pinto da Costa, and F.M.F. Sim˜oes, Some notes on friction and instabilities. In: Friction and Instabilities, J.A.C. Martins and M. Raous (eds.), Springer–Verlag, pp. 65-136, 2002.

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A energy conserving approximation for elastodynamic contact problems H. Khenous∗ , P. Laborde† , Y. Renard∗ ∗ MIP-INSA Complexe scientifique de Rangueil, 31077 Toulouse cedex4, France {khenous, renard}@insa-toulouse.fr † MIP, UPS 118 route de Narbonne, 31062 Toulouse cedex4, France [email protected]

ABSTRACT We are interested in the numerical solution of contact problem in elastodynamics. We present the strong formulation of problem and give the finite element discretization [2]. We prove that the last formulation is not a well posed problem. In order to overcome this difficulty, we propose then an original method based on a redistributed mass matrix. This new mass matrix is assembled conserving the total mass, the gravity center and the momentum inertia. The discrete elastodynamic contact problem expressed with the redistributed mass matrix is well posed, is energy conserving and has a Lipschitz continuous solution[1]. Finally, some numerical results are presented to coroborate the theoritical results. Simulations are done with and without the redistributed mass matrix for a Newmark scheme. We remark that the behaviour of the energy and the normal stress are improved using this new method. The energy is quasi-conserved and will be strictly conserved when time parameter goes to zero [1].

References [1] H.B. K HENOUS , P. L ABORDE & Y. R ENARD. Comparison of two approaches for the discretization of elastodynamic contact problems. Accepted for CRAS paris, 2006. [2] H.B. K HENOUS , J. P OMMIER & Y. R ENARD. Hybrid discretization of the Signorini problem with Coulomb friction. Theoretical aspects and comparison of some numerical solvers, Applied Numerical Mathematics, 2006, Vol 56/2 pp 163-192.

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On models of contact surfaces including anisotropy for friction and adhesion and their experimental validations. Alexander Konyukhov∗ , Karl Schweizerhof† , Peter Vielsack
∗ Institut

f¨ur Mechanik Englerstrasse 2, D-76131, Karlsruhe, Germany [email protected] † Institut f¨ ur Mechanik [email protected]
Institut

f¨ur Mechanik [email protected] ABSTRACT Smoothness and isotropy of contacting body surfaces can vary considerably for different contact problems. Classifying the surfaces roughness two types can be distinguished: a) surfaces with randomly distributed asperities, and b) asperities with algorithmic structure, e.g. the considered surface shows different macro properties in different directions. Mechanical characteristics for the associated contact problems of the first type a) are obtained via statistically distributed asperities. Constitutive modeling is applied for problems of the second type b). Such models are based on the generalization of Coulomb’s friction law into the anisotropic domain, see Zmitrowicz [1] and Curnier [2]. When looking at practical problems concerning friction there are some situations in which the tangential elasticity of the contact surfaces should be taken into account. Such a model including anisotropy for both friction and adhesion has been developed and analyzed numerically in Konyukhov and Schweizerhof [3]. In the current contribution we discuss the validation of this model with a particular experimental test. The contact surfaces are chosen to possess elastic properties, thus a corrugated rubber mat is taken. The results of the experiments show the necessity to use the model including anisotropy for both friction and adhesion. Thus, some originally surprising experimental phenomena, as e.g. geometrical isotropy despite obvious physical anistropies can be explained only within the proposed model, though the latter shows rather qualitative correlations then quantitative ones. It was shown in experiments that the classical model of orthotropic friction does not lead to the good correlation and cannot describe a particular phenomena when a sliding block shows isotropic behavior. A good qualitative result can be achieved with the model involving both orthotropy for adhesion and friction.

References [1] Zmitrowicz, A. A theoretical model of anisotropic dry friction. Wear. 73 (1981) 9–39. [2] Curnier, A. A theory of friction. International Journal of Solids and Structures. 20 (1984) 637– 647. [3] Konyukhov A., Schweizerhof K. Covariant description of contact interfaces considering anisotropy for adhesion and friction. Part 1. Part 2. Computer Methods in Applied Mechanics and Engineering. (submitted).

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Fast and Robust Solution Methods for Dynamic Contact Problems Rolf H. Krause∗ ∗ Institute

for Numerical Simulation, University of Bonn Wegelerstraße 6, D-53115 Bonn [email protected] ABSTRACT

For the numerical solution of dynamic contact problems, often implicit time discretization methods are used for stability reasons [2], giving rise to a nonlinear contact problem to be solved in each time step. We use a globally convergent non-smooth multigrid method which allows for solving multibody contact problems in linear elasticity as fast as linear elliptic problems. This multigrid method can be extended to the case of frictional contact problems leading to a nonlinear method for frictional contact problems with multigrid efficiency [1]. Using nonconforming domain decomposition methods, a stable discretization for the transfer of displacements and stresses at the interface between the bodies coming into contact can be developed, which is also capable of handling complicated geometries. In the framework of a Newmark-based time discretization scheme, this method is used for the construction of an efficient implicit time discretization scheme. Due to the inequality constraints at the contact interface, the time integration of dynamic contact problems often gives rise to oscillations in displacements and stresses. To remove these oscillations, we consider a stabilization based on a global L2 -projection which can be interpreted as a solution dependent correction of the velocities. The stabilization leads to an additional variational inequality to be solved in each time step, which can be done efficiently using our monotone multigrid method. We discuss the properties of the resulting method and illustrate its performance for a contact problem in biomechanics.

References [1] K. Fackeldey, R. Krause, Solving Frictional Contact Problems with Multigrid Efficiency Proceedings of the 16th International. Conference on Domain Decomposition Methods, to appear [2] A. Pandolfi, C. Kane, J. E. Marsden, M. Ortiz, Time–discretized Variational Formulation of non-smooth Frictional Contact International Journal for Numerical Methods in Engineering, 53, 1801–1829, 2002

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A Velocity-Based Time-Stepping Method for Frictional Dynamics Manuel D. P. Monteiro Marques∗ , Laetitia Paoli† ∗ C.M.A.F.,

Faculdade de Ciencias da Universidade de Lisboa Av. Prof. Gama Pinto, 2; 1649-003 Lisboa; Portugal [email protected]

† Universite de Saint-Etienne 23, rue Michelon; 42023 Saint-Etienne cedex 2; France [email protected]

ABSTRACT The dynamics of a mechanical system (a particle or a rigid body) submitted to one unilateral constraint possibly with friction may be formulated as a so-called measure differential inclusion. This was shown by J. J.Moreau (see e.g. [2]), who also introduced and used with great success several numerical timestepping methods for a variety of progressively more complex problems. A convergence proof and existence result may be found in [1], which concerns the case of a constant inertia matrix, equal to the identity. Here, we extend that study, by considering general state-dependent inertia matrices as well as anisotropic Coulomb friction. A convergence result for the corresponding time-stepping scheme, velocity-based ”a la Moreau”, is obtained. A thorough analysis of the limit friction law, including the situations leading to the well-known Painleve’s ”paradoxes”, is also given. For the single constraint case, the present study also extends and completes the results of D. E. Stewart [3].

References [1] M.D.P. Monteiro Marques, Differential inclusions in non-smooth mechanical problems: shocks and dry friction. Birkhauser, Boston, 1993. [2] J. J. Moreau, Unilateral contact and dry friction in finite freedom dynamics. Nonsmooth Mechanics and Applications (J. J. Moreau and P. D. Panagiotopoulos, eds.), Springer, New York, 1988. [3] D. E. Stewart, Convergence of a time-stepping scheme for rigid body dynamics and resolution of Painleve’s paradoxes. Arch. Rational Mech. Anal., 145, 215–260, 1998.

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Mechanical Modelling of Friction and Adhesion of Elastomers at Rough Interfaces T. Meyer, A. Le Gal, M. Klüppel Deutsches Institut für Kautschuktechnologie e. V., Eupener Straße 33, D-30519 Hannover [email protected]

ABSTRACT An advanced model of dynamic contact and sliding friction of elastomers at rough, self-affine interfaces is presented. It describes the frictional force via the dissipated energy, resulting from sliding stochastic excitations of the rubber by surface asperities on various length scales. The effect of surface roughness is considered by three surface descriptors: the fractal dimension and two cut-off lengths, which are obtained from a fractal analysis of the surface via stylus- or laser measurements. The hysteresis response of the rubber enters through viscoelastic master curves of the complex modulus up to high frequencies. Based on this concept stationary friction curves are estimated numerically over a broad velocity scale depending upon surface roughness and temperature. They are compared to experimental friction data found for filled elastomer systems. The obtained results provide a deeper insight into the role of adhesion forces under different contact conditions, e. g. by applying various lubricants. The investigations are found to be useful for a better understanding of the traction behavior of tires on dry and wet roads during ABS-braking of passenger cars. References [1] [2] [3] [4] [5] [6]

M. Klüppel and G. Heinrich, Rubber Chem. Technol. 73, 578 (2000); ibid. Paper No. 43, ACS Rubber Division Meeting, Chicago, 13.-16. May (1999) A. Le Gal, X. Yang and M. Klüppel, Journal of Chemical Physics, 123, 014704, 2005 M. Klüppel, A. Müller, A. Le Gal and G. Heinrich, “Dynamic contact of tires with road tracks“, Paper No. 49, ACS Meeting, San Francisco, April 28-30 (2003) J.A. Greenwood, J.B.P. Williamson, Contact of nominally flat surfaces, Proc. Of the Royal Soc. London, A 295, 300, 1966 H. Hertz, “Miscellaneous Papers”, Macmillan, London, 1896, p. 146 A. Le Gal, M. Klüppel, Kautschuk Gummi Kunststoffe, submitted (2006)

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A 9m Drop Test Simulation of a Dual Purpose Cask for Nuclear Research Reactors Spent Fuel Elements Carlos A. J. Miranda*, Miguel Mattar Neto†, Gerson Fainer†† CNEN-IPEN/SP - Nuclear Engineering Center Av. Prof. Lineu Prestes, 4492 - São Paulo, SP, Brazil * [email protected] † [email protected] †† [email protected]

ABSTRACT The qualification of casks for transportation or storage of nuclear spent fuel elements involves the evaluation of some conditions that simulate possible accidents. The cask should maintain its safety functions through its structural and functional integrity (in any condition, there should be the containment of the radioactive products inside it, the integrity of its biological shielding and assurance against criticality). The main conditions the cask should satisfy, mainly by test, to be qualified are: a 9m drop test against a rigid surface, a penetration test, 30min of fire under 800 oC and 200m immersion during one hour. The first condition is the most critical one. The regulatory bodies stress the qualification “by test” instead of “by analysis”. However, numerical simulations are important to determine, for instance, the most critical position for the free drop tests, saving a lot of money without reducing the project degree of safety. There is a multi-country project, sponsored by the IAEA, with the participation of Latin American countries with research reactors, to develop and qualify a shipping cask for their spent fuel elements. It involves, in its first phase, the project, construction, test and numerical simulation of a half scale model to establish parameters for the tests (mostly the 9m drop test). The cask is a stainless steel cylinder with flat heads, the bottom one is welded while the upper one has flanged threaded connections, and internal structures for the fuel elements. An external stainless steel cylinder contains the biological lead shielding. There are two impact limiters contained by steel shells, which are planned to be filled with a reconstituted wood. This work describes the cask project in details, the main hypothesis and some results obtained with the 9m drop test numerical simulation. Its purpose is to develop a modeling and results evaluation methodology to help the field tests, in order to be applied in future prototype design. In the simulations all non-linearities, mostly associated with the contacts among the cask several parts, the mechanical properties of the materials and the geometric changes, were considered.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A Discontinuous Galerkin Approach for the Numerical Treatment of Tractive Rolling Udo Nackenhorst∗ , Matthias Ziefle† ∗ Institute

of Mechanics and Computational Mechanics, University of Hanover Appelstr. 9A, 30167 Hannover, Germany [email protected]

† Institute

of Mechanics and Computational Mechanics, University of Hanover Appelstr. 9A, 30167 Hannover, Germany ziefl[email protected] ABSTRACT

Rolling contact problems are described in an Arbitrary Lagrangian Eulerian (ALE) kinematics for a numerically efficient treatment, see e.g. [1]. By this approach the motion is split into a pure rigid body motion, which is described in Eulerian coordinates, and into the deformation measured in Lagrangian coordinates relative to the formerly obtained reference configuration. One advantage of this relative kinematics framework is, that steady state rolling is described independent of time. An additional advantage concluded from the spatially fixed mesh is that a local mesh refinement can be introduced to the contact region as needed for a detailed contact analysis. However, one difficulty arises when inelastic material behavior has to be taken into account. Because within the ALE–framework the mesh points are neither fixed in space nor associated with the material particles, the path–lines of the particles have to be traced. This is solved within a fractional step approach, by which in a first (Lagrangian) step the local evolution of the inelastic variables is solved and in a second step the history is advected onto the motion of the particles. I has been proven that Time Discontinuous Galerkin methods possess of optimal stability and convergence behavior for the numerical solution of the related advection equations [2]. A further problem lies in the numerical solution of frictional contact, which yet seems not been solved satisfactory as more recently discussed in [3]. In this presentation on the basis of the experience made for the treatment of inelastic material properties within the ALE–framework of rolling, a fully implicit scheme for the computation of the spatial slip distribution is suggested. By a weak formulation of the stick conditions the spatial distribution of the slip–distances is computed directly. This enables the application of established implicit schemes for frictional contact. Numerical studies on rather simple examples so far show optimal convergence rated.

References [1] U. Nackenhorst, The ALE-Formulation of Bodies in Rolling Contact – Theoretical Foundations and Finite Element Approach. Computer Methods in Applied Mechanics and Engineering, 193(39–41), 4299–4322, 2004. [2] M. Ziefle and U. Nackenhorst, A new update procedure for internal variables in an ALEdescription of rolling contact. Proceedings in Applied Mathematics and Mechanics (PAMM), 5, 71–74, 2005. [3] T. A. Laursen and I. Stanciukescu, An algorithm for incorporation of frictional sliding conditions within a steady state rolling framework, Communications in Numerical Methods in Engineering, in press

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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An iterative method with BEM discretization for the friction contact problems P. Neittaanm¨aki∗ , A.S.Kravchuk† , I.G.Goryacheva† ∗ Department

of Mathematical Information Technology P.O. Box 35 (Agora), FIN-40014 University of Jyv¨askyl¨a, Finland [email protected].fi † Department

of Mechanics and Mathematics, Moscow State University 119899 Moscow State University, Moscow, Russia {kravchuk biocom, goryache}@mail.ru ABSTRACT

A new formulation for the friction contact problem is proposed. The feature of this formulation consists of using of the relative velocities in the friction law, and of taking into account the reciprocal influence of the normal and tangential components of contact stresses. This require a new constructions of the impenetrability condition and, secondly, to use the step–by–step type methods for the modelling of evolution of the contact forces. With a new definition of the kinematically admissible displacements and velocities including the unilateral constraints the local problem is transformed to a variational one, which is revealed as a quasi– variational inequality. To solve this quasi–variational inequality, a new iterative method is given. This method is based on the calculation of a solution increment using the calculation of the contact pressure from the previous iteration. Such a idea permits to reduce the problem to the sequence of a saddle–point problem. The convergence theorem is demonstrated. Numerical algorithm is based on the boundary elements approach. Numerical results, obtained for the 2D problems, give an estimates for the difference between the solutions corresponding to the different impenetrability conditions, and to different quantity of the steps with respect to a parameter defining the evolution of the system of contacting bodies. In particular, it is demonstrated that the normal displacements and contact pressure are defined with a good precision for all the formulations of the impenetrability condition, and for one step of the evolution parameter. But the analysis of the friction forces and relative sliding phenomena requires the non–linear (strict) impenetrability condition, and several steps for loads. This conclusion play a very important role for the fretting–wear investigation and for the reliability and life–time prediction. Analytical solution was obtained with the method developed in [1]. The work was partly supported by the grant No 05-01-00591a of the Russian Foundation for the Fundamental Researches, and TEKES MASI program.

References [1] I.G.Goryacheva, Contact Mechanics in Tribology. Kluwer, Dordrecht, 1998. 99

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Frictional Contact of Elastomer Materials on rough rigid Surfaces Jana Nettingsmeier∗ , Peter Wriggers∗ ∗ Institut

f¨ur Baumechanik und numerische Mechanik Universit¨at Hannover Appelstraße 9a 30167 Hannover [email protected] ABSTRACT

Within the analysis of many technical problems with frictional contact, Coulombs law is used, which implies a constant friction coefficient µ. This assumption is sufficient for many applications in structural mechanics; however in the special case of rubber friction on rough surfaces the resulting simplification cannot be accepted. The physical interactions between tire and road surface are very complex and still widely unknown. As it is apparent from experiments, the friction coefficient depends heavily on various parameters like sliding velocity, surface roughness, normal forces and temperature change. Our aim is now to derive a realistic friction law based on micromechanical observations. It can be shown that the energy dissipation in a rubberlike viscoelastic material implies a frictional behaviour for the whole specimen, even if we neglect any predetermined local friction on the microscale. The friction coefficient advances to zero for both very high and low sliding velocities, but reaches a maximum for middle speed. This effect is known as hysteretic friction and represents the main part of rubber friction. Due to the multiscale character of the surface roughness, we have to model the problem on different length scales. Therefore the frictional behaviour on the microscale is analyzed and the homogenized characteristics result in a friction law, which is projected onto the macroscopic problem in a staggered procedure. The fractal road surface is approximated by a superposition of several harmonic functions. The three-dimensional numerical problem will be modeled with a four-node contact element. Its G AUSS points are projected to the closest point on the rigid surface, which is given as an analytical function z = f (x, y). Even if we can use frictionless contact on the finest scale, we need to transport the frictional constitutive behaviour to the next scale. The friction coefficient will be adapted to the local conditions (normal stress, sliding velocity, etc.) obtained in the contact element. In addition to the described approach for pure hysteretic friction, adhesional effects between the surfaces will be included into the contact model. It has to be analyzed how the transmission of tensile stresses influences the global frictional behaviour of rubber. The properties of elastomer materials are sensitive to temperature changes. Rising temperatures have the same effect as a decreasing load frequency and vice versa. Because of this relation it is necessary to determine the magnitude of material heating due to internal energy dissipation.

References [1] B. Persson Theory of Rubber Friction and Contact Mechanics. Journal of Chemical Physics, 115,8, 2001. ´ [2] M. Raous, L. Cangemi, M. Cocu, A consistent model coupling adhesion, friction and unilateral contact. Computer Methods in Applied Mechanics and Engineering 177: 383-399, 1999. [3] P. Wriggers, Computational Contact Mechanics. Wiley, 2002.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Optimizing the Description of Forming Tools with Bézier Surfaces in the Numerical Simulation of the Deep Drawing Process M.C. Oliveira*, J.L. Alves †, L.F. Menezes * *

CEMUC, Department of Mechanical Engineering, University of Coimbra Pinhal de Marrocos, Polo II, 3030 Coimbra, Portugal [email protected]; [email protected] †

Department of Mechanical Engineering, University of Minho Campus de Azurém, 4810 Guimarães, Portugal [email protected]

ABSTRACT

In the simulation of the deep drawing process one of the challenges is the correct prediction of the actual contact surface and kind of contact established between the tools and the blank sheet, since they determine the boundary conditions. In the numerical simulation of the deep drawing process the contact conditions change continuously, increasing the importance of a correct evaluation of these parameters in each load step. In the finite element implicit code DD3IMP [1], devoted to the simulation of the deep drawing process, the tools are modeled with Bézier parametric surfaces, for which a frictional contact algorithm has been developed and continuously improved [1]. The description of the tools by parametric surfaces allows the direct use of the information provided by CAD software. However, in order to guarantee the contact algorithm convergence it is necessary to impose some continuity conditions between the surfaces that define the forming tools. In many situations the tools are defined by a set of plane surfaces connected by fillings with surfaces of constant or variable radius. To guarantee the correct tools’ modeling it is necessary to correctly model the curve that defines the radius. Previous works had shown that, to assure a correct correlation between tools’ geometry and Bézier description at least cubic segments should be used to define the radius [2]. The Bézier tools description must accurately represent the tools’ geometry, but also satisfy the continuity conditions that guarantee the contact algorithm convergence. In this work the influence of the degree of the polynomial functions used in the Bézier surfaces on the contact algorithm is evaluated. Also the error in the and continuity of the Bézier surfaces obtained with the CAD system is estimated, and its influence on the convergence of the contact algorithm is studied.

References [1] M.C. Oliveira, J.L. Alves and L.F. Menezes, Improvement of a Frictional Contact Algorithm for Strongly Curved Contact Problems. International Journal for Numerical Methods in Engineering, 58, 2083-2101, 2003. [2] M.C. Oliveira e L.F. Menezes, Optimização da Descrição das Ferramentas por Superfícies de Bézier na Simulação do Processo de Estampagem. Proceedings of the V Congresso Métodos Numéricos en Ingeniería, Ed. J.M. Goicolea, C. Mota Soares, M. Pastor and G. Bugeda, 2002 (CD-ROM edition).

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Exploring the dynamics of a simple system involving Coulomb friction Elaine Pratt, Alain L´eger CNRS, laboratoire de M´ecanique et d’Acoustique 31, chemin Joseph Aiguier, 13402 Marseille cedex 20, France [email protected] [email protected]

ABSTRACT The work presented here consists in an exhaustive study of a simple mass-spring system involving Coulomb friction. The aim was to gain some insight into the behaviour of a chain of masses in frictionnal contact. If it is simple to explicit the analytical solution of a single mass system, the analytical solution for a two mass system is already far more complicated. We thus consider two masses linked by a spring in bilateral contact with Coulomb friction and submitted to an external force applied onto one of the masses. The existence, uniqueness and regularity of the dynamics of the system is established through a recent paper [1]. Once the uniqueness is ensured it is simple to exhibit the explicit solution for certain values of the external force (i.e. when the amplitude of the force is either small or large). When the amplitude of the external force belongs to a certain intermediate range the dynamics turns out to be more interesting. The solution can be calculated analytically, however as the computation becomes rapidly tiresome, we use a symbolic calculus tool to compute a solution corresponding to a given external force. We thus observe that: • for a given value of the amplitude of the external force, the two masses oscillate for a certain time before coming to rest (these oscillations are obviously not periodic because of the non linearity due to the Coulomb friction), • for increasing values of the amplitude of the external force, the number of oscillations that the masses carry out before coming to a halt, grows each time that the amplitude of the external force passes through a critical value, • these critical values of the amplitude of the external force accumulate as the amplitude of the force reaches the upper bound of the interesting range. Extending these results to more than two masses may prove to be not such a simple task but some progress has been made and shall be presented.

References [1] P. Ballard et S. Basseville, Existence and uniqueness for dynamical unilateral contact with Coulomb friction: a model problem. Mathemetical Modelling and Numerical Analysis, Vol. 39, 1, 57-77, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Multibody Modeling of Pantographs for Catenary-Pantograph Interaction Frederico Grases Rauter1, João Pombo2, Jorge Ambrósio2, Manuel Seabra Pereira2 1

2

SNCF – Direction de l’Innovation et de la Recherche 45 Rue de Londres, 75008 Paris, France [email protected]

IDMEC – Instituto Superior Técnico, Technical University of Lisbon Av. Rovisco Pais, 1049-001, Lisboa, Portugal {jpombo,jorge,mpereira}@dem.ist.utl.pt

ABSTRACT In the great majority of railway networks the electrical power is provided to the locomotives by the pantograph-catenary system. From the mechanical point of view, the single most important feature of this system consists in the quality of the contact between the contact wire(s) of the catenary and the contact strips of the pantograph. Therefore not only the correct modeling of the catenary and of the pantograph must be achieved but also a suitable contact model to describe the interaction between the two systems must be devised. The work proposed here aims at enhancing the understanding of the dynamic behavior of the pantograph and of the interaction phenomena in the pantograph-catenary system. The potential contribution of this work to the railway community includes the decrease of the number of incidents related to this system and the reduction of the maintenance and interoperability development costs. The catenary system is described by a detailed finite element model of the complete subsystem while the pantograph system is described by a detailed multibody model. The dynamics of each one of these models requires the use of different time integration algorithms. In particular the dynamics of the finite element model of the catenary uses a Newmark type of integration algorithm while the multibody model uses a Gear integration algorithm, which is variable order and variable time step. Therefore, an extra difficulty that arises in study of the complete catenary-pantograph interaction concerns the need for the co-simulation of finite element and multibody models. As the gluing element between the two models is the contact model, it is through the representation of the contact and of the integration schemes applied for the finite and multibody models that the co-simulation is carried on. The work presented here proposes an integrated methodology to represent the contact between the finite element and multibody models based on a continuous contact force model that takes into account the co-simulation requirements of the integration algorithms used for each subsystem model. The discussion of the benefits and drawbacks of the proposed methodologies and of its accuracy and suitability is supported by the application to the real operation scenario considered and the comparison of the obtained results with experimental testing data. In the process future developments that include representing the pantograph model with flexible bodies, non-linear force elements and the inclusion of wear and cross-wind effects are also discussed.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Comparisons and coupling of algorithms for collisions, contact and friction in rigid multibody simulations Mathieu Renouf† and Vincent Acary∗ †

LaMCoS - TMI team CNRS/INSA de Lyon - UMR5514 18-20, rue des sciences F69621 VILLEURBANNE cedex - France [email protected] ∗ Bipop Project, INRIA Rhone–Alpes ZIRST Montbonnot, 655, avenue de l’Europe, 38664 Saint ISMIER, France [email protected]

ABSTRACT Numerous works in computational mechanics are dedicated to multi-body systems [1, 2]. This leads to the use of various methods to simulate the static or dynamic evolution of complex systems. The case of dense multi-contact assemblies is one of the more complex one: the problem have often a large number of unknown and have a infinity of solution due to the definition of the matrix of the system. Moreover this problem become harder when friction or more complex laws are introduced in the system. Thus we need fast and robust solvers to perform mechanical studies. These performances can be increased when the special problem structure is considered (sparse matrices, block structured problem). Our work is based on the Non Smooth Contact Dynamic framework introduced by Moreau [3]. The method uses a time-stepping integrator without explicit event-handling procedure and an unilateral contact impact formulation associated to Coulomb’s friction. In this paper we use and compare different iterative algorithms such as Gauss-Seidel, projected conjugate gradient and direct ones as Lemke and Quadratic programming solvers [4]. The efficiency of the methods is compared in terms of complexity, convergence criterion and of CPU time. To illustrate the results, we focus on granular assemblies. 3D frictional contact simulations are performed with ConF&TiS and the Numerics library of the siconos project.

References [1] M. Jean and J. J. Moreau, Unilaterality and dry friction in the dynamics of rigid bodies collection. In: Contact Mechanics International Symposium, A. Curnier ed., 1992. [2] C. Glocker and F. Pfeiffer. Multibody dynamics with unilateral contacts. John Wiley and Sons, 1996. [3] J.-J. Moreau, Numerical aspects of the sweeping process. Comp. Meth. Appl. Mech. Engrg, 177:329–349, 1999. [4] R. W. Cottle, J.-S. Pang and R. E. Stone, The linear complementarity problem. Academic Press, Inc., 1992.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Third body flow during wheel-rail interaction Mathieu Renouf, Aur´elien Saulot, Yves Berthier TMI Team - LaMCoS/INSA de Lyon - UMR5514 18-20, rue des sciences, F69621 VILLEURBANNE cedex - France {Mathieu.Renouf,Aurelien.Saulot,Yves.Berthier}@insa-lyon.fr ABSTRACT In a mechanism, when a contact occurs between two contactors, third body [1] is the generic name used to describe the material generated as a result of the contact interaction. Thus the wear phenomenom can be considered as the third body flow Qw definitely ejected from the contact area. Because the wear phenomenom involves as well the global scale (wheel and rail) as the local one (contact interface), the study of this phenomenom via numerical tools needs both continuum and discrete approaches [2, 3]. We propose here a two-dimensional analysis of the third body flow induced by the relative transverse

sliding motion between wheel and rail [2](Vp represents the micro transerve periodic velocity and F the normal load). Both Finite Element and Discrete Element Methods [4] are used. At the continuum level we take into account an elastoplastic behaviour for the two bodies. At the local level we use the Non Smooth Contact Dynamic method developped by Moreau and Jean [5], using non-smooth contact laws to describe body interactions. We use a non-smooth cohesive law to describe the behaviour of the third body.

References [1] M. Godet. The third-body approach : a mechanical view of wear. Wear, 100:437–452, 1984. [2] A. Saulot, S. Descartes, D. Desmyter, D. Levy and Y. Berthier. A tribological characterization of the ”damage mechanism” of low rail corrugation on sharp curved track. Wear, In Press. [3] N. Fillot, I. Iordanoff, and Y. Berthier. Simulation of wear through a mass balance in a dry contact. ASME J. Tribol.,127(1):230–237 [4] A. Munjiza The combinated finite-discrete element method. John Wiley and Sons, 2004. [5] M. Jean and J. J. Moreau. Unilaterality and dry friction in the dynamics of rigid bodies collection. In: Contact Mechanics International Symposium, A. Curnier ed., 1992.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Modelling thermal contact resistance on glass forming processes with special interface finite elements José César de Sá†, Sébastien Grégoire*, Philippe Moreau*, Dominique Lochegnies* †

Faculty of Engineering of the University of Porto Rua Dr. Roberto Frias, s/n, 4200-465 Porto, Portugal [email protected] *

Laboratoire d’Automatique, de Mécanique et d’Informatique Industrielles et Humaines – UMR CNRS 8530 Université de Valenciennes – Le Mont-Houy – Jonas 2 59313 Valenciennes Cedex 9 – France [email protected] , [email protected] , [email protected]

ABSTRACT The forming process of glass containers is a complex coupled thermal/mechanical problem with interaction between the heat transfer analysis and the viscous flow of molten glass. The transfer of heat and change of viscosity are fundamental phenomena in this process. The changes in temperature influence the very process of heat transfer since the thermal properties of glass change with temperature. On the other hand the great dependence of glass viscosity with the temperature influences dramatically the flow of the material and therefore the final product. The successive changes in shape produced by gravity and blow pressure, which depend on the actual properties that are influenced by temperature, affect subsequently the heat transfer process. The thermal contact between the glass and the metal moulds influences dramatically the glass thickness distribution of the final product. The heat flux at the interface is a function of the contact pressure, of the temperatures of both the glass and the mould, of the glass viscosity and of the presence of asperities and lubricants in the mould. Another decisive aspect for obtaining a good final product is the chosen procedure for the cooling of the moulds. In the modelling of theses processes with the finite element method a moving mesh, attached to the deforming glass, deals with the mechanical and thermal problems in the glass. Due to the low pressures involved only the heat transfer problem is addressed in the moulds that are therefore discretized with a fixed mesh. Consequently the thermal contact is dealt with non-matching meshes. A classical master/slave strategy may not be always adequate as there may be zones in the mould, where, due to large curvatures, the mesh is more refined than in the glass (typically the slave). A heat transfer contact element is therefore proposed, inspired on the contact “mortar method” developed for mechanical contact. Its formulation is obtained from a variational formulation in which the thermal contact is imposed in a penalised form, in which the penalty term is a function of the heat transfer coefficient. In the interface elements for some nodes the temperatures are constrained to have the same temperatures as the corresponding ones in the mesh of the glass and the surface temperatures are interpolated from the temperatures in the nodes of the mould. In other nodes of the interface elements the temperatures are interpolated from the node temperatures in the glass and the surface temperatures are the corresponding ones node in the mould. The heat transfer coefficient is evaluated for each node from the contact pressure and viscosity evaluated at the glass nodes. As a result an over constrained problem is avoided and a more effective thermal contact is obtained.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Numerical Investigation of Shakedown Residual Stresses Under Moving Surface Loads Jim S. Shiau Lecturer, Faculty of Engineering and Surveying University of Southern Queensland, Toowoomba, QLD, 4350, Australia [email protected]

ABSTRACT It can be shown theoretically that there is a load magnitude below which a protective residual stress will develop in a rolling and sliding contact of continuum structure, and above which it will undergo an incremental failure. This load is known as the `shakedown limit load' and the protective residual stresses associated with this shakedown limit load are the optimal residual stresses for the life of the structure. In his "Contact Mechanics" book, Professor Johnson described an analytical shakedown approach to predict the shakedown load limit and the associated residual stress distribution. This problem will be revisited in this paper using a numerical method proposed in the author's PhD Thesis. A numerical formulation based on Bleich-Melan shakedown theorem will be discussed by making use of finite element techniques and mathematical programming. The proposed numerical procedure can be used to solve for the shakedown limit load and the associated developed residual stresses of structures subjected to repeated moving surface loads. A series of results on shakedown residual stresses will be examined.

References [1] Jim S. Shiau (2001) “Numerical methods for shakedown analysis of pavements under moving surface loads". Ph.D. Thesis. The University of Newcastle, NSW, Australia [2] K. L. Johnson (1985) “Contact Mechanics”. Cambridge University Press.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Frictional Contact/Impact between a Hyperelastic Body and Moving Rigid Obstacles N. Str¨omberg∗ ∗ J¨ onk¨oping

University, Department of Mechanical Engineering P.O. Box 1026, 551 11 J¨onk¨oping, Sweden [email protected]

ABSTRACT In this paper a method for frictional contact/impact between a hyperelastic body and moving rigid obstacles is suggested and investigated. The work is a further development of the suggested method in [1]. The motion of an obstacle is defined by a prescribed translation vector and a prescribed rotation matrix. The geometry of the obstacles are defined by smooth functions. Each function is formulated in a moving frame which is governed by the translation vector and the rotation matrix. These functions are then included in new formulations of Signorini’s conditions and Coulomb’s law of friction. Instead of using contact forces, the mean value impulses are utilized in these formulations, which also are adopted in the law of motion which is given on velocity form. By following this approach, no search algorithm is needed, the normal and tangential directions are well defined and the treatment of non-constant transformation matrices in the law of motion is straight-forward. A total Lagrangian formulation of the system is given. The elastic properties of the body are defined by coupling the second Piola-Kirchhoff stress to the Green-Lagrange strain via the Kirchhoff-St.Venant law. The governing equations are solved by a nonsmooth Newton method. This is performed by following the augmented Lagrangian approach and deriving the consistent stiffness matrix as well as the contact stiffness matrices. The method is implemented in TriLab. TriLab is a user-friendly finite element toolbox for simulating contact and impact problems. TriLab is developed using Matlab and Visual Fortran. The Fortran code is linked to Matlab as mex-files. The code is vectorized and the sparsity is utilized. By using Trilab, the presented method will be demonstrated by solving two-dimensional problems.

References [1] N. Str¨omberg, An implicit method for frictional contact, impact and rolling, European Journal of Mechanics, A/Solids, 24, 1016–1029, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Incorrect Contact of Screw Surfaces and its Consequences Jaromír Švígler University of West Bohemia in Pilsen, Department of Mechanics, Czech Republic [email protected]

ABSTRACT The contribution deals with the analysis of the correct and incorrect contact of the screw surfaces which are represented with two-dimensional manifolds. One surface is defined by parametric equation and the conjugate surface is created in the direct envelope way according to the Distelli theorem. The solution which is presented in this contribution is founded on kinematical principles. The correct contact of both conjugate surfaces takes place in a contact line ch32, see Figure, for instantaneous time.

Correct line contact of conjugate screw surfaces σ 2 and σ 3 Temperature and force deformation bring about displacement of surfaces and the correct contact changes into the incorrect contact. Instead of the line contact of surfaces the contact at isolated points takes place. In this contribution the incorrect contact of surfaces is caused by a large parallel displacement of the axis of one of the surfaces. The inner deformation of screw surfaces is not involved into solution and likewise general position of rotor axes that arises by real deformation is not accepted. This simplification was necessary for the first step of the solution of this problem. The work is solved as three-dimensional problem. Theory of incorrect contact of screw surfaces is applied to the teeth of rotors of the screw machines. One of the most important parts of screw machines, i.e. screw compressors or expanders, is a work space with its boundary surface which is produced by the screw surfaces of teeth and an inner cylinder surface of a machine box. In course of working process the work space has a complicated, non-stationary shape. The tooth screw surfaces, which consist of different coupled surfaces, create the most important part of the work space. The original line contact between surfaces changes into the contact in four points along a length of teeth of rotors. The rise of a technically undesirable gap between both conjugated surfaces and disturbance of gear ratio are consequences of this change. The arisen gaps caused the increase in inner loss of screw machines. Geometrical visualization of the conjugated teeth with the arisen gap is performed in instantaneous time.

References [1] [2]

J. Švígler, Incorrect Contact of Screw Machine Rotors. International Conference on Compressors and their Systems, 3-12, John Wiley &Sons London Ltd., 2005. N. Stosic, I.K. Smith, A. Kovacevic, Rotor interference as a criterion for screw compressor design. Journal of Engineering Design, volume 14, No 2, 209-220, Taylor&Francis Ltd, 2003.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Three-Dimensional Rupture Instability of a Displacement-Softening Interface under Nonuniform Loading Koji Uenishi* *

Research Centre for Urban Safety and Security, Kobe University 1-1 Rokko-dai, Nada, Kobe 657-8501 Japan [email protected]

ABSTRACT In previous studies [1, 2], we investigated rupture instability on two-dimensional interfaces that follow a nonlinear displacement-softening (slip-weakening) relation and are subjected to a loading stress that is locally peaked spatially, the level of which changes quasi-statically in time. We showed that for the case in which the interface strength weakens linearly with slip (i.e., displacement gap), there exists a universal length of the slipping region at instability, independent of any length scales entering into the description of the shape of the loading stress distribution. Here, we study slip development and its (in)stability for three-dimensional planar interfaces that follow the linear displacement-softening relation. We employ an energy approach to give a Rayleigh-Ritz approximation for the dependence of size of the rupture zone and maximum slip on the level and shape of the loading stress distribution. The zone is assumed to be elliptical, x2/a2 + z2/b2 ≤ 1, with unknown axes a and b, where the interface coincides with the x-z plane (y = 0). A loading stress τp + Rt − κ (x2 + m2z2)/2 is assumed on that plane; κ and m are positive constants and Rt is the stress change from that for which the peak in the loading stress distribution equals the interface strength τp. The results, again independent of the curvature of the loading stress distribution κ, indicate that the loading induces a rupture zone whose aspect ratio a(t)/b(t) changes with time t except for the cases of circular cracks (m = a/b = 1) for the opening (tensile) mode and m = a/b ≈ 1/(1 − ν) for the sliding mode (unidirectional shear in the x-direction). Here, ν is Poisson’s ratio. These values of m minimize the area of critical rupture zone at instability (πacbc), and for the opening mode, the corresponding critical diameter is 2ac = 2bc ≈ 1.960 µ/[W(1 − ν)], with µ being shear modulus and W linear displacement-softening slope. For the sliding mode, the critical rupture size is written in terms of µ, ν and W (and the complete elliptic integrals of the first and second kinds) [3]. When, for example, ν = 0.25, those critical lengths are 2ac ≈ 2.598µ/W and 2bc ≈ 1.951µ/W. Comparison of these results with the two-dimensional ones (1.158µ/[W(1 − ν)] for modes I and II; 1.158µ/W for mode III) shows that the critical lengths are of the same order for all two- and three-dimensional cases.

References [1] K. Uenishi and J. R. Rice, Universal nucleation length for slip-weakening rupture instability under nonuniform fault loading. J. Geophys. Res., 108(B1), cn:2042, doi:10.1029/2001JB001681, pp. ESE 17-1 to 17-14, 2003. [2] K. Uenishi, Rupture instability on a displacement-softening interface under heterogeneous loading. In: Proc. ECCOMAS 2004 (edited by P. Neittaanmäki et al.), 14 pages, Department of Mathematical Information Technology, University of Jyväskylä, Finland, 2004. [3] K. Uenishi and J. R. Rice, Three-dimensional rupture instability of a slip-weakening fault under heterogeneous loading. Eos Trans. AGU, 85(46), Fall Meet. Suppl., Abstract S13E-04, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Contact of rough surfaces - A comparison of experimental and numerical results Kai I. Willner∗ , Daniel B. G¨orke∗ ∗ Universit¨ at

Erlangen-N¨urnberg Egerlandstraße 5, 91058 Erlangen, Germany {willner,goerke}@ltm.uni-erlangen.de ABSTRACT Measurements of rough surfaces show that the roughness topography can be described as a fractal over several length scales. A suitable description is then given by a discrete structure function. For a large class of typical surfaces measured structure functions can be approximated by a three-parameter function, employing the rms-value of the roughness, a transition length between fractal behaviour at high wavenumbers and stationary behaviour at low wavenumbers, and the fractal dimension in the fractal region, respectively, as intrinsic parameters to describe an isotropic rough surface. To study the normal contact behaviour of such fractal surfaces numerically, the first author presented in [?] a numerical model which allows to describe the elasto-plastic normal contact of isotropic fractal surfaces. This model is now tested against experimental data which are obtained from contact tests of several aluminum specimens. The paper gives a short review of the numerical model and describes than the experimental set-up for the contact tests. Numerical and experimental data for several aluminum specimens with different surface properties are shown and compared.

References [1] K. Willner, Elasto-plastic contact of three-dimensional fractal surfaces using halfspace theory. Journal of Tribology, 126, 28-33, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Continuum Mechanics Modelling and Simulation of Carbon Nanotubes Marino Arroyo1 and Ted Belytschko2 1 Universitat

` Politecnica de Catalunya E-08034 Barcelona, Spain [email protected]

2

Northwestern University Evanston, IL 60208, USA [email protected]

ABSTRACT The understanding of the mechanics of atomistic systems greatly benefits from continuum mechanics. One appealing approach aims at deductively constructing continuum theories starting from models of the interatomic interactions. This viewpoint has become extremely popular with the quasicontinuum method. The application of these ideas to carbon nanotubes presents a peculiarity with respect to usual crystalline materials: their structure relies on a two-dimensional curved lattice. This renders the cornerstone of crystal elasticity, the Cauchy-Born rule, insufficient to describe the effect of curvature. We discuss the application of a theory which corrects this deficiency to the mechanics of carbon nanotubes [1,2,3]. We review recent developments of this theory, which include the study of the convergence characteristics of the proposed continuum models to the parent atomistic models, as well as large scale simulations based on this theory. The latter have unveiled the complex nonlinear elastic response of thick multiwalled carbon nanotubes, with an anomalous elastic regime following an almost absent harmonic range.

References [1] Marino Arroyo and Ted Belytschko, An atomistic-based finite deformation membrane for single layer crystalline films, Journal of the Mechanics and Physics of Solids 50, 1941-1977, 2002. [2] Marino Arroyo and Ted Belytschko, Nonlinear mechanical response and rippling of thick multiwalled carbon nanotubes, Physical Review Letters, 91, 215505, 2003. [3] Marino Arroyo and Ted Belytschko, Finite element analysis of the nonlinear mechanics of carbon nanotubes, International Journal for Numerical Methods in Engineering, 59, 419-456, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Comparison of computational efficiency of modeling approaches to prediction of damping behavior of nanoparticle-reinforced materials Liya V. Bochkareva*, Maksim V. Kireitseu†, Geoffrey R. Tomlinson† *

UIIP National Academy of Sciences of Belarus Address: Filatova 7 – 28, Minsk 220026, Belarus [email protected] †

Department of Mechanical Engineering, the University of Sheffield Address: Mappin Street, Sheffield S1 3JD, the United Kingdom [email protected]

ABSTRACT Carbon nanotube-reinforced polymer-matrix composite materials (CNT-PMC) are now intensively studied; however, CNT-PMC damping behavior is rather contradictory result than plausible information. Therefore, it requires urgent investigations from multi-disciplinary viewpoint. The CNTreinforced material damping phenomenon is complex because of friction between nanotube and a matrix and the variety of other energy dissipation/fracture mechanisms involved, and because of the complex nature of the nanoparticles themselves, multi-walled structure etc. that are affect a damping. All of these mechanisms may be beneficial for dumping and/or add multifunctionality to engineering structures. It is worth noting that interfacial fracture energy is important and may play a great role for a total energy dissipated by the damping material. Particular advanced energy dissipation phenomena of CNT-reinforced polymeric materials can be explained by considerable interfacial fracture mechanics and bonging energy between CNT and polymeric molecular chains. Quantitative prediction of toughness would require a coupled and detailed modeling of the various damping / dynamic mechanisms and criteria for the different modes, which is at present still not feasible. Computational simulation and modeling tools called as a Virtual Reality Environment (VRE) can help to understand many physical effects and predict the behavior of materials and machine components via computer-generated media. In the present paper, multiscale computational approaches to modeling of nanoparticle-reinforced composite materials and virtual reality engineering tools have been used to describe/model an intuitive interface of some CNT-reinforced materials to enable efficient design and synthesis of next generation materials and nanoscale devices. The paper presents a comparison between computational approaches to modeling of damping/dynamics of CNTreinforced composite materials so as to estimate a validity of proposed methods. The underlying mechanics of material has been partially simulated by the use of energy dissipation mechanisms and programmed by using fast multipole method FMM-BIEM [1] accordingly. In the virtual working environment, the user can naturally grab and steer a nanoparticle, matrix and composite because the information flow between the user and the VRE is bidirectional and the user can influence the environment.

References [1] Y.J. Liu, N. Nishimura, Y. Otani, T. Takahashi et al., Fast multipole approach to modeling of nanocomposites, ASME J. of Appl. Mech., 72 (1), 140-160, 2005

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A Hybrid Atomistic-Continuum Finite Element Modelling of Nanoindentation – Test on Copper 1 , M. Mazdziarz 1 , G. Jurczak1 , P.Traczykowski 1 , ˙ ´ P. Dłuzewski S. Nagao2 , R. Nowak2 and K. Kurzydłowski3 1 Institute

2 Nordic

3 Warsaw

of Fundamental Technological Research,Warsaw,Poland [email protected]

Hysitron Laboratory, Dept. Materials Sci. and Engg., Helsinki University of Technology (TKK/HUT), Vuorimiehentie 2A, Helsinki FIN-02015, Finland [email protected].fi University of Technology, Faculty of Materials Science and Engineering (InMat), Wooska 141, Warszawa 02-507, Poland [email protected] ABSTRACT

Problem of locally disordered atomic structure is solved by using a hybrid formulation in which nonlinear elastic finite elements are linked with discrete atomic interaction elements. The continuum approach uses nonlinear hyperelasticity based upon the generalized strain while the atomistic approach employs the Tight-Binding Second-Moment Approximation potential to create new type of elements. The molecular interactions yielding from constitutive models of TB SMA were turned into interactions between nodes to solve a boundary value problem by means of finite element solver. Atomistic psudoelements are noting more than two-node atomic interaction. The application is used to deal with the problem of edge dislocation and its dissociation into two Schockley’s dislocation in Cu crystals. Three examples are shown. In the first, the whole considered crystal region has been discretized and solved by means of the nonlinear elastic FEs. In the second example, the same FE region has been discretized by means of the molecular lattice and solved by molecular dynamics and statics. Finally, in the third example the regions around the dislocations’ cores have been replaced by discretized atomic structure and linked on the boundary with corresponding nodes of continuum FEs. In this way, a single boundary value problem with two different types of discretization of the crystal structure has been solved in this example. A transition on the continuum/atomic interface is assured by taking into account a crystallographic data in the finite element mesh preparation. In this example the regions of dislocation cores were modelled using molecular interaction mesh while the ordered lattice was discretized by FEs. The obtained MD-FE model has been applied to simulate the nanoindentation test on nanocrystalline copper in order to conclude on the singularities observed in P-h curves.

References [1] P.Dłu˙zewski, G.Maciejewski, G.Jurczak, S.Kret and J.-Y.Laval, Nonlinear finite element analysis of residual stresses induced by dislocations in heterostructures. Computational Materials Science , 29, 379–395, 2004. [2] P.Dłu˙zewski, G.Maciejewski,Nonlinear finite element calculations of residual stresses in dislocated structures.Computational Materials Science, 30, 44–49, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Continuum Models for Composites Reinforced by Micro/Nano-Fibers Vladimír Kompiš*, Mário Štiavnický*, Michal Kaukiþ† *

Academy of Armed Forces of General M. R. Štefánik, Demänovská 393, 031 19 Lipt. Mikuláš, Slovakia [email protected], [email protected] †

University of Žilina, Faculty of Control and Informatics, VeĐký Diel, 010 24 Žilina [email protected]

ABSTRACT We will present a method of continuum models for composites reinforced by micro/nano-fibers. The fibers are supposed to be much stiffer than the matrix and their diameter can be much smaller than their length. The ideal cohesion between fiber and matrix is supposed in the present models. Interactions of a fiber with the matrix, with the other fibers and with the domain boundaries, are important parameters, which are decisive for the stiffening effect and for the bearing capacity of the composite material. Both near and far field effects are important for the material behavior. The near fields are important for simulation of the local effects like fracture and the far field effects determine the stiffening. For modeling of such problems the methods like FEM and BEM are not very efficient. BIE using distributed forces, dislocations and dipoles along the fiber axis (the source points) enable to model these effects much more efficiently. The intensity of the source functions simulates the interaction of the fiber with the other subjects. Because of very close distance of the source points to the fiber boundary and the quasi-singular form of the integrals, the integration is performed analytically in the fiber direction. The proposed models can be used in the multi-scale simulation of such composite structures.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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First Principles Evaluation of Ideal Strength of Cu Nanowire A. Kushima∗ , Y. Umeno∗ , T. Kitamura∗ ∗ Graduate

School of Engineering, Kyoto University Yoshidahonmachi Sakyo-ku Kyoto, 606-8501, Japan [email protected] ABSTRACT

Studying the ideal strength of the materials with high-symmetric structures can provide us with the fundamental mechanical property of the nano-structured materials. In this study, tensile simulations for cylindrical-shaped Cu nano-wires with different diameters are conducted and their ideal strengths are explicitly evaluated by the first principles calculation. And the results are compared with that of the atomic chain and the sheet, the elements constructing the wire. The ideal strengths of the wire is lower than that of the chain and the sheet because the fracture of the wires is caused by the formation of gap in the surface, while the direct breaking of inter-atomic bonds is observed in the case of the chain and the sheet. The ideal strength and the strain at which the fracture takes place of the wire with larger diameter are smaller than those of the wire with smaller diameter. This is due to the existence of the minute gaps in the surface of the wire, caused by the difference in the curvature between the surface layer and the layer inside it. In the tensile process, the gaps become larger and finally the fracture occurs to the wire. Fig.1 denotes the change in electronic structure of the wire under tension. The isosurface of 0.16×103 nm−3 is shown, and the contours of the surface charge distributions are extracted for detailed observation.

Figure 1: Change in atomic structure and charge density of the Cu nanowire under tension.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Variation Descriptions of Nano-Structured Media Svetlana A. Lisina1, Alexander I. Potapov 1,2 1 Mechanical Engineering Research Institute of the Russian Academy of Sciences 85 Belinsky str., 603024, Nizhny Novgorod, RUSSIA [email protected] 2 Nizhny Novgorod State Technical University, 24 Minina str., 603600, Nizhny Novgorod, RUSSIA [email protected]

ABSTRACT Medium microstructure (in particular, size and morphology of a grains) is the most important property of materials that directly influences their strength and other physical and mechanical characteristics. Complex dynamic behavior of materials is determined by the existence of intrinsic space-time scales - size of grain, lattice period, relaxation time, etc. The orientational effects that cannot be described by equations of the classic theory of continuum mechanics occur in nanostructured material [1]. In the phenomenological description of such media it is supposed that its representative volume contains discrete material microvolumes (structural elements). Two vector fields can describe the kinematics of oriented medium: field of particles displacements and field of microrotations. In this paper it has been shown that the variational approach allows one to construct effectively the nonlinear mathematical models of nanostructured media in terms of both Lagrange's and Euler's variables. The variational principle for oriented medium has been formulated, from which the variational equations of dynamics and their integrals of motion have been found. The first sets of variational equations describe macromotions (i.e. motions of mass centers of the particles), and the second one describe microrotations of structural elements [2]. Equivalence of the variational equations and local laws of conservation of the energy, momentum and angular momentum in terms of Euler's variables has been proved using an example of liquid crystal. It has been shown that the Ericsen's stress tensor and the molecular field in liquid crystal are defined as partial derivatives of free energy. Research was done under the Presidential Program of Support for Leading Scientific Schools of Russia, and at partial support by grants of RFBR (project N 04-02-17156). REFERENCES [1] Eringen, A.C., Microcontinuum Field Theories. 1: Foundation and solids. Springer Verlag, New York Inc. 1999. [2] V.B.Lisin, A.I.Potapov, Variational principle in mechanics of liquid crystals, Int. J. Non-Linear Mech., 1997, 32, ʋ 1, pp. 55-62.

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Homogenization of single-walled carbon nanotubes Tanguy Messager, Patrice Cartraud Institut de Recherche en Génie civil et Mécanique (GeM), UMR CNRS 6183 Ecole Centrale de Nantes, BP 92101, 44321 Nantes cédex3, France [email protected], [email protected]

ABSTRACT This work deals with the computation of the overall axial elastic behavior of single-walled carbon nanotubes (SWCNTs). The SWCNTs are modeled as space-frame structures, using beam elements to represent atomic bonds [1]. The application of homogenization theory [2] enables to derive rigorously the macroscopic anisotropic beam behavior of the SWCNT, from the solution of three-dimensional basic cell problems. Moreover, taking benefit of the two helical symmetries [3] of the microstructure, the basic cell can be reduced to only one half of an hexagon, as depicted in Fig.1 for a zigzag SWCNT. Therefore, the overall stiffness coefficients can be computed efficiently using very concise FE models (including only 3 beam elements): the helical symmetry properties of the displacement field lead to a set of linear relationships expressed in local cylindrical axes between the opposite nodes m and n (see Fig.1) then acting as boundary conditions [3]. The accuracy of this approach has been assessed with respect to reference solutions of the literature [1] and also from comparison with results given by large FE model (left part of Fig.1). This method has been applied for the computation of zigzag and armchair SWCNTs: as shown in Fig.2, the developed procedure allows to study the scale effects.

Fig.1: Overall structure and microscopic FE model

Fig.2: tensile stiffness evolution

References [1] K.I. Tserpes, P. Papanikos, Finite element modeling of single-walled carbon nanotubes. Composites Part B, 36, 468-477, 2005. [2] P. Cartraud, T. Messager, Computational homogenization of periodic beam-like structures. Int. J. of Solids and Structures, in press, 2005. [3] T. Messager, P. Cartraud, Homogenization of helical beam-like structures. Finite Elements in Analysis and Design, submitted, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Phonon scattering by perturbed multichannel waveguides M. S. Rabia Laboratoire de Mécanique, Structures et Energétique, Département de Génie Mécanique, Université M. Mammeri, Tizi-Ouzou 15000, Algérie. [email protected]

ABSTRACT During the last two decades, scattering and localization phenomena in disordered mesoscopic systems have been well established as well experimentally as theoretically. They are actually of renewed interest owing to advances in nanotechnologies, the basic motivation being the need to understand the limitations that structural disorder may have on the physical and mechanical properties of nanocrystalline materials. Our present knowledge of the related phenomena has been given by the work of Landauer who has related the conductance of the sample to its scattering matrix. His approach reveals further the essential difference between elastic and inelastic scattering regimes: the latter is responsible of dissipation of energy, whereas the no dissipative and phase conserving elastic process introduce quantum interference effects due to the coherent scattering between defects. This interpretation has stimulated many researchers to look for the effects of quantum coherence in dc transport particularly. Recently, several authors have shown that multiple scattering and quantum interference become very important to describe transport phenomena, localization of electron states in disordered media, coherent magnetotransport and to investigate structural properties of lowdimensional samples. In the present work, we concentrate on the influence of local defect on scattering properties of elastic waves in perturbed crystalline quasi-three-dimensional structure in the harmonic approximation. Our model consists of three infinite atomic plans, assimilated to a perfect waveguide in which a scatterer (or defect) is inserted in bulk or in surface. The numerical treatment of the problem, based on the Landauer approach, resorts to the matching method initially employed for the study of surface localized phonons and resonances. Numerical results show that the interferences between the multiple scattered waves give rise to a broad variety of structures in transmission (or conductance) spectra which can be regarded as identifying features of the specific defect structures and may therefore be used for their characterization. Some of these structures, resulting from the interferences between incidental and reflected waves, correspond to Fabry-Pérot oscillations and others, due to the coupling between propagating modes and a defect induced states, are identified to Fano resonances. The interaction between defect-induced states and propagating waveguide eigenmodes could provide an interesting alternative to investigate structural properties.

References [1] B. Kramer, Quantum Coherence in Mesoscopic Systems, (plenum, New York, 1991). [2] R. Landauer, Z. Phys. B 68, 217 (1987); J. Phys. Condens. Matter 1, 8099 (1989). [3] A Fellay, F. Gagel, K. Maschke, A. Virlouvet and A. Khater, Phys.Rev., B 55, 1707 (1997). [5] M. S. Rabia, H. Aouchiche and O. Lamrous, Eur. Phys. J. – A. P. 23, 95-102 (2003).

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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An Atomistic-Information-Based Continuum Inhomogeneous Material Model for Metal Nanorod H. A. Wu, X. X. Wang CAS Key Laboratory of Materials Behavior and Design, Department of Mechanics and Mechanical Engineering, University of Science and Technology of China, Hefei, Anhui 230026, China [email protected]

ABSTRACT Mechanical behaviors of materials and structures at nano scale are essentially different from those at macro scale, resulting from surface effect, size effect and time scale effect. To correctly predict the structure-property relations of elemental nanocomponents are very important for the design of nanodevices. Atomistic simulations have been widely used to investigate nanomechanics. The equivalent elastic modulus of a copper nanorod under extension can be obtained using molecular dynamics simulation [1]. In our previous work, it was found that the correct deflection could not be obtained when above equivalent elastic modulus was used to predict the bending behaviors of nanorod [2]. The error was even up to 50%, compared with the direct atomistic simulation. The aim of this work is to investigate the original mechanism of above discrepancy and to present a novel continuum model to predict the bending modulus correctly. We owe this difference mainly to the surface effect. The ratio of surface atoms to the totality is about 60% for copper nanorod with cross-section size of 2nm. In some approximation, the nanorod can be considered as continuum. However, it is not homogeneous across the section. In our continuum model, the nanorod is considered as inhomogeneous material. The material constants of surface atoms and inner atoms are calculated from atomistic simulation, so our material parameters for continuum model of metal nanorod are based on the atomistic information. A finite element analysis of bending is carried out. The result agrees with direct atomistic simulation well, which validates the continuum nanorod model. Further work is to incorporate the research output as a new material model into commercial finite element software. With the development of accurate inter-atomic potentials for a range of materials, classical MD simulations have become prominent as a tool for elucidating the mechanical behaviors of nano-structures. However, the length and time scales that can be probed using MD are still fairly limited. Our continuum metal nanorod model includes nano-effects and supplies another way to study nanomechanics.

References [1] H. A. Wu, Molecular dynamics study on mechanics of metal nanowire. Mechanics Research Communications, 33(1), 9-16, 2006. [2] H. A. Wu, Molecular dynamics simulation of loading rate and surface effects on the elastic bending behavior of metal nanorod. Computational Materials Science, 31(3-4), 287-291, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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On the Modeling of Deformation-Diffusion-Damage Coupling in Elastic Solids Fernando P. Duda∗ , Leonardo J. Guimar˜aes† , Angela C. Souza‡ , Jos´e M. Barbosa† ∗ Universidade

Federal do Rio de Janeiro PEM-COPPE, Cidade Universit´aria, Rio de Janeiro, RJ, Brasil [email protected] † Universidade Federal de Pernambuco CTG-Cidade Universit´aria, Recife, PE, Brasil {leonardo,jmab}@ufpe.br ‡ Universidade Federal Fluminense LMTA-PGMEC-TEM, Niteroi, RJ, Brasil [email protected]

ABSTRACT This paper deals with the formulation and numerical implementation of a fully coupled continuum model for deformation-diffusion-damage in elastic solids. The formulation is carried out within the framework of continuum mechanics, where, in addition to the standard fields, extra fields are introduced in order to describe diffusion and damage processes. The governing equations are then obtained after supplementing the basic balances with a thermodynamically consistent constitutive theory. The couplings are implemented via the free energy response and include both deformation and damage assisted diffusion. It is worth mentioning that a gradient damage theory is obtained, which allows the modeling of fracture problems. The numerical implementation is based on the finite element method and a Euler implicit scheme for spatial and temporal discretizations, respectively. A numerical algorithm is presented to solve the discrete system of equations. In order to illustrate the potentiality of the proposed model, applications in the context of hydrogen embrittlement are presented.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Frost Growth on Cold Flat Plate: a Correlation for the Diffusion Resistance Factors Jorge A. Gatica*, Vicente A. Pita*, Nisio de C. Lobo B.† *

Universidad del Bío-Bío Concepción, Chile [email protected] [email protected]

†

Universidade Federal do Rio de Janeiro Rio de Janeiro, Brasil [email protected]

ABSTRACT The condensation of the vapor of water contained in the humid air on low temperature surfaces is a very frequent phenomenon. When the surface is below zero degrees, an ablimation of the vapor of water phenomenon happens, depositing as frost on the cold surface. In this work a numeric computational model is developed for to simulate the frost formation and his growth process on low temperature plates. The frost formation is a common operational problem in fins of evaporators used in low temperature warehouses. This deposit affects the heat transfer reducing the refrigeration capacity; therefore, one better understanding of this phenomenon is of great interest. During this work several analytic-experimental investigations were revised. Of particular interest they were the proposals [1, 2] relative to the effective diffusion coefficient. Based on a control volumes analysis a non-linear system of differential partial equations is obtained: a mass diffusion equation and energy equation besides a group of auxiliary relationships that allow to close the system and defining interface and contour conditions. The finite differences method is used to solve the system. The equations are transformed to linear equations and resolved iteratively using the Thomas' algorithm for tridiagonal systems. The program allows obtaining in transient form the distribution of temperatures in the frost layer, its variation of thickness and growth, also the influence on the same process of the Reynolds' number , the humid air temperatures, etc. A strong influence of the diffusion resistance factors used in the reproduction of experimental data was observed. Then, is generated a correlation to determine these factors that it turns out to be function of the humid air and the cold flat plate temperatures.

References [1] Y.X. Tao, R.W. Besant and K.S. Rezkallah, A mathematical model for predicting the densification and growth of frost on a flat plate, Int. Journal of Heat and Mass Transfer, 36, n. 2, 353-363, 1993. [2] R. Le Gall, J.M. Grillot and C. Jallut, Modeling of frost growth and densification. Int. Journal of Heat and Mass Transfer, 40, n. 13, 3177-3187, 1997.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Discontinuous Space-Time Galerkin Finite Element Method in Linear Dynamic Fully Coupled Thermoelastic Problems with Strain and Heat Flux Discontinuities Dinara K. Khalmanova∗ , Francesco Costanzo† The Pennsylvania State University Engineering Science and Mechanics Department, 212 EES Building, University Park, PA, 16802 ∗ [email protected] † [email protected]

ABSTRACT A discontinuous space-time Galerkin finite element method has been developed by the authors for the study of linear elasto-dynamic and fully coupled thermoelastic problems with discontinuities in the displacement and temperature gradients. The method is proven to be unconditionally stable and capable of automatic adaptive mesh refinement. The developed formulation has been implemented to solve a number of model solid-solid phase transition problems in one and two dimensions. The results of numerical study are presented to illustrate the convergence to an analytical solution of a problem with smooth coefficients. The presented method is particularly suitable for the study of thermoelastic and elasto-dynamic moving boundary problems and can be applied to dynamic fracture problems in heterogeneous thermoelastic materials using cohesive zone models.

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Pore gas interaction in polymeric foams with respect to energy absorption Mats Landervik and Ragnar Larsson Department of Applied Mechanics Chalmers University of Technology SE-412 96 G¨oteborg, Sweden [email protected], [email protected]

ABSTRACT The main object of the present paper is to investigate the significance of gas-solid interaction in foamed polymers subjected to large deformations, particularly in compression, in combination with high rates thereof. The present development thus represents an extension of previous work in [1], where we developed a phenomenological model representing the response of the sole cellular network on the basis of a viscoplastic Perzyna model to account for the rate dependence of the cellular network response. In this context, a main feature of the paper concerns the establishment of gas-filled foams in the context of a two-phase porous material within the Theory of Porous Media [2], where the interaction between the phases is modeled in terms of a deformation dependent Darcian filter law [3]. Thereby, a continuum mechanical coupling between the gas pressure and the deformation of the cellular network is obtained. We propose to resolve this coupling in terms of a staggered solution procedure similar to the staggered handling of the thermo-mechanically coupled problem [4]. ”Constant pressure” decoupling, where the pressure is considered constant during the mechanical step, is used. The model is implemented in the finite element code LS-DYNA, and the paper is concluded by representative numerical simulations. As main feature of the paper, a parametric study in terms of the permeability properties of the foam is carried out to display the significance of the gas-solid interaction for polymeric foam used for energy absorption in vehicles.

References [1] M. Landervik and R. Larsson. Modeling of large inelastic deformations of foams with respect to energy absorption, Submitted for international publication, 2006. [2] R. de Boer. Highlights in the historical development of the porous media theory: Toward a consistent macroscopic theory. Appl. Mech. Rev., 49, 201–262, 1996. [3] W. Ehlers and B. Markert. A macroscopic finite strain model for cellular polymers. Int. J. Plast., 19, 961-976, 2003 [4] E. Armero and J.C. Simo. A priori stability estimates and unconditionally stable product formula algorithms for nonlinear coupled thermoplasticity, Int. J. Plast., 9, 749–782, 1993

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A thermo-hydro–damage model for the dehydration creep of concrete subjected to high temperature Fekri Meftah* and Hassen Sabeur*,† * Laboratoire de Mécanique – Université Marne La Vallée 5 boulevard Descartes – 77454 Marne La Vallée Cedex 2 – France e-mail: [email protected]

†

LAMI – Ecole Nationale des Ponts & Chaussées 8 avenue Blaise Pascal – 77455 Marne La Vallée Cedex 2 – France [email protected]

ABSTRACT When concrete is ascribed to combined mechanical loads and high temperature distributions, it exhibits strains which are conventionally [1] split to a set of additive components: - Stress-free components, referred to thermo-hygral strains, which include thermal expansion and hygral shrinkage due to both drying and dehydration. - Stress induced thermal strains which mainly consist in a temperature depend elastic strain, a micro-cracking strain and an additional component, commonly referred to as transient creep [1,2,3]. This additional component is generally related to the fact that physical transformations, such as drying and dehydration, are occurring under sustained stress fields, which lead to a rearrangement of the evolutionary microstructure and give rise to this macroscopically measured strain. In this contribution, a full coupled thermo-hydro-mechanical model is proposed for the modeling of the transient creep in the range of 105°C-400°C, which is referred here to as dehydration creep. In this model, a dehydration variable is introduced to describe chemical transformations due to the temperature increase. It also allows to govern the occurrence of the dehydration creep when the stress level does not exceeds 40% of the ultimate strength. In addition, the proposed model considers that the dehydration creep occurs with a kinetics controlled by the relaxation time of the dehydration process. Further, the model allows to describe an irrecoverable transient creep upon cooling or during a second heating to the same maximum dehydration level. This model have be implemented in a the finite element code CAST3M. The algorithm for the update, at the constitutive level, of the mechanical behavior is presented. A particular attention is the given to the treatment of the transient creep component. Numerical simulations are performed to assess the capability of the model to predict transient load induced thermal behavior of concrete.

