Numerical Simulation of Mechatronic Sensors and Actuators: Finite Elements for Computational Multiphysics [3 ed.] 978-3-642-40169-5, 978-3-642-40170-1

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Manfred Kaltenbacher

Numerical Simulation of Mechatronic Sensors and Actuators Finite Elements for Computational Multiphysics Third Edition

Numerical Simulation of Mechatronic Sensors and Actuators

Manfred Kaltenbacher

Numerical Simulation of Mechatronic Sensors and Actuators Finite Elements for Computational Multiphysics Third Edition

123

Manfred Kaltenbacher Institute of Mechanics and Mechatronics Vienna University of Technology Vienna Austria

ISBN 978-3-642-40169-5 DOI 10.1007/978-3-642-40170-1

ISBN 978-3-642-40170-1

(eBook)

Library of Congress Control Number: 2014960346 Springer Heidelberg New York Dordrecht London © Springer-Verlag Berlin Heidelberg 2004, 2007, 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer-Verlag GmbH Berlin Heidelberg is part of Springer Science+Business Media (www.springer.com)

Preface to the Third Edition

The third edition of this book fully preserves the character of the previous editions to combine physical modeling of mechatronic systems and their numerical simulation using the Finite Element (FE) method. Most of the text and general appearance of the previous editions were retained, while the topics have been strongly extended and the presentation improved. Thereby, the third edition contains the following main extensions: • • • • • •

Finite elements of higher order Flexible discretization towards non-conforming methods Computational fluid dynamics and coupled fluid-solid-interaction (FSI) Perfectly matched layer (PML) technique in time domain Comprehensive discussion of aeroacoustics with latest numerical schemes Advanced numerical schemes for piezoelectricity including macro- and micromechanical models.

We have enhanced the basic chapter concerning the Finite Element (FE) method by now providing the FE basis functions and integration points for all geometric elements (quadrilateral, triangle, tetrahedron, hexahedron, wedge and pyramid). Furthermore, we discuss in detail p-FEM (finite elements of higher order) both for nodal (see Sect. 2.9.1) and for edge finite elements (see Sect. 6.7.6) and also extend the scope to spectral elements (see Sect. 2.9.2). In addition, we provide a detailed discussion of non-conforming techniques, both the classical Mortar method and Nitsche type mortaring (see Sect. 2.10). As a new physical field, we now have also included the flow field (see Chap. 4), which allows us to derive linear acoustics by using a perturbation ansatz on the conservation of mass and momentum. The FE discretization of the Navier-Stokes equations leads to a stabilized FE formulation, more precisely we apply a Streamline Upwind Petrov Galerkin/Pressure Stabilized Petrov Galerkin (SUPG/PSPG) formulation. Furthermore, in Chap. 7 we discuss the fluid–solid interaction (FSI), and now have the capability to present aeracoustics by a more general approach. Thereby, we have extended the discussion of computational aeroacoustics by the

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classical vortex sound theory and by perturbation approaches, which allows a decomposition of flow and acoustic quantities within the flow region (see Chap. 9). Towards computational acoustics, we have extended our discussion for open domain problems by a new formulation for the time domain perfectly matched layer (PML) technique (see Sect. 5.5.2) and a mixed FE ansatz to solve the acoustic conservation equations in a quite efficient way using spectral elements (see Sect. 5. 4.2). Furthermore, we present latest piezoelectric models for precisely describing the polarization process (micro-mechanical model, see Sect. 12.4.2) and the whole operation range for actuators (hysteresis operator-based model, see Sect. 12.5.2). Finally, we present new industrial applications: (1) cofired piezoceramic multilayer actuators simulated with both the micro-mechanical and the hysteresis operator-based model (see Sect. 14.4); (2) simulation of the human phonation (see Sect. 14.7); (3) flow induced sound of obstacles in cross flow, edge tone and airframe noise (see Sect. 14.8). All presented numerical schemes have been implemented in our in-house research code CFS++ (Coupled Field Simulation) and most of the algorithms have also found their way to the commercial software NACS (see http://www.simetris.eu).

Acknowledgments The author wishes to acknowledge the many contributions that colleagues and collaborators have made to this third edition. First of all I would like to express my gratitude to all members of my research group at Vienna University of Technology. Amongst many, I wish to specially thank for contributions: Andreas Hüppe to computational acoustics and aeroacoustics as well as spectral finite elements; Stefan Zörner to fluid dynamics and human phonation; Simon Triebenbacher to nonconforming grid techniques and computational aeroacoustics; Andreas Hauck to finite elements of higher order. Furthermore, I would like to thank my former Ph.D. student Gerhard Link (now at CD-adapco Germany) for his contribution towards fluid dynamics and fluid-solid-interactions, and Stefan Becker and his team (Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany) for the fruitful cooperation on aeroacoustics. Much is owned by many discussions with my wife Barbara Kaltenbacher (Alpen-Adria-Universität Klagenfurt, Austria) on the modeling of piezoelectricity and the development of stable perfectly matched layers in time domain. For the exciting cooperation on flexible discretization within our common DFG (German Science Foundation)—FWF (The Austrian Science Foundation) project Numerical Simulation of Acoustics-Acoustics- and Structural Mechanics-Acoustics-Couplings on Nonmatching Grids I would like to thank Barbara Wohlmuth (Technische Universität München, Germany). Furthermore, many thanks to Michael Döllinger (Friedrich-Alexander-Universität ErlangenNürnberg, Germany) for the fruitful cooperation within the DFG-FWF project Physical basics of the human voice. In this context, also special thanks to Petr Sildof (Technical University of Liberec, Czech Republic) for the cooperation within

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our joint project on The Fundamentals of Human Voice: Hybrid methods in computational aeroacoustics funded by Science and Technology Cooperation Austria-Czech Republic. Furthermore, many thanks to Gary Cohen (INRIA ENSTA ParisTech, France) and Sebastien Imperiale (INRIA M3DISIM Paris, France) for the great cooperation towards spectral finite elements within our joint project Spectral Finite Elements for Wave Equations funded by Science and Technology Cooperation Austria-France. Moreover, I am grateful to György Paal (Budapest University of Technology and Economics, Hungary) for our research cooperation on the edge tone funded by Science and Technology Cooperation Austria-Hungary. Last but not least many thanks to Michael Nicolai (TU Dresden, Germany) for our cooperation on micro-mechanical models for the polarization process of piezoelectric materials. Finally, I want to thank Janet Sterritt-Brunner from Springer Heidelberg, Germany for her kind assistance, and V. Anandraj, M. Madhumetha as well as J. Anand (all from Scientific Publishing Services Pvt Ltd, Chennai, India), who did a great job in improving the layout of the book. November 2014

Manfred Kaltenbacher

Preface to the Second Edition

The second edition of this book fully preserves the character of the first edition to combine the detailed physical modeling of mechatronic systems and their precise numerical simulation using the Finite Element (FE) method. Most of the text and general appearance of the previous edition were retained, while the coverage was extended and the presentation improved. Starting with Chap. 2, which discusses the theoretical basics and computer implementation of the FE method, we have added a section describing the FE method for one-dimensional cases, especially to provide a easier understanding of this important numerical method for solving partial differential equations. In addition, we provide a section about a priori error estimates. In Chap. 3, which deals with mechanical fields, we now additionally discuss locking effects as occurring in the numerical computation of thin structures, and describe two well established methods (method of incompatible modes and of enhanced assumed strain) as well as a recently newly developed scheme based on balanced reduced and selective integration. The physical discussion of acoustic sound generation and propagation (see Chap. 5 has been strongly improved, including now also a description of plane and spherical waves as well as a section about quantitative measures of sound. The treatment of open domain problems has been extended and include a recently developed Perfectly Matched Layer (PML) technique, which allows to limit the computational domain to within a fraction of the wavelength without any spurious reflections. Recently developed flexible discretization techniques based on the framework of mortar FE methods for the numerical solution of coupled wave propagation problems allow for the use of different fine meshes within each computational subdomain. This technique has been applied to pure wave propagation problems (see Sect. 5.4.3) as well as coupled mechanical-acoustic field problems (see Sect. 8.3.2), where the computational grids of the mechanical region and the acoustic region can be independently generated and therefore do not match at the interface. Furthermore, we have investigated in the piezoelectric effect and provide in Chap. 9 an extended discussion on the modeling and numerical computation of nonlinear effects including hysteresis. In the last three years, we have established a research group on computational aeroacoustics to study the complex phenomenon of flow induced noise. Therewith, ix

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the totally new Chap. 10 contains a description of computational aeroacoustics with a main focus on a recently developed FE method for efficiently solving Lighthill's acoustic analogy. Within Chap. 12, which deals with industrial applications, we have rewritten Sect. 12.5 to discuss latest computational results on micromachined capacitive ultrasound transducers, and have added a section on high power ultrasound sources as used for lithotripsy as well as a section on noise generation by turbulent flows. Most of the formulations described in this book have been implemented in the software NACS (see http://www.simetris.eu/).

Acknowledgment The author wishes to acknowledge the many contributions that colleagues and collaborators have made to this second edition. First of all I would like to express my gratitude to the members of the Department of Sensor Technology and its head Prof. Reinhard Lerch for the pleasant and stimulating working atmosphere. Amongst many, I wish to specially thank M.Sc. Max Escobar, M.Sc. Andreas Hauck, M.Sc. Gerhard Link, Dipl.-Ing. Thomas Hegewald and Dipl.-Ing. Luwig Bahr for fruitful discussions and proof reading. Much is owned by many intensive discussions with my wife Prof. Barbara Kaltenbacher with whom I work on hysteresis models and parameter identification for electromagnetics and piezoelectrics. Special thanks are dedicated to Dr. Stefan Becker and his co-workers M.Sc. Irfan Ali and Dr. Frank Schäfer for the contribution on computational aeroacoustics and the intensive cooperation within the current research project Fluid-Structure-Noise founded by the Bavarian science foundation BFS. Furthermore, the author would like to thank Dr. Bernd Flemisch and Prof. Barbara Wohlmuth for the fruitful cooperation on nonmatching grids. A common research project on Numerical Simulation of Acoustic-Acoustic- and Mechanical-Acoustic-Couplings on Nonmatching Grids founded by German Research Foundation DFG has just started. Moreover, the author wants to acknowledge the excellent working environment at the Johann Radon Institute for Computational and Applied Mathematics in Linz, Austria, where the author stayed for one semester in 2005/06 as an invited lecturer for coupled field problems within a special semester on computational mechanics. Special thank is dedicated to Prof. Ulrich Langer, who organized this event, and who did a great job in bringing together different researchers from all over the world. During this time, I also started the cooperation with Prof. Dietrich Braess on enhanced softening techniques to avoid locking in thin mechanical structures, to whom I would like to express my gratitude for revealing new and interesting perspectives to me. February 2007

Manfred Kaltenbacher

Preface to the First Edition

The focus of this book is concerned with the modeling and precise numerical simulation of mechatronic sensors and actuators. These sensors, actuators, and sensor-actuator systems are based on the mutual interaction of the mechanical field with a magnetic, an electrostatic, or an electromagnetic field. In many cases, the transducer is immersed in an acoustic fluid and the solid–fluid coupling has to be taken into account. Examples are piezoelectric stack actuators for common-rail injection systems, micromachined electrostatic gyro sensors used in stabilizing systems of automobiles or ultrasonic imaging systems for medical diagnostics. The modeling of mechatronic sensors and actuators leads to so-called multifield problems, which are described by a system of nonlinear partial differential equations. Such systems cannot be solved analytically and thus a numerical calculation scheme has to be applied. The schemes discussed in this book are based on the finite element (FE) method, which is capable of efficiently solving the partial differential equations. The complexity of the simulation of multifield problems consists of the simultaneous computation of the involved single fields as well as in the coupling terms, which introduce additional nonlinearities. Examples are moving conductive (electrically charged) body within a magnetic (an electric) field, electromagnetic and/or electrostatic forces. The goal of this book is to present a comprehensive survey of the main physical phenomena of multifield problems and, in addition, to discuss calculation schemes for the efficient solution of coupled partial differential equations applying the FE method. We will concentrate on electromagnetic, mechanical, and acoustic fields with the following mutual interactions: • Coupling Electric Field—Mechanical Field This coupling is either based on the piezoelectric effect or results from the force on an electrically charged structure in an electric field (electrostatic force).

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• Coupling Magnetic Field—Mechanical Field This coupling is two-fold. First, we have the electromotive force (emf), which describes the generation of an electric field (electric voltage respectively current) when a conductor is moved in a magnetic field, and secondly, the electromagnetic force. • Coupling Mechanical Field—Acoustic Field Very often a transducer is surrounded by a fluid or a gaseous medium in which an acoustic wave is launched (actuator) or is impinging from an outside source towards the receiving transducer. In Chap. 2, we give an introduction to the finite element (FE) method. Starting from the strong form of a general partial differential equation, we describe all the steps concerning spatial as well as time discretization to arrive at an algebraic system of equations. Both nodal and edge finite elements are introduced. Special emphasis is put on an explanation of all the important steps necessary for the computer implementation. A detailed discussion on electromagnetic, mechanical, and acoustic fields including their numerical computation using the FE method can be found in Chap. 3–5. Each of these chapters starts with the description of the relevant physical equations and quantities characterizing the according physical field. Special care is taken with the constitutive laws and the resultant nonlinearities relevant for mechatronic sensors and actuators. In addition, the numerical computation using the FE method is studied for the linear as well as the nonlinear case. In Chap. 4, where the electromagnetic field is discussed, we explain the difficulties arising at interfaces of jumping material parameters (electric conductivity and magnetic permeability), and introduce two correct formulations adequate for the FE method. At the end of each of these chapters, we present an example for the numerical simulation of a practical device. In Chap. 6, we study the interaction between electrostatic and mechanical fields and concentrate on micromechanical applications. After the derivation of a general expression for the electrostatic force, applying the principle of virtual work, we focus on the numerical calculation scheme. The simulation of a simple electrostatic driven bar will demonstrate the complexity of such problems, and will show the necessity of taking into account mechanical nonlinearities. The physical modeling and numerical solution of magnetomechanical systems is presented in Chap. 7. In this chapter, we first discuss the correct physical description of moving and/or deforming bodies in a magnetic field. Later, we derive a general expression for the electromagnetic force, again (as for the electrostatic force) by using the principle of virtual work. The discussion on numerical computation will contain a calculation scheme for the efficient solution of magnetomechanical systems and, in addition, electric circuit coupling as arise for voltagedriven coils. Especially for the latter case, we give a very comprehensive description of its numerical computation. Chapter 8 deals with coupled mechanical-acoustic systems and explains the physical coupling terms and the numerical computation of such systems. The

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simulation of the sound emission of a car engine will illustrate different approaches concerning time-discretization schemes and solvers for the algebraic system. A special coupling between the mechanical and electrostatic field occurs in piezoelectric systems, which are studied in Chap. 9. After explaining the piezoelectric effect and its physical modeling, we concentrate on the efficient numerical computation of such systems. Whereas for sensor applications a linear model can be usually used, in many actuator applications nonlinear effects play a crucial role, which we here account for by applying an appropriate hysteresis model. Since the efficiency of the solution (both with respect to elapsed CPU time and computer memory resources) is of great importance, Chap. 10 deals with geometric and algebraic multigrid solvers. These methods achieve an optimal complexity, that is, the computational effort as well as memory requirement grows only linearly with the problem size. We present new especially adapted multigrid solvers for Maxwell's equation in the eddy current case and demonstrate their efficiency by means of TEAM (Testing Electromagnetic Analysis Methods) workshop problem 20 established by the Compumag Society [318]. After these rigorous derivations of methods for coupled field computation, Chap. 11 demonstrates the applicability to real-life problems arising in industry. This includes the following topics: Analysis and optimization of car loudspeakers Acoustic emission of electrical power transformers Simulation-based improvements of electromagnetic valves Piezoelectric stack actuators such as used, e.g., in common-rail diesel injection systems • Ultrasonic imaging system based on capacitive micromachined ultrasound transducers • • • •

The appendices provide an introduction to vector analysis, functional spaces, and the solutions of nonlinear equations. The structure of this book has been designed in such a way that in each of Chaps. 3–9 we first discuss the physical modeling of the corresponding single or coupled field, then the numerical simulation, followed by a simple computational example. If the reader has no previous knowledge of vector analysis, she/he should start with the first section of the Appendix. Chapter 2 can be omitted if the reader is only interested in the physical modeling of mechatronic sensors and actuators. The three chapters concerning the single field problems (mechanical, electromagnetic, acoustic) are written independently, so that the reader can start with any of them. Clearly, for the coupled field problems, the reader should have a knowledge of the involved physical fields or should have read the corresponding preceding chapters. Chapter 10 presents the latest topics on multigrid methods for electromagnetic fields and requires some knowledge on this topic. For a basic introduction of multigrid methods we refer to classical books [47, 256, 258]. Chapter 11 demonstrates the use of numerical simulation for industrial applications. For each of them, we first discuss the problem to be solved, followed by an analysis study applying numerical

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simulation to allow a better understanding of the different physical effects. For most applications, we also demonstrate measurements of the CAE-optimized prototype. Most in this book described formulations for solving multifield problems have been implemented in the software CAPA (see http://www.wissoft.de).

Acknowledgements The author wishes to express his gratitude to all the people who have inspired, increased, and sustained this work. Of course we feel obliged towards the Department of Sensor Technology in Erlangen and the previous Institute of Measurement Technology in Linz under the competent and generous leadership of Prof. Reinhard Lerch. The dynamic and stimulating atmosphere at the institute was certainly essential for this work; the author therefore thanks all his present and former colleagues among whom especially Dr. Reinhard Simkovics, Dr. Martin Rausch, Dr. Johann Hoffelner, Dr. Manfred Hofer, and Dipl.-Ing. Michael Ertl have to be mentioned. Much is owed to the long and fruitful cooperation with Dipl.-Math Hermann Landes and his company WisSoft. The author also thanks Dr. Stefan Reitzinger for many intensive and productive hours of work and discussion, and my wife Dozent Dr. Barbara Kaltenbacher for her assistance on mathematical problems and the cooperation on precise material parameter determination applying inverse methods. Moreover, we acknowledge the constructive working environment within the special research programs SFB 013 Numerical and Symbolic Scientific Computing in Linz (Prof. Ulrich Langer, Dr. Joachim Schöberl, Dr. Michael Schinnerl) funded by the Austrian science foundation FWF, and SFB 603 Modellbasierte Analyse und Visualisierung komplexer Szenen und Sensordaten in Erlangen (Dipl.-Math. Elena Zhelezina, Dr. Roberto Grosso, Dipl.-Inf. Frank Reck) funded by the DFG (Deutsche Forschungsgemeinschaft) (German Research Foundation) as well as the BMBF project Entwurf komplexer Sensor-Aktor-Systeme (Prof. Peter Schwarz, Dipl.-Ing. Rainer Peipp). Additionally, the author thanks the industrial partners involved in this work for the opportunity of doing research on real-life problems. Finally, I would like to thank my copyeditor Dr. Peter Capper for reading the book very carefully and pointing out many errors and misspellings. December 2003

Manfred Kaltenbacher

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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The Finite Element (FE) Method . . . . . . . . . . . . . . . . . . . 2.1 Finite Element Formulation . . . . . . . . . . . . . . . . . . . 2.2 Finite Element Method for a 1D Problem . . . . . . . . . 2.3 Nodal (Lagrangian) Finite Elements . . . . . . . . . . . . . 2.3.1 Basic Properties . . . . . . . . . . . . . . . . . . . . . 2.3.2 Quadrilateral Element in R2 . . . . . . . . . . . . 2.3.3 Triangular Element in R2 . . . . . . . . . . . . . . . 2.3.4 Tetrahedron Element in R3 . . . . . . . . . . . . . 2.3.5 Hexahedron Element in R3 . . . . . . . . . . . . . 2.3.6 Wedge Element in R3 . . . . . . . . . . . . . . . . . 2.3.7 Pyramidal Element in R3 . . . . . . . . . . . . . . . 2.3.8 Global/Local Derivatives . . . . . . . . . . . . . . . 2.3.9 Numerical Integration . . . . . . . . . . . . . . . . . 2.4 Finite Element Procedure . . . . . . . . . . . . . . . . . . . . . 2.5 Time Discretization . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Parabolic Differential Equation . . . . . . . . . . . 2.5.2 Hyperbolic Differential Equation. . . . . . . . . . 2.6 Integration over Surfaces . . . . . . . . . . . . . . . . . . . . . 2.7 Edge Nédélec Finite Elements . . . . . . . . . . . . . . . . . 2.8 Discretization Error . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Finite Elements of Higher Order . . . . . . . . . . . . . . . . 2.9.1 Legendre Polynomials and Hierarchical Finite Elements. . . . . . . . . . . . . . . . . . . . . . 2.9.2 Lagrange Polynomials and Spectral Elements . 2.10 Flexible Discretization . . . . . . . . . . . . . . . . . . . . . . . 2.10.1 Mortar FEM. . . . . . . . . . . . . . . . . . . . . . . . 2.10.2 Nitsche Type Mortaring. . . . . . . . . . . . . . . .

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2.10.3 Numerical Example. . . . . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Mechanical Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Navier’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Deformation and Displacement Gradient . . . . . . . . . 3.3 Mechanical Strain . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Plane Strain State . . . . . . . . . . . . . . . . . . . 3.4.2 Plane Stress State . . . . . . . . . . . . . . . . . . . 3.4.3 Axisymmetric Stress–Strain Relations . . . . . 3.5 Waves in Solid Bodies . . . . . . . . . . . . . . . . . . . . . 3.6 Material Properties . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Numerical Computation . . . . . . . . . . . . . . . . . . . . . 3.7.1 Linear Elasticity . . . . . . . . . . . . . . . . . . . . 3.7.2 Damping Model . . . . . . . . . . . . . . . . . . . . 3.7.3 Geometric Nonlinear Case . . . . . . . . . . . . . 3.7.4 Numerical Example. . . . . . . . . . . . . . . . . . 3.8 Locking and Efficient Solution Approaches . . . . . . . 3.8.1 Incompatible Modes Method . . . . . . . . . . . 3.8.2 Enhanced Assumed Strain Method . . . . . . . 3.8.3 Balanced Reduced and Selective Integration . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Flow 4.1 4.2 4.3

Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spatial Reference Systems . . . . . . . . . . . . . . . . . . . . Reynolds’ Transport Theorem. . . . . . . . . . . . . . . . . . Conservation Equations . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Conservation of Mass . . . . . . . . . . . . . . . . . 4.3.2 Conservation of Momentum . . . . . . . . . . . . . 4.3.3 Conservation of Energy . . . . . . . . . . . . . . . . 4.3.4 Constitutive Equations . . . . . . . . . . . . . . . . . 4.4 Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . 4.5 Characterization of Flows by Dimensionless Numbers . 4.6 Finite Element Formulation . . . . . . . . . . . . . . . . . . . 4.7 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . 4.7.1 Steady Channel Flow . . . . . . . . . . . . . . . . . 4.7.2 Unsteady Flow Around a Square . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Acoustic Field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Wave Theory of Sound . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Conservation of Mass (Continuity Equation) . 5.1.2 Conservation of Momentum (Euler Equation) .

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Contents

5.1.3 Pressure-Density Relation (State Equation) . . 5.1.4 Linear Acoustic Wave Equation . . . . . . . . . 5.1.5 Acoustic Quantities . . . . . . . . . . . . . . . . . . 5.1.6 Plane and Spherical Waves . . . . . . . . . . . . 5.2 Quantitative Measure of Sound. . . . . . . . . . . . . . . . 5.3 Nonlinear Acoustic Wave Equation . . . . . . . . . . . . . 5.4 Numerical Computation . . . . . . . . . . . . . . . . . . . . . 5.4.1 Linear Acoustic Wave Equation . . . . . . . . . 5.4.2 Linear Acoustic Conservation Equations . . . 5.4.3 Nonlinear Acoustics . . . . . . . . . . . . . . . . . 5.4.4 Non-conforming Grids. . . . . . . . . . . . . . . . 5.4.5 Discretization Error . . . . . . . . . . . . . . . . . . 5.5 Treatment of Open Domain Problems . . . . . . . . . . . 5.5.1 Absorbing Boundary Conditions . . . . . . . . . 5.5.2 Perfectly Matched Layer (PML) Technique . 5.6 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 Transient Wave Propagation in Unbounded Domains . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.2 Harmonic Wave Propagation in Unbounded Domains . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.3 Nonlinear Wave Propagation in a Channel . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

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Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Law of Ampère . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Law of Faraday . . . . . . . . . . . . . . . . . . . . . . . 6.1.3 Law of Gauss. . . . . . . . . . . . . . . . . . . . . . . . . 6.1.4 Solenoidal Magnetic Field . . . . . . . . . . . . . . . . 6.2 Quasistatic Electromagnetic Fields . . . . . . . . . . . . . . . . 6.2.1 Magnetic Vector Potential . . . . . . . . . . . . . . . . 6.2.2 Skin Effect. . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Electrostatic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Magnetic Permeability . . . . . . . . . . . . . . . . . . . 6.4.2 Electrical Conductivity . . . . . . . . . . . . . . . . . . 6.4.3 Dielectric Permittivity . . . . . . . . . . . . . . . . . . . 6.5 Electromagnetic Interface Conditions. . . . . . . . . . . . . . . 6.5.1 Continuity Relations for Magnetic Field . . . . . . 6.5.2 Continuity Relations for Electric Field . . . . . . . . 6.5.3 Continuity Relations for Electric Current Density 6.6 Numerical Computation: Electrostatics. . . . . . . . . . . . . . 6.7 Numerical Computation: Electromagnetics . . . . . . . . . . . 6.7.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . .

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227 227 229 230 233 234 235 235 236 238 239 239 242 243 244 244 245 247 247 249 249

xviii

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6.7.2 Discretization with Edge Elements . . . . . 6.7.3 Discretization with Nodal Finite Elements 6.7.4 Newton’s Method for the Nonlinear Case 6.7.5 Approximation of BH Curve . . . . . . . . . 6.7.6 Higher Order Edge Elements . . . . . . . . . 6.7.7 Modeling of Current-Loaded Coil . . . . . . 6.7.8 Computation of Global Quantities . . . . . . 6.7.9 Induced Electric Voltage . . . . . . . . . . . . 6.7.10 Voltage-Loaded Coil . . . . . . . . . . . . . . . 6.8 Numerical Examples . . . . . . . . . . . . . . . . . . . . . 6.8.1 Thin Iron Plate . . . . . . . . . . . . . . . . . . . 6.8.2 TEAM-13 Benchmark Problem. . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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255 257 260 263 265 271 272 275 275 277 277 280 282

7

Coupled Flow-Structural Mechanical Systems 7.1 Fluid-Solid Interaction . . . . . . . . . . . . . . 7.2 Coupling Types and Strategies . . . . . . . . 7.3 Grid Adaption . . . . . . . . . . . . . . . . . . . 7.4 Numerical Examples . . . . . . . . . . . . . . . 7.4.1 Solid Plunger . . . . . . . . . . . . . . 7.4.2 Flag in a Flow . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . .

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285 285 286 289 292 292 293 296

8

Coupled Mechanical-Acoustic Systems . . 8.1 Solid–Fluid Interface . . . . . . . . . . . 8.2 Coupled Field Formulation. . . . . . . 8.3 Numerical Computation . . . . . . . . . 8.3.1 Finite Element Formulation 8.3.2 Non-conforming Grids. . . . 8.3.3 Numerical Examples . . . . . References. . . . . . . . . . . . . . . . . . . . . . .

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297 297 299 300 300 302 303 308

9

Computational Aeroacoustics . . . . . . . . . . . . . . . . . . . 9.1 Requirements for Numerical Schemes . . . . . . . . . . 9.2 Lighthill’s Analogy . . . . . . . . . . . . . . . . . . . . . . . 9.3 Curle’s Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Vortex Sound . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Perturbation Equations. . . . . . . . . . . . . . . . . . . . . 9.6 Finite Element Formulation . . . . . . . . . . . . . . . . . 9.6.1 Lighthills’ Inhomogeneous Wave Equation 9.6.2 Perturbation Equations. . . . . . . . . . . . . . . 9.6.3 Source Term Treatment . . . . . . . . . . . . . . 9.7 Comparison of Different Aeroacoustic Analogies . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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309 309 312 317 322 324 327 327 330 333 334 337

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Contents

10 Coupled Electrostatic-Mechanical Systems . 10.1 Electrostatic Force. . . . . . . . . . . . . . . 10.2 Numerical Computation . . . . . . . . . . . 10.2.1 Calculation Scheme. . . . . . . . 10.2.2 Voltage-Driven Bar . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . .

xix

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339 339 346 347 349 351

11 Coupled Magnetomechanical Systems . . . . . . . . . 11.1 General Moving/Deforming Body . . . . . . . . . 11.2 Electromagnetic Force . . . . . . . . . . . . . . . . . 11.3 Numerical Computation . . . . . . . . . . . . . . . . 11.3.1 Force Computation Via the Principle of Virtual Work . . . . . . . . . . . . . . . 11.3.2 Grid Adaption Techniques . . . . . . . . 11.3.3 Calculation Scheme. . . . . . . . . . . . . 11.3.4 Moving Current/Voltage-Loaded Coil References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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353 353 355 357

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357 360 364 366 373

12 Piezoelectric Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . 12.2 Governing Equations: Linear Piezoelectricity . . . . . . . 12.3 Piezoelectric Material Properties . . . . . . . . . . . . . . . . 12.4 Models for Nonlinear Piezoelectricity . . . . . . . . . . . . 12.4.1 Macroscopic Model with Hysteresis Operators 12.4.2 Micro-mechanical Switching Model . . . . . . . 12.5 Numerical Computation . . . . . . . . . . . . . . . . . . . . . . 12.5.1 Linear Case . . . . . . . . . . . . . . . . . . . . . . . . 12.5.2 Macroscopic Hysteresis Based Approach . . . . 12.5.3 Micro-mechanical Switching Model . . . . . . . 12.6 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . 12.6.1 Computation of Impedance Curve . . . . . . . . . 12.6.2 Piezoelectric Disc Actuator . . . . . . . . . . . . . 12.6.3 Polarization and Depolarization Process . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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375 375 378 379 384 384 392 393 394 396 400 405 405 408 409 412

13 Algebraic Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Preconditioned Conjugate Gradient (PCG) Method 13.2 Multigrid (MG) Method. . . . . . . . . . . . . . . . . . . 13.3 Geometric MG Method . . . . . . . . . . . . . . . . . . . 13.3.1 Geometric MG for Edge Elements . . . . . 13.3.2 Case Study. . . . . . . . . . . . . . . . . . . . . . 13.4 Algebraic MG Method . . . . . . . . . . . . . . . . . . . 13.4.1 Auxiliary Matrix . . . . . . . . . . . . . . . . . . 13.4.2 Coarsening Process . . . . . . . . . . . . . . . .

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415 415 417 420 420 423 426 427 427

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xx

Contents

13.4.3 Prolongation Operators . . . . . . . . . 13.4.4 Smoother and Coarse-Grid Operator 13.4.5 AMG for Nodal Elements . . . . . . . 13.4.6 AMG for Edge Elements . . . . . . . . 13.4.7 AMG for Time-Harmonic Case . . . 13.4.8 Case Studies . . . . . . . . . . . . . . . . 13.5 Block Preconditioner for Higher Order Edge Element Discretization . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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431 431 432 433 436 437

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443 450

14 Industrial Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Electrodynamic Loudspeaker . . . . . . . . . . . . . . . . . . . 14.1.1 Finite Element Models. . . . . . . . . . . . . . . . . . 14.1.2 Verification of Computer Models . . . . . . . . . . 14.1.3 Numerical Analysis of the Nonlinear Loudspeaker Behavior . . . . . . . . . . . . . . . . . . 14.1.4 Computer Optimization of the Nonlinear Loudspeaker Behavior . . . . . . . . . . . . . . . . . . 14.2 Noise Computation of Power Transformers . . . . . . . . . 14.2.1 Finite Element Models. . . . . . . . . . . . . . . . . . 14.2.2 Verification of the Computer Models. . . . . . . . 14.2.3 Verification of the Calculated Winding and Tank-Surface Vibrations . . . . . . . . . . . . . 14.2.4 Verification of the Sound-Field Calculations. . . 14.2.5 Influence of Tap-Changer Position . . . . . . . . . 14.2.6 Influence of Stiffness of Winding Supports . . . 14.3 Fast-Switching Electromagnetic Valves . . . . . . . . . . . . 14.3.1 Modeling and Solution Strategy . . . . . . . . . . . 14.3.2 Actuator Characteristics . . . . . . . . . . . . . . . . . 14.3.3 Actuator Dynamics . . . . . . . . . . . . . . . . . . . . 14.3.4 Dynamics Optimization I: Electrical Premagnetization . . . . . . . . . . . . . . . . . . . . . 14.3.5 Dynamics Optimization II: Overexcitation . . . . 14.3.6 Switching Cycle . . . . . . . . . . . . . . . . . . . . . . 14.4 Cofired Piezoceramic Multilayer Actuators. . . . . . . . . . 14.4.1 Polarization of a Stack Actuator . . . . . . . . . . . 14.4.2 Stack Actuator: Hysteresis Based Approach . . . 14.5 Capacitive Micro-machined Ultrasound Transducers . . . 14.5.1 Requirements to Numerical Simulation Scheme 14.5.2 Single CMUT Cell . . . . . . . . . . . . . . . . . . . . 14.5.3 CMUT Array . . . . . . . . . . . . . . . . . . . . . . . . 14.5.4 Controlled CMUT Array . . . . . . . . . . . . . . . . 14.6 High-Intensity Focused Ultrasound . . . . . . . . . . . . . . . 14.6.1 Piezoelectric Transducer and Input Impedance .

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475 477 478 479 480 483 485 486 488 490 491 495 495

Contents

14.6.2 Pressure Pulse Computation . . . . . . . . . . 14.6.3 High-Power Pulse Sources for Lithotripsy 14.7 Human Phonation . . . . . . . . . . . . . . . . . . . . . . . 14.7.1 Mathematical Modeling . . . . . . . . . . . . . 14.7.2 2D Fully Coupled Simulation . . . . . . . . . 14.7.3 3D Driven Simulation . . . . . . . . . . . . . . 14.8 Aeroacoustics of Flow Around Obstacles . . . . . . . 14.8.1 Square Cylinder Geometries . . . . . . . . . . 14.8.2 Edge Tone . . . . . . . . . . . . . . . . . . . . . . 14.8.3 Airframe Noise. . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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497 498 503 505 505 510 515 515 523 530 533

15 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

537 538

Appendix A: Norms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

539

Appendix B: Scalar and Vector Fields . . . . . . . . . . . . . . . . . . . . . . . .

541

Appendix C: Tensors and Index Notation . . . . . . . . . . . . . . . . . . . . . .

557

Appendix D: Appropriate Function Spaces . . . . . . . . . . . . . . . . . . . . .

563

Appendix E: Solution of Nonlinear Equations . . . . . . . . . . . . . . . . . . .

569

Appendix F: Hysteresis Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

575

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

581

Notation

Mathematical Symbols e n t R r, R x ds C H ds C R dΓ Γ H dΓ Γ R dΩ

Unit vector Unit normal vector Unit tangential vector Set of real numbers Position vector Contour integral Closed contour integral Surface integral Closed surface integral Volume integral

Ω

r curl, r div, r
grad, r o=ox o=on d=dx ‖‖ || [] [I]

Nabla operator Curl Divergence Gradient Partial derivative Partial derivative in normal direction Total derivative Norm Semi-norm Tensor notation Identity tensor

xxiii

xxiv

Notation

Finite Element Method u; a, etc. C Fe K M nn nen ne neq nd Ni J jJ j x; y; z  Ω Ω ^ Ω Γ Γe Γn γP βH ; γ H ξ; η; ζ

Nodal vectors of displacement, acceleration, etc Damping matrix Mapping of element Stiffness matrix Mass matrix Number of nodes Number of nodes per finite element Number of elements Number of equations Space dimension FE basis function for node i Jacobi matrix Jacobi determinant Global coordinates Whole simulation domain Simulation domain without boundary Domain of reference element Boundary of simulation Dirichlet boundary Neumann boundary Integration parameter (parabolic PDE) Integration parameters (hyperbolic PDE) Local coordinates

Mechanics a [c] cL cT Em fV ½Fd Š ½Hd Š G m Pmech S [S] T [T]

Acceleration Tensor of mechanical modulus Velocity of longitudinal wave Velocity of shear wave Elasticity module Volume force Deformation gradient Displacement gradient Shear modulus Mass Mechanical power Linear strains (Voigt notation) Tensor of linear strains Second Piola-Kirchhoff stress (Voigt notation) Second Piola-Kirchhoff stress tensor

Notation

u υ V [V] αM ; αK ρ νp σ ½σŠ μL ; λL

xxv

Mechanical displacement Velocity Green-Lagrangian strain (Voigt notation) Green-Lagrangian strain tensor Damping coefficients Density Poisson ratio Cauchy stress (Voigt notation) Cauchy stress tensor Lamé-parameters

Flow e Eu Fr I Im Ma qh qT p P Re s St υ ½ǫŠ λf μf νf ½πŠ ρ ½σ f Š ½τŠ ω

Inner energy Euler number Froude number Momentum Molecular momentum Mach number Heat production per volume Heat flux Flow pressure Kinematic pressure Reynolds number Entropy Strouhal number Flow velocity Strain rate Bulk viscosity Dynamic viscosity Kinematic viscosity Momentum flux tensor Density Fluid stress tensor Viscous stress tensor Vorticity

Acoustics b B/A c cp

Diffusivity of sound Parameter of nonlinearity Speed of sound Specific heat by constant pressure

xxvi

Notation

cΩ e Ia k Ks Lpa ; SPL LIa LPa pa p0 Pa qh s υa wa xs Za ρa ρ0 κ λ λf μf ψ

Specific heat by constant volume Inner energy Sound-field intensity Wave number Adiabatic bulk modulus Sound-pressure level Sound-intensity level Sound-power level Acoustic pressure Mean pressure Acoustic power Heat production per volume Entropy Acoustic particle velocity Acoustic energy density Shock formation distance Acoustic impedance Acoustic density Mean density Adiabatic exponent Wavelength Bulk viscosity Dynamic viscosity Scalar acoustic potential

Electromagnetics A B D E Fel Fmag H I, i J Ji M qe Qe P R uind Ve

Magnetic vector potential Magnetic flux density Electric flux density Electric field intensity Electric force Magnetic force Magnetic field intensity Electric current Current density Impressed current density Magnetization Electric charge density Total electric charge Electric polarization Ohmic resistor Induced voltage Scalar electric potential

Notation

wel Wel wmag Wmag δ ρe γ ε μ ν σe φ ψm Ψ

xxvii

Electric energy density Total electric energy Magnetic energy density Total magnetic energy Skin depth Specific electric resistance Electrical conductivity Electric permittivity Magnetic permeability Magnetic reluctivity Electric surface charge Magnetic flux Reduced magnetic scalar potential Total magnetic flux

Chapter 1

Introduction

Each modern industrial process environment needs sensors to detect the physical quantities involved (e.g., electric current, mechanical torque, temperature, etc.), a signal-conditioning circuit, and an interface to computers, where the process parameters are controlled. According to the controlling signals, power electronic circuits supply the actuators, which will steer the process (see Fig. 1.1). As indicated by the gray boxes in Fig. 1.1, the main topic of this book is concerned with sensors and actuators, especially their sophisticated design. Therefore, we provide a comprehensive discussion on the precise physical modelling of sensors and actuators and, furthermore, on the numerical solution of the governing partial differential equations. To be precise, we concentrate our investigation on mechatronic sensors and actuators. Mechatronic sensors, actuators and sensor-actuator systems are based on the mutual interaction of the mechanical field and an electrostatic, an electromagnetic, or an acoustic field. Typical examples are: electromagnetic valves for injection systems in vehicles, capacitive micromachined ultrasound transducers (CMUTs) for medical imaging systems, electrodynamic loudspeakers, surface acoustic wave (SAW) devices for telecommunications (e.g., in a mobile phone), piezoelectric or magnetostrictive actuators for ultrasound cleaning, etc. (see Fig. 1.2). The main transducing mechanisms for mechatronic sensor and actuators are displayed in Fig. 1.3 (those in boldface will be discussed within this book). In most cases, the fabrication of prototypes within the design process of modern mechatronic transducers is a lengthy and costly task. The still widely used experimental-based design, as shown in Fig. 1.4a exhibits many disadvantages. First, it is very time-consuming, since for each change in the design a new prototype has to be fabricated and the relevant parameters have to be measured. In particular, if a design failure is recognized late, high costs result. Furthermore, not all parameters of interest can be measured (e.g., magnetic field or mechanical stresses inside a solid body) and the measurement setup may influence the (dynamical) behavior of the prototype. Since for the development of modern mechatronic transducers all the different coupling mechanisms of the involved physical fields have to be considered, the design process is a very complex task. Therefore, an increasing need for reliable and usable © Springer-Verlag Berlin Heidelberg 2015 M. Kaltenbacher, Numerical Simulation of Mechatronic Sensors and Actuators, DOI 10.1007/978-3-642-40170-1_1

1

2

1 Introduction

Process (physical, chemical, biological)

Actuator

Sensors

Any physical quantity

Any physical quantity

Interfacing signal conditioning

Electrical signal

Electrical signal

Computer controlling, information processing

Electronics, power amplifier

Analog/digital data

Fig. 1.1 Industrial process

(a)

(b)

Housing

Transducer cells

Yoke

Electronic Fuel inlet

Injection

Electrical connection

Bond pads Stator Coil

Fig. 1.2 Examples of mechatronic sensors and actuators. a Electromagnetic valve. b Capacitive micromachined ultrasound transducers (CMUT)

Mechanics

distance velocity acceleration force torque mass flow acoustic quantities

Sensor Actuator

piezoelectric piezoresistive electrostatic electrodynamic magnetomechanic optical

Electrics

voltage current charge impedance

Fig. 1.3 Transducing mechanisms of mechatronic sensors and actuators

computer modelling tools capable of precisely simulating the multifield interactions arises. Such appropriate computer-aided engineering (CAE) tools offer many possibilities to the design engineer [1]. Arbitrary modification of transducer geometry and selective variation of material parameters are easily performed and the influence on the transducer behavior can be studied immediately. Furthermore, an automatic optimization can be performed within the CAE-based design [2]. In addition,

1 Introduction

3

(a)

(b) Definition of concept

Definition of concept

Design

Computer aided design (CAD)

Optimization process

Change Prototype fabrication / test

No

O.K. Yes Manufacturing

Simulation

No

O.K. Yes Prototype fabrication / test

Fig. 1.4 Design process. a Experimental-based design. b CAE-based design

the simulation provides access to physical quantities that cannot be measured, and simulations strongly support the insight into physical phenomena. Thus, a CAE-based design as displayed in Fig. 1.4b can tremendously reduce the number of necessary prototypes within the design process. However, we want to emphasize that a direct physical control of the transducer design is possible only with the help of experiments, whereas the computer simulation is always based on a model of reality. Therefore, the quality of the results depends on the suitability of the physical model as well as the material parameters. Moreover, numerical effects may spoil the results unless proper computational methods are used. For these reasons, one should also be aware of the risk of trusting every result the CAE environment computes. These facts make it very important that the user of such tools has both a deep physical understanding of the ongoing processes and mathematical knowledge of the simulation algorithms. Summarizing, one can say that an appropriate use of CAE tools for the design of mechatronic transducers can tremendously reduce the number of necessary prototypes within the design process, but the user should always critically question each result. The use of these CAE tools in the design of mechatronic transducers started only some years ago. The main reason for the lack of CAE environments, capable of performing multifield simulations that arise in the design of mechatronic sensors and actuators, is the complexity of such problems. We will demonstrate this by discussing the requirements on a CAE tool for the numerical simulation of an electrodynamic loudspeaker. Figure 1.5 shows an electrodynamic loudspeaker with all its components. A cylindrical, small light voice coil is suspended freely in a strong radial magnetic field, generated by a permanent magnet. The magnet assembly, consisting of pole, back plate and top plate, helps to concentrate most of the magnetic flux within the magnet structure and, therefore, within the narrow radial gap. When the

4

1 Introduction

Surround

Cone diaphragm Dust cap Suspension Former Permanent magnet

Voice coil Magnet assembly

Fig. 1.5 Electrodynamic loudspeaker

coil is loaded by an electric voltage, the interaction between the magnetic field of the permanent magnet and the current in the voice coil results in an axial Lorentz force. The voice coil is wound onto an aluminum holder, which is attached to the rigid, light cone diaphragm in order to couple the forces more effectively to the air and, hence, to permit acoustic power to be radiated from the assembly. The dust cap does not usually fulfill any acoustic function, but prevents the penetration of dust into the air gap of the magnetic assembly. The main function of the suspension and the surround is to allow free axial movement of the moving coil driver, while nonaxial movements are suppressed almost completely. To obtain a precise description of an electrodynamic loudspeaker we have to model the electromagnetic, mechanical, and acoustic fields, and in addition, the following coupling mechanisms: • Mechanical Field—Acoustic Field The normal component of the surface velocity of the solid must meet the normal component of the particle velocity of the fluid. • Mechanical Field—Magnetic Field The interaction between these two physical fields is given by the Lorentz force and the motional electromotive force (emf) term leads to additional eddy currents in the coil. • Electric Circuit—Magnetic Field Due to the fact that the coil is voltage driven, the magnetic field equation has to be solved together with the electric circuit equation. For the design of electrodynamic loudspeakers, the frequency dependence of the axial pressure response at 1 m distance and the electrical input impedance of the voice coil are the two most important parameters. Since the overall physical model consists of a system of coupled, nonlinear partial differential equations a transient analysis has to be performed and then the results are transformed to the frequency domain. To obtain a good frequency resolution within the frequency range of interest (0−20 kHz) around 10,000 time steps have to be computed. Since the coupled system

1 Introduction

5

of partial differential equations cannot be solved analytically, a numerical scheme has to be applied. The finite element (FE) method has been established as the standard method for numerically solving the partial differential equations describing the physical fields including their couplings. Thus, a static, transient and time-harmonic analysis including nonlinearities (e.g., material nonlinearities, geometric nonlinearities, etc.) can be performed very efficiently. This method is currently used in most commercial computer codes. Of course it has to be mentioned that for different physical field problems (e.g., fluid dynamics, high-frequency electromagnetics, etc.) different numerical methods (e.g., finite volume, finite difference, finite integration, boundary element) might be the methods of choice [3–8].

References 1. M. Kaltenbacher, Simulationsbasierte Entwicklung von Sensoren. Tech. Mess. 79, 30–36 (2012) 2. F. Wein, M. Kaltenbacher, M. Stingl, Topology optimization of a cantilevered piezoelectric energy harvester using stress norm constraints. Struct. Multidiscip. Optim. 48, 173–187 (2013) 3. K.J. Binns, P.J. Lawrenson, C.W. Trowbridge (eds.), The Analytic and Numerical Solution of Electric and Magnetic Fields (Wiley, New York, 1992) 4. C.A. Brebbia, J. Dominguez, Boundary Elements, An Introductory Course (McGraw-Hill, New York, 1992) 5. J.H. Ferzinger, M. Peric, Computational Methods for Fluid Dynamics (Springer, New York, 2002) 6. T. Weiland, Time domain electromagnmetic field computation. Int. J. Num. Model. 9, 295–319 (1996) 7. M. Fischer, L. Gaul, Fast BEM-FEM mortar coupling for acoustic-structure interaction. Int. J. Numer. Methods Eng. 62(12), 677–1690 (2005) 8. G. Of, M. Kaltenbacher, O. Steinbach, Fast multipole boundary element method for electrostatic field computations. COMPEL 28(2), 304–319 (2009)

Chapter 2

The Finite Element (FE) Method

The finite element (FE) method has become the standard numerical calculation scheme for the computer simulation of physical systems [1–3]. The advantages of this method can be summarized as follows: • Numerical efficiency: The discretization of the calculation domain with finite elements yields matrices that are in most cases sparse and symmetric. Therefore, the system matrix, which is obtained after spatial and time discretization, is sparse and symmetric, too. Both the storage of the system matrix and the solution of the algebraic system of equations can be performed in a very efficient way. • Treatment of nonlinearities: The modeling of nonlinear material behavior is well established for the FE method (e.g., nonlinear curves, hysteresis). • Complex geometry: By the use of the FE method, any complex domain can be discretized by triangular elements in 2D and by tetrahedra in 3D. • Analysis possibilities: The FE method is suited for static, transient, harmonic as well as eigenfrequency analysis. The two essential disadvantages of the FE method are given by • Discretization: The effort for the discretization of the simulation domain is quite high, since in 2D the whole cross section and in 3D the whole volume has to be subdivided into finite elements. • Open domain problems: Models that need the treatment of an open boundary, e.g., the simulation of radiation characteristics of an ultrasound array, lead in the general case to errors due to the limitation of the simulation domain. One of several approaches to overcome this problem is the use of absorbing boundary conditions, perfectly matched layer (PML) techniques or so-called infinite elements (see Sect. 5.5). The general approach of the FE method is shown in Fig. 2.1. Starting from the partial differential equation (PDE) with given boundary conditions, we multiply it by appropriate test functions and integrate over the whole simulation domain. Performing an integration by parts, we arrive at the variational formulation, also © Springer-Verlag Berlin Heidelberg 2015 M. Kaltenbacher, Numerical Simulation of Mechatronic Sensors and Actuators, DOI 10.1007/978-3-642-40170-1_2

7

8

2 The Finite Element (FE) Method

Strong formulation of PDE

Weak (variational) form of PDE

Algebraic system of equations

Galerkin’s approximation method

Partial Integration

Fig. 2.1 From the strong formulation to the algebraic system of equation Fig. 2.2 FE method: discretization of the domain with quadrilateral finite elements

Domain

3

4 Finite element 1

2

1 4 Finite element nodes

called the weak formulation. Applying Galerkin’s approximation method using finite elements (FE) results in the algebraic system of equations. As already mentioned, the use of the FE method requires the discretization of the whole domain (see Fig. 2.2). For the discretization triangular as well as quadrilateral finite elements are used in 2D and tetrahedral as well as hexahedron finite elements in 3D. The physical quantity of interest (e.g., temperature, mechanical displacement, etc.) is approximated by so-called shape functions and the solution of the algebraic equation yields the physical quantity in the discretization points, the so-called finite element nodes, for Lagrangian finite elements and along the edges for Nédélec finite elements.

2.1 Finite Element Formulation In the following, all steps—from the strong formulation of the partial differential equation (PDE) to the algebraic equation—will be briefly described by means of the following simple PDE with the searched for quantity u(r, t), the known source term f (r, t) at each time t of the interval (0, T ), and the corresponding initial and boundary conditions. Given: f : Ω × (0, T ) → R u0 : Ω → R Find: u : Ω¯ × [0, T ] → R

2.1 Finite Element Formulation

9

∂u ∂t u ∂u ∂n u(r, 0)

(2.1)

= ∇ · ∇u + f = u e on Γe × (0, T ) = u n on Γn × (0, T ) = u 0 , r ∈ Ω.

In this so-called strong formulation of the initial-boundary value problem R denotes the set of real numbers, Ω¯ the simulation domain, Ω the simulation domain without the boundary Γ = Γe ∪ Γn , Γe the boundary with prescribed Dirichlet boundary condition, and Γn the boundary with prescribed Neumann boundary condition. Now, let us introduce for any t ∈ [0, T ] the space Tt Tt = {u(·, t) | u(·, t) ∈ H 1 (Ω), u(r, t) = u e (r, t) on Γe } ,

(2.2)

and G, the space of so-called test functions, as G = {w | w ∈ H 1 (Ω), w = 0 on Γe },

(2.3)

with H 1 the standard Sobolev space (see Appendix D). It has to be noted that the spaces Tt , t ∈ [0, T ] vary with time, whereas the space G is time-independent. In the first step, we multiply the partial differential equation with an arbitrary test function w and perform an integration over the whole domain Ω 

w Ω



∂u − ∇ · ∇u − f ∂t



d = 0.

Applying Green’s first integration theorem to the above equation results in 

∂u d + w ∂t Ω



∇w · ∇u d = Ω



Ω

w f d +



Γn

w

∂u dŴ. ∂n

(2.4)

Thus, the weak formulation (often also called variational formulation) for the initialboundary problem is as follows: Given: f u0

: Ω × (0, T ) → R :Ω→R

Find: u(t) ∈ Tt such that for all w ∈ G and t ∈ [0, T ]

10

2 The Finite Element (FE) Method



w Ω

∂u d + ∂t



∇w · ∇u d = Ω



Ω



w f d + Ω



wu n dŴ

(2.5)

Γn

u = u e on Γe × (0, T )  wu(0) d = wu 0 d. Ω

Since the Neumann boundary condition is now incorporated into the equation of the weak form, it is also called natural. The Dirichlet boundary condition on u still has to be explicitly forced, and is therefore called essential. Formally it can be proven that the two formulations according to (2.1) and (2.5) are mathematically equivalent, provided u is sufficiently smooth [4]. To discretize (2.5), which is still infinite dimensional, we now apply the approximation according to Galerkin’s method. Let us define the finite dimensional spaces Tth and Gh according to Tth ⊂ T

Gh ⊂ G.

Therefore, we perform the domain discretization (see Fig. 2.2), and approximate the searched for quantity u(t) as well as the test function w by u(t) ≈ u h (t)

w ≈ wh ,

(2.6)

with h the discretization parameter (defining the mesh size). Furthermore, we decompose u h (t) into the searched for value v h (t) and the known Dirichlet values u eh (t). For v h , w h , and u eh we choose the following ansatz h

v (t) =

n eq 

Na (r)va (t)

(2.7)

Na (r)ca

(2.8)

Na (r)u ea (t),

(2.9)

a=1 n eq

wh = u eh (t) =



a=1 ne  a=1

where Na (r) denotes appropriate shape functions (often also called interpolation or basis functions), n eq the number of unknowns, which is equal to the number of finite element nodes with no Dirichlet boundary condition, and n e the number of finite element nodes with Dirichlet boundary condition. Substituting (2.7)–(2.9) into (2.5) results in

2.1 Finite Element Formulation n eq   Ω

+



Ω

a=1  n eq

 a=1

11

n eq ∂  Nb vb Na ca ∂t

∂ Na ca ∂t

=

b=1 n e 

Nb u eb

b=1

n eq  



d +



∇

Ω



d +

Ω a=1

a=1  n eq



∇



n eq 

Ω

Na ca f (r a ) d +

 n eq  

Na ca

Na ca

a=1



·∇

 

 n eq 

· ∇

b=1 ne 

Nb vb



Nb u eb

b=1

Na ca u n (r a ) dŴ.

d



d

(2.10)

Γn a=1

Now, since we can put the sums before the integrals and having in mind that ∇ just operates on the shape functions N (r) (ca as well as u b are constants with respect to the space variables), we may write (2.10) for the 2D plane case as follows n eq  a=1

ca



n eq   

Ω

b=1

 

∂vb Na Nb d ∂t

∂ Na ∂ Nb ∂ Na ∂ Nb + ∂x ∂x ∂y ∂y  Ω  Na u n dŴ Na f d − − +

Ω



 d vb

Γn

  n e    ∂ Na ∂ Nb ∂ Na ∂ Nb + + dŴ u eb ∂x ∂x ∂y ∂y Ω b=1   n e   ∂u eb + Na Nb d = 0. ∂t Ω b=1

Since the equation has to be fulfilled for all coefficients ca , we obtain the defining equations for the searched for finite element node values vb : for each a (a = 1, .., n eq ) we have to solve an equation as follows n eq    b=1



  

∂ Na ∂ Nb ∂ Na ∂ Nb ∂vb d vb + + Na Nb d ∂t ∂x ∂x ∂y ∂y Ω Ω   − Na u n dŴ Na f d − Ω

Γn



n e    ∂ Na ∂ Nb ∂ Na ∂ Nb d u eb + + ∂x ∂x ∂y ∂y Ω b=1

n e   ∂u eb = 0. + Na Nb d ∂t Ω b=1

(2.11)

12

2 The Finite Element (FE) Method

Thus, the semi-discrete Galerkin formulation can be written in matrix form as follows (2.12)

Mv˙ + Kv = f , with v˙ = ∂v/∂t, v the nodal unknowns and f the right-hand side vector. • Mass matrix M: M = [Mab ]  Mab = Na Nb d

(2.13)

K = [K ab ]    ∂ Na ∂ Nb ∂ Na ∂ Nb d + K ab = ∂x ∂x ∂y ∂y Ω 1 ≤ a, b ≤ n eq

(2.14)

Ω

1 ≤ a, b ≤ n eq • Stiffness Matrix K:

• Right-hand side f : f = [ fa ]   fa = Na f d + Ω n e  



Na u n dŴ Γn

∂ Na ∂ Nb ∂ Na ∂ Nb + − ∂x ∂x ∂y ∂y Ω b=1 

n e  ∂u eb − Na Nb d ∂t Ω



d u eb (2.15)

b=1

1 ≤ a ≤ n eq 1 ≤ b ≤ ne

(2.16)

This example was supposed to illustrate the main steps of the FE method. Note that the mass and stiffness matrix may take different forms depending on the physical phenomena they model and on the material parameters (see Chaps. 3, 5, and 6). The resulting equation (2.12) is still infinite dimensional due to the time dependence. Therefore, in Sect. 2.5 we will discuss time-discretization schemes, in order to arrive at the algebraic system of equations. Before, we will provide a detailed description of the FE method by means of applying it to a 1D problem, followed by discussing all steps necessary for computer implementation.

2.2 Finite Element Method for a 1D Problem

13

2.2 Finite Element Method for a 1D Problem In order to illustrate the main idea of the FE method, we will consider the following 1D differential equation −

∂2u + c u = f (x) ∂x 2 u(a) = u a u(b) = u b ,

(2.17)

where [a, b] defines the computational domain. As described in Sect. 2.1, the first step is to derive the weak form of (2.17). For this purpose, we choose an appropriate test function v, multiply (2.17) by this test function and integrate over the whole domain  b  2 ∂ u v − 2 + c u − f (x) dx. (2.18) ∂x a

For the first term in (2.18) we perform an integration by parts b a

b ∂u b ∂v ∂u ∂2u − v 2 dx = v dx. ∂x ∂x a ∂x ∂x a

Provided that the test function v(x) vanishes on the Dirichlet boundary (first restriction on the test function, second one will be the existence of a first-order derivative in the weak sense, see Appendix D), we obtain for (2.18) b 

∂v ∂u + cvu ∂x ∂x

a



dx =

b

v f dx.

(2.19)

a

Therewith, the weak (variational) formulation reads as follows: Given: f, c : [a, b] → R Find: u ∈ V = {u ∈ H 1 (a, b)|u(a) = u a , u(b) = u b } such that for all v ∈ W = {v ∈ H 1 (a, b)|v(a) = v(b) = 0} a(u, v) =< f, v >

(2.20)

14

2 The Finite Element (FE) Method

with

a(u, v) =

b 

∂v ∂u + cvu ∂x ∂x

a

< f, v > =

b



dx

v f dx.

a

In (2.20) a(u, v) is called a bilinear form and < f, v > an inner product in the specified functional space. In the next step, we divide the computational domain into cells, so-called finite elements. Therewith, in our case, we divide the interval [a, b] into a set of smaller intervals [xi−1 , xi ], i = 1, . . . , M such that the following properties are fulfilled • Ascending order of node positions xi−1 < xi for i = 1, . . . , M • Complete covering of the domain [a, b] =

M

[xi−1 , xi ]

x0 = a, x M = b

i=1

• No intersection of intervals [xi−1 , xi ] ∩ [x j−1 , x j ] = 0 for i = j For simplicity, we choose an equidistant discretization, so that we obtain (see Fig. 2.3) xi = a + i h

h=

b−a M

i = 0, . . . , M.

The unknown quantity u(x) is now approximated by a linear combination of finite functions with local support, which means that these functions are just different from zero in a ‘small’ interval (see Fig. 2.4). Such a choice is given e.g., by piecewise linear hat-functions, defined as follows (see Fig. 2.5)

Ni (x) =

⎧ 0 ⎪ ⎪ ⎨ x−xi−1 h

xi+1 −x ⎪ ⎪ ⎩ h 0

a ≤ x ≤ xi−1 xi−1 < x ≤ xi xi < x ≤ xi+1 xi+1 < x ≤ b

(2.21)

2.2 Finite Element Method for a 1D Problem

15

h

x0 = a

xM = b

Fig. 2.3 Subdivision of the computational domain into finite elements

N (x)

α

x

β

Fig. 2.4 Finite element function with local support: suppN (x) = [α, β]

Ni (x)

1

N0

N1

Ni

NM

x Fig. 2.5 Piecewise, linear hat-functions

Our chosen ansatz (shape) functions fulfill the delta-property Ni (x j ) = δi j =



1 0

i= j i = j

(2.22)

and the approximation of the unknown u(x) is given by u(x) ≈ u h (x) =

M−1 

Ni (x)u i + N0 (x)u a + N M (x)u b

(2.23)

i=1

u h (x = xi ) = u i .

(2.24)

16

2 The Finite Element (FE) Method

The discretized weak (variational) formulation reads as Given: f, c : [a, b] → R Find: u h ∈ Vh = {u h (x) = v h ∈ Wh = {v h (x) =

M−1 

i=1 M−1 

Ni (x)u i + N0 u a + N M u b } such that for all Ni (x)vi }

i=1

a(u h , v h ) = < f, v h >

(2.25)

with h

h

a(u , v ) =

b 

∂v h ∂u h + cv h u h ∂x ∂x

a

< f, v h > =

b



dx

v h f dx.

a

For the finite dimensional functional spaces V h , W h we have the property V h ⊂ V , Wh ⊂ W. Now, we have to set up the algebraic system of equations in order to obtain the unknowns u i at all the finite element nodes within our grid. Using the approximation according to (2.23) for u as well as the test function v results in b a

⎞ ⎛ M−1 ∂ ⎝ Ni (x)vi N j (x)u j + N0 u a + N M u b ⎠ dx ∂x i=1 j=1 ⎞  ⎛ M−1 b  M−1   + c N j (x)u j + N0 u a + N M u b ⎠ dx Ni (x)vi ⎝ ∂ ∂x

a



 M−1 

i=1

j=1

−

b  M−1  a

i=1

Ni (x)vi



f dx = 0.

Considering that we can interchange the integrals and the sums, and that all vi as well as u j are constants (no function of x), we obtain

2.2 Finite Element Method for a 1D Problem M−1  i=1

⎛

M−1 

vi ⎝

uj

j=1

+

b 

∂ Ni ∂ N j + cNi N j ∂x ∂x

∂ Ni ∂x



a

b  a

−

b a

17



dx

  ∂ N0 ∂ NM ua + u b + cNi (N0 u a + N M u b ) dx ∂x ∂x ⎞

Ni f dx ⎠ = 0.

Letting vi with i = 1, . . . , M − 1 run through all unit vectors in R M , we obtain for each i an equation M−1 

i = 1, . . . , M − 1

Si j u j = f i

j=1

(2.26)

Su = f

with

Si j =

b 

∂ Ni ∂ N j + cNi N j ∂x ∂x

a

fi =

b

Ni f dx −

a

−

b a

b

∂ Ni ∂x





dx

(2.27)

∂ N0 ∂ NM ua + ub ∂x ∂x



dx

cNi (N0 u a + N M u b ) dx.

(2.28)

a

According to the properties of our chosen ansatz functions (see (2.22)), we get the following pattern for our system matrix S (see Fig. 2.6) ⎛

∗ ∗ 0 0 ··· ⎜∗ ∗ ∗ 0 ··· ⎜ ⎜0 ∗ ∗ ∗ ··· ⎜ S=⎜. . . . . ⎜ .. .. .. .. . . ⎜ ⎝0 · · · ··· 0 · · · ···

⎞ 0 0⎟ ⎟ 0⎟ ⎟ .. ⎟ .⎟ ⎟ ∗ ∗ ∗⎠ 0 ∗ ∗ 0 0 0 .. .

··· ··· ··· .. .

Si j =



0 j ∈ {i − 1, i, i + 1} ∗ j ∈ {i − 1, i, i + 1} ∗...nonzero entry

18

2 The Finite Element (FE) Method

Ni (x) Ni−1 Ni

1

Ni+1

x x0 = a

xM = b

Fig. 2.6 Shape functions for nodes xi−1 , xi and xi+1

Now, let us compute the nonzero entries of S by evaluating (2.27) and using (2.21)

Si,i−1 =

xi 

∂ Ni ∂ Ni−1 + cNi Ni−1 ∂x ∂x

xi−1



dx

    xi    1 −1 x − xi−1 xi − x = +c dx h h h h xi−1

−1 ch + h 6  xi+1  ∂ Ni ∂ Ni + cNi Ni dx = ∂x ∂x =

Si,i

xi−1

   xi     1 1 x − xi−1 x − xi−1 = +c dx h h h h xi−1

+

xi+1 

xi

−

1 h



−

1 h



+c



xi+1 − x h



xi+1 − x h



dx

2 2ch = + h 3 Si−1,i = Si,i−1 =

−1 ch + . h 6

For simplicity, we set our source term f (x) equal to 1 over the whole computational domain and assume u a = u b = 0, which results in

2.2 Finite Element Method for a 1D Problem

fi =

19

xi+1

Ni (x) dx

xi

x − xi−1 dx + h

xi−1

=

xi−1

xi+1

xi+1 − x dx = h. h

xi

Let us choose L = 10 and M = 5, so that our mesh size h is 2. For the parameter c we choose once the value 0 and once 0.5, and we set the boundary values u a as well as u b to zero. Substituting these values, results in

Kc=0

⎛

1 − 21 0

⎞

⎜ 1 ⎟ ⎜ − 2 1 − 21 0 ⎟ ⎜ ⎟ =⎜ 1⎟ 1 ⎝ 0 −2 1 −2 ⎠ 0

Kc=0.5

0

⎛

5 3

0 − 21 1

− 31 0

0

⎞

⎛ ⎞ 2 ⎜2⎟ ⎟ f =⎜ ⎝2⎠ 2

⎜ 1 5 ⎟ ⎜ − 3 3 − 13 0 ⎟ ⎜ ⎟ =⎜ ⎟ ⎜ 0 − 31 53 − 13 ⎟ ⎠ ⎝ 0 0 − 31 53

u(c = 0)

u(c = 0.5)

Solving the two algebraic systems of equations results in the solutions displayed in Fig. 2.7. It is worth mentioning that the FE solution corresponding to c = 0, which solves the 1D Poisson equation (Laplace operator, see (2.17)), is exact at the FE nodes. However, the case c = 0.5 already shows an error at the FE nodes. An a priori error estimate for the discretization error will be discussed in Sect. 2.8.

x Fig. 2.7 Solutions for the cases c = 0 and c = 0.5

x

20

2 The Finite Element (FE) Method

2.3 Nodal (Lagrangian) Finite Elements As previously discussed, the FE method subdivides the simulation domain into small elements (e.g., triangles, tetrahedra, etc.) and the unknown quantities are approximated by interpolation functions that have local support. After spatial and time discretization, we end up with an algebraic system of equations. Now our task is to discuss the computation of the matrices (stiffness, mass, etc.) as well as the right-hand side suitable for a computer implementation. The first important step is to rewrite the integration over the whole domain (see e.g., (2.14)) as a sum of integrations over the element domains, e.g., for the stiffness matrix    ∂ Na ∂ Nb ∂ Na ∂ Nb dΩ + K ab = ∂x ∂x ∂y ∂y Ω  ne    ∂ Na ∂ Nb ∂ Na ∂ Nb = d, (2.29) + ∂x ∂x ∂y ∂y e=1 Ω e

with n e the number of finite elements within the mesh. Therefore, we can introduce the so-called element stiffness (mass, etc.) matrix and obtain K =

ne 

ke

ke = [k pq ]

e=1

k pq =

 

Ωe

∂ N p ∂ Nq ∂ N p ∂ Nq + ∂x ∂x ∂y ∂y



d

1 ≤ p, q ≤ n en ,  with n en the number of element nodes and the assembly operator (for the assembling procedure see Sect. 2.4). In the second step, we shall briefly discuss the computation of the element matrices and the right-hand side. For this task, we need to compute the interpolation functions, their derivatives, and perform numerical integration. The easiest and most general strategy is to introduce transfer functions for the different geometric elements (quadrilateral, tetrahedral, etc.) to their parent elements (see Sect. 2.3). For these parent elements, we have to develop appropriate shape functions as well as numerical integration schemes. In addition, since we need the derivatives of the interpolation functions with respect to the global coordinates x, y, z (also called global derivatives), we have to develop a procedure for performing this task with the help of the transformation as well as local shape functions of the geometric elements (see Sect. 2.3.8). In the case that we choose for the transformation from the local to the global coordinate system

2.3 Nodal (Lagrangian) Finite Elements

21

x(ξ) =

n en 

Ni (ξ)x i

i=1

the same interpolation functions Ni as for the unknown quantity u h (ξ) =

n en 

Ni (ξ)u ih

i=1

we call these finite elements isoparametric.

2.3.1 Basic Properties Let us assume that we want to solve a partial differential equation with the scalar unknown u on the domain Ω with Dirichlet boundary Γe and Neumann boundary Γn as displayed in Fig. 2.8. After performing the domain discretization—in this case with triangular finite elements—we obtain 12 finite elements and 11 finite element nodes (see Fig. 2.9). The numbers in the parenthesis are the equation numbers, and as can be seen, for Dirichlet nodes this number is zero (compare with the decomposition of u into the unknown v and known values u e , see Sect. 2.1). Figure 2.10 displays the interpolation function N5 (r) for Eq. (5) (finite element node 9). As can be seen, the interpolation function has a local support, which means that it has the value 1 at

Fig. 2.8 Domain to be discretized

Γn

Ω

Γe Fig. 2.9 Discretization of the domain with finite elements

10(6)

11(0) 2(0)

7(3) 4(1)

9(5)

6(2) 8(4)

1(0) 3(0)

5(0)

22

2 The Finite Element (FE) Method

Fig. 2.10 Shape (basis) function for the unknown in FE node 9 9(5)

the node, decreases to zero approaching the neighboring nodes and is zero outside the neighboring elements. Therefore, the ansatz according to (2.7)–(2.9) is allowed, since, e.g., at any node b it exhibits exactly the value vb h

v (t)| r=r b =

n eq 

Na (r b )va (t) = vb (t).

a=1

Thereby, the interpolation function Na (also named shape or FE basis function) fulfilles the delta-property Na (r b ) =



1 for a = b 0 else

(2.30)

In addition, it is now clear that the mass, stiffness as well as effective system matrices show a sparse profile, since the integrals defining their entries (see (2.13) and (2.14)) are nonzero only if the supports of the interpolation functions Na and Nb overlap, which only happens if the functions belong to neighboring nodes. For our simple example we obtain the following matrix structure ⎛

a11 ⎜ a21 ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎝0 0

a12 a22 a32 0 0 a62

0 a23 a33 a43 a53 a63

0 0 a34 a44 a54 0

0 0 a35 a45 a55 a65

⎞ 0 a26 ⎟ ⎟ a36 ⎟ ⎟. 0 ⎟ ⎟ a56 ⎠ a66

The main properties to be fulfilled by any nodal finite element are: 1. Smoothness on each element interior Ω e 2. Continuity across each element boundary Γ e 3. Completeness Let us assume that we have to evaluate finite element matrices with partial derivatives of order m in the weak formulation. Then, properties 1 and 2 demand for any nodal finite element basis functions, which are m times differentiable in Ω e and (m − 1)

2.3 Nodal (Lagrangian) Finite Elements

23

times differentiable over the boundary Γ e . For example, for m = 1 any finite element with linear interpolation functions (C 0 finite element) fulfills property 1 and 2. In general we call finite elements satisfying property 1 and 2 conforming, or compatible. Now, let us illustrate the property completeness. We assume the following ansatz for the unknown quantity u n en  Na u ae , u h (x) = a=1

with n en the number of nodes for the finite element e, and the property u h (x ae ) = u ae . The finite element is said to be complete if u ae = c0 + c1 xae + c2 yae + c3 z ae

(2.31)

u h (x) = c0 + c1 x + c2 y + c3 z,

(2.32)

implies

with arbitrary constants c0 . . . c3 . This means the element interpolation functions are capable of exactly representing an arbitrary linear polynomial. Therewith, it is guaranteed that by reducing the mesh size h e , the approximated solution converges towards the exact solution.

2.3.2 Quadrilateral Element in R2 For the computation of the element matrices, it is convenient to transform each finite element to its reference element, where the numerical integration can be performed easily. Therefore, we introduce the bijective map FΩe from the reference element Ωˆ to the global finite element with domain Ωe , which is an element of the computational grid with index e (2.33) Fe : Ωˆ → Ωe . Let us investigate this transformation for the bilinear quadrilateral element in two space dimension as displayed in Fig. 2.11. The local coordinates   ξ ξ= η

(2.34)

24

2 The Finite Element (FE) Method

F −1 η 4 (-1,1)

3

3 (1,1)

4

ξ

y 1

2

x

1 (-1,-1)

2 (1,-1)

F Fig. 2.11 Bilinear quadrilateral element: global and local domain

are related to the global coordinates   x x= y

(2.35)

via the following transformation

x(ξ) =

⎧ 4  ⎪ ⎪ Ni (ξ, η)xie ⎨ x(ξ, η) = i=1

4  ⎪ ⎪ ⎩ y(ξ, η) = Ni (ξ, η)yie

(2.36)

i=1

Now, in order to compute explicitly the basis functions Ni , we choose the following bilinear ansatz x(ξ, η) = α0 + α1 ξ + α2 η + α3 ξη

(2.37)

y(ξ, η) = β0 + β1 ξ + β2 η + β3 ξη.

(2.38)

The shape functions have to be constructed in such a way that the relations x(ξi , ηi ) = xie y(ξi , ηi ) = yie are fulfilled. Since the local coordinates take only the following values node i 1 2 3 4

ξi

ηi

−1 −1 1 −1 1 1 −1 1

2.3 Nodal (Lagrangian) Finite Elements

25

N1 (ξ, η)

N2 (ξ, η)

N3 (ξ, η)

N4 (ξ, η)

Fig. 2.12 The four bilinear shape functions of a quadrilateral element

we obtain 8 equations for the eight unknowns α0 .. β3 . Using the solution for αi and βi in (2.37) and (2.38) and comparing the coefficients with (2.36), we arrive at the following explicit form of the shape function for node i (see Fig. 2.12) Ni (ξ) = Ni (ξ, η) =

1 (1 + ξi ξ)(1 + ηi η) . 4

(2.39)

Now, let us investigate the three mentioned properties for the quadrilateral element. 1. Smoothness on each element interior Ω e : The shape functions Ni define smooth functions, if each interior angle of the quadrilateral is less then 180◦ . 2. Continuity across each element boundary Γ e : Figure 2.12 displays the shape function Ni for node i defined by (ξi , ηi ), and it is easy to see that Ni (ξi , ηi ) = δi j (2.40) is fulfilled. Along the boundary, e.g., η = −1, we obtain Ni (ξ, −1) =

1 + ξi ξ , 2

26

2 The Finite Element (FE) Method

which is exactly the shape function for the 1D case. Since this shape function is typically the same for all edges, the quadrilateral element fulfills the continuity condition. 3. Completeness: uh =

=

n en 

i=1 n en 

Ni (ξ, η)u ie Ni (ξ, η)(c0 + c1 xie + c2 yie )

i=1

=

n en 



Ni (ξ, η) c0 +

i=1

n en 



Ni (ξ, η)xie c1 +

i=1





x(ξ,η)



n en 



Ni (ξ, η)yie c2

i=1





y(ξ,η)



Summing up all four shape functions results in n en  i=1

Ni (ξ, η) =

1 [(1 − ξ)(1 − η) + (1 + ξ)(1 − η) 4 +(1 + ξ)(1 + η) + (1 − ξ)(1 + η)]

= 1, which proves the completeness.

2.3.3 Triangular Element in R2 The linear triangular element is defined by its three nodes as displayed in Fig. 2.13. The local coordinates are as follows node i ξi ηi 1 2 3

0 0 1 0 0 1

Similar to the quadrilateral element (see above) we obtain the local shape functions given by N1 = 1 − ξ − η N2 = ξ N3 = η.

2.3 Nodal (Lagrangian) Finite Elements

27

F −1 η 3 3 (0,1)

y 1 2

x

1 (0,0)

2 (1,0)

ξ

F Fig. 2.13 Triangular element: global and local domain

2.3.4 Tetrahedron Element in R3 The linear tetrahedron element is defined by its four coordinates as shown in Fig. 2.14. node i ξi ηi ζi 1 2 3 4

0 1 0 0

0 0 1 0

0 0 0 1

Let us compute the transformation that maps any arbitrary tetrahedral element in the global (x, y, z)-domain to a parent tetrahedron in the local (ξ, η, ζ)-domain by choosing the following linear ansatz xi = α0 + α1 ξi + α2 ηi + α3 ζi

(2.41)

yi = β0 + β1 ξi + β2 ηi + β3 ζi z i = γ0 + γ1 ξi + γ2 ηi + γ3 ζi .

(2.42) (2.43)

We know that the transformation has to satisfy the following relations at the four nodes of a tetrahedron element x(ξ) = x ae Therefore, we obtain (see Fig. 2.14)

a = 1, . . . , 4.

28

2 The Finite Element (FE) Method

F −1 4

ζ 4 (0,0,1)

η 3 (0,1,0)

3

z

y

1

x

1

2

(0,0,0)

2 (1,0,0)

ξ

F Fig. 2.14 Tetrahedron element: global and local domain

y1 = β0

(2.44)

x2 = α0 + α1

y2 = β0 + β1

(2.45)

x3 = α0 + α2

y3 = β0 + β2

(2.46)

x4 = α0 + α3

y4 = β0 + β3 .

(2.47)

x1 = α0

Solving the above system of equations, we arrive at a general expression for the shape function Ni as a function of the local coordinates N1 = 1 − ξ − η − ζ N2 = ξ N3 = η N4 = ζ.

2.3.5 Hexahedron Element in R3 In many 3D applications hexahedron elements are used for the domain discretization, due to their good approximation property. Figure 2.15 displays the hexahedron element in its global and local coordinate system.

2.3 Nodal (Lagrangian) Finite Elements

29

F −1 8 (-1,1,1)

8

7 (1,1,1)

7 5 (-1,-1,1)

6

5

3

4

1

6 (1,-1,1) 4 (-1,1,-1)

3 (1,1,-1)

2 (1,-1,-1)

1 (-1,-1,-,1)

2

ζ

z

η

y F

x

ξ

Fig. 2.15 Hexahedron element: global and local domain

node i

ξi

ηi

ζi

1 2 3 4 5 6 7 8

−1 1 1 −1 −1 1 1 −1

−1 −1 1 1 −1 −1 1 1

−1 −1 −1 −1 1 1 1 1

For the element a trilinear mapping is applied between the global (defined by x) and the local (defined by ξ) element domain [4] x(ξ) = α0 + α1 ξ + α2 η + α3 ζ + α4 ξη + α5 ηζ + α6 ξζ + α7 ξηζ.

(2.48)

The coefficients αi are determined by the relations (see Fig. 2.15) x(ξa ) = x ae

a = 1, . . . , 8,

(2.49)

which results in Na (ξ) =

1 (1 + ξa ξ)(1 + ηa η)(1 + ζa ζ). 8

(2.50)

30

2 The Finite Element (FE) Method

Using a simple degeneration technique [4], one can obtain a pyramid, a wedge as well as a tetrahedron element from a hexahedron one.

2.3.6 Wedge Element in R3 The trilinear wedge element is defined by its six nodes as displayed in Fig. 2.16. The local coordinates are as follows node i 1 2 3 4 5 6

ξi ηi 0 1 0 0 1 0

ζi

0 −1 0 −1 1 −1 0 1 0 1 1 1

Thereby, the local shape functions compute as follows 1 (1 − ζ)(1 − ξ − η) 2 1 N2 = (1 − ζ) ξ 2 1 N3 = (1 − ζ) η 2 N1 =

F −1

ζ

6 4

6 (0,1,1) 4

5

(0,0,1)

5 (0,1,1) 3

z

1 (0,1,-1) 1

1

y x

η

(0,0,-1)

2

F

Fig. 2.16 Wedge element: global and local domain

2 (1,0,-1)

ξ

2.3 Nodal (Lagrangian) Finite Elements

31

1 (1 + ζ)(1 − ξ − η) 2 1 N5 = (1 + ζ) ξ 2 1 N6 = (1 + ζ) η. 2

N4 =

2.3.7 Pyramidal Element in R3 The first order pyramidal element is defined by its five nodes as displayed in Fig. 2.17. The local coordinates are defines as follows node i

ξi

1 2 3 4 5

ηi

ζi

1 1 0 1 −1 0 −1 −1 0 −1 1 0 0 0 1

This type of element is of great importance, when one performs a meshing of an inner domain with hexahedral elements and an outer domain with tetrahedrals. Then, pyramidal elements are necessary as transition elements. To obtain consistent basis functions for a pyramidal element is not trivial. Using the classical approach of a polynomial ansatz is not working. According to [5] the main

ζ

F −1 5

5 (0,0,1) 3

4

3 (-1 ,-1,0)

4 (-1,1,0)

η 2

1

z

2 (1,-1,0)

y x

1 (1,1,0)

ξ F

Fig. 2.17 Pyramidal element: global and local domain

32

2 The Finite Element (FE) Method

idea is to construct basis functions by adding a rational expression to the polynomial functions. The use of the rational term allows to achieve a non-singular Jacobian and continuity between triangular faces. Thereby, the basis functions compute as N1 = N2 = N3 = N4 = N5 =

 1 (1 + ξ)(1 + η) − ζ + 4  1 (1 + ξ)(1 − η) − ζ + 4  1 (1 − ξ)(1 − η) − ζ + 4  1 (1 − ξ)(1 + η) − ζ + 4 ζ.

 ξηζ 1−ζ  ξηζ 1−ζ  ξηζ 1−ζ  ξηζ 1−ζ

It has to be noted that the rational term at ζ = 1 is zero, since both ξ and η are zero.

2.3.8 Global/Local Derivatives For the computation we need to evaluate derivatives of the shape functions with respect to the global coordinate system (see e.g., (2.14)). Since the shape functions Ni depend on the local coordinates (ξ, η, ζ), we may write Na,x = Na,ξ ξ,x + Na,η η,x + Na,ζ ζ,x Na, y = Na,ξ ξ,y + Na,η η,y + Na,ζ ζ,y Na,z = Na,ξ ξ,z + Na,η η,z + Na,ζ ζ,z , with the notation, e.g., ξ,x = ∂ξ/∂x. In matrix form, we obtain ⎞ ⎤⎛ ⎞ ⎡ Na,ξ ξ,x η,x ζ,x Na,x ⎝ Na,y ⎠ = ⎣ ξ,y η,y η,y ⎦ ⎝ Na,η ⎠ . Na,ζ ξ,z η,z ζ,z Na,z ⎛

(2.51)

Now, we do not have the explicit form of the derivatives of the local coordinates with respect to the global coordinates. However, we can express this relation as follows ⎛

⎞ ⎤⎛ ⎞ ⎡ Na,ξ Na,x x,ξ y,ξ z ,ξ ⎝ Na,η ⎠ = ⎣ x,η y,η z ,η ⎦ ⎝ Na,y ⎠ . Na,z x,ζ y,ζ z ,ζ Na,ζ

(2.52)

2.3 Nodal (Lagrangian) Finite Elements

33

Comparing (2.51) and (2.52) we arrive at ⎤−1 ⎤ ⎡ ξ,x η,x ζ,x x,ξ y,ξ z ,ξ ⎣ ξ y η,y η,y ⎦ = ⎣ x,η y,η z ,η ⎦ , x,ζ y,ζ z ,ζ ξ,z η,z ζ,z    −1 −T T (J ) =J ⎡

(2.53)

with J the Jacobi matrix. The computation of J can be performed by the transformation between the global and local coordinate systems x(ξ, η, ζ) =

n en 

Na (ξ, η, ζ)xae

a=1

y(ξ, η, ζ) =

z(ξ, η, ζ) =

n en 

a=1 n en 

Na (ξ, η, ζ)yae Na (ξ, η, ζ)z ae .

a=1

Therefore, the explicit expression for the Jacobian reads as ⎡

n en 

Na,ξ xae

⎤ ⎢ a=1 ⎢n x,ξ x,η x,ζ en ⎢ ⎣ ⎦ Na,ξ yae J = y,ξ y,η y,ζ = ⎢ ⎢ a=1 ⎢ z ,ξ z ,η z ,ζ n en ⎣ Na,ξ z ae ⎡

a=1

n en 

a=1 n en 

a=1 n en 

a=1

Na,η xae Na,η yae Na,η z ae

n en 

a=1 n en 

a=1 n en 

a=1

⎤

Na,ζ xae ⎥ ⎥ ⎥

Na,ζ yae ⎥ ⎥.

Na,ζ z ae

(2.54)

⎥ ⎦

The algorithm can be summarized as follows (n int denotes the number of integration points, see next section): for l := 1, n int Determine: Wl , ξ˜l , η˜l , ζ˜l for a := 1, n en Calculate: Na , Na,ξ , Na,η , Na,ζ at (ξ˜l , η˜l , ζ˜l ) end Compute Jacobi matrix, determinant and its inverse Compute global derivatives Na,x , Na,y , Na,z at (ξ˜l , η˜l , ζ˜l ) end

34

2 The Finite Element (FE) Method

2.3.9 Numerical Integration For the computation of the element matrices as well as element right-hand sides we have to numerically evaluate an integral of the form 

f (x) d.

(2.55)

Ωe

Since we perform a transformation of each finite element to its parent element, (2.55) changes, e.g., for a hexahedron, to 1 1 1

f (x(ξ))|J | dξ dη dζ,

(2.56)

−1 −1 −1

with |J | the Jacobi determinant (see Sect. 2.3.8). In the 1D case a Gaussian quadrature formula is optimal, since by using n int integration points, we achieve an accuracy of order 2n int (see e.g., [4]) 1

g(ξ) dξ =

n int 

g(ξ˜l )Wl + E

(2.57)

l=1

−1

ξ˜l . . . zero positions of Legendre polynomial with order n int Wl . . . weighting factor for integration point l E . . . error. For our 3D case, we can write 1 1 1

1

f (x(ξ))|J | dξ dη dζ =

2

3

n int n int n int  

g(ξ˜l 1 , η˜l 2 , ζ˜l 3 )Wl 1 Wl 2 Wl 3 + E

l 1 =1 l 2 =1 l 3 =1

−1 −1 −1

=

n int 

g(ξ˜l , η˜l , ζ˜l ) Wl + E.

(2.58)

l=1

In the following, we give for each discussed geometric element the integration points as well as the weighting factors.

2.3 Nodal (Lagrangian) Finite Elements

35

• Quadrilateral elements (Gaussian quadrature, 2nd order): ξl

l

ηl

Wl

1 −0.57735026919 −0.57735026919 1.0 2 0.57735026919 −0.57735026919 1.0 3 0.57735026919 0.57735026919 1.0 4 −0.57735026919 0.57735026919 1.0 • Triangular elements (Gaussian quadrature, 2nd order): ξl

l

ηl

Wl

1 0.166 666 67 0.166 666 67 0.166 666 67 2 0.666 666 67 0.166 666 67 0.166 666 67 3 0.166 666 67 0.666 666 67 0.166 666 67 • Tetrahedron elements (Gaussian quadrature, 2nd order): l

ξl

ηl

ζl

Wl

1 2 3 4

0.585 410 0.138 196 0.138 196 0.138 196

0.138 196 0.585 410 0.138 196 0.138 196

0.138 196 0.138 196 0.585 410 0.138 196

0.041 666 7 0.041 666 7 0.041 666 7 0.041 666 7

• Hexahedron elements (Gaussian quadrature, 2nd order): l

ξl

ηl

ζl

Wl

1 2 3 4 5 6 7 8

−0.57735026919 0.57735026919 0.57735026919 −0.57735026919 −0.57735026919 0.57735026919 0.57735026919 −0.57735026919

−0.57735026919 −0.57735026919 0.57735026919 0.57735026919 −0.57735026919 −0.57735026919 0.57735026919 0.57735026919

−0.57735026919 −0.57735026919 −0.57735026919 −0.57735026919 0.57735026919 0.57735026919 0.57735026919 0.57735026919

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

• Wedge elements (Gaussian quadrature, 2nd order): l

ξl

ηl

ζl

Wl

1 2 3 4 5 6

0.16666666667 0.66666666667 0.16666666667 0.16666666667 0.66666666667 0.16666666667

0.16666666667 0.16666666667 0.66666666667 0.16666666667 0.16666666667 0.66666666667

−0.57735026919 −0.57735026919 −0.57735026919 0.57735026919 0.57735026919 0.57735026919

0.16666666667 0.16666666667 0.16666666667 0.16666666667 0.16666666667 0.16666666667

36

2 The Finite Element (FE) Method

• Pyramidal elements (Gaussian quadrature, 2nd order): ξl

l

ηl

ζl

Wl

1 0.0 0.0 0.69370598373 0.21333333333 0.28000 2 −0.4879500365 −0.4879500365 0.16548457453 0.28000 3 0.4879500365 −0.4879500365 0.16548457453 0.28000 4 0.4879500365 0.4879500365 0.16548457453 5 −0.4879500365 0.4879500365 0.16548457453 0.28000

2.4 Finite Element Procedure In the previous sections we discussed the basis function of different geometric finite elements, the global/local derivatives and the numerical integration. For a general definition of the computation of the element matrices as well as right-hand side, we use the concept of mapping (see Sect. 2.3.2). Therefore, the transformation to the integral can be written as 

f (x) dΩ =

Ωe



ˆ |Je | fˆ(ξ) dΩ.

(2.59)

Ωˆ

In (2.59) Je is the Jacobian of the grid element with index e. The FE basis functions are defined on the local element and do not need any further mapping. However, e.g. the nabla operator, which contains the derivatives with respect to the global coordinates, transforms with the Jacobian Je by ˆ ∇ → Je−T ∇.

(2.60)

Here, ∇ˆ denotes the nabla-operator with respect to the local coordinates. Summarizing, we can write the bilinear form of a simple Laplace-problem (see (2.29)) as follows   ˆ (2.61) ∇ Na · ∇ Nb dΩ = |Je |Je−T ∇ˆ Na · Je−T ∇ˆ Nb dΩ. Ωe

Ωˆ

Finally, we address the still-open question of the assembly procedure. In the first step we introduce the nodal equation array NE, which relates the global equation number P to the global node number A. ⎧ ⎨ P : if the quantity is unknown at A NE(A) = 0 : if the quantity is known at A ⎩ (e.g., Dirichlet boundary condition)

2.4 Finite Element Procedure

37

A . . . global node number P . . . global equation number. Since the whole simulation domain is discretized with finite elements, and we first compute the element matrices (right-hand side) and then assemble it to the global system matrix (right-hand side), we need information given by the following information element node array IEN IEN(a, e) = A a . . . local element node number e . . . element number A . . . global node number. Assuming that we solve a scalar PDE, we have just one unknown per FE node, and the local node number coincides with the local equation number. Combining the NE array with the IEN array results in the equation array EQ EQ(a, e) = N E (IEN(a, e)) = P . The EQ array connects the element node number (element equation number) a of element e with the global equation number P. The following simple example, displayed in Fig. 2.18, will demonstrate all the steps that have to be performed for the assembly process. The FE mesh consists of two finite elements with given Dirichlet boundary conditions u 1 , u 2 at node 1 and 2. Let us write the algebraic system of equations for some discretized PDE on this domain (e.g., the Poisson equation) in the following general form

Fig. 2.18 Example: global node numbers and in parenthesis the local node numbers

u2

2(4)

(3)4 (4)

(3)6

u1 1(1)

(2)3 (1)

(2)5

38

2 The Finite Element (FE) Method

⎡

K 11 ⎢ K 21 ⎢ ⎢ K 31 ⎢ ⎢ K 41 ⎢ ⎣ K 51 K 61

K 12 K 22 K 32 K 42 K 52 K 62

K 34 K 44 K 54 K 64

K 35 K 45 K 55 K 65

K 13 K 23 K 33 K 43 K 53 K 63

K 14 K 24 K 34 K 44 K 54 K 64

K 15 K 25 K 35 K 45 K 55 K 65

⎤ K 16 K 26 ⎥ ⎥ K 36 ⎥ ⎥ K 46 ⎥ ⎥ K 56 ⎦ K 66

Since we know u 1 and u 2 we can rewrite (2.62) as ⎡

K 33 ⎢ K 43 ⎢ ⎣ K 53 K 63

⎤ K 36 K 46 ⎥ ⎥ K 56 ⎦ K 66

⎤ ⎡ u3 ⎢ u4 ⎥ ⎢ ⎢ ⎥=⎢ ⎣ u5 ⎦ ⎣ u6 ⎡

The nodal equation array NE is given by

⎤ ⎡ u1 ⎢ u2 ⎥ ⎢ ⎢ ⎥ ⎢ ⎢ u3 ⎥ ⎢ ⎢ ⎥=⎢ ⎢ u4 ⎥ ⎢ ⎢ ⎥ ⎢ ⎣ u5 ⎦ ⎣ u6 ⎡

⎤ f1 f2 ⎥ ⎥ f3 ⎥ ⎥. f4 ⎥ ⎥ f5 ⎦ f6

⎤ f 3 − K 31 u 1 − K 32 u 2 f 4 − K 41 u 1 − K 42 u 2 ⎥ ⎥. f 5 − K 51 u 1 − K 52 u 2 ⎦ f 6 − K 61 u 1 − K 62 u 2

(2.62)

(2.63)

1 2 3456 0∗ 0∗ 1 2 3 4 ∗: at this node we already know the quantity (Dirichlet boundary condition) and using it for setting up the algebraic system, we recognize that we obtain a similar system as in (2.63) with four unknowns. Since there is no connection between the nodes 1, 2 and 5, 6 (see Fig. 2.18), the values of K 51 , K 52 , K 61 , and K 62 are zero. The number of nodes for the linear quadrilateral element is n en = 4 and the unknown quantity is a scalar. Therefore, we obtain the following element matrices as well as right-hand sides for the two finite elements ⎡

1 k11

⎡

2 ) (k 2 ) (k 2 ) (k 2 ) (k11 12 13 14

1 k12

1 k13

1 k14

⎤

⎡

f 11

⎤

⎤

⎡

< f 12 >

⎤

⎢ 1 1 ) (k 1 ) {k 1 } ⎥ ⎥ ⎢ ⎢ {k21 } (k22 < f 21 > ⎥ 23 24 ⎥ ⎥ f1 = ⎢ k1 = ⎢ ⎥ ⎢ ⎢ {k 1 } (k 1 ) (k 1 ) {k 1 } ⎥ ⎣< f 1 >⎦ ⎣ 31 32 33 34 ⎦ 3 1 1 1 1 f 41 k41 k42 k43 k44

⎢ 2 ⎢ ⎥ 2 ) (k 2 ) (k 2 ) ⎥ ⎢ (k21 ) (k22 ⎢ < f 22 > ⎥ 23 24 ⎥ 2 ⎢ ⎢ ⎥ ⎥ k =⎢ 2 f =⎢ 2 ) (k 2 ) (k 2 ) ⎥ 2 >⎥ ) (k (k < f ⎣ 31 ⎣ ⎦ 32 33 34 ⎦ 3 2 ) (k 2 ) (k 2 ) (k 2 ) 2 > (k41 < f 42 43 44 4 2

() ... contributes to the global stiffness matrix K < > ... contributes to the global right-hand side f {} ... contributes to the local right-hand side f e .

2.4 Finite Element Procedure Table 2.1 Information element node (IEN) and global equation array (EQ) for our exampled displayed in Fig. 2.18

39 IEN

el.nr.

Local node number

1 1 3 4 2

1 2 3 4

EQ 2 3 5 6 4

Local node number

el.nr. 1 2 3 4

1 0 1 2 0

2 1 3 4 2

The entries of the right-hand side f 1 for element 1 compute as ⎡

⎢ ⎢ f1 = ⎢ ⎣

⎤ f 11 1 u − k1 u ⎥ f 21 − k21 1 24 2 ⎥ . 1 1 1 u ⎥ f 3 − k31 u 1 − k34 2⎦ f 41

Combining the NE array with the element node array IEN, we arrive at the EQ array (Table 2.1). Assembling: EQ(1, 1) = 0 EQ(2, 1) = 1 EQ(3, 1) = 2 EQ(4, 1) = 0





1 K 11 ← K 11 + k22 1 K 12 ← K 12 + k23

f 1 ← f 21

1 K 22 ← K 22 + k33 1 K 21 ← K 21 + k32

f 2 ← f 31

⎧ 2 K 11 ← K 11 + k11 ⎪ ⎪ ⎪ ⎨ 2 K 13 ← K 13 + k12 EQ(1, 2) = 1 2 ⎪ K 14 ← K 14 + k13 ⎪ ⎪ ⎩ 2 K ← K 12 + k14 ⎧ 12 2 K 33 ← K 33 + k22 ⎪ ⎪ ⎪ ⎨ K ← K + k2 31 31 21 EQ(2, 2) = 3 2 ⎪ K ← K + k23 34 34 ⎪ ⎪ ⎩ 2 K 32 ← K 32 + k24 ⎧ 2 K 44 ← K 44 + k33 ⎪ ⎪ ⎪ ⎨ K ← K + k2 41 41 31 EQ(3, 2) = 4 2 ⎪ K 43 ← K 43 + k32 ⎪ ⎪ ⎩ 2 K 42 ← K 42 + k34

f 1 ← f 12

f 3 ← f 22

f 4 ← f 32

40

2 The Finite Element (FE) Method

⎧ K 22 ⎪ ⎪ ⎪ ⎨K 21 EQ(4, 2) = 2 ⎪ K 23 ⎪ ⎪ ⎩ K 24

← ← ← ←

2 K 22 + k44 2 K 21 + k41 2 K 23 + k42 2 K 24 + k43

f 2 ← f 42 .

Thus, we obtain the following algebraic system of equations ⎡ 1 ⎤ ⎡ 1 ⎤ 2 k1 + k2 k2 k2 f 2 + f 12 k22 + k11 23 14 12 13 ⎢ 1 ⎢ 1 ⎥ 2 k1 + k2 k2 k2 ⎥ ⎢ k32 + k41 ⎥ ⎢ f 3 + f 42 ⎥ 33 44 42 43 ⎥ ⎢ ⎢ ⎥. K=⎢ 2 2 2 k2 ⎥ f = ⎢ k24 k22 f 22 ⎥ ⎣ k21 ⎦ ⎣ 23 ⎦ 2 2 2 2 3 k31 k34 k32 k33 f2 Comparing the result with Fig. 2.18, we can set up the following rules • Diagonal entries: K 11 is the entry of K due to node 3. Since this node belongs to element 1 as well as 2, the entry in K will be a sum of the contributions from element 1 and element 2. The entry K 33 is due to global node 5, and therefore, only element 2 will contribute to it. • Off-diagonal entries: The same situation as described for the diagonal entries occurs. Figure 2.19 summarizes the whole assembly procedure, and the following pseudocode demonstrates the computer implementation. //loop over all finite elements for e := 1, n e //loop over all element nodes (equations) for a := 1, n en eq1 := EQ(a, e) if (eq1 > 0) //loop over all element nodes (equations) for b := 1, n en eq2 := EQ(b, e) if (eq2 > 0) K (eq1, eq2) := K (eq1, eq2) + ke (a, b) end end end end end

2.5 Time Discretization

41

Fig. 2.19 FE procedure

Start

Read input data Set up EQ - array Zero allocated storage for K and f

e =1

e

e

Compute k and f ; use EQ-array and assemble to global K and f

e=e+1 no

e> ne yes Stop

2.5 Time Discretization In the following we will describe single time step discretization algorithms for parabolic (e.g., electromagnetic field in the eddy current case) as well as for hyperbolic (mechanical as well as acoustic field) partial differential equations.

2.5.1 Parabolic Differential Equation Let us consider the following semi-discrete Galerkin formulation (discrete in space and continuous in time) Mu(t) ˙ + Ku(t) = f (t),

(2.64)

42

2 The Finite Element (FE) Method

Table 2.2 Finite difference schemes

γP

Method

0.0 0.5 1.0

Forward difference; forward Euler Trapezoidal rule; Crank-Nicolson Backward difference; backward Euler

with M the mass matrix, K the stiffness matrix, f the right-hand side, u the vector of unknowns at the finite element nodes and u˙ its derivative with respect to time. In a first step, let us subdivide the time interval [0, T ] into M sub-intervals, which are defined as follows [0, T ] =

M

[tn−1 , tn ] , t0 = 0 < t1 < t2 < .... < tn < .. < t M = T.

n=1

For simplicity, we assume the time step size Δt = tn − tn−1 = T /M to be constant over the whole time interval of interest. In a second step, we approximate the time derivative of the unknown u(t) by the forward time difference scheme u(t) ˙ ≈

u − un u(tn+1 ) − u(tn ) = n+1 . Δt Δt

(2.65)

In the third step, we have to decide at which point in time we evaluate the elliptic part Ku(t) and the right-hand side f (t). Applying a convex linear combination with the time integration parameter γP to these terms, will result in the following system of equations M

u(tn+1 ) − u(tn ) + γP Ku n+1 + (1 − γP )Ku n = γP f n+1 + (1 − γP ) f n . (2.66) Δt

Table 2.2 is listing the different time discretization methods depending on the value of γP . Rearranging the terms in (2.66), we arrive at & ' (M + γP ΔtK) u n+1 = Δt γP f n+1 + (1 − γP ) f n + (M − (1 − γP )ΔtK) u n . (2.67) We obtain the same system of algebraic equations, when we apply the general trapezoidal difference scheme [4], which is defined as follows ) ( u n+1 = u n + Δt (1 − γP )u˙ n + γP u˙ n+1 .

From (2.68) we can compute u˙ n+1 u˙ n+1 =

u n+1 − u n 1 − γP − u˙ n γP Δt γP

(2.68)

2.5 Time Discretization

43

and substituting it into the time discrete version of (2.64) at t = tn+1 leads to ) ( (M + γP ΔtK) u n+1 = γP Δt f n+1 + M u n + (1 − γP )Δt u˙ n .

(2.69)

To see that (2.69) and (2.67) are equivalent, we use the relation Mu˙ n + Ku n = f n and rewrite the right-hand side of (2.67) as follows

' & Δt γP f n+1 + (1 − γP ) f n + (M − (1 − γP )ΔtK) u n ' & = γP Δt f n+1 + Mu n + (1 − γP )Δt f n − Ku n    Mu˙ n

) ( = γP Δt f n+1 + M u n + (1 − γP )Δt u˙ n .

Applying the general trapezoidal difference method, we can write the solution process as a predictor-corrector algorithm, where we distinguish between an effective mass and effective stiffness formulation. 1. Effective Mass Formulation • Perform predictor step: u˜ = u n + (1 − γP )Δt u˙ n .

(2.70)

• Solve algebraic system of equations: M ∗ u˙ n+1 = f n+1 − K u˜

(2.71)

∗

M = M + γP Δt K . • Perform corrector step: u n+1 = u˜ + γP Δt u˙ n+1 .

(2.72)

2. Effective Stiffness Formulation • Perform predictor step: u˜ = u n + (1 − γP )Δt u˙ n .

(2.73)

44

2 The Finite Element (FE) Method

• Solve algebraic system of equations: 1 M u˜ γP Δt 1 M. K∗ = K + γP Δt

K ∗ u n+1 = f n+1 +

(2.74)

• Perform corrector step: u˙ n+1 =

u n+1 − u˜ . γP Δt

(2.75)

According to the choice of the integration parameter γP , one distinguishes between an explicit and implicit algorithm. 1. Explicit Algorithm: γP = 0 For this case, M∗ = M and lumping of the mass matrix might be considered (see e.g., [4]). Thus, the system matrix of the algebraic system is a diagonal matrix, and the only time-consuming part of the solution process are the matrix vector multiplications on the right-hand side. The disadvantage of this algorithm is clearly the lack of stability, which implies restrictions on the time step value Δt, depending on the material parameters and the quality of the mesh. In addition, one has to consider that the mass matrix must be regular. This is, for example, not the case within electromagnetic computations, where the entries in the mass matrix are zero for regions with zero electrical conductivity (e.g., air). A solution to this problem may be the application of a mixed explicit/implicit time discretization [4]. 2. Implicit Algorithm: 0 < γP ≤ 1 In this case, M∗ computes as the sum of the mass matrix M and the stiffness matrix K multiplied by γP Δt. Now M∗ is a sparse matrix and the algebraic system of equations has to be solved by any direct or iterative solver. For γP = 0.5 the time discretization is called the Crank-Nicolson scheme (second-order accurate), and for γP ≥ 0.5 the algorithm is absolute stable (independently of Δt). In order to investigate in the accuracy of the time discretization scheme, we will expand u(t) at time t = tk+γP in a Taylor series ˙ k+γP ) (t − tk+γP ) + u(t ¨ k+γP ) u(t) = u(tk+γP ) + u(t ' & +O (t − tk+γP )3 .

(t − tk+γP )2 2 (2.76)

By considering the relation tk+γP = tk + γP Δt we obtain for the evaluation of (2.76) at t = tk

2.5 Time Discretization

45

˙ k+γP ) (−γP Δt) + u(t ¨ k+γP ) u(tk ) = u(tk+γP ) + u(t ' & +O (−γP Δt)3

(−γP Δt)2 2

as well as at t = tk+1

˙ k+γP ) ((1 − γP )Δt) + u(t ¨ k+γP ) u(tk+1 ) = u(tk+γP ) + u(t ' & +O ((1 − γP )Δt)3 .

((1 − γP )Δt)2 2

Therewith, we obtain for the difference of u(tk+1 ) and u(tk ), which is used for approximating the time derivative (see (2.65)), the following expression u(tk+1 ) − u(tk ) = u˙ Δt + u¨

Δt 2 − 2γP Δt 2 + O(Δt 3 ). 2

(2.77)

Now, (2.77) clearly shows that only for γP = 0.5 (Crank-Nicolson), we arrive at a second order time discretization scheme, and for all other choices the scheme is first-order accurate.

2.5.2 Hyperbolic Differential Equation For the hyperbolic case, we arrive after the spatial discretization at the following system of second-order ordinary differential equations (still continuous in time) Mu(t) ¨ + Ku(t) = f (t).

(2.78)

Now, we will apply a finite difference scheme of second order to approximate u(t) ¨ u(t) ¨ ≈

− 2u n + u n−1 u u(tn+1 ) − 2u(tn ) + u(tn−1 ) = n+1 2 Δt Δt 2

(2.79)

with Δt the time step size. Similar to the parabolic case, we substitute the elliptic part K u(t) and the right-hand side f (t) by a convex, linear combination at t = tn−1 , tn and tn+1 weighted by the integration parameter αH . Therewith, we obtain the following algebraic system of equations M

u n+1 − 2u n + u n−1 + αH K u n+1 + (1 − 2αH )K u n + αH K u n−1 Δt 2 = αH f n+1 + (1 − 2αH ) f n + αH f n−1 , (2.80)

46

2 The Finite Element (FE) Method

which can be rewritten as (M + αH Δt 2 K )u n+1 = αH Δt 2 f n+1 + (1 − 2αH )Δt 2 f n + αH Δt 2 f n−1 + (2M − (1 − 2αH )Δt 2 K )u n

(2.81)

2

− (M + αH Δt K )u n−1 . Choosing αH = 0 and transforming M to a diagonal matrix (mass lumping, see e.g., [4]), results in an explicit time discretization scheme, which does not need any algebraic solver. However, such schemes will not be unconditionally stable. A stability analysis will exhibit the so-called CFL (Courant-Friedrich-Levi) condition [6] 2 Δt < * . λmax (M −1 K )

(2.82)

In (2.82) λmax denotes the largest eigenvalue of the matrix (M −1 K ). Since this value is of the order O(h −2 ), we should choose the time step size Δt and the mesh size h of the same order. For αH = 1/4 the above scheme is unconditional stable. For practical applications, especially if an additional damping matrix C is present, the Newmark schemes are mainly used. Let us start at the semi-discrete Galerkin formulation (2.83) M u¨ n+1 + C u˙ n+1 + K u n+1 = f n+1 , with M the mass matrix, C the damping matrix (e.g., C = αM M + αK K , see Sect. 3.7.2), K the stiffness matrix, f the right-hand side, u the vector of unknowns at the finite element nodes and u˙ as well as u¨ its first and second derivative with respect to time. Thus, we have (see e.g., [4]) ) Δt 2 ( (1 − 2βH )u¨ n + 2βH u¨ n+1 u n+1 = u n + Δt u˙ n + 2 ( ) u˙ n+1 = u˙ n + Δt (1 − γH )u¨ n + γH u¨ n+1 .

(2.84) (2.85)

In (2.84) and (2.85) n denotes the time step counter, Δt the time step value and βH , γH the integration parameters. Substituting u n+1 and u˙ n+1 according to (2.84) and (2.85) into (2.83) leads to the following algebraic system of equations ) ( M ∗ u¨ n+1 = f n+1 − C u˙ n + (1 − γH )Δt u¨ n   Δt 2 (1 − 2βH )u¨ n −K u n + Δt u˙ n + 2

(2.86)

M ∗ = M + γH Δt C + βH Δt 2 K .

According to the choice of the integration parameters βH and γH , one distinguishes similarly to the parabolic case between an explicit and implicit algorithm.

2.5 Time Discretization

47

1. Explicit Algorithm: βH = 0 and C = αM M Choosing for γH the value 0.5 we achieve a second-order accurate scheme. Similar to the parabolic case, it makes sense to lump the system matrix, so that again the time-consuming part of the solution process is the matrix vector multiplications on the right-hand side. Of course, the stability depends on the time step value Δt, the material parameters and the quality of the mesh. However, this kind of algorithm is used quite often in acoustic computations, especially when the discretization is performed by a mapped mesh. 2. Implicit Algorithm: βH = 0 , γH = 0 In this case, M ∗ computes as the sum of the mass matrix M, the damping matrix C and the stiffness matrix K with appropriate integration factors. The matrix M ∗ is now sparse and the algebraic system of equations has to be solved by any direct or iterative solver. For βH = 0.25 and γH = 0.5 the time discretization is second-order accurate with respect to time, if C vanishes. To keep the secondorder accuracy even in the damped case, one has to extend the Newmark scheme to the Hilbert-Hughes-Taylor scheme (see [4]). Writing the solution process for one time step as a predictor-corrector algorithm we arrive at the effective mass as well as effective stiffness formulations. 1. Effective Mass Formulation • Perform predictor step: u˜ = u n + Δt u˙ n + (1 − 2βH )

Δt 2 u¨ 2 n

u˜˙ = u˙ n + Δt (1 − γH )u¨ n .

(2.87) (2.88)

• Solve algebraic system of equations: M ∗ u¨ n+1 = f n+1 − K u˜ − C u˜˙

(2.89)

M ∗ = M + γH Δt C + βΔt 2 K . • Perform corrector step: u n+1 = u˜ + βH Δt 2 u¨ n+1 u˙ n+1 = u˜˙ + γH Δt u¨ n+1 .

(2.90) (2.91)

2. Effective Stiffness Formulation According to (2.84) and (2.85) we can express u¨ n+1 and u˙ n+1 as follows u¨ n+1 =

u n+1 − u˜ βH Δt 2

γH (u u˙ n+1 = u˜˙ + γH Δt u¨ n+1 = u˜˙ + − u). ˜ βH Δt n+1

(2.92) (2.93)

48

2 The Finite Element (FE) Method

Therefore, we obtain • Perform predictor step: u˜ = u n + Δt u˙ n + (1 − 2βH )

Δt 2 u¨ 2 n

(2.94)

u˜˙ = u˙ n + Δt (1 − γH )u¨ n .

