Model Validation and Uncertainty Quantification, Volume 3: Proceedings of the 38th IMAC, A Conference and Exposition on Structural Dynamics 2020 [1st ed.] 9783030476373, 9783030476380

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Conference Proceedings of the Society for Experimental Mechanics Series

Zhu Mao  Editor

Model Validation and Uncertainty Quantification, Volume 3 Proceedings of the 38th IMAC, A Conference and Exposition on Structural Dynamics 2020

Conference Proceedings of the Society for Experimental Mechanics Series Series Editor Kristin B. Zimmerman, Ph.D. Society for Experimental Mechanics, Inc., Bethel, CT, USA

The Conference Proceedings of the Society for Experimental Mechanics Series presents early findings and case studies from a wide range of fundamental and applied work across the broad range of fields that comprise Experimental Mechanics. Series volumes follow the principle tracks or focus topics featured in each of the Society’s two annual conferences: IMAC, A Conference and Exposition on Structural Dynamics, and the Society’s Annual Conference & Exposition and will address critical areas of interest to researchers and design engineers working in all areas of Structural Dynamics, Solid Mechanics and Materials Research

More information about this series at http://www.springer.com/series/8922

Zhu Mao Editor

Model Validation and Uncertainty Quantification, Volume 3 Proceedings of the 38th IMAC, A Conference and Exposition on Structural Dynamics 2020

Editor Zhu Mao Department, Dandeneau Hall 211 Univ of Massachusetts, Mechanical Engg Lowell, MA, USA

ISSN 2191-5644 ISSN 2191-5652 (electronic) Conference Proceedings of the Society for Experimental Mechanics Series ISBN 978-3-030-48778-2 ISBN 978-3-030-47638-0 (eBook) https://doi.org/10.1007/978-3-030-47638-0 © The Society for Experimental Mechanics, Inc. 2020 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, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

Model Validation and Uncertainty Quantification represents one of eight volumes of technical papers presented at the 38th IMAC, A Conference and Exposition on Structural Dynamics, organized by the Society for Experimental Mechanics, and held in Houston, Texas, February 10–13, 2020. The full proceedings also include volumes on Nonlinear Structures and Systems; Dynamics of Civil Structures; Dynamic Substructures; Special Topics in Structural Dynamics & Experimental Techniques; Rotating Machinery, Optical Methods & Scanning LDV Methods; Sensors and Instrumentation, Aircraft/Aerospace, Energy Harvesting & Dynamic Environments Testing; and Topics in Modal Analysis &Testing. Each collection presents early findings from experimental and computational investigations on an important area within Structural Dynamics. Model Validation and Uncertainty Quantification (MVUQ) is one of these areas. Modeling and simulation are routinely implemented to predict the behavior of complex dynamical systems. These tools powerfully unite theoretical foundations, numerical models, and experimental data which include associated uncertainties and errors. The field of MVUQ research entails the development of methods and metrics to test model prediction accuracy and robustness while considering all relevant sources of uncertainties and errors through systematic comparisons against experimental observations. The organizers would like to thank the authors, presenters, session organizers, and session chairs for their participation in this track. Lowell, MA, USA

Zhu Mao

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Contents

1

Variational Coupled Loads Analysis Using the Hybrid Parametric Variation Method . . . . . . . . . . . . . . . . . . . . . . . . . Daniel C. Kammer, Paul Blelloch, and Joel Sills

1

2

Bayesian Uncertainty Quantification in the Development of a New Vibration Absorber Technology . . . . . . . . . Nicolas Brötz and Peter F. Pelz

19

3

Comparison of Complexity Measures for Structural Health Monitoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hannah Donajkowski, Salma Leyasi, Gregory Mellos, Chuck R. Farrar, Alex Scheinker, Jin-Song Pei, and Nicholas A. J. Lieven

27

4

Selection of an Adequate Model of a Piezo-Elastic Support for Structural Control in a Beam Truss Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jonathan Lenz, Maximilian Schäffner, Roland Platz, and Tobias Melz

41

Impact Load Identification for the DROPBEAR Setup Using a Finite Input Covariance (FIC) Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Peter Lander, Yang Wang, and Jacob Dodson

51

5

6

Real-Time Digital Twin Updating Strategy Based on Structural Health Monitoring Systems . . . . . . . . . . . . . . . . . Yi-Chen Zhu, David Wagg, Elizabeth Cross, and Robert Barthorpe

55

7

On the Fusion of Test and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ibrahim A. Sever

65

8

Design of an Actuation Controller for Physical Substructures in Stochastic Real-Time Hybrid Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nikolaos Tsokanas and B. Stojadinovic

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9

Output-Only Nonlinear Finite Element Model Updating Using Autoregressive Process . . . . . . . . . . . . . . . . . . . . . . . . Juan Castiglione, Rodrigo Astroza, Saeed Eftekhar Azam, and Daniel Linzell

83

10

Axle Box Accelerometer Signal Identification and Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cyprien A. Hoelzl, Luis David Avendano Valencia, Vasilis K. Dertimanis, Eleni N. Chatzi, and Marcel Zurkirchen

87

11

Kalman-Based Virtual Sensing for Improvement of Service Response Replication in Environmental Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Silvia Vettori, Emilio Di Lorenzo, Roberta Cumbo, Umberto Musella, Tommaso Tamarozzi, Bart Peeters, and Eleni Chatzi

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12

Virtual Sensing of Wheel Position in Ground-Steering Systems for Aircraft Using Digital Twins . . . . . . . . . . . . . 107 Mattia Dal Borgo, Stephen J. Elliott, Maryam Ghandchi Tehrani, and Ian M. Stothers

13

Assessing Model Form Uncertainty in Fracture Models Using Digital Image Correlation . . . . . . . . . . . . . . . . . . . . . 119 Robin Callens, Matthias Faes, and David Moens

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Contents

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Identification of Lack of Knowledge Using Analytical Redundancy Applied to Structural Dynamic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Jakob Hartig, Florian Hoppe, Daniel Martin, Georg Staudter, Tugrul Öztürk, Reiner Anderl, Peter Groche, Peter F. Pelz, and Matthias Weigold

15

A Structural Fatigue Monitoring Concept for Wind Turbines by Means of Digital Twins . . . . . . . . . . . . . . . . . . . . . 139 János Zierath, Sven-Erik Rosenow, Johannes Luthe, Andreas Schulze, Christiane Saalbach, Manuela Sander, and Christoph Woernle

16

Damage Identification of Structures Through Machine Learning Techniques with Updated Finite Element Models and Experimental Validations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Panagiotis Seventekidis, Dimitrios Giagopoulos, Alexandros Arailopoulos, and Olga Markogiannaki

17

Modal Analyses and Meta-Models for Fatigue Assessment of Automotive Steel Wheels. . . . . . . . . . . . . . . . . . . . . . . . 155 S. Venturini, E. Bonisoli, C. Rosso, D. Rovarino, and M. Velardocchia

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Towards the Development of a Digital Twin for Structural Dynamics Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 Paul Gardner, Mattia Dal Borgo, Valentina Ruffini, Yichen Zhu, and Aidan Hughes

19

An Improved Optimal Sensor Placement Strategy for Kalman-Based Multiple-Input Estimation . . . . . . . . . . . . 181 Lorenzo Mazzanti, Roberta Cumbo, Wim Desmet, Frank Naets, and Tommaso Tamarozzi

20

Towards Population-Based Structural Health Monitoring, Part IV: Heterogeneous Populations, Transfer and Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 Paul Gardner, Lawerence A. Bull, Julian Gosliga, Nikolaos Dervilis, and Keith Worden

21

Feasibility Study of Using Low-Cost Measurement Devices for System Identification Using Bayesian Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 Alejandro Duarte and Albert R. Ortiz

22

Kernelised Bayesian Transfer Learning for Population-Based Structural Health Monitoring . . . . . . . . . . . . . . . . . 209 Paul Gardner, Lawrence A. Bull, Nikolaos Dervilis, and Keith Worden

23

Predicting System Response at Unmeasured Locations Using a Laboratory Pre-Test . . . . . . . . . . . . . . . . . . . . . . . . . . 217 Randy Mayes, Luke Ankers, and Phil Daborn

24

Robust Estimation of Truncation-Induced Numerical Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 François Hemez

25

Fatigue Crack Growth Diagnosis and Prognosis for Damage-Adaptive Operation of Mechanical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 Pranav M. Karve, Yulin Guo, Berkcan Kapusuzoglu, Sankaran Mahadevan, and Mulugeta A. Haile

26

An Evolutionary Approach to Learning Neural Networks for Structural Health Monitoring . . . . . . . . . . . . . . . . . 237 Tharuka Devendra, Nikolaos Dervilis, Keith Worden, George Tsialiamanis, Elizabeth J. Cross, and Timothy J. Rogers

27

Bayesian Solutions to State-Space Structural Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 Timothy J. Rogers, Keith Worden, and Elizabeth J. Cross

28

Analyzing Propagation of Model Form Uncertainty for Different Suspension Strut Models . . . . . . . . . . . . . . . . . . . 255 Robert Feldmann, Maximilian Schäffner, Christopher M. Gehb, Roland Platz, and Tobias Melz

29

Determining Interdependencies and Causation of Vibration in Aero Engines Using Multiscale Cross-Correlation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 Manu Krishnan, Ibrahim A. Sever, and Pablo A. Tarazaga

30

Dynamic Data Driven Modeling of Aero Engine Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 Manu Krishnan, Serkan Gugercin, Ibrahim Sever, and Pablo Tarazaga

31

Nonlinear Model Updating Using Recursive and Batch Bayesian Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 Mingming Song, Rodrigo Astroza, Hamed Ebrahimian, Babak Moaveni, and Costas Papadimitriou

Contents

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32

Towards Population-Based Structural Health Monitoring, Part I: Homogeneous Populations and Forms. . . 287 Lawerence A. Bull, Paul A. Gardner, Julian Gosliga, Nikolaos Dervilis, Evangelos Papatheou, Andrew E. Maguire, Carles Campos, Timothy J. Rogers, Elizabeth J. Cross, and Keith Worden

33

A Detailed Assessment of Model Form Uncertainty in a Load-Carrying Truss Structure . . . . . . . . . . . . . . . . . . . . . . 303 Robert Feldmann, Christopher M. Gehb, Maximilian Schäffner, Alexander Matei, Jonathan Lenz, Sebastian Kersting, and Moritz Weber

34

Recursive Nonlinear Identification of a Negative Stiffness Device for Seismic Protection of Structures with Geometric and Material Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 Kalil Erazo and Satish Nagarajaiah

35

Adequate Mathematical Beam-Column Model for Active Buckling Control in a Tetrahedron Truss Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 Maximilian Schaeffner, Roland Platz, and Tobias Melz

36

Site Characterization Through Hierarchical Bayesian Model Updating Using Dispersion and H/V Data . . . . 333 Mehdi M. Akhlaghi, Mingming Song, Marshall Pontrelli, Babak Moaveni, and Laurie G. Baise

37

BAYESIAN Inference Based Parameter Calibration of a Mechanical Load-Bearing Structure’s Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 Christopher M. Gehb, Roland Platz and Tobias Melz

38

Uncertainty Propagation in a Hybrid Data-Driven and Physics-Based Submodeling Method for Refined Response Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 Bhavana Valeti and Shamim N. Pakzad

39

Adaptive Process and Measurement Noise Identification for Recursive Bayesian Estimation. . . . . . . . . . . . . . . . . . 361 Konstantinos E. Tatsis, Vasilis K. Dertimanis, and Eleni N. Chatzi

40

Effective Learning of Post-Seismic Building Damage with Sparse Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 Mohamadreza Sheibani and Ge Ou

41

Efficient Bayesian Inference of Miter Gates Using High-Fidelity Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 Manuel A. Vega, Mukesh K. Ramancha, Joel P. Conte, and Michael D. Todd

42

Two-Stage Hierarchical Bayesian Framework for Finite Element Model Updating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 Xinyu Jia, Omid Sedehi, Costas Papadimitriou, Lambros Katafygiotis, and Babak Moaveni

43

Bayesian Nonlinear Finite Element Model Updating of a Full-Scale Bridge-Column Using Sequential Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 Mukesh K. Ramancha, Rodrigo Astroza, Joel P. Conte, Jose I. Restrepo, and Michael D. Todd

44

Optimal Input Locations for Stiffness Parameter Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 Debasish Jana, Dhiraj Ghosh, Suparno Mukhopadhyay, and Samit Ray-Chaudhuri

45

Modal Identification and Damage Detection of Railway Bridges Using Time-Varying Modes Identified from Train Induced Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 Ashish Pal, Astha Gaur, and Suparno Mukhopadhyay

46

Test-Analysis Modal Correlation of Rocket Engine Structures in Liquid Hydrogen – Phase II. . . . . . . . . . . . . . . . 413 Andrew M. Brown and Jennifer L. DeLessio

47

An Output-Only Bayesian Identification Approach for Nonlinear Structural and Mechanical Systems . . . . . . 431 Satish Nagarajaiah and Kalil Erazo

Chapter 1

Variational Coupled Loads Analysis Using the Hybrid Parametric Variation Method Daniel C. Kammer, Paul Blelloch, and Joel Sills

Abstract Time-domain coupled loads analysis (CLA) is used to determine the response of a launch vehicle and payload system to transient forces, such as liftoff, engine ignitions and shutdowns, jettison events, and atmospheric flight loads, such as buffet. CLA, using Hurty/Craig-Bampton (HCB) component models, is the accepted method for the establishment of design-level loads for launch systems. However, uncertainty in the component models flows into uncertainty in predicted system results. Uncertainty in the structural responses during launch is a significant concern because small variations in launch vehicle and payload mode shapes and their interactions can result in significant variations in system loads. Uncertainty quantification (UQ) is used to determine statistical bounds on prediction accuracy based on model uncertainty. In this paper uncertainty is treated at the HCB component-model level. In an effort to account for model uncertainties and statistically bound their effect on CLA predictions, this work combines CLA with UQ in a process termed variational coupled loads analysis (VCLA). The modeling of uncertainty using a parametric approach, in which input parameters are represented by random variables, is common, but its major drawback is the resulting uncertainty is limited to the form of the nominal model. Uncertainty in model form is one of the biggest contributors to uncertainty in complex built-up structures. Modelform uncertainty can be represented using a nonparametric approach based on random matrix theory (RMT). In this work, UQ is performed using the hybrid parametric variation (HPV) method, which combines parametric with nonparametric uncertainty at the HCB component model level. The HPV method requires the selection of dispersion values for the HCB fixed-interface (FI) eigenvalues, and the HCB mass and stiffness matrices. The dispersions are based upon component testanalysis modal correlation results. During VCLA, random component models are assembled into an ensemble of random systems using a Monte Carlo (MC) approach. CLA is applied to each of the ensemble members to produce an ensemble of system-level responses for statistical analysis. The proposed methodology is demonstrated through its application to a buffet loads analysis of NASA’s Space Launch System (SLS) during the transonic regime 50 s after liftoff. Core stage (CS) section shears and moments are recovered, and statistics are computed. Keywords Uncertainty quantification · Hurty/Craig-Bampton · Random matrix · Model form · Coupled loads analysis

Acronyms CLA CS DCGM DOF FEM FI HCB HPV

coupled loads analysis core stage diagonal cross-generalized mass metric degrees of freedom finite element model fixed-interface Hurty/Craig-Bampton hybrid parametric variation

D. C. Kammer () · P. Blelloch ATA Engineering, Inc., San Diego, CA, USA e-mail: [email protected]; [email protected] J. Sills NASA Johnson Space Center, Houston, TX, USA e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2020 Z. Mao (ed.), Model Validation and Uncertainty Quantification, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47638-0_1

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ICPS ISPE LSRB LVSA MC ME MEM MUF MPCV MSA NPV OTM PDF RINU RMS RMT RSRB SLS UQ VCLA

D. C. Kammer et al.

interim cryogenic propulsion stage integrated spacecraft payload element left solid rocket booster launch vehicle stage adapter Monte Carlo maximum entropy modal effective mass model uncertainty factor Multi-Purpose Crew Vehicle MPCV stage adapter nonparametric variation output transformation matrix probability distribution function replaceable inertial navigation unit root mean square random matrix theory right solid rocket booster Space Launch System uncertainty quantification variational coupled loads analysis

1.1 Introduction Uncertainty in structural responses during launch is a significant concern in the development of spacecraft and launch vehicles. Small variations in launch vehicle and payload mode shapes and their interactions can result in significant variations in system loads. Time-domain coupled loads analysis (CLA) is used to determine the response of a launch vehicle and payload system to transient forces, such as liftoff, engine ignitions and shutdowns, jettison events, and atmospheric flight loads. Atmospheric flight loads include static-aeroelastic, turbulence/gust, and buffet loads. CLA, using Hurty/CraigBampton (HCB) [1] component models, is the accepted method for the establishment of design-level loads for launch systems, but it is an expensive and time-consuming process that is performed a limited number of times using a sequence of increasingly more accurate system models. Ideally, a modal test of the integrated system is performed, and the resulting test-correlated system model is used in the final CLA. However, in many cases, an integrated-system test is not practical, so the accuracy of the system model is dependent on the accuracy of component models that are correlated and updated using component test results. Unfortunately, component test configurations are rarely, if ever, the same as component flight configurations, resulting in uncertainty in the component and, thus, system models. In addition, as a program advances, inevitable increasing cost and time constraints begin to limit the number of component tests that are actually performed before launch, which, again, increases the amount of uncertainty in the models and CLA results. In an effort to account for model uncertainties and their effect on CLA predictions, CLA sensitivity analyses could be performed, but due to the high cost of multiple CLAs, this has been impractical and typically not done. Another standard practice is to apply model uncertainty factors (MUF) to the predicted CLA load results. This is simple, but the MUF approach is not based on test results of the specific system it is being applied to, and such a blanket approach may be needlessly conservative in some instances, but unconservative in others. Uncertainty quantification (UQ) is used to determine statistical bounds on prediction accuracy based on model uncertainty. In the case of NASA’s Space Launch System (SLS), uncertainty is modeled at the HCB component-model level because it has the most direct link to modal test results. This work combines CLA with UQ in a process termed variational coupled loads analysis (VCLA) [2]. In the past, VCLA has not been feasible due to computational expense. But due to advances in computational speed, software, and mathematical approaches, VCLA is beginning to be practical, provided that care is taken in how the variations are modeled. The modeling of uncertainty using a parametric approach, in which finite element model (FEM) input parameters are represented by random variables, is common. The major drawback is that the resulting uncertainty is limited to the form of the nominal model. Uncertainty in model form is one of the biggest contributors to uncertainty in complex built-up structures. One way to treat model-form uncertainty is to use a nonparametric approach based on random matrix theory (RMT). In this work, UQ is performed using the hybrid parametric variation (HPV) [3] method, which combines parametric with nonparametric uncertainty at the HCB component-model level. The HPV method requires the selection of dispersion values for the HCB fixed-interface eigenvalues, and the HCB mass and stiffness matrices. The

1 Variational Coupled Loads Analysis Using the Hybrid Parametric Variation Method

3

dispersions are based upon component test-analysis modal correlation results. During VCLA, random component models are assembled into an ensemble of random systems using a Monte Carlo (MC) approach. CLA is applied to each of the systems to produce an ensemble of system-level responses for statistical analysis. The proposed methodology is demonstrated through its application to a buffet loads analysis of the SLS during the transonic regime at 50 s after liftoff. Core stage (CS) section shears and moments are recovered, and statistics, such as P99/90 non-exceedance values, are computed.

1.2 Theory The SLS consists of components that are assembled into the launch vehicle. In order to predict system performance, FEMs of the components are developed, reduced to HCB representations, and assembled to represent different phases of flight. There is always uncertainty in every model, which flows into uncertainty in predicted system results. For the SLS, it is natural to treat the model uncertainty at the HCB component-model level. The HCB component-model displacement vector is given T  by uH CB = xtT q T , where xt is the vector of physical displacements at the component interface, and q is the vector of generalized coordinates associated with the component fixed-interface (FI) modes. Given the assumption that the FI modes are mass normalized, the corresponding HCB mass and stiffness matrices have the form     MS Mtq KS 0 MH CB = = (1.1) K H CB T I Mtq 0 λ in which MS and KS are the component physical mass and stiffness matrices statically reduced to the interface, Mtq is the mass coupling between the interface and the fixed-interface modes, I is an identity matrix, and λ is a diagonal matrix of the FI mode eigenvalues. Details of the HCB component-model derivation can be found in reference [1]. In this work, uncertainty in the component HCB representations is quantified using the HPV approach, which combines parametric with nonparametric uncertainty. Purely parametric uncertainty approaches are the most common in the structural dynamics community. Component parameters that are inputs to the FEM, such as Young’s modulus, mass density, geometric properties, etc., are modeled as random variables. Parametric uncertainty can be propagated into the system response using a method such as stochastic finite element analysis [4]. The advantage of the parametric approach is that each random set of model parameters represents a specific random FEM. However, there are disadvantages associated with the parametric method: it can be very time consuming, there are an infinite number of ways to parameterize the model, and the selected parameter probability distributions are generally not available. The most significant drawback is that the uncertainty represented is limited to the form of the nominal FEM. It is known that most errors in a FEM stem from modeling assumptions or model-form errors, not parametric errors. Therefore, in practice, the parameter changes are merely surrogates for the actual model errors. In the case of HPV, the HCB components are parameterized in terms of the FI eigenvalues, not the inputs to the original FEM. While there is not a simple direct connection between the random FI eigenvalues and a random component FEM, there is a direct connection to the corresponding random HCB component. Uncertainty in model form is likely the largest contributor to uncertainty in complex built-up structures, as it cannot be directly represented by model parameters and thus cannot be included in a parametric approach. Familiar examples include unmodeled nonlinearities, errors in component joint models, etc. Instead, model-form uncertainty can be modeled using random matrix theory (RMT), where a probability distribution is developed for the matrix ensemble of interest. RMT was introduced and developed in mathematical statistics by Wishart [5], and more recently, Soize [6, 7] developed a nonparametric variation (NPV) approach to represent model-form uncertainty in structural dynamics applications. Soize’s approach was extended by Adhikari [8, 9] using Wishart distributions to model random structural mass, damping, and stiffness matrices. The nonparametric matrix-based approach to representing structural uncertainty has been used extensively in aeronautics and aerospace engineering applications [10, 11, 12]. Soize [7] employed the maximum entropy (ME) principle to derive the positive and positive-semidefinite ensembles SE+ and SE+0 that follow a matrix variate gamma distribution and are capable of representing random structural matrices. This means that the matrices in the ensembles are real and symmetric and possess the appropriate sign definiteness to represent structural mass, stiffness, or damping matrices. As the dimension of the random matrix n increases, the matrix variate gamma distribution converges to a matrix variate Wishart distribution. In structural dynamics applications, the matrix dimensions are usually sufficient to give a negligible difference between the two distributions. In letting ensemble member random matrix G be any of the random mass, stiffness, or damping matrices, it is assumed in this work that G follows a matrix variate Wishart distribution, G~Wn (p, ). In general, a Wishart distribution with parameters p and  can be thought of as the sum of the outer product of p independent random vectors Xi all having a multivariate normal distribution with zero mean and

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covariance matrix . Parameter p is sometimes called the shape parameter. The random matrix G can be written as G=

p i=1

Xi XiT

Xi ∼ Nn (0, )

(1.2)

where the expected value is given by E(G) = G = p

(1.3)

The dispersion or normalized standard deviation of the random matrix G is defined by the relation

2  E G − G F 2 2  δG = E G

(1.4)

F

in which ∗2F is the Frobenius norm squared, or trace(∗ T ∗ ). It can be shown that Eq. (1.4) reduces to the expression ⎡   2 ⎤ tr G 1 1 2 = ⎣1 + T  ⎦ = [1 + γG ] δG p p tr G G where γG =

  2 tr G  T tr G G

(1.5)

can be thought of as a measure of the magnitude of the matrix. The uncertainty in the random matrix

G is dictated by the shape parameter p, the number of inner products in Eq. (1.2). The larger the value of p, the smaller the dispersion δ G . There may be instances when it is desirable to have the same amount of uncertainty in two or more substructures. Suppose G1 and G2 represent structural matrices, such as stiffness, from two different system components. In order to have equivalent uncertainty in the two matrices, the shape parameter p must be the same for both ensembles. 2 = δ 2 , unless However, Eq. (1.5) shows that even if p1 = p2 = p, the dispersion values are not the same in general, δG G2 1 γ 1 = γ 2 . A more useful definition of matrix dispersion is the normalized dispersion δG δG 1 δGn = √ 1 =√ 2 =√ p 1 + γ1 1 + γ2

(1.6)

which is independent of the matrix magnitude γ G . It is important to realize, however, that just because two components have the same normalized dispersion, this does not mean that they will have the same uncertainty relative to a specific metric used to quantify component uncertainty, such as modal parameter uncertainty. Where the normalized dispersion is small, the modal statistics of the two components tend to be close, but as the normalized dispersion increases, the modal statistics for the two components become more disparate due to nonlinearity. Therefore, while Eq. (1.6) can be used to initially set a component matrix dispersion, it should ultimately be checked and adjusted based on the specific metric being used to quantify component uncertainty. Adhikari [8] referred to the random matrix method developed by Soize [6, 7] as Method 1. The Wishart parameters are selected as p and  = Go /p where Go is the nominal value of G. The mean of the distribution is given by Eq. (1.3) as G = p = p (Go /p) = Go . Therefore, Method 1 preserves the nominal matrix as the mean of the ensemble. In general, the nominal matrix can be decomposed in the form Go = LLT

(1.7)

In the case of a positive definite matrix, this would just be the Cholesky decomposition. Let (n × p) matrix X be given by   X = x1 x2 · · · xp

(1.8)

in which xi is an (n × 1) column vector containing standard random normal variables such that xi ~Nn (0, In ). Note that p ≥ n must be satisfied in order for G to be full rank. An ensemble member G~Wn (p, Go /p) can then be generated for MC analysis using the expression G=

1 LXXT LT p

(1.9)

1 Variational Coupled Loads Analysis Using the Hybrid Parametric Variation Method

5

It has been found that ensembles of random component mass matrices are best represented using Method 1. Adhikari [8] noted that Method 1 does not maintain the inverse of the mean matrix as the mean of the inverse; that is  −1 E G−1 = [E(G)]−1 = G (1.10) In some cases, the two can be vastly different, which is clearly not physically realistic. Instead, he proposed Method 3, in which the Wishart parameters are selected as p and  = Go /θ where θ=

1 [1 + γG ] − (n + 1) 2 δG

(1.11)

An ensemble member G~Wn (p, Go /θ ) can then be generated using the relation G=

1 LXXT LT θ

(1.12)

In this case, the inverse of the mean matrix is preserved as the mean of the ensemble inverses, where the mean matrix is now given by G = p = p (Go /θ ) =

p Go θ

(1.13)

In Method 3, the dispersion defined in Eq. (1.4) is now calculated with respect to the mean given in Eq. (1.13), while Eqs. (1.5) and (1.6) still hold. It has been determined that ensembles of random component stiffness matrices are best represented using Method 3. Therefore, the nonparametric portion of the HPV method is based on a Method 1 randomization of the component mass matrix and a Method 3 randomization of the component stiffness matrix. In this approach, the random component mass and stiffness matrices are assumed to be independent. Note that the Wishart matrix uncertainty model results in uncertainty both in mode shapes and in frequencies. However, MC simulation and analysis has shown that component mode shapes tend to be sensitive to the nonparametric matrix randomization provided by Methods 1 and 3, but the corresponding modal frequencies tend to be relatively insensitive. Therefore, a parametric component of uncertainty was added to the HPV approach in which the eigenvalues of the FI modes in the component HCB representation are also assumed to be random variables. ME is also used to derive the probability distribution function (PDF) for the HCB FI eigenvalues. The ME principle produces a PDF based solely on the available information, and in this case, there are two pieces of information available: First, the ith random eigenvalue must be strictly positive. Second, the expected value of the random eigenvalue is given by the nominal value. The application of ME yields a gamma distribution, λri ~G(ki , θ i ), where the shape parameter ki and the scale parameter θ i are given by ki = δi−2 and θi = λi δi2 , in which δ i is the corresponding coefficient of variation, or dispersion [13]. The FI eigenvalues are then independent random parameters within the HCB component stiffness matrix. The validity and impact of this assumption can be determined by considering only the FI component of the HCB substructure representation. The corresponding equation of motion in physical coordinates is given by Moo u¨ o + Koo uo = Fo

(1.14)

while the equation of motion in FI modal coordinates is I q¨ + λq = φ T Fo

(1.15)

where the modal stiffness λ is a diagonal matrix of FI modal eigenvalues from Eq. (1.1). The modal stiffness is given by λ = φ T Koo φ

(1.16)

The physical stiffness can be recovered by pre-multiplying Eq. (1.16) by Moo φ and post-multiplying by φ T Moo producing Moo φφ T Koo φφ T Moo = Moo φλφ T Moo

(1.17)

If all FI modes are retained, then φφ T Moo = I. In considering only the ith mode, Eq. (1.17) becomes Kooi = PiT Koo Pi = Moo φi λi φiT Moo

(1.18)

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where Pi = φi φiT Moo is an oblique projector [14] onto the column space spanned by the ith FI mode and Kooi is the contribution of the ith FI mode to the FI physical stiffness matrix. The physical stiffness can then be expressed as Koo =

no i=1

Kooi =

no i=1

 λi Moo φi φiT Moo

(1.19)

Therefore, the randomized physical stiffness matrix that corresponds to the randomized FI eigenvalues λri is given by Koor =

no i=1

αi Kooi =

no i=1

 n  αi λi Moo φi φiT Moo = λri Moo φi φiT Moo i=1

(1.20)

where α i is a random variable selected from a gamma distribution with a mean value of 1.0 and no is the number of FI modes. Equation (1.20) indicates that the parameterization of the FI stiffness in modal space using the eigenvalues λi corresponds to the parameterization of the FI stiffness in physical space using the matrices Kooi . The randomized stiffness parameters Koori = α i Kooi are independent of one another, meaning that changing the ith stiffness parameter has no impact on any of the other stiffness parameters. In summary, the FI eigenvalue parameterization used in the HPV method is equivalent to a parameterization within physical FI space where the stiffness components for the individual modes are scaled by no independent random parameters. The resulting sum produces a random physical stiffness matrix that preserves the nominal FI modes, but the generalized stiffness or eigenvalues vary. While the HPV FI eigenvalue parameterization preserves the model form of the HCB representation stiffness matrix, it does not preserve the model form of the physical FI stiffness matrix. Therefore, the variation of the HCB FI eigenvalues is a purely parametric variation with respect to the HCB representation, but it is not a purely parametric variation with respect to the physical stiffness partition Koo , meaning that it also results in modelform uncertainty. Also, if the corresponding random HCB stiffness matrix in its entirety is back transformed into physical coordinates, then not only is the Koo partition randomized as discussed above, but the Kaa and Kao partitions, corresponding to the interface DOF, are also randomized such that the sign-definiteness and the rigid body modes are preserved. The fact that this parameterization affects the FI eigenvalues but not the FI mode shapes allows it to be paired with the NPV method to produce a maximal impact on the HCB frequencies and a minimal impact on the HCB mode shapes. Therefore, the HPV approach provides the capability to almost independently adjust the uncertainty in the component frequencies and mode shapes, by adjusting the dispersion of the FI eigenvalues vs. the dispersions of the HCB mass and stiffness matrices. It is important to emphasize that this parameterization is not equivalent to the usual model input parameterization of the FEM. In contrast, if all of the random FI eigenvalues are varied in unison, such they are no longer independent, but perfectly correlated, then the random FI partition of the stiffness matrix in physical space is given by Koor =

n i=1

αi Kooi = α

n i=1

Kooi = αKoo

(1.21)

which is just a random scaling of the nominal stiffness. This randomization of the FI eigenvalues then preserves the model form of the FI partition of the physical stiffness matrix as well as the model form of the entire HCB stiffness matrix. As in the previous case, if the corresponding random HCB stiffness matrix is transformed back into physical coordinates, then the FI partition of the physical stiffness matrix is a scaled version of the nominal FI partition with the random scale factor α. The model form of the other random physical stiffness partitions involving the interface is not necessarily maintained. During each iteration in an MC analysis, a random draw of HCB FI eigenvalues is selected to generate a random HCB component stiffness matrix as described. The mean of this ensemble would just be the nominal HCB stiffness matrix. However, for each iteration, the parametrically randomized HCB stiffness is treated as the nominal matrix, and Method 3 is applied to provide model-form uncertainty on top of the FI eigenvalue parametric uncertainty. This is analogous to the approach proposed by Capiez-Lernout [10] for separating parametric and nonparametric uncertainty. Component frequency uncertainty can be based on component test-analysis frequency correlation, and the nonparametric mass and stiffness dispersion can be based on the corresponding orthogonality and cross-orthogonality results. The component slosh modes are likely to have much less uncertainty associated with them. In addition, there may be times when the component mass must be randomized while the rigid body mass is preserved. A special methodology has been developed within the HPV method to preserve the component slosh modes and rigid body mass when desired; the same methodology can be applied to any subset of the component modes. When the nominal matrix is positive semidefinite, such as in the case of an SLS flight component that has rigid body modes and a positive semidefinite stiffness matrix, special steps must be taken to decompose the nominal HCB stiffness matrix for subsequent randomization using Method 3 and Eq. (1.12). As in the case of component mass randomization, the slosh mode stiffness must be preserved. In addition, the rigid body stiffness must be preserved as the null matrix. Details of the methodology’s implementation within the HPV framework are given in reference [3].