References [1] G.A. Khoury, P.J.E. Sullivan and B.N. Grainger, Strain of concrete during first heating to 600°C under load. Magazine of Concrete Research, 37 (133), 1985. [2] Y. Anderberg, Fire-exposed Hyperstatic Concrete structures- An experimental and Theorical Study, Div. of Struct. Mech. And Concrete Constr., Inst. Of Tech., Lund,1976. [3] U. Schneider, Concrete at high temperature: A general review. Fire Safety Journal, 13, 55-68, 1988.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A time-space framework suitable for the LATIN computational strategy for multiphysics problems David Néron , Pierre Ladevèze

,∗

and Bernhard A. Schrefler2

LMT-Cachan (ENS de Cachan/CNRS UMR8535/Paris 6 University) 61, avenue du Président Wilson, F-94235 CACHAN CEDEX, France {neron, ladeveze}@lmt.ens-cachan.fr 2

Department of Structural and Transportation Engineering (University of Padova) Via Marzolo 9, I-35131 PADOVA, Italy [email protected] ∗

EADS Foundation Chair “Advanced Computational Structural Mechanics” ABSTRACT

Since the last few decades, the simulation of multiphysics phenomena has become one of the major issue in the design of computational methods. Indeed, these problems often lead to computationally intensive analyses and then strategies to keep these problems affordable are of special interest. In this context, a strategy based on the LArge Time INcrement (LATIN) method [1] was introduced in [2] and also defined in the general framework of discretized problems in [3]. The main idea of this new method was the concept of interface ‘between physics’, which can be viewed as an extension of the ‘material’ interface classically introduced between two substructures [4]. This concept made it possible to introduce in a simple way several improvements (use of different discretizations for space and time, radial loading approximation, treatment of nonlinearities [3, 5]). The proposed test case concerned the consolidation of saturated porous soils and the LATIN strategy was compared to ISPP, a standard partitioning scheme, showing a significant decrease in computational cost [2, 5]. The aim of this paper is to introduce a general time-space framework suitable for the LATIN computational strategy for multiphysics problems. This framework is based on the concept of generalized radial loading [6], namely on the representation of all fields in an algebra generated by sums of products of time functions per space functions. The treatment of multiphysics problems in this framework enables a important gain in terms of computational and storage costs.

References [1] P. Ladevèze. Nonlinear Computational Structural Mechanics – New Approaches and NonIncremental Methods of Calculation. Springer Verlag, 1999. [2] D. Dureisseix, P. Ladevèze and B. A. Schrefler. A computational strategy for multiphysics problems – application to poroelasticity. International Journal for Numerical Methods in Engineering, 56(10):1489–1510, 2003. [3] P. Ladevèze, D. Néron and B. A. Schrefler. A computational strategy suitable for multiphysics problems. Proceedings of the First International Conference on Computational Methods for Coupled Problems in Science and Engineering, 2005. [4] P. Ladevèze, D. Néron and P. Gosselet. On a mixed and multiscale domain decomposition method. Submitted to Computer Methods in Applied Mechanics and Engineering. [5] D. Dureisseix, P. Ladevèze, D. Néron and B. A. Schrefler. A multi-time-scale strategy for multiphysics problems: application to poroelasticity. International Journal of Multiscale Computational Engineering, 1(4):387–400, 2003. [6] P. Ladevèze. A technique relative to the LATIN method for the calculation of time and space integrals. Internal Report of LMT-Cachan, 193, 1997.

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Finite Element Analysis of the Thermomechanical Coupling in Quenching of Steel Cylinders Using a Constitutive Model with Diffusional Phase Transformations Wendell P. de Oliveira*, Marcelo A. Savi*, Pedro M. C. L. Pacheco†, and Luís F. G. de Souza† * UFRJ/COPPE – Department of Mechanical Engineering 21.945.970 – Rio de Janeiro – RJ, P.O. Box 68.503 - Brazil [email protected], [email protected] † 1 CEFET/RJ - Department of Mechanical Engineering Av. Maracanã, 229, 20271-110 - Rio de Janeiro - RJ - Brazil [email protected], [email protected]

ABSTRACT Quenching is a heat treatment usually employed in industrial processes. It provides a mean to control mechanical properties of steels as toughness and hardness. Phenomenological aspects of quenching involve couplings among different physical processes and its description is unusually complex. Basically, three couplings are essential: thermal, phase transformation and mechanical phenomena. This article deals with the modeling and simulation of quenching in steel cylinders using a multiphase constitutive model with internal variables formulated within the framework of continuum mechanics and the thermodynamics of irreversible processes. The energy equation thermomechanical coupling terms are exploited, considering two different models. The first one is an uncoupled model where thermo-mechanical couplings are neglected, corresponding to the rigid body energy equation. The second model considers the latent heat associated with phase transformation in order to represent thermomechanical coupling. A numerical procedure is developed based on the operator split technique associated with an iterative numerical scheme in order to deal with non-linearities in the formulation. With this assumption, the coupled governing equations are solved to obtain the temperature, stress and phase fields from four uncoupled problems: thermal, phase transformation, thermo-elastic and elastoplastic. Finite element method is employed for spatial discretization. The proposed general formulation is applied to the through hardening of steel cylinders. Numerical simulations present a good agreement with experimental data for temperature and phase transformation distributions, indicating some situations where it is important to consider the thermomechanical coupling in the description of quenching process.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A combined fracture-micromechanics model for tensile strain-softening in brittle materials, based on propagation of interacting microcracks Bernhard Pichler∗ , Christian Hellmich∗ , Herbert Mang∗ ∗ Vienna

University of Technology (TU Wien), Institute for Mechanics of Materials and Structures Karlsplatz 13/202, A-1040 Wien (Vienna), Austria [email protected], [email protected], [email protected]

ABSTRACT Strain-softening is the decline in stress at increasing strain. Although microcracking is a commonly accepted reason for strain-softening, the majority of theoretical developments involve macroscopic damage evolution laws [3, 2]. To improve this situation, we propose a micromechanics-based damage evolution law by coupling two seemingly separated scientific fields, i.e. by combining (i) the propagation criterion for a single penny-shaped crack embedded in an infinite matrix subjected to remote stresses (taken from linear-elastic fracture mechanics) and (ii) stiffness estimates for representative material volumes comprising interacting microcracks (taken from continuum micromechanics [4, 1]). This combination allows for modeling tensile strain-softening as a result of propagation of interacting microcracks, i.e. as a microstructural effect. The initial degree of damage, i.e. the initial microcrack size and the number of microcracks per unit volume, implies two different types of model-predicted tensile strain-softening behavior under strain control: (i) continuous strain-softening, which occurs in case of initial damage beyond a critical value, and (ii) an instantaneous stress drop at the peak load (”snap-back”), which occurs in case of initial damage below a critical value.

References [1] Y. Benveniste. On the Mori-Tanaka’s method in cracked bodies. Mechanical Research Communications, 13(4):193–201, 1986. [2] V. Pens´ee, D. Kondo, and L. Dormieux. Micromechanical analysis of anisotropic damage in brittle materials. Journal of Engineering Mechanics (ASCE), 128(8):889 – 897, 2002. [3] P.C. Prat and Baˇzant Z.P. Tangential stiffness of elastic materials with systems of growing or closing cracks. 45(4):611–636, 1997. See also: Addendum (with Errata). J. Mech. Phys. Solids 45(8):1419–1420, 1997. [4] A. Zaoui. Continuum micromechanics: Survey. Journal of Engineering Mechanics (ASCE), 128(8):808 – 816, 2002.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Transient Dynamic Response of Thermoelastic Cylindrical Layered Media Ö. ùEN*, D. TURHAN† *

The Scientific & Technical Research Council of Turkey Defense Industries Research & Development Institute 06261 Ankara, Turkey [email protected] †

Department of Engineering Sciences Middle East Technical University 06531 Ankara, Turkey [email protected]

ABSTRACT In this study, the transient dynamic response of thermoelastic, hollow circular cylindrical composites consisting of n-different isotropic, homogeneous and elastic layers is investigated. Thermomechanical behavior of each layer is governed by the equations of generalized thermoelasticity with two relaxation times predicting finite wave speeds for thermal disturbances [1-2]. The body is subjected to uniform dynamic mechanical and thermal inputs at inner and/or outer surfaces. The time dependence of the dynamic inputs may be arbitrary. The cylindrical composite is of finite thickness in the radial direction and extends to infinity in the axial direction. The layers of the body are assumed to be perfectly bonded to each other. Furthermore, the layered medium is assumed to be initially at rest. The governing field equations of generalized thermoelasticity with two relaxation times are applied to each layer and the solutions are required to satisfy the continuity conditions at the interfaces of the layers, the boundary conditions at the inner and outer surfaces, and quiescent initial conditions. Method of characteristics is employed to obtain the solutions [3]. The convergence and numerical stability of the method are well established, and different interface, boundary and initial conditions can be handled conveniently. The solutions reveal the existence of two wave fronts. The numerical results are displayed in curves denoting the variations of circumferential and radial stresses and temperature deviation with time at different locations and variations of stresses and temperature deviation along the thickness of the bodies at different times. The solutions reveal clearly the thermal and geometric dispersions in the wave profiles and the effects of refractions and reflections at the interfaces and at inner and outer surfaces of the body. The curves further display the severe variations in the field variables at the wave fronts.

References [1] A. E. Green and K. A. Lindsay, Thermoelasticity. Journal of Elasticity, 2, 1-7, 1972. [2] E. S. Suhubi, Thermoelastic Solids In Continuum Physics (A. C. Eringen, editor). New

York: Academic Press, 2, 279-265, 1975. [3] R. Courant and D. Hilbert, Methods of Mathematical Physics. New York: Interscience, 2, 1966.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Novel nonlocal continuum formulations. Part 1: gradient elasticity based on nonlocal displacements and nonlocal strains Harm Askes∗ , Miguel A. Guti´errez† , Antonio Rodr´ıguez-Ferran‡ ∗ University

of Sheffield Department of Civil and Structural Engineering Sheffield S1 3JD, United Kingdom h.askes@sheffield.ac.uk † Delft University of Technology Faculty of Aerospace Engineering Delft, The Netherlands [email protected] ‡ Universitat Polit` ecnica de Catalunya Department of Applied Mathematics III Barcelona, Spain [email protected]

ABSTRACT Nonlocal continuum formulations exist in integral formats and differential formats. The latter, also known as gradient-enriched continua, have successfully been applied in elasticity, plasticity and damage and provide a robust framework to analyse size effects and dispersive waves. Moreover, gradient enhancement can be used to remove singularities from elastic fields as well as guarantee well-posedness in the modelling of post-peak phenomena. In this paper, the focus will be on novel formulations for gradient elasticity. Aifantis and coworkers suggested a simple format of gradient elasticity in which the usual stress-strain is augmented with an additional term, namely the Laplacian of the strain. This then leads to a fourthorder differential equation in terms of the displacement and, hence, C1 continuity requirements in case of a numerical implementation. However, it is possible to split the fourth-order equations into two sets of second-order equations, whereby the first set coincides with the equations of classical elasticity and the second set comprises a set of diffusion-type equations that introduce the gradient enrichment. Hence, a staggered solution algorithm is obtained, whereby the displacements of classical elasticity are computed first and then used as input for the second set of equations in order to compute the gradient-enriched displacements. The staggered, displacement-based approach will be scrutinised together with two alternative formulations of gradient elasticity: (i) a staggered, strain-based approach, and (ii) another strain-based approach that can be derived from Pade approximations of the previous method. The three approaches will be presented with their boundary conditions, and it will be verified whether all singularities are removed from the strain field. Also, size effect tests will be reported. In the accompanying paper, the displacement-based approach will be used in the formulation of nonlinear gradient models.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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On the use of a damage model based on non-local displacements in the Element-Free Galerkin method Terry Bennett∗ , Sivakumar Kulasegaram† ∗ Cardiff

School of Engineering Cardiff University, UK [email protected]

† Cardiff

School of Engineering Cardiff University, UK [email protected] ABSTRACT

One of the ingredients of Element-Free Galerkin (EFG) shape functions are the weight functions as employed in Moving Least Squares interpolations [1]. They are similar to the weight functions used in non-local damage [2]. Therefore, it seems a natural choice to use non-local damage as a localisation limiter within the EFG method. This contribution investigates the need for regularisation techniques within the EFG method as meshfree approximations can be already considered to possess intrinsic non-local properties [3]. However, one must make a distinction between the mathematical non-local properties of the interpolation method (whether it be Finite Elements, EFG etc), and the mechanical non-local properties of the constitutive model. The recently developed displacement based non-local damage model [4] is utilised here to regularise damage within the EFG framework and provide further computational efficiency over strain based nonlocal methods due to (typically) less nodes being employed than integration points and less components for the displacements compared to the strains. The formulation of the displacement based non-local damage within the EFG framework is elucidated and examples of the new model’s regularisation capabilities is compared to a local damage formulation within the EFG method.

References [1] T. Belytschko, Element-Free Galerkin Methods. International Journal for Numerical Methods in Engineering, 37, 229–256, 1994. [2] G. Pijaudier-Cabot and Z.P. Bazant. Nonlocal damage theory. Journal of Engineering Mechanics (ASCE), 118(10), 1512–1533, 1987. [3] J-S. Chen, C-T. Wu and T. Belytchko, Regualrisation of material instabilities by meshfree approximations with intrinsic length scales. International Journal for Numerical Methods in Engineering, 47, 1303-1322, 2000. [4] A. Rodriguez-Ferran, I. Morata and A. Huerta, A new damage model based on non-local displacements. International Journal for Numerical and Analytical Methods in Geomechanics, 29, 473–493, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Modelling of Reinforced Concrete Beams Strengthened With Pre-Stressed CFRP 1

2

Thiago Domingues , J. Alfaiate

1 Instituto Superior Técnico and ICIST Av. Rovisco Pais 1049-001 Lisbon, Portugal [email protected] 2 Instituto Superior Técnico and ICIST Av. Rovisco Pais 1049-001 Lisbon, Portugal [email protected]

ABSTRACT In this work a numerical simulation on the behaviour of reinforced concrete beams, strengthened with pre-stressed CFRP at the lateral faces of the beams is presented. The numerical results are compared to experimental results obtained from a testing campaign made in 2004 by França [1]. In this simulation, several hypotheses are adopted related to the material and numerical models, namely: i) cracked concrete is modelled using a discrete cracking approach and strong discontinuities embedded in the finite elements; ii) an elastoplastic behaviour is adopted for concrete under compression; iii) reinforced bars and CFRP are also modelled using an elastoplastic stress-strain relationship; iv) the bond-slip relationship adopted between the concrete and reinforcement in tension is based on the MODEL CODE 1990 and v) a mode-II bond-slip relationship is adopted between the concrete and CFRP, ac-cording to the work presented by Costa[2]. Four beams with T section are studied. Two of them are reference beams, without CFRP pre-stressed and the other two beams are reinforced with pre-stressed CFRP, before and after the support, respectively. From this simulation some conclusions are obtained, related to the verification of the above hypotheses and the behaviour to the ultimate limit and serviceability states. From the numerical analysis it has been possible to verify the failure mechanism of each beam studied, such as crushing of concrete, bond slip and/or yielding of the tension reinforcement, as well as cracking of the interface between the concrete and the CFRP pre-stressed laminate. Cracking patterns and the deformed mesh are also an output of the analysis, allowing to determine the localization of the main cracks.

References [1] P. França, A. Costa and J. Appleton, Reforço de Estruturas com Laminados de CFRP Préesforçados. Encontro Nacional de Betão Estrutural, FEUP, Porto, Portugal, 759-766, 2004. [2] R. Costa, Modelação de Vigas de Betão Armado Reforçadas com Chapas Metálicas. Dissertação de Mestrado em Engenharia de Estruturas, IST, Lisboa, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Numerical and experimental studies of damage in porous materials ∗, , G.J. Creus‡ , J. Gr´ ¨ T. Fiedler∗ , L. Cunda† , A. Ochsner acio∗ ∗ Centre

†

for Mechanical Technology and Automation, University of Aveiro, Aveiro, Portugal Department of Mechanical Engineering, University of Aveiro, Portugal tfi[email protected], [email protected], [email protected]

Departamento de Materiais e Construc¸a˜ o, Fundac¸a˜ o Universidade Federal do Rio Grande, Rio Grande, Brasil. [email protected]

‡ Programa

de P´os-Graduac¸a˜ o em Engenharia Civil, Universidade Federal do Rio Grande do Sul, Porto Alegre, Brasil [email protected]

ABSTRACT Metallic foams comprise a steadily growing group of porous materials and are gaining more and more importance in industrial applications. Main advantages of this innovative material are high ability of energy adsorption and high specific stiffness. Accordingly, porous materials are utilised in lightweight construction and passive safety components which requires exact prediction of the evolution of damage and its impact on the mechanical behaviour of metallic foams. Two different approaches, namely finite element investigations and experiments, are applied. The effect of damage is mathematically described by the Gurson-Tvergaard model. This formulation is based on the isotropic damage parameter volumetric void fraction. The evolution of microscopic damage, respectively increase of volumetric void fraction, can be monitored by macroscopic parameters. Therefore, e.g. Young’s modulus is utilised for the determination of damage parameters. In order to compare the numerical findings to experimental results, the finite element analysis is confined on easy to manufacture, periodic porous geometries. The intention of this work is the development of computational and experimental procedures to analyse the damage in metallic foams. In the last instance, the determination of adequate damage parameters for foams is projected.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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The Development of a Continuum Damage Model for Fibre Metal Laminate Structures Ronan M. Frizzell, Conor T. McCarthy, Daire Cronin, Michael A. McCarthy, Ronan M. O’Higgins Composites Research Centre, Dept. of Mechanical & Aeronautical Engineering, University of Limerick, Limerick, Ireland. {ronan.frizzell, conor.mccarthy, daire.cronin, michael.mccarthy, ronan.ohiggins}@ul.ie

ABSTRACT Fibre Metal Laminates (FMLs) are a family of materials consisting of alternating layers of thin metal sheets and fibre-reinforced plastic. Glass composite based FMLs, commercially available under the name GLARE, have recently found application in the aircraft industry due to their excellent fatigue performance and impact properties. This work aims to develop a computational damage model for GLARE for use in finite element simulations. Unique challenges arise in modelling this material since it possesses both the elastic-plastic characteristics common to metals and the more brittle behaviour of glass based composite. Of particular interest to this project is the behaviour of GLARE in jointed configurations. The project proposes to develop a methodology for modelling damage initiation and growth in jointed GLARE structures and to validate the results against experimental data. An experimental investigation has been conducted on simple yet representative jointed GLARE structures. A pin-bearing test setup, without lateral constraints, has been used in order to model the central lap of a double-lap joint. Information has been gathered on the progression of damage and the failure mechanisms present in these joints. Specimens that promote bearing, net-tension and shear-out failure modes were examined. Damage in failed specimens and specimens tested to percentages of failure have been studied using microscopy. Results demonstrate that delamination between plies, matrix cracking and fibre failure are the dominant failure modes. In order to capture these complex failure modes, three-dimensional models of the pin-loaded GLARE specimens were developed in the non-linear finite element code ABAQUS. An in-house developed delamination model was used to predict the initiation and growth of delamination between the plies. This model uses three-dimensional failure criteria to predict delamination onset and a user-defined decohesion contact interface to model delamination growth. The model successfully predicted the initiation and growth of delamination in the joint configurations tested. A global-local sub-modelling approach was used to increase computational efficiency, and this is described. Current work is aimed at developing a continuum damage mechanics model to better capture the complex damage mechanisms that have been seen to occur in the composite plies. As a starting point, the Ladeveze [1] damage model has been implemented using a user-defined material subroutine UMAT, available in ABAQUS. This model is being extended to the three-dimensional case and its development and application will be described.

References [1]

P. Ladeveze, E. Le Dantec, Damage Modelling of the Elementrary Ply for Laminated Composites, Composites Science and Technology, 43, 257-267, 1992.

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Residual Strength of the Frost-Damaged Reinforced Concrete Beams Manouchehr Hassanzadeh* and Göran Fagerlund† *

Lund Institute of Technology, Division of Building Materials P.O. Box 118, SE22100, Lund, Sweden [email protected]

†

Lund Institute of Technology, Division of Building Materials P.O. Box 118, SE22100, Lund, Sweden [email protected]

ABSTRACT The most severe types of destruction mechanisms are those causing internal cracking and thereby loss of cohesion of the concrete, i.e. internal expansive attacks. The internal frost damage belongs to this category of destructive mechanisms. The frost attack causes a random system of cracks in the heart of the concrete together with cracks parallel to the surface of the concrete. In many cases there are also extended cracks parallel to joints and edges of the concrete, or emanating from corners. The amount of damage can vary from place to place in the same structure. Therefore, in most cases, series of data from many different places are required. Only in special cases data taken from one place can be used for the entire structure. Internal frost damage appears as loss in compressive and tensile strength, loss in E-modulus, and loss in the bond between the concrete and the reinforcement. Reductions in tensile and bond strength can be as high as 70%, or more. The effect on compressive strength is often limited to about 30% for normal grade concrete. The effect on E-modulus can be extremely high. Internal damage will affect the moment and shear capacity of slabs and beams, and the compression capacity of columns. It might seriously affect the structural capacity of pre-stressed concrete by significantly lowering the E-modulus of the concrete. It also changes the moment and force distribution in the structure by changing the stiffness in parts of the structure. In order to study the structural effects of the internal frost-damage large reinforced concrete beams were subjected to frost attack. The frost in combination with high degree of saturation induced internal cracks in concrete. The internal cracks reduced the strength of the reinforced beams and in some cases also changed the designed failure mode of the beams. For instance, beams which in an undamaged condition would fail due yielding of the reinforcement failed as a result of compression fracture of concrete caused by the loss of strength due to internal frost-damage. Results of this investigation show that internal frost damage, besides causing loss of strength, also causes reduction of the stiffness and extensive visible cracking. Furthermore, this investigation shows that the remaining load bearing capacity of the beams is remarkably high despite the big concrete destruction.

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Numerical and experimental evaluation of damage parameters for textile reinforced concrete under cyclic loading Martin Konrad∗ , Rostislav Chudoba ∗ , Bong-Gu Kang † ∗ Chair

of Structural Statics and Dynamics, RWTH-Aachen University Mies-van-der Rohe-Strae 1, 52074 Aachen, Germany [email protected]

† Institute

of Building Materials Research, RWTH-Aachen University Schinkelstr. 3, 52062 Aachen, Germany ABSTRACT

The textile reinforced concrete (TRC) has emerged in the last decade as a new composite material combining the textile reinforcement with cementitious matrix. Its appealing feature is the possibility to produce filigree high-performance structural elements that are not prone to corrosion as it is the case for steel reinforced concrete. In comparison with other composite materials, textile reinforced concrete exhibits a high degree of heterogeneity and imperfection that requires special treatment in the development of numerical models. The prerequisite for correct modeling of the bonding behavior is the proper representation of the damage process in the crack bridges. In a way the crack bridge can modelled in matters of a tensile test on yarn with an extremely short length extended with a shear-lag-like clamping of filaments. Due to the varying penetration profile along the yarn, the quality of the shear lag clamping exhibits high scatter. In the applied deterministic and stochastic bond layer models the scatter in the interface layer and the disorder in the filament bundle are reflected by distributions (1) of the bond quality ϕ, (2) of the bond free length λ and (3) of the delayed activation θ of filaments within the bond free length. The failure process in the bond layer next to the crack bridge involves both the damage of filaments and of the bond between the filaments and the matrix. In order to quantify the separate damage laws the bond layer model has been equipped with a material model that combines plastic softening with damage evolution.

The damage laws are derived using the measured load-displacement curves of the double-sided pull-out test with cyclic loading. In combination with the FILT-test (Failure Investigation using Light Transmission) these tests yield additionally the curve representing the instantaneous fraction of the broken filaments during the loading process. This important information will be utilized to verify the damage laws.

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Modeling the behavior of reinforced concrete beams strengthened with FRP P. Neto*, J. Alfaiate†, J. Vinagre† *

Escola Superior de Tecnologia do Barreiro/IPS Rua Stinville, nº 14, 2830-144 Barreiro [email protected] † Instituto Superior Técnico/UTL Av. Rovisco Pais, 1049-001 Lisboa [email protected] [email protected]

ABSTRACT The strengthening of reinforced concrete structures with fiber reinforced polymers (FRP) is particularly attractive due to their mechanical properties. The understanding of the premature failure modes is of great importance. Since rupture is frequently found to occur at the interface FRP-concrete, there is a clear need to study the nature of the bonding so as to develop techniques to permit its design modeling. The stress distribution in shear test models does not precisely match the one obtained in flexural reinforcement; in the latter, according to various authors [1], [2], in addition to the stresses tangential to the interface, normal stresses are also important. In this paper, a numerical model is presented to describe the behavior of reinforced concrete beams strengthened with FRP. This model is based on previous studies focused both on: i) the distribution of shear stresses at the interface FRP-concrete and on ii) the stress concentration at the plate ends in flexural models. Furthermore, the importance of the flexural cracks in the premature rupture of the element is also analyzed. The behavior of reinforced concrete beams strengthened with both FRP laminates and sheets is considered. The FRP-epoxy-concrete arrangement and the flexural cracks are modeled with interface elements with initial zero thickness, using a discrete approach and a localized damage model. A softening behavior is adopted to simulate the stress transfer along the FRP-concrete interface. The importance of considering the mixed mode of fracture is discussed. Mention is also made to some of the main mathematical models found in the literature.

References [1] A. M. Malek, H. Saadatmanesh, R. E. Mohammad, Prediction of failure load of r/c beams strengthened with FRP plate due to stress concentration at the plate end, ACI Structural Journal, 95(2), 142-152, 1998. [2] Z. S. Wu, H. D. Niu, Shear transfer along FRP-concrete interface in flexural members, Journal of Material, Concrete Structures and Pavements, JSCE, 49(662), 231-245, 2000.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Cracking Analysis in Concrete Dams using Isotropic Damage Models. Objectivity of Numerical Solutions 1

2

3

Sérgio Oliveira , Nelson Gaspar , Pedro Dinis 1

Laboratório Nacional de Engenharia Civil Av. do Brasil, 101, 1700-066, Lisboa, Portugal [email protected] 2 Instituto Superior de Engenharia de Lisboa R. Conselheiro Emídio Navarro, 1, 1950-062, Lisboa, Portugal

[email protected] 3 Instituto Superior Técnico Av. Rovisco Pais nº1 1049-001 Lisboa [email protected]

ABSTRACT The numerical simulation of cracking in large concrete structures can be made, in many cases, adopting the smeared cracking approach and using constitutive laws of continuous damage (with softening), in order to simulate the material tension ruptures. The consideration of a tension softening branch that depends on the value of the material fracture energy, implies the localization phenomena and requires the use of some specific numerical procedures in finite element analysis. Namely, consistent formulations evolving the energy dissipated during the rupture process must be used in order to obtain numerical results that do not dependent on the mesh discretisation – mesh objectivity. In this paper, a 3D finite element formulation and a constitutive law of isotropic damage, with two independent variables, conceived to model the tension and compression softening effects (independently), are presented. The finite element model is used in Fig A - Comparison of tensile the analysis of the Cabril Dam (the largest Portuguese arch dam) damages for the two meshes. when submitted to the self-weight and the hydrostatic pressure (water at crest level). Numerical results related with the cracks propagation for dif-ferent 3D finite element discretisations are presented, in order to analyze the solutions objectivity. These results consist of (i) the radial displacements, (ii) the principal stresses and (iii) the tensile damages at the dam (i) central cantilever and/or (ii) upstream and downstream faces (Fig.A).

References [1] Bazant, Z.P., Planas, J., Fracture and size effect in concrete and other quasi-brittle materials. CRC Press, USA, 1998. [2] Oliver, J., A consistent characteristic length for smeared cracking models. Int. J. of Numerical Methods in Engineering, Vol.28, pp. 461-474, 1989. [3] Oliveira, S.B., Modelos para análise do comportamento de barragens de betão considerando a fissuração e os efeitos do tempo. Formulações de dano (in Portuguese). Tese de doutoramento, FEUP, 2000.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Continuous-discontinuous modelling of dynamic failure of concrete using a viscoelastic viscoplastic damage model R.R. Pedersen, A. Simone and L.J Sluys Delft University of Technology, Faculty of Civil Engineering and Geosciences P.O. Box 5048, 2600 GA Delft The Netherlands [email protected]

ABSTRACT Concrete is a highly rate-dependent material at loading rates exceeding 15 GPa/s. This means that the apparent macroscopic mechanical properties of concrete depend on the applied loading rate. This has been determined, experimentally, for material strength and, to a smaller extent, for stiffness and fracture energy. The physical mechanisms responsible for the rate-dependency in high-rate dynamics are mainly inertia effects; moisture in nano- and micro pores contributes to an increase of the material parameters for moderate loading rates. Dynamic fracture of concrete is time dependent due to (i) viscoelastic material behaviour in the bulk material, and (ii) rate processes including inertia effects in the fracture process zone. In order to take the rate-dependency of concrete in dynamics into account we present a material model with viscous contribution to the bulk material in the elastic response and a viscous contribution to the cracked material. We elaborate the viscoelastic plastic model described in [2, 3] by coupling it to a viscoplastic damage model [4]. This model accounts for the strengthening effect associated with the viscous phenomenon due to moisture and includes retardation of micro-crack growth with an increase of strain rate. In the combined continuous-discontinuous approach, a crack opening is inserted after some degradation of the continuum material stiffness. Displacement discontinuities are incorporated via the partition of unity concept. The viscosity in the elastic bulk material is related to porosity and saturation level, while in the viscoplastic model the viscosity is linked to the width of the micro cracked zone. With this computational tool we examine the influence of the loading rate on the shape and size of the process zone as well as the crack velocity, and the increase in fracture energy for higher loading rates due to micro branching instabilities for crack velocities exceeding a critical velocity [1].

References [1] S.P. Gross E. Sharon and J. Fineberg. Energy dissipation in dynamic fracture. Physical Review Letters, 76:2117–2120, 1996. [2] J. Sercombe, F.-J. Ulm, and H.-A. Mang. Consistent return mapping algorithm for chemoplastic constitutive laws with internal couplings. International Journal for Numerical Methods in Engineering, 47:75–100, 2000. [3] J. Sercombe, F.-J. Ulm, and F. Toutlemonde. Viscous hardening plasticity for concrete in high-rate dynamics. Journal of Engineering Mechanics, 124:1050–1057, 1998. [4] A. Simone and L. J. Sluys. The use of displacement discontinuities in a rate-dependent medium. Computer Methods in Applied Mechanics and Engineering, 193:3015–3033, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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On the Formulation of Damage Constitutive Models for Bimodular Anisotropic Media José Julio de Cerqueira Pituba* *

Engineering Department, University of Western Sao Paulo - UNOESTE Rodovia Raposo Tavares, km 572, 19067-175, Presidente Prudente, São Paulo, Brazil [email protected]

ABSTRACT This work deals with the formulation of constitutive laws for elastic media that start to present different behaviors in tension and compression as well as some anisotropy degree when damaged. Initially, a formulation for bimodular and anisotropic elastic media proposed by [1] is reviewed. In this formulation, for the modeling of a bimodular hyperelastic material, the elastic potential energy density must be once continuously differentiable (whole wise), but only piecewise twice continuously differentiable. The stress-strain relationship derived from this potential is piecewise continuously differentiable leading to an elasticity tensor discontinuous referred to a hypersurface that contains the origin and divides the strain space into a compression and tension sub-domains. In this way, the modeling is able to reproduce different response in tension and compression. Soon after, the formulation proposed by [1] is extended to take into account the damage process. Accordingly with, the coefficients named bulk and shear modulus are considered as functions of the damage state, so that the stress-strain relationship would be influenced by damage variables. Moreover, the hypersurface taken as the criterion for the identification of the constitutive responses in compression or tension would be also influenced by the damage variables. It must be observed that the condition no tangential discontinuity of the elasticity tensor is also valid in this formulation. Fourth-order anisotropic tensors are requested by the formulation depending on the class of anisotropy that is assumed. The matricial form of those tensors is presented in the end of the work. Some damage models [2, 3] found in the literature are written according to the formalism proposed here in order to show its potentiality. It must be observed that the objective of this work is to show the potentialities of the formalism proposed here in way to be used as a proper tool in the formulation of the damage models. Therefore, some aspects related to criteria and evolution laws of damage were not discussed. This work also presents contributions related to matricial forms of the fourth-order tensors involved in the formulation of damage models. Those forms are quite interesting to the computational implementations. Finally, the proposal of the formulation for the damaged media can be used in future works for the development of many constitutive models depending of the involved phenomena.

References [1] A. Curnier, Q. He and P. Zysset, Conewise linear elastic materials. Journal of Elasticity, 37, 138, 1995. [2] J. J. C. Pituba, On the formulation of a damage model for the concrete (in portuguese). Sao Carlos School of Engineering, University of Sao Paulo, 2003. [3] C. Comi, A nonlocal damage model with permanent strains for quasi-brittle materials. A. Benallal ed. Continuous Damage and Fracture, Cachan, France, 221-232, 2000.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Soil-Structure Interaction - Case History Analysis Involving Structural Damage Luciana M. P. Rosa1, Eliane M. L. Carvalho1, Bernadete R. Danziger2 1

Universidade Federal Fluminense Niterói, RJ - Brazil [email protected] and [email protected] 2

Universidade do Estado do Rio de Janeiro Rio de Janeiro, RJ - Brazil [email protected]

ABSTRACT This thesis presents a case history contemplating a soil x structure interaction analysis of a construction that revealed unsatisfactory foundation performance with time, showing cracks due to excessive distortional settlements. The analysis includes settlement predictions with and without due consideration of soil x structure interaction with the foundation soil. The analyses contain the prediction of settlement development with time, its uniformization tendency, the columns load redistribution and the relevant changes in the stresses of some structure sections. The predicted distortional settlements with and without due consideration of soil x structure interaction were compared and confronted in conformity with the documented structure damages in different periods of the construction life time. The structure has been analysed with aid of a three-dimensional linear elastic FEM model, without considering the rheological deformation of concrete. The Kelvin model has been employed in order to properly represent the soil behaviour with time. It has been found that, notwithstanding the simplifications adopted in the analysis and tentatively justified in various phases of the study, the results obtained in the numerical analysis were capable of duplicating the damages that occurred. The conclusions emphasized the importance of a more real design conception of the structure, including the foundation soil, possible to be implemented with the computational tools available at present.

References [1] F. A. B. Danziger, Parecer Geotécnico sobre os Problemas Verificados no Edifício de Vitória, ES. COPPE/UFRJ, Rio de Janeiro, 21p, 2002. [2] F. E. Barata and B. R. Danziger, Compressibilidade de Argilas Marinhas Moles Brasileiras. Anais do VIII Congresso Brasileiro de Mecânica dos Solos e Engenharia de Fundações. Porto Alegre, 14p, 1986. [3] J. H. C. Reis and N. Aoki, Análise de interação solo-estrutura em maciço de argila mole. Seminário de Interação Estrutura-Solo em Edifícios. São Carlos, SP, 14p, 2000. [4] L. M. P. Rosa, Interação Solo-Estrutura – Análise de um Caso de Obra Envolvendo Danos Estruturais. Tese de M.Sc, Engenharia Civil, Universidade Federal Fluminense, Niterói, RJ, 117p, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

373

Evolution Equation of Creep Damage Under Stress Variation Yukio Sanomura*, Kzutaka Saitoh† * Tamagawa University 6-1-1 Tamagawa-gakuen, Machida, Tokyo, 194-8610 Japan [email protected] †

Honda R&D Co. Ltd 4630 Simotakanezawa, Haga-machi, Haga-gun, Tochigi, 321-3321 Japan [email protected]

ABSTRACT Design and assessment of structural components at elevated temperature are very significant for ensuring the safety. Lear damage accumulation (summation of creep time fraction) is widely used to predict creep rupture time under stress and temperature variation. Life prediction of creep under stress variation by creep damage mechanics of Kachanov-Rabotnov concides with that of linear damage accumulation model. However, creep rupture time under stress variation is essentially the nonlinear problem. A modified evolution equation of creep damage by Kachanob-Rabotnov is formulated by memory effect of the previous stress. The evolution equation of creep damage consists of tow terms as follows:(1) the power damage law of Kachanov-Rabotnov, (2) acceleration anad deceleration by memory effect of the previous stress. The memory effect of the previous stress gradually disappears under the present stress. Additional internal state variable describing this effect is defined and the evolution equation is formulated in order to approach the present stress value from previous stress. The evolution equation of creep damage can be extended to the multiaxial state of stress with isotropic creep damage (scalar) and anisotropic creep damage (2nd symmetric tensor) proposed by Murakami and Ohno. Creep constitutive equation (McVetty type) for damaged material is formulated by the conventional creep damage theory. The model is lacking in the representation of the transient creep after increased stress. Micrographs of specimen ruptured under constant stress and stress variation are observed and creep damage mechanisms are discussed. After material constants are identified by describing creep curves at constant uniaxial stress, the validity of the proposed theory is discussed by the theoretical predictions with the corresponding experiments on tough pitch copper under nonsteady uniaxial stress at 250℃. The prediction of creep rupture time by the present theory is fairly good with the experiment results under stress variation.

References [1] S. Murakami, Y. Sanomura and K. Saitoh: Formulation of Cross-Hardening in Creep and its Effect on the Creep Damage Process of Copper, Trasactions of ASME, Journal of Engineering Materials, 108, 167-173, 1986.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

375

Hybrid and Mixed Finite Element Formulations for Softening Materials Cristina M. Silva, Luís M.S.S. Castro Departamento de Engenharia Civil e Arquitectura, Instituto Superior Técnico Avenida Rovisco Pais, 1049-001 Lisboa {cmsilva,luis}@civil.ist.utl.pt ABSTRACT The conforming displacement elements are nowadays dominant in standard finite element applications. Nevertheless, they present some known limitations, particulary in what concerns accuracy and safety in stress estimates. With computational development and the motivation to model increasingly more complex structural problems, several alternative numerical techniques have been proposed to substitute or complement the tradicional displacement formulation, e.g. the boundary elements, meshless models and hybrid and mixed formulations. The hybrid and mixed finite element formulations adopted in this work [2] are developed from first-principles of Mechanics, namely, equilibrium, compatibility and constitutive relations. Recently, these formulations have been tested with continuum damage models in order to correctly simulate the behavior of softening materials such as concrete [3, 5, 6, 4]. The most promising formulations are the hybrid-mixed stress formulation, with an independent approximation of the effective stress field instead of an approximation of the stress filed [6], and the hybrid displacement formulation [4]. The objective of this communication is to compare the numerical performance of these two alternative numerical techniques with each other and also with the usually adopted displacement finite element formulation. A simple isotropic integral nonlocal damage model is adopted [1] and all the approximation functions of the hybrid-mixed formulations are chosen as complete sets of orthogonal Legendre polynomials. A set of benchmark tests are presented and discussed. It is shown that the alternative techniques may be competitive, namely in terms of stress estimates and computational effort.

References [1] C. Comi and U. Perego. Nonlocal aspects of nonlocal damage analyses of concrete strucutres. European Journal of Finite Elements, 10:227–242, 2001. [2] J. A. T. Freitas, J. P. M. Almeida, and E. M. B. R. Pereira. Non-conventional formulations for the finite element method. Computational Mechanics, 23:488–501, 1999. [3] C. M. Silva and L. M. S. S. Castro. Hybrid-mixed stress model for the nonlinear analysis of concrete structures. Computers & Structures, 83:2381–2394, 2005. [4] C. M. Silva and L. M. S. S. Castro. Modelos híbridos de deslocamento com dano contínuo. In J. L. P. Aparicio, J. César de Sá, R. Delgado, R. Gallego, J. Martins, M. Pasadas, and A. R. Ferran, editors, Congreso de Métodos Numéricos en Ingeniería. SEMNI, 2005. [5] C. M. Silva and L. M. S. S. Castro. Hybrid-displacement (trefftz) formulation for softening materials. Computers & Structures, accept for publication, 2006. [6] C. M. Silva and L. M. S. S. Castro. Hybrid-mixed stress formulation using continuum damage models. Communications in Numerical Methods in Engineering, accept for publication, 2006.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Theoretical and computational aspects of an elastoplastic damage gradient non local model Denis.L.S.Sornin1 , Kh´ema¨ıs.Saanouni1 1 Institut

Charles Delaunay. Universit´e de Technologie de Troyes. FRE CNRS 2848. LASMIS 12, rue Marie Curie- BP 2060- 10000 TROYES cedex FRANCE [email protected] , [email protected] ABSTRACT

FEM results of softening material problems are known to show a pathological mesh dependency. To avoid this problem, the mechanical behavior in each integration point must take into account the immediate vicinity. Models including the influence of neigbourhood give rise to non local formulation in the early 70th. Nowadays, numerous formulations have been proposed to ensure mesh independency in post peak zone. The majority of non local models have been initially developed for concrete or brittle materials and can’t be easily generalized to ductile elastoplasticity. However, the case of damaging ductile materials have been treated in some papers ([1],[4]).These models introduce simples higher gradient formulations, able to provide mesh independent FEM results. This paper proposes a non local elastoplastic model fully coupled to damage . A thermodynamically consistent formulation involving a scalar damage field and its first gradient in the Helmholtz free energy is used. The formulation is based on a classical material theory accounting for a strong damage-behavior coupling. The strong coupling to the elastic and isotropic and kinematic hardening modulus is based on the energy equivalence assumption [2]. The local damage evolution law depends on a non local variable associated to the damage driving force [3]. This variable is the solution of a non local condition solved in a coupled fashion with the standard equilibrium equation [5]. This gives rise to a new finite element with additional DOFs. The relevant numerical aspects related to the FEM development and material integration scheme are presented. The damage gradient model is coded using both UEL and UMAT subroutines of ABAQUS/standard. Comparison between the standard local and non local formulations are discussed.

References [1] B.Nedjar, Elastoplastic-damage modelling including the gradient of damage: formulation and computational aspects, Int Journal of solids ans structures 38 (2001), 5421–5451. [2] K.Saanouni and J.L.Chaboche, Comprehencive structural integrity, ISBN: 0-08-043749-4, 2003. [3] T. Liebe P.Steinmann and A.Benallal, Theoretical and computational aspects of thermodynamically consistent framwork for geometrically linear gradient damage, Comput. Methods Appl. Mech. Engrg 190 (2001), 6555–6576. [4] M.Geers R.L.J.M.Ubachs and R.A.B.Engelen, Strongly non-local gradient-enhanced finit strain elastoplasticity, Int Journal for Numerical Methods in Engineering 56 (2003), 2039–2068. [5] R.H.J.Peerlings T.J.Massart and M.G.D.Geers, A thermodynamical motivated implicit gradient damage framework and its application to brick masonry cracking, Compt Methods Appl. Mech. Engen. 193 (2004), 3403–3417.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

377

On a New Framework for Anisotropic Damage Model Jian-Ying Wu∗ , Jie Li† ∗ Department

of Civil Engineering, South China University of Technology Guangzhou, 510640, P.R. China [email protected]

† Department

of Building Engineering, Tongji University Shanghai, 200092, P.R. China [email protected] ABSTRACT

Damage induced anisotropy is crucial for those initially isotropic materials, e.g., quasi-brittle materials such as concrete, geomaterials, ceramics, etc. The modeling of anisotropic damage is not a straightforward task as that of isotropic one. Despite the substantial research efforts and the noteworthy recent contributions, this problem still remains a challenging issue, among which two major shortcomings of the existing damage models are to be resolved: (i)The damage variables are to a large extent arbitrarily selected without considering their physical meanings, regarding both the nature (scalar, vector, secondorder tensor, etc.) and the number; (ii)Directly using the concept of effective stress and the hypothesis of strain equivalence generally can not guarantee the major symmetry of the derived secant stiffness, leading to non-existence of an elastic potential. Introducing the energy equivalence hypothesis partially solves this problem but the physical definition of the damage variable is lost. In this paper a novel and rigorous theoretical framework for anisotropic damage model is developed to remedy the foregoing shortcomings. This framework rests on an improved version of representation theorem[1] on (virgin and damaged) secant stiffness tensors and the equivalent thermodynamical considerations. To be more specific, damaged elastic properties are represented in terms of Fourier series expansion of the shear and bulk modulus orientation distribution functions, where two damage variables respectively characterizing the damage mechanisms in the deviatoric and volumetric spaces are inherently selected. For different degree of approximation, the geometric characters and macroscopic effects of the microdefects (microcracks and microvoids) can be described and well controlled by the selected damage variables. Corresponding to the above method, the equivalent thermodynamical formulations are established based on the concept of effective stress and the hypothesis of strain equivalence, where the Helmholtz free energy is decomposed into its deviatoric and volumetric components respectively influenced by the selected damage variables. The problems of lacking uniformity and rigor in the selection of damage variables and the incompatibility between physics and thermodynamics resulted from introducing the hypothesis of energy equivalence are thus solved. To illustrate the proposed framework in modeling anisotropic (orthotropic) and isotropic damage, the special model with a second-order tensor for the deviatoric damage and a scalar for the volumetric damage, as well as the one with two damage scalars are exemplified. Then by considering the deviatoric-volumetric coupling Helmholtz free energy, a restrictive orthotropic damage model with a single second-order damage tensor is presented, demonstrating its capability of describing the shearbulk coupling effects experimentally evidenced in those quasi-brittle materials.

References [1] Q.-C. He and A. Curnier, A more fundamental approach to damaged elastic stress–strain relations. International Journal of Solids and Structures, 32, 1433–1457, 1995.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Shell Optimization under Constraint on Damage Accumulation Nikolay V. Banichuk*, Svetlana Yu. Ivanova*, Evgeniy V. Makeev*, Alexsander V. Sinitsin* *

Institute for Problems in Mechanics, Russian Academy of Scienses Prospect Vernadskogo 101, 119526 Moscow Russia {banichuk,ivanova,makeev,sin}@ipmnet.ru

ABSTRACT The optimization problems taken into consideration consist in finding the shape and thickness distribution of brittle or quasibrittle elastic shells with cracks, loaded by fixed statical forces in such a way that the cost functional (volume of the shell material or weight) reaches a minimum, while satisfying some fracture mechanics constraints. As constraints we use bounds on stress intensity factors or structural longevity. In the case of cycling loadings we consider the number of cycles before fracture as the constraint for optimization problem. Considered optimization problems are characterized by incomplete information concerning initial crack size, crack location and its orientation. In this context the paper presents some possible formulations of optimal structural design problems, based on guaranteed (minimax) and probabilistic approaches. Some examples of analytical and numerical solutions are presented.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Design of Acceptance-Sampling Plans under Bayesian Risk João M. Casaca*, A. Silva Gomes† National Laboratory for Civil Engineering Av. do Brasil 101, 1700-066 Lisbon, Portugal * [email protected], † [email protected]

ABSTRACT The paper deals with generic situations where a consumer acquires a product which he wants to be conforming to certain specifications. The product is delivered by lots and, on the reception of every lot, to be sure that the product is conforming to the specifications, the consumer inspects a sample of the lot and accepts or rejects it according to the results of the inspection. The consumer and the producer must negotiate previously an acceptance-sampling plan [1], contemplating the sampling strategy, the size of the sample and the acceptance rules. As the acceptance of a defective lot implies a loss for the consumer and the rejection of a lot, whether conform or defective, implies a loss for the producer, the acceptance plan must minimize simultaneously expected losses for both the consumer and the producer. The paper presents a model to design acceptance-sampling plans, taking into account the quality level of the lots and using Bayes risk [2] as a criterion to balance the expected losses for the consumer and for the producer. The model is based on an operational characteristic function [1] derived from the binomial distribution and uses acceptance rules similar to a statistical test of hypothesis. The quality level of the lot is modelled by a prior probability density function so that the significance level and the power of the acceptance-sampling plan, may be computed and incorporated in consumer and producer risk functions, for different quality backgrounds. The model has been previously applied to design acceptance-sampling plans for the objects and attributes of Geographic Information Systems and for positional errors of large scale Topographical Maps. The application of the model to the construction quality control of large fill dams is currently being studied. The fill dams are built by placing successive thin layers of compacted materials, each layer being inspected before placing the next one. The rejected layers must be removed. The consumer (the dam owner) does not want defective layers to be accepted. The producer (the contractor) does not want any layers to be rejected. The design of an acceptance-sampling plan minimizing the consumer’s and the producer’s expected losses is of utmost importance for both. References [1] D. Montgomery, Statistical Quality Control. John Wiley & Sons, New York, 1991. [2] V. Barnett, Comparative Statistical Inference. John Wiley & Sons, New York, 1975.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Optimal design for the worst case scenario Elena Cherkaev∗ , Andrej Cherkaev† ∗ University of Utah, Department of Mathematics 155 South 1400 East, JWB 233, Salt Lake City, UT 84112 [email protected] † University of Utah, Department of Mathematics 155 South 1400 East, JWB 233, Salt Lake City, UT 84112 [email protected]

ABSTRACT The talk discusses a problem of robust optimal design of elastic structures when the loading is unknown, and only an integral constraint for the loading is given [1]. The optimal design problem is formulated as minimization of the principal compliance of the domain equal to the maximum of the stored energy over all admissible loadings. The principal compliance is the maximal compliance under the extreme, worst possible applied force [2]. The robust optimal design is a min-max problem for the energy stored in the structure. The minimum is taken over the design parameters, while the maximum of the energy is chosen over the constrained class of loadings. It is shown that the problem for the extreme loading is reduced to an elasticity problem with mixed nonlinear boundary conditions; the last problem may have multiple stationary solutions. The optimization takes into account the possible multiplicity of extreme loadings and designs the structure to equally resist all of them. Continuous change of the loading constraint causes bifurcation of the solution of the optimization problem. It is shown that an invariance of the constraints under a symmetry transformation leads to a symmetry of the optimal design. Examples of optimal design are investigated; symmetries and bifurcations of the solutions are discussed.

References [1] A. Cherkaev and E. Cherkaev, Optimal design for uncertain loading conditions. In: Homogenization, V. Berdichevsky, V. Jikov, and G. Papanicolaou, eds., World Scientific, 193–213, 1999. [2] E. Cherkaev and A. Cherkaev, Principal compliance and robust optimal design, Journal of Elasticity, 72, 1-3, 71–98, 2003.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Dimension Reduction Method for Reliability-Based Robust Design Optimization K.K. Choi*, Ikjin Lee*, and David Gorsich† *

Department of Mechanical and Industrial Engineering The University of Iowa, Iowa City, IA 52242, U.S.A. [email protected] [email protected] † AMSTA-TR-N (MS 263) US Army National Automotive Center Warren, MI 48397-5000, U.S.A. [email protected]

ABSTRACT In reliability-based robust design optimization formulation, the product quality loss function is minimized subject to probabilistic constraints. Since the quality loss function is expressed in terms of the first two statistical moments, mean and variance, several methods have been proposed to accurately and efficiently estimate the statistical moments. However, it is computationally expensive to calculate the statistical moments of the output performance function using the multidimensional integral, especially when the number of the random input variables is relatively high. To overcome the shortcomings, three methods have been recently proposed: univariate dimension reduction method (DRM) [1], performance moment integration (PMI) method [2], and percentile difference method (PDM) [3]. In this paper, a reliability-based robust design optimization method is developed using DRM and compared to PMI and PDM for the robust design part. It is found that PDM cannot estimate the statistical moments of the performance function accurately. The PMI and DRM are also compared in terms of accuracy and efficiency in estimation of statistical moments of the performance function. Several numerical examples are used to compare accuracy and efficiency of these methods. The numerical results show that DRM is effective when the number of random variables is small, whereas PMI is more effective when the number of random variables is relatively large. For the inverse reliability analysis, the enhanced hybrid mean value (HMV+) method is used, whereas the enriched performance measure approach (PMA+) is used for reliability-based design optimization.

References [1] Xu, H., and Rahman, S., “A Univariate Dimension-Reduction Method for Multi-dimensional Integration in Stochastic Mechanics,” Probabilistic Engineering Mechanics, Vol. 19, No. 4, pp. 393-408, 2004 [2] Youn, B. D., Choi, K. K., and Yi, K., "Performance Moment Integration (PMI) Method for Quality Assessment in Reliability-Based Robust Optimization," Mechanics Based Design of Structures and Machines, Vol. 33, No. 2, pp. 185-213, 2005. [3] Du, X., Sudjianto, A., and Chen, W., “An Integrated Framework for Optimization Under Uncertainty Using Inverse Reliability Strategy,” ASME Journal of Mechanical Design, Vol 126, No. 4, pp. 562-570, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A Fatigue Life Reliability-based Design Optimization of a Slat Track using Mesh Morphing Roberto d’Ippolito1, Stijn Donders1, Luc Hermans1, Michael Hack2, Joost Van de Peer3 and Nick Tzannetakis3 1 LMS International Interleuvenlaan 68, B-3001 Leuven, Belgium [email protected], [email protected], [email protected] 2

LMS Deutschland GmbH Kaiserslautern, Germany [email protected] 3 Noesis Solutions Interleuvenlaan 68, B-3001 Leuven, Belgium [email protected], [email protected]

ABSTRACT Although the aerospace production process is much better controlled than the process in other industries, it remains true that very small manufacturing tolerances exist in the geometrical parameters (flange thicknesses, hole diameters, …). In the current design process, the effect of this manufacturing variability on the structural durability and safety cannot be accurately assessed and is hence compensated for by applying safety factors. This is not an ideal situation, as it may lead to slightly over-designed structures. A much more promising approach is to include probabilistic models of design variables into the mechanical simulation process. Then, with a new methodology based on reliability analysis, engineers can obtain a better understanding of the actual effect of the manufacturing tolerances. Based on the analysis results, the robustness and reliability of the design can be assessed and improved if needed. In this paper, the above-mentioned probabilistic approach is demonstrated on a slat track structure. Measurements of different geometrical properties have been collected during the manufacturing process and their variability has been characterized probabilistically with statistical models. Then, a reliability analysis has been carried out using morphing technology and fatigue life predictions with an industrial-sized FE model of the slat track to assess the reliability of the structure in terms of fatigue life. The outcome of the analysis consists of a probabilistic model of the fatigue life, given the variability in the geo-metrical parameters. This analysis not only provides a better insight in the effect of variability in the fatigue life prediction, but also provides sensitivity measurements of the design parameters on the final performance of the structure. These results provide guidelines to improve structural designs and manufacturing tolerances, by using a reliability-based design optimization procedure. A powerful tool is thus obtained to reduce design conservatism while maintaining and even improving structural safety.

References [1] R.E. Melchers, Structural Reliability Analysis and Prediction, 2nd Edition, John Wiley & Sons, UK, 1999. [2] B.D. Youn, K.K. Choi, Liu Du, Adaptive Probability Analysis Using An Enhanced Hybrid Mean Value Method, Journal of Structural and Multidisciplinary Optimization, vol. 29, no. 2, 2004, pp. 134-148, Springer Berlin Heidelberg.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

383

A New Approach of Robust Design Based on the Concept of Allowable Load Set Byung Man Kwak*, Jinho Chang*, Jae Hyun Kim† *Korea Advanced Institute of Science and Technology 373-1 Guseong-dong, Yusoeng-gu Daejeon, 305-701 Republic of Korea {bmkwak,jhchang}@khp.kaist.ac.kr † LG Electronics Co., Digital Appliance Research Lab. 327-23 Gasan-dong, Geumcheon-gu, Seoul, 153-802, Korea [email protected]

ABSTRACT A concept called “Allowable load set (ALS)” developed by the authors allows designers to view a design process in a completely different direction. In the usual structural design, a load is given at a point and the size or shape of a structure is to be found to support the given load. Now an allowable load set denotes the set of loads that are safe to a given structure. The design problem is to find the most suitable allowable load set by adjusting the size or shape. The ALS can be visualized graphically for simple cases, and the integrity of the structure for a given design load can be seen visually. To quantify the structural integrity, two measures are devised: one is a relative safety index (RSI) denoting the distance from the mean design load vector to the boundary of the ALS, and another is a normalized safety index (NSI) which is just the value of a performance function or a constraint function. This latter involves no optimization of finding a distance as required in the usual reliability index approach. Secondly ALS can be applied easily to multi-body mechanical systems, especially efficiently for linear elastic material. Examples thus include illustrations from three bar truss and torque arm design. A robust design is to obtain a design which is most insensitive to uncertainties. In robust design optimization, no probability information of uncertainties is assumed given. The ALS design approach is a new method of achieving this goal. Another challenging example is applying the ALS design method to find trajectory of motions of a biomechanical system. The philosophy is that a man will take a motion such that its configuration at any instant is as far as possible from danger when taken naturally. As will be shown, this gives natural motions of lifting a heavy object with or without low back pain. Especially the NSI formulation is very efficient. The ALS design concept is innovative and well applicable to real world problems. The RSI formulation is physically more meaningful but requires much larger computational cost than the NSI. One disadvantage of NSI is that a suitable scaling may be necessary sometimes to obtain right proportions among constraints, but no simple way is found yet. Comparative study of ALS and other methods of robust design will be given in addition to the theory and numerical applications.

References [1] B. M. Kwak and J. H. Kim, “Concept of allowable load set and its application for evaluation of structural integrity,” Journal of Mechanics of Structures and Machines, 30, 213-247, 2002

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A sampling technique enhancing accuracy and efficiency of metamodel-based RBDO: constraint boundary sampling Tae Hee Lee*, Jae Jun Jung*, Do Hyun Jung† * Hanyang University School of Mechanical Engineering, Sungdong-ku, Seoul 133-791, Korea [email protected] [email protected] † Korea Automotive Technology Institute Body and Chassis Engineering Center, Chonan, Chungnam 300-912, Korea [email protected]

ABSTRACT Reliability-based design optimization (RBDO) has been developed to consider uncertainty of input design variables during optimization process. To provide the reliability, reliability index approach (RIA) and performance measure approach (PMA) are often used.[1, 2] However, these reliability analyses usually require extremely expensive computational costs due to many simulation runs. Thus, it is necessary to reduce significantly the number of actual simulation runs during RBDO. Metamodels such as response surface model and kriging model are investigated for this purpose [3]. Metamodel for computer simulation is often built from space-filling sampling that evenly locates sample points within whole design domain. However, it requires considerably many sample points to approximate probabilistic constraints throughout whole design region when constraints reveal nonlinearity and when feasible region is small compared to whole design region. In this research, constraint boundary sampling technique is proposed to maximize accuracy and efficiency of metamodel-based RBDO. Constraint boundary sampling is sequentially to locate sample points around constraint boundary by using kriging metamodel and its mean squared error. To verify the proposed method, mathematical examples are performed and their accuracy and efficiency are compared to those obtained from classical space-filling design. Through this study, we learn that RBDO using kriging model under the constraint boundary sampling technique coincides precisely with the exact solutions. Moreover, the efficiency of RBOD is improved so that RBDO using constraint boundary sampling technique can reduced by about 50% compared to conventional RBDO in the number of actual response analysis.

References [1] Yu, X., Chang, K.H., and Choi, K.K., Probabilistic Structural Durability Prediction, AIAA Journal, 36, 628-637, 1998. [2] Tu, J. and Choi, K.K., A New Study on Reliability Based Design Optimization, Journal of Mechanical Design, ASME, 121, 557-564, 1999. [3] Choi, K. K., You, B.D., and Yang, R.J., Moving Least Square Method for Reliability-Based Design Optimization, 4th World Congress of Structural and Multidisciplinary Optimization, Dalian, China, June 4-8, 2001.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

385

Interval sensitivity analysis of dynamic response envelopes for uncertain mechanical structures David Moens∗ , Dirk Vandepitte∗ ∗ K.U.Leuven,

Dept. Mechanical Engineering, PMA Celestijnenlaan 300B, B-3001 Heverlee, Belgium [email protected]

ABSTRACT Non-deterministic approaches are gaining momentum in the field of finite element analysis. The ability to include non-deterministic properties is of great value for a design engineer. It enables realistic reliability assessment that incorporates the uncertain aspects of the design. Furthermore, the design can be optimised for robust behaviour under varying external influences. Recently, criticism arises on the general application of the probabilistic concept in this context. Especially when objective information on the uncertainties is limited, the subjective probabilistic analysis result proves to be of little value, and does not justify its high computational cost. Consequently, alternative non-probabilistic concepts are used for non-deterministic finite element analysis, as e. g. the fuzzy and interval concept. Recently, a fuzzy finite element methodology to calculate a fuzzy frequency response function (FRF) of uncertain undamped structures was developed by the authors. The procedure consists of the solution of a sequence of interval problems. The goal of each interval analysis is to calculate the envelope of the FRF taking into account that the input uncertainties can vary within the bounded space defined by their combined intervals. The resulting envelope response function gives a clear view on the possible variation of the response in the frequency domain. The applicability of this interval response analysis was proven on realistic case studies. In this result, all uncertain parameters are considered to act simultaneously. While this is often the most realistic representation of the physical condition of the actual product, for design purposes, it can be of great value to know the contribution of the individual uncertainties to the response range obtained from the interval analysis. This enables a designer to distinct between the non-deterministic influences that have an important contribution to the fuzziness on the dynamic behaviour of the design, and those that have little or no influence. This distinction can be very valuable in the definition of e. g. tight design tolerances or realistic allowable working conditions, and as such, could lead to less conservative designs. This paper introduces an interval sensitivity procedure that calculates the sensitivity of the envelope response function in the outcome of the interval FRF analysis to each individual interval model uncertainty. The procedure focusses on the calculation of the sensitivity of the bounds defining the FRF response range to the width of each individual uncertain input parameter. The approach differs from the classical sensitivity analysis in the fact that it does not calculate local changes on the output resulting from local changes in the input. The interval sensitivity result describes the change of the response interval width, taking into account a change in the parameter interval width. The paper first describes the theoretical background of the fuzzy and underlying interval FRF procedure. Next, it introduces the methodology for interval sensitivity analysis. Finally, the method is illustrated on a numerical example.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

386

Multi-Objective Robust Design Optimization of an Engine Crankshaft Carlo Poloni*, Paolo Geremia†, and Alberto Clarich† *

Università di Trieste-Dipartimento di Energetica via Valerio 10, 34100 Trieste [email protected]

† Esteco srl Area Science Park, Padriciano 99, 34100 Trieste {geremia, clarich}@esteco.it

ABSTRACT When designing a commercial product, engineers have to meet several requirements which boil down to finding the better performances and the higher reliability as possible. Another significant factor that determines product quality is its sensitivity to external or uncontrollable variations. This methodology of design is generally called Robust Design [1]. This paper shows an application of Robust Design methodology to a multi-disciplinary optimization of an engine crankshaft by considering uncertainties in terms of manufacturing errors over the shaft dimensions as well as dynamic loads variability. The application is run using ANSYS Workbench solver and modeFRONTIER [2], through a direct interface between the two codes that has been recently developed. A full Robust Design analysis is applied in order to check the stability of the best candidate solutions according to uncertainties in terms of both manufacturing errors and forcing loads The results obtained are very encouraging, and the procedure described can be applied, in principle, to even more complex problems.

References [1] Clarich A., Pediroda V., Poloni C., A competitive Game Approach for Multi Objective Robust Design Optimization, AIAA 2004-6511, Chicago, 20-22 September 2004 [2] www.esteco.com

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

387

Development and Application of A New Metropolis GA for the Structural Design Optimization Yeon-Sun Ryu Dept of Ocean Engineering, Pukyong National University, Busan, Korea [email protected]

ABSTRACT A Metropolis genetic algorithm (MGA) is developed and applied for the structural design optimization. In MGA, favorable features of Metropolis criterion in simulated annealing (SA) are incorporated in the reproduction operations of simple genetic algorithm (SGA). This way, the MGA maintains the wide varieties of individuals and preserves the genetic information of early generations. Consequently, the proposed MGA alleviates the disadvantages of finding imprecise solution in SGA and time-consuming computation in SA. Performances of MGA are compared with those of conventional algorithms such as Holland's SGA, Krishnakumar's micro genetic algorithm (µGA), and Kirkpatrick's SA. Typical numerical examples are used to evaluate the favorable features and applicability of MGA. The effects of population sizes and maximum generations are also evaluated for the performance reliability of MGA. From the theoretical evaluation and numerical experience, it is concluded that the proposed MGA is a reliable and efficient tool for structural design optimization.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

388

Stochastic Response Surface Using the Enhanced Dimension-Reduction (eDR) Method for Reliability-Based Robust Design Optimization Byeng D. Youn1, Zhimin Xi1, Lee J. Wells1, and David A. Lamb2 1

2

Department of Mechanical Engineering and Engineering Mecha Michigan Technological University Houghton, MI 49931 {bdyoun,zxi,ljwells}@mtu.edu

National Automative Center (NAC), U.S. Army RDECOM-TARDEC Warren, MI 48397-5000 [email protected]

ABSTRACT As the reliability analysis and design methodology has been advanced, its implementation becomes more complicated to improve computational efficiency and stability. Furthermore, most reliability analysis methods in RBDO require gradient (or sensitivity) information [1]. Therefore, this paper attempts to develop a stochastic response surface method. The method makes it possible to perform sensitivity-free RBDO using any deterministic optimizer. Recently, the dimension reduction (DR) method has been proposed [2]. Although the DR method is known to be an accurate and efficient method for the uncertainty quantification (UQ) of system responses, it may produce a relatively large error for the second-order or higher moments of nonlinear responses. Thus, this paper first proposes the enhanced dimension-reduction (eDR) method [3] by incorporating two alternative integration schemes and one-dimensional response approximations. Both moment based quadrature rule and an adaptive Simpson integration rule are alternatively used for numerical integration. The stepwise moving least squares (SMLS) method is proposed for response approximation. The SMLS is based on a moving least squares (MLS) method. Secondly, the paper proposes a stochastic response surface method. The stochastic response surface is built using the SMLS method with the results of the eDR method at sampled designs. In aid of the stochastic response surface method, RBDO or robust design optimization can be performed with commercial (deterministic) optimization softwares (e.g., Microsoft Excel, Matlab, etc.). In this paper, some examples are used to demonstrate the eDR method and further the stochastic response surface method for RBDO.