(2.95)

• Solve algebraic system of equations: 

1 γH C M+ βH Δt 2 βH Δt 1 γH C+ K∗ = K + M. βH Δt βH Δt 2

K ∗ u n+1 = f n+1 − C u˜˙ +



u˜

(2.96)

• Perform corrector step: u n+1 − u˜ βH Δt 2 = u˙˜ + γH Δt u¨ n+1 .

u¨ n+1 =

(2.97)

u˙ n+1

(2.98)

2.6 Integration over Surfaces Very often one has to evaluate an integral over a 3D surface or along a contour in 2D. In the first case the integration is performed within a 2D finite element in a 3D space and in the second case within a 1D finite element in a 2D space. Let us assume a scalar function f (x, y, z) as the integrand of a surface integral. According to [7], we can write 

f (x, y, z) dŴ =

Γ

with

 

f (x(ξ, η), y(ξ, η), z(ξ, η)) |x ξ (ξ, η) × x η (ξ, η)| dξ dη (2.99) ⎛ ∂x ⎞

⎜ xξ = ⎜ ⎝

∂ξ ∂y ∂ξ ∂z ∂ξ

⎟ ⎟ ⎠

⎛ ∂x ⎞

⎜ ∂∂ηy ⎟ ⎟ xη = ⎜ ⎝ ∂η ⎠ . ∂z ∂η

For the second case of a contour integral over a scalar function f (x, y), we obtain 

C

f (x, y) ds =



f (x(ξ), y(ξ)) |x ξ (ξ)| dξ,

(2.100)

2.6 Integration over Surfaces

49

with xξ =

 ∂x  ∂ξ ∂y ∂ξ

.

Therefore, with slight modifications we can evaluate such integrals by performing an integration in the local domain and instead of the Jacobian determinant we have to compute the expressions given in (2.99) and (2.100).

2.7 Edge Nédélec Finite Elements Edge finite elements belong to the family of vector finite elements (shape functions are vectors) and assign the degrees of freedom to the edges rather than to the nodes of the element. These types of elements were first introduced by Whitney (see e.g., [8]). Their importance in electromagnetics were realized by Nédélec (see e.g., [9]), who constructed edge elements on quadrilateral and tetrahedron elements. Many important studies followed for the further development of different electromagnetic field problems (see e.g., [10–14]). Within edge elements, a physical vector quantity A (e.g., the magnetic vector potential) is approximated by the following ansatz A ≈ Ah =

ne 

Aek N k .

(2.101)

k=1

In (2.101) n e defines the number of edges in the finite element mesh, N k the edge shape function associated with the k-th edge, and Aek the corresponding degree of freedom, namely the line integral of the physical vector quantity along the k-th edge Aek

=



A · ds.

(2.102)

k

For edge shape functions N k of lowest order, the following conditions have to be fulfilled [15]: 1. The tangential component of N k along the edge k has to be constant. 2. The tangential component of N k along edges l = k is zero. 3. The divergence of N k is zero inside the element. Since, in this work, Nédélec finite elements are exclusively used for 3D magnetic field computation, we will restrict ourselves to this case using tetrahedron elements. Let us consider the following vector ansatz function N 1 (∇ defines here the derivatives with respect to the global coordinates x) N 1 = N1 ∇ N2 − N2 ∇ N1

(2.103)

50

2 The Finite Element (FE) Method

F −1

4

ζ

4 (0,0,1)

6

η

3 (0,1,0) 3

3

1 z

2

1

2

(0,0,0)

4

5

2 1

(1,0,0)

ξ

y F x

Fig. 2.20 Tetrahedron element: global and local domain with orientation of the edges

along edge 1 defined by the nodes 1 and 2 (see Fig. 2.20) and N1 and N2 the nodal interpolation functions (see Sect. 2.3.4) in node 1 and 2, respectively. To check condition 1, we compute, e.g., the tangential component of N 1 along edge 1 N 1 · t 1 = N1 t 1 · ∇ N2 − N2 t 1 · ∇ N1 . It has to be mentioned that we have to compute the derivatives with respect to the global coordinate system (see Sect. 2.3.8). Since t 1 · ∇ N1 = −1/(2l1 ) and t 1 · ∇ N2 = 1/(2l1 ) (l1 denotes the length of edge 1), we obtain N1 · t1 =

1 . l1

Condition 2 is also fulfilled, since N1 vanishes along the edges (4,5,6), N2 along the edges (2,3,6) and t 2 · ∇ N1 , t 3 · ∇ N1 , t 4 · ∇ N2 , t 5 · ∇ N2 are zero. Therewith, shape function N 1 has no tangential component along the edges (2,3,4,5,6). Condition number 3 states that the divergence of N 1 has to vanish inside the element. Applying the divergence to N 1 results in ∇ · (N1 ∇ N2 − N2 ∇ N1 ) = N1 ∆N2 + ∇ N1 · ∇ N2 − N2 ∆N1 − ∇ N2 · ∇ N1 = N1 ∆N2 − N2 ∆N1 . Since the interpolation functions N1 as well as N2 are linear functions, the value of ∇ · N 1 is zero.

2.7 Edge Nédélec Finite Elements

51

To obtain for the tangential components of N k the value 1, we have to scale the vector function with the length of the corresponding edge length, which results in the following edge interpolation functions for a linear tetrahedron N 1 = (N1 ∇ N2 − N2 ∇ N1 ) l1 N 2 = (N1 ∇ N3 − N3 ∇ N1 ) l2 N 3 = (N1 ∇ N4 − N4 ∇ N1 ) l3 N 4 = (N2 ∇ N3 − N3 ∇ N2 ) l4 N 5 = (N4 ∇ N2 − N2 ∇ N4 ) l5 N 6 = (N3 ∇ N4 − N4 ∇ N3 ) l6 . These FE basis functions defines the lowest order Nédélec functions for a tetrahedron, which are of polynomial order p = 0, since they have a constant tangential component along each edge. Higher order elements are discussed in Sect. 6.7.6.

2.8 Discretization Error When applying the FE method, it is of great importance to have some knowledge of the discretization error. Since the error analysis is a quite sophisticated task, we will just provide a short overview containing important results. For a detailed discussion on this topic we refer to [6]. For our purpose of error analysis let us consider the following variational form: Given: f, c : Ω → R Find: u ∈ Vg such that for all v ∈ V0 (2.104)

a(u, v) = < f, v > with a(u, v) =



∇v · ∇u d +



v f d

Ω

< f, v > =



c v u d

Ω

Ω

Vg = {u ∈ H 1 |u = g on Γ } V0 = {v ∈ H 1 |v = 0 on Γ }.

52

2 The Finite Element (FE) Method

Applying the FE method to our problem, results in Given: f, c : Ω → R Find: u h ∈ Vgh such that for all v h ∈ V0h a(u h , v h ) = < f, v h >

(2.105)

with a(u h , v h ) =



∇v h · ∇u h d +



v h f d

c v h u h d

Ω

Ω

< f, v h > =



Ω

with Vgh ⊂ Vg and V0h ⊂ V0 . In general, we distinguish between a priori and a posteriori error estimates: • The a priori error estimate is expressed as ||u − u h || ≤ C1 (u) h α

(2.106)

with α > 0 and C1 (u) being a positive constant. In (2.106) u denotes the exact solution, u h our solution obtained by the FE method, h the discretization parameter (e.g., longest edge in the FE mesh) and || · || an adequate norm. The constant α depends on the smoothness of the solution u and the polynomial degree of the chosen FE basis (shape) functions. In general, this constant is not known. • The a posteriori error estimate reads as ||u − u h || ≤ C2 (u h , h).

(2.107)

As for the a priori error estimate, u denotes the exact solution, u h our solution obtained by the FE method, h the discretization parameter and || · || an adequate norm. The constant C2 depends on the FE solution and the discretization parameter h. Providing an optimal error estimator, it is possible to obtain C2 or at least sharp bounds for C2 (see e.g., [16]). In the following, we will concentrate on a priori estimates, and we assume that our bilinear form as defined in (2.105) is V0 -elliptic and V0 -bounded (for a proof see e.g., [6]). V0 -ellipticity means that there exists a constant C1 > 0 so that a(v, v) ≥ C1 ||v||2H 1

2.8 Discretization Error

53

for all v ∈ V0 . V0 -boundedness states that there exists a constant C2 > 0, such that |a(u, v)| ≤ C2 ||u|| H 1 ||v|| H 1 for all u, v ∈ V0 . In addition to the norms defined in Appendix D, we introduce the energy norm of the error e = (u − u h ) associated with the bilinear form a(u, v) ||u − u h ||a =

* * a(u − u h , u − u h ) = a(e, e).

Since V0h ⊂ V0 , we may rewrite (2.104) by

a(u, v h ) = < f, v h > for all v h ∈ V0h . Now, let us subtract from this equation the discrete weak form (see (2.105)). This operation results in a(u − u h , v h ) = 0 for all v h ∈ V0h , which is known as the Galerkin orthogonality of the error. This result implies that in a certain sense u h is the best approximation of u in Vgh . The most important theorems for the a priori estimation of the discretization error in H 1 (Ω) are Céa’s lemma (see Theorem 1) and the Bramble-Hilbert lemma [17]. The lemma of Cea ´ allows us to transform the estimation of the discretization error to an estimation of the approximation error [6]. Theorem 1 Let the bilinear form a(·, ·) be V0 -elliptic and V0 -bounded. Then, the error estimation reads as ||u − u h || H 1 ≤

C1 C2

inf ||u − w h || H 1 .

w h ∈Vgh

Since the approximation error inf ||u − w h || H 1

w h ∈Vgh

can be estimated from above by the interpolant (see Fig. 2.21) Π h (u) =

M  i=0

Ni (x) u i ⊂ Vgh ,

(2.108)

54

2 The Finite Element (FE) Method

u(x) u uh Π h (u)

x x0 = a

xM = b

Fig. 2.21 Exact solution u, interpolant Π h (u) and the FE solution u h

we may rewrite (2.108) by ||u − u h || H 1 ≤

C1 C2

inf ||u − w h || H 1 ≤

w h ∈Vgh

C1 ||u − Π h (u)|| H 1 . C2

According to this important result and using the Bramble-Hilbert lemma [17], we will obtain practical a priori error estimates. For our FE discretization we will use piecewise, continuous polynomial basis functions of order m, and can state the following theorem [4]: Theorem 2 Let a(·, ·) be a V0 -elliptic and V0 -bounded bilinear form. Then, for all u ∈ H r with r ≥ 0, there exists a discrete solution u h ∈ Vgh and a constant C indepent of u and the discretization parameter h, such that ||u − u h || H m ≤ C h α ||u|| H r ,

(2.109)

where α = min(k + 1 − m, r − m) is the rate of convergence. Thereby, m is the order of the highest derivative in the bilinear form a(·, ·), and k is the order of the FE basis functions. For lower H s norms, 0 ≤ s ≤ m, an error estimate can be developed by the Aubin-Nitsche method (see, e.g., [18]) resulting in ||u − u h || H s ≤ C h β ||u|| H r

(2.110)

with C being independent of u and h, and β = min(k + 1 − s, 2(k + 1 − m)). For our considered bilinear form (see (2.104)), the exact solution u is in H 2 (r = 2), and the order of the highest derivative in the bilinear form is m = 1. Using linear FE basis functions for which k = 1, we arrive at the following practical estimates using (2.109) and (2.110)

2.8 Discretization Error

55

||u − u h || H 1 ≤ C h|u| H 2 = O(h) ||u − u h || L 2 ≤ C˜ h 2 |u| H 2 = O(h 2 )

(2.111) (2.112)

with C, C˜ being constants and independent of the discretization. However, problems with discontinuous jumps in material parameters (e.g., magnetic permeability at an iron–air interface) usually do not permit solutions that are smooth enough to be in H 2 . In such a case, the bounds deteriorate to ||u − u h || H 1 ≤ C h α |u| H 2 = O(h α ) ||u − u h || L 2 ≤ C˜ h 2α |u| H 2 = O(h 2α )

(2.113) (2.114)

with 0 < α < 1. The value of α depends on the smoothness (regularity) of the solution u.

2.9 Finite Elements of Higher Order As explained in the previous sections, the standard FE method utilizes an isoparametric approximation, whereby the geometry and the unknown physical field are interpolated using polynomials of fixed low order (typically 1st or 2nd order). This approach is commonly referred to as h-FEM, as a further increase in accuracy is accomplished by refining the mesh and thus decreasing the mesh size h. In contrast, the p-version method achieves convergence by successively increasing the polynomial degree (commonly referred to as p), while keeping the spatial discretization fixed, as first introduced by Babuška and Szabó in [19]. It could be proven that in case of smooth problems exponential convergence can be achieved [20]. A distinct feature of the p-version is the usage of hierarchical polynomials, which keeps the polynomials of lower order when increasing p [21]. This is different from the classical Lagrange-type elements as the basis functions are not defined in terms of nodal basis functions anymore, which need distinct spatial supporting points, but originate from general hierarchical 1D polynomials (e.g. Legendre et al. [22]) which are then composed to shape functions associated with nodes, edges, faces and the interior of the elements. In other words, the basis functions are not interpolatory anymore, and in order to evaluate the field variable, one has to perform true interpolation involving all shape functions. To illustrate the convergence of linear finite elements, we consider the Helmholtz equation (wave equation in the frequency domain, see Sect. 5.4.5) ∆ pˆ + k 2 pˆ = 0

(2.115)

with pˆ the acoustic pressure in the frequency domain and k the wave number. Please note that this PDE has the same structure as our model equation in Sect. 2.2. As an example, we consider the simple case of an acoustic wave in a channel with given

56

2 The Finite Element (FE) Method

pˆ(x = 0) = 1 L pˆ(x = L) = 0 Fig. 2.22 Acoustic wave in a channel with given Dirichlet boundary conditions Fig. 2.23 Reduction of numerical error by h-refinement

Dirichlet boundary conditions on the left and right end (see Fig. 2.22). We use three different wave numbers k and preform for the computations a successively reduction of the element mesh size h. For comparison we compute the accumulated error E a + , nn ( )2 , h , pˆ − pˆ i , i=1 i Ea = , , nn  pˆ i2

(2.116)

i=1

with pˆ ih the FE solution at node i, pˆ i the analytical solution at node i, and n n the number of finite element nodes. The analytic solution pˆ computes by p(x) ˆ = cos(kx) −

cos(k L) sin(kx). sin(k L)

(2.117)

In Fig. 2.23 the obtained accumulated error is displayed. It can be seen that the numerical error decays only slowly even for the finest grid featuring approximately

2.9 Finite Elements of Higher Order

57

150 elements per wavelength for k = 0.5. To increase the rate of convergence we will now introduce two higher order polynomials which can be utilized within FEM. For both approaches, an exponential convergence rate can be achieved as demonstrated for our channel example in Sect. 5.4.5 together with the discretization error analysis.

2.9.1 Legendre Polynomials and Hierarchical Finite Elements In the FE context, any analytical function u gets approximated by a finite dimensional subset of interpolation functions, defined on a finite element mesh. In element local coordinates, this reads as u(ξ) ≈ u h (ξ) =

n eq 

Ni (ξ) u i ,

(2.118)

i=1

where u h (ξ) is the approximated function, Ni the ansatz functions (FE basis functions), u i the unknown physical quantity for equation i, and n eq the number of equations. In the case of standard Lagrangian elements, Ni are defined by the corner coordinates and u i is the related physical quantity at node i. The ansatz functions of first order on the unit domain Ωˆ ∈ [−1, 1] are defined as follows N1 (ξ) =

1−ξ 2

N2 (ξ) =

1+ξ . 2

(2.119)

However, one disadvantage of the Lagrangian basis is that for each polynomial degree, one needs a new set of shape functions (see Fig. 2.24a), which prevents the use of different approximation orders within one FE-mesh. In contrast, hierarchic ansatz functions are defined in such a way that every basis of order p is contained in the set of functions of order p + 1 (see Fig. 2.24b). We make use of the Legendre based interpolation functions as given by [20] 1 1 (1 − ξ) N2 (ξ) = (1 + ξ) 2 2 Ni (ξ) = φi−1 (ξ), i = 3, . . . , p + 1 ,

N1 (ξ) =

where φi denotes the integrated Legendre polynomials L i φi (ξ) =



ξ

L i−1 (x) d x

(2.120)

1 di 2 (x − 1)i . 2i ! d x i

(2.121)

−1

L i (x) =

58

2 The Finite Element (FE) Method

(a)

(b) N2

N1

N2

N1

p=1

p=1 -1

1

0

N1

-1

N3

N2

1

0

N2

N1

p=2

p=2 -1

1

0

-1

1

0

N3 N1

N3

N2

N4

N1

N2

p=3

p=3 N4 -1

0

1

-1

0

1

N3

Fig. 2.24 Lagrange and Legendre based ansatz functions up to order 3. a Lagrange functions. b Legendre functions

These polynomials are mutual orthogonal with respect to the H 1 semi-norm (see (D.11)) 1 φ′ i (ξ)φ′ j (ξ) dξ = 0 for i = j. (2.122) −1

Furthermore, the following recursive formula allows a fast evaluation for order i ≥ 2 [23] φ1 (ξ) = ξ ' 1 & 2 φ2 (ξ) = ξ −1 2 (i + 1)φi+1 (ξ) = (2i − 1)ξφi (ξ) − (i − 2)φi−1 (ξ) for i ≥ 2.

(2.123)

Note that these functions give only a non-zero value within the interval. Therefore they are also called internal modes or bubble modes.

2.9.1.1 Quadrilateral Element The definition of shape functions for quadrilateral elements is established by means of a 2D tensor product. The reference quadrilateral is depicted in Fig. 2.25, which

2.9 Finite Elements of Higher Order

59

Fig. 2.25 Reference quadrilateral on the unit domain Ωˆ ∈ [−1, 1] × [−1, 1]

e3

v4

v3

η

pη

e4

v1

ξ e1

e2

pξ

v2

is connected to the two spatial polynomial degrees ( pξ , pη ). For the quadrilateral element, the basis function can be splitted into the following types: 1. Vertex modes: The 4 nodal modes are the standard bilinear functions known from the first order Lagrange element (see Sect. 2.3.2) = Nv1,1 i

1 (1 + ξi ξ)(1 + ηi η), i = 1, . . . , 4 , 4

(2.124)

where ξi , ηi denote the local coordinates of the reference element vertices in [−1, 1]. Figure 2.26a displays the shape function for vertex v3 . 2. Edge modes: The edge modes are defined individually for each edge and their number depends on the approximation order in the direction of the edge. The modes corresponding to edge 1, e.g. are p ,1

Ne1ξ

=

1 φ p (ξ)(1 − η) , 2 ξ

(2.125)

where φ pξ denote the related one-dimensional integrated Legendre basis function as given in (2.120). 3. Internal mode: The internal modes are only defined within the element interior and vanish at its boundaries (see Fig. 2.26c). They are defined as p , pη

Nintξ

= φ pξ (ξ)φ pη (η) .

(2.126)

In contrast to the one-dimensional case, in 2D the notion of a polynomial degree involves now the spatial tuple ( pξ , pη ), spanning the global polynomial space, as sketched in Fig. 2.27. The single entries can be identified again as vertex ( pξ = 1, pη = 1), edge and interior degrees/face degrees of freedom (see Fig. 2.28). The set of shape functions can now be divided into the following spaces [24]: p

• Trunk Space Wtr (Ω): This space is spanned by the following set of monomials: – ξ i η j i, j = 0, 1, . . . , p; i + j = 0, 1, . . . , p – ξη for p = 1 – ξ p η, ξη p for p ≥ 2.

60 Fig. 2.26 Vortex, edge and bubble functions for quadrilateral element. a Basis function for vertex v3 . b Edge function of order 2 for edge e1 . c Interior shape function of order p = 2

2 The Finite Element (FE) Method

(a)

(b)

(c)

This set of functions is depicted in Fig. 2.29 and the dotted line marks the functions for the trunk space Wtr3 . The corresponding shape functions are shown in Fig. 2.30. p • Tensor Product Space Wtp (Ω): This space is spanned by all monomials ξ i η j with i = 0, 1, . . . , p and j = 0, 1, . . . , p. As an example, we display the polynomials for the tensor product space Wtr3 in Fig. 2.31 and the corresponding shape functions in Fig. 2.32.

2.9 Finite Elements of Higher Order

61

p

p

Fig. 2.27 2D polynomials and type of shape functions

Fig. 2.28 Shape functions according to the polynomials of Fig. 2.27 (scaled for display reasons) Fig. 2.29 Trunk space with p=3

p

p

It is obvious that due to the higher number of internal modes, the tensor product space supplies more FE basis functions. As the internal degrees of freedoms are purely local to the element, they can be eliminated by static condensation. This procedure increases the computational time on the element level. However, it dratsically

62

2 The Finite Element (FE) Method

Fig. 2.30 Shape functions of trunk space of order 3 (scaled for display reasons)

Fig. 2.31 Tensor product space with pξ = pη = 3

p

p

Fig. 2.32 Shape functions of tensor product space of order 3 (scaled for display reasons)

improves the overall performance due to the reduced size of the global system matrix and in addition due to the drastic decrease in the condition number (see Chap. 13), which is of high importance when applying iterative solvers.

2.9 Finite Elements of Higher Order

63

Fig. 2.33 Reference hexahedron with directional polynomial degrees pξ , pη , pζ

e11

v8 e12 v5

v7 e10

f6

e9 e8

pζ

v6

e7

ζ η

e5

v4

f4

ξ

e6

e3

e4 v1

f5 f

1

pη pξ

f3 v3 e2

e1

f2

v2

2.9.1.2 Hexahedral Element To build the ansatz functions in 2D and 3D, we simply take the Cartesian product of the one-dimensional basis functions. As we have three element local directions ξ, η, ζ, we can accordingly choose three different approximation orders pξ , pη , pζ . For the hexahedron the resulting basis functions can be distinguished into the following types (see Fig. 2.33): 1. Vertex modes: The 8 vertex modes are the standard trilinear functions known from the first order Lagrange element (see Sect. 2.3.5) = Nv1,1,1 i

1 (1 + ξi ξ)(1 + ηi η)(1 + ζi ζ), i = 1, . . . , 8 , 8

(2.127)

where ξi , ηi , ζi denote the local coordinates of the reference element vertices in [−1, 1]. 2. Edge modes: The edge modes are defined individually for each edge and their number depends on the approximation order in the direction of the edge. The modes corresponding to edge 1, e.g. are p ,1,1

Ne1ξ

=

1 φ p (ξ)(1 − η)(1 − ζ) , 4 ξ

(2.128)

where φ pξ denotes the related one-dimensional integrated Legendre basis function as given in (2.120). 3. Face modes: Also the face modes are defined for each face separately, e.g. for face 1 1 p , pη ,1 = (1 − ζ)φ pξ (ξ)φ pη (η) . (2.129) N f1ξ 2 4. Internal modes: The internal modes are only defined within the elements interior and vanish at its boundaries. They are defined as

64

2 The Finite Element (FE) Method p , pη , pζ

Nintξ

(2.130)

= φ pξ (ξ)φ pη (η)φ pζ (ζ) .

The number of nodal functions is always 8, whereas the number of edge, face and internal modes depends on the choice of pξ , pη and pζ . In order to guarantee a C 0 -continuous approximation, the inter-element orientation of edge and surface functions have to be considered.

2.9.1.3 Enforcing Conformity (Minimum Rule) In the general 1D case, the polynomial order of each element can be chosen individually, as only nodal and interior/bubble modes are present. In the 2D/3D case however, also edge and face degrees of freedoms are involved. In order to guarantee a conforming approach, the polynomial degrees of edges/faces between neighboring elements have to be set accordingly as demonstrated by the example in Fig. 2.34.

2.9.1.4 Duffy Transformation By viewing a triangle as a collapsed quadrilateral, one can also construct tensorialtype basis for triangles in a similar way as for quadrilateral elements [25]. Thereby, the Duffy transformation from a quadrilateral to a triangle reads as D : (u, v) → (ξ, η)

ξ = 41 (1 + u)(1 − v)

(2.131)

η = 21 (1 + v).

By using the inverse of the Duffy transformation, we can parameterize the triangle as follows N2 − N1 ξ −1= ∈ [−1, 1] 1−η N1 + N2 v = 2η − 1 = 2N3 − 1 = 1 − 2N1 − 2N2 ∈ [−1, 1]

u=2

Fig. 2.34 Application of minimum rule results in the displayed edge degrees

p=4

3

4

(2.132)

p=3 2 p =2

p=6

5

p =5

1 p=1

2.9 Finite Elements of Higher Order

65

with N1 , N2 , N3 the basis functions as defined in Sect. 2.3.3. In a similar way we can define the Duffy transformation for tetrahedral elements to use tensorial basis ξ = 81 (1 + u)(1 − v)(1 − w) D : (u, v, w) → (ξ, η, ζ) with η = 41 (1 + v)(1 − w)

(2.133)

ζ = 21 (1 + w). Here, the inverse map reads as ξ N2 − N1 −1= ∈ [−1, 1] 1−η−ζ N1 + N2 2N3 − (1 − N4 ) N3 − (N1 + N2 ) η −1= = ∈ [−1, 1] (2.134) v=2 1−ζ 1 − N4 N1 + N2 + N3 w = 2ζ − 1 = 2N4 − 1 ∈ [−1, 1] u=2

with N1 , N2 , N3 , N4 the basis functions as defined in Sect. 2.3.4. In [23] this approach has been used in order to construct higher order FE basis functions for all types of finite elements.

2.9.2 Lagrange Polynomials and Spectral Elements Spectral methods are high order techniques that are used to solve PDEs either in their strong or weak formulation. Common to all methods is the approximation of unknowns by a high order orthogonal polynomial expansion. In the context of FEM a higher order approximation is mostly limited to order three or four to balance computational effort and desired accuracy. In the context of spectral methods approximation orders up to one thousand are utilized. Examples for such methods are the Fourier Collocation Method or the Fourier Galerkin Method [26]. Although these methods feature excellent convergence properties their applicability towards complex geometries is difficult due to their global nature, the incorporation of discontinuities in e.g. material parameters is not straight forward and the computational costs increase when high polynomial degrees are needed to resolve all features of the problem under investigation. In contrast to these global or single domain methods, the spectral element method can be introduced for classical FEM [27]. The idea is to combine the accuracy and computational advantages of spectral methods with the geometrical flexibility of the finite element method. Within these methods one needs to find a trade-off between polynomial order and the number of finite elements. The error within a spectral element method will vary with E∼

e−α p , n e p+1

(2.135)

66

2 The Finite Element (FE) Method

in which we denote the polynomial order by p and the number of elements by n e [26]. For the definition of the spectral element method one needs to define the interpolating polynomial including its supporting points and a quadrature rule with abscissas at these supporting points with adequate accuracy for the order of the interpolation polynomial. Within the Galerkin spectral element method one chooses Lagrange polynomials of order p as interpolating polynomials. These polynomials feature the delta property and extensions towards quadrilateral and hexahedral elements are straight forward. Giving p supporting points at locations ξ j , distributed on the interval [ξa , ξa+1 ] one can associate to each supporting a polynomial as p

N j (ξ) =

p .

i=0,i = j

ξ − ξi . ξ j − ξi

(2.136)

We can already see, that for a higher degree of approximation, the complete set of basis functions changes (see Fig. 2.24). The hierarchical approach as done with the integrated Legendre polynomials is obviously no longer possible. The actual choice of supporting points for Lagrange polynomials is a critical one as we will show in the following. Without further consideration one could choose equidistant points on the given interval which will give rise to instabilities also known as Runges phenomenon. Let’s approximate the functional f (ξ) =

1 1 + 10ξ 2

(2.137)

on the interval [−1, 1] by sixth and fourteenth order Lagrange polynomials with different set of supporting points. The results are shown in Fig. 2.35. It can be seen that for the equidistant set of supporting points, the interpolated solution does not converge and shows heavy oscillations at the ends of the interval. In fact the interpolation error tends to infinity if the order is increased further. In contrast to this, the solution obtained by choosing the so called Legendre-Gauss-Lobatto (LGL) nodes approximates the exact solution smoothly with increasing polynomial order. The Legendre-Gauss-Lobatto points are located at the roots of the derivative of the Legendre polynomials. As already pointed out, the extension of Lagrange shape functions to quadrilateral and hexahedral elements is given by the tensor product. All nine second order shape functions on the quadrilateral element are displayed in Fig. 2.36 and some of the third order functions in Fig. 2.37. Another crucial point in the spectral element method is the choice of quadrature rule to use. In the case of LGL nodes the natural method would be the Gauss-Lobatto quadrature rule which is exact for polynomials of order 2 p − 1 which is lower than the optimal Gaussian quadrature but still sufficient. Summarizing, the abscissas and weight of the Gauss-Lobatto rule are given by [28]

2.9 Finite Elements of Higher Order

67

Fig. 2.35 Interpolation results for (2.137) by using Lagrange polynomials. Locations of supporting points are marked by red circles. a 6th order: Using equidistant supporting points. b 6th order: Using non-equidistant Legendre Gauss Lobatto (LGL) nodes. c 14th order: Heavy oscillations at the ends of the interval. d 14th order: Smooth approximation of exact curve

ξ j = +1, −1 and roots of L ′p (ξ) ; w j =

2 (

p( p + 1) L p (ξ j )

)2

(2.138)

with L p the Legendre polynom of order p (see (2.121)). The actual nodes and weights for arbitrary order of approximation can be found using a Newton method [29]. Explicitly computed values can also be found in various text books such as [28]. The big advantage of spectral element methods for time domain computations is that it gives rise to exact diagonal mass matrices which enables the use of explicit time stepping schemes. Although, these are limited in their time step size by the Courant-Friedrichs-Lewy (CFL) condition [30], they are preferable due to their low computational cost in each time step especially for problems with a high number of unknowns.

2.10 Flexible Discretization In many technical applications a sensor or/and actuator is immersed in an acoustic fluid, e.g. ultrasound transducers for nondestructive testing as well as medical diagnostic and therapy, ultrasound cleaning, electrodynamic loudspeakers, capacitive microphones, etc. For many applications, the numerical simulation of the actuator

68

2 The Finite Element (FE) Method

Fig. 2.36 All quadratic shape functions defined on the quadrilateral element

Fig. 2.37 Four of twelve cubic shape functions on the quadrilateral element

mechanism within the structure is quite complex, since in most cases we have to deal with a nonlinear coupled problem (e.g. the electrostatic-mechanical principle used in many micro-electromechanical systems), where in addition to the nonlinear coupling terms each single field is nonlinear (e.g., geometric nonlinearity in mechanics, moving body problem in the electrostatic field, see Chap. 10). Furtheron, in most cases the discretization within the structure has to be much finer than the one we need for the acoustic wave propagation in the fluid. A very similar problem arises in computational aeroacoustics, when solving the inhomogeneous wave equation according to Lighthill’s analogy (see Sect. 14.8.2). The inhomogeneous term of the wave equation is calculated by the fluid flow data within the fluid region and, to

2.10 Flexible Discretization

(a)

69

(b)

(c)

Fig. 2.38 Mesh types. a Uniform grid. b Unstructured mesh. c Non-conforming mesh

obtain reliable results, the discretization of the wave equation within this domain has to be very fine (up to 1,000 linear finite elements per wavelength). However, outside the flow region, the homogeneous wave equation is solved, and one could have a relatively coarse mesh (about 20 linear finite elements per wavelength). For standard FEM computational meshes are required to be geometrically conforming. This means that either uniform grids which result in many unknowns (c.f. Fig. 2.38a) or many transition elements between fine subdomain grids and coarse ones (c.f. Fig. 2.38b) need to be applied. This can lead to grave problems concerning the mesh quality. Meshes with a poor quality can even cause phenomena like numerical dispersion and reflection in acoustics or mechanics and can therefore lead to physically wrong results. A computational framework which does not suffer from the mentioned problems and provides the necessary flexibility is the Mortar Finite Element Method (Mortar FEM). In contrast to the traditional FEM, it provides the freedom to apply nonconforming grids across subdomains as Fig. 2.38c shows. The term Mortar finite elements refers to all methods where a Mortar method is used in a finite element context. Here, the coupling is between subdomains with different triangulations and associated finite element spaces. Due to the non-conforming grids, special attention has to be paid to the coupling of the physical quantities across the interfaces. The corresponding projection operation for the involved quantities is in general termed as mesh intersection problem, inter-grid communication problem or grid transfer problem.

2.10.1 Mortar FEM We start with the geometrical properties of the domain decomposition and the corresponding geometric operations to handle non-conforming meshes, followed by a detailed description of the numerical procedure [31–33].

70

2 The Finite Element (FE) Method

2.10.1.1 Geometrical Definitions and Terminology The coupled problem is defined on a decomposition of the global domain Ω into NΩ non-overlapping subdomains Ωk satisfying Ω=

NΩ

Ω k , Ωk ∩ Ωk ′ = ∅ for k = k ′ .

(2.139)

k=1

Such decompositions can either be geometrically conforming or geometrically nonconforming. Similar to the definition of conforming finite element meshes, the restriction for a geometrically conforming decomposition is that two subdomains are either allowed to touch along an entire common edge (2D and 3D) or face (3D) Γi , in one point, or not at all (cf. Fig. 2.39a). In contrast to that, hanging nodes (2D) respectively edges (3D) are allowed for the geometrically non-conforming case as Fig. 2.39b shows. In the example in Fig. 2.40 there are three subdomains NΩ = 3. The partition into subdomains is geometrically non-conforming in this example. There may exist i ∈ {1 . . . NΓ } inner interfaces in total. The skeleton S of the decomposition, S=

NΩ

(2.140)

∂Ωk \ ∂Ω

k=1

consists of the union of the inner interfaces which satisfy the condition Γi ∩ Γi ′ = ∅ for i = i ′ and i, i ′ ∈ {1 . . . NΓ }.

(2.141)

In Fig. 2.40 three inner interfaces NΓ = 3 are depicted as solid lines. The geometric realization of the skeleton of a d-dimensional domain is given by a number of injective parametric mappings

(a)

(b)

Ω1

Ω5

Ω6

Ω1

Fig. 2.39 Examples for geometrically conforming and non-conforming domain decompositions. a Conforming decomposition. b Non-conforming decomposition

2.10 Flexible Discretization Fig. 2.40 Example of a global domain decomposed into three subdomains. Exploded view of the subdomains with inner interfaces Γi and mappings γk, j

71

γ2,1 γ1,2

Ω1

γ1,1

Γ2

Ω2 γ2,2

Γ1

γ3,1

Ω3

Γ3

γk, j : Rd−1 → Rd k ∈ {1 . . . NΩ }, and j being the number of whole curves (d = 2) or faces (d = 3) of the subdomain Ωk . Therewith, the inner interfaces for the example in Fig. 2.40 can be written more precisely as Γ1 = γ1,1 ∩ γ3,1 , Γ2 = γ1,2 ∩ γ2,1 ,

(2.142) (2.143)

Γ3 = γ2,2 ∩ γ3,1 .

(2.144)

The set of all mappings γk, j admits a partition of the skeleton into a subset Gm of Nm mortar or master edges or faces Gm = S =

Nm

γ m , and γm ∩ γm ′ = ∅, m = m ′ , γm , γm ′ ∈ {γk, j }.

m=1

The subdomain Ωk associated with a γm and therefore also with the corresponding Γi is then denoted as mortar or master side Ωm(i) in respect to γm (c.f. Fig. 2.41). The choice of mortar faces γm is not unique in general. One possible choice for the given domain could be Gm = {γ3,1 , γ1,2 } (cf. Fig. 2.41d). Once this choice has been made the remaining subset of Ns faces which also make up a partition of the skeleton is fixed and the γk, j contained therein are called non-mortar or slave faces Gs = S =

Ns

γ s , and γs ∩ γs ′ = ∅, s = s ′ , γs , γs ′ ∈ {γk, j } \ Gm .

s=1

The function values on the slave side of the interface are determined by the values on the corresponding master side. For our example (see Fig. 2.41d) the remaining set of slave faces is Gs = {γ1,1 , γ2,1 , γ2,2 }. Depending on the actual choices, the tuple of (Nm , Ns ) for this example case, maybe either (2,3) or (3,2), since there are five curves γk, j . An algorithm for partitioning the faces γi, j of the skeleton into mortar

72

2 The Finite Element (FE) Method

(a)

(b)

(c)

(d)

Fig. 2.41 Four possible choices for partitioning the set of main edges making up the skeleton into master and slave sides. Master sides are marked by a thick line. a Gm = {γ2,1 , γ3,1 }, Gs = {γ1,1 , γ1,2 , γ2,2 }. b Gm = {γ1,1 , γ2,1 , γ2,2 }, Gs = {γ1,2 , γ3,1 }. c Gm = {γ1,1 , γ2,1 , γ2,2 }, Gs = {γ1,2 , γ3,1 }. d Gm = {γ1,2 , γ3,1 }, Gs = {γ1,1 , γ2,1 , γ2,2 } Fig. 2.42 Subdomains with cross points in the interior (filled dots) and on the Dirichlet boundary Γe (hollow dots). Master and slave sides in respect to Γi

Γn

Γe Ωm (i)

Cross point

Γi Ωs(i)

and slave faces on arbitrary geometrically non-conforming polygonal domains is proposed in [33]. The points of S, where the interfaces Γi meet and where multiple master sides are involved are called cross points or vertex points (cf. Fig. 2.42). At these points the function values for the slave sides are determined by multiple master sides, which would result in an overdetermined system of equations. Therefore a special treatment has to be applied for these points. Cross point treatment must also be applied to points where an interface touches a Dirichlet boundary, since the function value is already given there.

2.10 Flexible Discretization

73

2.10.1.2 Formulation of the Coupled Problem In the following the well-known time independent diffusion or Laplace problem for the unknown scalar function u on a domain Ω will be considered as a prototype for the introduction of the Mortar FEM −∇κ(x)∇u = f

in Ω,

u = ue

on Γ.

(2.145)

In order to reformulate the Laplace problem (2.145) on the decomposed domain with possibly discontinuous functions on the skeleton, physical coupling conditions have to be introduced. First of all, the physical quantity has to be continuous across the inner interfaces. Therefore, the jump [u] has to be zero [u] = 0 on S. This condition has to be reformulated as a weak condition using test functions µ from a suitable Lagrange multiplier space M (cf. [32, 34]) 

[u] µdΓ = 0 ∀ µ ∈ M and [u] ∈ H 1/2 (S).

S

Furthermore, the flux of the unknown is continuous across the interfaces S in normal direction. The condition on the jump of the normal derivatives is therefore [κ∂u/∂n] = 0, where n is defined in respect to the slave side. This can be achieved by introducing a Lagrange multiplier (LM) λ = −κs

∂u s ∂u m = −κm ◦ Φ on S . ∂n ∂n

(2.146)

Here Φ denotes a spatial mapping, which relates the points on the slave sides to the points on the master sides of the interfaces. The weak formulation of the Laplace problem can now be rewritten on many subdomains by substituting the definition of the Lagrange multiplier according to (2.146). The arising boundary integral over the outer boundary is set to zero for simplicity. Summing up, one arrives at the symmetric saddle point problem of finding (u i , λ) and i = 1 . . . NΩ such that ⎛ ⎞   NΩ  ⎟ ⎜ ⎝ κ(x)(∇u) · (∇v) d⎠ + [v] λ dŴ = 0 i=1

Ωi

S



S

[u] µ dŴ = 0,

(2.147)

74

2 The Finite Element (FE) Method

for all (µ, vi ). It shall be noted, that µ is a test function with the same basis as the LM λ. This property is responsible for the symmetry of the arising stiffness matrix.

2.10.1.3 Projection of Coordinates The jump of the unknown function across a single internal interface Γi from the master side Γm to the slave side Γs is given by [u]i = u s − u m ◦ Φi on Γi ,

(2.148)

where Φi denotes a bijective global coordinate mapping Φi : Γs → Γm on Γi . This mapping relates every point on a subset Γs of the curve γs to a point on its corresponding subset of Γm of the master curve γm along their common intersection Γi . In the continuous setting it is simply the identity mapping Φi (x) = x.

(2.149)

After the spatial discretization of the subdomains by finite elements the mapping gets more complicated since the discretizations Γs and Γm of the coincident curves γs and γm need not coincide any more as Fig. 2.43 shows for two linear elements. Here an initial point on the curve γs gets mapped onto a linear element on the slave side by the mapping φs and the same point on the curve γm gets mapped to an element on the master side by the mapping φm . As long as both mappings are known, i.e. the finite element code has information about the analytical description of the geometry of the inner interfaces, the mortar mapping Φi can be written as Φi (x) = φm ◦ φ−1 s (x).

(2.150)

The information about the analytical geometry may simply be lost, when only the discretized subdomain meshes generated by a preprocessor are available as input. In

slave φs Φi φm

master Fig. 2.43 Mortar mapping along curved interface

2.10 Flexible Discretization

75

such a situation it is necessary to find good local approximations of Φ. In practice this can be achieved by finding pairs of closely spaced elements and by defining a mapping in respect to these pairs. Algorithms for calculating such approximative mappings are described in [35]. 2.10.1.4 Mesh Intersection Operations In 2D the intersection of line elements has to be considered. If the interface is planar this amounts to do simple interval checks. If the interface is curved, the elements have to be projected onto a common line segment first for doing the interval checks there. These considerations also apply in a modified way for domains in 3D where interfaces are surfaces. It has to be noted however that the seemingly simple operation of finding the intersection domain of arbitrary surface elements is a highly non-trivial task even for first order elements with straight edges. • Intersection of Straight Line Elements If an intersection of two co-linear line elements exists, it is again a line element sharing two of the four endpoints of both parent elements in the co-linear case. To check for an intersection one has to project the endpoints [m1 , m2 ] of the element on the master side of the interface in two dimensional coordinates to the one dimensional local coordinate system defined by the endpoints of the slave element [s1 , s2 ]. The local coordinates of the slave nodes [s1 , s2 ] are trivially given by 0 and 1. The four local coordinates of the pair of lines are then brought into ascending order and therefore four possible cases for the intersection of two line elements may be identified as depicted in Fig. 2.44. 1. 2. 3. 4.

ξ1 < 0 ∧ 0 < ξ2 ≤ 1: the intersection is the line [s1 , m2 ] 0 ≤ ξ1 < 1 ∧ ξ2 > 1: the intersection is the line [m1 , s2 ] ξ1 ≤ 0 ∧ ξ2 ≥ 1: the intersection is the line [s1 , s2 ] ξ1 > 0 ∧ ξ2 < 1: the intersection is the line [m1 , m2 ]

Fig. 2.44 Four possible cases of two lines intersecting each other

master ξ1 0

0 ξ1

ξ2 1

case 1

1 ξ2

slave

case 2 master

ξ1

0

1 case 3

ξ2

0

ξ1

ξ2

case 4

slave 1

76

2 The Finite Element (FE) Method

• Intersection on Coplanar Interfaces The algorithm for finding intersections of lines can be extended in a straight forward manner to a 3D setting if only axis-parallel quadrilateral elements are to be considered on the interface. The term axis-parallel does not refer to the global coordinate axes in this context, but to the fact that the quadrilateral edges on both sides of the interface have to be parallel. This includes the case of parallelograms. The method is more thoroughly described in [32, 36]. Again, the local coordinates (ξ1 , η1 ) and (ξ2 , η2 ) of the first and third corner of the master element are computed in respect to the slave element (c.f. Fig. 2.45). After bringing the local coordinates for both directions into ascending order there are sixteen possible cases for the intersection of two quadrilaterals. The ordering is necessary due to the fact, that the order of nodes for elements is just guaranteed to be counter-clockwise, but the master element might have been rotated in respect to the slave element as a whole. The algorithm used for intersecting a pair of arbitrary planar convex polygons is a specialization of the Sutherland-Hodgman algorithm [37] and is designed to fit the needs of finite element grids. The polygons are considered to be cyclic directed graphs with the same orientation (i.e. clockwise or counter-clockwise). The basic idea of this algorithm is to find a cut of two edges of the polygons as a starting point and then to “walk” along the boundary of the intersection. Once the intersection polygon has been obtained it gets either triangulated or a quadrilateral element is generated if the number of corners is equal to four. The algorithm is described in detail in [38]. • Intersection on Arbitrarily Curved Interfaces The intersection calculation on curved interfaces is quite complicated since the elements have to be mapped to a common plane before the actual intersection calculation can be performed. The approach proposed in [32] relies on pair-wise mappings of the elements to the plane or line of the element on the slave side of the interface along the face normal vector. Figure 2.46 depicts the situation for two skewed triangle elements which only intersect in a common line element. In order to perform the intersection calculation triangle 2 gets projected into the plane of 1 resulting in triangle 3. The actual intersection takes place between the triangles 1 and triangle 3. It should not go unnoticed that this method still has the drawback of overlapping elements and gaps are not being treated properly. It may also lead to holes in the intersection mesh. This inevitably leads to poor results while evaluating the integrals and can just be counteracted, by increasing the resolution of the interface mesh.

2.10.1.5 Discrete Problem For some convergence estimates to hold, each triangulation Tk,h is usually assumed to be quasi-uniform. Quasi-uniformity for a triangular mesh means that there exist two

2.10 Flexible Discretization

77

4 master 3 sla ve

η2 1 η1

1

0

0

ξ1

1

2

ξ2

Fig. 2.45 Intersection of two axis-parallel parallelogram-shaped elements

(a)

(b)

(c)

Fig. 2.46 Projection of two triangles onto common plane (taken from [38]). a Intersection of two triangles. b Projection onto common plane. c Actual configuration for polygon intersection

positive constants τ and σ such that for all triangles T of Tk,h , τ h max ≤ h T ≤ σρT . In this relation h max and h T denote the maximum diameter of the triangles in Tk,h and the diameter of the triangle T respectively. The diameter of the inscribed circle is denoted by ρT . The associated finite element spaces with piecewise linear elements are denoted by Sh (Tk,h ). The unconfined product space with no continuity condition between the subdomains is then given by Xh =

NΩ .

Sh (Tk,h )

(2.151)

k=1

= {u ∈ L 2 (Ω) | u |Ωk ∈ Sh (Tk,h ) for k = 1, . . . , NΩ , u |Γn = u Γe }. Unfortunately this space is not suitable for the discretization. This fact can quickly be understood if the jumps of the functions across the skeleton are considered (cf. Fig. 2.47). The derivatives across those interfaces evaluate to indefinite values. Since the Lebesgue measure of S is not zero the functions u are elements of L 2 (Ω) but not of H 1 (Ω). Since the locations where the unknowns are defined are in general not the same across the inner interfaces, no pointwise continuity of the unknown function can be required. Therefore, a weak continuity condition is required to be fulfilled. After some necessary definitions this so-called mortar condition can be stated.

78

2 The Finite Element (FE) Method

(a)

(b)

Fig. 2.47 Illustration of the traces of finite element basis functions along a non-conforming interface. The basis functions for two degrees of freedom on the interface are depicted. A bilinear nodal basis is used on the quadrilateral elements, while a linear basis is used on the triangles. a Nonconforming meshes along coplanar interface. b Basis functions for two DOFs on the interface and their traces

Let Tp = Tk,h |γp denote the discretization inherited from the volume discretization in subdomain k to which the master or slave edge γp p = m, s belongs. On every discretization Ts a space of test functions is introduced which, for piece-wise (bi-)linear basis functions in the slave subdomains Ω(s), fulfill the following conditions which characterize the Mortar method among other domain decomposition methods: (i) Ms,h ⊂ Ss,h (Ts ) ⊂ H −1/2 (ii) dim(Ms,h ) = dim(Ss,h (Ts ) ∩ H01 (Ts )) (iii) Ms,h contains constants on ∂Ts

(2.152)

Condition (i) states that the space on interface Ts has to be a subspace of the trace space inherited from the triangulation on the slave side Sh (Tk,h ). The second condition (ii) states that the dimension of the space should be equal to the dimension of the trace space on the slave side with homogeneous Dirichlet conditions on the boundaries of the discretized interface Ts . The third condition (iii) makes sure, that the function value from the boundaries of the master triangulations get transferred to the boundary nodes of the slave side. For higher order basis functions on the slave

2.10 Flexible Discretization

79

Fig. 2.48 Basis functions for the standard Lagrange multiplier for one interface

side, this condition gets modified by stipulating that the space should be able to represent the first derivative of the functions. In Fig. 2.48 the standard nodal basis functions derived from the piece-wise linear basis of the unknown function on the slave side is depicted. It has to be noted that no LM degree of freedom is assigned to the first and last node of the discretization. Let q be the number of inner interfaces Γi geometrically corresponding to a single non-mortar edge γs . With these definitions it is possible to define the discrete Mortar bilinear form for all Ns slave side discretizations b(u h , µh ) :=

Ns   ( ) u s,h − u m,i,h ◦ Φi µh dŴ and i = 1...q. s=1 T

(2.153)

s

Therewith, a global function u h is a Mortar function if the discrete Mortar condition b(u h , µh ) = 0

(2.154)

is fulfilled. It states that the L 2 -projection of the jump has to be orthogonal to the Lagrange multiplier space. The domain bilinear form a(·, ·) and right hand side linear form (·, ·) are defined as a(u h , vh ) =

NΩ  

κ ∇u h · ∇vh d,

(2.155)

NΩ  

f vh d ,

(2.156)

k=1Ω k

( f, vh ) =

f ∈ L 2 (Ω).

k=1Ω k

The material parameter κ(x) is continuous in each Ωk and satisfies the following relation κ(x) ≥ κ0 > 0. For simplicity of the presentation κ(x) = κk = constant for each Ωk . The previous relation ensures the positive definiteness of the bilinear form a(·, ·). With these definitions, the mixed formulation of the Mortar coupled problem can be written as: Find (u h , λh ) such that

80

2 The Finite Element (FE) Method

a(u h , vh ) + b(vh , λh ) = ( f, vh ) b(u h , µh ) = 0 ,

(2.157) (2.158)

where uh ∈ X h , vh ∈ {v ∈ X h | v = 0 on Γe } λh , µh ∈ M h ⊂ H −1/2 (S) . For the special case of k = 2 subdomains the algebraic system arising from (2.157 and 2.158) has the following form ⎞⎛ ⎞ ⎛ ⎞ ⎛ f1 u1 Ku1 0 D ⎝ 0 Ku2 M⎠ ⎝u 2 ⎠ = ⎝ f 2 ⎠ . DT M T 0 λ 0

(2.159)

In (2.159) Ku1 , Ku2 are standard stiffness matrices of the subdomains. The new matrices D and M are due to the non-conforming interface and are formally mass matrices. They compute element-wise as D=

n es 

de ; de = [dab ];

e=1

dab = M=



Γe

n isec

Nas Nbs dŴ ,

(2.160)

me ; me = [m ab ];

e=1

m ab =



Γe

(Nam ◦ Fm−1 ◦ Φ)(Nbs ◦ Fs−1 ) dŴ .

(2.161)

Here n es is the number of surface elements on the slave side of the interface and n isec is the number of intersection elements on the interface. The finite element basis functions Nas and Nam denote the traces of the FE basis on the slave and on the master side of the interface, Nbs denotes the Lagrange multiplier basis given with respect to the slave side, Φ is the global coordinate mapping from slave to master side and F −1 is the coordinate mapping from global to element local coordinates (see Sect. 2.3.2). The integrals in (2.160) are defined in the usual manner with respect to Ts . The necessary steps for the computation of the integrals in (2.161) are discussed in the following.

2.10 Flexible Discretization

81

2.10.1.6 Evaluation of Coupling Integrals The feature which makes the Mortar FEM so flexible, namely the usage of nonconforming meshes in different subdomains, comes at the cost of a more elaborate implementation. Since the grids are allowed to be non-conforming on the interfaces of two subdomains, the integrals defined on these interfaces involving basis functions from both sides have to be evaluated with respect to two different meshes. It is therefore necessary to first compute the domains where pairs of elements on the interface intersect. In the developed implementation the intersection elements on curved interfaces are found by projecting the element on the master side to element on the slave side along the slave side normal vector. The coupling integrals defined in (2.161) are then evaluated over these intersection elements and it is up to the assembly operator to assemble the corresponding results into the correct positions of the coupling matrices. Once the intersection elements have been found, the coupling integral (2.161) can be evaluated on these elements by means of standard Gauss quadrature. For a single element the integral can be rewritten as follows 

Γe

(Nam ◦ Fm−1 ◦ Φ)(Nbs ◦ Fs−1 ) dŴ ≈ n int 

Wl Na (ξlm )Nb (ξls )J Fe (ξle ) .

(2.162)

l=1

Here n isec is the number of intersection elements, n int is the number of quadrature points, Wl are the quadrature weights and the determinant of the Jacobian J e accounts for the change in volume due to the element mapping. The difficulty which arises when this quadrature formula is applied, is that only the quadrature point ξle in respect to the local coordinates of the intersection element is known in advance and that the points ξlm in the master element and ξls in the slave element have to be projected into those elements, before the basis functions can be evaluated there (see Fig. 2.49). It is very important to notice that nodes of the intersection element do not carry any degrees of freedom by themselves. The intersection element is just an auxiliary geometrical entity which only serves as integration domain. The projection operation for general elements involves the following steps: 1. Map local coordinates ξle of quadrature point in intersection element to global coordinates using Fe 2. Project global coordinates of integration point on slave side to global coordinate on master side using Φ 3. Map global coordinates of quadrature point to local coordinates ξlm of master element using Fm−1 4. Map global coordinates of quadrature point to local coordinates ξls of slave element using Fs−1 .

82

2 The Finite Element (FE) Method

−1 Fm

ξlm

Element Nodes

master Φ

Integration Points

n Nodes of Intersection Element slave

Fs−1

Fe ξls

ξle

Fig. 2.49 Projection of quadrature points from the intersection element into the master element and into the slave element

Points 3 and 4 in general involve the application of a Newton algorithm. A linear mapping algorithm may only be used for 2-node isoparametric line elements, 3-node isoparametric triangle elements or higher order elements which just use a linear local-to-global mapping. Once the values of the basis functions Na and Nb have been obtained and (2.162) has been evaluated, the assembly operator has to make sure, that the contribution gets added to the corresponding entry in the coupling matrix.

2.10.2 Nitsche Type Mortaring In the previous section, we have discussed in detail the standard Mortar FEM, which fulfills the continuity of the trace (the physical quantity) in a weak sense and the flux (normal derivative of the physical quantity) in a strong sense by introducing the Lagrange multiplier (LM). The main drawback of this approach can be seen in the arising saddle point structure of the coupled system of Eq. (2.147). Therefore, we present a second approach, which is named Nitsche type mortaring. The method of Nitsche [39] was originally introduced to impose essential boundary conditions weakly. Unlike the penalty method, it is consistent with the original differential equation. The strong point of Nitsche’s method is that it retains the convergence rate of the underlying FE method, whereas the standard penalty method either requires a very large penalty parameter or destroys the condition number of the resulting algebraic system of equations. Let us consider the Laplace problem as stated in (2.145). For standard FEM, we will require a functional space with the constraint according to the Dirichlet (essential) boundary condition

2.10 Flexible Discretization

83

V = {ϕ ∈ H 1 | ϕ|Γ = u e }.

(2.163)

The weak formulation will then read as follows: Find u ∈ V such that   κ(x) ∇v · ∇u dΩ = f v dΩ Ω

(2.164)

Ω

for all v ∈ V0 = {ϕ ∈ H 1 | ϕ|Γ = 0}. Now, Nitsche’s approach incooperates the Dirichlet boundary condition and the functional space for u is free of constraints. Thereby the weak formulation reads as follows: Find u ∈ H 1 such that 

κ ∇v · ∇u dΩ −

Ω



Γ

=



Ω

  1  ∂u ∂v dΓ − κu dΓ + βκ κv uv dΓ ∂n ∂n hE E(Γ ) ΓE Γ   1  ∂v dΓ + βκ gv dΓ. (2.165) f v dΩ − u e ∂n hE E(Γ )

Γ

ΓE

Thereby, the boundary Γ is discretized into surface elements and the two terms with the sum penalize the deviation of u at the boundary from u e . Please note that the material parameter κ is explicitly within the penalty term. For this formulation, one can proof convergence for h → 0 and β > 0. We will now apply Nitsche’s ansatz for a computational domain consisting of two subdomains Ω1 and Ω2 with a common interface ΓI as display in Fig. 2.50. Thereby, we introduce two test functions v1 and v2 and write in a first step the weak formulation of each subdomain individually 

κ1 w1

Ω1



ΓI





∂u 2 dŴ = κ2 w2 ∂n2

κ1 ∇w1 · ∇u 1 d d − κ2 ∇w2 · ∇u 2 d −

Ω2

∂u 1 dŴ = ∂n1



w1 f 1 d

(2.166)



w2 f 2 d.

(2.167)

Ω1

Ω2

ΓI

For the sake of a simpler notation, we choose κ1 and κ2 constant in Ω1 and Ω2 , which is by no means a restriction. In general, the material parameter can even depend on Fig. 2.50 Domain with two sub-regions Ω1 and Ω2

Γ Ω1 n2 ΓI n1 Ω2

84

2 The Finite Element (FE) Method

the physical quantity u (nonlinear case). In a next step, we add the two Eq. (2.166) and (2.167), and explore the relation n = n1 = −n2 ; κ1

∂u 1 ∂u 1 ∂u 2 ∂u 2 = κ1 = −κ2 = κ2 ∂n1 ∂n ∂n2 ∂n

to arrive at    ∂u 1 dŴ (2.168) κ1 ∇w1 · ∇u 1 d + κ2 ∇w2 · ∇u 2 d − κ1 [w] ∂n Ω1 Ω2 ΓI   = w1 f 1 d + w2 f 2 d. Ω1

Ω2

In (2.169) the operator [ ] defines the jump operator, e.g., [w] = w1 − w2 . In order to retain symmetry, we add to (2.169) the following term −



κ1

∂w1 [u] dŴ with [u] = u 1 − u 2 . ∂n

ΓI

This operation is allowed, since we postulate on the interface u 1 = u 2 . In a final step, we add the penalization term β κ¯



E(ΓI )



1 hE

[w] [u] dŴ

ΓE

with β a penalty factor and κ¯ = (κ1 + κ2 )/2. Therewith, we arrive at the final formulation for Nitsche’s approach 

Ω1

κ1 ∇w1 · ∇u 1 d +



κ2 ∇w2 · ∇u 2 d



κ1 [w]

Ω2

−

ΓI



+ β κ¯

 ∂w1 ∂u 1 d − [u] dŴ κ1 ∂n ∂n ΓI     

Consistency



E(ΓI )

=





Ω1

1 hE

Symmetri zation



[w] [u] dŴ





ΓE

Penalt y/Stabili zation

w1 f 1 d +



Ω2

w2 f 2 d.

(2.169)

2.10 Flexible Discretization

85

If the penalty parameter β in (2.169) is chosen large enough, the bilinear form is coercive on the discrete space Vh ⊂ H 1 and one can derive optimal a priori error estimates in both the energy norm and the L 2 norm for polynomials of arbitrary degree (but same at both sides) [40]. 

2 

||∇(u − u h )||20,Ωi + h −1 ||[u]||2ΓI

i=1

 21

≤ Ch p ||u||Ω, p+1

(2.170)

||u − u h ||0,Ω ≤ Ch p+1 ||u||Ω, p+1 . (2.171)

In order to obtain the system of equations, we rewrite (2.169) as two equations with all individual terms according to the two test functions w1 and w2 

Ω1



∂u 1 dŴ − κ1 w1 ∂n

∂w1 u 1 dŴ ∂n ΓI ΓI   1  ∂w1 u 2 dŴ + β κ¯ w1 u 1 dŴ + κ1 ∂n hE E(ΓI ) ΓE ΓI   1  −β κ¯ w1 u 2 dΓ = w1 f 1 d hE

κ1 ∇w1 · ∇u 1 d −

E(ΓI )



Ω2



κ1

Ω1

ΓE

(2.172)  1  ∂u 1 κ2 ∇w2 · ∇u 2 d + κ1 w2 dŴ + β κ¯ w2 u 2 dŴ ∂n hE E(Γ ) I ΓI ΓE   1  −β κ¯ w2 u 1 dŴ = w2 f 2 d. hE 

E(ΓI )

Ω1

ΓE

We now assume a discretization with non-conforming meshes at the interface ΓI as displayed in Fig. 2.51. Furthermore, we perform the FE ansatz according to w1 ≈ w1h =



N1i w1i ; u 1 ≈ u 1h =

i

w2 ≈ w2h =

 i



N1 j u 1 j

(2.173)

N2 j u 2 j

(2.174)

j

N2i w2i ; u 2 ≈ u 2h =

 j

to arrive at the discrete system of equations. As in the case of Mortar FEM, we will also need all operations (projection of coordinates, intersection operations, etc.) between the two surface meshes in order to compute the entries of the matrices. Now, the matrix system of equations reads as follows

86

2 The Finite Element (FE) Method

*

*

*

Γ2

Ω2

... Interface nodes

*

*

... Coupling nodes

*

* *

* *

... Inner nodes

Γ1

Ω1

*

*

Fig. 2.51 Discretized subdomains Ω1 and Ω2



K 11 0 0 K 22



u1 u2



+



K Γ1 K Γ1 Γ2 K Γ2 Γ1 K Γ2



u1 u2



=



f1 f2



.

(2.175)

Thereby, the entries of the matrices as well as right hand side compute as ij K 11

=



κ1 ∇ N1i · ∇ N1 j d

(2.176)



κ2 ∇ N2i · ∇ N2 j d

(2.177)

Ω1 ij

K 22 =

Ω2 ij

 ∂ N1 j ∂ N1i dŴ − κ1 N1 j dŴ ∂n ∂n Γ1 Γ1  1  +β κ¯ N1i N1 j dŴ hE E(Γ1 ) ΓE   1  ∂ N1i N2 j dŴ − β κ¯ = κ1 N1i N2 j dŴ ∂n hE

K Γ1 = −

ij

K Γ1 Γ2



κ1 N1i

E(Γ1 )

Γ1

ij

K Γ2

&

ij

'T

= K Γ2 Γ1  = β κ¯

E(Γ2 )

f i1 =

1 hE

(2.178)

ΓE

(2.179) 

N2i N2 j dŴ

(2.180)

ΓE



N1i f 1 d

(2.181)



N2i f 2 d

(2.182)

Ω1

f i2 =

Ω2

2.10 Flexible Discretization

87

Here, we have already substituted ΓI by Γ1 as well as Γ2 , which are the discretized interfaces (see Fig. 2.51). Please note that not only nodes on the interfaces but also neighboring nodes in Ω1 and in Ω2 are involved, since the computation of some entries requires normal derivatives. Studying the structure of our coupled system of equations, we see that our formulation is symmetric and does not introduce any additional unknowns (we have no Lagrange multiplier as in case of Mortar FEM). However, we have a penalty parameter β, which has to be chosen large enough to guaranty u 1 = u 2 and on the other hand should not be too large to deteriorate the condition number of the system matrix. However, practical computations show that the result is not very sensitive to the choice of β and a value of about 100 is enough.

2.10.3 Numerical Example In order to compare the two non-conforming methods, we consider an electromagnetic computation in 2D, which can be described by (2.145) when using for κ the magnetic reluctivity ν and A z (z-component of magnetic vector potential) for u (see Chap. 6). The setup under investigation is a solenoid as displayed in Fig. 2.52. We have generated three different meshes: 1. Conforming mesh for reference computations; 2. Non-conforming mesh, where the interface is completely inside the air region (see Fig. 2.52); 3. Non-conforming mesh, where the interface includes a part of the surface of the yoke (see Fig. 2.53). For the iron core (yoke and anchor), we consider a nonlinear BH-curve (see Sect. 6.7.5) and perform computations on the conforming grid as well as the two nonconforming grids. Figure 2.54 shows the resulting magnetic flux density obtained by the classical Mortar approach. Comparing the computed fields of the different formulations, no visible differences can be seen. To perform a more detailed analysis,

Symmetry Coil

Yoke

Anchor Non-conforming interface (mesh)

Fig. 2.52 Computational setup of the solenoid and detail of non-conforming mesh with the nonconforming interface just in the air region

88

2 The Finite Element (FE) Method Coil Yoke

Anchor Symmetry

Non-conforming interface (mesh)

Fig. 2.53 Computational setup of the solenoid and detail of non-conforming mesh with the nonconforming interface including the iron core Fig. 2.54 Computed magnetic flux lines in case of non-conforming mesh using Mortar formulation

Table 2.3 Magnetic energy in Ws (computations with classical Mortar and Nitsche type mortaring on the non-conforming mesh according to Fig. 2.52)

Conform (Ws) Anchor Yoke

Anchor 1 Yoke

Anchor Yoke

Mortar (Ws) Error

5288.62 5348.27 8268.42 8372.63 Nitsche β = 20 (Ws) Error 5347.38 1.11 % 8371.11 1.24 % Nitsche β = 500 (Ws) Error 5275.94 0.24 % 8243.98 0.30 %

1.13 % 1.26 % Nitsche β = 100 (Ws) Error 5329.47 0.77 % 8339.55 0.86 % Nitsche β = 1000 (Ws) Error 5233.28 1.05 % 8167.48 1.22 %

we compute the magnetic energy in the yoke and the anchor region. Table 2.3 lists the results obtained by the conforming mesh as well as non-conforming mesh according to Fig. 2.52. We have also investigated in different penalty factors β for the Nitsche type mortaring method. As it can be seen, in case of β = 500 we obtain the smallest error compared to computations with the conforming mesh. In conclusion, we can state that the Nitsche type mortaring approach is quite robust w.r.t. the penalty factor β and can be even superior to the classical Mortar method if the optimal value for β

2.10 Flexible Discretization Table 2.4 Magnetic energy in Ws (computations with classical Mortar and Nitsche type mortaring on the non-conforming mesh according to Fig. 2.53)

89 Conform (Ws) Anchor Yoke

Anchor Yoke

Anchor Yoke

Mortar (Ws)

Error

5288.62 5347.8 1.12 % 8268.42 8371.03 1.24 % Nitsche Nitsche β = 20 β = 100 (Ws) Error (Ws) Error 5360.19 1.35 % 5340.34 0.97 % 8393.31 1.50 % 8356.8 1.07 % Nitsche Nitsche β = 500 β = 1000 (Ws) Error (Ws) Error 5303.95 0.29 % 5274.77 0.26 % 8288.53 0.24 % 8233.72 0.42 %

is chosen. These findings also hold in the case, where the non-conforming interface is not completely in air but also contains a part of the surface of the yoke, where the magnetic reluctivity ν is discontinuous (see Table 2.4). For further applications we refer to Sect. 5.4.4 concerning acoustics, to Chap. 11 for rotating systems in electromagnetics, to Sect. 8.3.2 for mechanical-acoustic systems, to Sect. 14.8.2 for computational aeroacoustics, and for computational mechanics we refer to [41].

References 1. K.J. Bathe, Finite Element Procedures (Prentice Hall, New Jersey, 1996) 2. N. Ida, Engineering Electromagnetics (Springer, 2004) 3. O.C. Zienkiewicz, R.L. Taylor, The Finite Element Method, vol. 1 (Butterworth-Heinemann, UK, 2003) 4. T.J.R. Hughes, The Finite Element Method, 1st edn. (Prentice-Hall, New Jersey, 1987) 5. F.X. Zgainski, J.L. Coulomb, Y. Marechal, A new family of finite elements: the pyramidal element. IEEE Trans. Magn. 32, 1393–1396 (1996) 6. M. Jung, U. Langer, Methode der Finiten Elemente für Ingenieure, 2nd edn. (Springer, 2013) 7. K. Meyberg, P. Vachenauer, Höhere Mathematik 1 (Springer, 1993) 8. H. Whitney, Geometric Integration Theory (Princeton University Press, Princeton, 1957) 9. J.C. Nédélec, Mixed finite elements in R 3 . Numer. Math. 35, 315–341 (1980) 10. M.L. Barton, Z.J. Cendes, New vector finite elements for three-dimensional magnetic field computation. J. Appl. Phys. 61(8), 3919–3921 (1987) 11. A. Bossavit, J.C. Verite, A mixed FEM-BIEM method to solve 3-D eddy current problems. IEEE Trans. Magn. 18, 431–435 (1982) 12. A. Kameari, Three dimensional eddy current calculation using edge elements for magnetic vector potential. Appl. Electromagn. Mater. 225–236 (1986)

90

2 The Finite Element (FE) Method

13. G. Mur, A.T. Hoop, A finite-element method for computing three-dimensional electromagnetic fields in inhomogeneous media. IEEE Trans. Magn. 21, 2188–2191 (1985) 14. J.S. Welij, Calculation of eddy current in terms of H on hexahedra. IEEE Trans. Magn. 21, 2239–2241 (1985) 15. P. Silvester, R. Ferrari, Finite Elements for Electrical Engineers (Cambridge, 1996) 16. M. Ainsworth, J.T. Oden, A Posteriori Error Estimation in Finite Element Analysis, (Wiley, 2000) 17. Ch. Großmann, H.G. Roos, Numerik Partieller Differentialgleichungen (Teubner, 1994) 18. G. Strang, G. Fix, An Analysis of the Finite Element Method (Cambridge Press, Wellesley, 2008) 19. I. Babuska, B. Szabo, I. Katz, The p-version of the finite element method. SIAM J. Numer. Anal. 18(3), 515–545 (1981) 20. B. Szabó, I. Babuška, Finite Element Analysis, 1st edn. (Wiley, 1991) 21. O.C. Zienkiewicz, J.P. De, S.R. Gago, D.W. Kelly, The hierarchical concept in finite element analysis. Comput. Struct. 16, 53–65 (1983) 22. G. Szegö, Orthogonal polynomials. American Mathematical Society Colloquium Publications, no. Bd. 23. American Mathematical Society (1959) 23. S. Zaglmayr, High order finite element methods for electromagnetic field computation. Ph.D. thesis, Johannes Kepler University, Linz (2006) 24. A. Düster, Lecture notes: high order FEM. 132 (2005) 25. G. Karniadakis, S.J. Sherwin, Spectral/HP Element Methods for CFD (Oxford University Press, 1999) 26. D.A. Kopriva, Implementing Spectral Methods for Partial Differential Equations (Springer, Dordrecht, 2009) 27. A.T. Patera, A spectral element method for fluid dynamics—laminar flow in a channel expansion. J. Comput. Phys. 54, 468–488 (1984) 28. G.C. Cohen, Higher-Order Numerical Methods for Transient Wave Equations (Springer, New York, 2002) 29. William H. Press, Saul A. Teukolsky, William T. Vetterling, Brian P. Flannery, Numerical Recipes 3rd edition: The Art of Scientific Computing, 3rd edn. (Cambridge University Press, New York, 2007) 30. R. Courant, K. Friedrichs, H. Lewy, On the partial difference equations of mathematical physics. IBM J. Res. Dev. 11, 215–234 (1967) 31. C. Bernardi, Y. Maday, F. Rapetti, Basics and some applications of the mortar element method. GAMM-Mitt. 28(2), 97–123 (2005) 32. B. Flemisch, Non-matching triangulations of curvilinear interfaces applied to electromechanics and elasto-acoustics, Ph.D. thesis, University of Stuttgart, (2006) 33. J. Danek, H. Kutakova, The mortar finite element method in 2D: implementation in MATLAB, in 16th Annual Conference Proceedings of Technical Computing (Prague, Czech Republic, 2008) 34. B.I. Wohlmuth, A mortar finite element method using dual spaces for the Lagrange multiplier. SIAM J. Numer. Anal. 38(3), 989–1012 (2000). MRMR1781212 (2001h:65132) 35. S. Triebenbacher, Nonmatching grids for the numerical simulation of problems from aeroacoustics and vibroacoustics. Ph.D. thesis, Alpen-Adria-Universität Klagenfurt, Austria (2012) 36. S. Triebenbacher, M. Kaltenbacher, B. Flemisch, B. Wohlmuth, Applications of the mortar finite element method in vibroacoustics and flow induced noise computations. Acta Acust. United Acust. 96, 536–553 (2010) 37. Ivan E. Sutherland, Gary W. Hodgman, Reentrant polygon clipping. Commun. ACM 17, 32–42 (1974) 38. J. Grabinger, Mechanical-acoustic coupling on non-matching finite element grids. Master’s thesis, University Erlangen-Nuremberg, June 2007 39. J. Nitsche, Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Tech. Report 36, Abh. Math. Univ. Hamburg, (1971)

References

91

40. A. Hansbo, P. Hansbo, M.G. Larson, A finite element method on composite grids based on nitsche’s method. ESAIM Math. Model. Numer. Anal. 37(3), 495–514 (2003) 41. A. Fritz, S. Hüeber, B. Wohlmuth, A comparison of mortar and Nitsche techniques for linear elasticity. CALCOLO (2004)

Chapter 3

Mechanical Field

3.1 Navier’s Equation Let us consider a solid body with prescribed volume force fV and support at equilibrium, which means that the sum of all forces as well as the sum of all moments are zero. In the first step we cut out a small part of this solid body so that the faces of this small body are parallel to the Cartesian coordinate system (see Fig. 3.1a). Now, we have to apply forces at the cutting planes to still guarantee equilibrium (see Fig. 3.1b). These forces correspond to inner forces acting within the solid body. Since the applied forces are distributed all over the cutting planes, we describe them by mechanical stresses (force per unit area). The stress state of each face is defined by its stress vectors σ x , σ y and σ z , where the index denotes the associated face. Thus, we can describe the stress state as follows σ x = σxx ex + σxy ey + σxz ez σ y = σyx ex + σyy ey + σyz ez σ z = σzx ex + σzy ey + σzz ez

(3.1) (3.2) (3.3)

with (σxx , σyy , σzz ) the normal stresses and (σxy , σxz , σyx , σyz , σzx , σzy ) the shear stresses and ei defining the unit vector in direction i. Now in a second case, we consider a body with an oblique cutting face and investigate on the mechanical stress σ α on the face dΓα (see Fig. 3.2). We define the normal vector nα by nα = n x e x + n y e y + n z e z with the property |nα | = of all forces results in



n 2x + n 2y + n 2z = 1. According to Fig. 3.2, the equilibrium

dΓα σ α − dΓx σ x − dΓ y σ y − dΓz σ z = 0. © Springer-Verlag Berlin Heidelberg 2015 M. Kaltenbacher, Numerical Simulation of Mechatronic Sensors and Actuators, DOI 10.1007/978-3-642-40170-1_3

(3.4) 93

94

3 Mechanical Field σz

(b)

(a)

σzz

σzx σy σzy σyx

σyz σxy

σyy

σxx σxz z

σx y x

Fig. 3.1 Mechanical stress. a Cutting out a small body. b Mechanical stress σ on a surface of the small body Fig. 3.2 Mechanical stresses on the surface of a small body with an oblique face

σα

σx

z y x

σy σz

In (3.4) dΓα denotes the infinite small surface of the oblique cutting face, and the surfaces of the three other faces can be expressed as dΓx = dΓα n x

dΓ y = dΓα n y

dΓz = dΓα n z .

Therewith, we can rewrite (3.4) by σα = n x σ x + n y σ y + n z σ z , which results by using (3.1)–(3.3) in     σ α = σxx n x + σ yx n y + σzx n z ex + σxy n x + σyy n y + σzy n z e y   + σx z n x + σ yz n y + σzz n z ez .

To obtain a more compact expression, we introduce the mechanical stress tensor [σ], called the Cauchy stress tensor, as

3.1 Navier’s Equation

95

⎤ σxx σxy σxz [σ] = ⎣ σyx σyy σyz ⎦ . σzx σzy σzz ⎡

(3.5)

The stress tensor now allows us to express any physical stress vector σ α acting on a face defined by its normal vector nα as follows σ α = [σ]T nα . After having defined the stress state, we can now investigate in the equation for translation

fV dΩ + [σ]T dΓ = 0 (3.6) Ω

Γ

and the equation for rotation

Ω

(r × fV ) dΩ +



Γ

(r × [σ]T n) dΓ = 0,

(3.7)

which have to be fulfilled, if the body is at rest. First, let us consider the equation for translation in x-direction given by

Ω

 fV · ex dΩ + [σ]T ex · dΓ = 0.

(3.8)

Γ

Rewriting [σ]T ex by σ x and applying the divergence theorem to the second term of (3.8), we obtain f x dΩ + ∇ · σ x dΩ = 0, (3.9) Ω

Ω

with fV = ( f x f y f z )T . Since this result has to hold for each volume Ω, we may write (3.9) in the following form f x + ∇ · σ x = 0.

(3.10)

Similar expressions are obtained for the y- and z-directions fy + ∇ · σy = 0 f z + ∇ · σ z = 0.

(3.11)

The equilibrium condition for the rotation around the x-axis with the position vector r = (x y z)T reads as

96

3 Mechanical Field



Ω

(y f z − z f y ) dΩ +



Γ

(yσ z − zσ y ) · n dΓ = 0.

(3.12)

Applying the divergence theorem and omitting the volume integral results in the following differential form y f z − z f y + ∇ · (yσ z ) − ∇ · (zσ y ) = 0.

(3.13)

Using the vector identity ∇ · (ξu) = ξ∇ · u + u · ∇ξ

(3.14)

y f z − z f y + y∇ · σ z + σ z · ∇ y − z∇ · σ y − σ y · ∇z = 0.

(3.15)

we obtain

With the result of (3.11) we can simplify (3.15) to σzy = σyz .

(3.16)

The relations for the remaining two axes of rotation yield σzx = σxz and σyx = σxy .

(3.17)

Therefore, the equilibrium equation for a body at rest can be expressed as follows1 fV + ∇ · [σ] = 0,

(3.18)

where [σ] denotes the Cauchy stress tensor. Since the stress tensor [σ] is symmetric, it is convenient to write it as a vector of six components using Voigt notation [1] ⎞ ⎛ ⎞ ⎛ ⎞ σ11 σ1 σxx ⎤ ⎡ ⎡ ⎤ ⎜ σyy ⎟ ⎜ σ22 ⎟ ⎜ σ2 ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ σxx σxy σxz σ11 σ12 σ13 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎣ σyx σyy σyz ⎦ = ⎣ σ21 σ22 σ23 ⎦ ; ⎜ σzz ⎟ = ⎜ σ33 ⎟ = ⎜ σ3 ⎟ . ⎜ σ yz ⎟ ⎜ σ23 ⎟ ⎜ σ4 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ σzx σzy σzz σ31 σ32 σ33 ⎝ σx z ⎠ ⎝ σ13 ⎠ ⎝ σ5 ⎠ σxy σ12 σ6 (3.19) By introducing the differential operator B ⎛

⎛

∂ ∂x

⎜ B=⎝ 0 0 1

0 ∂ ∂y

0

0 0 ∂ ∂z

For the divergence of a tensor see Appendix C.

0 ∂ ∂z ∂ ∂y

∂ ∂z

0 ∂ ∂x

∂ ∂y ∂ ∂x

0

⎞T

⎟ ⎠ ,

(3.20)

3.1 Navier’s Equation

97

(3.18) takes on the following form B T σ + fV = 0.

(3.21)

In the dynamic case the sum of all forces is equal to the inertia force and we arrive at Navier’s equations describing the dynamical behavior of mechanical systems fV + B T σ = ρa,

(3.22)

with ρ denoting the density of the medium and a the acceleration of the body.

3.2 Deformation and Displacement Gradient Figure 3.3 displays the relation between an arbitrary point P0 in Ω0 , the initial configuration of a body, and the corresponding point P in the deformed configuration defined by Ω. Furthermore, X defines the position of any material point in Ω0 (often called Lagrangian coordinates), and x the position of the corresponding point in Ω (often called Eulerian coordinates). The motion of the body shall be described by the unique map Φ x = Φ(X, t).

(3.23)

To locally compute the deformation, we introduce the deformation gradient [F d ], which maps the differential line element dX in Ω0 to the corresponding differential line element dx in Ω

Fig. 3.3 Body with volume Ω0 in the initial configuration and with volume Ω in the deformed configuration

Initial configuration P0 u(X, t)

Ω0

Deformed configuration P

X

z x(X , t)

y x

Ω

98

3 Mechanical Field

dx = [F d ] dX ∂x = ∇X Φ [F d ] = ∂X ⎡ ∂x ∂x ∂x ⎤ ∂X

⎢ = ⎣ ∂∂Xy ∂z ∂X

∂Y ∂y ∂Y ∂z ∂Y

∂Z ∂y ∂Z ∂z ∂Z

⎥ ⎦.

(3.24)

Since the map Φ is bijective, the Jacobi determinant |J | of [F d ] is different from zero, and in order to exclude intersections, it has to be greater than zero. By introducing the displacement vector u according to u(X, t) = x − X = Φ(X, t) − X,

(3.25)

we obtain F d = ∇X (X + u) = I + ∇X u = I + [H d ] ⎡ ∂u ∂u x

∂X

x

∂Y ∂u y ∂Y ∂u z ∂u z ∂ X ∂Y

⎢ ∂u y ∇X u = ⎢ ⎣ ∂X

∂u x ∂Z ∂u y ∂Z ∂u z ∂Z

(3.26) ⎤

⎥ ⎥, ⎦

(3.27)

with [H d ] the displacement gradient. The knowledge of F d allows the definition of transformations for differential quantities. In particular, the transformation between the differential surface element dΓ 0 in Ω0 to dΓ in Ω is given by [2] dΓ = n dΓ = |J |[F d ]−T n0 dΓ0 = |J |[F d ]−T dΓ 0 ,

(3.28)

and for a differential volume element by dΩ = |J | dΩ0 .

(3.29)

3.3 Mechanical Strain In order to first provide a basic physical understanding of the mechanical strain, we will derive the relation between mechanical displacement and strain in the linear case for the configuration shown in Fig. 3.4. Due to a given load case, a general point P0 (x, y) of the elastic body at the initial state will undergo a deformation according to the displacement components u x (x, y) and u y (x, y). We assume that the infinite

3.3 Mechanical Strain

99

ux (x, y + ∆y)

y

uy (x, y + ∆y)

ux (x, y)

ux (x + ∆x, y)

y

x

uy (x + ∆x, y)

α uy (x, y)

y + ∆y

β

x

x + ∆x

Fig. 3.4 Initial state and deformed state for a infinite small rectangle

small rectangle with side-length ∆x and ∆y will deform in a parallelogram with small angles α and β. Therewith, the side-length in x-direction computes for the deformed body as u x (x + ∆x, y) − u x (x, y) ≈ u x (x + ∆x, y) − u x (x, y) = ∆u x . cos α

(3.30)

Now, the elongation in x-direction computes for the limit ∆x → 0 u x (x + ∆x, y) − u x (x, y) . ∆x→0 ∆x lim

Expanding the term u x (x + ∆x, y) in a Taylor series u x (x + ∆x, y) = u x (x, y) +

∂u x ∆x + . . . higher order terms ∂x

and neglecting the higher order terms, results in sxx =

∂u x . ∂x

(3.31)

In (3.31) sxx is the unit elongation of the elastic body in x-direction and we call it the normal strain in x-direction. The same derivation can be applied to the unit elongation in y-direction, which will define the normal strain in y-direction syy =

∂u y . ∂y

Now, let us compute the shearing of the body, which means to obtain a relation between α, β and the displacements u x , u y . According to Fig. 3.4, we obtain

100

3 Mechanical Field

tan α =

u y (x+∆x,y)−u y (x,y) ∆x u x (x+∆x,y)−u x (x,y) ∆x

u y (x + ∆x, y) − u y (x, y) = ∆x + u x (x + ∆x, y) − u x (x, y) 1+

.

(3.32)

Expanding the terms u x (x + ∆x, y) and u y (x + ∆x, y) in a Taylor series up to the linear term, substituting these expressions into (3.32) and performing the limit ∆x → 0, results in tan α =

∂u y ∂x x 1 + ∂u ∂x

(3.33)

.

Assuming that the terms ∂u y /∂x and ∂u x /∂x are small compared to 1, the angle α will also be small, and we obtain the following approximation of (3.33) α=

∂u y . ∂x

Applying the same steps for the computation of β results in β=

∂u x . ∂y

The total shearing of our elastic body computes as the sum of α and β α+β =

∂u y ∂u x + = 2sxy , ∂x ∂y

and we call sxy the shear strain. For deriving the general relation between the mechanical strain and displacement, we consider the case as displayed in Fig. 3.5. The displacement u maps the initial configuration into the deformed one. The deformation state of a body is defined by considering the change of a line element between two neighboring points (P0 (X, Y, Z ), Q 0 (X, Y, Z )) in the initial configuration and (P(x, y, z), Q(x, y, z))

Fig. 3.5 Strain measurement

Initial configuration

dl0 P0

Ω0

Q0

Deformed configuration

Q z

P y x

dl

Ω

3.3 Mechanical Strain

101

in the deformed configuration (see Fig. 3.5). Since the metric of a space—the measure of the length and angle of the deformation—is defined by the square of the line element, we obtain for the differential element dl0 in the initial configuration dl02 = dX 2 + dY 2 + dZ 2 = dX T dX

(3.34)

and for dl in the deformed configuration dl 2 = dx 2 + dy 2 + dz 2 = dx T dx.

(3.35)

With the help of (3.24), we can express the difference as follows dl 2 − dl02 = dx T dx − dX T dX

= dX T [F d ]T [F d ] dX − dX T dX  = dX T [F d ]T [F d ] − I dX = dX T 2[V ] dX,

(3.36)

where [V ] denotes the Green–Lagrangian strain tensor. Thus, V measures the difference between the square length of an infinitesimal segment in the deformed configuration and in the initial configuration. Since we can express [F d ] by (I + ∇X u) the Green–Lagrangian strain tensor [V ] takes the form 1 (I + ∇X u)T (I + ∇X u) − I 2 1 1 ∇X u + (∇X u)T + (∇X u)T ∇X u . = 2 2

[V ] =

(3.37)

As can be easily seen, the first part in the above equation defines the linear strain, whereas the addition of the second part allows the description of large deflections. The explicit form of the Green–Lagrangian strain tensor written in vector notation is given as follows ⎡   2  2  ⎤ 2 ∂u y ∂u x 1 z + + ∂u ∂u x ⎥ ⎢2 ∂X ∂X ∂X ∂X ⎥ ⎢ ⎢ ⎥ ⎢  2  2  2  ⎥ ∂u y ∂u y ⎢ ⎥ ⎢ ⎥ ∂u z ∂u 1 x ∂Y + ∂Y + ∂Y ⎥ ⎢ ⎥ ⎢2 ∂u z ⎥ ⎢ ⎥ ⎢  ∂Y  ⎢ ⎢ ⎥ ∂Z ⎥ ⎢ 1  ∂u 2  ∂u y 2  ∂u 2 ⎥ ⎥ ⎢ ∂u y z x ∂u z ⎥ + ⎢ V =⎢ ⎥, + ∂Z + ∂Z ∂Z ⎥ ⎢ ∂ Z + ∂Y ⎥ ⎢ 2 ⎥ ⎢ ⎥ ⎢ ∂u y ∂u y ⎥ ⎢ ∂u z ∂u z ∂u z ∂u x ⎥ ⎢ ∂u x ∂u x ⎥ ⎢ ∂ X + ∂ Z ⎥ ⎢ ∂Y ∂ Z + ∂Y ∂ Z + ∂Y ∂ Z ⎥ ⎣ ⎦ ⎢ ∂u y ∂u y ∂u z ∂u z ∂u ∂u ⎢ ⎥ x x ∂u ∂u x y + ∂Z ∂X + ∂Z ∂X ⎣ ⎦ ∂ Z ∂ X + ∂Y ∂X ∂u y ∂u y ∂u z ∂u z ∂u x ∂u x    + + ∂ X ∂Y ∂ X ∂Y ∂ X ∂Y ⎡

⎤

S

(3.38)

102

3 Mechanical Field

with [S] the tensor of the linear strains in vector notation. The linear strain tensor [S] in vector form according to Voigt notation reads as ⎤ ⎤ ⎡ s11 s12 s13 sxx sxy sxz ⎣ syx syy syz ⎦ = ⎣ s21 s22 s23 ⎦ ; s31 s32 s33 szx szy szz ⎡

⎞ ⎛ ⎞ ⎞ ⎛ s1 s11 sxx ⎜ syy ⎟ ⎜ s22 ⎟ ⎜ s2 ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ szz ⎟ ⎜ s33 ⎟ ⎜ s3 ⎟ ⎟ ⎜ ⎟ ⎜ ⎟=⎜ ⎜ 2s yz ⎟ ⎜ 2s23 ⎟ = ⎜ s4 ⎟ . ⎟ ⎜ ⎟ ⎟ ⎜ ⎜ ⎝ 2sx z ⎠ ⎝ 2s13 ⎠ ⎝ s5 ⎠ s6 2s12 2sxy ⎛

(3.39)

The factor of two on the shear strains results from the requirement that the computed energy in Voigt notation and index notation must be the same. It should be noted that the Green–Lagrangian strain tensor [V ] defines the strains in the initial configuration, whereas the Cauchy stress tensor [σ] defines the stress for the deformed configuration. In the nonlinear case, we will introduce the second Piola–Kirchhoff stress tensor, which is also defined for the initial configuration, and therefore fits to the Green–Lagrangian strain tensor. However, in the linear case we do not have to distinguish between the initial and the deformed configuration and can relate the Cauchy stress tensor [σ] to the linear strain tensor [S].

3.4 Constitutive Equations The simplest and most frequently used relation between the stress and strain is the linear law of elasticity known as Hooke’s law [3, 4]. Assuming an isotropic material, it can be expressed by knowledge of the shear modulus G and the Poisson ratio νp only, as follows (sij denotes the components of the linear strain tensor [S])   νp σxx = 2G sxx + (sxx + syy + szz ) ; 1 − 2νp   νp (sxx + syy + szz ) ; σyy = 2G syy + 1 − 2νp   νp σzz = 2G szz + (sxx + syy + szz ) ; 1 − 2νp

σxy = 2Gsxy σyz = 2Gsyz

(3.40)

σzx = 2Gszx .

The often-used elasticity modulus E m is computed via G and νp by G=

Em . 2(1 + νp )

(3.41)

3.4 Constitutive Equations

103

By introducing the so-called Lam´e parameters λL and µL νp E m (1 + νp )(1 − 2νp ) Em µL = , 2(1 + νp ) λL =

(3.42) (3.43)

we obtain the following explicit form of the stress–strain relation for isotropic materials ⎞ ⎛ ⎤ ⎛ ⎞ ⎡ sxx λL + 2µL σxx λL λL 0 0 0 ⎜ ⎟ ⎜ σyy ⎟ ⎢ λL λL + 2µL λL 0 0 0 ⎥ ⎜ ⎥ ⎜ syy ⎟ ⎟ ⎢ ⎜ ⎟ ⎥ ⎜ σzz ⎟ ⎢ s λ λ λ + 2µ 0 0 0 L L L L ⎥ ⎜ zz ⎟ . (3.44) ⎜ ⎟ ⎢ ⎜ ⎟ ⎥ ⎜ σ yz ⎟ = ⎢ 2s 0 0 0 µ 0 0 L ⎥ ⎜ yz ⎟ ⎜ ⎟ ⎢ ⎝ σx z ⎠ ⎣ 0 0 0 0 µL 0 ⎦ ⎝ 2sx z ⎠ 2sxy 0 0 0 0 0 µL σxy Therefore, Navier’s equation can be expressed by µL ∇ · ∇u + (λL + µL )∇ ∇ · u + fV = ρ

∂2 u . ∂t 2

(3.45)

For the general anisotropic case, we introduce the 4th order tensor of elasticity moduli [c] in the form (for the contraction operator: see Appendix C.) [σ] = [c] : [S] σij = cijkl skl ,

(3.46)

which has the following properties cijkl = cijlk cijkl = cjikl

(3.47)

cijkl = cklij .

Due to symmetry, we can write the stress as well as the strain tensors as vectors. Thus, we can combine the four indices of the cijkl to two indices cIK as follows ij/kl I /K 11 1 22 2 33 3 23 4 13 5 12 6

104

3 Mechanical Field

Expressing the linear strain vector S by Bu and combining (3.22) with (3.46), results in B T [c]Bu + fV = ρ

∂2 u . ∂t 2

(3.48)

3.4.1 Plane Strain State The plane strain case as depicted in Fig. 3.6 is a simplification of the general case and can be used when the third dimension (assumed as the z-direction) is very large and within each cross section (in our case the x y-plane) the same boundary conditions as well as forces act on the body. Therefore, the dependence of the mechanical displacements u x and u y on the z-coordinate can be neglected szy = szx = 0 szz = 0.

(3.49) (3.50)

The stress–strain relation for the isotropic case simplifies to ⎛

⎞ ⎤ ⎛ ⎞ ⎡ λL + 2µL sxx σxx λL 0 ⎝ σyy ⎠ = ⎣ λL λL + 2µL 0 ⎦ ⎝ syy ⎠ . 2sxy 0 0 µL σxy

(3.51)

F

F

σxy = σyx ey

σxx

ex ez Fig. 3.6 Plane strain case

σyy

3.4 Constitutive Equations

105

3.4.2 Plane Stress State A demonstrative example for the plane stress case is shown in Fig. 3.7, where a thin sheet is loaded by mechanical forces at the boundary and the forces act within the defined plane. By cutting an infinitely small piece out of the whole thin sheet, we can assume to have stresses on its surface as displayed in Fig. 3.7. Therefore, the following relations are fulfilled σzx = σzy = 0 σzz = 0.

(3.52) (3.53)

By using (3.52) and (3.53), we immediately obtain for the isotropic case (see (3.44)) szx = szy = 0

(3.54)

λL szz = − (sxx + syy ), λL + 2µL

(3.55)

which leads to the following simplifications for the plane stress case assuming an isotropic material ⎞ ⎡ 2λL µL + 2µ L σxx λL +2µL 2λL µL ⎝ σyy ⎠ = ⎢ ⎣ λL +2µL σxy 0 ⎛

2λL µL λL +2µL 2λL µL λL +2µL + 2µL

0

⎤ ⎛ ⎞ sxx ⎥ ⎝ syy ⎠ . 0 ⎦ 2sxy µ 0

(3.56)

L

F3 ez

ey

σyx = σxy

F2 F1 Fig. 3.7 Plane stress case

σyy σxx

ex

106

3 Mechanical Field

3.4.3 Axisymmetric Stress–Strain Relations In a cylindrical-coordinate system, the displacement components read as u r displacement in radial direction (r -direction) u z displacement in axial direction (z-direction) u θ displacement in circumferential direction (θ-direction) Since in the axisymmetric case the mechanical displacements do not depend on the θ-coordinate, the following equations must hold uθ = 0

(3.57)

sr θ = szθ = 0.

(3.58)

Thus, the stress–strain relation for the isotropic case is given by ⎛

⎞ ⎡ σrr λL + 2µL λL ⎜ σzz ⎟ ⎢ λ λ + 2µL L L ⎜ ⎟ ⎢ ⎝ σrz ⎠ = ⎣ 0 0 λL λL σθθ

⎤ 0 λL ⎥ 0 λL ⎥ ⎦ µL 0 0 λL + 2µL

⎛

⎞ srr ⎜ szz ⎟ ⎜ ⎟ ⎝ 2srz ⎠ . sθθ

(3.59)

3.5 Waves in Solid Bodies In this section we want to investigate different wave types that can propagate within solids. In the general case, the mechanical displacement u can be decomposed into an irrotational part (curl u = 0) and a solenoidal part (div u = 0) (Helmholtz decomposition) u = grad ϕ + curl ψ.

(3.60)

By using this decomposition for (3.45) and setting fV to zero, we arrive at µL ∇ · ∇grad ϕ + (λL + µL )∇ ∇ · grad ϕ + µL ∇ · ∇curl ψ ∂ 2 curl ψ ∂ 2 grad ϕ + (λL + µL )∇ ∇ · curl ψ = ρ + ρ ∂t 2 ∂t 2

(3.61)



 ∂2ϕ grad ρ 2 − (λL + 2µL )∇ · ∇ϕ ∂t  2  ∂ ψ +curl ρ 2 − µL ∇ · ∇ψ = 0. ∂t

(3.62)

3.5 Waves in Solid Bodies

107

This relation is fulfilled if ϕ and ψ solve the two following equations ∂2ϕ λL + 2µL ∇ · ∇ϕ = 2 ∂t ρ ∂2ψ µL ∇ · ∇ψ. = 2 ∂t ρ

(3.63) (3.64)

First, we will choose for (3.63) an ansatz expressed by ϕ = f (ξ) = f (k · r − ωt),

(3.65)

which defines a wave propagation in direction of k with velocity c. By using this ansatz for (3.63), we obtain with    ∂ ∂ϕ ∂ϕ ∂ξ ∂2ϕ = −ω = ω2 2 ∂ξ ∂t ∂t ∂ξ ∂ξ     2 2 ∂ ∂ϕ ∂ξ ∂ ϕ ∂ϕ ∂ 2 ∂ ϕ k = k = = i i ∂xi ∂ξ ∂xi ∂t ∂ξ ∂ξ 2 ∂xi2

∂2ϕ ∂ = ∂t 2 ∂t



(3.66) (3.67)

the following result ω2

∂2ϕ = ∂ξ 2



=



λL + 2µL ρ

  3

λL + 2µL ρ



i=1

ki2

∂2ϕ ∂ξ 2

3 ∂2ϕ  2 ki . ∂ξ 2 i=1   

(3.68)

k2

Now, since the relation c2 = ω 2 /k 2 holds, (3.63) is fulfilled, if c2 takes on the value (λL + 2µL )/ρ. The mechanical displacement for the scalar component ϕ computes ∂ϕ ∂ϕ ∂ϕ ex + ey + ez ∂x ∂y ∂z ∂ϕ ∂ξ ∂ϕ ∂ξ ∂ϕ ∂ξ ex + ey + ez = ∂ξ ∂x ∂ξ ∂ y ∂ξ ∂z ∂ϕ . =k ∂ξ

u = grad ϕ =

(3.69)

Thus, (3.69) clearly shows that the mechanical displacements are in the direction of the wave propagation, which defines a longitudinal wave with velocity cL

108

3 Mechanical Field

cL =



λL + 2µL = ρ



(1 − νp )E m . (1 + νp )(1 − 2νp )ρ

(3.70)

By choosing the ansatz F = F(ξ) = F(k · r − ωt)

(3.71)

we can solve (3.64). In this case the mechanical displacement is computed by u = curl ψ ∂ψ , = k× ∂ξ

(3.72)

and we obtain a mechanical displacement that is perpendicular to the direction of propagation. The type of wave is called a shear wave, which propagates with velocity cT    µL Em G = = . (3.73) cT = ρ 2(1 + νp )ρ ρ The ratio of the two velocities cL = cT



λL + 2µL µL

(3.74)

leads to the following inequality cL >

√

2cT .

(3.75)

3.6 Material Properties The mechanical material properties are defined by the density ρ and the tensor of mechanical moduli [c]. In the general case, the entries of [c] depend on the mechanical stress σ. Figure 3.8 displays a typical stress–strain curve of a metallic material obtained by a tensile test. For the region defined by stresses up to σp , we find a strict proportionality between the stress and strain as used in linear elasticity computations. For stresses larger than σp , we can find a super-proportional increase of the strain till the stress reaches σy , the so-called yield stress. By a further increase of the applied force, the stress again strongly increases due to stiffening effects of the material until the sample breaks at σb . For a more detailed discussion, especially on material models (e.g., viscoelastic, viscoplastic, etc.) we refer to [1, 2].

3.6 Material Properties

109

σ σb σy σp

S Fig. 3.8 Stress–strain curve for a metallic material

Furthermore, heating up a solid body will also result in a mechanical deformation. The resulting thermal strain sijth can be modeled as follows sijth = αi (T − T0 ),

(3.76)

with αi the so-called thermal expansion coefficient in direction i and T0 the reference temperature. For a homogeneous and isotropic material, the value of α is the same for all directions, so that the shear strains are zero. Therefore, Hook’s law can be written as  (3.77) [σ] = [c] [S] − [Sth ]  α(T − T0 ) for i = j sijth = . (3.78) 0 for i = j In Table 3.1 the mechanical properties of some materials are summarized.

Table 3.1 Mechanical properties of some materials Material ρ (kg/m3 ) E m (N/m2 ) νp 3 (10 ) (1010 ) Aluminum Iron Copper PVC

2.7 7.7 8.9 1.1

7.20 21.6 12.5 0.30

0.34 0.29 0.35 0.48

α (1/T) (10−6 )

cL (m/s) (103 )

cT (m/s) (103 )

24 12 12 11

6.3 5.9 4.7 2.2

3.13 3.20 2.26 1.10

110

3 Mechanical Field

3.7 Numerical Computation 3.7.1 Linear Elasticity The strong formulation for linear elasticity problems reads as follows: Given: u0 u˙ 0 ρ, cij fV

:Ω :Ω :Ω :Ω

→ Rd → Rd →R → Rd .

Find: u(t) : Ω¯ × [0, T ] → Rd B T [c]Bu + fV = ρu¨ .

(3.79)

Boundary conditions T

u = ue on Γe × (0, T )

[σ] n = σ n on Γn × (0, T ). Initial conditions u(r, 0) = u0 , r ∈ Ω ˙ u(r, 0) = u˙ 0 , r ∈ Ω. For simplicity, we will set the boundary conditions to zero (ue = 0, σ n = 0). Multiplying (3.79) by an appropriate test function u′ and performing a partial integration will transform (3.79) to its variational formulation, which reads as follows: Find u ∈ H 10 such that

ρu′ · u¨ dΩ +

Ω



(Bu′ )T [c]Bu dΩ =

Ω



u′ · fV dΩ

(3.80)

Ω

for any u′ ∈ H 10 . Let us perform the spatial discretization with standard nodal finite elements, which approximate the continuous displacement u as follows ′

u ≈ uh =

nn nd   i=1 a=1

′

Na u ia ei =

nn  a=1

⎞ Na 0 0 N a ua ; N a = ⎝ 0 Na 0 ⎠ , 0 0 Na ⎛

(3.81)

with n d the space dimension and n ′n the number of finite element nodes with no Dirichlet boundary condition. Applying the same approximation to the test function u′ , we have the following semi-discrete Galerkin formulation for linear elasticity

3.7 Numerical Computation

111

⎛ n ′n n ′n   ⎝ ρN aT N b dΩ u¨ b + BaT [c]Bb dΩ ub a=1 b=1

Ω

Ω

−



Ω

with

⎛ ∂N

a

∂x

⎜ Ba = ⎜ ⎝ 0 0

⎞

N a fV (r a ) dΩ ⎠ = 0,

0

0

0

∂ Na ∂y

0

0

∂ Na ∂z

∂ Na ∂z ∂ Na ∂y

∂ Na ∂ Na ∂z ∂ y Na 0 ∂∂x ∂ Na 0 ∂x

⎞T

⎟ ⎟ . ⎠

(3.82)

(3.83)

In matrix form, we may write (3.82) as M u u¨ + K u u = f ,

(3.84)

with Mu = Ku = f =

ne 

meu ; meu = [m pq ]; m pq =

ne 

keu ; keu = [k pq ]; k pq =



ne 

f e ; f e = [ f p ]; f p =

e=1

e=1

e=1

Ωe





ρN Tp N q dΩ

(3.85)

Ωe

B Tp [c]Bq dΩ

(3.86)

N p fV (r p ) dΩ.

(3.87)

Ωe

In (3.84) u, ¨ u denote algebraic vectors containing all the three unknowns of acceleration and displacement at all nodes. The time discretization is performed by a standard Newmark method as explained in Sect. 2.5.2. Thus, we arrive at the following time stepping scheme for an effective mass formulation: • Perform predictor step: ∆t 2 (1 − 2βH ) u¨ n 2 u˜˙ = u˙ n + (1 − γH )∆t u¨ n .

u˜ = u n + ∆t u˙ n +

(3.88) (3.89)

• Solve algebraic system of equations: M ∗u u¨ n+1 = f n+1 − K u u˜ − C u u˜˙ M ∗u

(3.90) 2

= M u + γH ∆t C u + βH ∆t K u .

(3.91)

112

3 Mechanical Field

In (3.90) we have introduced a damping matrix C u according to a standard Rayleigh model (see Sect. 3.7.2). • Perform corrector step: u n+1 = u˜ + βH ∆t 2 u¨ n+1 u˙ n+1 = u˙˜ + γH ∆t u¨ n+1 .

(3.92) (3.93)

3.7.2 Damping Model In general, vibrating mechanical systems will always show a damped behavior. The reason for the damping is mainly to friction within the material and its mathematical model is usually an added velocity proportional damping term. Therefore, within the FE method a constant damping matrix C u is introduced and the term C u u˙ is added to the semi-discrete Galerkin formulation given in (3.84). In many FE formulations, the Rayleigh damping model is applied, so that C u is computed via a combination of the mass matrix M u and the linear stiffness matrix K u C u = αM M u + αK K u .

(3.94)

In (3.94) αM denotes the mass proportional and αK the stiffness proportional damping coefficients. As shown in [5], a mode superposition analysis including damping according to (3.94) leads to the following relation αM + αK ωi2 = 2ωi ξi ,

(3.95)

with ωi the ith eigenfrequency (in rad/s) and ξi the modal damping for the ith eigenmode. The modal damping ξi corresponds to the loss factor tan δi for ωi , so that we obtain αM + αK ωi2 tan δi = 2ξi = . (3.96) ωi In addition, it can be shown (see [5]) that (3.84) (with C u u) ˙ can be decomposed in a system of non-coupled single degree of freedom differential equations with unit mass as well as stiffness x¨i (t) + 2ξi ωi x˙i (t) + ωi2 xi (t) = f i (t),

(3.97)

with generalized displacements xi and forces f i . The technically relevant solution of (3.97) will be an exponentially (with amplitude ξi ) damped sine curve as displayed in Fig. 3.9. The relation between the logarithmic decrement Di and the modal damping factor ξi computes as

3.7 Numerical Computation

113

Fig. 3.9 Damped sine curve

xn+1 xn

Di = ln ξi =



2πξi = 1 − ξi2

xn xn+1

Di2

Di2 + 4π 2

.

(3.98)

(3.99)

Therefore, if we measure e.g., the decay of a mechanical wave excited with frequency ωi propagating within a solid body, we can compute the damping factor ξi and thus the loss factor tan δi . The computation of αM and αK for this ξi can then be performed using (3.96) as follows αM + αK (ωi + ∆ω)2 = 2(ωi + ∆ω) ξi 2

αM + αK (ωi − ∆ω) = 2(ωi − ∆ω) ξi .

(3.100) (3.101)

The value ∆ωi shall be kept small, so that we meet the prescribed ξi at ωi . However, if we have to model a wide frequency range by fixed αM and αK , which is, e.g., the case within a transient analysis, the actual ξ will differ from ξi according to (3.96). Let us suppose we perform a transient analysis of a thickness mode piezoelectric transducer with resonance frequency 1 MHz, and we set the damping coefficient ξ at resonance frequency to 0.005. Then, we will compute the Rayleigh damping coefficients in order to meet this damping at resonance frequency, which will exhibit αM = 3.1×104 and αK = 7.9×10−10 . For all other frequencies within the excitation signal, the damping ξ will be according to Fig. 3.10.

114

3 Mechanical Field

tan δ

Fig. 3.10 Loss factor tan δ as a function of frequency (tan δi = 0.01 at 1 MHz; computed αM = 3.1 × 104 and αK = 7.9 × 10−10 )

f (MHz)

3.7.3 Geometric Nonlinear Case We have derived the partial differential equation for the mechanical field by considering the equilibrium equations (translation as well as rotation) for an elastic body (for simplicity we just consider the static case) ∇ · [σ] + fV = 0 u = ue on Γe [σ]T n = σ n on Γn .

(3.102)

In (3.102) [σ] denotes the mechanical stress tensor, fV any volume force, and u the mechanical displacement. However, (3.102) is just applicable for linear mechanics, since we are mixing up quantities defined in the deformed configuration (e.g., Cauchy stress tensor [σ]) and quantities defined in the initial configuration (e.g., mechanical volume force fV ). Now, if we perform any computation, we always start at the initial configuration and aim at calculating the deformation of the body due to any prescribed boundary conditions and volume forces given for the initial configuration. Thus, we have to transform the Cauchy stress tensor [σ] from the deformed to the initial configuration using (3.28)

Γ

[σ] dΓ =



Γ0

|J |[σ][F d ]−T dΓ 0 =



[τ ] dΓ 0 .

(3.103)

Γ0

In (3.103), [τ ] stands for the 1st Piola–Kirchhoff tensor, which is a unsymmetric stress tensor. Therefore, we introduce the 2nd Piola–Kirchhoff tensor [T ], which represents no physical stresses, but is symmetric and computes as

3.7 Numerical Computation

115

[T ] = [F d ]−1 [τ ] = |J |[F d ]−1 [σ] [F d ]−T .

(3.104)

Thus, we can rewrite (3.102) as ∇X · ([F d ] [T ]) + fV = 0,

(3.105)

including only quantities defined on the initial configuration (no mixing of quantities defined in the deformed and initial state as in (3.102)). According to (3.105), we define the nonlinear operator F as a function of the mechanical displacement u F(u) = ∇X ([F d ] [T ]) + fV = 0.

(3.106)

Now, a Newton step can be written as (see Appendix E.2) uk+1 = uk + s with F ′ (uk )[s] = −F(uk ).

(3.107)

First, let us derive the weak formulation of (3.105). For an arbitrary test function u′ ∈ H 10 and assuming homogeneous Neumann boundary condition σ n = 0, we obtain u′ · (∇X · ([F d ] [T ]) + fV ) dΩ = 0 (3.108) Ω0



Ω0

with

 [T ] : [F d ]T ∇X u′ dΩ = u′ · fV dΩ,

(3.109)

Ω0

⎛ ∂u ′

x

∂X ⎜ ∂u ′ y ∇X u′ = ⎜ ⎝ ∂X ∂u ′z ∂X

∂u ′x ∂Y ∂u ′y ∂Y ∂u ′z ∂Y

∂u ′x ∂Z ∂u ′y ∂Z ∂u ′z ∂Z

⎞ ⎟ ⎟ ⎠

(3.110)

being in general a non-symmetric tensor, and : denotes the contraction operator (see Appendix C). Since [T ] is a symmetric tensor, the scalar product with [F d ] will always result in a symmetric tensor (see Appendix C). Therefore, we rewrite the first term in (3.108) as  1 [T ] : [F d ]T ∇X u′ dΩ = [T ] : [F d ]T ∇X u′ + ∇XT u′ [F d ] dΩ. 2 Ω0

Ω0

(3.111) According to (3.107), we need the Frechét derivative F ′ , which can be approximated by F(uk + s) − F(uk ) (see Appendix E.2). In its weak form, we have to compute

116

3 Mechanical Field



[T (uk + s)] :

Ω0

−



Ω0

1 [F d (uk + s)]T ∇X u′ + ∇XT u′ [F d (uk + s)] dΩ 2

[T (uk )] :

  1 [F d (uk )]T ∇X u′ + ∇XT u′ [F d (uk )] dΩ. 2

(3.112)

Now, let us remember the following relations [T ] = [c] : [V ] = [F d ] = I + ∇X u,

 1 [c] : [F d ]T [F d ] − I 2

(3.113) (3.114)

with [c] the tensor of the mechanical elasticity coefficients (in our case we assume constant entries). The evaluation of the term [T (uk + s)] leads to [T (uk + s)] = [c] : [V (uk + s)] 1 [F d (uk + s)]T [F d (uk + s)] − I [V (uk + s)] = 2  T  1  k k I + ∇X (u + s) I + ∇X (u + s) − I = 2   T  1 k k I + ∇X u + ∇X s I + ∇X u + ∇X s − I . (3.115) = 2 Neglecting all second-order terms, we arrive at   1 [F d (uk )]T [F d (uk )] − I [V (uk + s)] ≈ 2  T   1  I + ∇X uk ∇X s + ∇XT s I + ∇X uk + 2   1 k T T k k [F d (u )] ∇X s + ∇X s [F d (u )] . = [V (u )] + 2

(3.116)

In addition, the term [F d (uk + s)] can be expressed as follows [F d (uk + s)] = I + ∇X (uk + s) = [F d (uk )] + ∇X s.