1 Variational Coupled Loads Analysis Using the Hybrid Parametric Variation Method

7

1.3 Selection of Component Eigenvalue and Stiffness Matrix Dispersion Values The HPV approach for modeling component uncertainty requires the selection of dispersion values for the HCB component FI eigenvalues, mass matrix, and stiffness matrix. Ideally, these dispersion values are selected for each component based on component modal test results. This is because test-analysis modal correlation metrics are used to determine the dispersions. Test-analysis frequency error is used to identify the HCB FI eigenvalue uncertainties, but one of the biggest challenges in the propagation of component test-analysis frequency error into uncertainty in the HCB flight configuration FI modes is that the component test configuration and the component flight configuration boundary conditions and/or hardware are almost never the same. Because of this, it is difficult to match test configuration modes with flight configuration FI modes. The boundary condition mismatch can be alleviated using a mixed-boundary approach. In general, the HCB flight configuration FI modes will be over-constrained when compared to the test configuration modes. Therefore, the HCB stiffness matrix in Eq. (1.1) can be written as ⎡ ⎤  Kcc Kcb 0 KS 0 = = ⎣ Kbc Kbb 0 ⎦ 0 λ 0 0 λ 

KH CB

(1.22)

where the HCB flight configuration set of boundary degrees of freedom (DOF) have been divided into two subsets: the c-set contains all DOF that are free in the component test configuration, and the b-set contains the DOF that are constrained in the component test configuration. When the HCB flight configuration is constrained at the test configuration interface DOF (b-set), it produces the mass and stiffness matrices 

Mcc Mcq MC = Mqc Mqq





Kcc 0 KC = 0 λ

 (1.23)

T  T φT . These eigenvalues and with corresponding eigenvalues λC and mass normalized eigenvectors φC = φcc cq eigenvectors are consistent with the boundary conditions of the test configuration modes used in the component test-analysis correlation. Error or uncertainty in the analytical test configuration eigenvalues can be much more easily mapped onto uncertainty λC in the eigenvalues of the system in Eq. (1.23). The HCB representation of the component using λC and φ C as FI modal properties has the stiffness matrix and corresponding displacement vector given by  KB =

KSb 0 0 λC



T  uB = xbT qCT

(1.24)

where KSb is KS statically reduced to the b-set, xb is the physical displacement of the b-set, and qC are the modal coordinates of the FI modes with the c-set free. The transformation between displacement vector uB and the original HCB displacement vector uHCB is given by

uH CB

⎧ ⎫ ⎡ ⎤ ψ φcc   ⎨ xc ⎬ xb = T uB = xb = ⎣ I 0 ⎦ ⎩ ⎭ qC q 0 φcq

(1.25)

The relation between KB and KHCB is then KB = T T KH CB T

(1.26)

The test configuration HCB FI eigenvalues λC can be randomized (λCr ) based upon the component test-analysis correlation results, and the uncertainty can be propagated into the random flight configuration HCB component stiffness (KHCBr ) using the expression KH CBr = T

−T

KBr T

−1

=T

−T



 KSb 0 T −1 0 λCr

(1.27)

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Details of the procedure used to assign the uncertainty to the eigenvalues λC are discussed in the following section. Reference [3] details a procedure for identifying the HCB mass matrix dispersion based upon the component test mode selforthogonality matrix. Alternatively, the mass matrix dispersion could be based on engineering judgment, past experience, and historical results. In this work, it is assumed that the mass representation of the components is accurate, so the mass matrix is not dispersed. Once the eigenvalue dispersions have been identified, test-analysis cross-orthogonality is used to identify the dispersion of the component stiffness matrix. In this case, the root mean square (RMS) diagonal value of the component test-analysis cross-orthogonality matrix, referred to as the diagonal cross-generalized mass (DCGM), is used as the metric. For this, a series of MC analyses is performed in which the HCB stiffness matrix dispersion value is swept over a range, and the most probable value of DCGM is computed for each MC analysis. The goal is to select the stiffness dispersion value that gives the most probable DCGM value that is equal to the test result. Details of the identification procedure are discussed in the following section and in reference [3].

1.4 Nominal ICPS/LVSA Based on ISPE Configuration 3 Dispersion values for the combined nominal interim cryogenic propulsion stage (ICPS)/launch vehicle stage adapter (LVSA) HCB components are based on the integrated spacecraft payload element (ISPE) configuration 3 modal test-analysis correlation results. The FEM representation of ISPE configuration 3 is shown in Fig. 1.1. There are 11 FEM target modes matched to 11 of 19 test modes. Only these target modes are considered in this analysis, because the other eight modes are dominated by the MPCV stage adapter (MSA)/Multi-Purpose Crew Vehicle (MPCV) simulator, which is not part of the ICPS/LVSA component. The test-analysis frequency correlation results are listed in Table 1.1. The nominal model accurately predicts the first bending test mode frequencies. This is consistent with the static test results, which showed good agreement between the nominal model and the test results for overall bending and axial stiffness. Only one second-order bending test mode was identified in the test data, and no axial modes were identified. The nominal model does a poor job of predicting the second-order bending test mode frequency, and the LVSA shell test mode frequencies are even less accurately predicted. Note that the frequency error is calculated relative to the nominal FEM frequency rather than the test; this is done because the uncertainty analysis is performed relative to the FEM HCB representation. As mentioned, the SLS HCB flight components and the test components do not match in most cases. In the case of the ICPS/LVSA HCB flight component, there is no MSA nor MPCV simulator. In addition, the ISPE configuration 3 is not tested in the HCB flight configuration. At the time of this writing, the mixed-boundary approach discussed above has not yet been applied to the ICPS/LVSA HCB flight component. Instead, the test configuration modal frequency uncertainties are mapped directly onto the ICPS/LVSA HCB flight component FI modes. It is assumed that the component test correlation results can be used as an indicator of what level of uncertainty is expected in the corresponding HCB component model.

Fig. 1.1 ISPE configuration 3 FEM representation

1 Variational Coupled Loads Analysis Using the Hybrid Parametric Variation Method

9

The first element of uncertainty to be specified is the dispersion of the 33 non-slosh FI modal frequencies for the HCB flight component. These modes have the base of the LVSA and the top of the ICPS at the MSA interface constrained, whereas during the test, the base of the LVSA is attached to the core simulator, and the top of the ICPS is attached to the MSA/MPCV simulator. Test-analysis frequency error is mapped onto the FI modes using modal effective mass (MEM). The nominal FEM ISPE configuration 3 MEM is dominated by the fundamental bending and axial modes, and to a lesser extent, the second-order bending modes. The LVSA shell modes have little or no MEM. Based on MEM, the HCB FI modes are placed in three different bins of frequency uncertainty. Bin 1, associated with the FEM configuration 3 first bending pair, is assigned a frequency dispersion of 1.39%, corresponding to the RMS error in the prediction of the first bending configuration 3 test mode pair frequencies. The modal frequencies are assumed to follow the gamma distribution described previously. Bin 2 has a frequency dispersion of 9.01%, corresponding to the test-analysis frequency error of the second-order bending configuration 3 mode. FI modes that have little or no MEM, analogous to the LVSA shell test modes, are assigned to bin 3 with a frequency dispersion of 10.91%, corresponding to the RMS frequency error in the configuration 3 LVSA shell modes. Once the FI eigenvalue uncertainty is applied, the dispersion of the stiffness matrix is determined by computation of the DCGM metric based on cross-orthogonality, which is the RMS value of the diagonal after modes are matched and resorted accordingly. Based on the nominal ISPE configuration 3 cross-orthogonality matrix for the 11 target modes, the dispersion metric has the value DCGM = 90.44. The goal is to select a stiffness dispersion value for the MC analysis such that the test-based metric value is the most probable. The MC analysis is based on the first 37 nominal HCB elastic non-slosh modes and 3000 ensemble members. The selected FI eigenvalue uncertainties are applied, and then a series of MC analyses are performed with stiffness dispersion swept over a range of values. A stiffness dispersion of 7%, which corresponds to a normalized dispersion of 0.85%, produces the most probable DCGM value that agrees with the test value.

1.5 Updated ICPS/LVSA Based on ISPE Configuration 3 A posttest correlation was performed to update the ISPE configuration 3 FEM; thus, dispersion values for the updated ICPS/LVSA HCB component are based on the updated ISPE configuration 3 FEM. Like the nominal model analysis, there are 11 FEM target modes matched to 11 of 19 test modes. The test-analysis frequency correlation results are also listed in Table 1.1. Note that the frequency error for the LVSA shell modes and the second-order bending modes have been dramatically reduced. There are now 243 HCB FI non-slosh modes in the updated ICPS/LVSA fifty-seconds-of-ascent HCB component. The FI MEM is calculated and the same three-bin uncertainty assignment strategy is applied. Bin 1 is assigned a frequency dispersion of 2.04%, corresponding to the RMS error in the prediction of the first bending test mode pair. Bin 2 is assigned a frequency dispersion of 4.83%, corresponding to the test-analysis frequency error of the second-order bending test mode. As in the nominal model analysis, the remaining FI modes have little or no MEM, analogous to the LVSA shell test modes. Therefore, these FI modes are assigned to bin 3 with a frequency dispersion of 1.97%, corresponding to the RMS frequency error in the configuration 3 LVSA shell modes. Based on the updated ISPE configuration 3 cross-orthogonality matrix for the 11 target modes, the dispersion metric has the test value DCGM = 95.43. The selected FI eigenvalue uncertainties were applied, and then a series of MC analyses were performed with stiffness dispersion swept over a range of values. In this case, the MC analysis was based on the first 35

Table 1.1 Test-analysis frequency error for ISPE configuration 3 nominal model Test mode 1 2 3 4 5 6 9 10 14 15 19

Pretest mode 6 5 9 10 11 12 14 13 24 23 22

Error (%) −0.16 −1.94 10.13 9.95 7.21 6.97 12.90 12.60 15.29 14.98 −8.63

Posttest mode 6 5 7 8 9 10 14 13 20 19 24

Error (%) −0.89 −2.70 1.43 1.23 3.96 3.50 −0.23 −0.57 −0.01 −0.33 −4.72

Description First bending First bending LVSA shell ND 5 LVSA shell ND 5 LVSA shell ND 4 LVSA shell ND 4 LVSA shell ND 6 LVSA shell ND 6 LVSA shell ND 7 LVSA shell ND 7 Second bending

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nominal HCB elastic non-slosh modes to about 20 Hz and 3000 ensemble members. The DCGM metric was computed for each ensemble member cross-orthogonality matrix. The stiffness dispersion that produced the most probable DCGM metric that was in greatest agreement with the test value was 9%, which corresponds to a normalized dispersion of 1.01%. Further details on the procedure used to assign ICPS/LVSA HCB FI frequency and stiffness matrix dispersions for the pretest and updated models can be found in reference [3].

1.6 Propagation of Uncertainty into Response to Buffet Loads Buffet loads are forces on the system flight vehicle caused by the interactions between separated flow turbulence, attached boundary layer turbulence, and shock wave oscillations [15]. A buffet loads analysis can be performed in either the frequency or the time domain. The SLS program uses the time-domain approach. The forcing function time histories are generated from wind tunnel tests of scale models [16]. The time domain has several advantages over the frequency domain: time histories of various loads can be combined, and the phasing is automatically correct; the size of the problem is reduced because a frequency-domain analysis requires force cross-spectra; and the response time histories can be used directly in subsequent MC analysis [15]. The HPV approach for uncertainty quantification and propagation was applied to the SLS fifty-seconds-of-ascent (Ascent 50) configuration with buffet loads in the transonic regime at Mach 0.9 and 0.0 degrees of pitch and roll. The corresponding CLA model assembled from HCB components has over 12,000 DOF. This proved to be too large for VCLA, but through the judicious combination of interfaces, selection of residual vectors, elimination of internal DOF, and reduction in component mode frequency cutoff, the size of the assembled model was reduced to 4102 DOF, which is an acceptable size for VCLA. It requires approximately 33 s on a desktop computer to perform one MC iteration, which makes it feasible to run about 1000 iterations overnight. The SLS HCB components contained FI modes to 65 Hz plus 6 modal truncation vectors (MTVs) for each component interface. In addition, there is a residual vector for each buffet force application DOF, due to the fact that they are interior to the components. One percent modal damping was used for system modes up to 20 Hz, and 2% modal damping was used for system modes above 20 Hz. The buffet loads were applied to core and solid rocket booster (SRB) centerline nodes for 100 s with system modes to 50 Hz and system-level residual flexibility. Response due to buffet loading involves system oscillations with relatively small rigid body motion, so the aerodynamic damping and stiffness can be conservatively neglected [15]. The appropriate equation of motion in system modal coordinates is then given by I q¨S + 2ζS ωS q˙S + ωS2 qS = φST FB

(1.28)

where qS is a vector of system generalized coordinates, ζ S is a diagonal matrix of modal damping, ωS is a diagonal matrix of system natural frequencies, and FB is the vector of buffet forces. In this application, it is assumed that the buffet loads are zero mean and Gaussian, and that the time histories are long enough that the RMS values have converged. Figure 1.2 illustrates the RMS buffet forces per unit length applied along the SRB and CS centerline nodes. CS section loads were recovered using displacement and acceleration output transformation matrices (OTM), which were transformed to the reduced model used in the UQ analysis. Nominal model CS RMS section loads are shown in Fig. 1.3. The CS sections loads will also be zero mean Gaussian, and it is assumed that there is infinite data. For a narrowband response to zero-mean Gaussian inputs, the peaks and the corresponding envelope function will follow a Rayleigh distribution [17]. The quantile function, or inverse of the cumulative distribution function, QP for a Rayleigh distribution is given by the expression  (1.29) QP = σ −2 ln (1 − P ) = σ ξP For 99% enclosure, P = 0.99, and the P99 response value is given by  x.99 = σ −2 ln (1 − .99) = ξ.99 σ = (3.03485)σ

(1.30)

where σ is the Rayleigh distribution shape parameter, which is equivalent to the RMS value of the time-domain response x. Therefore, the P99 value for the response can be determined using x.99 = ξ.99 xRMS

(1.31)

RMS Force/Length - lbf/in

1 Variational Coupled Loads Analysis Using the Hybrid Parametric Variation Method

RMS Force/Length - lbf/in

1000

RMS Force/Length - lbf/in

1000

1000

11

RMS Buffet Force per Unit Length at RSRB CL Nodes

X Y Z

1500

2000

2500

3000

3500

4000

4500

5000

Axial Station (X) - in. RMS Buffet Force per Unit Length at Core CL Nodes

1500

2000

2500

3000

3500

4000

4500

5000

Axial Station (X) - in. RMS Buffet Force per Unit Length at LSRB CL Nodes

1500

2000

2500

3000

3500

4000

4500

5000

Axial Station (X) - in. Fig. 1.2 RMS force per unit length applied at SRB and CS centerline nodes

In the case of a more broadband response like the current situation, the Rayleigh assumption and Eq. (1.31) provide a conservative estimate of the P99 response. Unfortunately, the actual buffet loads that are applied are not quite Gaussian, and the resulting CS section load envelope functions [17] are not quite Rayleigh, therefore Eq. (1.31) does not bound the ∼ actual P99 peak value. Alternatively, the .99 quantile of the envelope function x could be computed. The P99 value based on the envelope function will always give bounding results regardless of whether the time histories are zero mean Gaussian. However, the current variational buffet loads analysis is for demonstration purposes only, so for simplicity, the Rayleigh approximation in Eq. (1.31) is used to compute P99 CS section loads. Further details on computing P99 peak response can be found in reference [17]. The Ascent 50 system configuration consists of seven different components: MPCV+MSA, ICPS+LVSA, CS, left (LSRB), right (RSRB), LSRB_CBAR, and RSRB_CBAR. All the components are HCB models except for LSRB_CBAR and RSRB_CBAR, which are simple bar representations of the connections between the boosters and the CS. The Ascent 50 system uncertainty model is based on the component uncertainty models derived for the pretest and updated ICPS/LVSA,

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CS RMS Shear Section Loads

Force - lbf

Fx Fy Fz

2300 2550 2800 3050 3300 3550 3800 4050 4300 4550 4800

Axial Station - in

CS RMS Section Moments

Moment - lbf-in

Mx My Mz

2300 2550 2800 3050 3300 3550 3800 4050 4300 4550 4800

Axial Station - in Fig. 1.3 Buffet CS section RMS shear forces and moments

which were generated based on the ISPE configuration 3 test-analysis correlation results presented in the previous section. In this buffet UQ analysis, the updated uncertainty model is used for the ICPS/LVSA HCB component. However, there is currently no test-analysis correlation data available for any of the other six components. Therefore, the uncertainty models for these components are based on the two uncertainty models derived for the ICPS/LVSA. The pretest level is based on the ICPS/LVSA pretest uncertainty model and corresponds to the assignment of three bins of frequency uncertainty (1.39%, 9.01%, 10.91%) to the HCB FI modes based on the modes’ MEM. The HCB stiffness matrix normalized dispersion is assigned so that the most probable DCGM value based on sweeping the dispersion over a range in a series of MC analyses corresponds to the pretest value of 90.44. This uncertainty model is applied to components that have no heritage and no available modal test-analysis correlation data, such as the MPCV/MSA. The updated uncertainty level is based on the updated ICPS/LVSA uncertainty model. It corresponds to the assignment of three bins of frequency uncertainty (2.04%, 4.83%, 1.97%) to the HCB FI modes based on MEM. The HCB stiffness matrix normalized dispersion is assigned so that the most probable DCGM value corresponds to the updated model value of 95.43. This uncertainty model is applied to components that have a heritage of previous use, testing, and model validation, such as the boosters. Table 1.2 summarizes the resulting uncertainty models for all seven components based on the two uncertainty levels.

1 Variational Coupled Loads Analysis Using the Hybrid Parametric Variation Method

13

Table 1.2 Ascent 50 configuration system uncertainty model Component MPCV/MSA ICPS/LVSA Core LSRB RSRB LSRB CBAR RSRB CBAR

Uncertainty level Pretest Updated Updated Updated Updated Updated Updated

Assigned HCB FI frequency dispersion 1.39%, 9.01%, 10.91% 2.04%, 4.83%, 1.97% 2.04%, 4.83%, 1.97% 2.04%, 4.83%, 1.97% 2.04%, 4.83%, 1.97% 0.0% 0.0%

Normalized stiffness dispersion 0.37% 1.01% 0.30% 1.53% 1.53% 3.52% 3.52%

Ascent 50 Seconds Frequency Uncertainty

12 11

RMS Frequency Uncertainty - %

10 9 8 7 6 5 4 3 2 1 0 30

40

50

60

70

80

90

100

110

120

130

Mode Number Fig. 1.4 RMS system frequency uncertainty for non-slosh elastic modes below 11 Hz

Note that the CS is assigned an updated level of uncertainty, even though the component has no history and the pretest HCB model is being used, because CS mode-shape uncertainty is very sensitive to both the assigned FI eigenvalue uncertainty and the stiffness matrix dispersion. The reduced uncertainty level is assigned to predict what is expected for CS section load uncertainty in the case where the CS model has been updated. The left and right boosters are assigned the same updatedlevel uncertainty model, even though the HCB representations are slightly different. The two booster CBAR components are relatively simple components; therefore, they are also assigned the updated level of uncertainty, although in this case, a slightly different approach must be taken to assign an uncertainty model. The two components are essentially triaxial springs with 12 rigid body modes and three elastic modes. They are not HCB component representations; therefore, FI eigenvalue uncertainty cannot be assigned. The stiffness dispersion was determined such that the mean RMS uncertainty of the three elastic eigenvalues had a value of 5%, corresponding to the assumed average level of eigenvalue uncertainty over all FI eigenvalues in the updated ICPS/LVSA model. An MC analysis was performed generating 1000 random system models using the HCB component uncertainty models listed in Table 1.2. Figure 1.4 shows the system RMS frequency uncertainty for the first 100 non-slosh elastic modes below 11 Hz. The average RMS uncertainty is 2.75%, with a maximum value of 11.25% for system mode 96. Mode matching

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RMS Uncertainty - %

60

RMS Uncertainty in Peak CS Section Forces Axial - X Shear - Y Shear - Z

50 40 30 20 10

0 2300 2550 2800 3050 3300 3550 3800 4050 4300 4550 4800

Axial Station - in.

40

RMS Uncertainty in Peak CS Section Moments

RMS Uncertainty - %

Torsion - X

30

Moment - Y Moment - Z

20

10

0 2300 2550 2800 3050 3300 3550 3800 4050 4300 4550 4800

Axial Station - in. Fig. 1.5 Percent RMS CS section load uncertainties for Ascent 50 uncertainty model

between the random and nominal models is not particularly accurate beyond 10 Hz. CS P99 section loads were calculated for each of the 1000 ensemble members using Eq. (1.31). The corresponding RMS uncertainties in the P99 loads with respect to the nominal section load values are shown in Fig. 1.5. The uncertainty in the CS axial and torsion section loads (X) is much higher than expected, but it is due to variations from relatively small nominal loads forward of the booster-CS connections and at the boattail. The uncertainty in the transverse section shears in the Y direction are also higher than expected between the booster-CS connections, the largest being 26.07% at section X = 4066.86 . While the RMS uncertainty relative to the nominal value is 26.07% for the Y shear force at X = 4066.86 , the standard deviation divided by the mean, or coefficient of variation, is only 4.82%. The large RMS uncertainty is due to a large offset between the nominal value and the mean of the ensemble. Figure 1.6 shows the P99/90 CS Y shear force compared to the nominal value and the ensemble mean by station. The P99/90 value is determined by computing the 0.90 quantile of the ensemble of random model P99 values. It generally would be expected that the 90th percentile of the P99 ensemble is greater than the nominal model P99 value, but that is not the case because of the large offset between the nominal values and the ensemble means at stations between X = 3125 , where the boosters connect to the CS, and X = 4472 . Large offsets occur at sections with large RMS uncertainty as shown in the top of Fig. 1.6. The same behavior occurs for the CS Z shear forces, but to a lesser extent, as shown in Fig. 1.7.

1 Variational Coupled Loads Analysis Using the Hybrid Parametric Variation Method

15

Force - lbf

CS Section Force - Y

P99 Nominal P99/90 P99 Mean

2300 2550 2800 3050 3300 3550 3800 4050 4300 4550 4800

Axial Station - in.

Moment - lbf-in

CS Section Moment - Z

2300 2550 2800 3050 3300 3550 3800 4050 4300 4550 4800

Axial Station - in. Fig. 1.6 P99 nominal, mean, and P99/90 CS Y section shears and Z bending moments

In contrast, RMS uncertainty in the CS section bending moments in Fig. 1.5 are all less than 10%, except toward the aft end where the moments get smaller. Figure 1.6 shows a tight dispersion among the nominal P99, P99/90, and mean P99 Z bending moment values. In the region forward of the booster-CS connection, the mean and nominal P99 values are very close, with the P99/90 value being larger as expected. Between the booster-CS connections, the nominal and P99/90 values are very close. Figure 1.7 illustrates the nominal P99, P99/90, and mean P99 Y bending moment values. In contrast with Z bending moments, the nominal and mean P99 values are very close over almost the entire length of the CS. It was determined that the CS transverse shear forces at sections between the booster and CS connections are very sensitive to uncertainties in the CS HCB FI frequencies. The large offset between the nominal and ensemble mean P99 Y shear forces in Fig. 1.6 is due to modes between 30 and 40 Hz, while the large offset between the nominal and ensemble mean P99 Z shear forces in Fig. 1.7 is due to modes between 15 and 20 Hz. The Y and Z buffet forces with the largest RMS magnitude acting on the core centerline have a significant amount of power in these two frequency regions, respectively. It is believed that the variability of the mode shapes in the 15–20 Hz. and 30–40 Hz. frequency bands due to the uncertainty in the CS

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CS Section Shear - Z

P99 Nominal

Force - lbf

P99/90 P99 Mean

2300 2550 2800 3050 3300 3550 3800 4050 4300 4550 4800

Axial Station - in.

Moment - lbf-in

CS Section Moment - Y

2300 2550 2800 3050 3300 3550 3800 4050 4300 4550 4800

Axial Station - in. Fig. 1.7 P99 nominal, mean, and P99/90 CS Z section shears and Y bending moments

HCB FI frequencies causes the RMS response of the transverse shear forces to drop significantly compared to the nominal response, causing large RMS shear force uncertainties relative to the nominal values. This was confirmed by computing the uncertainty in the transverse acceleration frequency response of the CS centerline nodes. While the reduced transverse shear response predicted for the random models within the MC analysis been confirmed using both time- and frequency-domain analysis, the physical cause of the reduced transverse shear response is currently unknown. In contrast, the structurally more significant CS section bending moments are much less sensitive to uncertainty in the CS, leading to reduced uncertainty levels. Future work will further investigate the source and validity of the reduced random response of the CS transverse section loads.

1 Variational Coupled Loads Analysis Using the Hybrid Parametric Variation Method

17

1.7 Conclusion Uncertainty in structural responses during launch is a significant concern in the development of spacecraft and launch vehicles, where small variations in mode shapes and their interactions can result in significant variation in system loads. CLA, using HCB component models, is the accepted method for determining the response of a launch vehicle/payload system to transient forces, such as liftoff, engine ignitions and shutdowns, jettison events, and atmospheric flight loads, such as buffet. UQ is used to determine statistical bounds on prediction accuracy based on model uncertainty. In this paper it is assumed that the uncertainty is most naturally represented at the component HCB level. In an effort to account for model uncertainties and their effect on CLA predictions, this work combines CLA and UQ in a process termed VCLA. The UQ component is performed using the HPV method, which combines parametric and nonparametric uncertainty at the HCB component model level. The HPV method requires the selection of dispersion values for the HCB FI eigenvalues, and the HCB mass and stiffness matrices. The dispersions are based on component test-analysis modal correlation results. In the past, there has been a problem of mapping uncertainty in test configuration modes into uncertainty in HCB flight configuration FI modes due to the fact that the boundary conditions for the two configurations are almost never the same. However, the new approach outlined here properly maps the uncertainty by accounting for the differences in the boundary conditions. VCLA has previously not been feasible due to computational expense, but the HPV approach of UQ combined with the use of judiciously selected HCB interfaces makes the VCLA practical within a MC analysis. The proposed methodology was demonstrated by successful application of the process to a buffet loads analysis of the SLS during the transonic regime at 50 s after liftoff. CS section shears and moments were recovered, and relevant statistics were computed. The system uncertainty model based on component test-analysis correlation results produces reasonable uncertainty results for the CS section moments. However, the transverse CS section shear loads were very sensitive to the uncertainty in the CS HCB FI frequencies from the standpoint of producing random responses that were significantly below the nominal values. This produced large RMS uncertainties relative to the nominal shear loads. It is believed that this phenomenon is an artifact of the models being used and not a function of the proposed VCLA method, but future work will further investigate the validity of the predicted results.