References [1] Youn, Byeng D., Choi, K. K., and Du, L., “Enriched Performance Measure Approach (PMA+) for Reliability-Based Design Optimization,” AIAA Journal, Vol. 43, No. 4, pp. 874-884, 2005. [2] Rahman, S. and Xu, H., "A Univariate Dimension-Reduction Method for Multi-Dimensional Integration in Stochastic Mechanics," Probabilistic Engineering Mechanics, Vol. 19, pp. 393-408, 2004. [3] Youn, B.D., Zhmin, X., Wells, L., and Lamb, D., "The Enhanced Dimension Reduction (eDR) Method for Reliability-Based Robust Design Optimization," 11th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Accepted, Portsmouth, Virginia, Sep. 6-8, 2006.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

389

DEM analysis of granular flow in pyramidal hoppers R. Baleviìius*, R. Kaìianauskas* *

Laboratory of Numerical Modelling, Vilnius Gediminas Technical University, Vilnius, Lithuania [email protected] [email protected]

ABSTRACT The processes of granular material handling in the hoppers are of a great importance in pharmaceutical, chemical, food and other industries. Theoretical treatments of such problems are usually based on simplified continuum models, which are useful to predict the stress field within the hopper, especially, on the walls at the end of filling process. The Continuum approach has some drawbacks for discharge state modeling when description of transient flow is required. Consequently, discrete element method (DEM) based on the application Newton’s and contact mechanics laws predicting dynamical parameters, such as position, velocity, etc., of individual particles, have been adopted in numerical analysis of granular flow in hoppers. Filling and discharge flow in pyramidal hoppers of different shape is considered by the discrete element method. Non-cohesive frictional visco-elastic spherical particles are applied in modelling. The boundary conditions are described by using locally oriented planes of a finite size enabling to handle different shapes of hoppers. Evolution of the system kinetic energy, discharge mass fraction as well as distribution of particle velocities and material porosity fields is considered. Geometry of the hopper, particle contact forces and the velocity fields during discharge are presented in Fig. 1.

a)

b)

Fig. 1 Hopper geometry, particle flow (a) and velocity fields (b) during discharge at t=1.5 s

The DEM concept is implemented into original software code DEMMAT [1], where the 5th-order Gear’s predictor-corrector scheme is used for numerical integration of equations of motion. Particular postprocessors for evaluation of various field variables are also developed.

References [1] Baleviìius R, Kaìianauskas R, Džiugys A, Maknickas A, Vislaviìius, DEMMAT code for numerical simulation of multi-particle dynamics. Information Technology and Control, 34(1), 71-78, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

390

DQEM and DQFDM for Computational Mechanics Problems Chang-New Chen Department of Systems and Naval Mechatronic Engineering National Cheng Kung University, Tainan, Taiwan, ROC [email protected]

ABSTRACT Because only problems having simple regular domains and under simple external environments can be solved by using DQ, the application of this method is very limited. The author has proposed the DQEM for solving a generic engineering or scientific problem having an arbitrary domain configuration. Like the FEM, in this method, the analysis domain of a problem is first separated into a certain number of subdomains or elements. Then the DQ, GDQ or EDQ discretization is carried out on an element-basis. The governing differential or partial differential equations defined on the elements, the transition conditions on inter-element boundaries, and the boundary conditions on the analysis domain boundary are in computable algebraic forms after the DQ, GDQ or EDQ discretization. By assembling all discrete fundamental equations an overall algebraic system can be obtained which is used to solve the problem. The DQFDM has also been proposed by the author. The finite difference operators are derived by DQ. They can be obtained by using the weighting coefficients for DQ discretizations. The derivation is straight and easy. By using different orders or the same order but different grid DQ discretizations for the same derivative or partial derivative, various finite difference operators for the same differential or partial differential operator can be obtained. Finite difference operators for unequally spaced and irregular grids can also be generated through the use of GDQ. DQEM and DQFDM have been used to develop solution algorithms for computational mechanics. In this paper, numerical results are presented to demonstrate these two discrete analysis methods.

References [1] R.E. Bellman and J. Casti, Differential quadrature and long-term integration. Journal of Mathematical Analysis and Applications, 34, 234-238, 1971. [2] C.N. Chen, A differential quadrature element method. Proceedings of the First International Conference on Engineering Computation and Computer Simulation, Changsha, China, 1, 25-34, 1995. [3] C.N. Chen, A differential quadrature finite difference method. Proceedings of the First International Conference on Advanced Computational Methods in Engineering, Gent, Belgium, 1, 713-720, 1998. [4] C.N. Chen, DQEM and DQFDM for the analysis of composite two-dimensional elasticity problems.Composite Structures, 59, 3-13, 2003.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

391

An innovative truly–mixed method for cohesive–crack propagation problems C. Cinquini∗ , M. Bruggi∗ , Paolo Venini∗ ∗ University

of Pavia, Department of Structural Mechanics via Ferrata 1, I27100 Pavia, Italy {cinquini,matteo.bruggi,paolo.venini}@unipv.it ABSTRACT

We propose a novel approach for the analysis of cohesive crack propagation in elastic media. Unlike all existing methods that move from continuous displacement formulations that are properly enriched to handle the discontinuity, see e.g. the extended finite element method (XFEM) [Mo¨es et al., 1999] or the embedded discontinuity [Jir´asek, 2000] approaches, inherently discontinuous displacements and H(div) stresses in a truly mixed setting are herein proposed. The formulation, originally introduced to handle incompressible materials in plane elasticity, is herein extended to the analysis of propagating cohesive cracks in elastic media thanks to a novel variational formulation that is enriched with an interface energy term. Notably, no edge element is introduced but simply the inherent discontinuity of the displacement field is taken advantage of. Furthermore, stress flux continuity is imposed in an exact fashion within the formulation and not as an additional weak constraint as classically done. Extensive numerical simulations are presented to complete the theoretical framework.

References [Jir´asek, 2000] Jir´asek, M., 2000. Comparative study on finite elements with embedded cracks, Computer Methods in Applied Mechanics and Engineering, 188, 307–330. [Mo¨es et al., 1999] Mo¨es, N., Dolbow, J., Belytschko, T., 1999. A finite element method for crack growth without remeshing. International Journal for Numerical Methods in Engineering, 46, 131– 150.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

392

Compactly Supported Fundamental Functions for Spline-Based Differential Quadrature ˜ P´erez† Domingo Barrera Rosillo∗ , Francisco Ib´anez ∗ Departamento de Matem´ atica Aplicada ETS de Ingenieros de Caminos, Canales y Puertos Universidad de Granada Campus Universitario de Fuentenueva, 18071-Granada, Spain [email protected] † Grupo Sacyr-Vallehermoso Paseo de la Castellana, 83, 28046-Madrid, Spain fi[email protected]

ABSTRACT The Differential Quadrature Method (DQM) is a numerical discretization technique for the approximation of derivatives by means of weighted sums of function values. It was proposed by Bellman and coworkers in the early 1970’s, and it has been extensively employed to approximate spatial partial derivatives (cf. [1], [4] for instance). The classical DQM is polynomial-based, and it is well known that the number of grid points involved is usually restricted to be below 30. Some spline based DQMs have been proposed to avoid this problem, but the construction of these schemes depends strongly on the degree of the considered B-spline (see for instance [2] and [5]). In this work we present a general DQM based on interpolation and quasi-interpolation. Firstly, we consider the construction of compactly supported cardinal functions L that interpolate the Kronecker sequence. They are linear combinations of translates of a B-spline Mn centered at the origin. Then, we revise some spline discrete quasi-interpolants defined from the same B-splines. We are interested in some recently defined and analyzed quasi-interpolants (cf. [3]). They are constructed by minimizing an error constant appearing in a particular expression of the quasi-interpolation error for regular enough functions. Finally, both the interpolants and the quasi-interpolants are used to define new interpolants having compactly supported fundamental functions again, and the maximal order of approximation, and the quintic case is described and compared with the results obtained in [5].

References [1] C. W. Bert and M. Malik, Differential quadrature method in computational mechanics: a review. Appl. Rev., 49, 1–27, 1996. [2] Q. Guo, H. Zhong, Non-linear vibration analysis y a spline-based differential quadrature method. Journal of Sound and Vibration, 269, 413–420, 2004. [3] M. J. Ib´an˜ ez-P´erez, Quasi-interpolantes spline discretos de norma casi m´ınima. Teor´ıa y aplicaciones. Doctoral Dissertation, University of Granada, 2003. [4] C. Shu, Differential Quadrature and its applications in Engineering. Springer-Verlag, London, 2000. [5] H. Zhong, Spline-based differential quadrature for fourth order differential equations and its applications to Kirchhoff plates. Applied Mathematical Modelling, 28, 353–366, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

393

Differential Quadrature Solution for Parabolic Structural Shell Elements Francesco Tornabene*, Erasmo Viola† *

DISTART - Department, Faculty of Engineering, University of Bologna Viale Risorgimento 2, 40136 Bologna, Italy [email protected]

†

DISTART - Department, Faculty of Engineering, University of Bologna Viale Risorgimento 2, 40136 Bologna, Italy [email protected]

ABSTRACT This work deals with the dynamical behaviour of complete parabolic shells of revolution and parabolic shell panels. The First-order Shear Deformation Theory (FSDT) is used to analyze the above moderately thick structural elements. The treatment is conducted within the theory of linear elasticity, when the material behaviour is assumed to be homogeneous and isotropic. The governing equations of motion, written in terms of internal resultants, are expressed as functions of five kinematic parameters, by using the constitutive and the congruence relationships. The boundary conditions considered are clamped (C) and free (F) edge. Numerical solutions have been computed by means of the technique known as the Generalized Differential Quadrature (GDQ) Method. The solution is given in terms of generalized displacement components of the points lying on the middle surface of the shell. At the moment it can only be pointed out that by using the GDQ technique the numerical statement of the problem does not pass through any variational formulation, but deals directly with the governing equations of motion. Referring to the formulation of the dynamic equilibrium in terms of harmonic amplitudes of mid-surface displacements and rotations, in this paper the system of second-order linear partial differential equations is solved, without resorting to the onedimensional formulation of the dynamic equilibrium of the shell. The discretization of the system leads to a standard linear eigenvalue problem, where two independent variables are involved. Several examples of parabolic shell elements are presented to illustrate the validity and the accuracy of GDQ method. The convergence rate of the natural frequencies is shown to be very fast and the stability of the numerical methodology is very good. The accuracy of the method is sensitive to the number of sampling points used, to their distribution and to the boundary conditions. The effect of the distribution choice of sampling points on the accuracy of GDQ solution is investigated. GDQ results, which are based upon the FSDT, are compared with the ones obtained using commercial programs such as Ansys, Femap/Nastran, Abaqus, Straus, Pro/Engineer.

References [1] E. Reissner, The effect of transverse shear deformation on the bending of elastic plates. Journal of Applied Mechanics ASME 12, 66-77, 1945. [2] E. Viola and E. Artioli, The G.D.Q. method for the harmonic dynamic analysis of rotational shell structural elements. Structural Engineering and Mechanics 17, 789-817, 2004. [3] C. Shu, Differential Quadrature and Its Application in Engineering. Springer, Berlin, 2000.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Multiple Crack Growth failure in Cortical Bone under Tension by the eXtended Finite Element Method Elisa Budyn∗ , Laurent Henry† , Thierry Hocr† ∗ University

of Illinois at Chicago 842 West Taylor Street, Chicago, IL 60607, USA [email protected] † Ecole Centrale Paris Grande Voie des Vignes, 92295 Chatenay Malabry, France [email protected], [email protected]

ABSTRACT A multi-scale analysis for multiple crack growth in unit cell of cortical bone is presented. The cracks are grown until complete failure of the cell. The initial cracks are placed in maximum strain locations. The stress intensity factors are computed at each crack tip and a load parameter is adjusted so that the stress intensity factors remain at the critical value. In the case of competitive crack tips, a stability analysis is performed by computing the second derivative of the potential energy for each crack. The load deflection behavior of the representative volume element is obtained until the point of complete failure. The model is fed with experimental geometrical, mechanical and damage parameters and validated through a comparison with experimental samples. The discretization utilizes the eXtended Finite Element Method and requires no remeshing as the cracks grow. The crack geometries are arbitrary with respect to the mesh, and are described by a vector level set. Special boundary conditions and the algorithm for detecting crack bridging and crack entering Haversian canals which allows the cracks to grow until maximum failure and/or percolation is presented.

References [1] E. Budyn, G. Zi, N. Mo¨es and T. Belytschko, A Method for Multiple Crack Growth in Brittle Materials without Remeshing, Int. J. for Num. Meth. in Eng., 61, Number 10, pp. 1741-1770,2004. [2] T. Belytschko and T. Black, Elastic Crack Growth in Finite Elements With Minimal Remeshing, Int. J. for Num. Meth. in Eng., 45, Number 5, pp. 601-620.1999.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Subdivision Shells Fehmi Cirak California Institute of Technology Center for Advanced Computing Research, Pasadena, CA 91125, U.S.A. [email protected] ABSTRACT Subdivision elements, as originally introduced by Cirak, Ortiz, and Schr¨oder [5], provide a new paradigm for thin-shell finite-element analysis based on the use of subdivision surfaces for (i) describing the geometry of the shell in its undeformed configuration, and (ii) generating smooth interpolated displacement fields. The displacement fields obtained by subdivision are H 2 and, consequently, have a finite Kirchhoff-Love energy. The displacement field of the shell is interpolated from nodal displacements only and no nodal rotations are used. The interpolation scheme induced by subdivision is nonlocal, i.e. the displacement field over one element depends on the nodal displacements of the three element nodes and all nodes of immediately neighboring elements. However, the use of subdivision schemes ensures that all the local displacement fields combine conformingly to define one single limit surface. Numerical tests, demonstrate the high accuracy and optimal convergence of the method even in highly nonlinear problems [4]. Furthermore, because of the unification of representations for mechanics and geometric modeling (i.e. CAD: Computer Aided Design), subdivision elements are ideally suited to applications in shape optimization [3]. Recently, specialized cohesive elements have been developed that account for in-plane tearing, shearing, and hinge modes of shell fracture [1]; and methods for coupling subdivision shells to gas dynamics [2].

References [1] F. Cirak, M. Ortiz and A. Pandolfi. A Cohesive Approach to Thin-Shell Fracture and Fragmentation. Computer Methods in Applied Mechanics and Engineering, 194, 2604–2618, 2005. [2] F. Cirak and R. Radovitzky. A Lagrangian-Eulerian Shell-Fluid Coupling Algorithm Based on Level Sets. Computers & Structures, 83, 491–498, 2005. [3] F. Cirak, M.J. Scott, E.K. Antonsson, M. Ortiz and P. Schr¨oder. Integrated Modeling, Finite-Element Analysis, and Engineering Design for Thin-Shell Structures Using Subdivision Computer-Aided Design, 34, 137–148, 2002. [4] F. Cirak and M. Ortiz. Fully C 1 -Conforming Subdivision Elements for Finite Deformation ThinShell Analysis International Journal for Numerical Methods in Engineering, 51, 813–833, 2001. [5] F. Cirak, M. Ortiz and P. Schr¨oder. Subdivision Surfaces: A New Paradigm for Thin-Shell FiniteElement Analysis International Journal for Numerical Methods in Engineering, 47, 2039–2072, 2000.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Capturing Slip Weakening and Variable Frictional Response in Localizing Geomaterials Using an Enhanced Strain Finite Element Craig D. Foster*, Ronaldo I. Borja† * Stanford University Blume Earthquake Engineering Center MC 4020 Stanford, CA 94305 [email protected] † Stanford University Terman Engineering Center MC 4020 Stanford, CA 94305 [email protected]

ABSTRACT The formation and propagation of faults, cracking of concrete structures, and progressive fracture in ceramics are examples of localized deformation in quasi-brittle geomaterials. In these materials, localization tends to take the form of a fractured surface rather than a deformation band of finite width. In order to capture the propagation and post-localization slip of these surfaces, we must properly model their behavior. The constitutive response along the surface exhibits two distinct phases. The first is slip weakening, in which the shear strength degrades in an approximately linear fashion with slip displacement. This degradation corresponds to a loss of cohesive strength as a coherent macrocrack forms, and takes place over small displacements, on the order of 0.5 mm. After the complete loss of cohesive strength, the response is purely frictional. The frictional response may vary with slip speed, wear on a changing population of contacts, temperature, and other factors. For many materials and applications, this response is captured well using a frictional model developed by Dieterich [1], Ruina, Rice, and others. Coupling this friction law to the slip-weakening model becomes challenging since it is difficult to predict the shear stress at the end of the slip-weakening phase while the coefficient of friction is changing. In this work, we embed this coupled slip weakening-frictional response into an enhanced strain element with an embedded strong discontinuity, similar to the one described in [2]. The weakening and frictional models are successfully coupled by making some simplifying assumptions, and embedded numerically in the slip response by means generalized trapezoidal method. The slip speed and state variable are solved via an element-level Newton iteration.

References [1] J.H. Dieterich and M.F. Linker, Fault stability under conditions of variable normal stress. Geophysical Research Letters, 19, 1691-1694, 1992. [2] R.I. Borja and R.A. Regueiro, Strain localization of frictional materials exhibiting displacement jumps. Computer Methods in Applied Mechanics and Engineering 190, 2555-2580, 2001.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A partition of unity finite element method applied to the study of viscoelastic sandwich structures L.Hazard*, Ph. Bouillard*, J.-Y. Sener† *

Structural and Material Computational Mechanics Department, CP 194/5 Université Libre de Bruxelles, Av. F.D. Roosevelt 50, 1050 Brussels, Belgium [email protected] †

Arcelor Innovation, Boulevard de Colonster, B57, 4000 Liège, Belgium

ABSTRACT The scope of this research concerns the passive damping of vibrations of structures by the use of viscoelastic layers. It is motivated by the need for efficient numerical tools to deal with the medium frequency behaviour of industrial viscoelastic sandwich products. The sandwich modelling technique is based on the use of an interface element: the two deformable plates are modelled by special plate elements while the intermediate dissipative layer is modelled with interface elements. This interface element is based on the first-order shear deformation theory and assume constant peel and shear stresses in the polymer thickness. This element couple the lower and upper layers without additional degrees of freedom. The partition of unity finite element method (PUFEM) is applied to the development of enriched Mindlin plate elements. The element shape functions are obtained as the product of partition of unity functions with arbitrary chosen enrichment functions. Polynomial enrichment leads to the generation of high-order polynomial shape functions and is therefore very similar to a p-FEM technique. Numerical examples illustrate the use of both PUFEM Mindlin plate elements and interface elements for the simulation of viscoelastic sandwich structures.

References [1] I. Babuška, J. Melenk, The partition of unity method, International Journal for Numerical

Methods in Engineering, 40, 727-758, (1997) [2] E. De Bel, P. Villon and Ph. Bouillard, Forced vibrations in the medium frequency range

solved by a partition of unity method with local information, International Journal for Numerical Methods in Engineering, 62, 1105-1126, (2005)

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An enriched space-time finite element method for fluid-structure interaction – Part II: Thin flexible structures A. K¨olke∗ and A. Legay† ∗ Institut f¨ur Statik Technische Universit¨at Braunschweig, Beethovenstraße 51, 38106 Braunschweig, Germany [email protected] † Chaire de M´ecanique Conservatoire National des Arts et M´etiers, 2 rue Cont´e, 75003 Paris, France [email protected]

ABSTRACT A new numerical approach for fluid-structure interaction of viscous fluid flow and flexible structures of negligible thickness (e.g. membranes, plates) on a topologically fixed fluid discretization is presented. The linear elastic and geometrically nonlinear structure is embedded [1] into the flow field described by the incompressible Navier-Stokes equations using locally enriched space-time (EST) finite elements [2, 3] to consider resulting strong and weak discontinuities in the fluid field appropriately [4]. Since the formulation of fluid, structure and coupling conditions uniformly uses velocities as unknowns and the integration of governing equations is perfomed on the deformed space-time mesh, the simulation of the strongly coupled physical domains becomes very comfortable and results in a monolithic system [5]. Numerical examples of fluid-stucture interaction processes show the ability of the developed numerical method to describe those situations of coupled systems for that common mesh moving strategies are not useful applicable and would require the introduction of time-consuming remeshing algorithms.

References [1] A. Legay, J. Chessa and T. Belytschko. An Eulerian-Lagrangian Method for Fluid-Structure Interaction Based on Level Sets. Computer Methods in Applied Mechanics and Engineering, in press, 2005 [2] A. K¨olke and D. Dinkler. Extended Space-Time Finite Elements for Two-Fluid Flows in FluidStructure Interaction. Proceedings of Sixth World Congress on Computational Mechanics, China, 2004 [3] A. K¨olke and D. Dinkler. Extended Space-Time Finite Elements for Boundary-Coupled MultiField Problems on Fixed Grids. Proceedings of International Conference on Computational Methods for Coupled Problems in Science and Engineering, Greece, 2005 [4] T. Belytschko, T.N. Mo¨es, S. Usui, and C. Parimi. Arbitrary Discontinuities in Finite Elements. International Journal of Numerical Methods in Engineering, 50:993–1013, 2001. [5] B. H¨ubner, E. Walhorn, and D. Dinkler. A Monolithic Approach to Fluid-structure Interaction using Space-time Finite Elements. Computer Methods in Applied Mechanics and Engineering, 193(23-26):2069–2086, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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An enriched space-time finite element method for fluid-structure interaction – Part I: Prescribed structural displacement A. Legay∗ and A. K¨olke† ∗

Chaire de M´ecanique Conservatoire National des Arts et M´etiers, 2 rue Cont´e, 75003 Paris, France [email protected] † Institut f¨ur Statik Technische Universit¨at Braunschweig, Beethovenstraße 51, 38106 Braunschweig, Germany [email protected]

ABSTRACT This contribution introduces a new approach to treat fluid-structure interaction problems. This presentation (part one) focuses on applications with prescribed and a priori known displacement of thin structures. The extension of the presented numerical method to flexible structures enables the approach to handle fully coupled fluid-structure interaction situations (part two). A velocity-pressure-based weak formulation of the governing equations of viscous and incompressible fluid flow (Navier-Stokes-Equations) is discretized by finite space-time elements using a discontinuous Galerkin-scheme for time integration. The resulting space-time slabs are computed sequentially. The location of infinite thin structures in the fluid domain is represented by the zero level set of a space-time defined level set function [1]. To capture the occuring moving discontinuities from embedding a thin solid body into the flow field, a locally enriched space-time (EST) finite element method [2] is applied to ensure a fluid mesh independent from the current configuration of the structure. Based on the concept of the extended finite element method [3] the space-time approximation of the pressure is enriched to represent strongly discontinuous solutions at the position of the structure. The velocity approximation is properly enriched to capture discontinuities in the gradient. The presentation concludes with several numerical examples including flow around flaps with large displacements and rotating blades.

References [1] A. Legay, J. Chessa and T. Belytschko. An Eulerian-Lagrangian Method for Fluid-Structure Interaction Based on Level Sets. Computer Methods in Applied Mechanics and Engineering, in press, 2005 [2] A. K¨olke and D. Dinkler. Extended Space-Time Finite Elements for Boundary-Coupled MultiField Problems on Fixed Grids. Proceedings of International Conference on Computational Methods for Coupled Problems in Science and Engineering, Greece, 2005 [3] T. Belytschko, T.N. Mo¨es, S. Usui, and C. Parimi. Arbitrary Discontinuities in Finite Elements. International Journal of Numerical Methods in Engineering, 50:993–1013, 2001.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Hybrid-Trefftz Finite Element Models for Bounded and Unbounded Elastodynamic Problems Ionut¸ D. Moldovan∗ , Jo˜ao A. Teixeira de Freitas† ∗† Departamento

de Engenharia Civil e Arquitectura, Instituto Superior T´ecnico, Avenida Rovisco Pais, 1049-001 Lisboa ∗ [email protected], † [email protected] ABSTRACT

The displacement and stress models of the hybrid-Trefftz finite element formulation are applied to the spectral analysis of bounded and unbounded elastodynamic problems [4, 3]. The displacement model is derived by constraining the approximation of the displacement field to satisfy locally the governing Navier differential equations and by approximating independently the tractions on the Dirichlet and inter-element boundaries and used to enforce, on average, the local displacement continuity conditions. Conversely, the stress model is derived by constraining the approximation on the stress field to satisfy locally the governing Beltrami differential equations. The displacements are approximated independently on the inter-element and Neumann boundaries and used to enforce, on average, the local flux continuity conditions. Two alternative approaches are used to extend the proposed formulations to semi-infinite (half-space) media, namely a finite element with absorbing boundary and an infinite element. The (fully localised) absorbing boundary condition is built for the displacement model through Dirichlet-to-Neumann (Neumann-to-Dirichlet, in the stress model) mapping [2]. The tractions (displacements) are approximated independently on the absorbing boundary and used to enforce the asymptotic approximation of the Sommerfeld radiation condition. The domain approximation used in the displacement (stress) model infinite element satisfies explicitly the Sommerfeld radiation condition, thus eliminating the uncertainties regarding possible spurious reflections in the vicinity of the absorbing boundary. The alternative stress and displacement models are used to model the response of a fluid saturated porous media, using the Biot’s theory of porous media [1]. Their performance in terms of convergence of the mechanical energy, stresses and displacement estimates and sensitivity to mesh distortion is emphasised.

References [1] M. A. Biot, Theory of propagation of elastic waves in a fluid saturated porous solid. I. Low frequency range. J. Acoust. Soc. America, 28, 168–178, 1956. [2] S. V. Tsynkov, Numerical solution of problems on unbounded domains. A review. Appl. Num. Math., 27, 465–532, 1998. [3] J. A. T. Freitas, C. Cismas¸iu, Hybrid-Trefftz displacement element for spectral analysis of bounded and unbounded media. Int. J. Sol. Struct., 40, 671–699, 2003. [4] J. A. T. Freitas, I. D. Moldovan and M. Toma, Trefftz spectral analysis of biphasic media. Proc. VI World Conf. on Comp. Mech. in conjunction with Asia-Pacific Conf. on Comp. Mech., Sept. 5-10, Beijing, China, Tsinghua University Press & Springer-Verlag, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Explicit dynamic with X-FEM to handle complex geometries P. Rozycki1, E. Bechet 2 , N. Moës1 1 Institut de Recherche en Génie Civil et Mécanique UMR CNRS 6183 Ecole Centrale de Nantes, 1 rue de la Noë, BP 92101, F-44321 Nantes Cedex 3 [email protected]; [email protected] 2

Laboratoire de Physique et Mécanique des Matériaux UMR CNRS 7554 Université de Metz, Ile de Saulcy, F-57045 Metz Cedex 1 [email protected]

ABSTRACT Although the calculation capacities have considerably increased these last years, the complexity of the numerical simulations in dynamic fields still induce many problems, essentially due to CPU time calculation. For instance, the use of explicit scheme yields a critical time step. It depends on the greatest structure eigenvalue [1], [2]. Commonly, rather than to identify this eigenvalue, an upper approximation computed corresponds to the smaller characteristic size (if all elements share the same behavior). These critical time steps are usually induced by mesh constraints. For complex geometries, very small sized elements may arise. One approach is then to optimize the mesh by removing elements or by using mass scaling to improve the critical time step. The work is based on the developments suggested for static problems, which are using the eXtended Finite Element Method [3], [4]. Thanks to the unity partition theory, it is possible to add some specific functions of enrichment to the conventional approximation field of the displacement. These added functions allow, for example, the treatment of cracks, material interfaces, holes, etc. Consequently, this approach authorizes the non-conformity between mesh and discontinuities. This paper presents the work carried out about the X-FEM finite elements, which are dedicated to the dynamic explicit problems including holes or external surfaces. In a first part, the developments are presented for 1D cases and for mono-material structures. The objective is to propose the theoretical framework of the X-FEM finite element: a reformulation of the stiffness matrix and an adapted mass matrix offer the possibility to increase time step calculation in the case of non-meshed surfaces. A generalization of the method is then proposed for the 2D and 3D cases. Validations for different type of structure are exposed. The results in comparison with ABAQUS software are relevant and allows us to present some outlooks.

References [1] M.N. Newmark, A method of computation for structural dynamics, Proc. ASCE 85, EM3, 1959. [2] T. Belytschko, T.J.R. Hughes, Computational methods for transient analysis, North-Holland, 1986. [3] N. Sukumar, D. L. Chopp, N. Moës and T. Belytschko, Modeling Holes and Inclusions by Level Sets in the Extended Finite–Element Method, Computer Methods in Applied Mechanics and Engineering, Vol. 190, Number 46–47, pp. 6183–6200, 2001 [4] C. Daux, N. Moës, J. Dolbow, N. Sukumar, and T. Belytschko, Arbitrary branched and intersecting cracks with the eXtended Finite Element Method, International Journal for Numerical Methods in Engineering, 48:1741-1760, 2000.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Prediction of macroscopic material failure based on microscopic cohesive laws Lidija Stankovi´c, J¨orn Mosler Institute of Mechanics Ruhr University Bochum, Universit¨atstr. 150, D-44780 Bochum, Germany {lidija,mosler}@tm.bi.ruhr-uni-bochum.de ABSTRACT A three–dimensional finite element formulation is applied to the process of determination of macroscopic material properties based on constitutive relationships characterising a microscale. More specifically, a macroscopic failure criterion is computed numerically. The adopted finite element model captures the localised fully nonlinear kinematics associated with the failure on the microscale by means of the Strong Discontinuity Approach (SDA). In contrast to classical continuum mechanics, the deformation gradient is additively decomposed into a conforming part corresponding to a smooth deformation mapping and an enhanced part reflecting the final failure kinematics of the microscale. The implementation of the Enhanced–Assumed–Strain (EAS) concept leads to the elimination of the additional degrees of freedom (displacement jump) on the material point level. More precisely, the applied numerical implementation is similar to that of standard (finite) plasticity. The model does not require any assumption regarding neither the type of the finite elements, nor the constitutive behaviour. Any traction–separation law, connecting the displacement jump to the traction vector, can be chosen. Based on the proposed finite element formulation, microscopic material properties (traction–separation laws) are then used for the computation of the macroscopic material failure. The applicability of the presented numerical model is demonstrated by means of rather academic examples.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Strict, sharp and practical bounds of computed outputs of interest for evolution problems L. Chamoin∗ , P. Ladev`eze∗

†

∗ LMT-Cachan

(ENS-Cachan / CNRS / Paris 6 University) 61, avenue du Pr´esident Wilson - 94235 Cachan Cedex - France [email protected] † EADS

Foundation Chair Advanced Computational Structural Mechanics ABSTRACT

A constant concern both in industry and in research has been the verification of the models used for simulation of physical phenomena. We particularly need to assess the quality of the numerical solutions we get using approximate methods such as the FEM. Effective tools had appeared for thirty years [1] [2], allowing to assess global error (in the energy norm) then local error for quantities of interest which are relevant data for design. For this latter topic, most of the works concern linear problems and yield relatively good bounds of the error for this kind of problem. However, local error estimation for more complex problems has not been mastered yet. Some works dealing with this issue do not yield guaranteed bounds, which is a serious drawback. Others tend to give strict upper and lower bounds but with prohibitive computer ressources. This paper focuses on a method that yields strict bounds for quantities of interest resulting from a finite element analysis of linear evolution problems. The method, developped over the time-space domain, leans on an extraction technique [3] leading to the solution of an adjoint problem. We use dissipation error which is a practical tool developped at the LMT-Cachan for more than ten years [4]. An important feature is the solution of the adjoint problem by means of techniques introducing numerical or analytical functions in space (Partition of Unity Method) and time. Thus, one gets good quality for the bounds on the error with a reasonable numerical cost and without changing the framework of finite element codes. Another aspect of the method is the way to deal with quantities of interest more or less sensitive to history. We take cumulative error effects into account to reach a reliable assessment of the local errors. First results are presented in this paper for linear viscoelasticity problems in 2D. In conclusion, this work which can be extended to non linear problems shows that we can get both good and strict bounds and seems to be therefore a new step forward for the issue of model verification.

References [1] I. Babus˘ka and T. Strouboulis, The finite element method and its reliability, Oxford university press, 2001. [2] P. Ladeve`ze and J-P. Pelle, Mastering calculations in linear and nonlinear mechanics, Springer NY, 2004. [3] R. Becker and R. Rannacher, An optimal control approach to shape a posteriori error estimation in finite element methods, A. Isereles (Ed.), Acta Numerica, 10, 1–120, Cambridge Uni. Press, 2001. [4] P. Ladeve`ze, Nonlinear Structural Mechanics - New Approaches and Non-Incremental Methods of Calculation, Springer NY, 1998.

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Error bounds on outputs of interest for linear stochastic problems. ´ . Florentin, P. Ladeve`ze(
), J. Bellec E LMT-Cachan ENS-Cachan/CNRS/Paris VI University 61, avenue du Pr´esident Wilson - 94235 Cachan Cedex - France e-mail: [email protected] - Web page: http://lmt.ens-cachan.fr (
) EADS Foundation Chair Advanced Computational Structural Mechanics. ABSTRACT This work deals with outputs of interest for linear elastic F.E. analysis in the presence of uncertainties (material, loads...). The objective of this paper is precisely to develop tools for the assessment of linear stochastic models. Our approach relies on an extension of the constitutive relation error method, which is a very effective verification tool in the deterministic case. In order to obtain bounds of outputs of interest, one must solve an adjoint problem. In order to do that, one must build for the direct and adjoint problems an associated admissible displacement-stress pair. Then, bounds corresponding to a given level of certainty can be calculated. Theses bounds take into account the errors due to the finite element discretization as well as the errors due to the stochastic approximation method. The method is illustrated through numerical tests. These tests demonstrate the capabilities of this new tool in providing bounds which can be of direct use to the designer. With such bounds, calculation can lead to certification, even in the case of uncertain loading cases.

References [1] P. Ladev`eze and J.P. Pelle Mastering calculations in linear and nonlinear mechanics Springer NY, 2004. [2] I. Babu˘ska and T. Strouboulis The finite element method and its reliability. Oxford university press, 2001. [3] R. Ghanem and P. Spanos Stochastic Finite Element : A special approach. Springer, 1991. [4] M.K. Deb, I. Babu˘ska, and J.T. Oden Solution of stochastic partial diffenrential equations using galerkin finite element techniques. Comp. Meth. in Applied Mech. and Engrg., 190:6359–6372, 2001. [5] Ghanem R. and Pelissetti M. Error estimation for the validation of model-based predictions. In Proc. of 5th World Congress on Computational Mechanics, 2002.

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An hp-adaptive analysis of some linear free vibration problems ´ ∗ , G. Zboinski ´ † M. Jasinski ∗ Institute

of Fluid-Flow Machinery of PASci Fiszera 14, 80-952 Gda´nsk, Poland [email protected] † [email protected]

ABSTRACT This paper concerns an hp-adaptive finite element analysis of the free vibration problems of linear elasticity. We have implemented the 3D formulation of the Reissner-Mindlin shell theory, higher order hierarchical shell models and the 3D-elasticity to analyse complex (shell-solid) structures. The idea is to use modified methods derived for elastostatics adaptive solutions i.e. Equilibrated Residual Method to estimate local error and Texas 3-step strategy (solutions for an initial mesh, h-refined mesh and p-enriched mesh) to reach the finite element space of desired properties. Basically the procedure for error estimation is as follows. Firstly, we solve the eigenproblem for a given h and p. Then, for a given frequency: we calculate equilibrated interelement stresses and solve local (elemental) problems with these stresses as boundary conditions with h, p+1. With the latter we estimate local error in the energy norm. Having local errors one can perform finite element space update. In this paper we show how the hp-adaptive technique for linear elastostatics can be used in case of free vibrations. Numerical examples conclude the paper.

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Moving Mesh Adaptivity applied to Railway Dynamics Håkan Lane, Per Kettil, Nils-Erik Wiberg Department of Applied Mechanics, Chalmers University of Technology 412 96 Gothenburg, Sweden [email protected]

ABSTRACT Elastic wave propagation induced by the passage of high-speed trains can be the cause of major nuisances for people living close to railway lines. Because of resonance, the phenomenon is especially visible in configurations with a soft material in the soil. When the speed of the train matches propagation velocities in the ground, the vehicle catches up with the wave, leading to greatly magnified displacements. A compound multi body – finite element model has been created to simulate the interaction between the train, the track and the soil [1]. A fixed mesh with a length of 89 m of straight track was used for these analyses. More comprehensive simulations in various conditions require a more flexible model, where the train and the track/soil mesh domain move together across long distances [2]. It will be possible to move the domain in the tangent direction representing straight track, in a linear or quadratic transition followed by a circular elevated curve or along a constant direction, e.g. a slope. As less degrees of freedom are used compared to a large, static mesh, the approach will lead to faster calculations and lower memory demands. Moving an entire FE domain has been used with success in other applications [3]. The magnitude of waves in the different geometries will be analysed and evaluated. It will also be investigated whether vibrations change over long distances in any configuration and/or velocity. Another issue is the question of how many elements are needed for accuracy.

References [1] H. Lane: Rail Vehicle – Track Structure – Subgrade Computational Analysis. Thesis for the degree of Licentiate Engineering. Department of Applied Mechanics, Chalmers University of Technology, Gothenburg, Sweden, 2005. [2] P. Kettil, H. Lane and N.-E. Wiberg: Moving Mesh Domain Adaptation Technique – Application to Train Induced Wave Propagation. Proceedings of the Eighth International Conference on Computational Plasticity held in Barcelona, Spain, 5th – 7th September 2005. [3] R. Gwynllyw, A.R. Davies and T. Phillips: A moving spectral element approach to the dynamically loaded journal bearing problem. Journal of Computational Physics, 2, pp. 476-494, 1996.

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Finite strain r-adaption based on a fully variational framework J. Mosler∗ , M. Ortiz† ∗ Institute of Mechanics Ruhr University Bochum, Universit¨atsstr. 150. D-44780 Bochum, Germany [email protected] † Graduate

Aeronautical Laboratories California Institute of Technology, Pasadena, CA 91125, USA [email protected] ABSTRACT A novel r-adaptive finite element strategy based on a fully variational framework is presented. Provided the underlying physical problem is characterized by means of a minimization principle, the proposed method seeks, for a fixed number of nodes, for the best finite element interpolation depending on the nodal positions with respect to the deformed (x) as well as the undeformed (X) configuration, cf. [1]. The existence of a minimization problem does not represent a very strong restriction, since for many physical problems such as standard dissipative media an incremental potential can also be recast, cf. [2]. While minimizing the potential considered by fixing the nodes within the undeformed configuration corresponds to classical N EWTONian mechanics, a variation with respect to (X) is associated with E SHELBY mechanics, cf. [3]. However, in contrast to the simplicity of the concept, its numerical implementation is far away from being straightforward. According to [4], the resulting system of equations is highly singular and hence, standard optimization strategies cannot be applied. In this paper, a viscous regularization is used. This approach is designed to render the minimization problem well-posed while leaving its solutions unchanged. Obviously, relocating the nodes within the undeformed configuration by fixing the triangulation (the connectivity) may lead to strong topological constraints. As a consequence, an energy based re-meshing strategy is advocated. Contrary to classical mesh-improvement methods based on geometrical quality measures, the novel concepts identifies local energy minimizers. That is, the energy of the new triangulation is always lower than that of the initial discretization. The performance of the resulting finite element model is demonstrated by fully three-dimensional examples.

References [1] P. Thoutireddy and M. Ortiz, A variational r-adaption and shape-optimization method for finitedeformations elasticity. International Journal for Numerical Methods in Engineering, 61, 1-21, 2004. [2] M. Ortiz and L. Stainier, The variational formulation of viscoplastic constitutive updates. Computer Methods in Applied Mechanics and Engineering, 171, 419-444, 1999. [3] M. Braun, Configurational forces induced by finite element discretization. Proc. Estonian Acad. Sci. Phys. Math., 46, 24-31, 1997. [4] J. Mosler and M. Ortiz, On the numerical implementation of Variational Arbitrary LagrangianEulerian (VALE) formulations. International Journal for Numerical Methods in Engineering, 2005 (accepted).

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Extension processes, adaptivity and remeshing for elasto-plastic problems Ernst Rank*, Vera Nübel†, Alexander Düster* * Lehrstuhl für Bauinformatik, Technische Universität München D-80290 München, Germany {rank,düster}@bv.tum.de †

Numerical Simulation, New Business & Technology Hilti Entwicklungsgesellschaft mbH D-86916 Kaufering, Germany [email protected]

ABSTRACT Convergence of the finite element method is obtained by a systematic extension of the Ansatz spaces approximating a given mathematical model. The classical h-version extends the approximation by (uniform or adaptive) mesh refinement using a fixed polynomial degree in each element. The pversion keeps the mesh fixed and increases the element polynomial degree, where again uniform or adaptive methods can be applied. The r-method reallocates nodes and elements and adjusts (if high order elements are used) the geometric shape of element edges and faces to certain criteria. All of the three methods (h-, p-, r-extension) can be combined in order to achieve optimized control over approximation error and computational resources. We will characterize these methods in this paper and compare their performance on benchmark problems for elasto-plastic computation. Rate independent as well as rate dependent elastoplastic problems will be investigated in two as well as three dimensions and it will be shown that an exponential rate of convergence can be obtained by a combination of r- and p-methods. Finally, we will give guidelines for practical computation using lower order elements as they are available in commercial finite element codes.

References [1]

[2]

[3]

[4]

A. Düster, E. Rank. The p-version of the finite element method compared to an adaptive h-version for the deformation theory of plasticity. Computer Methods in Applied Mechanics and Engineering. 190:1925-1935, 2001. A. Düster, E. Rank. A p-version finite element approach for two- and threedimensional problems of the J2 flow theory with non-linear isotropic hardening. International Journal for Numerical Methods in Engineering, 53:49-63, 2002. B. Szabo, A. Düster, E. Rank. The p-version of the finite element method. In: E. Stein, R. de Borst, T.J.R. Hughes, Editoren: Encyclopedia of Computational Mechanics, Volume 1: Fundamentals, Chapter 5, pp. 119-139, John Wiley & Sons, 2004. V. Nübel, A. Düster, E. Rank. An rp-adaptive finite element method for the deformation theory of plasticity. To appear in: Computational Mechanics (2006)

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Efficient implementation of domain decomposition methods using a hierarchical h-adaptive finite element program Juan J. Ródenas1, José Albelda1, Cristina Corral2 and José Mas2 1

Centro de Investigación en Tecnología de Vehículos Departamento de Ingeniería Mecánica y de Materiales Universidad Politécnica de Valencia Camino de Vera s/n. 46022-Valencia, Spain {jjrodena,jalbelda}@mcm.upv.es 2

Instituto de Matemática Multidisciplinar Universidad Politécnica de Valencia Camino de Vera s/n. 46022-Valencia, Spain {ccorral,jmasm}@imm.upv.es

ABSTRACT A previous contribution[1] showed the hierarchical relationships between parent and child elements that come out if these elements are geometrically similar. Under this similarity condition, the terms involved in the evaluation of element stiffness matrices (ke=³BtDB|J|dV), corresponding to parent and child elements, are related by a constant which is a function of the ratio of the element sizes (scaling factor). These relations were used in the basic implementation of a hierarchical h-adaptive Finite Element program based on element subdivision for the resolution of the 2-D linear elasticity problem. The program makes use of a hierarchical data structure to carry out the h-adaptive process, which significantly reduces the amount of calculations required for the evaluation of the problem stiffness matrix, element stresses, element error estimation,… The h-adaptive refinement process based on element splitting produces a natural decomposition of the domain which, together with the hierarchical data structure of the program directly produces an arrowhead stiffness matrix allowing for a decomposition of the global problem into smaller problems. Thus, a domain decomposition solver has been used in this paper to efficiently solve the linear system of equations arising during the analysis process. The numerical test presented in the paper clearly show a considerable improvement in memory requirements and solution times and suggest the use of recursive domain decomposition into the original subdomains.

References [1] J.J. Ródenas, J.E. Tarancón, J. Abelda, A. Roda, J. Fuenmayor, Hierarquical properties in elements obtained by subdivision: a hierarquical h-adaptivity program. In P. Díez and N.E. Wiberg, editors, Adaptive Modeling and Simulation 2005. CIMNE, Sept. 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Adaptive remeshing in transient impact processes with large deformations and nonlinear material behavior Wolfgang A. Wall *, Tobias Erhart†, Ekkehard Ramm†† *

Chair for Computational Mechanics, TUM Boltzmannstr. 15, 85748 Garching (b. München), Germany [email protected] †

††

DYNAmore GmbH, Industriestr. 2, 70565 Stuttgart, Germany [email protected]

Institute of Structural Mechanics, Paffenwaldring 7, 70550 Stuttgart [email protected]

ABSTRACT The present study is concerned with impact processes that appear in civil and military security technology, dynamic soil compaction, vehicle crash or fastening and demolition technology. They are characterized by varying non-linearities, as e.g. large deformations and strains, highly non-linear material behavior, frictional contact between multiple bodies and stress wave propagation. A combination of different methods in adaptivity, constitutive modeling, element technology, efficient time discretization and contact are essential for the reliable computation of practical relevant engineering tasks and for predictions in industrial applications. Accuracy, robustness and efficiency are the authoritative requirements for the solution of those complex problems. In this contribution, we will mainly focus on two issues, namely adaptive remeshing along with subcycling strategies and constitutive modeling aspects especially prepared for impact loading. Since large deformations occur in impact simulations and a Lagrangean description is used, repeated remeshing of individual domains is essential [1,2]. To achieve quality controlled solutions and an optimal distribution of used computational resources at the same time, an adaptive strategy is applied. The core of this strategy is the assessment of discretization errors by adequate error indicators. For this purpose, different possibilities are presented and new methods are developed, which are appropriate for the simulation of transient impact processes. Based on the theory of finite plasticity, constitutive models for thermoviscoplastic metals and cohesive as well as non-cohesive frictional materials are presented and developed. Here, the main focus will be on a formulation for loose, granular media under high pressure loadings [3]. Therefore, a Drucker-Prager-Cap model [4] is modified and enhanced. The properties and effects of the developing powder will be examined. The proposed methods are verified for model problems and their performance in practical relevant applications is evaluated.

References [1] G.T. Camacho and M. Ortiz, “Adaptive Lagrangian modelling of ballistic penetration of metallic targets”, Comp. Meth. Appl. Mech. Engrg., Vol. 142, pp. 269-301, (1997). [2] T. Erhart, L. Taenzer, R. Diekmann and W.A. Wall, “Adaptive remeshing issues for fast transient, highly nonlinear processes”, Proc. of ECCM 2001, Cracow, Poland, (2001). [3] T. Erhart, W.A. Wall and E. Ramm, “A robust computational approach for dry powders under quasi-static and transient impact loadings”, CMAME, 194, pp. 4115-4134, (2005). [4] G. Hofstetter, J.C. Simo and R.L. Taylor, “A modified cap model: Closest Point Solution Algorithms”, Comput. & Struct., Vol. 46, pp. 203-214, (1993).

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Particle Swarms in Engineering Design Problems Bogdan Bochenek, Paweá ForyĞ Institute of Applied Mechanics, Cracow University of Technology Jana Pawla II 37, 31-864 Krakow, Poland [email protected], [email protected]

ABSTRACT Many modern computational techniques are inspired by biological systems. For example, artificial neural network is a simplified model of the human brain whereas genetic algorithm is inspired by the evolution of living forms. Here we discuss another type of biologically based intelligence system called particle swarm. It is an artificial intelligence technique based on the study of social behaviour in the systems of self-organized population. Such a system is made of a group of simple agents which interact with one another and with their environment. The sum of local interactions results in a global behaviour of the population. Ant colonies, bird flocking and fish schooling can serve as examples of swarm intelligence systems found in nature. Nowadays, the most popular swarm inspired method in computational intelligence area is Particle Swarm Optimization (PSO). PSO shares many similarities with evolutionary computation techniques such as genetic algorithms. The system is initialized with a population of random solutions. Taking into account the best positions of particles at subsequent iteration steps the algorithm searches for optima by updating generations. However, unlike genetic algorithms, PSO has no evolution operators such as crossover and mutation. In PSO, particles being potential solutions fly through the problem space searching for the optimum configuration. In past several years, PSO has been successfully applied in many application areas being the basis of the most commonly used non-gradient based stochastic search algorithms. It is demonstrated that in many cases PSO leads to better results obtained in a faster, less expensive way compared with other methods. Another reason that makes PSO attractive is that there are only few parameters to adjust. Usually one version, with slight modifications, works well in a wide range of applications. In this paper a new improved algorithm based on the Particle Swarm Optimization concept is developed and its application to engineering optimization is presented. Many extensions to the original version of the Particle Swarm method introduced by Kennedy and Eberhart in 1995 are proposed. The amendments regard constraint handling as well as modification of the rules of velocities updating. The two-state version of the algorithm is developed in which two weighting factors are used and their values are selected according to the swarm performance. If a particle moves to a better position a history information is disregarded and the free move in this direction is allowed for - state 1, otherwise the new position is calculated based on the swarm performance history - state 2. This new switching technique allows natural creation of swarm leaders. Their behaviour has then great impact on other swarm members what finally speeds up search process convergence. In classic PSO algorithm velocities are limited by arbitrarily selected values what should be treated as a weakness of the algorithm. For example, if many local optima exist and the distance between them is larger than move limits imposed, finding global optimum among them may be even impossible. Here more flexible approach to limiting velocities values is proposed. The algorithm starts with large kinetic energy of particles, which is then limited while exploring search space. As for engineering optimization the implementation of mixed integer/continuous design variables is discussed in detail, and the effective application technique is proposed. The paper is illustrated by numerical results of selected engineering design problems.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Particle Swarm Optimization: efficient globally convergent modifications Emilio F. Campana∗ , Giovanni Fasano∗† , Daniele Peri∗ , Antonio Pinto∗ ∗ The Italian Ship Model Basin (INSEAN) via Di Vallerano 139, 00128 Rome, ITALY {e.campana,g.fasano,d.peri,a.pinto}@insean.it † Istituto

di Analisi dei Sistemi ed Informatica “A. Ruberti” (IASI) - CNR viale Manzoni 30, 00185 Rome, ITALY [email protected]

ABSTRACT In this paper we consider the Particle Swarm Optimization (PSO) algorithm [1, 2], in the class of Evolutionary Algorithms, for the solution of global optimization problems. We analyze a couple of issues aiming at improving both the effectiveness and the efficiency of PSO. In particular, first we recognize that in accordance with the results in [3], the initial points configuration required by the method, may be a crucial issue for the efficiency of PSO iteration. Therefore, a promising strategy to generate initial points is provided in the paper. Then, we address some very preliminary aspects of PSO global convergence towards stationary points, for some Ship Design problems. To this purpose observe that the class of Ship Design applications includes several challenging smooth problems, where expensive simulations provide information to the optimizer, and each function evaluation may require up to hours of CPU-time. In addition, the final solution provided by the optimization method is also required to be a stationary point.

References [1] M.Clerc, J.Kennedy, The Particle Swarm - Explosion, Stability, and Convergence in a Multidimensional Complex Space, IEEE Transactions on Evolutionary Computation, 6, 58–73, 2002. [2] J.Kennedy, R.C.Eberhart, Particle swarm optimization, Proceedings of the 1995 IEEE International Conference on Neural Networks (Perth, Australia), IEEE Service Center, Piscataway, NJ, IV, 1942–1948, 1995. [3] E.F.Campana, G.Fasano, A.Pinto, Dynamic system analysis and initial particles position in Particle Swarm Optimization, IEEE Swarm Intelligence Symposium (SIS2006), May 12-14, 2006, Indianapolis, Indiana, USA.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Evolutionary Optimization of Chemistry of Bulk Metallic Glasses George S. Dulikravich1, Igor N. Egorov2, Nenad Jelisavcic1 1

Florida International University, Department of Mechanical & Materials Engineering 10555 West Flagler Street, Room EC 3474, Miami, Florida 33174, U.S.A. [email protected] 2 IOSO Technology Center Kasatkina 13, Moscow, 129301, Russia [email protected]

ABSTRACT Metallic glass is basically an alloy whose metallic species are “frozen” in amorphous glassy state rather than forming a standard crystalline structure. Metallic glasses have no grain boundaries and no dislocations and stacking faults. They are several times stronger than steel and considerably harder and more elastic. Formation of metallic glasses by extremely high cooling (~105 K/sec) of the melt was first accomplished in 1960s. The resulting metallic glass thickness was limited to extremely thin ribbons. In the 1990s, researchers formed new classes of metallic glasses in bulk. The bulk metallic glasses (BMGs) are composed of three or more metals in the alloy melt and a few diatomatous earth ingredients in order to lower the cooling rate. Cooling rates of the new alloys are from 100 K/s to 1 K/s. The possible thickness of these newer metallic glasses increased from micrometers to centimeters. One of the keys to lowering the cooling speed and creating larger specimens is that bulk metallic glasses should have ingredients with atomic species having large size and chemical differences. Thus, multiple thermo-mechanical properties and the cooling speed of bulk metallic glass alloys depend strongly on the concentrations of each of the chemical elements in a given alloy. The proposed methodology for accurately determining concentration of each of the important alloying elements is based on the use of a combination of a robust multiobjective optimization algorithm and on traditional experimentation. Specifically, the proposed alloy design method combines an advanced stochastic multi-objective evolutionary optimization algorithm based on self-adapting response surface methodology and a relatively very small data set of thermo-mechanical properties and the corresponding concentrations of alloying elements. During the iterative computational design procedure, new metallic glass alloys need to be manufactured and experimentally evaluated for their properties in order to continuously verify the accuracy of the entire design methodology. This metallic glass alloy design optimization method thus minimizes the need for costly and time-consuming experimental evaluations of new metallic glass alloys to fewer than 200 new alloys. References 1. I. N. Egorov-Yegorov, G.S. Dulikravich, Chemical Composition Design of Superalloys for Maximum Stress, Temperature and Time-to-Rupture Using Self-Adapting Response Surface Optimization. Materials and Manufacturing Processes, 20 (3) (2005), 569-590. 2. G.S. Dulikravich, I.N. Egorov, Optimizing Chemistry of Bulk Metallic Glasses for Improved Thermal Stability. Symposium on Bulk Metallic Glasses. TMS 2006 Annual Meeting & Exhibition, eds: Liaw, P. K. and Buchanan, R. A., San Antonio, TX, March 12-16, 2006.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Introduction of Control Points in Splines for Synthesis of Optimized Cam Motion Program Tarun K. Naskar Department of Mechanical Engineering, Jadavpur University Kolkata 700032, India [email protected]

ABSTRACT One of the basic objectives of the synthesis of cam motion program is to minimize the peak values of the kinematic parameters ( KP ) – acceleration ( AP ) and jerk ( J P ) -- of the follower for ensuring smooth and noiseless follower action especially in high-speed machines using cams. Higher order polynomials are combined piecewise at knots for constructing splines and B -splines for cam displacement functions. In this work classical splines of 6th, 7th and 8th orders are taken as cam displacement functions. Multiple control points ( CP s), characterized by parameters like angular position ( AP ) and follower displacement ( FD ), are introduced. The AP and J P are minimized by manipulating the CP parameters. Two parameters of a CP are varied first independently and finally simultaneously with a view to minimizing the AP and J P . The acceleration and jerk of a cam follower are so interrelated that lowering of the value of one gives rise to the value of the other. A method is suggested for minimizing both by varying the values of AP and FD of each CP . A comparative study of AP and

J P obtained from different order spline functions is made. A searching procedure is adopted, based on genetic algorithm (GA) and fuzzy membership function, for obtaining goal function. CP s are also introduced in 6-order and 8-order B -splines. Minimization of the AP and J P is done for two situations – jerk finite and ping finite. CP s in B -splines are characterized by AP . For jerk finite AP and jerk at end position ( J e ) are manipulated, while for ping finite AP and ping at end position ( Pe ) are manipulated. -- first independently and finally simultaneously -- to minimize the

JP .

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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On the use of Differential Evolution in the Trajectory Modeling of Parallel Architecture Robot Giovana T. S. Oliveira*, Sezimária F. P. Saramago*, Plínio J. Oliveira† *

Federal University of Uberlândia 2160 João Naves de Ávila Av., CEP 38400-902, Uberlândia - MG, Brazil saramago @ ufu.br †

Federal University of Goias 1120 Lamartine Pinto Avelar Av., Catalão (GO), Brazil [email protected]

ABSTRACT The advance of the computational recourses has encouraged the utilization of the optimization techniques in the solution of complex problems. Thus, become very attractive the possibility of to join the feature of natural optimization methods to one algorithm which allow to work with small populations and large reduction of computational time. The Differential Evolution (DE) is a simple evolutionary algorithm and it has these advantages. The most distinct feature of DE is to perturb individuals of a population by weighted difference between random individuals of the population. The simplicity, efficiency and robustness of the Differential Evolution in terms of easy implementation are demonstrated by an engineering problem. Thus, to demonstrate the algorithm potentiality the trajectory optimization of a parallel structure is shown and discussed. The procedure is used for optimize the trajectory of a parallel manipulator named as CaPaMan (Cassino Parallel Manipulator) by applying a performance criterion that includes the mechanical energy and total traveling time. The multi-objective function is minimizing by using Differential Evolution. The trajectory is modeling by cubic B-splines. The results are compared with those obtained using Genetic Algorithms.

References [1] S. F. P. Saramago, G. Carbone, M. Ceccarelli, P. J. Oliveira, J. C. M. Carvalho, Optimum

path Planning of Capaman(Cassino Parallel Manipulator) by Using Inverse Dynamics. In: 2nd International Symposium On Multibody Systems And Mechatronics, Musme2005. IFToMM, 1, 332-343, 2005. [2] R. Storn, K. Price, Differential Evolution: a simple and efficient adaptive scheme for global optimization over continuous spaces. Technical Report TR-95-012, International Computer Science Institute, Berkeley, 1995.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Evolutionary Topologic Optimization using the Finite Element Method Mário M. R. Teixeira*, Maurício P. Brandão† *

Instituto Tecnológico de Aeronáutica Praça Marechal Eduardo Gomes, 50 Campus do CTA - Vila das Acácias 12228-901 São José dos Campos - SP – Brasil [email protected] †

Comando Geral de Tecnologia Aeroespacial Avenida Brigadeiro Faria Lima, 1941 Jardim da Granja 12227-000 São José dos Campos - SP - Brasil [email protected]

ABSTRACT Darwin’s Species Evolution Theory is presented as a suitable method to determine a structures optimum topology for a given loading and prescribed boundary conditions. Among other existents methodologies for this task, Genetic Algorithms are discussed with more detail, evidencing its difficulty in simulating biological or physical evolution processes. The Topological Evolutionary Method is applicable to every class of problems, without knowing the crystalline structure or any intrinsic material characteristics, just by the elimination of the elements chosen by the system itself as less useful according to a given criterion. The Finite Elements Method (FEM) is examined as a Solids Mechanics good solution finder. Its main characteristics, as accuracy, modelling capacity, and userfriendliness, are discussed. The applications use a hexahedral finite element to define the domain under analysis and a Von Mises stress value as criterion to select the to-be-eliminated elements. The operator defines approximated, but not necessarily exact, domain, exact loads, and nodes restrictions positioning. Then, the operator searches for a solution that can be achieved in two ways: best topological shape (positive-defined global stiffness matrix) or desired specific mass. The application, software developing, auxiliary computational tools, and microcomputer operational system are all based on free software. Some classical examples (truss structure, cantilever and Michel beam) in 3D and 2D formulations have solutions obtained by the application. The results are discussed and compared with solutions available in the literature. Checkerboards, as a side effect of approximated methods implementations, like the FEM, are discussed. Filters are applied for 2D formulations to decrease checkerboard effects and also to accelerate execution. Applications can be used to solve any structural topologic optimization problem with suitable geometric domain, actual loading, and nodes restrictions, by using adequate finite element grids.

References [1] C. Darwin. On the origin of species by means of natural selection, or the preservation of favoured races in the struggle for life. John Murray, Albemarle Street, London, 1859. [2] Technical Committee on Optimal Structural Design of the ASCE. Recent Advances in Optimal Structural Design. Chap. 1. Edited by Scott A. Burns. May, 2002. 384p.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A Multilevel Domain Decomposition Methodology for Solving Coupled Problems in Fluid-Structure-Thermal Interaction Eugenio Aulisa∗ , Sandro Manservisi∗,† , Padmanabhan Seshaiyer∗ ∗

Texas Tech University Department of Mathematics and Statistics, Lubbock, 79409-1042, TX {eugenio.aulisa, sandro.manservisi, padmanabhan.seshaiyer}@ttu.edu † University of Bologna DIENCA, Laboratory of Montecuccolino, via dei Colli 16, Bologna, 40136, Italy [email protected]

ABSTRACT Engineering analysis is constantly changing to develop novel techniques to solve coupled processes that arise in multi-disciplinary applications. Efficient solutions to complex coupled processes involving fluid-structure-thermal applications are still a challenging problem in computational sciences and engineering. Currently there exist numerous public-domain and commercial codes available for Computational Fluid Dynamics (CFD), Computational Structural Dynamics (CTD) and Computational ThermoDynamics (CTD). Different groups specializing in modeling individual process such as CSD, CFD, CTD often come together to solve a complex coupled application. The coupling of these solvers provide an insight for predictive capability for simulating complex nonlinear interactions that arise in several applications such as, hypersonic flight, where the structural deformation due to the aerodynamics and thermal loads leads to a significant flow field variation; MAVs (Micro Air Vehicles) where the geometry changes possibly due to thermal effects may lead to a transient phase in which the structure and the flow field interact in a highly non-linear fashion. Direct numerical simulation of the highly non-linear equations, governing even the most simplified fluid-structure-thermal interaction models depend on the convergence of iterative solvers which in turn relies heavily on the properties of the system coupling. Domain decomposition techniques have become increasingly popular in this regard, for obtaining fast and accurate solution. The the global domain (on which the coupled process evolves) is partitioned into several sub-domains over each of which, local problems are solved. The solution of the global problem is then constructed by suitably piecing together solutions obtained locally from independently modeled sub-domains. During this assembly process, it is often necessary to guide and coordinate non-matching grids arising over separate sub-domains. The purpose of this paper is to introduce a flexible multilevel algorithm with finite elements that can be used to study a coupled fluid-structure-thermal interaction (FSTI). The method relies on decomposing the complex global domain, into several local sub-domains; solving smaller problems over these subdomains and then gluing back the local solution in an efficient and accurate fashion to yield the global solution. Our numerical results suggest that the proposed solution methodology is robust and stable.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Moving Mesh Algorithm for Unstructured Grids Based on Interpolation with Radial Basis Functions A. de Boer, M. S. van der Schoot, H. Bijl Delft University of Technology Kluyverweg 2, 2629 HS, Delft, The Netherlands [email protected] ABSTRACT Fluid-structure interaction computations typically involve moving boundaries for the flow due to the deformation of the structure. Examples can be found in: flutter simulation of wings, blood flow through veins and stability analysis of bridges and tall buildings subjected to windloads. Because of these moving boundaries a fast and reliable method for deforming the computational grid is needed to be able to perform the unsteady flow computations accurately and efficiently. Structured grids can be deformed by fast and accurate algebraic techniques, but for the meshing of complex domains and grid adaptation the greater flexibility of unstructured grids is required. For unstructured grids two different mesh movement strategies are known. The first exploits the connectivity of the internal grid points. The connection between the grid points is represented for example by springs or as solid body elasticity. These methods involve solving a system of equations as large as the number of flow points involved and are therefore very expensive. Also special treatment is required for hanging nodes. The other strategy moves each grid point individually based on its position in space and are the so called point-by-point schemes. Hanging nodes are no problem and when radial basis function interpolation is used, a much smaller system, only involving the nodes on the boundary, has to be solved. Also the implementation for partitioned meshes, occuring in parallel flow computations, is straightforward. However, untill now point-by-point schemes are only applied to the boundary nodes of multi-grid blocks [1] (the structured interior mesh of the blocks is adapted with algebraic techniques) or the data transfer over the fluid-structure interface [2, 3]. In this paper a new point-by-point mesh movement algorithm based on interpolation with radial basis functions (RBF’s) is developed, which interpolates the displacements of the boundary nodes to the whole flow mesh, instead of over the fluid-structure interface only as is the case in [2] and [3]. The algorithm is tested with several RBF’s for a variety of problems. The new method can handle large translations, rotations and deformations, depending on the used RBF. The best accuracy and robustness are obtained with the thin plate spline [3]. However, when efficiency is more important, the C 2 RBF with compact support [2] is the best choice. Further research includes comparing the new method with existing methods on accuracy and efficiency and applying it to a real fluid-structure interaction problem.

References [1] M. A. Potsdam, G. P. Guruswamy, A parallel multiblock mesh movement scheme for complex aeroelastic applications, Tech. Rep. AIAA-2001-0716, 2001. [2] A. Beckert and H. Wendland, Multivariate interpolation for fluid-structure-interaction problems using radial basis functions, Aerospace Science and Technology, 0, 1–11, 2001. [3] M. J. Smith, C. E. S. Cesnik, D. H. Hodges, Evaluation of some data transfer algorithms for noncontiguous meshes, Journal of Aerospace Engineering 13 (2), 52–58, 2000.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

419

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Mechanical Engineering Department, School of Engineering, Shiraz 71345, Iran e-mail: [email protected] Ŕ

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Mechanical Engineering Department, School of Engineering, Shiraz 71345, Iran 5KCOCMPA"[CJQQEQWM

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III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

420

Fluid-Structure Interaction in FEM Journal Bearing Simulations Alex de Kraker∗ , Daniel J. Rixen† , Ron A.J. van Ostayen∗ ∗ Delft

University of Technology, Faculty of Mechanical, Maritime and Marine Engineering, Department of Precision & Microsystems Engineering, Mechatronic Design, laboratory of Tribology Mekelweg 2, 2628 CD Delft, The Netherlands [email protected] † Delft University of Technology, Faculty of Mechanical, Maritime and Marine Engineering, Department of Precision & Microsystems Engineering, Mechatronic Design, laboratory of Dynamics [email protected]

ABSTRACT

This paper describes a numerical method solving the mixed lubrication problem for stationary running elastic journal bearings. The problem is described by a coupled set of 2D Reynolds - and 3D structure deformation equations. An asperity contact model, relating the contact pressure to the film height, has been used to account for partial contact in mixed lubrication. With increasing loads, the bearing deformation becomes more important to the performance of the bearing system, especially with the introduction of polymers as a bearing material. This is due to the high sensitivity of the pressure solution from the Reynolds equation with respect to a variation in film height. Therefore, a strong coupling exists between the fluid and structure equations. The fluid and structure equations are solved seperately by finite elements. A direct iterative algorithm with various under relaxation strategies has been found insufficient to find a converged solution to the coupled journal bearing problem [1], even in full film lubrication where no contact occurs. For softer surfaces or higher loads, the bearing deformation becomes larger and the coupling between the equations becomes stronger. In the method presented in this paper, artificial dynamics have been added to the stationary structure deformation equations by the introduction of a damping term to the discretised set of equations. To enforce convergence of the problem, the ratio between the damping coefficient and the artificial time step can chosen. The so called Stribeck curve, depicting the bearing coefficient of friction at constant load as a function of the rotational frequency of the journal - or shaft -, is a useful tool to evaluate the bearing performance. Therefore, an additional computing loop iterating for the correct journal position that results into the target load, is needed. To calculate the complete Stribeck curve, at least 50 points are necessary and hence, the fluid structure problem including the constant load constraint has to be solved about 50 times at different journal frequency. Hence, three nested loops are necessary: a time-loop, solving the fluid structure equilibrium, a second loop, iterating for the target load and an outer loop that takes a number of velocity steps, computing the Stribeck curve. Reduction of computing time can be obtained by scaling the intermediate solutions in the fluid-structure - or time - loop. As we know that the final solution will meet the load constraint, intermediate solutions for the fluid - and contact pressure distribution in the time loop are scaled with respect to the target load. Hence, the total load that is applied to the bearing surface is constant and that we only iterate for the correct balance between fluid - and contact pressure. As a result, no large differences in the solution for the bearing deformation between successive iterations occur and convergence is obtained faster.