(3.117)

With the help of these expressions for [V (uk + s)] and [F d (uk + s)], we can rewrite the first term in (3.112) as    1 k k T T k [T (u )] + [c] : [F d (u )] ∇X s + ∇X s [F d (u )] (3.118) 2 Ω0

1 : 2

    k k T T ′ T ′ [F d (u )] + ∇X s ∇X u + ∇X u [F d (u )] + ∇X s dΩ.

3.7 Numerical Computation

117

Setting all second-order terms to zero, we arrive at   1 k T ′ T ′ k k [F d (u )] ∇X u + ∇X u [F d (u )] dΩ [T (u )] : 2 Ω0

+ +



Ω0

  1 T ′ T ′ ∇X s ∇X u + ∇X u ∇X s dΩ [T (u )] : 2



  1 k T T k [c] : [F d (u )] ∇X s + ∇X s [F d (u )] 2

Ω0

k

(3.119)

  1 k T ′ T ′ k [F d (u )] ∇X u + ∇X u [F d (u )] dΩ. : 2 Substituting this result into (3.112), we get   1 [c] : [F d (uk )]T ∇X s + ∇XT s [F d (uk )] 2 Ω0

  1 [F d (uk )]T ∇X u′ + ∇XT u′ [F d (uk )] dΩ 2   1 ∇XT s ∇X u′ + ∇XT u′ ∇X s dΩ. + [T (uk )] : 2 :

(3.120)

Ω0

Therefore, the Newton step can be evaluated as follows   1 [c] : [F d (uk )]T ∇X s + ∇XT s [F d (uk )] 2 Ω0

  1 [F d (uk )]T ∇X u′ + ∇XT u′ [F d (uk )] dΩ : 2   1 ∇XT s ∇X u′ + ∇XT u′ ∇X s dΩ + [T (uk )] : 2 Ω0 = u′ · fV dΩ Ω0

−



Ω0

u

k+1

  1 k T ′ T ′ k [F d (u )] ∇X u + ∇X u [F d (u )] dΩ [T (u ] : 2

= uk + s.

k

(3.121)

118

3 Mechanical Field

Before we go over to the discretized version of (3.121), let us apply some helpful transformations. Using the relation [F d (uk )] = [I + ∇X (uk )], we obtain   1 k T T k [F d (u )] ∇X s + ∇X s [F d (u )] 2 1 1 = [I + ∇X (uk )]T ∇X s + ∇XT s [I + ∇X (uk )] 2 2 1  1  = ∇X s + ∇XT s + ∇XT uk ∇X s + ∇XT s∇X uk .   2   2 Bs

Bnl (uk ) s

The differential operator B has already been defined (see (3.20)) and B nl computes as follows ⎛ ⎞ k k k ∂u x ∂ ∂X ∂X ∂u kx ∂ ∂Y ∂Y ∂u kx ∂ ∂Z ∂Z ∂u kx ∂ ∂Z + ∂Z ∂u kx ∂ ∂Z + ∂Z ∂u kx ∂ ∂u kx ∂ X ∂Y + ∂Y

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ nl B =⎜ k ⎜ ∂u x ⎜ ∂Y ⎜ ⎜ ∂u kx ⎜ ⎝ ∂X

∂u y ∂X ∂u ky ∂Y ∂u ky ∂Z

k ∂ ∂u y ∂Y ∂Y k ∂ ∂u y ∂X ∂X k ∂ ∂u y ∂X ∂X

∂ ∂Z ∂ ∂Z ∂ ∂Y

∂u z ∂X ∂u kz ∂Y ∂u kz ∂Z

∂ ∂X ∂ ∂Y

∂ ∂Z ∂u k + ∂ Zy ∂u k + ∂ Zy ∂u k + ∂Yy

k ∂ ∂u z ∂Y ∂Y k ∂ ∂u z ∂X ∂X k ∂ ∂u z ∂X ∂X

∂ ∂Z ∂ ∂Z ∂ ∂Y

∂ ∂X ∂ ∂Y

∂ ∂Z ∂u k + ∂ Zz ∂u k + ∂ Zz ∂u k + ∂Yz

∂ ∂Y ∂ ∂X ∂ ∂X

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(3.122)

Using the relation of double contraction of two tensors (see Appendix C), we can rewrite this term as follows [T (uk )] :

1 T ˜ TX [T ] B ˜ ∇X s ∇X u′ + ∇XT u′ ∇X s = B 2 ˜ = B



∂ ∂ ∂ ∂ X ∂Y ∂ Z

T

.

(3.123)

(3.124)

Approximating s and u′ by nodal finite elements (see (3.81)) will result in the following discrete Galerkin formulation k K NL u (u )s = f a − f i .

(3.125)

In (3.125) K NL u , f a (external applied forces) as well as f i (internal forces due to stresses) are calculated as follows

3.7 Numerical Computation

K NL u = k pq

ne 

e=1

119

keu ; keu = [k pq ]

(3.126)

  T  nl u nl u B˜ Tp [T ]B˜q I dΩ Bp + Bp [c] Bq + Bq dΩ + =

fa = fp= fi = fp=

Ωe ne 

e=1



Ωe ne 

e=1

Ωe

f e; f e = [ f p]

(3.127)

N Tp fV (r p ) dΩ f e ; f ei = [ f p ]

(3.128)

 T B up + B nl [T (uk )] dΩ, p

Ωe

with B up as given in (3.83), I the identity matrix and T the second Piola–Kirchhoff k tensor in vector notation. The operator B nl p depends on u and computes as follows ⎛

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ nl Bp = ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

∂u kx ∂ N p ∂X ∂X ∂u kx ∂ N p ∂Y ∂Y ∂u kx ∂ N p ∂Z ∂Z ∂u kx ∂ N p ∂u kx ∂Y ∂ Z + ∂ Z ∂u kx ∂u kx ∂ N p ∂X ∂Z + ∂Z ∂u kx ∂ N p ∂u kx ∂ X ∂Y + ∂Y

∂Np ∂Y ∂Np ∂X ∂Np ∂X

∂u ky ∂ N p ∂X ∂X ∂u ky ∂ N p ∂Y ∂Y ∂u ky ∂ N p ∂Z ∂Z ∂u ky ∂u ky ∂ N p ∂Y ∂ Z + ∂ Z ∂u ky ∂ N p ∂u ky ∂X ∂Z + ∂Z ∂u ky ∂u ky ∂ N p ∂ X ∂Y + ∂Y

∂Np ∂Y ∂Np ∂X ∂Np ∂X

∂u kz ∂ N p ∂X ∂X ∂u kz ∂ N p ∂Y ∂Y ∂u kz ∂ N p ∂Z ∂Z ∂u kz ∂u kz ∂ N p ∂Y ∂ Z + ∂ Z ∂u kz ∂ N p ∂u kz ∂X ∂Z + ∂Z ∂u kz ∂ N p ∂u kz ∂ X ∂Y + ∂Y

⎞

∂Np ∂Y ∂Np ∂X ∂Np ∂X

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(3.129)

with ∂u kα /∂β computed within each integration point by n

e  ∂ Na k ∂u kα = u ∂β ∂β a α

a=1

α, β ∈ {X, Y, Z }.

(3.130)

The operator B˜ p is given by B˜ p =



∂Np ∂Np ∂Np ∂ X ∂Y ∂ Z



.

(3.131)

The iterative solution process is stopped if the incremental error as well as the residual error fulfill

120

3 Mechanical Field

||u k+1 − u k ||2 < δa ||u k+1 ||2

− f a ||2 || f k+1 i || f a ||2

< δr ,

(3.132)

with appropriate δa and δr . To guarantee that the Newton method converges to the correct solution, a line search algorithm should be applied to obtain an optimal relaxation parameter η in each Newton step (see Appendix E) u k+1 = u k + ηs.

(3.133)

3.7.4 Numerical Example In order to show the influence of the discretization as well as the order of the shape functions used for the approximation, we will compute the deflection of a beam due to a mechanical load. Figure 3.11 displays the setup and since the beam is fully supported at both sides, we exploit the symmetry. The beam has a length of L = 1000 µm and a thickness of d = 4 µm, and we perform a plane strain analysis. On the left side (x = 0), where the beam is fully supported, we set u x = u y = 0 and on the right side (symmetry) we set u x = 0. The nodal force has a value of 1 N and is supplied at x = L/2 and y = d. In the first step we will perform a linear analysis using once linear and once quadratic shape functions for the approximation. The discretization in the y-direction is done just by one finite element and the discretization in the x-direction is varied. As displayed in Table 3.2, the obtained tip displacement for the linear shape functions converges very slowly to the correct value when increasing the number of finite elements n e . The reason for this effect is that linear shape functions exhibit a very poor approximation of the true solution when applied to thin walled structures. This effect is called locking (see e.g., [5, 6]), and will be discussed in detail in the subsequent section. In the second step we perform a nonlinear analysis and compare the results to the linear one (see Fig. 3.12). For the discretization, quadrilateral elements with quadratic shape functions have been used with one finite element in the y-direction and 40 finite

Force

y

L/2 d

x Fixed

Fig. 3.11 Setup

Symmetry

3.7 Numerical Computation

121

Table 3.2 Tip displacement u y at x = L/2 and y = 0

ne

Type of basis function Linear (µm) Quadratic (µm)

20 40 80 160 320

–0.145 –0.947 –2.182 –3.238 –3.683

Fig. 3.12 Displacement along the beam obtained by linear and nonlinear analysis

–4.436 –4.457 –4.465 –4.468 –4.468

l

l

elements in the x-direction. Figure 3.12 clearly demonstrates that due to the large deflection, which is of the order of the thickness of the beam, a nonlinear analysis has to be performed for our setup.

3.8 Locking and Efficient Solution Approaches For some problems in computational mechanics, we recognize that the straight forward application of the displacement-based finite element method (as described in Sect. 3.7.1) will provide inaccurate results and show slow convergence when reducing the mesh size. Being specific, the finite element solution results in too small displacements (see example in the previous section), and we will refer to this effect as locking. In general, the three main locking phenomena are as follows: • Shear locking: The shear locking effect occurs by the application of the displacement-based finite element method to thin walled structures, where especially the relation t ≪ h holds (t denotes the thickness and h the mesh size).

122

3 Mechanical Field

• Volumetric (Poisson) locking: For nearly incompressible materials (Poisson ration νP → 0.5, which means for the Lam´e parameters λL ≫ µL ) volumetric locking will occur when applying the displacement-based finite element method. • Membrane locking: This type of locking just occurs in curved beam and shell elements. We will concentrate ourself on the shear locking and start with a mathematical investigation (see e.g., [7]). We consider the following weak formulation for the unknown u in an appropriate Hilbert space X a0 (u, v) +

1 (Bu, Bv) =< f, v > . t2

(3.134)

In (3.134) t denotes a parameter with 0 < t ≤ 1, a0 : X × X → R a continuous, symmetric and coercitive bilinear form, B : X → L 2 a continuous mapping and f a given function in the computational domain Ω. Now, one can show that for a given f , there exists a v0 such that < f, v0 >= 0 and Bv0 = 0. Then, the solution u is bounded from below by ||u|| ≥ C|| f || with a constant C independent of t. However, in the FE setting, we obtain [8] ||u h || ≤ t 2 C(h −2 )|| f ||. Therewith, for small values of the parameter t the standard finite element method will result in too small displacements, and we call the phenomena locking. Let us demonstrate the shear locking effect by considering a thin Timoshenko beam. There, the assumption is made that a plane section original perpendicular to the neutral axis remains plane, but according to the shear deformations rotates as displayed in Fig. 3.13. The shear angle γ and curvature κ can be expressed as follows γ=

Fig. 3.13 Timoshenko beam assumptions

∂w −β ∂x

κ=

∂β . ∂x

γ β

Neutral axis

w

Beam cross-section

3.8 Locking and Efficient Solution Approaches

123

Therewith, the total potential energy E pot computes as (see e.g., [5])

E pot

Em I = 2

L 

∂β ∂x

0

2

G Ak dx + 2

L 

∂w −β ∂x

0

2

dx +

L

pw dx. (3.135)

0

In (3.135), E m denotes the elasticity modulus, G the shear modulus, A = bt the cross sectional area with t the thickness and b the width of the beam, I the moment of inertia (I = bt 3 /12), k the shear correction factor and p the external load. Therein, the first term defines the bending energy, the second term the shear energy and the last one the potential of the load. By normalizing E pot and neglecting for the further considerations the load term, we obtain E˜ pot =

L  0

∂β ∂x

2

1 dx + 2 t

L 

∂w −β ∂x

0

2

dx.

(3.136)

By applying the variation of the potential energy E˜ pot , we arrive at the weak formulation for the Timoshenko beam, which will be similar to (3.134) with B(β, w) =

∂w − β. ∂x

From (3.136) we can conclude that for the case of a very thin beam (t 1 Ma >> 1

Incompressible flow Subsonic Sonic (trans sonic) Supersonic Hyper sonic

with the ALE reference system correction term v c , as defined in Eq. (4.3). Under the condition of low Mach number flow (see Table 4.1) Ma =

||v|| , c

(4.39)

incompressibility may be assumed, which simplifies (4.38) to ∂v + (v c · ∇)v + ∇ P − ν f Δv = 0 ∂t ∇ · v = 0,

(4.40)

with the kinematic viscosity ν f = μ f /ρ, kinematic pressure P = p/ρ and outer forces f Ω = 0. For the sake of completeness, the whole set of incompressible NSE, together with boundary conditions read as ∂v + (v c · ∇)v + ∇ P − ν f Δv = 0 ∂t ∇ ·v =0

v = g0 P = P0

[σ f ] · n = h

in Ω × R,

(4.41a)

in Ω × R,

(4.41b)

on ΓDV × R, on ΓDP × R, on ΓN × R.

(4.41c) (4.41d) (4.41e)

The boundary conditions given in (4.41) are divided up into Dirichlet boundary conditions for velocity and pressure, (4.41c) and (4.41d) respectively, and a Neumann condition (4.41e). Here, ∂Ω = Γ DV ∪ Γ D P ∪ Γ N and the boundaries are disjointed.

4.5 Characterization of Flows by Dimensionless Numbers Referring the fluid field variables to characteristic flow quantities yields a dimensionless form of (4.41). Similarity considerations can be made among different flows with characteristic numbers. Two flows around geometric similar models are physically similar if all characteristic numbers coincide [4]. Especially for measurement

4.5 Characterization of Flows by Dimensionless Numbers

147

Table 4.2 Characteristic numbers of fluid mechanics Name Formulation Force ratio Stationary inertia vc lc Reynolds number Re = νf Viscous Pressure ΔP Euler number Eu = 1 2 Inertia 2 ρvc vc Inertia Froude number Fr = √ Gravity gc l c Transient inertia f c lc Strouhal number St = vc Stationary inertia The subscript c denotes characteristic value of the according physical quantity

setups these similarity considerations are important as it allows measuring of down sized geometries. This possibility is often used in the development of, e.g., ships and airplanes. In general it is not possible to fulfill all characteristic numbers so that only those describing the dominant flow characteristic are fulfilled and the model is only partially similar. Besides for down-sizing measurements, the characteristic numbers are also used to classify a flow situation. The Reynolds number is the best known. It is named after the physicist Osbourne Reynolds (1842–1912) and it provides the ratio between stationary inertia forces and viscous forces. In Table 4.2 all dimensionless numbers of incompressible flows are listed. The Euler number relates pressure and inertial forces and the Froude number relates inertial and gravitational forces. The Froude number is an important characteristic number in case of free surface flow. In unsteady problems, periodic oscillating flow structures may occur, e.g. the Kármán vortex street in the wake of a cylinder. The dimensionless frequency of such an oscillation is denoted as the Strouhal number.

4.6 Finite Element Formulation Most flow problems are treated with the finite-volume method (FVM), which has the advantage of being mass and momentum conserving on discrete volumes. However, ensuring discrete conservation is not equivalent to high precision. The velocity and pressure distributions have first priority, and hence the numerical error of these field variables is the most important property of a numerical discretization scheme. Therefore, we will apply the FE method to numerically solve the incompressible Navier-Stokes equations (NSEs). Thereby, the function spaces for the field variables are defined as (d denotes the space dimension) V = {v ∈ (H 1 )d (Ω)|v = g on ΓDV } Q = {P ∈ L 2 (Ω)} and for test functions w of the momentum equation W = {w ∈ (H 1 )d (Ω)|w = 0 on ΓDV }.

148

4 Flow Field

Therewith, the weak form of (4.41) read as follows: Find (v, P) ∈ (V × Q) such that2 (˙v , w) + (v c · ∇v, w) − (P, ∇ · w) + ν f (∇v, ∇w) − (∇ · v, q) = (h, w)ΓN

(4.42)

for all (w, q) ∈ (W × Q). In order to write (4.42) in a more compact way, the following differential operator A is introduced A(v c ; {v, P}, {w, q}) = (˙v , w) + (v c · ∇v, w)

+ ν f (∇v, ∇w) − (P, ∇ · w) − (∇ · v, q)

and (4.42) may be written as follows A(v c ; {v, P}, {w, q}) = (h, w)ΓN ∀ (w, q) ∈ (W × Q). The integral theorem of Green is applied to the viscous and the pressure term yielding a natural boundary condition containing ∂v/∂n = 0, which is known to be an appropriate open boundary condition [5]. If the pressure at the boundary is assumed to be zero, which is generally possible, open boundary conditions are obtained naturally by omitting the surface integral. As these open boundary conditions are obtained naturally, they are also called the do-nothing boundary conditions. Furthermore, the formulation also enables to prescribe pressure drop conditions. A pressure drop can be established by treating the inlet and the outlet as open boundaries and by prescribing different pressure values. The open boundary can therefore be understood as an inflow-outflow condition. The standard Galerkin finite-element approach applied to the incompressible Navier-Stokes equations possesses two sources of instabilities. The first one is a result of the convective term. With increasing Reynolds numbers the impact of the convective term becomes dominant and stabilization is needed. The second instability arises from the saddle point structure of the equations and standard finite elements do not fulfill the LBB (Ladysenskaja Babuska Brezzi) condition and therefore lead to an unstable discretization [6]. In context of Finite-Volume (FV) methods stabilization techniques are applied, as e.g. upwinding [7]. An enormous amount of publications exists regarding stabilized FEMs. A comprehensive overview is given in [8]. In order to circumvent the LBB condition and to still obtain a stable fluid mechanical solution, residual-based stabilization is applied herein for flows with moderate Reynolds numbers [9, 10]. Overviews of the development can be found, e.g., in [8, 11, 12]. The residual-based stabilized FEM represents a group of stabilized methods including Streamline Upwind Petrov Galerkin/Pressure Stabilized Petrov Galerkin (SUPG/PSPG) [13], Galerkin Least

A point over a letter denotes first-order time derivative, (a, b) is the short form of a · b dΩ, and Ω

(c, d)ΓN of c · d dΓ .

2

ΓN

4.6 Finite Element Formulation

149

Squares (GLS) [14] and Unusual Stabilized FEM (USFEM) [15]. In [16], a new sight on the residual-based stabilized FEM has been initiated by introducing the variational multiscale method (VMM). The VMM provides the physical background of the additional terms appearing in the stabilized FEM [10, 17]. The stabilization is based on the residual of the weak form of the incompressible Navier-Stokes equations. The idea of residual-based stabilization is to add terms including the residual of the momentum equation LM and the residual of the continuity equation LC given as LM = v˙ + v c · ∇v − ν f Δv + ∇ P LC = ∇ · v,

(4.43a) (4.43b)

which are tested with stabilization operators Lstab M,C . Now, we perform the spatial approximation of our unknowns, i.e. for the flow velocity (similar as for computational mechanics, see 4.44) ′

v ≈ vh =

nn nd   i=1 a=1

′

Na via ei =

nn  a=1

⎞ Na 0 0 N a v a ; N a = ⎝ 0 Na 0 ⎠ . 0 0 Na ⎛

(4.44)

However, due to the many bilinear forms obtained by the stabilization procedure, we will not substitute the approximation ansatz, but will keep the discrete unknowns, e.g., v h . The stabilized variational formulation with the two stabilization parameters τm , τc is therefore given by      A v ch ; {v h , P h }, {w h , q h } + τm LM , Lstab M

e

e

+

 e

(4.45)

  τc LC , Lstab = (hh , w h )ΓN . C e

The index e denotes the elements of the grid. The inner products of the stabilization terms only have to be solved in the element interior, which is denoted by (·, ·)e in (4.45). Depending on the stabilization operators Lstab M,C different methods are obtained. In Table 4.3 the operators of SUPG/PSPG, GLS and USFEM are listed for the momentum equation. The stabilization operator of the continuity equation denoted by graddiv stabilization was introduced in [8] and reads as follows h Lstab C =∇ ·w .

Table 4.3 Momentum stabilization operators Lstab M for different schemes

(4.46)

Method

Terms

SUPG/PSPG [9] GLS [8] USFEM [8]

v ch · ∇w h + ∇q h v ch · ∇w h − ν f Δw h + ∇q h v ch · ∇w h + ν f Δw h + ∇q h

150

4 Flow Field

In our scheme SUPG/PSPG together with grad-div stabilization is applied to gain a stable formulation. Thereby, the NSE with our chosen stabilization terms result in the following nonlinear semi-discrete Galerkin formulation M(v h )˙v h + K (v h )v h + G(v h )P h = F h ,

(4.47)

with matrices M, K and G defined as       M(v h )˙v h = v˙ h , ω h + τ M v˙ h , (v ch · ∇)ω h + τ M v˙ h , ∇q h

      K (v h )v h = (v ch · ∇)v h , ω h + ν f ∇v h , ∇ω h − τ M ν f Δv h , (v ch · ∇)ω h     + τ M (v ch · ∇)v h , (v ch · ∇)ω h + τ M (v ch · ∇)v h , ∇q h

      − τ M ν f Δv h , ∇q h − ∇ · v h , q h − τC ∇ · v h , ∇ · ω h

      G(v h )P h = − P h , ∇ · ω h + τ M ∇ P h , (v ch · ∇)ω h + τ M ∇ P h , ∇q h   F h = hh , ω h

ΓN

.

The crucial point of all stabilized FE methods is the appropriate choice of the stabilization parameters. In (4.45) two stabilization parameters are included (τm and τc ) in order to be able to stabilize momentum and continuity equations individually. A review on the different choices of the stabilization parameters can be found in [8]. Here, we use stabilization parameters suitable for second-order quadrilateral elements as follows τm =

he ζ 2||v||2

and

τc =

||v||2 h e ζ 2

(4.48)

with ||v|| √ 2 the L 2 -norm of v, h e a characteristic value of the discretization, e.g., h e = d Ωe , and ζ=



Ree , 0 ≤ Ree < 1 1,

Ree ≥ 1

.

(4.49)

Ree denotes the element based Reynolds number defined by Ree =

||v||2 h e . 24ν f

(4.50)

4.6 Finite Element Formulation

151

For the time discretization, the 2nd order backward difference formula (BDF2) has been shown to be reliable in combination with the NSE, as it has low numerical dissipation, is A-stable and even L-stable according to [18]. Thereby, using the relation of the B D F2 scheme for the time derivative according to h v n+1 − v nh

△t

results in 

=

h   1 v nh − v n−1 2 h , tn+1 . + F h v n+1 3 △t 3

      2 2 h h h h h M v n+1 v n+1 Pn+1 + Δt K v n+1 + Δt G v n+1 3 3   4  1 h 2 h h h = Δt F n+1 + M v n+1 v − v 3 3 n 3 n−1

(4.51)

(4.52)

with the superscripts n + 1, n, n − 1 denoting new, current and old time steps, respectively. Before the algebraic system can be solved, linearization needs to be applied. The nonlinearity stems from the convective term (v c ·∇v) and its linearization in the context of stabilized FEM can be found in [8, 19]. With k the fluid iteration counter, the linearization of the convective term is given by [8, 19] k+1 k+1 k+1 k k v k+1 c,n+1 · ∇v n+1 ≈ v c,n+1 · ∇v n+1 + λ1 v c,n+1 · ∇v n+1

− λ1 v kc,n+1 · ∇v kn+1 . For λ1 = 0 the fixpoint and for λ1 = 1 the Newton scheme is obtained. When a sufficient error bound ε is reached, calculated by  h,k+1  h,k+1 h,k  h,k  v  P n+1 − v n+1 2 n+1 − Pn+1 2 + < ε,  h,k+1   h,k+1     P v n+1

n+1

the iteration stops and the next time step can be processed. Further details on our formulation as well as applications can be found in [20, 21].

4.7 Numerical Examples 4.7.1 Steady Channel Flow The computational domain of the flow-through channel is shown in Fig. 4.6 together with the mesh, consisting of 16 × 8 second order quadrilateral elements. The fluid flows from left to right. At the top and at the bottom, the walls are fixed. In a fully

152

4 Flow Field

Γwall

(a)

(b)

Γin

H=1

L=2

Γout

y x Fig. 4.6 Flow-through a channel: setup and computational grid. a Computational domain of the channel. b Computational grid

developed plane flow, the y-velocity vanishes and the incompressible Navier-Stokes equations reduce to d 2 vx dp = μf . (4.53) dx dy 2 This equation can be solved analytically with appropriate boundary conditions. In the numerical simulation the vanishing y-velocity is incorporated by homogeneous Dirichlet conditions at the whole boundary. The density and the dynamic viscosity of the fluid are set to ρ = 1.0 kg/m3

and

μ f = 0.002 Pa s.

(4.54)

Case I: Velocity driven To obtain a velocity driven flow a parabolic inflow profile is assumed at the inlet boundary Γin   1 1 y − 0.5 2 vx (y) = − m/s. (4.55) 2 2 0.5 At the top and at the bottom boundary Γwall , the x-velocity is set to zero. To ensure the outflow at Γout a free x-velocity is set. The pressure is set to zero only at one node, the lower right corner node (2 m/0 m). Under these boundary conditions the analytic solution for the pressure is given by p(x) = 8 μ f vx max

L−x Pa. H2

(4.56)

The numerical results of pressure and x-velocity are shown by contour plots in Fig. 4.7. The analytical and numerical pressure results are consistent, as shown along the horizontal line (y = 0.5 m) in Fig. 4.8a.

4.7 Numerical Examples

153

(b)

(a)

(a)

(b)

1.8

0.6

Numerical Analytical

1.4

Velocity (m/s)

Pressure /(hPa)

Fig. 4.7 Numerical results of the channel flow. a Pressure p. b Velocity vx

1.0 0.6 0.2 0.0

1.0

x (m)

2.0

Numerical Analytical

0.4

0.2

0.0

0.0

0.2

x (m) 0.6

1.0

Fig. 4.8 Numerical and analytic results of the channel flow. a Pressure in x-direction along y = 0.5 m. b Velocity vx in y-direction along x = 1 m

Case II: Pressure driven The second set of boundary conditions is supposed to prescribe a pressure drop so that the parabolic x-velocity distribution develops. The wall conditions are the same as in the velocity driven case. At the in- and outlet the following pressure conditions are applied h = pn with p = 0.016 Pa on Γin and h = 0 on Γout ,

(4.57)

while the x-velocity is free there. The comparison of the numerically computed x-velocity along the vertical line at x = 1 with the analytic solution is shown in Fig. 4.8b. The spatial distribution of pressure and velocity is similar to the results depicted in Fig. 4.7.

154

4 Flow Field

4.7.2 Unsteady Flow Around a Square The flow around a square basically leads to similar results as the flow around a cylinder. The difference, however, is that the location of the flow separation point is now known to be a corner. Spatial distributed velocity values are available for the considered test case [22]. The geometrical setup is chosen according to [22] and is shown in Fig. 4.9a. The density and the dynamic viscosity of the fluid are thereby ρ = 1 kg/m3

and

μ f = 0.01 Pa s.

(4.58)

A parabolic profile with a maximal velocity of vx (y = 0 m) = 1 m/s is prescribed at the inflow  y 2 m/s on Γin (4.59) vx (y) = 1 − 4 leading to a Reynolds number of Re = 100. The time step size is set to △t = 0.03 s. The mesh is composed of 40.000 (500 × 80) equidistant quadrilaterals with basis functions of 2nd order (see Fig. 4.9b for the zoom in the square), which correspond to the roughest Finite-Volume (FV) mesh in [22]. The unsteady flow is again visualized with streamlines of three time steps, see Fig. 4.10. The first picture (t 1 ) represents the time step, at which at the upper downstream edge a small recirculation eddy just develops. This eddy grows (see t + T /4) and finally detaches from the edge at t +T /2. At the same time the upper eddy detaches, a new recirculation eddy develops at the lower downstream edge. The cycle is repeated. In order to quantify the spatial distribution of the computed results the velocity along three lines inside the flow field are compared with results of FV-method and the Lattice Boltzmann Automata (LBA) method. The time step at which the y-velocity changes its sign from minus to plus at the point (10.5 m/0 m) is chosen to compare the velocity along two vertical lines (x = 0 m and x = 4 m), as shown in Fig. 4.11. In Fig. 4.12 the velocity along the horizontal centerline is displayed for the same time step. The results of our flow solver matches the results of the FV-method and LBA-method.

(a)

(b)

1

Γwall 8

y x Γin

Γout

Γobstacle

12.5

Γwall 37.5

Fig. 4.9 Flow around a square: Geometry of the fluid domain (units in m) and FE mesh around the square. a Setup. b Mesh around obstacle

4.7 Numerical Examples

155

Fig. 4.10 Streamlines of a Kármán vortex street in the wake of a square at Re = 100 showing one half cycle of the eddy detaching in three time steps

156

4 Flow Field

(a) 1.4

(b)

FVM FEM

vy (m/s)

x=0m

vx (m/s)

0.5

FVM FEM

x=4m

x=4m

0

x=0m

0 -0.2

0

-4

y (m)

4

-0.3 -4

0

y (m)

4

Fig. 4.11 Velocity along two vertical lines in the flow field, through the center of the square (x = 0 m) and inside the wake (x = 4 m). a Stream-wise velocity vx . b Cross-stream velocity v y

(a) 1.2

(b)

vy (m/s)

vx (m/s)

0.6

LBA FVM FEM

0 -0.2 -10

0

y (m)

LBA FVM FEM

0

-0.6

20

-10

0

y (m)

20

Fig. 4.12 Velocity along the horizontal centerline (y = 0 m). a Stream-wise velocity vx . b Crossstream velocity v y

References 1. T.J.R. Hughes, W.K. Liu, T.K. Zimmermann, Lagrangian-eulerian finite element formulation for incompressible viscous flows. Compu. Methods Appl. Mechani. Eng. 29(3), 329–349 (1981) 2. F. Durst, Grundlagen der Strömungsmechanik (Springer, New York, 2006) 3. M.S. Howe, Acoustics of Fluid-Structur Interactions (Cambridge Monographs on Mechanics, Cambridge, 1998) 4. H. Schlichting, K. Gersten, Grenzschicht-Theorie (Boundary Layer Theory) (Springer, Berlin, 2006) 5. P.M. Gresho, R.L. Sani, Incompressible Flow and the Finite-Element Method (Wiley, Chichester, 2000) 6. F. Brezzi, M. Fortin, Mixed and Hybrid Finite Element Methods (Springer, New York, 1991) 7. J.H. Ferzinger, M. Peric, Computational Methods for Fluid Dynamics (Springer, New York, 2002) 8. W.A. Wall, Fluid-Struktur-Interaktion mit stabilisierten Finiten Elementen, Ph.D. thesis, University of Stuttgart (1999)

References

157

9. A.N. Brooks, T.J.R. Hughes, Streamline upwind/petrov-galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 32, 199–259 (1982) 10. V. Gravemeier, The variational multiscale method for laminar and turbulent incompressible flow, Ph.D. thesis, Universität Stuttgart (2003) 11. F. Brezzi, M.O. Bristeau, L.P. Franca, M. Mallet, G. Rogé, A relationship between stabilized finite element methods and the galerkin method with bubble functions. Comput. Methods Appl. Mech. Eng. 96, 117–129 (1992) 12. J. Donea, A. Huerta, Finite Element Methods for Flow Problems (Wiley, Chichester, 2003) 13. T.E. Tezduar, S. Mittal, S.E. Ray, R. Shih, Incompressible flow computations with stabilized bilinear and linear equal-order-interpolation velocity-pressure elements. Comput. Methods Appl. Mech. Eng. 95, 221–242 (1992) 14. L.P. Franca, S.L. Frey, Stabilized finite element methods: II the incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 99, 209–233 (1992) 15. L.P. Franca, C. Farhat, Bubble functions prompt unusual stabilized finite element methods. Comput. Methods Appl. Mech. Eng. 123, 299–308 (1995) 16. T.J.R. Hughes, Multiscale phenomena: green’s functions, the dirichlet-to-neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods. Comput. Methods Appl. Mech. Eng. 127, 387–401 (1995) 17. T.J.R. Hughes, G. Scovazzi, L.P. Franca, Encyclopedia of Computational Mechanics: Ch 2 Multiscale and Stabilized Methods (Wiley, New York, 2004) 18. E. Hairer, S.P. Norsett, G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, 2nd Revised edn. (Springer, Berlin, 1991) 19. R. Codina, A nodal-based implementation of a stabilized finite element method for incompressible flow problems. Int. J. Numer. Meth. Fluids 33, 737–766 (2000) 20. G. Link, A finite element scheme for fluid-solid-acoustics interactions and its application to human phonation, Ph.D. thesis, University Erlangen-Nuremberg (2008) 21. S. Zörner, Numerical simulation method for a precise calculation of the human phonation under realistic conditions, Ph.D. thesis, Vienna University of Technology (2014) 22. M. Breuer, Direkte Numerische Simulation und Large-Eddy Simulation Turbulenter Strömungen auf Hochleistungsrechnern (Shaker, Aachen, 2002)

Chapter 5

Acoustic Field

Acoustic waves can propagate in non-viscous media just in the form of longitudinal waves. Thus, the particles of the fluid move forwards and backwards in the direction (and opposite) of the propagation and produce locally a compression and expansion of the fluid (see Fig. 5.1). The driving force, which is essential for the propagation of the wave, is generated by the pressure change in the media.

5.1 Wave Theory of Sound Let us consider a loudspeaker generating sound at a fixed frequency f and a number of microphones recording the sound as displayed in Fig. 5.2. In a first step, we measure the sound with one microphone fixed at x 0 , and we will obtain a periodic signal in time with frequency f and period time T = 1/ f . In a second step, we use all microphones and record the pressure at a fixed time t0 . Drawing the obtained values along the individual position of the microphone, e.g. along the coordinate x, we again obtain a periodic signal in space. This periodicity is characterized by the wavelength λ and is uniquely defined by the frequency f and the speed of sound c via the relation c (5.1) λ= . f Assuming a frequency of 1 kHz, the wavelength in air takes on the value of 0.343 m (c = 343 m/s) and in water 1.5 m (c = 1,500 m/s). The propagation of waves is described by the time and the spatial variation of the density ρ ρ=

mass (kg/m3 ) volume

the pressure p p=

force (N/m2 ) surface

© Springer-Verlag Berlin Heidelberg 2015 M. Kaltenbacher, Numerical Simulation of Mechatronic Sensors and Actuators, DOI 10.1007/978-3-642-40170-1_5

159

160

5 Acoustic Field

Direction of propagation

Fig. 5.1 Compression and expansion of a fluid

z y x pa (x0 , t) T

pa (x, t0 ) λ t x

Fig. 5.2 Sound generated by a loudspeaker and measured by microphones

and the velocity v v=

distance (m/s). time

These quantities can be decomposed in their mean and alternating part according to ρ = ρ0 + ρa p = p0 + pa v = v0 + va ,

(5.2)

where we denote the acoustic density by ρa , the acoustic pressure by pa and the acoustic particle velocity by v a . For linear acoustics, we have the following relations between the mean and alternating quantities ρa ≪ ρ0

pa ≪ p0

va ≪ v0 .

(5.3)

5.1 Wave Theory of Sound

161

The acoustic field is fully described by the equation of mass conservation (continuity equation), the equation of momentum conservation (Euler’s equation, Newton’s law for fluids) and the pressure-density relation (state equation). In Chap. 4 we have already derived the general equations of mass (see (4.10)) and momentum (see (4.20)) conservation. The acoustic field can be modeled as a perturbation with relations according to (5.3). Therefore, we will start at these equations and will derive the linearized equation of mass and momentum conservation to model linear wave propagation. Here, we will restrict to the case of zero mean velocity v 0 .

5.1.1 Conservation of Mass (Continuity Equation) The propagation of an acoustic wave through a fluid (gas) causes local changes in the density. In its general form, this is described by the conservation of mass as given in (4.10). We now substitute the relation (5.2) into (4.10) and arrive at ∂(ρ0 + ρa ) + ∇ · ((ρ0 + ρa )v a ) = 0. ∂t

(5.4)

Thereby, we assume that the mean density ρ0 is independent of time and space. Furthermore, since we derive the linearized mass conservation, we can cancel second order terms such as ρa v a , and arrive at ∂ρa + ρ0 ∇ · v a = 0. ∂t

(5.5)

5.1.2 Conservation of Momentum (Euler Equation) The conservation of momentum in its general case is given in (4.20). Since we assume non-viscous media, we omit in a first step the viscous stress tensor [τ ] and arrive at ∂ρv + ∇ · (ρ v ⊗ v) = −∇ p + f Ω . ∂t

(5.6)

Furthermore, we set the external volume force f Ω to zero and apply the relation (5.2) to arrive at ∂(ρ0 + ρa )v a + ∇ · ((ρ0 + ρa ) v a ⊗ v a ) = −∇( p0 + pa ). ∂t

(5.7)

162

5 Acoustic Field

Again we can assume that the mean quantities ρ0 and p0 are independent of time and space. To achieve the linearized version of the momentum conservation, we further neglect all second order terms and obtain ρ0

∂v a + ∇ pa = 0. ∂t

(5.8)

5.1.3 Pressure-Density Relation (State Equation) According to the quasistatic theory of thermodynamics, which assumes that the fluid is in a thermodynamic equilibrium, we can describe the local state of a fluid at any time by two intrinsic state variables. When we apply specific heat qh to a fluid element, then the specific inner energy e increases and at the same time the volume changes by p dρ−1 . This thermodynamic relation is expressed by (5.9) de = dqh − p dρ−1 , where the second term describes the work done on the fluid element by the pressure. If the change occurs sufficiently slowly, the fluid element is always in thermodynamic equilibrium, and we can express the heat input by the specific entropy s and the temperature T (5.10) dqh = T ds. Therefore, we may rewrite (5.9) and arrive at the fundamental law of thermodynamics1 de = T ds − p dρ−1 p = T ds + 2 dρ. ρ

(5.11)

Towards acoustics, it is convenient to choose the mass density ρ and the specific entropy s as the two intrinsic state variables. Hence, the specific inner energy e is completely defined by a relation denoted as the thermal equation of state e = e(ρ, s).

(5.12)

Therefore, variations of e are given by de =

1

We have: dρ−1 = −(1/ρ2 )dρ.



∂e ∂ρ



s

dρ +



∂e ∂s



ρ

ds.

(5.13)

5.1 Wave Theory of Sound

163

A comparison with the fundamental law of thermodynamics (5.11) provides the thermodynamic equations for the temperature T and pressure p 2



p=ρ  ∂e T = ∂s

∂e ∂ρ 



(5.14) s

(5.15)

.

ρ

Furthermore, we may express p as a function of ρ and s and obtain dp =



∂p ∂ρ



dρ +

s

= c2 dρ +



where 2



c =

∂p ∂s



∂p ∂s



∂p ∂ρ



ds

(5.16)

ρ

ds ,

ρ



(5.17) s

is the square of the isentropic speed of sound. While (5.17) is a definition of the thermodynamic variable c(ρ, s), we can show that c is indeed a measure for the speed of sound. Since in many applications the fluid considered is air at ambient pressure and temperature, we may use the ideal gas law p = ρRT

(5.18)

with the universal gas constant R, which computes for an ideal gas as R = c p − cΩ .

(5.19)

In (5.19) c p , cΩ denote the specific heat at constant pressure and constant volume, respectively. Furthermore, the inner energy e depends for an ideal gas just on the temperature T via (5.20) de = cΩ dT. Substituting this relations in (5.11), assuming an isentropic state (ds = 0) and using (5.18) results in R dρ p dT = . (5.21) cΩ dT = 2 dρ → ρ T cΩ ρ By further exploring (5.18) in the form of d p = RT dρ + Rρ dT →

dρ dT dp = + p ρ T

164

5 Acoustic Field

and using (5.19), we arrive at c p dρ dp dρ = =κ p cΩ ρ ρ

(5.22)

with κ the specific heat ratio (also known as adiabatic exponent). A comparison of (5.22) with (5.17) yields √  (5.23) c = κ p/ρ = κRT .

We see that the speed of sound c of an ideal gas depends only on the temperature. For air κ has a value of 1.402 so that we obtain a speed of sound c at T = 15 ◦ C of 341 m/s. For most practical applications, we can set the speed of sound to 340 m/s within a temperature range of 5–25 ◦ C. Summarizing, we may write the acoustic pressure-density relation for linear acoustics according to (5.17) as (5.24) pa = c2 ρa and compute the speed of sound c for gases by (5.23). For liquids, such as water, the pressure-density relation is expressed by the adiabatic bulk modulus K s according to ∂ p(ρ, s) Ks = ∂ρ ρ  Ks . c= ρ

(5.25) (5.26)

The reciprocal 1/K s is known as the adiabatic compressibility. In the case of a liquid, a negligible error occurs, if we consider the state to be isentropic (adiabatic) or isotherm. The reason for this behavior is the little difference between the adiabatic bulk modulus K s , which is quite difficult to measure, and the isothermal bulk modulus K T (for details see [1]).

5.1.4 Linear Acoustic Wave Equation The linear acoustic wave propagation is fully described by the linear mass conservation (5.5), linear momentum conservation (5.8) and by the acoustic pressure-density relation (5.24). Applying the time derivative to (5.5) and interchanging time and space derivatives yields

5.1 Wave Theory of Sound

165

∇·

1 ∂ 2 ρa ∂v a =− . ∂t ρ0 ∂t 2

(5.27)

Furthermore, expressing the term ∂v a /∂t by (5.8) results in the following second order PDE   1 ∂ 2 ρa 1 ∇ · − ∇ pa = − . (5.28) ρ0 ρ0 ∂t 2 Now, we explore the pressure-density relation according to (5.24) and arrive at the acoustic wave equation for pa ∆pa =

1 ∂ 2 pa . c2 ∂t 2

(5.29)

By applying the curl operator to (5.8), it is easy to show that the acoustic velocity v a is irrotational   ∂v a + ∇ pa = 0 ∇ × ρ0 ∂t ∂ ρ0 (∇ × v a ) + ∇ × ∇ pa = 0 ∂t (5.30) ∇ × v a = 0. Thus, the acoustic velocity v a can be expressed as the gradient of a scalar potential, the so-called acoustic velocity potential v a = −∇ψ.

(5.31)

By introducing the acoustic velocity potential in the linearized momentum Eq. (5.8), we obtain the relation between ψ and pa − ρ0

∂ (∇ψ) = −∇ pa ∂t ∂ψ , pa = ρ0 ∂t

(5.32)

where the integration constant can be set to zero, since (5.31) defines ψ only up to a constant. With (5.32) we may write the wave equation (5.29) for the acoustic velocity potential as 1 ∂2ψ ∆ψ = 2 2 . (5.33) c ∂t

166

5 Acoustic Field

5.1.5 Acoustic Quantities Let us first consider the linear version of the momentum equation (see (5.8)) and take the dot product with the acoustic velocity v a v a · ρ0

∂v a = −v a · ∇ pa ∂t = −∇ · ( pa v a ) + pa ∇ · v a .

(5.34)

Using the linear version of the mass conservation and the linear pressure-density relation ρa = pa /c2 results in 1 ∂ρa ∂v a = −∇ · ( pa v a ) − pa ∂t ρ0 ∂t   2  pa 1 ∂ ∂ 1 . ρ0 v a · v a = −∇ · ( pa v a ) − ∂t 2 2 ∂t ρ0 c2 v a · ρ0

(5.35)

Therewith, we obtain ∂wa + ∇ · Ia = 0 ∂t 1 wakin = ρ0 v a · v a 2 pa 2 pot wa = 2ρ0 c2

(5.36) (5.37) (5.38)

with wa the total acoustic energy density and I a the acoustic energy flux, also called the acoustic intensity. Furthermore, we refer to wakin as the acoustic kinetic energy pot density and wa the acoustic potential energy density. Now let us assume that the acoustic source is driven with constant frequency f (angular frequency ω = 2π f ), so that we can express the acoustic field variables by pa (t) = pˆ cos(ωt) v a (t) = vˆ cos(ωt + ϕ). Therewith, we obtain the following expressions for the time averaged acoustic energy density wa as well as intensity I a (T = 1/ f is the period time) waav

1 = T =

t 0 +T t0

1 1 ρ0 vˆ 2 cos2 (ωt + ϕ) dt + 

 2 T 1 2 (1+cos(2(ωt+ϕ))

1 1 pˆ 2 ρ0 vˆ 2 + 4 4 ρ0 c2

t 0 +T t0

1 pˆ 2 2 ρ0 c2

cos2 (ωt)  

1 2 (1+cos(2ωt))

dt

5.1 Wave Theory of Sound

I av a

1 = T =

t 0 +T t0

167

pˆ vˆ cos(ωt) cos(ωt + ϕ) dt  

1 2

cos ϕ+ 21 cos(2ωt+ϕ)

pˆ vˆ cos ϕ. 2

By using these results and substituting it in (5.36), we arrive at ∇ · I av a = 0. This means that the spatial variation of I av a has to be zero, or expressing it in integral form   I av ∇ · I av dΩ = a · n dΓ = 0. a Ω

Γ (Ω)

However, if the considered volume encloses any acoustic sources, the above derivation does not apply, and we obtain Paav =



I av a · n dΓ.

(5.39)

Γ (Ω)

In (5.39) we denote by Paav the average acoustic power radiated by all sources enclosed by the surface Γ . Strictly speaking, each acoustic wave has to be considered as transient, having a beginning and an end. However, for some long duration sound, we speak of continuous wave (cw) propagation and we define for the acoustic pressure pa a mean square pressure ( pa )2av as well as a root mean squared (rms) pressure pa,rms

pa,rms



t0 +T

t0 +T 



1

1 2  pa2 dt. = ( p − p0 ) dt =  T T t0

(5.40)

t0

In (5.40) T denotes the period time of the signal or if we cannot strictly speak of a periodic signal, an interminable long time interval. Any acoustic media is defined by its acoustic impedance Z a , which computes as the quotient of sound pressure pa and the acoustic particle velocity v a in normal direction pa . (5.41) Za = va · n If the particles of the fluid have the same oscillation in the plane normal to the direction of propagation, the corresponding acoustic wave is called a plane wave.

168

5 Acoustic Field

For plane waves, the acoustic impedance is constant and is called the specific acoustic impedance pa = ρ0 c (Ns/m3 ). (5.42) Z0 = va · n In air the acoustic impedance Z 0 has a value of 408 Ns/m3 , in water a value of 1.5 × 106 Ns/m3 and in solids values between (3 − 110) × 106 Ns/m3 . For non-plane waves the sound field impedance is not constant, e.g., for a spherical wave Z a computes as Za =

jρ0 ckr pa jkr = = Z0 va · n 1 + jkr 1 + jkr

(5.43)

with k = ω/c = 2π/λ the wave number and r the distance from the source.

5.1.6 Plane and Spherical Waves In order to get some physical insight in the propagation of acoustic sound, we will consider two special cases: plane and spherical waves. Let’s start with the simpler case, the propagation of a plane wave as displayed in Fig. 5.3. Thus, we can express the acoustic pressure by pa = pa (x, t) and the velocity by v a = v(x, t)ex . Using these relations together with the linear pressure-density law, we arrive at the following 1D linear wave equation ∂ 2 pa 1 ∂ 2 pa − 2 = 0, (5.44) 2 ∂x c ∂t 2 which can be rewritten in factorized version as    ∂ ∂ 1 ∂ 1 ∂ pa = 0. − + ∂x c ∂t ∂x c ∂t

(5.45)

This version of the linearized, 1D wave equation motivates us to introduce the following two functions

Fig. 5.3 Propagation of a plane wave

pa = pa (x, t)

5.1 Wave Theory of Sound

169

ξ = t − x/c

η = t + x/c with two main properties ∂ ∂ ∂ = + ∂t ∂ξ ∂η

∂ 1 = ∂x c



∂ ∂ − ∂η ∂ξ



.

Therewith, we obtain for the factorized operator ∂ 1 ∂ 1 ∂ − =− ∂x c ∂t 2c ∂ξ

∂ 1 ∂ 1 ∂ + = ∂x c ∂t 2c ∂η

and the linear, 1D wave equation transfers to −

1 ∂ ∂ pa = 0. 4c2 ∂ξ ∂η

The general solution computes as a superposition of arbitrary functions of ξ and η pa = f (ξ) + f (η) = f (t − x/c) + g(t + x/c).

(5.46)

This solution describes waves moving with the speed of sound c in +x and −x direction, respectively. Now let us derive the relation between acoustic pressure pa and particle velocity v a . For this, we substitute the above relations in the linearized, 1D equations for momentum and mass conservation, and use the linear pressure-density relation ∂va ∂ pa + ρ0 c2 =0 ∂t ∂x  ∂ pa ∂ pa ∂va ∂va =0 + + ρ0 c − ∂ξ ∂η ∂η ∂ξ ∂ ∂ ( pa − ρ0 cva ) + ( pa + ρ0 cva ) = 0 ∂ξ ∂η ∂ pa ∂va + =0 ρ0 ∂x    ∂t 1 ∂ pa ∂va ∂va ∂ pa ρ0 + =0 + − ∂ξ ∂η c ∂η ∂ξ ∂ ∂ (ρ0 cva − pa ) + (ρ0 cva + pa ) = 0. ∂ξ ∂η

170

5 Acoustic Field

Once adding and once subtracting these two obtained equations will result in ∂ (ρ0 cva + pa ) = 0 ∂η

∂ (ρ0 cva − pa ) = 0 ∂ξ

so that (ρ0 cva + pa ) as well as (ρ0 cva − pa ) become functions of ξ and η, and we arrive at pa 1 ( f (t − x/c) + g(t + x/c)) = . (5.47) va = ρ0 c ρ0 c Therewith, the value of the acoustic pressure over acoustic particle velocity for a plane wave is constant and results in the specific acoustic impedance (see (5.42)). Furthermore, the acoustic energy density wa and the acoustic intensity I a simplifies to pa 2 ρ0 va 2 pa 2 = + 2 2ρ0 c2 ρ0 c2 2 pa ex = cwa ex . Ia = ρ0 c

wa =

(5.48) (5.49)

In (5.48) we realize that the acoustic kinetic and potential energy density are equal, and (5.49) demonstrates that the acoustic energy propagates with the speed of sound c. The second case of investigation will be a spherical wave, where we assume a point source located at the origin. In the first step, we rewrite the linearized wave equation in spherical coordinates and consider that the pressure pa will just depend on the radius r . Therewith, the Laplace-operator reads as ∆pa (r, t) = and we obtain

∂ 2 pa 2 ∂ pa 1 ∂ 2 r pa + = ∂r 2 r ∂r r ∂r 2

1 ∂ 2 r pa 1 ∂ 2 pa − 2 = 0. 2 2 r ∂r c ∂t 

(5.50)

1 ∂ 2 r pa r ∂t 2

A multiplication of (5.50) with r results in the same wave equation as obtained for the plane case (see (5.44)), just instead of pa we have r pa . Therefore, the solution of (5.50) reads as 1 pa (r, t) = ( f (t − r/c) + g(t + r/c)) , (5.51) r which means that the pressure amplitude will decrease according to the distance r from the source. The assumed symmetry requires that all quantities will just exhibit

5.1 Wave Theory of Sound

171

a radial component. Therewith, we can express the time averaged acoustic intensity I av a in normal direction n by a scalar value just depending on r av I av a · n = Ir

and as a function of the time averaged acoustic power Paav of our source Irav =

Paav . 4πr 2

(5.52)

According to (5.52), the acoustic intensity decreases with the squared distance from the source. This relation is known as the spherical spreading law. In order to obtain the acoustic velocity v a = va (r, t)er as a function of the acoustic pressure pa , we substitute the general solution for pa (see (5.51), in which we set without loss of generality g = 0) into the linear momentum equation (see (5.8)) 1 ∂va =− ∂t ρ0 1 va = − ρ0

  ∂ pa f (t − r/c) 1 ∂ =− ∂r ρ0 ∂r r   ∂ F(t − r/c) ∂r r

(5.53)

with f (t) = ∂ F(t)/∂t. Using the relation 1 ∂ F(t − r/c) ∂ F(t − r/c) =− ∂r c ∂t and performing the differentiation with respect to r results in F(t − r/c) 1 1 ∂ F(t − r/c) + ρ0 r ∂r ρ0 r 2 1 1 ∂ F(t − r/c) F(t − r/c) = + ρ0 c r ρ0 r 2  ∂t

va (r, t) = −

(5.54) (5.55)

f /r = pa

F(t − r/c) pa + . = ρ0 c ρ0 r 2

(5.56)

Therewith, spherical waves show in the limit r → ∞ the same acoustic behavior as plane waves. Now with this acoustic velocity-pressure relation, we may rewrite the acoustic intensity for spherical waves as Ir =

pa 2 pa 2 pa 1 ∂ F 2 (t − r/c) F(t − r/c) = + + , ρ0 c ρ0 r 2 ρ0 c 2ρ0 r 3 ∂t

172

5 Acoustic Field

which results for the time averaged quantity (assuming F(t − r/c) is a periodic function) in the same expression as for the plane wave Irav =

( pa 2 )av . ρ0 c

5.2 Quantitative Measure of Sound Sound is characterized by its pressure amplitude and its frequency spectra defining the tone color. For an average, young person (about 20 years) the sensible frequency range is from 16 Hz to 16 kHz. It is of great interest that the human perception concerning the frequency of sound signals is a relative one. This means that frequency differences in two sound signals are sensed as equal, when the ratio of sound is the same (strictly speaking this is true for frequencies above 500 Hz, see [2]). E.g., a change in the frequency from 500 to 600 Hz and from 5,000 to 6,000 Hz will be perceived as equal. Therewith, in most analysis of acoustic signals, 1/3-octave filters, and for some technical reasons also octave filters are used. Their specifications are as follows: • Filter specifications (see Fig. 5.4): fm = • Octave filter:



fl f u

fl . . . lower frequency limit f u . . . upper frequency limit f m . . . center frequency f u = 2 fl

Fig. 5.4 Filter function (band-pass)

Filter amplitude

Therewith, the center frequency computes√by f m = the filter is △ f = f u − fl = f u /2 = f m / 2.

√ 2 fl and the bandwidth of

1

fl

fm

fu

f

5.2 Quantitative Measure of Sound

• 1/3-Octave filter:

173

fu =

√ 3

2 fl

Therewith, the center√frequency calculates as f m = bandwidth is △ f = ( 3 2 − 1) fl ≈ 0.26 fl .

√ 6 2 fl ≈ 1.12 fl and the

The sound is often broadband and so it makes sense to display the frequency spectra in so-called frequency bands by applying the above mentioned filters. Furthermore, if we consider e.g., traffic noise, then a too resolved frequency spectra will lead in most cases to non-reproducible results, so that adequate frequency bands are used, instead. Figure 5.6 displays the frequency spectra of an acoustic signal, which is generated by a turbulent flow (see Fig. 5.5) over a forward-facing step (flow-induced sound). As can be clearly seen, the levels for the octave filtered spectra are the largest, followed by the 1/3-octave filtered one and the original signal (fully resolved spectra) (Fig. 5.6).

Fig. 5.5 Visualization of the flow structure near a forward-facing step

Fig. 5.6 Frequency spectra of acoustic pressure level SPL (see (5.57)): full resolution, 1/3-octave and octave filtered

174

5 Acoustic Field

Similar to the perception of the tone color is the perception of the human ear to pressure amplitude changes, e.g., a pressure amplitude change from pa to 2 pa is sensed as equal with a pressure amplitude change from 5 to 10 pa ( pa has to be larger than 40 dB, see [2]). Now, it has to be mentioned that the threshold of hearing of an average human is at about 20 µPa and the threshold of pain at about 200 Pa, which differs 107 orders of magnitude. Thus, logarithmic scales are mainly used for acoustic quantities. The most common one is the decibel (dB), which expresses the quantity as a ratio relative to a reference value. The sound pressure level L pa (SPL) is defined by pa,rms L pa = 20 log10 pa,ref = 20 µPa. (5.57) pa,ref The reference pressure pa,ref corresponds to the sound at 1 kHz that an average person can just hear. The sound-intensity level L Ia is then defined by L Ia = 10 log10

Iaav Ia,ref

Ia,ref = 10−12 W/m2 ,

(5.58)

with Ia,ref the reference sound intensity corresponding to pa,ref . Furthermore, the sound-power level L Pa computes as L Pa = 10 log10

Paav Pa,ref

Pa,ref = 10−12 W ,

(5.59)

with Pa,ref the reference sound power corresponding to pa,ref . In Tables 5.1 and 5.2 some typical sound pressure and sound power levels are listed. In general, an acoustic signal is not mono-frequent, but consists of an superposition of signals with different frequencies. For simplicity, let us analyze a pressure signal, which can be expressed as follows pa (t) = A1 cos(ω1 t) + A2 cos(w2 t). Table 5.1 Typical sound pressure levels SPL Threshold Voice Car Pneumatic hammer of hearing at 5 m at 20 m at 2 m 0 dB

60 dB

80 dB

100 dB

Accelerating motor at 5 m

Jet at 3 m

110 dB

140 dB

Table 5.2 Typical sound power levels and in parentheses the absolute acoustic power Pa Voice Fan Loudspeaker Jet airliner 30 dB (25 µW)

110 dB (0.05 W)

128 dB (60 W)

170 dB (50 kW)

5.2 Quantitative Measure of Sound

175

Then, the mean square pressure computes as

2 pa,rms

1 = T

=

t o +T t0

t o +T t0

+

A21 cos2 (ω1 t) + 2 A1 A2 cos(ω1 t) cos(ω2 t) + A22 cos2 (ω2 t)

A21 (1 + cos(2ω1 t)) dt + 2

t o +T t0



t o +T t0



dt

A22 (1 + cos(2ω2 t)) dt 2

A1 A2 (cos((ω1 − ω2 )t) + cos((ω1 + ω2 )t)) dt.

Assuming that T = n 1 (2π/ω1 ) = n 2 (2π/ω2 ) = n 3 (2π/(ω1 + ω2 )) = n 4 (2π/(ω1 − ω2 )) with n i ∈ IN, then all integrals of the form t o +T

Ai cos(ωi t) dt

t0

are zero, and we obtain pa,rms =



A2 A21 + 2. 2 2

Generalizing the above case, we can rewrite (5.58) by

L pa = 10 log10 L av ai = 10 log10

N 

k=1

2 pa,rms,k

2 pa,ref

2 pa,rms,k 2 pa,ref

= 10 log10

 N  k=1

10

L av ai /10



(5.60)

(5.61)

In (5.60) L pa is the overall sound pressure level (OSPL),and can be interpreted as the formula for adding sound pressure levels generated by different, incoherent sound sources. Figure 5.7 displays the increase of the OSPL achieved by two sound signals in comparison to the sound pressure level (SPL) of the individual signals. Therewith, due to the logarithmic scaling, which corresponds to the perception of the human ear, the sum of two equal sound pressure levels just increase the OSPL by 3dB, when both sound signals have the same SPL. For the case that one sound signal has a SPL of about 20 dB smaller than the other, we can neglect the contribution of this sound signal to the OSPL.

176

5 Acoustic Field

Fig. 5.7 Increase of the overall sound pressure level achieved by two sound signals as a function of the difference in SPL of the individual signals

An interesting point we want to stress here is the fact that the above formula can be also applied to frequency bands, as e.g., obtained by an octave-filtering.

5.3 Nonlinear Acoustic Wave Equation In this section we will consider a general formulation for nonlinear wave propagation in lossy and compressible fluid media. Furthermore, we assume zero mean velocity v 0 , so that v = v a . Thus, we will derive Kuznetsov’s equation for nonlinear acoustics [3], which is a second-order approximation for viscous heat-conducting fluids. For this case, we start at Navier-Stokes equations as derived in Chap. 4 ρ

 µ ∂v f + ρ(v · ∇)v + ∇ p = µ f ∆v + + λ f ∇(∇ · v) ∂t 3 ∂ρ + ∇ · (ρv) = 0 ∂t

(5.62) (5.63)

with the dynamic viscosity µ f (also denoted by shear viscosity) and the bulk viscosity λ f (see also Sect. 4.3.4). Since the following vector identities are fulfilled (domain is assumed to be convex) ∇(∇ · v a ) = ∆v a + ∇ × ∇ × v a 1 (v a · ∇)v a = ∇(v a · v a ) − v a × ∇ × v a , 2

(5.64) (5.65)

5.3 Nonlinear Acoustic Wave Equation

177

we can rewrite (5.62) as (using v = v a ) ρ ∂v a + ∇(v a · v a ) + ∇ p = ρ ∂t 2



 4µ f + λ f ∆v a 3  µ f + λ f ∇ × ∇ × v a + ρ0 v a × ∇ × v a . + 3 (5.66)

The last two terms in (5.66) describe acoustic streaming, and according to [4] decrease exponentially with the distance to solid walls. Since in our case we can still consider the acoustic particle velocity v a as irrotational and set ρ = ρ0 + ρa as well as p = p0 + pa (assume ∇ p0 = 0), (5.66) reads as ∂v a ∂v a ρ0 ρ0 + ρa + ∇(v a · v a ) + ∇ pa = ∂t ∂t 2



 4µ f + λ f ∆v a . 3

(5.67)

The state equation, defining the relation between the acoustic pressure pa and density ρa within the fluid, can be expressed for the nonlinear wave propagation as follows [4]   1 1 B 2 κ 1 ∂ pa pa − − p . (5.68) ρa = 2 − c ρ0 c4 2 A a ρ0 c4 cΩ cp ∂t In (5.68) B/A denotes the parameter of nonlinearity, κ the adiabatic exponent and cp , cΩ the specific heat capacitance at constant pressure as well as constant volume, respectively. Now, the goal is to combine the three Eqs. (5.63), (5.67), and (5.68) into a single wave equation with the acoustic scalar potential ψ as the unknown. Therefore, we will substitute any physical quantity in a second-order term by its linearized one, since the resulting errors will be of third order. This means that we will express second-order terms in the mass conservation (5.63) and momentum conservation (5.67) by the linearized ones obtained for the linear wave propagation (see (5.5) and (5.8)). Let us start with the continuity equation in the form ∂(ρ0 + ρa ) + ∇ · ((ρ0 + ρa ) v a ) = 0 ∂t ∂ρa + ρ0 ∇ · v a = −ρa ∇ · v a − v a · ∇ρa , ∂t where we have arranged all second-order terms on the right. By using ρa ≈

pa c2

∇ · va ≈ −

(linear state equation)

1 ∂ρa ρ0 ∂t

(linear continuity equation)

(5.69)

178

5 Acoustic Field

and the relation pa

1 ∂ pa2 ∂ pa = , ∂t 2 ∂t

we derive the following approximation − ρa ∇ · v a ≈

pa ∂ pa 1 ∂ pa2 ρa ∂ρa ≈ = . ρ0 ∂t ρ0 c4 ∂t 2ρ0 c4 ∂t

(5.70)

By using the linear state as well as the momentum equations, we transform the second term on the right-hand side of (5.69) to − v a · ∇ρa ≈ −v a ·



1 ∇ pa c2



  ρ0 ∂(v a · v a ) ρ0 ∂v a = −v a · − 2 . (5.71) = 2 c ∂t 2c ∂t

Thus, our modified equation of continuity reads as ∂ρa 1 ∂ pa2 ρ0 ∂(v a · v a ) + ρ0 ∇ · v a = + 2 . 4 ∂t 2ρ0 c ∂t 2c ∂t

(5.72)

Furthermore, the use of (5.68) for ρa leads to 1 ∂ pa 1 B ∂ pa2 κ − − 2 4 c ∂t ρ0 c 2 A ∂t ρ0 c4



=

1 1 − cΩ cp



∂ 2 pa + ρ0 ∇ · v a ∂t 2

1 ∂ pa2 ρ0 ∂(v a · v a ) + 2 . 2ρ0 c4 ∂t 2c ∂t

(5.73)

Now, let us apply a similar procedure to (5.67). For the term ρa ∂v a /∂t we find the following approximation ρa

pa ∂v a pa 1 ∂v a 1 ≈ 2 ≈ − 2 ∇ pa = − ∇ pa2 . ∂t c ∂t c ρ0 2ρ0 c2

(5.74)

In addition, we can rewrite according to [5] the term on the right of (5.67) within the second-order approximation by 

   4µ f ∂ pa 4µ f 1 + λ f ∆v a ≈ + λ . f ∇ 3 ρ0 c2 3 ∂t

(5.75)

Using (5.74) and (5.75) for (5.67), we arrive at our modified Euler equation ρ0

∂v a ρ0 1 1 + ∇ pa = ∇ pa2 − ∇(v a · v a ) − 2 ∂t 2ρ0 c 2 ρ0 c2



4µ f + λf 3



∇

∂ pa . (5.76) ∂t

5.3 Nonlinear Acoustic Wave Equation

179

Similar to the linear case, we apply the divergence operator to (5.76)     ∂v a ∂ pa 4µ f 1 ρ0 1 2 + ∆pa = + λf ∆ ρ0 ∇ · ∆pa − ∆(v a · v a ) − ∂t 2ρ0 c2 2 ρ0 c2 3 ∂t (5.77) and a time derivative to (5.73) 1 ∂ 2 pa 1 B ∂ 2 pa2 κ − − c2 ∂t 2 ρ0 c4 2 A ∂t 2 ρ0 c4 =



1 1 − cΩ cp



∂ ∂ 3 pa + ρ0 (∇ · v a ) ∂t 3 ∂t

1 ∂ 2 pa ρ0 ∂ 2 (v a · v a ) + . 2ρ0 c4 ∂t 2 2c2 ∂t 2

(5.78)

Subtracting (5.78) from (5.77), allowing to interchange time and spatial derivative (means that the term ρ0 (∂/∂t (∇ · v a )) cancels out) yields   1 1 ∂ 2 pa 1 B ∂ 2 pa2 κ 1 ∂ 3 pa ∆pa − 2 =− − − c ∂t 2 ρ0 c4 2 A ∂t 2 ρ0 c4 cΩ cp ∂t 3   4µ f ∂ pa 1 ρ0 + λf ∆ − ∆(v a · v a ) − 2 2 ρ0 c 3 ∂t −

1 ∂ 2 pa2 ρ0 ∂ 2 (v a · v a ) 1 − 2 + ∆pa2 . (5.79) 4 2 2ρ0 c ∂t 2c ∂t 2 2ρ0 c2

Furthermore, we use the relations between time and spatial derivatives according to the linear wave equation for pressure and acoustic velocity 1 ∂ 2 pa c2 ∂t 2 1 ∂ 2 va ∆v a = 2 . c ∂t 2 ∆pa =

(5.80) (5.81)

Applying (5.80) and (5.81) to the right-hand side of (5.79) yields ∆pa −

1 B ∂ 2 pa b ∂∆pa ρ0 ∂ 2 (v a · v a ) 1 ∂ 2 pa − =− 2 − 2 , (5.82) 2 2 4 2 c ∂t c ∂t ρ0 c 2 A ∂t c ∂t 2

with b the diffusivity of sound 1 b= ρ0



4µ f + λf 3



κ + ρ0



1 1 − cΩ cp



.

180

5 Acoustic Field

Since the relations between the pressure as well as velocity to the scalar velocity potential ψ still hold in the nonlinear case v a = −∇ψ ∂ψ pa = ρ0 , ∂t we can rewrite (5.82) as follows       ∂ψ 1 ∂2 b ∂ ∂ψ ∂ψ − 2 2 ρ0 =− 2 ∆ ρ0 ∆ ρ0 ∂t c ∂t ∂t c ∂t ∂t   2 ∂ψ 2 1 B ∂ ρ0 − ρ0 c4 2 A ∂t 2 ∂t −

ρ0 ∂ 2 (∇ψ · ∇ψ). c2 ∂t 2

(5.83)

Thus, the nonlinear wave equation, also called Kuznetsov’s equation, has the form ∂2ψ ∂ c ∆ψ − 2 = − ∂t ∂t 2



1 B b∆ψ + 2 c 2A



∂ψ ∂t

2



+ ∇ψ · ∇ψ .

(5.84)

Besides Kuznetsov’s equation, the following three partial differential equations modeling nonlinear wave propagation are widely used (see e.g., [4]): • Burger’s equation:

b ∂ 2 pa βa ∂ pa ∂ pa − 3 , = p 2 ∂x 2c ∂τ ρ0 c3 ∂τ

(5.85)

with τ = (t −x/c) the retarded time, b the diffusivity of sound and βa = 1+ B/2 A the coefficient of nonlinearity. The Burger’s equation allows to study the combined effect of dissipation and nonlinearity on progressive plane waves. • Westervelt equation: ∇ 2 pa −

b ∂ 3 pa βa ∂ 2 pa2 1 ∂ 2 pa + = − , c2 ∂t 2 c4 ∂t 3 ρ0 c4 ∂t 2

(5.86)

with b the diffusivity of sound. This partial differential equation can describe the propagation of plane waves including the nonlinearity effects as well as dissipation. This means that we can use this equation, when cumulative nonlinear effects dominate local nonlinear effects. For example, once the propagation distance is greater than a wavelength, the waveform distortion is dominated by the cumulative effect. The approximation is not appropriate for the simulation of standing waves.

5.3 Nonlinear Acoustic Wave Equation

181

• Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation: ∂ 2 pa βa ∂ 2 pa2 b ∂ 3 pa c 2 = , pa − 3 − ∇⊥ ∂x∂t 2 2c ∂t 3 2ρ0 c3 ∂t 2

(5.87)

with βa = 1 + B/2 A the coefficient of nonlinearity, b the diffusivity of sound and ∇⊥ the nabla operator including just partial derivatives with respect to the transversal coordinates (in our case y and z, since the wave propagates in the x-direction). The simplification (compared to Kuznetsov’s equation) is given by modeling directional wave propagation fulfilling k⊥ ≪ 1, kx with k⊥ the wave number in the transversal direction and k x the wave number in the direction of propagation. This approximation is mainly used for investigation of diffraction, nonlinearity and dissipative effects in directional sound beams, e.g., occurring in medical ultrasound, acoustic microscopy or non-destructive testing.

5.4 Numerical Computation Within this section we will discuss the FE formulation of the linear wave equation as well as Kuznetsov’s wave equation for the nonlinear case. Furthermore, we will derive an FE formulation, which allows different mesh sizes for the computational subregions.

5.4.1 Linear Acoustic Wave Equation The strong formulation for linear acoustics expressed by the acoustic pressure reads as follows2 Given: f : Ω → IR c : Ω → IR. Find: pa (t) : Ω × [0, T ] → IR, such that 1 p¨a − ∆ pa = f c2 2

Same formulation holds for the acoustic velocity potential ψ.

(5.88)

182

5 Acoustic Field

with boundary conditions pa ∂ pa ∂n and initial conditions pa (r, 0) p˙ a (r, 0)

= pa,e on Γe × (0, T ) = pa,n on Γn × (0, T ) = pa0 , r ∈ Ω = p˙ a0 , r ∈ Ω.

In (5.88) f denotes any excitation function for generating the acoustic wave. For simplicity we set the boundary as well as initial conditions to zero. To obtain the variational formulation, we multiply (5.88) by an appropriate test function ω and perform an integration by parts. Thus, the weak form reads as: Find pa ∈ H01 such that    1 w p¨a dΩ + ∇w · ∇ pa dΩ − w f dΩ = 0 (5.89) c2 Ω

Ω

Ω

for any w ∈ H01 . Using standard nodal finite elements, we approximate the continuous acoustic pressure pa as well as the test function w by pa ≈ pah =

n eq 

w ≈ wh =



Na paa

(5.90)

Na wa .

(5.91)

a=1 n eq

a=1

Thus, (5.89) is transformed to the following semi-discrete Galerkin formulation n eq  a=1

wa

n eq  b=1

⎛ ⎝



1 Na Nb dΩ p¨ab c2

Ω

+



(∇ Na ) · ∇ Nb dΩ pab −

Ω



Ω

⎞

Na f (r a ) dΩ ⎠ = 0.

(5.92)

In matrix form, (5.92) reads as M p¨a + K pa = f ,

(5.93)

5.4 Numerical Computation

183

with M= K = f =

ne 

e=1 ne 

e=1 ne 

e

e

m ; m = [m pq ] ; m pq = ke ; ke = [k pq ] ; k pq = f

e

e=1

e

; f = [ f p] ; f p =



Ωe





Ωe

1 N p Nq dΩ c2

(∇ N p ) · ∇ Nq dΩ N p f (r p ) dΩ.

Ωe

The time discretization is performed by applying a standard Newmark algorithm (see Sect. 2.5.2), which results in the following time stepping scheme (effective mass matrix formulation): • Perform predictor step: ∆t 2 (1 − 2βH ) p¨ an 2 + (1 − γH )∆t p¨ an .

p˜a = p an + ∆t p˙ an +

(5.94)

p˜˙ a = p˙ an

(5.95)

• Solve algebraic system of equations: M ∗ p¨ an+1 = f n+1 − K p˜ an ∗

(5.96)

2

M = M + βH ∆t K . • Perform corrector step: p an+1 = p˜ a + βH ∆t 2 p¨ an+1 p˙ an+1 = p˜˙ a + γH ∆t p¨ an+1 .

(5.97) (5.98)

The question if one should solve for the acoustic pressure or acoustic velocity potential is a question of the boundary conditions. • Acoustic pressure: pa = pa,e on Γ Clearly, in this case we will solve for pa . • Normal component of acoustic particle velocity: n · v a = van on Γ Due to the relation v a = −∇ψ we obtain n · v a = van = −

∂ψ , ∂n

and we will solve for ψ and compute pa according to (5.32).

184

5 Acoustic Field

• Mixed boundary conditions: pa = pa,e on Γe ; n · v a = van on Γn In an acoustic pressure formulation the boundary conditions will read (by use of (5.8)) pa = pa,e on Γe ∂vn ∂ pa = −ρ0 on Γn , ∂n ∂t whereas for the acoustic velocity potential ψ we obtain (using (5.32)) ∂ψ = pa,e on Γe ∂t ∂ψ = vn on Γn . − ∂n

ρ0

For the time harmonic case, we will always apply an acoustic pressure formulation with boundary conditions defined by pˆ = pˆ e on Γe ∂ pˆ = − jωρ0 vˆn on Γn . ∂n

5.4.2 Linear Acoustic Conservation Equations In a first step one proceeds analogously as we have done for the wave equation but with the difference that we have now two unknowns acoustic pressure pa and acoustic particle velocity v a . According to the classical theory of mixed finite elements [6], we need two different spaces V and U . Therefore, we choose two appropriate test functions ψ ∈ V and ϕ ∈ U , multiply (5.5) with ϕ and (5.8) with ψ, and integrate over the whole computational domain 

1 ∂ pa dΩ + ϕ 2 c ρ0 ∂t

Ω



Ω



∇v a ϕ dΩ = 0

(5.99)

Ω

ρ0 ψ ·

∂v a dΩ + ∂t





 I



∇ pa · ψ dΩ = 0.

(5.100)

Ω



 II



To ensure a stable and correct solution of the variational formulation, appropriate functional spaces are required in order to ensure the LBB (Ladysenskaja Babuska Brezzi) condition (see also Sect. 4.6). The first and natural choice would be to define

5.4 Numerical Computation

185

V = H (div) and U = L 2 and perform an integration by parts on the bilinear form I I as    ∇ pa · ψ dΩ = − pa ∇ · ψ dΩ + pa n · ψ dΓ. (5.101) Ω

Ω

Γ

Considering the boundary integral, we notice the occurrence of n · ψ which means that the normal component of the test function needs to be continuous not only on the boundary of the domain but also between two adjacent elements. Finite element ansatz functions, which are H (div) conform, ensure this continuity. A prominent example would be the Raviart Thomas element [7] which features at most second order accuracy. Higher order methods in this case are much more complicated to obtain and their evaluation becomes computationally more expensive [8]. The other choice of functional spaces which would fulfill the LBB condition is v a , ψ ∈ [L 2 ]d , with d the space dimension, and pa , ϕ ∈ H 1 . We immediately see the benefit that all standard types of finite elements can be used within these spaces although the space L 2 requires special attention from a practical point of view. To be able to choose these spaces an integration by parts is performed on the bilinear form I which yields 

ϕ ∇ · v a dΩ = −

Ω



v a · ∇ϕ dΩ +

Ω



v a · nϕ dΓ.

(5.102)

Γ

For a practical implementation, this means that the velocity unknowns are defined locally on each element due to v a ∈ [L 2 ]d . This situation is depicted in Fig. 5.8 for a simple two element example. On first sight the problem of requiring continuous normal velocity components between the elements remains as in the first choice of spaces. But one can cope with this problem on a discrete level as shown in [9] by introducing the well known Piola-transformation in the mapping from grid to reference element [7]. The Piola transformation of the velocity unknowns from the reference element vˆ a : Ωˆ → Rd to the ones on the grid element v a : Ωe → Rd can be given as (see e.g. [10]) va =

1 Je vˆ a . |J e |

(5.103)

If the Piola transform is applied to the variational form, two remarkable aspects become apparent. First, the boundary integral on the reference domain becomes Fig. 5.8 Location of acoustic pressure and particle velocity unknowns due to the mixed ansatz

pa4

pa1

pa6

pa5 va4

va3

va1

va2 pa2

va4

va3

va1

va2 pa3

186

5 Acoustic Field



Γ

v a · nϕ dΓ →



vˆ a · nˆ ϕˆ dΓ ,

(5.104)

Γˆ

which means that the boundary integral is identical in the complete domain with respect to the direction of the normal of the reference domain. This means, that this integral will always vanish when evaluated at the boundary between two elements. The second effect can be obtained when evaluating the stiffness integral on an element Ωe as   (5.105) v a · ∇ϕ dΩ → vˆ a · ∇ˆ ϕˆ dΩ. Ωe

Ωˆ

So not only the boundary integrals but also the stiffness integrals become identical throughout the complete domain. Unfortunately the structure of the global stiffness matrix still depends on the connectivity of the computational grid but it nevertheless enables many possibilities for code optimization in order to save computational time and reduce memory consumption. For example one can think about just storing the connectivity information in a sparse matrix consisting of integers which reduces the required memory by a factor of two compared to a matrix consisting of doubles. Employing these optimizations in combination with the spectral element method enables a very efficient numerical scheme which can compete in terms of efficiency with a computation using the wave equation as shown below. Finally we can give the formal definition of the functional spaces including the Piola transformation     ˆ and q = 0 on Γ Uhk = q ∈ H01  q|Ωe ◦ Fe ∈ Q k (Ω)   1  d  k ˆ Vh = w ∈ [L 2 ]d  Je w|Ωe ◦ Fe ∈ Q k (Ω) . |Je |

(5.106) (5.107)

ˆ represents some choice of polynomial ansatz functions, and Fe the Here, Q k (Ω) ˆ = Ωe (see Sect. 2.3.2). By discretization of the variational mapping such that Fe (Ω) formulation, e.g., by spectral finite elements (see Sect. 2.9) we obtain the semi discrete Galerkin formulation as follows         fp p˙ a pa D 0 0 −R T a + = , (5.108) 0 B fv R 0 v˙ a va a

where f p , f v model any right hand side due to boundary conditions. For the final a a time discretization, a multistage scheme as the Runge-Kutta scheme is appropriate. For further details on this formulation and its application we refer to [11].

5.4 Numerical Computation

187

5.4.3 Nonlinear Acoustics The strong formulation for the nonlinear wave equation is given as follows Given: f : Ω → IR c : Ω → IR B/A, b : Ω → IR. Find: ψ(t) 1 ∂2ψ 1 ∂ − ∆ψ = f + 2 c2 ∂t 2 c ∂t

: Ω × [0, T ] → IR , such that 

B/A b(∆ψ) + 2c2

with boundary conditions ψ ∂ψ ∂n and initial conditions ψ(r, 0) ˙ 0) ψ(r,



∂ψ ∂t

2

+ (∇ψ)2



(5.109)

= ψe on Γe × (0, T ) = ψn on Γn × (0, T ) = ψ0 , r ∈ Ω = ψ˙ 0 , r ∈ Ω.

For simplicity we set all boundary conditions as well as initial conditions to zero. Now, to obtain the variational formulation of (5.109), we multiply (5.109) by an appropriate test function and perform partial integration for the Laplace term on the left- and right-hand side. Thus, the variational formulation reads as follows: Find ψ ∈ H01 such that 

Ω

1 ¨ w ψ dΩ + c2



Ω

(∇w) · (∇ψ) dΩ =



ω f dΩ

Ω

−



Ω

∇



 b ˙ dΩ w · (∇ ψ) c2



2 B w ψ˙ ψ¨ dΩ c4 2 A Ω    2 ˙ dΩ + w ∇ 2 ψ · (∇ ψ) c +

Ω

(5.110)

188

5 Acoustic Field

for any w ∈ H01 . In the following, we will assume that c and b are constant over the computational domain. Using the same ansatz for the approximations as in the linear case (see (5.90)), we arrive at the following semi-discrete Galerkin formulation n eq 

wa

a=1

n eq   b=1

−



1 Na Nb dΩ ψ¨b c2



(∇ Na ) · (∇ Nb ) dΩ ψb

Ω

−

Ω



b (∇ Na ) · (∇ Nb ) dΩ ψ˙ b c2 Ω   n eq   2 B + Nc ψ˙ c dΩ ψ¨b Na Nb c4 2 A −

c=1

Ω



n eq  2 + N (∇ N ) · ∇ N c ψc a b c2 c=1 Ω   + Na f dΩ = 0.





dΩ ψ˙ b (5.111)

Ω

In matrix form, we may write (5.111) as ˙ ψ¨ − N 2 (ψ)ψ˙ = f , M ψ ψ¨ + K ψ ψ + C ψ ψ˙ − N 1ψ (ψ) ψ

(5.112)

with Mψ = Kψ = Cψ = N 1ψ =

ne 

meψ

;

e=1 ne 

e=1 ne 

ceψ

;

ceψ

= [c pq ] ; c pq =



1 N p Nq dΩ c2

(5.113)



(∇ N p ) · (∇ Nq ) dΩ

(5.114)



b (∇ N p ) · (∇ Nq ) dΩ (5.115) c2

= [m pq ] ; m pq =

keψ ; keψ = [k pq ] ; k pq =

e=1 ne 

meψ

Ωe

Ωe

Ωe

(n1ψ )e ; (n1ψ )e = [n 1pq ] ;

e=1

n 1pq

=



Ωe

n  en  2 B ˙ N p Nq Nc ψc dΩ c4 2 A c=1

(5.116)

5.4 Numerical Computation ne 

N 2ψ =

189

(n2ψ )e ; (n2ψ )e = [n 2pq ] ;

e=1

n 2pq ne 

f =

e=1

=



Ωe

 n  en  2 N p (∇ Nq ) · ∇ Nc ψc dΩ c2

(5.117)

c=1

f e ; f e = [ f p] ; f p =



N p f (r p ) dΩ.

(5.118)

Ωe

The time discretization is performed by a standard Newmark algorithm (see Sect. 2.5.2). Since (5.112) is a nonlinear equation, we have to apply an iterative scheme. By shifting all nonlinearities to the right-hand side of the equation system, we arrive at the following scheme: 1. Perform predictor step: ∆t 2 ψ˜ = ψ n + ∆t ψ˙ n + (1 − 2βH ) ψ¨ n = ψ kn+1 2 k . ψ˜˙ = ψ˙ + (1 − γH )∆t ψ¨ = ψ˙ n

n

n+1

(5.119) (5.120)

2. Solve algebraic system of equations: k+1 M ∗ψ ψ¨ n+1 = f n+1

M ∗ψ

− =

(5.121)

k k K ψ ψ˜ − C ψ ψ˜˙ + N 1ψ (ψ˙ n+1 )ψ¨ n+1 M ψ + γH ∆t C ψ + βH ∆t 2 K ψ .

+

k N 2ψ (ψ kn+1 )ψ˙ n+1

(5.122)

3. Perform corrector step: k+1 ψ k+1 = ψ˜ + βH ∆t 2 ψ¨ n+1 n+1

k+1 k+1 ψ˙ n+1 = ψ˜˙ h + γH ∆t ψ¨ n+1 .

(5.123) (5.124)

4. Test convergence: k+1 k ψ¨ n+1 − ψ¨ n+1 2 k+1 ψ¨ n+1 2

≤ δψ



fulfilled : perform next time step not fulfilled : k := k + 1 goto step 2.

(5.125)

190

5 Acoustic Field

5.4.4 Non-conforming Grids Within this section, we will investigate flexible discretization techniques for the approximate solution of acoustic wave propagation problems. In particular, we will introduce non-conforming grids, where we can allow totally different grids to exist at the interface of two subregions. This is in particular of great interest in computational aeroacoustics, where we need a much finer grid in the source region, where the turbulent flow generates the sound, than in the regions, where we just compute the propagation of sound (see Chap. 9). In order to keep as much flexibility as possible, we intend to use independently generated grids which are well suited for approximating the solution of decoupled local subproblems in each subdomain. Therefore, we have to deal with the situation of non-conforming grids appearing at the common interface of two subdomains. Special care has to be taken in order to define and implement the appropriate discrete coupling operators. We will investigate in two approaches (see Sect. 2.10) to handle non-conforming grids: (1) Mortar coupling and (2) Nitsche type coupling. In the first approach we guarantee the strong coupling of the numerical flux (normal derivative of the acoustic pressure) by introducing a Lagrange multiplier and coupling of the acoustic pressure in a weak sense. Nitsche type coupling does not need the additional Lagrange multiplier and handles the coupling by symmetrizing the bilinear form and adding a special jump term. The global domain and its decomposition is displayed in Fig. 5.9. Thus, in each subdomain we have to solve the wave equation for the acoustic pressure pa i : Ωi × (0, T ) → IR, 1 p¨a − ∆ pa i = f i , in Ωi × (0, T ), i = 1, 2 (5.126) c2 i completed by appropriate initial conditions at time t = 0 and boundary conditions on the global boundary Γa . According to the physical interface conditions, we have to impose continuity in the trace and flux of the acoustic pressure, i.e., pa1 = pa2 and

Γa

∂ pa2 ∂ pa1 = on ΓI , ∂n ∂n

ΓI

n

Ω1

Ω1 Ω2

ΓI Ω2

Fig. 5.9 Acoustic domain with two subregions Ω1 and Ω2 with different discretizations

5.4 Numerical Computation

191

5.4.4.1 Mortar Formulation The flux coupling condition will be enforced in a strong sense by introducing the Lagrange multiplier [12] ∂ pa2 ∂ pa1 =− . (5.127) λ=− ∂n ∂n However, the continuity of the trace will be understood in a weak sense 

ΓI

( pa1 − pa2 )µ dΓ = 0

(5.128)

for all test functions µ out of a suitable Lagrange multiplier space. We proceed with performing the weak formulation and obtain from (5.126), ignoring for the moment the boundary condition on Γa and the excitations f i , 

1 wi p¨ai dΩ + c2



∇wi · ∇ pai dΩ −

wi ni ·∇ pai dΓ = 0 ,

ΓI

Ωi

Ωi



for all test functions wi , i = 1, 2. Inserting the definition of the Lagrange multiplier (5.127) and summing up, we obtain the symmetric evolutionary saddle point problem of finding pa1 , pa2 and λ such that ⎛ ⎞   2  ⎟ ⎜ 1 ∇wi · ∇ pai dΩ ⎠ wi p¨ai dΩ + ⎝ 2 c i=1

Ωi

Ωi

+



(w1 − w2 )λ dΓ = 0

(5.129)



( pa1 − pa2 )µ dΓ = 0

(5.130)

ΓI

ΓI

for all µ and wi , i = 1, 2. We now face a primal-dual problem where the coupling is realized in terms of Lagrange multipliers. Now, we perform a spatial discreizations and assume the Lagrange multipliers to be chosen with respect to Ω1 . The resulting semi-discrete Galerkin formulation reads as ⎛ ⎞ ⎛ ⎛ ⎞ ⎞⎛ ⎞ p¨   a1 p a1 K1 0 D M1 0 0 ⎜ ⎟ ⎝ 0 M2 0 ⎠ ⎜ p¨ ⎟ + ⎝ 0 K2 M ⎠ ⎝ p ⎠ = 0 . (5.131) ⎝ a2 ⎠ a2 0 T MT 0 0 0 0 D λ λ¨ The coupling matrices D, M are given by

192

5 Acoustic Field

De = [D pq ]; D pq =



M = [M pq ]; M pq =

ΓIe



ΓI

N p1 φq dΓ,

(5.132)

N p2 φq dΓ,

(5.133)

where N p1 and N p2 denote the finite element basis functions on T1 and T2 , respectively, and φq denotes the Lagrange multiplier basis function associated with the node q. We note that the assembly of D poses no difficulty since all basis functions involved are defined with respect to the same grid T1 . However, the assembly of M is more involved, since N p2 and φq are defined with respect to different grids (see Sect. 2.10). 5.4.4.2 Nitsche Type Mortaring To handle the non-conforming discretization within Nitsche’s method, we start at the weak formulation for both subdomains Ω1 and Ω2 (again for simplicity we neglect the surface integral over the outer boundary Γa and set f i to zero) 

  1 ∂ pa1 w p ¨ dΩ + dΓ = 0 ∇w · ∇ p dΩ − w1 1 a1 1 a1 c2 ∂n1 Ω1 Ω1 ΓI    ∂ pa2 1 w2 p¨a2 dΩ + ∇w2 · ∇ pa2 dΩ − w2 dΓ = 0. c2 ∂n2 Ω2

Ω2

(5.134)

(5.135)

ΓI

In a next step, we add the two Eqs. (5.134) and (5.135), and explore the relation n = n1 = −n2 ;

∂ pa2 ∂ pa1 ∂ pa2 ∂ pa1 = = =− ∂n1 ∂n ∂n2 ∂n

to arrive at 

Ω1

  1 1 w1 p¨a1 dΩ + ∇w1 · ∇ pa1 dΩ + w2 p¨a2 dΩ c2 c2 Ω1 Ω2   ∂ pa1 dΓ = 0. + ∇w2 · ∇ pa2 dΩ − [w] ∂n Ω2

(5.136)

ΓI

In (5.136) the operator [ ] defines the jump operator, e.g., [w] = w1 − w2 . In order to retain symmetry, we add to (5.136) the following term −



ΓI

[ pa ]

∂w1 dΓ with [ pa ] = pa1 − pa2 . ∂n

5.4 Numerical Computation

193

This operation is allowed, since [ pa ] is forced to be zero at the interface. In a final step, we add along the interface ΓI the following penalty term β

 1  [ pa ] [w] dΓ , hE E

ΓE

with β the penalty factor. Therewith, we arrive at the following final formulation for the Nitsche’s approach 

1 w1 p¨a1 dΩ + c2

Ω1

+



Ω2



∇w1 · ∇ pa1 dΩ +

Ω1



1 w2 p¨a2 dΩ c2

Ω2

 ∂w1 ∂ pa1 dΓ − [ pa ] dΓ ∇w2 · ∇ pa2 dΩ − [w] ∂n ∂n ΓI ΓI  1  +β [ pa ] [w] dΓ = 0. hE 

E

(5.137)

ΓE

Now, performing a spatial discretization results in symmetric matrices. So, this method preserves the property of the original system. In order to compute the entries of the matrices obtained by the boundary integrals over ΓI , the same intersection operations as in the Mortar method are necessary. Furthermore, within the numerical computations it turned out that the formulation is quite robust with respect to the penalty factor β, and we chose a value of about 20.

5.4.4.3 Numerical Results To study both approaches, we perform numerical computations on a setup as displayed in Fig. 5.10. Thereby, the acoustic wave is excited on the inner circle by

Fig. 5.10 Computational setup

194

5 Acoustic Field

Fig. 5.11 Computational mesh (just a zoom)

Fig. 5.12 Computational result: acoustic pressure at different time steps

a pressure pulse. On the outer boundary we simply apply some absorbing boundary conditions of first order (see Sect. 5.5.1). The discretization of both subdomains is shown in Fig. 5.11. The inner domain Ω1 is discretized by triangles and we use first order finite elements. The computational grid of the outer domain Ω2 is quite coarse, however we use basis functions of 6th order on the quadrilaterals. Furthermore, the time discretization is performed by an implicit Newmark scheme, which is of second order accuracy in time. The time step size is chosen by Tm /20, where Tm is the length of the excitation pulse (see Fig. 5.10). For both simulations, we performed the computations in time domain and Fig. 5.12 displays for three characteristic time steps the acoustic pressure pulse traveling outwards. Here, the results of Nitsche’s formulation is shown. As can be seen, the pulse travels over the interface without any reflections. In order to compare the results of Mortar and Nitsche’s formulation, we recorded the computational results in the observation point (see Figs. 5.10, and 5.13) displays the acoustic pressure over time. One can observe that the two results agree very well. So we can demonstrate that both formulations can successfully handle non-conforming discretizations for acoustic wave propagations.

5.4.5 Discretization Error In this section, we will investigate the discretization error of the FE method applied to the wave equation and will provide practical rules for choosing the mesh size h as

5.4 Numerical Computation

195

Fig. 5.13 Acoustic pressure over time at the observation point

a function of the wave number k. In the first step we will restrict to the time harmonic case to get rid off the error introduced by the time discretization. The wave equation in the time harmonic case, also called Helmholtz equation, reads as follows ∆ pˆ + k 2 pˆ = 0

(5.138)

with k = ω/c = 2π f /c the wave number. The comparison of the FE solution to the analytic solution of this kind of PDE has already been discussed in Sect. 2.2 (just substitute there the parameter c by k). There, we have observed that the FE solution for the parameter c = 0 is no longer exact at the FE nodes (see Fig. 2.7). In the context of the Helmholtz equation we decompose the total discretization error into a local error eint (interpolation error) defined by eint = pˆ − pˆ I and into the pollution error epol computed by epol = pˆ I − pˆ h . This decomposition is illustrated in Fig. 5.14. A very important result concerning the error analysis has been reported in [13], which reads as follows   eh ≤ C1 (kh) + C2 k 3 h 2 eh =

kh < 1

(5.139)

pˆ h |

| pˆ − H1 | p| ˆ H1

with | · | H 1 the H 1 semi-norm (see Sect. D) and C1 , C2 constants independent of k as well as h. In (5.139) the first term represents the local error and the second term

196

5 Acoustic Field

pˆ(x)

pˆ pˆh pˆI epol

eint x Fig. 5.14 Decomposition of the discretization error into the interpolation error eint and pollution error epol

the pollution error. The main effect of the pollution error is that the wave number k h of the FE solution is different from the wave number k of the analytic solution. We call this dispersive and therefore speak of the dispersion error. The behavior of the dispersion error has been extensively studied in [14]. This analysis provides a concrete guideline for choosing the order q of the finite element basis functions and the mesh size h in order that the dispersion error is virtually eliminated. This relation is given by q+

kh 1 > + C(kh)1/3 2 2

(5.140)

with the constant C, which can in practice be set to one. In addition, we can precisely define the dispersion error in the small wave number limit kh > 1  q! 2 (kh)2q+2 + O(kh)2q+4 (2q)! 2q + 1  2q+1 khe sin(kh) cos(k h h) − cos(kh) = 2 2(q + 1) 1 cos(k h) − cos(kh) = 2 h



kh > 1

In Sect. 2.9, we have considered an acoustic wave in a channel with given Dirichlet boundary conditions (see Fig. 2.22). Thereby, we have demonstrated the poor convergence rate when using h-FEM (successively reduction of the element mesh size h). Here, we will apply FE basis function of higher order to increase the rate of convergence (see Sect. 2.9). The overall number of unknowns remain in a similar order of

5.4 Numerical Computation

197

Fig. 5.15 Accumulated error in computational domain (see (2.116)). Exponential convergence with increasing polynomial order for spectral (s-FEM) and hierarchical (p-FEM) finite elements

magnitude but the numerical error is tremendously reduced compared to h-FEM as depicted in Fig. 5.15. Both presented approaches p-FEM and s-FEM (FE method of higher order using spectral elements, see Sect. 2.9) show exponential convergence and perform almost equally well although the spectral method shows slightly better results. Also visible is that for the very coarse discretization of one element per wavelength, the numerical error remains almost constant even though the polynomial order is increased. This behavior is expected, since (5.140) is not fulfilled. In a second example, we consider a sine pulse traveling through a channel and record the wave signal at a distance of two times the fundamental wavelength. Now, in addition to the correct choice of the mesh size h we have to deal with the time discretization. Figure 5.16 displays the setup. The choice of the time and spatial discretization is taken as a function of the fundamental exciting frequency f 0 , which results in a fundamental wavelength λ0 = c/ f 0 . Fig. 5.16 Setup of the transient case study

Propagation direction

Excitation: sine pulse Recording of wave signals

pa (t) t

198

5 Acoustic Field

Table 5.3 Time and spatial discretization Discretization 1 1/(20 f 0 ) λ0 /20

Discretization 3

1/(40 f 0 ) λ0 /40

1/(60 f 0 ) λ0 /60

pa (t) (Pa)

h ∆t

Discretization 2

t (ms) Fig. 5.17 Recorded pulse signals at a distance of two times the fundamental wavelength

Table 5.3 lists the mesh size h as well as time step size ∆t for the three different computations. In Fig. 5.17 it can be clearly noticed that around the pulse wave parasitic waves arise due to the numerical dispersion. In fact, the generated parasitic waves occur since the discrete speed of sound ch is a function of the wave number k and therefore of the frequency of the wave. Clearly, this is a pure numerical effect, since in a homogeneous media the continuous speed of sound c is constant. For a detailed discussion on this topic we refer to [8]. Again, using FE formulations of higher order achieves quite better results. Figure 5.18 displays the results obtained by a mesh of 10 finite elements per wavelength and fourth order basis functions both for the p-FEM and s-FEM method.

5.5 Treatment of Open Domain Problems Many applications within computational acoustics are open-domain problems. In order to correctly compute such problems with the FE method, we have to define appropriate boundary conditions. Using a simple homogeneous Dirichlet or break Neumann boundary condition will result in a total reflection of the outgoing waves at the boundary. Therefore, special boundary conditions have to be developed for absorbing the waves impinging on the artificial boundary imposed on the acoustic domain.

5.5 Treatment of Open Domain Problems

199

1.0

s-FEM p-FEM

pa (t) (Pa)

0.5

0.0 -1.0 -0.5 14.0

14.5

15.0

15.5

16.0

16.5

17.0

t (ms) Fig. 5.18 Recorded pulse signals of p-FEM and s-FEM computations at a distance of two times the fundamental wavelength

The three main approaches handling open-domain problems with the FE method can be briefly described as follows: • Infinite finite elements: At the boundary an additional layer of finite elements, so-called infinite finite elements try to model the effect of waves in the outer region (non-computational domain). For a basic discussion we refer to [15], and for an advanced formulation see [16]. • Absorbing boundary conditions: The general idea of applying at the boundary a special condition, so-called absorbing boundary conditions, is that only outgoing waves can pass the boundary. • Perfectly matched layer (PML) technique: By this method the computational domain is surrounded by a damping layer, which is constructed in such a way that the acoustic impedance of this layer matches the impedance of the computational (wave propagation) domain and in which the waves are absorbed. In the following, we will provide details about ideas and computer implementation of local absorbing boundary conditions of first order and of the perfectly matched layer technique.

5.5.1 Absorbing Boundary Conditions In [17, 18] absorbing boundary conditions for general classes of wave equations are discussed. Subsequently, a hierarchy of highly absorbing local boundary conditions that approximate the theoretical non-local relation have been obtained. The locality of these conditions preserves the sparsity of the FE matrices. For an explanation of how absorbing boundary conditions work, let us consider a boundary located

200

5 Acoustic Field

p− a (x, t)

x

Fig. 5.19 Wave propagating in the direction of negative x values

at x = 0. A wave with one degree of freedom propagating in the direction of negative x values, as displayed in Fig. 5.19, can be expressed by pa− (x, t) = pˆ ej(ωt+kx)

(5.141)

with ω the angular frequency and k the wave number. By considering the relation for the speed of sound c = ω/k, it can be easily proven that the wave fulfills the following condition   ∂ ∂ pa = 0. −c (5.142) ∂t ∂x Thus, the wave can totally pass (no reflection) the boundary. However, a wave, which is propagating towards positive x values and that can be mathematically modeled by pa+ (x, t) = pˆ ej(ωt−kx)

(5.143)

will not fulfill (5.142) and will be totally reflected. Indeed, (5.142) defines a locally absorbing boundary condition of first order (see [17, 18]). An extension of this method to waves in solids (especially piezoelectric materials) can be found in [19]. To derive the correct formulation including the absorbing boundary condition as given in (5.142), we start from the weak form (see (5.89)) without setting the boundary integral to zero (which would correspond to homogeneous Neumann boundary condition) 

Ω

1 w p¨a dΩ + c2



Ω

∇w · ∇ pa dΩ −



Γ

w

∂ pa dΓ − ∂n



w f dΩ = 0. (5.144)

Ω

Now, in all directions we can express the 1D spatial derivative in (5.142) as the normal derivative in (5.144). By doing so, special attention has to be given to the direction of the normal vector relative to the positive direction of the x axis. In our case, the normal points out of the boundary towards the infinite domain that is in the same direction of the outgoing wave, whereas x is positive in the direction opposite to the propagation of the normal wave. Finally, changing the direction of the normal

5.5 Treatment of Open Domain Problems

201

vector and substituting (5.142) into (5.144) results in 

Ω

1 w p¨a dΩ + c2



∇w · ∇ pa dΩ +

Ω



1 ∂ pa w dΓ − c ∂t

Γ



w f dΩ = 0. (5.145)

Ω

The additional surface integral, including a first time derivative of the acoustic pressure pa , may be seen as a damping matrix C acting only on the surface of the computational domain. After performing the spatial discretization using finite elements, we obtain the following formula for the computation of C Γ

C=

ne 

e=1

e

e

c ; c = [c pq ] ; c pq =



Γe

1 N p Nq dΓ , c

(5.146)

with n Γe the number of surface elements (for the evaluation of the surface integral see Sect. 2.6). The matrix C is almost empty, since only terms along the boundary Γ of the domain contribute to its entries.

5.5.2 Perfectly Matched Layer (PML) Technique There has been much research work on the PML-technique since the first introduction of this technique [20] (see e.g., [21–24]). In this section we will focus on the acoustic wave propagation in the frequency domain and invest in basic ideas concerning impedance matching, describe the PML-technique and prove the perfect matching. For linear acoustics in the 1D case the relation between the acoustic particle velocity v a = vξ eξ and the acoustic pressure pa is given by (conservation of mass and momentum, see Sect. 5.1) ∂vξ ∂ pa = −ρ0 c2 ∂t ∂ξ

∂vξ 1 ∂ pa =− , ∂t ρ0 ∂ξ

which results in the linear wave equation for pa 1 ∂ 2 pa ∂ 2 pa − = 0. c2 ∂t 2 ∂ξ 2

(5.147)

For a plane wave the acoustic impedance Z a is calculated by ρ0 c, and the reflection coefficient R between two media as displayed in Fig. 5.20 computes as R=

Z a2 − Z a1 Z a2 + Z a1

with

Z a1 = ρ0 c

Z a2 = ρ˜c. ˜

(5.148)

202

5 Acoustic Field Interface

Interface Propagation-region

ρ0 , c

Propagation-region

Damping-region

ρ˜, c˜

Damping-region

ϕ1 ρ0 , c

ϕ2 ρ˜, c˜

Fig. 5.20 Plane wave impinging perpendicular (left) and at angle ϕ1 to the interface (right)

Clearly, to make R equal to zero, Z a2 has to match Z a1 . However, we are free in choosing the two quantities ρ˜ and c˜ in the damping region in such a way that just their product equals ρ0 c. Therefore, by setting ρ˜ = ρ0 (1 − jσξ )

c˜ =

c 1 − jσξ

(5.149)

in the damping region, we obtain for the acoustic impedance Z a2 = Z a1 . Furthermore, the wave Eq. (5.147) reads in the time-harmonic case as ω2 ∂ 2 pˆ p ˆ + =0 c˜2 ∂ξ 2

(5.150)

with ω the pulsation of the wave and pˆ the acoustic pressure amplitude. Without any restrictions, we choose σξ = σ0 (constant within the damping region) and obtain ω2 ∂ 2 pˆ (1 − jσ0 )2 pˆ + 2 = 0. 2 ∂ξ c 

(5.151)

k˜ 2

Therewith, we arrive at a wave equation with a complex wave number k˜ ω k˜ = (1 − jσ0 ) = k(1 − jσ0 ). c

(5.152)

Since the general solution of (5.151) is ˜

pˆ = pˆ 0 e j (ωt−kξ) = pˆ 0 e j (ωt−kξ) e−σ0 ξ , we achieve our goal that we have impedance matching and damping of the wave at the same time. However, if we consider a plane wave impinging at an angle ϕ to the normal vector of the interface (see Fig. 5.20), then the acoustic impedance compute by (see e.g., [1])

5.5 Treatment of Open Domain Problems

Z a1 =

203

ρ0 c cos ϕ1

Z a2 =

ρ˜c˜ cos ϕ2

and our method will not be working. Therefore, as introduced in [20] we need a splitting of the acoustic pressure pa into pax , pay and paz and introduce artificial damping. Therewith, the mass as well as momentum conservation equation for linear acoustics change to ∂vax 1 ∂ pa + σx vax = − ∂t ρ0 ∂x ∂vay 1 ∂ pa + σ y vay = − ∂t ρ0 ∂ y ∂vaz 1 ∂ pa + σz vaz = − ∂t ρ0 ∂z

∂vax ∂ pax + σx pax = −ρ0 c2 ∂t ∂x ∂ pay ∂vay + σ y pay = −ρ0 c2 ∂t ∂y ∂ paz ∂v az + σz paz = −ρ0 c2 ∂t ∂z

(5.153) (5.154) (5.155)

In the above equations σx , σ y and σz are damping functions, which are zero within the acoustic propagation domain and which are different from zero within the PML-layer enclosing the acoustic propagation domain (see Fig. 5.21). By applying a Fouriertransformation to (5.153)–(5.155) and rearranging the involved terms, we arrive at the following equations 1 jω + σx 1 pˆ y = −ρ0 c2 jω + σ y 1 pˆ z = −ρ0 c2 jω + σz pˆ x = −ρ0 c2

∂ vˆ x ∂x ∂ vˆ y ∂x ∂ vˆ z ∂z

1 ∂ pˆ x 1 ρ0 jω + σx ∂x ∂ pˆ y 1 1 vˆ y = − ρ0 jω + σ y ∂ y 1 ∂ pˆ z 1 vˆ z = − . ρ0 jω + σz ∂z vˆ x = −

(5.156) (5.157) (5.158)

As can be seen from (5.156)–(5.158), we can directly compute vˆ x , vˆ y and vˆ z as a function of pˆ x , pˆ y and pˆ z . Therewith, we obtain for the total acoustic pressure PML region

Propagation region

ΩPML

Ωprop

Fig. 5.21 Computational setup: propagation region surrounded by a PML-region

204

5 Acoustic Field

pˆ = pˆ x + pˆ y + pˆ z the following modified Helmholtz equation ∂ η y ηz ∂x



1 ∂ pˆ ηx ∂x



∂ + η x ηz ∂y



1 ∂ pˆ ηy ∂ y



∂ + ηx η y ∂z



1 ∂ pˆ ηz ∂z



+ ηx η y ηz k 2 pˆ = 0.

(5.159)

In (5.159) k = ω/c denotes the acoustic wave number and the functions η1 , η2 and η3 compute as follows ηx = 1 − j

σx ω

ηy = 1 − j

σy ω

ηz = 1 − j

σz . ω

(5.160)

This decomposition of the acoustic pressure pa into its x-component, y-component and z-component is the key point of the PML-technique. For a physical interpretation we will consider an interface between a propagation and a PML region, which is parallel to the y-axis and has a normal vector pointing in x-direction (see Fig. 5.22). Now, the x-component pax of the total acoustic pressure pa can be regarded as a plane wave propagating just in x-direction and which will be damped in the PMLregion with the damping coefficient σx . Therewith, we have the 1D case and achieve our goal of a reflection-less interface, since the impedance matches. This simple consideration already provides us with the information, where to choose the individual damping coefficients σx , σ y and σz different from zero. Figures 5.23 and 5.24 display the PML-technique for the 2D and 3D case. At this point we want to note that there is a second general method to derive (5.159). This is known as a mapping of the solution of Helmholtz equation in the real coordinate space to a complex coordinate space, e.g., an analytic continuation of the solution (see e.g., [21, 25]). Thereby, the following complex change of variables inside ΩPML is introduced 1 x˜i (xi ) = xi + jω

xi

σi (x) dx ;

xi ∈ {x, y, z} ,

(5.161)

0

Fig. 5.22 Physical interpretation of PML-technique

Interface

y

Propagation region

x

PML region

σx = 0 σy = 0

5.5 Treatment of Open Domain Problems

205

σx = 0, σy = 0 σx = 0 σy = 0 y σx = 0 σy = 0

x Fig. 5.23 Construction of PML layer in 2D

σx = 0, σy = 0, σz = 0 σx = 0, σy = 0, σz = 0 σx = 0, σy = 0, σz = 0 σx = 0, σy = 0, σz = 0

y x z σx = 0, σy = 0, σz = 0

σx = 0, σy = 0, σz = 0

σx = 0, σy = 0, σz = 0 Fig. 5.24 Construction of PML layer in 3D

where the damping function σi is positive inside ΩPML and vanishes in Ωprop . Hence, we obtain the following relations σi ∂ 1 ∂ ∂ x˜i = ηi and =1− j = . ∂xi ω ∂ x˜i ηi ∂xi Now, let us prove that the PML will have the same acoustic impedance as the propagation medium (a similar proof is given in [20] for electromagnetic waves). Since any general solution of the homogeneous wave equation can be expressed by the superposition of plane waves, we can restrict our investigation to plane waves impinging at an arbitrary angle ϕ on the interface, where the PML starts. Therewith, we consider a setup as displayed in Fig. 5.25. In this 2D consideration, the set of partial differential equations describing the plane wave in the layer are as follows

206

5 Acoustic Field

Fig. 5.25 Wave propagation in the PML region

Interface PML-region

Propagation-region

ϕ

vx

vy

∂vax ∂t ∂vay ∂t ∂ pax ∂t ∂ pay ∂t

1 ∂ ( pax + pay ) ρ0 ∂x 1 ∂ ( pax + pay ) =− ρ0 ∂ y ∂vax = −ρ0 c2 ∂x ∂v ay . = −ρ0 c2 ∂y

+ σx vax = − + σ y vay + σx pax + σ y pay

(5.162)

(5.163)

Our goal is to compute the acoustic impedance in the PML. Since we assume a plane wave, we can perform the following ansatz for the general solutions of the components of the acoustic particle velocity and pressure vax = v0 sin ϕ e j (ωt−k x x−k y y) vay = v0 cos ϕ e j (ωt−k x x−k y y)

˜

˜

˜

˜

(5.164)

pax = px0 e

.