References 1. Craig, R.R., Bampton, M.C.C.: Coupling of substructures for dynamic analysis. AIAA J. 6, 1313–1319 (1968) 2. Majed, A., Partin, K., Henkel, E., Sarafin, T.: Variational coupled loads analyses: reducing risk in development of space-flight hardware. J. Spacecr. Rocket. 42(1), 98–104 (2005) 3. Kammer, D., Blelloch, P., Sills, J.: Test-Based Uncertainty Quantification and Propagation Using Hurty/Craig-Bampton Substructure Representations. IMAC, Orlando (2019) 4. Ghanem, R., Spanos, P.: Stochastic Finite Elements: A Spectral Approach. Springer, New York (1991) 5. Wishart, J.: Generalized product moment distribution in samples. Biometrika. 20A(1–2), 32–52 (1928) 6. Soize, C.: A nonparametric model of random uncertainties for reduced matrix models in structural dynamics. Probab. Eng. Mech. 15(3), 277–294 (2000) 7. Soize, C.: Maximum entropy approach for modeling random uncertainties in transient elastodynamics. J. Acoust. Soc. Am. 109(5), 1979–1996 (2001) 8. Adhikari, S.: Generalized Wishart distribution for probabilistic structural dynamics. Comput. Mech. 45, 495–511 (2010) 9. Adhikari, S.: Wishart random matrices in probabilistic structural mechanics. J. Eng. Mech. 134(12), 1029–1044 (2008) 10. Capiez-Lernout, E., Pellissetti, M., Pradlwarter, H., Schueller, G.I., Soize, C.: Data and model uncertainties in complex aerospace engineering systems. J. Sound Vib. 295(3–5), 923–938 (2006) 11. Pellissetti, M., Capiez-Lernout, E., Pradlwarte, r.H., Soize, C., Schueller, G.: Reliability analysis of a satellite structure with a parametric and a non-parametric probabilistic model. Comput. Methods Appl. Mech. Eng. 198(2), 344–357 (2008) 12. Mignolet, M., Soize, C., Avalos, J.: Nonparametric stochastic modeling of structures with uncertain boundary conditions/coupling between substructures. AIAA J. 51(6), 1296–1308 (2013) 13. Soize, C.: Uncertainty Quantification: An Accelerated Course with Advanced Applications in Computational Engineering. Springer, Cham (2017) 14. Ben-Israel, A., Greville, T.: Generalized Inverses – Theory and Applications, 2nd edn. Springer, New York (2003) 15. Kabe, A.: Time Domain Buffet Loads Analysis. Aerospace Report No. TOR-2007(2207)-6078, El Segundo, CA, 2006. 16. Piatak, D., Sekula, M., Rausch, R., Florance, J., Ivanco, T.: Overview of the Space Launch System Transonic Buffet Environment Test Program. AIAA SciTech, Kissimmee (2015) 17. Sako, B., Kabe, A., Lee, S.: Statistical combination of time varying loads. AIAA J. 47(10), 2338–2349 (2009)

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Dr. Daniel C. Kammer has expertise in dynamics and identification of structural systems, analytical model validation, and uncertainty quantification. Professionally, he spent 2 years at GD/Convair, 5 years at SDRC, and 29 years as a Professor at the University of WisconsinMadison. He is currently a Senior Technical Advisor for ATA Engineering, Inc.

Chapter 2

Bayesian Uncertainty Quantification in the Development of a New Vibration Absorber Technology Nicolas Brötz and Peter F. Pelz

Abstract The engineer starts with paper and pencil. From the basic idea to the axiomatic model, this is all he or she needs. If the axiomatic model does not reproduce the results of the experiment due to too many simplifications, the axiomatic model shall be extended. The aim is to guarantee the desired functionality at an early design stage and thus ensure a safe design. Mathematical models of a new vibration absorber technology of different complexity are utilized in order to predict its dynamic response under different operation conditions. Such a prediction of the dynamic response is subject to model uncertainty. The focus of this paper is on model’s uncertainty resulting from model’s complexity. The model’s complexity is designed to be as simple as possible for an efficient optimisation approach. Since development is domain-specific, system and boundaries must first be defined. For this purpose, the modules are cut out of the overall system. The system boundaries, of the Fluid Dynamic Vibration Absorber have first to be found for a mathematical model but also for an experimental setup to get reliable empiric data. After this the proposed dynamic response of mathematical model is investigated. At a hydraulic transmission in an oscillating system, there are several approaches to modelling the oscillating flow and damping. Damping plays a decisive role in vibration absorbers. The occurring uncertainty of prediction of the dynamic response of the different models has to be quantified, especially if it represents a risk for vehicle occupants. Therefore, a Bayesian interval hypothesisbased method is used to quantify this uncertainty. It turns out that the choice of model boundary is a crucial one for model confidence. Keywords Bayesian interval hypothesis · Model validation · Vibration absorber · Uncertainty

2.1 Introduction Axiomatic models are used for calculation and optimisation. The models are designed to be as simple as possible so that the calculation time for potential estimates in which a large number of variables can be freely selected is as short as possible. Uncertainty in the modelling results from simplification of the underlying model. For a potential estimation with no concrete construction yet for example the friction is the first parameter which is neglected, the effect of the product should be predicted as exactly as possible. At the model boundaries, too, simplifications are made, since the model boundaries are either unknown or deliberately neglected [1]. The same applies to initial and boundary conditions [2]. Some model parameters are taken from the literature which also leads to uncertainty. Thus, models with varying complexity are created, which yield different results. A model evaluation must therefore be carried out. The result is a quantitative measure of the agreement between predicted and experimentally response [3]. In this contribution a novel hydraulically translated absorber [4] is investigated. This has a non-linear damping due to the hydraulic translation, which is represented by mathematical models based on linear and non-linear approaches. In addition, the model boundary of the systems differs. With the Bayesian interval hypothesis the different approaches are evaluated [4].

N. Brötz () · P. F. Pelz Chair of Fluidsystems, Technische Universität Darmstadt, Darmstadt, Germany e-mail: [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2020 Z. Mao (ed.), Model Validation and Uncertainty Quantification, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47638-0_2

19

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N. Brötz and P. F. Pelz

150 mm

853 mm

absorber spring

body spring

duct

piston

fluid

Fig. 2.1 The TU Darmstadt Fluid Dynamic Vibration Absorber with parallel-connected body spring [5]

2.2 Fluid Dynamic Vibration Absorber (FDVA) The essential requirements for the suspension strut of modern vehicles are high driving safety and high driving comfort. The control of the driver over the vehicle, and thus also the driving safety, can only be ensured if there is wheel-ground contact at any time. The driving comfort is influenced decisively by acceleration acting on the occupants. Thus, the suspension strut of a vehicle affects both driving safety and driving comfort. The adjustment of a classical shock absorber system leads to a compromise between safety and comfort. In contrast a dynamic vibration absorber is capable of reducing vibrations of the wheel in a defined frequency band without affecting the body acceleration. Therefore, energy of the accelerated wheel is redirected into the structure extension. The FDVA is a dynamic vibration absorber with a hydraulic transmission of inertia. Instead of a solid mass, hydraulic oil is moved by a piston through a duct. The inertia of the FDVA is translated by the hydraulic transmission and can therefore be built more lightly compared to a conventional dynamic vibration absorber [5]. Figure 2.1 shows the functional demonstrator. The functional demonstrator consists of an absorber spring connected to the piston rod of a double-acting hydraulic cylinder. The chambers of the hydraulic cylinder are connected via several ducts on the outer side. Mechanical valves can close the ducts in order to change the ratio α = A/a between the surface of the piston A and the ducts a. The FDVA is connected parallel to the body spring and wheel axle with the movement zW to the chassis with movement zB . When the wheel axle is moved, the movement is transmitted to the piston via the absorber spring. Due to the piston movement the fluid mass inside the ducts is accelerated. The transmission factor between fluid movement and piston movement is α2 . The light weight principle of the FDVA is illustrated by the ratio of heavy mass to the acting inert mass, which is of the order of magnitude of α−2 at a small piston mass mP . With a change in the active channel area, realized by closing individual duct valves, the inertia of the FDVA, as well as its natural frequency, can be adjusted.

2.3 Mathematical Modeling and Experimental Testing of FDVA The function demonstrator is used to validate the modelling. This is necessary to obtain reliable results when estimating the potential. A potential estimation requires models that are as simple as possible for short computing time. Three models are investigated, model A an analytical model, model B an axiomatic model with non-linear damping and model C a model adapted to the functional demonstrator which takes Coulomb’s friction and the mass of the sensor connection into account. Figure 2.2 shows the three models and the experimental setup. Each model and the experiment have a harmonic base excitation with a frequency f of 0.5–25 Hz in 0.1 Hz steps as input variable. The frequency interval is adapted to the relevant frequencies in the quarter vehicle studies, since the FDVA is investigated for this study. Each examination is repeated 5 times to consider the uncertainty of the variability of the measurement. The investigations in this paper are limited to tests with two open channels, since the damping in this configuration is close to the optimal damping according to Corneli [6]. Future work will also consider the number of open channels. Various output values can be extracted to validate the dynamics of the FDVA. Our output values for calculation and measurement are the piston movement and the force at the excitation point. The description of a vibration absorber is made by the transmissibility V and the phase shift according to Den Hartog [7]. These quantities are generated from the piston movement in relation to the excitation. Thus, the output quantities examined here are (i) the transmissibility V = zˆ /ˆz0 , (ii) the phase shift between piston movement z and excitation z0 and (iii) the force amplitude Fˆ .

2 Bayesian Uncertainty Quantification in the Development of a New Vibration Absorber Technology

MODEL A

MODEL B

MODEL C

A

21

EXPERIMENT

A displacement

force

Fig. 2.2 Overview of different models and the test setup

2.4 Model A An analytical model can be solved using the common mode approach [8]. It is necessary to know the damping constant. This is obtained from preliminary investigations of the hydraulic transmission. The spring constant k is also determined from a force-displacement measurement. The transmission of the inertia and thus the actual inertia θ are known from the model consideration. The transmissibility is given by 1 V (η) =  , 2 1 − η2 + (2Dη)2

(2.1)

√ with η = 2πf/ k/θ and the phase shift 2Dη ψ (η) = arctan 1 − η2

! (2.2)

can be calculated. The piston movement and also the force F = k (z0 − z)

(2.3)

are known. This model offers the lowest complexity.

2.5 Model B The second model is created with the axioms continuity equation, Bernoulli equation and momentum theorem. The continuity equation in integral form delivers with the translation α = A/a between surface of piston A and channel area a the velocity in the channel z˙ F = −αz.

(2.4)

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The Bernoulli equation gives the pressure at the piston p1 = −lα z¨ + Δpl ,

(2.5)

where the pressure loss  p l = − 2



 ! !2 1 1 2 32ηl −1 + 1− + ζ α 2 z˙ 2 − 2 α z˙ α0 α D

(2.6)

consists of constriction, impact, deflection and pipe friction. The constriction parameter α0 and the loss factor of the deflection ζ are determined from the geometry. The force F = al z¨ α 2 + (mP + 2Laα) z¨ − p l A

(2.7)

can be calculated using the principle of linear momentum at the piston. The resulting differential equations can only be calculated numerically.

2.6 Model C In model C, model B is extended by the friction force between spring guide and spring housing and the friction force at the piston to the cylinder. The friction force is modelled without static friction. In addition, a mass mA is considered when connecting the spring to the force sensor by nuts, connecting plate and the spring housing. This represents an additional inertial mass at the point of application of the excitation and only influences the force with the value mA (2π f )2 at this point. Model C has an adjustment of the model boundary, since the force cannot be measured directly at the spring.

2.7 Experiment For validation, experiments with harmonic path excitation are performed on a hydro pulse system. The displacement of FDVA’s piston is measured on the piston rod with a displacement transducer according to the linear variable differential transformer principle. The force is determined with a load cell ALF300 which is screwed onto the plunger of the hydro pulse system. The load cell works with a bending beam measuring cell. The measurement data is recorded with a DSPACE system and processed with MATLAB.

2.8 Quantification of Uncertainty with Bayesian Interval Hypothesis-Based Method The three models each deliver different output quantities. The resulting uncertainty is quantified with Bayesian interval-based hypothesis testing [9, 10]. The variability of the measurement and the uncertainty of the model are considered. The procedure (Fig. 2.3) starts with collecting the data. For this purpose, the three considered output quantities transmissibility, phase shift and force amplitude for the experiment and the model are calculated. The values are then normalized with the respective maximum of the experiment. You get the normalized matrices T ∈ Rv×n×rt for the experiment and M ∈ Rv×n×rm for the respective model, with v the number of output variables, n the number of frequency steps, and rt as well as rm the number of repetitions. It is assumed that the matrices have a normal distribution. With this assumption, the mean value T ∈ Rv×n can be formed over the number of repeated measurements. The same applies to M. These are shown in Fig. 2.4. Now, according to Zhan and Fu [9], the diagonal matrix of the repetition error of the experimental tests   t = diag ∗1t , ∗2t , . . . , ∗vt are defined.

(2.8)

2 Bayesian Uncertainty Quantification in the Development of a New Vibration Absorber Technology

23

start

collect and normalize data

quantify test data variability

calculate mean curves of test data

calculate mean curves of model data

quantify model data variability

calculate difference curves of test data calculate the variance and add data variability Bayesian interval hypothesis testing

select threshold

no

acceptable? yes stop

1.5

/

1

1

0.5

TRANSMISSIBILITY /

1.5 experiment model A model B model C

1

0.5

FORCE /

1.5

PHASE

Fig. 2.3 Procedure of Bayesian uncertainty quantification

0

0 0

10

20

FREQUENCY

0 0

30

0.5

in Hz

10

20

FREQUENCY

30 in Hz

0

10

20

FREQUENCY

30 in Hz

Fig. 2.4 Analytical, simulation and experimental results. Transmissibility (left), phase shift (middle) and force (right) normalized with occurring maximum of experimental test data Vmax, T , ψ max, T , F max, T

For this purpose, the deviation of the repeated measurements from the mean value is calculated, the error matrix reshaped and the mean value for each output variable ∗v formed and listed in the diagonal matrix. The same applies to the diagonal matrix of the repetition error of the model results. The next step is to create the difference matrix D = T − M between the mean matrix of the tests T and the model M. The covariance  D

1 (di − μD ) (di − μD )T n n

=

i=1

(2.9)

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N. Brötz and P. F. Pelz

is formed from the difference matrix D. μD are the mean values of the output variables over the individual frequency steps of the difference matrix D. The covariance now becomes the diagonal matrix of the repetition error of the experimental tests as well as of the model and one obtains the uncertainty of the tests  D ∗ =  D + Λt + Λc .

(2.10)

T  The individual lines yd of the difference matrix D = y d1 y d2 y d3 , thus the deviation of the respective output quantity, as well as the uncertainty Σ D ∗ of the experimental and simulative tests are used for the Bayesian interval hypothesis testing. In the framework of the Bayesian interval hypothesis approach, the null hypothesis H0 is defined by H0 : |yd | ≤ ε and the alternate hypothesis H1 by H1 : |yd | ≥ ε. The null and alternate hypothesis require a threshold



ε = b μe .

(2.11)

This is formed with the threshold factor b and μe  the maximum norm of the lines of the mean matrix of the tests T . The threshold factor is defined according to Zhan and Fu [9] by the model user and is assumed here to be 15%. The likelihood according to Jiang and Mahadevan [4] of the zero hypothesis p (yd |H0 ) =

n  n=1

 " " 1 T −1 " "  y exp −  ∗ y d , if y d ≤ ε d D 2 (2π )n | D ∗ | 

1

(2.12)

implies the probability that the difference between experiment and model is within threshold limits. In contrast to this is the likelihood of the alternate hypothesis n     p y d |H0 = n=1

 " " 1 T −1 exp − y d  D ∗ y d , if "y d " ≥ ε, n | ∗ | 2 (2π ) D 

1

(2.13)

whereby the difference lies outside the threshold here. The division of the two values is described as Bayes factor [11] B=

p (yd |H0 ) . p (yd |H1 )

(2.14)

The Bayes factor can be used to calculate the model confidence M=

B , B +1

(2.15)

which can be used to make a statement about the validation.

2.9 Model Confidence for Model A, B and C The model confidence of Model A is 2% what means that the analytical model does not show good agreement with the measured values. So, the next model is selected and the process is repeated. With model B, the confidence is already at 46%, not acceptable yet. Only the extended model C provides a confidence of 84%. The threshold plays a decisive role for the confidence but is not the focus of this paper, please refer to Mallapur and Platz [10]. For a better understanding, the result for the force amplitude is shown graphically in Fig. 2.5. The individual points represent the repeated simulation measurements of the models A, B and C over the individual excitation frequencies. It can be seen that there are no large deviations in comparison to the course over the frequency. Thus, the data uncertainty is small. The model uncertainty is shown when the threshold lines are exceeded, which can be seen in the analytical model A outside the interval of 2.5 to 10 Hz. The deviation of the model over a wide frequency range is larger than the pre-set threshold and thus the confidence is very low. The same applies to Model B. Model C is within the threshold over a wide frequency range and thus shows a high confidence. The most important point here is the model boundary, which has been adjusted in Model C. It shows that the choice of the model boundary plays an important role for the confidence analysis.

2 Bayesian Uncertainty Quantification in the Development of a New Vibration Absorber Technology

25

1.2

1

FORCE /

0.8

0.6

threshold experiment model A model B model C

0.4 0.2 0 0

5

10 FREQUENCY

15

20

25

in Hz

Fig. 2.5 Graphical presentation of the model confidence result

2.10 Conclusion In this contribution, the Bayesian interval hypothesis-based method was used to validate a novel vibration absorber. Three models are available to illustrate the function of the vibration absorber. A very simple analytical model, an axiomatic model with non-linear damping and a model with a model boundary adapted to the experiment. The respective model uncertainty is evaluated with the confidence model and only the third model shows an acceptable confidence. The precise design of the model boundary in such an investigation is therefore particularly important. It often happens in the experimental setup that measured variables cannot be measured at the location intended in the model. What remains open and is investigated in future work is how this uncertainty at the model boundary should be incorporated into the design of the model for optimisation calculations. Acknowledgements Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Projektnummer 57157498 – SFB 805. The authors especially would like to thank the project cooperation partner ZF Friedrichshafen AG for supporting this project.

References 1. Hedrich, P., Brötz, N., Pelz, P.F.: Resilient product development – a new approach for controlling uncertainty. AMM. 885, 88–101 (2018) 2. Kennedy, M.C., O’hagan, A.: Bayesian calibration of computer models. J. R. Stat. Soc. Ser. B. 63(3), 425–464 (2001) 3. Roy, C.J., Oberkampf, W.L.: A comprehensive framework for verification, validation, and uncertainty quantification in scientific computing. Comput. Methods Appl. Mech. Eng. 200(25–28), 2131–2144 (2011) 4. Jiang, X., Mahadevan, S.: Bayesian wavelet method for multivariate model assessment of dynamic systems. J. Sound Vib. 312(4–5), 694–712 (2008) 5. Brötz, N., Hedrich, P., Pelz, P.F.: Integrated fluid dynamic vibration absorber for mobile applications. In: 11th International Fluid Power Conference (11th Ifk), pp. 14–25, Aachen (2018) 6. Corneli, T., Pelz, P.F.: Employing hydraulic transmission for light weight dynamic absorber. In: 9th International Fluid Power Conference (9th Ifk), pp. 198–209, Aachen (2014) 7. Den Hartog, J.P.: Mechanical Vibrations. Dover Publications, New York (1985) 8. Markert, R.: Strukturdynamik: Skript Zur Vorlesung “Strukturdynamik Für Maschinenbauer”, 1st edn. Technische Universität, Fachgebiet Strukturdynamik, Darmstadt (2011)

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9. Zhan, Z., Fu, Y., Yang, R.-J., Peng, Y.: Bayesian based multivariate model validation method under uncertainty for dynamic systems. J. Mech. Des. 134(3), 2408 (2012) 10. Mallapur, S., Platz, R.: Quantification of uncertainty in the mathematical modelling of a multivariable suspension strut using Bayesian interval hypothesis-based approach. AMM. 885, 3–17 (2018) 11. Bailer-Jones, C.A.L.: Practical Bayesian Inference. Cambridge University Press, Cambridge (2017) Nicolas Brötz earned his Master’s degree from the TU Darmstadt, graduating in 2017. Since then he has been working as a research assistant at the Institute of Fluidsystems. His project deals with the investigation of a hydraulically translated absorber. This project is located in the Collaborative Research Centre 805.

Chapter 3

Comparison of Complexity Measures for Structural Health Monitoring Hannah Donajkowski, Salma Leyasi, Gregory Mellos, Chuck R. Farrar, Alex Scheinker, Jin-Song Pei, and Nicholas A. J. Lieven

Abstract The field of structural health monitoring (SHM) applies damage detection techniques to provide timely in-situ system condition assessment. Previously, researchers have suggested a fundamental axiom for SHM that states, “damage will increase the complexity of a system.” One way this increased complexity can manifest itself is in the increased complexity of sensor data recorded from the structure when damage occurs. The question then becomes how to best quantify the increase in complexity of those data. Information complexity is one such approach and within this framework various information entropy quantities have been proposed as measures of complexity. The literature has shown that there are multiple information entropy measures, including; Shannon Entropy, Rényi Entropy, Permutation Entropy, Sample Entropy, Approximate Entropy, and Spectral Entropy. With multiple measures proposed to quantify information entropy; a study to compare the relative effectiveness of these entropy measures in the context of SHM is needed. Therefore, the objective of this paper is to compare the effectiveness of entropy-based methods in distinguishing between “Healthy” and “Unhealthy” labeled datasets. The labeled datasets considered in this study were obtained from a 4DOF impact oscillator, a rotating machine with a damaged bearing, and an impact oscillator excited by a rotating machine. Furthermore, two methods were used in this study to classify the results from the different entropy measures; Naïve-Bayes classification, and K-means clustering. Effectiveness of a given entropy measure is determined by the number of misclassifications produced when compared to the true labels. The analysis showed that entropy measures obtained from data corresponding to sensors closer to the damage source had fewer misclassifications for the datasets tested. For the datasets considered in this study, the researchers concluded that each dataset had a different most effective entropy measure. The study would need to be expanded to include other classification methods and other datasets to define more precisely which entropy measure is the most effective in identifying the increase in complexity associated with damage and, hence, distinguishing between healthy and damaged data. Keywords Structural health monitoring · Vibration · Entropy methods · Damage detection · Thresholding · Rotating machinery · Structural dynamics

H. Donajkowski Department of Mechanical Engineering-Engineering Mechanics, Michigan Technological University, Houghton, MI, USA e-mail: donajkowski\[email protected] S. Leyasi School of Biological and Health Systems Engineering, Arizona State University, Tempe, AZ, USA e-mail: [email protected] G. Mellos School of Engineering, University of California Merced, Merced, CA, USA e-mail: [email protected] C. R. Farrar · A. Scheinker Los Alamos National Laboratory, Los Alamos, NM, USA e-mail: [email protected]; [email protected] J.-S. Pei University of Oklahoma, Norman, OK, USA e-mail: [email protected] N. A. J. Lieven () University of Bristol, Bristol, UK e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2020 Z. Mao (ed.), Model Validation and Uncertainty Quantification, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47638-0_3

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3.1 Introduction Structural health monitoring (SHM) is the process of identifying damage in near real time for various engineering systems such as aerospace, civil, and mechanical engineering infrastructure [1]. Most SHM approaches are based on the assumption that damage can be identified through changes in the measured dynamic response properties of the system [2]. Damage initiation and propagation cause changes in the geometric and material properties of the system, which in turn produce changes in vibration amplitude and frequency content [1]. In order to identify how damage affects the physical properties of a system, data from an undamaged system and a damaged system are required for comparison. Typically, this comparison is based on some damage-sensitive features that are extracted from the respective data sets. Once it is shown that appropriate features have been selected, they can then be used to predict damage for similar systems in the future. An axiom of SHM states: when damage is introduced to a system, the complexity of the system increases [3]. There is a diverse set of complexity measures that have been proposed for various natural and man-made systems. One definition of system complexity is the degree of difficulty in describing, organizing, or recreating the specific system [4]. Additionally, the complexity of a system can be described by the number of components and interconnections [5]. In the context of damage detection, there are three different forms of complexity to be considered: geometric complexity, material complexity, and information complexity. Because of the difficulty of measuring geometric and material complexity on in-service engineering systems, information complexity, which is a measure of how much information is needed to describe the behavior of a system [6] is typically used for SHM applications. Entropy is a complexity measure that describes the uncertainty in the next outcome of a random process. Previous research has shown various entropy measures have been effective for damage detection. Some entropy measures that have been applied in SHM studies are approximate entropy, Shannon entropy, Rényi entropy, spectral entropy, sample entropy, permutation entropy, and mutual information. Shannon entropy has been used for structural damage detection in bridges [7]. Chai et al. used a parameter based on the Shannon entropy to monitor fatigue crack growth in a CrMoV steel sample and a crack formed in a 316LN stainless steel beam during a three-point bending test [8]. Furthermore, the Shannon approach has also been adopted successfully by Ameri et al. [9]. Camarena-Martinez et al. proposed a method to detect broken rotor bar (BRB) faults in induction motors using Shannon entropy and clustering using a K-means algorithm [10]. The Rényi entropy was introduced as a generalization of the Shannon entropy, but it has not been used in SHM applications [11]. Spectral entropy as defined by Powell and Percival [12] has been more widely used for SHM, when compared to other entropy measures. Ceravolo, R. et al. used spectral entropy to detect damage in masonry buildings that were subject to mild earthquakes [13]. Additionally, spectral entropy has been used to monitor the degradation of roller bearings [14, 15], study the onsets of nonlinearities caused by simulated damage in a four-story structure [16] and to determine the health of a composite plate [17]. Approximate entropy [18] has been shown to work as a diagnostic tool for structural health monitoring. Yan and Gao demonstrated approximate entropy as an effective method to monitor the health of an electrical motor and rotating bearing systems [19]. To improve the computation efficiency of the approximate entropy, Richman and Moorman [20] created the sample entropy; which ignored self-similar patterns in the data when computing the entropy of the system. Multi-scale crosssample entropy extends the concept of sample entropy to the measurement of similarity between two different time series. An example of cross-sample entropy used in SHM comes from Lin and Liang, who used it to determine damage location in test structures [21]. Previous work has used mutual information to monitor the structural health of a bridge while in operation. This approach has been enacted by calculating the time-delayed mutual information, which is the mutual information between consecutive, time-delayed, windowed sections of the continuous time series signal from onboard sensors [22]. Despite the fact that different entropy measured have been proposed for SHM, each of the studies reported in the literature uses a single entropy measure to determine the health of a structure. While this method has produced credible results, understanding how different entropy measures performed on similar datasets would be beneficial. In this study, different entropy measures will be compared using data from three different experiments; an impact oscillator, a rotating machine with a damaged bearing at multiple speeds, and an impact oscillator excited by the vibration from a rotating machine. Specifically, the Shannon, sample, spectral, permutation, Rényi and approximate entropies along with the mutual information will be compared. The performance of the various measures will be assessed by the relative classification error when they are applied to data from known damage conditions using either a Naïve Bayes or K-mean classifier.

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3.2 Theoretical Background 3.2.1 Shannon, Rényi, and Spectral Entropy Shannon entropy is the statistic of a probability distribution function that measures the tendency of a random variable to approach a uniform distribution. It was developed in the dawn of computer science to determine how many bits were needed to quantify information [6]. The Shannon entropy is a common metric for studying the complexity of a data set. Shannon entropy, H(x), measures how uncertain the next outcome is for a set of N random variables. N H (x) = − pi (x) ∗ log2 (pi (x)) i=1

(3.1)

where x is the random variable and p(x) is the probability density function (pdf) of random variable x. For a uniform distribution there is complete uncertainty as to what value the next realization will take on and the Shannon entropy will take on a high value in this case. The Rényi entropy is a generalization of the Shannon Entropy, where the entropy order of is defined. It characterizes the amount of information that is needed to specify a realization of a random process with a prescribed precision when the pdf for that random variable is known. The entropy order affects the pdf by scaling the probabilities to emphasis high probability values. An entropy order of 0 represents the maximum entropy for a given discrete variable and an entropy order of 1 is the Shannon Entropy. As the entropy order (α) increases, differences in probabilities are amplified. ! N 1 α RE α (x) = log pi (x) i=1 1−α

(3.2)

Spectral entropy is an application of the Shannon entropy where a normalized power spectral density function replaces the pdf in the entropy calculation. The normalized power spectral density function, pi (f ), is created by dividing the energy of a frequency band by the total energy of the signal, to produce a function with unit area as given in Eq. 3.3. f (i) pi (f ) = #N J =1 f (i)

(3.3)

Where f (i) is the distribution of frequencies of signal X and pi (f ) is the probability distribution function of those frequencies. The spectral entropy is then calculated by [14]: N SE(f ) = − pi (f ) ∗ log (pi (f )) i=1

(3.4)

3.2.2 Approximate and Sample Entropy Approximate entropy (ApEn) quantifies the regularity of time series data by determining the unpredictability in pattern observations [18]. Higher values for approximate entropy indicate a more complex system. It reflects the likelihood that observations will not be followed by similar observations. Given a time series, S, made up of N data points, SN = {x(1), x(2) . . . x(N)}, the approximate entropy of the time series can be calculated using the following process: The time series is broken up into a series of vectors as follows: X(1) = {x(1), x(2) . . . x(m)} X(2) = {x(2), x(3) . . . x (m + 1)} X (N − m + 1) = {x (N − m + 1) , x( N − m + 2 }. . . x(N )}

(3.5)

Where m is the length of the vector. The difference between two vectors X(i) and X(j) is described as their maximum corresponding difference when i = 1, 2, 3 . . . N − m + 1 and j = 1, 2, 3 . . . N − m + 1.

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d (X(i), X(j )) = maxk=1,2...m (|x (i + k − 1) − x (j + k − 1) |)

(3.6)

Each vector is compared with all others to determine the maximum differences that meet the set r, where r is the criterion of similarity, value in all cases of i and j. The similarity between each vector and all other vectors is calculated as  N −m+1 1 1, x ≥ 0  {r − d [X(i), X(j )]} Where  {x} = (3.7) Cim (r) = i=1 0, x < 0 N − (m − 1) Then by defining φ m (r) =

   1 ln Cim (r) , i N −m+1

i = 1, 2, . . . .N − m + 1

(3.8)

The approximate entropy for discrete time-series data can be calculated as ApEn (m, r, N) = φ m (r) − φ m+1 (r)

(3.9)

The sample entropy is a modification of the approximate entropy developed by Richman and Randall [24]. The difference between the approximate and sample entropies is the sample entropy does not consider self-similar patterns when comparing vector groups. The difference between two vectors is described by their maximum corresponding difference (3.5) when i = 1, 2, 3 . . . N − m and j = 1, 2, 3 . . . N − m. Each vector is compared to all other to determine when the maximum difference is less than r, for all cases, as in Eq. 3.6, except when i = j. However, in the case of sample entropy φ m (r) is defined as: φ m (r) =

1 N −m m Ci (r), i=1 N −m

where i = 1, 2, . . . .N − m

Sample entropy can then be calculated as SampEnt (m, r, N) = − ln

φ m+1 φm

(3.10)

3.2.3 Permutation Entropy Permutation entropy (PE) looks for patterns in the data. A monotonic function will yield an entropy of zero. Systems with no perceivable pattern will yield a higher entropy. The normalized PE is derived below, where π i is the frequency of each ordinal pattern in the time series over the number of ordinal patterns present. P E m,norm =

−1 m! πi ∗ log2 (πi ) i=0 log2 (m!)

(3.11)

Both Zhao et al. [23] and Bandt and Pompe [24] described examples that are helpful in understanding permutation entropy.