References [1] Oh, K.P., Huebner, K.H., Solution of the Elastohydrodynamic Finite Journal Bearing Problem. Journal of Tribology, July, 342–352, 1973.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

421

Numerical simulation of wind-structure interaction for thin shells and membranes Alexander Kupzok, Roland Wüchner and Kai-Uwe Bletzinger Technical University Munich, Arcisstr. 21, 80290 München, Germany { kupzok, wuechner, kub}@bv.tum.de

ABSTRACT Modern architecture promotes light and efficient structures. With the use of innovative constructions and materials, the realization of wide-spanned and creative buildings is possible. However, increasing lightness and slenderness bring along a higher susceptibility to wind effects, which can become the decisive design factor. An accurate assessment of these wind effects with deterministic tools is complicated, in particular in the case of aeroelastic phenomena. In this regard, numerical multiphysics simulations are a promising complement and enhancement to elaborate experimental approaches. The long-term aim of this research is to propose a methodology for the analysis and improvement of light, thin-walled structures, such as thin shells and membrane roofs towards wind effects. The focus is on the appropriate combination of different physical and numerical disciplines to account for the relevant factors inherent to the simulation of light, thin-walled structures as well as highly turbulent air flows. To fulfill these requirements the occurring wind-structure interaction is accessed by a surface-coupled fluid-structure interaction (FSI) method. This is realized in a modular and flexible software environment with the use of a partitioned coupling approach: the structural field is solved by the in house finite element program CARAT using several finite element types and advanced solution strategies for form finding, nonlinear and dynamical problems. The fluid field is solved by the CFD software package CFX-5 of ANSYS Inc. Additional care towards the realistic modeling of physical wind is taken. A prerequisite to allow for the assessment of aeroelastic problems, beyond the mere exchange of data between the two physical fields, is the utilization of stable as well as efficient coupling strategies. Moreover, the comprehensiveness of this approach opens the possibility for multiphysics optimization. The contribution will present theory and realization of an implementation enhanced by illustrative examples. Strategies for the extension of the approach towards multiphysics optimization will be presented.

References [1] K.-U. Bletzinger, Roland Wüchner, Alexander Kupzok: “Towards FSI for light-weight structures subjected to wind“, in: Conference Proceedings of Int. Conf. on Computational Methods for Coupled Problems in Science and Engineering COUPLED PROBLEMS 2005 (M. Papadrakakis, E. Oñate and B. Schrefler ,eds) CIMNE, Barcelona, Spain, 2005 [2] M. Kuntz, J.Carregal Ferreira, F. R. Menter and G. N. M. Oudendijk, “Analysis of FluidStructure Interaction with an improved coupling strategy”, ECT Conference, Prague, (2002). [3] D.P. Mok, W.A. Wall, “Partitioned analysis schemes for the transient interaction of incompressible flows and nonlinear flexible structures”, in Trends in Computational Structural Mechanics (W.A. Wall, K.-U. Bletzinger, K. Schweizerhof, eds.), CIMNE, Barcelona, 689- 698, (2001).

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

422

Reliability Analysis of Prestressed Egg-shaped Digester Jie LI *, Hua-ming CHEN †, Jian-bing CHEN † *

School of Civil Engineering, Tongji University Siping Road 1239, Shanghai, P. R. China [email protected]

† Shanghai Municipal Engineering Design Institute Zhong Shan North Second Road 901, Shanghai, P. R. China [email protected] †

School of Civil Engineering, Tongji University Siping Road 1239, Shanghai, P. R. China [email protected]

ABSTRACT The fluid-solid interface of prestressed egg-shaped digester is conical, so the dynamic liquid pressure and the resulting reactions are rather difficult to gain[1]. Through the computer program ANSYS, seismic evaluation of the digester is performed using a 3D finite-element model that includes the effect of fluid-solid interaction[2]. Comparisons of calculated responses and those of shaking table tests show reasonably good agreement. By using the probability density evolution method for random vibration analysis of stochastic structures which has been developed in recent years [3,4], reliability of the prestressed egg-shaped digester is investigated. The results demonstrate that the proposed method is efficiency for larger structures.

References [1] Sutter,Gerhard; Hanskat, Charles S. World’s largest egg-shaped digesters. Water Environment and Technology, 1990, 29(40):52-55 [2] Mehdi S. Zarghamee, Atis A. Liepins, etc. Egg-shaped digester design and seismic evaluation. Restructuring: America and Beyond Structures Congrees - Proceedings. ASCE, New York, NY, USA. 1995:1766-1780 [3] Li J. Stochastic Structural System: Analysis and Modeling. Beijing: Science Press, 1996. [4]

Li J, Chen JB. Probability density evolution method for analysis of stochastic structural dynamic response. Actamechanica sinica. 2003, 35(6):716-722.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

423

The Vortex Structures in the Sphere Wakes in the Wide Range of the Reynolds and Froude Numbers Paul V. Matyushin, Valentin A. Gushchin Institute for Computer Aided Design of the Russian Academy of Sciences (ICAD RAS) 19/18, 2nd Brestskaya str., Moscow 123056, Russia [email protected], [email protected]

ABSTRACT At the present paper the homogeneous (at 200 ” Re ” 106, fig. 1 (a-b)) and stratified (at 50 ” Re ” 1000, 0.005 ” Fr ” 1, fig. 1 (c)) viscous incompressible fluid flows around a sphere are investigated by means of the direct numerical simulation (DNS) and the visualization of the 3D vortex structures in the wake (Reynolds number Re = Ud/v, where U is the free-stream velocity, d is the diameter of the sphere, and v is the kinematic viscosity; Froude number Fr = U/(N·d), where N is the buoyancy frequency). For DNS the Splitting on physical factors Method for Incompressible Fluid flows (SMIF-MERANGE) with hybrid explicit finite difference scheme (second-order accuracy in space, minimum scheme viscosity and dispersion, monotonous) is used [1]. For the visualization of the 3D vortex structures in the sphere wake the isosurfaces of Im(ı1,2) are drawing, where Im(ı1,2) is the imaginary part of the complex-conjugate eigen-values of the velocity gradient tensor (fig. 1). In spite of the set of the papers devoted to the homogeneous fluid flows around a sphere the detailed formation mechanisms of vortices (FMV) in the sphere wake are still unclear [2]. At the present paper for the homogeneous fluid flows the six basic FMV have been selected; the detailed FMV for the different unsteady periodical flow regimes are explained (270 < Re d 290, 290 < Re d 320, 320 < Re ” 400, 400 < Re < 700 and Re > 700); at 290 < Re d 320 a new flow regime has been found; the laminar-turbulent transition in the boundary layer on the sphere is discussed (9·104 < Re < 4·105). The numerical studies of the stratified fluid flows past a sphere are very rare. At the present paper four different flow regimes of the stratified fluid flows have been simulated: 1) “Two steady twodimensional attached vortices” (Re < 120, Fr < 0.2, fig. 1 (c)); 2) “Unsteady two-dimensional attached vortices” (Re > 120, Fr < 0.15); 3) “Lee-wave instability” (0.2 < Fr ” 0.4); 4) “Nonaxisymmetric attached vortex” (Re < 500, Fr > 0.4). The high gradient sheets of density have been observed near the poles of the resting sphere [3] and of the moving sphere (Fr ” 0.02). The lee waves, the recirculating zone and other vortex structures of the wake have been visualized (fig. 1 (c)). This work is supported by Russian Foundation for Basic Research (grant ʋ 05-01-00496); by the program “Mathematical Modeling” of the Presidium of the Russian Academy of Sciences (RAS); by the program ʋ 3 for Basic Research of the Department of the Mathematical Sciences of RAS.

a)

b)

c)

Fig. 1. a) Re = 350, Im(ı1,2) = 0.05; b) Re = 4.1·105, Im(ı1,2) = 2; c) Re = 100, Fr = 0.08, Im(ı1,2) = 0.005.

References [1] V. A. Gushchin, V. N. Konshin, Computational aspects of the splitting method for incompressible flow with a free surface. J. of Computers and Fluids, 21, ʋ 3, 345-353, 1992. [2] V. A. Gushchin, A. V. Kostomarov, P. V. Matyushin, 3D Visualization of the Separated Fluid Flows. Journal of Visualization, 7, ʋ 2, 143-150, 2004. [3] V.G. Baydulov, P.V. Matyushin, Yu.D. Chashechkin, Structure of a diffusion-induced flow near a sphere in a continuously stratified fluid. Doklady Physics, 50, ʋ 4, 195-199, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

424

Transient Analysis Methods for Hypersonic Applications with Thermo-Mechanical Fluid-Structure Interaction Reinhold Niesner*, Matthias Haupt*, Peter Horst* *

Institute of Aircraft Design and Lightweigh Structures, TU Braunschweig Hermann-Blenk-Str. 35, 38108 Braunschweig, Germany [email protected]

ABSTRACT A numerical framework developed for coupled fluid-structure interaction problems is presented, with emphasis on the numerical methods employed for time integration and equilibrium iteration. The framework has been succesfully used in the German IMENS project for the simulation of hypersonic applications. The framework uses a modular concept with stand-alone solvers for the disciplines involved (fluid and structure) with a thoroughly designed data interexchange interface [1]. The solution approach used for mechanical quasi-stationary and thermal transient coupled analyses is presented, taking into account different time scales of the physical domains and efficiency aspects, which are essential when dealing with complex models and solution strategies (e.g. Navier-Stokes codes). For time integration both iterative and simple staggered methods have been used to account for accuracy and efficiency demands of the different problem cases. Several means to accelerate the time integration have been studied. For iterative staggered methods the acceleration methods for the equilibrium iteration presented in [2] have been adopted and improved for the present study case. For example, the gradient method proposed in [2] has been modified to suit the modular concept approached here, where neither the Schur complements can be explicitly computed nor is it possible to easily switch off boundary conditions, as demanded in [2]. The control theory approach for adaptive time stepping presented in [3] has been tested and adapted for the present application area of hypersonic fluid-structure interaction (thermo-mechanical interaction with compressible NavierStokes flows and geometrically and physically non-linear structures). The benefits of this approach over the classical time stepping strategies have been evaluated on the basis of selected parameter studies. The different numerical methods have been tested, evaluated and compared using some generic example models from the IMENS project, e.g. a flap-gap configuration or a nosecap model, which are the subject of this presentation.

References [1] M. Haupt, R. Niesner, P. Horst, Coupling Techniques for Aero-Thermo-Elasticity. Proc. of Int. Conf. on Computational Methods for Coupled Problems in Science and Engineering, Coupled Problems 2005, Santorin, 2005. [2] W.A. Wall, D.P. Mok, E. Ramm, Interactive Substructuring Schemes for Fluid Structure Interaction. Analysis and Simulation of Multifield Problems (W.L. Wendland, M. Efendiev, eds.), 349-360, Springer, 2003. [3] A.M.P. Valli, G.F. Carey, A.L.G.A Coutinho, Control strategies for timestep selection in finite element simulation of incompressible flows and coupled reaction-convection-diffusion processes. International Journal for Numerical Methods in Fluids, 47, 201-231, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

425

Stall Induced Vibration & Flutter In A Symmetric Airfoil Sunetra Sarkar∗ , Hester Bijl∗ ∗ Delft

University of Technology [email protected] ABSTRACT

This paper investigates the aeroelastic stability of a wind turbine rotor in the dynamic stall regime. Increased flexibility of modern turbine blades make them more susceptible to aeroelastic instabilities. Further, complex oscillation modes like flap/lead-lag are of particular concern, which give way to potential structural damage [1], [2]. We study the stall induced oscillations in pitching direction and in combined flapwise, lead-lag wise directions. The aerodynamic loads acting on the rotor body in the stall regime are nonlinear. We consider a wide ranging parametric variation and underline their effect on the aeroelastic instability and overall nonlinear dynamical behavior of the system. An engineering dynamic stall model (Onera) [4], [3] has been used to calculate the aerodynamic loads. They represent the aerodynamic loads well in the dynamic stall regime and captures the bifurcation behavior and the chaotic routes of the aeroelastic system under study. The aerodynamic loads are given in terms of differential equations which are combined with the governing equations of the aeroelastic system; the resulting system of equations are solved by a 4/5th order variable step Runge-Kutta method. Parameters considered for the pitching oscillation case are nondimensional airspeed (U ), mean angle of attack (αm ), initial condition (αinit ), structural nonlinearity (Knl ) and reduced frequency (k) and amplitude (F¯0 ) of external forcing. Both the self excited and forced system reveal existence of routes to chaos. For different αm , period doubling routes to chaos have been obtained with different initial conditions. A cubic structural nonlinearity has been seen to alter the bifurcation pattern of the above system. Varying k as a bifurcation parameter in the forced system shows presence of period-3 orbits near chaos. The second case of flap/edgewise oscillation in the stall regime identifies nondimensional rotational speed of the rotor along with structural stiffnesses and nonlinearity as most important parameters of the self excited system. However, no chaotic response has been obtained. External forcing shows presence of higher harmonics and quasi-harmonics in the response. Once again, no chaotic attractor has been found.

References [1] Chaviaropoulos, P., Flap/lead-lag aeroelastic stability of wind turbine blade sections. Wind Energy, 2, 99–112, 1999. [2] Chaviaropoulos, P. , et al., Viscous and aeroelastic effects on wind turbine blades. Wind Energy, 6, 387–403, 2003. [3] Dunn, P. and Dugundji,J, Nonlinear stall flutter and divergence analysis of cantilevered graphite/epoxy wing. AIAA Journal, 30, 153–162, 1992. [4] Tran, C.T. and Petot, T., Semi-empirical model for the dynamic stall of airfoils in view of the application to the calculation of responses of a helicopter blade in forward flight. Vertica, 5, 35– 53, 1981.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

426

Application of Lagrange Multipliers for Computational Aeroelasticity Ralf Unger∗ , Matthias C. Haupt∗ , P. Horst∗ ∗ Institute

of Aircraft Design and Lightweight Structures Technical University Braunschweig Hermann-Blenk-Str. 35 38108 Braunschweig, Germany [email protected]

ABSTRACT The prediction of aeroelastic effects is one of the key problems during the design process of an aircraft. One challenging aspect of this goal is to compute space and time-accurate fluid and structural interactions. In the loose coupling approach, well-established CFD and CSD codes are taken and integrated in a flexible software environment. The main focus of the present work is on the state and load transfer over nonconforming grids on the coupling interface. To fulfill conservation in the overall solution process, a weak formulation of the continuity conditions on the common interface is used and Lagrange multipliers are introduced. This approach is based on a variational formulation of the scalar energy functional of the full system, which utilize Hamilton’s principle and which will be given here. Using an intermediate frame between the interfaces to be joined leads to the so-called three-field formulation, where an additional interface state variable and two Lagrange multipliers are used, [1]. The simplified and more common two-field formulation is utilized which is equivalent with a weighted residual formulation of the continuity condition. Using Galerkin’s method leads to a transfer scheme, which minimizes the L2 error norm. An extended transfer approach, which minimizes the more general Sobolev norm, [2], will be discussed and applied to aeroelastic problems and further the use of dual-Lagrange multipliers will be presented. Numerical results obtained from simulation of an oscillating one-dimensional plate in transonic flow and three-dimensional wing example will be presented to demonstrate the applicability and performance of the concepts and to compare the properties of the different coupling techniques and transfer methods.

References [1] K.C. Park, C.A. Felippa, R. Ohayon, Partitioned formulation of internal fluid-structure interaction problems by localized Lagrange multipliers. Comput. Meth. Appl. Mech. Eng., 190, 2989–3007, 2001. [2] X. Jiao, M.T. Heath, Common-refinement-based data transfer between non-mathing meshes in multiphysics simulations. Int. J. Num. Meth. Engng., 61, 2402–2427, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

427

Computer Simulation of Diffraction Technique Applied for Measurements of Surface Stress Gradients J.T. Assis, V.I. Monine, S.A. Philippov, and S.M. Iglesias

Instituto Politécnico/UERJ Nova Friburgo, Rio de Janeiro, Brazil [email protected]

ABSTRACT Modern surface treatment technologies, like laser treatment or surface modifications by ion beams, introduce high residual stresses characterized by strong gradients of stress distribution in the depth of treated surface. The diffraction methods of stress gradient determination based on analysis of nonlinearity of dϕ,ψ - sin2ψ dependency is not sufficient to practical using because experimental criterions predicting the existence and level of stress gradient are absent. Experimental attempts to develop the methodology of stress gradient determination are not successful because of the difficulty to prepare the samples with known parameters of stress gradient. Computer simulation of diffraction profile formed by reflection from surface layers with stress gradient allows to simulate experimental dependency dϕ,ψ = f(sin2ψ) permitting to obtain the relationships between the stress gradient parameters, non-linearity of dϕ,ψ = f(sin2ψ) and broadening of diffraction line caused by stress.

References [1] P. Nikravesh, Computer-aided analysis of mechanical systems. Prentice-Hall, Englewood Cliffs, New Jersey, 1988. [2] W. Schiehlen, Multibody system dynamics: Roots and perspectives. Multibody System Dynamics, 1, 149–188, 1997. [3] F. Armero and S. Glaser, Enhanced strain finite element methods for finite deformation problems. M. Doblaré, J.M. Correas, E. Alarcón, L. Gavete and M. Pastor eds. III Congreso de Métodos Numéricos en Ingeniería, SEMNI, Barcelona, Spain, 423-437, 1996.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

428

Numerical Assessment of a Micromorphic Model of Ductile Rupture Koffi Enakoutsa∗ , Jean-Baptiste Leblond∗ , Gilles Perrin† ∗ Universit´ e Pierre et Marie Curie (Paris VI) Laboratoire de Mod´elisation en M´ecanique, Tour 65-55, 4 place Jussieu, 75252 Paris Cedex 05, France [email protected], [email protected] † Institut

Franc¸ais du P´etrole Division de M´ecanique Appliqu´ee, 1-4 avenue de Bois-Pr´eau, 92852 Rueil-Malmaison Cedex, France [email protected] ABSTRACT All constitutive models involving softening predict unlimited strain localization, and the famous Gurson [1] model of ductile rupture is no exception. An improved variant of this model aimed at solving this problem was derived by Gologanu et al. [2] from some refinement of Gurson’s original homogenization procedure. They obtained a new model of “micromorphic” nature, involving the second gradient of the macroscopic velocity and generalized macroscopic stresses of “moment” type. In this paper, the practical relevance of this new model is investigated through study of its numerical predictions. Two criteria are used for this critical assessment: absence of mesh size effects in finite element computations and agreement of numerical and experimental results for some typical ductile fracture tests. The necessary implementation of the model into some finite element code raises two main problems. The first one is the apparent need for elements of class C 1 . This need is obviated through introduction of some new nodal variables representing the components of the strain rate. The second difficulty lies in the necessary operation of “projection” onto the sophisticated yield locus. An implicit algorithm similar in principle to the classical Nguyen [3] algorithm for the von Mises criterion, although much more complex in detail, is adopted for this purpose. Numerical simulations of the fracture of axisymmetric pre-notched and pre-cracked specimens reveal satisfactory independence with respect to mesh size and reasonable agreement with experimental results. This shows that the model of Gologanu et al. [2] may be regarded as a viable solution to the problem of unlimited localization in Gurson’s model of ductile rupture.

References [1] A.L. Gurson, Continuum theory of ductile rupture by void nucleation and growth: Part I - Yield criteria and flow rules for porous ductile media. ASME Journal of Engineering Materials and Technology, 99, 2-15, 1977. [2] M. Gologanu, J.B. Leblond, G. Perrin and J. Devaux, Recent extensions of Gurson’s model for porous ductile metals. P. Suquet ed., Continuum Micromechanics, CISM Courses and Lectures 377, Springer, 61-130, 1997. [3] Q.S. Nguyen, On the elastic plastic initial-boundary value problem and its numerical integration. International Journal for Numerical Methods in Engineering, 11, 817-832, 1977.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

429

Fatigue Crack Trajectory Analysis of Single-Side Repaired Thin Aluminum Panels with Various Composite Patch Lay-up Configurations H. Hosseini-Toudeshky, B. Mohammadi, S. Bakhshandeh Aerospace Engineering Department, Amirkabir University of Technology Hafez Ave., 424, Tehran, Iran {Hosseini, Bijan_Moh, Bakhshandeh}@aut.ac.ir

ABSTRACT In this paper experimental and numerical finite elements fatigue crack propagation analysis of the single-side repaired thin aluminium panels containing an initial inclined flaw of 450 are studied. These panels are repaired with the 4 layers glass/epoxy composite materials with the lay-up configurations of [90]4, [105]4, [-45]4, [-45/+45]2, and [902/02]. In the performed three dimensional analyses it was assumed that the crack-front remains perpendicular to the panels’ surfaces during its propagation. The effects of the patch lay-up configurations on the crack trajectories at patched and un-patched surfaces of the panels are presented. It will be shown that crack trajectories at patched surface is different from the un-patched surface and therefore leads to the existence of a three dimensional fatigue-fracture surface for all repaired panels. A typical fracture surface is shown in Figure 1-(a). Figure 1-(b) compare the typical crack trajectories of un-patched surface of the repaired panel with the patch layup of [-45]4 obtained from experiment with that obtained from finite elements analyses. The finite elements results show a comparable agreement with those obtained from the experiments. The finite elements results also show that using various lay-up configurations may lead to various crack propagation orientation with 50 to 100 differences.

2B FEM

Exp.

(a) (b) Figure 1. (a) Typical fracture surface; (b) Typical crack trajectories of un-patched surface of the repaired panel with the patch lay-up of [-45]4 obtained from experiment and FEM

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

430

Prediction of the Crack Initiation Life of Turbine Blade Sharadchandra D. Jog and Rajeshwar. Baddam Indian Institute of Technology, Bombay Mechanical Engineering Department IIT, Powai, Mumbai, 400076. [email protected] Indian Institute of Technology, Bombay Mechanical Engineering Department IIT, Powai, Mumbai, 400076. [email protected]

ABSTRACT High Cycle Fatigue of turbo machinery blades is a significant design problem because one of the turbine stages may operates very close to the resonant condition and lead to fatigue failures. In order to assess the crack initiation life of a turbine blade, it is essential to correlate vibration to fatigue. Often a crack initiates from the material imperfections under the combination of steady stresses and fluctuating stresses in high cycle fatigue phenomena. This work models a turbine blade as an untwisted, non tapered cantilever beam with asymmetric cross section. The natural frequencies of the turbine blade were determined by using modal analysis in ANSYS. Nozzle excitation frequencies and forces were determined from the analysis of flow path field between stator and rotor blades. The critical condition at which natural frequencies are coincident with nozzle excitation frequencies were spotted from the Campbell diagram. Steady stresses and dynamic stresses were calculated in ANSYS using excitation forces corresponding to the resonance condition. The stress results obtained were compared with the analytical approach. The true stresses in the vicinity of the defect were calculated by Neuber’s rule with dynamic stresses as input. Local strain around defect was calculated through the formulae given by Martin et al. Crack initiation life was predicted by solving strain life equation.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

431

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III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

432

Computed Analysis to Determine Service Life Criteria of Special Elements and Applications M. Kopecky Dept.of Physical Material Engineering Faculty of Industrial Technologies, University of Trencin SK-020 01 Puchov, Slovak Republic [email protected]

ABSTRACT A characteristic feature of new trends in development of new aggregates of mobile machinery is a continuous increase in manufacturing and operating costs. Simultaneously, transmitted outputs are also higher and a sufficient reliability has to be maintained. There is a tendency towards a higher use of materials, i.e. a relatively higher stress on particular parts of the aggregate. At the same time, a real safety of operation against the maximum admissible stress decreases. This all requires a further improvement of the method of designing and strength checking of a construction. The problem of fatigue strength and service-life, as the most important phenomena of strength reliability under those conditions, is connected more or less with a certain degree of uncertainty. The methods described in this paper are the ways to reach the solution goals by means of a characteristic curve of fatigue strength and reduced fatigue curve with the maximum use of computer technology.

References [1] M. Kopecky, F. Peslova. Assessment methodology of elements and constructions, reliability criteria for mobile machines and equipment. In: ISTLI special publication 2: Teaching and Education in Fracture and Fatigue, Imprint: E & FN SPON , London, England, pp.325-330, (1996). [2] V. Cuth, J. Tvaruzek, J. Vavro, S. Husar, B. Varkolyova. The stress analysis and the service life prediction on the low-power motorcycle. In: 4th Mini Conf. on Vehicle System Dynamics, Identification and Anomalies, Budapest, Hungary, pp.171-177, (1994). [3] I. Letko, O. Bokuvka, G. Nicoletto, M. Janousek, P. Palcek. Fatigue resistance of two tool steels. In: 11th Danubia-Adria Symposium on Experimental Methods in Solid Mechanics, Baden, Austria, pp.139-140, (1994). [4] J. Vavro. Optimisation of the Design of Cross-Sectional Quantities in Transport Machines and Equipments. In: Studia i materialy, Technika, Zelena Gora, Poland, pp.187-194, (1998). [5] W. Weibull. A statistical distribution function of wide applicability. In: Journal of Appl. Mechanics, No.3, (1951).

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

433

Analysis of Crack Initiation and Propagation in Polycrystalline Meso and Microstructures of Metal Materials Torsten Luther*, Carsten Könke† *

Institute of Structural Mechanics Marienstrasse 15, 99421 Weimar, Germany [email protected] † Institute of Structural Mechanics Marienstrasse 15, 99421 Weimar, Germany carsten.kö[email protected]

ABSTRACT Durability and life cycle analysis of engineering structures is often based on numerical simulations of macroscopic damage behaviour using phenomenological damage and fracture models. Therewith the true mechanisms of crack initiation and various crack propagation can not be covered. In order to integrate the physical material effects which are leading to crack initiation as well as crack propagation, simulations on the meso- or microstructure have to be performed. For metallic polycrystals we can assume that crack propagation on mesoscale (10-3m – 10-6m) occurs mainly along grain boundaries and depends strongly on atomic debonding on microscale (10-6m – 10-10m). The mutual dependence can be investigated by a multiscale analysis obtaining a reasonable damage model based on micro mechanical features. The current work is focused on the investigation of micro structural damage behavior on mesoscale using a two dimensional polycrystal model. The geometry of our mesoscale model describing the polycrystal material structure was first based on a Voronoi cell diagram, wherein each cell was assigned to a single crystal. A comparison of grain size distribution in generated Voronoi structures with grain size distribution measured in natural thin layers of metal materials has shown significant differences. Hence, a Voronoi geometry is not flexible enough to represent realistic grain size distributions. More suitable representations can be obtained by a Weibull or Lognormal distribution. In the current work an advanced algorithm is used to generate polycrystal material structures based on arbitrary distribution functions for a more realistic simulation on mesoscale. In order to take into account the dependency of material properties on crystal orientation we assign an orthotropic linear elastic material model to the single crystals. Extension to elastic plastic material model with realistic plasticity properties has shown no relevant improvements against the linear elastic model. In the analysis both, crystal orientation and material properties of each crystal are distributed in a statistical manner. To simulate crack propagation we apply a coupled cohesive zone model (CCZM) on the interface along grain boundaries. Therein, the peak strength of the CCZM depends directly on the missorientation between neighbouring single crystals. The polycrystal model is applied to analyse the crack initiation and propagation in static loaded representative volume elements (RVE) of metal materials on mesoscale without necessity of initial damage definition. The future research work is focused on the determination of constitutive relations for the CCZM from quasicontinuum (QC) simulations performed on a RVE on microscale and homogenized to the mesoscale. Therefor, we show first investigations of atomic debonding along grain boundaries based on the QC method.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

434

Analysis of displacement and stress distributions in riveted joints E. Szymczyk*, A.Derewońko*, J.Jachimowicz† *

Military University of Technology Kaliskiego Str, Warsaw, Poland [email protected] †

Institute of Aviation Al. Krakowska, Warsaw, Poland [email protected]

ABSTRACT This paper deals with the displacement and stress analyses in riveted joint. This is a stage of study of the local physical phenomena in riveted joint. The aim of the investigation is to improve the prediction method for the joint failure mode associated with a crack initiation and propagation at a rivet hole. Riveting is the most commonly used method of joining sheet metal components of the aircraft structures. The residual stress and strain state appears in the joint after the riveting process. Furthermore in service condition aircraft structures are subjected to variable fatigue loads. The riveted joints are critical areas of the aircraft structure due to severe stress concentrations, plastic strain and effects such as surface damage (fretting wear) and secondary bending. Therefore the fatigue crack initiation will start at the rivets holes. Fretting fatigue is recognised as a surface damage phenomenon and describes situation where microslip between contacting surfaces appears to give rise to reduction in fatigue life [2]. The object of the analysis is a tensile loaded lap joint. Fatigue tests are performed for the riveted joint specimens consisting of a steel rivet (shank diameter 5mm) and two D-16 aluminium alloy plates (thickness 2.8 mm). Steel and aluminium interaction due to their different properties causes fretting corrosion and decreasing in fatigue life. This feature is convenient to experimental tests. Numerical FE simulations are carried out with the MARC code for a single lap riveted joint specimen. Three-dimensional numerical model is used to analyse the resulting displacement fields at a hole and a rivet in the neighbourhood of a contact interface. Relative displacements between the rivet and the hole are investigated for various friction coefficients. The local numerical model describes a single rivet, a hole and its neighbourhood in two plates. This type of problem requires the use of contact between the elements assembled and non-linear geometric and elasto-plastic multilinear material models to simulate the behaviour of the rivet and plates [1]. The double sided contact for deformable bodies is defined between the two metal plates and between the rivet and the hole. Although the literature on the fatigue behaviour of riveted joints is quite abundant, many aspects are still not sufficiently understood and investigated and, therefore, they require a further study. The advantage of the numerical simulation is to limit development costs and to improve analysis by giving more complete information about contact stress compared to the pure experimental way.

References [1] B. Langrand at al., An alternative numerical approach for full scale characterisation for riveted joint design. Aerospace Science and Technology 6 (2002) 343–354 [2] D.Nowell, Recent developments in the understanding of fretting fatigue. EFM 73 (2006), 207222

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

435

The synergetic effects of hybrid crossover operators in structural optimisation Carlos Conceição António * *

Faculty of Engineering, University of Porto 4200-465 Porto, Portugal [email protected]

ABSTRACT The development of the crossover operator is based on three different mechanisms: mating selection mechanism, offspring generation mechanism and offspring selection mechanism. Most of the crossover operators are able to get exploration or exploitation of the domain depending on the way in which they handle the current diversity of the population. Each crossover operator directs the search towards a different zone in the neighbourhood of the parents. The quality of the elements that belong to the visited region depends on the particular problem to be solved. This is confirmed by the well known no free lunch theorems. The simultaneous use of diverse crossover operators on the population will induce more efficient algorithms. The aiming of this paper is to analyse and to study the complementary properties resulting from the synergetic effects using several crossover operators in genetic algorithms. The improvements reached in structural optimisation using hybrid crossover operators will be analysed through some standard examples.

References [1] C.A.C. António and I.A. Lhate, A hybrid crossover operator for structural optimization based on commonality in genetic search. Engineering Computations, International Journal for ComputerAided Engineering and Software, 20(4), 390–408, 2003. [2] C.A. Conceição António, A hierarchical genetic algorithm with age structure for multimodal optimal design of hybrid composites. Structural and Multidisciplinary Optimization, SpringerVerlag, Editors: G. Rozvany (Germany) and J. Sobieski (USA), in print, online January 2006.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

436

An adaptive mesh generation strategy for the solution of structural shape optimization problems using evolutionary methods Gabriel Bugeda‡, Juan José Ródenas †, Elke Pahl* , Eugenio Oñate * ‡

Escola Universitària d’Enginyeria Tècnica Industrial de Barcelona (EUETIB-UPC) C/ Comte d'Urgell, 187; 08036 Barcelona (Spain) [email protected] * International Center for Numerical Methods in Engineering (CIMNE-UPC) C/ Gran Capitán s/n; Campus Nord UPC; Módulo C1; 08034 Barcelona (Spain) {elkepahl,onate}@cimne.upc.edu

†

Departamento de Ingeniería Mecánica y de Materiales; Universidad Politécnica de Valencia (UPV) Camino de Vera s/n; 46022 Valencia (Spain) [email protected]

ABSTRACT Evolutive methods are a powerful and robust tool for the resolution of structural shape optimization problems. Nevertheless, the use of these methods requires the analysis of an important number of different designs. The computational cost and the quality of the solutions are very much dependent on the quality of the finite element meshes used for the analysis. One important ingredient of the numerical analysis is the strategy for the generation of a proper mesh for each design. Here we can see two types of strategies: 1. To adapt a single existing mesh to the geometries of all different designs. Some existing strategies allow adapting an existing mesh for very big modifications of the boundary shape preventing the elements from being too much distorted. Nevertheless, despite the fact that this type of strategies provides a valid mesh for each design, there is no control of the discretization error contained in the results of each analysis. 2. To perform a classical adaptive remeshing procedure for the analysis of each different design. Of course, this procedure ensures good quality results in the numerical analysis of each design, but the total computational cost grows significantly because each design is computed more than once. This work presents a new strategy that allows generating an adapted mesh for each design without the necessity of performing a full adaptive remeshing procedure. It is based on the use of sensitivity analysis of all magnitudes related with adaptive remeshing (location of nodes, error estimation,…) with respect to the design variables. This sensitivity analysis is performed only once using a geometry of reference and it is used to project the results of the corresponding analysis to all other designs to be analyzed. The projected information allows generating an appropriate adapted mesh for each new design usually in one shot, greatly reducing the computational cost compared with the described strategy 2. This method was developed and used in the context of the solution of shape optimization problems using deterministic methods.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

437

Multiobjective Optimization of Multibody Systems with Genetic Algorithms João P. Dias*, Ricardo M. Corrêa* *IDMEC - Instituto de Mecânica - Pólo IST Instituto Superior Técnico, Technical University of Lisbon, 1049-001 Lisbon, Portugal [email protected]; [email protected]

ABSTRACT The technological breakthroughs in the transportation industry have made it necessary to simultaneously develop vehicle’s structures, in order to achieve better results in impact situations. In the last few years, the passive safety of vehicles is becoming more important in society, and, in the specific case of automobiles, there are entities responsible for testing, analyzing and reporting impact studies conducted over these vehicles. In this context, engineers are required to achieve better results, in some cases with restricted design schedules. The problem of designing a vehicle’s structure is, in fact, a multiobjective optimization problem in which factors like deformations, accelerations, costs and others, conflict amongst themselves. The most rigorous way to test and study impact scenarios – excluding experimental tests because of the high costs they involve – is to use finite elements methods. However, finite element methods require not only big modeling periods, but also high computation times. This way, using finite elements to study multiobjective optimization problems is not the better approach, mainly in the first stages of the development of a vehicle’s structure, when only its major properties are to be determined. One of the alternatives to finite element methods is to use simplified models, based on multibody dynamics. These models have been capable in the past to produce good results in optimization problems, requiring at the same time small computation times [1]. Mainly in the 90s, genetic algorithms have been used in several engineering problems, and have shown to have numeral advantages over classical methods of optimization, specifically in cases of complex mathematical problems. In this work a methodology for the multiobjective optimization of general structures, with application to railroad vehicles, is presented. This methodology uses simplified 1D and 2D models of the structures, in association with several genetic and classical optimization algorithms [2-4], and a simple graphical interface to help defining the optimization problems. One of the implemented genetic algorithms has been developed specifically to deal in a more efficient way with impact problems. Several 1D and 2D problems are approached with the proposed methodology and presented in this work, showing satisfactory results when compared with past studies.

References [1] M. S. Pereira, J. P. Dias, Analysis and Design for Train Crashworthiness Using Multibody Models, Vehicle System Dynamics Supplement, 40, 107-120, 2004. [2] G. Vanderplaats, DOT – Design Optimization Tools, Version 3.0, VMA Engineering, Colorado Springs, 1992. [3] K. Deb, A. Pratap, S. Agarwal e T. Meyarivan, A Fast and Elitist Multi-Objective Genetic Algorithm: NSGA-II, Kanpur Gen. Algorithms Laboratory Report 200001, India, 2001. [4] Dias, J. P., Corrêa, R. e Antunes, F., “Crashworthiness Optimization of Train Structures with Evolutionary Algorithms”, EUROGEN 2003, 83, Barcelona, Spain, 15-17 September, 2003. .

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

438

Buckling Optimization of Grid Structures via Genetic Algorithms L. Iuspa*, V. Minutolo†, E. Ruocco† *

Dept. Aerospace and Mechanical Engineering Second University of Naples [email protected] †

Dept. Civil Engineering Second University of Naples [email protected], [email protected]

ABSTRACT Optimization methods are very useful tools in the design process of complex structural systems. Use of fiber-reinforced composites in mechanical, aerospace, advanced civil structures and other branches of engineering, all require some numerical technique to find the best configuration that satisfies the assumed goal and meets assigned constraints. In the present work a topological optimization of a generalised 2D grid structure is shown. Specifically, the minimum weight design of homogeneous/composite plates and stiffened panels subjected to buckling load is herein addressed. A closed-form solution was implemented in a numerical procedure and then used to analyze arbitrary geometries. That procedure has proved to solve efficiently the governing equations using coarse meshes with a high-speed convergence [1]. Starting from a bounded, variously arranged orthogonal grid, a prismatic shape is obtained by extrusion and then solved to give the main critical load under assigned boundary conditions. Grid assembly is based on a hybrid (continuous-discrete) parametric description. Discrete parameters are used to control the number of cells in both x and y planar directions. Continuous parameters are then used in NURBS based spatial distributions to alter locally grid points, leaving untouched the topology. Additional parametric distributions are finally added to define local thickness. To perform the optimization task, a bit-masking oriented genetic algorithm has been used. The evolutionary engine of this implementation improves efficiency and flexibility of the optimisation process, and provides also specific capability for handling properly the discretecontinuous domain[2]

References [1] V. Minutolo, E. Ruocco, On Initial Postbuckling of Composite plate Assemblies by semianalytical Procedure, Mechanics of Composite Materials and Structures, 8, 1-14, 2001. [2] L. Iuspa, F. Scaramuzzino, P. Petrenga, Optimal Design of an Aircraft Engine mount via bitmasking oriented Genetic Algorithms. Advances in Engineering Software, 34, 707-720, 2003.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

439

Multiscale Multiresolution Genetic Algorithm Using Diverse Population Groups Dae Seung Kim*, Yoon Young Kim† *

School of Mechanical and Aerospace Engineering, Seoul National University, Seoul, Korea Shillim-Dong San 56-1, Kwanak-Gu, Seoul, 151-744, Korea

[email protected] †

National Creative Research Initiatives Center for Multiscale Design, School of Mechanical and Aerospace Engineering, Seoul National University, Seoul, Korea Shillim-Dong San 56-1, Kwanak-Gu, Seoul, 151-744, Korea

[email protected]

ABSTRACT When globally optimal designs are important and required sensitivity analysis is complicated, genetic algorithms can be effective alternatives to gradient-based optimization methods. Genetic algorithms are being used more widely in various engineering fields. However, when the design variables become large, standard genetic algorithms become difficult to use because of excessive computation time needed to obtain converged solutions. In this investigation, we consider two-dimensional structural topology optimizations based on finite element models as a large-size optimization problem. To expedite the solution convergence by orders of magnitude, the genetic algorithm in the multiscale multiresolution setting is proposed. By the multiscale setting, standard single-scale binary design variables are represented in multiple scales by the so-called Binary Wavelet Transform. Note that all genetic operators such as crossover and mutation must be redefined or modified for multiscaled design variables. A key advantage of using multiscale design variables is that the exploration of solution becomes more effective with them. If the multiresolution setting based on the multiscale variables is used, the design optimization can proceed from low to high resolution over several resolution stages where the information passing over resolutions is very effective. In this multiresolution approach, a converged population at a certain resolution level is reused as a part of the initial population at the next higher resolution level. In fact, we form the initial population in three groups: the converged population in the previous resolution, a randomly generated population and the mixture of the converged population and the random population. The detailed technique to generate the three groups is presented, which is an important part of the developed multiscale multiresolution genetic algorithm. The convergence improvement by the multiscale multiresolution approach was verified numerically with several numerical case studies. G

References [1] D. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning. AddisonWesley, 1989 [2] Y.Y. Kim and G.H. Yoon. Multi-resolution multi-scale topology optimization - a new paradigm. International Journal of Solids and Structures, 37, 5529-5559, 2000.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

440

An Adaptive Correction Function for Structural Optimization with Genetic Algorithms Pascual Martí-Montrull*, Osvaldo M. Querin†, Concepción Díaz-Gómez* *

Department of Structures and Construction, Technical University of Cartagena Campus Muralla del Mar, 30202 Cartagena (Murcia), Spain [email protected] [email protected] †

School of Mechanical Engineering, The University of Leeds Leeds LS2 9JT, United Kingdom [email protected]

ABSTRACT This paper presents a new self adaptive correction function for the optimisation of size, geometry and topology of space truss structures using the Genetic Algorithm (GA) method, applied to both continuous and discrete design variables. This function guarantees the diversity of the population at the early stages of the optimisation process. In addition, this function moves the final solution to the feasible region. The adaptive correction function proposed is the product of two independent functions. The first is an individual correction function that corresponds to the increase of the objective function that will be necessary to move an unfeasible individual into the feasible region. The second is a penalty function that increases or decreases the imposed correction, achieving this based on the feasibility or infeasibility of the population members during recent generations. The application of this self adaptive correction function to structural optimization was made using a very simple GA algorithm [1] with binary codification, standard crossover, mutation and elitism. The adaptive correction function proposed was implemented in the optimal design system DISSENY [2]. Numerical experiments were carried out with different structural optimization problems (i.e. crosssectional size, topology and shape optimization of 3-D trusses), with continuous and/or discrete variables, stress, displacement, slenderness and buckling constraints. The obtained results demonstrate that this self adaptive correction function is effective and robust, relieving the user from the burden of having to determinate the penalty parameters for each new problem. The results produced using this self adaptive function are equal or better than those produced using penalty functions.

References [1] D.E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning. AddisonWesley, Reading, Massachusetts, 1989. [2] P. Martí and P. Company, An Integrated System for the Structures and Structural Elements Optimal Design, B.H.V. Topping (Ed.). Advances in Structural Engineering Optimization, CivilComp Press, 1996.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

441

Optimization of the Topology of Masonry Units from the Thermal Point of View using a Genetic Algorithm Luísa C. Sousa, Catarina C. Castro and Carlos C. António Faculty of Engineering, University of Porto Rua Dr. Roberto Frias s/n, 4200-465 PORTO {lcsousa, ccastro, cantonio}@fe.up.pt

ABSTRACT Masonry enclosures subjected to environment conditions have a considerable economic weight in building construction cost, and are also important when considering thermal and structural behaviour. Also, more accurate, methods of analysis of wall systems create an opportunity to design buildings where masonry walls can be more efficient. In Mediterranean countries like Portugal the use of thick single leaf envelope walls can be an interesting alternative to cavity walls because thermal insulation inserts are expensive components of masonry walls [1]. The use of single leaf external walls is acceptable, as long as the correct material and construction techniques are used and a satisfactory structural behaviour is attained [2]. The use of masonry blocks made with lightweight concrete with expanded clay aggregates is increasing in Portugal. Lightweight concrete expanded clay aggregates exhibit particular properties as good thermal and acoustic behaviour provided by the volume of voids. Unfortunately, compressive strengths of lightweight concretes are lower than normal density concretes and low compressive strength reduces the load that can be carried by walls made of lightweight concretes. Moreover the quantity of lightweight concrete must be limited so the production cost is proportional to its quantity and the cost of transport and laying increases with weight. In this paper we present a computational method to optimize the masonry unit topology according to thermal normative requests. Current techniques for the evaluation of the wall thermal performance are focused on the thermal resistance value of the clear wall area. Finite element twodimensional computer simulations are used to optimize the topology of vertically perforated lightweight concrete masonry units. A developed numerical evolutionary algorithm [3] iterates over the direct analysis performed by the commercial code ABAQUS. The optimal solution presented in this paper exhibits a thermal transmittance of the masonry unit U = 0.54 W/(m2 ºC) with an air percentage equal to 42.2% .

References [1] R.Veiga, F. Carvalho and H. Sousa, Experimental evaluation of watertightness of single leaf walls. 12th Int. Brick/Block Masonry Conf. Proc. Vol. IV, Madrid, 25-28 June, 2201-2214, 2000. [2] S.Alves e H.Sousa, Paredes exteriores de edifícios em pano simples, Lisboa - PortoCoimbra, Lidel – Edições Técnicas, Lda, 2003. [3] C.F. Castro, C.A.C. António e L.C. Sousa, Optimization of shape and process parameters in metal forging processes using genetic algorithms. Journal of Materials Processing Technology, 146, 356-364, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

442

Topology Optimization of bidimensional continuum structures by genetic algorithms and stress iso-lines Mariano Victoria, Pascual Martí Department of Structures and Construction, Technical University of Cartagena Campus Muralla del Mar, 30202 Cartagena (Murcia), Spain

[email protected] [email protected]

ABSTRACT In these last years, the algorithms based on the biological process of natural evolution have been confirmed as a potent and robust search procedure. Presently work is introduced a new algorithm for the topology optimization of bidimensional continuum structures. The topology and the external shape of the design depend on a genetic algorithm, which, through the stress iso-lines of Von Mises defines the number, forms and distribution of the contours. The analysis of the structure is carried out by a fixed mesh of finite elements. The genetic algorithm (GA) uses the operators: selection (binary tournament), crossover (single point), and mutation (multibit). The procedure has been implemented in the programming language FORTRAN 95, the versatility and flexibility of the algorithm has been proven through several examples, using for it different fitness functions (Fully Stressed Design, compliance, weight, strain energy, etc). The results have been contrasted with the obtained of the most recent bibliography. Due to the scheme of the procedure, the number of evaluations of the fitness function is inferior to the needful for other procedures of similar characteristics, as: Multi-GA, VCL-GA. The produced results confirm the robustness and efficiency of the procedure.

References [1] H. Kim, M.J. García, O.M. Querin, G.P. Steven and Y.M. Xie, Introduction of fixed grid in evolutionary structural optimisation. Eng. Comp., 17(4), 427–439, 2000. [2] S.Y. Woon, L. Tong, O.M. Querin, G.P. Steven, Optimising Topologies through a Multi-GA System. WCSMO 5, Venecia, 2003. [3] L.Y. Kim, O.L. Weck, Variable chromosome length genetic algorithm for progressive refinement in topology optimization. Struc. Multidisc. Optim., DOI 10.1007/s00158-004-0498-5, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

443

Dynamic Analysis of Folding Patterns for Multi-Folding Structures Ichiro Ario∗ , Piotr Pawlowski† and Jan Holnicki-Szulc† ∗ Dept.

† Institute

of Civil & Environmental Engineering, Hiroshima University Higashi-Hiroshima, 739-8527, Japan [email protected]

of Fundametal Technological Research, Polish Academy of Sciences Swietokrzyska 21, 00-049 Warsaw, Poland {ppawl,holnicki}@ippt.gov.pl

ABSTRACT We present the basic mechanisms for the folding of a multi-layered truss, such as a combination of pantographs under dynamic loading, from a post-buckling perspective. This problem is considered in terms of the large-deflection range that the truss is allowed to fold. It develops several element forms to be stronger for dynamic control in a mechanism reminiscent of a deployable structure. Although there are several kinds of folds at the critical points, we need to develop a new concept for a multifolding mechanism (MFM) that allows various different folds even through a simple structure. We explore dynamic critical and post-buckling effects through the concept of energy minimization and hidden symmetries. For comparisons with the final large-deflection folded patterns, we use the original dynamic program for truss analysis. We demonstrate that, as final buckling develops, the mode patterns must change depending on both the velocity of the dynamic loading and some of the imperfections of the geometry of structure through the behavior of the post-buckling aspects of the fold pattern.

References [1] J. Holnicki-Szulc, P. Pawlowski and M. Wiklo, High-performance impact absorbing materials the concept, design tools and applications, Smart Materials and Structures, 12 (2003), 461-467. [2] J.M.T. Thompson and H.B. Stewart, Nonlinear dynamics and chaos geometical methods for engineers and scientists, John Wiley & Sons Ltd., 1986. [3] J.M.T. Thompson & G.W. Hunt, A general theory of elastic stability, London: Wiley, 1973.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Mechanical Systems Design and Control Optimization with Varying Time Domain J. E. Barradas Cardoso *, P. Pires Moita †, Aníbal J. J. Valido † *

Instituto Superior Técnico, Departamento de Engenharia Mecânica Av. Rovisco Pais, 1049-001 Lisboa , Portugal [email protected] †

Escola Superior de Tecnologia, Instituto Politécnico de Setúbal Campus do IPS, Estefanilha, 2914-508, Setúbal, Portugal [email protected] , [email protected]

ABSTRACT This paper presents an integrated methodology for optimal design and control of nonlinear flexible mechanical systems. A design and control sensitivity analysis and multicriteria optimization formulation is derived. This formulation is implemented in an optimum design code and it is applied to the nonlinear behavior response. Damping and stiffness characteristics plus control driven forces are considered as decision variables. A conceptual separation between time variant and time invariant design parameters is presented, this way including the design space into the control space and considering the design variables as control variables not depending on time. By using time integrals through all the derivations, the design and control problems are unified. In the optimization process we can use both types of variables simultaneously or by interdependent levels. Total time is also considered as varying. For treating time domain variation, a unit time interval is mapped onto the original time interval, then treating equally time variant and time invariant problems. The dynamic response and its sensitivity are discretized via space and time finite elements, and are integrated either by at-once integration or step-by-step. Adjoint system approach is used to calculate the sensitivities. The response analysis and corresponding DSA are implemented into an optimal design code.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Modelling for the determination of the interaction force of impacted Structures. S. Pashah∗ , E. Jacquelin∗ , J.P. Lain´e† , M. Massenzio∗ ∗ LBMC

- Universit´e Claude Bernard Lyon I - IUT B - INRETS 17 rue de France - 69627 Villeurbanne - France sulaman.pashah,[email protected], [email protected] † Ecole

Centrale de Lyon - LTDS - 36 Av. Guy de Collongue - 69134 Ecully - France [email protected] ABSTRACT

The design of impacted structures requires the determination of the interaction force between the structures involved. This requires to take the local problem of contact into account and the global description of the structures as well: the cost of numerical calculations can be very high. Hence, a modelling with few degrees of freedom (dof) is required to minimize the durations of calculations. Usually, to determine the interaction force, structures are modelled either by a single degree-of-freedom system , or by a modal description [1], [2]. The modal description is not adapted for non-linear simulations and is slowly convergent: hence a lot of eigenmodes are required for a good accuracy. This interaction problem may also be studied by describing a structure with its “anti-oscillators” : they are an alternative of the traditional eigenmodes. In fact, the ideas which lead to anti-oscillators come from the component modal synthesis method, that is from Craigh and Bampton [3]. Indeed, they are based on: 1- the constraint modes of a structure; they are the eigenmodes of the structure with an extra boundary condition: the displacement at the impact location vanishes; 2- the static mode: it is the shape caused by a static load applied at the impact point in the direction of the impact, such that the displacement at the impact location is equal to one. It is possible to show that these modes lead to a single dof system: the mass of this latter is connected to a set of single dof systems referred to as the “anti-oscillators” because their natural frequencies are some antiresonances of the structure. This modelling is based on a modal approach, but its philosophy is very different because it uses the antioscillators which compel the structure to be motionless: the anti-oscillators have a physical meaning. Then this model allows not only a simulation of the impact with few dof, but also a better understanding of the phenomena involved. An application of cylinder to cylinder impact is given.

References [1] S Abrate Impact on Composite Structures. Cambridge University Press, Cambridge, 1998. [2] W. J. Stronge, Impact mechanics. Cambridge University Press, Cambridge, 2000. [3] R. Roy and J. Craig and MCC Bampton, Coupling of substructure for dynamic analysis. AIAA Journal, 6(7), 1313–1319, july 1968.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Evolutionary Identification and Optimization of Composite Structures W. Beluch Department for Strength of Materials and Computational Mechanics, Silesian University of Technology, Konarskiego 18a, 44-100 Gliwice, Poland [email protected]

ABSTRACT Composite materials, especially composite laminates, play a significant role in the modern industry. Laminate is a material built by joining two materials and it usually consists of two phases: the matrix and the reinforcement. The laminate is typically build of many plies (laminas) having different ply angles. Laminates are popular due to two main reasons: i) the high weight-strength ratio (in comparison with the conventional materials); ii) the possibility to tailor the material properties to the designer requirements by manipulating several parameters like: components material, stacking sequence, fibres orientation or layer thickness [1]. If laminas are composed of the different materials the laminate is called a hybrid one. Two aims of the present paper are: i) to identify the material constants of the laminates; ii) to find the optimum stacking sequence of the laminates. The standard laminates as well as the hybrid ones are considered [2]. Different optimization criteria connected with the modal analysis and free vibrations are taken into account. To solve global optimization tasks and discrete optimization tasks as well as to avoid difficulties with the objective function gradient computation, the evolutionary algorithm (EA) is employed as the optimization procedure. To reduce the computation time, the distributed version of the evolutionary algorithm is used [3]. The finite element method (FEM) professional software package with the laminate modeller is used to solve a direct eigenfrequency problem for the laminate plates. The numerical examples presenting the efficiency of the proposed attitude are attached. As it can be seen from the numerical examples, the proposed identification and optimization method gives positive results. Consequently, this method can be applied to different laminated structures in order to identify the material constants or to find the desired laminate properties for a given criteria.

References [1] S. Venkataraman, R.T. Haftka, Optimization of Composite Panels - A Review. Proc. of the 14th Annual Technical Conf. of the American Society of Composites, Dayton, OH. Sep. 27-29, 1999. [2] L. Grosset, S. Venkataraman and R. Haftka, Genetic optimization of two-material composite laminates. Proc. of ASC conference, Blacksburg, 2001. [3] T. BurczyĔski, W. KuĞ, Distributed and parallel evolutionary algorithms in optimization of nonlinear structures. 15th International Conference on Computer Methods in Mechanics CMM-2003, Wisáa, 2003.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Parallel evolutionary optimization of heat radiators by using MSC MARC/MENTAT software Adam Dđugosz Department for Strength of Materials and Computational Mechanics, Silesian University of Technology, Konarskiego 18a, 44-100 Gliwice, Poland [email protected]

ABSTRACT The paper deals with the application of parallel evolutionary algorithms [1] (PEA) and the finite element method [3] (FEM) in shape optimization of heat radiators. The fitness function is computed by means of the thermoelsticity problem modeled by MSC MARC/MENTAT software. In order to create mesh, boundary conditions and material properties of the model a preprocessor MENTAT is used. Internal script language implemented in MENTAT allows avoiding external mesher procedure. Another benefit of this approach is that MENTAT takes into account shadowing effect in radiation [2]. Figure 1a shows the main steps of evaluation of the fitness function. The aim of the optimization is to find the optimal shape of the heat radiator shown in Figure 1b. This problem is solved by the minimization of the different types of functionals.

Figure 1. a) Evaluation of the fitness function b) The geometry of the heat radiator In order to reduce the number of design parameters in evolutionary algorithms the shape of the structure is modeled by Bezier curves. Numerical examples for some shape optimization are included.

References [1] Michalewicz Z. Genetic Algorithms+Data Structures = Evolutionary Programs. Springer Verlag, Berlin and New York, 1996. [2] Siegel H., Howell J.R., Thermal Radiation Heat Transfer, 3rd ed. Hemisphere, Washington, 1992. [3] Zienkiewicz O.C., Taylor R.L. The Finite Element Method, Vol. 1-3, Butterworth, Oxford 2000.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Evolutionary algorithm and boundary element method for solving inverse problems of piezoelectricity G. Dziatkiewicz, P. Fedelinski Department for Strength of Materials and Computational Mechanics Silesian University of Technology Konarskiego 18A, 44-100 Gliwice, Poland [email protected], [email protected]

ABSTRACT The piezoelectric phenomenon is widely uitilized in many devices, for example, sensors and actuators, micro-electro-mechanical systems (MEMS), transducers [5]. An analysis of piezoelectric devices requires a solution of coupled mechanical and electrical partial differential equations. In this paper the boundary element method (BEM) is implemented to solve the coupled field problem in piezoelectrics. The method allows the analysis by discretization of the boundary only [1]. The piezoelectric material is modelled as two-dimensional: homogenous, transversal isotropic, linear elastic and dielectric [3]. In most boundary value problems, the governing equations have to be solved with the appropriate boundary conditions [3]. These problems are called direct. However, when the boundary conditions are incomplete on a certain boundary part, the boundary value problems are generally ill-posed, then the existence, uniqueness and stability of the solution is not always guaranteed. These problems are inverse problems. In this paper the Cauchy problem is considered [3]. Another inverse problem is identification of the material constants. For considered two-dimensional case, the physical properties of the piezoelectric material depend on the nine material constants: four elastic constants, three piezoelectric constants and two dielectric constants. A relatively big number of the constants and difficulties in obtaining the gradient information, cause, that the identification problem of the piezoelectric material constants is quite complicated. To solve the inverse problems the distributed evolutionary algorithm is used [2], [4]. Numerical examples will be presented and they will show that the boundary element formulation with evolutionary algorithm gives an efficient computational intelligence tool for solving inverse problems of piezoelectricity.

References [1] C.A. Brebbia, J. Dominguez, Boundary elements. An introductory course. Computational Mechanics Publications, McGraw – Hill Book Company, Southampton – Boston, 1992. [2] T. Burczyński, W. Kuś, Optimization of structures using distributed and parallel evolutionary algorithm, Lecture Notes on Computational Science, 3019, 572-579, 2004. [3] G. Dziatkiewicz, P. Fedelinski, Boundary element method for solving direct and inverse problems of piezoelectricity. In 10th International Conference on “Numerical Methods in Continuum Mechanics NMCM – 2005, CD- ROM Proceedings, University of Zilina, Zilina 2005. [4] Z. Michalewicz, Genetic Algorithms + Data Structures = Evolutionary Programs, Springer Verlag, Berlin, 1992. [5] H.F. Tiersten, Linear piezoelectric plate vibrations: Elements of the linear theory of piezoelectricity and the vibrations of piezoelectric plates, Plenum Press, New York, 1969.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Evolutionary Optimization of Preform and Die Shape in Forging Using Computational Grid Wacáaw KuĞ * *

Silesian University of Technology, Department for Strength of Materials and Computational Mechanics, Konarskiego 18a, 44-100 Gliwice, Poland wacá[email protected]

ABSTRACT The paper is devoted to shape optimization of perform and die in forging process[1]. The idea is to use evolutionary optimization in computational grid environment[2]. The shape optimization of structures can be solved using methods based on sensitivity analysis information or non-gradient methods based on genetic algorithms. Applications of evolutionary algorithms in optimization need only information about values of an objective (fitness) function. The fitness function is calculated for each chromosome in each generation by solving the boundary - value problem by means of the Finite Element Method. This approach does not need information about the gradient of the fitness function and gives the great probability of finding the global optimum. The main drawback of this approach is the long time of calculations. The applications of the distributed evolutionary algorithms can shorten the time of calculations. The computational grids allows to use distributed computational resources. The use of grid techniques in optimizations can lead to improvements in hardware and software utilization. The other advantages of grids are simple and uniform end user communication portals and programs.

References [1] S. Badrinarayanan, Preform and die design roblems in metalforming, PhD. thesis,

Cornell University, 1997. [2] KuĞ W., BurczyĔski T., Grid-based evolutionary optimization of structures, Proc. PPAM 2005, PoznaĔ, 2005 (the full paper will apper in Springer LNCS series)

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Optimization of mechanical structures using serial and parallel artificial immune systems Wacáaw KuĞ *, Tadeusz BurczyĔski *† *

Silesian University of Technology, Department for Strength of Materials and Computational Mechanics, Konarskiego 18a, 44-100 Gliwice, Poland wacá[email protected] †

Cracow University of Technology, Institute of Computer Modeling, Artificial Intelligence Division, Warszawska 24, 31-155 Cracov, Poland [email protected]

ABSTRACT A shape optimization problem of structures can be solved using methods based on sensitivity analysis information or non gradient methods based on genetic algorithms or on artificial immune systems. This paper is devoted to the method based on the serial and parallel artificial immune system. Artificial immune systems are developed on the basis of mechanism discovered in biological immune systems [?9]. An immune system is a complicated, distributed group of specialized cells and organs. The main purpose of the immune system is to recognize and destroy pathogens – funguses, viruses, bacteria and improper functioning cells. The artificial immune systems (AIS) [1] take only few elements from the biological immune systems. The most frequently used are mutation of the B cells, proliferation, memory cells, and recognition using the B and T cells. The artificial immune systems are used to optimization, classification and also computer viruses recognition. A parallel artificial immune system (PAIS) was introduced in [2] for classification problems. The applications of an artificial immune system in optimization need only information about values of an objective function. The objective function is calculated for each B cell in each iteration by solving the boundary – value problem of elasticity by means of the finite element method (FEM). The main drawback of this approach is the long time of calculations. The applications of the parallel artificial immune system can shorten the time of calculations but additional requirements are needed: a multiprocessor computer or a cluster of computers are necessary. The message passing paradigm of parallel computations is used in presented approach. An artificial immune system is implemented as one master process, other processes – workers evaluate objective functions for B cells.

References [1] L. N. de Castro, F. J. Von Zuben, Learning and Optimization Using the Clonal Selection Principle, IEEE Transactions on Evolutionary Computation, Special Issue on Artificial Immune Systems, 6, n. 3, 239-251, 2002. [2] A. Watkins, X. Bi, A. Phadke, Parallelizing an Immune-Inspired Algorithm for Efficient Pattern Recognition. Intelligent Engineering Systems through Artificial Neural Networks: Smart Engineering System Design, 13, 225-230, 2003.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Experiments of Damage Detection in Strips Based on Soft Computing Methods and Wave Propagation ´ ∗ Piotr Nazarko∗ , Leonard Ziemianski ∗ Rzesz´ ow

University of Technology, Department of Structural Mechanics, W.Pola 2, 35-959 Rzesz´ow, Poland [email protected], [email protected]

ABSTRACT All industry branches like aerospace, mechanical and civil engineering are interested in less intrusion and more accuracy failure assessment techniques. They are mostly interested in damages like cracks, delaminations, disbanding, corrosion, etc. Damage detection and assessment technique was developed in this paper. It uses variations in structural wave propagation for undamaged and damaged structure. This Structural Health Monitoring (SHM) method is useful especially in large, complex and inaccessible structures [1, 2]. Based on earlier promising results with this approach [3, 4] a set of laboratory tests were carried out on simple elements like strips. Two kind of materials were used: steel and plexy. Several failure cases were introduced by cutting or drilling the samples. Piezoceramics (PZT) elements were served as transmitters and receivers of elastic waves trough the monitored specimens. During these experiment different groups of excitation signals (continuous sine wave, one, four and six sine wave impulses) and frequency (frequency range from 2 to 50 kHz) were applied to introduce wave to the structure. The numerical models were also created using Finite Element Method (FEM). Defects in the form of a notch were simulated by the removal of selected finite elements from the model. This simulation gave possibility to extend set of damages cases and improved nets generalization properties. In both laboratory and numerical experiments advanced signal processing techniques were adopted. The measured signals were preprocessed by wavelet transform in order to remove noise. Frequency analysis was carried out by Fast Fourier transform (FFT). Replication technique was adopted to experimental data. To realize dependences between input (harmonic frequencies) and output data (height, width and localization of damage) Artificial Neural Networks (ANNs) were used. Several input combinations and nets architectures were tested. Results presented in this paper proved reliability and usefulness of proposed approach.

References [1] Doebling S.W., Farrar C.R., Prime M.B., A summary review of vibration-based damage identification methods, The Shock and Vibration Digest, 30(2), 91-105, 1998 [2] W. Ostachowicz, M. Krawczuk, M. Cartmell, M. Gilchrist, Wave propagation in delaminated beam, Computers and Structures 82, 475483, 2004 [3] S. Bhalla, C.K. Soh and Z. Liu, Wave propagation approach for NDE using surface bonded piezoceramics, NDT& International, 38, 143-150, Elsevier, 2005. [4] P. Nazarko, L. Ziemia´nski, The Use of Neural Networks in planning of experimental research on identification in rods. Proceeding of AI-METH, Gliwice, 2002.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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The Optimization and Identification Problems of Structures with Fuzzy Parameters Piotr Orantek Silesian University of Technology, Department for Strength of Materials and Computational Mechanics Konarskiego 18a, 44-100 Gliwice, Poland [email protected]

ABSTRACT This paper is devoted to optimization and identification problems of structures with fuzzy parameters. The elasticity problem is considered in the paper. The fuzzy shape of the body, boundary conditions and material parameters are assumed. The optimization and identification problems concern on finding the above parameters. The objective functions of optimization and identification problems are proposed. The objective function includes: (i) mass, (ii) stresses and (iii) displacements of the structure in the case of the optimization problems. The objective function includes the measured and computed displacements in selected sensor points placed on the boundary of the structure in the case of identification problem. The fuzzy numbers are expressed as the set of interval values. To solve the boundary value problem the interval finite element method (IFEM) is used. The optimization and identification problems are carried out using the Two-Stages Fuzzy Strategy (TSFS). The Fuzzy Evolutionary Algorithm is used in the first stage of TSFS. The fuzzy gradient method with neuro-computing of the sensitivity is applied in the second stage. Several tests are performed, and will be presented in the full paper.