(5.165)

j (ωt−k˜ x x−k˜ y y)

pay = p y0 e

j (ωt−k˜ x x−k˜ y y)

In the first step, we substitute these general solutions into (5.163) and obtain ρ0 c2 v0 sin ϕ k˜ x ρ0 c2 k˜ x v0 sin ϕ = jω + σx ηx ω 2 2 ρ0 c k˜ y v0 cos ϕ ρ0 c v0 cos ϕ k˜ y . = = jω + σ y ηy ω

px0 =

(5.166)

p y0

(5.167)

Using this intermediate results in (5.162) will lead us to the following two relations j k˜ x c2 ηx sin ϕ = ω2



k˜ x sin ϕ k˜ y cos ϕ + ηx ηy



j k˜ y c2 η y cos ϕ = ω2



k˜ x sin ϕ k˜ y cos ϕ + ηx ηy



(5.168)

,

(5.169)

5.5 Treatment of Open Domain Problems

207

so that we can e.g., express k˜ y by a function of k˜ x k˜ y =

η y cos ϕ k˜ x . ηx sin ϕ

Substituting this result into (5.168) leads to a quadratic equation for k˜ x ηx2 sin2 ϕ =

 k˜ x2 c2  2 2 sin ϕ + cos ϕ , ω2  

=1

form which we just take the positive solution

 σx  ω . k˜ x = ηx sin ϕ = k sin ϕ 1 − j c ω

(5.170)

Performing similar steps results in the expression for k˜ y  σy  ω . k˜ y = η y cos ϕ = k cos ϕ 1 − j c ω

(5.171)

Therewith, the substitutions of these terms into (5.166) and (5.167) lead to px0 = ρcv0 sin2 ϕ

p y0 = ρcv0 cos2 ϕ ,

so that the total pressure computes as   px0 + p y0 = ρcv0 sin2 ϕ + cos2 ϕ = ρcv0 . The acoustic impedance of the perfectly matched layer now results in the same expression as for a plane wave in the propagation media Za =

p = ρ0 c. v

The proper choice of the damping functions is of great importance, especially in order to obtain a very robust and efficient PML-technique. For this purpose, let us consider the case, in which a wave is propagating within the PML-layer in y-direction and having an angle of ϕ with respect to the y-axis. The total pressure pa computes as ˜ ˜ ˜ ˜ (5.172) pa = pax + pay = px0 e j (ωt−k x x−k y y) + p y0 e j (ωt−k x x−k y y) . Substituting the values for k˜ x , k˜ y according to (5.170), (5.171) and considering that for the chosen case σx = 0, we obtain

208

5 Acoustic Field

pa = ( px0 + p y0 ) e j (ωt−kx sin ϕ−ky cos ϕ) e−(σ y /c) cos ϕ = p0 e−(σ y /c) cos ϕ . (5.173) Assuming a layer thickness of L, the damped wave will be totally reflected at the outer boundary of the PML-region and this reflected wave at the interface between propagation and PML region takes the value

par = p0 e

−(2/c) cos ϕ

#L

σ y (y)dy

o

= p0 R.

(5.174)

A reasonable choice of the reflection factor R is 10−3 , since we have to take care that a too strong damping in a too small PML-region can strongly disturb the numerical solution. In addition, in order to get rid of the dependence of the overall damping on the speed of sound c, we will choose all damping functions σ y direct proportional to c (see (5.174)). In a first case, we will assume a constant damping σ y = σ0 . Therewith, we obtain from (5.174) the following relation for σ0 σ0 =

−c ln R . 2L cos ϕ

(5.175)

In a second case, we consider a quadratically increasing damping function, hence we set 2 q y σ y = σ0 2 , L and assume that y is equal to zero at the interface and is increasing within the PMLq region. Again, exploiting (5.174) we arrive at a relation for the constant factor σ0 q

σ0 =

3c ln R . 2L cos ϕ

(5.176)

In the last step, we will introduce a singular function, given by σy =

c , L−y

(5.177)

which means that we increase the damping inverse with the distance. By using this damping function, the PML is named unbounded and in [26] it could be proved that optimality for the Helmholtz equation can be achieved. Furthermore, we want to emphasize that the damping functions according to (5.175) as well as (5.177) are different from zero at the interface, and therefore will introduce a discontinuity at the interface. However, due to the properties of the PML-technique, no spurious reflections will occur. For the FE formulation of the wave equation within a PML region, we start at the strong setting, which reads as follows.

5.5 Treatment of Open Domain Problems

209

Given: fˆ : Ω → C I c, ρ : Ω → C. I : Ω¯ → C I , such that

Find: pˆ

η y ηz

∂ ∂x



1 ∂ pˆ ηx ∂x



+ η x ηz

∂ ∂y



1 ∂ pˆ ηy ∂ y



  1 ∂ pˆ ∂ ∂z ηz ∂z 2 η x η y ηz pˆ − fˆ = 0 (5.178) +ω c2

+ ηx η y

with boundary conditions ∂ pˆ = 0 on Γ. ∂n

(5.179)

I the set of complex In (5.178) and (5.179) fˆ denotes any acoustic source term, C numbers, Γ the outer boundary of the computational domain and ηx , η y , ηz the damping functions, which compute according to (5.160). In the first step, we multiply (5.178) by an appropriate test function v and integrate over the whole domain Ω 

Ω



∂ v η y ηz ∂x



1 ∂ pˆ ηx ∂x



  ∂ 1 ∂ pˆ + ηx η y ∂z ηz ∂z  η x η y ηz + ω2 pˆ − fˆ dΩ = 0. (5.180) c2

∂ + η x ηz ∂y



1 ∂ pˆ ηy ∂ y



Now, we apply Green’s integral theorem to the second-order spatial derivatives and incorporate the homogeneous Neumann boundary condition (5.179). These steps lead to the following weak formulation of (5.179): Find pˆ a ∈ H 1 such that   ηx η y ∂v ∂ pˆ η x η y ηz η y ηz ∂v ∂ pˆ ηx ηz ∂v ∂ pˆ + + dΩ − ω 2 v pˆ dΩ ηx ∂x ∂x ηy ∂ y ∂ y ηz ∂z ∂z c2 Ω Ω  = v fˆ dΩ (5.181) Ω

for any v ∈ H 1 . Using standard nodal finite elements, we arrive at the following discrete complex algebraic system of equations 

 K − ω 2 M pˆ = fˆ

(5.182)

210

5 Acoustic Field

with K the stiffness matrix, M the mass matrix, pˆ the nodal vector of complex acoustic pressure and fˆ the complex nodal vector of the right-hand side. For a stability investigation we refer, e.g., to [27, 28]. Performing an inverse Fourier transform of (5.178) to arrive at a time domain formulation, will lead to convolution integrals, see e.g. [29]. A method to avoid convolution integrals is the use of auxiliary variables as demonstrated, e.g., in [30–32]. We start at the modified Helmholtz equation as given in (5.178), and which has to be solved in ΩPML . Before we proceed, we use the fact that ∂ηl /∂xk = 0 for k = l to rewrite (5.178) by η x η y ηz



jω c

2

∂ pˆ − ∂x



η y ηz ∂ pˆ ηx ∂x



 ∂ η x ηz − ∂y ηy  ∂ ηx η y − ∂z ηz

 ∂ pˆ ∂y  ∂ pˆ = 0. ∂z

(5.183)

Now, we investigate in the first term of (5.183) and expand the terms ηi by (5.160) to obtain η x η y ηz



jω c

2

     σy σx σz jω 2 1+ 1+ 1+ pˆ pˆ = c jω jω jω 1  = 2 ( jω)2 + jω(σx + σ y + σz ) + σx σ y + σx σz c  σ x σ y σz p. ˆ (5.184) + σ y σz + jω 

For an inverse Fourier transformation of (5.184) we recognize, that the last term will result in an integral over time. Therefore, we introduce the first auxiliary variable according to pˆ . (5.185) v= jω In a next step, we analyze the second term in (5.183) and start with

II =

η y ηz ∂ pˆ = ηx ∂x

 1+

σy jω





1+

1+ 

σx jω

σz jω



∂ pˆ ∂x

Now, we perform the following rearrangements and use also (5.185)

(5.186)

5.5 Treatment of Open Domain Problems

II = =

(σ y + jω)(σz + jω) ∂ pˆ jω(σx + jω) ∂x σ y σz jω + (σ y + σz ) + jω ∂ pˆ

211

±

σx ∂ pˆ σx + jω ∂x

(σx + jω) ∂x + (σ y + σz − σx ) ∂ pˆ ∂ pˆ = + (σx + jω) ∂x ∂x   ∂v ∂ pˆ ∂ pˆ 1 σ y σz + + (σ y + σz − σx ) . = σx + jω ∂x ∂x ∂x σ y σz jω

(5.187)

The same procedure is performed on the third and fourth term in (5.183). Having an inverse Fourier transform in mind, we introduce a vectorial auxiliary variable u with the following relations   ∂v ∂ pˆ 1 σ y σz + (σ y + σz − σx ) σx + jω ∂x ∂x   ∂v 1 ∂ pˆ σ x σz + (σx + σz − σ y ) uy = σ y + jω ∂y ∂y   1 ∂v ∂ pˆ uz = σx σ y . + (σx + σ y − σz ) σz + jω ∂z ∂z

ux =

(5.188)

Now, we are ready to apply the inverse Fourier transform to (5.183), (5.185), (5.188) and achieve at the following coupled system of partial differential equations 1 ∂ 2 pa ∂ pa + β pa + γv − ∇ · ∇ pa − ∇ · u = 0 +α c2 ∂t 2 ∂t ∂u + A u + B ∇ pa − C ∇v = 0 ∂t ∂v = pa ∂t

(5.189) (5.190) (5.191)

with α=

σ x + σ y + σz σ x σ y + σ x σz + σ y σz σ x σ y σz ; β= ; γ= 2 2 c c c2

⎛

σx A=⎝ 0 0 ⎛ σx B=⎝

⎞ ⎛ ⎞ σ y σz 0 0 0 0 σ y 0 ⎠ ; C = ⎝ 0 σ x σz 0 ⎠ 0 0 σx σ y 0 σz

⎞ − σ y − σz 0 0 ⎠. 0 σ y − σ x − σz 0 0 0 σz − σ x − σ y

(5.192)

(5.193)

(5.194)

212

5 Acoustic Field

In [32], a full stability analysis has been performed. This analysis revealed a critical term, and this term could be a source for long time instabilities. Therefore, we also define the formulation, where C is set to zero (i.e. we omit C ∇v in (5.190)) and will denote this reduced formulation by rPML. We are aware of the fact that this rPML formulation will not achieve perfect matching. However, as numerical results will demonstrate, the additional error compared to the full PML formulation is small, and it strongly increases the stability in case of thin damping layers (see Sect. 5.6.1). Furthermore, we want to state that the rPML is a true PML in case of 1D as well as 2D computations. In these cases, we do not need the additional scalar auxiliary variable v and so C is not present. E.g., in 2D we just need the auxiliary vector variable u = (u x , u y )T leading to a total number of just three unknowns in the PML region. This can be also seen by analyzing (5.189)–(5.191), e.g., assuming waves in the x y−plane. Then γ and C get zero (σz = 0) resulting in just three scalar equations for pa , u x , u y . So an error just occurs in 3D, when waves propagate towards corners, where all three damping coefficients are active. The FE formulation of the derived system of PDEs is straightforward and will not be presented here, for details we refer to [32]. Furthermore, we refer to [33] concerning an efficient PML formulation for the acoustic concervations equations (see Sect. 5.4.2) both for time and frequency domain computations.

5.6 Numerical Examples 5.6.1 Transient Wave Propagation in Unbounded Domains The setup of our numerical example for investigating the PML formulation is displayed in Fig. 5.26. On the surface of the sphere we set a Dirichlet boundary condition for all nodes according to the following function f (t) =

d  −π2 ( f0 t−1)2  e . dt

We choose the propagation region to be λ/2 (λ being the wavelength given by λ = c/ f 0 with c the speed of sound) and use PML regions with λ/8, λ/4 and λ/2. The propagation as well as PML region are discretized with tetrahedra finite elements of first order having an average edge length of about λ/16. Thus, the PML is discretized in thickness direction with 2 finite elements for the λ/8 layer, with 4 finite elements for the λ/4 layer, and with 8 finite elements for the λ/2 layer, respectively. Furthermore, we use an implicit Newmark time stepping scheme (see Sect. 2.5.2) with β = 0.25 and γ = 0.5 and we choose a time step size of ∆t = 1/(80 f 0 ). For the first investigations concerning the accuracy, we use the inverse distance damping function. For this simple setup, we could compare our results to an analytical solution. However, to just measure the error introduced by the PML, we compute a reference

5.6 Numerical Examples

213

Observation point

Propagation region

PML-thickness

Surface excitation

Fig. 5.26 Computational setup

solution using the same computational mesh in the propagation region and use instead of the PML region an additional larger propagation region. We consider the evolution of the l 2 -error at each time step tn

Nprop

%

1  $ ref (t ) 2 .  pai (tn ) − pai Error PML (tn ) = n Nprop

(5.195)

i=1

In (5.195) pai denotes the solution at each FE node i using the PML formulation, ref the reference solution at FE node i and N pai prop the number of nodes within the propagation region. In a first investigation, we computed the acoustic pressure over time applying the PML formulation and compare the results with our reference solution at the three observation points P1 , P2 and P3 (see Fig. 5.26). The results with a PML thickness of λ/8 are displayed in Fig. 5.27, for a PML thickness of λ/4 in Fig. 5.28 and finally for a PML thickness of λ/2 in Fig. 5.29. From the three graphs, we can clearly see the improvement in the solutions with increasing PML thickness. The main difference, as expected, is given for the pressure at observation point P3 , which is the corner point between propagation region and PML region (see Fig. 5.26). Furthermore, the results demonstrate that the reduced PML formulation denoted by rPML just has slightly worse properties than the full PML formulation and the differences vanishes almost completely for a damping layer thickness of λ/2 (see Fig. 5.29). Based on the obtained results we further computed the l 2 -norm of the error as given in (5.195). In Fig. 5.30 we show the obtained error introduced by the PML

214

5 Acoustic Field P1

10

Reference PML rPML

P2

pa (t) (mPa)

P3 5

0

−5

−10 5

10

15

20

t (s)

25

30

Fig. 5.27 Pressure signal at the three observation points; PML-thickness was set to λ/8 P1

10

Reference PML rPML

P2 P3

pa (t) (mPa)

5

0

−5

−10 5

10

15

20

25

30

t (s) Fig. 5.28 Pressure signal at the three observation points; PML-thickness was set to λ/4 P1

10

Reference PML rPML

pa (t) (mPa)

P2

P3

5

0

−5

−10 5

10

15

t (s)

20

25

30

Fig. 5.29 Pressure signal at the three observation points; PML-thickness was set to λ/2

5.6 Numerical Examples

215

ErrorPML

0.16

0.12

0.08

0.04

0.0 0

5

10

15

20

25

30

35

40

t (s) Fig. 5.30 The l 2 -error graph over time

and rPML with a thickness of λ/8 and λ/4. Again, we observe the decrease in the error over time when increasing the PML thickness. We had to stop the computation of the error at t = 40 s, since at this time the reference solution already showed first reflections back to the propagation region. Therefore, in order to investigate the stability, we also computed the overall acoustic energy at each time step tn within the propagation region by the following formula    ρ0 pa (tn )2 v a (tn ) · v a (tn ) + dΩ. (5.196) E acoustic (tn ) = 2 2ρ0 c2 Ωprop

In (5.196) ρ0 denotes the mean density of the fluid and v a the acoustic particle velocity. For this long term stability study, we perform computations for all three cases of layer thicknesses (λ/8, λ/4 and λ/2) and use the two different damping functions—constant and inverse distance. The acoustic particle velocity v a is related to the acoustic pressure pa via the linear momentum conservation (5.8). Therewith, in a time discrete setting using a trapezoidal scheme, we obtain v a (tn ) = v a (tn−1 ) −

∆t ∇ pa (tn ) , ρ0

which is inserted into (5.196). Figure 5.31 displays the computed acoustic energy within the propagation region for the reference solution and the PML as well as rPML formulations. The first main observation is that for the λ/8 damping layer the results for both damping functions get instable. Canceling the critical term in the PML formulation (denoted by rPML) as revealed by the stability proof, leads to stable results (see Fig. 5.31b). The second main observation is that both damping functions results in stable long time computations for layer thicknesses of λ/4 and λ/2. Furthermore, applying the rPML formulation achieves in all cases a stronger

216

5 Acoustic Field

(a)

Energy (J)

10 10 10 10

(b)

6

4

10 2

0

10 10 40

60

10 10 10 10

100

120

140

4

2

0

-2

100

120

140

10 10

10

10 10

60

80 t (s)

100

120

140

160

rPML: inverse rPML: constant Reference

4

2

0

-2

20

40

60

80 t (s)

100

120

140

160

(f)

6

10

PML: inverse PML: constant Reference

4

10 Energy (J)

10

40

6

160

(e) 10

10

10 80 t (s)

0

-2

10

60

2

20

PML: inverse PML: constant Reference

40

rPML: inverse rPML: constant Reference

4

(d)

6

20

6

160

Energy (J)

10

80 t (s)

(c)

Energy (J)

10

-2

20

Energy (J)

10

PML: inverse PML: constant Reference Energy (J)

10

2

0

10 10

-2

10 20

40

60

80 t (s)

100

120

140

160

6

rPML: inverse rPML: constant Reference

4

2

0

-2

20

40

60

80 t (s)

100

120

140

160

Fig. 5.31 Acoustic energy over time for different layer thicknesses and damping functions. a PML with thickness λ/8. b rPML with thickness λ/8. c PML with thickness λ/4. d rPML with thickness λ/4. e PML with thickness λ/2. f rPML with thickness λ/2

damping behavior of the acoustic energy for increasing time. Furthermore, when comparing the energy curves with the reference solution, we may clearly state that the inverse distance damping function (both for PML and rPML) provides the most accurate results.

5.6.2 Harmonic Wave Propagation in Unbounded Domains In order to evaluate the PML method for the harmonic case (Helmholtz equation), we perform numerical simulations of cylindrical waves and compare it with analytic results. As displayed in Fig. 5.32, we prescribe the acoustic pressure on the surface of a cylinder and compute the radiated sound field. Although, the problem is 2D, we will perform a full 3D computation.

5.6 Numerical Examples

217 PML-region

y Propagation-region

L λ/2 z

x

Excitation

λ/2

L

Fig. 5.32 Computational setup: propagation region surrounded by a PML-region

L = λ/8

L = λ/2

L = λ/4

Fig. 5.33 Coarsest grid (h ≈ λ/20) of the three different setups concerning the thickness of the PML

The dimension of the propagation domain is fixed by λ/2 and the PML-thickness L is varied from L = λ/2 over λ/4 to λ/8 (see Fig. 5.33). Furthermore, we perform a discretization, where we use 20, 40, 60 and 80 bilinear hexahedron elements per wavelength λ. The analytical solution is given by pˆ = p(R ˆ 0)

H0(2) (kr ) (2)

(5.197)

H0 (k R0 )

(2)

with H0 the Hankel function, r = (x, y) the position vector, k the wave number, and R0 the radius, where we define the excitation p(R ˆ 0 ) = 1. We compute the relative accumulated error (sum over all finite element nodes N in the propagation region) as follows

N

 h

( pˆ − pˆ i )2

i=1 i

E total = 100 %. (5.198) N   2 pˆ i i=1

218

5 Acoustic Field

L = λ/2

L = λ/4

L = λ/8

Fig. 5.34 Iso-lines on the coarsest grid (h ≈ λ/20) of the three setups concerning the thickness of the PML (inverse distance weighted damping function)

In Fig. 5.34, we display the iso-lines for the three cases having different PMLthickness L and which was computed on the coarsest grid (mesh size h ≈ λ/20). Table 5.4, lists the computed error according to (5.198). It has to be noted that in addition to the inexact approximation of the free radiation condition by the PML, we measure with (5.198) the overall error (see e.g., [8, 13, 14, 34]). However, since this is a practical case, we are interested in the overall error of our numerical scheme. As can be seen from Table 5.4, we achieve a very consistent reduction of the error, by decreasing the mesh size h as well as increasing the layer thickness. The great superiority of the inverse distance damping function compared to the constant as well as quadratically increasing one can be clearly seen, when studying the error E total in Table 5.4 as well as the iso-lines in Fig. 5.35. We can summarize that one should use the same mesh size in the PML-region as used in the propagation region and the mesh should have at least 2 bilinear finite elements within the thickness L of the PML-layer.

5.6.3 Nonlinear Wave Propagation in a Channel Let us consider a plane wave in a semi-infinite channel as displayed in Fig. 5.36. The excitation is performed by a rigid-body motion on the left side, so that we obtain a defined acoustic particle velocity. The prescribed mechanical motion follows a sineburst signal as displayed in Fig. 5.37 with a frequency of 100 kHz and an amplitude of 100 µm. For the medium we use water with a parameter of nonlinearity B/A equal to 5 and a diffusivity of sound value b = 6 · 10−9 m2 /s. Now, as a function of the distance x from the source (vibrating mechanical solid body), analytical solutions exist (see e.g., [35]). Introducing the dimensionless distance σ as σ = x/x¯

λ/20 λ/40 λ/60 λ/80

2.9712 1.3537 0.9302 0.6163

1.3760 0.5378 0.3329 0.2338

0.4919 0.1817 0.1065 0.0982

2.8872 1.0470 0.3721 0.2152

1.1259 0.2376 0.0776 0.0413

0.3685 0.1238 0.0880 0.0787

Table 5.4 Computational error E total for different mesh sizes h and different PML-thickness L h Damping constant Damping quadratically increasing L = λ/8 L = λ/4 L = λ/2 L = λ/8 L = λ/4 L = λ/2 0.5576 0.3139 0.1580 0.1159

0.4137 0.1119 0.0557 0.0374

Damping inverse with distance L = λ/8 L = λ/4

L = λ/2 0.3174 0.0773 0.0409 0.0232

5.6 Numerical Examples 219

220

5 Acoustic Field

Constant damping

Quadratic damping

Inverse damping

Fig. 5.35 Iso-lines of the solution for a PML-thickness of λ/4 and the coarsest grid (h ≈ λ/20) using the three different damping functions

Wave propagation

Vibrating solid body

Acoustic channel

Fig. 5.36 Simulation model

Fig. 5.37 Signal form used for excitation (mechanical motion)

with x¯ the shock formation distance, we obtain the following two formulas • σ < 1.0: Fubini solution va (x, t) = vˆa

' ∞ &  2 Jn (nσ) sin[n(ωt − kx)] , nσ n=0

(5.199)

5.6 Numerical Examples

221

with va (0, t) = vˆa sin(ωt).

(5.200)

In (5.199) va denotes the acoustic particle velocity, k = ω/c0 the wave number, c0 the speed of sound, Jn for the Bessel function of order n, and x the coordinate for the propagation direction. Due to the continuity of the mechanical surface velocity and the acoustic particle velocity in normal direction, va (0, t) corresponds to the mechanical vibration velocity of the solid (see Chap. 8). • σ > 3.0: Fay solution va (x, t) = vˆa

∞ &  n=0

' 2/Γ sin[n(ωt − kx)] , sinh[n(1 + σ)/Γ ]

with Γ =

(5.201)

B/A . b

The shock formation distance x¯ itself computes for a harmonic excitation with velocity amplitude vˆa and the speed of sound c [36] x¯ =

c2 . ω vˆa (1 + B/(2 A))

(5.202)

For the acoustic part, we specify homogeneous Neumann boundary conditions on the whole boundary. For the mechanical solid body, we prescribe the mechanical displacement for all FE nodes with an amplitude of 100 µm, which varies with time according to the sine-burst signal. The FE mesh consists of linear quadrilateral elements with just one finite element for the width of the channel and 250 finite elements per fundamental wavelength for the length of the channel. In order to also resolve higher harmonics, which will arise due to the nonlinearity, this fine mesh has been used. For the time discretization a time step value of 20 ns is specified, which corresponds to 500 time samples per fundamental time period. Again, as for the spatial discretization, this small time step is necessary to be able to resolve higher harmonics. In Fig. 5.38 it is clearly demonstrated how the number of higher harmonics grows with the distance from the source. The comparison of the Fay solution with the numerical data is displayed in Fig. 5.39 for different distances. A detailed discussion on parameter variations (B/A, b, spatial and time discretization), especially how they influence the results, is given in [36].

222

5 Acoustic Field

(a)

(b) Analytic solution Numerical solution

100

A (dB)

pa (MPa)

50 0 −50

−100 40

45

50

55

60

f (MHz)

t (µs)

(d)

A (dB)

A (dB)

(c)

f (MHz)

f (MHz)

(e)

(f)

A (dB)

pa (MPa)

Analytic solution Numerical solution

t (µs)

f (MHz)

Fig. 5.38 Time signal and frequency spectra of Fubini solution and numerical computation for different distances. a σ = 0.2. b σ = 0.2. c σ = 0.6. d σ = 0.6. e σ = 1.0. f σ = 1.0

References

223

(b)

t (µs)

t (µs)

t (µs)

(f)

t (µs)

pa (MPa)

pa (MPa)

(e)

(d)

pa (MPa)

pa (MPa)

(c)

pa (MPa)

pa (MPa)

(a)

t (µs)

t (µs)

Fig. 5.39 Time signal of Fay solution for distances between σ = 3 and 8. a σ = 3.0. b σ = 4.0. c σ = 5.0. d σ = 6.0. e σ = 7.0. f σ = 8.0

References 1. A.D. Pierce, Acoustics—An Introduction to its Physical Principles and Applications (Acoustical Society of America, Woodbury, 1991) 2. E. Zwicker, H. Fastl, Psychoacoustics (Springer, Berlin, 1999) 3. V.P. Kuznetsov, Equations of nonlinear acoustics. Sov. Phys. Acoust. 16(4), 467–470 (1971) 4. M.F. Hamilton, D.T. Blackstock, Nonlinear Acoustics (Academic Press, San Diego, 1998)

224

5 Acoustic Field

5. S.I. Aanonsen, T. Barkvek, J.N. Tjotta, S. Tjotta, Distortion and harmonic generation in the near field of a finite amplitude sound beam. J. Acoust. Soc. Am. 75, 749–768 (1984) 6. F. Brezzi, M. Fortin, Mixed and Hybrid Finite Element Methods (Springer, New York, 1991) 7. P.A. Raviart, J.M. Thomas, A mixed finite element method for 2nd order elliptic problems, in Mathematical Aspects of the Finite Element Method. Lecture Notes in Mathematics, pp. 292–315 (1977) 8. G.C. Cohen, Higher-Order Numerical Methods for Transient Wave Equations (Springer, Berlin, 2002) 9. G. Cohen, S. Fauqueux, Mixed finite elements with mass-lumping for the transient wave equation. J. Comput. Acoust. 8, 171–188 (2000) 10. D.N. Arnold, D. Boffi, R.S. Falk, Quadrilateral H(div) finite elements. SIAM J. Numer. Anal. 42, 2429–2451 (2005) 11. A. Hüppe, Spectral Finite Elements for Acoustic Field Computation. Ph.D. thesis, Alpen-AdriaUniversität, Klagenfurt, (2013) 12. B. Flemisch, M. Kaltenbacher, S. Triebenbacher, B. Wohlmuth, Non-matching grids for a flexible discretization in computational acoustics. Commun. Comput. Phys. 11(2), 472–488 (2012) 13. F. Ihlenburg, I. Babuska, Finite element solution of the Helmholtz equation with high wave number part I: the h version of the finite element method. Comput. Math. Appl. 30, 9–37 (1995) 14. M. Ainsworth, Discrete dispersion relation for hp-version finite element approximation at high wave number. SIAM J. Numer. Anal. 42(2), 553–575 (2004) 15. L. Olson, K. Bathe, An infinite element for analysis of transient fluid-structure interactions. Eng. Comput. 2, 319–329 (1985) 16. D. Dreyer, O. von Estorff, Improved conditioning of infinite elements for exterior acoustics. Int. J. Numer. Methods Eng. 58, 933–953 (2003) 17. R. Clayton, B. Engquist, Absorbing boundary conditions for acoustic and elastic wave equations. Bull. Seismol. Soc. Am. 67, 1529–1540 (1977) 18. B. Engquist, A. Majda, Absorbing boundary conditions for the numerical simulation of waves. Math. Comput. 31, 629–651 (1977) 19. M. Hofer, Finite-Elemente-Berechnung von periodischen Oberflächenwellen-Strukturen. Ph.D. thesis, University of Erlangen-Nuremberg, (2003) 20. J.P. Berenger, A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114, 185 (1994) 21. F. Collino, P. Monk, The perfectly matched layer in curvilinear coordinates. SIAM J. Sci. Comput. 19, 2061–2090 (1998) 22. I. Harari, M. Slavutin, E. Turkel, Analytical and numerical studies of a finite element PML for the Helmholtz equation. J. Comput. Acoust. 8, 121–127 (2000) 23. Fang Q. Hu, Absorbing boundary conditions. Int. J. Comput. Fluid Dyn. 18, 513–522 (2004) 24. I. Singer, E. Turkel, A perfectly matched layer for Helmholtz equation in a semi-infinite strip. J. Comput. Phys. 201, 439–465 (2004) 25. F.L. Teixeira, W.C. Chew, Complex space approach to perfectly layers: a review and some developments. Int. J. Numer. Model. 13, 441–455 (2000) 26. A. Bermúdez, L. Hervella-Nieto, A. Prieto, R. Rodríguez, An optimal perfectly matched layer with unbounded absorbing function for time-harmonic acoustic scattering problems. J. Comput. Phys. 223(2), 469–488 (2007) 27. E. Becache, S. Fauqueux, P. Joly, Stability of matched layers, group velocities and anisotropic waves. J. Comput. Phys. 188, 399–433 (2003) 28. J.H. Bramble, J.E. Pasciak, Analysis of a finite PML approximation for the three dimensional time-harmonic Maxwell and acoustic scattering problems. Math. Comput. (2006) 29. T. Rylander, J.M. Jin, Perfectly matched layer for the time domain finite element method. J. Comput. Phys. 238–250 (2004) 30. Björn Sjögreen and, N. Anders Petersson, Perfectly matched layers for Maxwell’s equations in second order formulation. J. Comput. Phys. 209(1), 19–46 (2005)

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Chapter 6

Electromagnetic Field

6.1 Maxwell’s Equations The full system of partial differential equations describing the electromagnetic field was published for the first time by James Clerk Maxwell in his work A Treatise on Electricity and Magnetism, Vol. I,II [1, 2] in the year 1862. He based his theory on the work and experiments of Amp`ere, Gauss, and Faraday. His great contribution lies in the unification of the different equations to a set of partial differential equations. The introduction of the displacement current, which generalizes Ampère’s law, allowed him to foresee the physical phenomena of the propagation of electromagnetic waves. In general, we distinguish two domains in electromagnetism, both are of course included in Maxwell’s equations: • The high-frequency domain, which includes the study of electromagnetic waves and propagation of energy through matter. In general, we will define highfrequency domains, as domains, where the displacement currents cannot be neglected. • The low-frequency domain includes the major part of electromagnetic devices like motors, relays or transformers. These are all applications at frequencies below a few tens of kHz. Strictly speaking, any application in which displacement currents can be neglected is a low-frequency application. In these domains, corresponding to the quasistatic case, we can, in general, study electric and magnetic fields as separate quantities. At high frequencies, the electric and magnetic fields are interdependent. According to Fig. 6.1 we can divide electromagnetism into different cases, where each case represents a particular aspect of Maxwell’s equations. These equations are a set of partial differential equations, linear in space and time, applied to electromagnetic quantities. When electromagnetic fields interact with materials, the equations can assume nonlinear forms. The electromagnetic quantities involved in Maxwell’s equations are (Table 6.1):

© Springer-Verlag Berlin Heidelberg 2015 M. Kaltenbacher, Numerical Simulation of Mechatronic Sensors and Actuators, DOI 10.1007/978-3-642-40170-1_6

227

228

6 Electromagnetic Field ELECTROMAGNETISM (Maxwell’s equations)

ELECTROMAGNETISM

ELECTROMAGNETISM

High frequencies (waves)

Low frequencies (electromechanical)

MAGNETISM

ELECTROSTATICS

MAGNETOSTATICS

MAGNETODYNAMICS

Fig. 6.1 Classification of Maxwell’s equations [3] Table 6.1 Electromagnetic quantities Notation Unit E D H B J qe M P

(V/m) (As/m2 ) (A/m) (T) (A/m2 ) (As/m3 ) (T) (As/m2 )

Description Electric field intensity Electric flux density (electric induction) Magnetic field intensity Magnetic flux density (magnetic induction) Current density Charge density Magnetization Electric polarization

In addition, we define the following material parameters • • • •

Magnetic permeability µ (Vs/Am) Magnetic reluctivity ν = 1/µ (Am/Vs) Electric permittivity ε (As/Vm) Electric conductivity γ (1/m = S/m)

The four partial differential equations (PDEs), stated as Maxwell’s equations in differential form, fully describe all phenomena in electromagnetic fields ∂D ∂t ∂B ∇×E=− ∂t ∇ · D = qe ∇ · B = 0.

∇×H = J+

(6.1) (6.2) (6.3) (6.4)

6.1 Maxwell’s Equations

229

In addition, to get solvability of the four PDEs, the following constitutive equations are introduced J = γ(E + v × B)

(6.5)

D = εE = ε0 E + P B = µH = µ0 H + M.

(6.6) (6.7)

It has to be stated that (6.5) is a direct consequence of the force relation defined by Lorentz f V = qe (E + v × B).

(6.8)

Thus, the total electromagnetic force acting on the electric volume charge qe is given by a term defined by the product of the electric field E and a volume charge qe at rest and a term defined by the magnetic induction B and a volume charge qe at velocity v. As already mentioned, Maxwell’s equations are based on experiments and concluding empirical laws stated by Ampère, Faraday, and Gauss.

6.1.1 Law of Ampère With the help of experiments, Ampère was able to prove in the year 1820 that an electric current (dc or ac) generates a magnetic field (Fig. 6.2). This empirical law states that the line integral of the magnetic field intensity along a closed contour is equal to the electric current I  H · ds = I.

(6.9)

C(I )

The direction of the magnetic field intensity H is related to the direction of the electric current I via the right-hand rule. For general distributed currents within a cross section Γ with boundary curve C, the law of Ampère reads as

Fig. 6.2 Experiments of Ampère

I

H

230

6 Electromagnetic Field



H · ds =



J · dΓ ,

(6.10)

Γ

C

and by applying Stokes’ theorem (see Appendix B.9), we obtain 

(∇ × H) · dΓ =

Γ



J · dΓ .

(6.11)

Γ

Since this relation has to be independent of the surface, the law results in the differential form, which reads as follows ∇ × H = J.

(6.12)

Maxwell found that (6.12) does not fully describe the general case and added the so-called displacement current density ∂ D/∂t in order to obtain a description for the general case (see e.g., [4]). Since we are mainly concerned in the quasistatic case, we will not discuss this term in detail.

6.1.2 Law of Faraday According to his experiments, Faraday postulated that the relation between the induced voltage u ind in an open conductive loop and the magnetic flux φ through it (Fig. 6.3), is given by dφ . (6.13) u ind = − dt The magnetic flux φ is defined by φ=



B · dΓ ,

(6.14)

Γ

Fig. 6.3 Loop within a time-varying magnetic field

dφ/ dt ds 1 uind 2 iind

6.1 Maxwell’s Equations

231

with Γ the cross section passed by the magnetic induction B. A change of the magnetic flux with respect to time can be obtained in two different cases. First, the loop is at rest and the magnetic field is time dependent, and secondly, the loop changes its position as a function of time and the magnetic field is constant. Let us consider a surface Γ on which E and B are defined, then we can rewrite (6.2) as   ∂B · dΓ . (∇ × E) · dΓ = − ∂t Γ

Γ

Through the use of Stoke’s theorem, we obtain 

E · ds = −



∂B · dΓ , ∂t

Γ

C(Γ )

where C is the contour that limits the surface Γ . We now look at the application of this expression in an example as depicted in Fig. 6.4. Let us assume that the magnetic field is concentrated in the high-permeable cylinder and the magnitude of B is constant at each cross section of the cylindrical core. Furthermore, an electrical conductive loop, electrically isolated from the core, encloses the core such that it forms a contour C(Γ ). First, note that assuming the direction of dΓ as in Fig. 6.4a, the direction of d s must be as shown (right-hand rule). With the direction of ∂ B/∂t as shown in Fig. 6.4b, we obtain ∂B · dΓ < 0 ; E · ds < 0, − ∂t which makes E point in the direction opposite to ds.

(a)

B

(b)

ds

E uind

dΓ

∂B/∂t

uind

C(Γ )

Fig. 6.4 Induced voltage in an open conductive loop, caused by a time-varying magnetic field. a Idealized setup. b Induced voltage

232

6 Electromagnetic Field

The circulation of E along C leads to an electromotive force (emf), detectable by a voltmeter as an electric potential difference. The electromotive force computes as u ind =



E · ds.

(6.15)

C

It has to be noted that the electric field due to the time variation of the magnetic field is solenoidal, and thus, the measured voltage depends on the way the electric voltmeter is connected to the open loop [5]. Because of the fact that B only depends on time and not on its position, we conclude that ∂ B/∂t = d B/dt. Since the integration over the surface Γ and the time derivative are independent operations, we can write −



d ∂B · dΓ = − ∂t dt

Γ



B · dΓ = −

dφ , dt

Γ

and obtain the expression u ind = −

dφ . dt

(6.16)

Now let us consider the case of a conductive slab positioned in a uniform and timeindependent magnetic field B (see Fig. 6.5). As already mentioned in this section, the total electromagnetic force on a charge is given due to an electrostatic field acting on a charge in rest, and a magnetic field acting on a moving charge. Therefore, an electromagnetic force F mag will act on the electric charges Q e within the conductive slab (see Fig. 6.5) (6.17) F mag = Q e (v × B) , and the charges will distribute on the surface of the slab as depicted in Fig. 6.5. Due to this separation of positive and negative charges, an electrostatic field E arises, which will result in the restoring force

Fig. 6.5 Electrically conductive slab moving with velocity v in a time-constant magnetic field B

B

v

E

v×B

6.1 Maxwell’s Equations

233

F el = Q e E.

(6.18)

By assuming the equilibrium state, we obtain F mag + F el = 0 E = −v × B.

(6.19) (6.20)

If we now connect a resistor R to the slab in parallel by using a flexible cable (so that the resistor is at rest), then the electric field E will result in an electric current i. Therefore, the term v × B is called the motional electromotive force. By neglecting the resistance of the connections, the potential difference across the resistor, which we define by the voltage u, is given by u = Ri =



E · ds = l · (v × B) ,

(6.21)

with l the length of the conductive slab (direction points into the direction of the current). Hence, the mechanical power Pmech we apply to the slab (since we move it with velocity v) is converted into electric power Pel Pmech = F mech · v = Ri 2 = Pel .

(6.22)

Simply speaking, we have the case of an electric generator with an ohmic load.

6.1.3 Law of Gauss By means of experiments, Gauss postulated that the amount of electric flux density D crossing a closed surface Γ is equal to the total electric charges qe within the volume Ω (Fig. 6.6)   D · dΓ =

Γ

Fig. 6.6 Sources of the electric field

qe dΩ.

(6.23)

Ω

D qe

n

Γ

234

6 Electromagnetic Field

Applying the divergence theorem (Appendix B.6), the above equation reads as 

(∇ · D) dΩ =

Ω



qe dΩ

(6.24)

Ω

and, since the law has to hold for any volume Ω, we end up at the differential form ∇ · D = qe .

(6.25)

Therefore, the sources of the electric field are given by the electric charges, and according to this relation the electric field is irrotational.

6.1.4 Solenoidal Magnetic Field Since magnetic materials cannot be split into a piece only having a north pole or a south pole; no magnetic charges exist, and the magnetic field is always solenoidal (closed field lines). This property of the magnetic field can be stated by 

B · dΓ = 0 ,

(6.26)

Γ (Ω)

which means that the magnetic flux is conservative. Applying the divergence theorem 

(∇ · B) dΩ = 0,

Ω

and generalizing it for any volume Ω, we obtain the differential form ∇ · B = 0.

(6.27)

It is very interesting to note that this equation, which has been introduced by Maxwell, is necessary to guarantee that no magnetic charges exist. If we apply the operator ∇· to (6.2) ∂B (6.28) ∇ · (∇ × E) = −∇ · ∂t we achieve the following relation ∂ (∇ · B) = 0. ∂t

(6.29)

The solution of this equation without considering the relation ∇ · B = 0 would result in ∇ · B = const. = 0, and therefore, in the existence of magnetic charges.

6.2 Quasistatic Electromagnetic Fields

235

6.2 Quasistatic Electromagnetic Fields The most important case for electromagnetic sensors and actuators is the quasistatic case often referred to as the eddy current case. For quasistatic electromagnetic fields we can neglect the displacement current density term ∂ D/∂t, which transforms Maxwell’s equations including the constitutive equations to the following subset ∇×H = J

∂B ∇ × Es = − ∂t ∇·B=0

J = γ(E total + v × B) B = µH = µ0 H + M.

(6.30) (6.31) (6.32) (6.33) (6.34)

In (6.33) E total denotes the total electric field intensity given as a sum of any irrotational part E i (defined by (6.3)) and of any solenoidal part E s E total = E i + E s .

(6.35)

J = J i + γ(E s + v × B),

(6.36)

Now, we can rewrite (6.33) as

with J i an impressed current density due to a given electric potential difference (current- or voltage-loaded coil).

6.2.1 Magnetic Vector Potential According to (6.32) the magnetic field is solenoidal and therefore can be described by the curl of a vector B = ∇ × A. (6.37) The vector A is called the magnetic vector potential. This ansatz results for (6.2) in the following relation ∂ ∇ × E s = − (∇ × A) ∂t   ∂A ∇ × Es + = 0. ∂t

(6.38)

Since we are concerned with eddy current problems, the electric field is pure solenoidal. Thus, we set E s = −∂ A/∂t and arrive at

236

6 Electromagnetic Field

∇ × ν∇ × A = J i − γ

∂A + γ(v × ∇ × A). ∂t

(6.39)

It has to be noted that the introduction of the magnetic vector potential A leads to non-uniqueness of the solution. This can be seen very easily by adding the gradient of a scalar function to A, which leads to a new vector A∗ A∗ = A + ∇ψ.

(6.40)

Since the curl of any grad is zero, we arrive at B = ∇ × ( A∗ − ∇ψ) = ∇ × A∗ .

(6.41)

To obtain a unique solution, we have to gauge the magnetic vector potential A, which is, e.g., achieved by a Coulomb gauge [6] ∇ · A = 0.

(6.42)

In the 2D case the magnetic vector potential is given by A = A(x, y)ez

(6.43)

and therefore (6.42) is automatically guaranteed. The same holds for the axisymmetric case (6.44) A = A(r, z)eϕ . Furthermore, it has to be noted that we are interested in the quantity B computed via ∇ × A, and therefore, no physical need exists to gauge the magnetic vector potential A.

6.2.2 Skin Effect Supplying an electric conductor with cross section ΓC by an alternating current with frequency f C , we will find that the higher the frequency, the more the current will be concentrated near the surface of the conductor and tend to zero towards the center of the conductor. To investigate this effect, let us simplify the setup to an infinitely extended half-plane as depicted in Fig. 6.7. Therefore, the current density J and, respectively, the electric field intensity E only depend on the coordinate z and point towards the x-direction. Using (6.12), we obtain

6.2 Quasistatic Electromagnetic Fields Fig. 6.7 Infinite extended half-plane

237 x

y

J, E B, H

z

−

∂ Hy = Jx ∂z ∂ By ∂ Ex =− . ∂z ∂t

(6.45) (6.46)

By applying the operation ∂/∂z to (6.45) and using (6.46) as well as the constitutive laws (6.5) and (6.7), we arrive at −

∂ 2 Hy ∂ Hy ∂ Ex ∂ Jx =γ = −γµ . = 2 ∂z ∂z ∂z ∂t

(6.47)

Equation (6.47) is called the diffusion equation, because it models the diffusion of electromagnetic energy into the conductor. For the time-harmonic case with angular frequency ω, (6.47) gets the form H y = Hˆ y (z) ejωt

∂ 2 Hˆ y ∂z 2

(6.48)

= jωγµ Hˆ y ,

(6.49)

which has the general solution Hˆ y (z) = C1 e

√

jωγµz

√

+ C 2 e−

jωγµz

.

(6.50)

Since for physical reasons, the magnetic field decreases with distance z, the constant C1 has to be zero. By using the relation 

1+j j= √ 2

and the boundary condition Hˆ y (z = 0) = H0 , we arrive at the solution for the magnetic field inside the infinite half-plane (using (6.48)) H y (z) = H0 e−z/δ ej(ωt−z/δ) Hy (z) = Re(H y ) = H0 e

−z/δ

(6.51) cos(ωt − z/δ).

(6.52)

238

6 Electromagnetic Field

In (6.51) δ denotes the so-called skin (penetration) depth and as can be seen from (6.50), it is a function of frequency and material properties 1 . δ=√ π f γµ

(6.53)

Additionally, we obtain solutions for B, J, and E B y (z) = B0 e−z/δ cos(ωt − z/δ) −z/δ

Jx (z) = J0 e cos(ωt − z/δ) E x (z) = E 0 e−z/δ cos(ωt − z/δ).

(6.54) (6.55) (6.56)

Thus, the magnitudes of the electromagnetic quantities not only decay exponentially but also their phase changes with z/δ, so that at a certain depth, the field quantities are even in opposite directions. Concluding, it has to be noted that the expression for the skin depth δ is not restricted to this simplified example but can be considered as a general relation for eddy current problems (see e.g., [7, 8]).

6.3 Electrostatic Field For the static case, the physical quantities do not depend on time, and Maxwell’s equations with constitutive laws can be separated into a magnetic and an electric subsystem. For the electric subsystem, called the electrostatic case, we obtain the following system of partial differential equations ∇×E=0 ∇ · D = qe

D = εE.

(6.57) (6.58) (6.59)

Since the curl of the electric field intensity E is zero, we can express it by the gradient of a scalar potential Ve , which is called the electric scalar potential E = −∇Ve .

(6.60)

Thus, by combining (6.58), (6.59), and (6.60), we arrive at − ∇ · ε∇Ve = qe ,

(6.61)

which describes the electrostatic field within any media characterized by the material quantity ε.

6.4 Material Properties

239

6.4 Material Properties 6.4.1 Magnetic Permeability The constitutive law within the magnetic field is given by (6.7), which relates the magnetic induction B and the magnetic field intensity H via the magnetic permeability µ. In general, the magnetic permeability is a tensor of rank 2, but for most cases we can assume a scalar value, which can be decomposed as follows µ = µ0 µr

(6.62)

µ0 = 4π 10−7 (Vs/Am).

(6.63)

Therein, µr is called the relative permeability and characterizes the magnetic material. Often, instead of the magnetic permeability µ the magnetic reluctivity ν given by ν=

1 µ

(6.64)

is used. In general, we differentiate between two types of magnetic materials, namely • Soft magnetic materials, which can be classified into diamagnetic, paramagnetic, and ferromagnetic materials • Hard magnetic materials of which permanent magnets are fabricated By placing a diamagnetic material within an external magnetic field, the internal magnetization M will be opposite to the external field. Therefore, the total magnetic field within such a material will be smaller than the external applied field (see Fig. 6.8a). Within paramagnetic materials the internal magnetization will be directed already in the direction of the external magnetic field. However, the effect is small, as can be seen in Fig. 6.8b. Only for ferromagnetic materials will the resulting internal magnetization be large and, as shown in Fig. 6.8c, the magnetic field concentrates inside the cube. By switching off the external field, the magnetization almost disappears, and we call it a reversible process. However, if the amplitude of the external

(a)

(b)

(c)

Fig. 6.8 Cube within an external magnetic field. a Diamagnetic. b Paramagnetic. c Ferromagnetic

240

6 Electromagnetic Field

field is large, irreversible processes will take place, and after switching off the external field a remnant magnetic field can be detected. Therefore, the following definitions of permeabilities exists: • Reversible permeability:

µrev = lim

ΔH →0

ΔB . ΔH

(6.65)

This case is given if a large constant magnetic field is superimposed on a small alternating field. • Differential permeability: dB . (6.66) µdiff = dH This permeability is always larger than the reversible permeability, since it will also contain irreversible processes. • Starting permeability: dB (6.67) µs = | B=0,H =0 . dH By first applying an external magnetic field to a ferromagnetic material, the magnetic induction will rise strongly until the material is saturated. If we now decrease the amplitude of the external applied field, we will recognize that the magnetic induction inside the material will decrease along a curve different from the initial magnetization curve and at external field zero, we will have a remnant magnetic induction Br of the material. This effect is due to the already discussed irreversible processes inside the body. Finally, by driving the external magnetic field to a negative maximum amplitude and back to the positive amplitude, we will measure a similar hysteresis curve as displayed in Fig. 6.9. The quantity Hc is called the coercive magnetic field intensity (force), since we have to externally apply this value in order to obtain zero magnetic induction inside the material. In addition to the magnetic field dependency of the magnetic permeability, magnetic materials completely lose magnetization, if the temperature exceeds the

Fig. 6.9 Hysteresis and initially magnetization curve for a soft magnetic material

B

Br Hc

H

6.4 Material Properties

241

Fig. 6.10 Typical magnetization curve of hard magnetic material

B(T) Br

H(kA/m) Hc

Table 6.2 Properties of permanent magnets

Material

Br (T)

Hc (kA/m)

(B H )max (kWs/m3 )

Alnico Ferrite Ne-Fe-B

1.25 0.38 1.15

60 240 800

50 25 230

so-called Curie temperature TCurie , which can be modeled as follows B = µ(H )H

(6.68)

µ = f (T ) =



µ = µ(H ) : T < TCurie . µ = µ0 : T > TCurie

(6.69)

In general, Br is small for soft magnetic materials, which also means that the magnetization follows the external field almost with no essential retardation effect. This physical phenomenon is quite contrary for hard magnetic materials, which exhibit a large remnant magnetic field Br . A typical BH curve is given in Fig. 6.10 and Table 6.2 summarizes the data of important permanent magnets. Therefore, if we consider a magnetic assembly including permanent magnets, we can describe the magnetic field via the magnetic vector potential by the following partial differential equation (assuming a known magnetization M and no motional emf) ∇×

∂A 1 ∇ × A = Ji − γ + ∇ × M. µ ∂t

(6.70)

If we just consider a static field problem, and decompose the magnetic field intensity H into H J generated by a current density J and H M describing the magnetic material, we can introduce the so-called reduced magnetic scalar potential ψm according to H = HJ + HM

H M = −∇ψm .

(6.71) (6.72)

242

6 Electromagnetic Field

Therefore, by using (6.4) we obtain the describing equation for ψm ∇ · µ(H J + H M ) = 0 ∇ · µ∇ψm = ∇ · (µH J ),

(6.73) (6.74)

and H J is computed via Biot–Savart’s law (see e.g., [5]) H(x ′ , y ′ , z ′ ) =

1 4π



J × er dΩ, r2

(6.75)

Ω

with (x ′ , y ′ , z ′ ) the field (observation) point, Ω the volume of a current loaded structure and er the unit vector pointing from the volume Ω to the field point.

6.4.2 Electrical Conductivity In general, the conductivity is a tensor of rank 2, but for many applications it can be assumed to be a scalar value. Table 6.3 gives the values for some materials. The electrical conductivity γ mainly depends on the temperature T . In most cases the dependency is available for the specific electrical resistivity ρe = 1/γ and takes the relation  (6.76) ρe = ρ20 1 + α(T − T20 ◦ C ) + β(T − T20 ◦ C )2 , with α the linear and β the quadratic temperature coefficient. In (6.76) ρ20 denotes the specific resistivity at T = T20 ◦ C . Therefore, we obtain the electric conductivity (see Fig. 6.11)

with γ20 = 1/ρ20 .

γ20 , γ=

1 + α(T − T20 ◦ C ), +β(T − T20 ◦ C )2

Table 6.3 Electrical conductivity of some materials Iron Aluminum Conductivity (S/m)

1 × 107

3.5 × 107

(6.77)

Copper

Steel

Carbon

5.8 × 107

5 × 106

3 × 104

6.4 Material Properties

243

γ (MS/m)

Fig. 6.11 Dependency of the electrical conductivity of steel on temperature

T ( ◦ C)

6.4.3 Dielectric Permittivity The modeling of dielectric materials is quite similar to that of magnetic materials. Instead of the magnetization M, we define the polarization P for characterizing dielectric materials [5] D = ε0 εr E = ε0 E + P.

(6.78)

The scalar permittivity ε can be decomposed into the permittivity of vacuum ε0 = 8.854 × 10−12 (As/Vm) and the relative permittivity εr . The material can be described by a scalar value εr (see Table 6.4), and therefore D, E, and P have the same direction. However, materials with a large value of relative permittivity exhibit a strong anisotropy and are called ferroelectrics. This is the special case for piezoelectric materials. These materials also show a hysteresis as depicted in Fig. 6.12 with Dr denoting the dielectric remnant and E c the coercive electric field intensity. Table 6.4 Relative permittivity of dielectric materials εr

Air

Insulating paper

Glass

Water

PZT 5A (3-direction)

1

3

6

80

3,000

244

6 Electromagnetic Field

Fig. 6.12 Hysteresis and initial polarization curve

D

Dr Ec

E

6.5 Electromagnetic Interface Conditions 6.5.1 Continuity Relations for Magnetic Field According to (6.4) and by application of the divergence theorem, we obtain 

Ω

∇·B=0 (∇ · B) dΩ = 0  B · dΓ = 0.

(6.79)

Γ

At the interface between two materials with different permeabilities, we have to evaluate the surface integral according to Fig. 6.13 for b → 0. lim



b→0 Γ

B · dΓ = B 1 · n1 + B 2 · n2 = 0 n1 · (B 1 − B 2 ) = 0

B1n = B2n µ1 H1n = µ2 H2n .

Fig. 6.13 Continuity of the normal component of B

(6.80) (6.81)

B1 Γ1

b

Γ2 B2

6.5 Electromagnetic Interface Conditions

245

Fig. 6.14 Continuity of the tangential component of H

H1 s b

t

H2

Therefore, at an interface of changing permeability, the normal component of the magnetic induction B is continuous. By using (6.30) and assuming no current density at the interface (see Fig. 6.14), we obtain the continuity relation for the magnetic field intensity H (applying Stoke’s theorem)  (∇ × H) · dΓ = 0 lim b→0 Γ  lim H · ds = 0 b→0

H1 · s − H2 · s = 0

s t · (H 1 − H 2 ) = 0 H1t = H2t .

(6.82)

Therefore, at an interface of changing permeability, the tangential component of the magnetic field intensity H is continuous. By using the material relation between B and H we get the defining equation for the tangential component of the magnetic induction B2t B1t = . µ1 µ2

(6.83)

6.5.2 Continuity Relations for Electric Field Using the integral form of (6.3) 

Ω

(∇ · D) dΩ =



qe dΩ ,

Ω

with the relation dΩ = b dΓ and b → 0 (see Fig. 6.15) we obtain

(6.84)

246

6 Electromagnetic Field

Fig. 6.15 Continuity of the normal component of D

D1 Γ1

b

Γ2 D2

lim qe dΩ = σe dΓ  qe dΩ = σe dΓ ,

b→0

lim

b→0



Ω

(6.85) (6.86)

Γ

with σe the surface charge along the interface. By applying the divergence theorem 

Ω

(∇ · D) dΩ =



D · dΓ ,

(6.87)

Γ

and performing the step for b → 0, we arrive at lim

b→0



D · dΓ = D1 · n1 + D2 · n2 = n1 ( D1 − D2 ).

(6.88)

Γ

Therefore, for the normal components of the electric field quantities, the following relations have to hold D1n = D2n + σe ε1 E 1n = ε2 E 2n + σe .

(6.89) (6.90)

For the tangential component of the field quantities at an interface between two materials with different permittivity and no time-varying magnetic field (Fig. 6.16), we have to start at (6.2) and perform similar steps as for the tangential component of the magnetic field, and obtain E 1t = E 2t D1t D2t = . ε1 ε2

(6.91) (6.92)

6.5 Electromagnetic Interface Conditions

247

Fig. 6.16 Continuity of the tangential component of E

E1 s b

t

E2

6.5.3 Continuity Relations for Electric Current Density Similar to (6.80), we can rewrite (6.30) after applying the divergence on both sides as ∇ · J = 0, (6.93) which results in the following interface condition J1n = J2n .

(6.94)

Assuming a jump of the electrical conductivity γ at an interface, we obtain for the eddy current case (see Sect. 6.2.1 and v = 0) γ1

∂ A1n ∂ A2n = γ2 . ∂t ∂t

(6.95)

6.6 Numerical Computation: Electrostatics Let us consider the situation of a domain with given electric volume charges qe , where we want to compute the generated electrostatic field. The strong formulation of this problem will be stated as follows Given: qe : Ω → IR ε : Ω → IR. Find: Ve : Ω¯ → IR − ∇ · ε∇Ve = qe .

(6.96)

248

6 Electromagnetic Field

Boundary conditions Ve = 0 on Γ = Γe . To obtain the variational formulation, we multiply (6.96) with an appropriate test function ω ∈ H01 (see Appendix D) and perform a partial integration. Therefore, the weak form reads as: Find Ve ∈ H01 such that 

ε ∇w · ∇Ve dΩ −

Ω



wqe dΩ = 0,

(6.97)

Ω

for any w ∈ H01 . Using standard nodal finite elements, we approximate the continuous electric scalar potential Ve as well as the test function w by Ve ≈ Veh =

n eq

w ≈ wh =



Na Vea

(6.98)

Na wa ,

(6.99)

a=1 n eq

a=1

with n eq the number of equations. Therefore, (6.97) is transformed into the following discrete formulation ⎞ ⎛ n eq n eq   ⎝ ε (∇ Na )T ∇ Nb dΩ Veb − (6.100) Na qe (r a ) dΩ ⎠ = 0. a=1 b=1

Ω

Ω

In matrix form, (6.100) may be written as K Ve Ve = f V ,

(6.101)

e

with K Ve = fV = e

ne 

e=1 ne 

e=1

keVe ;

keVe

= [k pq ]; k pq =

f eV ; f eV = [ f p ]; f p = e

e



Ωe



Ωe

ε (∇ N p )T ∇ Nq dΩ N p qe (r p ) dΩ.

(6.102)

(6.103)

6.7 Numerical Computation: Electromagnetics

249

6.7 Numerical Computation: Electromagnetics The numerical computation of electromagnetic fields has been performed for more than 20 years. For the domain discretization nodal as well as edge finite elements have been used successfully. Nevertheless, in recent years inaccurate results at material parameter interfaces in the magnetostatic as well as in the eddy current case, and spurious modes in Maxwell’s eigenvalue problems have been reported. Therefore, we will perform a precise investigation in the formulation and further discretization of electromagnetic fields in the eddy current case.

6.7.1 Formulation Let us consider a domain Ω with boundary Γ consisting of two subdomains Ω1 and Ω2 with interface Σ12 as displayed in Fig. 6.17. The material relations are given by H = ν B and J = γ E with ν, γ the magnetic reluctivity (ν = 1/µ) and electrical conductivity of the material. On the boundary Γ = ∂Ω the normal component of the magnetic induction (B · n = 0 on Γ ) shall be prescribed with n being the normal unit outward vector on Γ . In addition, we consider the continuity conditions that have to be fulfilled at an interface Σ12 of changing magnetic permeability as well as conductivity [B · n] = B 1 · n − B 2 · n = 0 [H × n] = H 1 × n − H 2 × n = 0

(6.104) (6.105)

[ J · n] = J 1 · n − J 2 · n = 0.

(6.106)

The partial differential equation to be solved has been derived in Sect. 6.2.1, and assuming no moving bodies, reads as γ

∂A + ∇ × ν∇ × A = J i . ∂t

(6.107)

The boundary conditions change to n × A = 0 on Γ , and the interface conditions take on the form

Fig. 6.17 Domain with given material properties

Γ Ω1 (µ1 , γ1 ) Σ12

Ω2 (µ2 , γ2 )

250

6 Electromagnetic Field

[ A × n] = 0 [ν n × ∇ × A] = 0   ∂A γ n· = 0. ∂t

(6.108) (6.109) (6.110)

Therefore, the strong formulation for the eddy current case reads as follows: Given: A0 : Ω → IRd γ, ν : Ω → IR. Find: A(t) : Ω¯ × [0, T ] → IRd γ

∂A + ∇ × ν∇ × A = J i . ∂t Boundary conditions

(6.111)

n × A = 0 on Γ × (0, T ). Interface conditions A1 × n = A2 × n on Σ12 × (0, T ) ν1 n × ∇ × A1 = ν2 n × ∇ × A2 on Σ12 × (0, T ) ∂ A1 ∂ A2 = γ2 n · on Σ12 × (0, T ). γ1 n · ∂t ∂t Initial condition A(r, 0) = A0 , r ∈ Ω. Now, multiplying (6.111) by appropriate test functions A′ and applying Green’s first integral theorem in vector form (Appendix B.10) will transform the PDE into its variational formulation, which reads as follows: Find A ∈ H Σ 0 (curl) such that 

Ω

γ A′ ·

∂A dΩ + ∂t



∇ × A′ · ν ∇ × A dΩ =

Ω



A′ · J i dΩ,

(6.112)

Ω

for any A′ ∈ H Σ 0 (curl) with the Sobolev space 2 3 2 3 HΣ 0 (curl) = {u ∈ (L (Ω)) | ∇ × u ∈ (L (Ω)) ,

u × n|Γ = 0, [n × u]|Σ = 0}.

(6.113)

If the conductivity γ is positive, the problem has a unique solution in H 0 (curl), whereas for γ = 0 (globally, or only in some regions), the solution is unique only up to gradient fields in the case that Ω is simply connected. Since we are interested

6.7 Numerical Computation: Electromagnetics

251

in the quantity B computed via ∇ × A, there is no need to gauge the vector potential A, though. The fact that our variational formulation (6.112) automatically fulfills the interface conditions (6.108–6.110) can be shown as follows. Starting at (6.112) and performing partial integration (Green’s first integral theorem in vector form, see Appendix B.10), we obtain   

′  ′ ∂A ′ γA · dΩ + A ∇ × ν ∇ × A dΩ + A × ν1 ∇ × A · n1 dΓ ∂t    Ω

+

Ω1

Σ





A′ ∇ × ν ∇ × A dΩ +

Ω2

+



Γ

=



Σ



 A′ × ν∇ × A · n dΓ

A′ ·(n×ν1 ∇× A1 )



 A′ × ν2 ∇ × A · n2 dΓ    A′ ·(n×ν2 ∇× A2 )

A′ · J i dΩ

(6.114)

Ω

with n1 = −n2 = n. Now, let us choose a test function A′ being concentrated near the interface Σ and zero elsewhere. Furthermore, we assume A′ to point into normal direction, i.e., A′ = wn with w an arbitrary scalar function. Since the term (n × ν∇ × A) results in a vector orthogonal to n, the scalar product with A′ is zero. Moreover, due to (6.106) and the fact that the intersection of the support of A′ with the interior of Ω1 and Ω2 is made arbitrary small, and the interface Σ is at distance from the boundary Γ , only the first term in (6.114) remains 

γ1 wn ·

∂ A1 dΓ = ∂t

Σ



γ2 wn ·

∂ A2 dΓ. ∂t

(6.115)

Σ

This holds for any w, so that



γn·

 ∂A = 0. ∂t

(6.116)

On the other hand, when we choose A′ arbitrarily tangential to Σ and the support of A′ as concentrated to Σ, then the volume integrals in (6.114) are vanishing, while only the surface integrals over Σ can give nonzero values 

Σ

A′ · (n × ν1 ∇ × A1 ) dΓ =



A′ · (n × ν2 ∇ × A2 ) dΓ

Σ

[ν n × ∇ A] = 0.

(6.117)

252

6 Electromagnetic Field

Fig. 6.18 Hat function φε (z) with support [−ε, ε]

φ(z)

−ε

ε

z

The different asymptotic behaviors of volume and surface integrals, as it was exploited here for deriving (6.117), can more easily be made obvious in a special situation without losing generality. Let us specify Σ as a plane orthogonal to the z-axis Σ = {(x, y, 0)|(x, y) ∈ [0, 1]2 }. Then, our proof amounts to choose ⎛

wx (x, y)

⎞

⎟ ⎜ A′ (x, y, z) = ⎝ w y (x, y) ⎠ φε (z) 0

with φε (z) as a hat function having support [−ε, ε] and value1 at z = 0 (see Fig. 6.18). ε Now, observe that φε equals 1 on Σ for all ε, while, since −ε φε (z) dz goes to zero as ε → 0, volume integrals containing A′ in their integrand have to vanish as ε → 0. Note that this situation, although looking quite simple, is sufficiently significant, since any smooth interface can locally be smoothly transformed to a piece of a plane. The edge finite elements [9] for the spatial discretization of (6.112) are H 0 (curl)conform, and solving the resulting algebraic system of equations leads to correct results. Nevertheless, the solution of the algebraic system requires special care in order to obtain an optimal solver (see e.g., [10, 11]). We suggest to add a fictitious electric conductivity γ ′ to regions with zero electric conductivity to obtain a variational form, which is elliptic [12]. Of course, this fictitious conductivity γ ′ has to be chosen small compared to the reluctivity of the material. The proof of convergence even in the case of γ ′ → 0 is given in [13]. For the application of nodal finite elements, we have to perform additional steps. As shown in [14], the space H Σ 0 (curl) has the decomposition for any convex domain Ω  3 1 ⊕ grad H01 (Ω). (6.118) HΣ 0 (curl) = H0 (Ω) This is equivalent to splitting the magnetic vector potential A as follows A = w + ∇φ

∇ · w = 0,

(6.119)

6.7 Numerical Computation: Electromagnetics

253

3 with w ∈ H01 (Ω) and φ ∈ H01 (Ω). The same decomposition is done for the test function A′ = v + ∇ψ. Since J i is assumed to be divergence free, ψ does not enter into the right-hand side.  To guarantee ∇ ·w = 0, we may add the term Ω ν ∇ ·v ∇ ·w dΩ to the variational formulation (6.112), which corresponds to a penalty formulation (see e.g., [15]). The variational formulation changes to: Find (w, φ) ∈ (H01 (Ω)3 , H01 (Ω)) such that ν∇ × v · ∇ × w dΩ +



∂ γ(v + ∇ψ) · (w + ∇φ) dΩ ∂t



J i · v dΩ,

Ω

+





ν ∇ · v ∇ · w dΩ

Ω

Ω

=

(6.120)

Ω

for any (v, ψ) ∈ (H01 (Ω)3 , H01 (Ω)). Thus, the standard continuous nodal finite elements can be used to approximate both fields w and φ and we arrive at a conforming FE approximation. It should be noted that due to the decomposition of A into w and ∇φ the magnetic induction B as well as eddy current density J eddy compute as B = ∇ × A = ∇ × (w + ∇φ) = ∇ × w ∂w ∂∇φ ∂A = −γ −γ . J eddy = −γ ∂t ∂t ∂t

(6.121) (6.122)

As reported in many scientific contributions, the discretization of (6.120) with nodal finite elements leads to correct values in the eddy current case, where the permeability is constant all over the domain (see e.g., [16]). Inaccurate results have been demonstrated in the case of domains with materials of different magnetic reluctivity (e.g., iron–air interface). Therefore, we will concentrate on this problem and in the following investigate the magnetostatic case (∂/∂t = 0) 

ν∇ × v · ∇ × w dΩ +

Ω



ν ∇ · v ∇ · w dΩ



J i · v dΩ.

Ω

=

(6.123)

Ω

In the first step we will consider a domain as displayed in Fig. 6.17, where we have a jump of the reluctivity at the interface Σ between the subdomains Ω1 and Ω2 . If we perform similar steps as for the previous variational formulation (6.112), we will see that (6.123) fulfills the following interface conditions

254

6 Electromagnetic Field

Fig. 6.19 Ferromagnetic cube in air

ν2 → 0 ν1 ν2

[ν n12 × ∇ × w] = 0 [ν n12 ∇ · w] = 0.

(6.124) (6.125)

Of course, from the numerical point of view, the approximation of these interface conditions using linear interpolation functions will be poor, but applying quadratic interpolation functions will give acceptable results. So, the question arises, what is the reason for the poor results discretizing (6.123) with nodal finite elements? Let us consider the case of a ferromagnetic cube embedded in air (see Fig. 6.19). Assuming the case ν2 → 0 (permeability is very large in the iron), we arrive at a non-convex domain. Now, according to [17], it is known that for non-convex domains the discretization with nodal finite elements produces wrong solutions due to the nondensity of smooth fields. In [17] the authors could prove that by introducing a special weighting function inside the divergence integral, nodal finite elements can yet be used for the approximation. Therefore, the second term in the variational formulation (6.123) has to be changed to 

ν ∇ · v ∇ · w dΩ →

Ω



ν s ∇ · v ∇ · w dΩ,

(6.126)

raα .

(6.127)

Ω

with s=



a∈A

In (6.127) A denotes the set of all re-entrant corners, ra the distance to each re-entrant corner, and α an exponent. This means that w has to be in H Y H Y = {w ∈ H 0 (curl) | ∇ · w ∈ Y }.

(6.128)

By a correct choice of the weighted Sobolev space Y , the space H 10 is dense in H Y , and thus, nodal finite elements do lead to correct results, and we call it the weighted regularization method.

6.7 Numerical Computation: Electromagnetics

255

It has to be noted that this idea can be implemented in a simple way by setting the weighting function s to zero for finite elements near re-entrant corners [18]. This was exactly the treatment the authors in [19] employed, where a current vector potential formulation was used. Since the current vector potential is just defined in electric conductive regions, in most cases the boundary will exhibit re-entrant corners.

6.7.2 Discretization with Edge Elements Performing an edge finite element discretization of (6.112), we first define the approximation of the vector potential A as A≈

m eq

N b Ab .

(6.129)

b=1

In (6.129) m eq defines the number of edges with unknown magnetic vector potential in the finite element mesh, N b the edge shape function associated with the bth edge, and Ab the corresponding degree of freedom, namely the line integral of the magnetic vector potential along the bth edge Ab =



A · ds.

(6.130)

b

It is well known, that an edge FE-discretization of (6.112) is H0 (curl)-conform. Nevertheless, for problems with regions where the electric conductivity γ is zero, the variational formulation is not elliptic. Therefore, we suggest to transform the weak formulation to a regularized one by adding a fictitious electric conductivity γ ′ to regions with zero electric conductivity. Of course, this fictitious conductivity γ ′ has to be chosen small as compared to the reluctivity of the material. The proof of convergence even in the case of γ ′ → 0 is given in [13]. Furthermore, for higher order FE approximations, this regularization is just applied to the lowest order Nédélec functions (see Sect. 6.7.6). Applying the same discretization to the test function A′ , we arrive at (see (6.112)) m eq m eq a=1 b=1

⎛ ⎝



γ N a · N b dΩ A˙ b +

Ω



ν(∇ × N a ) · (∇ × N b ) dΩ Ab

Ω

−



Ω

⎞

N a · J i dΩ ⎠ = 0.

(6.131)

256

6 Electromagnetic Field

In matrix form, we obtain M A A˙ + K A A = f

(6.132)

with MA = KA = f =

ne 

e=1 ne 

e=1 ne 

e=1

e

e

γ N p · N q dΩ

(6.133)



T  Bqcurl dΩ ν B curl p

(6.134)



N p · J i dΩ.

(6.135)

m ; m = [m pq ] ; m pq = ke ; ke = [k pq ] ; k pq = e

e

f ; f =

[ f pe ]

;

f pe

=



Ωe

Ωe

Ωe

we know, according to Sect. 2.7 that the edge shape For the computation of B curl p function N p for a tetrahedra is given by N p = (Ni ∇ N j − N j ∇ Ni )l p ,

(6.136)

with p defining the edge and i, j the corresponding vertices. To find a more explicit form for B curl p , let us rewrite N p as follows

Np =

=

⎛

⎞

⎛ ∂Nj ∂x ⎟ ⎜ ⎜ ⎜ ∂N ⎟ l p Ni ⎜ ∂ y j ⎟ − l p N j ⎜ ⎝ ⎠ ⎝ ∂Nj ∂z ⎞ ⎛ ∂Nj Ni Ni ∂x − N j ∂∂x ⎟ ⎜ ⎟ ⎜ ∂N l p ⎜ Ni ∂ yj − N j ∂∂Nyi ⎟ . ⎠ ⎝ ∂N Ni Ni ∂z j − N j ∂∂z

∂ Ni ∂x ∂ Ni ∂y ∂ Ni ∂z

⎞ ⎟ ⎟ ⎠

(6.137)

(6.138)

Therefore, we obtain ⎛

⎜ ⎜ B curl = ∇ × N = 2l p p⎜ p ⎝

∂ Ni ∂ N j ∂ y ∂z ∂ Ni ∂ N j ∂z ∂x ∂ Ni ∂ N j ∂x ∂ y

− − −

∂Nj ∂y ∂Nj ∂z ∂Nj ∂x

∂ Ni ∂z ∂ Ni ∂x ∂ Ni ∂y

⎞

⎟ ⎟ ⎟. ⎠

(6.139)

The time discretization is performed by applying the trapezoidal method (see Sect. 2.5.1) using, e.g., an effective mass formulation, which results in the following scheme:

6.7 Numerical Computation: Electromagnetics

257

• Perform predictor step: A˜ = An + (1 − γP )Δt A˙ n .

(6.140)

• Solve algebraic system of equations: M ∗A A˙ n+1 = f n+1 − K A A˜ M ∗A

• Perform corrector step:

(6.141)

= M A + γP Δt K A .

(6.142)

An+1 = A˜ + γP Δt A˙ n+1 .

(6.143)

6.7.3 Discretization with Nodal Finite Elements According to (6.120), we have to compute the unknowns of the vector w and the scalar φ at all finite element nodes. To obtain four equations for the three unknowns of w and the one of φ, we set once the test function ψ and once the test function v to zero, which results in    

˙ + ∇ φ˙ dΩ ν∇ × v · ∇ × w dΩ + ν ∇ · v ∇ · w dΩ + γ v · w Ω

Ω

Ω

=



J i · v dΩ

(6.144)

Ω



Ω



˙ + ∇ φ˙ dΩ = 0, γ ∇ψ · w

(6.145)

˙ = ∂w/∂t and φ˙ = ∂φ/∂t. In the first step we will perform a spatial diswith w cretization of (6.144) and (6.145) by introducing the following approximations of the continuous functions w and φ ′

w ≈ wh =

nn nd i=1 a=1

′

Na wia ei =

n ′n

φ ≈ φh =



Na φa .

nn a=1

⎞ Na 0 0 N a wa ; N a = ⎝ 0 Na 0 ⎠ (6.146) 0 0 Na ⎛

(6.147)

a=1

In (6.146) and (6.147) n ′n denotes the number of nodes with no Dirichlet boundary condition, n d the space dimension and Na the interpolation function for node a. Applying the same approximations for the test functions A′ and ψ, we arrive at

258 n ′n

6 Electromagnetic Field n ′n

⎛

 ⎜ ⎜ γ N T N b dΩ w ˙ + γ N aT ∇ Nb dΩ φ˙ b b ⎝     a  a=1 b=1

Ω

+



Ω

−



Ω n ′n

n ′n

a=1 b=1

⎛ ⎝



Ω

1 Bab

2 Bab

ν (∇ × N a )T (∇ × N b ) dΩ w b +    (Bacurl )T



Ω

⎞

N a · J i dΩ ⎠ = 0

ν (∇ · N a )T (∇ · N b ) dΩ wb    (Badiv )T

(6.148)

˙b + γ(∇ Na )T N b dΩ w

Ω



Ω

⎞

γ∇ Na · ∇ Nb dΩ φ˙ b ⎠ = 0.

(6.149)

The different operators B have the following explicit form ⎛

0

Na − ∂∂z

∂ Na ∂y Na − ∂∂x

⎜ Bacurl = ⎜ ⎝

∂ Na ∂z − ∂∂Nya

Na Nb

0

0

⎜ 1 Bab =⎝

0

Na Nb

0

0

0

Na Nb

⎛

0 ∂ Na ∂x

0

⎞

⎛ ∂ Na

0

∂x

0

⎞

⎟ ⎜ ∂ Na ⎟ Badiv = ⎜ ⎝ 0 ∂y 0 ⎠ Na 0 0 ∂∂z ⎛ ∂ Nb ⎞ Na ∂x ⎜ ∂ Nb ⎟ 2 ⎜ Bab = ⎝ Na ∂ y ⎟ ⎠. Nb Na ∂∂z

⎟ ⎟ ⎠

⎞ ⎟ ⎠

(6.150)

(6.151)

Thus, we can write the spatially discretized matrix equation as 

M ww M wφ T M wφ M φφ



w˙ φ˙



+



K ww 0 0

0



w φ



=



f mag 0



,

(6.152)

with w, φ the unknowns at the nodes and the matrices as well as the right-hand side vector compute as M ww = M wφ =

ne 

e=1 ne 

e=1

meww ; meww = [m pq ]; m pq = mewφ ;

mewφ

= [m pq ]; m pq =



γB 1pq dΩ

(6.153)

Ωe



Ωe

γB 2pq dΩ

(6.154)

6.7 Numerical Computation: Electromagnetics

M φφ =

ne 

e=1

meφφ ; meφφ = [m pq ];

m pq = K ww =

ne 

e=1

ne 

e=1



Ωe

(6.155)

γ∇ Na · ∇ Nb dΩ

keww ; keww = [k pq ]

k pq = f mag =

259



Ωe

 T curl div T div dΩ ν (B curl ) B + (B ) B p q p q

(6.156)

f emag ; f emag = [ f a ];

fa =



(6.157)

(Na Jix , Na Jiy , Na Jiz )T dΩ,

Ωe

with n e the number of finite elements. The time discretization is performed by applying the trapezoidal method (see Sect. 2.5.1) both to w, φ using, e.g., an effective mass formulation, which results in the following scheme: • Perform predictor step: ˙n w ˜ = w n + (1 − γP )Δt w φ˜ = φn + (1 − γP )Δt φ˙ n .

(6.158) (6.159)

• Solve algebraic system of equations: 

M ∗ww M wφ T M wφ M φφ



w˙ n+1 φ˙



n+1 M ∗ww

=



f n+1 0



−



K ww 0 0

= M ww + γP Δt K ww .

0



w ˜ ˜φ



(6.160) (6.161)

• Perform corrector step: ˜ + γP Δt w ˙ n+1 w n+1 = w ˜ ˙ = φ + γP Δt φ . φ n+1

n+1

(6.162) (6.163)

260

6 Electromagnetic Field

6.7.4 Newton’s Method for the Nonlinear Case The main nonlinearity within magnetic field computation is due to the dependence of the permeability µ on the magnetic field, which can be modeled as follows B = µ(H )H H = ν(B)B,

(6.164) (6.165)

with µ denoting the magnetic permeability and ν the magnetic reluctivity. Since experimental setups for measuring the nonlinear behavior of magnetic materials exhibit a functional relation between H and B, we can derive the following alternative formulation H (B) = ν(B)B H (B) ν(B) = B ′ (B)B − H (B) H ν ′ (B) = . B2

(6.166) (6.167) (6.168)

For the magnetostatic case, we obtain from (6.112) the following weak formulation 

ν(|∇ × A|) (∇ × A) · (∇ × A′ ) dΩ =

Ω



J i · A′ dΩ,

(6.169)

Ω

with the test function A′ ∈ H 0 (curl). This equation can be expressed with the nonlinear operator F as F( A) = 0. Newton’s method is now defined by (see Appendix E.2) Ak+1 = Ak + S ,

where S solves F ′ ( Ak )[S] = −F( Ak ).

(6.170)

Therefore, in the first step we must derive the linearized form of F F ′ ( Ak )[S] = F( Ak + S) − F( Ak ) + O(||S||2 ).

(6.171)

The term F( Ak + S) − F( Ak ) for arbitrary test functions A′ ∈ H 0 (curl) computes in its weak form as

6.7 Numerical Computation: Electromagnetics



261

ν(|∇ × ( Ak + S)|) (∇ × ( Ak + S)) · (∇ × A′ ) dΩ

Ω

−



ν(|∇ × Ak |) (∇ × Ak ) · (∇ × A′ ) dΩ.

(6.172)

Ω

By adding the term ν(|∇ × Ak |)(∇ × ( Ak + S) · (∇ × A′ ) to (6.172) as well as at the same time subtracting it from (6.172) and combining similar terms leads to     ν(|∇ × ( Ak + S)|) − ν(|∇ × Ak |) ∇ × ( Ak + S) · (∇ × A′ ) dΩ

Ω

+



ν(|∇ × Ak |) (∇ × S) · (∇ × A′ ) dΩ.

(6.173)

Ω

Now we can perform the following approximations   ν(|∇ × ( Ak + S)|) − ν(|∇ × Ak |) ≈ ν ′ (|∇ × Ak |) |∇ × ( Ak + S)| − |∇ × Ak | |∇ × ( Ak + S)|2 − |∇ × Ak |2 |∇ × ( Ak + S)| + |∇ × Ak | 2 2 |∇ × Ak | + |∇ × S| + 2(∇ × Ak ) · (∇ × S) − |∇ × Ak |2 = |∇ × ( Ak + S)| + |∇ × Ak | (∇ × S) · (∇ × Ak ) ≈ . |∇ × Ak |

|∇ × ( Ak + S)| − |∇ × Ak | =

(6.174)

With the help of these approximations, (6.173) can be written as 

ν ′ (|∇ × Ak |)

Ω

(∇ × S) · (∇ × Ak ) (∇ × ( Ak + S) · (∇ × A′ ) dΩ |∇ × Ak |  + ν(|∇ × Ak |) (∇ × S) · (∇ × A′ ) dΩ. (6.175) Ω

Neglecting all terms of second order and retaining only terms linear in S, we arrive at 

V

ν ′ (|∇ × Ak |)

(∇ × S) · (∇ × Ak ) (∇ × Ak ) · (∇ × A′ ) dΩ |∇ × Ak |  + ν(|∇ × Ak |) (∇ × S) · (∇ × A′ ) dΩ. Ω

(6.176)

262

6 Electromagnetic Field

Therefore, the resulting Newton step computes as (see (6.170)) 

ν(Bk ) (∇ × S) · (∇ × A′ ) dΩ

Ω

+



ν ′ (Bk ) (∇ × S) · e Bk Bk e Bk · (∇ × A′ ) dΩ



J i · A′ dΩ



ν(Bk )(∇ × Ak ) · (∇ × A′ ) dΩ

Ω

=

Ω

−

(6.177)

Ω

Ak+1 = Ak + S,

(6.178)

with B k = ∇ × Ak , Bk = |B k | and e Bk = B k /Bk . Now, let us apply the FE method to (6.177) using, e.g., nodal finite elements. Therefore, we have to consider instead of (6.169) a weak formulation similar to (6.123). Since we have to ensure ∇ · w = 0 (see (6.119)), we apply a penalty formulation by adding  ν ∇ · v ∇ · w dΩ to (6.177) (having the decomposition A = w + ∇φ in mind)

Ω



ν(Bk ) (∇ × v) · (∇ × S) dΩ +

Ω

+



ν(Bk ) ∇ · v ∇ · S dΩ

Ω



ν ′ (Bk )Bk (∇ × v) · e Bk (∇ × S) · e Bk dΩ



J i · v dΩ

Ω

=

Ω

−



Ω

  ν(Bk ) (∇ × wk ) · (∇ × v) + (∇ · wk )(∇ · v) dΩ.

(6.179)

Performing the same procedure as in Sect. 6.7.3, we arrive at the following algebraic system of equations   K L (w k ) + K NL (wk ) S = f mag − K L (w k )w k = f res .

(6.180)

In (6.180) K L and f mag are calculated as in (6.156) and (6.157). The matrix K NL , also called the tangent stiffness matrix, is computed as follows

6.7 Numerical Computation: Electromagnetics

K NL =

ne 

e=1

keNL ;

kNL pq =



Ωe

263

keNL = [kNL pq ]  curl T dΩ, ν ′ (Bk ) (Bcurl p · e Bk ) Bk (Bq · e Bk )

(6.181)

with n e the number of finite elements and ν ′ (Bk ) as well as e Bk to be evaluated at each integration point during the numerical integration. The iterative solution process is performed, until the following two stopping criteria are fulfilled ||wk+1 − w k ||2 < δa ||wk+1 ||2

|| f r es ||2

|| f mag ||2

< δr ,

(6.182)

with appropriate δa and δr . Since a standard Newton method is not globally convergent, it is necessary to apply a technique that controls the updating by w k+1 = wk + η S,

(6.183)

with some relaxation parameter η. The scalar parameter η > 0 is introduced to control the convergence during early steps of the iteration process, or in the presence of nonmonotonic material relations. A common algorithm to compute η is a line search (see Appendix E) defined by approximately minimizing |G(η)| = |S T f r es (w k + ηS)|.

6.7.5 Approximation of BH Curve As can be seen from Sect. 6.7.4, not only values of the magnetic reluctivity ν as a function of the magnetic induction B but also first derivatives of ν are required for the solution of the nonlinear equation. Figure 6.20a shows a typical BH curve obtained by linear interpolation of the measured BH values. Since, from the physical point of view, the BH curve is always monotone, which is not the case for the ν(B) curve as displayed in Fig. 6.20b, it makes sense to find a functional expression for H (B) and compute ν as well as ν ′ as given in (6.167) and (6.168). At this point, the question arises if one should apply an interpolation or an approximation scheme to obtain the function H (B). Since the measured BH values will always just be accurate up to a measurement error, it is not appropriate to fix the curve to these values at the measurement points, as would be done by interpolation. Instead, one should apply an approximation to the measurement points combined with some regularization to avoid oscillations. We will use so-called smoothing splines [20] and minimize the following regularized least squares functional

264

6 Electromagnetic Field 2

(a)

1.8 1.6

B (T)

1.4 1.2 1 0.8 0.6 0.4 0.2 0 0

5

10

15

20

1.5

2

H (kA/m) 12000

(b) 10000

ν (Am/Vs)

8000

6000

4000

2000

0 0

0.5

1

B (T) Fig. 6.20 Typical BH curve and computed ν(B) curve. a Typical BH curve. b Corresponding ν(B) curve

α

J [H ] =

Bend 0



N 2 H ′′ (B) dB + α (H (Bk ) − Hk )2 ,

(6.184)

k=1

with H ′′ (B) the second derivative with respect to B and N the number of measured BH values. Therefore, we obtain for small values of α a very smooth approximation and for large values of α a solution, which will be very close to the discrete points. For determining the regularization parameter α a posteriori, we use a discrepancy principle with bisection, i.e., set α equal to the largest member of the geometrically decreasing

6.7 Numerical Computation: Electromagnetics

265

sequence 2− j such that for the minimizer B α of J α the residual ! N ! " (H (B ) − H )2 k k

(6.185)

k=1

is of the same order of magnitude as the measurement noise. Note that—unlike ν—the BH curve itself always has to be monotone for physical reasons, so in order to be able to monitor and preserve this property we prefer to apply N points. the described smoothing spline technique directly to the collection of (Bk , Hk )k=1 Practically, the monotonicity of the BH curve approximation is gained by putting restrictions on the coefficients of the spline curve. These restrictions arise naturally, if we demand on H ′ (B) > 0. Consequently, the task is to find a monotone approximation that fulfills the discrepancy principle for a given data noise level. Note additionally that the N points. If we would deduce an estimate for noise level is given in terms of (Bk , Hk )k=1 N points, this would, due to the strong variation the data noise in terms of the (νk , Bk )i=k in scale of both function and derivative values of the curve under consideration, lead to a locally too pessimistic noise estimate and hence to a poor curve approximation. For a detailed discussion on this topic we refer to [21, 22].

6.7.6 Higher Order Edge Elements The two key requirements for our choice of higher order H(curl)-conforming FE shape functions are the ability to choose the polynomial degree p independently in each local direction (ξ, η, ζ), as well as the availability of efficient iterative solution techniques (i.e. an efficient preconditioner). The first requirement leads to the use of hierarchical shape functions, which can be written by the following decomposition [23, 24] N(T ) =

e

f

N e,N0 ⊕ N f,∇ ⊕

e



N e,∇ ⊕

N int ⊕

f



Nf⊕

N int,∇ .

(6.186)

The shape functions N are composed of unknowns defined on edges, faces and in the interior (subscripts e, f and int), see Fig. 6.21 for a hexahedral element. Furthermore, we denote with the index N0 the lowest order Nédélec functions (see (6.189)) and with the index ∇ basis functions describing pure gradient fields (see (6.190), (6.194) and (6.195)). For the definition of the vectorial shape functions, it is convenient to introduce the bilinear shape functions αi and linear shape functions βi for every vertex vi as

266

6 Electromagnetic Field

e11

v8 e12 v5

v7 e10

f6

e9 e8

pζ

v6

e7

ζ

pξ

η e5

v4

ξ

f4

e6

f5 f

1

f3 v 3

e3

e4 v1

pη

e2

f2

e1

v2

Fig. 6.21 Degrees of freedom for the hexahedral element

α1 = α2 = α3 = α4 = α5 = α6 = α7 = α8 =

1 (1 − ξ)(1 − η)(1 − ζ); 8 1 (1 + ξ)(1 − η)(1 − ζ); 8 1 (1 + ξ)(1 + η)(1 − ζ); 8 1 (1 − ξ)(1 + η)(1 − ζ); 8 1 (1 − ξ)(1 − η)(1 + ζ); 8 1 (1 + ξ)(1 − η)(1 + ζ); 8 1 (1 + ξ)(1 + η)(1 + ζ); 8 1 (1 − ξ)(1 + η)(1 + ζ); 8

1 ((1 − ξ) + (1 − η) + (1 − ζ)) 2 1 β2 = ((1 + ξ) + (1 − η) + (1 − ζ)) 2 1 β3 = ((1 + ξ) + (1 + η) + (1 − ζ)) 2 1 β4 = ((1 − ξ) + (1 + η) + (1 − ζ)) 2 1 β5 = ((1 − ξ) + (1 − η) + (1 + ζ)) 2 1 β6 = ((1 + ξ) + (1 − η) + (1 + ζ)) 2 1 β7 = ((1 + ξ) + (1 + η) + (1 + ζ)) 2 1 β8 = ((1 − ξ) + (1 + η) + (1 + ζ)). 2 β1 =

This can be utilized to parametrize each edge e = [v1 , v2 ] as ξe = (βv1 − βv2 ) ∈ [−1, 1] , with the tangential vector τ e = ∇(βv1 − βv2 ) .

6.7 Numerical Computation: Electromagnetics

267

Fig. 6.22 Displayed are edge extension parameter λe and tangential direction τ e for edge 1

v3

v4 e = const.

Fig. 6.23 Displayed is the face extension parameter λ f and its gradient ∇λ f = −n f for face 1

e1

v4

v1

v2

1



e

v1

1

f

v3

1

f1

v2

f = const. 1

In addition, the extension operator λe = αv1 + αv2 is defined as 1 on the edge e and 0 on all other parallel edges (see Fig. 6.22). Analogously, we can parametrize each face f = [v1 , v2 , v3 , v4 ] as (ξ f , η f ) = (βv1 − βv2 , βv1 − βv4 ) ∈ [−1, 1] × [−1, 1] .

(6.187)

Every face can be extended into the element by the face extension parameter λ f = αv1 + αv2 + αv3 + αv4 , which is 1 on the face f and 0 on the opposite one (see Fig. 6.23). The normal direction of the face f (pointing out of the volume) can thus be written as n f = −∇λ f .

(6.188)

Thus the H(curl)functions on the hexahedral element are defined as1 1. Edge Functions: On every edge ei = [v1 , v2 ], i = 1, . . . , 12 there are two types of shape functions defined: • The lowest order Nédélec functions are defined as N ei ,N0 = τ ei λei = ∇(βv1 − βv2 )(αv1 + αv2 ). 1

φk denotes the integrated Legendre function of order k, see Sect. 2.9.1.

(6.189)

268

6 Electromagnetic Field

• The higher order shape functions up to order 0 ≤ m ≤ pξe − 1 are pure gradient fields and defined as Nm ei ,∇ = ∇(φm+2 (ξe ))λei = ∇(φm+2 (βv1 − βv2 ))(αv1 + αv2 ).

(6.190)

2. Face Functions: On every face f i = [v1 , v2 , v3 , v4 ], i = 1, . . . , 6 there are three different types of shape functions for every set of polynomial orders 0 ≤ m ≤ pξ f , 0 ≤ n ≤ pη f . On every face the local coordinates (ξ f , η f ) are defined as in (6.187). • The first type of shape functions is defined as 

′ ′ N m,n f i ,1 = φm+2 (ξ f ) φn+2 (η f ) ∇ξ f − φm+2 (ξ f ) φn+2 (η f ) ∇η f λ f . (6.191)

• The second set of shape functions is defined as

N 0,n f i ,2 = φn+2 (η f ) λ f ∇ξ f

N m,0 f i ,2 = φm+2 (ξ f ) λ f ∇η f .

(6.192) (6.193)

• The third set of shape functions consists only of gradients

 N m,n f i ,∇ = ∇ φm+2 (ξ f ) φn+2 (η f ) λ f .

(6.194)

3. Interior Functions: In the interior of each cell there are for every set of polynomial orders 0 ≤ m ≤ pξ , 0 ≤ n ≤ pη , 0 ≤ q ≤ pζ three types of shape functions: • The gradient based functions are defined as m,n,q

N int,∇ = ∇(φm+2 (ξ) φn+2 (η) φq+2 (ζ)).

(6.195)

• The rotated gradient shape functions N int,1 = diag(1, −1, 1) N int,∇

m,n,q

m,n,q

(6.196)

= diag(1, 1, −1)

m,n,q N int,∇ .

(6.197)

m,n,q N int,1∗

• The third independent set is defined as 0,n,q

N int,1 = φn+2 (η) φq+2 (ζ) eξ

(6.198)

= φm+2 (ξ) φq+2 (ζ) eη

(6.199)

= φm+2 (ξ) φm+2 (η) eζ .

(6.200)

m,0,q N int,1 N m,n,0 int,1

The lowest order Nédélec functions N ei ,N0 , which have a constant tangential component along one edge ( p = 0), are explicitly included. In addition, higher order gradient components on edges N ei ,∇ , faces N fi ,∇ and in the interior N int,∇ are represented separately. This key feature—also known as the local exact sequence property—is equivalent to fulfilling the so-called De-Rham complex already on the finite element level [25], i.e.

6.7 Numerical Computation: Electromagnetics

269

the gradients, forming the null-space of the curl-operator, can be completely omitted for each type of unknowns (edge, face, interior) separately if only the flux density B is of interest. In [23], this is denoted as the reduced basis and can be used to gauge the problem in the following way: • For the lowest order Nédélec functions N ei ,N0 , we add the fictitious conductivity γ ′ as described in Sect. 6.7.2. • For the higher order terms, we simply skip the gradient functions N ei ,∇ , faces N fi ,∇ and N int,∇ . Another unique advantage of this basis according to [26] is that shape functions of arbitrary order are available for all types of elements in 2-D and 3-D, utilizing any kind of hierarchical 1-D shape functions, e.g. Legendre functions. In Fig. 6.24 the basis functions of edge e1 are displayed for p = 0, 1, 2. The usage of hierarchical finite elements of higher order makes it possible to adapt the local accuracy in different spatial directions, especially applicable for the magnetic field computation in thin steel sheets. In general the magnetic flux density B is defined as ⎞ ⎛ ∂A ∂ Az ⎛ ⎞ − ∂zy ∂ y Bx ⎟ ⎜ ⎜ Ax Az ⎟ B = ⎝ B y ⎠ = ∇ × A = ⎜ ∂∂z (6.201) − ∂∂x ⎟ . ⎝ ⎠ Bz ∂ Ay ∂ Ax ∂x − ∂ y In case of thin structures, the in-plane components (Bx , B y ) are dominant (see Fig. 6.25). Additionally, the variation of the in-plane components of A in z-direction (∂ A x /∂z, ∂ A y /∂z) in (6.201) is already resolved accurately by the FE-discretization in thickness direction of the single sheet layers. Thus the dominant terms left are ∂ A z /∂ y and ∂ A z /∂x, i.e. the magnetic vector potential A should be approximated quite accurately in the inplane direction. Therefore, it is advantageous to reflect this behavior in the anisotropic polynomial degree as pη , pξ > pζ , (6.202) assuming that the global z and the local ζ direction coincide. The increase of the polynomial degree only affects the face N f and inner N int degrees of freedoms, as we skip higher order gradient functions N e,∇ and N f,∇ due to gauging. This leads to practical order templates like paniso = (2, 2, 1) or paniso = (3, 3, 1). Although, it seems that the lowest order anisotropic template should be paniso = (1, 1, 0), this does not lead to more accurate results, as only faces in (x, y)-direction get additional unknowns, which do not contribute to an improved resolution of A z . As the permeability in air µ0 is typically several orders of magnitude smaller compared to the one in the ferromagnetic core, the flux is mostly concentrated in the core. This allows us to choose a small isotropic polynomial degree of pair = 0 for the approximation. The last step is especially effective, if structured grids are utilized or if the air domain is significantly large. Finally, the explicit representation of gradients in the basis functions allows the application of a simple Schwarz-type block preconditioner with a conjugate gradient solver

270 Fig. 6.24 H(curl)basis functions on edge 1. a Lowest order function on edge 1 (N e1 ,N0 ). b Gradient function of order 1 on edge 1 (N 2e1 ,∇ ). c Gradient function of order 2 on edge 2 (N 2e1 ,∇ )

6 Electromagnetic Field

6.7 Numerical Computation: Electromagnetics Fig. 6.25 Flux/potential distribution on face in (x, z)-plane

271

Az(x) z Az(z) z

By(x)

y

Az(x) x

x

for an efficiently solution of the arising algebraic system. By adapting the block size automatically according to the aspect ratio, deterioration of convergence in case of thin elements can be prevented, as demonstarted in Sect. 13.5.

6.7.7 Modeling of Current-Loaded Coil In this section we will investigate the numerical computation of electromagnetic fields excited by a current-loaded coil. Figure 6.26 displays a typical setup of a coil with a ferrite core. To simplify the geometric modeling of the winding structure as well as the further meshing, we will substitute the complex structure by a hollow cylinder (see Fig. 6.26b). This simplification can be done since the change in the computed magnetic field can be neglected. The relation for the current density J in the coil is now defined by J = J eJ =

(a)

Ferrite core

I I Nc eJ = eJ . Γw κΓc Ferrite core

(b)

i(t)

(6.203)

i(t) Complex winding structure

Hollow cylinder

u(t)

Fig. 6.26 Coil with a ferrite core. a Structure of the winding. b Hollow cylinder

272

6 Electromagnetic Field

In (6.203) e J denotes the unit vector for the direction of J, I the current, Γw the cross section of one winding, Nc the number of turns, Γc the cross section of the hollow cylinder, and κ the filling factor.

6.7.8 Computation of Global Quantities In order to compare simulated results with measurements, we have to evaluate quantities derived from the computed field.

6.7.8.1 Magnetic Flux and Inductance In general, the inductance L for a magnetic assembly is defined by [5] L=

Ψ . I

(6.204)

In (6.204) Ψ denotes the total magnetic flux and I the electric current loading the coil. The total magnetic flux Ψ for a coil with Nc turns computes as Ψ = Nc φ, where φ is the flux defined by (see Fig. 6.27)  B · dΓ . φ=

(6.205)

Γ

Expressing the magnetic induction B via the magnetic vector potential A and using Stoke’s theorem (Appendix B.9), we arrive at  φ = (∇ × A) · dΓ Γ

=



A · ds.

(6.206)

C(Γ )

Fig. 6.27 Magnetic field crossing Γ

B

n

Γ C

ds

6.7 Numerical Computation: Electromagnetics

273

It has to be noted that the orientation of d s depends on the normal vector n according to the right-hand rule.

6.7.8.2 General 3D Case To obtain a very accurate numerical value for the magnetic inductance, we compute the magnetic flux over the whole coil volume and then normalize it to the cross section of the coil Γc (see Fig. 6.26). Thus, we obtain ⎞ ⎛ n  1 ⎜ ⎟ ( A · e J ) dx dy dz ⎠ , (6.207) φ= ⎝ Γc i=1 Ω e i

with n the number of finite elements within the coil region (hollow cylinder) and e J the unit vector defining the direction of the current density J in the coil.

6.7.8.3 2D Plane Case According to the 2D plane model (see Fig. 6.28), we have A = A z (x, y)ez

d s = dz ez . Thus, the evaluation of (6.206) gives us

φ = −A1z l + A2z l = l(A2z − A1z ). Since the magnetic vector potential can vary over the cross section, we compute the magnetic flux φ by the following averaging

(a)

(b) y

Cut l

Γc

Γc x

Region 1 Γc

Fig. 6.28 Example for a 2D plane case. a Full 3D setup. b 2D model

Region 2

274

6 Electromagnetic Field

⎛ ⎞ n2  n1  l ⎜ ⎟ A1z dx dy ⎠ , φ= A2z dx dy − ⎝ Γc i=1Ω e

i=1 Ω e

i

i

with n 1 and n 2 the number of finite elements in regions 1 and 2.

6.7.8.4 Axisymmetric Case The situation for the axisymmetric case is displayed in Fig. 6.29. In the axisymmetric case, just the circumferential component of the magnetic vector potential is different from zero, thus A = Aϕ (r, z)eϕ

ds = dϕ eϕ .

Here we also perform an averaging, which means to compute the integral over the volume of the coil and normalize it to the cross section Γc of the coil Ψ = Nc φ =

n  2π Nc r Aϕ dr dz, Γc

(6.208)

i=1 Ω e i

with n the number of finite elements in Γc .

Fig. 6.29 Example for a axisymmetric case. a Full 3D setup. b 2D model

(a)

Ferrite core

(b)

z

Ferrite core

i(t) Hollow cylinder Coil, Γc

r

6.7 Numerical Computation: Electromagnetics

275

6.7.9 Induced Electric Voltage Let us consider an electrically open coil, and the task of computing the induced electric voltage due to a time-varying magnetic field. Then, according to Sect. 6.1, we have to evaluate u ind = −Nc

dφ , dt

(6.209)

with Nc the number of turns. Thus, we can simply use (6.207) (or its 2D versions), replace ˙ and multiply the result by −Nc to arrive at (6.209). A by A

6.7.10 Voltage-Loaded Coil Let us assume the case as displayed in Fig. 6.30. The voltage source with amplitude u(t) is directly connected to the coil, which has an ohmic resistance defined by Rs . Therefore, the circuit equation reads as u(t) = Rs i(t) + Ψ˙ .

(6.210)

In (6.210) Ψ˙ = Nc φ˙ denotes the time derivative of the total magnetic flux, which is computed by the FE method using the magnetic vector potential A as follows Ψ˙ = Nc φ˙ = Nc



˙ · dΓ B

Γf

= Nc



Γf

i(t)

u(t)

 ˙ · dΓ = Nc ∇×A



˙ · ds, A

C(Γf )

Rs

Ψ˙

Fig. 6.30 Voltage-loaded coil with electric circuit

FE - model

(6.211)

276

6 Electromagnetic Field

with Γf the cross section of the ferrite core. The evaluation of the contour integral is usually done by averaging the magnetic vector potential A over the whole volume Ωc of the coil (see Sect. 6.7.7) 

 Nc ˙ · e J dΩ, Ψ˙ = A (6.212) κΓc Ωc

with e J a unit vector pointing into the direction of the current density J. The FE formulation for computing the electromagnetic field in the eddy current case reads as M A A˙ n+1 + K A An+1 − ˜f A,n+1 = 0.

(6.213)

In (6.213) A and A˙ denote magnetic vector potential as well as its time derivative at the FE nodes, M A the magnetic mass matrix, K A the magnetic stiffness matrix, ˜f A,n+1 the vector due to a current flowing in the coil and excited by a voltage u(t) (see (6.210)) and n the time step counter. Now, if we can assume that the skin effect in each winding can be neglected (see Sect. 6.2.2), we model the coil as in the current-loaded case, so that ˜f A computes as ˜f = i n+1 f A A,n+1 fA =

ne 

f eA

e=1

fp

Nc = κΓc

(6.214) f eA = [ f p ] 

(N p e J x N p e J y N p e J z )T dΩ.

(6.215)

Ωe

Therefore, we can set up the following matrix equation system using (6.210) and (6.213) ⎛

⎞       MA − f A  A˙ 0 An+1 KA 0 n+1 + ⎝  T ⎠ . = 0 0 u n+1 i n+1 i n+1 fA Rs

(6.216)

For the time discretization of A˙ n+1 we will apply a general trapezoidal method (see Sect. 2.5.1) 

An+1 = An + Δt (1 − γP ) A˙ n + γP A˙ n+1 .

(6.217)

˜ = An + (1 − γP )Δt A˙ n . A

(6.218)

Therefore, the following predictor/corrector algorithm results Predictor:

Algebraic system of equations:

6.7 Numerical Computation: Electromagnetics

277

⎞    M ∗A − f A  A˙ ˜ −K A A n+1 ⎠ ⎝  T = i n+1 u n+1 fA Rs ⎛

M ∗A = M A + γP Δt K A .

Corrector:

˜ + γP Δt A˙ n+1 . An+1 = A

(6.219) (6.220) (6.221)

Since, in the above equation, the system matrix is indefinite, it is preferable to solve the algebraic system by constructing a Schur complement. First, we can express A˙ n+1 using (6.219) as follows −1 −1



˜. A˙ n+1 = M ∗A (6.222) KA A f A i n+1 − M ∗A       d1

d0

−1

Instead of computing the inverse M ∗A , we will solve two algebraic systems, both ∗ having the same system matrix M A M ∗A d 0 = f A

(6.223)

˜ M ∗A d 1 = −K A A.

(6.224)

Now, since we know d 0 and d 1 , we can compute i n+1 as follows T T Rs i n+1 = u n+1 + f A d1 − f A d 0 i n+1

i n+1 =

Td u n+1 + f A 1 Td Rs + f A 0

.

(6.225)

Therefore, instead of solving (6.219), we solve (6.223) and (6.224), compute i n+1 via (6.225) and then calculate A˙ n+1 from (6.222).

6.8 Numerical Examples 6.8.1 Thin Iron Plate In a first example, we study the distribution of the magnetic induction within a thin iron plate (Fig. 6.31a). At the outer boundary the tangential component of the magnetic vector potential A has been set to zero, whereas at the symmetry planes the normal component has to vanish. The given current density within the coil generates the magnetic field. The finite element discretization using hexahedra elements is shown in Fig. 6.31b. The computational results by using nodal finite elements of 1st and 2nd order applying the standard regularization is displayed in Fig. 6.32. One can observe a quite non-physical behavior of the magnetic induction over the iron plate. Especially towards the left side the magnetic induction strongly drops to small values. This non-physical behavior is not observed by the results obtained by the weighted regularization as shown in Fig. 6.33.

278

6 Electromagnetic Field

(a)

(b)

J

Coil Plate

Plate

Coil

Fig. 6.31 Iron plate surrounded by a coil. a Geometry setup: Iron plate and coil. b Finite element mesh without ambient air

B (T)

B (T)

p=1

p=2

Fig. 6.32 Magnetic induction in the iron plate computed by nodal finite elements with the standard regularization

B (T)

B (T)

p=1

p=2

Fig. 6.33 Magnetic induction in the iron plate computed by nodal finite elements with weighted regularization

6.8 Numerical Examples

279

Here, we can also recognize an improvement when increasing the polynomial oder from p = 1 to p = 2. Finally, we display the results obtained by using edge elements of lowest order p = 0 and first order p = 1 in Fig. 6.34. Here, we observe a similar distribution of the magnetic induction as in the case of the weighted regularization. However, the maximal amplitude is about 20 % higher. Furthermore, we have computed the inductance of the coil (global quantity) for all three formulation. The results are listed in Table 6.5 and show between the two nodal finite element formulations just a difference of less than 3 %. This result indicates that the error occurring at interfaces of jumping permeability is a local one for the standard formulation. The difference to the edge finite element formulation computes to about 8 % and is less as the difference in the magnetic induction. Summarized, we can state that the weighted regularization can handle interfaces of jumping permeability and results in physical distributions of the magnetic field. However, the natural discretization for Maxwell’s equation are the H(curl)-conforming edge elements and already a low number of unknows compute the magnetic quantities quite accurately (see Table 6.5).

B (T)

B (T)

p=0

p=1

Fig. 6.34 Magnetic induction in the iron plate computed by edge elements Table 6.5 Computed inductances and degree of freedoms (DOFs) for each formulation Inductance (mH) DOFs Edge elements, p = 0 Edge elements, p = 1 Nodal elements, standard regularization, p = 1 Nodal elements, standard regularization, p = 2 Nodal elements, weighted regularization, p = 1 Nodal elements, weighted regularization, p = 2

1.7047 1.7149 1.5277 1.5386 1.5665 1.5813

17.787 104.685 18.375 149.646 18.375 149.646

280

6 Electromagnetic Field

6.8.2 TEAM-13 Benchmark Problem In computational electromagnetics the TEAM (Testing Electromagnetics Analysis Methods) workshops have a strong tradition. Thereby, results to existing benchmark problems are presented, and in addition new possible benchmark problems are discussed. TEAM problem 13 is a nonlinear magnetostatic problem consists of two U-shaped, disaligned steel channels with an additional vertical center plate (see Fig. 6.35). Due to the small air gap and the position of the plates a true three-dimensional field distribution is generated. The behavior of the steel plates is isotropic nonlinear, with a BH-curve as depicted in Fig. 6.36. The coil is loaded by a constant excitation of 1,000 A turns, which already saturates the steel sheets partially. The computational mesh is shown in Fig. 6.37 and

y coil 3.2

0 R5

25

channel (steel)

50

120

40

x

0

120

120

5

R2

10

coil

z

center plate (steel)

x channel

50

3.2

center plate

3.2

120

4.2

120

25

3.2

10

0.5

3.2 25

120

B (T)

Fig. 6.35 Setup of static, nonlinear TEAM 13 benchmark problem

H (A/m)

Fig. 6.36 Measured and interpolated BH-curve for TEAM 13 benchmark example

25

6.8 Numerical Examples

281

Fig. 6.37 Mesh of TEAM 13 setup (only steel parts shown) and lines for flux density comparison

line 2 line 1

z y x Table 6.6 Unknowns, runtime and memory consumption

Order p

Unknowns

Memory (MB)

Runtime (s)

0 1 2 0 (fine grid)

6.504 38.592 126.568 4.945.860

170 290 950 80.000

3 13 104 13.088

Fig. 6.38 Flux density |B| at lines of interest for p = 0, 1, 2. a Flux density along line 1. b Flux density along line 2

(a)

(b)

consists of 2.173 elements with 2.970 nodes. The mesh is strongly graded towards the air gaps and corners to resolve the large field variation. We are interested to compare the computed magnetic flux density against measurents along the two lines sketched in Fig. 6.37 for an increasing polynomial order p. Furthermore, we will compare the overall flux distribution against a numerical solution obtained

282

6 Electromagnetic Field

Fig. 6.39 Convergence of flux density distribution. a p = 0, b p = 1, c p = 2, d Fine grid, p = 0

on a very fine grid with 1.660.198 elements and 1.709.586 nodes. The runtime values in Table 6.6 were measured with a single CPU applying a sparse direct solver [27]. From Fig. 6.38 it is evident that the initial discretization with a polynomial degree of p = 0 results in strong deviations of the solution compared to the measured values. If p is increased to 1 or 2, the flux values increase and converge towards the measured values. The flux distribution in the complete steel channels is depicted in Fig. 6.39. All results were interpolated to the fine mesh mentioned earlier. One can see that for p = 0 the element boundaries are clearly visible due to the poor approximation of the flux at boundaries. With increasing polynomial order the solution gets much smoother and even more important the average flux values increase. For p = 2 the difference to the numerical solution obtained on the fine mesh is hardly visibly anymore.