3.2.4 Mutual Information The mutual information between two signals measures how much information can be inferred from one signal using the information from the second signal [25], shown in Eq. 3.12. !   pXY (x, y) (3.12) pXY (x, y) ln I (X; Y ) = y x pX (x) ∗ pY (y) Where X and Y are the signals, pXY (x, y) is the joint probability distribution of signal X and Y. pX (x) and pY (y) are the marginal probability distributions.

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3.2.5 Naïve Bayes Classifier The Naïve Bayes classifier uses conditional probability to classify data into groups; for this study two groups are used. The data are classified based on a series of feature vectors; each feature vector is assumed to be independent. Two normal distributions are then mapped to the data in the feature vector. A training data set is used to create the model for the structure, where the entropy vector is the feature vector. An evaluation data set is then used to determine the errors associated with the model; error is defined as the number of misclassifications from the labelled dataset.

3.2.6 K-Means Clustering The K-Means clustering algorithm is an iterative technique that sorts data into a predefined number of groups according to their distance from the cluster centroids. For structural health monitoring purposes, we anticipate two groups, “Healthy” and “Unhealthy.” For this analysis, the two centroids are given the initial positions of the minimum and maximum values from each set of complexity values. These constraints help to ensure that the data do not converge on local minima. For each iteration, the city block distance between a data point and each centroid is calculated. The city block distance is the sum of the absolute value of their Cartesian coordinates. The data point is assigned to the cluster with the closer centroid. After all the points are assigned to a cluster, the centroids move to the average value of the data assigned to its cluster. This process continues until a minimum movement criterion is met. K-Means then outputs a list associating each data point with a cluster. However, since the K-Means algorithm is unsupervised, an additional step is required to assign labels of “Healthy” or “Unhealthy” to each cluster. The indices associated with cluster 1 are compared with the true labels of the data with the same indices. Whichever cluster matches its associated true labels more closely is given the that label. For example, if cluster 1 most closely corresponds to the group “Healthy,” cluster 1 is renamed the “Healthy” cluster.

3.3 Experimental Design Three different mechanical structures were used to simulate common types of damage: damage that results in nonlinear system response (analogous to crack opening and closing) and defective bearing in a rotating machine. Nonlinearities in a system can be an indication of damage if the undamaged system exhibits linear response characteristics. The procedures used to collect “Healthy” and “Unhealthy” labelled data from each of the test structures are described in the following sections.

3.3.1 Bearing Defects in a Rotating Machine Bearing faults in rotating machinery are commonly analyzed in the SHM context. A damaged bearing will change the vibration response of the bearing housing, which can be monitored with an accelerometer. The Spectra Quest Machinery Fault Simulator (MFS) was used to simulate a bearing failure in rotating machinery. Tri-axial accelerometers were mounted on each bearing housing and secured to the structure using wax. Accelerometer measurements in the y- and z-directions (orthogonal to the axial direction) were collected using a National Instruments DAQ system at a sample rate of 1000 Hz. Figure 3.1 shows the fault simulator testing setup. Tests were conducted on the MFS to simulate damaged and healthy systems, at varying motor speeds. The first test set has a healthy bearing in both bearing housings. The motor is run at 1200, 2400, and 3600 RPM, for 60 s per test. The second test set replaces the healthy bearing farthest from the motor with a bearing that simulates a combination of inner race defects, outer race defects, and ball faults. The motor is again run at 1200, 2400, and 3600 RPM, for 60 s per test. The data collected from this experiment was analyzed using the complexity measurements defined earlier. The results of this analysis are presented in Sect. 3.4.

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Fig. 3.1 Spectra Quest Machinery Fault Simulator. Bearing Housing labels and axis labels are included. Channel 1 and 2 measures accelerometer data in the y-axis and z-axis on bearing 1. Respectively. Channel 3 and 4 measures accelerometer data in the y-axis and z-axis on bearing 2. Respectively

3.3.2 Four-Degree of Freedom Impact Oscillator Nonlinearities introduced to an initially linear system are an indication of damage [26]. The most common type of nonlinear damage is caused by cracks that open and close under different loading conditions [26]. A dataset from a four-degree of freedom (4DOF) system was used [27]. The system is a three-story structure, consisting of four aluminum plates (30.5 × 30.5 × 2.5 cm) and 12 aluminum columns (17.7 × 2.5 × 0.6 cm) with bolted joints and a rigid based. The structure forms a system with four degrees of freedom. Nonlinear behavior was introduced to the system through a center column (15.0 × 2.5 × 2.5 cm) that was suspended from the top floor that came into contact with a bumper on the second floor. The bumper’s position could be adjusted. The structure was only allowed to move in the x-direction due to rails on the base (Fig. 3.2). The structure was excited through an electromagnetic shaker mounted on the base plate (76.2 × 30.5 × 2.5 cm). Accelerometers were attached to each floor to measure the response. Data was collected at a sampling frequency of 322.58 Hz (sampling interval of 3.1 ms) with a frequency resolution of 0.039 Hz. A band-limited random signal excited the structure from 20 Hz to 150 Hz. Data from the structure was analyzed in its baseline condition when the bumper and suspended column are members of the structure. Additionally, data was analyzed from five other conditions, at different lengths of the gap between the hanging column and the bumper (0.20, 0.15, 0.13, 0.10, and 0.05 mm). These data sets were considered to determine how the introduction of nonlinearities effects the complexity measurements.

3.3.3 Impact Oscillator with Bearing Damage A combination of the previous experiments was performed to see how coupling two structures affects the vibration response. A cantilever beam was mounted to the MFS; both bearings in the are the baseline healthy bearings used in the experiment described in Sect. 3.1. Figure 3.3 shows the experimental setup for the hybrid structure experiment. The MFS was used to excite the cantilever beam at its resonance frequency. The MFS ran for 60 s at a speed of 714 RPM. Three accelerometers were mounted to the testing rig; one normal to bearing housing 1, one normal to bearing housing 2, and one normal to the free end of the cantilever beam. A 22.6 gram-mass was added the motor’s shaft to cause the shaft to become out of balance, increasing the stimulation to the cantilever beam. After a baseline test for both healthy and unhealthy

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Fig. 3.2 Four-degree of freedom test structure. The labels for each sensor is shown. Channel 1 is a force transducer to measure the shakers input into the system. Channels 2–5 measure uni-axial accelerometer data from each of the floors

Fig. 3.3 MFS and impact oscillator experimental setup. Channel 1 and 2 measure accelerometer data in the z-axis on both of the bearings respectively. Channel 3 measures uni-axial accelerometer data from the cantilever beam

bearing states, a bumper is introduced underneath the cantilever beam at distance of 168.49 ± 0.2166 mm. The introduction of the bumper will cause impacts between the cantilever beam and the bumper, resulting in a nonlinear response. The test was repeated for three different types of bumpers (soft, hard, and metallic) with both healthy and unhealthy bearings.

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Table 3.1 Parameters used to calculate the complexity measures Parameter Symbol Value Window size N/A 8192 Embedding dimension m 2 5 Tolerance R 0.2*standard deviation Bins per window N/A Sturges’ bin size formula (log2 (length(data))+1) Entropy order

α

2

Relevant complexity measures Justification All Data that was available Approximate, sample Permutation Approximate, sample Shannon, Rényi Optimal bin size to capture changing entropy Rényi

3.4 Data Analysis 3.4.1 Applying Complexity Measures to Time Series Data The rotating machine experiment had two tri-axial accelerometers, one per bearing housing that collected acceleration data from the given structures. The 4DOF structure had a force sensor mounted on the first floor closer to the stinger shaker rod, with a single accelerometer on each of the floors of the structure. The data was labeled “Healthy” or “Unhealthy”; the “Unhealthy” label was applied to the simulated damage data. The complexity measures described earlier were then applied to the time series to create a complexity vector for every sensor from the experiments. The complexity values can be influenced by user-defined input parameters; these parameters are: the embedding dimension, number of histogram bins, tolerance value, and entropy order. Table 3.1 defines the parameters and their adopted values used to calculate the complexity measures in this experiment and the complexity measures that are affected by these parameters.

3.4.2 Performing K-Means Clustering on the Labeled Complexity Measurements and Calculating Error The complexity values are then subjected to the K-Means clustering algorithm, calculated using the K-Means command in Matlab’s Statistics and Machine Learning Toolbox. The K-Means clustering label vector is then compared to the original labels from the experiment. The comparison between these two label vectors is done by creating a confusion matrix for each complexity vector. The confusion matrix measures how many correct and incorrect identifications there are between the true labels and the predicted labels [28]. An incorrect identification could be either a “Healthy” point labeled as “Unhealthy” or the reverse. From the confusion matrix, an error value is calculated by dividing the number of incorrect identifications by the total number of data points. The calculated error is then used to rank the efficiency of each complexity measurement.

3.5 Results and Discussions The total misclassification error for each complexity measurement for each experimental sensor are placed into a comparison table. The error tables for each classification method and system are shown in the following sub-sections.

3.5.1 Bearing Defects in Rotating Machine The error comparison charts show similar trends for both K-Means clustering and Naïve-Bayes classification. In this experiment data from when both of the bearings were healthy was labeled the “healthy” dataset. The data from once a combination fault bearing was introduced in bearing 2 was labeled the “Unhealthy” dataset. Channel 1 and Channel 2 (Bearing 1 y-axis and Bearing 2 z-axis) showed the least error in classification and clustering for all motor speeds (Tables 3.2 and 3.3). A trend among both K-Means clustering and Naïve-Bayes classification is that

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Table 3.2 Error charts for Naive-Bayes classification for the bearing defects in rotating machine experiment at (a) 3600 RPM, (b) 2400 RPM, (c) 2400 RPM

both permutation and spectral entropy had the least error in all motor speeds. The sample and approximate entropy produced high errors at lower speeds, but at 3600 RPM both measurements on bearing 2 along the y-axis produced useful findings.

3.5.2 Four-Degree of Freedom Impact Oscillator From the data used in this study, only the gap distance of 50 microns was labeled as damaged, with the other gap distances labeled as healthy. The 50 micron dataset was selected for the damaged label because, the severity of the nonlinearities was greatest for this gap distance. In the 4DOF structure experiment, the complexity measurements that worked best at identifying damage: spectral, permutation, and approximate entropy, could identity damage from the accelerometers on the third and fourth floors (channel 4 and 5) of the structure. On lower levels of the structure, the same complexity measures performed worse. Since the nonlinearity was introduced between the third and fourth floors of the structure, sensors closer to the nonlinearity should have better damage detection than more distant sensors. Sample, spectral, permutation and approximate entropy did the best in separating between damaged and healthy data sets (Tables 3.4 and 3.5).

3.5.3 Impact Oscillator with Bearing Damage In this experiment, there were eight different “Healthy”/“Unhealthy” states in the data: healthy, nonlinear damage-soft bumper, nonlinear damage-hard bumper, nonlinear damage- metal bumper, bearing damage, nonlinear damage-soft bumper with bearing damage, nonlinear damage-hard bumper with bearing damage, nonlinear damage-metal bumper with bearing

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Table 3.3 Error charts for K-Means clustering for the bearing defects in rotating machine experiment at (a) 3600 RPM, (b) 2400 RPM, (c) 1200 RPM

Table 3.4 Naive-Bayes classification error for the four-degree of freedom impact oscillator

Table 3.5 K-Means clustering error for the four degree of freedom impact oscillator

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Table 3.6 Naive-Bayes classification error for impact oscillator with bearing damage

Table 3.7 K-Means clustering error for impact oscillator with bearing damage

damage. The error charts shown in Tables 3.6 and 3.7 show the clustering/classification error on separating the data from the eight different states. Baïve-Bayes and K-Means clustering separation performed the best in differentiating data in all complexity measures from channel 3, beam accelerometer. This channel showed a direct response from the nonlinear impact introduced to the system. Approximate and sample entropy were able to separate the data the best in all channels.

3.6 Discussion The error comparison charts showed different results. The Shannon and Réyni entropy performed similarly for each experiment, and both produced high errors relative to the other measurements. The spectral entropy performed well using data from sensors closer to the simulated damage locations. Overall the permutation entropy did well in distinguishing between healthy and damaged data. Similarly, to spectral entropy, permutation entropy’s effectiveness depended on the physical location of the sensor relative to the damage site. The approximate entropy performed better than the sample entropy approach on the 4DOF impact oscillator. For the rotating machine experiment the sample and approximate entropy methods produced high errors at lower speeds, but at 3600 RPM both measurements on bearing 2 along the y-axis produced useful findings. The mutual information approach for the rotating machine experiments and the 4DOF structure produced high errors. Since the mutual information produced high classification errors across multiple classifiers and speed, the sensors used could not be used to predict damage. The error comparison charts showed two key findings; the sensor location can affect the sensitivity of these measurements, and spectral and permutation entropy seem to perform better for detecting damage on average. This observation can also be seen in the rotating machine experiments. For all speeds, the accelerometer mounted on the bearing 2 housing showed a lower error than the accelerometer located on bearing housing 1. Moreover, the accuracy of the spectral and permutation entropies increased when these complexity measurements were performed using the accelerometer on bearing 2. Across the experiments, spectral and permutation entropy showed to be effective methods for distinguishing between the healthy and damaged datasets. The spectral entropy uses the power spectral density of the signal. The initial hypothesis is that damage in structures can cause a nonlinear response in structures [1]. The introduction of nonlinearities creates side band excitation in the frequency domain, which changes the pdf of the frequency response. These changes to the pdf will change the spectral entropy value for healthy and unhealthy structures. As defined previously, the permutation entropy identifies frequency of patterns occurring in time series data. When damage is introduced into a system, the dynamic response of the structure changes, which would change the frequency patterns present in the time series. These changes in time series patterns, allow for easier identification of healthy and damaged permutation entropy values.

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3.7 Conclusion For the datasets used in this study, different complexity measures have been applied to distinguish between health and unhealthy labeled datasets. Spectral entropy and permutation entropy had lower classification errors on the tested datasets. The complexity measurement presented in this research were a subset used in the SHM literature. Future work will compare a larger group of entropy measures to determine the relative effectiveness of complexity measures for a given structure. Moreover, with different sensors providing different complexity measurements that can best identify damage, determining the optimal sensor location on a structure will aid in damage detection. By identifying which complexity measurements perform best for common SHM structures, damage detection methods can be improved by using the most effective entropy value.

References 1. Farrar, C.R., Worden, K.: An Introduction of Structural Health Monitoring CISM Courses and Lectures, pp. 1–17. Springer, New York (2011) 2. Doebling, S.W., Farrar, C.R., Prime, M.B., Shevitz, D.W.: Damage Identification and Health Monitoring of Structural and Mechanical Systems from Changes in Their Vibration Characteristics: A Literature Review LA-13070-MS, Los Alamos. United States Department of Energy, Washington, DC (1996) 3. Farrar, C.R., Worden, K.: Structural Health Monitoring. Chichester, Wiley (2012) 4. Llyod, S.: Measures of complexity: a Nonexhaustive list. IEEE Control. Syst. Mag. 21(4), 7–8 (2001) 5. Min, B.-K., Chang, S.H.: System complexity measure in the aspect of operational difficulty. IEEE Trans. Nucl. Sci. 38(5), 1035–1040 (1991) 6. Shannon, C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27(3), 379–423 (1948) 7. Bao, Y., Li, H.: Application of Information Fusion and Shannon Entropy in Structural Damage Detection. SPIE Smart Structures and Materials + Nondestructive Evaluation and Health Monitoring, San Diego (2007) 8. Chai, M., Zhang, Z., Duan, Q.: A new qualitative acoustic emission parameter based on Shannon’s entropy for damage monitoring. Mech. Syst. Signal Process. 100, 617–629 (2017) 9. Amiri, M., Modarres, M., Droguett, E. L.: AE entropy for detection of fatigue crack initiation and growth. In: 2015 IEEE Conference on Prognostics and Health Management (PHM), Austin, TX, pp. 1–8 (2015). https://doi.org/10.1109/ICPHM.2015.7245038 10. Camerena-Martinez, D., Valtierra-Rodriguez, M., Amezquiita-Sanchez, J.P., Granados-Lieberman, D., Romero-Troncoso, R.J., Garcia-Perez, A.: Shannon entropy and K-means method for automatic diagnosis of broken rotor bars in induction motors using vibration signals. Shock Vib. 2016, (2016) 11. Reyni, A.: On measures of entropy and information. In: Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability. 547–561 (1961) 12. Powell, G.E., Percival, I.C.: A spectral entropy method for distinguishing regular and irregular motion of Hamiltonian systems. J. Phys. A Math. Gen. 12, 2053–2071 (1979) 13. Ceravolo, R., Lenticchia, E., Miraglia, G.: Use of spectral entropy for damage detection in masonry buildings in the presence of mild seismicity. In: International Conference on Experimental Mechanics, Brussels, Belgium, 2018 14. Pan, Y.N., Chen, J., Li, X.L.: Spectral entropy: a complementary index for rolling element bearing performance degradation assessment. J. Mech. Eng. Sci. 223, 1223–1231 (2009) 15. Yu, H., Li, H., Xu, B.: Rolling bearing degradation state identification based on LCD relative spectral entropy. J. Fail. Anal. Prev. 16, 655–666 (2016) 16. West, B. M., Locke, W. R., Andrews, T. C., Scheinker, A., Farrar, C. R.: Applying concepts of complexity to structural health monitoring. In: IMAC 2019 (2019) 17. Castro, E., Moreno-Garcia, P., Gallego, A.M.N.: Damage detection in CFRP plates using spectral entropy. Shock Vib. 2014, (2014) 18. Pincus, S.M.: Approximate entropy as a measure of system complexity. Proc. Natl. Acad. Sci. USA. 88(6), 2297–2301 (1991) 19. Yan, R., Gao, R.X.: Approximate entropy as a diagnostic tool for machine health monitoring. Mech. Syst. Signal Process. 21, 824–839 (2005) 20. Richman, J.S., Randall, M.J.: Physiological time-series analysis using approximate entropy and sample entropy. Am. J. Physiol. 278, (2000) 21. Lin, T.-K., Liang, J.-C.: Application of multi-scale (cross-) sample entropy for structural health monitoring. Smart Mater. Struct. 24(8), (2015) 22. Zhou, J., Yang, J.: Analysis on non-linear characteristics of bridge health monitoring based on time-delayed transfer entropy and mutual information. In: Fifth International Joint Conference on INC, IMS, and IDC (2009) 23. Zhao, L.-Y., Wang, L., Yan, R.-Q.: Rolling bearing fault diagnosis based on wavelet packet decomposition and multi-scale permutation entropy. Entropy. 17, 6447–6461 (2015). https://doi.org/10.3390/e17096447 24. Bandt, C., Pompe, B.: Permutation entropy: a natural complexity measure for time series. Phys. Rev. Lett. 88(17), (2002) 25. Bishop, C.M.: Pattern Recognition and Machine Learning. Springer, New York (2006) 26. Farrar, C.R., Worden, K., Todd, M.D., Park, G., Nicholas, J., Adams, D.E., Bement, M.T., Farinholt, K.: Nonlinear System Identification for Damage Detection. Los Alamos National Laboratory, Los Alamos (2007) 27. Figueiredo, E., Park, G., Figueiras, J., Farrar, C.R., Worden, K.: Structural health monitoring algorithm comparisons using standard data sets. Los Alamos National Labs Report. LA-14393, (March 2009) 28. Worden, K., Manson, G.: The application of machine learning to structural health monitoring. Philos. Trans. R. Soc. A: Math. Phys. Eng. Sci. 365(1851), (2006)

3 Comparison of Complexity Measures for Structural Health Monitoring

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Hannah Donajkowski, Salma Leyasi, and Gregory Mellos attended the 2019 Los Alamos Dynamics Summer School. The summer School combines lectures in the field of Structural health monitoring alongside a dedicated 10 week research project. The research under the supervision of Dr Chuck Farrar, Dr Alex Scheinker, Prof Jin-Song Pei and Prof Nicholas Lieven is the result of the project.

Chapter 4

Selection of an Adequate Model of a Piezo-Elastic Support for Structural Control in a Beam Truss Structure Jonathan Lenz, Maximilian Schäffner, Roland Platz, and Tobias Melz

Abstract Axial and lateral loads of lightweight beam truss structures e.g. used in automotive engineering may lead to undesired structural vibration that can be reduced near a structural resonance frequency via resonant piezoelectric shuntdamping. In order to tune the electrical circuits to the desired structural resonance frequency within a model-based approach, an adequate mathematical model of the beam truss structure is required. Piezo-elastic truss supports with integrated piezoelectric stack transducers can transfer the axial and lateral forces and may be used for vibration attenuation of single beams or whole beam truss structures. For usage in a single beam test setup, the piezo-elastic support’s casing is clamped rigidly and is connected to the beam via a membrane-like spring element that allows for rotation as well as axial and lateral displacements of the beam. In this contribution, the piezo-elastic support is integrated into a two-dimensional beam truss structure comprising seven beams, where its casing is no longer clamped rigidly but is subject to axial, lateral and rotational displacements. Based on the previously verified and validated model of the single beam test setup, two different complex mathematical models of the piezo-elastic support integrated in the two-dimensional beam truss structure are derived in this contribution. The two mathematical models differ in their number of degrees of freedom for the piezo-elastic support as well as in the assumption of rigid or compliant casing. By comparing numerically and experimentally determined structural resonance frequencies and vibration amplitudes, the model that more adequately predicts the truss structure’s vibration behavior is selected on basis of the normalized root mean squared error. For future works, the more adequate model will be used to tune electrical circuits for resonant piezoelectric shunt-damping in a three-dimensional truss structure. Keywords Model selection · Structural control · Piezo-elastic beam support · Truss structure

4.1 Introduction In mechanical and civil engineering, truss structures are commonly used as lightweight load-bearing complex structures comprising beams connected to each other via beam supports. Such beam truss structures are required to withstand static and dynamic loads in axial and lateral directions resulting in vibrations, which may lead to fatigue, reduced comfort or reduced durability. Classical passive solutions for vibration attenuation are mechanical vibration absorbers, tuned mass dampers or vibration isolation techniques. An alternative (semi-)active approach is resonant shunt- damping with piezoelectric transducers, which has been investigated for several decades, [1–3]. In principle, a piezoelectric transducer converts mechanical energy from a vibrating structure into electrical energy that is dissipated via an electrical resistance connected to the electrodes of the transducer. If the transducer is connected to an electrical resistance and inductance, an RL-shunt, the resulting electromechanical system behaves similar to a mechanical vibration absorber i.e. it is tuned to attenuate vibrations in a frequency range near a structure’s resonance frequency, [4]. Various possible shunt layouts for piezoelectric shunt-damping exist with different advantages and disadvantages, but all have to be tuned precisely to the structure’s resonance frequency to achieve optimal vibration attenuation, [3, 5, 6]. Most often, the shunt parameters are tuned

J. Lenz () · M. Schäffner · R. Platz System Reliability, Adaptive Structures and Machine Acoustics SAM, Technische Universität Darmstadt, Darmstadt, Germany e-mail: [email protected]; [email protected]; [email protected] T. Melz System Reliability, Adaptive Structures and Machine Acoustics SAM, Technische Universität Darmstadt, Darmstadt, Germany Fraunhofer Institute for Structural Durability and System Reliability LBF, Darmstadt, Germany e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2020 Z. Mao (ed.), Model Validation and Uncertainty Quantification, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47638-0_4

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within a model-based approach. Thus, a mathematical model of the structure with connected piezoelectric transducer that adequately predicts the system’s output is required. In the case that various models of the same system exist, which differ in their prediction due to different simplifications or negligence of physical behavior, a selection between them for the most adequate model is necessary, [7]. In [6], Götz used a piezo-elastic support with integrated shunted piezoelectric stack transducers for attenuation of lateral beam vibrations at the first resonance frequency in a single beam test setup subjected to uncertainty from varying static axial loads as well as from production and assembly of the piezo-elastic support. There, a model of the piezo-elastic support with a rigidly clamped casing was verified and validated. In this contribution, for vibration attenuation with piezo-elastic supports in a two-dimensional beam truss structure, the casing is no longer clamped rigidly and a new mathematical model is required for the subsequent shunt tuning. Two mathematical models of the piezo-elastic support with different complexity are presented. The models differ in their assumption of rigid and compliant casing as well as in their number of degrees of freedom. The mathematical model that more adequately predicts the vibrational behavior is selected by comparing numerical and experimental data. Thereby, the normalized root mean squared error (NRMSE) is used to quantify the goodness of fit between the models and the measurements.

4.2 System Description Figure 4.1a shows the two-dimensional truss structure comprising seven beams with circular cross-section that is investigated in this contribution. It is derived from the upper truss structure of the Modular Active Spring Damper System (German acronym MAFDS), Fig. 4.1b. This is an academic exemplary system at the German Collaborative Research Center (CRC) 805 to investigate methods and possible (semi-)active approaches to control uncertainty while adapting load-paths, minimizing vibrations and preventing buckling of beams, [8]. An adequate mathematical model of the piezo-elastic support is required for vibration attenuation within the upper truss structure via piezoelectric shunt-damping of the MAFDS. As a first step, in this contribution the two-dimensional truss structure from Fig. 4.1a is used to select from two competing models of the piezo-elastic support with different complexity. The seven beams 1 to 7 of the two-dimensional truss structure are mounted to five spheres A to E with connector elements. Spheres A and C are connected rigidly to the frame structure, which is considered to be rigid. The piezo-elastic support F used for vibration attenuation by shunt-damping connects beam 2 and sphere B instead of a connector element. A detailed description of the piezo-elastic support is given in Sect. 4.3. The dynamic excitation in x- and y-direction of the two-dimensional truss structure is realized with an electrodynamic shaker G that acts on a lever. An additional mass ml may be attached to spheres D and E to vary the static load supported by the two-dimensional truss structure. Figure 4.2 shows the experimental test setup. The signal processing chain that consists of an anti-aliasing filter for the sensor signals, an amplifier for the electrodynamic shaker and a dSpace system connected to a computer for data acquisition is not shown in Fig. 4.2. The beams in the experimental test setup are made of aluminum alloy EN AW-7075 with the length lb = 0.4 m, radius rb = 0.005 m, Young’s modulus Eb = 71.4 · 109 N/m2 and density ρb = 2.8 · 103 kg/m3 . The connection spheres with radius rs = 0.03 m and mass ms = 0.268 kg are made of aluminum as well. In contrast, the connector elements are made of hardened steel 1.2312 with length lu = 0.045 m, radius ru = 0.0118 m, young’s modulus

A

G

1

F

C

2

y z

3

4 D ml

a)

upper truss

B

x

5

7

6 spring damper

E

lower truss

ml

b)

Fig. 4.1 Design model of (a) the simplified two-dimensional beam truss structure derived from (b) the Modular Active Spring Damper System (German acronym MAFDS) with upper truss structure

4 Selection of an Adequate Model of a Piezo-Elastic Support for Structural Control in a Beam Truss Structure

G

A 1

2

4

D

C

F

B

3

43

5

6

E

7

ml

ml

Fig. 4.2 Experimental test setup of the two-dimensional truss structure with electrodynamic shaker G, accelerometers at beams 1-7, the piezoelastic support F and mounts for the additional masses ml at spheres D and E

Eu = 210 · 109 N/m2 and density ρu = 7.8 · 103 kg/m3 . The local vibrational behavior of all seven beams is measured with seven accelerometers each positioned slightly off-centered at length 0.4625 lb from a sphere. In this contribution however, only the measured acceleration at beam 2 is investigated.

4.2.1 Mathematical Model of the Two-Dimensional Truss Structure The two-dimensional truss structure is modeled as a finite-element (FE) model with the beam in Fig. 4.3a discretized by N Euler-Bernoulli elements. The n-th finite beam element is shown in Fig. 4.3b with the three degrees of freedom, the displacements u and v in the i-th beam’s local axial and lateral x˜i - and y˜i -directions and the rotational displacement ϕ at each node. The well known Galerkin method with Hermitian cubic shape functions is used to derive the beam’s [6 × 6] element mass Mel and stiffness Kel matrices [9]. For the connector elements and connection spheres, simplified models are designed by approximating them as Euler-Bernoulli beam elements, Fig. 4.3a. The FE modeling of an Euler-Bernoulli beam with circular solid cross-section has been discussed in detail in various works e.g. [9–11]. Thus, here the authors only present the resulting FE equation of motion Mb ψ¨ b (t) + Db ψ˙ b (t) + Kb (Fx˜ )ψ b (t) = fb (t)

(4.1)

with the beam’s mass matrix Mb , damping matrix Db and stiffness matrix Kb (Fx˜ ), its beam displacement vector ψ b = [u1 v1 ϕ1 · · · uN+1 vN+1 ϕN+1 ]T and excitation vector fb (t). In (4.1), Kb (Fx˜ ) = Kb, e − Fx˜ Kb, g

(4.2)

is the beam’s stiffness matrix with elastic and geometric beam stiffness matrices Kb, e and Kb, g that takes the axial force Fx˜ in local x-direction ˜ due to variable static load of the truss structure, Fig. 4.2, into account. Figure 4.3c shows the two-dimensional truss structure with seven discretized beams and static loads Fl = ml g at spheres D and E due to the additional masses, Fig. 4.2. The excitation force F (t) from the electrodynamic shaker acts on sphere B via the lever with length lle at angle β resulting in vibrations of the truss structure. The model’s output is the lateral displacement vout in y-direction at length 0.4625 lb of beam 2. Since the local coordinate systems, x˜i , y˜i and z˜ i , of beams i = 3, 4, 5, 6 differ from the global truss structures coordinate system, x, y and z, the transformations Mb,i = Ti Mb T−1 i ,

Db,i = Ti Db T−1 i ,

and

Kb,i (Fx˜ ) = Ti Kb (Fx˜ ) T−1 i

(4.3)

are used to transform from local to global coordinates. Thus after transformation and assembly, the truss structure’s global equation of motion

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lle rs

lu

lb

lu

rs

z˜i

r u , E u , ½u

rb , Eb , ½b

vn 'n

un

ms

C

2 B

3

vn+1 'n+1 un+1 n+1

vout

1

®

x ˜i

a)

n

¯

A

y˜i

4

5

6

y x z

7 D

b) c)

E Fl

Fl

Fig. 4.3 (a) single beam with connector elements and connection spheres at each end, (b) n-th FE beam element between node n and n+1 with local beam coordinates and (c) schematic sketch of the two-dimensional truss structure with discretized beams, additional static loads Fl , excitation force F (t) and output lateral displacement vout

Mgl ψ¨ gl (t) + Dgl ψ˙ gl (t) + Kgl (Fax, gl ) ψ gl (t) = bin F (t)

(4.4)

is derived with the global mass matrix Mgl , damping matrix Dgl , stiffness matrix Kgl (Fax, gl ) and global displacement vector ψ gl (t). The axial forces Fax, gl are assumed to be static forces resulting from the different additional masses ml , Fig. 4.2, which are realized in the model as static loads Fl at spheres D and E, Fig. 4.3. In this contribution, however, no additional masses, ml = 0 kg, are attached to spheres D and E, thus Fl = 0 N and Fax, gl = 0 N. In (4.4), the input vector bin transforms the excitation force F (t) into separate excitations of the three degrees of freedom at sphere B. The FE output equation y(t) = vout (t) = bTout ψ gl (t)

(4.5)

calculates the displacement vout from the global displacement vector via the output vector bTout = [0 0 · · · 0 1 0 · · · 0 0]. The interested reader is referred to [10, 12] for a more detailed account on the assembling process of the global equation of motion.