References [1] H.D.Bui, Inverse Problems in the Mechanics of Materials: An Instroduction. CRC Pres, Bocca Raton 1994. [2] T. Burczynski, W. Beluch, A.Długosz, P. Orantek, M. Nowakowski. Evolutionary methods in inverse problems of engineering mechanics. In: Inverse Problems in Engineering Mechanics II (eds. M. Tanaka and G.S. Dulikravich), Elsevier, 2000, pp. 553-562. [3] P. Orantek, Fuzzy evolutionary algorithms and neural networks in uncertain optimization problems. International Symposium on Neural Networks and Soft Computing NNSC 2005, Cracow 2005. [4] W. Pedrycz, Fuzzy evolutionary computing. Soft Computing 2 (1998), Springer-Verlag 1998. [5] R. Schaefer. The basis of the genetic global optimization. WUJ, Krakow, 2002. (in Polish)

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Topology optimization of the 3-D structures for various criteria using evolutionary algorithm Arkadiusz Poteralski Silesian University of Technology, Department for Strength of Materials and Computational Mechanics, Konarskiego 18a, 44-100 Gliwice, Poland [email protected]

ABSTRACT Evolutionary methods have been applied in mechanics, especially in structural optimization. The main aim of these methods is the simulation of biological processes based on heredity (genetics) and on the natural selection (the theory of evolution) to create the optimal individuals (solutions) presented by single chromosomes. Evolutionary models of computation can be performed by using genetic algorithms. Evolutionary computations are performed on a population of individuals. The main advantage of the evolutionary algorithm is the fact that this approach does not need any information about the gradient of the fitness function and gives a strong probability of finding the global optimum. The main drawback of this approach is the long time of the calculations. In order to eliminate this disadvantage the distributed evolutionary algorithm can be used to speedup the computations. The shape, topology and material of the structure are generated for various optimization criteria like: the minimization of mass with constraints imposed on equivalent stresses and resultant displacements of the structure, the maximization of the first eigenfrequency, the maximization of difference between first, second and third eigenfrequency and the maximization of the difference between first, second, third eigenfrequencies and forced vibration frequency with constraint imposed on mass of the structure. The interpolation based on the neighborhood of elements which aim at the appropriate selection of mass densities is used. For this interpolation several kinds of number and distribution of control points is considered. Dependence between Young’s modulus and mass density is used in this paper [1]. The optimization process is controlled by evolutionary changing of the mass density. After optimization the procedure, which smoothes an external and internal boundary of threedimensional structure, is used. As a tool for solving the direct problems concerning displacement and stress analysis problems of 3-D elastic structures the Finite Element Method is chosen.

References [1] M. P. Bendsøe: Optimal shape design as a material distribution problem, Struct. Optim. Vol. 1, s. 193-202, 1989

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Computational intelligence system in non-destructive identification of internal defects Antoni Skrobol*, Tadeusz Burczyński*† *

†

Silesian University of Technology, Department for Strength of Materials and Computational Mechanics, Konarskiego 18a, 44-100 Gliwice, Poland [email protected]

Cracow University of Technology, Institute of Computer Modeling, Artificial Intelligence Division, Warszawska 24, 31-155 Cracov, Poland [email protected]

ABSTRACT There are many ways to identify the internal defects in the body, however the most interesting are non-destructive methods of identification. Using these methods, the internal defect can be found only on the basis of the knowledge that can be acquire without destroying of the considered structure. The application of the computational intelligence system (CIS) in non-destructive identification of the internal defects on the basis of the knowledge about displacements in several sensor points on the boundary of the structure is presented. The CIS is composed of an evolutionary algorithm (EA) coupled with the adaptive fuzzy inference systems with pseudo-gaussian membership functions (PGFISs) [2]. The aim of the EA is to minimize the fitness function that is formulated as a weighted sum of difference between the measured boundary displacements of the considered structure and displacements computed for the numerical model of the body with assumed number and kind of defects. The displacements are approximated by the PG-FISs. The values of the position and size of defects are put on the inputs of the systems, the approximated values of the boundary displacements are obtained on the outputs of the PG-FISs. During the tests the CIS had to identify the number, position and size of defects in a two-dimensional elastic body under dynamic load. It had also to select the actual kind of defect (circular void, crack or circular inclusion). The PG-FISs were trained using the steepest descent method and conjugate gradient method. The training and testing data were obtained by the boundary element method (BEM). The CIS is able to identify the kind, position, size and the number of the internal defects in the 2D body on the basis of the knowledge about boundary displacements in elastodynamics [1][2]. The time of identification is very short and depends very slightly on the geometry of the body [1].

References [1] T. Burczyński , P. Orantek , A. Skrobol, Fuzzy-neural and evolutionary computation in identification of defect, Journal of Theoretical and Applied Mechanics, Vol. 42, No. 3, 445-460, 2004. [2] T. Burczyński, A.Skrobol, Internal defect identification by the computational intelligence system. Recent Developements in Artificial Intelligence Methods, Gliwice, 4 pages, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Optimization of topology and stiffeners locations in 2-D structures using evolutionary methods Mirosław Szczepanik Silesian University of Technology, Department for Strength of Materials and Computational Mechanics, Konarskiego 18a, 44-100 Gliwice, Poland [email protected]

ABSTRACT The paper deals with an application of the distributed evolutionary algorithm and the finite element method to the optimization problems of 2-D structures in respect of topology and stiffeners arrangement. Recently, evolutionary methods [2] have been wide applied in mechanics, especially in structural optimization [1]. The main feature of those methods is to simulate biological processes based on heredity principles (genetics) and the natural selection (the theory of evolution) to create optimal individuals (solutions) presented by single chromosomes. The main advantage of the evolutionary algorithm is the fact that this approach does not need any information about the gradient of the fitness function and gives a strong probability of finding the global optimum. The main drawback of this approach is the long time of calculations. The applications of distributed evolutionary algorithms can shorten the time of calculations. The fitness function is calculated for each chromosome in each generation by solving the boundary-value problem by means of the finite element method. The main feature of the first proposed optimization method is an evolutionary distribution of the material in the construction by the change of its material properties (Youngs moduli, densities) or by the change of thickness. This process leads to the elimination of the part of material from the construction and in effect the new shape and topology of the construction emerges. Using interpolation surfaces reduces the number of design variables and shortens the time of the computation. The main feature of the second proposed optimization method is an evolutionary change of the stiffeners arrangement and their shape. The application of the professional program of the finite element method MSC NASTRAN in both methods enables optimization of the complex mechanical systems. The numerical examples confirm the efficiency of the proposed optimization method and demonstrate that the method based on evolutionary computation is an effective technique for solving computer aided optimal design problems.

References [1] Burczyński T., Osyczka A. (eds): Evolutionary Methods in Mechanics. Proc. IUTAM Symposium, Kluwer, Dordrecht 2003. [2] Michalewicz Z. : Genetic Algorithms + Data Structures = Evolutionary Programs. Springer Verlag, Berlin 1992.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A Structural Optimization Problem Formulation for Design of Compliant Gripper Using a Genetic Algorithm Nianfeng Wang , Kang Tai School of Mechanical and Aerospace Engineering Nanyang Technological University 50 Nanyang Ave Singapore 639798 [email protected] , [email protected]

ABSTRACT This paper demonstrates the automatic design of a compliant gripper by structural optimization using genetic algorithms. Compliant mechanisms are single-piece jointless structures that use compliance (elastic deformation) as a means to achieve motion. As such, they have many advantages compared to conventional rigid-link mechanisms and so can be created as a replacement for their rigid-link counterparts, especially when the applications are in the micro-dimensional scale. Recently, relatively simple compliant mechanisms have been successfully synthesized by applying structural optimization methods because these methods automatically determine the topology and shape of structures based on any given desired structural criteria. The mechanism designed in this paper is meant to be able to grip an object and convey it from one point to another. Such a mechanism has useful applications in MEMS and various automation devices, but it is relatively complex. They are difficult to design mainly because their motion has to be analyzed by finite element methods and the relationship between their geometry and their elastic behavior is highly complex and non-linear. The synthesis of such a mechanism is to be achieved by a structural optimization approach using a genetic algorithm as the optimizer and a special morphological representation for defining the design geometry. The problem is formulated as a discrete multiobjective constrained optimization problem. The genetic algorithm is a multiobjective algorithm with constraint handling, based on maintaining separate non-domination (Pareto) rankings for objectives and constraints satisfaction, thus enabling an intelligent selection of solutions for cooperative mating which eliminates the need to prescribe penalty function parameters commonly used for constraint handling. In addition, a recently developed morphological geometric representation scheme is used to define the topology and shape of the structure via an arrangement of skeleton and surrounding material. This technique facilitates the transmission of topological/shape characteristics across generations in the evolutionary process, and will not render any undesirable geometric features such as disconnected segments, checkerboard patterns or single-node hinge connections. A non-linear finite element program has also been used for the large-displacement analysis of the structure, and the program is integrated with the genetic algorithm to form an overall working framework for structural optimization. Some tentative formulations of the optimization problem needed to achieve the required elastic/structural behavior of the design have been developed. The resulting geometries obtained here are clearly defined due to the discrete nature of the geometry representation, unlike in the homogenization or material density methods of topology optimization which require the prescription of some threshold point to interpret whether the resulting material density values in the elements indicate solid material or void. Optimal resulting designs have been obtained that are valid and practical for realization/fabrication.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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The prediction of bankruptcy using weighted fuzzy classifiers R. J. Almeida∗ , S. M. Vieira∗ , J. M. C. Sousa∗ and U. Kaymak† ∗ Technical

University of Lisbon, Instituto Superior T´ecnico Dept. of Mechanical Engineering GCAR - IDMEC Av. Rovisco Pais, 1049-001 Lisbon, Portugal [email protected], [email protected]

† Erasmus

University Rotterdam, Faculty of Economics Department of Computer Science P.O. Box 1738, 3000 DR Rotterdam, The Netherlands, Portugal [email protected] ABSTRACT In real-world databases sometimes one of the classes is more difficult to classify then the others. This can happen, for instance, when one of the class is much bigger than the other. To cope with this problem, this paper proposes to assign specific weights to each class in the model evaluation criterion of the feature selection algorithm. The proposed technique is applied to a real world classification problem: the prediction of bankruptcy. The data set used in this study has missing values and extreme values. The data set also presents a much smaller bankruptcy class than the not bankruptcy class. Feature selection is used to choose the number of relevant features, using the so called correctly classified criterion. In the case study, less features are selected using this criterion, consequently less computational time is taken. Moreover, this paper compares five different fuzzy clustering algorithms in terms of model accuracy and computational burden. These clustering algorithms are also used and compared during the feature selection procedure. For this bankruptcy data set, the classification rate with only 16 features is 80% for the companies that bankrupt, whereas the percentage of companies that do not bankrupt is 95.4%. If we take in consideration that the number of obtained rules is 6, only 16 features are used, as opposed to 70 rules and 35 features as in [?], where the accuracy is 81%, with all of the interpretability issues that this carries, then it can be considered that these are promising results.

References [1] Jerzy Stefanowski and Szymon Wilk, Evaluating business credit risk by means of approachintegrating decision rules and case-based learning. International Journal of Intelligent Systems in Accounting, Finance and Management, 10(2), 97114, June 2001.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Optimization of a logistic process by ant colonies, wasp swarms and genetic post-optimization Pedro C. Pinto∗,† , Thomas A. Runkler∗ , Jo˜ao M. C. Sousa† ∗ Siemens

AG, Corporate Technology, Information and Communications, CT IC 4 81730 Munich - Germany {pinto.pedro.ext,Thomas.Runkler}@siemens.com † Technical

University of Lisbon, Instituto Superior T´ecnico Av. Rovisco Pais, 1049-001 Lisbon - Portugal [email protected]

ABSTRACT This paper addresses the problem of system optimization using meta-heuristical algorithms inspired by biological processes. In particular, we focus on Ant Colony Optimization (ACO), Wasp Swarm Optimization (WSO) and Genetic Algorithms (GA), where GA is used alone and for post-optimization. The meta-heuristics inspired by social insects share a common background, and it is interesting to see how they compare, how they can be applied to a problem, and to which problem each algorithm is more adequate. The table shows the main structure of both ACO and WSO algorithms. As it can be seen, the processes behind them have many similarities, with the pheromones of the ants being the equivalent to the force based hierarchy of the wasps. x is the system variable, computed in both algorithms from a probability matrix. ACO and WSO basic algorithm Initialization of the pheromone matrix τ Computation of the forces f (x) Computation of the probability matrix p(τ ) Computation of the probability matrix p(f ) Computation of the solution x(p) Computation of the solution x(p) Update of the pheromone matrix τ Update of the forces f (x) ACO is based on how unsupervised colony agents cooperate to achieve a common goal, and as such it is a natural way of optimizing systems where cooperation is advantageous. With some bigger or smaller modifications it can be applied to several different problems. WSO is based on how wasps compete between themselves, and as the optimization of logistic systems or network routing usually imply some kind of competition, for resources, power, etc, WSO can be applied successfully in many cases. The paper has four main sections. First, we do a brief introduction. In the second part we present a point by point comparison between ACO, WSO and GA, analyze their potential in optimizing different kinds of example systems and how to determine which one is the best option for a given situation. We follow this study with a comparison of the five variants ACO, WSO, GA, ACO-GA and WSOGA when applied to two systems, a theoretical benchmark problem and a real world logistic system at Fujitsu-Siemens Computers. We end the paper with a global conclusion of the matters discussed, and a summary of the future work. We conclude that WSO produces better results than ACO and GA for the considered logistic problem. We also conclude that the hybrid WSO/ACO-GA performs better than their stand-alone counterparts, and that WSO/GA performs better than ACO/GA due to the GA’s part of the algorithm having a more diversified population.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Kriging-based estimation with noisy data S. Sakata*, F. Ashida*, M. Zako† *

Department of Electronic Control Systems Engineering, Interdisciplinary Faculty of Science and Engineering, Shimane University, 1060 Nishikawatsu-cho, Matsue, Shimane, Japan

[email protected] [email protected] †

Department of Management of Industry and Technology, Graduate School of Engineering, Osaka University, 2-1, Yamada-Oka, Suita, Osaka, Japan

[email protected]

ABSTRACT Several approximate optimization methods will be effective for a recent engineering optimization, especially more flexible methods such as neural network, radial basis function or Kriging method are applicable to complex problem, for example, a non-convex and multi-peaked solution space. These methods will generate an estimated surface which is fit well to the sampling results It is sometimes difficult, however, that these methods are applied to noisy data because of its flexibility. Especially the ordinary Kriging will give an exact interpolation [1], therefore some improvement will be required to be used for approximate optimization with noisy data. In this study, the ordinary-type Kriging method will be originally improved to apply to both of precise and noisy data. A formulation of Kriging estimation and empirical semivariogram will be arranged from the viewpoint of dispersion of sampling results. We choose different types of semivariogram function for sampling data and estimated surface in the case of using the semivariogram model, and the Gaussian type semivariogram model is adopted in this study. The nugget effect is included in the semivariogram model to consider some noises. The nugget effect will cause discontinuity of estimated surface, but this approach enables to eliminate the discontinuity. To take a dispersion of sampling results into account, the empirical semivariogram and Cressie’s weighted least squares criterion is re-defined. The effect of changing empirical semivariogram on estimated results will be also investigated. As a test problem, a surface is estimated with using noisy data, which are generated by giving random noises to a known mathematical function. By comparing the results obtained by the proposed Kriging system with the exact ones or the results obtained by other approximation method, estimation quality of the proposed method is investigated. Influence of noises in sampling results on estimated results is investigated. The proposed method will give a better estimation, and numerical examples illustrate validity and effectiveness of the proposed approach.

Reference [1] Hans Wackernagel, Multivariate Geostatistics, Springer Verlag, 1995 (Japanese Translated edition).

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Distributed Optimization using ACO for Concrete Delivery C. A. Silva*, J. M. Faria*, D. Naso† *

Instituto Superior Técnico IDMEC / GCAR [email protected] †

Politecnico de Bari Dep. Electrotechnics and Electronics [email protected]

ABSTRACT The timely production and distribution of rapidly perishable goods such as concrete is a complex combinatorial optimization problem in the context of supply chain management [1]. The problem involves several tightly interrelated scheduling and routing problems that have to be solved considering a trade-off of production and delivery costs. Different approaches have been developed for this problem: a hybrid meta-heuristic method combining genetic algorithms with constructive heuristics [1]; a hybrid approach combining genetic algorithms and ant colony optimization [2]. However, all these approaches consider the optimization problems as separate problems. This paper introduces a novel approach, based on he distributed optimization paradigm proposed in [3], which obtained good results for other supply chains examples. In the distributed optimization framework, both problems are optimized in parallel and exchange information during the optimization process through the pheromone matrix. In this way, it is possible to bias the solution of one of the system in order to improve the performance of the other and thus achieve a better global solution for the supplychain. The simulation results show that this approach globally improves the supply chain results.

References [1] D.Naso, M. Surico, B. Turchiano, U. Kaymac. Justi-in-Time Production and Delivery in Supply Chains: a hybrid evolutionary approach. Proceedings ofIEEE SMC 2004. [2] C.A Silva, J.M.Faria, P. Abrantes, J.M. Sousa, M.Surico, D. Naso, Concrete Delivery using GA and ACO. Proceedings of the 44th IEEE Conference on Decision and Control and European Control Conference ECC 2005, Seville, Spain. [3] C.A Silva, J.M. Sousa, T. Runkler, J.M.G. Sá da Costa, Distributed Optimization of a Logistic System using Ant Colonies. Accepted in International Journal of Systems Science, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Tuning a Vibrating Blade Dynamic Vibration Absorber by using Ant Colony Optimization and Finite Element Modeling Felipe Antonio Chegury Viana *, Giovanni Iamin Kotinda*, Valder Steffen, Jr* *

Federal University of Uberlândia, School of Mechanical Engineering 2121 João Naves de Ávila Av., Campus Santa Mônica, CEP 38400-902, P.O. Box 593, Uberlândia, Brazil [email protected], [email protected], [email protected]

ABSTRACT The present contribution deals with the optimal tuning of a vibrating blade dynamic vibration absorber (VBDVA). To achieve this aim, the natural optimization technique named Ant Colony Optimization (ACO) is applied to the finite element model of the system. Dynamic vibration absorbers (DVAs) are systems constituted by mass, spring and damping elements (secondary structure), which are coupled to a mechanical system (primary structure). The main idea behind the DVAs is the generation of a force, which has the same intensity of the excitation force but in the opposite phase [1]. This phenomenon is known as anti-resonance. The tuning of the DVAs is the procedure that sets the anti-resonance frequency to a given value by changing the DVA parameters (mass, spring and damping values). VBDVA was studied in this work, which is composed by a blade that is subjected to an initial tension and fixed lumped mass. These three parameters (the mass value and its position and the initial tension) are responsible for the VBDVA tuning [2]. Supported by this theory, the optimization problem is described as the minimization of the objective function that relates the difference between the resonance frequencies of the primary system and the VBDVA. The optimum tuning defines the minimum difference respecting the design constraints. To solve the optimization problem it was used ACO [3]. This population-based technique is inspired in the behavior of real ants and their communication scheme by using pheromone trail. A moving ant lays some pheromone on the ground, thus marking the path. The collective behavior that emerges from the participating agents is a form of positive feedback where the more the ants follow a trail, the more attractive that trail becomes for being followed. In the early nineties, when the Ant Colony algorithm was first proposed, it was used as an approach for the solution of combinatorial optimization problems, such as the traveling salesman problem. However, the extension for continuous variables is recent and it is still under development. In this context, this paper presents an engineering application of a continuous domain problem of ACO. Numerical results are reported, illustrating the success of using the methodology presented, as applied to mechanical systems.

References [1] J. P. Den Hartog, Mechanical Vibrations, 4th edition. McGraw-Hill, New York, 1956. [2] G. I. Kotinda, Vibrating Blade Dynamic Vibration Absorber (in Portuguese), M. Sc. Dissertation, Federal University of Uberlândia, Brazil, 2005. [3] K. Socha, ACO for Continuous and Mixed-Variable Optimization, Proc of 4th International Workshop on Ant Colony Optimization and Swarm Intelligence – ANTS’2004, Brussels, Belgium, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Energy pumping of systems connected to a nonlinear energy sink device. C.H. Lamarque∗ , E. Gourdon∗ ∗ LGM / DGCG / ENTPE Rue Maurice Audin 69518 Vaulx en Velin Cedex FRANCE [email protected], [email protected]

ABSTRACT The present study deals with energy pumping phenomenon which consists in passive irreversible transfer of energy from a linear system to a strong nonlinear attachment [1, 2, 3]. The aim is to be able to design efficient nonlinear energy sink devices (for example with cubic nonlinearity [4]) in particular to attenuate modal responses for transient and steady vibrations. The main point is the strong nonlinear coupling. Contrary to the case of standard tuned mass dampers, energy transfer is irreversible because of modal localization which prevents the energy from being released back to the main structure. Various results (theoretical, numerical and experimental) about energy pumping based on recent works are given. Thus, the phenomenon is studied for different excitations: transient and periodical. For transient excitations, the case of seisms is also considered with an indicator of efficiency (i.e. the Arias Intensity). Moreover, advantages of such a system are carried out. The theoretical findings are tested and verified experimentally using appropriately designed reduced scale buildings with one or four floors. In particular, a comparison with an optimized classical linear tuned mass damper (like Frahm damper) can be done. With strong cubic coupling, the features (in particular the modes) of the system are not modified. Indeed, experimental verification has shown the efficiency of energy pumping compared to classical linear tuned mass damper. Not only is the phenomenon robust theoretically but it is possible to implement it practically with a small realistic building model.

References [1] O. Gendelman, Transition of Energy to a Nonlinear Localized Mode in a Highly Asymmetric System of Two Oscillators Nonlinear Dynamics, 25, 237-253, 2001. [2] A.F. Vakakis, Inducing passive nonlinear energy sinks in linear vibrating systems. ASME J. Vibr. Acoust., 123, 324-332, 2001. [3] A.F. Vakakis, O. Gendelman, Energy Pumping in Nonlinear Mechanical Oscillators II: Resonance Capture. ASME J. Appl. Mech., 68, 42-48, 2001. [4] E. Gourdon, C.H. Lamarque, Energy Pumping with various Nonlinear Structures : Numerical Evidences. Nonlinear Dynamics, 40, 281-307, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

463

Stress analysis of curved elastic bar and elastic wedge under bending load; infinite systems and asymptotic Vyacheslav V. Lyakh, Vyacheslav V. Meleshko Kyiv National Taras Shevchenko University 01033 Kyiv, Ukraine [email protected] ABSTRACT The paper addresses the classical problem of plane elasticity – bending of a curved elastic bar a≤r≤b, −α≤ϑ≤α (r, ϑ – polar coordinates, α – the opening angle), and an infinite elastic wedge a≤r@ 2XU REMHFWLYH KHUH LV WR VKRZ WKH XVHIXOQHVV RI FRPSXWDWLRQDO 7')' PHWKRGV FRXSOHG ZLWK ' UHDO FRUWLFDO RU WUDEHFXODU ERQH VWUXFWXUHV IURP FRPSXWHG WRPRJUDSK\  RU PLFURVWUXFWXUHV IURP 65—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

5HIHUHQFHV

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III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

596

Microstructural model of the viscoelastic behaviour of biological tissues ∗ ¨ Fanny Moravec∗ , Milan Muller ∗ Department

of Mechanics, University of West Bohemia in Pilsen Univerzitni 22, 306 14 Pilsen, Czech Republic [email protected], [email protected] ABSTRACT

A model of viscoelastic material, whose elastic and dissipative potentials are build on a microscopic restructuring, is proposed. The continuum body is viewed as a collection of ‘elementary structures’ whose borders’ deformation is governed by the gradient of deformation F. Each elementary structure is constituted of a central cell floating in a viscous matrix to mimic, in a simple way, the micro-structure of biological tissues. The application of an external load leads to the restructuring of the inner geometry of the elementary structure by stretching or contracting the fibers present in cell and in matrix, and by running the flow of the viscous fluid around the central cell. Simulations are restricted to the two-dimensionnal unidirectional traction test. In this case, the inner configuration of the elementary structure is completely characterized by one size length of the cell, let say c1 . The law governing the time evolution of the internal variable c1 is determined solving an ordinary differential equation, resulting from the compensation between elastic and viscous forces within the elementary structure f (c1 ) + g(c1 , c˙1 ) = 0, where f derives from the elastic potential, and g derives from the dissipative potential. The time dependence is transmitted to the macroscopic behaviour of material. A simulation test applying an instantaneous strain is done and the relaxation properties of the material are discussed.

Figure 1: Traction test and material microstructure.

Figure 2: Response of the microstructure to an instantaneous strain loading

Figure 3: Response of the macro-stress to an instantaneous strain loading (relaxation phenomenon).

References [Holeˇcek 2005] M. Holeˇcek, F. Moravec, Hyperelastic model of a material whose microstructure is formed by “balls and springs”, in submission to the International Journal of Solids and Structures. [Pioletti 2000] D.P. Pioletti, L.R. Rakotomanna (2000), Non-Linear viscoelastic laws for soft biological tissues, Eur. J. Mech. A/Solids 19, 749-759

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

597

Scaling views on strength of soft/hard composites Ko Okumura Department of Physics, Graduate School of Humanities and Sciences Ochanomizu University, 2-1-1, Otsuka, Bunkyo-ku, Tokyo 112-8610, Japan [email protected] ABSTRACT We frequently find composite structures, especially comprised of soft and hard elements, in many strong materials in nature: timber, teeth etc. For example, in nacre, found inside of certain seashells, hard and brittle aragonaite plates are glued together by soft and thin protein layers. Toughness of nacre is about 3000 times as high as that of pure aragonite, although the volume fraction of the soft protein is only 1/100! This lecture concerns, on a macroscopic level of continuum models, fracture properties of nacre and similar layered structures. We show that one of the important mechanisms of the enhancement of strength is a weakened stress concentration around the tip due to the structure [1-4]. Our approach may be quite unusual in the light of computational solid and structural mechanics; we work with the vision of impressionists: ignoring many details to capture simple views from complex systems; in many cases, discussing only on the level of scaling laws (i.e., power laws), which is one of the standard strategies of soft matter physics. The mechanism of the above less effective stress concentration works in the regime of linear elasticity. With the above-mentioned strategy, we also discuss how viscoelastic effects can play a vital role. This predicts an unusual crack shape, different from the conventional parabolic one in the linear elastic fracture mechanics: the shape is like that of a trumpet! [5] In addition, we may discuss some of the following topics: (1) Possible views on an isotropic composite [6]: doublenetwork gel consisting of two independently cross-liked networks, composites especially promising for artificial joints that are synthesized recently [7]. (2) Elastic particle-reinforced composites [6]. We propose a continuum elastic model to extract the controlling parameters of toughness of the composites. (3) Strength of perforated paper [8], related to freezing effects of apples and potatoes in food science [9]. (4) a direct experimental confirmation of Griffith’s scaling law in linear-elastic polymer foam [10].

References [1] P.-G. de Gennes and K. Okumura, C. R. Acad. Sci. Paris t.1, Ser. IV (2000) 257. [2] K. Okumura and P.-G. de Gennes, Eur. Phys. J. E 4 (2001) 121. [3] K. Okumura, J. Phys.: Condens. Matter 17 (2005) S2879. [4] K. Okumura, Eur. Phys. J. E 7 (2002) 303. [5] K. Okumura, Europhys. Lett. 63 (2003) 701. [6] K. Okumura, Europhys. Lett. 67 (2004) 470. [7] J. P. Gong, Y. Katsuyama, T. Kurokawa, and Y. Osada, Advanced Materials 15 (2003) 1155. [8] S. Nakagawa, K. Okumura and J. F.V. Vincent, submitted. [9] A. Khan and J. F.V. Vincent, J. Texture Studies 27 (1996) 143. [10] Y. Shiina, Y. Hamamoto, and K. Okumura, in preparation.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A micro-Macro strategy for ship structural analysis with FETI-DP method A. Mobasher Amini∗ , D. Dureisseix† , P. Cartraud∗ , N. Buannic‡ ∗ G` ´ eM, CNRS UMR 6183 / Ecole Centrale de Nantes 1, rue de la No¨e, BP 92101 - 44321 NANTES CEDEX 3 - France [email protected] [email protected] † LMGC, CNRS UMR 5508 / Universit´ e Montpellier 2 CC048,Place E. Bataillon, 34095 MONTPELLIER CEDEX 5, France [email protected] ‡ Principia Marine 1, rue de la No¨e, BP 22112 - 44321 NANTES CEDEX 3 - France [email protected]

ABSTRACT In the analysis of ship structures at small scale, with structural details heterogeneities and because there is only one prototype produced, which is the final product, the designers rely on finite element simulations. The finite element discretization of such structure, leads to a huge global numerical model, that suffers for computational cost and memory resource that may be unaffordable. In such a case, a multi-scale analysis should be performed. The classical local-global analysis that is used by engineers has several limitations such as:

• structure details are not periodic, therefore classical homogenization methods are not easily applicable; • edge effects are not taken into account; • zooming techniques are not easy to use: the gluing they require with the global scale often introduces artificial edge effects. This paper presents a micro-macro strategy based on the domain decomposition FETI-DP method as the solver in analysis of ship structure. With this approach, the two scales (micro and macro) are coupled during the iterations of the solver and we can consider the structural details in areas of interest, area where the fine mesh is used and a sub-domain is located. Performances are discussed and results in term of convergence are presented for several examples.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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On the use of Fourier expansions for the simulation of elastic composite pipes with defects E. Baranger∗ , O. Allix∗ , L. Blanchard† ∗ LMT-Cachan 61 Avenue du Pr´esident Wilson 94235 Cachan Cedex France [email protected], [email protected] † Alcatel Alenia Space 100 Boulevard du midi BP 99 Cannes la Bocca Cedex 60156 France [email protected]

ABSTRACT Due to the manufacturing process, defects such as delamination or matrix cracking are present in composite pipes used in satellite applications. To determine if these parts must be rejected, an experimental approach is used at the present time. The purpose of this study is to provide the Alcatel Alenia Space engineers with a dedicated numerical decision-making tool and, thus, reduce both the cost of verification and the number of rejected pipes. The behavior of the damaged pipes is modeled at the scale of the elementary ply with the model described in [1] following the work of [2]. A major issue at this scale is the computational cost, because the ply thickness is 0.2 mm. This leads to solve a several million degrees of freedom problem to catch edge effects. The proposed strategy is divided in two steps. First, due to the damage location, the exact beam theory developed in [3] is used to build the elastic solution at the middle of the pipe [4] while the ends are treated using a fine non linear modeling. Second, the elastic problem defined on the end zone and used in a non linear resolution scheme is solved using special finite elements based on Fourier expansions in the hoop direction. This allows, by the use of a preconditioned conjugate gradient method, to uncouple the resolution of the non axisymmetric problems [5]. A major question is then tackled, it is concerning the expansion order needed to get a given quality in the hoop description. Industrial cases of several millions of degrees of freedom with defects have been treated in elasticity with a prototype code developed in Matlab.

References [1] C. Hochard, P. A. Aubourg and J. P. Charles, Modelling of the mechanical behaviour of wovenfabric CFRP laminates up to failure, Composites Science and Technology, 61, 221–230, 2001. [2] P. Ladev`eze and E. Le Dantec, Damage modelling of the elementary ply for laminated composites, Composites Science and Technology, 43(3), 257–267, 1992. [3] P. Ladev`eze and J. Simmonds, New concepts for linear beam theory with arbitrary geometry and loading, European Journal of Mechanics and Solids, 17(3),377–402,1998. [4] E. Baranger, O. Allix and L. Blanchard, A dedicated computational strategy for composite pipes: basic principle and illustration, Science and Engineering of Composite Materials, 12(1-2), 2005. [5] O. Allix, E. Baranger and L. Blanchard, An efficient strategy for the calculation of end effects on composite pipes : the thermoelastic case, Composite Structures, accepted.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A multigrid approach for non-linear structural analysis in explicit dynamics B. Bourel∗,† , A. Combescure∗ , L. Di Valentin† ∗ Laboratoire

de M´ecanique des Contacts et des Solides, UMR CNRS 5514, INSA Lyon 20 Av. Albert Einstein, 69621 Villeurbanne, France [email protected], [email protected] † PSA

Peugeot Citro¨en Route de Gisy, F-78943 V´elizy-Villacoublay Cedex, France [email protected] ABSTRACT This study deals with a method to change the space-time scales for multi-domains calculations in explicit dynamics. The interest of such a method inspired by the techniques of mesh refinement [1], is to improve only when necessary the space (and time) discretization of one or more subdomain, during a significant phase of calculation. The method is based on a mesh refinement or coarsening which is activated according to a predefined criterion. Here, the different meshes used for the same domain are defined before the calculation. This is why we will refer to switch or mesh change rather than remeshing. Although this work is a part of researches on multi-domains approaches [2,3], we will concentrate here on refinement of one of the subdomains omitting the interactions with the other subdomains. After the mesh change, the different mechanical fields associated with the new mesh must be projected from the old mesh and must allow us to continue the calculation on the new mesh. This continuation is correct if the projected fields, associated with the new mesh, satisfy the equilibrium equations as well as possible. The transfer of fields by simple interpolation does not ensure, in general, this condition, in particular for problems with high non-linearities. Moreover, these simple interpolation methods become particularly unstable when used in an dynamic explicit scheme. So, the main difficulty will be to maintain stability and precision of calculation. That is why we will concentrate on the transition step and on the way of equilibrating the solution on the new discretization in the linear and non-linear case (material non-linearity). Finally, the decision to switch from a coarse mesh to a fine mesh is controlled here by a physical criterion based on the maximum plastic strain. Currently the algorithm developed in c has been tested on several simple geometries. These first examples allowed us to validate CASTEM the method and to show its efficiency in the linear and non-linear case.

References [1] P. Cavin, A. Gravouil, T. Lubrecht, A. Combescure. Efficient FEM calculation with predefined precision through automatic grid refinement, Finite Elements Anal. Des. 41 : 1043-1055, 2005. [2] B. Herry, L. Di Valentin, A. Combescure. An approach to the connection between subdomains with non-matching meshes for transient mechanical analysis, Int. J. Numer. Meth. Engng. 55 : 973-1003, 2002. [3] A. Gravouil, A. Combescure. Multi-time-step and two-scale domain decomposition method for non-linear structural dynamics, Int. J. Numer. Meth. Engng. 58 : 1545-1569, 2003.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Mechanical behaviour of textile structures: two-scales approach V. Carvelli , C. Corazza, C. Poggi Department of Structural Engineering, Technical University (Politecnico) of Milan Piazza Leonardo da Vinci 32, 20133 Milano (Italy) {valter.carvelli, corazza, carlo.poggi}@polimi.it

ABSTRACT Technical textiles are widely used in different industrial fields for applications like screen printing and filtration. In order to predict the mechanical response of textile structural components, the knowledge of the textile mechanical features is indispensable in the design and the development activities. The prediction of the mechanical properties of textiles has been item of several researchers by different approaches. In particular the investigations can be grouped in three main methodologies: experimental (see e.g. [1]), analytical (see e.g. [2]) and numerical (see e.g. [3]). This paper presents a numerical approach based on two scale modelling. The first scale deals with the prediction of the mechanical behaviour of the plane weave monofilament textile by the numerical analysis of a Representative Volume (RV size of the order 10-5 m) assuming a regular distribution of the fibres in the warp and weft directions (Figure 1a). In the second scale modelling the numerical analysis of the entire textile component (size of the order 1m) is performed employing the homogenized textile mechanical behaviour obtained in the first step. The first scale analysis (textile scale) is validated by experimental results including uniaxial and biaxial tensile tests. The second scale analyses deals with a cylindrical component employed in a screen printing technique (Figure 1b). Different configurations of the cylinder are considered in order to understand the influence of the textile and the geometric parameters on the deformed shape (i.e. the print quality).

(a) (b) Figure 1. (a) First scale modelling, analysis of the RV. (b) Second scale modelling, analysis of the structural component.

References [1] A. Gasser, P. Boisse, S. Hanklar; Mechanical behaviour of dry fabric reinforcements. 3D simulations versus biaxial tests, Comp. Mat. Science, vol. 12, 7-20, (2000). [2] S.V. Lomov, G. Huysmans, I Verpoest; Hierarchy of textile structures and architecture of fabric geometric models, Textile Research Journal, vol. 71, 534-543, (2001). [3] M Tarfaoui, J. Y. Drean; Predicting the stress-strain behaviour of woven fabrics using the finite element method, Textile Research Journal, vol. 71, 790-795, (2001).

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

602

On multilevel strategies for nonlinear computations with domain decomposition: application to post-buckling Philippe Cresta†,*, Olivier Allix*, Christian Rey*, Stéphane Guinard† *Laboratory of Mechanics and Technology (LMT) ENS Cachan, 61 av. du Président Wilson, 94235 Cachan Cedex, France [email protected] †EADS Corporate Research Center (CRC) 12 rue Pasteur, BP 76, 92152 Suresnes Cedex, France [email protected]

ABSTRACT
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     [2] P. Le Tallec, J. Mandel, M. Vidrascu, A Neumann-Neumann Domain Decomposition Algorithm for Solving Plate and Shell Problems, J. Numer. Anal. , Vol. 35, No. 2, 836-867, 1998. [3] C. Farhat, K. Pierson, M. Lesoinne, The second generation FETI methods and their application to the parallel solution of large-scale linear and geometrically non-linear structural analysis problems, Comput. Meth. In Appl. Mech. and Engrg., 18, 333-374, 2000. , 833"'9

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III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

603

Textile fabric simulator: collisions handling at the level of yarns M. Kuprys*, R. Barauskas† *

Department of System Analysis, Kaunas University of Technology Studentu str. 50-313b, LT-51368 Kaunas, Lithuania [email protected]

†

Department of System Analysis, Kaunas University of Technology Studentu str. 50-407, LT-51368 Kaunas, Lithuanian [email protected]

ABSTRACT The paper deals with the modeling of the physical behavior of the woven structures imitating the textile fabrics. The model is focused on the mezzo-layer of the woven fabric, which gives us ability to investigate these structures in more precisely view. The model is based on a combined approach which presents longitudinal elastic properties of each yarn by a system of non-volumetric structural elements (springs), which us note as combined particles (CP-s), while the cross-deformation of the volumetric yarn is evaluated in a 3D space with the help of tight-fitting of oriented bounding boxes. Collision detection and response is an essential part of the simulation process. Dealing with deformable bodies it is the main time consuming stage of the computational model covering both collisions detection and response stages among colliding parts of the yarns. At a present time several techniques suitable for collision handling of deformable objects exist, so the brief discussion of the mostly suitable for level of yarns is presented. The focus of this paper is to develop an efficient model of the fabric as weaved structure. The collision detection algorithm implemented in this work is based on the idea of tracking the closest pairs of colliding elements known as temporal coherence together with the stochastic approach that is used to generate a new pairs of potentially colliding elements anywhere on the two approaching yarns. After local minima of a distance are reached the collision response scheme between two closest CP-s is applied. Collisions are updated at each time step during the simulation process, thus avoiding interpenetrations of the yarns while the cross-sectional deformations of the yarns are assumed not to exceed the half of the initial radius. An empiric model for evaluating deformations is used by assuming the geometrical shape of the cross-section being always elliptic with changing the length of the axes. The approach is a compromise between the simplified uni-dimensional rod system and a fully volumetric model of a yarn in a weave. It enables to achieve good performance along with the possibility to analyze the deformable yarn structure in 3D space. The advantage in comparison with traditional models presenting a yarn as a full volumetric deformable body is the significantly reduced number of degrees of freedom of the structure while preserving the “volumetric” behavior. Numerical examples considering the generation of the initial woven structure by tension of the structure of crimped yarns and the failure at shooting-through the fabric are presented.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

604

A computational strategy for contact simulation Julien Pebrel∗ , Pierre Gosselet∗ , Christian Rey∗ ∗ LMT Cachan ENS Cachan, 61 av du pr´esident Wilson, 94235 Cachan cedex, France {pebrel,gosselet,rey}@lmt.ens-cachan.fr

ABSTRACT We consider the simulation of structures undergoing frictional contact conditions. The chosen formulation [1] coupled with a Newton-like solver, leads to the resolution of a sequence of non-symmetric linear systems with non-invariant matrices. We propose a computational strategy based on non-overlapping domain decomposition method [2, 3, 4] and augmented Krylov iterative solvers [5]. After each Newton iteration, numerical information on the condensed interface problem is stored inside so called Krylov subspace; we propose to reinject most significant part of this information inside a second scale problem in order to accelerate the resolution of following systems. This strategy can be viewed as an extension of previous works [6] to non-symmetric problems: new strategies to build reused information and relevance estimators are assessed.

References [1] P. Alart and A. Curnier, A mixed formulation for frictional contact problems prone to Newton like solution methods Comp. Meth. Appl. Mech. Eng., 92, 353–375, 1991. [2] J. Mandel, Balancing domain decomposition Comm. in Appl. Num. Meth. and Eng., 9, 233–241, 1993. [3] C. Farhat and F.-X. Roux, Implicit parallel processing in structural mechanics Computational Mechanics Advances, 2 (1), 1-124, 1994. [4] P. Alart, M. Barboteux, P. Letallec and M. Vidrascu, M´ethode de Schwartz additive avec solveur grossier pour probl`emes non sym´etriques C.R. Acad. Sci. Paris, t. 331, S´erie I, 399-404, 2000. [5] Y. Saad, Iterative methods for sparse linear systems, 2000. [6] P. Gosselet and C. Rey, On a selective reuse of Krylov subspaces in Newton-Krylov approaches for nonlinear elasticity Proceedings of the 14th conference on domain decomposition methods, 419–426, 2002.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

605

A simulation strategy for life time calculations of large, partially damaged structures Christian Rickelt∗ , Stefanie Reese∗ ∗ Institute

of Solid Mechanics TU Braunschweig, D-38023 Braunschweig, Germany [email protected], [email protected] ABSTRACT The present paper is motivated by the increasing concern to accomplish more realistic lifetime estimations of complex engineering structures. For this purpose the entire system and additional mechanisms like damage evolution or changes in the loading situation as well as in the surroundings have to be incorporated into such a long-term computation. Undoubtedly the finite element method represents a suitable tool to this end. But inspite of the fast development of computer technology the life time computation is still to complex to be carried out without advantageous and effective strategies to reduce computational cost. In this contribution we present a discretisation strategy which takes into account that only small parts of a structure demand a non-linear analysis. Accordingly we strictly decompose our system on the structural level into non-linear and linear subsystems by an exact substructure technique. We are finally able to determine the entire system response by the solution of a number of small non-linear subsystems. Additionally, if it is required, the linear subsystems may be evaluated in a post processing calculation. Further we may reduce the number of degrees-of-freedom of the linear subsystems, because they influence the evolution of damage only indirectly. Hence we join our proposed substructure strategy with projection-based model reduction techniques for linear second order systems, like modal truncation, Ritz vectors and the proper orthogonal decomposition [1]. An alternative approach of partial model reduction is e.g. presented by [2]. In the non-linear substructures we model the evolution of damage for ductile damage behaviour of metals taking into consideration large inelastic strains by the material model of [3]. At the material level we exploit the advantages of a formulation in principle axes in combination with the exponential mapping algorithm. This material model is implemented into the computationally efficient Q1SP finite element formulation of [4], which is based on the concept of reduced integration with hourglass stabilisation.

References [1] P. Holmes, J.L. Lumley and G. Berkooz, Turbulence, coherent structures, dynamical systems and symmetry. Cambridge University Press, 1996. [2] J.E. Barbone, D. Givoli and I. Patlashenko, Optimal modal reduction of vibrating substructures. Int. J. Numer. Meth. Engrg., 57, 341–369, 2003. [3] A. Eckstein and Y. Bas¸ar, Ductile damage analysis of elasto-plastic shells at large inelastic strains. Int. J. Numer. Meth. Engrg., 47, 1663–1687, 2000. [4] S. Reese, On a physically stabilized one point finite element formulation for three-dimensional finite elasto-plasticity. Comput. Methods Appl. Mech. Engrg., 194, 4685–4715, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

606

Finite Element Method In First-principles Calculation Yoshinori Shiihara*, Osamu Kuwazuru†, Nobuhiro Yoshikawa† * The University of Tokyo 4-6-1, Komaba, Meguroku, Tokyo, 153-8505 Japan [email protected] †

Institute of Industrial Science, The University of Tokyo 4-6-1, Komaba, Meguroku, Tokyo, 153-8505 Japan {kuwa, yoshi}@iis.u-tokyo.ac.jp

ABSTRACT We propose a finite element implementation for the first-principles calculation based on the Density Functional Theory (DFT). The atomic-scale simulation based on the DFT plays an important roll to predict various material properties such as the physical strength. Such simulation seems contribute much to design of new materials of useful functions without loborious compricated experiments. Practical complex atomic systems, such as interfaces of metal/ceramics, contain huge number of atoms and it certainly results in large-scale calculation. The traditional DFT scheme based on the plane-wave basis is not higly recommended for the large-scale calculation because the plane-wave basis scheme requires the Fast Fourier Transforms (FFT). The FFT requires all-node communications, which result in reduction of the parallel-computing performance. An advantage of the DFT scheme based on the Finite Element Method (FEM) is its parallelability. The localization of the finite elements in real space corresponds to the localization of components in the global matrix. This feature is suitable for the massively parallel computation. We formulate the DFT scheme based on the norm-conserving pseudo-potential technique by the FEM. The Kohn-Sham equation as the governing equation of the DFT is discretized by the Galerkin’s weighted residual method. The Kohn-Sham equation is treated as a integral form in the FEM scheme and the nonlocal pseudo-potential can be easily estimated by the integral form. In our finite-element formulation the divergence term in the periodic local potential is treated by the Ewald scheme[1]. Inadequate setting of parameters employed in the Ewald scheme gives wrong potential, yield incorrect results and also causes the inefficient calculation. We carry out the parameter setting for the Ewald scheme in the real-space method. The optimized parameters are systematically obtained and the computational efficiency and the numerical accuracy is conserved. Test calculation for a silicon dimer is performed. Through the calculation, we show the fact that 1) Our FEM formulation follows the variational principle. 2)The free energy obtained by our FEM formulation is consistent with the highly-converged value obtained by the established plane-wavebased package to the meV/atom order. 3) The order of the error in the free energy is O(element size4) in our FEM formulation. 4) The equilibrium bonding length predicted by our FEM code is consistent with that of the plane-wave-based package to the mÅ order.

References [1] M.C. Payne, M. P. Teter, D. C. Allan, T. A. Arias and J. D. Joannopoulos, Iterative minimization techniques for ab-initio total-energy calculations: molecular dynamics and conjugate gradient, Rev. Mod. Phys., 64, 1045-1097, 1992.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

607

Modelling and Simulation of Earthquake Ground Motion via Functional Series TARMA Models with Wavelet Basis Functions Minas D. Spiridonakos and Spilios D. Fassois1 Stochastic Mechanical Systems & Automation (SMSA) Laboratory Department of Mechanical & Aeronautical Engineering University of Patras, GR 265 00 Patras, Greece E-mail: {mspirid,fassois}@mech.upatras.gr Internet: //www.mech.upatras.gr/∼sms ABSTRACT The present study explores non-stationary Functional Series Time-dependent AutoRegressive Moving Average (FS-TARMA) models with wavelet basis functions for the modelling and simulation of earthquake ground motion. FS-TARMA models constitute conceptual extensions of their conventional (stationary) counterparts, in that their parameters are time-dependent belonging to functional subspaces [1]. Wavelets, with their scaling and localization in time, comprise a promising functional basis for “fast” evolutions in the dynamics. The study focuses on the assessment of wavelet based FS-TARMA modelling and simulation for two California earthquake ground motion signals: an El Centro accelerogram recorded during the 1979 Imperial Valley earthquake, and a Pacoima Dam accelerogram recorded during the 1994 Northridge earthquake. A systematic analysis leads to a TARMA(2, 2) model for the El Centro case and a TARMA(3, 2) model for the Pacoima Dam case. Both models are formally validated and their analysis and simulation (synthesis) capabilities are demonstrated via Monte Carlo experiments focusing on important ground motion characteristics.

Figure 1: (a) 2-D plot of the El Centro accelerogram non-parametric STFT-based time-dependent PSD estimate, and (b) 2-D plot of the TARMA(2, 2)[2,2] -based parametric Melard-Tjøstheim time-dependent PSD estimate.

References [1] A. Poulimenos and S. Fassois, Parametric time-domain methods for non-stationary random vibration modelling and analysis – A critical survey and comparison. Mechanical Systems and Signal Processing, 20, 763–816, 2006. 1

Corresponding author.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

608

Identification of Damage in Multispan Beams Using Parameter-dependent Frequency Changes and Neural Networks ´ ∗ Artur Borowiec∗ , Leonard Ziemianski ∗ Rzesz´ ow

University of Technology, Department of Structural Mechanics, W.Pola 2, 35-959 Rzesz´ow, Poland [email protected], [email protected] ABSTRACT

Nowadays the knowledge of the structure condition is considered to be more and more important. The state of the structure and its safety strongly depends on the degradation of the structure elements (beams, connections, etc.). Nondestructive methods predict the location and the extent of damage in existing engineering structures. The publications on the identification of damages present mainly the approach which implies the knowledge of eigenfrequencies and eigenmodes of an undamaged structure. The damage is identified on the basis of the variations of dynamic parameters with respect to the initial values [1]. Some methods require the introduction of external perturbations to the structure. The detection method, which provides the global assessment of damage, is usually not sensitive to the size of the damage. In paper by Dems et al. [2] to increase the accuracy of identification an additional parameter is introduced (namely concentrated elastic or rigid support, additional mass elastically or rigidly attached to the structure, boundary constraint). This paper is intended to provide the analysis of eigenvalues with respect to the additional mass [4] and the application of ANNs [3] to the damage identification. ANN is applied for the analysis of dynamic response of a structure and for the assessment of the structure condition. Herein three examples are discussed, in all of them ANNs are applied to develop a new method of identification. The assessment of the state of the considered structure relies, in the case of application of the proposed extended identification method, on the comparison of structure eigenfrequencies obtained from the systems with additional masses placed in different nodes. The differences in the source of information employed to identify the location and the extent of the damage. The additional parameter introduced to the structure increases the identification accuracy. The Artificial Neural Networks are able to locate the damage and the extent of the structure degradation. The obtained results show that it is possible to identify the damage using the dynamic responses of the structure. The results presented in this paper are very promising, in the next step more complicated structure will be taken into account. Moreover other perturbations should be also considered.

References [1] Deobeling S.W, Farrar C.R, Prime M.B, Sheritz D.W. Damage identification and health monitoring of structural and mechanical systems from changes in their vibration characteristic: a literature review. Los Alamos Natl. Lab, 1996. [2] Dems K, Mr´oz Z. Identification of damage in beam and plate structure using parameter-dependent frequency changes. Engineering Computations, 18, 96-120, 2001. [3] Waszczyszyn Z., Ziemia´nski L., Neural Networks in Mechanics of Structures and Materials — New Results and Prospects of Applications, Computers&Structures, 79: 2261–2276, 2001. [4] Ziemia´nski L, Pia¸tkowski G. The detections and localizations of an attached mass in plates Proceedings of third European Conference on Structural Control, Vienna, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

609

Microplane model parameters estimation using neural networks Anna Kuˇcerov´a, Matˇej Lepˇs, Jan Zeman Czech Technical University in Prague Th´akurova 7, Prague 6, 166 29, Czech Republic {anicka, leps, zemanj}@cml.fsv.cvut.cz ABSTRACT Concrete is one of the most frequently used material in Civil Engineering. Nevertheless, as a highly heterogeneous material, it shows very complex non-linear behavior, which is extremely difficult to describe by a sound constitutive law. As a consequence, a numerical simulation of response of complex concrete structures still remains a very challenging and demanding topic. One of the most promising approaches to modelling of concrete behavior is based on the microplane paradigm [1]. It is a fully three-dimensional material law that incorporates tensional and compressive softening, damage of the material, supports different combinations of loading, unloading and cyclic loading along with the development of damage-induced anisotropy of the material. As a result, the material model [1] is fully capable of predicting behavior of real-world concrete structures, once provided with proper input data. The major disadvantages of this model are, however, a large number of phenomenological material parameters and a high computational cost associated with structural analysis even in a parallel implementation [2]. The authors of the microplane model proposed a heuristic calibration procedure [1], that is based on the trial-and-error method but is computationally inefficient. Therefore, a new procedure based on artificial neural networks is proposed in the present contribution. In order to asses the reliability of identified material parameters, results of a stochastic sensitivity study based on the Latin Hypercube Sampling (LHS) method are presented first. Different tests, proposed in [2], are simulated numerically and used to determine, which model parameters can be reliably identified from these tests. In the next step, a neural network-based procedure is presented for identification of material parameters. A crucial point is the generation of a training set used to determine weights of individual neurons. To this end, the LHS method is again employed as it allows using a limited number of computational simulations while ensuring the representativeness of the generated training set. The training procedure itself is based on a real-coded genetic algorithm SADE [3]. Finally, the application of the proposed identification procedure to the back analysis of laboratory experiments is presented.

References [1] Z.P. Baˇzant, F.C. Caner, I. Carol, M.D. Adley, S.A. Akers, Microplane model M4 for concrete. Part I: Formulation with work-conjugate deviatoric stress, Part II: Algorithm and calibration, Journal of Engineering Mechanics-ASCE, 126, (2000), 944-953, 954–961. [2] J. Nˇemeˇcek, B. Patz´ak, D. Rypl, Z. Bittnar, Microplane models: Computational aspects and proposed parallel algorithm, Computers and Structures, 80, (2002), 2099–2108. [3] O. Hrstka, A. Kuˇcerov´a, Improvements of real coded genetic algorithms based on differential operators preventing the premature convergence, Advances in Engineering Software, 35, (2004), 237–246.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

610

ANNs and Linguistic Variables in the Analysis of Mine Induced Rockbursts Transmission to the High Building Krystyna Kuzniar Pedagogical University of Cracow ul. Podchorazych 2, 30-084 Krakow, Poland [email protected]

ABSTRACT Rockbursts are one of the negative phenomena accompanying underground mining. A large quantity of energy is released during a tremor. The energy causes propagation of seismic waves that reach the surface of the earth. They induce the building vibrations subsequently. Although these tremors are strictly connected with human activity, they differ considerably from other paraseismic vibrations. They are not subject to human control and they develop in an uncontrolled manner. In Poland, mining tremors resulted from underground raw mineral material exploitations in Legnica-Glogow Copperfield (LGC) induce the surface horizontal vibrations reaching even 0.2 acceleration of gravity (g) and vertical components reaching 0.3g. The large scale of the effects might be shown by the fact that the intensity of surface vibrations is greater than the predicted (and taken into consideration in structural design) intensity of vibrations from earthquakes in neighbouring countries: Slovak Republic, Czech Republic and Germany. Soil-structure interaction is a very important problem from the engineering point of view. The prognosis of vibration influences on structures as well as estimation of the way of ground vibrations transmission to building basements are essential. The comparison of maximal values (amplitudes) of vibrations (accelerations, velocities and displacements) recorded at the same time on the ground and on the basement level is the simplest and very often employed way of estimation of the vibrations transmission from the ground to the building. The paper deals with an application of artificial neural networks (ANNs) for evaluation of soilstructure interaction in case of the transmission of ground vibrations from mining tremors to building basement. The problem is analysed with respect to typical prefabricated eleven-storey building with load bearing walls. The influence of mining tremors parameters as mining tremor energy and epicentral distance on the soil-structure interaction effect is also discussed. The parameters are estimated as approximate values found experimentally. Therefore the linguistic variables associated with the fuzzy character of the parameters are introduced in the neural network analysis. From the obtained results it can be stated that application of simple neural networks enables us to predict the building foundation vibrations with satisfactory accuracy, thus effects of the transmission of ground vibrations to building foundation (soil-structure interaction) may be analysed using neural networks.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

611

Differential Flatness of Aircraft Flight Dynamics and Neural Inversion Wen C. Lu *, †, Lili Duan†, Félix Mora-Camino*, †, and Roger M. Faye# * LAAS du CNRS 7 avenue du Colonel Roche, 31077, Toulouse, France {wclu, mora}@laas.fr † ENAC 7 avenue Edouard Belin, 31055, Toulouse, France {wenchi.lu, lili.duan, felix.mora}@enac.fr #

Ecole Supérieure polytechnique BP5085, UCAD, Dakar, Sénégal [email protected]

ABSTRACT Differential flatness, a property of some dynamic systems, introduced by Fliess et al., has made possible the development of new tools to control complex nonlinear dynamic systems. Many dynamic non linear systems have been proved to be differentially flat. Some authors have investigated the differential flatness of conventional aircraft dynamics, although none of them has considered separately the flatness property of the flight guidance and the attitude dynamics of a rigid aircraft. In this paper, it is shown that the inertial position coordinates of an aircraft can be considered as differential flat outputs for its flight guidance dynamics. Since this differential flatness property is implicit, a neural network is introduced, as a numerical device, to deal with the inversion of the guidance dynamics. It is shown also that, once conveniently structured and trained, the neural network is able to generate in real time directives to conventional autopilot systems concerned with attitude and engine regime control so that the reference trajectory can be tracked. Numerical simulation results are displayed and discussed.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

612

Finite Element Analysis of an Energy Absorbing Crush Zone Using Expanded Metal A. Ayestarán*, C. Graciano† Universidad Simón Bolívar, Departamento de Mecánica, Caracas 1080-A, Venezuela. * [email protected] † [email protected]

ABSTRACT The design of crashworthy structures is a very important issue in automotive design. Structures able to absorb a great amount of energy during impact are a challenge for automotive engineers nowadays. Currently, computer simulation is an efficient and costeffective manner of designing structures and the understanding of their structural behavior can be studied in depth. This paper is aimed at studying the nonlinear behavior of the crush zone of a vehicle used for the Annual Formula SAE competition. The crush zone is made of expanded metal sheets and solid corners joined by welding. The use of expanded metal sheets reduces considerably the weight of the structure. In addition the energy absorbing properties of the crush zone are improved due to the imperfection sensitive nature of structures made of expanded metal sheets. A finite element model is built taking into material and geometrical nonlinearities using the commercial software MSC.Marc. In the numerical analysis, the crush zone is subjected to axial compression in order to obtain the energy absorbing properties of this crush zone. A sensitivity analysis is performed on the influence of the following parameters: material properties, gauge thickness, size of the expanded metal pattern and orientation of the pattern (longitudinal or transversal). The results show the enhancement in the energy absorbed by the crush zone obtained by means of using expanded metal.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

613

Design of Satellite Control System using the Optimal Nonlinear Theory Luiz Carlos Gadelha DeSouza National Institute for Space Research – INPE. Av. dos Astronautas, 1758, 12201-940 - São José dos Campos - SP-Brasil [email protected] †

Institution Second and Third Author Address [email protected]

ABSTRACT Nowadays, attitude control systems of satellites with rigid and flexible components are demanding more and more better performance resulting in the application of new advanced nonlinear control theory. This is the case when the dynamics of the plant that describe the attitude motion of the satellite is nonlinear and the mission involves stringent pointing accuracy. As a result, control designs methods presently available, needs more investigation to know their capability and limitations. In that context, the guaranty of the controller performance depends not only on its good design but also on the knowledge of the nonlinear characteristics of the system in order to improve the overall control system efficiency. In this paper, a new nonlinear control law for satellite attitude control (SAC) is presented. It is based on an extension of the linear quadratic regulator (LQR) theory to the case where the dynamics is described by a nonlinear system of equation (Euler´s equation). The control law performance is investigated in frequency domain evaluating the compromise between the gain level and bandwidth length. In time domain the control law performance is observed by its capability of shifting the overshot to the origin direction. By and large, one observes that the nonlinear terns in the control law are able to deal with the nonlinear term in the model, which reveals that the nonlinear control law is more efficient than the control law based on the linear theory, even in the presence of the parameters variation.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

614

Design for Crashworthiness of Train Structures with Simplified Multibody Models João P. Dias*, Filipe Antunes† and Manuel S. Pereira† *IDMEC - Instituto de Mecânica - Pólo IST Instituto Superior Técnico, Technical University of Lisbon, 1049-001 Lisbon, Portugal [email protected] *IDMEC - Instituto de Mecânica - Pólo IST Instituto Superior Técnico, Technical University of Lisbon, 1049-001 Lisbon, Portugal [email protected]; [email protected]

ABSTRACT In this paper simplified models for the design of vehicle structures under impact are presented. The use of multibody system dynamics based models in structural crashworthiness problems proved to be useful and accurate enough in simulating train collisions and in the optimization of energy absorption devices [1]. The presented simplified computational models are based on multibody rigid-flexible systems, where flexibility is included using the finite element method. Structural damping and contact models are also considered. In structural impact, members can be subjected to axial or bending tensions that usually result in components’ plastic deformation in areas known as plastic elements. Plastic elements can be modeled associating cinematic joints with non-linear springs, whose constitutive relationship correspond to the components’ collapse behavior. The constitutive relationship is computationally defined using parameter identification techniques. Sometimes not all the areas where plastic deformation occurs can be predicted, so, a methodology to detect plasticity and automatically insert new plastic elements in simplified models, called remodelling, is proposed. Since simplified models are based on multibody systems, crashworthiness simulations are much faster when compared to finite element commercial software, enabling the use of genetic algorithms for the design process [2]. Dynamic analysis formulations are integrated with the multiobjective optimization evolutionary algorithm NSGAII [3] resulting in a 2D mechanical systems multiobjective optimization tool, used to perform optimization simulations where parameters most frequently used in crashworthiness problems are studied. The presented methodologies and formulations have been implemented computationally in order to develop a tool for the first stages of vehicle design. Remodelling and multiobjective optimization examples are presented to demonstrate the presented methodologies.

References [1] J.P. Dias and M.S. Pereira, Optimal Design of Train Structures for Crashworthiness using a Multi-load Approach, I. J. of Crashworthiness, 7, 331-343, 2002. [2] Dias, J. P., Corrêa, R. e Antunes, F., “Crashworthiness Optimization of Train Structures with Evolutionary Algorithms”, EUROGEN 2003, 83, Barcelona, Spain, 15-17 September, 2003. . [3] K. Deb, Multi-Objective Optimization Using Evolutionary Algorithms, John Wiley and Sons, LTD, 2001.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

615

Analysis of Stress and Strain in the Absolute Nodal Coordinate Formulation with Nonlinear Material Behavior Johannes Gerstmayr*, Marko K. Matikainen† *

†

Institute for Technical Mechanics, Johannes Kepler University of Linz Altenbergerstr. 69, 4040 Linz, Austria [email protected]

Department of Mechanical Engineering, Institute of Mechatronics and Virtual Engineering Lappeenranta University of Technology, P.O.Box 20, FIN-53851, Lappeenranta, Finland [email protected]

ABSTRACT The present paper deals with the analysis of strain and stress in the absolute nodal coordinate formulation (ANCF). An accurate stress distribution is needed for the evaluation of comparative strains in nonlinear material behavior. The ANCF has been recently developed and studied by many investigators in the field of flexible multibody dynamics. The ANCF focuses on the modeling of beams and plates including large deformation and represents exact rigid body inertia. The derivation of the equations of motion for an ANCF element is usually based on a solid finite element formulation and thus leads to finite elements that show locking behavior. While the problem of locking in the ANCF might be solved by means of different techniques [1, 2], the accuracy of stress and strain quantities within the element is still poor and needs to be improved in order to incorporate nonlinear material behavior. In the present paper, a higher order element is presented where locking is prevented by means of standard selective reduced integration techniques and the improved order and accuracy of stress and strain quantities is shown in comparison with the original formulation. As an example of nonlinear material behavior, Prandl-Reuss plasticity is included to the absolute nodal coordinate formulation. The results of stress and strain components for the improved higher order element are compared to the solution of fully tree-dimensional computations performed with the commercial software ABAQUS. Good agreement of the ANCF is found with the results of ABAQUS as well as with examples of elasto-plastic multibody systems available from the literature.

References [1] J. Gerstmayr, A.A. Shabana, Efficient Integration of the Elastic Forces and Thin ThreeDimensional Beam Elements in the Absolute Nodal Coordinate Formulation, Proceedings of the Multibody Dynamics 2005 ECCOMAS Thematic Conference, Goicolea, Cuadrado, García Orden (eds.), Madrid, Spain, 2005. [2] A.L. Schwab, J.P. Meijaard, Comparison of three-dimensional flexible beam elements for dynamic analysis: finite element method and absolute nodal coordinate formulation, Proceedings of the ASME 5th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, paper number DETC2005-85104, ASME, New York, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Nonlinear Multimode Dynamics of a Moving Microbeam For Noncontacting Atomic Force Microscopy S. Hornstein and O. Gottlieb

Department of Mechanical Engineering Technion – Israel Institute of Technology Haifa 32000, Israel. [email protected], [email protected]

ABSTRACT Atomic force microscopy (AFM) is a modern imaging technique that is used to map surfaces down to atomic resolution and enables a quantitative estimation of atomic interaction forces. This is obtained by measuring a van der Waals like atomic interaction between a sample and a vibrating microcantilever, which has a sharp tip at its free end. Of particular importance are biological and nonconducting materials that cannot be mapped by alternative methods, without destruction of their surfaces by a conducting coating layer. The growing demand for detection of sub-atomic features and industrial use, increases the need for faster and more accurate scanning. Various methodologies have been proposed to speed up the scan rate. These include individual moving microbeam control strategies and use of an array of probes. However, the accuracy of force estimation from measured data crucially depends on the quality of the mathematical model in use. A typically used model is that of a lumped mass system that reduces the microbeam to a linear spring with a nonlinear force, derived from a candidate tip-sample interaction. This model does not incorporate the dynamic boundary condition of the scan process and cannot resolve the rich spatio-temporal dynamic response of the nonlinear dynamical system. Thus, the objectives of this research include theoretical derivation and analyses of a continuous model for the moving and vibrating AFM microbeam that consistently incorporates the nonlinear atomic interaction and the dynamic conditions of the scan process. A nonlinear initial boundary-value problem is derived using the extended Hamilton's principle. The continuum system is then reduced to a multimode dynamical system using a Galerkin procedure. Numerical analysis of a three mode system reveal that below the 'jump-to-contact' stability threshold, there exist a dense set of coexisting bounded periodic (ultrasubharmonic) solutions. This complex bifurcation structure is augmented by quasiperiodic solutions that are found to correspond to a 3:1 internal resonance between the third and second microbeam modes.

‘Jump to contact’ stability threshold

A quasiperiodic time-series (top) and power spectra (bottom).