References 1. 2. 3. 4. 5.

J.C. Maxwell, A Treatise on Electricity and Magnetism, vol. I (Dover, New York, 1954) J.C. Maxwell, A Treatise on Electricity and Magnetism, vol. II (Dover, New York, 1954) N. Ida, Engineering Electromagnetics (Springer, New York, 2004) Electromagnetic Theory (McGraw-Hill Inc, New York, 1941) H. Hofmann, Das elektromagnetische Feld (Springer, New York, 1986)

References

283

6. K.J. Binns, P.J. Lawrenson, C.W. Trowbridge (eds.), The Analytic and Numerical Solution of Electric and Magnetic Fields (Wiley, New York, 1992) 7. G. Lehner, Elektromagnetische Feldtheorie (Springer, New York, 1996) 8. W.R. Smythe, Static and Dynamic Electricity (Taylor & Francis, Philadelphia, 1989) 9. J. Nédélec, A new family of mixed finite elements in R 3 . Numer. Math. 50(1), 57–81 (1986) 10. D. Arnold, R. Falk, R. Winther, Multigrid in H(div) and H(curl). Numer. Math. 85, 197–218 (2000) 11. R. Hiptmair, Multigrid method for Maxwell’s equations. SIAM J. Numer. Anal. 36(1), 204–225 (1999) 12. M. Schinnerl, J. Schöberl, M. Kaltenbacher, Nested multigrid methods for the fast numerical computation of 3D magnetic fields. IEEE Trans. Magn. 36(4), 1557–1560 (2000) 13. S. Reitzinger, J. Schöberl, Algebraic Multigrid for Edge Elements. Numer. Linear Algebra Appl. 9, 223–238 (2002) 14. V. Girault, P.-A. Raviart, Finite Element Approximation of the Navier-Stokes Equations (Springer, Berlin, 1979) 15. Ch. Großmann, H.G. Roos, Numerik Partieller Differentialgleichungen (B.G. Teubner, Stuttgart, 1994) 16. P.J. Leonard, D. Rodger, Comparision of methods for modelling jumps in conductivity using magnetic vector potential based formulations. IEEE Trans. Magn. 33(2), 1295–1298 (1997) 17. M. Costabel, M. Dauge, Weighted regularization of Maxwell equations in polyhedral domains. Techincal Report IRMAR 01–26, IRMAR, Institut Mathematique Universite de Rennes 1, (2001) 18. M. Kaltenbacher, S. Reitzinger, Appropriate finite element formulations for 3D electromagnetic field problems. IEEE Trans. Magn. 38(2), 513–516 (2002) 19. K. Preis, O. Biró, I. Ticar, Gauged current vector potential and reentrant corners in the FEM analysis of 3D eddy currents. IEEE Trans. Magn. 36(4), 840–843 (2000) 20. C. de Boor, A Practical Guide to Splines (Springer, New York, 1987) 21. S. Reitzinger, B. Kaltenbacher, M. Kaltenbacher, A note on the approximation of B-H curves for nonlinear computations. Technical Report, SFB F013: Numerical and Symbolic Scientific Computing (Linz, Austria, 2002). September 22. C. Pechstein, B. Jüttler, Monotonicity-preserving interapproximation of B-H curves. Sfb report, Johannes Kepler University Linz, SFB Numerical and Symbolic Scientific Computing, (2004) 23. J. Schöberl, S. Zaglmayr, High order Nédélec elements with local complete sequence properties. COMPEL - Int. J. Comput. Math. Electr. Electron. Eng. 24(2), 374–384 (2005) 24. S. Zaglmayr, High Order Finite Element Methods for Electromagnetic Field Computation, Ph.D. thesis, Johannes Kepler University, Linz. (2006) 25. A. Bossavit, Computational Electromagnetism: Variational formulation, complementary, edge elements (Academic Press Inc, London, 1989) 26. R.D. Graglia, A.F. Peterson, F.P. Andriulli, Curl-conforming hierarchical vector bases for triangles and tetrahedra. IEEE Trans. Antennas Propag. 59(3), 950–959 (2011) 27. O. Schenk, K. Gärtner, On fast factorization pivoting methods for symmetric indefinite systems. Elec. Trans. Numer. Anal. 23, 158–179 (2006)

Chapter 7

Coupled Flow-Structural Mechanical Systems

The field interactions are realized through boundary conditions as well as source terms and can be generally separated into volume and surface coupled phenomena.

7.1 Fluid-Solid Interaction The fluid-solid interaction takes place at the domain interface Γfs and is a surface coupled interaction. At this interface kinematic and dynamic continuity have to be ensured. The complete coupled problem has to fulfill the condition that the location of the fluid-solid interface coincides for both fields x f = x s,0 + u

on (0, T ) × Γfs .

(7.1)

Here, x f defines the position of the fluid, x s,0 the position of the solid at initial state and u the mechanical displacement (all at the interface Γfs ). As the fluid-solid interface is impermeable and the fluid adheres to the solid, no-slip/no-penetration is assumed. The fluid-solid interaction boundary condition is of an inhomogeneous Dirichlet type v=

∂d ∂t

on (0, T ) × Γfs

(7.2)

with d = u on Γfs . The movement of the fluid-solid interface leads to a change of the computational fluid domain. In order to keep the topology of the fluid grid, we will need an appropriate grid adaption, which will result in a movement of the fluid grid governed by the grid velocity v g . By applying the ALE (ArbitraryLagrangian-Eulerian) formulation (see Sect. 4.1), the convective term in (4.28) will change to   v − v g · ∇v = v c · ∇v.

© Springer-Verlag Berlin Heidelberg 2015 M. Kaltenbacher, Numerical Simulation of Mechatronic Sensors and Actuators, DOI 10.1007/978-3-642-40170-1_7

(7.3) 285

286

7 Coupled Flow-Structural Mechanical Systems

The fluid-solid boundary condition for the solid is given by an inhomogeneous Neumann condition of the form [σ s ] · n = [σ f ] · n

(7.4)

describing the equivalence of fluid stresses [σ f ] and solid stresses [σ s ] in normal direction n. The fluid action on the solid according to (7.4) is equivalent to a surface force density f fluid , which can be split into a pressure and a shear component  f fluid = − pI · n + µ f ∇v + (∇v)T · n .       Pressure

(7.5)

Shear

7.2 Coupling Types and Strategies Depending on the physics, we distinguish between surface and/or volume couplings, as displayed in Fig. 7.1. E.g., within fluid-solid-acoustic interactions both types are present. The fluid-solid interaction is a surface coupled phenomena, whereas the fluid-acoustics couple within the volume (see Chap. 9). In surface coupled problems the meshes of the involved fields are treated separately and their intersection defines the field interface. The conditions describing the field interaction are defined along

1

Field 1

2

Field 2

1

2

Interaction

(b) Field 2

U

Interface

(a)

Interaction

2

Field 1 1

=

Fig. 7.1 Field coupling types: surface and volume coupling. a Setup for surface coupling. b Setup for volume coupling

7.2 Coupling Types and Strategies

287

the interface Γ . In volume coupled phenomena the intersection generally is given through a part of the involved domains. One field is often completely enclosed by the other field, see Fig. 7.1. This is especially true for aeroacoustic computations, where the flow is computed in Ω1 (also named acoustic source region) and the acoustic field is computed in Ω1 ∪ Ω2 . In general, two main coupling strategies exist to treat multifield problems numerically. These are: • the simultaneous, monolithic, and • the partitioned one. Figure 7.2 displays the two approaches along the time scale. The simultaneous concept is based on the coupling at the PDE level. All participated fields are computed as one unit. This monolithic approach resolves all possible field interactions in crossdiagonal terms. Only one system matrix is built, which is solved in one step. Depending on the problem at hand, badly conditioned matrices may occur which are hardly solvable. With a partitioned approach, the PDEs are handled separately and each field is solved individually. The interaction between the respective fields is introduced by a synchronization of the involved fields. Whether the synchronization is realized iteratively or staggered, the partitioned scheme is either called strong or weak coupled. The main advantages of the partitioned concept compared to the simultaneous one can be summarized in the following way [1]: • The optimal discretization method can be chosen independently for each field (e.g. FE method can be coupled with Finite Volume (FV) method). • The optimal mesh can be chosen independently for each physical field. E.g., the coupling can be realized with the concept of non-conforming grids (see Sect. 2.10). • The software as well as the development process can be treated modularly. • The condition numbers of the single system matrices are more suitable to be solved numerically.

(a)

(b)

Monolithic

Partitioned

Field 1

Field 1 & 2

tn−1

tn

tn+1

Field 2

tn−1

tn

tn+1

Fig. 7.2 Monolithic and partitioned coupling scheme. a Monolithic scheme. b Partitioned scheme

288

7 Coupled Flow-Structural Mechanical Systems

1. Solve NSE for the flow (see (4.41)); a) Determine the mechanical velocity along the coupling interface Γfs (see (7.2)) b) Set this velocity as a Dirichlet condition for the flow c) Compute the new flow field d) Determine fluid forces on common boundary Γfs (see (7.5)). 2. Solve for the mechanical field (see (3.80)) a) Set the fluid forces along the common boundary Γfs b) Compute the new mechanical displacement. 3. Check convergence. Fig. 7.3 Computational scheme for fluid-solid coupling within one time step

Because of these advantages the partitioned concept is chosen to realize the fluid-solid coupling. The algorithmic structure for each time step is listed in Fig. 7.3. As already mentioned, the partitioned approach can be further classified into strongly coupled or weakly coupled, also known as loosely coupled or staggered scheme (see, e.g., [2–4]). A weak approach means, that no iteration is applied within each time step. So, the convergence test, which is performed in step 3, is omitted and one immediately proceeds to the next time step. However, a strong coupled scheme iterates between the physical fields until an equilibrium is reached approximating the monolithic solution. The error bound may be, e.g., an incremental error check of the fluid field, the structural field or both. From the computational point of view, the sequential staggered coupled scheme is clearly favorable, since no iterations are performed and therewith the computational time reduces strongly. It is obvious that one needs a reduced time step size in order to ensure convergence. However, many practical investigations showed that decreasing the time step size leads to instabilities, also known as added mass effect. These are inherent and for simplified coupled fluid-solid equations a mathematical investigation can be found [2, 3]. The name added mass effect comes from the fact that the major part of the fluid acts as an added mass on the solid. Thereby, the fluid forces at the coupling interface can be expressed by [3] f ΓI = m f M v ΓI

(7.6)

with m f a characteristic fluid mass (e.g., a nodal mass of a lumped mass element), M the added mass operator and v ΓI the flow velocity at the coupling interface ΓI . Now, as shown in [3], the sequential staggered coupled scheme becomes instable

7.2 Coupling Types and Strategies

289

in case of a BDF2 time discretization scheme for the fluid part when the following inequality holds 3 mf λmax > . (7.7) ms 2 In (7.7) m s is the characteristic solid mass, and λmax the largest eigenvalue of M. Summarizing, we can conclude that the sequential staggered coupled scheme is attractive from the computational point of view but needs adaption for each computational setup. Therefore, it is more appropriate to use a strong formulation to achieve a convergent solution. In order to speed-up or to make the fluid-solid convergence possible at all, the partitioned coupled scheme is treated with a Richardson iteration scheme [5]. Thereby, the following relaxation is introduced for the displacement at the interface k ˆk+1 k k d k+1 n+1 = ω d n+1 + (1 − ω )d n+1

(7.8)

k+1 with the un-relaxed interface displacement dˆn+1 . The relaxation parameter ω can thereby be chosen to be fixed or flexible during the simulation time. The flexible relaxation is according to Aitken [6] because it is known to be efficient and robust [5]. In each fluid-solid iteration k the relaxation parameter ω k is computed based on the change of interface displacements k ˆk+1 △d k+1 n+1 = d n+1 − d n+1 .

(7.9)

k With the Aitken factor ηn+1

k ηn+1

T   △d kn+1 − △d k+1 △d k+1 n+1 n+1 k−1 k−1 = ηn+1 + ηn+1 −1 k+1 k △d n+1 − △d n+1

(7.10)

the relaxation parameter computes as k k ωn+1 = 1 − ηn+1 .

(7.11)

Alternative relaxation parameter estimations are, e.g., the reduced order model [7] and the gradient approach [5, 8].

7.3 Grid Adaption As already mentioned, the fluid-solid interaction is a surface coupled problem and realized in a partitioned approach. In order to prescribe the coupling conditions, all elements need to be defined along the fluid-solid interface. The fluid forces according

290

7 Coupled Flow-Structural Mechanical Systems

to (7.5) cause a deformation of the solid. The resulting solid field deformation leads to a domain update for the fluid field. In order to interpolate the deformation inside the fluid domain, a grid adaption is performed. The movement of the fluid domain is governed by the interface displacements d. As soon as the interface displacements are larger than the fluid element size along the fluid-solid interface, it is necessary to interpolate the interface displacements inside the whole fluid domain in order to avoid overlapping elements. Hereby, the fluid elements should be deformed as little as possible to keep the numerical errors small. Different strategies exist to perform this grid adaption, e.g.: • linear interpolation [9], • elliptic smoothing [10] and • the pseudo-structure approach [8, 11]. The first two approaches can be implemented very efficiently. However, especially linear interpolation is restricted to simple deformations. Elliptic smoothing schemes already allow a more complex deformation state. The most powerful grid adaption scheme is given by the pseudo-structure scheme as it allows very complex deformations. Hereby, the fluid domain is assumed to be a pseudo-solid which is loaded by inhomogeneous Dirichlet boundary conditions at the interface, namely the interface displacements d. The pseudo-domain deformation finally provides the grid adaption needed. If grid adaption fails, re-meshing may be necessary. However, the required projection of field variables from the old to the new mesh introduces projection errors and should therefore be avoided as long as possible [11]. The pseudo-solid approach is based on static solid mechanics, where a pseudo-deformation field r has to fulfill ∇ · [σ g ] = 0 on Ωf with r = d on Γfs .

(7.12)

The pseudo stresses [σ g ] can be computed via the pseudo strains [Sg ] = 21 (∇ r + (∇ r)T ) and the constitutive equation of a Hook body [σ g ] = [cg ][Sg ]. Generally, the computational domain for the grid mechanical field is the same as for the flow. As soon as the fluid domain deformation is bounded locally, it is possible to split the fluid domain Ωf into a moving ΩALE and a fixed part ΩEuler with Ωf = ΩEuler ∪ ΩALE . In Fig. 7.4 such a local domain decomposition is sketched. Thereby, calculations necessary for the grid adaption can be restricted to the ALE domain. The pseudo-solid field is discretized with the standard Galerkin FEM, yielding the following algebraic system of equation Kr = f .

(7.13)

Here, the force vector f consists only of contributions from inhomogeneous Dirichlet boundary conditions at the fluid-solid interface Γfs . Furthermore, r denotes the vector of unknown grid displacements. A time discretization is hereby not needed. However,

7.3 Grid Adaption

291

Fig. 7.4 ALE and Euler domains of fluid

ΩEuler ΩALE Ωs

as the mesh velocities are required due to the ALE description of the fluid field, the grid velocities v g have to be computed based on the mesh displacements r . The following approximations of the grid acceleration a g and grid velocities v g are applied to obtain second-order accurate time discretization ag =

r n+1 − 2r n + r n−1 , △t 2

(7.14)

vg =

r n+1 − r n △t − a . △t 2 g

(7.15)

In order to avoid badly shaped elements during grid adaption, the elasticity modulus varies from element to element. An exponential elasticity decrease provides good results for many applications. The elasticity modulus E dist of the pseudo solid is thereby reversely proportional to the distance |x − xΓfs | between a certain fluid element and the fluid-solid interface Γfs E dist = E 0



xmax |x − x Γfs |

q

.

(7.16)

In (7.16) xmax denotes the distance to the fluid-solid interface. E 0 stands for the initial elasticity modulus (assumed constant for all elements). Numerical tests showed that the best grid adaption is obtained with an exponent q = 1.5 [12]. As shown in Fig. 7.5, the element elasticity E elem is set reverse proportional to the distance between each element and the fluid-solid interface Γfs according to (7.16). In a second step, a straindependent contribution to the stiffness is computed, based on the pseudo strains [Sg ]. The strain is computed and the elasticity is modified again in such a way that elements with a higher strain energy become even stiffer

E strain ∝



2 2 sg,11 + sg,22

2

.

The pseudo-solid computation is now based on the composite elasticity distribution E elem = E dist + E strain . The stopping criterion is reached as soon as the maximal strain is below a prescribed threshold.

292

7 Coupled Flow-Structural Mechanical Systems

Fig. 7.5 Grid-adaption algorithm

tn Edist = E0 (xmax /|x − xΓfs |)q Eelem = Edist (Estrain = 0) Eelem + = Estrain Estrain ∼ |Sg |

Pseudo Solid

Converged?

no

yes

tn+1

7.4 Numerical Examples 7.4.1 Solid Plunger For the first verification example of the fluid-solid coupling a channel flow [13] is considered as displayed in Fig. 7.6. The fluid domain is discretized with (10 × 1) second order quadrilateral elements. A plunger is positioned on the left side of that channel. At both horizontal walls slip conditions are applied to obtain a 1d flow, for which the momentum conservation reduces to ρ

∂p ∂vx + = 0. ∂t ∂x

(7.17)

The plunger moves during the considered time span (0 s till 2 s) 1 m from left to right with a time step size of △t = 0.002 s. The x-displacement is prescribed by

sin(0.5πt) u x = 0.5 t − 0.5π

Fig. 7.6 A moving plunger inside a channel to verify the fluid-solid coupling (units in m)



m/s.

(7.18)

7.4 Numerical Examples

293

Fig. 7.7 Fluid-solid test case: moving plunger. a Prescribed plunger displacement, velocity and acceleration; b numerical and analytical pressure at the fluid-solid interface

In Fig. 7.7a the prescribed displacement, velocity and acceleration of the plunger is displayed. The fluid density is set to ρ = 1.0 kg/m3 . Thereby, (7.17) can be solved analytically for the described plunger movement p = ρ(10 − u x )u¨ x .

(7.19)

The action from the solid to the fluid field is verified with this example, as the plunger displacement is prescribed, no action of the fluid on the solid field is considered. In Fig. 7.7b the computed pressure distribution at the fluid-solid interface is compared to the analytical solution (7.19).

7.4.2 Flag in a Flow To validate the scheme with respect to the fluid-solid coupling, we present computational results of the DFG benchmark setup displayed in Fig. 7.8. Concerning the Γwall

Γout

g

22

(71/0)

4

240

x

x

Γin

Sheet

y

0.04

Cylinder

y

10 60

Γwall 55

284

Fig. 7.8 Geometry of the benchmark problem for fluid-solid interaction (units in mm)

Weight

294

7 Coupled Flow-Structural Mechanical Systems

Table 7.1 Material parameters

Material

E (Pa)

νp

ρs (kg/m3 )

Cylinder Sheet Weight

1.5 × 1011

0.3 0.3 0.3

4,300 7,855 7,800

2.0 × 1011 2.0 × 1011

experimental measurements, we refer to [14] and for simulations to [10]. As depicted in Fig. 7.8 a thin sheet (flag) is placed in the wake of a cylinder in a channel filled with a mixture of polyglycol and water. The gravitational acceleration g is acting in x-direction. The maximal deflection of the trailing edge of the flag is measured to be approximately 20 mm in y-direction, so that the artificially added mass effect and mechanical geometric nonlinearities play a significant role. The clamping boundary conditions of the solid field are restricted to the center of the cylinder, which is therefore able to rotate. The solid is composed of three materials and their material parameters are listed in Table 7.1. The fluid mechanical material parameters for the polyglycol-water mixture are µ f = 1.64 × 10−4 Pa · s for the dynamic viscosity and ρ = 1,050 kg/m3 for the density. The time step size is set to △t = 5 ms for the fluid-solid simulation. As sketched in Fig. 7.8, a velocity-driven flow is considered with the following inflow profile

vx = 1.45 1 −



|y| 120

30 

m/s.

(7.20)

20

y (mm)

10

0

−10

−20 0

20

x (mm)

60

80

Fig. 7.9 DFG fluid-solid benchmark; (x, y)-displacements of the trailing edge of the flag

7.4 Numerical Examples

295

At the top and the bottom boundary, wall conditions are applied and the outlet is realized with open boundary conditions. The stopping criterion of the incremental change of the fluid field is 10−3 m/s. The nonlinear solid is assumed to be converged if the error in the strain energy is less than 10−6 J. The stopping criterion of the absolute change of interface displacement of the fluid-solid interaction is set to 10−2 mm. The fluid-solid interaction simulation is restarted on a fluid simulation with a small y-velocity of 0.05 m/s at the flag in order to speed up the onset of oscillations. During the fluid-solid coupled run the flag velocity is in accordance with the fluidsolid interaction condition. The simulated and measured (x, y)-displacements of the trailing edge correlate in an acceptable range, as shown in Fig. 7.9. The velocity magnitudes and the deformed fluid domain are shown for one cycle in Fig. 7.10.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

Fig. 7.10 Velocity magnitude (0 − 2.5 m/s) for one cycle (a–o)

296

7 Coupled Flow-Structural Mechanical Systems

References 1. W.A. Wall, Computational fluid-solid-mechanics in bio-medicalengineering—where to go from here? ECCOMAS 2012 CD-ROM Proceedings, (2012) 2. P. Causin, J.F. Gerbeau, F. Nobile, Added-mass effect in the design of partitioned algorithms for fluid-structure problems. Comput. Methods Appl. Mech. Eng. 194, 4506–4527 (2005) 3. Ch. Förster, W.A. Wall, E. Ramm, Artificial added mass instabilities in sequential staggered coupling of nonlinear structures and incompressible viscous flows. Comput. Methods Appl. Mech. Eng. 196(7), 1278–1293 (2007) 4. S. Zörner, Numerical simulation method for a precise calculation of the human phonation under realistic conditions. Ph.D. thesis, Vienna University of Technology, (2014) 5. D.P. Mok, Partitionierte Lösungsansätze in der Strukturdynamik und der Fluid-StrukturInteraktion (Partitioned analysis schemes in structural dynamics and fluid-structureinteraction). Ph.D. thesis, University of Stuttgart, (2001) 6. B.M. Irons, R.C. Tuck, A version of the Aitken accelerator for computer iteration. Int. J. Numer. Methods Eng. 1, 275–277 (1969) 7. J. Vierendeels, K. Dumontb, P.R. Verdonckb, A partitioned strongly coupled fluid-structure interaction method to model heart valve dynamics. J. Comput. Appl. Math. 218, 602–609 (2008) 8. W.A. Wall, Fluid-Struktur-Interaktion mit stabilisierten Finiten Elementen. Ph.D. thesis, University of Stuttgart, (1999) 9. F. Schäfer, S. Kniesburges, T. Uffinger, S. Becker, J. Grabinger, G. Link, M. Kaltenbacher, Numerical simulation of fluid-structure- and fluid-structure-acoustic interaction based on a partitioned coupling scheme. High Performance Computing in Science and Engineering Munich 2007, Transactions of the Third Joint HLRB and KONWIHR Result and Reviewing Workshop, Leibniz-Rechenzentrum München (LRZ), Garching (Springer, Heidelberg, 2007) 10. S. Yigit, M. Schäfer, M. Heck, Grid movement techniques and their influence on laminar fluid structure interaction computations. J. Fluids Struct. 24(6), 14 (2008) 11. T.E. Tezduyar, Finite element methods for fluid dynamic with moving boundaries and interfaces, Chapter 17, in Encyclopedia of Computational Mechanics, ed. by E. Stein, R. de Borst, T.J.R. Hughes (Wiley, New York, 2004) 12. G. Link, A Finite Element Scheme for Fluid-Solid-Acoustics Interactions and its Application to Human Phonation. Ph.D. thesis, University Erlangen-Nuremberg, (2008) 13. B. Hübner, Simultane Analyse von Bauwerk-Wind-Wechselwirkung (Simultaneous Analysis of Construction-Wind Interaction). Ph.D. thesis, Universität Braunschweig, (2003) 14. J.P. Gomes, H. Lienhart, Experimental study on a fluid-structure interaction reference test case, Fluid-Structure Interaction—Modelling Simulation, Optimization (Springer, Heidelberg, 2006)

Chapter 8

Coupled Mechanical-Acoustic Systems

In many technical applications, the sensor/actuator is immersed in an acoustic fluid. Therefore, mechanical vibrations will generate acoustic waves, which itself will act as a surface pressure load on the vibrating structure. In general, we distinguish between the following two situations concerning mechanical-acoustic systems [1]: • Strong Coupling: In this case, the mechanical and acoustic field equations including their couplings have to be solved simultaneously. A typical example is a piezoelectric ultrasound array immersed in water (see Fig. 8.1). • Weak Coupling: If the pressure forces of the fluid on the solid are negligible, a sequential computation can be performed. For example, the acoustic sound field of an electric transformer as displayed in Fig. 8.2 can be obtained in this way. Thus, in a first simulation the mechanical surface vibrations are calculated, which are then used as the input for an acoustic field computation.

8.1 Solid–Fluid Interface At a solid–fluid interface, the continuity requires that the normal component of the mechanical surface velocity of the solid must coincide with the normal component of the acoustic velocity of the fluid (see Fig. 8.3). Thus, the following relation between the velocity v of the solid expressed by the mechanical displacement u and the acoustic particle velocity v a expressed by the acoustic scalar potential ψ arises ∂u v a = −∇ψ ∂t n · (v − v a ) = 0 ∂ψ ∂u = −n · ∇ψ = − . n· ∂t ∂n v=

© Springer-Verlag Berlin Heidelberg 2015 M. Kaltenbacher, Numerical Simulation of Mechatronic Sensors and Actuators, DOI 10.1007/978-3-642-40170-1_8

(8.1) 297

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8 Coupled Mechanical-Acoustic Systems

Fig. 8.1 Acoustic sound field of a piezoelectric ultrasound array antenna

Fig. 8.2 Acoustic sound field of an electric transformer due to the Lorentz forces acting on its winding Fig. 8.3 Solid–fluid interface

v

n

Fluid

Solid

In addition, one has to consider the fact that the ambient fluid causes on the surface a mechanical stress σ n ∂ψ σ n = −n pa = −nρ0 , (8.2) ∂t

8.1 Solid–Fluid Interface

299

which acts like a pressure load on the solid. In (8.2) ρ0 denotes the mean density of the fluid. When modelling special wave phenomena, we often arrive at a partial differential equation for the acoustic pressure. Therewith, we will also derive the coupling conditions between the mechanical displacement and acoustic pressure at a solid–fluid interface. For the first coupling condition, the continuity of the velocities, we have to establish the relation between the acoustic particle velocity v a and the acoustic pressure pa . According to the linearized momentum equation (see 5.8), we can express the normal component of v a by n·

∂v a 1 ∂ pa =− . ∂t ρ0 ∂n

(8.3)

Therewith, since n · v = n · v a holds, we get the relation to the mechanical displacement by ∂2 u 1 ∂ pa . (8.4) n· 2 =− ∂t ρ0 ∂n The second coupling condition as defined in (8.2) is already established for an acoustic pressure formulation.

8.2 Coupled Field Formulation Let us consider a setup of a coupled mechanical-acoustic problem as shown in Fig. 8.4, where at the interface Γ I we have to consider the solid–fluid coupling. Now, within the domain Ωs the partial differential equation for the mechanical field (see 3.80), within the domain Ωf the partial differential equation for the acoustic field (see 5.88) and along the interface Γ I the coupling conditions according to (8.1) and (8.2) have to be fulfilled. In a first step, let us transform the partial differential equations to their weak form without setting the boundary integral (obtained by using integration by parts) to zero. Therefore, we obtain for the mechanical system 

′

ρu · u¨ dΩ



′ T

(Bu ) [c]Bu) dΩ −

′

u · σ n dΓ =

ΓI

Ωs

Ωs





u′ · f V dΩ, (8.5)

Ωs

and for the acoustic system (acoustic excitation f being zero) 

Ωf

1 w ψ¨ dΩ + c2



Ωf

∇w · ∇ψ dΩ +



ΓI

w n · ∇ψ dΓ −



w n f · ∇ψ dΓ = 0.

Γf

(8.6)

300

8 Coupled Mechanical-Acoustic Systems

Fig. 8.4 Setup of a coupled mechanical-acoustic problem

Γf

nf

n

Ωf

Ωs

ΓI

It should be noted that the plus sign in (8.6) in front of the boundary integral over Γ I is due to the choice of n (see Fig. 8.4). In a second step, we will now incorporate the coupling conditions. With the help of (8.2), we can rewrite the boundary integral in (8.5) as follows    ∂ψ dΓ. (8.7) u′ · σ n dΓ = − u′ · n pa dΓ = − u′ · n ρ0 ∂t ΓI

ΓI

ΓI

Using (8.1), we obtain for the boundary integral along the interface Γ I (at the outer boundary Γ f we set for simplicity ∂ψ/∂n = 0) in (8.6) the following form   ∂ψ ∂u dΓ = − w n · dΓ. (8.8) w ∂n ∂t ΓI

ΓI

Thus, we arrive at the following coupled system of equations 

ρu′ · u¨ dΩ +



(Bu′ )T [c]Bu) dΩ +

u′ · n ρ0

∂ψ dΓ = ∂t



u′ · f V dΩ

Ωs

ΓI

Ωs

Ωs



(8.9) 

Ωf

1 w ψ¨ dΩ + c2



∇w · ∇ψ dΩ −

Ωf



∂u dΓ = 0. wn· ∂t

(8.10)

ΓI

8.3 Numerical Computation 8.3.1 Finite Element Formulation Before we perform a domain discretization, we multiply (8.10) by −ρ0 in order to obtain in (8.10) a boundary integral similar to the one in (8.9). Thus, the matri-

8.3 Numerical Computation

301

ces, occurring from an FE discretization of these two boundary integrals, will be transposed to each other and hence symmetry of the overall system matrix will be obtained. Using nodal finite elements, we approximate the mechanical displacement u as well as the scalar acoustic potential ψ as follows u ≈ uh =

nd  n1 

h Na u ia ei =

i=1 a=1

ψ ≈ ψh =

n2 

n1  a=1

⎛

⎞ Na 0 0 N a uah ; N a = ⎝ 0 Na 0 ⎠ (8.11) 0 0 Na

Na ψah ,

(8.12)

a=1

with n 1 the number of nodes with unknown mechanical displacement and n 2 the number of nodes with unknown acoustic potential. Following the same procedure as described in Sect. 3.7.1 for the mechanical equation and in Sect. 5.4.1 for the acoustic equation we obtain a coupled system of equations 

Mu 0 0 −M ψ

     0 C uψ u¨ u˙ u Ku 0 + + T ¨ ˙ ψ C 0 ψ ψ 0 −K ψ uψ  fu . (8.13) = 0

The new matrix C uψ computes as follows

C uψ =

n Ie 

C euψ ; C euψ = [C pq ] ; C pq =

e=1



Γe

⎞ N p Nq n x ρ0 ⎝ N p Nq n y ⎠ dΓ, N p Nq n z ⎛

(8.14)

with n Ie the number of finite elements along the interface. At this point, it should be emphasized that the coupled system of equations remains symmetric. This is not the case if instead of the acoustic velocity potential an acoustic pressure formulation is used. The time discretization is performed by a standard Newmark algorithm (see Sect. 2.5.2), which reads for the effective mass matrix formulation as follows • Perform predictor step: ∆t 2 (1 − 2βH ) u¨ n 2 u˜˙ = u˙ n + (1 − γH )∆t u¨ n u˜ = u n + ∆t u˙ n +

ψ˜ = ψ n + ∆t ψ˙ n +

∆t 2 2

(1 − 2βH ) ψ¨ n

ψ˜˙ = ψ˙ n + (1 − γH )∆t ψ¨ n .

(8.15) (8.16) (8.17) (8.18)

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8 Coupled Mechanical-Acoustic Systems

• Solve algebraic system of equations:



M ∗u C ∗uψ ∗ T (C uψ ) −M ∗ψ



u¨ n+1 ψ¨

n+1



=



fu 0



 u˜ Ku 0 − 0 −K ψ ψ˜ u˜˙   0 C uψ − T C uψ 0 ψ˜˙ 

(8.19)

M ∗u = M u + βH ∆t 2 K u M ∗ψ = M ψ + βH ∆t 2 K ψ

(8.20) (8.21)

C ∗uψ = γH ∆t C uψ .

(8.22)

• Perform predictor update: u n+1 = u˜ + βH ∆t 2 u¨ n+1 u˙ n+1 = u˜˙ + γH ∆t u¨ n+1 2

(8.23) (8.24)

ψ n+1 = ψ˜ + βH ∆t ψ¨ n+1

(8.25)

ψ˙ n+1 = ψ˜˙ + γH ∆t ψ¨ n+1 .

(8.26)

8.3.2 Non-conforming Grids In many practical cases, we wish to perform the spatial discretization within the elastic body independent to the discretization of the surrounding fluid. A typical example is given in Fig. 8.5, where on the one hand we need a much finer grid within the mechanical structures (plates) and on the other hand wish to have a regular grid within the fluid domain.In particular, the advantages of using non-conforming grids are demonstrated in Fig. 8.5, where the mechanical regions have to be resolved by a substantially finer grid than the fluid region. Therewith, we will investigate in such flexible discretization techniques for the approximate solution of coupled mechanical-acoustic problems. For the mechanical-acoustic coupling, the problem formulation for non-conforming grids remains essentially the same as for the matching situation. This means, in contrast to the problem setting considered in Sect. 5.4.4 (acousticacoustic coupling over non-conforming grid), no additional Lagrange multiplier has to be introduced. For the spatial discretization, we use two independently generated triangulations Ts and Tf on Ωs and Ωf , respectively, and approximate the displacement u on Ts and the potential ψ on Tf by finite elements. The two triangulations inherit two (d−1)-dimensional grids ∂Ts and ∂Tf on Γ I . Due to the flexible construction of both grids, the finite element nodes on ∂Ts and ∂Tf will in general not coincide. On the

8.3 Numerical Computation

303

Fig. 8.5 Non-conforming discretization of mechanical structure and surrounding fluid

contrary, motivated by different spatial scales required for the resolution of the local subproblems, the difference in the mesh sizes can become quite large. The discretized version of (8.9), (8.10) results in the same matrix-equations as for the matching grid approach (see 8.13). The coupling between the two grids is represented by the matrices C uψ and C Tuψ which realize the boundary integrals in (8.9) and (8.10). Their entries are given by C uψ = [C pq ]; C pq =



ΓI

ρf N ps Nqf n dΓ ∈ IRd ,

(8.27)

where N ps is the scalar basis function associated with the node p on ∂Ts , and Nqf is the one for node q on ∂Tf . For a detailed discussion of the element wise assembly of C uψ we refer to [2].

8.3.3 Numerical Examples 8.3.3.1 Noise Radiation from an Oil Pan and a Car Engine A typical problem that is encountered in numerical acoustics of automotive applications consists of the calculation of the sound radiated from machine parts. In the standard approach, first the structural response due to dynamic loads is calculated based on the FE method. In a second step, the acoustic radiation is calculated, again applying the FE method. The surface grid is displayed in Fig. 8.6, whereas a detail of the finite element model, showing the embedding of the oil pan structure in a sphere consisting of acoustic finite elements, is shown in Fig. 8.7. The total finite element model consisted of approx. 500,000 3D acoustic finite elements and the resulting sound field at a driving frequency of 600 Hz as computed by this model is shown in Fig. 8.8. Next, this modelling scheme was applied in the simulation of a complete car engine. Here, the surface grid consists of 8,200 2D elements. This model was embedded in a sphere of radius 2 m and the mesh generator NETGEN [3] was used to mesh the unfilled volume with acoustic finite elements. The embedding of the

304

8 Coupled Mechanical-Acoustic Systems

Fig. 8.6 Surface grid of an oil pan

Fig. 8.7 Embedding of oil pan into finite element grid

Fig. 8.8 Sound field radiated at 600 Hz

Fig. 8.9 Embedding of car engine into finite element grid

engine in the finite element mesh is displayed in Fig. 8.9, whereas the innermost layer of acoustic elements (tetrahedra) is shown in Fig. 8.10. The results of the transient structural simulation were used as excitations for the acoustic calculations. In the simulations, two different models have been considered, consisting of 210,000 and 1,340,000 finite elements, respectively. For the numerical computation three different approaches have been performed:

8.3 Numerical Computation

305

Fig. 8.10 Innermost tetrahedra layer around car engine

• Direct, implicit: An implicit Newmark time-integration scheme has been used and the resulting algebraic system of equation has been solved by a direct solver. • Iterative, implicit: Again, an implicit Newmark time-integration scheme has been used, but now we have applied an iterative solver (GMRES with ILU as preconditioner, see [4]). • Explicit: An explicit Newmark time-integration scheme with lumped mass matrix has been used. Therewith, the system matrix is a pure diagonal and the algebraic solver just divides the right-hand side by the diagonal entries. In Tables 8.1 and 8.2, the required computer resources are summarized. The direct, implicit solution was not available for the large model, since it required 10 GB of main memory. For the iterative solver, convergence was achieved for an accuracy of 10−8 (relative residual norm) within six iterations per time step. Of course, this convergence behavior strongly depends on the quality of the generated grid. For the small model, no difference was found between the solution for the direct and the iterative solver. However, as can be seen from Figs. 8.11 and 8.12, due to the diagonal mass matrix formulation for the explicit time scheme, the solution based on the explicit time stepping differs from both solutions obtained with implicit time stepping. This difference vanishes, when an even finer time discretization is used for the explicit scheme (for our examples we applied a time step of ten times smaller as used for the implicit scheme). Table 8.1 Computer resources, small model

Solver

Time steps

Memory (MB)

Direct, implicit Explicit Iterative (GMRES)

2,000 20,000 2,000

364 20 38

306 Table 8.2 Computer resources, large model

Fig. 8.11 Comparison of normalized acoustic pressure at evaluation point for small model

8 Coupled Mechanical-Acoustic Systems Solver

Time steps

Memory (MB)

Direct, implicit Explicit Iterative (GMRES)

– 20,000 2,000

– 102 206

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2

Implicit

0.1 0

Explicit

-0.1 0

Fig. 8.12 Comparison of normalized acoustic pressure at evaluation point for large model

1

2

3

4

5 6 t (ms)

7

8

9

10

8

9

10

1 0.9 0.8

Implicit

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

Explicit

-0.1 0

1

2

3

4

5 6 t (ms)

7

8.3.3.2 Acoustic Wave Generation by a Multiple Plate Structure To demonstrate the applicability of the non-conforming grid technique, we present the generation and radiation of acoustic waves by multiple structures, which admits the steering of the waves by exciting the structures in a specified chronological order. In particular, we use for the structure Ωs 25 cylindrical silicon plates with diameter 50 µm and height 1 µm. They are placed as a (5 × 5)-array, each plate having a distance of 50 µm to its nearest neighbors. An excitation force with frequency f = 1 MHz is applied on their lower end. For the acoustic domain Ωf , which is assumed to be water, a cuboid of length and width 1,200 µm and height 420 µm is

8.3 Numerical Computation

307

chosen. Due to symmetry reasons, we use as computational domain one quarter of the original one. In Fig. 8.13 a part of the finite element meshes is shown, for which a uniform grid of 40 × 40 × 28 cubes is used to discretize the acoustic domain and a grid of 768 hexahedra is employed for each plate. Thus, having a mesh width of h a = 600 µm/40 = 15 µm, we use ten elements per wavelength for Ωf . If one had to employ matching grids, it would be quite difficult to generate them, and if the meshwidth could not be very small over the whole domain, the resulting element shapes would possibly result in a poor approximation of the solution. The nonconforming approach admits to use the grid desired for each subdomain regardless of the grids for the other subdomains. Figure 8.14 shows snapshots, taken every ten time steps of

Fig. 8.13 Cylindrical plates attached to the fluid domain and isosurfaces of the acoustic potential, deformed plates

Fig. 8.14 Evolution of the acoustic velocity potential and of the deformed structures, successive excitation: snapshots after 10, 20, . . . , 80 time steps

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8 Coupled Mechanical-Acoustic Systems

3.5 ns, of the evolution of the acoustic velocity potential ψ along with the deformation of the structures (magnified by a factor of 1,000). For the results presented in Fig. 8.14, the plates are excited successively, and as can be seen, the waves emitting from the structures add up as expected to constitute the superposed global sound beam. Given a target point, it is possible to optimally steer the acoustic wave towards this point by appropriately adjusting the chronological order of excitation of the silicon chips. This principle is used in so-called capacitive micro-machined ultrasound transducers (CMUTs) (see Sect. 14.5). There, the deformation of the structure is induced by an electrostatic surface force acting on the loaded electrodes.

References 1. M. Kaltenbacher, Computational acoustics in multi-field problems. J. Comput. Acoust. 19, 27–62 (2011) 2. B. Flemisch, Non-matching triangulations of curvilinear interfaces applied to electro-mechanics and elasto-acoustics. Ph.D. thesis, University of Stuttgart, (2006) 3. J. Schöberl, An advancing front 2D/3D-mesh generator based on abstract rules. Comput. Vis. Sci. 1(1), 41–52 (1997) 4. C.T. Kelly, Iterative Methods for Linear and Nonlinear Equations (SIAM, Philadelphia, 1995)

Chapter 9

Computational Aeroacoustics

A large amount of the total noise in our daily lives is generated by turbulent flows (e.g., airplanes, cars, air conditioning systems, etc.). The physics behind the generation process is quite complicated and still not fully understood. The use of numerical simulation tools is one important way to analyze the generation of flow-induced sound, e.g., [1].

9.1 Requirements for Numerical Schemes Since the beginning of computational aeroacoustics (CAA) several numerical methodologies have been proposed, each of these trying to overcome the challenges that the specific problems under investigation pose for an effective and accurate computation of the radiated sound. The difficulties which have to be considered for the simulation of flow noise problems include [2, 3]: • Energy disparity and acoustic inefficiency: There is a large disparity between the energy in the flow and the radiated acoustic energy. In general, the total radiated power of a turbulent jet scales with O(v 8 /c5 ) (v is the characteristic flow velocity and c the speed of sound), and for a dipole source arising from pressure fluctuations on surfaces inside the flow scales with O(v 6 /c3 ). This shows that an aeroacoustics process at low Mach number is rather a poor sound emitter. • Length scale disparity: Large disparity also occurs between the size of an eddy in the turbulent flow and the wavelength of the generated acoustic noise. Low Mach number eddies have a characteristic length scale l, velocity v, a life time l/v and a frequency f . This eddy will then radiate acoustic waves of the same characteristic frequency, but with a much larger length scale, which scales as follows λ∝c

l l = . v Ma

© Springer-Verlag Berlin Heidelberg 2015 M. Kaltenbacher, Numerical Simulation of Mechatronic Sensors and Actuators, DOI 10.1007/978-3-642-40170-1_9

(9.1)

309

310

9 Computational Aeroacoustics

In (9.1) Ma denotes the Mach number, which is defined by the ratio of the characteristic velocity v over the speed of sound c Ma =

v . c

• Preservation of multipole character: The numerical analysis must preserve the multipole structure of the acoustic source in order to resolve the whole structure of the source. • Dispersion and dissipation: The discrete form of the acoustic wave equation cannot precisely represent the dispersion relation of the acoustic sound. Numerical discretization in space and time converts the original non-dispersive system into a dispersive discretized one, which exhibits wave phenomena of two kinds: 1. Long wavelength components approaching the solution of the original PDE as the grid is refined. 2. Short wavelength components (spurious waves) without counterpart in the original PDE evolving in the numerical scheme disturbing the solution. The wave equation shows a non-dissipative behavior; as such, dissipative errors must be avoided by a numerical implementation, in which both the amplitude and phase of the wave are of crucial importance. • Difficulties in nonlinear wave phenomena: In turbulent flows at high speed, nonlinear effects will play a role since the wave equation has to be solved over a long range, which induces dissipation of acoustic energy and refraction. • Flows with high Mach and Reynolds number: Aeroacoustic problems often involve both high Mach and Reynolds numbers. Flows at a high Mach number may induce new nonlinear sources and convective effects while flows at a high Reynolds number introduce multiple scale difficulties due to the disparity between the acoustic wavelength λ and the size of the energy dissipating eddies. • Simulation of unbounded domains: As a main issue for the simulation of unbounded domains using volume (interior) discretization methods remains the boundary treatment which needs to be applied to avoid the reflection of the outgoing waves on the truncating boundary of the computational domain (see Sect. 5.5). Currently, available aeroacoustic methodologies overcome only some of these broad range of numerical and physical issues, which restricts their applicability, making them, in many cases, problem dependent methodologies. In a Direct Numerical Simulation (DNS), all relevant scales of turbulence are resolved and no turbulence modeling is employed. The application of DNS is becoming more feasible with the permanent advancement in computational resources. However, due to the large disparities of length and time scales between fluid and acoustic fields, DNS remains restricted to low Reynolds number flows. Therefore, although some promising work has been done in this direction [4], the simulation of practical problems involving high Reynolds numbers requires very high resolutions which are still far beyond the capabilities of current supercomputers [5]. Hence, hybrid methodologies have been

9.1 Requirements for Numerical Schemes

311

established as the most practical methods for aeroacoustic computations, due to the separate treatment of the fluid and the acoustic computations. In these schemes, the computational domain is split into a nonlinear source region and a wave propagation region, and different numerical schemes are used for the flow and acoustic computations. Herewith, first a turbulence model is used to compute the unsteady flow in the source region. Secondly, from the fluid field, acoustic sources are evaluated which are then used as input for the computation of the acoustic propagation. In these coupled simulations it is generally assumed that no significant physical effects occur from the acoustic to the fluid field. Figure 9.1 shows typical numerical methods which are employed when using any of these hybrid methodologies. Among the group of hybrid approaches, integral methods remain widely used in CAA for solving open domain problems like airframe noise, landing gear noise simulation, fan (turbines) noise, rotor noise, etc. One reason which motivates the use of integral formulations in such applications is that, in general, their acoustic sources can be considered to be compact and only an extension of the acoustic solution at a few points in the far field is of interest. Therefore, in such cases, integral methods based on Lighthill’s acoustic analogy, Curle’s formulation, Ffowcs Williams and Hawkings (FW-H) formulation, Kirchhoff method or extension thereof are computationally cheaper than interior methods where a whole discretization of the acoustic domain is required (see e.g., [6–9]). On the other hand, for interior aeroacoustic problems, where non-compact solid boundaries are present, or if structural/acoustic effects are considered, it is more appropriate to use an acoustic interior method to account for the interactions between the solid surfaces and the flow-induced noise directly in the acoustic simulation. In such cases, integral formulations would require a priori knowledge of a hardwall Green’s function that is not known for complex geometries [10]. Furthermore, integral methods do not allow for a straightforward inclusion of the elastic effects of structures in the flow. An additional advantage of interior methods is that they can also be used to include the effects of wave propagation in non-uniform background flows. Among the interior methods we find those based on Linearized Euler Equations (LEE) [11, 12], Acoustic Perturbation Equation (APE) [13–15], FE formulations of Lighthill’s acoustic analogy [10, 16], as well as the linearized perturbed compressible equations (LPCE) [17] (for a discussion on these methods see [18–22]). Figure 9.1 depicts the general configuration when using these methods. Herewith, Ω f denotes the subdomain, where the flow field is firstly computed and where the acoustic sources are interpolated from the fluid mesh to the acoustic mesh. In order to accurately resolve the source terms, unsteady computational fluid dynamics (CFD) schemes are required. Mainly used turbulence models are LES (Large Eddy Simulation), DES (Detached Eddy Simulation) and SAS (Scale Adaptive Simulation). For a detailed discussion on  these methods we refer to [23]. The acoustic propagation region is given by Ω f Ωa , where the acoustic field is computed in the second step by solving the inhomogeneous wave equation or a corresponding set of equations depending on the CAA methodology followed. Since interior methods require the whole discretization of the propagation domain, usually they are used to compute the radiated sound until an intermediate region in the far field (i.e., until Γa in Fig. 9.1),

312

9 Computational Aeroacoustics Far field observation point at large distances

Acoustic propagation using Interior Methods: Lighthill’s analogy Euler eqs. APEs, LPCEs

Integral methods: - Lighthill’s analogy - Curle’s formulation - FW-H formulation - Kirchhoff surfaces

Ωa Turbulent Flow

Flow

Γs

Ωf Γa

Γf

Non-linear region: Unsteady CFD (LES, URANS, SAS)

Fig. 9.1 Schematic depicting some of the possible strategies when using an aeroacoustic hybrid approach

before moving to an integral formulation in which the acoustic solution from the interior method at the interface is used as input for computing pressure levels at the far field. Such a combined scheme has been presented in [24] using LEE for the intermediate solution and a Kirchhoff method for the far field noise.

9.2 Lighthill’s Analogy The sound generated by turbulence in an unbounded fluid is usually called aerodynamic sound. Most unsteady flows in technical applications are of high Reynolds number, and the acoustic radiation is a very small by-product of the motion. Thereby, the turbulence is usually produced by fluid motion over a solid body and/or by flow instabilities. Lighthill transformed the general equations of mass and momentum conservation to an exact inhomogeneous wave equation whose source terms are important only within the turbulent region [25, 26]. Lighthill was initially interested in solving the problem, illustrated in Fig. 9.2a, of the sound produced by a turbulent nozzle and arrived at the inhomogeneous wave equation. However, at this time a volume discretization by numerical schemes was not feasible and so a transformation of the PDE into an integral representation was performed, which can just be achieved for a free field setup, for which Green’s

9.2 Lighthill’s Analogy

313

sound

(a)

(b)

sound

v

turbulent nozzle flow Fig. 9.2 Sound generation by turbulent flows. a Turbulent nozzle flow. b Isolated turbulent region

function is available. Therefore, Lighthill’s theory in its integral formulation just applies to the simple situation as given in Fig. 9.2b. This avoids complications caused by the presence of the nozzle. The fluid is assumed to be at rest at the observer position, where a mean pressure, density and speed of sound are respectively equal to p0 , ρ0 and c0 . So Lighthill compared the equations for the production of density fluctuations in the real flow with those in an ideal linear acoustic medium (quiescent fluid). For the derivation, we start at Reynolds form of the momentum equation, as given by (4.22) neglecting any force density f Ω ∂ρv + ∇ · [π] = 0 , ∂t

(9.2)

with the momentum flux tensor πij = ρvi v j + ( p − p0 )δij − τij , where the constant pressure p0 is inserted for convenience. In an ideal, linear acoustic medium, the momentum flux tensor contains only the pressure πij → πij0 = ( p − p0 )δij = c02 (ρ − ρ0 )δij

(9.3)

and Reynolds momentum equation reduces to  ∂  2 ∂ρvi + c0 (ρ − ρ0 ) = 0. ∂t ∂xi

(9.4)

Rewriting the conservation of mass in the form ∂ρvi ∂ =0 (ρ − ρ0 ) + ∂t ∂xi

(9.5)

allows us to eliminate the momentum density ρvi in (9.4). Therefore, we perform a time derivative on (9.5), a spatial derivative on (9.4) and substract the two resulting equations. These operations leads to the equation of linear acoustics satisfied by the perturbation density

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9 Computational Aeroacoustics



   1 ∂2 2 c − ∇ · ∇ (ρ − ρ ) = 0. 0 0 c02 ∂t 2

(9.6)

Because the turbulence is neglected, the unique solution of this equation satisfying the radiation condition and we obtain ρ − ρ0 = 0. It can now be asserted that the sound generated by the turbulence in the real fluid is exactly equivalent to that produced in the ideal, stationary acoustic medium forced by the stress distribution   L ij = πi j − πij0 = ρvi v j + ( p − p0 ) − c02 (ρ − ρ0 ) δij − τi j ,

(9.7)

where [L] is called the Lighthill stress tensor. Indeed, we can rewrite (9.2) as the momentum equation for an ideal, stationary acoustic medium of mean density ρ0 and speed of sound c0 subjected to the externally applied stress L ij  ∂πij0 ∂ρvi ∂  + πij − πij0 , =− ∂t ∂x j ∂x j

(9.8)

 ∂ L ij ∂ρvi ∂  2 + c0 (ρ − ρ0 ) = − . ∂t ∂x j ∂x j

(9.9)

or equivalent

By eliminating the momentum density ρvi using (9.5) we arrive at Lighthill’s equation 

   ∂ 2 L ij 1 ∂2 2 c − ∇ · ∇ . = (ρ − ρ ) 0 0 ∂xi ∂x j c02 ∂t 2

(9.10)

Therefore, the problem of calculating the flow generated sound is equivalent to solving this wave equation, which is possible when the source term ∂ 2 L ij /∂xi ∂x j is provided, e.g., by a CFD computation. This type of source term can be interpreted as a quadrupole term. Therefore, the free field turbulence is an extremely weak sound source, and so in low Mach number flows just a very small portion of the flow energy is converted into sound. However, in the presence of walls the sound radiation by turbulence can be dramatically enhanced. In the next section, we will see that compact bodies will radiate a dipole sound field associated to the force which they exert on the flow as a reaction to the dynamic force of the flow applied to them. Sharp edges are particularly efficient radiators. In the definition of the Lighthill tensor according to (9.7) the term ρvi v j is called the Reynolds stress. It is a nonlinear term and can be neglected except where the

9.2 Lighthill’s Analogy

315

  motion is turbulent. The second term ( p − p0 ) − c02 (ρ − ρ0 ) δij represents the excess of moment transfer by the pressure over that in the ideal fluid of density ρ0 and speed of sound c0 . This is produced by wave amplitude nonlinearity, and by mean density variations in the source flow. The viscous stress tensor τi j properly accounts for the attenuation of the sound. In most applications the Reynolds number in the source region is high and we can neglect this contribution. The solution of (9.10) for free field radiation condition with outgoing wave behavior can be rewritten in integral form as follows [27] c02 (ρ − ρ0 )(x, t)

1 ∂2 = 4π ∂xi ∂x j

∞

L ij ( y, t − |x − y|/c0 ) dΩ. |x − y|

(9.11)

−∞

Thereby, y defines the source coordinate and x the coordinate at which we compute the acoustic density fluctuation. This provides a useful prediction of the sound, if L ij is known. Please note that the terms in L ij not only account for the generation of sound, but also includes acoustic self modulation caused by • • • •

acoustic nonlinearity, the convection of sound waves by the turbulent flow velocity, refraction caused by sound speed variations, and attenuation due to thermal and viscous actions.

The influence of acoustic nonlinearity and thermoviscous dissipation is usually sufficiently small to be neglected within the source region. Convection and refraction of sound within the flow region can be important, e.g., in the presence of a mean shear layer (when the Reynolds stress will include terms like ρv0i v ′j , where v 0 and v ′ respectively denote the mean and fluctuating components of v). Such effects are described by the presence of unsteady linear terms in L ij . Furthermore, since for practical applications the source term is obtained by numerically solving NavierStokes equation, the question of how accurate the source term is resolved, is always present. Now, let’s consider the situation for which the mean density and speed of sound are uniform throughout the fluid. The variations in the density ρ within a low Mach number, high source flow are then of order O(ρ0 Ma2 ). Thus,  Reynolds2 number  ρvi v j = ρ0 1 + O(Ma ) vi v j ≈ ρ0 vi v j . Furthermore, if c(x, t) is the local speed of sound in the source region, it can be shown that c02 /c2 = 1 + O(Ma2 ), so that we obtain p − p0 − c02 (ρ − ρ0 ) = ( p − p0 )(1 − c02

ρ − ρ0 ) ≈ ( p − p0 )(1 − c02 /c2 ) ∼ O(ρ0 v 2 Ma2 ). p− p

 0 1/c02

(9.12)

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9 Computational Aeroacoustics

Therefore, if viscous dissipation is neglected, we may approximate the Lighthill tensor by (9.13) L ij ≈ ρ0 vi v j for Ma2 ≪ 1. Please note that with this assumptions, the divergence of (4.22) provides the following equivalence (assuming an incompressible flow ∇ · v = 0 and f Ω = 0) ∇ · ∇ p = −ρ0

∂ 2 vi vi . ∂xi ∂x j

(9.14)

With this result we obtain for the pressure fluctuation using the isentropic pressuredensity relation the following integral representation ρ0 vi v j ( y, t − |x − y|/c0 ) ∂2 ′ dΩ (9.15) p (x, t) ≈ ∂xi ∂x j 4π|x − y|   xi x j |x| x·y ∂2 ≈ y, t − dΩ. (9.16) + v v ρ 0 i j c0 c0 |x| 4πc02 |x|3 ∂t 2 To obtain (9.16), we have used the far field approximation, which allows the following substitution [27] ∂ 1 xj ∂ =− . (9.17) ∂x j c0 |x| ∂t Now, we want to derive the order of the magnitude of the acoustic pressure as a function of the flow velocity v. In doing so, we introduce a characteristic velocity v and length scale l of a single vortex as displayed in Fig. 9.3. Fluctuations in vi v j occurring in different turbulent regions by distances larger than O(l) will be treated to be statistically independent. So the sound may be considered to be generated by a collection of Ω f /l 3 independent vortices. The characteristic frequency of the

Fig. 9.3 Single vortex in a turbulent flow region at its acoustic radiation towards the far field

x

vortex

l

turbulent source region Ωf

9.2 Lighthill’s Analogy

317

turbulent fluctuations can be estimated by f ∼ v/l so that the wavelength λ of sound will result in c0 l l c0 ∼ = ≫ l for Ma = v/c0 ≪ 1. λ= f v Ma Hence, we arrive at the quite important conclusion that the turbulent vortices are all acoustically compact. This means that in the relation (9.16) the retarded time variation x · y/(c0 |x|) can be neglected. Therefore, the value of the integral over one source vortex in (9.16) can be estimated to be of order ρ0 v 2 l 3 . The order of the magnitude for the time derivative in (9.16) is estimated to be v ∂ ∼ . ∂t l Collecting all this estimates, we may now state that the acoustic pressure in the far-field, generated by one vortex, satisfies pa ∼

l l ρ0 v 4 ρ0 v 2 Ma2 . = |x| c02 |x|

(9.18)

The acoustic power defined by Pa =



pa v a · ds =

Γ



pa v a · n dΓ

(9.19)

Γ

can be computed in the far-field with the relation v a · n = pa /(ρ0 c0 ) as follows Pa =



pa 2 dΓ. ρ0 c0

(9.20)

Γ

This formula allows us to estimate the acoustic power generated by one vortex Pa ∼ 4π|x|2

l 2 ρ0 v 8 pa 2 = ρ0 l 2 v 3 Ma5 . ∼ ρ0 c0 c05

(9.21)

This is the famous eighth power law.

9.3 Curle’s Theory The main restriction of Lighthill’s integral formulation is that it can just consider free radiation. Therewith, it can not consider situations where there is any solid body within the region. In [28] this problem was solved by deriving an integral formulation

318

9 Computational Aeroacoustics

(a)

(b)

f >0 n

Lij = 0 Ω

dΓ

Ω f 0 for x in Ω.

This surface may either be a solid body, or just an artificial control surface used to isolate a fixed region of space containing both solid bodies and fluid or just fluid. To derive Curle’s equation we start with the momentum equation according to (9.9) and multiply it with the Heaviside function H ( f ) H( f )

 ∂ L ij ∂  2 ∂ρvi + H( f ) c0 (ρ − ρ0 ) = −H ( f ) . ∂t ∂x j ∂x j

(9.22)

Now, we use the product rule for differentiation (noting that the time derivative of the Heaviside function, which just depends on space, is zero) and obtain by also writing L ij by its individual components  ∂ ∂  2 ∂H( f ) c0 (ρ − ρ0 )H ( f ) − c02 (ρ − ρ0 ) (ρvi H ( f )) + ∂t ∂x j ∂xi  ∂H( f )      ∂ . L ij H ( f ) + ρvi v j + ( p − p0 ) − c02 (ρ − ρ0 ) δij − τi j =− ∂x j 

∂x j

L ij

(9.23)

We can cancel out the term c02 (ρ−ρ0 ) ∂ H ( f )/∂xi being at both sides of the equation and arrive at

9.3 Curle’s Theory

319

 ∂ ∂  2 c0 (ρ − ρ0 )H ( f ) (ρvi H ( f )) + ∂t ∂x j    ∂H( f ) ∂  =− . L ij H ( f ) + ρvi v j + (( p − p0 )) δij − τi j ∂x j ∂x j

(9.24) (9.25)

The same procedure is now applied to the mass conservation according to (9.5) and so we obtain ∂ ∂H( f ) ∂ = 0. (ρvi H ( f )) − ρvi ((ρ − ρ0 )H ( f )) + ∂t ∂xi ∂xi

(9.26)

Now, we perform a time derivative to this equation and rearrange it for ρvi H ( f ) ∂2 ∂ (ρvi H ( f )) = ∂t∂xi ∂t

  ∂2 ∂H( f ) ρvi − 2 ((ρ − ρ0 )H ( f )) . ∂xi ∂t

(9.27)

In a last step, we apply the divergence operation to (9.25) and substitute the expression for ρvi H ( f ) from (9.27) 

  1 ∂2 2 − ∇ · ∇ c (ρ − ρ )H ( f ) 0 0 c02 ∂t 2

(9.28)

∂ 2 L ij H ( f ) ∂xi ∂x j    ∂H( f ) ∂  ρvi v j + ( p − p0 )δij − τi j − ∂xi ∂x j   ∂ ∂H( f ) + ρv j . ∂t ∂x j =

This equation is now valid throughout the space, including the region enclosed by Γs . Furthermore, comparing to Lighthill’s equation, we have obtained two additional terms on the right hand side of the wave equation including space derivatives of the Heaviside function H ( f ). Thereby, according to our previous investigation the second term on the right hand side corresponds to a dipole and the third term to a monopole with the following interpretation: • Γs is the boundary of a solid body: In this case the surface dipole represents the production of sound by the unsteady force that the body exerts on the exterior fluid, whereas the monopole is responsible for the sound generated by volume pulsations (if any) of the body. • Γs is just an artificial control surface: The dipole and monopole sources account for the presence of solid bodies and turbulences within Γs (when L ij is different from zero in Γs ) and also for the interaction of sound generated outside Γs with the fluid and solid bodies inside Γs .

320

9 Computational Aeroacoustics

To transform (9.28) to the corresponding integral representation is a straight forward operation. According to the wave equation and its integral representation we obtain for the monopole term 1 4π

∞

−∞

1 ∂   ∂H( f ) ρv j dΩ , ∂t ∂ y j |x − y|

(9.29)

where  indicates that the term has to be evaluated at ( y, t − |x − y|/c0 ). For the dipole term we obtain the following integral representation 1 ∂ 4π ∂xi

∞

−∞

  ∂H( f ) 1 dΩ. ρvi v j + ( p − p0 )δij − τi j ∂ y j |x − y|

(9.30)

The last step is the handling of the Heaviside function, which has according to generalized function theory the following property for an arbitrary smooth function Φ(x) [27] ∞

−∞

∂H( f ) Φ( y) dΩ = ∂yj



Φ( y)n j dΓ =



Φ( y) dΓ j .

(9.31)

Γs

Γs

Exploring this property, we finally arrive Curle’s equation in integral form c02 (ρ − ρ0 )

∂2 H( f ) = ∂xi ∂x j



 L ij dΩ 4π|x − y|



Ω

∂ − ∂xi

   ρvi v j + ( p − p0 )δij − τij dΓ j ( y) 4π|x − y|

Γs

∂ + ∂t



Γs

  ρv j dΓ j ( y). 4π|x − y|

(9.32)

Now, let us restrict to a rigid body for which the flow velocity in normal direction on this body is zero, so that (9.32) reduces to c02 (ρ − ρ0 )

∂2 H( f ) = ∂xi ∂x j



Ω



 L ij dΩ 4π|x − y|

   ( p − p0 )δij − τij ∂ dΓ j ( y). − ∂xi 4π|x − y| Γs

(9.33)

9.3 Curle’s Theory

321

Furthermore, we assume the body to be acoustically compact, which means that Ma = v/c0 ≪ 1. In the following investigation, we want to estimate the order of the sound generation by the dipole term. With the characteristic velocity v and scale length l of a vortex, we have the following relations v ( p − p0 ) ∼ ρ0 v 2 ; τ ∼ µ f . l

(9.34)

Therefore, we can compute the ratio ρ0 vl vl p − p0 ∼ = τ µf νf with ν f = µ f /ρ0 the kinematic viscosity, which is just the definition of the Reynolds number Re = vl/ν f . Since in turbulent flows Re is quite high, we can neglect the viscous contribution. In the far-field, the acoustic pressure pa is equal to c02 (ρ − ρ0 )H ( f ) (H ( f ) is there just 1), and exploring the compactness which allows us to neglect the retarded time variation x · y/(c0 |x|) results for the second term in (9.33) using (9.17) in pa ≈

∂ xi 2 4πc0 |x| ∂t



Γs

    |x| ∂ Fi xi |x| dΓi = t − ( p − p0 ) y, t − c0 4πc0 |x|2 ∂t c0

(9.35) with F the total unsteady surface force. For a surface element with a diameter of l, the contribution to the acoustic pressure pa can be estimated by l 1 v ρ0 v 2 l 2 = ρ0 v 2 Ma. c0 |x| l |x|

(9.36)

Assuming to have Γs /l 2 independently radiating surface elements, we can estimate the acoustic power (see also (9.21))   Γs l2 p a 2 Γs 2 2 4 2 = 4π|x| ρ0 v Ma Pa ∼ 4π|x| 2 2 ρ0 c0 l ρ0 c0 |x| l2 2

∼ ρ0 Γs v 3 Ma3 .

(9.37)

So, we see that in case of a dipole we arrive at a sixth power law and compared to the quadrupole we have a factor of 1/Ma2 being stronger. At this stage, it should be noted that there was and still is a lot of scientific discussions about Lighthill’s acoustic analogy and the extended versions of Curle and Ffowcs Williams and Hawkings including surfaces in arbitrary motion. In [29] it was argued that forces acting on a rigid body due to fluctuations of the flow pressure cannot produce acoustic sound as the boundary is fixed and does not vibrate. Indeed, in [30] the authors successfully demonstrated that the surface distribution of Curle’s

322

9 Computational Aeroacoustics

analogy is equivalent to the scattering of sound waves by the rigid surface, which are originally generated by the volume distribution of quadrupoles (second spatial derivative of Lighthill’s tensor). Furthermore, a quite interesting approach has been presented in [21]. Here, Lighthill’s tensor [L] is used to compute in a first step the fluctuating incident pressure field by solving i

p (x, ω) =



∂ 2 L ij ( y, ω) ˆ G(x, y) dΩ ∂ yi ∂ y j

(9.38)

Ω

ˆ with G(x, y) = e jk|x− y| /(4π|x − y|) as the harmonic free field Green’s function. Furthermore, the gradient of this fluctuation pressure is calculated by ∂ p i (x, ω) = ∂xk



ˆ ∂ 2 L ij ( y, ω) ∂ G(x, y) dΩ. ∂ yi ∂ y j ∂xk

(9.39)

Ω

In a second step, the scattered field p ′ is computed by using the results of (9.38), (9.39) and evaluating (for details see [21]) ′

α( y) p ( y) +



Γs

ˆ ∂ G(x, y) ′ ∂ p′ ˆ p (x) dΓ = dΓ + p i ( y). (9.40) G(x, y) ∂n(x) ∂n(x) Γs

In (9.40) α( y) is a free-term coefficient and equals 1 in the domain and 0.5 on a smooth surface. Therefore, we can summarize that Lighthills’ inhomogeneous wave equation is a quite general model to describe flow-induced sound. Solving this partial differential equation by a volume discretization method includes all sources of the sound. The additional source terms, as given in (9.32) just come up, because the partial differential equation is converted to an integral representation for which Greens’ function is needed.

9.4 Vortex Sound Restricting to low Mach number flows and neglecting combustion and entropy sources of sound, we arrive at the classical theory of vortex sound [27]. For a real fluid, we may decompose the flow velocity v according to Helmholtz by v = ∇ × ξ + ∇φ = v ic + ∇φ ,

(9.41)

where v ic contains the solenoidal part of the flow velocity v and has the property ∇ · v ic = 0 (defines the incompressible part of v). Furthermore, φ denotes the scalar

9.4 Vortex Sound

323

velocity potential and ξ the vector potential. Thereby, the vorticity ω is defined by ω = ∇ × v = ∇ × v ic .

(9.42)

Knowing v, we can compute the scalar potential by ∇ · ∇φ = ∇ · v − ∇ · ∇ × ξ = ∇ · v ,

(9.43)

and by taking the curl, we obtain for the vector potential ∇ × ∇ × ξ = ∇ × v − ∇ × ∇φ = ω.

(9.44)

Furthermore, by using the vector identity ∇ · ∇ξ = ∇∇ · ξ − ∇ × ∇ × ξ we may write ∇ · ∇ξ = −ω.

(9.45)

To find φ we can take φ = 0 at infinity. To obtain ξ we can use Green’s function for the Laplace equation and arrive at ξ=



ω d y. 4π|x − y|

(9.46)

Ω

On the other hand, knowing the vorticity ω, we may compute the incompressible velocity by ω( y, t) dy, (9.47) v ic = ∇ × 4π|x − y| Ω

which is a pure kinematic relation. Now, because vorticity is transported by convection and diffusion, an initially confined region of vorticity will tend to remain within a bounded body, so that it may be assumed that ω → 0 as |x| → ∞ [27]. This result allows the following estimate ic

v ∼O



1 |x|3



.

(9.48)

So the flow field as described by v ic (pure solenoidal part of the flow) decays with O(1/|x|3 ) towards the far-field. Furthermore, by assuming constant density ρ0 the main part of Lighthills’ source term for low Mach number flows may be rewritten by

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9 Computational Aeroacoustics

ρ0

∂ 2 viic v icj ∂xi ∂x j

    1 ic ic = ρ0 ∇ · ω × v ic − ρ0 ∇ · ∇ v ·v . 2

(9.49)

So, the overall fluctuating pressure can be seen as a superposition obtained by the two source terms       1 ic ic p ′ (x, t) = p1′ ρ0 ∇ · ω × v ic + p2′ −ρ0 ∇ · ∇ . v ·v 2 As shown in [27], one can derive the following estimates for the far field l ρ0 u 2 Ma2 |x| l l ρ0 u 2 Ma2 ρ0 u 2 Ma4 + . ∼ |x| |x| Re

pa1 ∼

(9.50)

pa2

(9.51)

Therefore, we can state pa2 ≪ pa1 in turbulent flows, where Ma ≪ 1 and Re ≫ 1. Furthermore, we can conclude that in such cases the component ρ0 ∇ · (ω × v ic ) of the Lighthill source term ρ0

∂ 2 viic v icj ∂xi ∂x j

(9.52)

is the principle source of sound.

9.5 Perturbation Equations The acoustic/viscous splitting technique for the prediction of flow induced sound was first introduced in [31], and afterwards many groups presented alternative and improved formulations for linear and non linear wave propagation [13, 14, 32, 33]. These formulations are all based on the idea, that the flow field quantities are split into compressible and incompressible parts. In [33], the authors directly started at the compressible Navier Stokes equations and obtained a system describing nonlinear wave propagation which they called perturbed compressible equations (PCE). For an incompressible flow they linearized their approach to arrive at linearized perturbed compressible equations (LPCE) [17]. A slightly different approach is utilized in [13]. Here, the incompressible limit for the flow simulation was dropped by starting directly from the linearized Euler equations (LEE) using a temporal average of the flow quantities and ensured pure acoustic wave propagation by a technique they call source term filtering. The main difference of the two approaches is related to the notion of splitting of the flow field quantities. Table 9.1 summarizes the splittings used for the different approaches in which we denote incompressible quantities by a superscript ic.

9.5 Perturbation Equations Table 9.1 Splitting of flow field quantities for perturbation equations

325 LEE

Viscous/Acoustic

v = v¯ (x) + v ′ (x, t) p = p(x) ¯ + p ′ (x, t) ρ = ρ¯ (x) + ρ′ (x, t)

v = v ic (x, t) + v c (x, t) p = p ic (x, t) + p c (x, t) ρ = ρ0 + ρ1 (x, t) + ρc (x, t)

We notice that the density in the viscous/acoustic case is split not into two but three components. In [31] the authors referred to the additional term ρ1 as a density correction. This density variation is not acoustic but has to be taken into account due to the fact, that the fluctuating parts of the incompressible pressure p1 need to be balanced by a density change even in case of incompressible flows in order to keep the resulting field description isentropic, i.e. to ensure constant entropy in the fluid. In this context, adiabatic reversible processes are isentropic whereas the converse does not need to be true. One can give the relation according to the equation of state for an homotropic medium ρ1 =

p ic − p ic 1 p1 = . 2 c0 c02

(9.53)

The last equality can be seen as an estimation of this density correction. In [14] a different correction term based on the spatial mean pressure in combination with a Mach number scaling which leads to an alternative yet similar system of perturbation equations has been proposed. Even though the introduced splittings subtract obviously non-acoustical components from the field variables it is by no means guaranteed that the perturbation quantities are purely acoustic quantities. Therefore, not only the source terms on the right hand side but also the left hand side of the resulting system of equations has to be changed such that only acoustic waves can propagate. Therefore, we introduce a generic splitting of physical quantities to the Navier Stokes equations. For this purpose we choose a combination of the two splitting approaches introduced above and define the following p = p¯ + p ic + p c = p¯ + p ic + p a v = v¯ + v ic + v c = v¯ + v ic + v a ρ = ρ¯ + ρ1 + ρa .

(9.54) (9.55) (9.56)

Thereby the field variables are split into mean and fluctuating parts just like in the LEE. In addition the fluctuating field variables are split into acoustic and non-acoustic components. Finally also the density correction is build in as introduced above. This choice is motivated by the following assumptions • The acoustic field is a fluctuating field. • The acoustic field is irrotational, i.e. ∇ × v a = 0.

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9 Computational Aeroacoustics

• The acoustic field requires compressible media and a incompressible pressure fluctuation is not equivalent to an acoustic pressure fluctuation. By doing so, we arrive for an incompressible flow at the following perturbation equations1  ∂ pa −1 Dp ic a + ∇ · ( p v) + ∇ · v a = ∂t ρ0 c02 Dt a ∂v 1 + (v · ∇)v a + (v a · ∇)v + ∇ p a = 0 ∂t ρ0 1 ρ0 c02



(9.57)

with mean speed of sound c0 and density ρ0 . This system of partial differential equations corresponds to [13]. The source term is the substantial derivative of the incompressible flow pressure Dp ic /Dt = ∂ p ic /∂t + v · ∇ p ic . For completeness, we also present the linearized perturbed compressible equations (LPCE) [17], which reads as follows Dρa + ρ0 ∇ · v a = 0 Dt   ∂v a 1 + ∇ v a · v ic + ∇ p a = 0 ∂t ρ0 Dp a Dp ic + κ p ic ∇ · v a + v a · ∇ p ic = − . Dt Dt

(9.58) (9.59) (9.60)

Again, the substantial derivative of the incompressible flow pressure is the source term. A practical drawback of the LPCE approach is related to the fact, that the convective terms also contain transient flow field variables. In a finite element scheme this would result in a reassembly of the matrices in each time step which is computationally expensive. Nevertheless it is interesting to notice that the coupling between the transient solenoidal field and the acoustic field is fully taken into account in difference to the formulations given in [13]. Finally, it is of great interest that by neglecting the mean flow v¯ in (9.57), we arrive at the linearized conservation equations of acoustics with ∂ p ic /∂t as a source term 1 ∂ pa −1 ∂ p ic a + ∇ · v = ρ0 c02 ∂t ρ0 c02 ∂t a ∂v 1 + ∇ p a = 0. ∂t ρ0

(9.61) (9.62)

As in the standard acoustic case, we perform a time derivative on (9.61) and a spatial derivative on (9.62) and subtract the two equations to arrive at 1

For a detailed derivation of perturbation equations both for compressible as well as incompressible flows, we refer to [20].

9.5 Perturbation Equations

327

1 ∂ 2 pa −1 ∂ 2 p ic a − ∇ · ∇ p = . c02 ∂t 2 c02 ∂t 2

(9.63)

We call this new partial differential equation the aeroacoustic wave equation (AWE). Please note, that this equation can also be arrived by starting at Lighthills’ inhomogeneous wave equation for incompressible flow, where we can substitute the second spatial derivative of Lighthill’s tensor by the Laplacian of the incompressible flow pressure (see (9.14)) 1 ∂ 2 p′ − ∇ · ∇ p ′ = −∇ · ∇ p ic . (9.64) c02 ∂t 2 Using the decomposition of the fluctuating pressure p ′ p ′ = p ic + p a . results again into (9.63).

9.6 Finite Element Formulation 9.6.1 Lighthills’ Inhomogeneous Wave Equation We will perform a volume discretization of Lighthill’s equation (see 9.10) by applying the finite element method (FEM). Therewith, any solid–elastic body will be implicitly taken into account, and there is no need to use the extended form of Lighthill’s equation as given by (9.28). At this stage we assume that the Lighthill tensor [L] is a known quantity, e.g., obtained by a flow computation using a large eddy simulation (LES). In the first step, we multiply (9.10) (with p ′ = c02 (ρ − ρ0 )) by an appropriate test function w and integrate over the whole domain Ω (corresponding in Fig. 9.1 to Ωf ∪ Ωa )   ∂ 2 L ij 1 ∂ 2 p′ ∂ 2 p′ w 2 − − dΩ = 0. (9.65) ∂xi ∂x j c0 ∂t 2 ∂xi2 Ω

Now, we apply Green’s integral theorem to the second spatial derivative of p ′ as well as L ij . This operation will result in the following relations

Ω



Ω

w

w

∂ 2 p′ dΩ = ∂xi2

∂2 L

ij

∂xi ∂x j

dΩ =



w

∂ p′ dΓ − ∂n

Γs ∪Γa



Γs

∂ L ij w n i dΓ − ∂x j



∂w ∂ p ′ dΩ ∂xi ∂xi

(9.66)



∂w ∂ L ij dΩ. ∂xi ∂x j

(9.67)

Ω

Ωf

328

9 Computational Aeroacoustics

We want to emphasize that the boundary integral (9.67) is just over the surface Γs of any solid–elastic body, whereas in (9.66) we have to integrate over Γs as well as over Γa , which limits the computational domain (see Fig. 9.1). Now we can substitute ∂ L ij /∂x j within the first term on the right-hand side of (9.67) by (9.9) and obtain   ∂ L ij ∂ 2 ′ ∂ρvi − w w − n i dΓ = (c ρ ) n i dΓ. (9.68) ∂x j ∂t ∂xi 0 Γs Γs Since on a solid surface vi n i = 0 is fulfilled, we see that the surface integral term over Γs reduces to ∂ L ij ∂ρ′ ∂ p′ dΓ = − dΓ. (9.69) w c02 w w n i dΓ = − ∂x j ∂n ∂n Γs Γs Γs Therewith, we can rewrite (9.65) as ∂w ∂ p ′ 1 ∂ 2 p′ w dΩ + dΩ − ∂xi ∂xi c02 ∂t 2 Ω

Ω

=−



w

∂ p′ dΓ ∂n

Γs ∪Γa

∂w ∂ L ij dΩ − ∂xi ∂x j

Ωf





w Γs

∂ p′ dΓ. ∂n

(9.70)

Combining the surface integrals results in a single boundary integral just performed over the outer boundary Γa , on which we, e.g., apply absorbing boundary conditions of first order (see Sect. 5.5.1). Therewith, we utilize the relation c0

∂ p′ ∂ p′ =− ∂n ∂t

(9.71)

and arrive at the weak form: Find p ′ ∈ H 1 such that

Ω

1 ∂ 2 p′ + w c02 ∂t 2



Ω

∂w ∂ p ′ dΩ + ∂xi ∂xi



1 ∂ p′ w dΓ = − c ∂t

Γa



∂w ∂ L ij dΩ (9.72) ∂xi ∂x j

Ωf

for any w ∈ H 1 . Using standard nodal finite elements, we approximate the continuous acoustic pressure p ′ as well as the test function w by h

p′ ≈ p′ =

n eq 

Na pa′

(9.73)

Na wa .

(9.74)

a=1

w ≈ wh =

n eq  a=1

9.6 Finite Element Formulation

329

Thus, (9.72) is transformed to the following semi-discrete Galerkin formulation M p¨′ n+1 + C p˙′ n+1 + K p ′ n+1 = f n+1

(9.75)

with p¨′ = ∂ 2 p ′ /∂t 2 , p˙′ = ∂ p ′ /∂t, p ′ the nodal unknowns of the acoustic pressure and n the time step counter. The matrices as well as right-hand side vector compute as follows: M=

ne 

e

e

m ; m = [m pq ] ; m pq =

e=1



ce ; ce = [c pq ] ; c pq =

e=1

K =

f n+1 =

1 N p Nq dΩ c02

Ωe

n ΓI

C=



ne 



1 N p Nq dΓ c0

ΓIe e

e

k ; k = [k pq ] ; k pq =



e=1

Ωe

ne 



e=1

f e ; f e = [ f p] ; f p =

Ωe

∂ N p ∂ Nq dΩ ∂xi ∂xi n+1 ∂ N p ∂ L ij dΩ. ∂xi ∂x j

In the above equations n e is the number of finite elements, n ΓI the number of finite  the finite element assembly surface elements along the outer boundary ΓI and operator. The time discretization is performed by applying a standard Newmark algorithm as described in Sect. 2.5.2. By performing a harmonic analysis, it is possible to compute the sound radiation for specific frequency components present in the acoustic sources. In this way, we obtain the complex acoustic pressure at each node in the computational domain. For deriving the harmonic formulation of the implementation, we can simply apply a Fourier-transformation to the semi-discrete Galerkin formulation from (9.75), since the matrices M, C and K are frequency independent. The resulting complex algebraic system of equations is given by   − ω 2 M + iωC + K pˆ = fˆ

(9.76)

where the source term fˆ represents the complex nodal acoustic sources, which are obtained by applying a Fourier transformation to the data set of transient nodal sources interpolated from the fluid grid to the acoustic grid (see Sect. 9.6.3).

330

9 Computational Aeroacoustics

9.6.2 Perturbation Equations In comparison to the conservation equations of linear acoustics (see Sect. 5.4.2) we notice that the only differences are the convective terms in (9.57). Those terms are usually handled within finite element methods by employing some kind of upwinding schemes such as the Streamline-Upwind-Petrov-Galerkin (SUPG) as used for the Navier-Stokes equations (see Sect. 4.6). However, motivated by its good performance properties as obtain by the mixed variational ansatz as utilized for the conservation equation of linear acoustics, we will follow here a similar approach. For a shorter notation we assume for now a spatially constant mean flow v 0 and neglect all source terms. The variational form for v a , ψ ∈ L 2 and pa , ϕ ∈ H 1 is given as ∂ pa ϕ dΩ + (v 0 · ∇ pa ) ϕ dΩ − v a · ∇ϕ dΩ = 0 ∂t Ω Ω Ω ∂v a · ψ dΩ + (v 0 · ∇)v a ·ψ dΩ + ∇ pa · ψ dΩ = 0.

 ∂t

Ω

Ω

B

(9.77)

(9.78)

Ω

Three major difficulties in the mixed finite element formulation are evident: 1. The choice of the discontinuous space L 2 for the acoustic particle velocity v a is in contradiction with the term B (see (9.78)) in which the existence of a scaled divergence is required. 2. Due to the convective term B, the global stiffness matrix is no longer skew symmetric which gives rise to eigenvalues with positive real part and thereby unstable solutions. 3. The convective character of the above equation usually requires an upwinding scheme. Therefore, in a first step we introduce a numerical flux similar as done for Discontinuous Galerkin (DG) schemes. Let’s assume two elements K 1 and K 2 with a common boundary Γ12 as pictured in Fig. 9.5. For the x-component of the vector quantity v a we can define the average as {{vax }} =

1 (vax,1 + vax,2 ), 2

(9.79)

in which u ax,1 is the x-component of the particle velocity associated with the element K 1 and u ax,2 the corresponding unknown associated with the element K 2 . Accordingly we define the jump between the elements as [[vax ]] = vax,1 n1 + vax,2 n2 ,

(9.80)

where n1 and n2 denote the outward normal with respect to the element K 1 and K 2 respectively. In order to introduce a numerical flux for the convective bilinear form

9.6 Finite Element Formulation

331

Fig. 9.5 Two elements with common boundary and element local normal vectors as well as the unique normal vector n12 associate to the edge Γ12

in (9.78), we initially perform an integration by parts which reads as

(v 0 · ∇)v a · ψ dΩ = −

Ω



v a · (v 0 · ∇)ψ dΩ +

Ω



(n · v 0 )v a · ψ dΓ. (9.81)

Γ

Due to the discontinuous choice for the unknowns and test functions, the boundary term does not necessarily vanish between two interior elements and a flux term is introduced on the boundary Γ12 . Due to the structure of the convective term it is now possible to write the flux for each component of v a individually. The flux for the 2D problem for the edge Γ12 can then be defined as a12 (v a , ψ) =



{{v 0 vax }} · [[ψx ]] dΓ +

Γ12



{{v 0 vay }} · [[ψ y ]] dΓ.

(9.82)

Γ12

The extension to the three dimensional case is now straight forward. The bilinear form including numerical fluxes reads as

(v 0 · ∇)v a · ψ dΩ = −

 e

Ω

v a · (v 0 · ∇)ψ dΩ +



ai j (v a , ψ).

(9.83)

Γij

Ωe

Even with the introduction of the flux term, the stiffness matrix is not skew symmetric which is desirable due to energy conservation and stability issues as pointed out e.g. in [34]. In order to restore skew symmetry we perform another integration by parts on the right hand side of (9.83) and obtain −

 e

v a · (v 0 · ∇)ψ dΩ +



ai j (v a , ψ) =

e

Γij

Ωe

−





(v 0 · ∇)v a · ψ dΩ

Ωe

(ni · v 0 )v ai · ψ i + (n j · v 0 )v a j · ψ j dΓ

Γij Γ ij

+

 Γij

ai j (v a , ψ).

(9.84)

332

9 Computational Aeroacoustics

Averaging of the right hand sides of (9.83) and (9.84) yields the final bilinear form

Ω

⎞ ⎛  1 ⎟ ⎜ (v 0 · ∇)v a · ψ dΩ = ⎝ (v 0 · ∇)v a · ψ dΩ − v a · (v 0 · ∇)ψ dΩ ⎠ 2 e Ωe

Ωe

⎞  1 ⎟ ⎜ − ⎝ (ni · v 0 )v ai · ψ j dΓ + (n j · v 0 )v a j · ψ i dΓ ⎠ 2 ⎛

Γij

Γij

Γij

(9.85)

Now, all derivatives are defined only locally on each element which ensure existence of the derivative and the matrix entries from this bilinear form will be skew symmetric. This becomes apparent when one thinks about the direction of the element normals. As ni = −n j it is obvious that the entries are skew symmetric. In a last step, a penalization upwinding as used in full DG methods (e.g. in [35]) will be introduced. Due to the different structure of the convective terms in the case under investigation one can define the upwind in more intuitive way with respect to the unique edge normal n12 . Such an upwind flux is usually defined as ⎧ if v 0 · n12 > 0 ⎨ v 0 vax,1 if v 0 · n12 < 0 (9.86) {{v 0 vxa }}v = v 0 vax,2 ⎩ {{v 0 vax }} if v 0 · n12 = 0. As stated in [36] one can alternatively introduce a penalization term between two adjacent elements which is equivalent to (9.86). Such a penalization was also used in [35] to remove spurious waves in the context of a full DG formulation. Indeed, one can show that {{v 0 vax }}v · n12 = ({{v 0 vax }} + α[[vax ]]) · n12 .

(9.87)

Here, the parameter α denotes the upwinding parameter. By setting α = |v 0 · n|/2, the penalization is exactly equivalent to (9.86). The penalization is applied to the convective bilinear form which yields the transformed formulation of the initial bilinear form as ⎛ ⎞ 1⎜ ⎟ (v 0 · ∇)v a · ψ dΩ → ⎝ (v 0 · ∇)v a · ψ dΩ − v a · (v 0 · ∇)ψ dΩ ⎠ 2 Ω

∀e

Ωe

Ωe

⎛

 ⎜   (v 0 · n j vax, j ψx,i + (v 0 · ni vax,i ψx, j dΓ − ⎝ ∀Γij

+



Γij

Γij

⎞

⎟ α[[vax ]] · [[ψx ]] dΓ + · · · ⎠ .

(9.88)

9.6 Finite Element Formulation

333

Thereby, by considering all the above mentioned steps, we arrive at a stable and efficient FE formulation for the perturbation equations. For further details, we refer to [20].

9.6.3 Source Term Treatment A crucial point is the transformation of the acoustic sources from the computed flow data to the acoustic grid. In order to preserve the acoustic energy, we perform an integration over the source volume (corresponding to the computational flow region) within the FE formulation and project the results to the nodes of the fine flow grid, which has to be interpolated to the coarser acoustic grid (see Fig. 9.6). Thereby, our interpolation has to be conservative in order to preserve the total acoustic energy. As illustrated in Fig. 9.7, we have to find for each nodal source f kf in which finite

Nodal acoustic loads on the CFD grid Nodal acoustic loads on the acoustic grid

CFD grid

Conservative interpolation

Acoustic grid

Fig. 9.6 Conservative interpolation

η f1a F

f2a

f3a

ξ

(ξk , ηk )

fkf (xk , yk )

y

1

2

f4a

x Fig. 9.7 Standard conservative interpolation on a quadrilateral mesh

3

4

334

9 Computational Aeroacoustics

element of the acoustic grid it is located. Then, we compute from the global position (xk , yk ) its local position (ξk , ηk ) in the reference element. This is in the general case a nonlinear mapping and is solved by a Newton scheme. Now, with these data we can perform a bilinear interpolation and add the contribution of f kf to the nodes of the acoustic grid by using the standard finite element basis functions Ni f ia = Ni (ξk , ηk ) f kf .

(9.89)

For details concerning the algorithms for performing the location of the CFD nodes within the acoustic elements as well as the global to local mapping, which is in general a nonlinear mapping [37].

9.7 Comparison of Different Aeroacoustic Analogies As a demonstrative example to compare the different acoustic analogies, we choose a cylinder in a cross flow, as displayed in Fig. 9.8. Thereby, the computational grid is just up to the height of the cylinder and together with the boundary conditions (bottom and top as well as span-wise direction symmetry boundary condition), we obtain a pseudo two-dimensional flow field. The diameter of the cylinder D is 1 m resulting with the inflow velocity of 1 m/s and chosen viscosity in a Reynolds number of 250 and Mach number of 0.2. From the flow simulations, we obtain a shedding frequency of 0.2 Hz (Strouhal number of 0.2). The acoustic mesh is chosen different from the flow mesh, and resolves the wavelength of two times the shedding frequency with 10 finite elements of second order. At the outer boundary of the acoustic domain we add a perfectly matched layer to efficiently absorb the outgoing waves. For the acoustic field computation we use the following formulations: • Lighthills’ acoustic analogy with Lighthill’s tensor [L] according to (9.13) as source term • Lighthills’ acoustic analogy with the Laplacian of the incompressible flow pressure p ic as source term (see (9.14)) • the aeroacoustic wave equation (AWE) according to (9.63) • perturbation equations (PE) according to (9.57); for comparison, we set the mean flow velocity v¯ to zero. Figure 9.9 displays the acoustic field for the different formulations. One can clearly see that the acoustic field of PE (for comparison with the other formulations we have neglected the convective terms) meets very well the expected dipole structure and is free from dynamic flow disturbances. Furthermore, the acoustic field of AWE is quite similar and exhibits almost no dynamic flow disturbances. Both computations with Lighthill’s analogy show flow disturbances, whereby the formulation with the Laplacian of the incompressible flow pressure as source term shows qualitative better result as the classical formulation based on the incompressible flow velocities.

9.7 Comparison of Different Aeroacoustic Analogies

335

Fig. 9.8 Computational setup for flow computation

Fig. 9.9 Computed acoustic field with the different formulations

Lighthill: ∇ · ∇ · [L]

Lighthill: ∇ · ∇pic

AWE: 1/c20 ∂ 2 pic /∂t2

PE: ∂pic /∂t

To obtain a more detailed and quantitative analysis, we have saved the acoustic pressure along a circle of 20 times the diameter of the cylinder (outside the main flow region) and performed a Fourier transformation. Figure 9.10 displays the obtained results in a plot over the angle of the circle. We can observe very similar results for all formulations, although PE and AWE exhibits most clearly the expected dipole structure. Performing a further analysis at a circle with just two times the diameter of the cylinder results in the plots displayed in Fig. 9.11. The obtained results confirms the previous findings according to Fig. 9.9 and concludes that a perturbation ansatz is necessary to obtain reliable acoustic pressure results in the flow region. Lighthill’s acoustic analogy (both incompressible flow velocity or pressure based source terms)

336 Fig. 9.10 Fourier transformed acoustic pressure along a circle with a diameter of 20 times the diameter of the cylinder

9 Computational Aeroacoustics

Lighthill: ∇ · ∇pic

Lighthill: ∇ · ∇ · [L] (Hz)

(Hz)

Angle -60

SPL (dB) 0

60

AWE: 1/c20 ∂ 2 pic /∂t2

PE: ∂pic /∂t

Angle

Angle

Fig. 9.11 Fourier transformed acoustic pressure along a circle with a diameter of 2 times the diameter of the cylinder

Angle

Lighthill: ∇ · ∇pic

Lighthill: ∇ · ∇ · [L] (Hz)

(Hz)

Angle

Angle -60

SPL (dB) 0

AWE: 1/c20 ∂ 2 pic /∂t2

Angle

60

PE: ∂pic /∂t

Angle

models an overall fluctuating pressure in the flow region and contains frequency components, which are not present outside the flow region, where all formulations result in very similar acoustic pressure values.

References

337

References 1. C.K.W. Tam, Computational aeroacoustics. AIAA J. 33, 1788–1796 (1995) 2. J.C. Hardin, M.Y. Hussaini (eds.), Computational Aeroacoustics, ch. Computational Aeroacoustics for Low Mach Number Flows (Springer, New York, 1992), pp. 50–68 3. J.C. Hardin, M.Y. Hussaini (eds.), Computational Aeroacoustics, Ch. Regarding Numerical Considerations for Computational Aeroacoustics (Springer, New York, 1992), pp. 216–228 4. J.B. Freund, S.K. Lele, P. Moin, Direct numerical simulation of a Mach 1.92 turbulent jet and its sound field. AIAA J. 38, 2023–2031 (2000) 5. A. Uzun, A.S. Lyrintzis, G.A. Blaisdell, Coupling of Integral Acoustics Methods with LES for Jet Noise Prediction. Proceedings of AIAA Aerospace Sciences Meeting and Exhibit, no. 2004–0517, Reno, January 2004 6. P. Di Francescantonio, A new boundary integral formulation for the prediction of sound radiation. J. Sound Vib. 202(4), 491–509 (1997) 7. K. Brentner, F. Farassat, An analytical comparison of the acoustic analogy and Kirchhoff formulation for moving surfaces. AIAA J. 36(8), 1379–1386 (1998) 8. F. Farassat, Acoustic radiation from rotating blades—the Kirchhoff method in aeroacoustics. J. Sound Vib. 239(4), 785–800 (2001) 9. D. Lockard, J. Casper, Permeable Surface Corrections for Ffwocs Williams and Hawkings Integrals. Proceedings of 11th AIAA/CEAS Aeroacoustics Conference, Monterey, May 2005 10. A. Oberai, F. Ronaldkin, T. Hughes, Computational procedures for determining structuralacoustic response due to hydrodynamic sources. Comput. Methods Appl. Mech. Eng. 190, 345–361 (2000) 11. C. Bailly, D. Juvé, Numerical solution of acoustic propagation problems using linearized Euler equations, AIAA J. 38(1), 22–29 (2000) 12. M. Dumbser, C.-D. Munz, ADER discontinuous Galerkin schemes for aeroacoustics. Comptes Rendus Mecanique 333(9), 683–687 (2005) 13. R. Ewert, W. Schröder, Acoustic perturbation equations based on flow decomposition via source filtering. J. Comput. Phys. 188, 365–398 (2003) 14. C.D. Munz, M. Dumbser, S. Roller, Linearized acoustic perturbation equations for low Mach number flow with variable density and temperature. J. Comput. Phys. 224, 352–364 (2007) 15. A. Hüppe, M. Kaltenbacher, Spectral finite elements for computational aeroacoustics using acoustic pertrubation equations. J. Comput. Acoust. 20(2) (2012) 16. M. Kaltenbacher, M. Escobar, I. Ali, S. Becker, Numerical simulation of flow-induced noise using LES/SAS and Lighthill’s acoustics analogy. Int. J. Numer. Method Fluids 63(9), 1103– 1122 (2010) 17. J. Seo, Y. Moon, Linearized perturbed compressible equations for low Mach number aeroacoustics. J. Comput. Phys. 218(2), 702–719 (2006) 18. M. Escobar, Finite Element Simulation of Flow-Induced Noise Applying Lighthill’s Acoustic Analogy (Universität Erlangen-Nürnberg, Ph.D., 2007) 19. W. De Roeck, Hybride Methodologies for the computational aeroacoustic analysis of confined, subsonic flows, Ph.D. thesis, KTU Leuven (2007) 20. A. Hüppe, Spectral Finite Elements for Acoustic Field Computation, Ph.D. thesis, AlpenAdria-Universität Klagenfurt, Austria, (2013) 21. P. Croaker, A particle acccelerated hybrid CFD-BEM method for low Mach number flow induced noise, Ph.D. thesis, University of New South Wales, Australia, (2014) 22. A. Schell, Entwicklung einer Berechnungsmethode zur Vorhersage der Schallausbreitung im Nahfeld eines umströmten Kraftfahrzeugs, Ph.D. thesis, University of Stuttgart, (2014) 23. M. Breuer, in Direkte Numerische Simulation und Large-Eddy Simulation turbulenter Strömungen auf Hochleistungsrechnern (Shaker-Verlag, 2002) 24. E. Manoha, C. Herrero, P. Sagaut, S. Redonnet, in Numerical Prediction of Airfoil Aerodynamic Noise. Proceedings of 8th AIAA/CEAS Aeroacoustics Conference, Breckenridge, June 2002 25. M.J. Lighthill, On sound generated aerodynamically I. General theory. Proc. Roy. Soc. Lond. 211, 564–587 (1952)

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26. M.J. Lighthill, On sound generated aerodynamically II. Turbulence as a source of sound. Proc. Roy. Soc. Lond. 222, 1–22 (1954) 27. M.S. Howe, Theory of Vortex Sound, Cambridge Texts in Applied Mathematics (Cambridge University Press, New York, 2003) 28. N. Curle, The influence of solid boundaries upon aerodynamic sound. Proc. Roy. Soc. Lond. 231, 505–514 (1955) 29. A. Powell, Aerodynamic noise and the plate boundary. Acoust. Soc. Am. 32, 982–990 (1960) 30. X. Gloerfelt, F. Perot, C. Bailly, D. Juve, Flow-induced cylinder noise formulated as a diffraction problem for low Mach numbers. Sound Vib. 287, 129–151 (2005) 31. J.C. Hardin, D.S. Pope, An acoustic/viscous splitting technique for computational aeroacoustics. Theoret. Comput. Fluid Dyn. 6, 323–340 (1994) 32. W.Z. Shen, J.N. Sørensen, Aeroacoustic modelling of low-speed flows. Theoret. Comput. Fluid Dyn. 13, 271–289 (1999) 33. J.H. Seo, Y.J. Moon, Perturbed compressible equations for aeroacoustic noise prediction at low mach numbers. AIAA J. 43, 1716–1724 (2005) 34. Y. Morinishi, Skew-symmetric form of convective terms and fully conservative finite difference schemes for variable density low-mach number flows. J. Comput. Phys. 229, 276–300 (2010) 35. N. Castel, G. Cohen, M. Duruflé, Application of Discontinuous Galerkin spectral method on hexahedral elements for aeroacoustic. J. Comput. Acoust. 17(2), 175–196 (2009) 36. F. Brezzi, L.D. Marini, E. Suli, Discontinuous Galerkin methods for first-order hyperbolic problems. Math. Models Meth. Appl. Sci. 14, 1893–1903 (2004) 37. A. Düster, Lecture Notes: High order FEM, p. 132 (2005)

Chapter 10

Coupled Electrostatic-Mechanical Systems

In electrostatic-mechanical systems, the structure is subject both to rigid motions and elastic deformations by means of the electrostatic force. Thus, the change on the geometry in turn may strongly influence the electric field and thus the electrostatic force distribution. A typical application is a micro-machined pump [1], shown in Fig. 10.1. If an electric voltage is applied to the electrodes, the elastic pump diaphragm is deformed by the electrostatic force and bends towards the counter electrode. Thereby, fluid will be sucked in through the inlet valve. When the supply voltage is switched off, the relaxation of the diaphragm will push the fluid through the outlet valve.

10.1 Electrostatic Force An elegant method for deriving a general formula for the electrostatic force F el is obtained by applying the principle of virtual work (displacement). Thus, a virtual displacement δr of a body due to a force F el will result in a variation of the electrostatic energy δWel δWel = F el · δr. (10.1) The electrostatic energy density wel is computed by the scalar product of the electric field intensity E and the electric flux density D wel =

1 D · E. 2

(10.2)

By using the constitutive law D = εE (assuming constant ε), we obtain the following expression 1 ε wel = E · E = D · D. (10.3) 2 2ε © Springer-Verlag Berlin Heidelberg 2015 M. Kaltenbacher, Numerical Simulation of Mechatronic Sensors and Actuators, DOI 10.1007/978-3-642-40170-1_10

339

340

10 Coupled Electrostatic-Mechanical Systems

Counterelectrode

Pump diaphragm

Actuation unit

Valve unit

Inlet

Outlet

Fig. 10.1 Schematic view of an electrostatically driven micropump [1]

The total electrostatic energy Wel is computed by integration of the energy density wel over the volume Ω  Wel = wel dΩ. (10.4) Ω

Now, the variation of the electrostatic energy δWel computes as ⎛ ⎞  δWel = δ ⎝ wel dΩ ⎠ Ω

=

 

Ω

 D · δD D 2 δε dΩ, − ε 2ε2

(10.5)

with D = | D|. By using the scalar electric potential Ve (E = −∇Ve ), the first term in (10.5) can be rewritten as 

D · δD dΩ = − ε

Ω



(∇Ve · δ D) dΩ.

Ω

Furthermore, the vector identity ∇ · (Ve δ D) = Ve ∇ · δ D + δ D · ∇Ve and ∇ · δ D = δqe (variation of the free charges) lead to

(10.6)

10.1 Electrostatic Force

−



341

(∇Ve · δ D) dΩ =

Ω



(Ve δqe ) dΩ −

Ω



∇ · (Ve δ D) dΩ.

(10.7)

Ω

The use of the divergence theorem (see Appendix B.6) transforms the third integral in a surface integral over Γ (Ω) 

∇ · (Ve δ D) dΩ =

Ω



Ve δ D · dΓ .

(10.8)

Γ (Ω)

Now, we know that the electric potential Ve is proportional to 1/r (with r denoting the distance), and therefore, the electric flux density D is proportional to 1/r 2 [2]. Since the surface only can grow proportional to r 2 , the surface integral will tend to zero for increasing Ω. The variation of the volume charge δqe is equal to the scalar product of the virtual displacement δr and the gradient of qe δqe = δr · ∇qe . In addition, we can use the relation ∇(Ve qe ) = qe ∇Ve + Ve ∇qe in order to modify the second term in (10.7) 

Ve δqe dΩ = δr ·

Ω



(Ve ∇qe )dΩ

Ω

= δr ·

   ∇(Ve qe ) − qe ∇Ve dΩ

Ω

⎛

⎜ = δr · ⎝



Ve qe dΓ −



Ω

Γ (Ω)

⎞

⎟ qe ∇Ve dΩ ⎠ .

(10.9)

The surface integral in (10.9) will tend to zero for increasing volume Ω for the same reason as discussed above. Therefore, the differential energy δWel computes by using (10.5), (10.9), and the relation δε = δr · ∇ε 1 D2 δε dΩ 2 ε2 Ω Ω ⎛ ⎞    2 E qe E − = δr · ⎝ ∇ε dΩ ⎠ . 2

δWel = δr ·



qe EdΩ −

Ω



(10.10)

342

10 Coupled Electrostatic-Mechanical Systems

h n12

ε2 ε1

r2 ∆Ω

ε1

r1 ∆Γ

fΓel

∆Ω

ε2 ε2 < ε 1

Fig. 10.2 Interface between two materials with different permittivity

Comparing the above expression with (10.1), we obtain the equation for the total electrostatic force F el F el =

   E2 qe E − ∇ε dΩ , 2

(10.11)

Ω

and for the electrostatic volume force f Ωel f Ωel = qe E −

E2 ∇ε. 2

(10.12)

The first term models the electric force acting on electric volume charges qe in an electric field E. The second term exhibits a force at surfaces of changing permittivity, and therefore, we are looking for a surface force expression. Let us assume a small volume ∆Ω in form of a cylinder with height h, which is located between the two materials with permittivity ε1 and ε2 (see Fig. 10.2). Having no volume charge qe within ∆Ω, we can compute the electrostatic force ∆F as follows

∆F =



f Ωel dΩ = ∆Γ

f Ωel dr

r1

∆Ω

∆Γ =− 2

r2

r2 (E 2 ∇ε)dr.

(10.13)

r1

The scalar product of the force ∆F with the normal vector n and the relation dε = dr · ∇ε = n · dr ∇ε, lead to

10.1 Electrostatic Force

343

∆Γ n · ∆F = − 2

∆F12 = −

∆Γ 2

r2

r1 ε2

E2

dε dr dr

E 2 dε.

(10.14)

ε1

Now, we know that at an interface of changing permittivity the normal component of electric flux density and the tangential component of electric field intensity are continuous (see Sect. 6.5.2). With the help of the normal vector n and the tangential vector t, we can decompose E and D as follows E = (E · t) t + (E · n) n = E t + E n D = ( D · t) t + ( D · n) n = Dt + Dn . These relations allow us to write the term E 2 = E · E in the form     D2 Dn Dn + Et · + E t = 2n + E t2 . E·E= ε ε ε The above result in combination with (10.14) leads to the term for the surface force density f Γ el in the direction of n (∆F 12 → d F 12 , ∆Γ → dΓ )

f Γ el

⎞ ⎛ ε2 ε2 1 ⎝ 2 dε dF12 =− = + E t2 dε⎠ Dn dΓ 2 ε2 ε1

f Γ el

ε1

  D2 1 1 E2 = n − + t (ε1 − ε2 ) 2 ε2 ε1 2 = f Γ el n.

(10.15)

Now, within the FE computation we can evaluate (10.15) to obtain at each FE node on the boundary the corresponding electrostatic force. However, the more accurate and general approach is to use the principle of virtual work and apply it to the FE formulation. According to (10.1), we can calculate the force in a direction r by δ Fr = δr



1 E · D dΩ. 2

(10.16)

Ω

Let us consider the setup as displayed in Fig. 10.3, where a virtual displacement of the micro-mechanical cantilever will just lead to a deformation of the surrounding finite elements in the air gap and thus in a change of the electrostatic energy just within these elements. Therewith, the force Fr can be expressed by

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10 Coupled Electrostatic-Mechanical Systems

Top electrode

Cantilever Node i

Bottom electrode Fig. 10.3 Micro-mechanical cantilever with FE discretization; nodes with a point may move during a virtual deformation and nodes with a cross are fixed

  ne  ǫ0 δ E · E dΩ δr Ωe 2 e=1   n e  δ(dΩ) ǫ0 δE . dΩ + E·E = ǫ0 E δr δr Ωe 2 Ωe

Fr =

(10.17)

e=1

Here the first term takes account of the changing electric field within the domain, whereas the second term deals with the geometric distortion of the element. Now, since in the coupling to the mechanical field, we are interested in the local force, we may rewrite (10.17) for the total force acting on a node i located at the surface of the electromechanical cantilever i

Fri

=

n e  e=1

δE dΩ + ǫ0 E δr Ωe



Ωe

δ(dΩ) ǫ0 E·E 2 δr



.

(10.18)

In (10.18) n ie is the number of finite elements to which node i belongs (see Fig. 10.3). In the next step, we will now transform the integration over the global space of the finite elements to an integration over the reference elements. Thus, we substitute dΩ by | J|dξdηdζ with | J| the determinant of the Jacobi matrix J i

Fri

=

n e  e=1

δE |J|dξdηdζ + ǫ0 E · δr Ωe



Ωe

 δ|J| ǫ0 E·E dξdηdζ . 2 δr

(10.19)

After the solution process of the electrostatic field, we obtain in all FE nodes the electric potential Ve , form which we can compute the electric field intensity

10.1 Electrostatic Force

345

E = −∇Ve = −

n en

∇ Na (x)Vea

(10.20)

a=1

with n en the number of element nodes per finite element and Na the shape function at node a. Performing the computation at the reference element, we may rewrite (10.20) by (see Sect. 2.3.8) E = −(J T )−1

n en

∇ Na (ξ)Vea .

(10.21)

a=1

According to (10.19), we have to evaluate the variation of E with respect to the direction r , which computes by using (10.21) n en δE δ(J T )−1 =− ∇ Na (ξ)Vea . δr δr

(10.22)

a=1

Applying the variation to (J T )−1 will lead to quite complex terms. In order to arrive at a more convenient expression, we utilize the following relation (I denotes the unit matrix) δJT −1 T δI δ(J−1 )T δ(J−1 · J) = (J ) + JT = =0 δr δr δr δr δ(J−1 )T δ(J)T −1 T = −(J−1 )T (J ) . δr δr Therewith, the variation of the electric field intensity E in the direction r can be computed from the FE results as follows δE δ(J)T = −(J−1 )T E. δr δr

(10.23)

Substituting this result into (10.19) allows us to directly compute the force acting on a node i by i

Fri

=

n e  e=1

Ωe

δ|J| ǫ0 E·E dξdηdζ − 2 δr



ǫ0 E (J Ωe

 δJT E |J| dξdηdζ . ) δr

−1 T

For any integration point, the transposed Jacobian matrix computes as

(10.24)

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10 Coupled Electrostatic-Mechanical Systems

JT

n en

n en

n en

⎞ N z a,ξ a ⎟ ⎞ ⎜ a=1 ⎛ a=1 a=1 ⎟ ⎜n xξ xη xζ n n en en en ⎟ ⎜

⎜ ⎠ ⎝ N z N y N x = yξ yη yζ = ⎜ a,η a ⎟ a,η a a,η a ⎟, a=1 a=1 ⎟ ⎜ a=1 zξ zη zζ n n n en en en ⎠ ⎝

Na,ζ z a Na,ζ ya Na,ζ xa ⎛

Na,ξ xa

Na,ξ ya

a=1

a=1

a=1

and its variation with respect to δr as ⎛ n en n en n en

δ ya δxa N N Na,ξ δzδra a,ξ a,ξ δr δr ⎜ a=1 a=1 a=1 ⎜n n en n en en ⎜

δJ T a Na,η δx Na,η δδrya Na,η δzδra =⎜ δr ⎜ δr a=1 a=1 a=1 ⎜ n en n en n en ⎝

a Na,ζ δδrya Na,ζ δzδra Na,ζ δx δr a=1

a=1

a=1

⎞

⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎠

The term δxa /δr (δ ya /δr , δz a /δr ) obtains the value 1, if and only if node a is equal with node i, which belongs to the surface and is virtually moved, and the variation is in the x-direction (y-direction, z-direction). In all other cases, we apply 0 to those terms. The last term in (10.24) to be evaluated, is the variation of the Jacobian determinant, which computes as follows δxξ δxζ δxη δ|J| = (yη z ζ − z η yζ ) + (yξ z ζ − z ξ yζ ) + (yξ z η − z ξ yη ) δr δr δr δr δ yξ δ yζ δ yη (xη z ζ − z η xζ ) + (xξ z ζ − z ξ xζ ) + (xξ z η − z ξ xη ) + δr δr δr δz ξ δz ζ δz η + (xη yζ − yη xζ ) + (xξ yζ − yξ xζ ) + (xξ yη − yξ xη ). δr δr δr

10.2 Numerical Computation The challenges in the simulation of electrostatic-mechanical transducers can be summarized as follows. First, one has to deal with at least two different physical fields, usually the electrostatic and mechanical field. In many cases the transducer is immersed in an acoustic fluid (e.g., in ultrasound applications), and therefore, the acoustic wave propagation also has to be computed. In addition, all nonlinearities of the involved physical fields have to be considered (e.g., geometric nonlinearity within the mechanical field due to large displacements and/or strains). Furthermore, the coupling mechanisms between the physical fields have to be taken into account, which are highly nonlinear in nature. Therefore, even in the case when all single fields are governed by linear partial differential equations, the coupled system is nonlinear and has to be solved by an iterative method.