4.3 Piezo-Elastic Support for Structural Control Figure 4.4a shows the piezo-elastic support that is investigated in this contribution and connects sphere B and beam 2, Fig. 4.2. The sectional view of the piezo-elastic support with the membrane spring, two piezoelectric stack transducers positioned opposite to two helical disc springs that are used to mechanically prestress the transducers is shown in Fig. 4.4b. The beam’s rectangular axial extension made of hardened steel 1.2312 with length lext = 0.0087 m connects the beam orthogonally to the transducers transforming the beam’s lateral vibrations in y˜ and z˜ -direction into the transducers axial deflections. The transducers are PI-885.51 piezoelectric stack transducers with capacitance Cp = 1.65·10−6 F and mechanical stiffness kp = 20.5 · 106 N/m for short circuited electrodes. The support has a load-bearing function realized by the membrane spring made of spring steel 1.248 that bears the beam’s axial and lateral forces in local x-, ˜ y˜ and z˜ -directions and allows for rotational deflections. In this contribution, however, a two-dimensional truss structure is investigated, Fig. 4.2. Therefore, two two-dimensional models of different complexity of the piezo-elastic support with only the lateral, axial and rotational degrees of freedom u, v and ϕ in Fig. 4.3b, are derived and investigated. Figure 4.5 shows the two models of the piezo-elastic support that are derived and investigated in this contribution. Model 1 of the piezo-elastic support, is modeled as a single finite element with two nodes, 6 degrees of freedom, and assumed rigid casing with the mechanical model shown in Fig. 4.5a. For the model 2, the piezo-elastic support’s casing is no longer assumed to be rigid, but to be compliant and split into two separate finite Euler-Bernoulli elements and an extra finite element for the axial extension resulting in 5 nodes and 15 degrees of freedom, Fig. 4.5b. Table 4.1 presents the characteristic differences between both models of the piezo-elastic support. The detailed description of the models and derivation of their stiffness and mass matrices follows in Sects. 4.3.1 and 4.3.2.

4 Selection of an Adequate Model of a Piezo-Elastic Support for Structural Control in a Beam Truss Structure

45

y˜ piezoelectric transducer

z˜ membrane spring

axial extension

beam x ˜ helical disc spring

a)

b)

Fig. 4.4 Piezo-elastic support: (a) photo of the support integrated into the two-dimensional truss structure, Fig. 4.2, (b) sectional view lext

Kc,2

Kc,1

2 krot,1 +lext kp,1

krot,2 node n+1

node n kax,1

node n

klat,1

node n+4

node n+3 klat,2 kax,2

kp

Kc,1 y˜ x ˜

a)

node n+1

y˜ x ˜

lu

z˜

b)

Kc,2 node n+2

lu

z˜

Fig. 4.5 Mechanical model of the piezo-elastic support: (a) model 1 and (b) model 2 at the position in Fig. 4.2 with the axial stiffness kax, m , 2 k of model m with m = 1, 2 lateral stiffness klat, m and rotational stiffness krot, m +lext p Table 4.1 Characteristics of the piezo-elastic support models 1 and 2

Property Casing Degrees of freedom Complexity

Model 1 Model 2 (Sect. 4.3.1) (Sect. 4.3.2) Rigid Compliant 2 · 3=6 5 · 3 = 15 Low High

4.3.1 Piezo-Elastic Support Model 1 Figure 4.5a shows the mechanical model 1 of the piezo-elastic support with discrete axial, lateral and rotational stiffness, kax, 1 , klat, 1 and krot, 1 , connecting the beam 2 at node (n + 1) with the sphere B at node n via the casing. The axial stiffness kax, 1 = 2 · 106 N/m, lateral stiffness klat, 1 = 26.5 · 106 N/m and rotational stiffness krot, 1 = 105.5 Nm/rad represent the membrane spring. The piezoelectric stack transducer is positioned between both nodes at the

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2 k at distance lext from node (n + 1), Fig. 4.4. Thus, its mechanical stiffness is represented in the rotational stiffness lext p node (n+1). From the mechanical model, Fig. 4.5b, the piezo-elastic support’s stiffness matrix

⎡

Kps, 1

kax, 1 ⎢ 0 ⎢ ⎢ ⎢ 0 =⎢ ⎢ −kax, 1 ⎢ ⎣ 0 0

0

0

klat, 1 klat, 1 lu 0 −klat, 1 0

klat, 1 lu 2 k klat, 1 lu2 + krot, 1 + lext p 0 −klat, 1 lu 2 k −krot, 1 − lext p

−kax, 1 0 0 kax, 1 0 0

0 −klat, 1 −klat, 1 lu 0 klat, 1 0

⎤ 0 ⎥ 0 ⎥ ⎥ 2 −krot, 1 − lext kp ⎥ ⎥ ⎥ 0 ⎥ ⎦ 0 2 krot, 1 + lext kp

(4.6)

of the low complexity model is derived. The mass matrix ⎡

Mps, 1

mps, 1 ⎢ 0 ⎢ ⎢ ⎢ 0 =⎢ ⎢ 0 ⎢ ⎣ 0 0

0

0 0

mps, 1 0 0 0 0

0 0 0 0 0 0

ps, 1 0 0 0

0 0 0 0 0 0

⎤ 0 0⎥ ⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎦ 0

(4.7)

is formed with the assumption that the mass mps, 1 and rotational inertia ps, 1 of the piezo-elastic support’s casing completely assigned to the node n in Fig. 4.5a. The derivation is presented in a previous work [12].

4.3.2 Piezo-Elastic Support Model 2 For the piezo-elastic support model 2, the stiffness and mass element matrices of the casing’s finite elements are derived by approximating them with circular hollow Euler-Bernoulli beam elements with constant inner and outer radii, [9]. Figure 4.5b shows the corresponding mechanical model with the element stiffness matrices Kc, 1 and Kc, 2 representing the stiffnesses of the casing elements between the casing nodes n and (n+1) as well as nodes (n+1) and (n+2). Furthermore, node (n+1) is coupled to node (n+3), the end position of the axial beam extension, Fig. 4.4, via the discrete lateral transducer stiffness kp . Additionally, the membrane spring’s discrete lateral stiffness klat, 2 = 20.5 · 106 N/m, rotational stiffness krot, 2 = 105.5 Nm/rad and axial stiffness kax, 2 = 2 · 106 N/m couple nodes (n+2) and (n+4). Node (n+4) represents the end point of the beam, where its attached to the spring membrane, Fig. 4.4. The derivation of the [15 × 15] matrix Kps, 2 and mass matrix Mps, 2 of model 2 is more complicated compared to model 1. First, the [6 × 6] element stiffness and mass matrices Kel and Mel of the two casing elements and the axial extension, Fig. 4.5b, are derived analogous to the beam’s element matrices in Sect. 4.2.1. The two casing elements are coupled to each other via node (n + 1) while the axial extension’s element matrices are not coupled directly to the casing’s matrices. Figure 4.6 shows the assembly scheme for the stiffness matrix Kps, 2 of model 2 with the gray squares representing the element stiffness matrices. In case of the stiffness matrix Kps, 2 , the piezoelectric transducers stiffness and membrane spring’s axial, lateral and rotational stiffness, kp , kax, 2 , klat, 2 and krot, 2 , couple the casing with the axial extension, Fig. 4.5b. This is realized in Fig. 4.6 by the piezoelectric transducer stiffness matrix ⎡

0 Kp = ⎣ 0 0

0 kp 0

⎤ 0 0⎦ 0

(4.8)

and the membrane spring’s stiffness matrix ⎡

Kms

kax, 2 =⎣ 0 0

0 klat, 2 0

⎤ 0 0 ⎦. krot, 2

(4.9)

4 Selection of an Adequate Model of a Piezo-Elastic Support for Structural Control in a Beam Truss Structure Fig. 4.6 Assembly scheme for the [15 × 15] FE stiffness matrix Kps, 2 the piezo-elastic support model 2

47

casing element 1 node n casing element 2 node n+1 +Kp

−Kp node n+2 +Kms

−Kms node n+3 +Kp

−Kp −Kms

node n+4 +Kms axial extension

4.3.3 Transfer Function of the Two-Dimensional Truss Structure To investigate the vibration behavior of the two-dimensional truss structure for the sensor position on beam 2, Fig. 4.3c, and to select the most adequate model of the piezo-elastic support, the state space model of (4.4) with the output vout is derived. This provides a model with clearly defined inputs and outputs. The general state space model for a single-input-single-output system without feedthrough x˙ (t) = Am x(t) + bm u(t)

(4.10)

y(t) = cTm x(t)

is described by the system matrix Am , the input vector bm and the output vector cTm of model m, [13]. The state vector x(t), input signal u(t) and output signal y(t) are defined as   ψ gl (t) x(t) = ˙ , ψ gl (t)

u(t) = F (t) and

y(t) = vout (t).

(4.11)

With the definitions in (4.11), the state space model ⎡ ⎢ x˙ (t) = ⎣ &

⎤ 0

I

⎥ ⎢ ⎦ x(t) + ⎣

−1 −M−1 gl Kgl −Mgl Dgl '( )





Am

⎤

⎡

&

0

⎥ ⎦ u(t)

−M−1 gl bin '( ) bm

(4.12)

y(t) = bTout 0 x(t). & '( ) cTm

is derived from (4.4) and (4.5). The state space model (4.12) is transformed in Laplace domain resulting in the m-th model transfer function Gv,m (s) =

vout (s) y(s) = = cTm (s I − Am )−1 bm u(s) F (s)

(4.13)

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from excitation force F (s) to the displacement vout (s), [13]. In order to compare the experimentally measured acceleration signals with the numerical simulations, the acceleration transfer function Ga, m (s) = s 2

vout (s) F (s)

(4.14)

is derived.

4.4 Comparison of the Two Models The selection for the more adequate model of the piezo-elastic support is conducted by comparing numerical and experimental amplitude and phase responses from the shaker’s excitation force to the lateral acceleration at beam 2 (4.14), Fig. 4.3c. Since piezoelectric shunt-damping will be applied to attenuate vibrations at the first resonance frequency f1 in future works, it is of particular interest for the model to adequately predict f1, m and the amplitude at f1, m . The normalized root mean squared error NRMSE = 1 −

||Ga, exp (s) − Ga, m (s)||

(4.15)

||Ga, exp (s) − Ga, exp (s)||

40

40

0

0

-40

-40

0

0

-90

-90

phase in

◦

amplitude in dB

is used to evaluate the goodness of fit of the models and experiment in order to determine the more adequate model. In (4.15), Ga, m (s) is the numerical acceleration transfer function of model m (4.14) and Ga, exp (s) the experimental acceleration transfer function with its arithmetic mean Ga, exp (s) for s = j 2 π f . Figure 4.7a shows the amplitude and phase responses from experimental and numerical simulation with the piezo-elastic support model 1, Sect. 4.3.1. The measured and simulated responses differ from each other in the observed frequency range, especially for frequencies f > 100 Hz the discrepancy increases. For the first resonance frequency, the numerical simulation predicts f1, 1 = 70.9 Hz, which is 0.6 Hz higher than the measured frequency f1, exp = 70.3 Hz while the model underestimates the amplitude with 28.1 dB compared to the measured amplitude 28.6 dB. Figure 4.7b shows the amplitude and phase responses from experimental and numerical simulation with the piezo-elastic support model 2, Sect. 4.3.2. The general transfer behavior is adequately predicted by the model for f < 100 Hz but the discrepancy between the numerical simulations and experiment increases with increasing frequency f ≥ 100 Hz. The numerically simulated first resonance frequency f1, 2 = 70.7 Hz overestimates the measured frequency f1, exp = 70.3 Hz and the amplitude at the first resonance frequency is overestimated as well with 28.8 dB. The discrepancy between model and measurement around the first resonance frequency is smaller for the piezo-elastic support model

-180 25

100 a) frequency fi n Hz

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Fig. 4.7 Amplitude and phase responses of the two-dimensional truss structure from the experiments ( piezo-elastic support (a) model 1 ( ), Sect. 4.3.1, and (b) model 2 ( ), Sect. 4.3.2

) and numerical simulation with the

4 Selection of an Adequate Model of a Piezo-Elastic Support for Structural Control in a Beam Truss Structure

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2, furthermore its NRMSE value is higher. Thus, the piezo-elastic support model 2 more adequately predicts the twodimensional truss structures vibrational behavior and is selected to be used in future works.

4.5 Conclusion In this paper, a model selection from two models of different complexity of a piezo-elastic support for piezoelectric shuntdamping of lateral vibrations in a two-dimensional truss structure is conducted. The two models differ in the assumption of a rigid or compliant casing and in their number of degrees of freedom. Precise prediction of the vibration behavior is necessary within a model-based approach to tune electrical circuits for the piezoelectric shunt-damping to maximize the lateral vibration attenuation. The discrepancy between the two models and the experiments is insignificant for a frequency range around the system’s first resonance frequency but increases with higher frequencies. Furthermore, no significant difference in the discrepancy between the two models is observable, which would easily indicate the more adequate model. Therefore, the normalized root mean squared error (NRMSE) is utilized to compare the numerical and experimental transfer functions. A NRMSE value closer to 1 is determined for the model with assumed compliant casing, thus, it is selected as the more adequate model of the piezo-elastic support. In future works, the two-dimensional truss structure will be subjected to different static loads, leading to a shift of resonance frequencies and a change in the vibrational behavior. Therefore, a model verification and validation is necessary to investigate the axial load-dependency of the two-dimensional truss structure. Acknowledgments The authors like to thank the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) for funding this project within the Sonderforschungsbereich (SFB, Collaborative Research Center) 805 “Control of Uncertainties in Load-Carrying Structures in Mechanical Engineering” – project number: 57157498.

References 1. Forward, R.L.: Electronic damping of vibrations in optical structures. Appl. Opt. 18(5), 690–697 (1979) 2. Hagood, N.W., Crawley, E.F.: Experimental investigation into passive damping enhancement for space structures. J. Guid. Control Dyn. 14(6), 1100–1109 (1991) 3. Moheimani, S.O.R., Fleming, A.J.: Piezoelectric Transducers for Vibration Control and Damping, 1st edn. Springer, London (2006) 4. Neubauer, M., Oleskiewicz, R., Popp, K.: Comparison of damping performance of tuned mass dampers and shunted piezo elements. Proc. Appl. Math. Mech. 5, 117–118 (2005) 5. Neubauer, M., Oleskiewicz, R., Popp, K., Krzyzynski, T.: Optimization of damping and absorbing performance of shunted piezo elements utilizing negative capacitance. J. Sound Vib. 298(1–2), 84–107 (2006) 6. Götz, B.: Evaluation of uncertainty in the vibration attenuation with shunted piezoelectric transducers integrated in a beam-column support, Dissertation, Technische Universität Darmstadt (2019) 7. Smith, R.C.: Uncertainty Quantification: Theory, Implementation and Applications, 1st edn. Society for Industrial and Applied Mathematics, Philadelphia (2014) 8. Feldmann, R., Platz, R.: Assessing Model Form Uncertainty for a Suspension Strut using Gaussian Processes. In: Proceedings of the 3rd International Conference on Uncertainty Quantification in Computational Sciences and Engineering, pp. 1–8 (2019) 9. Klein, B.: FEM, 9th edn. Vieweg+Teubner Verlag, Wiesbaden (2012) 10. Chopra, A.K.: Dynamics of Structures, 1st edn. Prentice-Hall, Englewood Cliffs (1995) 11. Schäffner, M.: Quantification and evaluation of uncertainty in active buckling control of a beam-column subject to dynamic axial loads. Dissertation, Technische Universität Darmstadt (2019) 12. Lenz, J., Holzmann, H., Platz, R., Melz, T.: Vibration attenuation of a truss structure with piezoelectric shunt-damping for varying static axial loads in the truss members. In: Proceedings of the International Conference on Structural Engineering Dynamics (2019) 13. Lunze, J.: Regelungstechnik 1 Systemtheoretische Grundlagen, Analyse und Entwurf einschleifiger Regelungen, 7th edn. Springer, Berlin (2008) Jonathan Lenz, M.Sc. – Work: Structural Health Control of load-carrying Mechanical Systems (Collaborative Research Center 805 – Control of Uncertainty in Load-Carrying Mechanical Systems – Subproject C7) – Graduate research assistant and PhD student, research group System Reliability, Adaptive Structures and Machine Acoustic SAM, Technische Universität Darmstadt

Chapter 5

Impact Load Identification for the DROPBEAR Setup Using a Finite Input Covariance (FIC) Estimator Peter Lander, Yang Wang, and Jacob Dodson

Abstract Various applications in structural dynamics may require the real-time estimation of unknown input. A recently developed joint input-state estimator for linear systems treats the unknown input as white Gaussian noise with finite covariance. The performance of this finite input covariance (FIC) estimator is validated using simulated data from a finite element model of the Air Force Research Laboratory’s experimental testbed called the DROPBEAR (Dynamic Reproduction of Projectiles in Ballistic Environments for Advanced Research). The estimator performance is compared with a few wellknown estimators, including the augmented Kalman filter (AKF) and the weighted least squares (WLS) estimators. The results show that the FIC estimator is capable of accurately estimating an impact load applied to the beam when acceleration is measured at a small number of locations. Additionally, the results show that the FIC estimator eliminates the low-frequency drift error that other well-known estimators are susceptible to. Keywords Input estimation · Dynamic testing · Linear stochastic system · Impact load identification

5.1 Introduction The objective of this research is to assess the performance of an input and state estimation framework for structures undergoing high-speed dynamic events. In such scenarios, oftentimes the structure is subject to an unknown and unmeasurable impact load that needs to be identified/estimated. The proposed joint input-state estimator is validated with a test setup named DROPBEAR (Dynamic Reproduction Of Projectiles in Ballistic Environments for Advanced Research) at the Air Force Research Laboratory Munitions Directorate, located at Eglin Air Force Base [1]. The technical approach focuses on the input and state estimation from acceleration response data using a finite input covariance (FIC) estimator. The performance of the FIC estimator is compared to two other estimators, one using an augmented Kalman filter (AKF) and another using weighted least squares (WLS) estimation.

5.2 Background The simulations performed for this research are based on the DROPBEAR testbed [1]. The testbed consists of a cantilever steel beam with an actuated roller that serves as either a moving or stationary pin support along the length of the beam. Additionally, an electromagnet can be attached to the beam and programmed to fall at a certain time, acting as a changing

P. Lander () School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA, USA e-mail: [email protected] Y. Wang School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA, USA School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA, USA e-mail: [email protected] J. Dodson Air Force Research Laboratory, Munitions Directorate, Eglin AFB, FL, USA e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2020 Z. Mao (ed.), Model Validation and Uncertainty Quantification, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47638-0_5

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Fig. 5.1 DROPBEAR setup with detaching electromagnet (mass change) and rollers (moving support condition) [2]

mass condition. DROPBEAR provides the ability to repeatedly collect data on a structure with changing parameters, and this makes it ideal for validating input estimation algorithms (Fig. 5.1). Toward input estimation, the mass, stiffness, and damping matrices from the FE model are first used to construct a statespace model assuming a discrete-time stochastic linear system with direct feedthrough of input. At every time step, the estimators execute four stages of computation: measurement update of input, measurement update of state, time update of state, and time update of input.The well-known AKF estimator is premised on a Gaussian random walk input model, i.e.  uk + 1 = uk + ξ k with ξk ∼ N 0, ξ . The approach augments the state vector with the unknown input; and the resulting augmented state can then be estimated using a regular Kalman filter. This AKF estimator is known to suffer from an unobservability issue when only acceleration measurements are available [3]. The focus of this research is the finite input covariance estimator (FIC), which assumes uk to be zero-mean white Gaussian with uk ∼ N (0, u ). Previous studies demonstrated that the FIC estimator does not suffer from a drift issue [4], which is an advantage to be validated with the DROPBEAR setup. When the input covariance  u of the FIC estimator approaches infinity ( ∞ ) and the feedthrough matrix D has full column rank, the FIC estimator is proved to be equivalent to the WLS estimator proposed by Gillijns and De Moor in [5]. Therefore, the results using the WLS estimator are also presented for performance comparison.

5.3 Analysis To study input estimation with the DROPBEAR testbed, a finite element model is first constructed using dimensions of the beam and typical material properties for steel. The model consists of 100 Euler-Bernoulli beam elements, with a vertical and rotational degree of freedom at each node. The beam has a fixed support condition at one end, and the roller is located three quarters of the beam length away from the fixed end. An impact u is applied in the vertical direction at the free end of the beam. The impact has a triangular shape with a total duration of 0.4 milliseconds and a peak magnitude of 500 N. A sampling frequency of 50 kHz is used to obtain the vertical acceleration response at quarter points along the beam. The three estimators mentioned above, AKF, FIC, and WLS are applied to the input estimation problem with DROPBEAR. The estimators use a reduced order model of the beam that is created using modal decomposition and keeps only the first 10 vibration modes. Using the acceleration measurements at each sampling time step, each estimator recursively provides an estimation of the input load u together with the state x, which contains displacements and velocities in modal coordinates. Figure 5.2a plots the actual (correct) input history of u (black solid line), together with the three estimated uˆ histories, for the entire 5 s of simulation. Both the AKF (blue dashed) and WLS (blue dotted) estimates show noticeable slow drift as time progresses, but the FIC (red dashed) estimate does not suffer from this drift issue. The drift is more clearly visible in Fig. 5.2b, which shows a close-up view around the last second of the simulation. Figure 5.3a plots the actual u and FIC estimate uˆ over the duration of the impact, ramping up from 0 to 500 N and back down to 0 N. No visible difference can be observed between the two in this plot, and the RSM error for the FIC input estimation over this time period is 2.75 N. Also included in the figure is the 99.7% confidence interval of the estimation, which shows a very tight bound that is not easily visible in this view. Figure 5.3b zooms into the peak area, where the difference between the actual history and the FIC estimate becomes visible. Here it can be seen that the actual input mostly lies within the confidence interval.

5 Impact Load Identification for the DROPBEAR Setup Using a Finite Input Covariance (FIC) Estimator

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Fig. 5.2 Input estimation results. (a) The entire 5-s duration. (b) Zoomed in to last second

Fig. 5.3 Detailed FIC input estimation results. (a) Impact duration. (b) Around the peak

5.4 Conclusion The numerical simulations performed for this research demonstrate that the FIC estimator outperforms conventional approaches, including an augmented Kalman filter and a weighted least squares estimator. The FIC estimator provides a more accurate input estimate and eliminates the drift seen from the other two estimators. This research provides the fundamental and theoretical framework that can be deployed to various applications that require real-time input estimation. These include applications with high-speed dynamic events on structures subject to unknown/unmeasurable loads, commonly found in the aerospace and defense industries. Also applicable are civil structures, such as bridges, where measuring the loading can be of use to the owner. Through incorporating state-of-the-art estimation theory and structural dynamics, this interdisciplinary work is also of significant interest to the broader academic communities involving structural engineering. Acknowledgements This research was supported by the Air Force Research Lab (AFRL) Summer Faculty Fellowship Program. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the sponsor.

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References 1. Joyce, B., Dodson, J., Laflamme, S., Hong, J.: An experimental test bed for developing high-rate structural health monitoring methods. Shock. Vib. 2018, 10 (2018) 2. Joyce, B.S., Greenoe, K., Dodson, J., Wolfson, J., Abramczyk, S., Karsten, H., Markl, J., Minger, R., Passmore, E.: An experimental test bed with time-varying parameters for developing high-rate structural health monitoring methods. In: Proceedings of the 36th International Modal Analysis Conference (IMAC XXXVI), February 12–15, Orlando (2018) 3. Lourens, E., Reynders, E., De Roeck, G., Degrande, G., Lombaert, G.: An augmented Kalman filter for force identification in structural dynamics. Mech. Syst. Signal Process. 27, 446–460 (2012) 4. Liu, X., Wang, Y.: Input estimation of a full-scale concrete frame structure with experimental measurements. In: Proceedings of the 37th International Modal Analysis Conference (IMAC XXXVII), January 28–31, Orlando (2019) 5. Gillijns, S., De Moor, B.: Unbiased minimum-variance input and state estimation for linear discrete-time systems with direct feedthrough. Automatica. 43(5), 934–937 (2007) Peter Lander has spent the last three years pursuing a PhD in Civil Engineering at the Georgia Institute of Technology. As a member of Dr. Yang Wang’s Laboratory for Smart Structural Systems, Peter’s research is centered on structural health monitoring.

Chapter 6

Real-Time Digital Twin Updating Strategy Based on Structural Health Monitoring Systems Yi-Chen Zhu, David Wagg, Elizabeth Cross, and Robert Barthorpe

Abstract In structural health monitoring (SHM), model updating is concerned with identifying and updating system parameters (e.g. stiffness and mass) based on the measured response data of the monitored structure. With the increasing number of SHM systems deployed on modern structures in recent years, real-time model updating has become possible. This allows a digital twin of the monitored structure to be built, such that the structural behaviour can be monitored and predicted simultaneously throughout its life-cycle. In real applications, the structural response data are normally measured under operational conditions where the environment and loading condition cannot be directly controlled, which leads to significant identification uncertainty. The system model can also be complex, meaning that identifying system parameters directly from measured response data is challenging and time consuming. Focusing on the above concern, a real-time updating strategy for a SHM digital twin is proposed in this work. An intermediate model is used for environmental condition estimation and divergence analysis in order to increase the updating efficiency. A Bayesian system identification approach is adopted so that the identification uncertainty can be fully accounted for. Synthetic and laboratory examples are presented to illustrate the proposed updating strategy. Keywords Digital twin · Structural health monitoring · Model updating · Operational modal analysis · Bayesian method

6.1 Introduction Model updating focuses on improving the current mathematical models for the structure based on the observed measurements [1, 2]. In structural health monitoring (SHM), system identification is concerned with identifying and updating the model parameters (e.g., stiffness and mass) based on the measured response data of the structure of interest. It has become an important task which provides information for downstream applications such as serviceability assessment, damage detection and maintenance management [3–7]. Various forms of system identification methods have been developed over the past decades [8, 9]. Furthermore, the Bayesian approach [10, 11] provides a rigorous means for system identification using probability logic where the parameters are viewed as uncertain variables and updated in terms of their posterior distribution given measured data [12, 13]. Without a built-in SHM system, conventional system identification (especially for civil structures) can only be conducted from time to time due to the cost of sensor deployment. However, with the increasing number of modern SHM systems installed on structures in recent years, real-time system identification becomes possible. Using the streamed SHM data, system identification can be performed continuously. The resulting system model is hence up-to-date and its structural behaviour though its life-cycle can be monitored and predicted at the same time. This also allows the digital twin model [14, 15] to be built. In real applications (especially for a full-scale civil structure), there are a number of challenges where a digital twin is to be constructed for SHM purposes. First, the structure is normally complex and hence the corresponding system model is complicated (e.g. a complicated FE model possibly with non-linear elements involved). Identifying the system model parameters directly from the measured structural response data can be challenging and time consuming. Second, the structural response data are measured under operational conditions where the input information is unknown and cannot be directly controlled. The associated identification uncertainties are significant and cannot be neglected. Third, the dynamic properties of the structure can be affected by both environmental conditions (e.g. temperature and excitation condition) and

Y.-C. Zhu () · D. Wagg · E. Cross · R. Barthorpe Dynamics Research Group, Department of Mechanical Engineering, The University of Sheffield, Sheffield, UK e-mail: [email protected]; [email protected]; [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2020 Z. Mao (ed.), Model Validation and Uncertainty Quantification, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47638-0_6

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system changes (e.g. stiffness change due to ageing or damage). The changing rate of the model parameters may also vary under different conditions. Focusing on vibration based SHM data, this paper proposes a model updating strategy in order to address the forgoing concerns. Assuming a complex, computationally expensive system model of the structure is available and in use, instead of updating the system parameters directly every time the SHM data is streamed, the proposed method coverts the problem into three sub-steps where a modal model is used and divergence analysis is conducted as intermediate steps. Compared to a full system identification which needs heavy computational effort, modal analysis is efficient and can be usually done in seconds. A divergence analysis step is conducted to judge whether the system model needs to be updated or not. The divergence is calculated based on the posterior distribution of the identified modal parameters where the identification uncertainty can be assessed. A Bayesian approach is adopted so that uncertainty associated with modal analysis can be considered rigorously when updating the system model.