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Model Reduction with Mean-Axes in Deformable Multibody Dynamics Yi-shih Lin and Parviz E. Nikravesh Department of Aerospace and Mechanical Engineering University of Arizona, Tucson, AZ 85721 {yishih, pen}@email.arizona.edu

ABSTRACT Dynamic analysis of multibody systems containing deformable bodies is computationally time consuming due to the large number of deformation degrees-of-freedom. Transforming the equations of motion to modal space and then truncating a significant number of higher frequency modes can greatly reduce the problem size and hence improve the computational time. There is a misconception that if mean-axes are adopted as the floating reference frame for a deformable body, modal truncation will yield inaccurate results. This may be due to the misunderstanding that only free-free modes are compatible with the mean-axes and, therefore, mean-axes cannot be used for constrained systems. This paper attempts to clarify this issue and shows that it is perfectly fine to use the standard static and normal modes, as they are obtained in structural analysis, in conjunction with the mean-axis conditions.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Impacts With Friction In Flexible Multibody Dynamics Juana M. Mayo Department of Mechanical and Materials Engineering University of Sevilla Camino de los Descubrimientos s/n, 41092 Sevilla [email protected]

ABSTRACT Most formulations on normal impacts of flexible multibody systems use the momentum balance equations. Newton’s hypothesis is usually employed, where the restitution coefficient is defined by the relative normal velocities of the impacting bodies before and after the collision. When friction is present in the impact, the process becomes more complicated. Different impact modes are possible: sliding, sticking or reverse sliding. The variables in the momentum balance equations are the changes in the velocity and the two components of the impulse, one in the normal direction to the common tangent of the contact surfaces and other in the tangent direction. To solve the equations two additional conditions are needed, one comes from the Coulomb law, and the other from the definition of the restitution coefficient. The use of the Newton hypothesis for the definition of the restitution coefficient leads to wrong results in the simulation of impacts with friction. Therefore, it is necessary to use the Poisson hypothesis. The restitution coefficient is defined through the normal impulses of the compression and restitution periods. In this paper a formulation of impacts with friction of planar flexible multibody systems is presented. The floating frame of reference formulation is used to model the flexible bodies. The normal and tangential impulses in the contact point are calculated by a computational algorithm based on the graphics techniques developed by Routh. Lankarani and Pereira used this technique to analyse impacts with friction of planar rigid multibody systems.

References [1] E.J. Routh, Dynamics of a system of rigid bodies. MacMillan, London, 1891. [2] H.M. Lankarani and M.Pereira, Treatment of impact with friction in planar multibody mechanical systems. Multibody System Dynamics, 6, 203-227, 2001.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A conservative augmented Lagrangian algorithm for the dynamics of constrained mechanical systems Juan C. Garc´ıa Orden∗ , Roberto A. Ortega∗ ∗ Computational

Mechanics Group. School of Civil Engineering. Universidad Polit´ecnica de Madrid. c/ Profesor Aranguren s/n 28040 Madrid, Spain [email protected], [email protected] ABSTRACT

The motion of many practical mechanical systems is often constrained. An important example is the dynamics of multibody systems, where these constraints arise from the modeling of joints that connect different bodies. The numerical solution of the dynamics of this type of systems faces several difficulties, mainly due to stability problems [1]. Different methods have been proposed in the literature to overcome these problems, based on different strategies for the constraints formulation. One of these strategies is the augmented Lagrange formulation, which allows the use of numerical integrators for Ordinary Differential Equations, combined with an update scheme for the algebraic variables, accomplishing exact fulfillment of the constraints. In this context, this work focuses on the design of a conservative version of this augmented Lagrangian formulation for holonomic constraints, proposing a numerical procedure that exhibits excellent stability, thus providing an interesting alternative for the dynamical analysis of these type of systems. A point of departure is a conservative formulation based in the penalty method [2], which exhibits good stability but does not accomplish exact fulfillment of the constraints.

References [1] K. E. Brenan, S. L. Campbell, and L. R. Petzold. Numerical Solution of Initial-Value Problems in Differential-Algebraic Numerical Solution of Initial-Value Problems in Differential-Algebraic. SIAM, 1996. [2] J. C. Garc´ıa Orden and J. M. Goicolea. Conserving properties in constrained dynamics of flexible multibody systems. Multibody System Dynamics, 4:225–244, 2000.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Non Linear Model for Coupled Axial/Torsional/flexural Vibrations of Drill-strings. Marcelo T. Piovan∗† , Rubens Sampaio† ∗ Mechanical

Systems Analysis Group, Universidad Tecnol´ogica Nacional - Fac. Reg. Bah´ıa Blanca, 11 de Abril 461, Bah´ıa Blanca, BA, 8000, Argentina [email protected]

† Department

of Mechanical Engineering, Pontif´ıcia Universidade Catˆolica - Rio de Janeiro, Rua Marquˆes de S˜ao Vicente 225, Rio de Janeiro, RJ, 22453-900, Brasil [email protected], [email protected]

ABSTRACT In the present work a continuous model is presented to study, by means of finite element discretization, the coupling of extensional, flexural and torsional vibrations under a state of initial stresses on a drillstring, which is described as a vertical slender beam under axial rotation [1]. The structure is subjected to distributed loads due to its own weight, the reaction force and perturbation moments at the lower end. The beam structure is also confined to a move inside a rigid cylinder, which simulates the borehole [2]. The impacts and friction of the drill-string with the borehole are modeled employing simplified forms. It is known that the state of initial stresses (which implies accounting for geometrical non-linearities) affects the dynamics of slender beams. The vibrations of drill-strings are frequently analyzed by means of lumped parameter models [3]. Normally, these models employ equivalent lumped parameters which are obtained from experimental field data or from continuous models assuming one-mode approximation for extensional, flexural and torsional vibrations. However, the lumped parameter models do not include dynamical effects due to geometrical non-linearities. In this context, the objective of present work is to analyze the effects of geometrical non-linearities due to initial stresses in the vibration of drill-strings together with the patterns of vibroimpact and comparing the results with the predictions of linear models. The beam model is discretized using a finite element with 12 degrees of freedom. The results have shown an important influence of the geometric non-linearities (when compared with the predictions of a linear model) in the dynamic responses of the drill-strings, especially when the beam undergoes impact patterns with the borehole or the rock formation. This influence can be observed in the calculation of reaction forces at top position as well as the time histories of radial displacements.

References [1] Sampaio, R., Piovan, M.T. and Venero Lozano, G., 2005, ”Non Linear model for Coupled Axial/Torsional Vibrations of Drill-Strings”, Proceedings COBEM 2005. [2] Trindade, M.A., Wolter, C. and Sampaio, R., 2005, ”Karhunen-Lo`eve descomposition of coupled axial/bending vibrations of beams subjected to impact”, Journal of Sound and Vibration, Vol.279, pp. 1015-1036. [3] Yigit, A.S. and Christoforou, P., 2003, ”Fully coupled vibrations of actively controlled drillstrings”, Journal of Sound and Vibration, Vol.267, pp. 1029-1045.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Identification of Time-Varying Structures Under Unobservable Excitation: An Overview and Experimental Comparison of Parametric Methods Aggelos G. Poulimenos, Minas D. Spiridonakos, and Spilios D. Fassois Stochastic Mechanical Systems & Automation (SMSA) Laboratory Department of Mechanical & Aeronautical Engineering University of Patras, GR 265 00 Patras, Greece E-mail: {poulimen,mspirid,fassois}@mech.upatras.gr Internet: //www.mech.upatras.gr/∼sms ABSTRACT This paper addresses the problem of parametric time-domain identification and dynamic analysis for time-varying mechanical structures under unobservable random excitation. The identification uses Time-dependent AutoRegressive Moving Average (TARMA) models (or state-space equivalents), which are conceptual extensions of their conventional (time-invariant) ARMA counterparts in that their parameters and innovations variance are varying with time. TARMA methods may be classified according to the type of mathematical structure imposed upon the evolution of the model parameters as follows: (a) Unstructured parameter evolution methods, (b) stochastic parameter evolution methods, and (c) deterministic parameter evolution methods [1]. The characteristics and relative merits of each class are outlined. A representative method from each class is then applied to the modelling of a time-varying (moving) laboratory structure. This structure consists of a beam with a cylindrical mass sliding on it at a selected speed. The setup is meant to model a “bridge-like” structure with a heavy vehicle travelling along its length. The beam is subject to broadband random force excitation, while structural identification is solely based upon its vertical acceleration response. The methods applied are: The Recursive Maximum Likelihood TARMA (RML-TARMA) method (unstructured parameter evolution), the Smoothness Priors TARMA (SP-TARMA) method (stochastic parameter evolution), and a Functional Series TARMA (FS-TARMA) method (deterministic parameter evolution) [1]. The models obtained by each method are shown to accurately describe the time-varying structural dynamics in terms of the “frozen-time” power spectral density function and the modal quantities. Modelling accuracy is judged based upon “frozen-configuration” (fixed mass location) characteristics of the experimental structure also extracted through “frozen-configuration” (multiple stationary experiments) identification (baseline modelling). The highest tracking accuracy and model parsimony (economy) is exhibited by the deterministic parameter evolution (FS-TARMA) method, followed by the unstructured parameter evolution (RML-TARMA) method, which is in turn followed by the stochastic parameter evolution (SP-TARMA) method. Overall, the results demonstrate the parametric methods’ applicability, effectiveness, high potential for parsimonious and accurate identification, and model-based dynamic analysis of time-varying structures under unobservable excitation.

References [1] A. Poulimenos and S. Fassois, Parametric time-domain methods for non-stationary random vibration modelling and analysis – A critical survey and comparison. Mechanical Systems and Signal Processing, 20, 763–816, 2006.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Variational integrators for the rigid body dynamics Ignacio Romero ∗ E.T.S.I. Caminos Universidad Polit´ecnica de Madrid Profesor Aranguren s/n, 28040 Madrid, Spain [email protected]

ABSTRACT In recent years, variational integrators have emerged as an original, powerful, and promising family of methods for integrating in time the equations of nonlinear solid dynamics. See, for example, [1, 2, 3, 4]. The idea underlying these methods is strikingly simple: instead of discretizing the equations of motion (as traditional time stepping methods do), variational integrators seek the stationary solution of a discrete action functional. In order to do so, only an approximation of the Lagrangian is required which must be expressed in terms of the discrete positions of the system. The formulation of such a Lagrangian is trivial for mechanical systems with linear configuration spaces. In contrast, if the mechanical system has a nonlinear configuration space the task is not so simple. Rigid bodies and some models of rods and shells fall into the later case and they are of obvious interest in Computational Mechanics. The first two are formulated in SO(3), the space of proper rotation tensors, and the last one in S2 , the unit sphere. In this work we will present variational integrators for the Euler equations of rigid body dynamics formulated directly on SO(3), without equations of restriction. We will discuss the advantages and drawbacks of these methods when compared with traditional ODE solvers, paying special attention to the conservation properties.

References [1] J. M. Wendlandt and J. E. Marsden. Mechanical integrators derived from a discrete variational principle. Phys. D, 106(3-4):223–246, 1997. [2] C. Kane, J. E. Marsden, M. Ortiz, and M. West. Variational integrators and the newmark algorithm for conservative and dissipative mechanical systems. International Journal for Numerical Methods in Engineering, 49:1295–1325, 2000. [3] J. E. Marsden and M. West. Discrete mechanics and variational integrators. Acta Numerica, pages 357–514, 2001. [4] A. Lew, J. E. Marsden, M. Ortiz, and M. West. Asynchronous variational integrators. Archives of Rational Mechanics and Analysis, 167:85–146, 2003.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Planning and Optimization of Maneuver Strategy of Large Flexible Space Structures Evtim V. Zahariev Institute of Mechanics, Bulgarian Academy of Sciences Acad. G. Bonchev Sy., bl. 4, Sofia 1113, Bulgaria [email protected]

ABSTRACT The nowadays space missions are implement by large lightweight structures which flexibility causes significant influence over the mission from launching till landing. Space ships, stations and satellites caring large flexible devices, for example, long booms, tethered satellites, solar arrays, antennae and many others implement complex motion in space. The reliable dynamic model and preliminary analysis of the tasks, as well as, of the possible scenarios and casual events are the preconditions for the mission success. Major undesirable phenomena are the large flexible deviations and vibrations exaggerated because of implementation of deployment, folding, maneuvers, etc. Planning of motion strategies for minimization and passive and active damping of flexible deviations and vibrations is of crucial importance. Solution of these problems is a challenging realm for scientific investigations [1]. The concepts developed are based on the modal and eigenvale analysis of the flexible systems. This approach considers small deviations around the equilibrium position and very often it is inapplicable for extremely large flexible systems. Large flexible deflections and vibrations are subject of the contemporary multibody system investigations [2]. The present paper regards the problems of path planning and optimization of maneuver motion of large space flexible structures as long booms and tethered satellites. The approach is based on the multibody system methodology for dynamics simulation and forward analysis. The optimization problem is defined as nonlinear programming problem. Polynomial approximation of the input motion is used, its coefficient being changed using optimization techniques. The principle for minimization of the total energy of the deformable system is applied in an algorithm for planning of controlled motion and suppression of flexible deviations. Admissible velocities of the maneuvers are estimated. The process of exaggeration of high order vibrations is analyzed. Examples of maneuver implementation of extremely flexible structures and damping of vibrations are presented.

References [1] D. Izzo, L. Pettazzi and C. Valente, A comparison between Models Representing Flexible Spacecrafts. 6th International Conference on Dynamics and Control of Systems and Structures in Space, Riomaggiore, Italy, 18-22 July, 2004. [2] W. Schiehlen, Multibody system dynamics: Roots and perspectives. Multibody System Dynamics, 1, 149-188, 1997.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Free plain motion of flexible beams in MBS – A comparison of models R. Zander∗ , H. Ulbrich∗ ∗ Institute of Applied Mechanics · Department of Mechanical Engineering Technical University of Munich · Boltzmannstr. 15 · 85 748 Garching · Germany [zander,ulbrich]@amm.mw.tu-muenchen.de

ABSTRACT In multibody systems (MBS) structural elasticities are usually described using one floating frame of reference for each body. This leads to a compact mathematical formulation for the described physics, but it is restricted to small deflections relative to the floating frame. To overcome this limit, we combine the MBS ideas with concepts of the finite element (FE) method. Single elements are treated in a formulation of hybrid MBS giving exact rigid body movements for single elements. The compact form of equations is maintained and independent discretisations for longitudinal and transversal deformations are possible. To permit the assembly of several elements to one structure, the equations of motion are transformed to a coordinate set motivated by FE, whereas global positions are used as nodal coordinates instead of displacements. This approach allows for large translations and geometrically large deformations of the entire structure. The ‘absolute nodal coordinate formulation’ (ANCF) [2] also uses global coordinates and provides exact rigid body motions for single elements, thus allowing for large structural deflections. Instead of rotations, the derivatives of coordinates are used. In contrast to our approach, the ANCF uses equal discretisations for longitudinal and bending deformations in the case of a one-dimensional continuum and gives a constant mass matrix. Based on two examples, both models are compared with respect to their relative accuracy and numerical efficiency. As a first model a simple crank-shaft with a highly flexible long crank is investigated. To minimise the influence of the time discretisation on the results, an error-controlled time integration is used for these simulations. For fine spatial discretisations both models converge to one solution which is used as a reference for the evaluation of the model. The main focus lays on the number of degrees of freedom and the computational effort that is required for each method to obtain a desired accuracy. Moreover, the relative error using equal numbers of degrees of freedom is studied. The second example – an elastic rocking rod – is built by a highly flexible and initially straight beam bouncing on two point obstacles. Impacts and constraints are treated following modern methods for non-smooth rigid MBS dynamics [1] as an extention to flexible systems. In this context, the computational effort for contactsolving together with the robustness of the formulations are investigated.

References [1] F. Pfeiffer, M. F¨org, and Heinz Ulbrich. Numerical aspects of non-smooth multibody dynamics. Computer Methods in Applied Mechanics and Engineering, 2005. In Press, Available online 10 October 2005. [2] A. A. Shabana. Computer implementation of the absolute nodal coordinate formulation for flexible multibody dynamics. Nonlinear Dynamics, 16:293–306, 1998.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Numerical computation of Non Linear Modes of elastic structures Arquier R.∗ , Cochelin B.† Laboratoire de mecanique et d’acoustique CNRS UPR7051 31 chemin Joseph-Aiguier 13402 Marseille cedex 20 FRANCE ∗ [email protected] † [email protected]

ABSTRACT This paper concerns the computation of nonlinear modes of elastic structures under large displacements. We present a numerical method that we have implemented in a general purpose finite element code. Bifurcation of modes will be also addressed. We begin by introducing a general simple quadratic framework that is suitable for most elastic models (beam, plate, shell) and most classical finite elements. We define the non linear modes as two dimensional invariants of the phase space which are tangent to the eigenspaces of the associated linear system [1]. Theses invariant subsets are determined by making continuation of one dimensional families of periodic orbits. The periodic solutions are computed using the periodic orbit approach [2]. We use the exact energy-conserving Simo scheme [3] to time-discretise the periodic orbits. We do not use the classical shooting method to compute the periodic orbits but another one which consist to write the governing equation at each time step in a whole system. This lead to a large system of algebraic equations containing the displacement of the degres of freedom at each time step. The nonlinear modes (or their approximations) are obtained by making the continuation with respect to adequate parameters. We use the asymptotic-numerical method [4] for this purpose, since it is particularly efficient for such difficult problems with quite complex bifurcation diagrams.

References [1] S.W. Shaw and C. Pierre, Non-linear normal modes and invariant manifolds. Journal of Sound and Vibration, 150(1), pp. 170-173, 1991. [2] R. Seydel, Practical Bifurcation and Stability Analysis, from equilibrium to chaos. SpringerVerlag, second edition , 1994 [3] J. C. Simo and N. Tarnow, The discrete energy-momentum method. Conserving algorithms for nonlinear elastodynamics. Z angew Math Phys, 43,pp. 757-792, 1992 [4] B. Cochelin, N. Damil and M. Potier-Ferry, Asymptotic numerical methods and Pade approximants for non-linear elastic structures. International journal for numerical methods in engineering, 37, pp 1187-1213, 1994

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Nonlinear modes: amplitude-phase formulation and bifurcation analysis Sergio Bellizzi, Robert Bouc Laboratoire de M´ecanique et Acoustique - CNRS 31 chemin Joseph Aiguier 13402 Marseille France {bellizzi,bouc}@lma.cnrs-mrs.fr

ABSTRACT Nonlinear Modes (NMs) are efficient tools for analysing the behaviour of dynamical mechanical systems[1]. The objective of this contribution is to show how this concept can be used to characterize periodic orbits and limit cycles. Following Shaw and Pierre[2] the concept of NMs is introduced here in the framework of the invariant manifold theory for dynamical systems. A NM is defined in terms of amplitude, phase, frequency, damping coefficient and mode shape with the distinctive feature that the last three quantities are amplitude and phase dependent. An amplitude-phase transformation is performed to give as well the time evolution of the NM motion (through the two first order differential equations governing the amplitude and phase variables) as the geometry of the invariant manifold. The conservative case was considered in [4]. The formulation is extended here to autonomous mechanical systems including gyroscopic and/or nonlinear damping terms. Our approach differs of that in [3] where the amplitude-phase transformation is based on the frequency of the linearized system. Bifurcation analysis, existence and stability of periodic orbits on the associated invariant manifold can be studied from the differential equations governing the amplitude and phase variables.The procedure is illustrated on a 2 DOF van der Pol mechanical system.

References [1] A.F. Vakakis, Nonlinear normal modes (NNMs) and their applications in vibration theory: an overview. Mechanical Systems and Signal Processing, 11(1), 3–22, 1997. [2] S.W. Shaw, C. Pierre, Normal modes for nonlinear vibratory systems. Journal of Sound and Vibration, 164(1), 85–124, 1993. [3] E. Pesheck, C. Pierre, S.W. Shaw, A new Galerkin- based approach for accurate non-linear normal modes through invariant manifolds. Journal of Sound and Vibration, 249(5), 971–993, 2002. [4] S. Bellizzi, R. Bouc, A new formulation for the existence and calculation of nonlinear normal modes. Journal of Sound and Vibration, 287(3), 545–569, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Multi-Modal Non-linear Free Vibration of Thin Isotropic Circular Plates M. Haterbouch1 , P. Ribeiro2 , R. Benamar3 1 LMCS,

Facult´e des Sciences et Techniques d’Errachidia BP 509 Boutalamine, Errachidia, Morocco [email protected]

2

IDMEC/DEMEGI, Faculdade de Engenharia, Universidade do Porto Rua Dr. Roberto Frias, 4200-465 Porto, Portugal [email protected] 3

GDNLS/LERSIM, Ecole Mohammadia d’Ing´enieurs BP 765 Agdal, Rabat, Morocco [email protected] ABSTRACT

The large-amplitude periodic free axisymmetric vibration of clamped immovable thin isotropic circular plates is investigated using the energy method and a multimode approach. Von K´arm´an’s non-linear strain-displacement relationships are employed and the middle plane in-plane displacements are included in the model. The equations of motion are derived by applying Lagrange’s equations. Using the harmonic balance method (HBM), the equations of motion are converted into a non-linear algebraic form and are solved by a continuation method. Interesting results for the plate’s fundamental mode shape such as the occurrence of internal resonance, resulting in a complicated backbone curve, and the variation of the mode shape with the amplitude of vibration and during the period of vibration are obtained.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Computational approaches to prediction of damping behavior of nanoparticle-reinforced coatings and foamy structures Maksim V. Kireitseu†, Geoffrey R. Tomlinson† †

Department of Mechanical Engineering, the University of Sheffield Address: Mappin Street, Sheffield S1 3JD, the United Kingdom [email protected]

ABSTRACT This paper concerns advanced computational engineering approach based on finite element modeling and fundamental physical phenomena of energy dissipation mechanisms related to vibration damping [1]. Nanoparticle/tube-reinforced composite materials are relatively new class of engineering materials and their vibration damping application is commonly unknown from both computational and experimental sides [2]. The novel concept of nanoparticle-based damping technology shows that a molecule-level mechanism can considerably enhance vibration damping and dynamic of aerospace components (fan blades) via enhanced energy dissipation because of large surface-to-volume aspects in nanoparticle-reinforced composite material, large damping energy sources for friction and slipstick motion at interfaces of matrix and nanoparticle. Therefore, to add some knowledge our group is working on computational characterization approach and modeling technique that describe relationships between structure and damping/dynamic properties of the materials, formalize the set of structural mechanical approaches to build a bridge between macro and nanoscales. Structural micro to nanomechanical approach has been developed to predict damping (dynamic) behavior of carbonnanotube-reinforced composite material. The model is based on ‘‘stick-slip’’ frictional motion to address the damping characteristics of SWNT-reinforced composite material. It is worth noting that SWNT can be represented as a shell hollow frame-like structure with a simple nanoscale damping spring characteristics. Thus the developed model can be assembled into entire engineering workbench. A comparison of available modeling strategies is presented. Carbon nanotube-reinforced material is particularly illustrated via advanced numerical codes, using a hollow shell representation of the individual nanotubes. Comparing to the FEM, the new technique may introduce further reduction of both computer time and storage requirement. Thus results of the project will potentially create fundamental basis for investigation and development of 3-D reinforced composite structures with high nanoscale structures volume content, using nano-scale reinforcement architecture to reduce component weight and dimension.

References [1] M. Kireitseu, V. Kompiš, H. Altenbach, L. Bochkareva, D. Hui, S. Eremeev, Continuum mechanics approach and computational simulations of submicrocrystalline and nanoscale composite materials. Fullerenes, Nanotubes and Carbon Nanostructures, Marcel Dekker Press, 13(4), 313-329, 2005. [2] M. Kireitseu, G. Tomlinson, G. Rongong and D. Hui (INVITED), Next generation damping systems: nanoparticle reinforcement design concepts and computational modeling tools, in Proceedings of ICCE-12 Twelve Int.-l Conference on Composites/Nano Engineering, ed. by D. Hui, in Tenerife, Canary Islands, Spain, 280-284, August 1-6, 2005 (WEB: www.uno.edu/~engr/composite/)

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Buckling under conservative and nonconservative load Attila Kocsis , and Gy¨orgy K´arolyi† Department of Structural Mechanics Budapest University of Technology and Economics M˝uegyetem rkp. 3., H-1111 Budapest, Hungary [email protected] † Department of Structural Mechanics, and Center for Applied Mathematics and Computational Physics Budapest University of Technology and Economics M˝uegyetem rkp. 3., H-1111 Budapest, Hungary [email protected]

ABSTRACT Since the first invention of chaos theory, it has been found to play a very important role in many different fields, ranging from physics through biology to engineering, among others. We deal with a phenomenon called spatial chaos which is a special form of spatial complexity, when the governing equations are reminiscent of a chaotic dynamical system, but the role of time is taken over by a spatial coordinate (e.g. arc-length). Many examples of spatial chaos have been addressed recently in general mathematical studies, in fluid dynamics, in the case of buckling of elastic rods or linkages. It also plays an important part in biology where biological filaments – like DNA, (bio)polymers, or tendrils – may exhibit complicated spatial patterns. It has been shown that the elastic linkage provides both a mathematical discretization of Euler’s buckling problem and a mechanical discretization of a continuous rod.The discrete problem is in the state of spatial chaos: it has much more complicated equilibrium shapes than has the continuous Euler-problem. The reason of this is that the governing equations of the continuous problem coincide with a non-chaotic initial value problem, the mathematical pendulum, while the equations of the linkage are the same as the well-known chaotic map, the standard map. We deal with the buckling problem of a cantilever under a quite general set of loads which can be either conservative or non-conservative. We assume that the material behavior can be nonlinear and the rod can be non-prismatic. Using a discrete model, an elastic linkage we show that the static stability is related to a chaotic map, which is conservative both in case of conservative or non-conservative loads. It proves that conservative spatial chaos is not a unique feature of conservative buckling problems. We detail some special examples and construct their global bifurcation diagrams.

References [1] G. Domokos and P. Holmes: Euler’s problem, Euler’s method, and the standard map; or, the discrete charm of buckling. Journal of Nonlinear Science 3 (1993) 109–151. [2] A. Kocsis, Gy. K´arolyi: Buckling under nonconservative load: conservative spatial chaos. Periodica Polytechnica 49/2 (2006) 86–101.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

630

Bifurcation of Periodic Solutions in the Two-Degree-of-Freedom System With Clearances N. Kranjcevic, M. Stegic, N. Vrankovic

Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb P.O. Box 102, 10002 Zagreb, Croatia {nkranjce, mstegic, nvrankov}@fsb.hr

ABSTRACT Clearances exist in many mechanical systems either by design or due to manufacturing tolerances and wear. The characteristics of such systems include abrupt variation of stiffness usually approximated as piecewise linear. It is well-known that the stiffness discontinuity can be a source of the instabilities in the dynamic behavior of the system. In this paper, periodic solutions of the two-degree-of-freedom mechanical system with clearances subjected to periodic excitations are studied. The periodic solution may lose its stability via a static bifurcation (cyclic-fold or flip), or via a Neimark bifurcation. The bifurcation depends on the eigenvalues of the Jacobian matrix of the nonlinear vector field. By applying Hurwitz criterion on the Jacobian matrix, the bifurcation can be classified. For the analyzed dynamical system with clearances, a Neimark bifurcation occurs. The analytical results are compared with the numerical solutions obtained by the finite element in time method. The bifurcation analysis in time finite element procedure is performed by using Poincaré map. The system is assumed to be controlled by the excitation frequency (codimension-one bifurcation). Imposing small increments in the excitation frequency, the critical point is found from which the Neimark bifurcation takes place. The qualitatively different phase portraits, prior to and after the critical point, confirm the Neimark bifurcation.

References [1] J.M.T. Thompson and H.B. Stewart, Nonlinear dynamics and chaos: Geometrical methods for engineers and scientists. John Wiley, Chicester, UK, 1986. [2] A.H. Nayfeh and B. Balachandran, Applied nonlinear dynamics. John Wiley, New York, 1995. [3] N. Kranjcevic, M. Stegic and N. Vrankovic, Stability and bifurcation analysis of a twodegree-of-freedom system with clearances. K.J. Bathe ed. Computational Fluid and Solid Mechanics 2005, Elsevier, Boston, USA, 297-301, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

631

Perturbation Method for Analysis of Strongly Non-Linear Free Vibration of Beams R. Lewandowski* *

Poznan University of Technology, Institute of Structural Engineering 60-695 Poznan, ul. Piotrowo 5, Poland [email protected]

ABSTRACT The perturbation method is one of the oldest methods used to analyse the dynamic behaviour of nonlinear systems. There are many versions of the perturbation method but most of them apply to weakly non-linear cases only. To overcome this limitation, new techniques have been proposed. Cheung et al. [1], Lim et al. [2] and Hu [3] proposed some modifications which make possible the analysis of strongly non-linear systems but with one degree of freedom only. In this paper, the possibility of application of the perturbation method to the dynamic analysis of strongly non-linear free vibrations of beams is discussed. Beams are treated as geometrically nonlinear systems. The von Karman theory is used to describe non-linear effects. The finite element method is adopted to discetize the beam and the motion equation is written in a matrix form.

The first order perturbation equation is solved and the obtained solution is compared with the solution found with the help of the harmonic balance method which is widely used and applicable to the analysis of strongly non-linear dynamic systems. It was proved that both solutions are almost identical and differences are negligibly small. On the basis of similarities discovered in the both solutions, it was concluded that, for the value of small parameter ε = 1 , the solution obtained by means of the perturbation method is almost identical to the one given by the harmonic balance method. The results of typical calculations confirm these observations. Finally, it is concluded that the perturbation solution has also enough accuracy when the strongly non-linear systems are considered. It is believed that the reason of success of the presented perturbation method comes from a feedback which is taken into account when the tangent stifness matrix is introduced. The numerical procedure enabling determination of backbone curves is also briefly described. Theoretical results are supplemented by a description of the results of typical calculations.

References [1] Y.K. Cheung, S,H, Chen, L.S. Lau, A modified Lindsteadt-Poincare method for certain strongly non-linear oscillators, International Journal of Non-Linear Mechanics, 26, 367-378, 1991. [2] C.W. Lim, B.S. Wu, A modified Mickens procedure for certain non-linear oscillators, Journal of Sound and Vibration, 257, 202-206, 2002. [3] H. Hu, A classical perturbation technique which is valid for large parameters, Journal of Sound and Vibration, 269, 409-412, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

632

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III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

633

Resonant Non-Linear Dynamic Responses of Horizontal Cables via Kinematically Non-Condensed/Condensed Modeling Narakorn Srinil and Giuseppe Rega Department of Structural and Geotechnical Engineering, University of Rome ‘La Sapienza’, via A. Gramsci, 53 Rome 00197, Italy [email protected], [email protected]

ABSTRACT

This paper focuses on a comparison of cable non-linear dynamic responses obtained with the kinematically non-condensed and condensed modeling. Planar non-linear interactions involving simultaneous primary external and internal resonance in horizontal suspended cables are analytically investigated. 1:1 or 2:1 internal resonance is considered. The governing partial-differential non-linear equations of motion of the non-condensed cable model account for the effects of dynamic extensibility, i.e., dynamic tension spatio-temporal variation, and capture the non-linear coupling and contributions of longitudinal/transversal modal displacements. On the contrary, in the condensed cable model, a single integro-partial-differential equation of motion is obtained by neglecting the longitudinal inertia according to a quasi-static stretching assumption of cable in motion. This entails linking the longitudinal displacement to the transversal one and considering a space-independent dynamic tension. This simplified model is typically considered in the literature involving cable nonlinear dynamics. Based on a multi-dimensional Galerkin-based discretization and a second-order multiple scales approach accounting for higher-order non-linear effects and resonant/non-resonant modal contributions, the ensuing dynamic responses and their stability are evaluated by means of force- and frequency-response diagrams with stability analyses. Moreover, the corresponding spacetime non-linear coupled configurations and dynamic tension distributions are analyzed. The numerical explorations highlight that, depending on cable elasto-geometric properties, internal resonance condition and system control parameters, the condensed model may lead to quantitative and/or qualitative discrepancies in the non-linear dynamic responses, with respect to the non-condensed model. The results allow us to point out such meaningful effects of disregarding the system longitudinal dynamics via the kinematic condensation procedure, and to identify cases where the parametric investigation has to be pursued with the more accurate non-condensed model.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

634

Computing Effective Properties of Nonlinear Structures Exposed to Strong High-Frequency Loading at Multiple Frequencies Jon Juel Thomsen Technical University of Denmark MEK - Solid Mechanics Nils Koppels Alle, Building 404 DK-2800 Kgs. Lyngby, Denmark [email protected]

ABSTRACT Effects of strong high-frequency excitation at multiple frequencies (multi-HFE) are analyzed for a class of generally nonlinear systems. The effects are illustrated for a simple pendulum system with a vibrating support, and for a parametrically excited flexible beam. For the latter, theoretical predictions are supported by preliminary experimental results. The main effect of strong multi-HFE is to change the effective or apparent stiffness in a manner similar to that of mono-HFE (e.g. [1-5]), provided the HFE frequencies are well separated and non- resonant. Then the change in effective stiffness is proportional to the mean-square velocity of the excitation velocities, and the corresponding changes in equilibriums, equilibrium stability, and natural frequencies can be computed as for the mono-HFE case. When there are two or more close excitation frequencies, an additional contribution of slowly oscillating stiffness appears. This may cause strong parametrical resonance at conditions that might not appear obvious, i.e. when the difference in two HFE-frequencies is near twice an effective natural frequency of the system, which due to the HFE itself is shifted away from the natural frequency without HFE. Also, it is shown that strong multi-HFE can stabilize otherwise unstable equilibriums, but only if the frequencies are well separated; thus continuous broadband and random HFE does not have a stabilizing effect paralleling that of mono-HFE, or multi-HFE with non-close frequencies. The general results may be used to investigate general effects, or as a short cut to calculate effective properties for specific systems, or to calculate averaged equations of motion that may be much faster to simulate numerically.

References [1] I.I. Blekhman, Vibrational Mechanics - Nonlinear Dynamic Effects, General Approach, Applications. World Scientific, Singapore, 2000. [2] A. Fidlin, Nonlinear Oscillations in Mechanical Engineering. Springer-Verlag, Berlin Heidelberg, 2005. [3] J.J. Thomsen, Vibrations and Stability: Advanced Theory, Analysis, and Tools. Springer-Verlag, Berlin Heidelberg, 2003. [4] J.J. Thomsen, Some general effects of strong high-frequency excitation: stiffening, biasing, and smoothening. Journal of Sound and Vibration, 253, 807-831, 2002. [5] J.J. Thomsen, Slow high-frequency effects in mechanics: problems, solutions, potentials. International Journal of Bifurcation and Chaos, 15, 2799-2818, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

635

Implementation of a Vapour Cavitation into Computational Models of Rotors Supported by Long Journal Bearings Jaroslav ZapomČl VSB-Technical University of Ostrava Centre of Inteligent Systems and Structures - branch of IT ASCR in Ostrava 17.listopadu 15, Ostrava-Poruba, 708 33, Czech Republic [email protected]

ABSTRACT Dynamical behaviour of rotors supported by hydrodynamical bearings is considerably influenced by presence of a gas phase in the lubricating oil. The reason for its occurance is a vapour cavitation. The observations show that pressure of the two-phase medium in cavitated regions remains approximately constant [1]. At long bearings ( most of the bearings ) the pressure gradient in the axial direction is insignificant and the pressure distribution in bearing gap can be described by a simplified Reynolds equation. To be satisfied the continuity of flow and incompressibility of oil the pressure gradient must be zero at the entrance into the cavitated region. To determine edges of the cavitated area a new algorithm has been developed. First the pressure distribution between two nodes corresponding to two adjacent oil inlets into the bearing is calculated. If the minimum pressure drops below the critical value, a vapour cavitation occurs. Then the border nodes are successively chosen from the node of the pressure minimum in the direction opposite to the rotor rotation and the Reynolds equation is solved for the boundary conditions : pressure at the oil inlet, zero pressure gradient at the chosen border node. This process continues until the pressure in the border node is equal to the cavitation one. In the next step the border nodes are chosen from the node of the pressure minimum in the direction of the rotor rotation and the Reynolds equation is calculated for the boundary conditions : pressure in the cavitated area, pressure at the oil inlet. The border node at which the difference between the flow rate and the flow rate through the inlet edge of the cavitation area is minimum is considered to be the outlet edge of the cavitated region. This procedure was implemented into the algorithms for investigation of the transient response of rotors excited by force and kinematic effects ( rotor unbalance, earthquake excitation, etc. ). A modified Newmark method has been chosen [2] for solution of the equation of motion. The modification consists in continuous linearization of the vector of hydraulical forces in the neighbourhood of the current rotor position.

References [1] F.Y. Zeidan, J.M. Vance, Cavitation regimes in squeeze film dampers and their effect on the pressure distribution. STLE Tribology Transactions, 33, 447-453,1990 [2] J.Zapomel, E.Malenovsky, Approaches to numerical investigation of the character and stabilityof the forced and self-excited vibration oof flexible rotors with non-linear supports, IMechE Conference Transactions, 7th International Conference on Vibrations in Rotating Machinery, University of Nottingham, 691-700, 2000.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

636

Multi-criteria optimizations and robustness estimations for problems of crashworthiness, structural dynamics, and acoustics of car bodies. Fabian M. E. Duddeck Queen Mary College, London University Department of Engineering Mile End Road, London E1 4NS, UK [email protected]

ABSTRACT Recent work on optimization for crashworthiness have shown that evolutionary algorithms, e.g. [1] perform superior compared to other optimization strategies when meta-modelling fail in representing the physics of the problems [2]. This is valid especially for simultaneous optimization of several disciplines, i.e. for multi-disciplinary optimization (MDO). These problems are normally driven by the most non-linear problem (e.g. a frontal impact). The high non-linearity of the optimization problems may lead to optimal designs which are loosing their optimality when the design variables are altered only slightly. Unfortunately, this will happen in all real industrial applications – either by additional constraints, by manufacturing irregularities, or by uncertainties inherent to the system. Thus, a robustness analysis should be integrated in the overall optimization scheme. For many of the industrial-sized problems, computing time is not a negligible criterion for the successful implementation of the optimization into the product development process [3]. Evolutionary algorithms require a high amount of CPU time; an additional robustness analysis is thus problematic. Nevertheless, multi-criteria optimization problems have recently been solved successfully for acoustics, structural dynamics and even for crashworthiness. Because a real robustness analysis is often too time demanding, a first estimate of the robustness of the chosen design on the Pareto front can be obtained by regarding the neighbouring points on the front. The variance of the design variables of these points along the Pareto front indicates sensitivities and therefore robustness. This will be demonstrated in first examples taken from industrial studies on acoustics, structural dynamics, and crashworthiness.

References [1] T. Bäck, Evolutionary Algorithms in Theory and Practice. Oxford University Press, New York, 1996. [2] F. Duddeck, K. Volz, Evaluation of optimization algorithms for crash and NVH problems. In: K.J. Bathe (ed.): Computational Fluid and Solid Mech. 2005, Elsevier, 2005. [3] J. Lescheticky, F. Duddeck, L. Willmes, S. Girona, Efficient Product Development of Car Bodies Using Multi-disciplinary Optimization. In: Numerical Analysis and Simulation, VDI Conference, Würzburg, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

637

Process Robustness in Sheet Metal Forming by an Integrated Engineering Strategy K. Grossenbacher*, F. Duddeck†, P. Hora#, M. Ganser* *

BMW Group, Product and Process Development Knorrstrasse 147, D-80788 Munich, Germany [email protected], [email protected] †

#

Queen Mary College, London University Mile End Road, London E1 4NS, UK [email protected]

Institute for Virtual Production, ETH Zurich (Center) CLA F 9 Tannenstrasse 3, CH-8092 Zurich [email protected]

ABSTRACT In the modern automotive industry the development of car bodies depends considerably on the use of computer-aided tools. Herewith one can meet the challenges of rising product complexity and growing number of variants. About fifteen years ago simulation of sheet metal forming was used for the first time in industrial application. However, the calculation times were long and the quality of the results was often unsatisfactory. Today, enabled by improved material models and new numerical methods, those simulations have become essential for the evaluation of press-tools before they are manufactured. Due to the high number of varying influences on the production process of sheet metals, the resulting quality of the parts is not always stable. In most cases, these variances are lying in predefined limits of tolerance. Otherwise additional efforts and costs for testing the parts and for reworking them are required leading to higher reject costs in total. By the means of existing highly qualitative methods for numerical simulation combined with standardized statistical methods one can identify these varying influences, their interconnection and effects on car body parts. On the basis of such an analysis appropriate optimization algorithms will lead to an improved overall part quality along with higher robustness. In this paper, the realization of an integrated engineering strategy as mentioned above within the forming department of the BMW Group, the combination of corresponding engineering tools, and their reasonable cooperation in a planning process will be described. The essential steps of these processes and the methods and tools used are presented illustrated by industrial-sized examples.

References [1]

E.Dietrich, A.Schulze, Statistische Verfahren zur Maschinen- und Prozessqualifikation. Carl Hanser Verlag München Wien, 5. Auflage, 2005

[2]

J. Meinhardt, W. Volk, H. Schmidt, “Virtuelle Prozessentwicklung von Presswerk-zeugen im industriellen Umfeld”, in E. Doege (Editor), Umformtechnik – Erschließung wirtschaftlicher und technologischer Potenziale, 17. Umformtechnisches Kolloquium Hannover, 271-284 (2002).

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

638

Topology Optimization of Robots Using Mapping Techniques Michael R. Hansen*, Torben O. Andersen†, John M. Hansen††, Ole Ø. Mouritsen* *

Institute of Mechanical Engineering, Aalborg University Pontoppidanstræde 101, 9220 Aalborg E, Denmark [email protected], [email protected] †

Institute of Energy Technology, Aalborg University Pontoppidanstræde 105, 9220 Aalborg E, Denmark [email protected]

†† Man B&W Diesel Teglholmsgade 41, 2450 Copenhagen SV, Denmark [email protected]

ABSTRACT This work is an extension of previous work that utilizes mapping techniques to handle the discrete nature of robot design. The developed design approach utilizes dimensionless parameters that map data bases of commercially available components. The important properties are derived by applying interpolation on the data bases. The handling of both distinct components sub-groups as well as a method for the gradual conversion of the dimen-sionless variables into integers has already been presented. In this work the approach is further developed by introducing a parameter that maps a set of possible system topologies. The possible topologies include three different spatial mechanisms and the dynamic time domain simulation model that is used to evaluate a design is developed so that it can analyze each of the possible topologies. The mapping of the robot topologies has been done in such a way that the performance of a design varies continuously despite a change in topology during the optimization. The design criteria include costs of drives and structural components, tool point precision, fatigue in welded details, over heating and stalling of the motors and gears as well time of operation. A number of examples are given.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

639

Optimization Strategies for Highly Non-Linear FE-Applications as Crashworthiness Applications ¨ 1 , M. van den Hove2 , and B. Mlekusch2 ¨ H. Mullersch on 1

DYNAmore GmbH - www.dynamore.de [email protected]

2 AUDI AG - www.audi.de {marcel.vandenhove,bernd.mlekusch}@audi.de

ABSTRACT The purpose of this paper is to explore some interesting aspects of optimization for crashworthiness occupant safety applications and to propose optimization strategies for highly nonlinear problems. In the first part of the paper different optimization strategies are discussed and pros and cons are compared. In addition, a methodology to get a reliable surrogate model using neural networks is introduced. The surrogate model (Meta-Model or Response Surface Model) approximates the relationship between design parameters and a physical response and can be used to visualize and explore the design space. In the second part the application of the Successive Response Surface Scheme (SRSM) for the optimization of an adaptive restraint system is conducted. For this, several front crash load cases are considered. This is performed using LS-OPT (Stander et al. [11]) as optimization software and PAM-Crash as solver for the finite element occupant safety simulations. The procedure of generating an advanced meta-model to get an approximation of the global design space using neural networks is demonstrated for this example. Furthermore, the visualization of multi-dimensional meta-models in two- and three-dimensional design space is illustrated

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

640

Local and Global Searches of Approximate Optimal Designs of Regular Frames Makoto Ohsaki∗ ∗ Department

of Architecture and Architectural Engineering, Kyoto University Kyotodaigaku-Katsura, Nishikyo, Kyoto 615-8540, Japan [email protected] ABSTRACT

In this paper, methods of local and global searches of approximate optimal designs are presented for regular frames subjected to static loads. Constraints are given for stresses and displacements. Recently, it has been pointed out that there may exist many fully stressed frames with almost the same total structural volume [1]. Therefore, obtaining only one solution will not be enough for practical purpose, where several solutions satisfying stress constraints should be compared in view of other performance measures such as eigenfrequencies and requirements in construction process. Furthermore, the objective function may not be strictly minimized; i.e., it will be helpful for the designers if several approximate solutions with different distributions of cross-sectional areas are obtained. Jog and Haber [2] suggested that the nonuniqueness of the solution to a compliance optimization problem can be detected by the singular values of the matrix defined as the gradients of the equivalent force vector with respect to the design variables. However, they did not show how the singular vectors are used for finding approximate optimal solutions. In this paper, we first demonstrate nonuniquesness of the optimal solution by a continuous beam with periodic boundary conditions for uniform loads. The optimal solutions are locally searched from a solution found from an arbitrary generated initial solution. The search direction is determined by singular value decomposition of the stiffness matrix with respect to the cross-sectional areas or the sensitivity matrix of the constraints. Approximate optimal solutions are next globally and consecutively found so as to maximize the distance from the already found solutions under stress and displacement constraints. The distance between the solutions is defined by the Euclidean norm of the differences in the cross-sectional areas. The constraint is given for the total structural volume, where the specified upper bound is slightly larger than the objective value of an optimal solution. The effectiveness of the proposed methods is demonstrated in application to a 10 × 10 and 3 × 27 plane frames. It is shown that approximate optimal solutions have been successfully found using the singular vectors of the stiffness matrix with respect to the crosssectional areas. However, accuracy of the solutions can be improved using the singular vectors of the sensitivity matrix of the constraints.

References [1] K. M. Mueller, M. Liu and S. A. Burns, Fully stresses design of frame structures and multiple load paths. J. Structural Engineering, 128(6), pp. 806–814, 2002. [2] C. S. Jog and R. B. Haber, Stability of finite element models for distributed-parameter optimization and topology design, Comp. Meth. Appl. Mech. Eng., 130, pp. 203–226, 1996.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

641

Optimisation of car body parts regarding equivalent radiation power using a genetic algorithm and morphing J. Reger*, T. Schneider*, C. Ehlert* * P+Z Engineering München, Germany [email protected] [email protected] [email protected]

ABSTRACT The continuing demand for light weight car body structures and the increasing awareness of acoustic comfort is a conflict of objectives. Excitations introduced by the driveline or the drivetrain can not be avoided. The body structure will radiate sound to the passenger compartment. Consequently, reducing the radiated sound power in car body structures develops towards a main topic in the car body development process. The common approach to stiffen the body structure parts by introducing beads using an empirical approach is not sufficient for reducing the sound pressure level. Stiffening, that decreases the radiated sound power at certain frequencies, often leads to increased radiation at other frequencies. To improve the overall acoustic behaviour the complete frequency response of the structure has to be taken into account. Amplitudes of radiated sound power have to be decreased over the complete frequency range of interest. State-of-the-Art optimisation methods can help to find solutions. Presently in this context optimisation is normally used with shell thickness parameters to increase eigenvalues or to decrease frequency dependent accelerations of dedicated points in the structure. However, the gradient methods used are inadequate for the optimisation of body structures concerning equivalent radiation power. The reasons for the poor performance of gradient methods are highly nonlinear objective functions when minimising the equivalent radiation power. To find a topography of the body structure which radiates significantly less sound a global optimisation has to be run. The introduced method changes the topology of the structure by morphing the FE-Mesh. A genetic algorithm with discrete variable representation is applied for global optimisation. In order to perform successful optimizations with genetic algorithms a large number of function evaluations is required. To reduce the overall computation time the use of substructuring and parallelisation is necessary. The acoustic behaviour of the complete car body is taken into account.

The example shows a successful optimization of the topography of a car body part. The equivalent radiation power can be decreased significantly over the complete frequency range of interest.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Grid-Based Topology Optimization of Rigid Body Mechanisms Using Different Problem Formulations Kai Sedlaczek, Peter Eberhard Institute of Engineering and Computational Mechanics University of Stuttgart Pfaffenwaldring 9, 70569 Stuttgart, Germany {sedlaczek,eberhard}@itm.uni-stuttgart.de

ABSTRACT In the design process of rigid body mechanisms two main development steps can be distinguished, namely type (topology) and dimensional synthesis. Whereas much work has already been done on solving the problem of dimensional synthesis, optimization based approaches to topology design of rigid body mechanisms are rare. Unlike solving the discrete combinatorial problem of optimal mechanism topology by means of genetic algorithms [1], we investigate in this work a ground structure approach similar to [2] but based solely on rigid bars. A relaxed formulation of the kinematic constraint equations allows an almost straightforward kinematic analysis despite the over-determined system of equations due to redundant bars in the ground structure. Similar to cross sectional areas in topology optimization of truss structures, the bars in the ground structure are parameterized by continuous design variables that can have intermediate values between 0 and 1. This continuous description allows a solution with efficient gradient-based optimization methods. However, the problem is of discrete (binary) nature and intermediate values are physically meaningless so that appropriate problem formulations must be found in order obtain a 0-1 design. This work investigates different problem formulations and solution techniques and their ability to solve the intrinsically discrete problem of mechanism topology optimization. All presented formulations are using the continuously parameterized truss-like structure with rigid bars, but they are based on different (separation) constraints in order to achieve a 0-1 design. This includes a simple quadratic penalization as well as the power-law (SIMP) method for the solution of the path generation and output maximization problem. The functionality, the advantages and drawbacks of the grid structure approach with respect to the problem of rigid body mechanism design are discussed and illustrated by example problems.

References [1] K. Sedlaczek, T. Gaugele, P. Eberhard, Topology Optimized Synthesis of Planar Kinematic Rigid Body Mechanisms. In: J.M. Goicolea, J. Cuadrado, J.C. Garcia Orden (Eds.), Advances in Computational Multibody Dynamics. Proceedings of the ECCOMAS Thematic Conference on Multibody Dynamics, Madrid, Spain, 2005. [2] A. Kawamoto, M.P. Bendsøe, O. Sigmund. Articulated Mechanism Design with a Degree of Freedom Constraint. International Journal for Numerical Methods in Engineering, 61, 1520-1545, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Combining Optimization and Robust Engineering Methods in the Engineering Product Design Process Holger Wenzel∗ ∗ Engineous

Software GmbH Lichtenbergstrasse 8 85748 Garching [email protected]

ABSTRACT Many engineers that work in the simulation based product creation process are faced with a multitude of different and often conflicting demands: The product needs a better performance, it should be more reliable, yet less expensive and on top of all, the design time is shortened dramatically. These challenges have driven the use of process integration and design optimization (PIDO) tools into all stages of product development. This trend has been fueled further by the increase of computational power that spreads the application of automatic design improvement methods even to areas that are extremely expensive to simulate. One of the inherent challenges of the use of optimization algorithms in the design of industrial products is the fact that they tend to drive the design to extreme points, where even very small changes in the setup can cause the product to fail. This effect can be counteracted by the application of robust engineering methods, which can analyze and automatically improve the robustness and reliability of the product. Nonetheless, a systematic coupling of optimization and robust engineering methods is a relatively new and still emerging field. This work gives an overview of four recent examples from the aerospace and automotive industry: a blade-disc connection of a gas turbine, the robust optimization of the idle shake of a passenger car and two applications of passenger car occupant safety. Owing to the newness of this approach, there are no established methods or best practices and the presented examples show a large variety in the solution of the problem. The most straightforward way is to combine an optimization with a robust assessment of the optimum, either via a Monte-CarloAnalysis or a robustness-estimation method like FORM (First-Order-Reliability-Method). A very interesting approach is established in the idle shake application. Here a multi objective optimization is performed, using the NCGA method, and then the robustness of the points of the Pareto-front is used to select the design used for production. The other examples show more different methods, the application of the Taguchi robust design method, or the direct use of the output variation as part of the objective function, in order to achieve a robust design. Although the combination of optimization and robust engineering is a too young field to already propose finalized routines and fixed solutions, the examples shown in this work illustrate the great benefit that already can been realized by this approach. Some of the designs are in production and one design system for passenger safety is implemented based on one of the presented applications.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Combining topological and shape derivatives in structural optimization G. Allaire∗ , F. Jouve∗ , F. de Gournay∗ & A.-M. Toader† ∗ Centre

de Math´ematiques Appliqu´ees (UMR 7641), Ecole Polytechnique 91128 Palaiseau, France

† CMAF,

Faculdade de Ciˆencias da Universidade de Lisboa Av. Prof. Gama Pinto 2, 1699 Lisboa, Portugal [email protected] ABSTRACT

Two recent methods in shape and topology optimization of structures are combined in order to obtain an efficient optimization algorithm that benefits of advantages from both methods. The level set method, based on the classical shape derivative, is known to easily handle boundary propagation with topological changes. However, in practice it does not allow for the nucleation of new holes (at least in 2-d). The bubble or topological gradient method of Schumacher, Masmoudi, Sokolowski and their co-workers, is precisely designed for introducing new holes in the optimization process. Therefore, the coupling of these two methods yields a robust algorithm which can escape from local minima in a given topological class of shapes. The method we propose is a logical sequel of our previous work [1], [2] where we proposed a numerical method of shape optimization based on the level set method and on shape differentiation. The novelty is in the coupling and in the robustness of the proposed numerical implementation. Our basic algorithm is to iteratively use the shape gradient or the topological gradient in a gradient-based descent algorithm. The tricks are to carefully monitor the decrease of the objective function (to avoid large changes in shape and topology) and to choose the right ratio of successive iterations in each method. We provide several 2-d and 3-d numerical examples for compliance minimization and mechanism design. The main advantage of our coupled algorithm is to make the resulting optimal design largely independent of the initial guess, although local minima may still exist (even in the class of shapes sharing the same topology). Similar numerical results where discussed in [3].

References [1] Allaire G., Jouve F., Toader A.-M., A level set method for shape optimization, C. R. Acad. Sci. Paris, S´erie I, 334, 1125-1130 (2002). [2] Allaire G., Jouve F., Toader A.-M., Structural optimization using sensitivity analysis and a level set method, J. Comp. Phys., Vol 194/1, pp.363-393 (2004). [3] Allaire G., De Gournay F., Jouve F., Toader A.-M., Structural optimization using topological and shape sensitivity via a level set method, Control and Cybernetics, 34, 59-80 (2005).

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A Comparison of Displacement and Mixed Finite Element Formulations for Variational Design Sensitivity Analysis Franz-Joseph Barthold∗ , Karin Wiechmann† ∗ Numerical

Methods and Information Processing, University of Dortmund August-Schmidt-Str. 8, D-44227 Dortmund, Germany [email protected]

† Institute

of Mechanics and Computational Mechanics, University of Hannover Appelstr. 9A, D-30167 Hannover, Germany [email protected] ABSTRACT

The authors formulate design sensitivity analysis in form of a variational approach based on a novel local representation of continuum mechanics, see the publications of the first author, e.g. [1], for a summarising overview on the concept and the publications of the second author, e.g. [3], for the subsequent application to elasto-plastic material behaviour. The central idea is to trace and separate the influence of geometry mappings from the influence of deformation mappings on all field quantities. Thus, a reformulation of continuum mechanics following the intrinsic formulation by Noll [2] but using two independent mappings defined on a local parameter space is advocated. Consequently, the consistent linearisation concept of computational mechanics used to derive tangent stiffness matrices, should also be applied to the geometry mappings, i.e. to compute tangent geometry sensitivity matrices. Firstly, the fundamentals of the advocated treatment are described in general terms. Secondly, the consequences for the finite element development procedure are outlined on the theoretical as well as computational level. Here, the parallelism of sensitivity and stiffness computation are highlighted for different finite elements, i.e. for standard displacement and mixed formulations. Thirdly, hints are given to guarantee correct variational sensitivity information by a general comparison strategy using finite differences. The outlined theoretical and computational framework is seen to be an efficient method to investigate the influence of different element formulations on the solution of the optimisation problem.

References [1] F.-J. Barthold, Zur Kontinuumsmechanik inverser Geometrieprobleme. Braunschweig Series on Mechanics. Report No. 44-2002, Braunschweig, ISBN 3-920395-43-3, 2002. [2] W. Noll, A new mathematical theory of simple materials, Archive of Rational Mechanics, 48(1), 1-50, 1972. [3] K. Wiechmann, Theorie und Numerik zur Berechnung und Optimierung von Strukturen mit elastoplastischen Deformationen. Institute of Mechanics and Computational Mechanics, Report No. F01/8, Hannover, ISBN 3-935732-02-3, 2001.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

646

Inverse acoustic scattering by small-obstacle expansion of misfit function Marc Bonnet∗ ∗ Solid Mechanics Laboratory (UMR CNRS 7649) Ecole Polytechnique, F-91128 Palaiseau cedex, France [email protected]

ABSTRACT The present study is set in the general framework of inverse scattering of scalar (e.g. acoustic) waves. To identify hidden obstacles from external measurements (e.g. overspecified boundary data) associated with the scattering of known incident waves by the unknown object(s), it is customary to invoke iterative algorithms such as gradient-based optimization procedures. The numerical solution of the forward scattering problem associated with an assumed obstacle configuration is often a computationally demanding task. Besides, iterative inversion algorithms are sensitive to the choice of initial “guess” (number of components, initial location, shape and size of obstacle(s)). This has prompted the definition of preliminary probing techniques, which aim at delineating in a computationally fast way the hidden obstacle(s), namely the linear sampling [2], not pursued here, or the concept of topological sensitivity [1, 3]. If J denotes the cost function used for solving the inverse problem, then in 3D situations the topological derivative T3 (xs ) associated with the nucleation of a small obstacle of volume O(ε3 ) and specified shape appears through the expansion J(ε, xs ) − J(0) = ε33 T (xs ) + o(ε3 )

(1)

In this communication, an extension of the topological derivative is presented, whereby J(ε, xs ) is expanded further in powers of ε. Specifically, the expansion to order O(ε6 ) for 3D acoustic scattering by a hard obstacle of size ε is presented. The choice of order O(ε6 ) is important for cost functions J of least-squares format. In particular, the expansion of J for any centrally-symmetric infinitesimal hard obstacle of radius ε centered at xs is found to have the form J(ε, xs ) − J(0, xs ) = ε3 T3 (xs ) + ε5 T5 (xs ) + ε6 T6 (xs ) + o(ε6 ) = J(0, xs ) + J6 (ε, xs ) + o(ε6 ) (2) The previously known topological derivative T3 (xs ) and the new coefficients T5 (xs ), T6 (xs ) have explicit expressions in terms of the relevant acoustic Green’s function. Expansions of the form (2) offer the option of minimizing the approximate polynomial expression J6 (ε, xs ). This is a simple and inexpensive task, which can be performed for locations xs spanning a search grid, thereby defining a (approximate) global search procedure. The values of xs and ε leading to an absolute minimum of J6 (ε, xs ) over the search grid then constitute the best estimate of the hidden scatterer furnished by this procedure, and might provide e.g. a useful initial guess for an iterative inversion algorithm. Results of numerical experiments in 3D conditions based on this idea will be presented at the conference.

References [1] B ONNET, M., G UZINA , B. B. Sounding of finite solid bodies by way of topological derivative. Int. J. Num. Meth. in Eng., 61, 2344–2373 (2004). [2] C OLTON , D., K IRSCH , A. A simple method for solving inverse scattering problems in the resonance region. Inverse Problems, 12, 383–393 (1996). [3] G UZINA , B. B., B ONNET, M. Topological derivative for the inverse scattering of elastic waves. Quart. J. Mech. Appl. Math., 57, 161–179 (2004).

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

647

Second Order Topological Sensitivity Analysis J.R. Faria∗ , R.A. Feijo´o∗ , A.A. Novotny∗ , E. Taroco∗ & C. Padra† ∗ Laborat´ orio

Nacional de Computac¸a˜ o Cient´ıfica LNCC/MCT, Av. Get´ulio Vargas 333, 25651-075 Petr´opolis - RJ, Brasil [email protected], [email protected], [email protected], [email protected] † Centro

At´omico Bariloche, 8400 Bariloche, Argentina [email protected] ABSTRACT

The topological sensitivity analysis provides an asymptotic expansion of a given cost function with respect to the insertion of a small hole at an arbitrary point of the domain. This sensitivity results in a real function called topological derivative [3] that has been used as a descent direction to solve several problems, among others: topology optimization and inverse problems. In order to present the basic idea, let us consider an open bounded domain Ω ⊂ R 2 with a smooth boundary ∂Ω and a cost function ψ (Ω) = JΩ (u), where u denotes the solution of a state equation defined in Ω. If the domain Ω is perturbed by introducing a small hole B ε of radius ε at an arbitrary point x ˆ ∈ Ω, we have a new domain Ω ε = Ω − B ε , whose boundary is denoted by ∂Ωε = ∂Ω ∪ ∂Bε . Therefore, the topological asymptotic expansion of the cost function may be expressed as following ψ(Ωε ) = ψ(Ω) + f (ε)DT ψ + g(ε)DT2 ψ + O(g(ε)) ,

(1)

where f (ε) and g (ε) ∈ O(f (ε)) are positive functions that decreases monotonically such that f (ε) → 0 and g(ε) → 0 with ε → 0+ . The term DT ψ is classically defined as the first order topological derivative [1] of ψ. In addition, if we divide eq. (1) by g (ε) and after taking the limit ε → 0, we can recognize the term DT2 ψ as the second order topological derivative of ψ. In fact, DT2 ψ = lim

ε→0

ψ(Ωε ) − ψ(Ω) − f (ε)DT ψ , g(ε)

(2)

which can be used to devise optimality conditions in the context of topology optimization and inverse problems, for instance. In this work, we apply the Topological-Shape Sensitivity Method developed in [2] as a systematic approach to compute first as well as second order topological derivative for the Poisson’s equation, taking into account homogeneous Neumann and Dirichlet boundary condition on the hole.

References [1] J. C´ea, S. Garreau, Ph. Guillaume & M. Masmoudi. The Shape and Topological Optimizations Connection. Computer Methods in Applied Mechanics and Engineering, 188:713-726, 2000. [2] A.A. Novotny, R.A. Feij´oo, C. Padra & E. Taroco. Topological Sensitivity Analysis. Computer Methods in Applied Mechanics and Engineering, 192:803-829, 2003. ˙ [3] J. Sokolowski & A. Zochowski. On the Topological Derivative in Shape Optimization. SIAM Journal on Control and Optimization, 37:1251-1272, 1999.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Sensitivity Analysis of Shape Memory Alloy Shells Matthijs Langelaar*and Fred van Keulen† Department of Precision and Microsystems Engineering, Delft University of Technology, Mekelweg 2, 2628 CD Delft, The Netherlands * [email protected] † [email protected]

ABSTRACT Shape memory alloys (SMAs) are active materials with a high power density capable of producing comparatively large actuation strains and stresses. However, designing effective multi-dimensional SMA actuators is a challenging task, due to the complex behavior of the material and the fact that often electrical, thermal and mechanical aspects have to be considered simultaneously. For this reason, interest in the application of systematic computational design approaches, such as design optimization techniques, to the design of SMA structures is increasing. To enable efficient SMA design optimization procedures, the availability of sensitivity information is crucial. This paper presents the formulation and computation of design sensitivities of SMA shell structures using the direct differentiation method, in a steady state electro-thermo-mechanical finite element context. Semi-analytical and refined semi-analytical approaches are considered. The SMA constitutive model used in this study is particularly intended for design optimization of SMA structures and actuators. In contrast to the majority of SMA models, the formulation of this model is history-independent, which simplifies the sensitivity analysis considerably. The model is specifically aimed at the description of the superelastic behavior of NiTi alloys, based on the R-phase transformation. This behavior is characterized by its negligible hysteresis, which is very attractive for actuator applications. This research is aimed at SMA shell structures, which can generate large actuator displacements through bending. The most general case of actuation is considered, where the SMA effects are initiated by temperature changes induced by Joule heating. This implies that a coupled electrothermo-mechanical finite element analysis is required. As a consequence, the sensitivity analysis includes coupling terms between three different physical fields. The most challenging aspect of this work lies in particular in the fact that the constitutive model in the considered plane stress setting contains implicit equations, which lead to complications in the sensitivity analysis. This problem manifests itself in the thermo-mechanical sensitivity coupling terms and in sensitivities of derived mechanical responses such as stresses or equivalent strains. Solutions for this difficulty based on finite difference and analytical approaches are discussed. Finally, the accuracy of the sensitivities is evaluated on a representative finite element model of a miniature gripper, as a function of relative design perturbations in thickness, shape, loading and material parameters. A comparison is given between results obtained by finite-difference, semi-analytical and refined semi-analytical methods.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

649

Stress Constrained Optimization using X-FEM and Level Set Description L. Van Miegroet*, T. Jacobs†, E. Lemaire* & P. Duysinx* * University of Liège LTAS - Department of Mechanics and Aerospace Chemin des chevreuils 1, 4000 Liège Belgium [email protected], [email protected] † Université Laval Département de Génie civil, Pavillon Adrien-Pouliot, Québec Canada [email protected]

ABSTRACT Topology optimization has experienced an incredible soar since 1988 and is now available within several commercial finite element (FE) codes. Meanwhile, parametric shape optimization has found few industrial applications. This is may be due to its inherent difficulties to deal with mesh management with boundary modifications. Recently the extended finite element method (X-FEM) has been proposed (see [1] for a review) as an alternative to remeshing methods. The X-FEM method is naturally associated with the Level Set description of the geometry to provide an efficient treatment of problems involving discontinuities and propagations. Up to now the X-FEM method has been mostly developed for crack propagation problems, but the potential interest of the X-FEM method and the Level Set description for other problems like shape and topology optimization was identified very early (see [2]). In this paper, the authors present an intermediate approach between parametric shape and topology optimization by using the X-FEM and Level Set Description. The method benefits from fixed mesh work using X-FEM and from smooth curves representation of the Level Set description. One major characteristic of the approach is to be able to model exactly void and solid structures. The statement of the optimization problem is similar to classical shape optimization: Design variables are the parameters of basic Level Set features (circles, rectangles, super ellipse, etc.) or could be NURBS control points, while various global (compliance) and local responses can be considered in the formulation. Conversely to shape optimization, structural topology can be modified since basic Level Sets can merge or separate from each other. The sensitivity analysis (related to the compliance and/or the stresses) and the way it can be carried out efficiently is detailed. A special attention is paid to stress constrained problems which are often neglected in other Level Set Methods works. Numerical applications revisit some classical 2D (academic) benchmarks from shape optimization and illustrate the great interest of using X-FEM and Level Set description. The paper presents the results of minimum compliance and stress constrained problems using the proposed method.

References [1] T. Belytschko. C. Parimi, N. Moes, N. Sukumar & S. Usui. Structured extended finite element methods for solids defined by implicit surfaces. Int. J. Numer. Meth. in Engng 2003; 56 : 609-635. [2] T. Beltschko, S. Xiao & C. Parimi. Topology optimization with implicit functions and regularization. Int. J. Numer. Meth. in Engng 2003; 57 : 1177-1196.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Level Set Method for Optimization of Contact Problems ´ † Andrzej My´slinski † Systems Research Institute ul. Newelska 6, 01 - 447 Warsaw, Poland [email protected]

ABSTRACT This paper deals with the numerical solution of structural optimization problems of an elastic body in unilateral contact with a rigid foundation. The contact problem with a given friction is described by an elliptic inequality of the second order governing a displacement field. The optimization problem consists in finding, in a contact region, such topology and shape of the boundary of the domain occupied by the body that the normal contact stress is minimized. Level set methods [3, 4] are numerically efficient and robust procedures for the tracking of interfaces, which allows domain boundary shape changes in the course of iteration. The evolution of the level set function is governed by the Hamilton Jacobi equation. The speed vector field driving the propagation of the level set function is given by the Eulerian derivative [2] of an appropriately defined cost functional with respect to the free boundary. In this paper the necessary optimality condition for the shape and topology optimization problem of this contact problem is formulated. The paper extends results of [1] to contact problems with a given friction. The level set method, based on the classical shape gradient, is coupled with the bubble or topological derivative method, which is precisely designed for introducing new holes in the optimization process. The holes are supposed to be filled by weak phase mimicking voids.Since both methods capture a shape on a fixed Eulerian mesh and rely on a notion of gradient computed through an adjoint analysis, the coupling of these two method yields an efficient algorithm. Moreover the finite element method is used as the discretization method. Numerical examples are provided and discussed.