10.2 Numerical Computation

347

10.2.1 Calculation Scheme The FE formulation of the electrostatic field and the mechanical field has already been derived in Sect. 6.6 as well as Sect. 3.7. Using predictor values for the electric potential in order to compute the electrostatic force and predictor values for the mechanical displacement to update the configuration for the computation of the electrostatic field, we can split the coupled system of partial differential equations into a mechanical part and an electric part. To guarantee a full coupling between the two fields, we perform an iterative solution process. The resultant decoupled discrete matrix system of equations reads as follows 

M ∗u

0

0 K Ve (u kn+1 )



u k+1 n+1 Ve k+1 n+1



⎛

=⎝

f u (Ve kn+1 ) f V (u kn+1 ) e

⎞

⎠.

(10.25)

In (10.25) M ∗u denotes the effective mechanical mass matrix, f u (Ve ) the nodal vector of external mechanical and electrostatic forces, u the nodal vector of mechanical displacement, f V (u) the electric source vector and Ve the nodal vector of scalar e electric potential, n the time step, and k the iteration counter within each time step. Since the system matrix K Ve and the nodal vector f V are updated by the mechanical e

displacement u kn+1 , we use a moving-mesh technique to avoid large mesh deformations in the FE grid (see Sect. 7.3). Therefore, the entries of the system matrix K Ve change as functions of u kn+1 . The solution algorithm of the fully coupled system of equations is displayed in Fig. 10.4. The outer iteration loop controls the iterative coupling process between the mechanical and electrostatic equation, which is performed in the simplest case by a fixed-point method. The convergence test is based on the following stopping criterion k ||u k+1 n+1 − u n+1 ||2 < δo , (10.26) ||u k+1 n+1 ||2 with u the nodal vector of mechanical displacements, δo an adjustable accuracy, || ||2 the L2 -norm, k the iteration counter for the outer loop (electrostatic-mechanical iteration) and n the time step number. Due to a movement and/or deformation of the mechanical structures in electrostatic devices, the finite elements computing the electrostatic field will be deformed (see Fig. 10.5 for a simple voltage-driven bar), and thus will lead to numerical inaccuracy. These deformations have to be controlled, and, before an intersection of finite elements occurs (see Fig. 10.5), a remeshing of the simulation domain has to be performed. To avoid this problem, we use a special kind of moving-mesh technique. Therefore, the finite elements for the electric field are modified in order to be able to handle mechanical degrees of freedom, too. Of course, the mechanical stiffness of these elements has to be chosen very small compared to the stiffness of the bar. Now, the deformation of the bar is not only acting on the first layer of finite elements

348

10 Coupled Electrostatic-Mechanical Systems

(b) (a)

Fig. 10.4 Coupled electrostatic-mechanical simulation schemes. a Linear mechanics. b Nonlinear mechanics

Fig. 10.5 Intersection of finite elements by using the standard method

around the bar, but on all finite elements between the bar and the counter electrode (see Fig. 10.6). Since the deformations of the mechanical structures especially in micro-machined capacitive transducers are often large compared to the geometric dimensions, a nonlinear formulation for the mechanical field has to be utilized. This leads to the

10.2 Numerical Computation

349

Fig. 10.6 No intersection of finite elements by using moving-mesh technique

calculation scheme as shown in Fig. 10.4b. An inner iteration loop is introduced for the computation of the mechanical field. The stopping criterion for this loop should be based on an error and on a residual principle j+1

j

||u n+1 − u n+1 ||2 j+1

||u n+1 ||2

j+1

< δia

||r n+1 ||2 || f u,n+1 ||2

< δir .

(10.27)

In (10.27) r denotes the nodal residual vector of the mechanical system, f V the nodal e vector of external mechanical and electrostatic forces, δia as well as δir adjustable accuracy values and j the iteration counter for the inner (mechanical) iteration loop.

10.2.2 Voltage-Driven Bar To investigate the behavior of electrostatic transducers, we will perform simulations of a voltage-loaded bar. Figure 10.7 displays the setup of a two-sided clamped bar (for symmetry reasons just half of the bar is shown). The goal is to compute the deflection of the bar as a function of the applied voltage till snap-in occurs. To account for typical geometries of micro-machined electrostatic transducers, we set the thickness of the bar to 1 µm, the length to 100 µm and the air-gap distance to 3 µm.

y x

Symmetry

Fixed Lower electrode Fig. 10.7 Setup of the voltage-loaded bar

Upper electrode

350

10 Coupled Electrostatic-Mechanical Systems

Fig. 10.8 Tip displacement as a function of the applied electric voltage (linear mechanics with dashed line and nonlinear mechanics with solid line)

For the discretization, quadrilateral elements with quadratic interpolation functions have been used. The thickness of the electrodes has been neglected. Therefore, all nodes belonging to the upper electrode are assigned inhomogeneous Dirichlet boundary conditions with a value according to the applied voltage. Furthermore, homogeneous Dirichlet boundary conditions are applied to all nodes belonging to the lower electrode. At the left end of the bar u x as well u y are set to zero to account for the full support and u x is set to zero at the right end of the bar to account for the symmetry. In the first step, we perform the simulation of the coupled electrostatic-mechanical system with linear mechanics. The obtained results are displayed in Fig. 10.8. For this setup, it can be clearly seen that the mechanical field has to be computed by the nonlinear formulation. Due to the strong deflection of the bar, inner stresses arise, which increase the stiffness of the bar. In a second computation, we have also modeled a prestressing of the bar. Due to the fabrication of micro-electromechanical systems (MEMS), inner stresses will remain within the structure after the fabrication process. We have assumed a value of 10 MPa, which is a typical value for such structures, and performed a nonlinear analysis for the mechanical field. As displayed in Fig. 10.9, the influence of the prestressing is quite strong. Now the voltage can reach values of about 10 V until snap-in occurs.

References

351 0 −0.2

Tip displacement (µm)

−0.4 −0.6 −0.8 −1 −1.2 −1.4 −1.6 −1.8 0

2

4

6 Voltage (V)

8

10

12

Fig. 10.9 Tip displacement as a function of the applied electric voltage using a nonlinear mechanical formulation with (dashed line) and without (solid line) prestressing

References 1. R. Zengerle, S. Kluge, M. Richter, R. Richter, A bidirectional silicon micropump, Proceedings of Sensor95 (Nuremberg) (1995), pp. 727–732 2. G. Wunsch, Feldtheorie, 2 (Verlag Technik, Berlin, 1975)

Chapter 11

Coupled Magnetomechanical Systems

Let us consider a magnetic solenoid valve as depicted in Fig. 11.1. The valve consists of a ferromagnetic pot with a copper coil and a ferromagnetic armature that is held in its position by a compression spring. If the copper coil is loaded by a current pulse, a magnetic force on the armature is generated that causes a motion of the armature, when its amplitude is larger than the spring force. Due to the time-varying magnetic field, eddy currents are generated in the magnet pot as well as armature. As a consequence, a time delay between the applied current and the magnetic force will occur. Milling cuts into the magnet pot will reduce these eddy currents, and therefore will lead to a shorter switching time.

11.1 General Moving/Deforming Body Due to the movement of an electric conductive body in a magnetic field, the so-called motional emf term (electromotive force term) γv × B,

(11.1)

with γ the electric conductivity and v the velocity of the body has to be taken into account. The magnetic induction B can be expressed by the magnetic vector potential via B = ∇ × A resulting in the expression for the eddy current density J v induced in an electrically conductive body J v = γv × B = γv × ∇ × A.

(11.2)

The total partial differential equation in the quasistatic case including moving bodies reads as ∂A + γv × ∇ × A. (11.3) ∇ × ν∇ × A = J i − γ ∂t © Springer-Verlag Berlin Heidelberg 2015 M. Kaltenbacher, Numerical Simulation of Mechatronic Sensors and Actuators, DOI 10.1007/978-3-642-40170-1_11

353

354

11 Coupled Magnetomechanical Systems

Magnetic pot

Coil Armature Shaft Fig. 11.1 Principal setup of a magnetic solenoid valve

If the velocity is a priori known, the additional term remains linear, but will lead to a so-called convective term. Therefore, the numerical computation will need some upwind technique for stability reasons (see e.g., [1, 2]). A typical example, which can be modeled by (11.3), is the investigation of an eddy current sensor as displayed in Fig. 11.2, where the material under test can be assumed to be infinitely wide and its velocity to be known. However, if the magnetic field significantly changes due to moving and/or deforming parts (e.g., moving parts within a magnetic valve or current/voltage-loaded coils), we have to apply an updated Lagrangian formulation of the electromagnetic field equations [3]. Let us consider a general point P of the moving body at time t and

Ferrite core

Exciting coil

Sensing coil 2

Sensing coil1

Material under test Fig. 11.2 Eddy current sensor used for NDE (nondestructive evaluation) Fig. 11.3 Moving point P at time t and t + ∆t

P (x + ∆x, t + ∆t)

P (x, t) z x y

11.1 General Moving/Deforming Body

355

t + ∆t as shown in Fig. 11.3. Now, we can express the change of the magnetic vector potential ∆ A within ∆t according to the fixed coordinate system (x, y, z) ∆A A(x + ∆x, t + ∆t) − A(x, t) = . ∆t ∆t

(11.4)

By assuming small ∆x and ∆t, we can express the term A(x + ∆x, t + ∆t) by a Taylor series up to the linear term ∂A ∆t ∂t ∂ A ∂x ∂ A ∂y ∂ A ∂z + ∆t + ∆t + ∆t ∂x ∂t ∂ y ∂t ∂z ∂t ∂A ∆t + (v · ∇) A ∆t, = A(x, t) + ∂t

A(x + ∆x, t + ∆t) ≈ A(x, t) +

(11.5)

with v = (∂x/∂t, ∂ y/∂t, ∂z/∂t)T . Now by letting ∆t → 0, we obtain from (11.4) using (11.5) ∂A dA = + (v · ∇) A. (11.6) dt ∂t On the other hand, we can rewrite the term γv × ∇ × A as follows γv × ∇ × A = γ∇ A (v · A) − γ(v · ∇) A,

(11.7)

where ∇ A denotes the operation of the nabla operator just on the magnetic vector potential A. By comparing (11.6) and (11.7), we can rewrite (11.3) for the updated Lagrangian formulation ∇ × ν∇ × A = J i − γ

dA + γ∇ A (v · A). dt

(11.8)

It has to be noted that the term γ∇ A (v · A) in (11.8) will always be zero in the 2D plane as well as axisymmetric case, since the vectors v and A are orthogonal. However, in the general 3D case, one has to consider this term.

11.2 Electromagnetic Force Applying the principle of virtual work (displacement) as used for deriving the electrostatic force will lead to a general expression for the magnetic force F mag . The variation of the magnetic energy δWmag relates the virtual displacement δr and magnetic force F mag as follows δWmag = δr · F mag .

(11.9)

356

11 Coupled Magnetomechanical Systems

Using the two magnetic quantities B and H, we can compute the electromagnetic energy density wmag by 1 wmag = B · H. (11.10) 2 Assuming an isotropic body with magnetic permeability µ we obtain for (11.10) wmag =

H·H B·B =µ . 2µ 2

Performing an integration over the whole volume Ω gives us the total magnetic energy Wmag , stored within the body Wmag =

1 2





1 2

B · H dΩ =

Ω

B·B dΩ. µ

(11.11)

Ω

Thereby, the variation of the magnetic energy leads to δWmag =

 

B · δB µ

Ω



dΩ −

1 2

 

B·B δµ µ2

Ω



dΩ.

(11.12)

Since δµ can be expressed by δµ = δr · ∇µ we obtain for the second term in (11.12) 1 2

 

B·B δµ µ2

Ω



1 dΩ = δr · 2

 

Ω

1 = δr · 2



 B·B ∇µ dΩ µ2

(H · H ∇µ) dΩ.

(11.13)

Ω

According to [4], we may rewrite the first term in (11.12) as  

B · δB µ

Ω



⎛

dΩ = δr · ⎝



Ω

⎞

( J × B) dΩ ⎠ .

This result, in combination with (11.13) and (11.9), leads to the total electromagnetic force F mag F mag =



Ω

1 ( J × B) dΩ − 2



Ω

(H · H ∇µ) dΩ,

(11.14)

11.2 Electromagnetic Force

357

and the expression for the magnetic volume force f mag f mag = J × B −

1 H · H ∇µ. 2

(11.15)

The first term describes a magnetic force that arises due to the current in a conductor (can also be a conductive body carrying eddy currents) placed in a magnetic field. The magnetic force described by the second term acts on all those interfaces where the magnetic permeability µ changes. Analogously to the electrostatic case, we can rewrite the second term in (11.15) in the form of a surface force density f Γmag acting at the interface between two materials with permeability µ1 and µ2 f Γmag

1 = 2



Bn2



1 1 − µ2 µ1



+

Ht2 (µ1

− µ2 )



n.

(11.16)

11.3 Numerical Computation The precise numerical simulation of magnetomechanical systems involves the computation of the magnetic field, the mechanical field and the nonlinear coupling terms. Since, in the general case, the single fields themselves are described by nonlinear partial differential equations (magnetization curve in the magnetic field and geometric nonlinearity in the mechanical field), the solution of large-scale problems demands for a very efficient calculation scheme.

11.3.1 Force Computation Via the Principle of Virtual Work In this section we will demonstrate the use of the principle of virtual work for the local force calculation within the FE method [5]. The magnetic force is obtained by the derivative of the magnetic energy while keeping the magnetic flux constant. Thus, the force in the direction of δr computes as δ Fr = − δr



Ω

⎛ ⎝

B 0

⎞

H · d B ⎠ dΩ.

(11.17)

The minus sign in (11.17) is explained as follows. We assume that the variation δr and the force Fr point in the same direction. The magnetic force always moves/deforms a body in such a way that the magnetic energy becomes a minimum. Therefore, the variation of the energy will be negative, and we have to apply a minus in (11.17).

358

11 Coupled Magnetomechanical Systems

Fig. 11.4 Movable body with discretization including the layer of virtually distorted finite elements

Moving body

FE - layer around moving body Moving / deforming node Fixed node

Now let us assume an FE mesh of the simulation domain as displayed in Fig. 11.4. The total force acting on the movable/deformable body computes as a sum over all finite elements n e surrounding the body ⎞ ⎛ B   ne δ ⎝ H · d B ⎠ dΩ. Fr = − δr e=1

Ωe

(11.18)

0

Since the magnetic permeability within the virtually distorted elements is that of air, we may write B

H · d B dΩ =

0

B·B . 2µ0

The variation now leads to ⎞ ⎛   ne B · B δ( dΩ) ⎟ ⎜ B δB Fr = − dΩ + · ⎠. ⎝ µ0 δr 2µ0 δr e=1

Ωe

(11.19)

Ωe

The local force Fri , which will act at the node i located on the surface of the body (see Fig. 11.4), computes as a sum over all elements surrounding the body and containing node i ⎞ ⎛ n ie   B · B δ( dΩ) ⎟ ⎜ B δB (11.20) dΩ + · Fri = − ⎠. ⎝ µ0 δr 2µ0 δr e=1

Ωe

Ωe

11.3 Numerical Computation

359

First, we will change the integration over the global elements to an integration over the reference elements, which means that we have to substitute dΩ by | J| dξ dη dζ with | J| the determinant of the Jacobi matrix J . Applying this substitution to (11.20), we arrive at ⎛ ⎞ n ie   B · B δ|J | ⎜ B δB ⎟ Fri = − · |J | dξ dη dζ + dξ dη dζ ⎠ . (11.21) ⎝ µ0 δr 2µ0 δr e=1

Ωe

Ωe

Since we will evaluate the integral numerically, we need the expression for δ B/δr for each integration point. After the computation of the magnetic vector potential Ah , we can compute B as follows B=

nd n en

h )= ∇ × (Na Aai

i=1 a=1

n en

∇ Na × Aah ,

(11.22)

a=1

with n en the number of nodes per finite element, Na the shape function at node a and Aah the calculated vector potential at node a (all three components). Performing the computation at the reference element, (11.22) changes to B=

n en (J T )−1 ∇ξ Na × Aah , a=1

with ∇ξ = (∂/∂ξ ∂/∂η ∂/∂ζ)T , and the variation of B leads to n en δ(J T )−1 δB = ∇ξ Na × Aah . δr δr

(11.23)

a=1

Since the variation of the term δ((J T )−1 J T )/δr is zero, we obtain δ(J T )−1 T δJ T J + (J T )−1 =0 δr δr δ(J T )−1 δJ T = −(J T )−1 (J T )−1 . δr δr Using this relation, we may write (11.23) as follows δB δJ T = −(J T )−1 B, δr δr

(11.24)

360

11 Coupled Magnetomechanical Systems

which changes (11.21) to ⎞ ⎛ n ie   T T B · B δ|J | δJ ⎟ ⎜ B Fri = B |J | dξ dη dζ − dξ dη dζ ⎠ . · (J T )−1 ⎝ µ0 δr 2µ0 δr e=1

Ωe

Ωe

(11.25)

The computation of ∂δJ T /δr and δ|J |T /δr is analogous to the electrostatic case (see Sect. 10.1).

11.3.2 Grid Adaption Techniques The computation of the magnetic field by applying the FE method requires the discretization of the whole simulation domain. Therefore, in an updated Lagrangian formulation any moving/deforming parts cause mesh distortion of the surrounding finite elements. In addition, if the displacements exceed the element size, a new mesh of the simulation domain is required (see also Sect. 7.3). In recent years, several approaches have been proposed to overcome this problem special developed for coupled magnetomechanical system: 1. Coupled Finite-Element–Boundary-Element Method [6–10]: Applying a finite-element–boundary-element method, where the solids are discretized by finite elements and the magnetic field in the surrounding air with boundary elements, results in a total separation of stationary and moving parts (see Fig. 11.5). This method requires no remeshing of the domain, as long as no contacts between the different parts occur. Since only the solid parts have to be discretized, the preprocessing, especially in 3D, is easier and open-domain problems can be computed exactly. Due to the updating of the boundary element

Fig. 11.5 FE–BE discretization of stationary and moving part as well as of the surrounding air

Body in motion

Boundary elements

Finite elements

Body at rest

11.3 Numerical Computation

361

matrices according to the mechanical displacements, the motional emf term is implicitly taken into account. However, the main disadvantage of this method is the large amount of memory and CPU time needed by the BEM. 2. Motional EMF-Term Method [11–15]: Since only nonmagnetic parts move within the electrodynamic loudspeaker, the motional emf term γ(v × B) (11.26) fully describes the movement of parts with conductivity γ and velocity v in a magnetic field with flux density B. Therefore, no remeshing of the calculation domain has to be performed. It has to be pointed out that the velocity of a moving part is in general not constant, but changes from time step to time step resulting in a new setup of the system matrices. Furthermore, the motional emf term leads to an unsymmetric system matrix and, depending on the conductivity, the velocity as well as the mesh size, an upwind technique has to be utilized. In addition, the change of the overall magnetic field due to the different positions of the moving part is not taken into account. 3. Coupling via Lagrange multiplier [2, 16–18]: This technique can be considered as a general method for coupling independent meshes. As displayed in Fig. 11.6 for the example of a rotating machine, the rotormesh and stator-mesh have a common interface Γc within the air gap. The two meshes are coupled by forcing the normal component of the magnetic induction [B · n] and the tangential component of the magnetic field intensity [H × n] at Γc of the rotor as well as stator to be equal. A common method for solving this problem is a Lagrange multiplier formulation. Thus, the velocity term is fully included, and no remeshing is necessary. The main disadvantage of this method is that the condition of the system matrix becomes poor. Furthermore, this method is restricted to problem cases in which the common interface Γc does not change its geometry, as is the case in rotary and translational electric machines. For such

Fig. 11.6 Common interface in an air gap of a rotary electric machine

Stator

Air gap

Γc

Rotor

362

11 Coupled Magnetomechanical Systems

cases, the Mortar FEM or Nitsche-type mortaring as described in Sect. 2.10 is beneficial. To demonstrate the capabilities of these two non-conforming methods for rotating systems, we consider a gear-wheel sensor as displayed in Fig. 11.7, consisting of a permanent magnet, a coil, and the gear wheel. Since the wheel is rotating and the sensor is fixed, we just have to perform a fine grid in the vicinity of the sensor. As can be seen in Fig. 11.7, the other regions have a quite coarse mesh. Here, we have to treat curved interfaces, when applying the methods for nonconforming meshes. Details of the mesh are displayed in Fig. 11.8. In a first investigation, we have filled the air gaps of the gear wheel also with iron, so that instead of the gear-wheel we have a iron cylinder. When rotating the iron cylinder, we should obtain no changes in the magnetic flux detected by the sensor. Thereby, any change in the computed flux has to be caused by the numerical scheme. Applying the classical Mortar as well as Nitsche type mortaring and computing the flux for a rotation of 60 rpm, we obtain a constant flux for all positions of the iron sphere. This clearly demonstrates the robustness of both methods and our implementation of the intersection operations on curved interfaces. In a second investigation, we rotate the gear-wheel at 60 rpm, apply both nonconforming formulations and compute the magnetic fields. Figure 11.9 displays the magnetic flux lines at a certain position of the gear-wheel w.r.t. the sensor. One can observe smooth flux lines over the non-conforming interface without any perturbations. Furthermore, the computed flux is displayed in Fig. 11.10 and shows no numerical disturbances. This example strongly demonstrates the advantages

Fig. 11.7 Computational setup and mesh of the gear-wheel with sensor

Fig. 11.8 Detail of non-conforming mesh

Permanent magnet Coil Gear wheel

Permanent magnet Gear wheel

Non-conforming interface

11.3 Numerical Computation

363

Fig. 11.9 Magnetic flux lines

Fig. 11.10 Magnetic flux change detected by the measurement coil

of the non-conforming methods. One can perform the geometric modeling and meshing individually for all subdomains and then glue them together by classical Mortar or Nitsche type mortaring. Furthermore, in case of rotating systems, e.g., in electrical drives, gear-wheel sensors, etc. we do not need any remeshing, which mostly introduces strong numerical errors. 4. Moving-Mesh Method [16, 19–24]: For this method, the whole volume (moving as well as stationary parts) is discretized by finite elements. If any movements occur, the coordinates of the finite elements, discretizing the moving parts, are updated in such a form that the distortions of the elements remain small. This can be achieved by also solving the mechanical equation within the air gap (of course with a very low mechanical stiffness and no mass) together with that of the moving parts. Therefore, the structure of system matrices will remain and the motional emf term is implicitly

364

11 Coupled Magnetomechanical Systems

z

z

z

z

∆z

Moving part Surrounding air 0

1 f (z)|t

r

0

1 f (z)|t+∆t

r

Fig. 11.11 Moving-material method: configuration at time step t and t + ∆t. The function f (z) defines a spatial weighting function for the position of the moving part

taken into account. However, the number of unknowns within the mechanical equation increases, the entries of the magnetic system matrix change and, if the movement is too large a remeshing has to be performed. 5. Moving-Material Method [25, 26]: The whole simulation domain is discretized by finite elements and will be unchanged throughout the simulation. Any movements of parts within the simulation domain are considered by changing the material values within each finite element according to the motion (see Fig. 11.11). Thus, the structure of the finite element matrices will not change and no remeshing is required. However, the entries of the system matrices change, and the motional emf term is not implicitly taken into account.

11.3.3 Calculation Scheme Applying the FE method to both partial differential equations, we achieve the following semi-discrete Galerkin formulation of the problem M u u¨ n+1 + C u u˙ n+1 + K u (u n+1 )u n+1 − f u (An+1 , A˙ n+1 ) = 0

(11.27)

M A (u n+1 ) A˙ n+1 + K A (An+1 , u n+1 )An+1 − f A,n+1 (An+1 , u n+1 , u˙ n+1 ) = 0.

(11.28)

In (11.27) and (11.28) M u denotes the mechanical mass matrix, C u the mechanical damping matrix, K u the mechanical stiffness matrix, f u the mechanical force vector, u n+1 the nodal displacements, M A the magnetic mass matrix, K A the magnetic stiffness matrix, An+1 the nodal magnetic vector potentials, and f A the nodal magnetic source vector.

11.3 Numerical Computation

365

For the time discretization, we apply the Newmark scheme for the mechanical equation and the general trapezoidal scheme for the magnetic equation (see Sect. 2.5). Using predictor values for the magnetic vector potential to calculate f u and predictor values for the mechanical displacement to compute M A , K A and f A a decoupling into a mechanical and a magnetic matrix equation is achieved. To still ensure a strong coupling between the magnetic and mechanical quantities, we have to solve these equations within each time step iteratively (see Fig. 11.12). The outer iteration loop controls the iterative coupling process between the magnetic and mechanical equation, which is performed in the simplest case by a fixed-point method. The convergence test is based on the following displacement stopping criterion k+1 − u kn+1 ||2 ||u n+1

< δo ,

(11.29)

(b)

(a)

Start

Boundary conditions

Start

Geometry

Magnetic vector potential

Boundary conditions

Geometry

No

Iteration i

k+1 ||2 ||u n+1

Converged? Magnetic vector potential

Yes

Magnetics

Mechanical deformation

Linear Mechanic

Displacement changed?

No

Solution

Yes

Nonlinear Magnetics

Mechanical deformation

Converged?

Yes

No

Iteration k

Coupling forces

Iteration j

Magnetic field

Iteration k

Coupling forces

Nonlinear Mechanics

Displacement changed?

Yes

No

Solution

Fig. 11.12 Coupled magnetomechanical simulation. a Linear magnetics and mechanics. b Nonlinear magnetics and mechanics

366

11 Coupled Magnetomechanical Systems

with u the nodal vector of mechanical displacements, δo an adjustable accuracy, || ||2 the L2 -norm, k the iteration counter for the outer loop (magnetomechanical iteration) and n the time step number. When we have to consider a nonlinear magnetization curve, we have an additional inner loop for the magnetic equation denoted by the iteration counter i, which has to fulfill the following convergence criteria i ||Ai+1 n+1 − An+1 ||2

||Ai+1 n+1 ||2

mag

< δia

||r i+1 n+1 ||2 || f A,n+1 ||2

mag

< δir .

(11.30)

In (11.30) r i+1 n+1 denotes the nodal residual vector of the magnetic equation, i the mag mag iteration counter for the inner (magnetic) iteration loop, and δia as well as δir adjustable accuracy values. The first convergence criterion in (11.30) is based on i the magnitude of the magnetic vector potential increments ∆A = Ai+1 n+1 − An+1 , the second criterion on the magnitude of the residual vector r (right-hand side criterion). By applying also a nonlinear formulation for the mechanical field, we need an additional inner loop for the mechanical equation. This loop will be denoted by the iteration counter j and has to fulfill the following stopping criteria j+1

j

||u n+1 − u n+1 ||2 j+1

||u n+1 ||2

j+1


Ec

E =0

Fig. 12.4 Orientation of the polarization of the unit cells: a initial state; b at strong external electric field; c after switching off the external field

Fig. 12.5 Ferroelectric hysteresis: polarization P as a function of the electric field intensity E

P 2

Psat 3

4 −Ec

1

7 Ec

E

6 5

−Psat

cells such that the spontaneous polarization will be more or less oriented towards the direction of the externally applied electric field as displayed in Fig. 12.4. Now, when we switch off the external electric field the ceramic will still exhibit a nonvanishing residual polarization in the macroscopic mean (see Fig. 12.4). We call this the irreversible or remnant polarization and the described process is termed poling. Now let us consider a mechanically unclamped piezoceramic disc at virgin state and load the electrodes by an increasing electric voltage. Initially, the orientation of the polarization within the unit cells is randomly distributed as shown in Fig. 12.5 (state 1). The switching of the domains starts when the externally applied electric field reaches the so-called coercive intensity E c .1 At this state, the increase of the polarization is much faster, until all domains are switched (see state 2 in Fig. 12.5). A further increase of the external electric loading results in an increase of the polarization with a quite smaller slope and the occurring micro-mechanical process remains reversible. Reducing the external applied voltage to zero will preserve the poled 1

It has to be noted that in literature E c often denotes the electric field intensity at zero polarization. According to [3] we define E c as the electric field intensity at which domain switching occurs.

12.3 Piezoelectric Material Properties

381

domain structure, and we call the resulting macroscopic polarization the remnant polarization Pr . If the previous excitation has aligned all domains, Pr corresponds to the saturation polarization. Psat . Loading the piezoceramic disc by a negative voltage of an amplitude larger than E c will initiate the switching process again until we arrive at a random polarization of the domains (see state 4 in Fig. 12.5). A further increase will orient the domain polarization in the new direction of the externally applied electric field (see state 5 in Fig. 12.5). Measuring the mechanical strain during such a loading cycle as described above for the electric polarization results in the so-called butterfly curve depicted in Fig. 12.6. Here we also observe that an externally applied electric field intensity E > E c is required in order to obtain a measurable mechanical strain. The observed strong increase is a superposition of two effects: First, we achieve an increase of the strain due to the alignment of the c-axes in the direction of the external electric field. Secondly, the orientation of the domain polarization leads to the macroscopic piezoelectric effect yielding the reversible part of the strain. As soon as all domains are switched (see state 2 in Fig. 12.6), the further increase of the strain just results from the macroscopic piezoelectric effect. A separation of the switching (irreversible) and the piezoelectric (reversible) strain can be best achieved by decreasing the external electric load to zero. Only the strain induced by the alignment of the c-axes remains and we denote this part the saturation strain Ssat . In addition to this electric loading, we can perform a mechanical loading, which will also result in switching processes [3]. According to Fig. 12.7, we start the process at the depolarized state. By applying a mechanical pressure load to a piezoelectric material, we obtain a linear, elastic behavior up to coercive mechanical stress σc . By further increasing the pressure load, first 90◦ switching of domains occur, resulting in a negative, irreversible strain in the direction of the load. Thereby, the mechanical

S

5

2

6 3 Ssat

4 −Ec

1

7 Ec

E

Fig. 12.6 Ferroelectric hysteresis: mechanical strain S as a function of the electric field intensity E

382

12 Piezoelectric Systems

Fig. 12.7 Ferroelastic hysteresis: purely mechanical loading

σ

3 σc −Ssat,c

1 Ssat,c

S

−σc 2

stiffness of the material reduces as long as these switching processes occur. Afterwards, the material behaves linearly elastic. In contrast to the electric loading, the mechanical induced switching has no preferred direction, so that no macroscopic polarization can be observed. By removing the pressure load, a irreversible strain Ssat,c remains. By applying a tensile pressure load and increasing it above σc , the individual domains will change their preferred direction by again 90◦ switching processes. Thereby, the domains switch in the direction of the tensile load, and a macroscopic polarization can be observed. After all domains are switched, we again arrive at a linear, elastic behavior. However, it should be noted that piezoelectric materials are quite brittle and will break, when the tensile load is to large. Furthermore, it can be noticed in Fig. 12.7 that the value of Ssat,c for pressure and tensile load is different. For a tetragonal monocrystal the ratio is 1:2, since there exists more equilibrium states by pressure than by tensile loading [3]. In a last case, we consider the coupled electric-mechanical loading. Thereby, we first electrically polarize the piezoelectric material, switch off the electric loading and then apply a mechanical pressure load. The horizontal line from 1 to 2 in Fig. 12.8 corresponds to the electric polarization process. After setting the electric excitation to zero, we start at Psat and Ssat . At point 2, we start with the mechanical pressure loading and till 3 the material behaves linearly elastic. By exceeding σc , switching processes are initiated and the domains start to align orthogonally to the pressure load. Since, different equilibrium states are possible, the overall, macroscopic polarization reduces. At 4 all domains have switched and the material again behaves linearly elastic. By releasing the pressure load, we arrive at a depolarized piezoelectric material as demonstrated in Fig. 12.8. In practical applications, a combined

12.3 Piezoelectric Material Properties Fig. 12.8 Ferroelastic hysteresis: mechanical depolarization of an initially electrically polarized piezoelectric material

383

1

2 Psat

P

3

4

−σ

electric and mechanical loading occurs and so it is of great importance to arrive at a mathematical model, which is capable to fully model ferroelectric and ferroelastic behavior of piezoelectric materials. When a piezoelectric material is polarized and we assume just small electric and mechanical loads, the linear model derived in Sect. 12.1 can be applied. Thereby, the working point is at Psat , Ssat . The linear material tensors [c E ], [ε S ], and [e], which relate the mechanical and electric quantities, show a certain sparsity pattern according to the crystal structure and polarization of the piezoelectric material. The 6 mm crystal class, which also represents the equivalent class for piezoelectric ceramics, has the following pattern: ⎞ c11 c12 c13 0 0 0 ⎟ ⎜c 0 0 ⎟ ⎜ 12 c11 c13 0 ⎟ ⎜ ⎟ ⎜ c c c 0 0 0 13 13 33 E ⎟ ⎜ [c ] = ⎜ ⎟ 0 0 c44 0 0 ⎜ 0 ⎟ ⎜ ⎟ ⎝ 0 ⎠ 0 0 0 c44 0 0 0 0 0 0 (c11 − c12 )/2 ⎛

⎛

⎞ 0 0 0 0 e15 0 ⎜ ⎟ 0 0 e15 0 0 ⎠ [e] = ⎝ 0 e31 e31 e33 0 0 0

⎞ ε11 0 0 ⎟ ⎜ [ε S ] = ⎝ 0 ε11 0 ⎠ . 0 0 ε33 ⎛

(12.28)

(12.29)

Properties of some widely used piezoelectric materials are summarized in Table 12.1 (PZT 5A/5H from [2], 3202 (Motorola) from [4]).

384

12 Piezoelectric Systems

Table 12.1 Material data for some materials of class 6 mm E (N/m2 ) E (N/m2 ) E (N/m2 ) E (N/m2 ) c11 c12 c13 c33 PZT-5A PZT-5H 32032

12.1 × 1010 12.6 × 1010 14.6 × 1010 e15 (C/m2 )

75.4 × 109 79.5 × 109 96.2 × 109 e31 (C/m2 )

75.2 × 109 84.1 × 109 10.2 × 1010 e33 (C/m2 )

PZT-5A PZT-5H 3202

12.3 17.0 15.3 ε11 (As/Vm)

–5.4 –6.5 –11.5 ε33 (As/Vm)

15.8 23.3 20.4

PZT-5A PZT-5H 3202

919 ε0 1730 ε0 1378 ε0

824 ε0 1437 ε0 1290 ε0

11.1 × 1010 11.7 × 1010 13.8 × 1010

E (N/m2 ) c44

E (N/m2 ) c66

21.1 × 109 23.0 × 109 25.5 × 109

22.8 × 109 23.2 × 109 24.9 × 109

12.4 Models for Nonlinear Piezoelectricity In most actuator applications, piezoceramic materials, e.g., PZT are used, which exhibit a strong nonlinear behavior for large signal excitations. This nonlinear behavior is characterized by the hysteresis loop of the polarization (see Fig. 12.5) and the butterfly curve of the mechanical strain (see Fig. 12.6). In general, we can divide the physical/mathematical models into three categories: 1. Thermodynamically consistent models These models are based on a macroscopic view to describe microscopical phenomena, see e.g., [5–7]. 2. Micro-mechanical models These models are sometimes also based on thermodynamic fundamentals, however they are constructed by breaking the material down to the size of single grains, see e.g., [8–10]. 3. Models with hysteresis operator These models are mostly restricted to the actuator working range and consider either the strain or the polarization hysteresis, see e.g., [11–13]. In the following, we will describe two advanced models: (1) a macroscopic model based on hysteresis operators [14, 15] and (2) a micro-mechanical model based on works of Belov et al. (see, e.g., [16]).

12.4.1 Macroscopic Model with Hysteresis Operators We follow [5] and decompose the physical quantities into a reversible and an irreversible part. For this purpose, we introduce the reversible part D r and the irreversible

12.4 Models for Nonlinear Piezoelectricity

385

part Di of the dielectric displacement according to D = Dr + Di .

(12.30)

In our case, using the general relation between dielectric displacement D, electric field intensity E, and polarization P we set Di = P i (irreversible part of the electric polarization). Analogously to (12.30), the mechanical strain S is also decomposed into a reversible part Sr and an irreversible part Si S = Sr + Si .

(12.31)

The decomposition of the strain S is done in compliance with the theory of elasticplastic solids under the assumption that the deformations are very small [17]. That assumption is generally valid for piezoceramic materials with maximum strains below 0.2 %. The reversible parts of mechanical strain Sr and dielectric displacement Dr are described by the linear piezoelectric constitutive law. Now, in contrast to the thermodynamically motivated approaches in, e.g., [5, 6, 17], we compute the polarization from the history of the driving electric field E by a scalar hysteresis operator H P i = H[E] eP ,

(12.32)

with the unit vector of the polarization eP , set equal to the direction of the applied electric field. Taking this into consideration, we currently restrict our model to uniaxially loaded actuators. A more general approach, the multi-axial loading, would require a vector hysteresis operator. For the realization of H we use a scalar Preisach hysteresis operator. This model uses an infinite number of relay operators γ(α, β) with up and down switching thresholds α, β and output values ±1 according to the amplitude of the input value E (see Fig. 12.9). The output of the Preisach hysteresis operator then computes as [18]  (12.33) H[E](t) = ℘ (α, β)γ(α, β)[E](t) dα dβ. S

In (12.33) ℘ (α, β) defines the Preisach weight function, which has to be identified from measurements. Since in our case the hysteresis loops are closed loops, it is natural to restrict the set of possible up and down switching values to S = {(α, β) ∈ IR2 | β ≤ α } (see Appendix F). The Preisach hysteresis model is superior to other hysteresis operators due to its fast evaluation, its capability to fully describe minor loops and since there exist well established procedures to adapt it to measured data [18]. The butterfly curve for the mechanical strain can also be modeled by an enhanced hysteresis operator. Nevertheless, as seen in Fig. 12.10, the mechanical strain S33 appears to be proportional to the squared dielectric polarization P3 (S ∝ P 2 ). The

386

12 Piezoelectric Systems

S

Fig. 12.9 Preisach plane and two examples of relay operators

Fig. 12.10 Measured mechanical strain S33 and squared irreversible polarization P3i on a piezoceramic actuator on different axis

relation S i = β · (H[E])2 , with a model parameter β, seems obvious. However, to keep the model more general, the ansatz S i = β1 · H[E] + β2 · (H[E])2 + · · · + βl · (H[E])l

(12.34)

is chosen. Similarly to [5] we define the tensor of irreversible strains as follows  [Si ] = 23 β1 · H[E] + β2 · (H[E])2 + · · ·   + βn · (H[E])n e P e P T − 13 [I] . (12.35)

With this, we account for the volume preserving property (no shear strains and the sum of the normal strains equals zero) of the irreversible strain (see, e.g., [5]). The

12.4 Models for Nonlinear Piezoelectricity

387

parameters β1 . . . βn need to be fitted to measured data. Furthermore, the entries of the tensor of piezoelectric moduli are now assumed to be functions of the irreversible electric polarization P i . Here the underlying idea is that the piezoelectric properties of the material only appear once the material is poled. Without any polarization, the domains in the material are not aligned and therefore, coupling between the electric field and the mechanical field does not occur. If the polarization is increased, the coupling also increases. Hence, we define the following relation Pki

[e( P)] =

i Psat

[e].

(12.36)

Herein, Pki denotes the element in polarization direction of the vector P i (in many i the irreversible saturation polarization, [e] the tensor of cases Pki = P3i ) and Psat constant piezoelectric moduli and [e( P i )] the tensor of variable piezoelectric moduli. Therewith, we model an uni-axial electric loading along the fixed polarization axis. Furthermore, the current model does not take into account any ferroelasticity. Finally, the constitutive relations for the electromechanical coupling can be established and written in e-form S = Sr + Si

(12.37)

P i = H[E]e P E

(12.38)

r

i

t

σ = [c ] S − [e( P )] E i

r

(12.39)

S

D = [e( P )] S + [ε ] E + P

i

(12.40)

or equivalently in d-form S = Sr + Si

(12.41)

i

P = H[E]e P

(12.42)

S = [sE ] σ + [d( P i )]t E + Si i

σ

i

D = [d( P )] σ + [ε ] E + P .

(12.43) (12.44)

Due to the symmetry of the mechanical tensors, we use Voigt notation and write the mechanical stress tensor [σ] as well as strain tensors [S] as six-component vectors. The relations between the different material tensors are given in (12.19). Combining the constitutive relations (12.37)–(12.40) with the governing equations as given in Sect. 12.2, we arrive at the following nonlinear coupled system of PDEs     (12.45) ρu¨ − B T [c E ] Bu − Si + [e( P i )]t ∇Ve = 0     ∇ · [e( P i )] Bu − Si − [ε S ]∇Ve + P i = 0

(12.46)

388

12 Piezoelectric Systems

with P i = H[−∇Ve ]e P    l

3 1 T i i eP eP − I . βi (H[−∇Ve ]) [S ] = 2 3 i=0

To summarize, our model fully describing the nonlinear behavior of piezoelectric materials on a macroscopic level consists of the following components: • Decomposition of the mechanical strain S and dielectric displacement D into a reversible and an irreversible part. • Description of the irreversible dielectric displacement Di = P i by a Preisach hysteresis operator. • Representation of the irreversible mechanical strain Si by a polynomial function of the irreversible polarization P i . • Using the governing equations for the mechanical and electrostatic field, and substituting the extend constitutive relations as given in (12.37)–(12.40) we arrive at a system of coupled nonlinear PDEs. At this point, we want to describe a possible extension to a multi-axial piezoelectric model. First of all, we have to apply a vector Preisach hysteresis model (see, e.g., [18]), which for each electric field intensity vector E provides a vector for the irreversible polarization P i P i = H(E).

(12.47)

Furthermore, we compute the coupling tensor [e( P i )] as in (12.36) and rotate it in the direction of the irreversible polarization P i . Similar as in the scalar case, we define the irreversible strains by [Si ] =

3 β1 · | H[E]| + β2 · |HE]|2 + · · · 2   + βn · |H[E]|n e P e P T − 31 [I]

(12.48)

with the unit vector of the irreversible polarization defined by eP =

Pi . | P i|

The determination of all material parameters for our nonlinear piezoelectric model is a quite challenging task. Since we currently restrict ourselves to the uni-axial case, two experimental setups suffice to obtain the necessary measurement data for the fitting procedure.

12.4 Models for Nonlinear Piezoelectricity

389

According to our ansatz (decomposition into a reversible and an irreversible part of the dielectric displacement and mechanical strain) we have to determine the following parameters: • entries of the constant material tensors, • weight function of the hysteresis operator (see (12.32) and (12.33)), • polynomial coefficients for the irreversible strain (see (12.34)). The determination of the linear material parameters of [sE ], [e], [εσ ] is performed by our enhanced inverse scheme [19, 20]. To do this, we carry out electric impedance measurements on the actuator and fit the entries of the material tensors by full 3d simulations in combination with the inverse scheme. Figure 12.11 displays the experimental setup, where it can be seen that we electrically pre-load the piezoelectric actuator with a DC voltage. The amplitude of the DC voltage source is chosen in such a way that the piezoelectric material is driven into saturation. The reason for this pre-loading is the fact, that the irreversible physical quantities show saturation and a further increase is then just given by the reversible physical quantities. These reversible quantities however, are modeled by the linear piezoelectric equations using the corresponding material tensors. The data for fitting the hysteresis operator and for determination of the polynomial coefficients for the irreversible strain are collected by a second experimental setup as displayed in Fig. 12.12. A signal generator drives a power amplifier to generate the necessary input voltage. Thereby, we use a voltage driving sequence as shown in Fig. 12.12 to provide appropriate data for identifying the hysteretic behavior [18]. The first peak within the excitation signal guarantees the same initial polarization for every measurement. The electric current i(t) to the actuator is measured by an ampere-meter, the electric voltage u(t) at the actuator by a voltmeter and the

DC voltage source (controlable)

IEEE-488

Impedance analyser

U= C1 C2 Sample PC

Fig. 12.11 Experimental setup for measuring the electric impedance at saturation of piezoelectric actuators

PC

2.0 m

Electric field intensity (kV/mm)

2

12 Piezoelectric Systems

1.5 1.0 0.5 0.0

-10

0

10

20

30 40 Time (s)

50

60

70

80

Measureable displacement D 3 (C/m )

390

0.07 0.06 0.05 0.04 0.03 0.02 0.01 0

0.5

1.0

1.5

2.0

Electric field intensity (kV/mm)

Sample

Laser vibrometer

Signal generator

A

IEEE-488

~

V Power amplifier

Oscilloscope

Fig. 12.12 Principle experimental setup for measuring the hysteresis curves of piezoelectric actuators

mechanical displacement x(t) by a laser vibrometer. Now in the first step, we can compute the total electric displacement D3 by D3 =

i Prem

1 + A

tend

i + D3m . i(t) dt = Prem

(12.49)

0

In (12.49) A denotes the surface of the electrode, D3m the measurable electric displacei ment and Prem accounts for the fact, that for unipolar excitations the dielectric displacement does not return to zero for zero electric field (instead it return to the remnant polarization), which cannot be determined by the current measurement. Furthermore, we compute the electric field intensity E 3 just by dividing the applied electric voltage u by the distance between the actuator’s electrodes. With the linear material parameters d33 and εσ33 we can now compute a first guess for the irreversible polarization P3i = D3 − d33 σ3 − εσ33 E 3 .

(12.50)

Here σ3 accounts for any mechanical preloading as in the case of the stack actuator or is set to zero as in the case of the disc actuator (stress-free boundary conditions). In the case of a clamped actuator, one will need an additional force sensor to determine σ3 . By simply iterating between the following two equations d33 (P3 ) =

P3i i Psat

d33

P3i = D3 − d33 (P3i )σ3 − εσ33 E 3

(12.51) (12.52)

12.4 Models for Nonlinear Piezoelectricity

391

we achieve at d33 (P3i ) and the final value of P3i . Using (12.43) we obtain the irreversible strain E S3i = S3 − s33 σ3 − d33 (P3i )E 3 ,

(12.53)

where S3 has been computed from the measured displacement x and the geometric dimension of the actuator. Since S3i is now a known quantity, we solve a least squares problem to obtain the coefficients βi according to our relation for the irreversible strain (see (12.34))

min β

nT

i=1

⎛ ⎝

l

j=1



β j P3i

j

⎞2

(ti ) − S3 (ti )⎠

(12.54)

collocated to n T discrete time instances. Once the input E 3 (t) and the output P3 (t) of the Preisach operator H are directly available, the problem of identifying the weight function ℘ amounts to a linear integral equation of the first kind 

℘ (α, β)γ(α, β)[E 3 ](t) dα dβ = P3 (t)

S

  t ∈ 0, t¯ .

(12.55)

Using a discretization of the Preisach operator as a linear combination of elementary hysteresis operators Hλ

aλ Hλ (12.56) H= λ∈Λ

and evaluating the output at n T discrete time instances 0 ≤ t1 < t2 < · · · < tn T ≤ t¯, we approximate the solution of (12.55) by solving a linear least squares problem for the coefficients a = (aλ )λ∈Λ min a

nT

i=1

aλ Hλ [E 3 ](ti ) − P3 (ti )

λ∈Λ

2

.

(12.57)

In (12.56), Hλ may be chosen as simple relays, Hλ = γ(αi , β j ), with (α j , βi ) for i = 1, . . . , n α , j = 1, . . . , n β . The solution of (12.57) provides the coefficients aλ (see Fig. 12.13), which corresponds to a piecewise constant approximation of the weight function. In that case, obviously the set Λ consists of index pairs λ = (i, j) corresponding to different up- and down-switching thresholds αi , β j and the array λ is supposed to be reordered in a column vector to yield a reformulation of (12.57) in standard matrix form. For further details of the fitting procedure, we refer to [14, 21].

392

12 Piezoelectric Systems

Fig. 12.13 Discretization of the Preisach plane by piecewise constants aλ within each element

α αλ

1

1

−1

β

−1

12.4.2 Micro-mechanical Switching Model We consider the individual switching states and define the constitutive law as follows [22] 

S D



=

=



M



I =1



[s E ] I [d]TI [d] I [εσ ] I

T [s E ]eff [d]eff [d]eff [εσ ]eff





ξI

σ E





+



+

Si Pi

.

σ E





M

I =1



SiI P iI



ξI

(12.58)

In (12.58) M denotes the number of variants and ξ I the volume fraction for variant I . The volume fractions compute by solving a system of ordinary differential equations, which couple the different switching states ξ˙I =

K

J =1,J = I

  c J I ξ αJ − c I J ξ αI .

(12.59)

The coefficients c I J compute by the transformation rates ω I J , which we define according to [23] ω I J = c I J ξ αI = ω0 e−

ΔH ( f I J ) kT

  fI J ξ αI ; ΔH ( f I J ) = ΔH0 1 − f∗

(12.60)

with ΔH the enthalpy of activation. In (12.60) f ∗ denotes the driving force at which switching takes place and the actual driving forces f I J compute by f I J = E · ΔPi + σ T ΔSi .

(12.61)

12.4 Models for Nonlinear Piezoelectricity

393

Fig. 12.14 From the real structure to the homogenized switching model [23]

Here, ΔPi and ΔSi define the differences of the irreversible polarization as well as strain between two variants Δ P i = P iJ − P iI ; ΔSi = SiJ − SiI . In the most commonly used ferroelectrics there exists a tetragonal phase with six distinct crystal variants and a rhombohedral phase with eight distinct crystal variants. In order to perform homogenization, 14 distinct variants are used [23] as displayed in Fig. 12.14. These 14 variants have been chosen in such a way that the model achieves almost isotropy in the un-poled state (for details, we refer to [23]). Now, we reformulate (12.58) in such a way, that the dependent physical quantities are the mechanical stress σ and electric displacement D. Using the governing equations for the mechanical and electrostatic fields (see Sect. 12.2) and incorporating our reformulated constitutive relation, results in   ˜ e=0 (12.62) ρu¨ − B T [c E ]eff Bu − Si − B T [e]eff BV     (12.63) ∇ · [e]eff Bu − Si − [ε S ]eff ∇Ve + P i = 0,

where the material tensors [ ]eff are the effective tensors obtained by the sum over all variants weighted by the volume fractions, and B is the differential operator according to (3.20).

12.5 Numerical Computation In the following we will first derive the discrete form of the linear piezoelectric partial differential equations applying the FE method. In the second step, we will investigate the numerical modeling of ferroelectric hysteresis within piezoelectric materials and finally, we will present simulation results.

394

12 Piezoelectric Systems

12.5.1 Linear Case Let us consider a simple setup (without loss of generality) as displayed in Fig. 12.15. The strong formulation for the piezoelectric system reads as follows: Given: : Ω → IRd

u0

u˙ 0 : Ω → IRd ρ, ci j , ei j , εi j : Ω → IR. Find: u(t), Ve (t) : Ω¯ × [0, T ] → IRd   ρ u¨ − B T [c E ]Bu + [e]T ∇Ve = 0   ∇ · [e]Bu − [ε S ]∇Ve = 0.

(12.64) (12.65)

Boundary conditions u = 0 on Γeu × (0, T ) Ve = 0 on ΓeV1 × (0, T ) Ve = V0 on ΓeV2 × (0, T ) n · [σ] = 0 on Γnσ × (0, T ) n · D = 0 on ΓnD × (0, T ) . Initial condition u(r, 0) = u0 , r ∈ Ω ˙ u(r, 0) = u˙ 0 , r ∈ Ω.

Γnσ, ΓeV2

Loaded electrode

Γnσ , ΓnD

Γnσ , ΓnD

Γeu , ΓeV1

Grounded electrode Fig. 12.15 Setup for the formulation

12.5 Numerical Computation

395

The variational formulation for this case with u′ and ψ defining appropriate test functions is    ˜ e dΩ = 0 ρu′ · u¨ dΩ + (Bu′ )T [c E ]Bu dΩ + (Bu′ )T [e]T BV Ω

Ω

Ω





˜ T [e](Bu) dΩ − (Bψ)

Ω

˜ T [ε S ] BV ˜ e dΩ = 0 (Bψ)

(12.66)

Ω

with B˜ = (∂/∂x, ∂/∂ y, ∂/∂z)T . Now, using standard nodal finite elements for the mechanical displacement u and electric potential Ve (n n denotes the number of nodes with unknown displacement and unknown electric potential) ⎞ ⎛ nd nn nn Na 0 0

Na u ia ei = N a ua ; N a = ⎝ 0 Na 0 ⎠ (12.67) u ≈ uh = 0 0 Na a=1 i=1 a=1 Ve ≈

Veh

=

nn

(12.68)

Na Vea

a=1

as well as for the test functions u′ and ψ, we obtain [24, 25] ⎛   nn nn

⎝ ρN aT N b dΩ u¨ b + BaT [c E ]Bb dΩ ub a=1 b=1

Ω

Ω

+



BaT [e]T B˜b dΩ Veb ⎠ = 0

Ω nn nn

a=1 b=1

⎛ ⎝



T

B˜a e Bb dΩ ub −



Ω

Ω

⎞

(12.69)

⎞

B˜aT [ε S ] B˜b dΩ Veb ⎠ = 0.

In (12.69) Ba , B˜a compute as ⎛ ∂ Na ∂x

⎜ Ba = ⎜ ⎝ 0 0

0

0

∂ Na ∂z

∂ Na ∂y

0

0

0

∂ Na ∂z

∂ Na ∂z ∂ Na ∂y

0

∂ Na ∂x

∂ Na ∂y ∂ Na ∂x

0

⎞T

⎛ ∂N ⎞ a

⎟ ⎜ ∂x ⎟ ⎟ ; B˜a = ⎜ ∂ Na ⎟ . ⎝ ∂y ⎠ ⎠

(12.70)

∂ Na ∂z

Introducing damping with a damping matrix C u (see Sect. 3.7.2) we may write (12.69) and (12.70) in matrix form            0 u¨ u˙ K u K uV u Mu 0 Cu 0 + + = , (12.71) T −K fe Ve 0 0 0 0 V¨e V˙e K uV V

396

12 Piezoelectric Systems

with the matrices M u , C u , K u as given in Sect. 3.7.1, K V in Sect. 6.6 and f e a right hand side due to the electric Dirichlet boundary conditions. The matrix K uV computes as K uV =

ne 

keuV

;

keuV

= [k pq ] ; k pq =

e=1



B Tp [e]T B˜q dΩ

Ωe

For the time discretization, a Newmark algorithm as described in Sect. 2.5.2 is used.

12.5.2 Macroscopic Hysteresis Based Approach A straight forward procedure to solve (12.45) and (12.46) is to put the hysteresis dependent terms (irreversible electric polarization and irreversible strain) to the right hand side and apply the FE method. Therewith, one arrives at a fixed-point method for the nonlinear system of equations. However, convergence can only be guaranteed if very small incremental steps are made within the nonlinear iteration process. A direct application of Newton’s method is not possible, due to the lack of differentiability of the hysteresis operator. Therefore, we apply the so-called incremental material parameter method, which corresponds to a quasi Newton scheme applying a secant like linearization at each time step. For this purpose, we decompose the dielectric displacement D and the mechanical stress σ at time step tn+1 as follows Dn+1 = Dn + Δ D;

σ n+1 = σ n + Δσ.

(12.72)

Since we can assume, that Dn and σ n have fulfilled their corresponding PDEs at time step tn , we have to solve ρΔu¨ − B T Δσ − Δ f = 0

B˜ T Δ D = 0

(12.73)

with B˜ = (∂/∂x, ∂/∂ y, ∂/∂z)T . Now, we perform this decomposition also for our constitutive equations as given in (12.43) and (12.44) Sn + ΔS = [sE ] (σ n + Δσ) + Sin + ΔSi   + [d n ]T + [Δd]T (E n + ΔE)

  Dn + Δ D = [d n ] + [Δd] (σ n + Δσ)

(12.74)

(12.75)

+ [εσ ] (E n + ΔE) + P in + Δ P i .

Again assuming equilibrium at time step tn , we arrive at the equations for the increments ΔS = [sE ] Δσ + [d n+1 ]T ΔE + ΔSi + [Δd]T E n σ

i

Δ D = [d n+1 ]Δσ + [ε ] ΔE + Δ P + [Δd]σ n .

(12.76) (12.77)

12.5 Numerical Computation

397

Now, we rewrite the two equations above as ΔS = [sE ] Δσ + [ d˜ n+1 ]T ΔE + [Δd]T E n

(12.78)

Δ D = [d n+1 ]Δσ + [˜ε] ΔE + [Δd]σ n ,

(12.79)

thus incorporating the hysteresis dependent quantities in the material tensors. The coefficients of the newly introduced effective material tensors compute as follows ε˜ j j = εσj j +

ΔP ji ΔE j

j = 1, 2, 3

(12.80)

  d˜31

= (d31 )n+1 +

ΔS1i ΔE 3

(12.81)



= (d32 )n+1 +

ΔS2i ΔE 3

(12.82)

  d˜33

= (d33 )n+1 +

ΔS3i ΔE 3

(12.83)

n+1



d˜32

n+1

n+1

  d˜15

n+1

= (d15 )n+1 .

(12.84)

Since we need expressions for σ and D in order to solve (12.73), we rewrite (12.78) and (12.79) and obtain Δσ = [cE ]ΔS − [cE ][ d˜ n+1 ]T ΔE − [cE ][Δd]T E n   Δ D = [d n+1 ][cE ]ΔS + [˜ε] − [d n+1 ][cE ][ d˜ n+1 ]T ΔE − [d n+1 ][cE ][Δd]T E n + [Δd]σ n .

(12.85)

(12.86)

To simplify the notation, we make the following substitutions [en+1 ]T = [cE ][d n+1 ]T [˜en+1 ]T = [cE ][ d˜ n+1 ]T [Δe]T = [cE ][Δd]T [εn+1 ] = [˜ε] − [d n+1 ][cE ][ d˜ n+1 ]T . Substituting (12.85) and (12.86) into (12.73) results in ˜ e = Δ f + B T [Δe]T BV ˜ en ρΔu¨ − B T [cE ]BΔu − B T [˜en+1 ]T BΔV

(12.87)

˜ en ˜ e = − B˜ T [d n+1 ][Δe]T BV B˜ T [en+1 ]BΔu − B˜ T [εn+1 ]BΔV − B˜ T [Δd]σ n .

(12.88)

398

12 Piezoelectric Systems

This coupled system of PDEs with appropriate boundary conditions for u and Ve defines the strong formulation for our problem. We now introduce the test functions u′ and ψ, multiply our coupled system of PDEs by these test functions and integrate over the whole computational domain Ω. Furthermore, by applying integration by parts,2 we arrive at the weak (variational) formulation: Find u ∈ (H01 )3 and Ve ∈ H01 such that    ′ ′ T E ˜ e dΩ ρ u · Δu¨ dΩ + (Bu ) [c ]BΔu dΩ + (Bu′ )T [˜en+1 ]T BΔV Ω

Ω

Ω

=



u′ · Δ f dΩ



˜ en dΩ (12.89) (Bu′ )T [Δe]BV

Ω

−

Ω



˜ T [en+1 ]BΔu dΩ − (Bψ)

Ω



˜ T [εn+1 ]BΔV ˜ e dΩ (Bψ)

Ω



˜ en dΩ ˜ T [d n+1 ][Δe]T BV (Bψ)



˜ T [Δd]σ n dΩ (Bψ)

=−

Ω

−

(12.90)

Ω

for all test functions u′ ∈ (H01 )3 and ψ ∈ H01 . Now, using standard Lagrangian (nodal) finite elements for the mechanical displacement u and the electric scalar potential Ve (n n denotes the number of nodes with unknown displacement and unknown electric potential) Δu ≈ Δuh =

nn d

Na Δu ia ei =

i=1 a=1

⎞ Na 0 0 N a = ⎝ 0 Na 0 ⎠ 0 0 Na ⎛

ΔVe ≈ ΔVeh =

nn

Na ΔVea

nn

N a Δua

a=1

(12.91)

(12.92)

a=1

2

For simplicity we assume a zero mechanical stress condition on the boundary. Furthermore, Ve satisfies homogeneous boundary conditions, since we assume the potential difference to be incorporated in the right hand side.

12.5 Numerical Computation

399

as well as for the test functions u′ and ψ, we obtain the spatially discrete formulation 

M uu 0 0 0



Δu¨ ΔV¨e





+

K uu K˜ uVe K Ve u −K Ve Ve





Δu ΔVe

=



fu fV

e



.

(12.93)

In (12.93) the vectors Δu and ΔVe contain all the unknown mechanical displacements and electric scalar potentials at the finite element nodes. The FE matrices and right hand sides compute as follows K uu =

ne 

keuu ; keuu = [k pq ] ; k pq =

e=1

K˜ uVe =

ne 



K Ve u =

e e k˜ uVe ; k˜ uVe = [ k˜ pq ] ; k˜ pq =

ne 

keVe u ; keVe u = [k pq ] ; k pq =

fu =

B Tp [˜en+1 ]T B˜q dΩ (12.95)



B˜ Tp [en+1 ]Bq dΩ

(12.96)

Ωe

keVe Ve ; keVe Ve = [k pq ] ; k pq =

e=1 ne 



Ωe

e=1

K Ve Ve =

(12.94)

Ωe

e=1 ne 

B Tp [cE ]Bq dΩ



B˜ Tp [εn+1 ]B˜q dΩ

(12.97)

Ωe

f eu ; f eu = [ f p ]

(12.98)

e=1

fp=



N p Δ f dΩ −

Ωe

fV = e

ne 

e=1



˜ n dΩ B Tp [Δen+1 ]Bψ

Ωe

f eV ; f eV = [ f p ] e

(12.99)

e

fp=−



˜ en dΩ − B Tp [d n+1 ][Δe]T BV

Ωe



B˜ Tp [Δd]σ n dΩ.

Ωe

 In (12.94)–(12.99) n e denotes the number of finite elements, the FE assembly operator (assembly of element matrices to global system matrices) and B p , B˜ p compute as ⎛ ∂N

p

∂x

⎜ ⎜ Bp = ⎜ 0 ⎝ 0

0

0

∂Np ∂y

0

0

∂Np ∂z

0 ∂Np ∂z ∂Np ∂y

0

∂Np ∂y ∂Np ∂x

∂Np ∂x

0

∂Np ∂z

 T B˜ p = ∂ N p /∂x, ∂ N p /∂ y, ∂ N p /∂z .

⎞T ⎟ ⎟ ⎟ ⎠

400

12 Piezoelectric Systems

Time discretization is performed by the Newmark scheme choosing respectively the values 0.25 and 0.5 for the two integration parameters β and γ to achieve 2nd order accuracy. Substituting this relation into (12.93) leads to a nonlinear system of algebraic equations. The solution for each time step (n + 1) is obtained by iteratively solving this fully discrete system of equations until the following incremental stopping criterion is fulfilled n+1 ||Δun+1 k+1 − Δuk ||2 n+1 ||2 ||Δuk+1

+

n+1 − Δϕn+1 || ||Δϕk+1 2 k n+1 ||2 ||Δϕk+1

< δrel

(12.100)

with k the iteration counter. For the practical computations, we have set δrel to 10−4 .

12.5.3 Micro-mechanical Switching Model Also here, a straight forward procedure to solve (12.62) and (12.63) is to put the irreversible electric polarization and irreversible strain to the right hand side and apply the FE method. Therewith, one arrives at a fixed-point method for the nonlinear system of equations, which only converges, if very small steps are taken, though. Thus, we apply the same incremental material parameter strategy as in Sect. 12.5.2. For this purpose, we decompose the electric displacement D and the mechanical stress σ at time step tn+1 as follows Dn+1 = Dn + Δ D;

σ n+1 = σ n + Δσ.

(12.101)

Since we can assume that Dn and σ n have fulfilled their corresponding PDEs at time step tn , we arrive at ρΔu¨ − B T Δσ − Δ f = 0

B˜ T Δ D = 0

(12.102)

with B˜ = (∂/∂x, ∂/∂ y, ∂/∂z, )T . Now, we perform this decomposition also for our constitutive equations as given in (12.58)   Sn + ΔS = [sEn ]eff + [ΔsE ]eff (σ n + Δσ) + Sin + ΔSi (12.103)   T T + [d n ]eff (E n + ΔE) + [Δd]eff   Dn + Δ D = [d n ]eff + [Δd]eff (σ n + Δσ) (12.104)   + [εσn + [Δεσ ]eff (E n + ΔE) + P in + Δ P i . Again assuming equilibrium at time step tn , we arrive at the equations for the increments T ΔS = [sEn+1 ]eff Δσ + [d n+1 ]eff ΔE + ΔSi + [ΔsE ]Δσ n T + [Δd]eff En

(12.105)

12.5 Numerical Computation

401

Δ D = [d n+1 ]eff Δσ + [εσn+1 ]eff ΔE + Δ P i + [Δd]eff σ n + [Δεσ ]eff E n .

(12.106)

Like in (12.78), (12.79), the main idea is to incorporate the irreversible quantities in the material tensors. E.g., for terms in (12.105) we write [sin+1 ]Δσ + [d in+1 ]T ΔE = ΔSi ,

(12.107)

which results in an under-determined system of equations T T   i i i i i Ax = b; x = s11 ; b = ΔS1i ΔS2i . . . ΔS6i . (12.108) s12 s13 . . . d11 ..d36

Therefore, we look for a solution of minimal norm, i.e. we consider a minimization problem ||x||2 = min! s.t. Ax = b, (12.109) which results in

A AT z = b; x = AT z.

(12.110)

Now, we may rewrite (12.105) and (12.106) as T T ΔS = [˜sEn+1 ]eff Δσ + [ d˜ n+1 ]eff ΔE + [ΔsE ]eff σ n + [Δd]eff En σ σ Δ D = [ d¯ n+1 ]eff Δσ + [¯ε ]eff ΔE + [Δd]eff σ n + [Δε ]eff E n n+1

(12.111) (12.112)

with, e.g., [ d˜ n+1 ]eff = [d n+1 ]eff + [d in+1 ]eff . Since we need expressions for σ and D in order to solve (12.102), we rewrite (12.111) and (12.112) and obtain T T Δσ = [˜cEn+1 ]eff ΔS − [˜cEn+1 ]eff [ d˜ n+1 ]eff ΔE − [˜cEn+1 ]eff [Δd]eff En

− [˜cEn+1 ]eff [ΔsE ]eff σ n   T ΔE Δ D = [ d¯ n+1 ]eff [˜cEn+1 ]eff ΔS + [¯εn+1 ]eff − [ d¯ n+1 ]eff [˜cEn+1 ]eff [ d˜ n+1 ]eff   T [Δεσ ]eff − [ d¯ n+1 ]eff [˜cEn+1 ]eff [Δd]eff En   + [Δd]eff − [ d¯ n+1 ]eff [˜cEn+1 ]eff [ΔsE ]eff σ n (12.113)

with [˜cEn+1 ]eff = [˜sEn+1 ]−1 eff . To simplify the notation, we make the following substitutions T T [˜en+1 ]eff = [˜cEn+1 ]eff [ d˜ n+1 ]eff

[¯en+1 ]eff = [ d¯ n+1 ]eff [˜cEn+1 ]eff T T [Δe]eff = [˜cEn+1 ]eff [Δd]eff T [¯εn+1 ]eff = [¯εσn+1 ]eff − [ d¯ n+1 ]eff [˜cEn+1 ]eff [ d˜ n+1 ]eff .

402

12 Piezoelectric Systems

Finally, we use (12.113) and substitute these relations into (12.102) to obtain T ˜ BΔVe ρΔu¨ − B T [˜cEn+1 ]eff BΔu − B T [˜en+1 ]eff T ˜ BVen − B T [˜cEn+1 ][Δs E ]σ n = Δ f + B T [Δe]eff (12.114)   T ˜ e = −B˜ T [¯en+1 ]eff [Δd]eff ˜ en B˜ T [¯en+1 ]eff BΔu − B˜ T [¯εn+1 ]eff BΔV − [Δεσ ] BV

  − B˜ T [Δd]eff − [¯en+1 ]T [ΔsE ] σ n . (12.115)

This coupled system of PDEs with appropriate boundary conditions for u and Ve defines the strong formulation for our problem. We now introduce the test functions u′ and ψ, multiply our coupled system of PDEs by these test functions and integrate over the whole computational domain Ω. Furthermore, by applying integration by parts,3 we arrive at the weak (variational) formulation: Find u ∈ (H01 )3 and Ve ∈ H01 such that    T ˜ BΔVe dΩ ρ u′ · Δu¨ dΩ + (Bu′ )T [˜cEn+1 ]eff BΔu dΩ + (Bu′ )T [˜en+1 ]eff Ω

Ω

Ω

=



′

u · Δ f dΩ −

Ω



T ˜ BVen dΩ (Bu′ )T [Δe]eff

Ω

+



(Bu′ )T [˜cEn+1 ][Δs E ]σ n dΩ

Ω

(12.116) 

˜ T [¯en+1 ]eff BΔu dΩ − (Bψ)



˜ e dΩ ˜ T [¯εn+1 ]eff BΔV (Bψ)

Ω

Ω

=−



  T ˜ ˜ en dΩ [¯en+1 ]eff [Δd]eff (Bψ) − [Δεσ ] BV

−



Ω

T

Ω

  ˜ T [Δd]eff − [¯en+1 ]T [ΔsE ] σ n dΩ (Bψ) (12.117)

for all test functions u′ ∈ (H01 )3 and ψ ∈ H01 . Now, using standard Lagrangian (nodal) finite elements for the mechanical displacement u and the electric scalar

3

For simplicity we assume a zero mechanical stress condition on the boundary. Furthermore, Ve satisfies homogeneous boundary conditions, since we assume the potential difference to be incorporated in the right hand side f ϕ .

12.5 Numerical Computation

403

potential Ve (n n denotes the number of nodes with unknown displacement and unknown electric potential) Δu ≈ Δuh =

nn d

Na Δu ia ei =

i=1 a=1

nn

N a Δua

a=1

⎞ Na 0 0 N a = ⎝ 0 Na 0 ⎠ 0 0 Na ⎛

ΔVe ≈ ΔVeh =

nn

(12.118)

(12.119)

Na ΔVea

a=1

as well as for the test functions u′ and ψ, we obtain the spatially discrete formulation      fu Δu¨ Δu M uu 0 K uu K˜ uVe + = . (12.120) fV ΔVe 0 0 ΔV¨e K Ve u −K Ve Ve e

In (12.120) the vectors Δu and ΔVe contain all the unknown mechanical displacements and electric scalar potentials at the finite element nodes. The FE matrices and right hand sides compute as follows K uu =

ne 

keuu ; keuu = [k pq ] ; k pq =

e=1

K˜ uVe =

ne 



e e k˜ uVe ; k˜ uVe = [ k˜ pq ] ; k˜ pq =

keVe u

ne 

keVe Ve

ne 

f eu ; f eu = [ f p ]

keVe u

;

= [k pq ] ; k pq =

T ˜ Bq dΩ B Tp [˜en+1 ]eff

(12.122)



B˜ Tp [¯en+1 ]eff Bq dΩ

(12.123)

Ωe

;

keVe Ve

= [k pq ] ; k pq =

e=1

fu =



Ωe

ne 

e=1

K Ve Ve =

(12.121)

Ωe

e=1

K Ve u =

B Tp [˜cEn+1 ]eff Bq dΩ



B˜ Tp [¯εn+1 ]eff B˜q dΩ

(12.124)

Ωe

(12.125)

e=1

fp=



N p Δ f dΩ −

Ωe



T ˜ BVen dΩ + B T [Δe]eff

Ωe

fV = e

ne 

e=1

B T [˜cEn+1 ][Δs E ]σ n dΩ

Ωe

f eV ; f eV = [ f p ] e



e

(12.126)

404

12 Piezoelectric Systems

fp=−



  T ˜ en dΩ B˜ T [¯en+1 ]eff [Δd]eff − [Δεσ ] BV

−



Ωe

Ωe

  B˜ T [Δd]eff − [¯en+1 ]T [ΔsE ] σ n dΩ.

 the FE assemIn (12.121)–(12.126) n e denotes the number of finite elements, bly operator (assembly of element matrices to global system matrices) and B p , B˜ p compute as ⎛ ∂N

p

∂x

⎜ ⎜ Bp = ⎜ 0 ⎝ 0

0

0

∂Np ∂y

0

0

∂Np ∂z

0 ∂Np ∂z ∂Np ∂y

∂Np ∂z

0

∂Np ∂y ∂Np ∂x

∂Np ∂x

0

 T B˜ p = ∂ N p /∂x, ∂ N p /∂ y, ∂ N p /∂z .

⎞T ⎟ ⎟ ⎟ ⎠

Time discretization is performed by the Newmark scheme choosing respectively the values 0.25 and 0.5 for the two integration parameters β and γ to achieve 2nd order accuracy. The solution for each time step (n +1) is obtained by iteratively solving this fully discrete system of equations until the following incremental stopping criterion is fulfilled n+1 ||Δuk+1 − Δukn+1 ||2 n+1 ||2 ||Δuk+1

+

n+1 − Δϕn+1 || ||Δϕk+1 2 k n+1 ||2 ||Δϕk+1

< δrel

(12.127)

with k the iteration counter. For the practical computations we have set δrel to 10−4 . In order to get the effective material tensors [ ]eff as well as the irreversible electric polarization Pi and mechanical strain Si , we have to solve the system of ordinary differential equations (ODEs) as given in (12.59). A simple approach would be to apply an explicit first order time stepping, which reads as ξ n+1 = ξ nI + Δt I

K

J =1,J = I

  α  α  c J I ξ nJ − c I J ξ nI .

(12.128)

However, this scheme allows just very small time step sizes Δt in order to converge. Therefore, we apply an implicit time stepping using the Rosenbrock method, which is of fourth order. Furthermore, we perform an automatic step size adjustment (RungeKutta-Fehlberg method according to Kaps and Rentrop [26]) and use an analytically computed Jacobian. This scheme for solving the ODEs allows a quite large time step size Δt for the overall scheme (PDEs with ODEs).

12.6 Numerical Examples

405

12.6 Numerical Examples In the following, we will discuss computations of the electric impedance of a piezoelectric disc as a first example. In the second example, we will perform computations for large signal loading of a piezoelectric disc actuator applying the macroscopic hysteresis model. Finally, we investigate in the micro-mechanical switching model and prove its functionality towards modeling the polarization process.

12.6.1 Computation of Impedance Curve We consider a transducer as shown in Fig. 12.16, i.e., a rotational symmetric disc with electrodes on top and bottom. In order to obtain the whole impedance characteristics in one simulation run, a special technique must be used. The model of the transducer is excited by an electric voltage pulse and its charge response is computed by means of transient analysis. In general, the following equation for the impedance characteristics Z (ω) is valid Z (ω) =

U (ω) U (ω) = , I (ω) jω Q(ω)

(12.129)

where U (ω), I (ω), and Q(ω) are Fourier spectra of the voltage, the current, and the charge time signal, respectively. One can see that the impedance characteristics can be calculated by dividing the spectrum of the voltage signal (excitation) by the j ωmultiple of the computed spectrum of the electric charge. Within a postprocessing step we can compute the electric charge as a function of time q(t) =



Γe

Fig. 12.16 Piezoelectric transducer

D · dΓ =

   ˜ e · dΓ . [e]Bu − [ε S ] BV

(12.130)

Γe

Symmetry axis Upper electrode

Lower electrode

406

12 Piezoelectric Systems

In (12.130) Γe denotes the electrode surface, e.g., of the loaded electrode and u and Ve the FE solution quantities. After performing the Fourier transformation of q(t) we obtain Q(ω). The voltage signal must obviously have a frequency spectrum that is able to excite the transducer in the interesting frequency range. It is recommended to choose a signal whose spectrum is nonzero up to a frequency of at least 10 f r (one order higher than the resonance frequency f r of the transducer). Due to the rotational symmetry, the transducer is modeled by an axisymmetric formulation. In addition, the planar symmetry of the transducer is utilized in order to reduce the size of the model (see Fig. 12.17). The geometrical dimensions of the electrode are neglected, i.e., it is modeled as an infinitely thin layer of surface nodes forming an equipotential area. The well-known piezoelectric ceramic material PZT-5A is assumed (see Table 12.1). According to Fig. 12.17, the radius of the transducer is R = 10 mm and its thickness is D = 2 mm. Since the estimated resonance frequency can be computed by c , (12.131) fr = 2D we obtain, with c ≈ 4,000 m/s, a value of 1 MHz. Hence, the excitation voltage signal should have a nonzero spectrum up to 10 MHz. The selected signal has the form of a triangular pulse (see Fig. 12.18). Due to the fact that a linear analysis is applied here, the magnitude of the pulse has no influence on the resulting impedance characteristics. Therefore, a unit pulse is taken for simplicity. The boundary conditions applied to the model are illustrated in Fig. 12.19. No displacement in the horizontal direction is allowed along the symmetry axis and, correspondingly, no displacement in the vertical direction is allowed at the symmetry plane. Furthermore, the symmetry plane serves as the reference surface for electric potential, i.e., zero electric potential is prescribed there. Finally, the upper surface, which represents the electrode, is an equipotential area, and the electric load is applied there. For the discretization of the simulation domain, linear finite elements have been used. Due to the simple domain, a mapped meshing with five elements in thickness

z

Symmetry axis R = 10 mm

D = 2 mm

r

Symmetry plane

Fig. 12.17 Model of the piezoelectric transducer

407

Electric voltage signal g(t)

12.6 Numerical Examples

t (µs) Fig. 12.18 Electric voltage excitation of the transducer

z Ve = g(t) ur = 0 uz = 0, Ve = 0

r

Fig. 12.19 Boundary conditions applied to the model of the transducer

and 50 elements in the radial direction is performed. The time step size Δt is set to 20 ns and a total number of 8,192 time steps have been computed. Therefore, we achieve a frequency resolution of about 3 kHz and ten time samples for the triangular excitation charge. After the simulation, a Fourier transformation for both the computed charge signal and the excitation voltage signal is performed. The computed impedance according to (12.129) is displayed in Fig. 12.20. One can clearly see the resonance and antiresonance points of the various vibration modes of the transducer. Furthermore, the principal thickness mode of the transducer appears at the frequency of about 1 MHz, but it is disturbed by additional modes. When designing such a piezoelectric transducer, this effect can be suppressed by increasing the diameter/thickness ratio, which leads to a better decoupling of the particular vibration modes.

12 Piezoelectric Systems

Absolute value of electric impedance (Ω)

408

Frequency (Hz) Fig. 12.20 Computed impedance characteristics of the piezoelectric transducer

12.6.2 Piezoelectric Disc Actuator We consider a simple disc actuator made of SP53 (CeramTec material) with a diameter of 35 mm and a thickness of 0.5 mm (see Fig. 12.21a), and will apply our hysteresis based macromodel as described in Sect. 12.4.1. We exploit both rotational and axial symmetry and end up with a two-dimensional axi-symmetric FE model (see Fig. 12.21b). Along the z-axis we set the radial and along the r-axis the axial displacement to zero. Furthermore, we set the electric potential to zero along the r-axis and apply half the measured electric voltage along the top electrode (since we model the disc actuator just by its half thickness). First we perform an impedance measurement of the piezoelectric disc with an electric preloading (see Fig. 12.11) and apply our inverse scheme to obtain the entries of the material

(a)

(b) z Electrode

Top electrode

r

Piezoelectric layer Symmetry

Fig. 12.21 Geometric setup and axisymmetric geometry used for FE simulation: a Geometric setup of the disc actuator; b FE model exploring rotational symmetry as well as axial symmetry (for display reasons not at scale)

12.6 Numerical Examples

409

Table 12.2 Model parameters for the single disc actuator s11 s33 s12 s13 s66 (m2 /N) (m2 /N) (m2 /N) (m2 /N) (m2 /N) 1 · 82 · 10−11 2· 04 · 10−11 −4 · 85 · 10−12 −5· 71 · 10−12 6· 33 · 10−11

d31 d33 d15 (C/N) (C/N) (C/N) −1· 74 · 10−10 4· 30 · 10−10 4 · 87 · 10−10

(a) ν 1 2 3 4

2

βν / (m ·C 1·13 · 10−2 2·33 · 10−1 −8·70 · 100 7·06 · 101

ε11 (F/m) 7· 39 · 10−9

ε33 (F/m) 1· 68 · 10−8

(b) −1 ν

)

2 1 α

0 −1 −2 β

(a) material parameters and polynomial coefficients for the irreversible mechanical strain, (b) logarithmic values of the Preisach weight function for M = 25

tensors [20]. Second, we do measurements according to the experimental setup in Fig. 12.12 and apply the fitting procedure as described in Sect. 12.4.1. The results for the parameters, the polynomial coefficients for approximating the irreversible strain and the Preisach weight function are listed in Table 12.2.4 A FE simulation has been performed with these fitted data, using the above described boundary conditions and a triangular excitation voltage similar to the one used for the fitting procedure. The average number of nonlinear iterations within each time step to achieve the stopping criterion of (12.100) with an accuracy of δrel = 0.01 % was about two Fig. 12.22 displays the comparison of the measured and FE simulated data in detail. This example clearly demonstrates, that using the fitted model parameters our FE scheme reproduces quite accurately the measured data in the experiment.

12.6.3 Polarization and Depolarization Process We consider a quarter model of a simple cube as displayed in Fig. 12.23, and will apply the developed micro-mechanical switching model as described in Sect. 12.4.2. This model allows to simulate both the ferroelectric and the ferroelastic behavior. 4

M, the discretization parameter for the Preisach plane, defines the number of discrete Preisach weights as M(M + 1)/2.

410

12 Piezoelectric Systems

(a)

(b)

x 10 −4

0.05

D (Measurement)

3

D (FE simulation)

0.045

S (FE simulation)

3

3

8

0.04

7

0.035

S3

D3 (C/m2)

S (Measurement)

9

3

0.03 0.025

6 5 4

0.02

3

0.015 0.01

2

0.005

1 0

0 0

50

100

150

200

250

0

50

100

t (s)

(c)

200

250

(d)

x 10 −4

15

0.05

S3 (Measurement) S (FE simulation)

0.045

3

0.04 0.035

S3

D3 (C/m2)

150

t (s)

0.03

10

0.025 0.02

5

0.015 0.01 D3 (Measurement)

0.005

D (FE simulation) 3

0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0

0.5

E3 (kV/mm)

1

1.5

2

E3 (kV/mm)

Fig. 12.22 Comparison of the measured and FE simulated data for the piezoelectric disc actuator: a dielectric displacement over time; b mechanical strain over time; c dielectric displacement over electric field intensity; d mechanical strain over electric field intensity

Fig. 12.23 Quarter model of a cube for proofing the functionality of our piezoelectric formulation

Top electrode

Bottom electrode

Symmetry planes

However, it is restricted to principle investigations and does not allow to model a whole transducer in actuation mode as in the example of Sect. 12.6.2. Electrically, we excite by an electric potential at the top electrode and set the electric potential at the bottom electrode to zero. Mechanically, we set the appropriate boundary conditions at the symmetry planes and furthermore fix the mechanical degrees of freedom in 3-direction at the bottom.

12.6 Numerical Examples

411

In a first computation, we electrically drive our piezoelectric cube to obtain the electric hysteresis and mechanical butterfly curve. We start our computations by setting all fourteens variants equal to 1/14, and no mechanical preloading is applied. The results are displayed in Fig. 12.24. In a second step, we apply compressive stress to the poled piezoelectric cube. Therewith, we first polarize the cube (see Fig. 12.25a), and then apply a strong

(a)

(b)

Fig. 12.24 Computed electric hysteresis and mechanical butterfly curve. a Electric hysteresis. b Mechanical butterfly curve

(a)

(b)

(c)

Fig. 12.25 Depolarization due to applied mechanical stress: after initial poling a compressive stress ins applied to depolarize the piezoelectric material. a Poling process. b Electric displacement response due to depolarization. c Mechanical strain response due to depolarization

412

(a)

12 Piezoelectric Systems

(b)

Fig. 12.26 Electric hysteresis and mechanical butterfly curves for different mechanical prestressing. a Electric hysteresis. b Butterfly curve

increasing mechanical compressive stress to depolarize the cube. In Fig. 12.25b we display the normal mechanical stress in 3-direction over the electric displacement in 3-direction, and in Fig. 12.25c the mechanical strain response is shown. In a last step, we electrically drive the piezoelectric cube at different mechanical prestressing. Figure 12.26 shows the electric hysteresis and mechanical butterfly curves for zero, 40 and 80 MPa mechanical prestressing.

References 1. J. Tichý, G. Gautschi, Piezoelektrische Meßtechnik (Springer, Berlin, 1980) 2. R. Lerch, Sensorik und Prozessmesstechnik, Vorlesungsskript. Universität Erlangen-Nürnberg 3. M. Kamlah, Ferroelectric and ferroelastic piezoceramics—modeling of electromechanical hysteresis phenomena. Contin. Mech. Thermodyn. 13, 219–268 (2001) 4. M. Kaltenbacher, B. Kaltenbacher, R. Simkovics, R. Lerch, Determination of piezoelectric material parameters using a combined measurement and simulation technique. In: Proceedings of the IEEE Ultrasonics Symposium (2002) 5. M. Kamlah, U. Böhle, Finite element analysis of piezoceramic components taking into account ferroelectric hysteresis behavior. Int. J. Solids Struct. 38, 605–633 (2001) 6. C.M. Landis, Non-linear constitutive modeling of ferroelectrics. Curr. Opin. Solid State Mat. Sci. 8, 59–69 (2004) 7. J. Schröder, H. Romanowski, A simple coordinate invariant thermodynamic consistent model for nonlinear electro-mechanical coupled ferroelectrica, ECCOMAS 2004 Proceedings of The European Community on Computational Methods in Applied Sciences, vol. 2 (Department of Mathematical Information Technology, University of Jyväskylä, Jyväskylä, Finnland, 2004) 8. A. Fröhlich, Mikromechanisches Modell zur Ermittlung effektiver Materialeigenschaften von piezoelektrischen Polykristallen, Ph.D. thesis, University of Karlsruhe (TH), Forschungszentrum Karlsruhe (2001) 9. W. Seemann, A. Arockiarajan, B. Delibas, Modeling and Simulation of Piezoceramic Materials Using Micromechanical Approach (European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS), Jyväskylä, 2004) 10. A. Arockiarajan, B. Delibas, A. Menzel, W. Seemann, Studies on nonlinear electromechanical behavior of piezoelectric materials using finite element modeling, International Workshop

References

11. 12. 13. 14.

15.

16. 17. 18. 19. 20.

21. 22. 23. 24. 25. 26.

413

on Piezoelectric Materials and Applications in Actuators, IWPMA (Heinz Nixdorf Institute, University of Paderborn, Paderborn, 2005) P.N. Sreeram, G. Salvady, N.G. Naganathan, Hysteresis prediction for a piezoceramic material system. ASME Winter Annu. Meet. 35, 1 (1993) D.C. Hughes, J.T. Wen, Preisach modeling and compensation for smart material hysteresis. Proc.: Act. Mater. Smart Struct. 2427, 50–64 (1995) K. Kuhnen, Inverse Steuerung piezoelektrischer Aktoren mit Hysterese-, Kriech- und Superpositionsoperatoren. Ph.D. thesis, University of Saarbrücken (2001) T. Hegewald, B. Kaltenbacher, M. Kaltenbacher, R. Lerch, Efficient modeling of ferroelectric behavior for the analysis of piezoceramic actuators. J. Intell. Mater. Syst. Struct. 19(10), 1117–1129 (2008) M. Kaltenbacher, B. Kaltenbacher, T. Hegewald, R. Lerch, Finite element formulation for ferroelectric hysteresis of piezoelectric materials. J. Intell. Mater. Syst. Struct. 21, 773–785 (2010) A.Y. Belov, W.S. Kreher, Simulation of microstructure evolution in polycrystalline ferroelectrics ferroelastics. Acta Mater. 54, 3463–3469 (2006) E. Bassiouny, A.F. Ghaleb, Thermodynamical formulation for coupled electromechanical hysteresis effects: combined electromechanical loading. Int. J. Eng. Sci. 27(8), 989–1000 (1989) I.D. Mayergoyz, Mathematical Models of Hysteresis and Their Applications (Elsevier, New York, 2003) B. Kaltenbacher, T. Lahmer, M. Mohr, M. Kaltenbacher, PDE based determination of piezoelectric material tensors. Eur. J. Appl. Math. 17, 383–416 (2006) T. Lahmer, M. Kaltenbacher, B. Kaltenbacher, R. Lerch, FEM-based determination of real and complex elastic dielectric and piezoelectric moduli in piezoceramic materials. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 55(2), 465–475 (2008) B. Kaltenbacher, M. Kaltenbacher, Modelling and iterative identification of hysteresis via Preisach operators in PDEs. Radon Series Comp. Appl. Math. (2007) J.E. Huber, N.A. Fleck, Multi-axial electrical switching of a ferroelectric: theory versus experiment. J. Mech. Phys. Solids 49, 785–811 (2001) M. Nicolai, Polarisierungsverhalten von Piezokeramik unter kombinierter elektrischer, mechanischer und thermischer Beanspruchung. Ph.D. thesis, University Dresden (2012) H. Allik, T.J.R. Hughes, Finite element method for piezoelectric vibration. Int. J. Numer. Methods Eng. 2, 151–157 (1970) H. Allik, T.J.R. Hughes, Simulation of piezoelectric devices by two- and three-dimensional finite elements. IEEE Trans. UFFC 37, 233–247 (1990) W.H. Press, S.A. Teukolsky, W.T. Vettering, B.P. Flannery, Numerical Recipes in C++: The Art of Scientific Computing, 3rd edn. (Cambridge University Press, Cambridge, 2007)

Chapter 13

Algebraic Solvers

In recent years, many different formulations using Lagrange (nodal) as well as Nédélec (edge) finite elements for the numerical computation of Maxwell’s equations have been published, e.g., [1, 2]. The resulting algebraic system of equations is mostly solved by applying the conjugate gradient method with incomplete Cholesky factorization as preconditioner (ICCG). However, the number of necessary iterations of ICCG increases strongly with the number of unknowns. Recently, investigations have been done to adapt multigrid (MG) methods for the fast solution of 3D electromagnetic field problems (e.g., [3–5]). In this section, we will give a detailed discussion on geometric and algebraic multigrid methods specially adapted for Maxwell’s equations in the quasistatic case. Since a most robust solution strategy is a preconditioned conjugate gradient (PCG) solver with an appropriate multigrid method as preconditioner, we will start with a brief description of the PCG method. For a basic introduction into multigrid methods we refer to [6–8].

13.1 Preconditioned Conjugate Gradient (PCG) Method Let us consider the algebraic system of equations of the form K h uh = f h .

(13.1)

Therein K h ∈ IRn h ×n h denotes the system matrix, f h ∈ IRn h the right-hand side and u h ∈ IRn h the solution vector of the unknown nodal (edge) quantity (usually the magnetic vector potential). The entries of K h are given by ki j = (K h )i j ∈ IR p× p with p defining the number of unknowns per node (edge). The number of unknowns n h is related to the usual discretization parameter h by n h = O(h −d ), with d = 2, 3 the spatial dimension. The system matrix K h is supposed to be sparse and symmetric positive definite (SPD) as is, in fact, the case for the used discretization of Maxwell’s © Springer-Verlag Berlin Heidelberg 2015 M. Kaltenbacher, Numerical Simulation of Mechatronic Sensors and Actuators, DOI 10.1007/978-3-642-40170-1_13

415

416

13 Algebraic Solvers

equations. In general, n h is quite large and due to limited memory resources, iterative solvers have to be used instead of direct ones. However, the convergence of iterative solvers strongly depends on the condition number κ of the system matrix K h κ(K h ) =

λmax (K h ) , λmin (K h )

(13.2)

with λmax and λmin the largest and the smallest eigenvalue of K h , respectively. In general, the convergence rate decreases when κ gets large. Since K h stems from an FE discretization of a second-order partial differential equation (PDE), the condition number κ(K h ) typically behaves like O(h −2 ). In order to cope with large condition numbers, we apply a symmetric preconditioner C h to (13.1), i.e., −1 C −1 h K h uh = C h f h ,

(13.3)

with the properties • C −1 h is an approximate inverse of K h , and −1 • C h can be applied very fast Consequently, the condition number of the preconditioned system is much smaller than the original one. Furthermore, the preconditioned system (13.3) is solved via a Krylov subspace method, i.e., conjugate gradient (CG) or quasi-minimal residual (QMR) method, see [9]. The standard method for solving (13.1) is to apply the preconditioned conjugate gradient (PCG) method as given in Algorithm 1. Algorithm 1: Preconditioned conjugate gradient (PCG) method. k=0 r 0 = Kh u 0h − f h Solve Ch d 0 = −r 0 s 0 = −d 0 r1 = r0 while r k+1  > ǫr 0  do (r k )T s k (d k )T Kh d k u k+1 = u kh + αk d k h r k+1 = r k + αk Kh d k Solve Ch s k+1 = r k+1 (r k+1 )T s k+1 β k = (r k )T s k d k+1 = −s k+1 + β k d k

αk =

k =k+1 end while

13.1 Preconditioned Conjugate Gradient (PCG) Method

417

The following result gives a bound on the number of iterations that are sufficient for a prescribed desired error reduction ε: Theorem 13.1 Let Kh ∈ IRn h ×n h and Ch ∈ IRn h ×n h be SPD with the relation γ˜ 1 Ch v, v ≤ Kh v, v ≤ γ˜ 2 Ch v, v ∀v ∈ IRn h , and γ˜ 1 , γ˜ 2 > 0. The starting solution error ||u − u 0h || K h is reduced by a factor ε applying        I (ε) = ln ε−1 + (ε−2 + 1) / ln ρ˜−1  PCG iterations with ρ˜ =



γ˜ 2 γ˜ 1

    γ˜ 2 −1 + 1 . γ˜ 1

In Theorem 13.1, || · || K h denotes the energy norm induced by the energy inner product, computed as ||w||2K h = Kh w, w for w ∈ IRn h . According to Theorem 13.1 the factor γ˜ 2 /γ˜ 1 , which is equivalent to κ(C−1 h Kh ) should be as close as possible to 1 in order to obtain fast convergence. Of course the theoretically best choice would be Ch = Kh , yielding γ˜ 2 /γ˜ 1 = 1, but this would have the consequence to solve Kh s k+1 = r k+1 in the PCG method. Therefore, we have to find a preconditioner Ch with γ˜ 2 /γ˜ 1 ≈ 1 such that Ch s k+1 = r k+1 can be solved in a very fast way. The conventional choice is to use incomplete Cholesky (IC) factorization, i.e., Ch = R T R with R containing the entries of the upper triangular matrix of the factorization of Kh but with the same structure as Kh (possible low fill in is allowed). In [10] it is shown that under special assumptions, the condition number κ(C−1 h Kh ) when −1 using IC as preconditioner behaves like O(h ). However, κ still depends on the discretization parameter h. This fact leads us to MG methods for which it can be shown that the number of necessary iterations does not depend on the mesh parameter h [11]. Furthermore, it was shown in [12] that a most robust solution strategy (concerning the quality of the FE mesh) for (13.1) is PCG with a geometric multigrid preconditioner.

13.2 Multigrid (MG) Method Multigrid methods improve the convergence by using information not only on the computational grid on which the system of equations is supposed to be solved, but also on a (usually hierarchical) sequence of coarser grids. In order to outline the construction of a MG preconditioner we explain this by means of a two-grid method. The indices H and h are related to the coarse and fine grid of an FE discretization, respectively. The linear mappings (with n h > n H and n h , n H the number of unknowns on the fine and coarse grid, respectively) I hH : IRn h → IRn H

and

h IH : IRn H → IRn h

(13.4)

418

13 Algebraic Solvers

are called restriction and prolongation operators. Therefore, the two-grid algorithm is performed as follows: 1. Smooth ν1 times on the fine grid Kh , u h , f h 2. Calculate the defect d h = f h − Kh u h 3. Restrict the defect d h onto the coarse grid d H = IhH d h 4. Solve the coarse grid problem K H v H = d H 5. Prolongate the coarse grid correction v H to the fine grid h vH vh = I H

6. Correct u h by v h , i.e., u h = u h + v h 7. Smooth ν2 times on the fine grid Kh , u h , f h By replacing the exact solution of the coarse grid problem in step 4 itself by a twogrid approximation, we arrive at the recursive definition of a multigrid cycle (see Fig. 13.1). The motivation for this approach comes from examining the error of the numerical solution in the frequency domain. High-frequency errors, which include local variations in the solution, are well eliminated by simple iterative smoothing methods (e.g., Gauss–Seidel smoother). Once this is achieved, further fine-grid iterations would only result in a convergence degradation. Therefore, the solution is transferred to a coarser grid by using an appropriate projection operator IhH . On this grid, the low-frequency errors of the fine grid manifest themselves as relatively high-frequency errors, and are thus eliminated efficiently again using simple iterative smoothing methods. If the coarsest grid has been reached, the equation has to be solved exactly (e.g., direct solver). Consequently, each grid level is responsible for eliminating a particular frequency bandwidth of the error. Grid l Presmoothing Grid l-1

Postsmoothing Direct solver

Grid 2

Grid 1 (coarse) Fig. 13.1 MG solution algorithm (V-Cycle)

Prolongation Restriction

13.2 Multigrid (MG) Method

419

The MG iteration operator Mh mapping the kth MG iteration error ek = u h − u kh (u h being the exact solution of (13.1)) onto the (k + 1)th MG iteration error ek+1 = u h − u k+1 h ek = Mh ek+1 (13.5) is in the two-grid case given by     pre ν post ν2 h −1 H Mh = Sh I h − IH K H Ih Kh Sh 1 ,

(13.6)

provided that the coarse grid system (step 4) is solved exactly (see e.g., [11]). In (13.6) pre post I h ∈ IRn h ×n h denotes the identity matrix, and Sh , Sh the smoothing operators, e.g., Gauss–Seidel forwards and backwards, respectively pre

Sh

post

Sh

= I h − τh (Lh + Dh )−1 Kh = I h − τh (Lh + Dh )−T Kh ,

with Lh the lower triangular part of Kh , Dh = diag(Kh ) and τh a relaxation factor. In the general situation, the MG iteration operator Mh on the finest grid can be defined in the following iterative process (by denoting the coarsest level by 1 and the finest by ℓ) Mh = Ml     post ν2 q I q − Iq−1 Iq−1 − Mq−1 Mq = Sq  pre ν1 −1 q−1 , Kq−1 Iq Kq Sq

with q = 2, 3, . . . , ℓ, M1 := 0. As already mentioned in the previous section, a most robust solution strategy for (13.1) is PCG with a geometric MG preconditioner. This means that solving Ch s k+1 = r k+1 (see Algorithm 1) corresponds to applying p MG cycles to Kh s k+1 = r k+1 . Using the introduced iteration operator Mh and setting the starting value s k+1 0 to zero, we obtain the iteration error according to (13.5) Kh−1r k+1 − s k+1 = (Mh ) p (Kh−1r k+1 − s k+1 p 0 )  −1 k+1 p k+1 s p = I h − (Mh ) Kh r .

(13.7)

k+1 = C−1 r k+1 , the preconditioner By setting the so-obtained solution s k+1 p equal to s h Ch takes the form  −1 . (13.8) Ch = Kh I h − (Mh ) p

420

13 Algebraic Solvers

13.3 Geometric MG Method In contrast to standard FE techniques, geometric MG methods are not based on a fixed FE mesh that describes the unknown field variable accurately enough. Geometric MG techniques start at a very coarse spatial discretization T1 of the computational domain. By dissecting the elements of T1 , a finer discretization T2 is obtained as shown in Fig. 13.2. This refinement process can include either all elements (uniform refinement) or only an appropriately selected part of the elements (adaptive refinement). By repeating the refinement, we obtain a hierarchy of FE discretizations T1 , . . . , Tℓ for which the systems Kq u q = f q q = 1, 2, . . . , ℓ

(13.9)

are assembled and a MG cycle as described in Sect. 13.2 can be performed. The generation of an appropriate coarse mesh and the adaptive refinement for the subsequent finer FE discretizations levels is a challenging task, which needs the full data exchange between the geometric modeler, the mesh generator and the error estimator [13].

13.3.1 Geometric MG for Edge Elements The two essential parts for a successful application of geometric MG to Maxwell’s equations are the choice of the prolongation and of the smoothing operator. In order q+1 to determine the prolongation operator Iq , we consider the refinement of a face Γ q on an edge tetrahedron element at level q. Dissecting the tetrahedron at level q into 8 q+1 q+1 tetrahedra at level (q + 1), each face Γ q is divided into 4 new faces Γ1 , . . . , Γ4 q+1 (Fig. 13.3). The prolongation operator Iq must guarantee that the magnetic flux

Re

fin

em

en

t

Fig. 13.2 Adaptive refinement of an initial mesh

13.3 Geometric MG Method

421

Prolongation

Fig. 13.3 Prolongation of the edge degrees of freedom from level q to level q + 1 q+1

across Γ q is equal to the one across Γ1

(∇ × u) · dΓ =

q+1

+ · · · + Γ4

4



[5]

(∇ × u) · dΓ .

(13.10)

k=1 q+1 Γk

Γq

q+1

By applying Stoke’s theorem (Appendix B.9), we obtain for each face Γk the relation 1 u · ds . (13.11) u · ds = 4 q

q+1

Γk

Γk

By exploiting the degrees of freedom of the FE formulation (see Fig. 13.3), we obtain q+1

− u7

q+1

+ u3

q+1

− u9

q+1

+ u8

u1

u2 u5

u7

q+1

+ u6

q+1

q+1

− u8

q+1

+ u4

q+1

+ u9

q+1

q+1 q+1

1 q u 4 1 1 q u = 4 1 1 q u = 4 1 1 q u = 4 1

=

q

q

(13.12)

+ u2 + u3

q

(13.14)

q q + u2 + u3 .

(13.15)

+ u2 + u3 q

q

+ u2 + u3 q

(13.13)

In addition, since the magnetic vector potential u is constant along the edge, we may write q+1

= u2

q+1

= u4

q+1

= u6

u1 u3 u5

q+1

q+1 q+1

1 q u 2 1 1 q = u2 2 1 q = u3 . 2 =

(13.16) (13.17) (13.18)

422

13 Algebraic Solvers

Combining the above results allows us to define the transfer operator as follows ⎞T ⎛ 0.5 0.5 0.0 0.0 0.0 0.0 0.25 0.25 −0.25 q+1 Iq = ⎝ 0.0 0.0 0.5 0.5 0.0 0.0 −0.25 0.25 0.25 ⎠ , 0.0 0.0 0.0 0.0 0.5 0.5 0.25 −0.25 0.25 q

which fulfills the requirements of flux conservation. The restriction operator Iq+1 is q+1

chosen to be that transposed to Iq

q

, i.e., q+1 T

Iq+1 = (Iq

(13.19)

) .

For the construction of the smoothing operator, we have to consider the fact that the Sobolev space H 0 (curl) has a Helmholtz decomposition of the form H 0 (curl) = N (curl) + N (curl)⊥ ,

(13.20)

with N (curl), N (curl)⊥ the kernel of the curl operator and its orthogonal complement. The smoothing operator has to damp out errors in both spaces efficiently, see [4]. In [3] it has been shown that overlapping block-smoothers, which collect all edges sharing a common vertex in a block, have this property and show a convergence rate independent of the error reduction value ǫ. Therefore, a simple Gauss–Seidel method does not behave well for Maxwell problems, but due to [3, 4] it is known that properly designed block-Gauss–Seidel iterations do the job. Each block is assigned to a vertex of the mesh and connects all edges sharing this node. Since each of the edges of the mesh is associated to more than one node, no standard block-Gauss–Seidel smoother but an overlapping technique must be applied. Therefore, all degrees of freedoms belonging to edges with a common node are smoothed together. This can be achieved by first introducing a connectivity matrix R j , whose entries are zeros and ones that allows the correspondj ing subblock K q to be picked out of our system matrix K q (q denotes the MG level). The dimension of R j is (n j × n e ) with n j the number of edges belonging to node j and n e the total number of edges (unknowns) in the mesh. Using this matrix, we can j pick out the quadratic sub-blocks Kq of the matrix Kq as follows j

Kq = R j K q R Tj .

(13.21)

Each of these small matrices has to be inverted in the preparation phase of the multigrid method. One step of the block-Gauss–Seidel iteration with initial approximation j u q,i is defined as j

j

j

j

u q,i+1 = u q,i + R Tj (Kq )−1 R j ( f q − Kq u q,i )

j = 1, . . . , n ,

(13.22)

13.3 Geometric MG Method

423

with i the iteration counter. It has to be mentioned that not the whole residual j f q − Kq u q,i has to be computed at each step, but only the few components picked out with R j . Therefore, one block-Gauss–Seidel step is not much more expensive than a simple Gauss–Seidel step.

13.3.2 Case Study In order to demonstrate the advantages of the presented scheme, (Testing Electromagnetic Analysis Methods) TEAM Workshop problem 20 is considered. Therefore, the convergence behavior of the multigrid solvers is compared to standard approaches. The TEAM Workshop problem 20 is a 3D, nonlinear, and static magnetic field problem [14]. The structure of this problem, consisting of a center pole, a yoke, and a coil, is displayed in Fig. 13.4.

13.3.2.1 Nested Multigrid First, the structure is discretized with a coarse mesh T1 of linear edge tetrahedron elements, which is shown in Fig. 13.4. Thereby, the symmetries of the problem in the x z-plane and the yz-plane are exploited. By dissecting each edge tetrahedron element of level 1 into 8 elements of level 2, a new refined mesh T2 is obtained. This procedure is repeated until a mesh T4 is produced that is accurate enough to describe the magnetic field. In Table 13.1 the generated hierarchy of FE meshes is shown. In order to achieve a good initial guess for the nonlinear iteration procedure on the finer levels, the problem is first solved on the coarser grids and the solution is prolongated to the finer meshes and used as a start approximation for the nonlinear iteration process. By this nested-multigrid approach, the number of costly iterations at the finer grids is considerably reduced [15].

Fig. 13.4 Coarse FE discretization of TEAM problem 20 (without air)

yoke

coil center pole

424 Table 13.1 FE hierarchy of TEAM problem 20

13 Algebraic Solvers Grid level

Edge elements

Edges (dofs)

1 2 3 4

3,050 24,500 196,000 1570,000

3,800 30,500 236,000 1,900,000

13.3.2.2 Convergence of the MG-PCG Solver The most time-consuming part of the computation process is the solution of the matrix equation system at the finest mesh T4 . Thereby, the MG solver is compared to a standard solution technique, based on a CG method with adapted block preconditioning (PCCG). In Fig. 13.5 the convergence behavior of both methods for a matrix equation system at the finest level with 1,900,000 edges is displayed. The MG solver achieves the requested normalized residual of 10−6 after 13 iterations and 890 s, whereas the PCCG with block preconditioning needs 180 iterations and 9,860 s. Thereby, an SGI ORIGIN with a RS12000 processor (300 MHz) is used. If applying a conventional CG solver with incomplete Cholesky preconditioning (ICCG) to the matrix equation, more than 1,000 iterations would be necessary and therefore much higher simulation times arise [16].

Fig. 13.5 Number of iterations versus normalized residual

13.3 Geometric MG Method

425

Table 13.2 Necessary MG-PCCG iterations and simulation times to reduce the normalized residuum of algebraic system of equations to 10−6

Level

Iterations

Time (s)

1 2 3 4

10 12 12 13

1.5 11 105 890

13.3.2.3 Optimal Complexity To show the optimal complexity of the MG solver, the number of necessary iterations and the solution time to achieve a normalized residual of 10−6 at each discretization level are displayed in Table 13.2. Since the number of MG iterations is almost independent of the size of the FE meshes and the time for a single MG iteration increases linearly with the number of unknowns, a linear dependency between the degrees of freedom and the solution time can be detected.

13.3.2.4 Accumulated Solution Times of Nested MG and a Conventional Approach In Table 13.3 the convergence behavior of the nonlinear iteration process is compared for different excitations Θ (given in Ampere-turns). For small Θ the number of necessary iterations at the coarser levels is low, but if the excitation increases, a higher number of iterations is necessary. On the other hand, due to the nested MG approach, the number of iterations on the finer levels is almost independent of the strength of the nonlinearity. Since the iterations on the finer grids are the most time-consuming parts of the computation process, also the accumulated simulation time is almost independent of the strength of the magnetic nonlinearity. In the seventh column of Table 13.3 the accumulated simulation time of a conventional approach, which means PCCG solvers for the resultant matrix equation

Table 13.3 Necessary nonlinear iterations for different excitations Θ (Computer: SGI, RS12,000 processor 300 MHz) Θ (Ampere-turns) Iterat. at level Accum. time Accum. time 1 2 3 4 nested MG (s) conventional (s) 1,000 3,000 4,500 5,000

4 11 15 17

3 6 6 6

3 5 5 5

2 3 3 3

8,300 9,570 9,600 9,600

45,400 114,500 154,000 173,600

426

13 Algebraic Solvers

systems, without exploiting the coarse grid information at the finer grids, is displayed. Thereby, a considerable advantage of the nested MG technique can be clearly seen.

13.4 Algebraic MG Method In contrast to geometric MG, algebraic multigrid (AMG) needs no FE discretization with hierarchical grids. The matrices Kq with q = 1, . . . , ℓ on the different levels are set up only by knowledge of the matrix Kh = Kl obtained from the FE discretization. In recent years, a lot of different approaches were published, e.g., [8, 17, 18], which mostly concern the scalar case of matrix equation systems arising from a nodal finite element discretization. Geometric MG methods suffer from the inherent need of a hierarchical FE mesh (see [11]), and thus algebraic multigrid (AMG) methods are of special interest, if at least one of the following cases arises: • The discretization provides no hierarchy of FE meshes, which would be essential for the geometric MG method. This is the case for many FE codes, especially commercial ones. • The coarsest grid of a geometric multigrid method is too large to be solved efficiently by a direct or classical iterative solver. • Classical iterative solvers are not efficient enough. AMG methods try to mimic their geometric counterpart, but only rely on the information available on a given single grid (for the pioneering work on AMG see [8]). While within a geometric MG solver the construction of a matrix hierarchy is rather simple if a hierarchy of grids is available (see e.g., [11]), this task is not as easy if either the matrix only or the information on the finest grid is available. The classical AMG approach assumes an SPD system matrix that is additionally an M matrix [8]. For such matrix classes a matrix hierarchy can be constructed, imitating the geometric counterpart well. It can be easily shown that the information of an SPD system matrix is not enough in order to construct an efficient and robust AMG method. Therefore, we assume the knowledge of the underlying PDE, the FE discretization scheme and additional information on the given FE mesh. Therefore, such enhanced AMG methods are able to reproduce the behavior of geometric MG methods even for Maxwell’s equation, although the system matrices are not M matrices here. For a detailed discussion we refer to [19]. First, we have to perform the coarsening process to extract from the given system matrix (arising from the FE discretization) matrices with decreasing dimension. The key point of the coarsening process is to construct an auxiliary matrix on which the coarsening is performed. Therefore, we can always guarantee a coarsening that is appropriate and, in addition, very fast. Furthermore, we have to define the smoothing operator and the restriction (prolongation) operator for the transfer of data between the different hierarchy levels based on the auxiliary matrix.