6.2 Updating Strategy For a full-scale structure, updating the overall system model based on the measured SHM data directly can be challenging and time consuming. It may also be unnecessary to update the system model every time the data is streamed when there are no significant changes in the system. In view of this, an intermediate model is introduced in the proposed updating strategy. Divergence analysis then can be conducted based on the intermediate model to determine whether updating the system model is needed. Generally, the intermediate model can be any surrogate of the overall system whose parameters can be readily identified from the SHM data. In this work, the modal model is used as the intermediate model in the updating strategy. This is because modal parameters (e.g., natural frequencies, damping ratios and mode shapes) are important properties in SHM that characterise the dynamic behaviour of the system of interest. They can also be extracted efficiently from the SHM data. A Bayesian framework is adopted in order to propagate the uncertainty between the measured data, the modal model and the system model in a principled manner. Figure 6.1 outlines the proposed updating strategy. For a set of streamed data Di with a duration of Ti , modal analysis is first conducted to determine the posterior distribution of the parameters of the modal model ω given the streamed data (i.e., p(ω|Di )). The divergences analysis then can be conducted in two parts. A system change detection step is first conducted to investigate whether the change of modal parameters is due to environmental condition or system changes. The system model should only be updated due to system changes. The discrepancy di between the posterior distribution of the modal parameters of the reference model and the one based on the current data set is then quantified. If the discrepancy is larger than the threshold set (i.e., dtol ), system identification then needs to be further conducted to update the model. A two-stage Bayesian system identification framework is applied for model updating [16], which allows the system parameters to be identified. The framework strictly obeys the Bayes’ theorem without heuristic formulations involved. The identification uncertainty of modal parameters can also be fully accounted for. It can be shown that the system parameters θ can be updated through the posterior distribution of θ given the measured data (i.e., p(θ|Di )) based on the posterior distribution of modal parameters (i.e., p(ω|Di )) as: * p (ω |Di ) p (ω |θ ) dω (6.1) p (θ |Di ) = p (θ) p (ω) This is the general formulation for model updating. A number of simplifying assumptions can be made to facilitate computation, which are outlined below. If given the system parameters θ, the modal parameters ω can be completely ˜ (θ) without any prediction error involved, then p(ω|θ) becomes a delta function, determined through the model prediction ω i.e., ˜ (θ)) p (ω |θ ) = δ (ω − ω

(6.2)

Substituting Eq. (6.2) into Eq. (6.1) and assuming a constant prior of modal parameters (i.e., p(ω) = cω ) gives −1 ˜ (θ) |Di ) p (θ |Di ) = cω p (θ) p (ω

(6.3)

For a globally identifiable problem (otherwise it will not be involved for model updating), distribution of ω   " the posterior ˆ i , Ci , where ω ˆ i is the most given Di can be well-approximated by a Gaussian distribution [17], i.e., p (ω |Di ) ≈ N ω "ω probable value of ω and Ci is the posterior covariance. Equation (6.3) then can be rewritten as

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Fig. 6.1 Proposed updating strategy

−1 p (θ |Di ) ∝ cω p (θ) exp (−Lθ )

(6.4)

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T   1 ˜ (θ) − ω ˆ i C−1 ˜ (θ) − ω ˆi ω ω i 2

(6.5)

The system parameters then can be updated by maximising Eq. (6.4), or equivalently minimising Eq. (6.5) if a uniform prior p(θ) is assumed. At the same time, the time duration of streamed data used for next modal analysis can also be determined based on the quantified discrepancy di . The data with a longer measurement duration can be used for modal analysis when the discrepancy is small in order to further save the computational effort (which also decreases the identification uncertainty). For a large discrepancy, a short measurement duration can be applied in order to track the model changes more actively. A minimum and maximum measurement duration can also be set to ensure the identifiability and the stationarity of the data.

6.3 Divergence Analysis The key step in the updating strategy is the divergence analysis, which determines whether the overall digital twin model shall be updated or not based on the current streamed data. The change in identified modal parameters may be due to either environmental conditions or system changes. The first part of the divergence analysis is to detect system changes; model updating will only be conducted for such changes. The second part is to quantify the discrepancy, where the posterior distribution of the identified modal parameters can be fully accounted so that the updating threshold can be set. In this section, the technical details of these two parts will be discussed. Techniques used to distinguish the change of modal parameter due to environmental conditions and system changes may vary depending on the available information from the SHM systems. If the environmental conditions (e.g., temperature,

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pressure and so on) are also monitored in the SHM system. Benchmark tests can be conducted when calibrating the reference model and modal parameters can be investigated under the interested range of environmental conditions. Regression models then can be used to describe the functional behaviour between the environmental conditions and the modal parameters. Predictions from such models can then be subtracted from subsequent data and hopefully the remaining is sensitive to system changes. Regression models are often referred as response surface models in this context with associated techniques ranging from simply polynomials [18] to state-of-art machine learning methods such as Gaussian process [19] and neural networks [20]. When environmental monitoring data is not available from the SHM system, one possible way to detect system change is to use anomaly detection techniques [21, 22]. Benchmark tests can be conducted when the structure is under normal working conditions. The distribution of modal parameters under normal working condition then can be obtained and anomaly condition can be detected. The benchmark tests for the interested structure under normal working conditions could also provide information in setting the acceptable tolerance of model discrepancies. Various forms of measures have been developed to quantitatively describe the difference between two probability distributions. They can be mainly divided into two categories, i.e., the f -divergence and integral probability metrics. The Kullback-Leibler divergence is the most widely used f -divergence which quantifies the discrepancies based on the ratio of two probability distributions [23]. On the other hand, Kolmogorov distance and its variants [24–26] are commonly used integral probability metrics where the differences are calculated based on the cumulative distribution function. Among others, the Hellinger distance is a good candidate to quantify the discrepancies of the posterior distributions of modal parameters given by + di =

1 2

*



!   "  2 " p (ω |Di ) − p ω Dref dω

(6.6)

This is because it accounts for the complete form of the probability distributions and is bounded by [0,1] (zero when two distributions are identical). Noting that the posterior distribution of the modal parameters can be approximated as a Gaussian distribution, the analytical expression of the Hellinger distance can then be given by , 0 / 1/4  ! T Ci + Cref −1   det (Ci )1/4 det Cref 1 ˆi −ω ˆi −ω ˆ ref ˆ ref di = .1 − ω exp − ω  8 2 Ci +Cref 1/2 det 2

(6.7)

which is interpretable and computationally efficient.

6.4 Synthetic Data Illustration A synthetic data example is presented in this section to illustrate the proposed model updating strategy. Consider a threestorey shear building structure with uniform floor mass of 6*105 kg. The inter-storey stiffness of the first and second floor is set to be 3000 kN/mm and remains unchanged. The inter-storey stiffness of the third floor has a time degrading feature following kt = 1500(exp(−10−7 t2 ) + 1)kN/mm, where t is the time. The damping ratios of the structural modes are set to be 1%. The system model used in this example is a simple three degrees of freedom model with the mass and damping matrix assumed to be known and unchanged. System updating here focuses on identifying the stiffness matrix of the structure. The system model of a real structure is normally much complicated. Nevertheless, a simply model is used in this example for the purpose of demonstrating the overall procedure of the proposed strategy. The structure is subjected to i.i.d. Gaussian white noise excitation with a root power spectral density (PSD) of 9.81 N/Hz1/2 in the lateral direction of each floor. The measurement noise is taken to be i.i.d. Gaussian white noise with a (root) PSD of 2*10−6 g/Hz1/2 for all data channels. The sampling rate of the data is set to be 100 Hz. Three hours of streamed data are analysed and the proposed modal updating scheme are applied. The time duration for the first steamed data set is 200 s and the minimum time duration of the streamed data is set to be 50s. The time duration of the next streamed based   data is determined  on the time duration of the latest streamed data set as well as the quantified discrepancy as Ti+1 = 0.5 + 1.5 × 1 − di10 Ti . In this example, the natural frequencies and mode shapes (identified using [27]) are used for discrepancy comparison and model updating. Figure 6.2 shows the root power spectrum of the first set of streamed data. The natural frequencies of the three modes shown in Fig. 6.2 are identified to be 4.89 Hz, 14.00 Hz and 20.26 Hz. The mode shapes are shown in Fig. 6.3

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where the solid line represents the true value and square represents the identified value based on the measured data. These two values almost coincide, indicating good identification quality. Figure 6.4 shows the natural frequencies of the three modes of the structure against the time where the solid lines denote the true values and the cross markers denote the modal analysis results. The modal analysis results used for model updating are shown as squares in the figure. The identification results are close to the true values used for data generation. It can be seen that the proposed method is capable of adjusting the time duration used for analysis based on the changing rate of the modal parameters. Data sets with small time duration are used when there is a rapid change in the natural frequencies (between 2000 and 4000 s in the figure). The system parameters are also updated more frequently. When there are only limited changes, the time duration gradually increases (after 8000 s) and the system model remains unchanged. Figure 6.5

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shows the stiffness of the three storeys against time. It can be seen that the identified stiffness is quite close to the true values which are used for data simulation. The proposed method can also track the stiffness degrading feature on the third floor.

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6.5 Divergence Analysis Based on Laboratory Data A laboratory example is present in this section to illustrate the effect of environmental condition on divergence analysis. The instrumented structure is a three-storey shear building (see Fig. 6.6) and its modal properties are investigated under different excitation amplitudes and temperature in this test. It is 900 mm*900 mm in plan with a uniform inter-storey height of 700 mm. The weight of each floor is 21.02 kg, 21.06 kg and 21.04 kg from first floor to third floor, respectively. The cross section of each steel column is 80 mm*10 mm. The ground floor of the structure is bolted on a 6-axis shake table. The horizontal vibration response of the structure at four corners of each storey was measured using uniaxial accelerometers. The accelerometer has a measurement range of ±500 g with a sensitivity around 10 mV/g. The structure was excited using a MAST (Multi-Axis Shake Table) system. The input force was white noise with a constant PSD of 0.01(m/s2 )2 /Hz ranging from 4 Hz to 120 Hz. The structure was first excited with different amplitude of this input force from 10% to 100% (with an interval of 10% each) at room temperature (about 25 ◦ C). The excitation amplitude was then set at 50% and the vibration response was measured under different temperature from 0 ◦ C to 40 ◦ C (with an interval of 10 ◦ C). Ten minutes of data were recorded for each setting with a sampling rate of 2048 Hz. The data were decimated to 512 Hz for analysis. Figure 6.7 shows the root power spectrum of the measured data set under 50% excitation amplitude and room temperature for reference. Figures 6.8 and 6.9 shows the natural frequencies and damping ratios of the second mode (the mode around 10 Hz, identified using [27])) of the structure under different environmental conditions. The error bar denotes ±2 standard deviation of the posterior identification uncertainty. The natural frequencies decrease with the increase of excitation amplitude. Increasing the temperature can also lead to a decrease of natural frequencies. The damping ratio generally increase with the excitation amplitude except for Mode 7. Increasing the temperature can also decrease the damping ratio, although it is not significant compared to that of natural frequencies. Figure 6.10 illustrates an example of divergence analysis. The squares in the figure denote the identified natural frequencies based on the measured data. A Gaussian process model is applied to investigate the functional behaviour of natural frequency against different excitation amplitudes and temperature, where the surface in the figure denotes its predicted mean values. The sphere in the figure denotes the natural frequency range based on a set divergence threshold. When the identified modal parameters are not within the sphere (i.e., the discrepancy is larger than the threshold) and away from the surface (i.e., may not be due to environmental change) for a certain degree (e.g., larger than 2 posterior standard deviation), the system model then needs to be updated. Fig. 6.6 Test setup of three-storey shear building

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g/√ (Hz)

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-1

-2

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Mode2 10.4 Room Temp 0 oC 10.35

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PSD of Modal Force (g /Hz)

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Fig. 6.8 Identified natural frequencies under different excitation amplitudes and temperature

6.6 Conclusion In this work, a real-time digital twin updating strategy is proposed for vibration-based structural health monitoring systems. The strategy introduces an intermediate model between the real structure and its digital twin model to increase the computational efficiency, which allows the updating strategy to be conducted in a real-time manner. Divergence analysis including environmental condition estimation and divergence quantification is conducted based on the associated parameters of the intermediate model to determine whether system model updating is needed or not. The whole strategy is encapsulated within a Bayesian framework so that the uncertainty can be accounted for rigorously. A synthetic data example is presented to illustrate the overall updating procedure. It is shown that the proposed strategy is capable of tracking the stiffness change accurately and actively. A laboratory example is also presented to illustrate the divergence analysis where the dynamic properties of a shear building structure are investigated under different excitation levels and temperature. Further divergence

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Mode2 0.02 Room Temp 0 oC

0.018

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Fig. 6.9 Identified damping ratios under different excitation amplitudes and temperature

Fig. 6.10 Illustrative plot for divergence analysis

investigation will be conducted for this test. A linear actuator will be installed on the shear building structure in order to change the inter-storey stiffness of the structure. The effect of environmental change and system change can then be investigated simultaneously. Acknowledgements This paper is supported by UK Engineering & Physical Research Council (EP/R006768/1 and EP/S001565/1). The financial support is gratefully acknowledged.

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References 1. Åström, K.J., Eykhoff, P.: System identification – a survey. Automatica. 7, 123–162 (1971). https://doi.org/10.1016/0005-1098(71)90059-8 2. Reynders, E.: System identification methods for (operational) modal analysis: review and comparison. Arch. Comput. Methods Eng. 19, 51– 124 (2012). https://doi.org/10.1007/s11831-012-9069-x 3. Chang, P.C., Flatau, A., Liu, S.C.: Review paper: health monitoring of civil infrastructure. Struct. Heal. Monit. 2, 257–267 (2003). https:// doi.org/10.1177/1475921703036169 4. Farrar, C.R., Worden, K.: An introduction to structural health monitoring. Philos. Trans. A. Math. Phys. Eng. Sci. 365, 303–315 (2007). https:/ /doi.org/10.1098/rsta.2006.1928 5. Brownjohn, J.M.W.: Structural health monitoring of civil infrastructure. Philos. Trans. A. Math. Phys. Eng. Sci. 365, 589–622 (2007). https:// doi.org/10.1098/rsta.2006.1925 6. Sohn, H., Farrar, C.R., Hemez, F., Czarnecki, J.: A review of structural health monitoring literature 1996–2001. Third World Conf. Struct. Control. 1–7 (2002) 7. Salawu, O.S.: Detection of structural damage through changes in frequency: a review. Eng. Struct. 19, 718–723 (1997). https://doi.org/10.1016/ S0141-0296(96)00149-6 8. Brownjohn, J.M.W., Xia, P.-Q.: Dynamic assesment of curved cable-stayed bridge by model updating. J. Struct. Eng. (2000). https://doi.org/ 10.1061/(Asce)0733-9445(2000)126:2(252 9. Yin, T., Lam, H.F., Chow, H.M., Zhu, H.P.: Dynamic reduction-based structural damage detection of transmission tower utilizing ambient vibration data. Eng. Struct. (2009). https://doi.org/10.1016/j.engstruct.2009.03.004 10. Cox, R.T.: The Algebra of Probable Inference. Johns Hopkins Press Baltimore, Baltimore (1961) 11. Malakoff, D.: Bayes offers a “new” way to make sense of numbers. Science. 80, (1999). https://doi.org/10.1126/science.286.5444.1460 12. Beck, J.L.: Bayesian system identification based on probability logic. Struct. Control Heal. Monit. 17, 825–847 (2010). https://doi.org/10.1002/ stc.424 13. Yuen, K.-V.: Bayesian Methods for Structural Dynamics and Civil Engineering. Wiley, New York (2010) 14. Wagg, D.J., Gardner, P., Barthorpe, R.J., Worden, K.: On key technologies for realising digital twins for structural dynamics applications. Conf. Proc. Soc. Exp. Mech. Ser. (2020). https://doi.org/10.1007/978-3-030-12075-7_30 15. Worden, K., Cross, E.J., Gardner, P., Barthorpe, R.J., Wagg, D.J.: On digital twins, mirrors and virtualisations. Conf. Proc. Soc. Exp. Mech. Ser. (2020). https://doi.org/10.1007/978-3-030-12075-7_34 16. Au, S.-K., Zhang, F.-L.: Fundamental two-stage formulation for Bayesian system identification, Part I: General theory. Mech. Syst. Signal Process. 1–12 (2015). https://doi.org/10.1016/j.ymssp.2015.04.025 17. Beck, J.L., Katafygiotis, L.S.: Updating models and their uncertainties. I: Bayesian statistical framework. J. Eng. Mech. 124, 455–461 (1998). https://doi.org/10.1061/(ASCE)0733-9399(1998)124:4(455 18. Cross, E.J., Koo, K.Y., Brownjohn, J.M.W., Worden, K.: Long-term monitoring and data analysis of the Tamar Bridge. Mech. Syst. Signal Process. 35, 16–34 (2012). https://doi.org/10.1016/j.ymssp.2012.08.026 19. Worden, K., Cross, E.J.: On switching response surface models, with applications to the structural health monitoring of bridges. Mech. Syst. Signal Process. 98, 139–156 (2018). https://doi.org/10.1016/j.ymssp.2017.04.022 20. Ni, Y.Q., Zhou, H.F., Ko, J.M.: Generalization capability of neural network models for temperature-frequency correlation using monitoring data. J. Struct. Eng. (2009). https://doi.org/10.1061/(ASCE)ST.1943-541X.0000050 21. Manson, G., Worden, K., Allman, D.: Experimental validation of a structural health monitoring methodology. Part II. Novelty detection on a Gnat aircraft. J. Sound Vib. (2003). https://doi.org/10.1006/jsvi.2002.5167 22. Worden, K., Manson, G., Allman, D.: Experimental validation of a structural health monitoring methodology: Part I. Novelty detection on a laboratory structure. J. Sound Vib. (2003). https://doi.org/10.1006/jsvi.2002.5168 23. Blei, D.M., Kucukelbir, A., McAuliffe, J.D.: Variational inference: a review for statisticians. J. Am. Stat. Assoc. (2017). https://doi.org/10.1080/ 01621459.2017.1285773 24. Wang, Z., Fu, Y., Yang, R.-J., Barbat, S., Chen, W.: Validating dynamic engineering models under uncertainty. J. Mech. Des. (2016). https:// doi.org/10.1115/1.4034089 25. Xu, H., Jiang, Z., Apley, D.W., Chen, W.: New metrics for validation of data-driven random process models in uncertainty quantification. J. Verif. Valid. Uncertain. Quantif. (2015). https://doi.org/10.1115/1.4031813 26. Oberkampf, W.L., Roy, C.J.: Verification and Validation in Scientific Computing. Cambridge University Press, New York (2011). https:// doi.org/10.1017/CBO9780511760396 27. Au, S.-K., Zhang, F.-L., Ni, Y.-C.: Bayesian operational modal analysis: theory, computation, practice. Comput. Struct. 126, 3–14 (2013). https://doi.org/10.1016/j.compstruc.2012.12.015 Dr Yi-Chen Zhu is a research associate in the Department of Mechanical Engineering at the University of Sheffield, focusing on the development of verification and validation techniques for digital twin models via Bayesian inference and machine learning techniques. His research also involves operational modal analysis in structural health monitoring

Chapter 7

On the Fusion of Test and Analysis Ibrahim A. Sever

Abstract When designing complex engineering machines, such as an aero-engine, there are many factors that need to be kept in consideration. However in essence they boil down to two important elements: Safety and operational efficiency. Essential though these two factors are, they are also at odds with each other in terms of their demands on eventual designs. The need for operational efficiency drives engineering solution towards ever-so lighter structures with optimised complex designs often requiring the utilisation of novel materials. All important safety on the other hand requires that a deeper understanding of underlying physical behaviour is acquired to ensure that the mechanical integrity is never at risk. The job of engineers operating in this field is to make sure that they utilise all available tools to map the solution space for an optimal balance of these two conflicting demands. Keywords Fusion of test and analysis · FTA · Model validation

One such important tool is the use of simulation models. Considerable effort is put into the generation and execution of such models at the component, sub-assembly and whole engine levels for simulation of their dynamic behaviour. Since the results obtained from these simulations feed into critical decisions that will have to be made in defining operational boundaries, it is of paramount importance that underlying models are validated adequately to ensure they represent the actual physics they are put together to simulate. Model validation in this context refers to a structured process that uses carefully made measurements to gauge the proximity of the measured results to those predicted by the model, and, a process that identifies corrections that need to be made to the model (updating of model parameters or upgrading of features) if the degree of correlation is not suitable. Although this process is easy to identify with, it contains the fundamental assumption that understanding of the physics is in place and all that needs to be done is to include the right features and tune parameters accordingly. However, this assumption is not always justified. Today we are at a level where in some areas, such as in materials, the technological advances are well ahead of our understanding of their dynamic behaviour. We are compelled to use these materials but we lack the understanding to assess them in reliable simulation models. The same is also true for structures working in harsh environments with many complex interfaces, invoking strong nonlinearities. Although it is desirable to obtain cost-effective simulation models in the long term, not having such models should not be a reason for not using novel materials or complex designs. At this stage the potential the intelligent testing can offer in building the required understanding has to be recognised, rather than insisting on simulations that are not capable. The answer undoubtedly lies in the Fusion of Test and Analysis (FTA) where a clear migration of insights is achieved from test to simulation and the overall understanding is encapsulated, such as in surrogate models. The objective is to achieve products that are safe to operate and efficient to run. This talk will focus on an all-encompassing strategy incorporating structured integration of test and simulation to achieve these objectives. Benefits of the approach, as well as practical applications from various stages of the process will be presented.

I. A. Sever () Rolls-Royce Plc., Derby, UK e-mail: [email protected] © Copyright Rolls-Royce plc 2020 Z. Mao (ed.), Model Validation and Uncertainty Quantification, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47638-0_7

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Fig. 7.1 Progression of engineering understanding

Fig. 7.2 Experimental investigations that are appropriately combined with corresponding predictions provide a powerful means of demonstrating structural integrity and compliance

As alluded to, the engineering insight is generated either through observation of behaviour through experimental investigations or through simulation models that are created at the back of a particular theoretical understanding. Figure 7.1 presents the FTA approach as a systematic utilisation of these two means and depicts the evolution of a concept from a stage at which it is not understood at all to a stage where basic simulation models are put together in the light of preceding experimental explorations. In the scenario depicted in Fig. 7.1, the 2 dimensional space represents the whole of the engineering understanding. Typically the Analysis Only region is small and works where the result show that the condition is inside an empirically derived threshold with some acceptable level of margin or safety. Analysis and Test region corresponds to traditional Model Validation where the test is used as guide to improve analysis capabilities or simply confirm that the design intent is met. The Test Only region is often utilised when technology runs ahead of our understanding of its dynamics and testing is the only means available to us for verification of the compliance. Sophisticated novel materials are a good example to this. Inherent uncertainties in analysis, and, testing make it difficult for us to get the designs right first time. We can bring these uncertainties to an acceptable level by intelligently utilising test and analysis throughout the design process, and not just at the end of it. Smart testing plays a significant role and the general perception of testing being more expensive is false. Analysis can be just as expensive. We should be ready for “Test Only” option when it is more pragmatic (e.g. new technologies for which analysis capability is not developed yet), and not shy away from “Analysis Only” option if there is sufficient confidence in the physics being simulated. The reality is that significant improvements are needed to both the testing and the analysis areas for them to stay relevant to demands of the new designs. The ultimate objective is to meet the Structural Performance requirements, and, a balanced approach in utilisation of Test and Analysis offers an effective solution (Fig. 7.2).

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Ibrahim A. Sever graduated from METU in Ankara and received his PhD from Imperial College in London. He is a Rolls-Royce Engineering Associate Fellow in Experimental Vibration working in Derby, England. His current research interests are model validation, damping estimation, dynamics of composite components, nonlinear response characterisation, engine data analysis, digital signal processing and non-contact measurement systems.

Chapter 8

Design of an Actuation Controller for Physical Substructures in Stochastic Real-Time Hybrid Simulations Nikolaos Tsokanas and B. Stojadinovic

Abstract Real-time hybrid simulation is a method to obtain the response of a system subjected to dynamic excitation by combining loading-rate-sensitive numerical and physical substructures. The interfaces between physical and numerical substructures are usually implemented using closed-loop-controlled actuation systems. In current practice, the parameters that characterize the hybrid model are deterministic. However, the effect of uncertainties may be significant. Stochastic hybrid simulation is an extension of the deterministic hybrid simulation where the parameters of the system are treated as random variables with known probability distributions. The results are probability distributions of the structural response quantities of interest. The arising question is to what extent does the actuation control system at the interface between physical and numerical substructures affect the outcomes of stochastic hybrid simulations. This question is most acute for real-time hybrid simulations. The response of a benchmark stochastic prototype to random excitation will be computed. Then, a part of the prototype will be replaced by a hybrid model whose substructure interfaces are actuated in closed-loop control. A controller that guarantees robustness and stability of the interfaces will be designed. The parameters of this hybrid model will be treated as random variables in repeated real-time hybrid response simulations to the same random excitation. The difference between the prototype and hybrid model responses will be used to determine if the controller design has an effect on the simulation outcomes, to predict such effects, and to propose guidelines for real-time controller design such that it has a predictable effect on the hybrid simulation. Additional criteria based on peak and root mean square tracking errors, as well as energy errors, are addressed in order to assess the overall system performance. Based on simulation data, surrogate models will be developed. Multiple additional runs of the surrogate models will give insight into the robustness and performance of the control system under uncertainties. Global sensitivity analysis of the overall system response will also be performed, identifying the most sensitive stochastic input variables. Cross-check validation of the results will take place using different meta-modeling techniques. Keywords Stochastic hybrid simulation · Uncertainty quantification · Surrogate modelling · Sensitivity analysis · Model-predictive control

8.1 Introduction Hybrid simulation (HS), formerly also called the hardware-in-the-loop (HIL) simulation or the online computer-controlled test method [1], is an experimental testing technique in which a simulation is executed based on a step-by-step numerical solution of the governing equations of motion for a model developed considering both loading-rate-sensitive numerical and physical components of a structural system. It’s a very effective technique since it incorporates the flexibility and cost-effectiveness of computer-aid simulations with the veracity of experimental testing to offer a novel and elegant tool for investigating the dynamic response of structural systems whose dimensions and complexities exceed the capacity of conventional testing facilities. One of the major benefits of HS is that the components that compromise critical parts of the overall system and are difficult to model can be tested experimentally, thereby revealing response that is not necessarily foreseen in simulations [2, 3].

N. Tsokanas () · B. Stojadinovic ETH Zürich, IBK, D-BAUG, Zürich, Switzerland e-mail: [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2020 Z. Mao (ed.), Model Validation and Uncertainty Quantification, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47638-0_8

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Real-time hybrid simulation (RTHS) [4] has been developed in the past two decades enabling researches to run experiments in real-time. Albeit attractive, real-time hybrid simulation comes with numerous challenges since it requires all necessary procedures, such as experimental data acquisition and computational calculations, to be made in the range of milliseconds or even less [5]. Furthermore, boundary conditions need to be applied on the physical substructures in the same time intervals in order to obtain representative results, meaning that dynamic actuators performing on such time rates are needed. Hence quick hardware and software are crucial elements needed to perform RTHS. In cases that such equipment is not available or scaling has to be made, similitude laws should be taken into consideration when designing a RTHS or a fast hybrid simulation. Using similitude allows us to extend the notion of real-time hybrid simulation from strictly conducting simulations at the time scale of one to conducting simulations such that the rate-sensitive phenomena are correctly reproduced and to account for time scale distortion [6, 7]. Apart from time-sensitive phenomena, another critical issue in RTHS is time delays due to the kinematics of the actuator. Delays are inevitably introduced to the system due to the dynamics of the transfer system in combination with the timeconsuming procedure of data acquisition and interaction with the physical substructure [8–10]. These time delays have been shown to be equivalent to adding negative damping to the simulated structure [8]. As a result, instabilities in RTHS are caused when the negative damping is greater or equal to the numerical damping of the system. Therefore, time delays should be effectively compensated to guarantee robust performance of the hybrid model. Lately, control engineering theory was used to develop time delay compensation techniques to guarantee desirable performance of the interfaces between physical and numerical substructures [5, 11–13]. An additional crucial aspect in RTHS, and HS in general, is the fact that the parameters of the hybrid model whose dynamic response is being simulated, are often treated as deterministic. The values of these parameters are regularly determined through deliberate simplifications, ignoring the associated uncertainties. However, the effect of uncertainties may be significant. Stochastic hybrid simulation (SHS) is a significant extension of the state-of-the-art HS to address the dynamic response of uncertain structural systems under uncertain operating conditions and uncertain excitation. Under this concept, the parameters of the hybrid model are treated as random variables with known probability distributions. The results are probability distributions of the structural response quantities of interest. Robust control theory techniques are employed to deal with the presence of uncertainties [9, 11]. The arising question is to what extent does the closed-loop actuation control system at the interface between physical and numerical substructures affect the outcomes of SHS, while maintaining high fidelity of the simulation in the presence of uncertainties. Performing stochastic modelling and simulations requires adequate computational power and large dataset capacity since repeated testing of the stochastic hybrid model is essential. Surrogate modelling, also called meta-modelling, aims at decreasing the high costs of stochastic simulation by replacing the original computationally expensive model with simplified surrogates. Acquiring enough simulation data enables global sensitivity analysis for the simulated stochastic hybrid model. Sensitivity analysis aims at quantifying the relevant effects of the input random variables onto the variance of the stochastic hybrid model response considering the entire input space [14]. In this paper, model predictive control (MPC) will be implemented in the scope of the benchmark problem defined by Silva et al. [15] to evaluate the performance and robustness of a virtual RTSHS (vRTHS) and also a stochastic framework will be developed in order to conduct SHS. The outcomes of vRTHSs with the MPC and the proposed PID controller will be compared. Definition of the benchmark problem, the reference structure, and the SHS model parameters and their probability distributions is presented first. Then, the response of the benchmark stochastic prototype to random excitation is computed and treated as the reference model. Thereafter, the prototype is replaced by a virtual SHS model whose substructure interfaces are actuated in closed-loop control using MPC. Repeated real-time simulations are conducted to assess the performance and robustness of the tracking controller of the SHS model. Result post-processing is performed using uncertainty quantification techniques. Based on simulation data, three different surrogate models are developed: (1) based on polynomial chaos expansion (PCE), (2) Kriging, and (3) polynomial chaos kriging (PCK). Different meta-modelling techniques are used to cross-check and validate the results. Finally, global sensitivity analysis is performed using Sobol indices in order to identify the most sensitive stochastic input variables.

8.2 Problem Definition The reference model of the RTHS benchmark problem [15] is shown in Fig. 8.1. It is a three-story, two-bay, planar steel moment frame. A linear-time-invariant (LTI) numerical model with 3 lateral degrees-of-freedom (DOF) at the floor locations is provided for the reference structure. Its equation of motion (EOM) follows:

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Fig. 8.1 Reference model of the RTHS benchmark problem. (From [15]) Table 8.1 Different cases of the RTHS reference model

Case 1 2 3 4

M r x¨ + C r x˙ + K r x = −M r i x¨g

M (kg) 1000 1100 1300 1000

ζ (%) 5 4 3 3

(8.1)

where Mr , Cr and Kr are the mass, damping and stiffness matrices of the reference structure model, respectively, x, x˙ and x¨ are the displacement, velocity and acceleration, respectively, all relative to the ground motion, x¨ g is the ground acceleration and i is the influence coefficient vector, which contains a value of one for each mass that develops an inertial force when the system is accelerated horizontally. Additionally, zero initial conditions are considered, i.e. x(0) = x˙ (0) = {0, 0, 0}T . Using modal damping ratios, the classical damping matrix of the reference structure follows: −1 Cr = ΦT diag (2ζ1 ω1 m1 , 2ζ2 ω2 m2 , 2ζ3 ω3 m3 ) Φ −1

(8.2)

where Φ is the modal matrix, ζ i is the modal damping ratio, ωi is the natural frequency related to the i-th mode of vibration and mi is the generalized modal mass of the i-th floor. Four cases of different deterministic structural parameters are proposed in the RTHS benchmark problem (Table 8.1). In this study instead, in order to investigate the robustness of the RTHS closed-loop actuation system in a wide range of parameter variability, specific deterministic parameters of the reference structure are selected and they are replaced with stochastic variables. These parameters are: 1. 2. 3. 4. 5. 6.