References [1] A. My´sli´nski, Topology and Shape Optimization of Contact Problems using a Level Set Method, Proceedings of 6 WCSMO Conference, Rio de Janeiro, Brasil, 2005. ˙ [2] J. Sokołowski and A. Zochowski, Topological optimization of contact problems. Research Report No 25/2005, Institute Ellie Cartan, 2005. [3] Y. H. Tsai, Y. Giga and S. Osher, A Level Set Approach for Computing Discontinuous Solutions of Hamilton - Jacobi Equations, Mathematics of Computation, 72, 159 - 181, 2002. [4] M.Y. Wang, X. Wang, D. Guo, A Level Set Method for Structural Topology Optimization, Computer Methods in Applied Mechanics and Engineering, 192, 227 - 246, 2003.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

651

An Iterative Procedure for Model Updating Based on Selective Sensitivity Hoang Anh Pham∗ , Christian Bucher† ∗ Institute

of Structural Mechanics, Bauhaus-University Weimar Marienstr. 15, D-99423 Weimar, Germany [email protected]

† Institute

of Structural Mechanics, Bauhaus-University Weimar Marienstr. 15, D-99423 Weimar, Germany [email protected]

ABSTRACT Model updating of a structural system may require a large number of parameter to be identified simultaneously. Due to the ill-conditionedness, large errors in identified parameter values will occur when errors are present in the measurements. One solution for this problem is using the concept of selective sencitivity [1]. The method allows to reduce ill-conditioning by providing specific excitations causing model responses sensitive to a small number of model parameters. Thus, only a few parameters are estimated at a time. However, defining such excitations generally involves the knowledge of all parameters to be identified. Therefore, an iterative experiment procedure is suggested (e.g the method of multi-hypothesis testing [2]) which is normally a time-consuming process. This paper presents the theory for an alternative iterative procedure for dynamical excitation of undamped, linear structures. The approach is developed using the concept of predictive control and then is incorporated into a Bayesian updating methodology to reduce the uncertainty in the system parameters. Simulation examples of a multi-storey frame structure and a continuously supported beam under hamonic excitation demonstrate the potential of the proposed method.

References [1] Y. Ben-Haim, Robust reliability in the mechanical sciences. Springer, Berlin-Heidelberg-New York, 1996. [2] U. Prells and Y. Ben-Haim, Selective sensitivity in the frequency domain-II. Applications. Mechanical System and Signal Processing, 7(6), 551–574, 1993.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Extension, bending and torsion of cylindrical Cosserat shells made from a porous elastic material Mircea Bˆırsan ‘A.I. Cuza’ University of Ias¸i, Faculty of Mathematics Bvd. Carol I, no. 11, 700506 Ias¸i, Romania [email protected]

ABSTRACT We consider the relaxed Saint–Venant’s problem in the linear theory of elastic shells made from a porous material. For our purpose, we use the model of Cosserat surfaces and the Nunziato–Cowin theory of elastic materials with voids [1]. We extend the method employed in [2, 3] for the case of Cosserat shells with two porosity fields: one field characterizes the volume fraction variations along the middle surface of the shell, while the other accounts for the changes in volume fraction along the shell’s thickness. We obtain the solution of the extension, bending and torsion problem in closed form, for both open and closed cylindrical shells of arbitrary cross–section. The solutions determined are shown to be minimizers of the strain energy functional on certain classes of solutions to the relaxed Saint–Venant’s problem for shells, by analogy with the well–known characterizations of the classical Saint–Venant’s solutions from the three–dimensional theory of elasticity. We observe that the torsion of cylindrical shells does not affect the porosity fields, so that the torsion problem reduces to the previously known results for the purely elastic case. On the other hand, for the extension and bending problem, we remark the influence of the material’s porosity on the deformation of the shell. In the particular case of porous Cosserat plates, we compute the solution of the extension and bending problem and show that this solution is in agreement with the corresponding results obtained in [1] and [4] using three–dimensional approaches. The solutions determined in this paper are exact and they prove useful in solving many practical problems and for the comparison with related results obtained by various computational methods.

References [1] S.C. Cowin and J.W. Nunziato, Linear elastic materials with voids, Journal of Elasticity, 13, 125– 147, 1983. [2] M. Bˆırsan, The solution of Saint–Venant’s problem in the theory of Cosserat shells, Journal of Elasticity, 74, 185–214, 2004. [3] M. Bˆırsan, Saint–Venant’s problem for Cosserat shells with voids, International Journal of Solids and Structures, 42, 2033-2057, 2005. [4] M. Ciarletta and D. Ies¸an, Non–classical Elastic Solids, Pitman Research Notes in Mathematics, no. 293, Longman Scientific & Technical, London, 1993.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A study of discontinuous Galerkin methods for thin bending problems N.T. Dung∗ , G.N. Wells† ∗ Faculty

of Civil Engineering and Geosciences, Delft University of Technology P.O. Box 5048, 2600 GA Delft [email protected]

† Faculty

of Civil Engineering and Geosciences, Delft University of Technology P.O. Box 5048, 2600 GA Delft [email protected]

ABSTRACT In this work, various continuous/discontinuous Galerkin (C/DG) formulations are examined for the analysis of thin plates. The continuous/discontinuous Galerkin method allows fourth-order partial differential equations to be solved using standard C 0 finite element shape functions. The concept was presented by Engel et al. [1], who utilised an interior-penalty type formulation for solving thin beam and plate problems. The interior-penalty method has some drawbacks, such as the conditional stability, the loss of accuracy for large penalty values and ambiguities when making the extension to nonlinear problems. Here, we draw on developments in discontinuous Galerkin methods for second-order elliptic equations, for which several unconditionally stable methods are known [2], and present continuous/discontinuous Galerkin formulations for thin bending problems inspired by these methods. Two important aspects that have been studied are the stability condition and the convergence rate. For each approach, benchmark simulations have been performed and compared. Also, conclusions are drawn on to the computational efficiency of the different methods. The presented numerical examples have been produced using the FEniCS Form Compiler (FFC) [3, 4]. FFC takes the variational form in a format which reassembles standard mathematical notation, and generates optimised finite element code for arbitrary order shape functions. It allows for the rapid development of efficient finite element code and it simplifies significantly the implementation of DG methods, an issue which is often cited as a disadvantage of DG methods. Examples of FFC functionality and use for DG problems are presented.

References [1] G. Engel, K. Garikipati, T. J. R. Hughes, M. G. Larson, L. Mazzei, and R. L. Taylor, Continuous/discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beam and plates, and strain gradient elasticity. Comput. Methods Appl. Mech. Engrg., 191, 3669-3750, 2002. [2] D. N. Arnold, F. Brezzi and B. Cockburn, and L. D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Number. Anal., 39 (5), 1749-1779, 2002. [3] FEniCS Form Compiler. www.fenics.org. [4] R. C. Kirby, and A. Logg, A Compiler for Variational Forms. Submited.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Discrete Strain Gap (DSG) solid finite elements at large deformations for non-linear analysis of shells and solids Moritz Frenzel∗ , Manfred Bischoff† , Wolfgang A. Wall∗ ∗ Chair

of Computational Mechanics Technical University of Munich Boltzmannstr. 15, 85747 Garching b. M¨unchen, Germany [email protected] † Chair of Structural Analysis Technical University of Munich [email protected]

ABSTRACT We present a non-linear solid finite element for structural analysis, based on the Discrete Strain Gap (DSG) method, involving large-deformations and non-linear material laws. For the linear case, the DSG method has proven to eliminate all geometric locking effects, such as shear locking, membrane locking, and trapezoidal locking, with one unique approach [1]. Satisfaction of the patch test – which is violated in the original DSG formulation for solids – may be ensured via a decomposition of strain modes into a constant part and higher order modes. However, this modification goes along with the appearance of trapezoidal locking (cf. “MacNeal’s Dilemma” [2]). In the particular case of thin shell analysis with solid or solid-shell elements, this is of noticeable relevance. A way out of this dilemma is to choose the original, locking-free formulation for the “out-of-plane” strain components while ensuring consistency via the mesh layout in transverse direction. The proposed method allows for an easy switch from 3d-structures to thin structures within one common element technology, and it is both consistent and locking-free in either case [3]. In order to remedy the material locking phenomenon of volumetric locking the well-established Enhanced Assumed Strain (EAS) method is used. For the presented element only three additional internal parameters are necessary. In the present contribution this element is extended to non-linear dynamic problems. Both geometric and material non-linearities are taken into account. From the material model point of view special attention will be drawn to hyperelastic and viscoelastic materials. Numerical examples involving static and dynamic analysis with both material and geometric non-linearities are given and compared with existing results obtained with other shell elements.

References [1] F. Koschnick, M. Bischoff, N. Camprubi, K.-U. Bletzinger, The discrete strain gap method and membrane locking. Computer Methods in Applied Mechanics and Engineering, 194,2444–2463, 2005. [2] R. H. MacNeal, A theorem regarding the locking of tapered four-noded membrane elements. International Journal for Numerical Methods in Engineering, 24:1793–1799, 1987 [3] M. Frenzel, M. Bischoff, K.-U. Bletzinger, W. A. Wall, Performance of Discrete Strain Gap (DSG) finite elements in the analysis of three-dimensional solids. In: Proceedings of the 5th International Conference on Computation of Shell and Spatial Structures, Salzburg, Austria, June 1-4, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A Simple Co-Rotational Geometrically Non-Linear Membrane Finite Element Wrinkling Analysis E. Gal*, M. Zelkha†, R. Levy† * Ben-Gurion University of the Negev Department of Structural Engineering, 84105 Beer-Sheva, Israel [email protected] †

Technion- Israel institution of Technology Faculty of environmental and Civil Engineering, 32000 Haifa, Israel [email protected]

ABSTRACT Thin pre-tensioned membranes are often used in many technological applications such as fabric constructions, marine and space technology. The advantages of these structures are the lightness of the membrane, which facilitates coverage of large spans and the ability to create a variety of shapes. Fabric membranes by definition have little compression resistance and have no bending stiffness, hence is very easy to wrinkle. This research presents the analysis of wrinkled membranes using a geometrically nonlinear membrane finite element by Levy and Spillers [1] as the core of the proposed analysis procedure. The wrinkle state of a membrane finite element can be defined as follows: Taut state: V 1 ! 0 and V 2 ! 0 ; Wrinkled state: V 1 ! 0 and V 2 d 0 and Slack state: V 1 d 0 and V 2 d 0 where V 1 and V 2 are the maximum and the minimum principle stresses of the element respectively. The physical interpretation of this definition is that elements in the Taut state are fully tensioned in both directions and therefore follow the isotropic constitutive equation, elements in the Wrinkled state are in tension the V 1 direction only and therefore follow the orthotropic constitutive equation and elements in the Slack state are in full compression and therefore inactive (e.g. see Tabarrok and Qin [3] and Miller and Hedgepeth [2]). The proposed analysis is performed in an incremental iterative fashion. At each increment a two step convergence criterion is imposed. The first step handles equilibrium in the final configuration of the membrane using the Newton-Rapson method. The second step takes care of the final wrinkled-status of the elements where stresses are updated (if necessarily) according to the element state i.e. negative principle stress are set to zero when the unbalanced force vector is computed due to these changes. Convergence of the second step is achieved when none of the elements changes its wrinkled-status. Finally validation and verification of the proposed analysis is performed by comparing results to those of several benchmark problems.

References [1] [2] [3]

R. Levy and W.R. Spillers, Analysis of geometrically nonlinear structures. Chapman & Hall, New York:,1994. R.R. Miller and J.M. Hedegepth, An Algorithm for finite element analysis for partly wrinkled membranes. AIAA Journal, 20(12), 1761-1763, 1982. B. Tabbarrok and Z. Qin. Nonlinear analysis of tension structures. Computer and Structures, 45, 973-984, 1992M. Stein and J. M. Hedegepth, Analysis of partly wrinkled membranes. NASA Technical Note D-813 July 1961.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Co-rotational System Definitions for Large Displacement Triangular and Quadrilateral Shell Elements Bassam A. Izzuddin Imperial College Lodnon London SW7 2AZ, United Kingdom [email protected]

ABSTRACT The co-rotational approach offers exceptional benefits for large-displacement structural analysis problems with deformations of the bending type, particularly when accounting for arbitrarily large rigid body rotations. A principal issue in any co-rotational approach is associated with the specific choice of the local reference system in relation to the current deformed element configuration. Whilst an arbitrary choice that closely follows the current element configuration, for example using the current positions of any two of the element sides, does not significantly affect the large displacement response predictions for small strain problems, this often leads to local system definitions which are not invariant to the specified order of the element nodes. It has been previously argued that this invariance characteristic would be desirable for extending the co-rotational approach to large strain problems [1] and, more recently, for identifying the bifurcation points of perfectly symmetric structures [2]. Two approaches were employed to achieve the invariance of the local system to nodal ordering [1,2], but these suffered from complexity associated with application to material points within the element domain, and resulted in an asymmetric consistent tangent stiffness matrix which leads to further computational disadvantages. In this paper, new definitions of the local co-rotational system are proposed for quadrilateral and triangular shell elements, which achieve the invariance characteristic to nodal ordering in a relatively simple manner, and importantly result in a symmetric tangent stiffness matrix. The proposed definitions utilise only the nodal coordinates in the deformed configuration, where two alternative definitions are outlined for each of the quadrilateral and triangular element shapes. The first is a bisector definition utilising alignment along the bisectors of one or more internal element angles, while the second is a zero-‘macro spin’ definition considering the minimisation of the spin for the triangular/quadrilateral shape as described by the element nodes. The paper presents the co-rotational transformations linking the local and global element freedoms for both definitions, and provides a numerical example to demonstrate their relative accuracy in large displacement analysis of plates and shells. It is shown that both definitions are equally accurate for small strain problems, but that the zero-‘macro spin’ definition has more general potential application in large strain problems.

References [1] M.A. Crisfield and G.F. Moita, A Unified Co-rotational Famework for Solids, Shells and Beams, Int. J. Solids Struct., 33, 2969-2992, 1996. [2] J.M. Battini and C. Pacoste, On the Choice of Local Element Frame for Corotational Triangular Shell Elements, Commun. Numer. Meth. Engng., 20(10), 819-825, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Non-Linear Analysis of Composite Plates and Shells Using a New Shell Element Peyman Khosravi∗ , Rajamohan Ganesan† , Ramin Sedaghati‡ Department of Mechanical and Industrial Engineering, Concordia University Montreal, Quebec, H3G 1M8, Canada email∗ : peyma [email protected] email† : [email protected] email‡ : [email protected] ABSTRACT One of the most popular approaches in the finite element analysis of plates and shells is using an assemblage of facet triangular elements built by combining a membrane and a plate bending element to model the curved surface. Due to the lack of a drilling degree of freedom in most triangular membrane elements, these elements may cause rotational singularity in the stiffness matrix. One approach to overcome this problem is using membrane elements with in-plane (or drilling) rotational degree of freedom. Although some elements with drilling degree of freedom have been derived, most of them suffer from aspect ratio locking. Recently, Felippa [1] developed an optimal membrane element with drilling degree of freedom. Its response for in-plane pure bending is not dependent on the aspect ratio. There are several triangular plate bending elements to combine with a membrane element. Batoz et al. [2] studied several triangular Kirchhoff plate bending elements and showed that Discrete Kirchhoff Triangle (DKT) [3], is the most reliable triangular element for analysis of thin plates. Katili [4] developed a discrete Kirchhoff-Mindlin triangular plate bending element called DKMT which is capable to include the transverse shear effects in thick plates, and coincides with the DKT element in case of thin plates. As a result, both thin and thick plates can be modeled with this element. In the present work, a new shell element for both thin and thick plates is developed by combining the DKMT plate bending element and the optimal membrane triangular element (also called OPT). The membrane-bending coupling effect of composite laminates is incorporated in the formulation, and inconsistent stress stiffness matrix and tangent stiffness matrix are formulated. Using co-rotational method and the tangent stiffness matrix, this new shell element is used to solve problems with geometric nonlinearity and the results are compared with analytical solutions or those available in the literature. The behavior and advantages of the new element are studied.

References [1] CA. Felippa, A study of optimal membrane triangles with drilling freedoms. Computer Methods in Applied Mechanics and Engineering, 192(16), 2125–2168, 2003. [2] JL. Batoz, An explicit formulation for an efficient triangular plate-bending element. International Journal for Numerical Methods in Engineering, 18, 1077–1089, 1982. [3] JL. Batoz, KJ. Bathe, LW. Ho, A study of three-node triangular plate bending elements. International Journal for Numerical Methods in Engineering, 15, 1771–1812, 1980. [4] I. Katili, A new discrete Kirchhoff-Mindlin element based on Mindlin-Reinssner plate theory and assumed shear strain fields- Part I: An extended DKT element for thick-plate bending analysis. International Journal for Numerical Methods in Engineering, 36, 1859–1883, 1993.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Total and Updated Lagrangian Geometrically Exact Beam Elements Jari Mäkinen*, Heikki Marjamäki† Tampere University of Technology, Applied Mechanics and Optimization P.O. Box 589, FIN-33101 Tampere, FINLAND * [email protected], †[email protected]

ABSTRACT In the paper [1] is given finite element implementations for a geometrically exact beam element with different updating procedures. The formulations are named Eulerian, total Lagrangian, and updated Lagrangian. The updated Lagrangian formulation can bypass the well-known singularity problem of the total Lagrangian formulation which is singular at the rotation angle 2π and its multiples. The u pdated Lagrangian formulation has additional benefits such as a fully symmetrical stiffness tensor when applying a conservative loading. In addition, any time integration algorithm can be used because the changes of the rotation vector belong to the same tangent space of the rotation manifold SO(3). The updated Lagrangian formulation requires some secondary storage variables for the curvature and rotation vectors, at every spatial integration point. A total Lagrangian formulation in static cases with the consistent stiffness tensor is given in [2]. Generally, a total Lagrangian formulation in a static case with a conservative loading has an important property that is path-independence, whereas an updated Lagrangian formulation is path-dependent. Lagrangian formulations have a consistent interpolation while in Eulerian formulations the interpolation has to apply within an approximate, inconsistent, way. As we have noted earlier, total Lagrangian formulations have singularity at the rotation angle 2π and its multiples that are a remarkable restriction, especially in dynamic cases. We consider a total Lagrangian updating formulation [3] for a material rotation vector and compare it with an updated Lagrangian formulation. The total Lagrangian formulation preserves the pathindependence and it can be regarded as a consistent updating formulation. The major drawback is the singularities at the rotation angle 2π and its multiples, but we overcome this difficulty by the complement rotation vector that is the change of parametrization. Numerical examples are also given.

References [1] A. Cardona, M. Géradin, A Beam Finite Element Non-Linear Theory with Finite Rotations, International Journal for Numerical Methods in Engineering, 26, 2403-2438, 1988. [2]

A. Ibrahimbegović, F. Frey, I. Kozar, Computational Aspects of Vector-Like Parametrization of

Three-Dimensional Finite Rotations, International Journal for Numerical Methods in Engineering, 38, 3653-3673, 1995. [3] Mäkinen, J., A Formulation for Flexible Multibody Mechanics – Lagrangian Geometrically Exact Beam Elements using Constraint Manifold Parametrization, Doctoral Thesis, TUT, Applied Mechanics and Optimization, 2004, 89 pp. http://www.tut.fi/~jmamakin/vk.pdf

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Large Displacement Analysis of Plates using Hybrid Equilibrium Elements Edward A. W. Maunder* , Bassam A. Izzuddin† * University of Exeter Harrison Building, North Park Road, Exeter, EX4 4QF, UK [email protected] †

Imperial College, University of London South Kensington Campus, London SW7 2AZ, UK [email protected]

ABSTRACT Small strains can coexist with large displacements, particularly for structures composed of thin plates. This paper is concerned with finite element modelling of such non-linear behaviour when the material retains its linear elastic constitutive relations and Reissner-Mindlin theory can be invoked. This type of behaviour is of particular interest when for example in-plane stiffening becomes significant in a floor slab, or when elastic buckling due to geometric imperfections can occur. The co-rotational concept is invoked whereby the rigid body translations and rotations of elements are accounted for by the associated movements of element local reference axes, and the linear elastic stiffness matrices of elements are maintained unchanged with respect to current local configurations. Such formulations have been described by Izzuddin [1] where a more conventional conforming type of quadrilateral element is used. Due to the inherent generality of this approach, it has now been extended to hybrid equilibrium flat shell quadrilateral elements, and this is described in the paper. The formulation of this element is also presented and use is made of local element axes which are defined so as to lead to symmetric tangent stiffness matrices. The hybrid equilibrium model [2,3] enables solutions to be determined which satisfy equilibrium in a strong pointwise sense. This feature is believed to be a novel one in the case of modelling geometrically non-linear behaviour with finite elements, and it allows new meaning to be given to the presentation of equilibrium paths. Thus it may be expected that a plot of a quantity of interest versus a load parameter is more accurate when the quantity is a stressresultant. Numerical examples are presented to compare the performance of the equilibrium models with conventional conforming models, and to investigate the potential advantages of having dual solutions to a geometrically non-linear problem. Theoretical questions regarding the significance of fully equilibrated solutions have not yet been answered for such problems, e.g. can such solutions lead to upper bounds to the energy of the errors in a conforming solution? Nevertheless in practice it may be argued that two solutions are always better than one, and useful bounds may be achieved.

References [1] B.A. Izzuddin, An enhanced co-rotational approach for large displacement analysis of plates. International Journal for Numerical Methods in Engineering, 64, 1350-1374, 2005. [2] E.A.W. Maunder, Hybrid elements in the modelling of plates. B.H.V.Topping, ed., Finite Elements: Techniques and Developments, Civil-Comp Press, 165-172, 2000. [3] E.A.W. Maunder, J.P. Moitinho de Almeida, A triangular hybrid equilibrium plate element of general degree, International Journal for Numerical Methods in Engineering, 63, 315-350, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Buckling Modes of Large-Scale Shell Structures Automatically Detected from Linearized Stiffness by Iterative Solvers Hirohisa Noguchi *, Fumio Fujii†, Yoshikazu Ishihara†† *

†

Department of Systems Design Engineering Keio University, Yokohama, Japan [email protected]

Department of Mathematical and Computational Engineering Gifu University, Gifu, Japan [email protected] ††

Safety Engineering and Technology Department Mitsubishi Research Institute, Inc., Tokyo, Japan [email protected]

ABSTRACT This study presents a novel method to automatically detect bifurcation buckling modes of large scale shell structures during solving a set of linearized stiffness equations in geometrically nonlinear problems by iterative solvers, such as the CG method or the Lanczos method. The proposed method is based on the LDLT decomposition method for direct solvers proposed by the authors [1][2] and is extended for the iterative solvers in order to handle large-scale problems. First, the proposed method detects the approximate buckling mode during the simultaneous process of tri-diagonalization and the LDLT decomposition of the stiffness matrix, utilizing the fact that the Lanczos algorithm still preserves the eigenvalue properties of original matrix during the tri-diagonalization. Second, it is also shown that the correction vector in the direction of solution in the CG method can approximate the buckling mode. The proposed method can avoid a time-consuming eigenanalysis, which is usually necessary for detecting bifurcation modes at critical points, and can compute the approximate bifurcation modes closed to the critical points very efficiently and accurately. Several numerical examples of bifurcation buckling of shells demonstrate the potential of the proposed method

References [1] F. Fujii, F. H. Noguchi, The buckling mode extracted from the LDLT-decomposed large-order stiffness matrix, Communication in numerical methods in engineering, 18, 459-467, 2002. [2] H. Noguchi, F. Fujii, Eigenvector-free indicator, pinpointing and branch-switching for bifurcation, Communications in Numerical Methods in Engineering, 19, 445-457, 2003.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Finite Element analysis of the wrinkling of orthotropic membranes Igor P. Oliveira, Eduardo M. B. Campello, Paulo M. Pimenta Department of Structural and Foundation Engineering, University of São Paulo P.O. Box 61548, São Paulo, Brazil [email protected], [email protected], [email protected]

ABSTRACT This work presents a fully nonlinear formulation for the analysis of the wrinkling on orthotropic membranes. Our approach describes the membrane kinematics as a thin shell motion, whose bending stiffness comes naturally from the shell assumptions. We combine the geometrically-exact isotropic shell model of [1,2] with an orthotropic constitutive equation for the membrane strains (see [3,4]), so that both bending and typical membrane capabilities are present in a totally consistent way. The strain energy function is split into an isotropic and an orthotropic part, the first one being relative to the shell (hyperelastic) behavior and the latter to the membrane deformations. The model is discretized under the light of the finite element method using the six-node triangular element of [2], and the performance of the formulation is assessed in several numerical examples (see e.g. Fig. 1). Unstructured meshes are deliberately employed whereas small geometrical imperfections are imposed for the wrinkles to be initiated. Experimental data from the membrane tests of [5] are also taken into account for comparison with our results.

Fig.1. Stretching of two orthotropic membranes. Deformed configurations.

References [1] P. M. Pimenta, On a geometrically-exact finite-strain shell model, Proceedings of the 3rd PanAmerican Congress on Applied Mechanics, III PACAM, São Paulo, 1993. [2] E. M. B. Campello, P. M. Pimenta and P. Wriggers, A triangular finite shell element based on a fully nonlinear shell formulation, Computational Mechanics, 31, 505-518, 2003. [3] Oliveira I. P., Campello E. M. B. and Pimenta P. M., “Wrinkling of nonlinear orthotropic membranes by the finite element method”, submitted to Comput. Mech., 2006. [4] Oliveira I. P., Análise não-linear de membranas: ortotropia e enrugamento. Master of Science Dissertation, Department of Structural and Foundation Engineering, University of São Paulo, 2006. [5] Shelter Rite£ by Seaman Corporation, High Performance 9032 – Architectural Fabric, Biaxial Stretch test, available at www.architecturalfabrics.com/9032.html, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Theory and Numerics of a Surface–Related Shell Formulation Rainer Schlebusch∗ , Bernd W. Zastrau† ∗ Technische

Universit¨at Dresden 01062 Dresden, Germany [email protected] † Technische

Universit¨at Dresden 01062 Dresden, Germany [email protected] ABSTRACT The solution of structural analysis problems, especially of shell structures, demands an efficient numerical solution strategy. Since one–sided contact problems are investigated, the shell model is formulated with respect to one of the outer surfaces, i.e. the shell formulation is surface–related. In particular the investigation of textile reinforced strengthening layers [2] will be carried out by this approach. Since even shells are three–dimensional structures, i.e. bodies, the basic field equations of continuum mechanics must be the starting point. This set of partial differential equations with pertinent boundary conditions has to be solved. An efficient numerical solution of this problem becomes easier, if the problem is reformulated against a background of variational calculus. The discretization of the resulting variational formulation is, among others, the source of several locking phenomena. The presented shell formulation uses linear shell kinematics with six displacement parameters, circumventing a rotational formulation. This low–order shell kinematics produces parasitical strains and stresses, leading to wrong or even useless results. Therewith an extension and/or adjustment of well– known techniques to prevent or reduce locking like the assumed natural strain (ANS) method [3] and the enhanced assumed strain (EAS) method [4] has to be performed. Using these adapted methods, a reliable and efficient solid–shell element with tremendously reduced locking properties is obtained. This concept comprises the utilization of unmodified three–dimensional constitutive relations by a minimal number of kinematical parameters [1]. With the aid of two nonlinear examples, the reliability and the efficiency of the new solid–shell element is demonstrated.

References [1] N. B¨uchter, E. Ramm, 3d–Extension of Nonlinear Shell Equations Based on the Enhanced Assumed Strain Concept, in C. Hirsch, ed., Computational Methods in Applied Sciences, Elsevier, 55–62, 1992. [2] R. Schlebusch, J. Matheas, B. Zastrau, On Surface–Related Shell Theories for the Numerical Simulation of Contact Problems, Journal of Theoretical and Applied Mechanics, 41(3), 623–642, 2003. [3] J.C. Simo, T.J.R. Hughes, On Variational Foundations of Assumed Strain Methods, Journal of Applied Mechanics, 53, 52–54, 1986. [4] J.C. Simo, M.S. Rifai, A class of mixed assumed strain methods and the method of incompatible modes, International Journal for Numerical Methods in Engineering, 29, 1595 - 1638, 1990.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Modelling and optimization of sails M. Spalatelu-Lazar, F. L´en´e, N. Turb´e Laboratoire Mod´elisation, Mat´eriaux, Structures (LM2S) Universit´e Paris VI, CC 161, 4 Place Jussieu, 75252 Paris Cedex 05, France [email protected] ABSTRACT The sails fabrication is an old activity principally based on the practice and the experience of sailmakers and users in constant search of performances and safety. The sails design registered a significant development under the impulse of sailing races like America’s Cup [1]. These competitions require the use of advanced technologies to increase the sails performances: optimization of the fibres orientation, of the weight, of the sails shapes in navigation. The objective of this paper is to shed some light on how to improve the quality and the performances by a control of the fibres orientation calling, for modelling, numerical experimentation and optimization methods. A sail is a lightweight flexible structure, made up of an assembly of panels, reinforced in its critical points by doublings or straps, and often rigidified by battens [2]. The efforts acting on the sails involve significant deformations according to the force of the wind and the nature of sailcloth. The parameters of the problem are thus very numerous, related to the definition of the sail and the loading cases. The sail is modelled here by a triangular membrane, submitted to large displacements and small strains [3]. Initial pre-tension load is required [4]. The behaviour law of the structure enters in the framework of the linear orthotropic elasticity modelling long fibres materials. The equilibrium equations, formulated on the midsurface, are solved by a modified Newton-Raphson method. The sailmaker can modulate the mechanical behaviour according to three parameters: the nature of the components, their proportion and the orientation of fibres. The presence of strong stresses zones usually leads to a sail cut-out in pieces with rectilinear fibres of constant orientation. Recent progress in sails fabrication allow the realization of structures with curvilinear supports and variable density rates. It then becomes essential to determine with precision which are, in each point, the orientation and the optimal rate of fibres. The present study is focused on the optimal fibres orientation. The mathematical problem of optimization is related to the displacement in the transverse direction of the sail. The optimization method uses the Nelder-Mead algorithm, efficient to solve non-linear problems. The numerical results are always independent of the initial fibres distribution and of the mesh.

References [1] C. A. Marchaj, Aero-hydrodynamics of sailing. Adland Coles limited, Granada Publishing, 1979. [2] V. Boh´e, P. Casari, F. L´en´e, P. Davies Comportement des mat´eriaux a` voiles de bateau. Revue des Composites et des Mat´eriaux Avanc´es , 3, 2003. [3] F. Muttin, Structural analysis of sails. European Journal of Mechanics, 10 (5) 517-534, 1991. [4] M. Quadrelli, S. Sirlin Modeling and control of membranes for gossamer spacecraft. AIAA Journal, 1201, 2001.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Instability Analysis of thin-walled structures using Incompressible Hyperelastic Shell Elements Masato Tanaka*, Hirohisa Noguchi† *

School of Science for Open and Environmental Systems Keio University, Yokohama, Japan [email protected] †

Department of Systems Design Engineering Keio University, Yokohama, Japan [email protected]

ABSTRACT In the present study, a new hyperelastic shell element is proposed which can be applied to the numerical simulation of instability analyses of thin-walled structures made of rubber-like materials. The recent research on the shell problem also involves flexible structures. This refers to many manmade rubber-like structures such as pneumatic membranes, automobile tires, hydraulic hoses and various biological soft tissues such as blood vessels, lung pleura and cardiac muscles. The numerical simulation of such materials is not possible under the assumption of small elastic strains and requires a hyperelastic constitutive formulation. Therefore, in order to analyze these materials, the effect of change in the shell thickness due to the normal strains should be taken into account. In recent years, these finite strain shell elements accounting for thickness change have been presented [1]. The elements use additional degrees of freedom for thickness change and three dimensional constitutive relations. Therefore, their shell elements are addressing the solid models and furthermore introduce one extra degree of freedom for pressure in the case of incompressible materials. However, these additional degrees are a drawback in the recent advances of large scale computational analysis. Contrary to the above theories, the aim of this study is to derive a nonlinear formulation of a shell element for incompressible hyperelastic materials undergoing large elastic strains without any additional degrees of freedom. The uniform deformation in the normal direction due to stretching of the shell middle surface is assumed in the element through the incompressibility condition. The additional state variable, hydrostatic pressure, which occurs for incompressible materials, is eliminated on the element level using the plane stress condition. Thus, the present formulation includes thickness change and hydrostatic pressure implicitly, and uses only 5 d.o.f per node; three translation degrees and two rotation degrees. The presented element provides a consistent tangent stiffness during deformation very efficiently according to this assumed normal strain formulation. The present shell element is based on the method of MITC4[2], and we derive formulation of the assumed shear and normal strains using the mixed variational principle. Several numerical instability analyses such as tube bending and balloon analysis are conducted to illustrate the performance of the shell elements developed herein.

References [1] M. Braun, M. Bischoff, E. Ramm, “Nonlinear Shell Formulation for Complete ThreeDimensional Constitutive Laws Including Composites and Laminates”, Computational Mechanics, 15, 1-18 (1994) [2] D. Chapelle, K.J. Bathe, “The finite element analysis of shells - fundamentals”,

Springer,(2003)

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Stochastic simulation of pitting corrosion J. L´opez De La Cruz∗ , M. A. Guti´errez∗ , L. Koene∗ ∗ Delft

University of Technology, Department of Aerospace Engineering P.O. Box 5058, 2600 GB Delft, The Netherlands [email protected] [email protected] [email protected]

ABSTRACT The simulation of pit initiation is the starting point for the simulation and understanding of stress corrosion cracking (SCC). The corrosion process is a phenomenon where few variables are deterministic. It has been found in previous research that not only the pitting potential is a random variable with a known distribution [1, 2] but also the pit growth rate follows a distribution. The simulation of SCC is a further step. After pit initiation and pit growth are successfully simulated by including their stochastic properties in the analysis , it is possible to obtain reliable insights with respect to the SCC nature. In this paper, the simulation of pit initiation applying the Bernoulli lattice process and the finite elements method (FEM) is shown. As supporting electrolyte, NaCl is selected and iron is the metal undergoing localized corrosion. Anodic and cathodic sites are determined beforehand taking into account the location of the pit and the area surrounding it. In the electrochemical model the reactions considered are the oxidation of iron, oxygen reduction and hydrogen reduction. The Nernst-Planck equation is employed to model the mass transport in the domain (region between the metal and the electrolyte). In the metal boundaries the flux of species is determined by the Butler-Volmer equation. In the electrolyte boundaries the current density in the normal direction is restricted to zero. The current density distribution calculated by the model is used to compute the corrosion intensity (loose of material per unit time). A deterministic point process [3] is used to sample pitting spots and the probability of pitting corrosion is computed assuming that all the points over the metal surface are uniformly distributed with equal probability of becoming anodic. Restrictions are imposed to the metal and physical variables affecting it. The results obtained from this study are a key tool for the analysis of SCC initiation and further behavior.

References [1] Hong, H..P. ”Application of the stochastic process to pitting corrosion”. Corrosion, vol. 55, No. 1, January (1999). [2] Digby D. Macdonald and Mirna Urquidi-Macdonald . ” Distribution functions for the breakdown of passive films”. Electrochimica acta, vol. 31, No. 8, pp. 1079-1086, (1986). [3] Dietrich Stoyan, Wilfrid S. Kendall & Joseph Mecke. Stochastic geometry and its applications. John Wiley & Sons Ltd, Second Print (1995).

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

666

Non Gaussian Response of Bridges Subjected to Turbulent Wind – Effect of the non Linearity of Aerodynamic Coefficients Vincent Denoël*, Hervé Degée* *

University of Liège Chemin des chevreuils, 1, B-4000 Liège (Belgium) [email protected]

ABSTRACT Wind loads acting of bluff bodies like bridge decks are complex functions of the components of the turbulence and of the structural displacements and velocities. In order to simplify the representation of these loads, approached models are generally considered. Since a convenient linear approximation gives accurate results in many cases, such a model has been widely used during last decades. Some researchers have however showed that it is possible, and even necessary, to account for the non linearity of this kind of loading. Such a non linearity is likely to come either from the squared velocity or from the shape of the aerodynamic coefficients as functions of the wind angle of attack. Non linear loading of the first kind (squared velocity) have already been studied in rather recent researches. Some of them are referenced at the end of this abstract. It has been showed that this so-called non linear quasi-steady aerodynamic loading leads to a non Gaussian response of the structure although the components of the turbulence are Gaussian. This has of course significant consequences on the structural design. This paper aims at showing that the second origin of non linear loading terms, i.e. the non linearity of the aerodynamic coefficients, can also have significant consequences on the design of wind-loaded structures. Developments are carried out for bi- and tri-dimensional turbulence fields, and the main conclusions are that these effects are of prime importance of course when the non linearity of the aerodynamic coefficients is important, but also when the transverse component of the turbulence is important. In a more detailed way, the proposed paper intends at presenting two main features. The first one consists in the determination of the statistical characteristics of the loading. In particular it is showed that the traditional linearization of the aerodynamic coefficients may lead to a significant inaccuracy of these statistical characteristics. The second feature is the presentation of a simplified method to derive statistical characteristics of the response from those of the loading. This very simple method is compared with a heavy rigorous analysis. It is concluded that this simple approach can give good estimations of the response that could be used, for instance, for the pre-design of a structure.

References [1] P.D. Spanos, Spectral moment calculation of linear system output. Journal of Applied Mechanics ASLE, 50, 901-903, 1983. [2] Kareem A. and al., Modeling and analysis of quadratic term in the wind effects on structures. Journal of Wind Engineering and Industrial Aerodynamics, 74, 1101-1110, 1998. [3] Gurley K. and al., Analysis and simulation tools for wind engineering. Probabilistic Engineering Mechanics, 12, 9-31, 1997.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

667

Simulation of non-Gaussian stochastic processes and fields with applications to structural engineering problems Mircea Grigoriu Cornell University Ithaca, NY, USA [email protected]

ABSTRACT Three classes of non-Gaussian functions are defined and Monte Carlo algorithms are developed for generating samples of the random functions in these classes. These classes consist of translation random functions, diffusion processes and memoryless transformations of these processes, and spectral
representation based non-Gaussian processes. A translation random function X(t) ∈ Rd , t ∈ Rd , is defined by     Xi (t) = Fi−1 ◦ Φ Gi (t) = hi Gi (t) , i = 1, . . . , d,


where Φ is the distribution of the standard Gaussian variable N (0, 1) and G(t), t ∈ Rd , is an Rd valued stationary Gaussian function  with coordinates  Gi (t), i
= 1, . . . , d, of mean 0, variance 1, and covariance functions ρij (τ ) = E Gi (t + τ ) Gj (t) , τ ∈ Rd . Hence, X is specified by its marginal distribution and second-moment properties. Diffusion processes can be viewed as outputs of dynamic systems to Gaussian white noise. For example, X is said to be a diffusion process if it is defined by the stochastic differential equation dX(t) = a(X(t)) dt + b(X(t)) d(t), t ≥ 0, ¯ where a and b denote the drift and diffusion of X and B is a Brownian motion. Spectral representation based non-Gaussian real-valued processes are defined by  ∞   X(t) = cos(ν t) dM1 (ν) + sin(ν t) dM2 (ν) , t ≥ 0, 0

where M1 and M2 are square integrable martingale. Monte Carlo simulation algorithms are developed for all non-Gaussian functions considered in the paper. The algorithms are simple, efficient, and can be based on MATLAB functions. Numerical examples are used to illustrate the implementation of some of the Monte Carlo simulation algorithms presented in the paper. It is shown that translation functions are versatile and their construction involves relatively relatively simple concepts.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

668

A novel approach for the efficient simulation of highly skewed non-Gaussian stochastic fields Nikolaos D. Lagaros, George Stefanou, Manolis Papadrakakis Institute of Structural Analysis & Seismic Research National Technical University of Athens, 9, Iroon Polytechniou Str., Zografou Campus, GR-15780 Athens, Greece {nlagaros,stegesa,mpapadra}@central.ntua.gr

ABSTRACT The problem of simulating non-Gaussian stochastic processes and fields has received considerable attention recently in the field of stochastic mechanics. This is due to the fact that several quantities involved in practical engineering problems (e.g. material and geometric properties of structural systems, soil properties in geotechnical engineering applications, wind loads, waves) exhibit nonGaussian probabilistic characteristics [2]. In this paper, a novel, computationally efficient method is presented for the simulation of homogeneous non-Gaussian stochastic fields with prescribed target marginal distribution F and spectral density function S ff (N ) [3]. The proposed approach is based on the translation field concept [2], but uses the extended empirical non-Gaussian to non-Gaussian mapping f ( x) F 1 ˜ F [ g ( x)] introduced in [1] for the generation of a non-Gaussian field f ( x ) having the prescribed characteristics. In this way, the possible incompatibility between the marginal distribution and the correlation structure of a translation field is surpassed and an algorithm covering a wider range of non-Gaussian fields is produced. The new algorithm is spectral representation-based as it makes use of the spectral representation method in order to generate sample functions of the underlying Gaussian field g ( x ) . It retains the accuracy characteristics of the method proposed in [1] while drastically reduces the computational effort of the simulation by reliably predicting the unknown Gaussian spectrum S gg (N ) . Precisely the function fitting ability of Neural Networks (NN) is employed to approximate the Gaussian spectrum and the resulting methodology matches the prescribed target marginal distribution and spectral density function with remarkable accuracy even in the case of narrow-banded fields with very large skewness. Various features of the new algorithm are demonstrated with the simulation of two stochastic fields having different correlation structure and following a highly skewed lognormal distribution.

References [1] G. Deodatis, R. C. Micaletti, Simulation of highly skewed non-Gaussian stochastic processes. J. of Engineering Mechanics (ASCE), 127, 1284-1295, 2001. [2] M. Grigoriu, Applied non-Gaussian processes. Prentice-Hall, Englewood-Cliffs New Jersey, 1995. [3] N. D. Lagaros, G. Stefanou, M. Papadrakakis, An enhanced hybrid method for the simulation of highly skewed non-Gaussian stochastic fields. Computer Methods in Applied Mechanics and Engineering, 194, 4824-4844, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

669

The Wave Propagation in a Vertically Inhomogeneou Soil with a Random Dynamic Shear Modulus Mattias Schevenels, Geert Lombaert, Geert Degrande, Daan Degrauwe, Brecht Schoors Department of Civil Engineering K.U.Leuven, Kasteelpark Arenberg 40, B-3001 Leuven, Belgium [email protected] ABSTRACT Vibrations induced by road and rail traffic are a common source of discomfort to people. Numerical models have been developed for the prediction of traffic induced vibrations in the free field or in the built environment. These models consist of a finite element formulation for the vehicles and the buildings and a boundary element formulation that accounts for the wave propagation in the soil. The latter is based on the Green’s functions of a horizontally layered halfspace. The experimental validation of these models reveals a discrepancy between the predicted and measured response in the higher frequency range. Given the crucial role of the Green’s functions in the prediction model, the dynamic soil characteristics governing these functions are a possible source of the discrepancy. Common techniques for the in-situ measurement of the dynamic soil characteristics such as the spectral analysis of surface waves (SASW) test and the seismic cone penetration test (SCPT) are based on local averages of the soil characteristics and have a limited resolution. The small scale variations of the soil characteristics are not revealed. In this paper, the influence of the small scale variations of the dynamic shear modulus on the Green’s functions of a vertically inhomogeneous soil is studied. A probabilistic approach is followed where the mean soil is modelled using the results of the aforementioned measurement techniques. Superimposed on the mean profile is a zero mean random process that represents the small scale variations of the dynamic shear modulus. This process is characterized by a marginal probability density function and a correlation function, estimated by means of a cone penetration test (CPT). The resolution of the CPT test is high as compared to the SASW and the SCPT tests. The non-Gaussian random process is discretized by means of a Hermite polynomial expansion and a Karhunen-Loeve decomposition [1]. The methodology of the stochastic finite element method [2] is applied to a hybrid thin layer – direct stiffness formulation [3] in order to assemble the stochastic system equations. These are solved by means of a Monte Carlo simulation to obtain the stochastic Green’s functions. The results of the simulation are in good correspondence with the discrepancy observed in the validation of the deterministic vibration prediction models. In the low frequency range, the small scale variations of the dynamic shear modulus are not resolved by the waves and all realizations of the stochastic Green’s functions tend to the Green’s functions of the mean soil. In the high frequency range, the waves do resolve the small scale variations. As a result, phase shifts and variations of the amplitude occur between different realizations of the stochastic Green’s functions.

References [1] B. Puig, F. Poirion, and C. Soize. Non-Gaussian simulation using Hermite polynomial expansion: convergences and algorithms. Probabilistic Engineering Mechanics, 17:253–264, 2002. [2] R.G. Ghanem. Ingredients for a general purpose stochastic finite elements implementation. Computer Methods in Applied Mechanics and Engineering, 168:19–34, 1999. [3] E. Kausel and J.M. Ro¨esset. Stiffness matrices for layered soils. Bulletin of the Seismological Society of America, 71(6):1743–1761, 1981.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

670

On the Karhunen-Loeve expansion and spectral representation methods for the simulation of Gaussian stochastic fields George Stefanou, Manolis Papadrakakis Institute of Structural Analysis & Seismic Research National Technical University of Athens, 9, Iroon Polytechniou Str., Zografou Campus, GR-15780 Athens, Greece {stegesa,mpapadra}@central.ntua.gr

ABSTRACT The parameters describing a structure are uncertain quantities and the analysis and safe design of most engineering systems must take into account these uncertainties. Uncertain structural parameters are usually modeled as random fields. Despite the fact that most of the uncertain quantities appearing in practical engineering problems are non-Gaussian in nature (e.g. material and geometric properties, wind loads), the Gaussian assumption is often used due to the lack of relevant experimental data. From the wide variety of methods developed for the simulation of Gaussian stochastic processes and fields, two are most often used in practice: the spectral representation method and the KarhunenLoeve (K-L) expansion. The K-L expansion can be seen as a special case of the orthogonal series expansion where the orthogonal functions are chosen as the eigenfunctions of a Fredholm integral equation of the second kind with the autocovariance as kernel. In this paper, a wavelet-Galerkin scheme is adopted for the efficient solution of the Fredholm equation. A one-dimensional homogeneous Gaussian random field with two types of autocovariance function (power spectrum), exponential and square exponential, is used as test example. The numerical instabilities arising in some cases during the calculation of eigenvalues of both kernels at high wavelet levels ( m ≥ 6 ) are reported. The influence of the scale of correlation on the simulation quality is quantified by using several values of correlation length parameter b. In this work, a comparison of the accuracy achieved and the computational effort required by the K-L expansion and the spectral representation for the simulation of the stochastic field is pursued. The accuracy obtained by the two methods is examined by comparing their ability to produce sample functions that match the target correlation structure and the Gaussian probability distribution or, alternatively, its low order statistical moments (mean, variance and skewness).

References [1] R. Ghanem, P. D. Spanos, Stochastic finite elements: A spectral approach. Springer-Verlag, Berlin, 1991. [2] K. K. Phoon, S. P. Huang S. T. Quek, Implementation of Karhunen-Loeve expansion for simulation using a wavelet-Galerkin scheme. Probabilistic Engineering Mechanics, 17, 293-303, 2002. [3] M. Shinozuka, G. Deodatis, Simulation of stochastic processes by spectral representation. Applied Mechanics Reviews (ASME), 44, 191-203, 1991. [4] G. Stefanou, M. Papadrakakis, Spectral representation versus Karhunen-Loeve expansion for the simulation of Gaussian stochastic fields: A comparative study. Submitted for publication, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

671

Study of Mechanical Properties of Human Skin 1 1 1 1 J.T. Barbosa , R.M. Natal Jorge , M.P.L. Parente , A.A. Fernandes , 2 2 T. Mascarenhas , B. Patrício 1 IDMEC – Pólo FEUP Faculdade de Engenharia da Universidade do Porto Rua Dr. Roberto Frias, 4200-465 Porto {jtrigo,rnatal,mparente,aaf}@fe.up.pt 2 Faculdade de Medicina/Hospital de S. João Alameda Prof. Hernâni Monteiro, 4200-319 Porto [email protected],[email protected]

ABSTRACT Human skin is a complex tissue consisting of several distinct layers, each consisting of their own components and structure. These several layers can be grouped into four layers, namely: stratum corneum dermis, living epidermis, dermis and subcutaneous fat. The main goal of this work is the development of numerical-experimental procedure to evaluate the elasticity of human skin in vivo. Experimental tests are carried out applying a non invasive method, the cutometer aspiration technique, and numerical tests using the ABAQUS software are employed for comparison. Regarding the numerical tests, both three-dimensional and axisymmetric finite element formulations were employed, and the Mooney-Rivlin, Yeoh and Neo-Hookean constitutive material models were found suitable for the present analysis.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

672

Thermophysical Properties of Different Samples of Tissue-Mimicking Materials for Ultrasound Hyperthermia Phantoms 1

2

1

Rodrigo L. Q. Basto , Cláudio S. P. Costa , Marco A.V. Krüger , Wagner C. A. 1 2 2 Pereira , Henrique M. Fonseca and Helcio R. B. Orlande 1

2

Biomedical Engineering Program, COPPE/UFRJ, P. O. Box 68510, 21941-972, Rio de Janeiro, RJ, Brazil [email protected]

Department of Mechanical Engineering - POLI/COPPE/UFRJ P. O. Box 68503, 21941-972, Rio de Janeiro, RJ,Brazil [email protected]

ABSTRACT In the biomedical ultrasonic domain, it is common to use test-objects made of materials that can mimic the main ultrasonic properties of biological tissues (acoustic impedance, wave propagation velocity and attenuation). These devices are called “phantoms” and they are used with several purposes, being the most common the study of image quality control of ultrasonographic equipment. In the last two decades, ultrasound therapy, as a mean to deliver energy to biological tissue in order to produce healing became common practice. This application raises an obvious concern about tissue safety and as a consequence, another kind of phantom is required. Such sort of phantoms must be able to mimic the tissue ultrasound propagation properties and also the heating produced by irradiation of tissues by ultrasound (absorption and thermal conductivity). The present work is aimed towards the development of phantoms for assessment of performance and safety of ultrasonic therapeutic equipments. It consists in measuring and adjusting the thermophysical properties of materials already employed in mimicking acoustic properties of biological tissues. The thermo physical properties of three composition samples measured with a Netzsch Nanoflash LFA 447/1 are presented. These samples are composed of water, glycerin and agar. PVC powder and graphite powder are added to generate the ultrasonic backscattering increasing the attenuation coefficient in order to mimic human soft tissue. The result closest to human tissue was achieved with a mixture containing 5% graphite and 80% PVC.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

673

Different computational approaches in the modeling of wrinkling of biological membranes Cavicchi A.*, Gambarotta L. **, Massabò R. ***

Department of Structural and Geotechnical Engineering, University of Genova, Italy * [email protected] [email protected] *** [email protected]

**

ABSTRACT In [1,2], the authors examined the problem of the wrinkling of plane isotropic biological membranes following the approach of Pipkin [3] and treating the out of plane geometric nonlinearities as constitutive nonlinearities through a modification of the elastic potential. The problem has been solved within the framework of finite strain hyperelasticity for a material characterized by a Fung type constitutive law in biaxial tension. All assumptions of classical Tension Field theory emerge as a result of such formulation. The model formulated in [1,2] is able to identify the distinct regions of behavior that characterize the response of stretched membranes: taut (biaxial tension), wrinkled (uniaxial tension) and slack (inactive). However, the assumption of zero bending stiffness does not allow for detailed predictions of the deformation fields of real membranes where wrinkles with finite magnitude and wavelength develop. This aspect of the problem has been highlighted by Cerda [4] where the limits of the Tension Field Theory [5] in the description of the wrinkling phenomenon have been discussed referring to the problem of a stretched annular membrane. In this paper simulations of reconstructive surgical procedures are presented where wrinkling of the skin occurs during and after the suture of the wound edges leading to unaesthetic scars with dog-ear formations. The effects of the natural tension of the skin on the wrinkling extension and final stress field are highligthed. A critical evaluation of the Tension Field assumptions is then made by considering the problem of the annular membrane subject to inner and outer uniform tractions, already examined by Cerda [4]. The results of the Tension Field theory are compared with the results of a buckling analysis. In the problem studied and under certain loading conditions, the extension of the wrinkled region and the number of circumferential waves predicted by the buckling analysis prove to be independent of the elastic constants and the bending stiffness of the membrane. This allows for a quantitative comparison with the results of the Tension Field theory. The comparison shows analogies and differences between the onset of membrane instability and the simplified description of the post-buckling configuration given by the Tension Field theory.

References [1]

[2] [3] [4] [5]

Gambarotta, L., Massabò, R., Morbiducci, R., Raposio, G., and Santi, P., In vivo experimental testing and model identification of human scalp skin, Journal of Biomechanics, 38, 2237-2247, 2005. Gambarotta, L., Massabò, R., Wrinkling of plane isotropic biological membranes, Journal of Applied Mechanics, in press, 2006. Pipkin, A.C., Relaxed Energy Density for large deformations of membranes, IMA J. Appl. Math., 52, 197-308, 1994. Cerda, E., Mechanics of scars, Journal of Biomechanics, 38, 1598-1603, 2005. Mansfield, E.H., The bending and stretching of plates, Second ed.Cambridge University Press., Cambridge, 1989.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

674

Dynamic study of the middle ear F Gentil(1), RM Natal Jorge(2), AJM Ferreira(3), MPL Parente(4), M Moreira(5), E Almeida(6) (1)

(2) (4)

Escola Superior de Tecnologia da Saúde do Porto Clínica ORL - Dr Eurico Almeida, Widex [email protected]

IDMEC-Polo FEUP, Faculdade de Engenharia, Universidade do Porto (2) [email protected] (4) [email protected] (3) (5)

INEGI, Faculdade de Engenharia, Universidade do Porto (3) [email protected] (6)

Clínica ORL - Dr Eurico Almeida [email protected]

ABSTRACT The human ear is a complex biomechanical system and is divided by three parts: outer, middle and inner ear. When a sound is made outside the outer ear, the sound waves travel down the external auditory canal and strike the eardrum. It vibrates and the vibrations pass through three tiny bones in the middle ear called the ossicles (malleus, incus and stapes). The ossicles amplify the sound and send the sound waves to the inner ear and into the fluid filled hearing organ (cochlea), by the oval window. However, the ossicles can suffer from several damages, for example, the Otosclerosis, being a need the application of mechanical prosthesis, by chirurgic intervention, keeping the right travel of sound wave. In order to study the implementation of prosthesis, it is very important to achieve a correct modelling of the vibro-acoustic behaviour of middle ear. In this work, a finite element modelling of the middle ear will be done. For this proposes, a dynamic study will be presented by using the ABAQUS program. The model will include the different ligaments of the support structure. A hyperelastic behaviour of this ligaments will be taken into account [1] and for the ossicles and eardrum the mechanical properties available in the literature [2] will be considered. The connection between ossicles will be done by using contact formulation. The eigenvalues and the eigenvectors will be carried out.

References [1] P.A.L.S. Martins, R.M. Natal Jorge, A.J.M. Ferreira, A.A. Fernandes, M. Figueiredo, R. Silva, “Modelling the mechanical behavior of soft tissues using hyperelastic constitutive models”, Proc. of II Int. Conf. on Comp. Bioengineering, ICCB2005, H Rodrigues et al. (Eds.), pp.403-410, Instituto Superior Técnico, Lisboa, 2005. [2] PJ Prendergast, P Ferris, HJ Rice, AW Blayney, Vibro-acoustic modeling of the outer and middle ear using the finite element method, Audiol Neurootol, 4, 185-191, 1999.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

675

Image-based Inverse Problems to Identify Three-dimensional Displacement Field Shouji Kuzukami*, Nobuhiro Yoshikawa†, Osamu Kuwazuru† *

Graduate Student, The University of Tokyo 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505 [email protected] †

Institute of Industrial Science, The University of Tokyo 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505 [email protected]

ABSTRACT An identification methodology of three-dimensional displacement within a biological soft tissue is presented in line with non-invasive testing by means of X-ray CT images. The high-compliance and heterogeneity of biological soft tissue induces its complex and nonlinear displacement field hardly measured by conventional methods, although the displacement field is necessary for determinations of its material properties. The full-field digital image correlation method[1] with the LevenbergMarquardt method [2] has been established to identify the displacement field by using a pair of twodimensional digital images with crisp contrast in its intensity distribution. The proposed approach is an extension of the method to three-dimensional problems. A pair of three-dimensional images, which are constituted from multi-slice CT images captured with small intervals, is obtained from the deformed and undeformed states of a soft tissue. To identify the displacement field, the undeformed state image is virtually deformed by a tentative displacement field described by the tri-cubic B-Spline basis functions with unknown parameters initially set to tentative values. The error of this identification is evaluated in terms of intensity difference between actually and virtually deformed images. The unknown parameters are successively modified to minimize the error. Primary obstacle against the three-dimensional image correlation is an explosion of computational cost for solving the huge simultaneous equations successively constructed during the error minimization procedure. We reduce the calculation time to a reasonable one by utilizing the numerical symmetry of the coefficient matrix and the locality of the basis functions. Thus, a fast parallel solver for the inverse problem is proposed to minimize the computational time. In this problem, moreover, an indeterminacy of the displacement field arises from noisy images with obscure contrast in its intensity distribution, as is the common case with the X-ray CT images. In this study, therefore, we propose a formulation stabilized by imposing incompressibility of the materials. By using an experimental specimen under compression load, the validity of the algorithm is checked.

References [1] P. Cheng, et. al. , Full-field Speckle Pattern Image Correlation with B-Spline Deformation Function. Experimental Mechanics, 42-3, 344-352, 2002. [2] W. H. Press, et. al. , Numerical Recipes, The Art of Scientific Computing. Cambridge University Press, U.S.A., 1986.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

676

Experimental Study of the Middle Ear Biological Support Structures Pedro S. Martins∗1 , Renato N. Jorge∗2 , Antonio M. Ferreira† , Fernanda Gentil‡ ∗ IDMEC-Polo

FEUP, Faculdade de engenharia da Universidade do Porto Rua Dr.Roberto Frias, 4200-465 Porto, Portugal 1 [email protected] 2 [email protected]

† DEMEGI,

Faculdade de engenharia da Universidade do Porto Rua Dr.Roberto Frias, 4200-465 Porto, Portugal [email protected]

‡ Escola Superior de Tecnologia da Saude do Porto Rua Joo de Oliveira Ramos, 87, 4000-294 Porto, Portugal [email protected]

ABSTRACT During the last few years, the effort for a better understanding of biological systems from a structural and physiological perspective, lead to the appearance of several scientific works concerning the biomechanical study and modeling of different human, and also animal structures. Biological systems are in general, the assembly of a given number of distinguishable components. In this context, the middle ear can be modeled as a mechanical system composed by several linked elements [1] (ossicles-Stirrup, Anvil and Hammer, and Eardrum). The simulation’s accuracy rely on the material parameters that are fed to the numerical approximation method (ex. FEM, BEM,...). The material parameters however, can only be obtained by a direct or indirect [2] measurement procedure, otherwise, the simulations may lead to results widely away from the physiological reality of the system in focus. In previous studies of the middle ear [3], the mechanical properties have been established and used in linear behavior context, assumption that is questionable due to widely common nonlinearity of biological materials [4]. The authors propose in this paper a experimental approach for the determination of the mechanical properties of middle ear’s support structures. This procedure assumes from a starting point the (possible) nonlinearity of these structures, which is an important step to increase the accuracy of the mechanical simulations, and ultimately may lead to a better understanding of middle ear’s physiology.

References [1] F. Gentil, R. Natal, A. Ferreira, M. Parente, M. Moreira and E. Almeida, Biomechanical Study of Middle Ear. Proc. COMPLAS VIII, 2, 785–788, Barcelona, 2005. [2] J. Fay, S. Puria, W. F. Decraemer and C. Steele, Three approaches for estimating the elastic modulus of the tympanic membrane. Journal of Biomechanics, 38, 1807–1815, 2005. [3] P. Ferris and P. J. Prendergast, Middle-ear dynamics before and after ossicular replacement. Journal of Biomechanics, 33, 581,2000 [4] Y. C. Fung, BIOMECHANICS-Mechanical Properties of Living Tissues, Second Edition. SpringerVerlag, 1993

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

677

A Real-time FEM Simulation for Cutting Operation using Haptic Device Tomoyuki Miyashita*, Hiroshi Yamauchi†, Masatomo Inui†, Hiroshi Yamakawa* *

Department of Mechanical Engineering, WASEDA University in JAPAN 3-4-1, Okubo, Shinjuku-ku, Tokyo 169-8555, JAPAN [email protected], [email protected] †

Department of Systems Engineering, IBARAKI University in JAPAN 4-11-1, Nakanarusawa-cho, Hitachi-shi, Ibaraki 316-8555, JAPAN [email protected], [email protected]

ABSTRACT A Computational simulation for cutting of materials is one of the difficult problems mainly because of time consumable calculation of finite elements method. Especially, the boundary and loading conditions may change during simulation and re-mesh procedures are often necessary to adopt above situations. However, considering haptic device to obtain dynamical information from FEM analysis, it is difficult to apply the formal FEM procedures described above from the point view of response time. There were previous studies to propose the approximation procedures to omit remesh procedures and obtain the approximated dynamical response using FEM analysis within allowable computer resources. Recently, the computer power is gradually improved and then we can propose new approximation procedure to make full use of computer resources to treat the cutting simulation. The cutting simulation is useful to discuss about surgical simulation from the point of view of dynamical properties of human organs or tissue structure. In this study, we will review the previous studies treating same problem using FEM analysis and propose the method to treat the cutting simulation considering real time computation and simulation results are directly transferred to the user through the haptic device. The hex and tetra elements were used to model the structure using FEM analysis and dynamical response was calculated using newmark E method. Here, the elements matrix was normalized according to the distance between a cutting device and nodes to omit the re-mesh procedures. The proposed method was implemented using three threads that handle graphics for display, dynamical calculation and model construction to improve the response for user operation. Then, we have developed the simulation system composed of the haptic device (PHANToM force feedback device) using ToolKit and the proposed method including graphics animation. Using the haptic device, we have been able to discuss about the obtained feeling through the device and obtain experimental results and compared from the points of view of the previous simple calculation method and the proposed method, the obtained reaction forces. We could confirm that the performances of the developed system using thread is very good and effective for the further improvement using detail calculation on FEM, the qualitative feelings and quantitatively obtained reaction forces are different among the compared method. Then, we could confirm the some properties and the effectiveness of the proposed method and the developed system.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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The biomechanical behavior of the pelvic floor muscles during a vaginal delivery M. Lages Parente1,2 , R. Natal Jorge1 ,T. Mascarenhas3 A.A.Fernandes1 , J.A.C. Martins4 1 Department

2

of Mechanical Engineering, Faculty of Engineering, University of Porto Rua Dr. Roberto Frias, 4200-465 Porto, Portugal {mparente,rnatal,aaf}@fe.up.pt

Faculty of Architecture and Arts, University Lusiada of Porto Rua Dr. Lopo de Carvalho 4369-006 Porto, Portugal 3 Faculty of Medicine, University of Porto Al. Prof. Hernni Monteiro 4200 - 319 Porto, Portugal [email protected] 4 Department

of Civil Engineering and ICIST Instituto Superior Tcnico Av. Rovisco Pais, 1049-001 Lisboa, Portugal [email protected]

ABSTRACT The women pelvic floor extends itself on the lower part of the pelvic cavity like a net to give support to the organs in the abdominal cavity. One of the most important muscles of the pelvic floor, that gives support to the pelvic organs, namely, to the urethra, vagina and the rectum, is the levator ani. The levator ani muscles actively support the pelvic contents, compressing the urethra and vagina by elevating the pelvic floor. Relaxation of these muscles allows evacuation of the bladder and rectum [1]. Using a finite element model for the pelvic floor, built using the dataset made public in [2], we present the preliminary results for the simulation of a birth trough vaginal delivery, in vertex presentation [3]. We present the results obtained for the stresses that appear on the the ligaments that connect the pelvic floor muscles to the coccyx, before the movement of extension [3]. The objective of this research is to help predict the damage to the pelvic floor that can occur during childbirth.