13.4 Algebraic MG Method

427

Fig. 13.6 Clipping of an FE mesh in 2D i

eij

j

13.4.1 Auxiliary Matrix Let us assume that the system matrix K h stems from an FE discretization on the FE mesh ωh = (ωhn , ωhe ), with ωhn , |ωhn | = Mh being the set of nodes and ωhe being the set of edges (see Fig. 13.6). An edge is defined as a pair of indices for which the connection of the two corresponding points is a geometric edge. For instance, let i, j ∈ ωhn be the indices of the nodes xi , x j ∈ IRd then the edge is given by ei j = (i, j) ∈ ωhe , and the corresponding geometric edge vector can be expressed by ai j = xi − x j ∈ IRd .

(13.23)

The first task we are concerned with is the construction of an auxiliary matrix B h ∈ IR Mh ×Mh with the following properties (Bh )i j =



bi j ≤0 1 − j =i bi j ≥ 0

if i = j, if i = j .

(13.24)

The entries of B h should be defined in such a way that the distance and parameter jumps of the variational forms are reflected. The matrix pattern of B h can be constructed via different objectives: B h reflects the geometric FE mesh, which is of importance for an edge element discretization, or B h reflects the matrix pattern of the system matrix K h , which is useful for nodal FE discretization.

13.4.2 Coarsening Process The auxiliary matrix B h is a sparse M matrix and therefore the coarsening process for B h is straightforward and can be done in a robust way. We know that B h represents a virtual FE mesh ωh = (ωhn , ωhe ). Such a virtual FE mesh can be split into two disjoint sets of nodes, i.e., ωhn = ωCn ∪ ω nF , ωCn ∩ ω nF = ∅ ,

428

13 Algebraic Solvers

Fig. 13.7 Illustration of coarsening

“Fine grid”

“Coarse grid”

Coarse-grid node

Fine-grid node

with sets of coarse grid nodes ωCn and fine grid nodes ω nF . The splitting is usually performed such that no coarse grid nodes are connected directly and that the number of coarse grid nodes is as large as possible (see Fig. 13.7). In order to perform a coarsening algorithm, let us introduce the following sets Nhi = { j ∈ ωhn : |bi j | = 0 , i = j} , Shi = { j ∈ Nhi : |bi j | > coarse (Bh , i, j) , i = j} , j

Shi,T = { j ∈ Nhi : i ∈ Sh } , where Nhi is the set of neighbors for node i, Shi denotes the set of strong connections and Shi,T is related to the set of nodes that have a strong connection to node i, respectively. The cutoff (coarsening) function is chosen as, e.g., ⎧  ⎨ θ · |bii ||b j j | , coarse (Bh , i, j) = θ · maxl =i |bil | , ⎩ θ,

see [20] , see [8] , see [21] ,

(13.25)

with an appropriate θ ∈ [0, 1]. In addition, we define the local sets ωCi = ωCn ∩ Nhi , ω iF = ω nF ∩ Nhi

(13.26)

E hi = {(i, j) ∈ ωhe : j ∈ Nhi } .

(13.27)

and

13.4 Algebraic MG Method

429

The coarsening algorithm is described in Algorithm 2. Algorithm 2: Coarsening phase. ωCn ← ∅,

ω nF ← ∅

while ωCn ∪ ω nF = ωhn do i ← Pick(ωhn \ (ωCn ∪ ω nF )) if |Shi,T | + |Shi,T ∩ ω nF | = 0 then ω nF ← ωhn \ ωCn else ωCn ← ωCn ∪ {i} ω nF ← ω nF ∪ (Shi,T \ ωCn ) end if end while Therein the function i ← Pick(ωhn \ (ωCn ∪ ω nF )) returns a node i for which the number |Shi,T | + |Shi,T ∩ ω nF | is maximal. Example Let us consider the FE mesh of Fig. 13.8. The auxiliary matrix is defined on an finite element r by the setting birj =

νr i = j , ai j 2

with νr the material parameter and ai j the geometric edge vector (see (13.23)). Let us assume the following entries for row 5 of the assembled auxiliary matrix b51 = −202

b54 = −400

b52 = −2 b53 = −200

b55 = 10n09 b58 = −101 b56 = −2 b59 = −1 .

b57 = −100

Using the coarsening function of [8] (see (13.25)) with θ = 0.25, we obtain

Fig. 13.8 Example with anisotropic mesh and parameter jump (material 1 in elements with number 1 and 2; material 2 in elements with numbers 3 and 4)

3 (3)

2

1 (2)

(1) 1

8

5 4

2

7

4 (4)

3 6

9

430

13 Algebraic Solvers

Nh5 = {1, . . . , 4, 6, . . . , 9} Sh5 = {1, 3, 4, 7, 8} Sh5,T = {4, 6, 7} . For the construction of set Sh5,T we assumed Sh1 = {3}

Sh6 = {1, 2, 5, 8}

Sh2 = {1}

Sh7 = {4, 5, 8}

Sh3 = {1}

Sh8 = {7, 9}

Sh4 = {1, 3, 5, 7, 8} Sh9 = {7, 8} . A special coarsening algorithm is the agglomeration technique, where θ is set to 0. Consequently, Nhi = Shi = Shi,T for all i = 1, . . . , Mh . Furthermore, we call MH (Ihi )i=1 (|ωCn | = M H < Mh ) a disjoint splitting for the agglomeration method if Ihi

∩

j Ih

= ∅,

MH 

Ihi = ωhn ,

i=1

is valid, see Fig. 13.9. If an appropriate prolongation Qh for B h is defined then a coarse auxiliary matrix is computed by B H = (Qh )T B h Qh , and B H represents again a virtual FE mesh ω H = (ω nH , ω eH ), with ω nH = ωCn . It can be shown that B H is again an M-matrix if the prolongation operator Qh fulfills certain criteria [8]. Thus the coarsening process can be applied recursively. Finally, it is assumed that the degrees of freedom on the coarse grid are numbered first. For

Fig. 13.9 Virtual FE mesh with a feasible agglomeration

Ihi

j

Ih Fine-grid node

Coarse-grid node

13.4 Algebraic MG Method

431

instance, the nodes are reordered like ωhn = (ωCn , ω nF ) (similarly for edges) and as a consequence the system matrix can be written as   K CC K CF Kh = . T K K CF FF

13.4.3 Prolongation Operators For a given splitting ωhn = ωCn ∪ ω nF the optimal prolongation operator is given by the Schur complement, i.e., ˜ ˜T K H = K CC − K CF K −1 FF K FC = Ph K h Ph with

−1 T P˜ h = (I H , −K CF K FF ) .

The prolongation operator P˜ h can hardly be realized in practice since the expression −K CF K −1 FF involves the inverse of K FF , which in turn implies a global transport of information. In addition, the coarse grid matrix K H becomes dense. The goal of an AMG method is to approximate P˜ h by some prolongation operator Ph , which acts only locally and therefore produces a sparse coarse-grid matrix.

13.4.4 Smoother and Coarse-Grid Operator An essential point in MG methods is the smoothing operator Sh ∈ IR Nh ×Nh that reduces the high-frequency error components. Typically, a particular smoother works for certain classes of matrices. It is shown in [21] that a point Gauss–Seidel or point Jacobi smoother is appropriate for FE discretizations with Lagrange FE functions for scalar elliptic PDEs of second order. Analogously, the block-Gauss–Seidel and block Jacobi smoother work well for the block counterpart, e.g., discretization of Maxwell’s equation with nodal finite elements. For the edge FE discretization we use a patch smoother. The coarse grid matrix K H is usually constructed by Galerkin’s method, i.e., K H = PhT K h Ph .

(13.28)

432

13 Algebraic Solvers

After a successful setup, an AMG-cycle can be performed as usual (see e.g., [11]). For instance in Algorithm 3 a V (ν F , ν B )-cycle with variable pre- and postsmoothing steps is described. The variable CoarseLevel stores the number of levels generated by the coarsening process until the size of the system is smaller than CoarseGrid. In the following subsections we specialize the abstract algorithms, define the components for nodal and edge FE discretization and additionally propose a method for complex symmetric systems. Prior to that we mention that static, transient, and nonlinear analysis of a given problem results in the solution of linear systems (13.1). Therefore, we restrict the discussion to the linear analysis. Other applications can be found in [22–26].

13.4.5 AMG for Nodal Elements First we consider (6.120) and use nodal elements for discretization. Note that the following approach includes the scalar case ( p = 1; e.g., scalar potential equation). Construction of virtual FE meshes: The definition of the auxiliary matrix B h plays an important role for this problem class. The classical approach uses Algorithm 3: V(ν F , ν B )-cycle Kℓ ← K,

fℓ ← f,

AMGStep(K , u, f , ℓ).

uℓ ← u

if ℓ = CoarseLevel then u ℓ ← CoarseGridSolver (Lℓ LℓT , f ℓ ) Return else d ℓ ← 0, w ℓ+1 ← 0 u ℓ ← Sℓν F (u ℓ , f ℓ ) d ℓ ← f ℓ − K ℓuℓ d ℓ+1 ← (Pℓ )T d ℓ AMGStep(K ℓ+1 , w ℓ+1 , d ℓ+1 , ℓ + 1) w ℓ ← Pℓ w ℓ+1 u ℓ ← u ℓ + wℓ u ℓ ← Sℓν B (u ℓ , f ℓ ) end if

(B h )i j = −ki j ∞ for i = j , with  ∞ the maximum norm. The diagonal entries of B h are computed according to (13.24). Now the degrees of freedom per node of the system matrix have to be related to an entry in the auxiliary matrix, which in turn implies that the matrix pattern of K h and of B h has to be equal, i.e., ki j ∞ = 0 ⇔ |bi j | = 0 .

13.4 Algebraic MG Method

433

Construction of coarse FE spaces: The simplest prolongation operator is given by

(Ph )i j =

⎧ Ip ⎪ ⎪ ⎪ ⎨

if i = j ∈ ωCn ,

i,T

|Sh ⎪ ⎪ ⎪ ⎩ 0

1 ∩ωCn |

· Ip

if i ∈ ω nF , j ∈ Shi,T ∩ ωCn ,

(13.29)

else ,

with I p ∈ IR p× p the p-dimensional identity matrix. The AMG method shows a better convergence behavior as compared to (13.29) with the following discrete harmonic extension ⎧ n ⎨ Ip  if i = j n∈ ωC , i −1 (Ph )i j = −kii ki j + ci j (13.30) if i ∈ ω F , j ∈ ωC , ⎩ 0 else , with

ci j =

  

p∈ω iF

q∈ωCi

k pq

−1

ki p k pj .

However, the increasing memory requirement and the slower application compared to the prolongation (13.29) is the major drawback of the discrete harmonic extension. Note that the entries of the prolongation operators are matrix valued, e.g., (Ph )i j ∈ IR p× p , like the entries of the system matrix K h . Smoothing operator: We use a block-Gauss–Seidel method as smoothing operator, e.g., [21], or a patch-block Gauss–Seidel method, e.g., [18]. The latter should be used for anisotropic problems.

13.4.6 AMG for Edge Elements The second class originates from an FE discretization with edge FE functions of the variational form (6.112). Construction of virtual FE-meshes: According to [4], the refinement of the FE mesh can be performed on the nodes as is usually done for Lagrange FE functions. We use this fact and base our coarsening on an auxiliary matrix B h , which is constructed for instance by the finite element wise setting birj = −

νr i = j and (i, j) ∈ ωhe , ai j 2

with νr the reluctivity of the material. Again the diagonal elements are computed via (13.24).

434

13 Algebraic Solvers

Example Let us consider the FE mesh of Fig. 13.8 and choose element r = 1. We get the following element matrix ⎛

2.5 ⎜ −0.5 B 1h = 100 · ⎜ ⎝ −1 0

−0.5 2.5 0 −1

−1 0 2.5 −0.5

⎞ 0 −1 ⎟ ⎟. −0.5⎠ 2.5

The entries (Bh1 )14 , (Bh1 )23 , (Bh1 )41 , and (Bh1 )32 are zero, i.e., there is no diagonal edge in the virtual FE mesh. Let us recall that an FE mesh is represented by ωh = (ωhn , ωhe ) , i.e., the set of nodes ωhn and the set of edges ωhe . The coarse grid is defined by identifying each coarse grid node j ∈ ωCn with an index k ∈ ω nH . This is expressed by the index map ind (.) as ω nH = ind (ωCn ) . A useful set of coarse grid edges ω eH can be constructed if we invest in a special prolongation operator Q h for the auxiliary matrix B h . The prolongation operator Qh is constructed such that each fine grid node prolongs exactly from one coarse grid node, so that one arrives at a partition of ωhn into clusters, each of them being represented by a coarse grid variable. We extend the index map ind : ωCn → ω nH defined above onto the whole fine set ωhn by assigning to all fine grid nodes of a cluster the coarse grid index of the representative ind : ωhn → ω nH . A consequence is that ind (i) = ind ( j) iff i, j ∈ ωhn prolongate from the same coarse grid variable. We define an agglomerate (cluster) Ihi of a grid point i ∈ ωhn by (see Fig. 13.10) Ihi = { j ∈ ωhn | ind ( j) = ind (i)} ⊂ Nhi , and hence the set of coarse grid nodes can be written as ω nH = {ind (i) | i ∈ ωhn } . The prolongation operator Qh has only 0 and 1 entries by construction, i.e.,  1 i ∈ ωhn , j = ind (i) (Qh )i j = (13.31) 0 otherwise .

13.4 Algebraic MG Method

435

Fig. 13.10 Virtual FE mesh with a feasible agglomeration and coarse-grid edges

j

Ih i

(i,j)

Ihi

j

r s) (r,

k s k

Ih Fine-grid node

Coarse-grid node

Coarse-grid edge

Now, a coarse-grid edge only exists if there is at least one fine edge connecting the agglomerates Ihi and Ihk with i = k (see Fig. 13.10), i.e., ∃r ∈ Ihi , ∃s ∈ Ihk such that (r, s) ∈ ωhe . Note that a decrease of the number of edges in the coarsening process is not proved in general, but a decrease is heuristically given, if the average number of nonzero entries of B h does not grow too fast. Construction of coarse FE spaces: The construction of the prolongation operator Ph : IR N H → IR Nh , is delicate because of the kernel of the curl -operator consisting of all gradient fields. Ph is defined for i = (i 1 , i 2 ) ∈ ωhe , j = ( j1 , j2 ) ∈ ω eH as ⎧ ⎨1 (Ph )i j = −1 ⎩ 0

if j = (ind (i 1 ), ind (i 2 )), if j = (ind (i 2 ), ind (i 1 )), otherwise ,

(13.32)

by assuming a positive orientation of an edge j = ( j1 , j2 ) from j1 to j2 if j1 < j2 holds. The constructed prolongation operator Ph has full rank, because the coarse grid edges prolongate to N H distinct fine-grid edges by construction. For a detailed discussion see [27]. Smoothing operator: To complete the components for an AMG method for edge element FE discretizations, we need an appropriate smoother. We consider two different types of smoothers for K h . The first one was suggested in [3]. This is a block-Gauss–Seidel smoother where all edges that belong to E hi (see (13.27)) are smoothed simultaneously for all i ∈ ωhn (see Fig. 13.11). Another kind of smoother was suggested in [4]. A mathematically equivalent ∈ IR Nh is defined by formulation is outlined in Algorithm 4. Therein the vector g e,i h

436

13 Algebraic Solvers

Fig. 13.11 Detail view of a virtual FE mesh

i

(i,j)

j

Algorithm 4: Hybrid smoother. u h ← GaussSeidel(K h , f h , u h ) for all i ∈ ωhn do  uh ← uh +

end for

( f h −K h u h ),g e,i h  e,i K h g e,i h ,g h

g e,i h

=



· g e,i h

grad h g n,i h

⎧ ⎨1 = −1 ⎩ 0

if j < i (i, j) ∈ E hi , if j > i (i, j) ∈ E hi , otherwise ,

∈ IR Mh , (g n,i ) = δi j . with a vector g n,i h h j

13.4.7 AMG for Time-Harmonic Case In the harmonic case the time derivative of the magnetic vector potential is substituted by ∂A ˆ, → jω A ∂t ˆ the complex magnetic with j the complex number, ω the angular frequency and A vector potential. Therefore, we have to apply the AMG method to a complex valued and symmetric algebraic system of equations with system matrix K h = K rhe + j K im h .

(13.33)

In (13.33) K rhe denotes the real part and K im h the imaginary part of the system matrix. The application to scalar potential equations has been presented in [28], and adaption to the magnetic vector potential formulation is straightforward as shown below.

13.4 Algebraic MG Method

437

Construction of virtual FE-meshes: The auxiliary matrix is defined to be real valued. This means that we set up B h for an edge element discretization as defined in Sect. 13.4.6. For a nodal element discretization we can use the procedure described in Sect. 13.4.5 for K rhe . Construction of coarse FE spaces: For the construction of a coarse-grid operator K H we define the system prolongation to be real valued and computed as defined in Sect. 13.4.5 as well as Sect. 13.4.6. Therefore, we get re im K H = PhT K h Ph = PhT K rhe Ph + jPhT K im h Ph = K H + j K H .

The prolongation Qh is also taken from the real-valued realization correspondingly. Smoothing operator: In the case of an algebraic system of equations arising from a nodal FE discretization we apply a block Jacobi or Gauss–Seidel smoother in the complex variant. The complex version of the smoother proposed in [3] is used for an edge FE discretization.

13.4.8 Case Studies In order to gain robustness and efficiency, the proposed AMG methods were used as a preconditioner in the conjugate gradient (CG) method for the static and eddy current case (in the time domain) and in the quasi-minimal residual (QMR) method for the time-harmonic case. The iteration was stopped as soon as the error in the preconditioner energy norm has been reduced by a factor 10−6 for the PCG method. In the time-harmonic case (QMR solver) we use the stopping criterion as follows  f h − K h u h 2 ≤ 10−6  f h 2 . For all calculations, a V (2, 2)-cycle has been applied and the coarsest matrix equation is solved by a Cholesky factorization, if the degrees of freedom less than 500. All computations were done on a PC with a Pentium 1.7 GHz chip. A good measure for the speed of coarsening is the so-called grid complexity, which is given by L  Mi i=1 , (13.34) GC(K h ) = M1 with L the number of levels and Mi the number of nodes (edges) for level i. This number is close to 1, if the reduction of unknowns is done very fast. If the number is very large then the coarsening is usually very slow. A second measure that is more related to the memory consumption and arithmetic costs is the operator complexity, i.e.,

438

13 Algebraic Solvers

OC(K h ) =

L 

NMEi · Ni

i=1

NME1 · N1

(13.35)

,

where NMEi denotes the average number of nonzero system matrix entries on level i and Ni the number of unknowns on this level. This number gives an idea of how much memory is used with respect to the finest grid. The same applies for the arithmetic costs. Again this number is close to 1 if only a small amount of memory is required. The abbreviation MB denotes the amount of memory used. The computations with Lagrange and Nédélec FE functions were always done on the same FE mesh. We want to emphasize that Nh = p |ωhn | for node (static case: p = 3; eddy current case: p = 4) and Nh = |ωhe | for edge FE discretization. 13.4.8.1 Static Analysis For the computational domain we consider the geometry of TEAM 20 (see Fig. 13.12), which has been discretized by tetrahedron elements. Table 13.4 displays evaluated grid complexity GC and operator complexity OC as well as the required memory for the nodal and edge case. Nh defines the number of unknowns. It can be clearly seen that the required memory scales optimally with the number of unknowns and the values for OC and GC are close to 1. The number of iterations as well as elapsed CPU times are shown in Table 13.5. The short time for performing the setup makes the AMG solvers very attractive for nonlinear problems.

13.4.8.2 Transient Analysis In order to show the performance of the proposed enhanced AMG methods for an eddy current problem, we present results of 3D magnetic field computations for a Fig. 13.12 FE mesh of TEAM 20 (without air region)

Yoke

Center pole

Coil

13.4 Algebraic MG Method

439

Table 13.4 TEAM 20: complexities and memory requirement Nh GC OC Nodes Edges Nodes Edges Nodes 1.263 8.022 56.673

2.253 16.217 122.762

1.4 1.3 1.3

1.2 1.2 1.2

1.07 1.06 1.07

Table 13.5 TEAM 20: CPU times and number of iterations Nh Setup (s) Solve (s) Nodes Edges Nodes Edges Nodes 1.263 8.022 56.673

2.253 16.217 122.762

0.2 0.4 1.7

0.2 0.5 2.9

0.2 2.0 30.7

Edges

MB Nodes

Edges

1.02 1.02 1.03

1.5 10 65

3 24 173

Edges

Iter Nodes

Edges

0.2 2.8 35.3

18 27 48

9 16 24

simplified MRI scanner with a z-gradient coil, as shown in Fig. 13.13 [29]. Here, gradient and magnet coils are assumed as smeared cylindrical coils. Furthermore, only the three inner cryostat cylinders are modeled. Table 13.6 displays the values for the grid complexity GC, the operator complexity OC and the required memory. Again a very good performance with optimal memory requirement can be found.

Fig. 13.13 FE mesh of a simplified MRI scanner (not the full air region is displayed) Gradient coil Shields

Magnet coil

Table 13.6 MRI scanner: complexities and memory requirement Nh GC OC Nodes Edges Nodes Edges Nodes Edges 27.834 88.053 162.882

61.342 197.375 368.131

1.2 1.2 1.2

1.2 1.2 1.2

1.06 1.06 1.05

1.03 1.03 1.03

MB Nodes

Edges

75 250 510

94 330 639

440

13 Algebraic Solvers

Table 13.7 MRI scanner: CPU times and number of iterations Nh Setup (s) Solve (s) Nodes Edges Nodes Edges Nodes 27.834 88.053 162.882

61.342 197.375 368.131

2.5 7.7 14.6

1.5 5.8 10.1

10.3 31.2 64.1

Edges

Iter Nodes

Edges

7.6 38.2 75.2

15 15 16

10 15 15

The performance of the proposed AMG solvers concerning the number of iterations and CPU time is shown in Table 13.7. Since in this example we have no parameter jump in the reluctivity, the number of iterations remains quite constant, which results in an optimal convergence rate.

13.4.8.3 Time-Harmonic Analysis Finally, we show the performance of the AMG method for the harmonic analysis. As a case study, we have chosen a configuration of a coil and a centered iron core surrounded by air. The established mesh can be seen in Fig. 13.14. The core diameter was fixed at 2 mm and the coil thickness was set to 1 mm. At an excitation frequency f of 500 Hz, a relative permeability μr = 1,000 and a conductivity of γ = 107 S/m one calculates a penetration depth of δ = 0.22 mm, which corresponds to approximately 1/5 of the core radius. For a different conductivity of γ = 105 S/m one gets an eddy current penetration depth of δ = 2.2 mm, which is equal to a full penetration of the core. For the discretization, we have used at least ten finite elements per penetration depth. In the first step we performed calculations on varying grids and have listed the evaluated characteristic solver data in Table 13.8. Therein, the operator and grid complexity are close to 1, which indicates a fast coarsening, and in addition small memory requirements for the AMG preconditioner. Furthermore, it can be noticed that the solution time scales approximately linearly with the number of edges Nh . Fig. 13.14 FE mesh of an iron core and surrounding air (broken open at yz-plane element boundaries, coil not displayed)

13.4 Algebraic MG Method Table 13.8 Case study: Performance and complexity for different FE meshes

441 Nh

Iter

Setup (s)

Solver (s)

OC

GC

18.708 107.793 245.418

30 30 32

0.39 2.55 6.04

4.81 31.33 81.35

1.026 1.031 1.029

1.216 1.263 1.211

In the second step we performed computations for different material parameters to investigate on the robustness of our new solver. In Figs. 13.15 and 13.16 the convergence behavior of our QMR-AMG solver is displayed. The typical QMR behavior—a short remaining at a constant relative error over a few iteration steps— can be detected clearly. Summarizing, we can note that the proposed QMR-AMG solver is very robust against parameter jumps. As a practical example, the results of a 3D magnetic field computation for an electric transformer are shown. In Fig. 13.17 the model including the finite element discretization is displayed.

Fig. 13.15 Convergence behavior with different conductivities γ and fixed relative permeability μr = 1,000 for the iron core

Fig. 13.16 Convergence behavior with different conductivities γ and fixed relative permeability μr = 1 for the iron core

442

13 Algebraic Solvers

Fig. 13.17 FE mesh of an electric transformer (no air region is displayed)

Air

Iron core

Windings

Due to symmetries, it is only necessary to simulate one quarter of the full configuration by applying proper boundary conditions. The core is made of iron (μr = 1,000) and has a conductivity of γ = 106 S/m. The number of coil windings, respectively, the size of the current have been chosen in way to ensure a maximum current density of 4 A/mm2 to avoid an unnecessary heating of the coil. Therefore, with 100 windings one gets a current amplitude of 30 A, considering an inner coil radius of 55 mm, an outer coil radius of 60 mm and a coil length of 150 mm. Figure 13.18 displays the computed magnetic induction in the iron core. In Table 13.9 one can see the characteristic solver data, e.g., setup and solution times as well as operator and grid complexity for different FE meshes. Again, the grid and operator complexity values are close to 1. However, it should be mentioned that for an optimal multigrid method, the iteration numbers are independent of the number of unknowns. In our case, a small increase of the iterations can be determined. The presented AMG solvers are well suited for the efficient solution—both concerning CPU time and memory requirements—of algebraic systems of equations arising from nodal as well as edge FE discretizations of Maxwell’s equations. In particular, the presented algorithms for the coarsening strategy make the solvers very attractive also for nonlinear electromagnetic field problems, since the setup time can be kept very small.

Table 13.9 Electric transformer: Performance and complexity for different FE meshes

Nh

Iter

Setup (s)

Solver (s)

OC

GC

9.115 17.977 32.835 485.451

22 24 30 52

0.24 0.34 0.63 11.35

1.38 3.2 7.47 274.23

1.017 1.020 1.021 1.024

1.196 1.206 1.195 1.181

13.4 Algebraic MG Method

443

Fig. 13.18 Main magnetic induction in the iron core of the electric transformer

Further research is concentrated on improvements of the prolongation operators to obtain even better convergence rates. One possible method was proposed in [20] (so-called smoothed aggregation), which could be applied for our problem classes. If even more speedup is required for practical applications, the presented AMG methods can be parallelized on distributed-memory computers. The first promising results can be found in [30, 31].

13.5 Block Preconditioner for Higher Order Edge Element Discretization We assume a FE discretization with hierarchical higher order edge elements utilizing anisotropic adaptive polynomial order as described in Sect. 6.7.6. The resulting system of equations after FE discretization may be written as K (A)A = f ,

(13.36)

and in more detail as ⎞⎛ ⎞ ⎛ A N0 K N0 ,N0 K N0 , f K N0 ,int ⎝ K f,N0 K f, f K f,int ⎠ ⎝ A f ⎠ = ⎜ ⎝ K int,N0 K int, f K int,int Aint ⎛

⎞ fN 0 ff ⎟ ⎠ . f int

(13.37)

The interior unknowns Aint can be eliminated by static condensation as Aint = K −1 int,int



f int − K int,N0 AN0 − K int, f A f



,

(13.38)

444

13 Algebraic Solvers

where K −1 int,int can be inverted on the element level. Substituting this result back in (13.37) results in the reduced system 

Kˆ N0 ,N0 Kˆ N0 , f Kˆ f,N0

Kˆ f, f



A N0 Af



⎛

=⎝

ˆf N ˆf

0

f

⎞

⎠ ,

(13.39)

with the modified matrices and RHS vectors as Kˆ N0 ,N0 = K N0 ,N0 − K N0 ,int (K −1 int,int )K int,N0

(13.40)

Kˆ N0 , f = K N0 , f − K N0 ,int (K −1 int,int )K int, f

(13.41)

Kˆ f,N0 = K f,N0 − K int, f (K −1 int,int )K N0 ,int

(13.42)

Kˆ f, f = K f, f − K int,f (K −1 int,int )K f,int

(13.43)

ˆf = f N − K N0 ,int (K −1 int,int ) f int N0 0

(13.44)

ˆf

f

= f

f

− K f,int (K −1 int,int ) f int .

(13.45)

The two main effects of the static condensation are: • The number of unknowns is reduced significantly, as with increasing polynomial degree p only face and interior unknowns are added. • The condition number κ of the reduced system (13.39) is much smaller compared to the one of the full system (13.37), causing less iterations of the CG solver or any other iterative solver. In order to solve the reduced system (13.39), we apply a Preconditioned Conjugate Gradient (PCG) method. It was shown in [32], that a robust preconditioner C −1 can be simply defined by a block Jacobian preconditioner (i.e. an additive Schwarz method, ASM), defined by C=



Kˆ N0 ,N0

0

0

B Kˆ f, f



,

(13.46)

where the single blocks are formed as follows: • Kˆ N0 ,N0 : The lowest order N´ed´elec functions can be either solved by a sparse direct solver [33] or by a suitable iterative method, respecting the Helmholtzdecomposition of the magnetic vector potential (see e.g. [34]). B • Kˆ f, f : For every face, all unknowns are grouped in one block (superscript B). The application of the preconditioner basically is just the inversion of the single face blocks K −1 f, f .

13.5 Block Preconditioner for Higher Order Edge Element Discretization

445

If the preconditioner (13.46) is applied for structures with a very high aspect ratio (AR), as is the case for magnetic field computations in thin steel sheets, the convergence of the iterative solver deteriorates. This can be explained by an increase in the condition number (see (13.2)), as the entries in the stiffness matrix K scale with 1/ h 2 , where h is the mesh size, leading to strongly coupled entries for nearly parallel edges/faces and thus to nearly singular systems with high condition numbers [35]. The idea proposed in [35] is based on a singularity decomposition technique, where new unknown variables are introduced and assigned to groups of parallel edges with small distance. However, this method introduces new matrix entries, as all edges in one group couple via the auxiliary variable. In addition, the method is only applicable to 1st order elements, as only edge degrees of freedom are considered. An alternative approach is proposed in [36], where a plane smoother for nodal and edge components of the A-φ-formulation, respecting the Helmholtz-decomposition, is applied within a geometric multigrid (MG) solver. Again, explicit knowledge of the anisotropic direction is needed a priori. In our approach, the idea of [36] is extended to the p-version of the FEM. As the lowest order edge contributions K N0 ,N0 are already solved with a direct solver and the inner degrees of freedom K int,int are eliminated by static condensation, only the face contributions Kˆ f, f are affected by the anisotropy B [37]. Thus we can modify the initial face blocks Kˆ f, f of the preconditioner matrix C (13.46) by grouping all unknowns of the faces perpendicular to the thin direction Bai in one diagonal block Kˆ f, f , if the aspect ratio of the element exceeds a user-defined threshold ARth . We can even generalize the idea, allowing for two anisotropic/thin directions within one element, e.g. in Fig. 13.19 the faces f 4 to f 10 couple strongly, as the size in both, ξ- and ζ-direction, is small compared to the extend in η-direction. This is especially useful in meshes with tensor-product structure.

f3

ζ η

f6

f2

f8 f5

f 10

f1

f7 f4

f9

ξ thin direction

Fig. 13.19 Thin structure with 2 distinct face groups {f1, f2, f3} and {f4, . . . , f10}, where η is the long direction

446

13 Algebraic Solvers

The modified preconditioner matrix is then defined as C=



Kˆ N0 ,N0 0

0 Bai Kˆ f, f



(13.47)

.

Bai The procedure for computing Kˆ f, f without explicit knowledge of the thin direction(s) is sketched in Algorithm 5. It collects strongly coupled (thin) faces in a graph Bai and determines the blocks of Kˆ f, f by calculating the set of connected components of it. As the only information needed for the algorithm is the size of the elements in each direction, the procedure can be applied to general 3-D elements (tetrahedra, wedges, pyramids) [37]. The applicability of the method is demonstrated for a typical 45◦ -multi-step-lap joint region of a transformer core (see Fig. 13.20) with 4 layers of steel sheet, each 0.24 mm in thickness, with a step-lap of 2 air gaps (width: 1 mm) in each sheet. As excitation, we apply a prescribed flux density B 0 = 0.1 − 2.5 T in y-direction. The model is discretized by 2,568 hexahedral elements and 3,172 nodes. The used nonlinear BH curve is depicted in Fig. 13.21.

30 cm

air gaps (exploded view)

15 cm 0.96 mm

z y

x

1 layer

B0

Fig. 13.20 Model of step-lap core without air domain (scale factor 30 in thickness direction)

Fig. 13.21 Approximation of measured BH data (see Sect. 6.7.5)

13.5 Block Preconditioner for Higher Order Edge Element Discretization

447

Fig. 13.22 Concentration of magnetic flux lines in corner (45◦ -view) for uniform p = 0 (top) and p = 3 (bottom) with B0 = 1.0 T (scale factor 30 in thickness direction)

Initially, we choose an isotropic polynomial degree p = 0, . . . , 3 and compare the spatial resolution of the magnetic flux density near the air gaps. Here, the Newton algorithm (see Sect. 6.7.4) takes between 3 and 9 iterations. The results of the simulation are visualized on a very fine postprocessing mesh. In Fig. 13.22 it is clearly visible that the continuation of the flux lines across the air Algorithm 5: Definition of anisotropic face blocks. Input: elements e of mesh T Output: groups of thin faces Gi , i = 1, . . . , n G Data: graph of connected anisotr. faces G = (v, e) foreach e ∈ T do if A Rmax (e) ≥ A Rth then compute size of element w.r.t. local directions (h ξ , h η , h ζ ) h max = max(h ξ , h η , h ζ ) foreach d ∈ {ξ, η, ζ} do if h d / h max ≥ A Rth then get faces f 1 , f 2 perpendicular to d-direction insert ( f 1 , f 2 ) in G n G = # of non-connected components of G for i = 1 to n G do Gi = connected faces in i-th component of G

gaps is poorly approximated for p = 0 and that the curvature of the streamlines is non-physical between the air gaps. In contrast, the simulation using p = 3 resolves accurately the flux concentration above and below the air gaps (depicted in red). The same observation holds true for the absolute value of the magnetic flux density in Fig. 13.23, where the flux concentration between the air gaps is smeared over a large area for p = 0. From Fig. 13.24 we deduce, that the iterative method is by a factor 2–10 slower compared to the direct method for all excitation values B 0 . In contrast, the memory consumption is only about 50 % compared to the direct one for higher polynomial degrees p, as seen in Table 13.10 (SC denotes the use of static condensation).

448

13 Algebraic Solvers

Fig. 13.23 Flux distribution for uniform p = 0 (top) and p = 3 (bottom) with B0 = 1.0 T (scale factor 30 in thickness direction)

Fig. 13.24 Simulation time for direct and iterative solution approach without anisotropic block preconditioner

Table 13.10 Memory requirement and DOFs for different polynomial degrees (SC: static condensation) Polynomial degree piso 0 1 2 3 # Total DOFs # Inner DOFs Direct solver Direct solver (SC) Iterative 1-step (SC)

7.824 43.956 − 12.840 Memory usage (GB) 0.24 0.39 0.24 0.37 0.24 0.28

141.840 71.904

332.292 208.008

1.12 0.92 0.53

3.61 2.32 1.61

The poor runtime performance of the iterative 1-step scheme in Fig. 13.24 can be explained by the extremely high aspect ratios up to 1:1,000 (see Fig. 13.25). All elements within the steel sheets have aspect ratios higher than 1:400, leading to more than 3,000 CG iterations on average. If we utilize the anisotropic preconditioner (13.47) for varying aspect ratio thresholds ARth = {1,000, 500, 100, 50, 10} the iteration numbers and time for solving the linear equation system drops significantly (see Fig. 13.26). The results are compared

13.5 Block Preconditioner for Higher Order Edge Element Discretization

449

Fig. 13.25 Aspect ratio of step-lap setup (not shown for elements in air)

Fig. 13.26 Reduction of CG iterations (left) and solution time (right) for varying aspect ratio threshold ARth

for the iterative 1-step solver with B 0 = 1.0 T. From an initial CG iteration count of about 3,000 (ARth = 1,000) we achieve an average reduction to 100–150 iterations for ARth = 10, corresponding to a factor of 20–30, depending on the polynomial degree. The effect on the solution time is similar, where a reduction by a factor of 9 ( p = 3) to 25 ( p = 1) can be achieved, making it comparable in runtime to the direct solver. The increase in memory for storing larger diagonal blocks is very moderate, being in the range of 5–15 % compared to the non-blocked version. The rate of reduction in iterations is not heavily depending on the polynomial degree, making the preconditioner a p-robust method for practical applications. For all the following results, we apply the preconditioner with a default threshold of ARth = 10. Finally, we utilize the strategy as explained in Sect. 6.7.6 by reducing the polynomial degree anisotropically in thickness direction pζ < pξ , pη ,. The results can be seen in Fig. 13.27. The drop in runtime for switching globally to the anisotropic degree paniso is already between 25 and 50 %. An even larger reduction is achieved by choosing pair = 0, i.e. the lowest order N´ed´elec elements only, which reduces the runtime by a factor of 3. In Table 13.11 a similar tendency can be observed for

450

13 Algebraic Solvers

Fig. 13.27 Total runtime (iterative 2-step approach) for different choices of polynomial degrees and ARth = 10 Table 13.11 Comparison of isotropic and anisotropic polynomial degrees (2-step iterative solver, ARth = 10, B0 = 1.0 T) Polynomial degree Total Inner Memory Total DOFs DOFs (GB) time (s) piso = 2 paniso = (2, 2, 1) psteel = (2, 2, 1), pair = 0 piso = 3 paniso = (3, 3, 1) psteel = (3, 3, 1), pair = 0

141.840 95.655 24.224 332.292 168.044 37.448

71.904 46.743 9.232 208.008 75.476 14.840

0.56 0.43 0.29 1.61 0.75 0.39

193 151 48 1.050 470 141

the memory consumption, where only 25 % of the memory is needed compared to the isotropic version for the finest model. For all variations of the polynomial degree, there was no visible difference between the isotropic and the anisotropic version in the distribution of the flux density.

References 1. O. Biró, K. Preis, On the use of the magnetic vector potential in the finite element analysis of three-dimensional eddy currents. IEEE Trans. Magn. 25(4), 3145–3159 (1989) 2. Y. Kawase, O. Miyatani, T. Yamaguchi, Numerical analysis of dynamic characteristics of electromagnets using 3-d finite element method with edge elements. IEEE Trans. Magn. 30(5), 3248–3251 (1994) 3. D. Arnold, R. Falk, R. Winther, Multigrid in H(div) and H(curl). NUMEM 85, 197–218 (2000) 4. R. Hiptmair, Multigrid method for Maxwell’s equations. SIAJN 36(1), 204–225 (1999) 5. M. Schinnerl, J. Schöberl, M. Kaltenbacher, Nested multigrid methods for the fast numerical computation of 3D magnetic fields. IEEE Trans. Magn. 36(4), 1557–1560 (2000) 6. W.L. Briggs, A Multigrid Tutorial, SIAM, 1987

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7. U. Rüde, Mathematical and computational techniques for multilevel adaptive methods, (SIAM, 1993) 8. J.W. Ruge, K. Stüben, In Algebraic Multigrid (AMG), Multigrid Methods, Frontiers in Applied Mathematics, ed. by S. McCormick (SIAM, Philadelphia, 1986), pp. 73–130 9. Y. Saad, Iterative Methods for Sparse Linear Systems (PWS Publication, Company, 1996) 10. D. Braess, Finite Elemente (Springer, Berlin, 2003). (3. korregierte Auflage) 11. W. Hackbusch, Multigrid Methods and Application (Springer, Berlin, New York, 1985) 12. M. Jung, U. Langer, A. Meyer, W. Queck, M. Schneider, Multigrid Preconditioners and their Application. In: Proceedings of the 3rd GDR Multigrid Seminar held at Biesenthal, KarlWeierstraß-Institut für Mathematik, pp. 11–52, May (1989) 13. J. Schöberl, An advancing front 2D/3D-mesh generator based on abstract rules, Comput. Vis. Sci. no. 1, 41–52, (1997) 14. T. Nakata, N. Takahashi, H. Morishige, J.L. Coulomb, C. Sabonnadiere, Analysis of 3D Static Force Problem. In: Proceedings of the TEAM Workshop on Computation of Applied Electromagnetics in Materials, pp. 73–79, (1993) 15. B. Heise, Mehrgitter-Newton-Verfahren zur Berechnung nichtlinearer magnetischer Felder, Ph.D. thesis, TU Chemnitz, 1994 16. K. Fujiwara, T. Nakata, H. Ohashi, Improvement for convergence characteristic of ICCG method for the A- method using edge elements. IEEE Trans. Magn. 32(3), 804–807 (1996) 17. D. Braess, Towards algebraic multigrid for elliptic problems of second order. COMPU 55, 379–393 (1995) 18. F. Kickinger, U. Langer, A note on the global extraction element-by-element method. ZAMM, no. 78, 965–966 (1998) 19. S. Reitzinger, Algebraic Multigrid Methods for Large Scale Finite Element Equations. Reihe C—Technik und Naturwissenschaften, no. 36, Universitätsverlag Rudolf Trauner, (2001) 20. P. Vanek, J. Mandel, M. Brezina, Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems. COMPU 56, 179–196 (1996) 21. A. Brandt, Algebraic multigrid theory: The symmetric case. APPMC 19, 23–56 (1986) 22. M. Kaltenbacher, H. Landes, S. Reitzinger, R. Peipp, 3D simulation of electrostatic-mechanical transducers using algebraic multigrid. IEEE Trans. Magn. 38(2), 985–988 (2002) 23. M. Kaltenbacher, S. Reitzinger, Algebraic Multigrid for Solving Electromechanical Problems. Multigrid Methods VI, Springer, Lecture Notes in Computational Science and Engineering, (1999) 24. M. Kaltenbacher, S. Reitzinger, Nonlinear 3D magnetic field computations using Lagrange FE-functions and algebraic multigrid. IEEE Trans. Magn. 38(2), 1489–1496 (2002) 25. M. Kaltenbacher, S. Reitzinger, J. Schöberl, Algebraic multigrid method for solving 3D nonlinear electrostatic and magnetostatic field problems. IEEE Trans. Magn. 36(4), 1561–1564 (2000) 26. S. Reitzinger, M. Kaltenbacher, Algebraic multigrid methods for magnetostatic field problems. IEEE Trans. Magn. 38(2), 477–480 (2002) 27. S. Reitzinger, J. Schöberl, Algebraic multigrid for edge elements. Numer. Linear Algebra Appl. 9, 223–238 (2002) 28. S. Reitzinger, U. Schreiber, U. van Rienen, Algebraic Multigrid for Complex Symmetric Matrices. Numerical Study, Technical Report 02–01, Johannes Kepler University Linz, SFB: Numerical and Symbolic Scientific Computing, (2002) 29. M. Rausch, M. Gebhardt, M. Kaltenbacher, H. Landes, Magnetomechanical field computation of a clinical magnetic resonance imaging (MRI) scanner. COMPEL—Int. J. Comput. Math. Electr. Electron. Eng. 22(3), 576–588 (2003) 30. G. Haase, M. Kuhn, U. Langer, S. Reitzinger, J. Schöberl, Parallel Maxwell solvers, scientific computing in electrical engineering. U. Van Rienen, M. Günther, and D. Hecht, (eds.), Lecture Notes in Computational Science and Engineering, vol. 18, Springer, (2000) 31. G. Haase, M. Kuhn, S. Reitzinger, Parallel AMG on distributed memory computers. SIAM J. Sci. Comput. 24(2), 410–427 (2002)

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32. J. Schöberl, S. Zaglmayr, High order Nédélec elements with local complete sequence properties. COMPEL—Int. J. Comput. Math. Electr. Electron. Eng. 24(2), 374–384 (2005) 33. O. Schenk, K. Gärtner, On fast factorization pivoting methods for symmetric indefinite systems. Elec. Trans. Numer. Anal. 23, 158–179 (2006) 34. O. Schenk, K. Gärtner, Multigrid method for Maxwell’s equations. SIAM J. Numer. Anal. 36(1), 204–225 (1999) 35. A. Kameari, Improvement of ICCG convergence for thin elements in magnetic field analyses using the finite-element method. IEEE Trans. Magn. 44(6), 178–1181 (2008) 36. Ch. Chen, O. Bíró, Geometric multigrid with plane smoothing for thin elements in 3-D magnetic fields calculation. IEEE Trans. Magn. 48(2), 443–446 (2012) 37. A. Hauck, M. Ertl, J. Schöberl, M. Kaltenbacher, Accurate magnetostatic simulation of step-lap joints in transformer cores using anisotropic higher order FEM. COMPEL 32(5), 1581–1595 (2013)

Chapter 14

Industrial Applications

14.1 Electrodynamic Loudspeaker The electrodynamic loudspeaker to be investigated is shown in Fig. 14.1. A cylindrical, small, light, voice coil is suspended freely in a strong radial magnetic field, generated by a permanent magnet. The magnet assembly, consisting of pole plate and magnet pot, helps to concentrate most of the magnetic flux within the magnet structure and, therefore, into the narrow radial air gap. When the coil is loaded by an electric voltage, the interaction between the magnetic field of the permanent magnet and the current in the voice coil results in an axial Lorentz force. The voice coil is wound onto a former, which is attached to the rigid, light, cone diaphragm in order to couple the forces more effectively to the air and, hence, to permit acoustic power to be radiated from the assembly. The main function of the spider and the surround is to allow free axial movement of the moving coil driver, while non-axial movements are suppressed almost completely. Since in the case of a loudspeaker the interaction with the ambient fluid must not be neglected, the electrodynamic loudspeaker represents a typical coupled magnetomechanical system immersed in an acoustic fluid. This is, why for the detailed finite element modeling of these moving-coil drivers the magnetic, the mechanical as well as the acoustic fields including their couplings have to be considered as one system, which cannot be separated. Furthermore, electrodynamic loudspeakers in the low-frequency range under large-signal conditions show a strongly nonlinear behavior, which is caused mostly by two factors—the inhomogeneity of the magnetic field in the air gap, i.e., magnetic nonlinearities, and the nonlinearity of the suspension stiffness, i.e., mechanical nonlinearities. These nonlinearities are caused by the large vibration amplitudes, especially at low frequencies. For large input powers the distortions increase rapidly and reach the same order of magnitude as the fundamental. To reduce the efforts in the development of electrodynamic loudspeakers, precise and efficient computer modeling tools have to be used. With these computer simulations, the costly and lengthy fabrication of a large number of prototypes, required in optimization studies by conventional experimental design, can be reduced tremen© Springer-Verlag Berlin Heidelberg 2015 M. Kaltenbacher, Numerical Simulation of Mechatronic Sensors and Actuators, DOI 10.1007/978-3-642-40170-1_14

453

454

14 Industrial Applications Magnet pot Surround

Cone diaphragm Permanent magnet Former Pole plate

Spider Voice coil

Magnet holder

Fig. 14.1 Schematic of an electrodynamic cone loudspeaker

dously. For many applications an equivalent electromechanical circuit model has been developed (see e.g., [1]). However, the main drawback of these simulation models is that the circuit-element parameters have to be determined empirically by measurements on a prototype. Therefore, we will demonstrate that a model based on the partial differential equations including all coupling terms and solved by an appropriate FE method (see Chaps. 8 and 11) can totally fulfill the needs of an engineer. Such a method just requires the geometry of the loudspeaker as well as material data of each part. In the following, we will first discuss finite element models for the small- and large-signal behavior. Later, comparisons between simulation results and accordingly measured data are shown for verification purposes. The main focus will be on the demonstration of the practical usability of this scheme within the industrial computeraided engineering of electrodynamic loudspeakers.

14.1.1 Finite Element Models 14.1.1.1 Small-Signal Computer Model The finite element discretization of the electrodynamic loudspeaker under smallsignal conditions is shown in Fig. 14.2. Here, the voice coil is discretized by socalled magnetomechanical coil elements based on the motional emf-term method (see Sect. 11.3.4), which solves the equations governing the electromagnetic and mechanical field quantities and takes account of the full coupling between these fields. Due to the concentration of the magnetic flux within the magnet assembly, the magnet structure and only a small ambient region have to be discretized by magnetic finite elements. Furthermore, the surround, spider, diaphragm, and former are modeled by mechanical finite elements. Finally, the surrounding fluid region in front of the loudspeaker is discretized by acoustic finite elements. The fluid region is surrounded by infinite elements for allowing open-domain computation (see Sect. 5.5.1), which

14.1 Electrodynamic Loudspeaker

455

Rotational axis Magnetic-acoustic finite elements

Acoustic finite elements

Magnetomechanical coil elements based on motional emfterm method

A=0

Magnetic finite elements Coupling acoustic fieldmechanical field

Acoustic finite elements

Infinite elements

Linear mechanical finite elements Loudspeaker

Fig. 14.2 Small-signal finite element model of an electrodynamic loudspeaker

have to be located in the far field of the moving-coil driver in order to work correctly. The input level of these simulations is 1 W referred to 4 .

14.1.1.2 Large-Signal Computer Model The finite element discretization of the electrodynamic loudspeaker under largesignal conditions is shown in Fig. 14.3. The following modifications have been performed in comparison to the above-explained small-signal computer model:

Rotational axis Magnetic elements for movingmaterial method Magnetomechanical coil elements based on movingmaterial method Magnetic finite elements

A=0 Mechanical finite elements without nonlinearities Spring - elements

Mechanical finite elements with geometric+material nonlinearities

Fig. 14.3 Large-signal finite element model of an electrodynamic loudspeaker

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14 Industrial Applications

1. To take into account the variation of the force factor for a coil under large excursions (i.e., magnetic nonlinearities), the magnetomechanical coil elements discretizing the voice coil of the loudspeaker are based on the moving material method (see Sect. 11.3.4). The force factor is defined by α f = Bl N , with B the magnetic induction in the air gap, l the length of one winding and N the number of turns of the voice coil, which are located in the homogeneous magnetic field. 2. Furthermore, first simulation results showed that the mechanical nonlinearities, i.e., the geometric nonlinearity as a result of large displacements and the material nonlinearity due to a nonlinear stress–strain relationship have to be taken into consideration only for the spider. Therefore, to allow a more efficient computation of the large-signal behavior the diaphragm and the surround are discretized by finite elements solving linear mechanics. 3. Finally, measurements have shown that the distortion factors of the near field and diaphragm acceleration are in excellent agreement. Due to this correlation a modified axisymmetric finite element model has been applied, in which acoustic elements were eliminated completely (see Fig. 14.3). The influence of the surrounding air, which consists of mass-loading effects and damping due to the sound emission, is now realized by so-called spring elements. These elements have been located on the outside boundary of the surround and diaphragm.

14.1.2 Verification of Computer Models The verification of the computer models described above has been performed by comparing simulation results with corresponding measured data. In the first step, the most important small-signal results (frequency dependencies of the electrical input impedance, diaphragm acceleration and axial sound pressure levels as well as ThieleSmall parameters [2]) were considered. As can be seen in Fig. 14.4, good agreement between simulation results and measured data was achieved. Next, the force–displacement characteristics were measured and compared with simulations (see Fig. 14.5). After this basic validation of the large-signal computer model, the total harmonic distortion (THD) factors of the voice coil currents and diaphragm accelerations at large-signal conditions have been calculated, which compute as

THD = 



pˆ 22 + pˆ 32 + · · ·

pˆ 12 + pˆ 22 + pˆ 32 + · · ·

,

(14.1)

14.1 Electrodynamic Loudspeaker

457

|Z| ( )

SPL (dB/Watt (4

(a) 25

(b) 100

20

80

10 5 0

Measurement

90

Finite element simulation

15

Measurement 20 100 1k Frequency (Hz)

))

Finite element simulation

70

5k

20

100 1k Frequency (Hz)

8k

Fig. 14.4 Comparison of simulated and measured small-signal results. a Frequency dependency of electrical input impedance Z. b Axial small-signal sound response level SPL at 1 m distance

Displacement (mm)

(a)

8

THD at 32 W (%)

(b) 100 Measurement

Measurement

4 0

10

FE - Simulation with geom. and material nonlinearities

-4 -8 -20

-10

0 10 Force (N)

FE - Simulation with all nonlinearities 1

20

20

50 100 Frequency (Hz)

200

Fig. 14.5 Comparison of simulated and measured large-signal results. a Force-displacement characteristic of the loudspeaker. b Total harmonic distortion (THD) of diaphragm acceleration at an input power of 32 W

with pˆ i the amplitude of the ith harmonic. In addition, we define the pˆ 2

k2 = 

pˆ 12 + pˆ 22 + pˆ 32 + · · ·

k3 = 

pˆ 12

pˆ 3 +

pˆ 22

(14.2)

.

(14.3)

+ pˆ 32 + · · ·

The input level of these simulations was 32 W referred to 4 . As can be seen in Fig. 14.5, the good agreement of measured and simulated results over a wide frequency range validates the large-signal model depicted in Fig. 14.3.

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14.1.3 Numerical Analysis of the Nonlinear Loudspeaker Behavior Measurements as well as simulation results show that at frequencies f < 60 Hz the odd-order harmonics and at higher frequencies the even-order harmonics dominate. The large advantage of computer modeling is the separation of the different nonlinearities for the different components of the loudspeaker. In this way, the influence of the different nonlinear effects on the loudspeaker behavior can be very efficiently extracted and researched in the simulation. For example, simulations showed that the magnetic nonlinearities cause notable quadratic distortion factors at frequencies f > 60 Hz, whereas the mechanical nonlinearities cause the rapid increase of the even-order harmonics in the lower-frequency range (see Fig. 14.6). Furthermore, both mechanical and magnetic nonlinearities are responsible for the cubic distortion factor. In the next step of the numerical analysis, the influence of design parameters of the magnet system on the distortion factors has been investigated. Simulations considering only magnetic nonlinearities showed that large coil flux variations result in notable odd-order harmonics. On the other hand, a unsymmetric magnetic field in the air gap causes large coil offsets resulting in significant even-order harmonics. Further computations showed that the position of the permanent magnet has a big influence on the symmetry of the magnetic field in the air gap and therefore can be used in the optimization of the system (see Fig. 14.7b). Furthermore, it could be shown that the transient magnetic field of the current-carrying voice coil must not be neglected at large-signal conditions. To reduce the influence of the coil field under large-signal conditions on the symmetry of the force factor, the whole magnet pot has to be saturated and the upper air gap above the pole plate has to be increased (see Fig. 14.7b). These design modifications result in a much more symmetric decrease of

k 3 (%)

k 2 (%)

(b) 100

(a) 100

Both nonlin.

Both nonlin. Magnetic nonlin.

Magnetic nonlin.

10

10

Mechanical nonlin.

1 20

50 100 Frequency (Hz)

200

1 20

Mechanical nonlin. 50 100 Frequency (Hz)

200

Fig. 14.6 Numerical investigation of distortion factors of diaphragm acceleration at an input power of 32 W. a Quadratic distortion factor k2 . b Cubic distortion factor k3

14.1 Electrodynamic Loudspeaker

(a)

(b)

459

New permanent magnet position Increase of upper air gap Saturation of magnet pot Shorter gap design Increase of magnet width

(c)

Fig. 14.7 Finite element models. a Original magnet system. b Optimized magnet system. c Original and optimized spider

Normalized force - factor Bl (%)

Displacement (mm)

(a) 0

(b) 8

-10 -20

4

-30 -40

Optimized

Original -50 -60 4 6 -10 -6 -4 0 10 Axial coil displacement (mm)

Original Optimized

0 -4 -8 -20

-10

10 0 Force (N)

20

Fig. 14.8 Comparison of the original and optimized loudspeaker. a Simulated coil flux variation (normalized to the original small-signal value). b Simulated force-displacement characteristic

the force factor (see Fig. 14.8a). Furthermore, to minimize the variation of the force factor, i.e., to raise the so-called jump-out excursion, the thickness of the pole plate has to be reduced (see Figs. 14.7b and 14.8a). Finally, since this design modification results in a smaller efficiency of the loudspeaker, the width of the permanent magnet has to be increased. After the above-explained numerical analysis of magnetic nonlinearities, the influence of design parameters of the spider on the distortion factors caused by the mechanical nonlinearities has been investigated. Simulations considering magnetic and mechanical nonlinearities showed that a larger spider height results in a more linear force–displacement characteristic and significantly smaller odd-order harmonics Furthermore, a continuous displacement of each midpoint of the spider grooves causes a more symmetric force–displacement characteristic resulting in smaller even-order harmonics. For a more detailed discussion we refer to [2].

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14 Industrial Applications k 2 (%)

THD (%)

(a) 100

(b) 100 Measurement: Original Optimized

10

Measurement: Original Optimized

10

FE - simulation 1 20

k 3 (%)

50 100 200 Frequency (Hz)

1 20

(c) 100 Measurement: Original Optimized 10

FE - simulation 50 100 200 Frequency (Hz)

1 20

FE simulation

50 100 200 Frequency (Hz)

Fig. 14.9 Comparison of simulated and measured distortion factors of the optimized loudspeaker (at an input power of 16 W). a Total harmonic distortion (THD). b Quadratic distortion factor k2 . c Cubic distortion factor k3

14.1.4 Computer Optimization of the Nonlinear Loudspeaker Behavior In the course of this computer optimization, the knowledge of the sensitivity studies explained in the previous section was put into a new prototype to reduce the even- and odd-order harmonics under large-signal conditions. As can be seen in Fig. 14.9, significant smaller distortion factors were achieved. In particular, cubic distortion factors could be reduced tremendously. For example, at a frequency of 20 Hz the improvement is 70 % with respect to the original loudspeaker. This significant reduction of cubic harmonics is in accordance with studies concerning the subjective perception of low-frequency distortions [3]. According to [3], odd-order harmonics are above all responsible for the deterioration of the sound quality. Furthermore, the important ancillary condition of a similar small-signal behavior with respect to the original loudspeaker must be fulfilled. Small-signal simulations resulted in an acceptable reduction in efficiency of 0.5 dB. Furthermore, the numerically predicted improvements in the large-signal behavior of the loudspeaker could be successfully confirmed by measurements on the new prototype (see Fig. 14.9). Therefore, it can be stated that the presented simulation scheme is well suited to the industrial computer-aided design of electrodynamic loudspeakers, since an optimization with a significantly reduced number of prototypes can be achieved.

14.2 Noise Computation of Power Transformers The sound emission of power transformers conflicts more and more with tightened low emission standards, which must be fulfilled, especially at night. Therefore, the prediction and reduction of these sound emissions is of increasing interest for the

14.2 Noise Computation of Power Transformers

461

electrical power industry. The transformer noise is mainly caused by the following sources [4, 5]: 1. The no-load noise caused by magnetostrictive strain of core laminations. 2. The noise produced by fans or oil pumps. 3. The load-controlled noise caused by Lorentz forces resulting from the interaction between the magnetic stray field of one current-carrying winding and the total electric currents in the conductors of the other winding. These forces cause vibrations of the winding and result in acoustic radiations with twice the line frequency (100 or 120 Hz). During recent decades the magnetic noise caused by magnetostrictive strain of the core laminations and the noise of fans have been investigated and considerably decreased [6–8]. Therefore, the coil-emitted noise (see item 3 above) is of increasing interest. At the moment, approximate empirical prediction formulas, which primarily depend only on the rated power of the transformer, represent the state-of-the-art. However, the main disadvantage of these prediction formulas is that accurate parameters on the load-controlled noise are not available. Thereby, we have developed a special adapted computational scheme as displayed in Fig. 14.10.

Voltage Axisymmetric finite element model of one winding

Outermost winding surface displacements

3D finite element model of oil-filled tank

Tank surface displacements

3D finite element model of tank and test hall

Acoustic pressure Fig. 14.10 Overview of the developed calculation scheme

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14 Industrial Applications

14.2.1 Finite Element Models The goal of the numerical simulation is to predict very precisely the emitted sound of loaded transformers within a test hall. In the first step of the modeling scheme the outermost winding surface displacements are calculated by an acousticmagnetomechanical finite element model of one winding of the oil-filled transformer. Due to rotational symmetry of the winding and symmetric load (as a result of the ideal measurement condition in the factory test field), a 2D finite element model based on axisymmetric elements can be used (see Fig. 14.11). In the finite element model, the voltage-loaded conductors of the winding are discretized using magnetomechanical coil elements. These elements solve the equations governing the electric circuit, the magnetic as well as the mechanical field equations, and take account of the full coupling between these fields (see Chap. 11). All winding clamping and insulation materials between each coil as well as the winding support platforms are modeled using magnetomechanical finite elements, which solve the coupled magnetic and mechanical field equations. Instead of a complete model of the highly permeable core by magnetic finite elements, this computer model was simplified by applying Neumann boundary conditions at the boundary of the winding window (see Fig. 14.11). Furthermore, the surrounding oil within the tank is discretized using purely acoustic finite elements and magnetic-acoustic finite elements (solving the

Rotational axis

Mechanical finite elements for tank Acoustic finite elements for surrounding oil Neumann - boundary conditions Magnetic-acoustic finite elements for surrounding oil within the winding window Spring - elements Magnetomechanical finite elements for winding clampings and insulation tubes

LV

HV

Magnetomechanical coil elements for conductors (LV, HV, coarse and fine tapping winding)

Fig. 14.11 Axisymmetric acoustic-magnetomechanical finite element model of one winding of the oil-filled power transformer

14.2 Noise Computation of Power Transformers

463

magnetic as well as the acoustic partial differential equation without any coupling). Finally, the tank is modeled using standard mechanical finite elements. Additionally, the following aspects have to be considered for the precise computer simulation of the winding vibrations of loaded power transformers: • To measure the load-controlled noise in a factory test field, the transformer has to be operated at short-circuit and at rated currents. In this case, due to the small voltage during the short-circuit test, the core-emitted noise can be neglected and, therefore, a clear distinction between the no-load noise and the load-controlled noise can be achieved. In the simulations, this effect was taken into account by modeling the innermost low-voltage (LV) winding as a voltage-loaded coil with an external voltage of zero. The high-voltage (HV) winding and both in-series connected, outermost tapping windings, however, are loaded with the measured short-circuit voltage in a star connection. • Furthermore, the core clamping supports have been ignored in the finite element model to reduce the effort. Therefore, the influence of these supports, which consists of an additional axial stiffness of the winding, is realized by so-called spring elements. As shown in Fig. 14.11, these spring elements have been located at the outside boundary of the upper and lower winding support platform. In the simulations, a stiffness of 85 MN/m has been used, which is in accordance with the experience of the transformer manufacturers. • Finally, since measurement results revealed a big influence of the tap-changer position on the measured vibrations and sound pressure levels, the following simulations have been performed for three nominal positions: – Tap-changer position 1: The HV winding and both tapping windings are connected in series. – Tap-changer position 2: The HV winding and the coarse tapping winding are connected in series. – Tap-changer position 3: Only the HV winding is connected. For the computation of the amplitude of the winding surface displacements, a dynamic analysis using a sinusoidal 50 Hz (or 60 Hz) excitation signal for the voltage between the two supply terminals of the high-voltage winding was performed. It should be noted that further input parameters are the geometry of the power transformer, the density, modulus of elasticity, Poisson’s ratio and loss factor for the mechanical materials (tank, conductors, insulation and clamping materials), the electrical conductivity for the conductors as well as the density and bulk modulus for the surrounding oil. After the computation of the response signals (current in the conductors of both windings as well as mechanical displacements of the outermost winding), the Fourier transform of the output signals has to be calculated. Finally, the 100-Hz (or 120-Hz) component of the spectrum must be extracted. In the second step, the previously calculated winding surface displacements are now taken as mechanical excitation in a 3D acoustic-mechanical finite element model of the complete oil-filled tank (see Fig. 14.12). Furthermore, in the 3D finite element model the 120◦ phase shift between the three windings is taken into account. In this

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14 Industrial Applications Mechanical finite elements for tank

Acoustic finite elements for surrounding oil Mechanical finite elements for core clampings Mechanical excitations using results of previous 2D-simulation (120° phase shift between 3 windings)

Fig. 14.12 Three-dimensional acoustic-mechanical finite element model of an oil-filled transformer

model the tank, the core clamping as well as the connections between core clamping and top of the tank, are modeled using mechanical finite elements. Furthermore, the surrounding oil within the tank is discretized using acoustic finite elements. Finally, the additional stiffness of the core clamping supports is again implemented using spring elements. Since the load-controlled noise is primarily a simple 100-Hz (or 120-Hz) tone, a harmonic analysis has been performed. In the last step the previously calculated tank-surface vibrations are now applied as mechanical excitation in the final acoustic simulations to calculate the radiated transformer noise. Here, the radiation within closed rooms such as a high-voltage laboratory are calculated (see Fig. 14.10). For the computation of the sound radiation within a high-voltage laboratory, a 3D acoustic-mechanical finite element model has been set up (see Fig. 14.13). Here, the tank of the transformer is discretized using

Acoustic finite elements for surrounding air

Mechanical finite elements for tank

Fig. 14.13 Three-dimensional acoustic-mechanical finite element model of the transformer tank and the high-voltage laboratory

14.2 Noise Computation of Power Transformers

465

mechanical finite elements. Furthermore, the surrounding air within the test hall is modeled using acoustic finite elements. The walls of a typical high-voltage laboratory are not covered with any absorbing material. Therefore, in these simulations, the transformer was assumed to be positioned within a hall with ideally reflecting walls. In this computer simulation 640,000 3D finite elements have been used.

14.2.2 Verification of the Computer Models The verification of the computer models described above has been performed by comparing simulation results with corresponding measured data. It should be noted that due to the complexity of the sound emission of the loaded power transformer, analytic calculations are unavailable and therefore cannot be used for verification purposes.

14.2.3 Verification of the Calculated Winding and Tank-Surface Vibrations In the first step, the axisymmetric finite element model has been verified by comparing measured and calculated short-circuit currents, mechanical eigenfrequencies as well as winding surface accelerations. In Table 14.1, the short-circuit currents obtained by measurements as well as simulations are shown for two tap-changer positions. The good agreement between measured and calculated values (the deviation is within 1.25 %) validates again the developed coil-modeling scheme. Next, the measured and calculated transfer functions and mechanical eigenfrequencies of the complete winding system mounted on the core have been compared (see Fig. 14.14). The deviation at the second eigenfrequency is due to the fact that the actual winding does not show an exact axisymmetric construction. Furthermore, it should be noted that the winding was axially excited at the upper winding support platform and the resulting radial coil acceleration of the outermost winding was measured as well as simulated.

Table 14.1 Measured and simulated short-circuit currents (LV low voltage; HV high voltage) Measurement (A) Simulation (A) Current in LV winding, at tap-changer position 1 Current in HV winding, at tap-changer position 1 Current in LV winding, at tap-changer position 2 Current in HV winding, at tap-changer position 2

825 159 825 180

832 161 825 181

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14 Industrial Applications Transfer function

Radial coil acceleration Axial excitation force

(

2

m/s N

Mechanical eigenfrequencies

)

Measurement with piezoelectric accelerometer Finite element simulation

2. Res. 3. Res.

Measured

Simulated

1. Resonance

75 Hz

75 Hz

2. Resonance

135 Hz

145 Hz

3. Resonance

190 Hz

188 Hz

1. Res.

50

70

100 Frequency (Hz)

150

200

Fig. 14.14 Comparison of measured and simulated transfer functions and mechanical eigenfrequencies of the winding mounted on the core

In a final verification of the axisymmetric finite element model, the simulation results have been compared with corresponding measured winding surface accelerations. Here, the axial accelerations on the upper winding support platform and the radial vibrations on the outermost fine tapping winding were measured using an oilresistant piezoelectric accelerometer. The measurements showed that the sensitivity of these sensors against electromagnetic interferences of the high-voltage winding is negligible. Furthermore, the deviations of subsequent measurements were within a range of ±1.5 %. In Table 14.2, the winding vibrations of the transformer without a tank, in Table 14.3, the normalized accelerations of the transformer with an oil-filled

Table 14.2 Transformer without oil-filled tank: measured and simulated winding accelerations (m/s2 ) (m/s2 ) Radial coil acceleration at tap-changer position 1 Axial clamping acceleration at tap-changer position 1 Radial coil acceleration at tap-changer position 2 Axial clamping acceleration at tap-changer position 2

0.047 0.036 0.021 0.031

0.046 0.037 0.019 0.032

Table 14.3 Transformer with oil-filled tank: measured and simulated winding accelerations (normalized to the corresponding result without an oil-filled tank (see Table 14.2)) Measurement Simulation Radial coil acceleration at tap-changer position 1 Axial clamping acceleration at tap-changer position 1

0.59 0.96

0.62 0.99

14.2 Noise Computation of Power Transformers

467

Table 14.4 Measured and simulated tank accelerations Position Measurement Simulation Position (m/s2 ) (m/s2 ) 1 2 3 4 5

0.16 0.063 0.14 0.055 0.04

0.13 0.075 0.1 0.07 0.05

6 7 8 9

Measurement (m/s2 )

Simulation (m/s2 )

0.029 0.01 0.02 0.04

0.03 0.015 0.025 0.045

tank are compared, respectively. In the case of the transformer with an oil-filled tank (see Table 14.3), measurements as well as simulations reveal that the surrounding oil does not influence the axial accelerations of the winding support platform. However, due to the mass-loading effect of the surrounding oil, the radial coil acceleration amplitudes are nearly halved when compared to the vibrations ignoring the oil-filled tank. In summary, it can be stated that an axisymmetric finite element model precisely predicts the winding surface accelerations of a loaded power transformer for both configurations, with and without an oil-filled tank. Finally, the 3D acoustic-mechanical finite element model of the oil-filled tank has been verified by comparing the calculated tank side wall accelerations with values measured at nine different positions (see Table 14.4).

14.2.4 Verification of the Sound-Field Calculations After these basic validations of the computational models, the A-weighted soundpower level of the short-circuited transformer was measured in accordance with the European standard EN 60551 [9] and compared with the corresponding acoustic simulations. This standard requires that the A-weighted sound-pressure levels around the transformer have to be measured at a distance of 0.3 m from the tank surface and at half the tank height. Furthermore, these measurements have to be performed in a typical high-voltage laboratory at a transformer manufacturer. Due to this fact, the 3D finite element model for the calculation of the radiated noise within closed rooms, as shown in Fig. 14.13, has been used. An A-weighted sound power level of 66.5 dB(A) was calculated from the simulated sound-pressure levels. Considering the fact that the reproducibility of the sound-pressure measurements lies within a range of ±1 dB, a good agreement between measurement and simulation was achieved (see Table 14.5). Furthermore, as can be seen from Table 14.5, in this case the deviation between measured and calculated sound power level is considerably smaller than those resulting from the current-prediction formulas for the load-controlled noise. Therefore, it

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14 Industrial Applications

Table 14.5 Radiation within closed rooms: measured, simulated, and predicted sound-power levels Sound-power level (dB(A)) Sound-pressure measurement according to [9] Finite element simulation Empirical prediction formula according to [9] Empirical prediction formula according to [10]

68 66.5 63.5 60.5

can be concluded that this pure finite element scheme is well suited to the computation of the load-controlled noise of oil-filled power transformers, which are operated within a typical high-voltage laboratory.

14.2.5 Influence of Tap-Changer Position As a first application, the influence of the tap-changer position on the winding and tank surface vibrations as well as on the A-weighted sound-power level has been investigated. Table 14.6 shows the simulation results at tap-changer position 1 (HV winding and both tapping windings are connected in series) and at tap-changer position 3 (only the HV winding is connected). Simulations as well as measurements reveal that the radial vibrations of the outermost, fine tapping winding are greatly decreased at tap-changer position 3 as compared to position 1. With these simulations it was found that the radial magnetic volume forces acting on the innermost LV and HV winding are almost independent of the tap-changer position. Therefore, this reduction of the radial coil vibrations at tap-changer position 3 is based on the noncurrent-carrying, outermost, fine tapping winding. On the other hand, simulation results revealed that the axial magnetic volume forces acting on the LV and HV winding are larger at tap-changer position 3 when compared to position 1. This is caused by the edge-fringing effect and is responsible for the fact that the axial winding clamping vibrations are almost independent of the tap-changer position (see Table 14.6). Furthermore, due to the decrease of the radial surface accelerations of the outermost winding at tap-changer position 3, the

Table 14.6 Influence of the tap-changer position obtained by simulation Tap-changer position 1 Tap-changer position 3 Radial coil acceleration (m/s2 ) Axial winding clamping acceleration (m/s2 ) Tank side wall acceleration (m/s2 ) A-weighted sound power level (dB(A))

0.033 0.044 0.025 59

0.009 0.033 0.009 45

14.2 Noise Computation of Power Transformers

469

Table 14.7 Influence of stiffness of winding supports obtained by simulation (SPL sound-pressure level) Radial Axial winding SPL in coil acceleration (m/s2 ) (m/s2 ) 0.3 m (dB) With stiffness of winding support 85 MN/m 0.046 Without stiffness of winding support 0.044

0.037 0.62

56.8 82.3

tank-surface vibrations and, therefore, the calculated A-weighted sound-power level are greatly reduced. Therefore, in contrast to [10], where it is assumed that only axial winding vibrations are responsible for the load-controlled noise, these simulations clearly show that the radial coil vibrations also have a significant influence on the coil-emitted noise.

14.2.6 Influence of Stiffness of Winding Supports In a second application, the influence of the stiffness of the winding and core supports on the load-controlled noise has been investigated. This stiffness has been modeled in the finite element simulations by applying mechanical spring elements, which were located at the outside boundary of the upper and lower winding support platform (see Fig. 14.11). As expected, the simulations reveal that neglecting this axial stiffness causes significantly increased axial clamping accelerations and slightly decreased radial coil accelerations. Therefore, greatly increased sound-pressure levels result (see Table 14.7). These results indicate that this stiffness has a strong influence on the radiated transformer noise effect, which can be used in the optimization of the system.

14.3 Fast-Switching Electromagnetic Valves Electromagnetic high-pressure direct injection valves for gasoline motors are electromechanical devices. Loading the bobbin-mounted solenoid by an electric current results in a dynamic magnetic field. The later generates electromagnetic forces which act on the armature, and therefore changes the state of the valve. A mechanical spring element holds the valve in its default closed position. The gap size between the armature and stator corresponds to the maximal valve stroke. The valve motion is restricted by the end positions within this gap. The basic functional components of solenoid valves are the valve body, armature, plunger, spring and the coil (see Fig. 14.15). Modern fast-acting solenoidal valve applications demand further improvements of the operation speed and reproducibility of the opening and closing phase. The

470

14 Industrial Applications Fuel Intake

Bin T

Return Spring

2

Housing

Coil Stroke Gap Armature Valve Needle

Coil

Armature

Magnetic Saturation

1

Injection Nozzle

0

Fig. 14.15 Assembly of the electromagnetic valve and distribution of the magnetic flux density

development goes towards lightweight construction in combination with sophisticated energizing concepts. This gives rise to structural vibration as well as soundemission problems whose elimination by means of passive damping is not sufficient. A dynamic analysis of the switching behavior with sufficient precision taking into consideration all significant physical effects can be done only by a numerical analysis. The reproduction of the dynamics of fast-switching solenoidal valves proves to be highly nonlinear. Both the magnetization state of the ferromagnetic armature material and the induced eddy current distribution inside the armature during its accelerated motion within an inhomogeneous magnetic field are accounted for. In addition to the numerical calculation scheme presented in Sect. 11.3, we have to take impact dynamics into account to fully model a valve-switching cycle. Impact/contact problems are different from Neumann and Dirichlet boundary problems since the contact constraints are unknown in time and space and have to be determined as a part of the solution.

14.3.1 Modeling and Solution Strategy To precisely compute the dynamic behavior of a solenoidal actuator a numerical calculation scheme has to be able to handle the electromagnetic field, the mechanical field as well as the coupling terms. The strong coupling is necessary because of the interdependency between the position and velocity of moving mechanical parts and the inductivity of the solenoid as well as eddy current induction. Therefore, we are able to reproduce the dominant influence of eddy currents on the dynamics as well as the damping effect during the bouncing period at valve opening and closing.

14.3 Fast-Switching Electromagnetic Valves

471

14.3.1.1 Magnetic Field The governing equation describing the magnetic field can be derived from Maxwell’s equations for the quasistatic case (neglecting the displacement currents) using the magnetic vector potential as state variable (see Sect. 6.2.1). To precisely compute the magnetic field within a solenoidal actuator, the following physical phenomena have to be taken into account: • B–H curve: The measured B–H curve data are approximated by an enhanced smoothing spline technique to guarantee a smooth approximation of the curve as well as of its derivative. The resulting nonlinear magnetic equation is solved by a Newton method with a line search algorithm (see Sect. 6.7.5). • Voltage-loaded coil: For a voltage-loaded coil the additional circuit equation is simultaneously solved with the partial differential equation of the magnetic field (see Sect. 6.7.10).

14.3.1.2 Mechanical Field and Contact Mechanics For the mechanical field, Navier’s equation is solved with the mechanical displacement as state variable (Sect. 3.7). The impact of moving parts on the static parts of the valves at the switching of the valve state results in a noisy and unreliable bouncing, known as hard landing. A correct physical modeling of the impact as well as the overcoming of the inherent convergence problem of dynamic contact problems can be solved only by using a contact algorithm that satisfies all kinematic and kinetic conservation laws as can be found, e.g., in [11]. Including frictionless contact mechanics into the system starts with a contact-detection algorithm based on the normal distance of the two contacting bodies. In the case of satisfied contact conditions, the contact pressure in the normal direction is applied by using an exponential contact pressure–displacement relation pc (u) = p0 (l − g(u))m .

(14.4)

Here, p is the contact pressure and g the gap length, both in the normal direction and is dependent on the nodal displacements u within the discretized system. The surface hardness p0 can be considered as a penalty parameter to incorporate the contact constraints. The exponent m can be derived analytically using a statistical treatment of Hertzian micro-contacts and is verified by measurements within a range of 2.0–3.3 [12]. The constant l can be interpreted as the surface roughness and can be used to realize contact at a finite gap length as described later. Applying now the FE method to the continuum mechanics of contact, we have to add the contact force vector

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14 Industrial Applications

f c (u) =

ne 

f e (u) with f e =

e=1



pc (u)N dΩ

(14.5)

Ωe

to the right-hand side of the algebraic system of equations. The nodal contact force vector f c is given by the assembly of the element contact force vectors f e over all n e contact elements currently in closed contact. With pc we denote the contact pressure determined in (14.4). To achieve quadratic convergence in solving the nonlinear mechanical system with the Newton method we have to add further to the linear stiffness matrix (see Sect. 3.7.1) the tangential contact stiffness matrix [13]  f c (u n+1 )  Kc =   ∂u

,

(14.6)

u n+1

which is obtained by an exact and complete linearization of the contact force vector using a directional derivative with respect to the nodal displacements u at time step (n + 1). For further details we refer to [13].

14.3.2 Actuator Characteristics The direct acting and normally closed solenoid valve investigated can be characterized as a short-stroke valve with a compact design at a volume of 28 cm3 and an air-gap cross-sectional area of 61 mm2 . Valve opening is forced by the electromagnet against the force of the return spring with a spring constant of 6.7 kN/m and a pretensioning of 18 N. The limited stroke of max. 45 µm causes only a small dependency of the static actuator properties on the armature position in the complete operation range of the actuator. With increasing coil current and magnetic saturation, the inductivity of the electromagnet is decreasing in compliance with the decreasing permeability. At the same time, the gain in magnetic force diminishes due to the saturation. Investigating, furthermore, the actuator dynamics, the eddy-current-induced hysteresis can be acquired by using a high-level signal harmonic excitation at different frequencies (Fig. 14.16). The B–H magnetization curves of ferromagnetic valve components used for the numerical analysis have been adapted to the hysteresis measurement results. This procedure is necessary since some components are manufactured with particular mechanical and heat treatment whereby changing their original magnetic properties and a direct measurement of the saturation curves of small parts is full of uncertainty. Comparing simulation results using the adapted B–H magnetization curves with measurement results at higher frequencies, it can be shown that magnetic hysteresis effects are negligible in comparison to the eddy-current-induced hysteresis (Fig. 14.16).

14.3 Fast-Switching Electromagnetic Valves

473

Fig. 14.16 Eddy-current-induced hysteresis of magnetic flux, magnetic force, and coil inductivity

Eddy current induction increases with rising frequency and affects the diffusion rate and extension of the magnetic field inside the armature as well as its temporal expansion, known as the skin effect (see Sect. 6.2.2). Eddy currents are therefore a critical factor concerning the dynamic behavior of electromagnetic actuators since they constrain the operation range of the actuator. The magnitude and phase of the actuator frequency response (magnetic force over applied coil voltage, respectively current) is shown in Fig. 14.17. The transfer function shows a cutoff frequency of

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14 Industrial Applications 10 Normalized amplitude

Valve-operation range

Current-fed coil Voltage-fed coil

1

0.1

0.01

Phase ( O )

0 -40 -80 -120 1

10

100

1k f (Hz)

10 k

100 k

Fig. 14.17 Effect of eddy current induction on the frequency response

860 Hz for the current-fed coil, which is beyond the operation range of the valve. In summary, the dynamic magnetic force is a complex function of the system’s operation point defined by the parameters: • • • •

Armature position and velocity Applied coil current Magnetization state of the ferromagnetic material Induced eddy current distribution inside the armature.

14.3.3 Actuator Dynamics Investigating the actuator dynamics, it is instructive to consider first two types of coil energizing: A current-fed coil and a voltage-fed coil. Connecting a current source to the coil results in the shortest pull-in time since, due to the applied current, no voltage feedback effect can become active and the electromagnetic compensation process can be neglected. This forces the fastest movement of the armature and can be approximately realized by a digital controller unit. On the other hand, the use of a voltage source leads to much longer time constants due to the delayed rise of the coil current. As a result, applying a current, respectively voltage, Heaviside function to the coil, the current (voltage)-fed coil has minimal pull-in time of 0.16 ms (0.35 ms) and a minimum response delay time of 0.08 ms (0.26 ms) as shown in Fig. 14.18.

14.3 Fast-Switching Electromagnetic Valves

Current source

475

Voltage source

Pull-in time Response-delay time

Pull-in time Response-delay time

1.5

t (ms)

Strike velocity

v (m/s)

1.0

0.5

0 0

1 2 3 Normalized current I/IN, Normalized voltage U/UN

Fig. 14.18 Simulation results: valve-opening times and strike velocity

No further reduction of the pull-in time can be achieved without using a more sophisticated energizing concept or reduction of eddy current induction through design features. But although eddy currents are undesired, since they increase the switching times, they are desired to damp the bouncing of the armature. A short pull-in time comes with high strike velocity at the end of the valve-opening process (Fig. 14.18). Therefore, there is an optimization conflict between obtaining short valve-opening times and reducing bouncing effects caused by high impact velocities to achieve a reproducible valve function without reopening.

14.3.4 Dynamics Optimization I: Electrical Premagnetization The actuator dynamic can be affected basically by • Forcing the mechanical compensation process by enhancement of the dynamic magnetic force • Forcing the electrical compensation process by enhancement of the energizing power during the pull-in time. One method to accelerate the pull-in time of the valve is to premagnetize the actuator by a permanent magnet or electrically using a coil-current control unit. Premagnetization causes a magnetization state inside the armature at increased magnetic field intensity and reduced permeability. The lower permeability again leads to an increased magnetic field diffusion velocity and therefore to an accelerated rise of the magnetic force at a following coil excitation spike. The magnetic force caused

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by premagnetization itself as well as its accelerated rise will exceed the level of the return spring force at an earlier time stage (Fig. 14.19), resulting in a reduced response delay time. In contrast, the actual time of movement (valve needle flight time) itself is almost not influenced (Fig. 14.20).

Stroke (µm)

50 40

0.8 mA 0.6 mA

30

0.4 mA 20

0.2 mA 0.0 mA

10 0 0

2

4

0

2

4

6

8

10

6

8

10

v (mm/s)

150 100 50 0 -50 t (ms) Fig. 14.19 Stroke and velocity profile for several levels of electrical premagnetization Fig. 14.20 Effect of electrical premagnetization on the valve dynamics

10 Pull-in time Delay time Needle flight time

8

t (ms)

6 4 2 0 0.0

0.2 0.4 0.6 Premagnetization current (A)

0.8

14.3 Fast-Switching Electromagnetic Valves

477

14.3.5 Dynamics Optimization II: Overexcitation Further improvements in the valve dynamics can be achieved by applying a high current peak to the coil at the start of the valve-opening phase, known as overexcitation. The current spike acts until the end of the opening process. Afterwards, the coil current is reduced and controlled to the nominal current. The profit in higher magnetic force is low, since we get into the region of magnetic saturation, but due to the fact that the permeability decreases in this region, the diffusion velocity of the magnetic field into the armature can be increased. As a consequence thereof we obtain an accelerated rise of the magnetic force, which results in shorter pull-in times (Fig. 14.21). Applying different levels of overexcitation, the accelerated magnetic field diffusion into the armature material can be made visible by simulation (Fig. 14.22).

Overexcitation 5A 7A 9A 11 A

12

40 Time of first impact 20

I (A)

Magn. Force (N)

60

8

4

0

0 0.0

0.1

0.2

0.3 t (ms)

0.4

0.5

0.0

0.1

0.2

0.3 t (ms)

0.4

0.5

Fig. 14.21 Effect of coil overexcitation on the valve dynamics

Magn. Flux density (T)

1.6 1.2

11 A

9A

7A 5A

0.8

0.4

t = 240 µs

0.0 2.5

3.0

3.5

4.0

4.5

Armature radius (mm)

Fig. 14.22 Magnetic field diffusion inside the armature for different levels of overexcitations at one point in time

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The level of overexcitation is mainly limited by the thermal capacity of the actuator system.

14.3.6 Switching Cycle Modern applications of solenoidal actuators use a digital controller, which executes complex algorithms to provide a wide variation of the switching times and precise actuator control. The modulation of the current according to a predefined profile permits control of the electromagnetic force and thus of the valve state. In the dosing valve system under investigation, the objective is to shorten the opening and closing time, since in these periods an undefined fluid dosage takes place. The coil current is controlled in an ON/OFF mode by a pulse width modulation generated by the controller using a so-called soft switching technique where the lower coil voltage level is zero (Fig. 14.23). Soft switching gives the coil current lower ripple and achieves a smoother output profile of the magnetic force. The complete control profile for a valve operation cycle can be subdivided as follows: 1. Electrical premagnetization: Using the concept of electrical premagnetization as presented in the previous section with a coil current of 1 A, the valve-opening process can start from a higher magnetization level. 2. Valve opening: The control accomplishes the valve opening by applying a momentary highvoltage overexcitation spike to build a high-density magnetic flux field almost instantaneously, resulting in a steeper magnetic force gradient to shorten the needle flight time. 3. Hold phase: During the hold phase—the actual dosing period—the nominal coil current of 3 A creates the holding force at the end of the stroke. 4. Valve closing: After turning off the driving coil current with a damping negative voltage spike, the spring force returns the armature to its initial position. In a final simulation, the complete switching cycle of the valve was reproduced (Fig. 14.23). Combining the concepts of premagnetization as well as overexcitation to optimize the actuator dynamics, the pure valve needle flight time at valve opening can be reduced to 200 µs.

14.4 Cofired Piezoceramic Multilayer Actuators

Coil voltage

60 30 U (V)

Fig. 14.23 Valve-switching cycle: control profile and system response

479

0 -30

3

g sin Clo

6

ing

Coil current Ho ld

I (A)

9

Op en

ma Pre gn eti zat ion

-60

0 0

1

2

3

4

5

Magn. Force

F (N)

60 40 20 0 Stroke

Stroke (µm)

40 20 0 150 P (W)

Eddy current losses 100 50 0 0

1

2

3

4

5

t (ms)

14.4 Cofired Piezoceramic Multilayer Actuators Piezoceramic actuators are widely used for high-precision positioning systems. Their almost infinite resolution (in the nanometer range) and their very good repeatability predestine these actuators for the usage in linear stages, camera shutters or printer heads [14]. Rapid improvements in the performance of the ceramic materials used, make new smart designs possible and offer new application fields for piezoceramic actuators. The advantage of the piezoelectric actuator lies in the enormous force density and the high dynamics. In order to achieve improved deflections, hundreds

480 Fig. 14.24 Piezoceramic multilayer stack actuator

14 Industrial Applications Inactive top part

Active layers Interdigital electrodes

Inactive bottom part

of thin piezoceramic layers are stacked. Therefore, the mechanical displacement of the individual layers sum up, while the electric driving levels can be reduced due to a parallel switching of the ceramic sheets. The setup is similar to that used for multifoil-capacitors, as shown in Fig. 14.24. The highly dynamic deflection of these complex structures necessitates a sophisticated design in order to guarantee an effective operation and a large number of duty cycles. Due to the thin ceramic layers, with a thickness of about 100 µm, strong electric fields are established in the ceramic material at typical driving voltages of about 200 V. Therefore, these actuators show a strongly nonlinear response mainly caused by the ferroelectric nature of the ceramic materials. These effects, which are responsible for the actuator’s performance, have to be considered during the design process. Therefore, nonlinear material relations as well as ferroelectric hysteresis effects have to be considered.

14.4.1 Polarization of a Stack Actuator In a first investigation, we are interested in the polarization process of such a cofired stack actuator. Therefore, we apply our micro-mechanical switching model (see Sect. 12.4.2), and consider a small part of the piezoelectric stack as displayed in Fig. 14.25. We are especially interested to obtain the electric polarization as well as the mechanical stress at the tip of the center electrode. Therewith, we apply the electric potential excitation at the loading electrode (center electrode), which increases to a maximal value and then decreases back to zero (poling excitation). In 2-direction we just use one finite element and with the applied symmetry boundary condition the computed results in 3d are equivalent to just performing a 2d computation in the 13-plane. The results of the poling process towards the electric polarization is shown in Fig. 14.26 for different states of loading. Therewith, one can clearly see that at the beginning of the poling process, the electric polarization has its maximum at the

14.4 Cofired Piezoceramic Multilayer Actuators

481

Grounded electrode

No electrode

Loaded electrode

3 2

Grounded electrode 1

Fig. 14.25 Model for a small part of a piezoelectric stack with two layers and three electrodes (not at scale)

40% of full excitation 100% of full excitation

Fig. 14.26 Polarization process at 40 and 100 % of loading

tip of the center electrode. Finally, at the end of the polarization process, the inner part is fully and homogeneous polarized. However, almost the whole region, where there is no inner electrode, is almost not polarized and therefore a region of no active material. This fact has a strong impact on the mechanical stress distribution, as displayed in Fig. 14.27. For a detailed analysis we display in Fig. 14.28 the evolution of the electric polarization and mechanical stress for two different positions: (el1 ) in the homogeneous polarized region (el2 ) at the tip of the center electrode. We can clearly see the importance of compressive prestressing for a piezoelectric stack during polarization. Otherwise, as shown in Fig. 14.28b we obtain mechanical tensile stresses up to 280 MPa. Furthermore, we see that the region at the tip of the middle electrode does not get poled correctly (see Fig. 14.28a).

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1

125MPa 138MPa 54MPa 0MPa 0MPa 0MPa -94MPa -134MPa -59MPa

Fig. 14.27 Mechanical stress after the polarization process Fig. 14.28 Electric polarization and mechanical stress at the homogeneous region (el1) and at the tip of the middle electrode (el2). a Electric polarization. b Mechanical stress

(a)

(b)

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14.4.2 Stack Actuator: Hysteresis Based Approach Here, we consider a complete stack actuator (EPCOS AG) as displayed in Fig. 14.29a, and apply the macroscopic hysteresis based model as described in Sect. 12.4.1. The stack consists of 360 layers, each having a thickness of 80 µm and cross section of 6.8 × 6.8 mm2 . The overall length of the stack actuator is 30 mm and it exhibits a maximal stroke of 40 µm.

(a) Interdigital electrodes

Active ceramic layers

(b) Inactive layers Common electrode

Connectors

Fig. 14.29 Geometric setup and FE model of stack actuator. a Geometric setup. b FE model Table 14.8 Model parameters for the stack actuator

(a) Material parameters and polynomial coefficients for the irreversible mechanical strain; (b) Logarithmic values of the Preisach weight function for M = 30

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For the FE simulation we choose the full 3d setup and model the whole stack as one homogenized block (see Fig. 14.29b). Since we currently restrict ourselves to the uni-axial electric load case, it makes no sense to fully resolve the inter-digital structure of the electrodes. Furthermore, we set the electric potential at the top surface to the measured voltage multiplied by the number of layers, since we do not resolve the layered structure. To get all entries of the material tensors, we do an impedance measurement at the electrically preloaded stack actuator (see Fig. 12.11) and use our inverse scheme. Next we apply our measurement setup according to Fig. 12.12 and excite the stack actuator with a triangular signal. The material tensor entries as well as the polynomial coefficients for the irreversible strain and the Preisach weight function for the hysteresis operator are provided in Table 14.8. Now in a second step, we have excited the stack actuator by a sine-function and compare the measured and simulated data. The results are displayed in Fig. 14.30, and demonstrate that the FE results accurately predict the measured data even for other excitation sequences than those used for the fitting procedure.

(b)

x 10−4 15

0.06 0.05 0.04 D (Measurement) 3

0.03

D (FE simulation) 3

0.02 0.01

Mechanical strain S3

Dielectric displacement D3 (C/m2)

(a)

10 S (Measurement) 3

S (FE simulation) 3

5

0

0 0

5

10

15

20

25

30

35

40

0

5

10

Time t (s)

(d)

25

30

35

40

x 10−4 15

D (Measurement) 3

S (Measurement) 3

D3 (FE simulation)

S3 (FE simulation)

Mechanical strain S3

Dielectric displacement D3 (C/m2)

20

Time t (s)

(c) 0.06

15

0.05 0.04 0.03 0.02

10

5

0.01 0

0 0

0.5

1

1.5

Electric field intensity E3 (kV/mm)

2

0

0.5

1

1.5

2

Electric field intensity E3 (kV/mm)

Fig. 14.30 Comparison of the measured and FE simulated data for the piezoelectric stack actuator: a Dielectric displacement over time; b Mechanical strain over time; c Dielectric displacement over electric field intensity; d Mechanical strain over electric field intensity

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14.5 Capacitive Micro-machined Ultrasound Transducers 3D ultrasound imaging is an area of extensive research and of great interest for many industrial applications ranging from nondestructive testing to medical-imaging systems (see e.g., [15]). To obtain a 3D ultrasound image, a 2D array of transducer elements with electronic focusing and beam steering is needed. Using standard piezoelectric transducer elements will result in difficulties concerning fabrication as well as electronic connections. Capacitive micro-machined ultrasound transducers (CMUTs) may overcome many of the drawbacks of piezoelectric transducers, since CMUTs can be fabricated by adding a few technological steps to a standard CMOS process [16–18]. Due to small size and the possibility of integrating signalprocessing electronics on a chip [19], these transducers may be an attractive alternative to standard ultrasound transducers. Figure 14.31 shows the top view of a CMOS test-chip with 4 transducer arrays, each containing 19 capacitive transducers. The transducers are used in the transmitting as well as receiving mode, so that a classical pulse-echo mode can be performed. The principle setup of such an array is shown in Fig. 14.32 and the detail of one cell in Fig. 14.33. By applying a short voltage signal to the electrodes of each transducer cell, the membranes are deflected by the resulting electrostatic force, and an acoustic pulse is generated that propagates into the fluid. Now, the same transducer cells are used for measuring the reflected acoustic pulse, since the membranes will be deflected according to the fluid loading, resulting in a change of their capacitances.

Fig. 14.31 Top view of a CMOS chip with four arrays, each containing 19 capacitive transducers

Transducer cells

Electronic

Bond pads

Fig. 14.32 Principle setup of cells

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Fig. 14.33 Detailed view to a capacitive cell

Electrode (Al) Membrane (SiN) Air-gap SiN Si-Wafer

14.5.1 Requirements to Numerical Simulation Scheme The design of such micro-machined transducers strongly depends on the availability of appropriate computer-aided engineering (CAE) tools, since the fabrication of each prototype is quite costly. In addition, the signal-to-noise ratio is still too low compared to piezoelectric transducers, and a lot of research has still to be done [20]. Therefore, precise computer simulations are needed to analyze and, furthermore, optimize the dynamic behavior as well as efficiency of such transducers. However, the precise numerical computation of CMUTs is a quite complicated task, since one has to deal with several challenging problems, which can be summarized as follows: • Multifield-Problem: We have to deal with a multifield problem consisting of the electrostatic, mechanical and acoustic field including their couplings (see Chaps. 8 and 10). • Geometric aspect-ratio: A typical electrostatic cell has the following dimensions: Thickness of cell-membrane : 400 nm−1 µm Side-length of cell-membrane : 20−40 µm Air-gap : 200 nm−1 µm Assuming an air-gap of 200 nm and the wavelength λ in water at f = 1 MHz (λ = c/ f = 1.5 mm), we achieve at an aspect-ratio of 1.5 · 10−3 wavelength = = 7500, air-gap 200 · 10−9 which clearly results in a big problem concerning the computational grid generation. An enhanced solution approach for this problem will be the use of nonmatching grids as described in Sect. 8.3.2. • Nonlinearities: According to the setup, different nonlinearities of the involved physical fields have to be considered. Furthermore, the coupling mechanisms between the physical fields have to be taken into account, which are highly nonlinear in practice.

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487

– Large deflections: If the deflections are in the range of the thickness of the structure, we have to consider the geometric nonlinear case for the mechanical computation (see example in Sect. 3.7.4). – Stress-Stiffening effect The fabrication process of CMUTs leads to a large prestressing within the structure, which results in a stiffer structure behavior. This effect is fully considered within the geometric nonlinear case, see Sect. 3.7.3. – Electric field The electrostatic force leads to a deformation of the electrode-membrane structure, and therefore, introduces a kind of geometric nonlinearity in the electrostatic field PDE. This means that we always have to compute the electrostatic field on the actual configuration by performing an updated Lagrangian formulation (see Chap. 10). – Electrostatic force In a first case, let us consider a parallel plate capacitor, for which the electrostatic force computes as follows

FelC =

ε0 A U 2 . 2 d2

(14.7) In (14.7) U denotes the applied electric voltage, A the electrode surface, d the distance between the electrodes and ε0 the permittivity of the air. In a second case, we consider a parallel plate capacitor with insulation layer having a relative permittivity of εr . The electrostatic force at the interface between the insulation layer and air calculates by

FelI =

ε0 A U2 (εr −1) . 2 (d2 + d1 /εr )2

(14.8) Therewith, the ratio of the two forces is FelI FelC

≈

(d1 + d2 )2 εr ≈ εr d22

and we clearly see the importance of considering both parts of the electrostatic force, the force due to the electric charges on the electrodes as well as the interface force at changing permittivity. As described in Sect. 10.1, we will apply

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the method of virtual work, which considers the overall electrostatic force by evaluating (10.24) within an FE formulation. In the following, we will analyze the dynamic behavior of such transducers, especially their problems concerning crosstalk.

14.5.2 Single CMUT Cell The first numerical case study concentrates on a single CMUT cell. The computational domain consisting of the cell, the silicon wafer and the fluid domain as displayed in Fig. 14.34. The focus of the investigation in this section is the snap-in curve of an individual cell and its sensitivity to the variation of the geometric dimensions. Within a static analysis, we obtain a snap-in voltage of 194 V (see Fig. 14.35) compared to a measured snap-in voltage of 192 V [21]. However, the measurements

Fig. 14.34 Setup of the CMUT cell

Absorbing BCs

Fluid Insulation layer 1 (t=0.75µm, R=20µm) Top electrode (t = 0.45µm, R=16µm))

Insulation layer 2 (t=0.35µm, R=20µm) Bottom electrode

Fig. 14.35 Center displacement of membrane

Air-gap (0.35µm)

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489

of the geometric dimensions are not very accurate and in addition due to the fabrication process the dimensions will vary from cell to cell. In order to obtain the above mentioned agreement between measured and simulated snap-in voltage, we had to adapt the air-gap size as well as the membrane radius in the range of about 8 %. The change of the snap-in voltage as a function of the air-gap size as well as membrane radius is displayed in Figs. 14.36 and 14.37. Later we will emphasize that the material parameters will also have a strong influence on the computed results. Applying a transient analysis to our FE model (now also taking into account the fluid region) by exciting the CMUT cell by a short pulse u(t), we can compute the electric input impedance as follows

Fig. 14.36 Center displacement of membrane: variation of air-gap

Fig. 14.37 Center displacement of membrane: variation of membrane radius

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Fig. 14.38 Input impedance in air and water

Z (ω) =

FFT(u(t)) . j ω FFT(Q(t))

(14.9)

In (14.9) ω denotes the angular frequency, j the imaginary unit, FFT() the fast Fourier transformation and Q(t) the obtained total charge at the top electrode. Figure 14.38 shows the amplitude of the electric input impedance as a function of frequency. It can be clearly seen that the immersion of the CMUT cell into water strongly damps the resonance and antiresonance peaks and furthermore shifts them to lower frequencies due to the large mass loading of the water as compared to air.

14.5.3 CMUT Array In a second numerical case study we consider a CMUT array consisting of 9 cells. Due to symmetry, we just have modeled 4.5 cells (see Fig. 14.39). Each individual cell has the geometric setup as described in the previous section (see Fig. 14.34). The main focus of this investigation is the analysis of the cross-talk between the individual cells. Therefore, we just excite the center cell (cell 1) with a short sineburst and compute the mechanical vibration at the center of cells 2 to cell 5. With this result, we compute the cross-talk as follows cross-talk = 20 log10

u irms u 1rms

i ∈ [2, 3, 4, 5].

(14.10)

In (14.10) u rms denotes the root mean square of the mechanical displacement u(t). By neglecting the fluid domain, the occurring cross-talk must be purely mechanical, and as demonstrated in Table 14.9, the resulting values are quite small.

14.5 Capacitive Micro-machined Ultrasound Transducers

491

Absorbing BCs

Fluid Obstacle Cell 1 Cell 2 Cell 3 Cell 4 Cell 5

Cell spacing = 55µm

Fig. 14.39 Setup of the CMUT with nine cells Table 14.9 Mechanical cross-talk

Level (dB)

Table 14.10 Total cross-talk Level (dB)

Cell 2

Cell 3

Cell 4

Cell 5

−60.1

−63.2

−66.5

−70.8

Cell 2

Cell 3

Cell 4

Cell 5

−20.2

−22.1

−23.3

−23.8

However, if we additionally consider the fluid domain (CMUT array immersed in water) the cross-talk increases tremendously (see Table 14.10). Therefore, we can conclude that the dominant cross-talk between the individual cells is due to the acoustic coupling.

14.5.4 Controlled CMUT Array For the 3D simulation one of the four arrays (each consisting of 19 capacitive transducer cells), as shown in Fig. 14.31, was considered. The used finite element model consists of a quarter of one array (see Fig. 14.40). The thickness of the membranes

Fig. 14.40 Detail of the finite element model; the membranes are marked by the numbers 1–7

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is 1 µm and the gap between the electrodes has a size of 500 nm. We apply a dc voltage of 10 V to the electrodes and for excitation a single period of a sine burst with a frequency of 5 MHz and amplitude of 10 V. To study the acoustic crosstalk between the individual membranes, only the center membrane is excited and the deformations of the neighboring membranes are computed. As an example of the simulation results the center displacements of membranes 4 and 7 are shown in Fig. 14.41 together with the dynamic response of the driving membrane 1. The observed significant crosstalk between the individual elements leads to a reduction of the effective membrane deflections when all membranes are driven in parallel. This can be seen from Fig. 14.42, where the corresponding center displacements of membrane 1 are shown, and is in agreement with the results obtained for the 2D simulations. In the next step, the radiated pressure signal of the array is investigated. Of course, the long ring-down time of the membranes also shows up in the pressure signal. In the Fig. 14.41 Center displacements of driving membrane 1 (solid line), membrane 4 (dashed line) and membrane 7 (dotted line)

Fig. 14.42 Center displacements of membrane 1: all seven membranes (solid line) and just membrane 1 excited (dashed line)

14.5 Capacitive Micro-machined Ultrasound Transducers

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case that all membranes are driven in parallel, this results in a main, high-amplitude signal, which is followed by a secondary signal of lower amplitude (Fig. 14.43). When only the center membrane is excited, however, this secondary signal is of the same order in amplitude as the primary signal (Fig. 14.43). As a consequence, the bandwidth is significantly reduced for the single driven membrane. In order to decrease the ring-down time of the membranes, we have designed and applied controllers to each capacitive transducer of the array. Due to the quadratic dependency of the electrostatic force on the deflection of the membranes, no standard PID controller can be used. Instead, a nonlinear controller has been designed. In each outer iteration step k (see Sect. 10.2.1) the change of the capacitance of each transducer is computed from the mechanical displacements u kn+1 and used as the input of the controller. The controller algorithm then calculates the voltage for each transducer, which is a direct input value for the electric source vector f u . In Fig. 14.44

Fig. 14.43 Pressure signal of the array: all membranes (solid line) and just membrane 1 excited (dashed line)

Fig. 14.44 Center displacements of uncontrolled (solid line) and controlled (dashed line) membrane 1, when all membranes are excited

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the dynamic response of the center membrane, when all membranes are driven in parallel, is depicted. As can be seen, the controller strongly decreases the ring-down time of the membrane. Furthermore, the use of the controller strongly decreases the acoustic crosstalk between the membrane elements. This can be seen in Fig. 14.45, where the center deflection of membrane 2 is shown, in the case that only membrane 1 is excited. Therefore, the secondary signal in the acoustic pressure, as observed for the uncontrolled case, is no longer present for the controlled membrane array. This is shown in Fig. 14.46 for the case that all membranes are driven in parallel. As a consequence a smoothing effect of the controller is also observed in the frequency spectrum (Fig. 14.47).

Fig. 14.45 Center displacements of uncontrolled (solid line) and controlled (dashed line) membrane 2, when just membrane 1 is excited

Fig. 14.46 Pressure signal of uncontrolled (solid line) and controlled (dashed line) array, when all membranes are excited

14.6 High-Intensity Focused Ultrasound

495

Fig. 14.47 Frequency spectrum of uncontrolled (solid line) and controlled (dashed line) array, when all membranes are excited

14.6 High-Intensity Focused Ultrasound High-power ultrasound sources have found their way into a wide variety of applications, ranging from medical ultrasound, like lithotripsy or HIFU-therapy (HighIntensity Focused Ultrasound), to ultrasonic cleaning or welding and sonochemistry [22]. In contrast to ultrasonic applications with sources which radiate low amplitude pressure waves, the appearance of nonlinear effects like sawtooth and shock formation is observed at high-power ultrasonic sound generators. In order to speed up the design process of these high-power sources appropriate numerical simulation tools are required. These have to take into account not only the electro-mechanical and the fluid–solid coupling, but also must be able to simulate the propagation of finite amplitude waves through lossy and compressible media. Previous investigations on the numerical simulation of nonlinear wave propagation problems are mainly based on the Khokhlov-Zabolotskaya-Kuznetsov (KZK), nonlinear progressive wave equation (NPE), or the Burgers equation, to mention only the most popular ones. Various methods for the solution of the above equations can be found in the literature, ranging from spectral [23] to time domain algorithms [24], finite difference schemes (FD) and finite element approaches (FEM) [25] as well as operator splitting methods [26]. We will apply the computational scheme as described in Sect. 5.4.3, which numerically solves Kuznetsov’s equation by an enhanced FE formulation.

14.6.1 Piezoelectric Transducer and Input Impedance The principle setup of the acoustic power transducer is shown in Fig. 14.48. Due to the geometric focusing of the lens, high acoustic intensity can be achieved in the

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Fig. 14.48 Principle setup of high-power ultrasound source

Fig. 14.49 Impedance of the M453 piezoelectric transducer

focus region. The piezoelectric transducer has a diameter of 60 mm and the radius of curvature of the lense is 55 mm, which results in a focal distance of 70 mm. The operating frequency of the transducer is at 1.7 MHz. As the starting point in our investigations impedance calculations have been performed for the piezoelectric disc transducer. This was mainly performed to establish all necessary and unknown material parameters. The piezoelectric disc has a thickness of 1.2 mm and a diameter of 60 mm. The simulated and the measured electric impedance of the piezoelectric transducer is displayed in Fig. 14.49. Due to good agreement over a wide frequency range, we can trust our used material parameters. Next, the impedance of the whole HIFU source was simulated with water load. The

14.6 High-Intensity Focused Ultrasound

497

Fig. 14.50 FE model of the HIFU source used for the impedance calculations

Fig. 14.51 Electrical input impedance of the HIFU source loaded with water

finite element model used in the impedance simulation of the complete HIFU source is shown in Fig. 14.50. The computed electric impedance in water is displayed in Fig. 14.51.

14.6.2 Pressure Pulse Computation In the numerical simulation of the HIFU antenna, the piezoelectric and the fluid– solid coupling as well as the nonlinear wave propagation in the fluid must be taken into account. Therefore, a complete finite element model has been setup in which the piezoelectric transducer, the lens, the matching layer and the water has been discretized (see Fig. 14.52). The FE model consisted of about one million elements for both linear and nonlinear acoustic simulations. Near the sound source 20 elements per wavelength have been used. To account for the generation of higher harmonics, the spatial discretization was increased in propagation direction to 40 elements per wavelength in the focus region. Therewith, at least 8 elements per wavelength λ are guaranteed for the first 4 harmonics. For the excitation of the piezoelectric transducer a sine burst at 1.7 MHz and varying amplitude, as shown in Fig. 14.53, has been used.

498 Fig. 14.52 Finite element mesh (for display reasons, just a coarse mesh is shown)

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Axis of rotation

Fig. 14.53 Excitation voltage for pressure measurements

The simulation results were observed at several points on the rotational axis between the source and the focus region. Transient analysis were performed with 13,500 time steps with a time step size of 4 ns. First, we considered pressure pulses due to a low excitation voltage of U = 9Vpp . As can be seen from Figs. 14.54 and 14.55, linear simulation results and measurements compare very well. For the high-intensity measurement, an excitation voltage of U = 133Vpp has been used. It should be noted that also for the higher excitation voltage the piezoelectric transducer still behaved totally linear. Therefore, any distortions in the pressure signal stem from nonlinear effects in the fluid only. The comparison of measurements with nonlinear simulation results is shown in Figs. 14.56 and 14.57. The nonlinear distortion of the sine wave due to the generation of higher harmonics is clearly visible. In the simulations a damping value of 0.22 dB/MHz2 m has been used.