Beam length, Lb Column length, Lc second moment of area x beam. Ib second moment of area x column. Ic Floor mass, Mass Modal damping ratio, Zeta

The above parameters are treated as random variables. Probability distributions assigned to these random variables are summarized in Table 8.2 and illustrated in Fig. 8.2. All of those six stochastic input variables are modeled to be independent. The selection of these specific parameters was made in order to introduce a stochastic nature to the three main variables that represent the reference structure; mass, damping and stiffness.

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Table 8.2 Stochastic input variables Input variable Lb Lc Ib Ic Mass Zeta

Probability distribution Beta Beta Beta Beta Gaussian Beta

Nominal value (μ) 0.762 0.635 2.5527e-07 1.0489e-06 1000 0.05

Standard dev. (σ) 0.3 0.3 7.6580e-08 3.1467e-07 20 0.015

Coefficient of variation (CV) 0.3937 0.4724 0.3 0.3 0.02 0.3

Bounds [min, max] [Lb −0.005, Lb +0.005] [Lc −0.005, Lc +0.005] – – – –

Units m m m4 m4 kg −

Fig. 8.2 Probability distributions of the stochastic input variables listed in Table 8.2

After computing the response of the reference model, substructuring techniques are employed to divide the reference model into two substructures, as shown in Fig. 8.3. The left part of the first story is treated as the experimental substructure, while the remaining structure is the numerical substructure. The mass, stiffness and damping matrices of the reference structure are restated as sums of the numerical and experimental substructures, given by M r = M e + M n, K r = K e + K n, Cr = Ce + Cn

(8.3)

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Fig. 8.3 Reference, numerical and experimental substructrured models. (From [15])

Combining (8.1) and (8.3) the EOM of the hybrid model is obtained and follows: M n x¨ + C n x˙ + K n x = −M r i x¨g − M e x¨ + C e x˙ + K e x

(8.4)

where M e = diag (me , 0, 0) , K e = diag (ke , 0, 0) , C e = diag (ce , 0, 0) .

M e x¨ + C e x˙ + K e x = f e

(8.5)

corresponds to the force feedback vector associated with the experimental substructure.

8.3 Dynamics of Control Plant and Actuation System In this benchmark problem the control plant consists of a transfer system (a hydraulic actuator for this case) and the virtual experimental substructure as shown in Fig. 8.4. The transfer function which governs the motion of the control plant is given by: B0 + A3 s 3 + A2 s 2 + A1 s + A0

(8.6)

B0 = a1 β0 A0 = ke a3 β2 + a1 β0 A1 = ke a3 β1 + (ke + ce a3 + a2 ) β2 A2 = ke a3 + (ke + ce a3 + a2 ) β1 + (ce + me a2 ) β2 A3 = (ke + ce a3 + a2 ) + (ce + me a3 ) β1 + me β2 A4 = ce + me a3 + me β1 A5 = me

(8.7)

GP (s) =

A5

s5

+ A4

s4

where

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Fig. 8.4 Control plant, consisting of the actuator coupled with the experimental substructure. (From [15])

Fig. 8.5 Block diagram of the control plant. (From Ref. [15]) Table 8.3 Parameters of the control plant. (From Ref. [15]) Parameter a1 β 0 a2 a3 β1 β2 me ke ce

Component Servo-valve CSI Actuator Servo-valve Servo-valve Exp. sub. Exp. sub. Exp. sub.

Probability distribution – – Gaussian Gaussian Gaussian – – –

Nominal value (μ) 2.13 × 1013 4.26 × 106 3.3 425 10 × 104 29.12 1.19 × 106 114.6

Standard dev. (σ) – – 1.3 3.3 3.3 × 103 – – –

Units mPa/s mPa 1/s – 1/s kg N/m kg/s

The block diagram of the control plant is shown in Fig. 8.5. It consists of the individual dynamics of the servo-valve, the actuator, the experimental substructure and the control-structure interaction (CSI), represented in a controllable canonical form by Eq. 8.6 [16]. The servo-valve is modeled as a second-order linear system with the parameters a1 β 0 , β 1 and β 2 , the hydraulic actuator as a first-order transfer function with the parameter a3 , and the CSI as a first-order linear system with the parameter a2 . To account for simplifications on the modeling process, model uncertainties in the control plant are also introduced and are summarized in Table 8.3 along with the description of each parameter used in the control plant model. The hybrid model examined in this paper consists of the numerical substructure, the tracking controller and the control plant, as shown in Fig. 8.6. The input excitation to the numerical substructure is represented by the ground motionx¨g , and the displacement, yn , is the input signal in the tracking controller. The output of the controller is the compensated signal yGC and this is the command to the actuator in order to apply the desired displacement in the physical substructure. The restoring force, fe , and the displacement of the physical substructure, xm , are acquired from the data acquisition system. The measurement of the restoring force is sent back to the numerical substructure to compute the next step of the hybrid simulation, while the measurement of the displacement is sent to the tracking controller to close the loop and complete the control feedback.

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Fig. 8.6 Block diagram of the hybrid model for vRTHSs

8.4 Model Predictive Control Model predictive control (MPC) was employed to design the tracking controller. The main difference between conventional control and MPC is that the first one is using a pre-calculated control law, which remains the same for the whole simulation, while in the latter a new one is calculated at every time step. More specific, at each time interval a finite horizon optimization problem is being solved on-line, providing the plant with the new control action by using the current state of the plant as the initial state. This results in a control sequence that is applied to the hybrid model at every time step [17–19]. As mentioned before, uncertainties are inevitable in hybrid simulation. They originate either from the physical substructure (aleatory uncertainties, i.e. natural variability or epistemic uncertainties, i.e. lack of knowledge) or from experimental errors (systematic uncertainties, i.e. calibration errors or random uncertainties, i.e. measurement noise). Making use of a control methodology that accounts for all uncertainties that enter into the hybrid model and computes a control law which compensates for these uncertainties at every time step is a significant extension that can guarantee high fidelity of hybrid simulation as well as robustness to uncertainties that are introduced during the simulation process. In addition, MPC enables solving on-line optimization problems under hard constraints. In hybrid simulation, this is often useful, since experimental equipment is naturally limited in the capabilities it can offer, e.g. actuators are limited in the forces or displacements they can apply. The implementation of MPC in the benchmark problem is shown in Fig. 8.7. Specifically, the PID controller of the RTHS benchmark problem was replaced by an MPC. The phase lead compensation was kept the same as proposed in the RTHS benchmark problem. It’s described by the following transfer function: Gpn =

50.83s + 8570 s + 8570

(8.8)

The frequency rate of vRTHS was set to fs = 4096 (Hz). For the MPC case, an additional sampling frequency is needed for computing the new control law at every time step. This was set to fMPC = 1024 (Hz). In order to have representative results fMPC should be smaller than fs . The control plant is represented by a Single-Input-Single-Output (SISO), LTI model with the following state-space equations: x˙ = Ax + By GC xm = Cx + Dy GC

(8.9)

In this paper, the following formulation of the MPC problem is used: P     x  2 w m yp (k + i|k) − xm (k + i|k) J ∗ yp , xm zk = i=1

(8.10)

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Fig. 8.7 Schematic of the MPC implementation

where k represents the current control interval, P the prediction horizon, wxm the tuning weight of the measured displacement, xm , yp (k + i| k) the reference value to be tracked at the i-th prediction horizon step, xm (k + i| k) the predicted value  of the measured displacement at the i-th prediction horizon step and zkT = u(k|k)T u(k + 1|k)T . . . u(k + 1 − P |k)T consists of the control laws for every control interval k. The term, J∗ (yp , xm zk ), serves as the cost that penalizes the tracking error. This cost is minimized at every control interval k. A new control command is computed at every interval and it’s provided to the actuation system. The prediction horizon is set to P = 10, the tuning weight to wxm = 13.53, the next reference value yp (k + i| k) is being calculated from the numerical substructure and the xm (k + i| k) is predicted from a Kalman filter, designed for optimal state estimation. The control law is derived in every interval k, from the solution of the following optimization problem:   minJ ∗ yp , xm zk

(8.11)

−7 ≤ yp (k + i − 1|k) ≤ 7 (mm), i = 1, . . . , P ,

(8.12)

zk

which is subjected to the following constraints:

−25 ≤ y˙p (k + i − 1|k) ≤ 25

mm  sec

, i = 1, . . . , P

(8.13)

The above constraints correspond to the natural limitations of the physical actuator. Namely, the actuator has a maximum stroke of ±7 mm and a maximum velocity of 25 mm/sec.

8.5 Surrogate Modelling Classical uncertainty propagation methods demand large computational cost and extensive simulation data based on iterative simulations (e.g. in a Monte Carlo framework), even for simple hybrid models. On the other hand, surrogate modeling can provide simplified approximations of the original model with acceptable validation errors and accurate model responses at a much lower computational cost. In this paper, three different surrogate techniques are used; polynomial chaos expansion (PCE), Kriging, and polynomial chaos Kriging (PCK). In PCE, the system response is approximated by a sum of multivariate orthogonal polynomials [20]. Let’s define with Y = M(X) the finite-variance response of a computational model where X = {X1 , . . . , XM } correspond to independent random input variables. The PCE approximation of the system response follows: Y P CE =

P −1 

yi i (X) + ερ := y T  (X) + ερ

(8.14)

i=0

! M +p is the truncation scheme, with AM, p = {a ∈ NM :| a| ≤p}, M equal to the number p of input random variables and p the total degree corresponding to all polynomials, Ψ i (X) are the multivariate orthogonal polynomials, ερ the truncation error and yi are scalar coefficients. The latter coefficients are calculated using the Least Angle Regression (LARS) methodology, a technique for adaptive sparse PCE. The minimization problem reads:   where card AM,p := P =

8 Design of an Actuation Controller for Physical Substructures in Stochastic Real-Time Hybrid Simulations

yˆ = argminE



y∈R|A|

T

y  (X) − M (X)

2 

+ λy1

77

(8.15)



# where yˆ 1 = α∈A | yα | is a regularization term to select low rank solutions of the minimization problem. The solution reads: −1 yˆ = AT A AT Y

(8.16)

where Y = {M (X1 ) , . . . , M (XM )} and Aij = Ψ j (x(i) ), i = 1, . . . , M, j = 0, . . . , P − 1 is the experimental matrix containing the values of all the orthogonal polynomials at the experimental design points. In Kriging meta-modelling, the system response is approximated with a realization of a Gaussian process: Y Krig = β T f (x) + σ 2 Z (x, ω)

(8.17)

where β T f(x), the trend, corresponds to the mean of the Kriging meta-model with {β j , j = 1, . . . , P} the regression coefficients and {fj , j = 1, . . . , P} the regression functions, Z(x, ω) is a zero-mean, unit variance, stationary Gaussian process and σ 2 is the variance of the Gaussian process [21]. Changes in trend types result in different Kriging meta-models. Common trend types are the constant (also called ordinary Kriging), linear, quadratic and polynomial. The trend of the Kriging meta-model used for this study is the quadratic and the polynomial. The latter corresponds to the PCK case and is addressed later on. The stationary process Z(x, ω) is determined by a probability space ω and a correlation function R = R(xi , xj ; θ ). The latter characterize the correlation between two samples based on the hyperparameters θ of the correlation function. The Kriging meta-model is trained with a set of experimental design points {X, Y} = {{x i , i = 1, . . . , n} , {M (x i ) , i = 1, . . . , n}} and, based on that, provides predictions of the model response for new points x. The mean and variance of the Kriging predictor read: μYˆ (x) = f (x)T β + r(x)T R −1 (y − F β)  −1  u (x) σ 2ˆ (x) = σ 2 1 − r T (x)R −1 r (x) + uT (x) F T R −1 F

(8.18)

Y

where r(x) = R(x, xi ; θ ), Fij = fj (xi ), β = (FT R−1 F)−1 FT R−1 y is obtained by generalized least-squares estimate and u(x) = FT R−1 r(x) − f(x). PCK is a combination of the above two surrogate techniques [22]. The Kriging methodology is used and PCE is employed as its polynomial trend type. The system’s response approximation follows: Y P CK =



yα α (X) + σ 2 Z (x, ω)

(8.19)

α∈A

For the calculation of the PCE coefficients, the LARS methodology is used again.

8.6 Global Sensitivity Analysis Global sensitivity analysis (GSA) aims at quantifying to what extent each one of the stochastic input variables affects the model response. The outcome of such analysis provides engineers with viable information regarding the system design. If, for instance, if a stochastic input variable has a negligible influence to the output, then this variable could be replaced by a deterministic value, reducing the model complexity and the computational cost of simulating the model. Moreover, if a variable has a significant effect on the system response, then the variability of this variable should be, if possible, decreased in order to achieve higher robustness to uncertainties. In this paper, PCE-based Sobol’ indices, also called ANOVA, are employed in order to conduct GSA. The idea behind Sobol’ indices is to decompose the full computational model into submodels which depend on an increasing number of input variables [23]. By rearranging the PCE model we get:

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M (x) =



1

{i1 ,...,is }⊂{1,...,M} Ai1 ,...,is

yα α (X)

(8.20)

{i1 ,...,is }⊂{1,...,M} α∈Ai1 ,...,is

α∈A

where





yα α (X) = M0 +

= NM . The Sobol decomposition then reads: 

M (x) = M0 +

Mi1 ,...,is (x1 , . . . , xs )

(8.21)

{i1 ,...,is }⊂{1,...,M}

By unicity of the Sobol’ decomposition we get: 

Mi1 ,...,is (x1 , . . . , xs ) :=

yα α (X)

(8.22)

ya2

(8.23)

α∈Ai1 ,...,is

The variance of the model reads: D = V ar [M (X)] =

 α∈A α = 0

The first-order Sobol’ indices are obtained as follows: 2 3  ya2 /D, Ai = a ∈ NM : ai > 0, aj =i = 0 Si =

(8.24)

α∈Ai

The higher-order indices: Si1 ,...,is =

2 3 ya2 /D, Ai1 ,...,is = a ∈ NM : k ∈ {i1 , . . . , is } ⇐⇒ aj = 0



(8.25)

α∈Ai1 ,...,is

And the total order indices: SiT =



2 3 ya2 /D, ATi = a ∈ NM : ai > 0

(8.26)

α∈ATi

The first-order Sobol’ indices quantify the effect of each single input variable at the output response, without considering the interactions between the input variables. On the contrary, the higher-order indices provide information on how the interactions of every input variable affect the output response. The total order indices quantify the overall contribution of each variable, both independently and the interactions with the others. A large first-order index indicates that the corresponding input variable has, by itself, a significant effect on the output, while a combination of small first-order and large higher-order, indicates that this variable has an effect on the output only when it interacts with the other variables.

8.7 Results For each one of the six stochastic input variables, 100 samples were generated from the probability distributions described in Table 8.2. Based on the above samples, 100 hybrid simulation runs were conducted using the same excitation. The same experiments were repeated replacing the MPC controller with the RTHS benchmark PID controller, in order to compare their performance and robustness. In the RTHS benchmark problem [15] nine evaluation criteria were developed, J1-J9, to quantify the controller performance. An additional J10 criterion was developed here, based on energy errors [24, 25], as follows: J10 :=

j  i=1

Ei =

j  i=1

EiBE − EiE

(8.27)

8 Design of an Actuation Controller for Physical Substructures in Stochastic Real-Time Hybrid Simulations

79

where j is the number of integration steps followed by the hybrid simulation, EiBE is the best estimate of the energy in the experimental substructure and EiE the energy in the experimental substructure computed by the numerical integrator in the j-th integration step of the hybrid simulation. The last two follow respectively: EiBE = EiE =

  1 e fi−1 + fie xmi − xmi−1 2

(8.28)

  1 e fi−1 + fie yn i − yn i 2

(8.29)

The results from the 100 runs, Fig. 8.8, showed that only criteria J2 and J3 have a relevant representative probability distribution that it can be used to investigate the robustness of the controller. The remaining criteria have a rather predictable response with respect to the selected stochastic inputs, and were therefore excluded from the following analysis. These criteria will likely have a larger variability due to the inelastic behavior of the structure and due to any potential variability in the excitation. Moreover, criterium J3, the peak tracking error, was also rejected from the analysis. Therefore, only on

Fig. 8.8 Response distributions of criteria J1-J10 based on 100 runs with the MPC controller

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Table 8.4 Statistical moments of criterion J2 obtained from hybrid and surrogate models; MPC – PID comparison

Mean value (μ) (%) Standard deviat. (σ) (%)

HS with MPC controller Hybrid model PCE 6.202 6.285 0.953 0.999

Kriging 6.257 0.954

PCK 6.263 0.998

HS with PID controller Hybrid model PCE 9.707 9.832 1.092 1.106

Kriging 9.811 1.131

PCK 9.831 1.117

Fig. 8.9 Kernel densities of criterion J2: RMS tracking error

criterium J2, the normalized root mean square (RMS) tracking error, was considered. Criterium J3 was rejected because extreme values as of peak tracking error are not well represented by the selected surrogate techniques and the validation error was quite large to ensure representative results. On the contrary, mean values, such as RMS, are very well represented by PCE and Kriging with acceptable validation errors. Based on the simulation data, the PCE, the quadratic Kriging and the PCK surrogate models were developed. The inputs of these meta-models are the same as in Table 8.2 and the output is criterium J2. Each of these models was simulated for one million sampling points of the stochastic input variables. The Latin Hypercube Sampling (LHS) method was used for sampling the points. More simulation data gives a better insight on how the response is affected by the stochastic input variables. The results of these runs are probability distributions of criterium J2. Figure 8.9 illustrates the kernel densities of these distributions both for the hybrid simulation with MPC and PID tracking controller. The validation errors were 2.6%, 1.83% and 1.65% for PCE, Kriging and PCK, respectively. In Table 8.4 the mean and standard deviations values of J2 acquired by the hybrid model and the PCE, Kriging and PCK meta-models for both MPC and PID case are presented. The values of the hybrid model are obtained after 100 runs while the values of the meta-models after one million runs. From Fig. 8.9 and Table 8.4 we can acknowledge that the MPC controller can maintain a higher level of simulation fidelity under uncertainties with less variability and more robustness. In Fig. 8.10 the total Sobol’ indices based on the PCE meta-model are illustrated for the MPC and the PID controller cases. The second-order indices are omitted since they are numerically very small. As a result, the first-order indices are almost identical to the total indices. It is clear from Fig. 8.10 that the two controllers cannot be compared because their variances are not the same [Eqs. 8.23. 8.24, 8.25 and 8.26] and, as a result, they are not computed using the same scaling. However, what we can observe is that neither controller could compensate sufficiently well for uncertainty in the moments of inertia of the beam Ib and the column Ic , since these two input variables have the largest effect in the output. Designing a controller to focus on compensating more for the uncertainties relating to these two inputs will result in better overall tracking performance of the hybrid model. The implementation of the surrogate modelling and the GSA was performed with the UQLab software framework developed by the Chair of Risk, Safety and Uncertainty Quantification in ETH Zurich [26].

8 Design of an Actuation Controller for Physical Substructures in Stochastic Real-Time Hybrid Simulations Fig. 8.10 Total Sobol’ indices for the MPC and PID controller case

81

Total Sobol’ indices for J2:RMS 1 MPC PID 0.8

STi

0.6

0.4

0.2

0 Lb

Lc

Ib Ic Mass Zeta Input variable

8.8 Conclusion The RTHS benchmark problem [15] was used to investigate the effect of uncertainties and controller design on the outcomes of hybrid simulations. The dynamic response of a deterministic hybrid model was measured and treated as a reference. This hybrid model was then replaced with a stochastic hybrid model, assigning probability distributions to selected inputs variables. The simulation results are compared in terms of the probability distribution of the RMS tracking error. An MPC controller was developed and compared to the PID controller of the RTHS benchmark problem. A SHS framework was developed in order to conduct stochastic simulation and to appropriately perform the aforementioned comparison. Three different surrogate models were developed and trained on simulation data. It was found that an MPC can guarantee better tracking performance of stochastic hybrid simulations under uncertainties, proving its robust characteristics. Sensitivity analysis pointed out that uncertainty in the bending stiffnesses of the RTHS benchmark problem structural elements affect the output the most. The findings presented in this paper are the basis for design better tracking controllers to ensure robustness and high fidelity simulations stochastic hybrid simulations under uncertainties. Acknowledgements This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 764547. The sole responsibility of this publication lies with the author(s). The European Union is not responsible for any use that may be made of the information contained herein.

References 1. Schellenberg, A.H., Mahin, S.A., Fenves, G.L.: Advanced Implementation of Hybrid Simulation. Technical Report 2009/104, Pacific Earthquake Engineering Research Center (2009) 2. Al-Mahaidi, R., Hashemi, J., Kalfat, R., Burnett, G., Wilson, J.: Multi-Axis Substructure Testing System for Hybrid Simulation. Springer, Singapore (2018) 3. Abbiati, G., Marelli, S., Bursi, O.S., Sudret, B., Stojadinovic, B.: Uncertainty propagation and global sensitivity analysis in hybrid simulation using polynomial Chaos expansion. In: Kruis, J., Tsompanakis, Y., Topping, B.H.V. (eds.) Proceedings of the Fourth International Conference on Soft Computing Technology in Civil, Structural and Environmental Engineering, Stirlingshire, Scotland. Civil-Comp Press (2015) 4. Nakashima, M., Kato, H., Takaoka, E.: Development of real-time pseudo dynamic testing. Earthq. Eng. Struct. Dyn. 21(1), 79 (1992) 5. Fermandois, G.A.: Application of model-based compensation methods to real-time hybrid simulation benchmark. Mech. Syst. Signal Process. 131, 394–416 (2019) 6. Stojadinovic, B., Mosqueda, G., Mahin, S.A.: Event driven control system for geographically distributed hybrid simulation. J. Struct. Eng. 132(1), 68–77 7. Kumar, S., Itoh, Y., Saizuka, K., Usami, T.: Pseudodynamic testing of scaled models. J. Struct. Eng. 123(4), 524–526 (Apr 1997) 8. Horiuchi, T., Inoue, M., Konno, T., Namita, Y.: Real-time hybrid experimental system with actuator delay compensation and its application to a piping system with energy absorber. Earthq. Eng. Struct. Dyn. 28(10), 1121–1141 (1999) 9. Ning, X., Wang, Z., Zhou, H., Wu, B., Ding, Y., Xu, B.: Robust actuator dynamics compensation method for real-time hybrid simulation. Mech. Syst. Signal Process. 131, 49–70 (2019) 10. Wang, Z., Xu, G., Wu, B., Bursi, O.S., Ding, Y.: An effective online delay estimation method based on a simplified physical system model for real-time hybrid simulation. Smart Struct. Syst. 14(6), 1738–1991 (2014)

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11. Carrion, J.E., Spencer, B.F.: Model-Based Strategies for Real-Time Hybrid Testing. Technical report, Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign (2007) 12. Ou, G., Ozdagli, A.I., Dyke, S.J., Wu, B.: Robust integrated actuator control: experimental verification and real-time hybrid-simulation implementation. Earthq. Eng. Struct. Dyn. 44(3), 441–460 (2015) 13. Gao, X., Castaneda, N., Dyke, S.J.: Real time hybrid simulation: from dynamic system, motion control to experimental error. Earthq. Eng. Struct. Dyn. 42(6), 815–832 (May 2013) 14. Sudret, B.: Global sensitivity analysis using polynomial chaos expansions. Reliab. Eng. Syst. Saf. 93, 964–979 (2008) 15. Silva, C.E., Gomez, D., Maghareh, A., Dyke, S.J., Spencer Jr., B.F.: A benchmark control problem for real-time hybrid simulation. Mech. Syst. Signal Process. (2019) 16. Maghareh, A., Silva, C.E., Dyke, S.J.: Servo-hydraulic actuator in controllable canonical form: identification and experimental validation. Mech. Syst. Signal Process. 100, 398–414 (Feb 2018) 17. Mayne, D.Q., Rawlings, J.B., Rao, C.V., Scokaert, P.O.M.: Constrained model predictive control: stability and optimality. Automatica. 36(6), 789–814 (2000) 18. Limon, D., Alamo, T., de la Peña, D.M., Zeilinger, M.N., Jones, C.N., Pereira, M.: MPC for tracking periodic reference signals. IFAC Proc. 45(17), 490–495 (2012) 19. Zeilinger, M.N., Jones, C.N., Morari, M.: Real-time suboptimal model predictive control using a combination of explicit MPC and online optimization. IEEE Trans. Autom. Control. 56(7), 1524–1534 (2011) 20. Marelli, S., Sudret, B.: UQLAB User Manual – Polynomial Chaos Expansions. Report UQLab-V1.2-104, Chair of Risk, Safety & Uncertainty Quantification, ETH Zurich (2019) 21. Lataniotis, C., Marelli, S., Sudret, B.: UQLAB User Manual – Kriging (Gaussian Process Modelling). Report UQLab-V1.2-105, Chair of Risk, Safety & Uncertainty Quantification, ETH Zurich (2019) 22. Schobi, R., Marelli, S., Sudret, B.: UQLAB User Manual – Polynomial Chaos Kriging. Report UQLab-V1.2-109, Chair of Risk, Safety & Uncertainty Quantification, ETH Zurich (2019) 23. Marelli, S., Lamas, C., Konakli, K., Mylonas, C., Wiederkehr, P., Sudret, B.: UQLAB User Manual – Sensitivity Analysis. Report UQLabV1.2-106, Chair of Risk, Safety & Uncertainty Quantification, ETH Zurich (2019) 24. Mosqueda, G., Stojadinovic, B., Mahin, S.A.: Real-time error monitoring for hybrid simulation. Part I: methodology and experimental verification. J. Struct. Eng. 133(8), 1100–1108 (2007) 25. Mosqueda, G., Stojadinovic, B., Mahin, S.A.: Real-time error monitoring for hybrid simulation. Part II: structural response modification due to errors. J. Struct. Eng. 133(8), 1109–1117 (2007) 26. Marelli, S., Sudret, B.: UQLab: a framework for uncertainty quantification in Matlab. In: Proceedings 2nd International Conference on Vulnerability, Risk Analysis and Management (ICVRAM2014), Liverpool, United Kingdom, pp. 2554–2563 (2014) Nikolaos Tsokanas received his diploma from the faculty of Electrical Engineering, University of Patras, Greece. During his studies, he undertook internships in Airbus Defence & Space, Germany, and in the faculty of Aerospace Engineering in TU Delft. Currently, he is a PhD student in ETH focusing on real-time hybrid testing under uncertainties.

Chapter 9

Output-Only Nonlinear Finite Element Model Updating Using Autoregressive Process Juan Castiglione, Rodrigo Astroza, Saeed Eftekhar Azam, and Daniel Linzell

Abstract A novel approach to deal with nonlinear system identification of civil structures subjected to unmeasured excitations is presented. Using only sparse global dynamic structural response, mechanics-based nonlinear finite element (FE) model parameters and unmeasured inputs are estimated. Unmeasured inputs are represented by a time-varying autoregressive (TAR) model. Unknown FE model parameters and TAR model parameters are jointly estimated using an unscented Kalman filter. The proposed method is validated using numerically simulated data from a 3D steel frame subjected to seismic base excitation. Six material parameters and one component of the base excitation are considered as unknowns. Excellent input and model parameter estimations are obtained, even for low order TAR models. Keywords Model updating · Input estimation · Finite element model · Kalman filter · Auto-regressive model

9.1 Introduction Nonlinear finite element (FE) models properly calibrated with data recorded during damage-inducing events provide a powerful approach for damage identification (DID) and structural health monitoring (SHM). Distefano et al. [1] conducted a pioneering work in this subject for civil structures subjected to seismic excitation. In recent years, different methods have been proposed to update nonlinear mechanics-based FE models using input-output measured data (e.g., [2–4]). However, a limited number of studies have tackled the problem of calibrating FE models of structures subjected to non-stationary and non-broadband excitation. Most of these studies have focused on linear FE models (e.g., [5–7]), and just a few efforts have dealt with nonlinear FE models under unmeasured excitation (e.g., [8–10]). In this paper, a new approach to update mechanics-based nonlinear FE models is proposed. The unknown parameters of the FE model and the unknown parameters of a time-varying auto-regressive (TAR) model used to characterized the unmeasured excitation are estimated using the recorded response of the structure and the unscented Kalman filter (UKF) [11]. The method is validated using numerically simulated data from realistic streel frame structure subjected to seismic load.

9.2 Proposed Method The equation of motion of a nonlinear FE model can be written as:      M θf em q¨ k+1 θf em + C θf em q˙ k+1 θf em + rk+1 q1:k+1 θf em = fk+1 + gk+1

(9.1)

J. Castiglione · R. Astroza () Faculty of Engineering and Applied Sciences, Universidad de los Andes, Santiago, Chile e-mail: [email protected]; [email protected] S. E. Azam · D. Linzell Department of Civil Engineering, University of Nebraska-Lincoln, Lincoln, NE, USA e-mail: [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2020 Z. Mao (ed.), Model Validation and Uncertainty Quantification, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47638-0_9

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˙ q¨ are the vectors of displacement, velocity, and acceleration With θfem is the vector with the unknown model parameters; q, q, responses, respectively; M and C are the mass and damping matrices, r is the vector of internal resisting forces; and f and g are the known and unknown external force vectors. In the case of uniform base excitation, the external force vectors can be expressed, at time step (k + 1) as fk+1 = −MLu u¨ k+1 and gk+1 = −MLs s¨k+1 , where Lu and Ls are the corresponding influence matrices, and u¨ and s¨ are the known and unknown (unmeasured) ground acceleration time histories. The latter can be represented using a TAR model as shown in Eq. (9.2): (j )

s¨k+1 = −

p 

(j )

ak+1,i s¨k+1−i + ek+1 = sˆ¨k+1 + ek+1 (j )

(j )

(j )

(j )

(9.2)

i=1

Then, an augmented vector containing FE model parameters (θ fem ) and TAR model parameters (θ inp ) can then be defined: θ=

 T T T θinp θf em

(9.3)

where θ is the vector of parameters to be estimated. From Eq. (9.1), it is observed that the response of the nonlinear FE model at time step (k + 1) can be written as a nonlinear function of θ, u¨ 1:k+1 , and q0 , q˙ 0 (initial conditions) as  yˆ k+1 = lk+1 θf em , θinp , u¨ 1:k+1 , q0 , q˙ 0

(9.4)

where yˆ k+1 is the FE-predicted response vector and lk + 1 denotes a nonlinear response function. The real response (yk + 1 ) of the structure can be measured using different types of sensors and is associated with the FE-predicted response by: yk+1 = yˆ k+1 + vk+1

(9.5)

Where vk + 1 is the simulation error vector, assumed Gaussian white, stationary with zero-mean and covariance matrix R. Then, the following nonlinear state-space model can be formulated: θk+1 = θk + wk y1:k+1 = hk+1 (θ, u¨ 1:k+1 , q0 , q˙ 0 ) + v1:k+1 ,

(9.6)

Here, wk is the process noise, assumed Gaussian white and stationary with zero-mean and covariance matrix Q. An UKF is employed for estimating θ.