References [1] P.E. Papa Petros, The Female Pelvic Floor, Function, Dysfunction and Management Accordind to the Integral Theory, Springer Medizin Verlag, 2004. [2] S. Janda, F.C.T. Van der Helm, and S.B. de Blok, Measuring morphological parameters of the pelvic floor for finite elements modelling purposes. Journal of Biomechanics, 36, 749–757, 2003. [3] Alan H. DeCherney, Lauren Nathan, Current Obstretic & Gynecologic, Diagnosis & Treatment, Ninth Edition. Lange Medical Books/McGraw-Hill, 2003.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

679

Buckling Analysis of Unbranched Thin-Walled Members: Generalised Beam Theory and Constrained Finite Strip Method Sándor Ádány1, Nuno Silvestre2, Ben Schafer3 and Dinar Camotim2 1

2

Budapest University of Technology and Economics, Department of Structural Mechanics 1111 Budapest, MĦegyetem rkp. 3, Hungary [email protected]

Technical University of Lisbon, Department of Civil Engineering and Architecture, ICIST/IST Av. Rovisco Pais, 1049 Lisboa, Portugal {nunos, dcamotim}@civil.ist.utl.pt 3

Johns Hopkins University, Department of Civil Engineering Latrobe Hall 210, Baltimore, MD 21218, USA [email protected]

ABSTRACT The load-carrying capacity of thin-walled members is often governed by buckling phenomena. Usually, three main families of buckling phenomena/modes are considered: (i) global buckling, in which the member axis deforms (e.g., flexural or lateral-torsional buckling), (ii) local-plate buckling, involving only plate (wall) bending, and (iii) distortional buckling, combining wall bending with crosssection distortion  the last two phenomena are sometimes jointly described as “local buckling”. Although there exist several numerical and/or analytical methods to determine the buckling load/moment values and the associated buckling mode shapes, it is fair to state that only generalised beam theory (GBT) and the constrained finite strip method (cFSM) are able to perform this task for isolated (“pure”) or arbitrarily combined (“coupled”) modes. Although both methods lead to very similar solutions, (i) GBT is a generalisation of classical beam theories that includes additional degrees of freedom to allow for cross-section deformation, whilst (ii) cFSM is a specialisation of the classical plate theory that carefully selects constraints in order to force the member to deform (buckle) according to pre-defined configurations. This paper provides an in-depth comparison between the fundamentals of the two above approaches (GBT / cFSM), focusing on (i) their mechanical assumptions and domains of application, and (ii) the procedures adopted. This will contribute to a better understanding of both methods and the phenomena that they aim to uncover, thus paving the way to the development of more efficient tools for the analysis and design of thin-walled members. In order to illustrate the GBT / cFSM comparison, the local, distortional and global buckling behaviours of lipped channel columns (see Figure 1) and beams are analysed in detail. As one would expect, there is a virtually perfect coincidence between the two sets of buckling results. Figure 1: Variation of the buckling load Pb with the column length L

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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Lateral-Torsional Buckling Analysis of Web-Tapered I-Beams Using Finite Element and Spline Collocation Methods Anísio Andrade*, Paulo Providência e Costa*, Dinar Camotim† *

Civil Engineering Department, FCT, University of Coimbra Rua Luís Reis Santos – Pólo II da UC, 3030-788 Coimbra, Portugal {anisio,provid}@dec.uc.pt †

Department of Civil Engineering and Architecture, ICIST/IST, Technical University of Lisbon Av. Rovisco Pais, 1049-001 Lisboa, Portugal [email protected]

ABSTRACT Tapered members are widely used in the steel construction industry, because of their structural efficiency, ability to meet architectural and functional requirements and competitive fabrication costs. According to the current design codes, the load-carrying capacity of laterally unsupported beams (either prismatic or tapered) bent in their stiffer principal plane is estimated on the basis of their cross-sectional and elastic lateral-torsional buckling resistances. Bearing this in mind, two of the authors have recently proposed and validated a one-dimensional model to characterise the elastic lateral-torsional buckling behaviour of tapered thin-walled open beams [1]. This model can be described as a kinematically constrained version of a nonlinear shell model and, from a mathematical viewpoint, it amounts to a self-adjoint eigenvalue problem for a system of ordinary differential equations and separated boundary conditions. Numerous numerical methods are available for the solution of such an eigenvalue problem. Among the structural engineering community, the finite element method (FEM) is unquestionably the most popular one. The FEM is based on a variational (or weak) form of the problem, which involves lower-order derivatives than the classical (or strong) form and, therefore, poses less stringent continuity requirements. Moreover, it allows for a very simple and flexible geometrical description of irregular-shaped domains. However, these two features, which are responsible for the key role played by the FEM in solving boundary value problems for partial differential equations, are not essential in the case of ordinary differential equations: the construction of a high-order spline basis, for instance, is more or less straightforward and special geometrical flexibility is not needed. Therefore, it appears to be worth investigating other numerical approaches and, in this context, the collocation method seems particularly promising. The paper begins by specialising the one-dimensional mathematical model for doubly symmetric webtapered I-section cantilevers acted by tip point loads. Two distinct numerical approaches are then considered, namely (i) the development of a conforming displacement finite element model, tailor-made for this specific problem, and (ii) the use of a general purpose code (COLSYS) based on spline collocation at Gaussian points and designed to solve nonlinear multi-point boundary value problems for mixed-order systems of ordinary differential equations. The paper closes with the presentation of some numerical results and an appraisal of the relative merits of the two approaches.

References [1] A. Andrade and D. Camotim, Lateral-torsional buckling of singly symmetric tapered beams: theory and applications, Journal of Engineering Mechanics (ASCE), 131(6), 586-597, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

681

Stability of telescopic props for temporary structures João André, António M. Baptista Laboratório Nacional de Engenharia Civil (LNEC) Av. Brasil 101 - 1700-066 Lisboa - Portugal [email protected] Laboratório Nacional de Engenharia Civil (LNEC) Av. Brasil 101 - 1700-066 Lisboa - Portugal [email protected]

ABSTRACT Telescopic props represent one of the most common temporary structural elements used in the construction of buildings, to support the formwork. The design of these props is often associated to high safety factors, due to insufficient information about their real behaviour at the construction site, under the influence of load eccentricities and geometric imperfections. A research project is now being developed at the Portuguese National Laboratory of Civil Engineering (LNEC), involving experimental and numerical studies of the props behaviour and, in particular, of the effects of the geometric imperfections and corresponding tolerances on their stability. The numerical studies will take in account the geometric and material nonlinearities affecting the props resistance. The influence of the base plates stiffness will be analysed. This paper describes the results obtained in numerical studies carried out during an initial stage of this project, as well as their interpretation and subsequent conclusions.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

682

Formulation of a GBT-Based Finite Element to Analyse the Global Buckling Behaviour of Plane and Spatial Thin-Walled Frames Cilmar Basaglia, Dinar Camotim and Nuno Silvestre

Department of Civil Engineering and Architecture, ICIST/IST, Technical University of Lisbon, Av. Rovisco Pais, 1049-001 Lisboa, Portugal {cbasaglia, dcamotim, nunos}@civil.ist.utl.pt

ABSTRACT This paper deals with the use of Generalised Beam Theory (GBT) to analyse the global buckling behaviour of plane and spatial thin-walled frames. After a brief presentation of the main concepts and procedures involved in the performance of a GBT buckling analysis, one presents in detail the formulation and numerical implementation of a GBT-based beam finite element that includes only the four rigid-body deformation modes  namely, one describes the determination of the elementary and frame linear and geometric stiffness matrices (the latter incorporate the influence of the frame joints and boundary conditions). Particular attention is paid to issues concerning (i) the quantification of the warping transmission at the frame joints, (ii) effects stemming from the non-coincidence of the member centroidal and shear centre axes (cross-sections without double symmetry), and (iii) the definition of joint elements that relate the connected member GBT degrees of freedom to the joint generalised displacements. Next, one addresses kinematical models to simulate the warping transmission at frame joints connecting two or more non-aligned U and I-section members and exhibiting two different configurations (diagonal-stiffened and box-stiffened). Finally, in order to illustrate the application and capabilities of the proposed GBT-based finite element formulation, one presents and discusses numerical results concerning the global buckling behaviour of (i) an “L–shaped” frame (see Fig. 1), (ii) a pitched-roof plane frame (in-plane and spatial behaviours) and (iii) a three-bar simple spatial frame, acted by loadings that cause only member compression. Both diagonal-stiffened and box-stiffened joints are considered and, for validation purposes, most of the GBT-based results are compared with values yielded by beam finite element analyses carried out in the commercial code ANSYS. An excellent correlation, involving both the frame critical buckling loads and mode shapes, was found in all cases.

Figure 1: “L– shaped” plane frame global buckling: deformed configurations of the member mid-span cross-sections.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

683

GBT-Based Finite Element Formulation to Analyse the Buckling Behaviour of Thin-Walled Members Subjected to Non-Uniform Bending Rui Bebiano, Nuno Silvestre and Dinar Camotim Department of Civil Engineering and Architecture, ICIST/IST, Technical University of Lisbon Av. Rovisco Pais, 1049-001 Lisboa, Portugal e-mails: {rbebiano, nunos, dcamotim}@civil.ist.utl.pt

ABSTRACT In this paper, one investigates the local-plate, distortional and global buckling behaviour (critical bifurcation loads and buckling mode shapes) of thin-walled steel beams subjected to non-uniform bending moment diagrams, i.e., under the presence of longitudinal stress gradients. In order to achieve this goal, one begins by developing and numerically implementing a beam finite element formulation based on Generalised Beam Theory (GBT), which (i) can handle beams with arbitrary open cross-sections and (ii) incorporates all the effects stemming from the presence of longitudinally varying stress distributions. After presenting the main concepts, procedures and assumptions involved in the above formulation, one addresses the derivation of the equilibrium equation system that needs to be solved in the context of a GBT buckling analysis. Particular attention is devoted to the main steps involved in the determination of the elementary linear and geometric stiffness matrices, as they must incorporate the stiffness reduction stemming from the presence of the non-uniform bending moments (longitudinal stress gradients) and also of the pre-buckling shear stresses caused by them – the inclusion of this last effect constitutes an original contribution within the context of GBT buckling analyses. Then, in order to illustrate the application and capabilities of the proposed GBT-based finite element formulation, one presents and discusses numerical results concerning thin-walled steel Ibeams acted by various (uniform and non-uniform) bending moment diagrams. In particular, one analyses (i) cantilevers subjected to uniform major axis bending (Fig. 1(a)), tip point loads (Fig. 1(b)) and uniformly distributed loads (Fig. 1(c)), as well as (ii) simply supported lipped beams subjected to uniform major axis bending, mid-span point loads and uniformly distributed loads í by taking full advantage of the GBT modal features, one is able to acquire a much deeper understanding about the influence of the longitudinal stress gradients and shear stresses on the beam local and global buckling mode shapes. For validation purposes, some GBT-based critical loads/moments and buckling mode shapes are compared with values either (i) yielded by shell finite element analyses, performed in the code ANSYS, or (ii) reported in the literature. Finally, one assesses the computational efficiency of the buckling analyses carried out using the proposed GBT-based beam finite element, by comparing the number of degrees of freedom involved with those required to obtain equally accurate results with discretisations in shell finite elements (note that “uniform stress” GBT-based beam finite elements are no longer applicable).

(a)

(b)

(c)

Figure 1: Local-plate buckling mode shapes of I-section cantilevers subjected to (a) uniform major axis bending, (b) a tip point load and (c) a uniformly distributed load.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

684

A Non-Linear 3-D Beam Finite Element for the Study of Steel Frames with Tapered Members N. Boissonnade*, H. Degée† * University of Liège, M&S Department Chemin des Chevreuils, n°1, Bât B52/3 B-4000 Liège Belgium [email protected] †

University of Liège, M&S Department Chemin des Chevreuils, n°1, Bât B52/3 B-4000 Liège Belgium [email protected]

ABSTRACT In recently build structures, instability problems have become of prime importance, mainly because of the general tendency to increase the structural and members slenderness, coupled with the development of high strength steels. In that way, tapered I-members become more and more usual since they allow significant material savings and a consistent design. Indeed, because such members are built-up from several plates, designers get the possibility to spread out the material in the most efficient way over the cross-section and over the beam length, thus leading to a rather economical distribution of the material. Such members are particularly efficient in steel portal frames of medium and long span with no in-plane bracing system, where the increase in the fabrication cost is more than compensated by the decrease in weight when compared to rolled sections. Despite these advantages, the use of tapered members suffers from the lack of appropriate simple but accurate design formulae in most codes of practice. Then, design solutions resorting to tapered elements are often given up, because the only available approaches and/or recommendations consist in rough elastic design formulae, where taper effects are rather badly accounted for. Consequently, the derivation of successful sets of design formulae for tapered members would be helpful and useful. Checking both accuracy and safety of such rules over a wide range of parameters implies the availability of a specific numerical tool (i. e. through the finite element method). Once such a set of reference results would be made available, then the ability of several proposals to be efficient in daily use could be easily checked. Present paper intends at filling this gap. First, the background of a fully geometrically and materially non-linear tapered 3-D finite element is given. It consists in a 2-nodes beam finite element, with 7 degrees of freedom per node, written in total lagrangian corotational description. Then, as a validation study, results for all types of static analyses are presented, from the basic case of linear elastic behaviour to full non-linear analysis. They show that the use of the new beam element provides accurate results, in close agreement with those provided by shell elements, and that resorting to a so-called “segmentation technique” (stepped beams) can lead to significant mistakes. Finally, a numerical comparison between shell and beam modelling of a tapered steel frame is presented. The use of the beam tapered elements is found to be accurate and convenient: the number of degrees of freedom is limited when compared to a shell model, while accuracy remains reasonable for engineering purposes. The new tapered beam element then allows an easy meshing, a short computation time and an straightforward results interpretation.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

685

Non-Linear Dynamical Response of Steel Portal Frames with Semi-Rigid Connections Rafael A. Castro1, José Guilherme S. da Silva2, Pedro C. G. da S. Vellasco3, Sebastião A. L. de Andrade4, Luciano R. O. de Lima5 and Luis F. da C. Neves6 1

MSc Student in Civil Engineering, Faculty of Engineering State University of Rio de Janeiro, UERJ, Brazil [email protected] 2 Mechanical Engineering Department State University of Rio de Janeiro, UERJ, Brazil [email protected]

3,4,5 Structural Engineering Department State University of Rio de Janeiro, UERJ, Brazil [email protected]; [email protected]; [email protected] 6

Civil Engineering Department University of Coimbra, Portugal [email protected]

ABSTRACT Traditionally, the steel portal frame design assumes that beam-to-column joints are rigid or pinned. Rigid joints, where no relative rotations occur between the connected members, transfer not only substantial bending moments, but also shear and axial forces. Alternatively, pinned joints are characterised by an almost free rotation movement between the connected elements preventing the bending moment transmission. Despite these facts, it is largely recognised that the great majority of joints does not exhibit such idealised behaviour. These joints, called semi-rigid, should be designed according to their actual structural behaviour. Considering all these facts one of the main objectives of this investigation is to propose a modelling strategy to represent the dynamical behaviour of semirigid joints under dynamic actions. The developed finite element model included geometric nonlinearities and considered the influence of non-linear and hysteretic joint stiffness. The updated Lagrangean formulation is used to model the geometric non-linearity. The mathematical model calibration was made based on comparisons to semi-rigid tests and other numerical models [1,2] and proving to be in accordance to them. However, it must be emphasized that cautions should be taken on the direct use of the results in structural design. The main reasons for this affirmative are related to the occurrence of very important distortions due to the consideration of the semi-rigid joints geometric non-linearity effects on the steel portal frames dynamical response.

References [1] P.P.T Chui and S.L. Chan, Transient response of moment-resistant Steel frames with flexible and hysteretic joints. Journal of Constructional Steel Research, Vol 39, pp 221-243, 1996. [2] J.G.S. da Silva, P.C.G. da S. Vellasco, S.A.L. de Andrade, L.R.O. de Lima and R. de K.D Lopes, A dynamical parametric analysis of semi-rigid portal frames, The Ninth Int. Conference on Civil and Structural Engineering Computing, CC 2003, Netherlands, Civil Comp Press, pp. 1-17, 2003.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

686

Buckling Analysis of Stiffened Composite Panels Nian-Zhong Chen, C. Guedes Soares Unit of Marine Technology and Engineering, Technical University of Lisbon, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal {chennianzhong, guedess}@mar.ist.utl.pt

ABSTRACT In view of high specific strength and specific modulus, stiffened composite panels have been widely used in modern engineering fields. However, stiffened composite panels can develop buckling failure under service conditions as they are generally very thin. The buckling strength of stiffened composite panels is usually sensitive to the variation of boundary conditions, stacking sequences and lamina thickness. In order to permit stiffened composite panels to be designed efficiently with high reliability and safety against buckling, a parametric study to investigate the effects of boundary conditions, stacking sequences and lamina thickness on buckling strength of stiffened composite panels with various types of stiffeners is presented in the paper. The accurate buckling analysis of stiffened composite panels can be achieved by an incremental nonlinear finite element analysis. However, the complete incremental nonlinear solution of a stiffened composite panel up to buckling is in general expensive and thus a linearized buckling analysis for lowest buckling loads based on an updated Lagrangian formulation with the degenerated shell element and an explicit through-thickness integration scheme is presented in the paper. In this method, it is assumed that the pre-buckling deformations of the structure are small and buckling analysis of stiffened composite panels is considered as an eigenvalue problem. The numerical accuracy of the linearized buckling analysis has been proven to be effective by comparison with the experimental data of three graphite-epoxy stiffened panels with “ blade ” and “ I ” section stiffeners and two GRP hat section stiffened panels. After analysis, the results of the parametric study shows that boundary conditions, stacking sequences and lamina thickness generally have significant and distinct influence on buckling strength of stiffened composite panels. However, the buckling strength of stiffened composite panels will increase significantly with the increase of the lamina thickness while the effects of stacking sequences on buckling strength vary with various boundary conditions and the influence of boundary conditions on buckling strength varies with various stacking sequences.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

687

Interactive buckling of thin-walled rectangular hollow sections – a comparison between modified beam models and shell finite elements Hervé Degée*, Nicolas Boissonnade* *

University of Liège Chemin des chevreuils, 1, B-4000 Liège (Belgium) [email protected]

ABSTRACT In nowadays steel construction, thin-walled members have become very common through the use of both welded and cold-formed profiles. This implies that phenomena such as local buckling of the walls or distortional buckling of the section require a due attention. As soon as it comes to computational approaches, different methods are available to model thinwalled members and study their behavior. Among others, we can point out the finite strip method, the use of shell finite elements or the recently revived Generalized Beam Theory (GBT). All of these have shown their ability to provide accurate results, but exhibit of course typical advantages and drawbacks. In particular, almost none of them is suitable for an efficient analysis of a whole structure, such as for example a bridge or a tall building. Indeed only the use of shell finite elements would allow such an analysis, but with an unavoidable increase of the size of the problem leading to difficulties in the elaboration of the model, very long computation time and great amount of results to manage. In this context, a special beam finite element accounting for a possible deformation of the cross section is developed. In classical beam elements, the cross section is assumed to be fully rigid and the consecutive displacement field of the element is well known. In this proposal, it is assumed that the displacement of any point of the element is described by the classical beam displacement together with an additional local displacement field representing the local and distortional behavior. In this paper, we present an application of this special element to the modeling of thin-walled rectangular hollow profiles, in comparison with the use of shell FE models. In particular, the following topics are discussed: - Computation of critical bifurcation loads; - Geometrically non linear analysis of an elastic member susceptible to local and global buckling; - Geometrically and materially non linear analysis of a member susceptible to local and global buckling. In the case of non linear computations, some simplifications are proposed in order to avoid the need of an explicit description of the second order local membrane displacement field. The following conclusions can be drawn: - The results show a good agreement between the different models until the maximum load regarding both the evaluation of this maximum load but also the evaluation of the loss of axial stiffness. In particular, the modeling of short columns is very accurate; - The proposed simplifications lead to underestimate the coupling between local and global transverse effects in members with high global slenderness in both pre- and post-buckling range.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

688

Design and Analysis of Composite Panels R. Degenhardt, A. Kling, K. Rohwer DLR, Institute of Composite Structures and Adaptive Systems, Lilienthalplatz 7, 38108 Braunschweig, Germany [email protected]

ABSTRACT European aircraft industry demands for reduced development and operating costs, by 20% and 50% in the short and long term, respectively. The European Commission project POSICOSS, which lasted from January 2000 to September 2004 and the 4-year follow-up project COCOMAT, which started in January 2004, contribute to this aim [1]. Both projects are under the co-ordination of DLR, Institute of Composite Structures and Adaptive Systems. They reduce structural weight by exploiting considerable reserves in primary fibre composite fuselage structures through an accurate and reliable simulation of postbuckling and collapse. The POSICOSS team developed fast and reliable procedures for postbuckling analysis of fibre composite stiffened panels, created experimental data bases and derived design guidelines. The COCOMAT project builds up on the POSICOSS results and goes beyond by simulation of collapse. The project improves existing tools as well as design guidelines for stiffened panels taking skin stringer separation and material degradation into account and it creates a comprehensive experimental data base. The improved tools, developed within the POSICOSS and COCOMAT project, have to be validated by test results. Since appropriate test data bases were not available, both projects were constrained to create new experimental data bases for curved stringer stiffend CFRP panels. To that end suitable panels are designed, manufactured, inspected and tested under own project objectives. Each project differentiates between verification panels and industrial panels. The verification panels are designed as to specific limiting aspects of application of the software to be verified, e.g. small or large stiffness reduction in the postbuckling regime. These panels should have a significant postbuckling range up to collapse and have an early onset of degradation. The industrial panels were designed in regard to industrial applications, mainly by existing procedures used in day-to-day industrial design practice. For the analysis of the panels the partners utilized different finite element software tools. This paper focuses on the experience of DLR on the design and analysis of stringer stiffened CFRP panels gained in the frame of the POSICOSS and COCOMAT projects. Geometrical nonlinear computations up to collapse were performed applying the software ABAQUS/Standard. The material was assumed linear elastic. The onset of degradation of the structure and the skin-stringer connection was determined using different failure criteria. Results achieved so far will be presented and an outlook towards future activities will be given.

References [1] Degenhardt R., Zimmermann R., Rolfes R., Rohwer K., “Improved Material Exploitation of Composite Airframe Structures by Accurate Simulation of Postbuckling and Collapse – The projects POSICOSS and COCOMAT”, Proceedings of the 11th Australian International Aerospace Congress“, Melbourne, Australia, 13-17 March, 2005

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

689

On the Use of Shell Finite Element Analysis to Assess the Local Buckling and Post-Buckling Behaviour of Cold-Formed Steel Thin-Walled Members Pedro B. Dinis and Dinar Camotim Civil Engineering Department, ICIST/IST, Technical University of Lisbon Av. Rovisco Pais, 1049-001 Lisboa, Portugal {dinis,dcamotim}@civil.ist.utl.pt

ABSTRACT This paper deals with the use of shell finite element analyses to assess the (i) elastic bifurcation and (ii) elastic and elastic-plastic local-plate and distortional post-buckling behaviours of cold-formed steel thin-walled members (mostly columns, i.e., uniformly compressed members)  all the geometrically and physically non-linear analyses are performed using the code ABAQUS and adopting 4-node isoparametric shell elements to discretise the members. First, one addresses several relevant issues concerning (i) the member discretisation (shell element type and mesh refinement), (ii) the simulation of the member end support conditions (a key aspect in numerical structural analysis), (iii) the modelling of the applied loading and material behaviour, (iv) the incorporation of member initial geometrical imperfections and residual stresses, (v) the assessment of buckling mode interaction effects and (v) the methods employed to solve either the eigenvalue problem or the system of non-linear algebraic equilibrium equations. Then, in order to illustrate the concepts and issues mentioned above and, at the same time, illustrate the power and versatility of the shell finite element analyses, one presents and thoroughly discusses a fairly large number of numerical results concerning the buckling and post-buckling behaviour of lipped channel (mostly), Zed-section and Rack-section cold-formed steel members  some of the post-buckling analyses include interaction effects between local-plate and distortional buckling modes. These results consist of (i) buckling curves providing the variation of the critical stress with the member length (see Fig. 1(a)), (ii) elastic and elastic-plastic non-linear (post-buckling) equilibrium paths (see Figs. 1(b)-(c)), (iii) figures providing the evolution, along those equilibrium paths, of the elastic and elastic-plastic member deformed configurations, and (iv) figures showing the spread of plasticity along the members up to failure (see Fig. 1(d)) and conveying relevant information about the nature of their collapse mechanisms.

Figure 1: Lipped channel simply supported columns: (a) elastic buckling, (b) elastic distortional post-buckling and (c) elastic-plastic distortional post-buckling results, and (d) distortional post-buckling plastic strain evolution.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

690

A Finite Volume Method for Plate Buckling Analysis Nosrat A. Fallah Department of Civil Engineering, Faculty of Engineering, University of Guilan Rasht P. O. Box 3756, Iran [email protected]

ABSTRACT The problem of buckling of plates has been the subject of numerous investigations because of its relevance to structural, mechanical, aeronautical, nuclear, offshore and ocean engineering. The buckling of plates can be solved analytically, which is useful for plates having simply supported edges and simple configurations. The application of the analytical method for plates with complex configurations, which are of practical importance, may be quite tedious and difficult. In such cases the energy method is used to obtain the approximate buckling loads. Alternatively, the numerical methods such as finite difference method, finite element method and finite strip method can be used for solving the problem. However, researchers still present new methods from the viewpoint of computational modeling of the problem. The application of finite volume method to the analysis of structural problems has been growing in recent years. This is due to the simplicity of the method and its capability in accurate predictions of structural behavior of the problems investigated so far. For instance it has been observed that the displacement finite volume formulation based on the MindlinReissner plate theory behaves well in the bending analysis of thin to thick plates [1]. In the other hand, the displacement finite element formulation based on the same theory locks in the thin plate analysis due to the shear locking phenomena. It is well known that treating this form of locking in the displacement finite element formulation is due to a number of works that have been devoted to the problem. In this paper a formulation is developed for computing the buckling loads of isotropic plates. The implementation of finite volume technique for the instability analysis of plates is quite new and has not been reported so far. To obtain the buckling load of a plate, it is idealized by the mesh of elements. The elements are regarded as control volumes or cells. Equilibrium equations of the cells are expressed explicitly and an approximate variation of section rotations and transverse displacements are assumed and introduced to the equilibrium equations. These approximated equations with boundary conditions are such expressed to yield a system of linear equations. The eigenvalue equation is then derived with standard form that is solved to obtain the buckling load of the plate. The formulation is verified on two test problems. This testing demonstrates the capability of the method in terms of accurate predictions and wide range of applicability.

References [1] N. Fallah, Cell vertex and Cell centred finite volume methods for plate bending analysis, Computer methods in applied Mechanics and Engineering, 193, 3457-3470, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

691

Design of slender steel shear panels: a numerical study A. Formisano*, G. De Matteis†, M. Maruzzelli*, F.M. Mazzolani* *

University of Naples “Federico II” - Department of Structural Analysis and Design P.le Tecchio, 80 – 80125 Naples (Italy) [email protected]; [email protected]; [email protected] †

Università of Chieti/Pescara G. d’Annunzio - PRICOS V.le Pindaro, 42 – 65127 Pescara (Italy) [email protected]

ABSTRACT In the framework of passive control devices for the seismic protection of new and existing buildings, in the last years large attention has been focused on shear wall systems. They are based on the use of a series of steel plates which realise a stiffened central nucleus able to absorb the horizontal action effect. These devices, which are obtained by inserting a steel panel within an external reaction steel frame, have a low realization cost and high speedy of erection. They can be classified in two main categories: - panels acting on the stiffness and the strength of the main structure; - panels having a dissipative function. The Steel Plate Shear Walls (SPSW), belonging to the first typology, are characterised by slender steel plates. They have been largely used in the last years in North America and Japan as an effective device against seismic actions. The behaviour of such system is strongly conditioned by buckling phenomena occurring at the early stages of the loading process. Such phenomena may have a significant influence also on the ultimate strength of the system, despite the development of stable post-critical behaviour due to the well known tension field mechanism. The theoretical and numerical studies on the behaviour of these devices confirm the reliability of the structural system when specific geometrical ratios of the panel are respected, i.e. for panels having b/d ratio raging from 0.8 to 2.5 [1]. In this paper, the theoretical behaviour of steel panels in shear, based on existing simplified methodologies (strip model theory) [2] is analysed and then compared with the results obtained by an extensive numerical study carried out using sophisticate finite element models (implemented in the ABAQUS non linear code). The comparison between theoretical and numerical results has been developed with reference to different values of the late thickness and varying the b/d ratio. In addition, the influence of intermediate stiffeners is analysed. In the whole the obtained results provide useful information for the correct design of slender steel plates in shear to be used as stiffening devices in new and existing framed structures.

References [1] Canadian Standards Association (CSA), Limit states design of steel structures. CAN/CSA S1601, Canadian Standards Association, Willowdale, Ont., Canada, 2001. [2] S. Sabouri-Ghomi, C. Ventura, M.H.K. Kharrazi, Shear analysis and design of ductile steel plate walls. Proc. of the 4th International Conference STESSA 2003, Behaviour of Steel Structures in Seismic Areas, Naples, 9-12 June, 2003.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

692

Plastic Bifurcation of Thin-Walled Members: Thin Shell Elements vs. GBT-Based Beam Elements Rodrigo Gonçalves*, Philippe Le Grognec† and Dinar Camotim‡ *

†

Escola Superior de Tecnologia do Barreiro, Polytechnic Institute of Setúbal R. Stinville 14, 2830-114 Barreiro, Portugal [email protected]

Ecole des Mines de Douai, Dépt. Technologie des Polymères et Composites et Ingénierie Mécanique 941, rue Charles Bourseul - BP 10838, 59508 DOUAI Cedex, France [email protected] ‡

Civil Engineering Department, ICIST/IST, Technical University of Lisbon Av. Rovisco Pais, 1049-001 Lisbon, Portugal [email protected]

ABSTRACT In this paper, one compares plastic bifurcation results concerning thin-walled members made of non-linear elastic-plastic materials, which are obtained by means of two independent approaches, namely (i) a Total Lagrangian thin shell finite element formulation, developed by the second author, and (ii) a computationally efficient beam formulation based on Generalised Beam Theory (GBT), developed by the remaining two authors  it is worth mentioning that the latter neglects the effect of pre-buckling deflections. Initially, one addresses the fundamental concepts, procedures and underlying assumptions involved in the application of the above two formulations, focusing on the similarities and differences existing between them. Then, one presents and thoroughly discusses a set of numerical results, determined through analyses based on the two alternative approaches and concerning (i) aluminium lipped channel and (ii) stainless steel rectangular hollow section (RHS) thin-walled columns (i.e., uniformly compressed members). In the first case, a very good correlation was found between the results (bifurcation loads/stresses and buckling mode shapes) yielded by the two formulations (e.g., see Fig. 1). In the second case, a non negligible discrepancy was observed, as the bifurcation loads provided by the shell formulation consistently lay below the GBT values and the differences, due to the combined influence of relevant pre-buckling deflection and a high imperfectionsensitivity, reached 19%  however, the RHS buckling mode shapes exhibited again an excellent agreement.

Figure 1: Plastic distortional buckling mode shapes of a lipped channel column of length L=55cm (shell FEA and GBT).

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

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A Large Displacement and Finite Rotation Thin-Walled Beam Finite Element Formulation Rodrigo Gonçalves*, Manuel Ritto-Corrêa† and Dinar Camotim† *

Escola Superior de Tecnologia do Barreiro, Polytechnic Institute of Setúbal R. Stinville 14, 2830-114 Barreiro, Portugal [email protected]

†

Civil Engineering Department, ICIST/IST, Technical University of Lisbon Av. Rovisco Pais, 1049-001 Lisbon, Portugal {mcorrea,dcamotim}@civil.ist.utl.pt

ABSTRACT In this paper, one presents, implements and validates a total Lagrangian beam finite element formulation capable of handling large displacements and finite rotations, as well as cross-section in-plane (distortion and local bending) and out-of-plane (warping) deformations. This formulation can be viewed as a generalisation of the well-known Reissner-Simo beam theory that includes warping/transverse bending deformation modes and in which the member walls can undergo in-plane finite relative rotations. When compared with a shell finite element discretisation, the proposed beam formulation has the distinct advantage of drastically reducing the number of degrees-of-freedom required to achieve equally accurate results. Initially, the kinematical description of the member is addressed, devoting special attention to the assessment of the transverse plate bending effects associated with cross-section distortion. Next, the equilibrium equations and the corresponding symmetric tangent operator are obtained, by assuming an additive update of the rotational parameters. In order to illustrate the application, provide validation and show the capabilities of the proposed formulation, several numerical results are presented, discussed and compared with values yielded by large displacement shell finite element analyses (see Fig. 1)  one obtains a good correlation in all cases, which clearly demonstrates the vast potential of the proposed formulation.

(a)

(b)

Figure 1: Beam deformed configurations yielded by the (a) proposed beam formulation and (b) shell element model

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

694

A new method to assess the rotation capacity of structural hollow sections based in multibody theory *

R. Goñi*, E. Bayo† Assistant Professor. Department of Structural Analysis and Design. School of Architecture. University of Navarra. Spain [email protected] †

Chairman. Department of Structural Analysis and Design. School of Architecture. University of Navarra. Spain [email protected]

ABSTRACT When modern structural codes are used for plastic analysis, it is important to define the class of the section. One of the main issues for the class classification is the rotation capacity of the section, that is, the inelastic rotation that the section can sustain after the plastic hinge is formed. EC3 requires class1 to perform plastic analysis, unless the rotation capacity is known. Sometimes, the election of class 1 is too restrictive and plastic analysis could be performed with class 2 provided that the section has sufficient inelastic rotation [1]. For this reason, it is important to know the inelastic rotation of a section, and it could be provided in property tables of sections. There are two reliable ways to calculate the rotation capacity: experimental tests and simulation by finite element analysis. In general, both of them are expensive. This paper presents a new and reliable method to obtain the rotation capacity of square and rectangular hollow sections. The method expands the original work of Kecman [2], simulating the plastic hinge model by a complete multibody system. The method performs successive static equilibriums, as described [3], for each angle of rotation of the plastic hinge. The forces considered in the model are in agreement with the elasto-plastic theory of materials The work developed by Kecman only considers the post-critical behavior of plastic hinge, however the proposed method, provides the complete moment-rotation curve. This method allows performing a quick simulation to obtain the inelastic rotation instead of large FEM simulations or expensive execution of experiments.

References [1] B. Gil, J. M. Cabrero, R. Goñi, E. Bayo, An Assessment of the Rotation Capacity Required for Structural Hollow Sections for Plastic Analysis, Tubular Structures X. Edit. Swets & Zeiltlinger Publishers. pp. 277 – 284, 2003 [2] D. Kecman, Bending Collapse of Rectangular and Square Section Tubes, Int. J. Mech. Sci. Vol. 25, No. 9-10, pp. 623 – 636, 1983 [3] J. Garcia de Jalon and E. Bayo, Kinematic and Dynamic Simulation of Multiboy Systems – The Real Time Challenge, Springer-Verlag, New York, 1993

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

695

On the Stability Analysis of Thin Walled Shell Structures Containing Gas or Fluid Marc Haßler∗ , Karl Schweizerhof ∗ Institut

f¨ur Mechanik Englerstrasse 2, D-76131, Karlsruhe, Germany {Haßler,Schweizerhof}@ifm.uni-karlsruhe.de ABSTRACT Thin shell or membrane structures containing gas or fluid are widely standard, such as oil and water tanks, gas containers or even atmospheric balloons, pressurized girders or inflatable dams. For such thin walled structures the gas or fluid can be considered either as support or loading. It may have a major influence on the stability behavior under other external loading as for example in the Tensairity-concept [2], where internal air pressure in combination with some external strengthening is used to overcome buckling of thin walled girders. The goal of this contribution is to present some investigation of the influence of such a gas or fluid support on the stability, here the eigenvalues and eigenmodes of the stiffness matrix of shell or membrane-like structures undergoing large displacements. For this purpose an analytical meshfree or lumped parameter description for the fluid/gas (see also [1], [3], [4])is taken, which yields a special structure of the nonlinear equations representing the change of the gas or fluid volume or alternatively the change of the wetted part of the shell surface. Finally this procedure leads first to the so-called load-stiffness matrix [5], to which several rank-one updates depending on the volumes containing either gas or fluid or both are added. These rank updates are a key part in the stability analysis: They describe the different coupling of the fluid or gas volume modification with the structural displacements in addition to the deformation dependence of the standard pressure. The specific rank-one updates allow the derivation of a very efficient algorithm to compute the change of the eigenvalues and eigenmodes of the original stiffness matrix without gas or fluid loading or support.

References [1] Bonet J, Wood RD, Mahaney J and Heywood P. Finite element analysis of air supported membrane structures. Computer Methods in Applied Mechanics and Engineering. 190 (2000) 579–595. [2] Luchsinger RH, Pedretti A, Steingruber A, Pedretti M. The new structural concept Tensairity: Basic principles. Proceedings of the Second Conference of Structural Engineering, Mechanics and Computation; A.A. Balkema/Swets Zeitlinger, Lisse. 2004. [3] Rumpel T and Schweizerhof K. Volume-dependent pressure loading and its influence on the stability of structures. International Journal for Numerical Methods in Engineering. 56 (2003) 211– 238. [4] Rumpel T and Schweizerhof K. Hydrostatic Fluid Loading in Non-Linear Finite Element Analysis. International Journal for Numerical Methods in Engineering. 59 (2004) 849–870. [5] Schweizerhof K and Ramm E. Displacement Dependent Pressure Loads in Nonlinear Finite Element Analyses. Computers & Structures. 18 (1984) 1099–1114.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

Sensitivity Analysis on Ultimate Strength of Stiffened Aluminum Plates under Combined Inplane Compression and Lateral Pressure M.R.Khedmati*, M.R.Zareei† * Faculty of Marine Technology Amirkabir University of Technology, Tehran, Hafez Ave., Iran [email protected] †

Faculty of Marine Technology Amirkabir University of Technology, Tehran, Hafez Ave., Iran

ABSTRACT Aluminum structures for marine applications have normally been built by welding. It is well recognised that welding significantly affects the behaviour of aluminum alloys. In particular, heat affected zone (HAZ) is softened by welding, and this reduces the ultimate strength of welded aluminum structures. It is of vital importance for structural designers to better understand how fabrication by welding affects the aluminum panel ultimate strength characteristics. It is commonly accepted that the collapse characteristics of welded aluminum structures are similar to those of welded steel structures until and after the ultimate strength is reached, regardless of the differences between them in terms of material properties. However, it is also recognised that the ultimate strength design formulae available for steel panels cannot be directly applied to aluminum panels even though the corresponding material properties are properly accounted for. One of the major reasons for this is due the fact that the softening in HAZ reduces the ultimate strength behaviour of welded aluminum structures, whereas it can normally be neglected in welded steel structures. There are some research works on the ultimate strength behaviour of aluminum unstiffened/stiffened panels under longitudinal inplane compression. In spite of that, studies on the collapse behaviour of such panels under the combined action of lateral pressure and axial compressive loads are rarely published. Aluminum stiffened panels applied in the construction of high speed crafts are under big magnitudes of lateral hydrostatic and hydrodynamic loads. It is aimed in this paper to perform numerical collapse simulations on the aluminum stiffened panels under combined lateral pressure and axial inplane compression, applying a series of non-linear finite element analyses on such plate elements. Both material and geometric nonlinearities are taken into account. The values of lateral load, panel geometric properties and HAZ width are changed in a systematic manner. Buckling, post-buckling, ultimate strength and post-ultimate strength characteristics of the panels are investigated in details.

References [1]J.K. Paik, A. Duran, Ultimate strength of aluminum plates and stiffened panels for marine applications, Marine Technology, Vol.41, No.3, 2004.

696

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

697

Large Deflection Behavior of Functionally Graded Plates Under Pressure Loads, Using Finite Strip Method H. R. Ovesy, S. A. M. GhannadPour Aerospace Engineering Department, Amirkabir University of Technology Tehran, Iran [email protected]

ABSTRACT The application of the same FSM as that developed earlier by the authors in Ref. [1] is extended to the analysis of large deflection behavior of functionally graded plates subjected to the normal pressure loading. The fundamental equations for the plates of functionally graded material (FGM) are obtained by discretizing the plate into some functionally graded strips (FGS), which are developed by combining the Von-Karman theory for large transverse deflection and the concept of functionally graded material. The material properties of the functionally graded strips are assumed to vary continuously through the thickness of the plate, according to the simple power law distribution of the volume fractions of the constituents. The solution is obtained by the minimization of the total potential energy. The Newton-Raphson method is used to solve the resulting non-linear equilibrium equations. Numerical results for square functionally graded plates subjected to normal pressure loading are generated by varying the combination of the constituents. The effects of material properties on the stress field through the thickness and on the variation of the central deflection at a given value of normal pressure loading are determined and discussed. The results are also compared with those available in the literature, wherever possible. Some representative results are given in the table below. The good comparison of the results verifies the current large deflection FSM analysis for the case of functionally graded plates. Non-dimensional central deflection w/h at a non-dimensional normal pressure of Q=-400 for an aluminum-alumina FGM plate with A/h=20. w/h n=0 n=0.5 n=2 n=Infinity Present -1.95 -2.40 -3.05 -4.52 Ref [2] -1.98 -2.44 -3.16 -4.71 In the above table, the non-dimensional normal pressure Q is defined as Q = qA4/(Emh4), where q is a uniformly distributed pressure load, h is the thickness of the plate, A is the length of the plate and Em 70 GPa . The description of parameter n, which is used within simple power law distribution, is

given in reference [2].

References [1] Ovesy H.R., GhannadPour S.A.M. Geometric Nonlinear Analysis of Imperfect Composite Laminated Plates, Under End Shortening and Pressure loading, Using Finite Strip Method. Accepted for publication in special issue of the composite structures (iccs/13-14 November 2005-Australia-Monash University), 2006. [2] Woo J., Meguid S.A. Nonlinear analysis of functionally graded plates and shallow shells. International Journal of Solids & Structures, 38, 7409–7421, 2001.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

698

Buckling Analysis of Laminates with Multiple Through-The-Width Delaminations by Using Spring Simulated Model H. R. Ovesy, H. Hosseini-Toudeshky, M. Kharazi

Aerospace Engineering Department, Amirkabir University of Technology Tehran-Iran {ovesy,hosseini}@aut.ac.ir

ABSTRACT Fiber-reinforced composite materials have been increasingly used over the past few decades in a variety of applications in which a fairly high ratio of stiffness/strength to weight is required. However, these materials are prone to wide range of defects and damages that can cause significant reductions in stiffness and strength. In particular, when the laminated composites are subjected to compressive loads, delamination becomes a constraint in the design process. Various methods have been proposed for the analysis of a plate that contains through-the-width delaminations. In the current paper, a continuous method of analysis is developed for determining the buckling loads of delaminated plates. The system is modeled as a plate on an elastic foundation. The elastic adhesive layer between the buckled sublaminates is represented by some parallel springs. The plate on a discontinuous foundation is treated as a continuous foundation but with added transverse forces at a number of discrete points in the delamination regions to make the net transverse force at each of these points to vanish. The delaminated plates which contain one or two through-the-width delaminations are analyzed. In the cases where two delaminations are located at different depths across the thickness of the laminate, the governing differential equations of each sublaminates become coupled, resulting in a more challenging analysis. Some representative results are shown in the figure below.

Buckling load of a simply supported laminate with a single delamination (Pc: Buckling load of a laminate with a centrally located delamination)

As shown in the figure, the buckling loads obtained in the current study are in good agreement with those available ref. [1]. It is worth noting that the so-called spring simulated model, outlined above, is fairly simple, and it can be used in a variety of applications.

References [1] James Ting-Shun Wang and Shou-Hsiung Cheng, Local buckling of delaminated beams and plates using continuous analysis, Journal of Composite Materials, 29, 10, 1374-1402, 1995.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

Numerical Validation of the Eurocode 3 Design Rules for LateralTorsional Buckling of I-Beams Vila Real P. M. M. †, Lopes N. †, Simões da Silva L. *, Rebelo C.*, † Universidade de Aveiro Dep. Civil Engineering, 3810 Aveiro, Portugal [email protected]; [email protected] *

Universidade de Coimbra Dep. Civil Engineering, 3030 Coimbra, Portugal [email protected]; [email protected]

ABSTRACT This work presents a numerical study of the behaviour of steel I-beams subjected to lateral torsional buckling. The results obtained are compared with the beam stability design curves from Eurocode 3. The EN version of part 1-1 of this Eurocode introduces significant changes in the evaluation of the lateral torsional buckling [1] resistance of unrestrained beams, that improve the too conservative approach of part 1-1 of Eurocode 3 from 1992 in case of non uniform bending. The EN version of the Eurocode 3 [2] presents two methods for the evaluation of the design buckling resistance moment of laterally unrestrained I-beams subject to major axis bending. One of these methods is similar to the one already prescribed in Eurocode 3 of 1992, while the other presents some differences in the buckling curves used. Further, it explicitly allows taking into account the influence of the loading type, through the introduction of a correction factor. The present study aims at evaluating the behavior of these two methods and to analyse the influence of the loading type in the first method through the use of that correction factor. Aiming the safety evaluation of the design formulae, a statistical analysis of the results is performed on the basis of the EN 1990-Annex D [3] in a companion paper [4].

References [1] Boissonnade, N., Greiner, R. and Jaspart, J.P., Rules for member stability in EN 1993-1-1. Background documentation and design guidelines, ECCS Technical Committee 8 – Stability, 5th draft, 2005 [2] CEN, European Committee for Standardisation, Eurocode 3: Design of steel Structures – Part 1-1: General Rules and Rules for Buildings. EN 1993-1-1, Brussels, Belgium, 2005.

[3] CEN, European Committee for Standardisation, Basis of Structural Design, EN 1990, Brussels, Belgium, 2002. [4] Rebelo C., Simões da Silva L., Vila Real P. M. M., Lopes N., Statistical Evaluation of the Eurocode 3 Design Rules for Lateral-Torsional Buckling of IBeams, III European Conference on Computational Mechanics, Solids, Structures and Coupled Problems in Engineering, C.A. Mota Soares et.al. (eds.), Lisbon, Portugal, 5–8 June 2006.

699

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

700

Statistical Evaluation of the Eurocode 3 Design Rules for LateralTorsional Buckling of I-Beams Rebelo C.†, Simões da Silva L.†, Vila Real P. M. M.*, Lopes N.*, †Universidade de Coimbra Dep. Civil Engineering, 3030 Coimbra, Portugal [email protected]; [email protected] * Universidade de Aveiro Dep. Civil Engineering, 3810 Aveiro, Portugal [email protected]; [email protected]

ABSTRACT This work reports the statistical evaluation of resistance design models for lateral-torsional buckling of I-beams according to the EN1993(eurocode 3) [3] and EN1990(eurocode 0) [2]. Aiming at the preparation of the Portuguese National Annex of EC3 part 1-1 and the establishment of the corresponding NDP's (Nationally Determined Parameters) it becomes necessary to define the partial coefficients of safety for beams design formulae when lateral-torsional buckling is a potential failure mode. In this paper the methodology for the resistance evaluation of beam elements subjected to instability is briefly described and the results are compared with FEM numerical results for the same elements obtained in a companion paper [4]. Aiming the safety evaluation of the design formulae, a statistical analysis of the results is performed on the basis of the EN 1990-Annex D. A methodology is proposed for definition of the partial safety factors concerning the uncertainties in the resistance model. Results are presented for a wide set of beam geometries and loading cases. Main conclusions can be summarized as follows: (i) all the design methods give better results for rolled sections; the scatter and the mean value of the results grow with growing slenderness; only the Special Case method can be considered an exception, since the mean diminishes in the range of low slenderness (0,2 – 0,5) assuming values lower then the unity, therefore unsafe; (ii) the safety factor is lower for the General Case and higher when the Special Case is applied; particularly for the most common medium slenderness the safety factor for the Special case is substantially higher than for the other two methods; if the factor 1.1 is applied, it becomes equivalent to the f-modified General Case method..

References [1] Background Document for EC3, Doc. 5.01, Background document for the justification of safety factor gM0 = 1.0 for rolled beams in bending about the strong axis, 1989 [2] CEN, European Committee for Standardization, EN 1990:2002, Basis of Structural Design, April 2002, Brussels, 2002. [3] CEN, European Committee for Standardisation, EN 1993-1-1:2005, Eurocode 3: Design of steel Structures – Part 1-1: General Rules and Rules for Buildings, Brussels, Belgium, 2005. [4] Vila Real P. M. M., Lopes N., Simões da Silva L., Rebelo C., Numerical Validation of the Eurocode 3 Design Rules for Lateral-Torsional Buckling of I-Beams, III European Conference on Computational Mechanics, Solids, Structures and Coupled Problems in Engineering, C.A. Mota Soares et.al. (eds.), Lisbon, Portugal, 5–8 June 2006.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

701

On the Interpolation of Rotations and Rigid-Body Motions in Nonlinear Beam Finite Elements Manuel Ritto-Corrˆea, Dinar Camotim Civil Engineering Department, ICIST/IST, Technical University of Lisbon Av. Rovisco Pais, 1049-001 Lisboa, Portugal {mcorrea,dcamotim}@civil.ist.utl.pt ABSTRACT The formulation of 3D geometrically exact beam finite elements relies heavily on the interpolation scheme used for the unidimensional variables describing the rotations or the rigid-body motions of the beam cross-sections. Since these rotations and positions belong to the non-commutative Lie groups SO(3) and SE(3), they are not easily amenable to a direct discretisation. Several schemes to interpolate the rotations have been suggested in the last two decades, each of them with specific advantages and drawbacks. Although the subject is far from settled, it has reached a stage of maturity which makes it possible to clearly identify several properties that an interpolation scheme should preferably display: it ought to (i) preserve orthogonality, (ii) be independent from the iterative process adopted, (iii) be path-independent, (iv) be frame-invariant, (v) be applicable to an arbitrary number of nodes and (vi) lend itself to a computationally implementable linearisation. Until very recently, none of the interpolation schemes for the rotation field in structural finite elements fulfilled all these requirements. However, the interpolation rule proposed by Buss and Fillmore [1] and Merlini and Morandini [2] seems to gather all the aforementioned desirable properties. This paper discusses the implementation of a (geometrically exact) Reissner-Simo beam element adopting this interpolation scheme for the rotation field. In addition, a growing trend in the development of geometrically exact beam finite elements involves the so-called helicoidal interpolation, in which the displacement and rotation fields are assumed to be coupled – this designation stems from the fact that a linear helicoidal interpolation between two adjacent nodes is an helix. In view of some mathematical analogies between groups SO(3) and SE(3), it is believed that the interpolation scheme applicable to rotations can be extended to the case of rigidbody motions.

References [1] Samuel R. Buss and Jay P. Fillmore. Spherical averages and applications to spherical splines and interpolation. ACM Transactions on Graphics, 20(2):95–126, 2001. [2] T. Merlini and M. Morandini. The helicoidal modeling in computational finite elasticity. Part II: Multiplicative interpolation. International Journal of Solids and Structures, 41:5383–5409, 2004.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

702

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III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

703

Equivalent Geometric Imperfections for Steel Shell Structures Subject to Combined Loading Werner Schneider*, Marco Gettel† University of Leipzig, Institute for Structural Mechanics Marschnerstr. 31, 04109 Leipzig, Germany * [email protected] †[email protected]

ABSTRACT The inevitable deviations from the nominal data of the resistance parameters have to be included in a numerical calculation of the load-bearing capacity of shells, because these structures are very imperfection-sensitive. In addition to simpler methods, the new European code for the resistance verification of steel shell structures EN 1993-1-6:2005 allows a geometrically and materially nonlinear analysis with imperfections included. The assumed imperfections are fundamental for this most sophisticated numerical buckling strength verification, because they have to cover the influence of all accidental imperfections of the structure in a consistent manner. According to the Eurocode, the influence of all various deviations should be included by only geometric equivalent imperfections. In spite of the intensive research efforts in the last decades, many problems are still residual, which have to be solved in order to apply the mentioned most realistic basic principle of the Eurocode to shell buckling cases, which are not yet sufficiently investigated. Equivalent geometric imperfections are called consistent, if nonlinear numerical analyses including these imperfections result in the experimentally based buckling resistance. Consistent equivalent geometric imperfections have been developed during the last years for the basic buckling cases of the circular cylindrical steel shell ([1], [2]). The situation at shells subject to combined loading is more difficult, because not so much experimental data are available. Fundamental problems and previous proposals for assuming equivalent imperfections at combined loading are discussed in the contribution. In particular, it is mooted, if the equivalent geometric imperfections have to be chosen without regard to the loading case, because imperfections of real shells are caused by manufacturing and not by loading. Reasons are given for, why this is not the case at equivalent imperfections of a numerical simulation. The conception of quasi-collapse-affine imperfections [3], which has already been proved at the basic buckling cases, can also be applied to shells under combined loading. General information for the application is given on the basis of two relevant buckling cases under combined loading.

References [1] W. Schneider, Stimulating Equivalent Geometric Imperfections for the Numerical Buckling Strength Verification of Steel Shell Structures. Proc. 6th World Congress on Computational Mechanics - WCCM VI, Beijing, China, 2004. [2] W. Schneider; A. Brede, Consistent equivalent geometric imperfections for the numerical buckling strength verification of cylindrical shells under uniform external pressure. Thin-Walled Structures, 43/2, 175-188, 2005. [3] W. Schneider, I. Timmel, K. Höhn, The Conception of quasi-collapse-affine imperfections. A new approach to unfavourable imperfections of thin-walled shell structures. Thin-Walled Structures, 43/8, 1202-1224, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

704

Shear Buckling of Thin Plates with Constant In-Plane Stresses Igor Shufrin*, Moshe Eisenberger*,** *

Faculty of Civil and Environmental Engineering Technion – Israel Institute of Technology Technion City, 32000, Israel **

Department of Building and Construction City University of Hong Kong, Tat Chee Ave., Kowloon Hong Kong {shufrin, cvrmosh}@techunix.technion.ac.il

ABSTRACT This work presents highly accurate numerical calculations of the buckling loads for thin elastic rectangular plates with known constant uni-axial in-plane loading, and in-plane shear loading that is increased until the critical load is obtained and the plate losses its stability. The solutions are obtained using the multi term extended Kantorovich method. The solution is sought as the sum of multiplications of two one dimensional functions. In this method a solution is assumed in one direction of the plate, and this enables to transform the partial differential equations of the plate equilibrium into a system of ordinary differential equations. These equations are solved exactly by the exact element method [1], and an approximate buckling load is obtained. In the second step, the derived solution is now taken as the assumed solution in one direction, and the process is repeated to find an improved buckling load. This process converges with a small number of solution cycles. For shear buckling this process can only be used if two or more terms are taken in the expansion of the solution. As an example the shear buckling load of a simply supported square plate with different levels of constant compressive load, as shown in Figure 1, is given. In Figure 2 the variation of the normalized shear buckling load as a function of the compression level is shown. Many more new results will be given. 10.00

Lx/Ly=1.0 h=constant Nx=DNcr

9.00 8.00

Nxy O

7.00

SS SS

6.00 5.00

SS

4.00 3.00

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2.00

Nx=DNcr

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 D

References [1] M. Eisenberger, Buckling loads for variable cross-section members with variable axial forces, Int. Jour. Solids Structures, 27, 135-143, 1991.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

705

GBT Formulation to Analyse the Buckling Behaviour of FRP Composite Branched Thin-Walled Members Nuno Freitas Silva, Nuno Silvestre and Dinar Camotim

Department of Civil Engineering and Architecture, IST, Technical University of Lisbon, Av. Rovisco Pais, 1049-001 Lisbon, Portugal [email protected], {nunos, dcamotim}@civil.ist.utl.pt

ABSTRACT In this paper, one develops a Generalised Beam Theory (GBT) formulation to analyse the local and global buckling behaviour of composite thin-walled members with branched cross-sections made of laminate plate FRP (fibre-reinforced plastic), which takes into account the shear deformability effects. After briefly reviewing the most noticeable differences between the GBT formulations applicable to members with branched and unbranched open thin-walled cross-sections, the paper presents in some detail the steps and procedures involved in performing a GBT cross-section analysis of an arbitrarily branched composite (laminate plate) thin-walled member, which include (i) the identification and characterisation of conventional and shear deformation modes (the latter are not relevant in isotropic members) and (ii) the determination of the corresponding modal mechanical properties. Then, one addresses the numerical implementation of the proposed GBT formulation, which is carried out by means of the finite element method (GBT-based beam element)  particular attention is devoted to derivation of the elementary generalised stiffness and geometric matrices, which incorporate all the material coupling effects. Finally, in order to illustrate the application and capabilities of the proposed formulation/implementation, one presents and discusses numerical results concerning the local and global buckling behaviour of shear deformable FRP composite I-section columns with different ply orientations and stacking sequences (see Fig. 1)  in particular, one takes advantage of the modal features of GBT to acquire a deeper insight on complex buckling mode interaction phenomena, such as the ones due to bending-torsion or local-shear coupling effects. For validation purposes, some of the above results are also compared with values recently reported in the literature and obtained by means of numerical implementations of analytical models developed independently by several researchers  with a single exception (addressed in detail in the paper), an excellent correlation was found between the buckling load values (the differences are always below 2%).

Figure 1: Composite I-section columns local buckling mode and variation of global buckling loads with fibre angle ș.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

706

On the Influence of Material Couplings on the Buckling Behaviour of FRP Thin-Walled Columns – a GBT-Based Approach N. Silvestre and N. Freitas Silva Department of Civil Engineering and Architecture, IST, Technical University of Lisbon, Av. Rovisco Pais, 1049-001 Lisboa, Portugal

[email protected] ; [email protected] ABSTRACT The paper began by describing the main concepts involved in a GBT formulation to analyse the buckling behaviour of thin-walled composite members. Then, a beam finite element is derived in order to solve the system of differential equations. For validation purposes, some GBT-based results are compared with experimental and theoretical values available in literature. Finally, in order to illustrate the application and capabilities of the above formulation, the results of a study concerning the local and global buckling behaviour of fully fixed I-section columns is presented and discussed. Among the several conclusions drawn from this study, the following ones deserve to be specially mentioned: (i) In the context of linear (first order) analyses of I-section beams characterized by material couplings, the GBT-based results agree very well with the experimental and theoretical estimates. (ii) In the context of stability analyses of an I-section column characterized by material couplings, it is found that local buckling modes might be critical. Unlike columns made of isotropic materials, columns characterized by material couplings exhibit buckling modes with very unusual deformed configurations. In fact, it is unveiled that the shear modes play a relevant role in the mechanics of coupling between the different conventional modes (all shear undeformable). (iii) It is found that the incorporation of both matrices H and F, accounting for the modal couplings, is indispensable to achieve reliable results. Thus, neglecting these matrices may lead to nonconservative buckling load values (up to 25%) and to very different buckling mode shapes, as it can be observed from figures 1(a) (buckling mode from the exact analysis) and 1(b) (buckling mode from the analysis without matrices H and F).

(a)

(b)

Figure 1: Buckling mode configuration: (a) exact and (b) approximate (analysis without matrices H and F)

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

707

Ultimate Strength of Plate Assemblies with Localized Imperfection Subjected to Compressive Loads Rui M. Luís, C. Guedes Soares Unit of Marine Technology and Engineering, Technical University of Lisbon, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa [email protected]; [email protected]

ABSTRACT In a ship usually there are many plates, they are load carrying elements and they must be designed to resist the loading. The critical condition is the strength of deck structures under in-plane compressive loads leading to elasto-plastic buckling collapse. Plates are not perfect elements and their imperfections must be taken into account during the design process. In [1] it was shown that the strength of rectangular plates was governed by the amplitude of the buckling mode. In [2] a design process was proposed to take into account the global imperfections. However, localized imperfections can appear in a ship in any number of ways, the most commonly found are caused by welding or local damage. So, it is important for the designer to take into account the ability of the plate to resist to these imperfections. Most of the studies of the effect of imperfections have concentrated on individual plate elements or in stiffened plates. In [3] the effect of localized imperfections on long plates was studied and in [4] the study was extended to smaller plates and to combined imperfections. In welded plates the generated or induced initial imperfections, tend to have similar patterns in adjacent plates. Therefore the study of individual elements may be representative of the behavior of panels made of various plates. Localized imperfections related with local damage are not periodic and one cannot assume that adjacent plates have similar pattern of localized imperfections. This study analyzes the behavior of panels made up of three individual plates with localized imperfections. The effect in the ultimate collapse load of the level of imperfection and of its spatial location was studied. It was found that local imperfections clearly changed the strength of the panel when combined with the global ones. The effects of changing the position of the localized imperfection depend much on the final shape of the imperfections (local plus global). The effect of the localized imperfections can not be ignored by the designer and must be taken into account.

References [1] M. Kmiecik, Behavior of axially loaded simply supported long rectangular plates having initial deformations. Ship Research Institute, Report No. 84, Trondheim, 1971. [2] C. Guedes Soares, Design Equation for Ship Plate Elements under Uniaxial Compression. Journal of Constructional Steel Research, 22: 99–114, 1992. [3] R. S. Dow and C. S. Smith, Effects of Localized Imperfections on Compressive Strength of Long Rectangular Plates. J. Construct. Steel Research 4: 51-76, 1984. [4] C. Guedes Soares, A. P. Teixeira, R. M. Luís, T. Quesnel, P. I. Nikolov, E. Steen, I. A. Khan, C. Toderan, V. D. Olaru, A. Bollero and M. Taczala, Effect of the shape of localized imperfections on the collapse strength of plates. C. Guedes Soares, Y. Garbatov and N. Fonseca eds. Maritime Transportation and Exploitation of Ocean and Coastal Resources, IMAM, Lisbon, Portugal, 429-437, 2005.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

708

Higher Order Analysis of a Thin-Walled Beam Vieira, R., Virtuoso, F., Pereira, EBR. DECivil - Civil Engineering and Architecture Department Technical Superior Institute Technical University of Lisbon Av. Rovisco Pais, 1049-001 Lisboa, Portugal {ricardov, fvirtuoso, eduardo}@civil.ist.utl.pt

ABSTRACT A thin walled beam model formulation for the analysis of two dimensional problems is presented in this paper. The underlying concept of the model is to separate the correspondent two-dimensional elasticity problem into two parts: i) an approximation of the displacement field over the cross section and ii) a set of governing differential equations defined along the beam axis. A set of basis functions that uncouples to the most possible form the governing equations of the problem is obtained, which permits to consider explicitly higher order modes of the cross section deformation, in particular, warping and transverse shear effects. This process of uncoupling has the advantage of permitting a better physical understanding of the beam structural behaviour. An implementation of a numerical model for the solution of the orthogonal governing equation is developed within the framework of the finite element method, interpolating the coordinates of the deformation modes basis functions by a set of Hermite functions. Some numerical examples are presented in order to verify the model capabilities in modeling the non classical effects associated with high order deformation modes within the scope of a two dimensional thin-walled beam analysis. Acknowledgements ˜ para a Ciencia ˆ This work has been partially supported by FEDER and FCT (Fundac¸ao e Tecnolo´ gia) through the funding of research unit ICIST (Instituto de Engenharia de Estruturas Territorio e ˜ Construc¸ao).

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

709

Structural Optimization Using OPTIMIZER Program M.H. Abolbashari1, M. Majdi, and M.R. Mahpeykar 1

Manufacturing & Automotive Engineering Research Center, Ferdowsi University of Mashhad, PO Box 91775-1111, Mashhad, Iran [email protected]

ABSTRACT OPTIMIZER is a user-friendly design optimization study tool that helps users to optimize almost any optimization problems. There are several optimization algorithms in OPTIMIZER program such as Genetic Algorithm (GA), Constraint Steepest Descent (CSD) and Constraint Quasi-Newton (SQP). The OPTIMIZER can only solve problems that have an explicit mathematical expression both for cost function and constraints. To extend the OPTIMIZER capability for other applications, it is linked with an analysis software like ANSYS. In this paper, several structural optimization problems are solved using OPTIMIZER and the results are compared with other reported solutions. Furthermore, the effectiveness of the above-mentioned methods for the selected problems is presented.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

710

Optimization of dissipative characteristics of structures on the basis of problems on natural vibrations of viscoelastic solids Dmitry V. Babkin*, Eugeny P. Kligman†, Valery P. Matveyenko†, Natalya A. Yurlova† *†

Institute of Continuous Media Mechanics of Ural Branch of RAS Academician Korolev str., 1 Perm, 614013, Russia [email protected]

ABSTRACT The damping ability of a material plays an important role in the dynamical behavior of structures. It is responsible for decay of free vibrations, drastic decrease in amplitudes of the displacements and stresses arising in structures subjected to dynamical actions. To date, a lot of approaches have been developed which describe the mechanism of internal friction of materials, which causes the energy dissipation under vibrations [1]. The Boltzmann-Volterra theory [2] is the most general linear one, reflecting practically all peculiarities of the quasistatic and dynamic behavior of viscoelastic materials. The damping for structures can be positive or negative factor. A quantitative assessment of the dissipative properties of structures is generally based on the results of solving free vibrations. In this case the dissipation of a system leads to the decay of vibrations, and the rate of decay estimates quantitatively the dissipative properties of a system. The higher is the decay rate of vibrations, the greater are the dissipative properties. The problem of natural damped vibrations is formulated using a complex analog to the Boltzman-Volterra equations. The transition to the complex analog is made under some assumptions. To confirm the validity of the algorithm for optimization of the dissipative properties of a construction, the optimization search by solving the problem of forced steady-state vibrations is also performed. The application of the finite element method to the stated problem reduces it to the algebraic problem of eigenvalues for complex matrices. To have assurance that the lowest vibration modes defining the damping properties of the system will be determined and to decrease significantly the volume of calculations, the method is proposed which is based on expansion of the viscoelastic problem solution into finite series with respect to eigenforms of vibrations of the corresponding elastic structure. The optimization problem is solved within the framework of nonlinear programming. Mechanical and geometrical parameters of the system are used as optimization parameters. The performed numerical experiments showed that the optimal structure could be achieved even in the case when the real viscous properties of the material are given in rather rough manner.

References [1] A.A. Iliushin, and B.Y.Pobedrja, Fundamentals of mathematical theory of thermoviscoelasticity. Nauka, Moscow, 1970. [2] A. Nashif, D. Johnes and J. Henderson, Vibration damping., Mir, Moscow, 1988.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

711

Accuracy of Design Sensitivity Analysis for Optimization of Structures Using Small Strain Theory by Finite Element Method Kiran A. Bhagate1, Prashant Pawar2, Arbind Kumar Singh3, Jianqiao Ye1 1

2

School of Civil Engineering, University of Leeds, UK. [email protected]

Department of Aerospace Eng., IISc, Bangalore, India. 3 Department of Civil Eng, IIT Guwahati, India.

ABSTRACT Optimization of structures is always a fantasizing area for many researchers from past few decades. Many efforts have been taken in reducing the errors from the optimization process especially after advancement in computer faclities. The resulting structures are more efficient, economical and reliable. In traditional optimization techniques, the most important factor affecting on optimization process is the search direction which is the derivative of change in respose of structure due to change in design variables. This derivative is called as sensitivity derivative. Accuracy of sensitivity analysis is very much dependent on the method of structural analysis, technique of sensitivity calculation, computational efficiency etc. In this work, accuracy of sensitivity derivatives in elastic and plastic analyses are investigated on the basis of small strain theory. Combined with the Finite element method, which provides an excellent tool for the analysis of complex structures, the different techniques used for sensitivity calculations are finite difference method, semianalytical method and analytical method. Detail discussion of formulation and implementation of these methods are presented. Comparative study shows the relative error, cost of computation and efficiency of the above methods. From the results obtained, it can be stated that the finite difference method is the simplest technique that does not require access to the finite elemen analysis code and hence requires less efforts. However, this method is inefficient and less accurate. Analytical method is the most accurate method but its formulation and implementation is difficult as compared to other two metods. Semianalytical method is found to be a compromise of the two that results in more accurate solutions than from the finite dif erence method and is easy to implement as compared to the analytical method. The comparisons provide useful information for design engineers to decide a suitable method in the calculation of sensitivity erivatives for different structural optimization problems.

References [1] K. K. Choi and N. Kim, Structural Sensitivity Analysis and Optimization 1, Springer, 2005. [2] R. T. Haftka, Semi-Analytical Static Nonlinear Structural Sensitivity Analysis, AIAA Journal, 31, 1307-1312, 1993. [3] G. Cheng, Y. Gu and Y. Zhou, Accuracy of Semi-Analytical Sensitivity Analysis, Finite Elements in Analysis and Design, 6, 113-128, 1989.

III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et al. (eds.) Lisbon, Portugal, 5–8 June 2006

712

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