14.6.3 High-Power Pulse Sources for Lithotripsy In this section we will discuss the numerical computation of two high-power pulse sources used for lithotripsy application: a piezoelectric driven pulse source and an electromagnetic pulse source. In such applications we have up to 80 MPa in the focus,

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499

Fig. 14.54 Pressure pulse signal for low intensity measurement and linear simulation at the focal distance 0 lin. simulation measurement

−10

A (dB)

−20 −30 −40 −50 −60 −70

1

2

3 f (MHz)

4

5

6

Fig. 14.55 Frequency spectra of pressure pulse for low intensity measurement and linear simulation at the focal distance

and we need a quite fine grid within the acoustic domain in order to resolve the higher harmonics forming the high-pressure pulse. We will start with the piezoelectric high power pulse source as shown in Fig. 14.58 (for a discussion on the physics we refer to [27]). In parallel connected piezoceramic discs are placed on a spherical surface. The driving voltage is provided by discharging a capacitor. Due to geometrical focusing, high-power ultrasound is achieved in the

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Fig. 14.56 Pressure pulse signal for high-intensity measurement and nonlinear simulation at the focal distance

Fig. 14.57 Frequency spectra of pressure pulse for high-intensity measurement and nonlinear simulation at the focal distance

focal region of the source. A simulation of the nonlinear sound field was performed for the whole fluid domain. To account for the higher harmonics, we have chosen about 200 finite elements per fundamental wavelength, resulting in about 2.9 million bilinear quadrilateral elements for the axisymmetric setup. The sound pressure signal at the surface of the source is found by measuring the radiated pressure signal near the surface of the source, and is used for the simulation (see Fig. 14.59). Measured,

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Fig. 14.58 Piezoelectric high-power pulse source DL100

Fig. 14.59 Sound pressure signal at the surface of the source used in the simulation

linear and nonlinear simulated pressure signals in the focus region of the source are compared in Fig. 14.60. The maximum sound pressure in the measurement is 75.6 MPa, the maximum sound pressure in the nonlinear simulation is 69.7 MPa. In comparison to the nonlinear result, the linear simulation has been performed using the same simulation parameters. The maximum sound pressure here is 46.2 MPa. The sound pressure is therefore raised by a factor of 33.7 % due to the nonlinear behavior of the sound wave in the fluid domain. The second power source for lithotripsy is based on an electromagnetic principle, and its schematic setup is displayed in Fig. 14.61. When the slab coil is loaded by a capacitor discharge, eddy currents are induced in the metallic membrane. The interaction between these eddy currents and the overall magnetic field results in a magnetic volume force (Lorentz force) acting on the membrane. Therewith, the membrane-rubber structure is deformed and an acoustic pulse is radiated into the fluid and focused by the lens. For the numerical simulation a finite element grid width of

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60

40

20

0

−20 38

39

40

41

42

43

44

45

46

47

Fig. 14.60 Comparison between measured, nonlinear- and linear-simulated sound pressure level in the focal region of the source

Fig. 14.61 Schematic of an electromagnetic driven acoustic power source

90 µm (corresponds to about 70 finite elements per fundamental wavelength) was used for the acoustic domain. Since in this case, we have to consider the nonlinearities within the electromagnetic transducer, we perform the numerical simulation in two steps: 1. Transducer Computation: Since the nonlinearities of the acoustic field near the transducer can be neglected, we compute the acoustic pressure using the linear acoustic wave equation. Therewith, we fully take into account the fluid loading of the transducer. For modeling the electromagnetic transducer we consider all relevant nonlinearities (updated

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Fig. 14.62 Comparison between measured and simulated sound pressure level in the focal region of the electromagnetic pulse source

Lagrangian formulation for the magnetic field, geometric nonlinearity for the aluminum membrane and the nonlinear electromagnetic force term, see Chap. 11). 2. Nonlinear Wave Propagation Computation: In a second run, we fully solve Kuznetsov’s nonlinear wave equation using the computed pressure near the transducer obtained from the first simulation step. The measured and simulated pressure signals in the focus region of the source are shown in Fig. 14.62. The dispersion at the beginning and the decreasing slope of the simulated pressure pulse as compared to the measured one indicates that the mesh size and the time step have to be further reduced for a more precise computer simulation.

14.7 Human Phonation The human voice is the basis for verbal communication, i.e., speech. Its primary signal is generated inside the larynx by the two opposing oscillating vocal folds (VFs) [28]. The entire process of the human voice is a complex interaction of fluid mechanics, solid mechanics and acoustics. In a healthy voice, the process is initialized by a tensioning of the VFs, which seals the larynx. As the lungs compress, air pressure builds up causing a pressure gradient between the subglottal and the supraglottal region. An air stream is thereby induced through the trachea onto the VFs, forcing them to open. The air stream equalizes subglottal and supraglottal pressure which, in turn reduces the velocity of the air stream itself, enabling the VFs to close up again, thus repeating the process. This creates a pulsating jet, which together with VFs vibrations and supraglottal air vortices form the source of the perceived acoustic sound. For adult females the fundamental frequency is in the range of 165–255 Hz and for adult males, from 85–155 Hz [29].

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The human larynx is part of the respiratory system. It is located inside the neck and measures about 5 cm long. Besides phonation it has several tasks, as it connects the trachea with the pharynx. It is also a passage for air to reach the lungs and the epiglottis protects it from stray food particles by acting as a lid during swallowing. Figure 14.63a displays a coronal cut of the larynx. It is comparable to a simple tube with a constriction half way along it, the glottis. The form of the glottis is determined by the positioning of the two opposing VFs. Superior (above) to the VFs are the ventricular folds also known as false vocal folds (FVFs). The mucosa forms a protuberance between the VFs and the FVFs known as the laryngeal sinus or ventricle. Figure 14.63b is a superior view of a transversal cut through the larynx and exposes the VFs and their “V”-like position. Posterior, the VFs are attached to the pyramidal shaped arytenoid cartilages, which are linked by joints to the cricoid cartilage (see Fig. 14.63b). The cricoid cartilage is a ring shaped and forms the basis of the larynx, as depicted in Fig. 14.64. Two joints (cricothyroid joints) join the ring shaped cricoid cartilage to the thyroid cartilage, which surrounds the inner parts of the larynx, thereby protecting them. The cricoid and thyroid cartilages can be tiled against each other and put the VFs under tension. However, the main mechanism to control the VFs is via the arytenoid cartilage. These pivot-mounted cartilages can be turned,

(a)

(b)

Fig. 14.63 Larynx structure [30]. a Posterior view of coronal laryngeal cut. b Superior (top) view Fig. 14.64 Anterior-lateral view of larynx (Larynx external by Olek Remesz)

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mainly by the posterior cricoarytenoid muscles, which also causes the cartilages to be pushed and pulled in the anterior and posterior direction. These adjustments regulate the distance between the VFs and can close the glottis completely to initiate the phonation process. Furthermore, they control the tension of the VFs, which modulates the pitch of the voice.

14.7.1 Mathematical Modeling Our modeling approach is to resolve the physical details of the phonation process in space and time by means of partial differential equations (PDEs) within the larynx and adjacent regions. Inside the larynx fluid forces act on the vocal folds, which are deformed and thereby influence the velocity of the adhering fluid particles. Furthermore, these deformations alter the fluid domain, which has to be adapted. The fluid-acoustic interaction is described by aeroacoustics (see Chap. 9), which is the main acoustic source, and the solid-acoustics coupling by claiming coincident surface velocity (acceleration). This physical phenomenon is outlined in Fig. 14.65 and solved via PDEs of air flow, structural mechanics, acoustics and their interactions [31]. A fully coupled 3D fluid-structure simulation is not feasible regarding computational costs, since several iterations are necessary in each time step to reach a state of equilibrium for the two physical fields. Therefore, in a first step we perform a fully coupled simulation in 2D and investigate in strong and weak (sequential staggered) coupling schemes for flow and structural mechanics. In a second step, we then present 3D computations applying prescribed VFs vibrations, and analyze the generated acoustic sound in the larynx as well as outside the mouth.

14.7.2 2D Fully Coupled Simulation The human phonation process is a prefect application to investigate fluid-solidacoustic interaction phenomena. The computational setup as displayed in Fig. 14.66

Fig. 14.65 Couplings within fluid-solid-acoustic interaction: the solid lines defines the dominant couplings occurring in human phonation. Dashed lines define couplings, which can be neglected in human phonation

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Fig. 14.66 Fluid regions and boundary conditions

38mm

Monitoring point ΓG

16mm

Trachea wall

Fluid-Structure Interface

y Inflow x ALE region

Euler region

Outflow

is a realistic cross section of the trachea. The geometry is in accordance to Magneto Resonance Imaging (MRI) data [32], which showed that the vibration of the vocal fold covers only a part of trachea. In our investigations, this fact is considered by narrowing the channel width at the vocal fold position as shown in Fig. 14.66. Furthermore, we take the rather complex structure of the vocal folds into consideration by modeling them with four layers: muscle, ligament, lamina propria and the epithelium, which is a very thin cover of 0.03 mm. The thin and complex structure makes experimental measurements difficult. Therefore, the used values rely on good estimation and comparison with different models. We use a density of 103 kg/m2 for all layers, and choose their elasticity modulus according to Table 14.11. In order to accurately resolve the flow structure, we performed a grid study, which finally resulted in 45,000 finite elements with quadratic basis function and therefore in about 400,000 degrees of freedom for the flow velocity and pressure. As an inflow condition, we prescribe a pressure of 1 kPa and choose zero pressure at the outflow, which are realistic values also obtained by measurements. At the trachea walls we set non slip boundary conditions (v = 0). Furthermore, to define the mechanical field, we fix the vocal folds at the trachea (u = 0). The acoustic field is computed on the same mesh as the flow, and we set along the trachea walls and the vocal folds (since we do not consider vibrational sound) an acoustic hard wall condition (v a · n = 0). Furthermore, to avoid reflections at the in- and outflow boundaries, we extend the domain by a perfectly matched layer. Table 14.11 Elasticity modulus for the different layers to model the vocal folds

Layer

Elasticity (Pa)

Epithel Lamina propria Ligament Muscle

50 · 103 20 · 103 25 · 103 30 · 103

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We first performed computations with a strong fluid-solid coupling using a time step size of 2.5 · 10−5 s and performed 5,000 time steps (tend = 0.125 s). As stopping criterion, we choose for solving the nonlinear Navier-Stokes equations an incremental L2 -norm (both for velocity and pressure) with accuracy 10−5 . Concerning the strong coupling, we perform iterations between the flow and structural mechanical field until the following incremental L2 -norm for the mechanical displacement is fulfilled k+1 − u kn+1 ||2 < 10−6 ||u n+1

(14.11)

with n the time step counter and k the iteration counter. Convergence of the coupled scheme was usually achieved within 5 iterations. In a second step we performed the computations with the weak (staggered) coupling using different time steps [33]. All computations showed instabilities and after a certain amount of time steps the two vocal folds even started to vibrate asynchronously (see Fig. 14.67). For all chosen time step sizes we obtained instable results before achieving tend . Using the same time step size as in case of strong coupling, we just could perform 2,900 time steps. The time step size, for which we could obtain the longest simulation time , was 1.25 · 10−5 s resulting into tend = 0.10125 s. Therefore, for all comparisons we choose the results obtained by this time step. We start our investigation by computing the volume flux Q V at the glottis given by Q V (t) =



v · n dΓ.

(14.12)

ΓG (t)

In (14.12) ΓG denotes the integration path inside the glottis as depicted in Fig. 14.66, which varies in time due to the movement of the vocal folds. Figure 14.68a shows the resulting volume flux in frequency domain with the main peak at about 226 Hz. The difference between the two approaches is mainly observed by the amplitude, which is approximately 10 % lower for the weak coupling. A larger difference is obtained when we compare the fluid velocity in x-direction in the center of ΓG . The reason for the lower amplitude in the weak coupling case is that the glottis does not open as wide and close as narrow compared to the strong coupling scheme, as displayed in Fig. 14.69. This fact is also shown in Fig. 14.70, which displays the flow field for the two extreme vocal fold position—open and closed glottis. The time step of

Fig. 14.67 Asynchronous vibration of vocal folds in case of weak (staggered) coupling

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Fig. 14.68 Comparison of the fluid flow for the two coupling schemes by means of volume flux through the glottis and fluid velocity in the center of the glottis. a Volume flux through the glottis. b Fluid flow in x-direction in the center of the glottis Fig. 14.69 Frequency spectra of the mechanical displacement in y-direction at the monitoring point (see Fig. 14.66)

the corresponding phase are the same, but the glottis opening is slightly larger for the strong coupling. More evidently is the closing phase, where we observe a much smaller glottis opening for the strong coupling. Overall, the flow structure strongly differs between the strong and weak coupling especially for the divergent phase as displayed in Fig. 14.70. In a next step, we compare the acoustic results obtained by solving the perturbation equations with the acoustic source terms obtained from the flow computations (see Sect. 9.5). For this purpose, we compute the Fourier-transform of the sources and perform an analysis in the frequency domain. Here, we first investigate in the source term distribution at the main frequency (226 Hz) as displayed in Fig. 14.71. For both computations we find the main source terms within the glottis, and even with very similar distribution. Just the amplitude of the source terms in the center of the glottis are about 20 % smaller in case of the weak coupling. Furthermore, one can observe that the source terms move more and more out of the glottis as the frequency increases. As an example, we display in Fig. 14.72 the source terms for both computations at 2 kHz. Similar to the main frequency, we can observe a smaller amplitude in case of the weak coupling. In addition, the structure of the source terms differ strongly.

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(a)

(b)

(c)

(d)

Fig. 14.70 Comparison of the fluid velocity and structural displacement at two characteristic vocal fold position for both schemes. a Convergent phase of phonation cycle for strong coupling. b Convergent phase of phonation cycle for weak coupling. c Divergent phase of phonation cycle for strong coupling. d Divergent phase of phonation cycle for weak coupling

(a)

(b)

Fig. 14.71 Acoustic source terms at the main frequency for both computations. For a better comparison we have normalized the source term strength to the maximum of the data obtained by the strong coupled scheme. a Strong coupling. b Weak coupling

Finally, we choose a monitoring point at the outflow and compute the SPL (see Sect. 5.2) p (f) (14.13) SPL( f ) = 20 log a pref

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Fig. 14.72 Acoustic source terms at 2 kHz for both computations. For a better comparison we have normalized the source term strength to the maximum of the data obtained by the strong coupled scheme. a Strong coupling. b Weak coupling

Fig. 14.73 One third octave Sound Pressure Level (SPL) as obtained from the fluid-structure simulation with weak and strong coupling

from the data of both computations. Figure 14.73 displays the 1/3 octave SPL obtained for the weak as well as strong coupling. Therewith, the weak coupling causes a reduction of about 5 dB at the main frequency as expected due to the reduced amplitude of the source terms, and furthermore also reduced values at higher frequencies.

14.7.3 3D Driven Simulation The 3D model used for the simulations consists of three domains, the larynx, the vocal tract and an acoustic propagation region in which the radiated sound is monitored (see Fig. 14.74b) [34]. The flow field simulation is constricted to the larynx as this is the region in which the sound is generated. The coronal section of the CFD domain is given in Fig. 14.74a, and consists of a short subglottal region of length T0 =

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(a)

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(b)

Fig. 14.74 Geometry and mesh for the simulation. a Geometric model of the human larynx in coronal section. Comparison of the fine CFD grid and coarse acoustic grid. b Larynx, vocal tract, propagation region, perfectly matched layer (PML) regions

2.8 mm, the vocal folds of length TV F = 7.2 mm, and a supraglottal region of length T1 = 40 mm. In depth and height the laryngeal channel has the same dimension of 2H0 = 12.0 mm—the shape does not change along the anterior-posterior axis. The geometry of the vocal folds are modeled according to the “M5” model proposed by [35] with a medial surface convergence angle of ψ = −20◦ . To simulate the vocal folds oscillation a sinusoidal motion is prescribed in inferior and superior direction, given by w = A sin(2πf ).

(14.14)

Thereby, a frequency of f = 100 Hz and a oscillation amplitude of A = 4 mm is used, ensuring that the minimal gap between the two vocal folds of g = 0.2 mm is kept. For the acoustics, the vocal tract model is attached to the larynx and consists of multiple frustums concatenated one after another. The number of frustums and their radius determines the resulting sound radiating from the artificial mouth. In this case the vocal tract models the sound /u/ (“who”), according to the works of [36], who acquired 3D vocal tract shapes using magnetic resonance imaging (MRI).

14.7.3.1 CFD Model Here, we perform 3D simulations using prescribed motion of the VFs. The pressure at the inlet is set to pin = 300 Pa (i.e. lung pressure), pout = 0 Pa at the outlet and ∂ p/∂n = 0 for the channel walls. The boundary conditions of the fluid velocity are set to ∂uin /∂n = 0 at the inlet and no-slip condition u = 0 on the fixed walls. On the vocal fold surfaces, the flow velocity is set to the velocity of the moving vocal fold. In Fig. 14.75 four time instants of a full phonation cycle are shown. The flow field is dominated by a pulsating jet, which is gradually cut off by the VFs during the

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(a)

(b)

(c)

(d)

Fig. 14.75 Flow field at the cross section along glottis mid-line. Vorticity contours at four time instants during 18th (180–190 ms) cycle of vocal fold vibration. a Start of closing phase (180 ms). b Minimal closing gap of glottis (182.5 ms). c Opening phase of glottis (185 ms). d Maximal opening of glottis (187.5 ms)

closing stage (see Fig. 14.75a, b). As the VFs open again (Fig. 14.75c) the jet starts to impinge into the supraglottal region and is fully developed at the maximal opening phase (Fig. 14.75d). The deflection of the jet is caused by interacting with a large scale vortex downstream. This redirection of the jet repeats itself from cycle to cycle, since the closing phase is too short for the large scale vortex to completely disperse.

14.7.3.2 CAA Model I: Lighthill’s Analogy For Lighthill’s approach the computed fluctuated pressure p ′ is a superposition of flow and acoustic parts, and just outside the flow region this quantity approaches the acoustic pressure pa . For the current application to the human phonation, we can assumed that no heat conduction takes place, the viscous stress may be neglected and the flow is incompressible. Therefore, an approximation of [L] can be given as (see (9.14)) (14.15) L i j ≈ ρ0 viic v icj . Furthermore, as shown in Sect. 9.4, the divergence of the Lamb vector is the main acoustic source at low Mach numbers which radiates into the far field. Furthermore, we found that the second derivative of Lighthill’s tensor leads to stronger sound in the acoustic source structures, as shown in Fig. 14.76. It needs to be pointed out, that

(a)

(b)

Fig. 14.76 Visualization of different aeroacoustic source models on a cross section of the human larynx, at time step 180 ms. Results have been normed to their maximum value. a Second derivative in space of the Lighthill tensor. b Divergence of the Lamb vector

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the scale has been adjusted to make the sources in the turbulent region visible. Inside the glottis are the dominant source terms, which are more than an order higher. According to the rule of thumb, we discretized the acoustic domain by 20 linear finite elements. The grid used for the flow simulation (characteristic length 0.15 mm) is unnecessarily small for computation of acoustic wave propagation. Therefore, the acoustic sources are conservatively interpolated onto a coarser grid. Since the maximum frequency encountered in this application is about 3.5 kHz a characteristic length of 1.3 mm is sufficient as it has a frequency resolution of up to 12 kHz. Once the acoustic sources are determined, the sound propagation is calculated. We add the so called perfectly matched layers (PML) at the inflow as well as around the propagation region to efficiently damp outgoing waves without reflections (see Fig. 14.74b).

14.7.3.3 CAA Model II: Perturbation Equations (PE) To correctly monitor the acoustics inside the flow field, we use a perturbation equations (PE) as discussed in Sect. 9.5. The splitting of the field variables allow a separation of fluctuating flow and acoustic quantities. Since the perturbation equations also consider the mean flow velocity v convection and refraction effects are accounted for, which is not done for Lighthills’ wave equation. The right hand side, or source term, is determined with the help of the CFD results, which provides the incompressible variables p ic and, by time averaging, the mean flow field v. Figure 14.77a presents a coronal section of the flow field, showing significantly less noise in the structures as Lighthill source term does (see Fig. 14.76). The 3D iso-surfaces in Fig. 14.77b show the 3D structure, which correlates to the jet (see Fig. 14.75). Clearly visible are the significantly higher source term amplitudes inside the glottis.

(a)

(b)

Fig. 14.77 Visualization of aeroacoustic sources of PE at time step 180 ms. Results have been normed to their maximal value. a Coronal cross section of the human larynx. b 3D iso-surfaces of acoustic sources

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14.7.3.4 Aeroacoustic Results and Comparison For the evaluation of the acoustic pressure and the comparison of the hybrid approaches, two monitoring points are considered. The first “MIC 1” is situated at the end of the larynx, inside the flow domain. “MIC 2”, the second monitoring point, is 1 cm behind the vocal tract, in the acoustic propagation region. For these two monitoring points the acoustic sound pressure (SPL) is plotted in Fig. 14.78. As expected, Lighthill’s approach based on the fluctuation pressure p ′ has significantly stronger amplitudes inside the flow region, as evident in Fig. 14.78a. Comparing the splitting of the field quantities, between Lighthill’s approach and the PE approach, it is clear that the computed pressure for the wave equation case is superimposed by the incompressible flow pressure p ic . However, this superimposition does not propagate into the far field for all frequencies, as Fig. 14.78b shows. Here, at “MIC 2”, formants are aligned and are comparable to the formants recorded in natural speech, as given in Table 14.12. Both analogies, PE and Lighthill’s approach are in very good agreement, except for the main frequency (100 Hz) and its harmonics. Here, Lighthill’s wave equation again shows an over-prediction of the SPL for these frequencies.

(a)

(b)

Fig. 14.78 Acoustic pressure for both monitoring points, for the wave equation and the PE. a Acoustic pressure at “Mic 1”. b Acoustic pressure at “Mic 2”. Vertical black lines indicate the formant frequency Table 14.12 Recorded formants from natural speech “N”, as given by [36], and simulated by our reproduced vocal tract “S” (Lighthill’s analogy and PE are identical) /u/ N (Hz) /u/ S (Hz) F1 F2 F3

389 987 2,299

270 1,000 2,484

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14.8 Aeroacoustics of Flow Around Obstacles In the following, we will consider low Mach number flows around different obstacles and will investigate in the generated acoustic sources and the propagation of the acoustic waves.

14.8.1 Square Cylinder Geometries We consider the generation of sound due to two different cylinders mounted on a plate in cross-flow. Although the geometries are simple, it is nonetheless interesting to analyze since variations of it are very common sources for flow induced noise, e.g., antennas on cars. Understanding the mechanisms of sound generation for this geometry may therefore give important hints to engineers on how to reduce noise levels for similar settings. For the geometrical variations of the cylinder under investigation, it is known that the first occurring main frequency for a mean fluid flow velocity of 10 m/s is in the range from 50 to 60 Hz. Given the speed of sound in air at standard conditions (c = 343 m/s results in a wave length λ of about 5.72 m which should be resolved at least with 20 degrees of freedom. This means that the edge length of the finite elements for computational acoustics should be about h = 28.58 cm. This length is in the range of the dimensions of the source domain and is clearly too coarse to resolve the acoustic sources. The two different wall-mounted cylinder geometries are shown in Fig. 14.79. The general set-up for the hybrid simulations is depicted in Fig. 14.80. The region Ω1 denotes the domain where the flow field is computed and where the acoustic sources are interpolated from the fluid grid to the acoustic grid. The region encompassing Ω1 and Ω2 corresponds to the region where the acoustic field is computed.

14.8.1.1 Flow-Induced Sound from the Wall-Mounted Square Cylinder

20 mm

The domain, on which the flow is computed, is displayed in Fig. 14.81. The flow field was computed with the CFD code FASTEST-3D [37] and with the commercial CFD code ANSYS-CFX [38] using an Scale Adaptive Simulation (SAS) turbulence model [39]. The boundary conditions used in the fluid computation with respect to the

10 m/s

10 m/s

20 mm

Fig. 14.79 Profiles of wall-mounted cylinder geometries

20 mm

30 mm

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Fig. 14.80 Schematic representation of the hybrid domain as used for the 3D computations

Fig. 14.81 Numerical domain used for fluid computations depicting dimensions. D = 20 mm

Table 14.13 Boundary conditions used for fluid computations

Position

Boundary condition

X=0 X = 40 D Z = 11 D Y = 0, Y = 11 D Wall and bottom

Inlet profile based on experiments Convective exit boundary Symmetry boundary condition Symmetry boundary condition No slip boundary condition

configuration from Fig. 14.81 are described in Table 14.13. Therewith, we have used a measured inflow profile with a mean velocity of 10 m/s resulting in a Reynolds number of about 13,000. The LES simulations were performed on a SGI-ALTIX system using 8 processors with the code FASTEST-3D. We did computations on different fine grids, and at the end achieved at a grid with 3.1 million cells. The final grid was successively refined at the critical regions closed to the cylinder and the wall. The nearest grid point in dimensionless wall coordinates was at y + = 2. All further LES calculations were performed with this quite fine grid, which allowed us to reduce the grid sensitivity to a minimum. The time step size was set to ΔtfLES = 10 µs, which guaranteed a resolution of up to 10 kHz, and which resulted in a CFL-number of 2.1127. All the selected discretization parameters have been based on the experiences as reported in [40].

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For the simulation of the flow using the code ANSYS-CFX, a turbulence modeling approach based on Scale Adaptive Simulation (SAS) was employed. The SAS approach allowed us to coarsen the grid of the LES computations to about 1.1 million, which resulted in a shorter computational time and less memory usage. Regarding the time discretization, a time step size of ΔtfSAS = 2ΔtfLES = 20 µs was used. To get an impression about the flow field, we show in Fig. 14.82a the flow structure as obtained by ANSYS-CFX for a characteristic time step. The displayed results are iso-surfaces of ω 2 − ǫ2 = 100,000 s−2 colored with the eddy viscosity (here ω representing the vorticity and ǫ the strain rate). One can clearly see the horseshoe, the roof and spanwise vortex structure. In studying animations of the flow structure, one can observe a strong interaction between the roof and spanwise vortex, which

(a)

(b)

Fig. 14.82 Instantaneous visualization of transient flow field using SAS turbulence modeling and the used grid for computing the acoustic field. a Transient flow field at a characteristic time step. b Grid for acoustic computation

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results in a reduced vortex street behind the cylinder. For a detailed discussion we refer to [41]. For a quantitative comparison between LES and SAS computations we show in Fig. 14.83 the averaged streamwise component of velocity at the centerline together with measured data [41]. As can be seen, the computed data, both for LES and SAS, are in a good agreement with the experiments. At the centerline, 0.5 D behind the cylinder, the SAS results show a slightly higher velocity as compared to experiments and LES results. For the profiles at the region 1 D behind the cylinder the velocity distribution in LES data is slightly smaller, whereas the SAS data matches well with the experiments. In addition, we present in Fig. 14.84 the spectra of the wall pressure fluctuations at different monitor points as listed in Table 14.14. In both simulations, pressure fluctuations on the side walls (monitoring points P1 and P2) of the cylinder show the characteristic vortex shedding frequency of about 55 Hz, which are in good agreement with experiments [41]. In addition, the pressure fluctuations at monitoring point P3, which is located on the bottom behind the cylinder, exhibit in both simulations a dominant frequency at twice the vortex shedding frequency. This

(a)

(b) LES

10

LES

10

SAS

SAS

Experiments

8

6

Z/D

Z/D

6

4

4

2

2

0

-0.2

0

0.2

0.4

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8

0.6

0.8

1

0

1.2

-0.2

0

0.2

U/U0

0.4

0.6

0.8

1

U/U0

Fig. 14.83 Comparison of averaged streamwise component of the velocity at the centerline. a At 0.5 D after cylinder. b At 1.0 D after cylinder

(b)

−2

10

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P1 P2 P3

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P1 P2 P3

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Power Spectral Density in (Pa 2/Hz)

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4

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Fig. 14.84 Frequency spectra of the wall pressure fluctuation at different monitor points. a LES simulation. b SAS simulation

14.8 Aeroacoustics of Flow Around Obstacles Table 14.14 Points at which we have evaluated the wall pressure spectra (see Fig. 14.81)

519

Position

P01 (D)

P02 (D)

P03 (D)

X Y Z

10.5 5.0 3.0

10.5 6.0 3.0

15.0 5.5 0.0

is in accordance both with theory and measured data [41]. Overall, the frequency spectra of the pressure fluctuations are quite similar for LES and SAS computations. For these low-frequency tonal noise problems, which exhibit a main frequency at about 50 Hz resulting in an acoustic wavelength λ of about 6.5 m, a harmonic analysis is appropriate. In order to perform an acoustic computation in the frequency domain, first all the interpolated acoustic nodal sources are computed in the time domain and stored in a file. From the resulting data set a Fourier transformation is performed, producing the corresponding acoustic nodal sources in frequency domain. The main advantage of computing in the frequency domain is that the acoustic field is solved only for relevant frequency components thus avoiding any non-physical frequency components originating from the CFD computations. Extensive studies have been performed towards the acoustic grid sensitivity. In general, as known from acoustic computations, one needs about 10 second order finite elements per acoustic wavelength to obtain a precise spatial resolution. However, in the context of computational aeroacoustics, our numerical experiments showed, that within the source region (region of fluid computation) we need a discretization, which is approximately 5 times finer. For our concrete example, we arrived at a grid of about 20,000 finite elements within the source region. However, compared to the flow grid (3.1 million in case of LES and 1.1 million in case of SAS turbulence model) the acoustic computational grid is quite coarse. It should be noted, that this large difference in the two grids can just be chosen, since we perform a conservative interpolation of the acoustic nodal loads from the flow to the acoustic grid. In Fig. 14.85, the configuration of the simulation domain for the harmonic computation showing the monitoring points used for acoustic directivity analysis is presented. For all computations we used second order FE basis functions and computed the acoustic pressure field on the grid as displayed in Fig. 14.82. The iso-surfaces of the acoustic pressure, clipped at the yz-plane, is shown for the main frequency ( f = 55 Hz) component present in the computation in Fig. 14.86. The values of the iso-surfaces range from 5 to 54 mPa, the outermost corresponding to about 48 dB. These results represent the characteristic dipole-like radiation expected for this tonal noise problem. At this point it should be noted that for both LES- and SAS-based data no significant differences were found in the acoustic field. Thereby, the computation of the flow field for the wall-mounted cylinder with elliptic profile was just performed by applying the SAS turbulence model.

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Cylinder

Acoustic propagation region

Acoustic source region

x

PML region

4m z 0m

Fig. 14.85 Schematic drawing of the acoustic domain used for the harmonic computation showing points used for directivity analysis. Distance scale in m

Fig. 14.86 Iso-surface of acoustic pressure for wall-mounted square cylinder at f = 55 Hz, clipped at the yz-plane. Dotted region represents PML

14.8.1.2 Flow-Induced Sound from the Wall-Mounted Cylinder with Elliptic Profile The configuration of the numerical domain used for the flow computation and boundary conditions is analogous to that used for the square cylinder case, presented in Fig. 14.81 and in Table 14.13, respectively. The dimensions of the cylinder are identical with the square cylinder, except for the extension with the semi-elliptical shape in the downstream direction. Using the same inflow velocity profile as in the square cylinder case, the Reynolds number is again about 13.000 based on the cross-flow side length D = 20 mm. Similarly as in the results for the square cylinder, Fig. 14.87a visualizes the turbulent structures from the SAS computation. Now, the roof vortex is of less importance

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Fig. 14.87 Instantaneous visualization of transient flow field using SAS turbulence modeling and used grid for computing the acoustic field. a Instantaneous visualization of fluid field computation at a characteristic time step. b Grid for acoustic computation

for the development of the vortex street, because flow that separates at the leading edge at the top reattaches at the cylinder’s afterbody (ellipse) and thereby does not disturb the vortex street [41]. The acoustic computation was performed using the configuration for the computational domain depicted in Fig. 14.85, except for the cylinder geometry. Figure 14.87b presents a close-up of the acoustic grid showing the cylinder with the elliptical profile. In the harmonic acoustic computation, the main frequency component found for this problem was f = 39 Hz. For this frequency value, iso-surfaces of the acoustic pressure clipped at the yz-plane are presented in Fig. 14.88. Values of the iso-surfaces range from 22 to 100 mPa and the outermost iso-surface corresponds in this case to

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Fig. 14.88 Iso-surface of acoustic pressure for wall-mounted square cylinder with elliptical profile at f = 39 Hz, clipped at the yz-plane. Dotted region represents PML

61 dB. Similarly as in the case square cylinder profile, the directivity pattern for this main frequency value results in a dipole-like acoustic field.

14.8.1.3 Comparison In this section acoustic results from the harmonic computations for the two wallmounted cylinder geometries are compared with experimental results (for details see [41]). Table 14.15 presents the main frequency components found in the computations and the frequency values found in the measurements carried out in the anechoic wind tunnel [41]. The higher values obtained in the simulations for both cylinder profiles appear first in the CFD results and are then carried over to the acoustic simulations. This is attributed to the used symmetry boundary condition (see Table 14.13). In Table 14.16 a comparison of the SPL values between the numerical and the experimental results is presented. It can be seen, that for both geometries slightly higher SPL values are obtained in the numerical simulations. At this point it is important to note that, for practical reasons, in the experimental case the cylinders were mounted at the center of a wall with a cross-flow length L exp = 0.66 m. However, in the acoustic simulations we have used a quite larger computational domain for the acoustic propagation region (see Fig. 14.85), and have applied acoustic hard wall boundary conditions. Therewith, reflections of acoustic waves on this larger wall Table 14.15 Comparison of main frequency values (Hz) found in simulations and measurements

f main

Square profile

Elliptical profile

Simulation Measurements

55 53

39 36

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Table 14.16 Numerical and experimental SPL values (dB) of the flow-induced sound for the two cylinder profiles evaluated at 0◦ on the yz-plane SPL value Square profile Elliptical profile Relative SPL difference Simulation Measurements

47 44

61 60

14 16

Fig. 14.89 Directivity patterns for two cylinder profiles at radius r = 1.0 m on the cross-flow yz-plane

from the simulation domain is one of the reasons for the higher SPL values at the monitoring points compared. Furthermore, the relative difference in the SPL values from both profiles compare well with the measured value. In Table 14.16 this difference is observed to be 2 dB; 16 dB in the experiments and 14 dB in the numerical computations. Finally, directivity plots of the SPL levels at a radius r = 1 m away of the flowinduced sound for the two wall-mounted cylinders are presented in Fig. 14.89. In this plot it can be observed that significantly higher amplitudes at all monitoring points obtained in the flow-induced sound computation for the cylinder with the elliptical profile.

14.8.2 Edge Tone Edge tones can be encountered in several areas. Musical woodwind instruments like a flute or an organ pipe all share the principle of the sound generation by an air jet which oscillates around a wedge-shaped object. In addition to this sound generation mechanism the mentioned instruments exhibit also an attached Helmholtz resonator for the amplification of the generated sound. In general however, no resonator is required for the generation of sound (see, e.g., [42]). The air jet oscillates with

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(a)

(b)

Fig. 14.90 Schematic of edge tone configuration. a Edge tone configuration (stage 1). b Smoke visualization of a jet in stage 2 (taken from [43])

relatively stable frequencies which depend on the distance d of the wedge to the slit of the nozzle, the mean inflow velocity v mean , the viscosity of the medium ν f and the inflow profile of the jet. Under certain circumstances audible tones are emitted. By varying the inflow velocities or the slit-edge distance the emitted frequencies exhibit regions with continuous changes but there exist also jumps in the frequency. The continuous regions between jumps are referred to as stages of the edge tone and are assigned ordinal numbers. The first stage of an edge tone corresponds to a situation where there is one half-wave present in the shape of the jet between slit and edge. Higher stages are defined correspondingly (Fig. 14.90). At constant jet speed and nozzle-wedge distance a feedback mechanism stabilizes the flow. It is however not completely clear, whether this mechanism is based on acoustic or hydrodynamic principles (cf. [44] and references therein). The incompressible CFD simulations suggest however, that the mechanism is at least partly driven by hydrodynamics.

14.8.2.1 Problem Description The results presented here have been obtained during a study of whether it can be justified to calculate 2D-CFD results and extrude these results into the third dimension in order to avoid a complete 3D CFD simulation. A series of full 2D-2D coupled aeroacoustic simulations has been followed by 2D-3D coupled computations in which the source terms obtained from the 2D CFD have been extruded into 3D. The final simulations were fully 3D-3D CFD-aeroacoustic ones. The presentation shall be restricted to the latter configuration, since it is the numerically most demanding one and demonstrates the powerful applicability of non-conforming meshes (other details can be found in [45]). The main motivation for Mortar FEM (see Sect. 2.10) in conjunction with the edge tone is the fact, that the whole CFD domain fits into one element of the acoustic propagation mesh at the main frequency of 130 Hz. The geometrical setup for the computational CFD domain can be seen in Fig. 14.91. Only half models are depicted since the setup is symmetric with respect to the x-z-plane. The Reynolds number for this problem is defined in the following manner

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(a)

525

(b)

Fig. 14.91 CFD domain setup for edge tone. a View from top. b View from front

Re =

vδ = 225, νf

where δ = 1 mm is the slit width of the nozzle. The applied physical properties and boundary conditions are as follows: • Material properties Air at 25 ◦ C with a dynamic viscosity of ν f = 1.545 · 10−5 m2 /s is used. • Inflow profile A top-hat jet profile with an inlet velocity calculated from the Reynolds number is applied at the slit of the nozzle. In addition to the main jet a velocity inlet condition with 1 % of the velocity of the jet is applied at the left surface (“behind” the nozzle). This is meant to blow off the first few vortices created during the initial transient of the edge tone, before the periodic oscillation starts. • Boundary conditions A free slip wall condition is applied at the nozzle wall to avoid vortex generation. On the wedge wall a no slip condition is applied. On all other surfaces opening conditions are used. • Time stepping The time step is 170 µs and velocity and pressure fields are written out to an output file at every second step. • Turbulence model No turbulence model is applied and the flow is treated as laminar. • Input for aeroacoustic simulation The transient simulation is run until the oscillation of the jet reaches a stationary state. Beginning from the corresponding time step, 300 transient output steps, corresponding to a time interval of 102 ms length and to 13 stage 1 oscillations, are used for the aeroacoustic investigations. The transient sources are then interpolated to different acoustic grids of the source region and a FFT is performed to obtain harmonic sources at the main frequency of 130 Hz.

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(a)

(b)

Fig. 14.92 Slice through CFD meshes at z = 0. a Structured mesh. b Unstructured mesh

The spatial discretization for the CFD computation in the vicinity of the nozzle is depicted in Fig. 14.92a. The mesh is symmetric to the x-z-plane and the smallest edge-length (corresponding to the discretization of the wedge-wall-boundary layer) is about 5 µm whereas the coarsest one is about 5 mm. The mesh solely consists of hexahedra elements and the number of nodes is 715,268. Computations with an unstructured and unsymmetrical CFD mesh (cf. Fig. 14.92b), which also contained wedge elements, resulted in unsymmetrical dipole characteristics in the acoustic simulations. The computational domain for acoustics just consists of the source domain Ωs and three cube-shaped domains Ωi , Ωp , ΩPML of different sizes which are centered around the origin. The material air with the speed of sound c = 340 m/s is used. The dimensions and properties of the domains are therefore defined as follows: • Source domain Unlike in the CFD mesh, the nozzle volume is completely discretized by volume elements. Otherwise the geometry of the source domain is the same as in Fig. 14.91. • Intermediate domain Since elements in the propagation mesh can be bigger than the whole source domain an intermediate domain for bridging the gap between the propagation mesh and the source mesh with a structured mesh is introduced. The intermediate mesh has non-matching interfaces with Ωs as well as with Ωp . With the variable parameter w it is defined as Ωi = [−w . . . w]3 \ Ωs • Propagation domain The acoustic propagation domain is discretized with a structured mesh corresponding to the main frequency component of 130 Hz. The elements in this domain have the same edge length in all three dimensions. The domain is defined as Ωp = [−3 m . . . 3 m]3 \ (Ωs ∪ Ωi )

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• Damping Layer In order to simulate free-field conditions an additional PML layer is attached to the computational domain. Its thickness of 80 cm nearly corresponds to one third of the wave length λ = 2.62 m at 130 Hz. The discretization from Ωp is carried on conforming across the domain boundary. The PML domain is defined as ΩPML = [−3.8 m . . . 3.8 m]3 \ (Ωs ∪ Ωi ∪ Ωp ) With the given geometrical and physical setup a few test cases are defined in order to evaluate the sensitivity of the edge tone configuration to different grid resolutions. • ETFINE: This case is defined to be the reference for all other test cases. It features a very fine mesh in the source region (cf. Fig. 14.93). The width of the intermediate domain is defined as w = 40 cm. Tri-linear hexahedra are used in all domains except in Ωs . There, 6-node prism elements are used to cover the height of the nozzle. Above and below the nozzle 4-node tetrahedrons are used. The coarsest edge length in this mesh is about 6.66 cm which corresponds to a minimal value of about 39 degree of freedoms per wavelength. • ETCOFA: The same fine source mesh as in the ETFINE case is extended with a very coarse mesh in Ωp . The edge length of 26.66 cm still guarantees a resolution of about 10 degree of freedoms per wavelength. Since the whole source domain fits into one element of Ωp (depicted by the outer dashed white line in Fig. 14.94a), the width of the intermediate mesh w = 13.33 cm in Ωi is chosen in a way so that it just bridges this gap. In all domains except Ωs linear hexahedra are used. • ETCONE: The same mesh for the outer domains as in the ETCOFA case is used. The difference is however, that quadratic serendipity hexahedra are applied, which account for about 20 degree of freedoms per wavelength. In the source domain Ωs a coarser unstructured mesh is applied compared to the previous cases (cf. Fig. 14.94b).

Fig. 14.93 Cut through intermediate and propagation meshes at z = 0 for ETFINE case

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(a)

(b)

Fig. 14.94 Slice through acoustic meshes at z = 0. a Meshes for ETCOFA case. b Meshes for ETCONE case

14.8.2.2 3D Results In order to better understand the structure of the flow, color-contour and streamline plots of the source domain for one time step in the steady state are depicted in Fig. 14.95. One can clearly see, that the flow emanating from the whole height of the nozzle gets pushed towards the xy-plane by a large eddy. Already after about one third of the length of the wedge, the flow is mainly concentrated around the xy-plane. After the interpolation to the dedicated acoustic grid, the same behavior can also be observed for the harmonic Lighthill source term at the main frequency of 130 Hz (cf. Fig. 14.96). A cut through the source domain mesh of the ETFINE and the ETCOFA cases is shown. In addition to that, isocontours and a color mapped plot at z = 0 of the real part of the acoustic source terms and are shown.

(a)

(b)

Fig. 14.95 Structure of the fluid velocity field. a Fluid velocity field at z = 0. b Stream lines

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Fig. 14.96 Source domain for ETFINE and ETCOFA cases. Iso-surfaces for the real part of the nodal Lighthill sources are depicted

The expected dipole field for the whole acoustic domain can be seen in Fig. 14.97. The real part of acoustic pressure field at 130 Hz for ETFINE case is depicted. Also isocontours of the acoustic pressure at ±50 µPa are shown. A comparison of the directivity patterns of the sound pressure levels in the xyand yz-planes is depicted in Fig. 14.98. It reveals that coarser meshes, either in the source or the propagation domain, tend to under-predict the sound pressure levels. However, the deviations are just in maximum range of 1 dB. Again considering the fact, that the whole source domain is enclosed in a single element of the far-field domain, the obtained results match the reference case ETFINE quite well. For the number of unknowns as well as the wall clock times of the test cases refer to Table 14.17.

Fig. 14.97 Acoustic pressure plot over complete domain

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(a)

(b)

Fig. 14.98 Acoustic sound pressure levels in xy- and yz-planes. a Directivity plot xy-plane. b Directivity plot yz-plane Table 14.17 Number of unknowns and wall clock times for edge tone cases Test case Ωs Ωi Ωp ∪ ΩPML LM Total ETFINE ETCOFA ETCONE

144,788 144,788 11,905

35,892 380 1,343

217,952 30,176 114,824

285 76 220

398,917 175,420 128,292

WCT (s) 259.0 71.0 79.0

14.8.3 Airframe Noise For simulation of airframe noise we investigate the NACA 0012 test-case with an inflow velocity of Mach 0.2. An intensive investigation on this setup was carried out in [46], in which Lighthill’s analogy was successfully applied. For modeling the turbulent vortices within the Unsteady Reynolds Averaged Navier-Stokes (URANS) calculation, a transitional k − ω turbulence model is utilized. Even though the flow field of this setup is computed by the solution of the compressible Navier-Stokes equations, the low Mach number of 0.2 suggests that the resulting field is nearly incompressible and we apply the perturbation equations (9.57) for computing the acoustic field. The setup under investigation has the purpose to show the influence of convective effects on the wave propagation. Therefore, the acoustic pressure is also evaluated at certain microphone positions by the Ffowcs Williams and Hawkings (FWH) method which does not include mean flow effects on the wave propagation. The mean flow field of the setup is depicted in Fig. 14.99. Due to the higher mean flow velocity, the convective effects on the wave propagation are expected to be strong. One observes an inhomogeneous mean flow field with a maximum flow velocity magnitude of Mach 0.3. The contour plots of the right hand side of PE are depicted in Fig. 14.100a. Clearly visible are the vortex related sources on the top and bottom of the airfoil which arise due to vortex shedding in the shear layer. In the wake of the airfoil, lower

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Fig. 14.99 Contour plot of mean flow velocity field around the airfoil

Fig. 14.100 Contours of sources and acoustic pressure field obtained with PE. a Contours of PE right hand side density. Normed to maximum amplitude. b Contour plots of acoustic pressure at end of simulation run

amplitude vortex sources are visible. The location of the maximum in the source terms is right behind the tip of the airfoil. The contour plot of the acoustic pressure field is depicted in Fig. 14.100. One can see the decreasing amplitude of the acoustic waves traveling in downstream direction as well as their amplification in upstream direction. Still, the field computed by perturbation equations shows minor distortions which can be related to the compressible flow data used for computation of sources. The compressible flow field already contains acoustic components which falsify the solution of of the perturbation equations. One approach would be to apply a spatial filter to the source terms in order to suppress spurious sources as done in [47]. Another possibility would be to identify the vorticity of the flow field and to compute a filtered hydrodynamic pressure for calculating the source terms. A procedure which can be seen as a Helmholtz decomposition of the flow velocity field. To investigate the results in more detail we evaluate the third octave band spectrum at three characteristic points located on a half circle around the trailing edge with radius twice the length of the airfoil as depicted in Fig. 14.101. The locations are chosen such that one microphone is positioned right above the trailing edge whereas the other two are in up and downstream direction of the mean flow field, respectively. Therefore, similar results are expected for PE and FWH for microphone number two right above the cylinder because the mean flow field has only minor impact here. Bigger differences due to convective effects should be visible for the other microphones. In Fig. 14.102 one can observe the third octave spectra for the different microphone positions, verifying the initial assumption. The

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Fig. 14.101 Microphone positions surrounding the trailing edge of the airfoil

(a)

Perturbation equation Ffowcs Williams and Hawkings

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100

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SPL (dB)

SPL (dB)

100

80

60

80

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60 0.1

1

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Frequency (Hz)

Fig. 14.102 Sound pressure level at different microphone positions. a Microphone 1. b Microphone 2. c Microphone 3

results of perturbation equations and FWH coincide well at microphone position two, even though the FWH result shows higher amplitudes in the range from 4 to 5 kHz. For the microphone in downstream direction (Mic 3), the differences are much higher which is also expected because the amplitudes of acoustic waves decrease in downstream direction as also visible in Fig. 14.100b. For the upstream microphone (Mic 1), we can see slightly higher amplitudes in case of perturbation equations which

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corresponds to the steepening of acoustic waves propagating in opposite direction to the mean-flow. As a conclusion we can state that the results for perturbation equations seem to be reasonable with respect to the FWH solution and show the expected effects due to the mean flow.

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References

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46. M. De Gennaro, H. Kuehnelt, M. Kaltenbacher, A numerical investigation of the laminar instability multi-tonal noise of aerofoils, in Proceedings of the 18th AIAA/CEAS Aeroacoustics Conference (33rd AIAA Aeroacoustics Conference) (2012) 47. R. Ewert, W. Schröder, Acoustic perturbation equations based on flow decomposition via source filtering. J. Comput. Phys. 188, 365–398 (2003)

Chapter 15

Summary and Outlook

The precise numerical simulation of mechatronic sensors and actuators leads to so-called multifield problems, whose efficient solution—both with respect to CPUtime and memory—is a great challenge. The requirements on a CAE environment for the design of mechatronic sensors and actuators include the capability to simulate all involved single fields as well as all their relevant couplings. Furthermore, since in most cases we have to account for the nonlinearities within the single fields (e.g., nonlinear material dependencies, geometric nonlinearity, etc.) and for the nonlinearities due to the coupling terms (e.g., moving body in a magnetic/electric field, magnetic/electrostatic forces, etc.), the solution process is quite complicated. We have demonstrated that the development of sophisticated algorithms for the coupling of different physical fields within the numerical simulation process as well as the application of the latest algebraic solvers (geometric and/or algebraic multigrid methods) lead to simulation times that are acceptable for industrial design processes. However, although the performance of computer hardware has increased tremendously, which is particularly the case for PCs with Linux systems (e.g., Linux-cluster systems), real-life problems always ask for a further improvement of numerical algorithms. In the last years, Discontinuous Galerkin (DG) schemes have become very popular. Thereby, intelligent combinations of the Finite Volume (FV) and the Finite Element (FE) method have been explored, which utilize a local basis that mimics the FE method and at the same time includes the capability to solve conservation equations as the FV method. DG methods use completely discontinuous basis functions, and thus have a much higher flexibility than continuous FE methods, such as the allowance of arbitrary triangulation with hanging nodes, complete freedom in changing the polynomial degrees in each element independent of that in the neighbors (p-adaptivity), and extremely local data structure (elements only communicate with immediate neighbors regardless of the order of accuracy of the scheme) and the resulting embarrassingly high parallel efficiency (usually more than 99 % for a fixed mesh, and more than 80 % for a dynamic load balancing with adaptive meshes which often change during time evolution), see e.g. [1]. © Springer-Verlag Berlin Heidelberg 2015 M. Kaltenbacher, Numerical Simulation of Mechatronic Sensors and Actuators, DOI 10.1007/978-3-642-40170-1_15

537

538

15 Summary and Outlook

One step further are space-time DG methods. Here, the philosophy is to treat the time as an additional spatial coordinate. Therefore, the PDE is discretized in the space-time domain on so-called space-time slabs. A major advantage of these spacetime DG methods is that one can treat moving domains in a natural way. Thereby, the moving boundary is continuously given with respect to time and no projections between different deformed meshes is necessary. Furthermore, the method allows to apply local refinements in the space-time domain to resolve the local behavior of the exact solution. E.g., local singularities may occur when problems with moving boundaries have to be solved, or when problems with nonlinearities are considered, like in the case of fluid dynamics [2].

References 1. X. Feng, O. Karakashian, Y. Xing, Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations (Springer, Heidelberg, 2014) 2. M. Neumüller, Space-time methods. Ph.D. thesis, Technical University of Graz, Austria, 2013

Appendix A

Norms

A.1 Vector Norms Definition A.1 Vector norm: A vector norm on IRn is a function || || : IRn → IR, which fulfills the following properties: (i) (ii) (iii) (iv)

||x|| ≥ 0 for all x ∈ IRn ||x|| = 0 iff x = 0 ||x + y|| ≤ ||x|| + || y|| ||αx|| = |α| ||x||

We use the Hölder or p-norms, which are defined by  n 1/ p  p ||x|| p = |xi | with p ≥ 1 .

(A.1)

i=1

Therefore, we compute, e.g., the 1-norm, 2-norm, and ∞-norm as follows ||x||1 = ||x||2 =

n  i=1

|xi |

 n  i=1

|xi |

(A.2)

2

1/2

||x||∞ = maxi |xi | .

(A.3) (A.4)

The p-norms have the following useful properties • • • • •

|x T y| ≤ ||x|| p || y||q , 1p + q1 = 1 (Hölder inequality) |x T y| ≤ ||x||2 || y|| √2 (Cauchy–Schwarz inequality) ||x||2 ≤ ||x||1 ≤ √n||x||2 ||x||∞ ≤ ||x||2 ≤ n||x||∞ ||x||∞ ≤ ||x||1 ≤ n||x||∞

© Springer-Verlag Berlin Heidelberg 2015 M. Kaltenbacher, Numerical Simulation of Mechatronic Sensors and Actuators, DOI 10.1007/978-3-642-40170-1

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540

Appendix A: Norms

With the help of norms we can define a distance on a vector space, and furthermore, we call a vector space with a norm a normed space.

A.2 Matrix Norms Definition A.2 Matrix norm: A matrix norm on IRn×m is a function || || : IRn×m → IR, which fulfills the following properties: (i) (ii) (iii) (iv)

|| A|| ≥ 0 for all A ∈ IRn×m || A|| = 0 iff A = 0 || A + B|| ≤ || A|| + ||B|| for all A , B ∈ IRn×m ||α A|| = |α| || A|| for all α ∈ IR and A ∈ IRn×m

The matrix norm associated to the vector p-norm is defined by the operator norm || A|| p = sup x=0

|| Ax|| p . ||x|| p

(A.5)

Other matrix norms are ⎛ ⎞1/2 n  m  || A|| F = ⎝ |ai j |2 ⎠

Frobenius or F-norm

(A.6)

|ai j |

column sum norm

(A.7)

|ai j |

row sum norm .

(A.8)

i=1 j=1

|| A||1 = max j || A||∞ = maxi

n  i=1

m  j=1

Appendix B

Scalar and Vector Fields

Definition B.1 (Scalar field) If we assign to each point in IR3 defined by the vector r a scalar quantity V (r) (e.g., electric potential, temperature, acoustic velocity potential), then V is called a scalar field. For the illustration of scalar fields we use iso-lines in the 2D case and iso-surfaces in the 3D case, where the scalar quantity V (r) is constant (Fig. B.1). Definition B.2 (Vector field) If we assign to each point IR3 defined by the vector r a vector quantity F(r) (e.g., electric field, magnetic field, mechanical deformation), then F is called a vector field. Vector fields are divided into irrotational vector fields (e.g., electrostatic field) and solenoidal vector fields (e.g., magnetic field) as shown in Fig. B.2 (see also Sects. B.12 and B.13). The lines of force (see Fig. B.3) are defined by F(r) × dr = 0 ,

(B.1)

which means that in each point of the lines the field vector F is parallel to the tangential vector. In the following, we try to compute the lines of force for the vector field F(r) =

r r3

(B.2)

with the help of (B.1). Since we are just interested in the direction, we have to solve r × dr = 0 .

(B.3)

By using a Cartesian coordinate system, we obtain r = x ex + ye y + zez

dr = dx ex + dye y + dzez , © Springer-Verlag Berlin Heidelberg 2015 M. Kaltenbacher, Numerical Simulation of Mechatronic Sensors and Actuators, DOI 10.1007/978-3-642-40170-1

(B.4) (B.5) 541

542

Appendix B: Scalar and Vector Fields

V3 V1 V2

r2

r1 r0

Fig. B.1 Illustration of a scalar field V with the help of equipotential surfaces

(a)

y

y

(b) H

E

x

x

Fig. B.2 Vector fields. a Solenoidal vector field, b irrotational vector field

dr

F (r)

F (r1 ) r + dr r2

r1 0 Fig. B.3 Lines of force for the vector field F(r)

Appendix B: Scalar and Vector Fields

543

Fig. B.4 Lines of force of the vector field F(r) = r/r 3

and

⎛ ⎞ e x e y ez y dz − z dy r × dr = x y z = ⎝ z dx − x dz ⎠ . dx dy dz x dy − y dx

(B.6)

Therefore, we can formulate the following three relations y dz = z dy z dx = x dz

x dy = y dx .

(B.7) (B.8) (B.9)

We now search for the line of force including point P0 (x0 , y0 , z 0 ). Integration of (B.7) results in y z ln = ln (B.10) y0 z0 and z 0 y = y0 z .

(B.11)

Analogously, we can compute the solutions of the other two differential equations z 0 x = x0 z

y0 x = x0 y .

(B.12) (B.13)

From (B.12) a plane through point P0 containing the y-axis, and from (B.13) a plane through point P0 containing the z-axis is defined. The intersection of the two planes leads to a straight line through the origin, and therefore we obtain the vector field drawn in Fig. B.4, which corresponds, e.g., to the vector field of an electric charge.

B.1 The Nabla (∇) Operator First, we recall that a scalar function may depend on one or more variables, e.g., using Cartesian coordinates, a function can be denoted by f = f (x, y, z) .

544

Appendix B: Scalar and Vector Fields

The partial derivatives read as ∂f ∂f ∂f , , . ∂x ∂ y ∂z The nabla operator ∇ is defined in Cartesian coordinates by ∇=

⎛

⎜ ∂ ∂ ∂ ex + ey + ez = ⎜ ⎝ ∂x ∂y ∂z

∂ ∂x ∂ ∂y ∂ ∂z

⎞

⎟ ⎟, ⎠

(B.14)

where ex , e y and ez are the unit vectors in x-, y-, and z-directions. The interaction between the nabla operator and a scalar or a vector field yields its geometric significance.

B.2 Definition of Gradient, Divergence, and Curl We introduce a scalar function V with nonzero first-order partial derivatives with respect to the coordinates x, y, and z, and a vector field F with components Fx , Fy , and Fz . Then, the following operations are defined: 1. Gradient of a scalar:

⎛

⎜ grad V = ∇V = ⎜ ⎝

∂V ∂x ∂V ∂y ∂V ∂z

⎞

⎟ ⎟. ⎠

As can be seen, the result of this operation is a vector. 2. Divergence of a vector: div F = ∇ · F =

∂ Fy ∂ Fx ∂ Fz + + . ∂x ∂y ∂z

Therefore, the result of this operation is a scalar value. 3. Curl of a vector: ⎛ ∂ Fz ∂ Fy ex ey ez ∂ y − ∂z ⎜ ∂ Fx curl F = ∇ × F = ∂/∂x ∂/∂ y ∂/∂z = ⎜ − ∂∂xFz ⎝ ∂z Fx ∂ Fy ∂ Fx Fy Fz ∂x − ∂ y

The result of taking the curl of a vector is again a vector.

⎞

⎟ ⎟. ⎠

Appendix B: Scalar and Vector Fields

545

B.3 The Gradient We will consider the scalar function V (x, y, z) with its partial derivatives ∂V /∂x, ∂V /∂ y, ∂V /∂z and dependent on a point P = (x, y, z). In the first step we calculate the total differential of V dV =

∂V ∂V ∂V dx + dy + dz . ∂x ∂y ∂z

(B.15)

Now, we define a point P ′ infinitely close to P by P ′ = (x + dx, y + dy, z + dz). By calculating the vector dP = P ′ − P, which has the components dP = (dx, dy, dz)T , we can write (B.15) as

  ∂V ∂V ∂V dV = ex + ey + ez · dx ex + dye y + dzez ∂x ∂y ∂z = ∇V · dP.

(B.16) (B.17)

For the geometrical illustration of the gradient, consider an equipotential surface, i.e., a surface with V = const. (see Fig. B.5). Hence, for all differential displacements from P to P ′ on this surface d V = 0 holds, and therefore, ∇V · dP = 0 .

(B.18)

From the definition of the scalar product it is clear that ∇V and dP are orthogonal. In this situation the displacement from P to P ′ points into the direction of increasing V , as shown in Fig. B.6, and the scalar product ∇V · dP is positive. From the foregoing arguments, we conclude that ∇V is a vector, perpendicular to the surface on which V is constant and that it points in the direction of increasing V . As an example we consider a function r (x, y, z), which defines the distance of a point P from the origin (0, 0, 0). The surface r = const. is a sphere of radius r with center (0, 0, 0), whose equation is given by r=



x 2 + y2 + z2 .

Fig. B.5 The gradient is orthogonal to a constant potential surface

∇V

dP .

P V = const.

P′

546

Appendix B: Scalar and Vector Fields

Fig. B.6 Geometrical representation of the gradient

V = V2 ∇V P′ dP P V = V1

Therefore, the gradient calculates as x ∂r x =  = 2 2 2 ∂x r x +y +z ∂r y = ∂y r z ∂r = ∂z r x ex + ye y + zez r ∇r = = . r r Geometrically speaking, ∇r points in the direction of increasing r , or towards spheres with radii larger than r .

B.4 The Flux Definition B.3 (Flux) The vector field F(r) and a corresponding surface Γ as shown in Fig. B.7 are given. The vector n denotes the normal unit vector of the differential surface dΓ . Therefore, the differential flux dψ through dΓ is defined by dψ = F · dΓ = F · ndΓ .

(B.19)

The total flux ψ computes as ψ=



Γ

F · dΓ .

(B.20)

Appendix B: Scalar and Vector Fields

547

Fig. B.7 Flux through the surface Γ

n

Γ F dΓ

Fig. B.8 Flux ψ through the square with area h 2

ψ Γ

z

y

h h

x −h

In the following, we want to compute the flux ψ of the vector field F(r) = r through the square Γ with side length h according to Fig. B.8. With the normal unit vector n = ex and dΓ = dy dzex we obtain ψ=

h h 0

0

= −h

  −hex + ye y + zez · ex dy dz

h h

0 3

dydz

0

= −h .

(B.21)

The total flux ψ through a closed surface S is given by ψ=



Γ

F · dΓ

and defines whether we have sources (ψ > 0) or sinks (ψ < 0) within Γ .

(B.22)

548

Appendix B: Scalar and Vector Fields

Fig. B.9 Flux through the closed surface Γ1 ∪ Γ2

dΓ0 dΓ2 Γ1

dΓ1 Γ2

A very important property of the flux ψ defined by a closed surface is given by (see Fig. B.9)    F · dΓ + F · dΓ = F · dΓ . (B.23) Γ1 ∪Γ0

Γ2 ∪Γ0

Γ1 ∪Γ2

B.5 Divergence Definition B.4 (Divergence) The vector field F(r) is given. If we divide the flux ψ, defined by a closed surface Γ , by the corresponding volume Ω and let the volume Ω tend to zero, then the obtained value is called the divergence (source density)  1 dψ div F = lim . (B.24) F · dΓ = Ω→0 Ω Γ dΩ r Let us now consider the closed surface of a differential cube (see Fig. B.10) and the general vector field F(r) = Fx ex + Fy e y + Fz ez . In the first step, let us compute the differential flux through the hatched surfaces F · dΓ = [F(x + dx/2, y, z) − F(x − dx/2, y, z)] · ex dy dz

  ∂ Fx dx ∂ Fx dx dy dz − Fx (x, y, z) − ≈ Fx (x, y, z) + ∂x 2 ∂x 2 ∂ Fx dx dy dz . (B.25) = ∂x Analogously, we obtain the contribution of the other two directions, and thus, the differential flux

∂ Fy ∂ Fz ∂ Fx dx dy dz . (B.26) + + dψ = ∂x ∂y ∂z Since the differential volume dΩ is equal to dx dy dz, we end up with the following expression for the divergence of a vector field in Cartesian coordinates div F =

∂ Fy ∂ Fx ∂ Fz + + , ∂x ∂y ∂z

(B.27)

Appendix B: Scalar and Vector Fields

549

dx (x − dx/2), y, z) (x + dx/2, y, z) dz

(x, y, z)

dy r z y

x Fig. B.10 Flux through a cube

or, by using the nabla operator, div F = ∇ · F .

(B.28)

B.6 Divergence Theorem (Gauss Theorem) By the definition of the divergence (see B.24) we get dψ = ∇ · F dΩ  ∇ · F dΩ . ψ=

(B.29) (B.30)

Ω

On the other hand, we have the relation for the flux ψ according to (B.22). Combining these two expressions for the flux results in ψ=



Ω

∇ · F dΩ =



Γ (Ω)

F · dΓ .

(B.31)

This equality between the two integrals tells us that the flux of the vector F through the closed surface Γ is equal to the volume integral of the divergence of F over the volume Ω enclosed by the surface Γ . Consider a radial vector field F as shown in Fig. B.11, and assume that the magnitude of F is constant in all points on a sphere centered at P. To compute the flux of the vector field F through a spherical shell of radius R, we note that d s and

550

Appendix B: Scalar and Vector Fields

Fig. B.11 Radial vector field

F

P

F are collinear and in the same direction   ψ= F · dΓ = F dΓ = 4π R 2 F . S

Γ

From the divergence theorem, (the flux is nonzero) we conclude ∇ · F = 0 .

B.7 The Circulation The circulation of a vector field F(r) along a closed contour C is given by the closed-line integral  F · ds . (B.32) Z= C

Therefore, the important property follows (see Fig. B.12) 

C1 ∪C0

F · dr +



C2 ∪C0

F · dr =



C1 ∪C2

F · dr .

(B.33)

If the circulation along a closed curve C is not equal to zero, then we say the closed line contains eddies.

Fig. B.12 Circulation along the closed line C1 ∪ C2

C1

dr2

C0 dr1 C2

Appendix B: Scalar and Vector Fields

551

B.8 The Curl Definition B.5 (Curl) We consider a point defined by r (Fig. B.13), in which the curl of the vector field F has to be computed. Furthermore, we define a closed line C enclosing the area Γ and consider the circulation along C. If the area Γ tends to zero, we obtain the definition of the curl by n · curl F = lim

Γ →0



C

F · ds dZ = . Γ dΓ

(B.34)

The vector curl F is obtained by a separation in the three directions of the unit vectors ex , e y , and ez curl F = (ex · curl F)ex + (e y · curl F)e y + (ez · curl F)ez .

(B.35)

The circulation for the differential square in Fig. B.14 is given by dZ x = (F(x, y, z − dz/2) − F(x, y, z + dz/2)) · e y dy + (F(x, y + dy/2, z) − F(x, y − dy/2, z)) · ez dz

∂ Fy ∂ Fz ≈ dydz . − ∂y ∂z

(B.36)

Therefore, we obtain the x-component of curl F with dΓ = dy dz ex · curl F =

∂ Fy ∂ Fz − . ∂y ∂z

Fig. B.13 Curl in a point defined by r

(B.37)

n

Γ

r

0

552

Appendix B: Scalar and Vector Fields

Fig. B.14 x-component of curl F

dy

e1

dz z r y x

Analogously, the y- and z-component of curl F can be computed, and the full vector in Cartesian coordinates reads as ⎛ ∂ Fz ∂ Fy ⎞ ∂ y − ∂z ⎜ ∂F ⎟ ∂ Fz ⎟ x (B.38) curl F = ⎜ ⎝ ∂z − ∂x ⎠ , ∂ Fy ∂ Fx ∂x − ∂ y or with the help of the nabla operator

ex ∂ curl F = ∇ × F = ∂x F

x

ey ∂ ∂y

Fy

B.9 Stoke’s Theorem

ez ∂ ∂z . Fz

(B.39)

We consider the vector field F on the surface Γ with fixed oriented contour C as shown in Fig. B.15. For a differential surface dΓν , we obtain according to (B.34) dZ ν = n(r ν ) · curl F(r)dΓν  Z ν = curl F(r) · dΓ ν ,

(B.40)

Γν

and Z=



Γ

curl F · dΓ .

(B.41)

Furthermore, according to the definition of the circulation Z (see B.32), we get the following relation   F · dr = curl F · dΓ . (B.42) Z= C

Γ

Appendix B: Scalar and Vector Fields

553

Fig. B.15 Vector field F on the surface Γ with fixed oriented contour C

n

Γ Γν Cν

ds

rν 0

For a radial vector field F as shown in Fig. B.11, the closed-line integral along a circle C of constant radius  F · ds C

is zero, and therefore, the curl of this vector field ∇ × F is zero, too.

B.10 Green’s Integral Theorems The integral theorems of Green can be derived from the divergence theorem. For this purpose, we first introduce the Laplace operator by ∆=∇ ·∇ =

∂2 ∂2 ∂2 + + . ∂x 2 ∂ y2 ∂z 2

(B.43)

This differential operator can be applied to scalar as well as vector quantities ∆V = div (grad V )

(B.44)

∆F = (∆Fx )ex + (∆Fy )e y + (∆Fz )ez .

(B.45)

Setting a vector F equal to V1 ∇V2 and using the divergence theorem, we obtain according to (B.31) 

Ω

∇ · (V1 ∇V2 )dΩ =



Γ

(V1 ∇V2 ) · dΓ .

(B.46)

554

Appendix B: Scalar and Vector Fields

Since the term ∇ · (V1 ∇V2 ) can be expressed by (see B.53 below) ∇ · (V1 ∇V2 ) = V1 ∆V2 + ∇V1 · ∇V2 ,

(B.47)

we get the following integral theorem, called Green’s first integral theorem    ∂V2 V1 ∆V2 dΩ + ∇V1 · ∇V2 dΩ = V1 dΓ . (B.48) ∂n Ω Ω Γ By substituting V1 with V2 and vice versa in (B.46) and subtracting the resulting equation from (B.46), we achieve Green’s second integral theorem

   ∂V1 ∂V2 dΓ . (B.49) − V2 V1 V1 ∆V2 dΩ − V2 ∆V1 dΩ = ∂n ∂n Ω Ω Γ In addition, Green’s first integral theorem in vector form is  (∇ × u · ∇ × v − u · ∇ × ∇ × v) dΩ Ω

=



(u × ∇ × v) · ndΓ ,

(B.50)

Γ

and Green’s second integral theorem in vector form reads as  (u · ∇ × ∇ × v − v · ∇ × ∇ × u) dΩ Ω

=



(v × ∇ × u − u × ∇ × v) · ndΓ .

(B.51)

Γ

B.11 Application of the Operators By using the definitions of gradient, divergence, and curl in Cartesian coordinates, the following relations hold: ∇(V1 V2 ) = V1 ∇V2 + V2 ∇V1 ∇ · (V F) = V ∇ · F + F · ∇V

∇ · (F 1 × F 2 ) = F 2 · ∇ × F 1 − F 1 · ∇ × F 2 ∇ × (V F) = V ∇ × F − F × ∇V ∆F = ∇(∇ · F) − ∇ × (∇ × F) .

(B.52) (B.53) (B.54) (B.55) (B.56)

These relations combine the essential differential operators and build up a basis for the description of physical fields.

Appendix B: Scalar and Vector Fields

555

B.12 Irrotational Vector Fields We consider a vector field F, which is given as the gradient of a scalar potential F = ∇V . The computation of a line integral from point A to point B yields B

(∇V ) · dr = V (B) − V (A) .

(B.57)

A

Therefore, for any closed contour within this vector field, the following relation holds 

(∇V ) · dr = 0 .

(B.58)

This result proves that any vector field that can be expressed by the gradient of a scalar potential is irrotational. Furthermore, the local quantity, given by the curl of the vector field, is zero ∇ × ∇V = 0 . (B.59)

B.13 Solenoidal Vector Fields We will consider the solenoidal vector field ∇ × F for a domain as displayed in Fig. B.16. This domain shall consist of two subdomains defined by their surfaces Γ1 and Γ2 with their related contours C1 and C2 . By using Stoke’s theorem, we obtain the following relation 

Γ

(∇ × F) · dΓ = =



Γ1



C1

(∇ × F) · n1 dΓ + F · dr +



C2



Γ2

(∇ × F) · n2 dΓ

F · dr

= 0.

(B.60)

Fig. B.16 Domain for solenoidal vector field

Γ2

C2 C1 Γ1

556

Appendix B: Scalar and Vector Fields

Thus, the total flux (global quantity) is zero, and, furthermore, the local solenoidality, too ∇ · (∇ × F) = 0 . (B.61)

Appendix C

Tensors and Index Notation

Tensors are simply speaking a linear mapping. E.g., a second order tensor [ A] is a linear mapping that associates a given vector u with a second vector v by v = [S]u . The term linear in the above relation implies that given two arbitrary vectors u and v and two arbitrary scalars α, β, then the following relation holds [S] (αu + βv) = α[S]u + β[S]v . The extension to tensors of higher rank is straight forward. E.g., Hook’s law maps the mechanical strain tensor [S] by the 4th order elasticity tensor [c] to the mechanical stress tensor [σ] [σ] = [c] [S] . Now, index notation is a powerful tool to write complex operations of vectors and tensors in a more readable way. However, there are times when the more conventional vector notation is more useful. It is therefore important to be able to easily convert back and forth between the two notations. Table C.1 describes our notation.1 An index can be a free or a dummy index. For free indices, the following rules are defined: • The number of free indices equals the rank as displayed in Table C.1. Thereby, a scalar is a tensor with rank 0, and a vector is a tensor of rank 1. Tensors may assume a rank of any integer greater than or equal to zero. Please note that it is just allowed to sum together tensors with equal rank. • A free index appears once and only once within each additive term and remains within the expression after the operation has been performed, e.g. ai = ǫi jk b j ck + Ai j d j . 1

(C.1)

Our notation does not differ between tensors of different orders.

© Springer-Verlag Berlin Heidelberg 2015 M. Kaltenbacher, Numerical Simulation of Mechatronic Sensors and Actuators, DOI 10.1007/978-3-642-40170-1

557

558

Appendix C: Tensors and Index Notation

Table C.1 Vector and index notation

Scalar Vector Tensor (2nd order) Tensor (3rd order) Tensor (4th order)

Vector

Index

Rank

ξ u [ A] [B] [C]

ξ ui Ai j Bi jk Ci jkl

0 1 2 3 4

• The same letter must be used for the free index in every additive term. • The first free index in a term corresponds to the row, and the second corresponds to the column. Thus, a vector (which has only one free index) is written as a column of three rows ⎛ ⎞ u1 u = ui = ⎝ u2 ⎠ u3 and a second order tensor as ⎞ A11 A12 A13 [ A] = Ai j = ⎝ A21 A22 A23 ⎠ A31 A32 A33 ⎛

A dummy index defines an index, which does not appear in the final expression any more. The rules are as follows: • A dummy index appears twice within an additive term of an expression. For the example above (see (C.1)), the dummy indices are j and k. • A dummy index implies a summation over the range of the index, e.g. aii = a11 + a22 + a33 . For many operations we use the Kronecker delta (2nd order tensor) ⎞ ⎛ 1 0 0 δi j = ⎝ 0 1 0 ⎠ 0 0 1

and the alternating unit tensor (3rd order tensor) ⎧ ⎨ 1 if i jk = 123, 231 or 312 0 if any two indices are the same ǫi jk = ⎩ −1 if i jk = 132, 213 or 3321 Thereby, the following relation can be explored ǫi jk =

1 (i − j) ( j − k) (k − i) . 2

(C.2)

(C.3)

Appendix C: Tensors and Index Notation

559

With these definitions, we may write vector and tensor operations using index notation. Here, we list the most useful ones: • Scalar product of two vectors a · b = c → ai bi = c

(C.4)

• Vector product of two vectors a × b = c → ǫi jk a j bk = ci • Gradient of a scalar

∇φ = u →

• Gradient of a vector

⎛ ∂a

⎜ ∇a = ⎜ ⎝

1

∂x1 ∂a1 ∂x2 ∂a1 ∂x3

∂a2 ∂x1 ∂a2 ∂x2 ∂a2 ∂x3

∂φ = ui ∂xi

∂a3 ∂x1 ∂a3 ∂x2 ∂a3 ∂x3

• Gradient of a second order tensor ∇ [ A] = • Divergence of a vector

⎞

⎟ ⎟ → ∂ai ⎠ ∂x j

3  ∂ Ai j ∂[ A] = ei ⊗ e j ⊗ ek ∂x ∂xk

(C.5)

(C.6)

(C.7)

(C.8)

i, j,k=1

∂ai =b ∂xi

(C.9)

3  ∂ Ai j ei ∇ · [ A] = ∂x j

(C.10)

∇·a=b →

• Divergence of a second order tensor

i, j=1

• Curl of a vector

∇ × a = b → ǫi jk

∂ak = bi ∂x j

(C.11)

• Double product or double contraction of two second order tensors [ A] : [B] = c → Ai j Bi j = c

(C.12)

or of a fourth order tensor with a second order tensors, e.g. Hooks law (see Sect. 3.4) [σ] = [c] : [S]

(C.13)

560

Appendix C: Tensors and Index Notation

• Dyadic or tensor product a ⊗ b = [C] → ai b j = Ci j

(C.14)

[ A] ⊗ b = [C] → Ai j bk = Ci jk

(C.15)

[ A] ⊗ [B] = [ D] → Ai j Bkl = Di jkl

(C.16)

• Product of two tensors [ A][B] = [C] → Ai j B jk = Cik

(C.17)

Note that only the inner index is to be summed. • Vector product of a second order tensor and a vector a[B] = c → ai Bi j = c j

(C.18)

A typical example is obtaining out of the mechanical stress tensor [σ] the stress vector in normal direction, e.g. n[σ] = σ n . Please note that n[σ] = [σ]T n = [σ]n • Trace of a tensor tr ([ A]) = b → Aii = b

(C.19)

The transpose of a tensor [ A] is defined as the tensor [ A]T , which for any two vectors a and b satisfies (C.20) a · [ A]b = b · [ A]T a . The definition of the transposed for a tensor [c] of 4th order reads as [ A] : [c][B] = [B] : [c]T [ A] .

(C.21)

Furthermore, the following relations hold ([ A][B])T = [B]T [ A]T ; (a ⊗ b)T = b ⊗ a ; ([ A] ⊗ [B])T = [B] ⊗ [ A] . (C.22) Tensors that satisfy the property [ A]T [ A] = [I] ,

(C.23)

where [I] is the identity tensor, are said to be orthogonal. In fact, any second order tensor [T ] can be decomposed into a symmetric tensor [S] and into a skew tensor [ A] [T ] = [S] + [ A] ,

(C.24)

Appendix C: Tensors and Index Notation

561

which compute as follows [S] =

  1 1 [T ] + [T ]T ; [ A] = [T ] − [T ]T . 2 2

(C.25)

Thereby, the following properties for a general tensor [T ] and a symmetric tensor [S] are fulfilled [S] : [T ] = [S] : sym ([T ]) ; a · [T ]a = a · sym ([T ]) a .

(C.26)

Appendix D

Appropriate Function Spaces

Let us define the derivative of order α with respect to the multi-index α, with |α| = i αi and αi ∈ IN, as follows D α v :=

∂ |α| v . ∂x1α1 · · · xnαn

(D.1)

For example, the partial derivatives of order 2 in IR2 can be written as D α v with α = (2, 0), α = (1, 1) or α = (0, 2), since |α| = α1 +α2 = 2 is fulfilled for all three cases α = (2, 0) α = (1, 1) α = (0, 2)

Dαv =

∂2v ∂x12

∂2v ∂x1 ∂x2 ∂2v Dαv = . ∂x22 Dαv =

Definition D.1 Continuously differentiable functions: Let Ω be a closed domain in IRn and let C(Ω) denote the space of continuous functions on Ω. Now, the space of up to order m continuously differentiable functions is given by   C m (Ω) = v : Ω → IR | D α v ∈ C(Ω), |α| ≤ m . (D.2) If the function v is infinitely often continuously differentiable on Ω, we write v ∈ C ∞ (Ω). For the function u(x) shown Fig. D.1 the following inclusions hold (with v(x) = u ′ (x)) v ∈ C0 u ∈ C1 . © Springer-Verlag Berlin Heidelberg 2015 M. Kaltenbacher, Numerical Simulation of Mechatronic Sensors and Actuators, DOI 10.1007/978-3-642-40170-1

563

564

Appendix D: Appropriate Function Spaces

u′ (x)

u′′ (x)

2

2

2

1

1

1

u(x)

x

x

x 1

1

2

2

1

2

Fig. D.1 Example of a C 1 function

Definition D.2 Square integrable functions: Let Ω be a closed domain in IRn . Then, the function u is called square integrable, if it fulfills the following relation 

|u(x)|2 dx < ∞ .

(D.3)



(D.4)

Ω

We denote L 2 (Ω) = {u : Ω → IR |

| u(x) |2 dx < ∞} .

Ω

For example, the function f (t) with the definition ⎧ ⎪ ⎨ 1 for 0 < x < 2 0 for x = 0 f (t) = ⎪ ⎩ −1 for −2 ≤ x < 0

(D.5)

belongs to the space L 2 (−2, 2) (see Fig. D.2). Analogously to the above definition, we obtain the definition for L p (Ω)-spaces for p ∈ [1, ∞). Fig. D.2 Function u(x) = sgn(x) in the interval (–2,2)

u(x)

1

x −2

0 −1

2

Appendix D: Appropriate Function Spaces

565

Definition D.3 L p (Ω)-spaces: Let Ω be a closed domain in IRn . Then, the space of p-integrable functions is given by L p (Ω) = {u : Ω → IR|



|u(x)| p dx < ∞} .

(D.6)

Ω

Let us assume that the function u has a continuous derivative u ′ . According to the formula for partial integration, we have for each continuously differentiable function ϕ with ϕ(a) = ϕ(b) = 0 the following relation b a

′

u(x)ϕ (x) dx = −

b

u ′ (x)ϕ(x) dx .

(D.7)

a

With the help of (D.7), we can define the derivative of functions, which have no finite derivative in the classical sense. If u and w denote integrable functions that fulfill the following relation b a

′

u(x)ϕ (x) dx = −

b

w(x)ϕ(x) dx

(D.8)

a

for all differentiable functions ϕ with ϕ(a) = ϕ(b) = 0, then the function w is called the derivative of u in the weak sense (with respect to x). The function u defined by (see Fig. D.3) x + 1 for −1 ≤ x ≤ 0 u(x) = 1 − x for 0 < x ≤ 1 will have no derivative in the classical sense at x = 0. Applying partial integration for differentiable functions ϕ(x) with ϕ(−1) = ϕ(1) = 0, we obtain 1

−1

′

u(x)ϕ (x) dx =

0

−1

=−

′

(x + 1)ϕ (x) dx +

(1 − x)ϕ′ (x) dx

0

0

ϕ(x) dx + (x + 1)ϕ(x)|0−1

1

(−1)ϕ(x) dx + (1 − x)ϕ(x) |10

−1

−

1

0

⎡

= −⎣

0

−1

ϕ(x) dx +

1 0

⎤

(−1)ϕ(x) dx ⎦ + ϕ(0) − ϕ(0) . % &' ( =0

566

Appendix D: Appropriate Function Spaces

Fig. D.3 Example of a function in H 1 (a, b)

u(x) 1

−1

0

x

1

Therefore, in the weak sense of differentiation we obtain ) 1 for −1 ≤ x < 0 ′ u (x) = −1 for 0 < x ≤ 1 with an arbitrary value for u ′ (0). Definition D.4 Sobolev space: Let Ω be a domain in IRn . The functional space W pm (Ω) = {u ∈ L p (Ω)|D α u ∈ L p , |α| ≤ m}

(D.9)

is called Sobolev space W pm (Ω). The partial derivatives of u are defined in the weak sense. The appropriate norms on Sobolev spaces are defined by ⎛

||u||W pm (Ω) = ⎝

 

Ω |α|≤m

⎞1/ p

|D α u| p dx ⎠

(D.10)

and its semi-norm by ⎛

|u|W pm (Ω) = ⎝

 

Ω |α|=m

⎞1/ p

|D α u| p dx ⎠

.

(D.11)

If we restrict p to two, then we obtain a Hilbert space (W2m (Ω) = H m (Ω)) with the scalar product

(u, v) =



Ω

⎛ ⎝



|α|≤m

⎞

D α u D α v ⎠ dx .

(D.12)

Appendix D: Appropriate Function Spaces

567

For example, the function u(x) is in the space H 1 (a, b), if u ′ (x) exists and is within the space L 2 (a, b). The norm is computed via

||u|| H 1 (a,b)

* + b b + + = , (u(x))2 dx + (u ′ (x))2 dx ,

(D.13)

* + b + + = , (u ′ (x))2 dx ,

(D.14)

a

a

its semi-norm by |u| H 1 (a,b)

a

and the scalar product as follows

(u, v) H 1 (a,b) =

b a

u(x)v(x) dx +

b

u ′ (x)v ′ (x) dx .

(D.15)

a

Definition D.5 Let Ω be a domain in IRn and denote by C0∞ (Ω) the space of infinitely often differentiable functions with zero boundary values. Then we write for the closure of C0∞ (Ω) with respect to the H 1 norm H01 (Ω) = C0∞ (Ω)

H 1 (Ω)

⊂ H 1 (Ω) .

(D.16)

Definition D.6 Partial Integration: Let Ω ⊂ IRn , n = 2, 3 be a domain with smooth boundary Γ . Then, for any u, v ∈ H 1 (Ω) the following relation holds 

∂u v dx = ∂xi

Ω



Γ

uv n · ei ds −



u

∂v dx . ∂xi

(D.17)

Ω

In (D.17) n denotes the outer normal and Ω¯ the considered domain Ω with boundary Γ . By a multiple application of (D.17), we arrive at Green’s formula 

Ω

∆u v dx =



Γ

for all u ∈ H 2 (Ω) and v ∈ H 1 (Ω).

∂u v ds − ∂n



Ω

(∇u)T ∇v dx

(D.18)

Appendix E

Solution of Nonlinear Equations

In this section we are concerned with the solution of systems of nonlinear equations. As an example, we will consider the nonlinear Poisson equation, given as follows − ∇ · ε(|∇u|)∇u − f = 0 u=0

(E.1) on Γ .

(E.2)

This defines a nonlinear operator F that allows us to rewrite (E.1) and (E.2) as F(u) = 0 .

(E.3)

The weak formulation of (E.1) and (E.2) for all test functions v ∈ H01 reads as 

Ω

ε(|∇u|)∇v · ∇u dΩ −



v f dΩ = 0 .

(E.4)

Ω

By applying the finite element method, we arrive at the following algebraic system K (u)u = f ,

(E.5)

with the matrix K ∈ IRn×n , f ∈ IRn , u ∈ IRn and n the number of unknowns. Since we cannot solve (E.5) explicitly, we have to establish an approximate solution by setting up a series u k (k = 0, 1, 2, 3, ..) that is supposed to converge to the correct solution. Concerning the rate of convergence, we will restrict the discussion to the following types: Definition E.1 Convergence: Let u ∗ ∈ IRn be the exact solution. Then

• u k converges towards u ∗ q-quadratically (q stands for quotient), if there exists a C such that ||u k+1 − u ∗ || ≤ C||u k − u ∗ ||2 . © Springer-Verlag Berlin Heidelberg 2015 M. Kaltenbacher, Numerical Simulation of Mechatronic Sensors and Actuators, DOI 10.1007/978-3-642-40170-1

(E.6) 569

570

Appendix E: Solution of Nonlinear Equations

• u k converges towards u ∗ q-linearly with the q-factor σ ∈ (0, 1), if ||u k+1 − u ∗ || ≤ σ||u k − u ∗ || .

(E.7)

In general, a q-quadratically convergent algorithm is preferable to a q-linearly convergent one. However, we always have to take into account the numerical cost for one iteration. Therefore, in some cases the method with the slower convergence rate can even be faster. Since we solve (E.5) numerically by computing a series of approximating solutions u k , the question of the stopping criterion is of great importance. In general, we distinguish between the following two types of stopping criteria: (1) Error criterion: We take the solutions of two successive iteration steps and define an absolute accuracy εabs by ||u k+1 − u k ||2 < εabs ,

(E.8)

and a relative accuracy εrel by ||u k+1 − u k ||2 < εrel ||u k+1 ||2 ,

(E.9)

which has to be achieved. However, in some analysis the true solution may still be far away, although the above-defined stopping criteria are fulfilled. This may particularly occur in the solution methods that have to use a line search (see Sect. E.1) to avoid possible divergence during early steps of the iteration process or due to nonmonotonic material relations. Then, it can happen that the control parameter becomes very small, which results in almost no difference between u k+1 and u k . (2) Residual criterion: By computing the residual of the obtained solution, we can define an absolute accuracy εabs res by ||K (u k+1 )u k+1 − f ||2 < εabs res ,

(E.10)

as well as a relative accuracy εrel res by ||K (u k+1 )u k+1 − f ||2 < εrel res || f ||2 .

(E.11)

As shown in Fig. E.1, according to the problem type, this stopping criterion may also be reached too early. As a consequence of the above discussion, it is preferable to check both stopping criteria.

Appendix E: Solution of Nonlinear Equations

571

||K(uh )uh − f h ||2

εabs res u∗

uk+1

Fig. E.1 Obtained solution uk+1 is still far away from the true solution u∗

E.1 Fixed-point Iteration The simplest method of solving (E.5) is to rewrite it as a fixed-point equation u = K −1 (u) f .

(E.12)

This will result in the following sequence u k+1 = K −1 (u k ) f

K (u k )u k+1 = f .

(E.13) (E.14)

Thus, we can write the damped fixed-point iteration method as follows K (u k )∆u = f − K (u k )u k = r (u k ) u k+1 = u k + η∆u .

(E.15) (E.16)

The nodal vector r (u) is known as the residual of the problem and a solution is given by the set of nodal values u, for which the residual is zero. The scalar parameter η ∈ [0, 1] is introduced to control the possible divergence during the early steps of the iteration process or due to nonmonotonic material relations. A common algorithm to compute η is a line search (see [1]) defined by |G(η)| → min ,

(E.17)

G(η) = ∆u · r (u k + η∆u) .

(E.18)

with

572

Appendix E: Solution of Nonlinear Equations

K(u)u

f

u2

u1

u3

u4

u

Fig. E.2 Graphical interpretation for solving a nonlinear equation using the fixed-point method

One simple method of approximating the optimal η is as follows 1. Evaluate g1 = G(0.1) and g2 = G(1.0) 2. Calculate the straight line l(g1 , g2 ) between g1 and g2 10g1 −g2 3. Calculate the value η = 10·(g for which l(g1 , g2 ) = 0 holds 2 −g1 ) A graphical interpretation of the fixed-point method is given in Fig. E.2.

E.2 Newton’s Method Let us introduce the following linearization of the nonlinear operator F(u) at u k F(u) ≈ F(u k ) + F ′ (u k )[s]

(E.19)

with u k+1 = u k + s. The term F ′ (u k )[s] denotes the Frechét—derivative of the nonlinear operator F at u k in the direction of s and is defined as follows Definition E.2 Frechét—derivative: Let X and Y be two normed vector spaces and D ⊂ X an open domain. The operator F : D → Y is differentiable in the sense of Frechét at x, iff there exists an operator A : X → Y , so that for all y ∈ D F(y) = F(x) + A(y − x) + R(x, y) ,

Appendix E: Solution of Nonlinear Equations

with lim

y→x

573

||R(x, y)|| =0 ||y − x||

is fulfilled. A is the Frechét derivative F ′ (x). Therefore, Newton’s method reads as F ′ (u k )[s] = −F(u k ) u k+1 = u k + s .

(E.20) (E.21)

Analogously to the fixed-point method, a line-search parameter may accelerate the convergence, and in addition may guarantee a global convergence of the Newton method. A graphical interpretation of Newton’s method is displayed in Fig. E.3. To derive the Frechét derivative F ′ and Newton’s method for the nonlinear Poisson equation given in (E.1), we first compute the difference between F(u + s) and F(u) in the weak formulation for arbitrary test functions v ∈ H01   ε(|∇(u + s)|)∇v · ∇(u + s) dΩ − ε(|∇u|)∇v · ∇u dΩ . (E.22) Ω

Ω

Now, we will add to and at the same time subtract from (E.22) the term Ω ε(|∇(u)|) ∇v · ∇(u + s) dΩ, and obtain   ε(|∇u|)∇v · ∇s dΩ . (ε(|∇(u + s)|) − ε(|∇u|)) ∇v · ∇(u + s) dΩ + Ω

Ω

K(u)u

f

u1

u2

u3

u

Fig. E.3 Graphical interpretation for solving a nonlinear equation using Newton’s method

574

Appendix E: Solution of Nonlinear Equations

The term ε(|∇(u + s)|) − ε(|∇u|) can be approximated as follows ε(|∇(u + s)|) − ε(|∇u|) ≈ ε′ (|∇u|) (|∇(u + s)| − |∇u|) .

(E.23)

Now, let us investigate the term (|∇(u + s)| − |∇u|) |∇(u + s)| − |∇u| = = ≈

|∇(u + s)|2 − |∇u|2 |∇(u + s)| + |∇u|

(E.24)

∇u · ∇u + ∇s · ∇s + 2∇u · ∇s − ∇u · ∇u |∇(u + s)| + |∇u| ∇u · ∇s . |∇u|

(E.25) (E.26)

With this result, we can write  (ε(|∇(u + s)|) − ε(|∇u|)) ∇v · ∇(u + s) dΩ Ω

≈



ε′ (|∇u|)

Ω

∇u · ∇s ∇v · ∇u dΩ . |∇u|

(E.27)

Summarizing the above results, we conclude that the Frechét derivative F ′ (u k )[s] in the weak formulation of the PDE for a test function v is given by 

ε(|∇u k |)∇v · ∇s dΩ +

Ω



ε′ (|∇u k |)

Ω

∇u k · ∇s ∇v · ∇u k dΩ . |∇u k |

(E.28)

Therefore, by using (E.20) as well as (E.21), we obtain Newton’s method for the nonlinear Poisson equation 

ε(|∇u k |)∇v · ∇s dΩ +

Ω



ε′ (|∇u k |)



v f dΩ

Ω

=

−



∇u k · ∇s ∇v · ∇u k dΩ |∇u k |

ε(|∇u k |)∇v · ∇u k dΩ

∀v ∈ H01 (Ω)

Ω

u k+1 = u k + s .

(E.29)

By apply the finite element method to the above equation, we will arrive at the appropriate algebraic system of equations.

Appendix F

Hysteresis Model

One of the most general hysteresis models used is named after F. Preisach, who developed it in 1935. Preisach’s approach was purely intuitive and was based on plausible hypotheses concerning magnetic material behavior [2]. A mathematicalbased investigation was performed by M. Krasnoselskii in the 1970s (see e.g., [3]). In order to get some physical as well as mathematical understanding, let us investigate some properties of Preisach’s hysteresis model. Thus, we consider an infinite set of elementary hysteresis operators Rβ,α , where each of them can be represented by a rectangular loop (see Fig. F.1). Since we want to model the hysteresis within dielectric materials, we choose for the input quantity the normalized electric field intensity e and for the output quantity the normalized polarization p according to e=

E E sat

p=

P . Psat

(F.1)

In (F.1) E sat denotes the saturated electric field intensity and Psat the saturated electric polarization. In Fig. F.1 α and β are the up and down switching values and according to these switching values, the input will lead to an output value +1 or −1. Restricting the switching values to α ≥ β and |α|, |β| ≤ 1 leads to the following set S (see Fig. F.2) (α, β) ∈ S with S = {(α, β) ∈ IR2 , |α|, |β| ≤ 1, β ≤ α} .

(F.2)

Therewith, we describe the class of hysteresis loops with closed major loop [4]. Now, the Preisach operator for the electric polarization p computes as p(t) =



℘ (α, β)Rβ,α (e(t)) dαdβ .

(F.3)

S

© Springer-Verlag Berlin Heidelberg 2015 M. Kaltenbacher, Numerical Simulation of Mechatronic Sensors and Actuators, DOI 10.1007/978-3-642-40170-1

575

576

Appendix F: Hysteresis Model

Fig. F.1 Rectangular hysteresis loop

p

1 β

α

e

−1

Fig. F.2 Set S for possible switching values α and β

α 1 S

−1

1

β

−1

In (F.3) ℘ denotes the Preisach function, which defines the shape of the hysteresis loops and fulfills the following properties [4] . ≥ 0 for (α, β) ∈ S (F.4) ℘ (α, β) = 0 for (α, β) ∈ /S  ℘ (α, β) dαdβ = 1 (F.5) S

℘ (−β, −α) = ℘ (α, β) .

(F.6)

Now let us assume that the input e(t) increases monotonically up to a value of e1 at t = t1 . Thus, all Rβ,α operators with α less than e1 switch up, which means that their outputs take on the value of +1. Within the set S of possible (α, β) values, we will obtain a straight line parallel to the β-axis with α = e1 (see Fig. F.3a). In the next step, we assume that the input e(t) starts to decrease monotonically to a value of e2 at t2 . Now, all Rβ,α operators with down-switching values β larger than e2 will turn back, so that their output takes on the value of −1. This leads to a straight line parallel to the α-axis with value β = e2 , which is illustrated in Fig. F.3a. Therefore, as illustrated in Fig. F.3a, we can subdivide the region S into S + ( p takes on the value of +1) and S − ( p takes on the value of −1). For the general case, a staircase line

Appendix F: Hysteresis Model

577

α

(a)

α

(b)

e1

S−

S−

e3

e1

e5 e2

e2 e4

β

S+

β

S+

Fig. F.3 Decomposing into S + and S − . a e(t) increases till e1 and decreases till e2 , b Staircase line L(t)

L(t) will subdivide S into S + and S − (see Fig. F.3b) according to the following two rules: • A monotonically increasing input signal e(t) defines a straight line parallel to the β-axis with value e(t). • A monotonically decreasing input signal e(t) defines a straight line parallel to the α-axis with value e(t). Therefore, the horizontal lines represent relative maxima and the vertical lines relative minima. In addition, by storing the local maxima and minima, the hysteresis can be uniquely constructed. Due to the wiping-out property, not all relative maxima and minima have to be stored. This property states that each local input maximum wipes out the vertical L(t) whose α values are below this maximum, and each local minimum wipes out the vertices whose β values are above the minimum [4]. The wiping out is best illustrated by an input signal e(t) as displayed in Fig. F.4. Only the relative maxima e1 and e3 Fig. F.4 Input signal e(t) for illustration of the wiping out property

e(t) e1 e3

t

e4 e2

578

Appendix F: Hysteresis Model

as well as relative minima e2 and e4 have to be stored. All other maxima (minima) will be intermediately stored during the process in a list, but will be deleted due to the wiping-out property. Furthermore, the Preisach model fulfills the congruence property [4], which states that all minor hysteresis loops corresponding to back-and-forth variations of inputs between the same two consecutive extrema are congruent (see Fig. F.5). For the numerical computation of the Preisach operator, the following efficient evaluation has been developed. With e1 , ..., en those relative input extrema that have not been wiped out yet at time t, the value of the output at time t computes as p(t) = E(−e1 , e1 ) + 2

n−1 

E(ei , ei+1 ) ,

(F.7)

i=1

with E(ei , ei+1 ) the Everett function (see Fig. F.6) E(e1 , e2 ) =



℘ (α, β) dαdβ .

(F.8)

T (e1 ,e2 )

Fig. F.5 Congruency property of the hysteresis model

Fig. F.6 Computation of the Everett function E (e1 , e2 )

α 1 S

e2 T

−1

1

e1

−1

β

Appendix F: Hysteresis Model

579

For the simplest Preisach function ℘ (α, β) = 1/2, the Everett function computes as 1 (F.9) E(e1 , e2 ) = (e2 − e1 )2 sgn (e2 − e1 ) . 4 Thus, we have an efficient model for taking into account ferroelectric hysteresis within piezoelectric materials. For a detailed discussion concerning hysteresis operators in PDEs, and especially their identification from measured data, we refer to [5].

References 1. O.C. Zienkiewicz and R.L. Taylor, The Finite Element Method, vol. 2, Butterworth - Heinemann, 2003. 2. E. Preisach, über die magnetische Nachwirkung, Z. Phys. (1935), no. 94, 277–302. 3. M. Krasnoselskii, A. Pokrovskii, Systems with Hysteresis (Nauka, Tech. report, 1983) 4. I.D. Mayergoyz, Mathematical Models of Hysteresis (Springer Verlag, New York, 1991) 5. B. Kaltenbacher, M. Kaltenbacher, Modelling and iterative identification of hysteresis via Preisach operators in PDEs (Radon Series Comp. Appl, Math, 2007)

Index

A Absorbing boundary condition, 200 Acoustic averaged energy density, 166 averaged intensity, 166 averaged power, 167 density, 160 energy density, 166 energy flux, 166 field, 159 impedance, 167, 168 intensity, 166 linear wave equation, 164 nonlinear wave equation, 176 overall sound pressure level (OSPL), 175 particle velocity, 160 pressure, 160 quantities, 166 sound-field impedance, 167 sound-intensity level, 174 sound-power level, 174 sound-pressure level (SPL), 174 spherical spreading law, 171 velocity potential, 165 Acoustic perturbation equations, 324 Actuator, see mechatronic Adiabatic bulk modulus, 164 compressibility, 164 Aeroacoustic wave equation, 327 Aeroacoustics, 309 aeroacoustic wave equation, 327 airframe noise, 530 conservative interpolation, 333 Curle’s theory, 317 edge tone, 523 eighth power law, 317

Lighthill’s analogy, 312 linearized perturbed compressible equations(LPCE), 326 perturbation equations, 324 sixth power law, 321 vortex sound, 322 Agglomeration technique, see coarsening Airframe noise, 530 Aitken scheme, 289 ALE system, 139 Algebraic multigrid, see multigrid Ampère, 229 Approximation BH curve, 263 Auxiliary matrix, 427

B Balanced reduced and selective integration, 128 BH curve, see approximation Bijective map, 23 Biot–Savart’s law, 242 Boundary condition, 9 Dirichlet, 9 essential, 10 natural, 10 Neumann, 9 Burger’s equation, 180 Butterfly curve, 381

C Circulation, 550 CMUT, see micro-machined Coarse grid operator, see grid Coarsening

© Springer-Verlag Berlin Heidelberg 2015 M. Kaltenbacher, Numerical Simulation of Mechatronic Sensors and Actuators, DOI 10.1007/978-3-642-40170-1

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582 agglomeration technique, 430 function, 428 process, 427 Coil current-Loaded, 271 voltage-Loaded, 275 Condition number, 416 Configuration deformed, 97 initial, 97 Congruency, 578 Conjugate gradient (PCG) method, see preconditioned Conservation of energy, 144 of mass, 140, 161 of momentum, 141, 161 Contact mechanics condition, 471 pressure-displacement relation, 471 tangent stiffness matrix, 472 Continuity equation, 161 Convergence, 569 Coulomb-gauge, see gauge Coupling aeroacoustics, 309 electromagnetics-mechanics, 353 electrostatics-mechanics, 339 flow-acoustics, 309 flow-structural mechanical, 285 mechanics-acoustics, 297 piezoelectrics, 375 Coupling mechanisms, 4 Coupling strategy, 286 monolithic, 287 partitioned, 287 Courant-Friedrich-Levi (CFL) condition, 46 Crank-Nicolson scheme, 44 Curl, 544, 551 Curle’s theory, 317

D Damping, 112 modal, 112 Rayleigh model, 112 Delta-property, 15, 22 Density, 97 Derivative Frechét, 572 global/local, 32 weak sense, 565 Design process, 1

Index CAE-based, 1 experimental-based, 1 Diamagnetic, 239 Dielectric remnant, 243 Differential operator, 96 Diffusion equation, 237 Diffusivity of sound, 179 Displacement current density, 230 Divergence, 544, 548 theorem, 549 Duffy transformation, 64

E Edge tone, 523 Elasticity modulus, 102 Electric charge density, 228 conductivity, 228, 242 current, 229 current density, 228 field intensity, 228 flux density, 228 permittivity, 228 polarization, 228, 379 scalar potential, 238 specific resistivity, 242 Electrodynamic loudspeaker, see loudspeaker Electromagnetic energy, 356 field, 227 force, 355 interface conditions, 244 quasistatic field, 235 Electromagnetic-mechanical system, 353 calculation scheme, 364 Electromotive force, 232 Electrostatic energy, 339 field, 238 force, 339 Electrostatic-mechanical system, 339 calculation scheme, 347 Enhanced assumed strain method, 126 Entropy, 145 Error a posteriori, 52 a priori, 52 discretization, 51 dispersion, 196 interpolation, 195 pollution, 195

Index Euler equation, 161 Euler number, 147 Eulerian coordinate, 97 system, 139 Everett function, 579

F Faraday, 230 Fay solution, 221 Ferroelasticity, 381 Ferroelectricity, 243, 379 Ferromagnetic, 239 Filter 1/3 octave, 172 octave, 172 Finite element, 7 Nédélec , 265 compatible, 23 conforming, 23 delta-property, 15, 22 edge, 49 formulation, 8 hexahedral, 28 hierarchical elements, 57 higher order, 55 higher order edge elements, 265 infinite element, 199 isoparametric, 21 Lagrange polynomials, 20, 65 Legendre polynomials, 57 method, 7 Nédélec, 49 nodal, 20 non-conforming, 69 pyramidal, 31 quadrilateral, 23 spectral elements, 65 tensor product space, 60 tetrahedral, 27 triangular, 26 trunk space, 59 wedge, 30 Finite element/boundary element method, 360 Finiteelement assemblingprocedure, 36 Flexible discretization, 67 Mortar method, 69 Nitsche, 82 Flow bulk viscosity, 145

583 convective velocity, 139 density, 140 dynamic viscosity, 145 field, 137 kinematic pressure, 146 kinematic viscosity, 146 momentum flux tensor, 143 pressure, 142 strain tensor, 145 stress tensor, 143 velocity, 139 viscous stress tensor, 143, 145 Flow-Structural mechanical systems, 285 Fluid-solid-interface, 285 Flux, 546 Force electromagnetic, 355 electrostatic, 339 Formulation strong, 9 variational, 9 weak, 9 Froude number, 147 Fubini solution, 220 Functional spaces, 563 L p , 564 continuously differentiable, 563 Hilbert, 566 Sobolev, 566 square integrable, 564 weighted Sobolev, 254

G Galerkin, 10 method, 10 semi-discrete formulation, 12 Gauge, 236 Gauss, 233 Gauss theorem, see divergence Geometric multigrid, see muiltigrid Gibbs free energy, 376 Gradient, 545 deformation, 97 displacement, 98 of a scalar, 544 of a vector, 559 Green’s integral theorem, 553 scalar form, 553 vector form, 553 Grid adaption, 289, 360

584 coarse, 428 coarse-grid operator, 431 complexity, 437 fine, 428

H Harmonic distortion, 456 Helmholtz decomposition, 106, 422 Hooke’s law, 102 Hu-Washizu principle, 126 Hysteresis, 575 Preisach model, 575

I Incompatible modes method, 124 Index notation, 557 Induced electric voltage, 275 Inductance, see magnetic Infinite finite elements, 199 Interpolation conservativ, 333 function, 22 Irrotational, 234, 541 vector field, 555 Isentropic, 163 Isoparametric, 21

J Jacobi, 33 matrix, 33

K Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation, 181 Kuznetsov’s equation, 176

L Lagrange multiplier, 361 Lagrange polynomials, 65 Lagrangian coordinate, 97 system, 138 updated formulation, 354 Lam´eparameters, 103 Legendre polynomials, 57 Lighthill’s analogy, 312 Line search, 572 Litotripsy, 498 Local support, 21

Index Locking, 121 effect, 120 membrane, 122 Poisson, 122 shear, 121 Lorentz force, 4, 229 Loss factor, 112 Loudspeaker, 3, 366, 453

M Magnetic field intensity, 228 flux, 230, 272 hard material, 239 hysteresis, 240 inductance, 272 induction, 228 permeability, 228, 239 reluctivity, 239 remnant field, 240 scalar potential, 241 soft material, 239 vector potential, 235 Magnetic valve, 353, 469 overexcitation, 477 premagnetization, 475 switching cycle, 478 Magnetization, 228 Magnetomechanical system, see electromagnetic-mechanical system Maxwell’s equations, 227 Mechanical acceleration, 97 axisymmetric stress–strain, 106 contact, 471 damping, see damping field, 93 plane strain, 104 plane stress, 105 strain, 98 stress, 93 stress-stiffening effect, 487 yield stress, 108 Mechanical-acoustic system, 297 calculation scheme, 300 Mechatronic, 1 actuator, 1 sensor, 1 Micro-Machined capacitive ultrasound array (CMUT), 485 Motional electromotive force, 233, 353

Index method, 361, 367, 454 Moving body electric field, 347 magnetic field, 353 Moving coil current-loaded, 366 voltage-loaded, 366 Moving-material method, 364, 369, 456 Moving-mesh method, 347, 363 Multigrid, 415 algebraic, 426 geometric, 420 method, 417 nested, 423 Multilayer actuator, see piezoelectric

N Nabla operator, 543 Navier’s equations, 97 Navier-Stokes equations, 145 compressible, 145 incompressible, 146 Newmark scheme, 46 Newton method, 572 electromagnetics, 260 mechanics, 115 Newtonian fluid, 145 Non-Conforming grid magnetics, 87 Non-conforming grid, 69 acoustics, 190 magnetics, 362 mechanics-acoustics, 302 Mortar method, 69 Nitsche method, 82 Non-matching grid, see non-conforming grid Norms, 539 Hölder, 539 matrix, 540 p-norms, 539 vector, 539 Numerical computation aeroacoustics, 327 electromagnetics, 249 electromagnetics-mechanics, 364 electrostatics, 247 electrostatics-mechanics, 346 fluid-solid, 285 geometric nonlinear case, 114 linear acoustics, 181 linear elasticity, 110

585 mechanics-acoustics, 300 nonlinear acoustics, 187 nonlinear electromagnetics, 260 nonlinear mechanics, 114 piezoelectrics, 393 Numerical integration, 34 Gaussian quadrature, 34 O Operator complexity, 437 nonlinear, 569 P P-FEM, 55 anisotropic, 269 Paramagnetic, 239 Parameter of nonlinearity, 177 Partial differential equation, 8 hyperbolic, 45 parabolic, 41 Patch test, 125 Penalty formulation, 253 Penetration depth, see skin depth Perfectly matched layer (PML), 201 frequency domain, 203 reduced (rPML), 212 time domain, 210 Permeability, see magnetic permeability Piezoelectric, 375 ceramics, 379 cofired multilayer, 479 direct effect, 375 ferroelasticity, 381 ferroelectricity, 379 inverse effect, 375 macroscopic model, 384 micro-mechanical model, 392 switching, 392 systems, 375 Piola-transformation, 185 Poisson ratio, 102 Polarization irreversibel, 380 permanent, 380 saturation, 381 Poling, 380 Polymers, 379 Power transformer, 460 Preconditioned conjugate gradient (PCG) method, 415 Predictor-corrector algorithm, 43, 47

586 Preisach function, 576 model, 575 operator, 575 Pressure-density relation, 162 Prestressing, 350 Principle of virtual work, 339, 343, 355, 357 Prolongation, 418 operator, 417, 431

R Rayleigh damping model, see damping Remnant magnetic field, see magnetic Restriction, 418 operator, 417 Reynolds number, 147 Reynolds’ transport theorem, 139

S S-FEM, 65 Saturation strain, 381 Scalar acoustic velocity potential, 165 electric potential, 238 field, 541 magnetic potential, 241 Schur complement, 277 Sensor, see mechatronic Shape function, see interpolation Shear modulus, 102 Shock-formation distance, 221 Single crystals, 379 Skin depth, 238 effect, 236 Smoothing overlapping block-smoothers, 422 block-Gauss-Seidel, 422 Gauss-Seidel backforward, 419 Gauss-Seidel forward, 419 hybrid, 435 operator, 431 post, 418 pre, 418 Sobolev space, see functional spaces Solenoidal, 234, 541 vector field, 555 Solid/fluid interface, 297 Sound velocity, 159 Spherical spreading law, see acoustic SPL, see acoustic

Index Stabilized FEM, 149 Stack actuator, see piezoelectric State equation, 162 Stoke’s theorem, 552 Stopping criterion, 570 error, 570 residual, 570 Strain, see mechanical Strain tensor Green–Lagrangian, 101 linear, 102 Stress tensor 1st Piola–Kirchhoff, 114 2nd Piola–Kirchhoff, 114 Cauchy, 94 Stress-stiffening effect, 487 Strouhal number, 147 SUPG/PSPG, 148 Surface integration, 48

T TEAM (Testing Electromagnetic Analysis Methods), 423 Tensor basics, 557 of dielectric constants, 378 of elasticity moduli, 103, 378 of piezoelectric moduli, 378 scalar product, 559 tensor product, 560 Test function, 9 Thermal strain, 109 Time discretization, 41 effective mass formulation (hyperbolic), 47 effective mass formulation (parabolic), 43 effective stiffness formulation (hyperbolic), 47 effective stiffness formulation (parabolic), 43 explicit (hyperbolic), 47 explicit (parabolic), 44 implicit (hyperbolic), 47 implicit (parabolic), 44 Transducing mechanisms, 1 Transformer, see power transformer Trapezoidal difference scheme, 42 U Ultrasound high intensity focused (HIFU), 495

Index litotripsy, 498 V Vector field, 541 irrotational, 541 solenoidal, 541 Virtual work, see principle Voigt notation, 96, 102 Vortex sound, 322 W Wave

587 longitudinal, 107, 159 number, 168 plane, 168 shear, 108 spherical, 170 Weighted regularization, 254 Westervelt equation, 180 Wiping-out, 577

Y Yield stress, 108