9.3 Validation 9.3.1 Structure, FE Model and Measured Response A 3D steel frame (Fig. 9.1a) subjected to the ground motion recorded at the Sylmar Country Hospital station during the 1994 Northridge earthquake is considered as application example. A nonlinear FE model (Fig. 9.1b) was implemented in OpenSees using nonlinear, fiber-section, beam-column elements. One element with seven integration points was considered for each beam and column, while the Giuffré-Menegotto-Pinto model was used to characterized the nonlinear uniaxial response of the steel fibers. A total of six material model parameters, i.e., θ fem = [Ecol , fcol , bcol , Ebeam , fbeam , bbeam ]T , and the transverse input component (i.e., 90◦ component of the Sylmar record) are assumed unknown. The acceleration response at every level of the building was generated with the following true model parameters θ fem,true = [200GPa, 345 MPa, 0.04, 200GPa, 250 MPa, 0.03]T , contaminated with additive white Gaussian noise with root-mean-square amplitude of 0.5%g, and assumed measured (see green arrows in Fig. 9.1a).

9 Output-Only Nonlinear Finite Element Model Updating Using Autoregressive Process

b

6

e (Z) Transvers

3

4

5

) Time(sec

7

8

9

10

a

85

0

(g c. A c (Z)

)

Fig. 9.1 (a) 3D steel frame structure, (b) FE model Table 9.1 Estimation results

Model parameter Case ID

p

1 2 3

4 6 10

Input RRMSE

Ecol

f col

bcol

Ebeam

fbeam

true Ecol

true f col

true bcol

true Ebeam

true fbeam

bbeam true bbeam

Unfiltered

Filtered

1.00 1.00 1.00

1.00 1.00 1.00

1.11 1.09 1.21

1.00 1.00 1.00

1.01 1.01 1.01

0.80 0.80 0.76

11.1 10.7 11.7

4.1 3.9 4.3

9.3.2 Estimation Results Three different orders of the TAR model are analyzed, p = 4, 6, and 10. Table 9.1 summarizes the estimation results of the model parameters and the relative root-mean-square error of the estimated input. Note that a low-frequency component was observed in the estimated input, therefore, the results obtained after applying a low-pass Butterworth filter of order 7 are also shown in Table 9.1. As observed, good estimation results are achieved for unknown model parameters and input. Figure 9.2 summarizes the estimation results for case ID 1. Figure 9.2a shows the time history of the estimates of the six model parameters characterizing the steel constitutive model. It is observed that excellent estimation results are obtained for Young modulus (Ecol and Ebeam ) and yield stress (fcol and fbeam ) of beam and column steel parameters. Strain hardening parameters (bcol and bbeam ) are not well estimated because a limited amount of information about these parameters is contained in the measured responses. Very good estimation is also seen for the seismic input excitation.

9.4 Conclusions A novel approach for updating nonlinear finite element (FE) models under unknown seismic excitations is presented. The approach is validated using data simulated from a realistic 3D steel frame structure with six unknown model parameters and one unknown component of the seismic excitation. Very good estimation results are obtained with the proposed method, including parameter and input estimates, and also measured and unmeasured response quantities. Acknowledgements The authors acknowledge the support from the Chilean National Commission for Scientific and Technological Research (CONICYT), FONDECYT project No. 11160009. SEA and DL would like to also acknowledge the support provided by NSF Award #1762034 BD Spokes: MEDIUM: MIDWEST: Smart Big Data Pipeline for Aging Rural Bridge Transportation Infrastructure (SMARTI).

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(a)

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Fig. 9.2 Estimation results for case ID1. (a) model parameters, (b) input

References 1. Distefano, N., Rath, A.: Sequential identification of hysteretic and viscous models in structural seismic dynamics. Comput. Methods Appl. Mech. Eng. 6, 219–232 (1975) 2. Astroza, R., Ebrahimian, H., Conte, J.: Material parameter identification in distributed plasticity FE models of frame-type structures using nonlinear stochastic filtering. J. Eng. Mech. 04014149 (2014) 3. Ebrahimian, H., Astroza, R., Conte, J.P.: Extended Kalman filter for material parameter estimation in nonlinear structural finite element models using direct differentiation method. Earthq. Eng. Struct. Dyn. 44, 1495–1522 (2015) 4. Astroza, R., Nguyen, L.T., Nestorovi´c, T.: Finite element model updating using simulated annealing hybridized with unscented Kalman filter. Comput. Struct. 177, 176–191 (2016) 5. Gillijns, S., De Moor, B.: Unbiased minimum-variance input and state estimation for linear discrete-time systems with direct feedthrough. Automatica. 43, 934–937 (2007) 6. Lourens, E., Papadimitriou, C., Gillijns, S., Reynders, E., De Roeck, G., Lombaert, G.: Joint input-response estimation for structural systems based on reduced-order models and vibration data from a limited number of sensors. Mech. Syst. Signal Process. 29, 310–327 (2012) 7. Eftekhar Azam, S., Chatzi, E., Papadimitriou, C.: A dual Kalman filter approach for state etimation via output-only acceleration measurements. Mech. Syst. Signal Process. 60, 866–886 (2015) 8. Astroza, R., Ebrahimian, H., Li, Y., Conte, J.P.: Bayesian nonlinear structural FE model and seismic input identification for damage assessment of civil structures. Mech. Syst. Signal Process. 93, 661–687 (2017) 9. Erazo, K., Nagarajaiah, S.: An offline approach for output-only Bayesian identification of stochastic nonlinear systems using unscented Kalman filtering. J. Sound Vib. 397, 222–240 (2017) 10. Ebrahimian, H., Astroza, R., Conte, J.P., Papadimitriou, C.: Bayesian optimal estimation for output-only nonlinear system and damage identification of civil structures. Struct. Control. Health Monit. 25, e2128 (2018) 11. Julier, S.J., Uhlmann, J.K., Durrant-Whyte, H.F.: A new approach for filtering nonlinear systems. In: American Control Conference. Seattle, Washington (1995) Rodrigo Astroza is an Assistant Professor at the Faculty of Engineering and Applied Sciences of the Universidad de los Andes, Chile. His research interest are structural health monitoring, system identification, and nonlinear finite element modeling.

Chapter 10

Axle Box Accelerometer Signal Identification and Modelling Cyprien A. Hoelzl, Luis David Avendano Valencia, Vasilis K. Dertimanis, Eleni N. Chatzi, and Marcel Zurkirchen

Abstract Critical to railway infrastructure assessment is the tracing of interaction between railway vehicle and track.This is a non-trivial task characterized by non-stationary dynamics appearing due to changing operational conditions. To ensure reliable dynamic characterization of rail infrastructure, we propose a modeling methodology of locally non-stationary parametric time-series models to accommodate long-term variability. In the proposed approach, the non-stationary vibration, as measured from axle-box accelerometers, is modelled by means of a parametric time-series model with explicit dependence on the vehicle velocity and variations due to the infrastructure type. The postulated time-series model is demonstrated through real-world data stemming from the on-board measurement system of a Swiss Railway diagnostic vehicle, where the main drivers of uncertainty are the vehicle speed, the track type and its condition. The model will be exploited to build real-time indicators for railway infrastructure condition assessment. Keywords Axle box acceleration · Signal identification · Spectral analysis · Auto-regressive · Linear parameter varying auto-regressive

10.1 Introduction Nowadays inspection of railway infrastructure is carried out via field inspections or alternatively via dedicated diagnostic vehicles, which are equipped with multiple optical and inertial sensors. These inspections are carried out periodically to collect infrastructure condition data, which is essential for Life-Cycle Assessment (LCA). The drawbacks of this approach lie in its relatively high cost, the high organizational effort and limited availability only allow for measurements at relatively sparse intervals, as well as the risk to disrupt other services and regular operation. The alternative of utilizing in-service trains, equipped with on-board vibration monitoring systems has been investigated as an alternative approach to monitor railway track condition [1–4]. A typical on-board monitoring setup relies on the use of Axle Box Accelerometers (ABA), which typically record lateral and vertical accelerations at a sampling frequency typically ranging from 1 kHz up to 24 kHz. Irregularities on the running surfaces of the wheel and on the track, as well vehicle dynamics cause periodic or transient excitations of the vehicle axles. The excitation types, summarized in Table 10.1, illustrate that the excitation may be generated both by defects and by the inherent properties of the track. Previous investigations have shown that via the application of appropriate models and algorithms on ABA measurements, a wide range of condition monitoring tasks such as the identification of the wheel and track roughness [5], longitudinal level defects [3, 6], defects on rails [2] and defects at switches [7] may be achieved.

C. A. Hoelzl () · V. K. Dertimanis · E. N. Chatzi Institute of Structural Engineering, ETH Zürich, Zürich, Switzerland e-mail: [email protected]; [email protected]; [email protected] L. D. Avendano Valencia The Maersk Mc-Kinney Moller Institute, SDU, Odense, Denmark e-mail: [email protected] M. Zurkirchen Measurement and Diagnostics, SBB, Bern, Switzerland e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2020 Z. Mao (ed.), Model Validation and Uncertainty Quantification, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47638-0_10

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Table 10.1 Summary of excitation mechanisms [8] Excitation Track and rail irregularities Wheel irregularities Parameter excitation Self-excitation

Periodic/nearly periodic Corrugation, Waves Eccentric and polygonal wheels, flat spots Sleeper Crossing High-frequency sinusoidal motion

Transient Track alignment, Turnouts, Squats, Welds – Hollows, bridge ramps, crossings –

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In this work, vertical ABA response measurements are examined for identifying the source of excitation listed in Table 10.1. Non-parametric identification is initially employed to characterize the response to different excitation mechanisms. Subsequently, parametric identification is adopted. In this case, it is assumed that the system is described by a Linear Parameter Varying (LPV) model, where the main scheduling variable is the vehicle speed. Identification is performed in discrete time by means of LPV-AR models [9], in an output-only fashion, namely, only the response of the system (acceleration) is used in the identification scheme.

10.2 Non-parametric Response Signal Identification The data-driven non-parametric identification of the signal relies on multiple data streams to qualitatively identify the vehicle response. Namely each acceleration measurement channel can be mapped to the vehicle speed, track geometry and location during the measurement (online). The location of the vehicle is in turn linked to multiple other databases such as the fixed asset database, encompassing all track infrastructure components, as well as their history, and the condition monitoring database, which comprises measured or observed infrastructure defects (offline). One of the main drivers of non-stationarity of the ABA signal is the variance of the vehicle response due to changing speed. Figure 10.1 illustrates the distribution of the magnitude of measured acceleration, which was sampled uniformly with respect to the scheduling variable, i.e., the velocity. As a consequence, the frequency distribution of the response to the excitation mechanisms exposed in Table 10.1 is dependent on the vehicle speed. For instance, the peak response frequency to a wheel defect grows directly in proportion to the wheel revolution speed. The dynamic response of the track largely varies depending on construction type (slab or ballasted), sleeper type (wood, concrete, steel), sleeper padding type (none, soft, medium or stiff). The magnitude of the sleeper passage frequency is altered by the track receptance, which is a function of the superstructure elasticity and damping properties. For instance, a stiffer track has a larger magnitude of axle response due to its lower receptance. As illustrated in Fig. 10.2, the sleeper passage frequency magnitude lines up with the infrastructure type, whose color coding is summarized in Table 10.2 and where the sleeper passage frequency is equal to the vehicle speed divided by the spacing of the sleepers, spanning 50. . . 75 [Hz]. The equivalent conicity associated to the vehicle-rail contact interaction geometry plays a critical role in the dynamics of the railway vehicle. As the rail experiences load cycles, the geometry of the contact surface is altered due to the wear of the rail and wheel profile. The profiles of both rail and wheel are periodically corrected by griding and milling maintenance, which is performed with maintenance vehicles for the former and wheel profiling equipment for the latter. On the same

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section, the amplitude of the acceleration response in the frequency range between 500 and 800 Hz coincides with rail installation year as illustrated by Fig. 10.3. It is worth to note that all rails of the considered section have been grinded at least once since 2018, and have the same profile of type UIC60. The only difference lies in the steel grade used on some of the most recently replaced rails (R260 and R350HT). On the segment considered here, we observe that the track segments with more recently installed rails experience larger high frequency accelerations and thus larger surface contact stresses. The cause of the larger excitations measured on some segments will be investigated in further work. Transient effects may be caused by track geometry defects (low frequency) and due to point discontinuities (high frequency). High frequency transient effects such as insulation joints, turnouts and defects to the railhead can cause high impulses in the ABA measurement. The identification of these defects is impeded by many uncertainties, such as the contact geometry during defect roll-over. Previous research has shown that the magnitude of these impulses at a certain speed is related to the severity of the defect [2, 7], and that combining multiple measurement rides can increase the prediction rate of defects. In this analysis of ABA measurement signals and infrastructure data, the influence of the vehicle speed, rail condition and track type on the dynamic response of the system was illustrated. Particularly in the high frequency domain, effects such as the rail condition can not be neglected to accurately monitor and diagnose transient effects.

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10.3 Parametric Response Signal Identification Non-stationary vibration response may be modelled via Linear Parameter Varying AR when the dynamics are governed by an external scheduling variable. LPV-AR is a class of time-dependent AR models, where the values of the AR parameters ai (β[t]) are function of a scheduling variables β[t]; in this case the vehicle speed. The regression vector of LPV-AR models is defined as [9]: y[t] = −

na 

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(10.3)

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where w[t] are zero mean NID innovations with variance σw2 , F stands for a functional subspace of the respective quantity, ba(j ) and bs(j ) are the indices of the specific basis functions that are included in each functional subspace, while ai,j and sj stand for the coefficients of projection of the AR parameters and innovations variance. The raw ABA signal is sampled at 24 kHz, passed through a lowpass Butterworth filter at 150 Hz and down- sampled to 300 Hz in order to focus the analysis on the lower frequency component of the vehicle-track interaction. The LPV-AR models are subdivided by track type, as the response in the selected frequency band is mainly influenced by the track stiffness and the vehicle speed. The training is performed on the first axle ABA signal and the subsequent validation of the LPV-AR models are performed on the fourth axle.

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Standard model order selection is carried out, where range of model orders is calculated, while a plausible value of the functional basis order is selected. Different performance criteria are compared to determine the optimal model order. From Fig. 10.4 it can be observed that a basis order of 80 sufficient to reach less than 1.5% RSS/SSS error, while minimizing the BIC criterion and maximizing the log-likelihood. The dynamics of the identified LPV-AR models is subsequently analysed. Figure 10.5 illustrates the PSD of the LPVAR model as a function of vehicle speed for the different track type segments of Table 10.2. The sleeper passage frequency, evidenciated in Fig. 10.2 is captured by the LPV AR model, and as expected varies in function of the superstructure type. The highlighted in green in particular experiences much larger vibrations compared to the tracks with more elastic superstructures highlighted in red and blue.

10.4 Conclusions The present signal identification emphasizes that vehicle speed, rail condition and track type influence the vehicle axle box response. In the high frequency domain, effects such as the rail condition are not negligible for the detection of defects on the infrastructure type. Encouraging results from parametric identification substantiate future studies, integrating overall infrastructure type and condition into the estimation process. Acknowledgments The authors acknowledge the support of the Swiss Federal Railways (SBB) via the Future Mobility Research Program Grant (ETH Mobility Initiative) on the topic of On Board Monitoring for Integrated Systems Understanding & Management Improvements in Railways.

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References 1. Lederman, G., Chen, S., Garrett, J.H., Kovaˇcevi´c, J., Noh, H.Y., Bielak, J.: A data fusion approach for track monitoring from multiple in-service trains. Mech. Syst. Signal Process. 95, 363–379 (2017) 2. Molodova, M., Li, Z., Nunez, A., Dollevoet, R.: Automatic detection of squats in railway infrastructure. IEEE Trans. Intell. Transp. Syst. 15(5), 1980–1990 (2014) 3. Real, J., Salvador, P., Montalbán, L., Bueno, M.: Determination of rail vertical profile through inertial methods. Proc. Inst. Mech. Eng. Part F: J. Rail Rapid Transit 225(1), 14–23 (2011) 4. Bocciolone, M., Caprioli, A., Cigada, A., Collina, A.: A measurement system for quick rail inspection and effective track maintenance strategy. Mech. Syst. Signal Process. 21(3), 1242–1254 (2007) 5. Carrigan, M., Talbot, J.: Extracting information from axle-box accelerometers: on the derivation of rail roughness spectra in the presence of wheel roughness. In: IWRN 13 (2019) 6. Liu J., Wei Y., et al.: Detecting anomalies in longitudinal elevation of track geometry using train dynamic responses via a variational autoencoder. In: Proceedings of SPIE 10970, Sensors and Smart Structures Technologies for Civil, Mechanical, and Aerospace Systems (2019) 7. Wei, Z.: Modelling and monitoring of dynamic wheel-rail interaction at railway crossing. PhD thesis, Delft Technical University, Delft (2018) 8. Knothe, K.: Gleisdynamik. Berlin: Ernst & Sohn (2001) 9. Avendaño-Valencia, L.D., Fassois S.D.: Damage/fault diagnosis in an operating wind turbine under uncertainty via a vibration response Gaussian mixture random coefficient model based framework. Mech. Syst. Signal Process. 91, 326–353 (2016) Cyprien A. Hoelzl obtained his MSc Civil Engineering degree from ETHZ and is pursuing his PhD Studies. His research focuses on modelling the dynamic interaction between railway vehicle and track to develop health indicators pertaining the state of the railway infrastructure.

Chapter 11

Kalman-Based Virtual Sensing for Improvement of Service Response Replication in Environmental Tests Silvia Vettori, Emilio Di Lorenzo, Roberta Cumbo, Umberto Musella, Tommaso Tamarozzi, Bart Peeters, and Eleni Chatzi

Abstract Environmental tests are typically conducted in order to reproduce the operational response of a system. Nonrealistic excitation mechanisms and mismatches between operational and test boundary conditions, represent relevant limitations of the currently adopted testing procedures. The Boundary Condition Challenge (BCC) addresses the assessment of a testing practice that allows to reproduce structural responses, which better represent the operational environment, thereby allowing for a more precise prediction of potential failure mechanisms. In this framework, Virtual Sensing (VS) can be used to estimate the complete response field of the component in operation, and compare this to the one delivered via testing. If the full-field strain is estimated, stress fields can be derived and component failure can be predicted. This work focuses on employing VS for augmenting the information acquired from physical sensors during environmental tests on the Box Assembly with Removable Component (BARC) benchmark. In order to apply VS techniques such as Kalman-type filters, a Reduced Order Model (ROM) of the system has been built taking into account the boundary conditions employed during testing. Moreover, an existing Optimal Sensor Placement (OSP) strategy has been adopted for configuring the positioning of sensors to be used during environmental tests. Keywords Virtual sensing · Augmented Kalman filter · Optimal sensor placement · Environmental testing · Boundary condition challenge

11.1 Introduction Environmental testing is a standard procedure in spacecraft engineering used for qualification of the spacecraft’s mechanical design. These tests, crucial for ensuring and demonstrating the spacecraft integrity against the dynamical launch environment, are performed for verifying the resistance of the system and all its components to the random excitations to which they are subjected throughout their operational life. Environmental testing therefore represents an essential specification procedure. During these tests, the spacecraft is placed on a big electrodynamic or hydraulic shaker testing facility, which provides a controlled excitation with the purpose of replicating the in-service structural response of the tested structure. Besides the difficulties in the execution of these tests due to the structure’s dimensions, the most critical aspect is related to the poor operational environment representation which may be achieved while testing. Indeed, limitations in the adopted vibration S. Vettori () Siemens Industry Software NV, Leuven, Belgium Institute of Structural Engineering, ETH Zürich, Zürich, Switzerland e-mail: [email protected] E. Di Lorenzo · U. Musella · B. Peeters Siemens Industry Software NV, Leuven, Belgium e-mail: [email protected]; [email protected]; [email protected] R. Cumbo · T. Tamarozzi Siemens Industry Software NV, Leuven, Belgium KU Leuven, Leuven, Belgium e-mail: [email protected]; [email protected] E. Chatzi Institute of Structural Engineering, ETH Zürich, Zürich, Switzerland e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2020 Z. Mao (ed.), Model Validation and Uncertainty Quantification, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47638-0_11

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control strategies and the interaction between the structure under test and the testing facility, can give rise to undesired issues, e.g. over- or under testing, errors in failure modes and time to failure estimation or even damage. In this framework, a collaboration between Kansas City National Security Campus (managed by Honeywell Federal Manufacturing & Technology) and Sandia National Laboratories introduced the Boundary Condition Challenge (BCC). The goal of this project is to improve the in-service environment replication at a component level, leading to the establishment of component failure mechanisms closer to the ones that may arise during its operational life. The challenge focuses on the Box Assembly with Removable Component (BARC) setup, a simple mock-up that can yet represent the relevant challenges from the BCC point of view. Several BARC specimens have been distributed among the large number of research institutes that have accepted the BCC and that are now investigating the BARC with the purpose of providing new solutions [1–3]. In this view, Virtual Sensing (VS) techniques such as Kalman-type filters, can be used to combine information from simulated models and test data in order to calculate an estimate of the quantity of interest, e.g. excitations or responses of the system at locations that may not be easily instrumented via physical sensors. VS techniques can be employed in environmental testing applications to retrieve the complete strain field on the tested component both in operation and during tests, allowing for a more complete comparison of the two conditions. Entire stress fields can be derived from strain information, based on which component failure can be predicted. Moreover, forces are usually not measured during this type of tests. Therefore, a simultaneous estimation of states and inputs can be used to retrieve such relevant quantities. The first step in facing the BCC is to estimate the full field response of the BARC during tests. This works proposes the application of the Augmented Kalman Filter (AKF) [4] for simultaneously estimating the BARC dynamic response at unmeasured locations and the forces applied during testing. A Reduced Order Model (ROM) of the BARC, used for retrieving the reduced state-space matrices that are needed for the AKF implementation, has been built following the Component Mode Synthesis (CMS) method proposed in [5]. This method allows use of a model, which can account for the boundary conditions the BARC is subjected to when placed on a shaker for environmental testing. Estimation results from a fully simulated data set are firstly presented in this work. The AKF has been employed on data acquired during a measurement campaign, which has been performed by exciting the BARC via a monoaxial electrodynamic shaker. In this paper, the mentioned campaign is described and the estimation results are explored. For both the applications, i.e., AKF applied on either fully simulated data or measured data, the sensors locations have been configured using the Optimal Sensor Placement procedure presented in [6, 7].

11.2 The Boundary Conditions Challenge The BCC addresses the development of a methodology for assessing the design of environmental testing procedures, leading to more realistic response reconstruction for systems in-operation. When looking at environment replication at the component level, differences in boundary conditions between single component testing and full assembly testing must be taken into account. Removing the component from its original subassembly and testing it on a shaker, rather than directly testing the entire assembly, obviously gives rise to shifts in boundary conditions, thereby impacting the structural dynamics. Common practice consists in overtesting the structure, which typically results in undesired oversizing of the component and an erroneous environmental vibration replication. The ultimate goal of the BCC is to propose an alternative to overtesting, looking for new solutions able to improve the in-service environment replication, e.g. studying the influence of different excitation mechanisms and boundary conditions.

11.2.1 The Box Assembly with Removable Component The BCC makes use of a test bed comprising a relatively simple aluminum structure, the BARC shown in Fig. 11.1a. It is composed of a “component”, made from assembly of two C-channels connected by a beam, and a “subassembly”, consisting of a cut box-beam, on which the component rides. The component plays the role of the unit under testing, while the subassembly represents a generic operational support. In the BARC specimen provided to Siemens Industry Software, four M5 Holes have been drilled at the box base to directly connect the BARC to a commonly used 75 lbf shaker and an M8 hole, in order to permit connection to a larger shaker. Figure 11.1b illustrates the 2D Finite Element (FE) model developed in Simcenter 3D for the mentioned BARC specimen. The model was validated with test data coming from impact testing of the BARC in free-free conditions (in Fig. 11.2a),

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processed with Simcenter Testlab. Figure 11.2b reports the MAC diagram between simulated and experimental normal mode sets achieved by model updating through the Simcenter 3D Model Update tool, which makes use of NX Nastran SOL 200.

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11.3 Virtual Sensing for Environmental Tests on the BARC In order to investigate and evaluate new possible practices for environmental testing, a full comparison between operational and test component responses, both when the sole component or the entire assembly are tested, must be established. In this sense, VS techniques can be used to expand the information pertaining to the system’s response, i.e., to infer response at locations that are hard to instrument or even to yield the full field response of the tested structure. In case of strain measurements, the full stress field can be retrieved and failure can be precisely estimated. Moreover, joint input-state estimation techniques can be used for estimating inputs applied to the system, given that they are not usually measured during environmental tests. This paper addresses the derivation of quantities of interest when testing the entire BARC on a monoaxial shaker, as shown in Fig. 11.6a. The test setup foresees the clamping of the BARC base on a commonly used shaker of comparable size, implying that specific boundary conditions have to be taken into account when building a simulation model of the system.

11.3.1 BARC Reduced Order Model The BARC has been modeled according to the equation of motion: M¨z(t) + C˙z(t) + Kz(t) = Bu(t)

(11.1)

where z(t) ∈ Rndof is the FE model Degrees of Freedom (DoFs) vector, M,C,K are the mass, damping and stiffness matrices respectively; u(t) ∈ Rni (with ni number of loads) is the input vector and B ∈ Rndof ×ni is the Boolean input shape matrix that selects the DoFs where the inputs are applied. A Model Order Reduction (MOR) technique, described in [5], has been applied to reduce the model size. According to this method, the dynamic behavior of the BARC can be formulated as a superposition of modal contributions: z = q

(11.2)

where  ∈ Rndof ×nr is the reduction basis and q ∈ Rnr is the vector of the generalized coordinates of the system. Inserting the reduction basis into equation (11.1), it results in: ¨ + Cr q(t) ˙ + Kr q(t) = Br u(t) Mr q(t)

(11.3)

where the mass, damping, stiffness and input shape matrices of the reduced system are respectively Mr =  T Mr , Cr =  T Cr , Kr =  T Kr  and Br =  T B. The adopted reduction basis can be expressed as:    =  n  RI RA ;

(11.4)

where  n ∈ Rndof ×nk is the matrix of the normal modes to be included in the Reduced Order Model (ROM), i.e., the eigenmodes in the frequency range of interest, and  RI RA ∈ Rndof ×na is the Residual Inertia-Relief Attachment (RIRA) modes matrix. Standard residual attachment modes are usually inserted in this kind of reduction bases in order to include a representation of the static response of the structure to a specific input location. Typically, for each applied load, the relative residual attachment mode is included in the basis. Each of them is computed as a static mode of the system when a unitary input at the correspondent location is applied. Therefore, na is equal to the number of loads applied to the system and nr = nk + na . However, as mentioned before, the test configuration this works refers to foresees the BARC to be attached on a shaker as in Fig. 11.6a. In this configuration, the constraint applied to the system demands the base connection points to move together with the shaker. Therefore, no fixed constraint is provided in the global frame and rigid body motion is present. As a consequence, the static solution, and hence the residual attachment modes, can not be computed. A workaround to this issue consists in making use of the RIRA modes, i.e., specific attachment modes computed through the following steps: 1. Constrain the minimum number of random DoFs needed for having K non-singular, i.e., 6 DoFs in the case of the BARC 2D model; 2. Compute the static solution, i.e., obtain the Inertia-Relief Attachment modes  I RA as :

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11.3.2 Augmented Kalman Filter for Joint Input-State Estimation Kalman-based strategies require equation (11.1) to be written into the following conventional state-space form: 4

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Therefore, introducing (11.7) into (11.6) and passing from the continuous to the time-discrete form, the following formulation can be derived: 4 xak = Fa xak−1 + wak−1 (11.8) yk = Ca xak + vk where all the inputs applied to the system have been assumed to be unknown and mutually uncorrelated Gaussian noises w and v have been introduced to respectively take into account model uncertainties and measurement  The associated  noise. T Q 0 ≥ 0 for process noise covariance matrices are R = E{vl vk } ≥ 0 for measurement noise and Qa = E{wal wak } = 0 Qu noise, where matrix Q properly takes into account model uncertainties, while matrix Qu represents uncertainties on the input to be estimated. Equations (11.7) and (11.8) indicate that the location of the input has to be known in order to apply the AKF. Moreover, a model for the inputs dynamics must be introduced. In this work, a zeroth-order random walk model [6, 8, 9] has been adopted: uk = uk−1 + wuk−1

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11.3.3 Optimal Sensor Placement Strategy In this work, the OSP strategy presented in [6, 7] has been adopted. This procedure, based on the fullfilment of the observability requirements, allows for a smart selection of sensors quantity, type and location. The strategy can be summarized in the following steps: 1. Selection of a large number of random nodes (for position and acceleration responses) and elements (for strain responses) of the FE model; 2. Training of the FE model, i.e., its simulation when static and dynamic loads with different frequencies and amplitudes are applied at the known locations of the loads to be estimated; 3. Removal of sensors with low Signal/Noise ratio for each training scenario. Closely spaced sensors are discarded if they are of the same type; 4. Merging of different types of sensors, computation of matrix Ca and scaling of rows in order to avoid differences in the order of magnitude of its elements; 5. Selection of sensors with the highest contribution to system observability. For this check, the following observability indicator [8] is taken into account: O=

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11.4 Input-State Estimation: Simulated Data As a primarily step to prove the validity of the AKF for the hereby discussed application, a first estimation performed on a fully simulated data set is reported in this section. For this purpose, a BARC Reduced Order Model (ROM) has been built following the procedure outlined in Sect. 11.3.1 and taking into account the boundary conditions to which the system would have been subjected during tests. The idea is indeed to simulate a test representative model in order to obtain time histories of the responses at locations selected by OSP (shown in Fig. 11.3) and use them to update the AKF. For this application, only strain-type sensors have been used. The ROM reduction basis has been therefore created taking into account the first ten normal modes (frequency range of interest: 0–1200 Hz) and 4 RIRA modes, one for each of the forces applied at the BARC base. The discrete-time state-space matrices have been then computed and used to simulate the model when 4 random multisine inputs are applied at the base. Responses at locations shown in Fig. 11.3 have been obtained and noise has been added in order to simulate actual measurements. The noise covariance adopted for this purpose, equal for each strain measurement, has been computed using the data acquisition device absolute accuracy and some strain gauges parameters such as the gauge factor. The parameters needed for the AKF implementation have been set as follows: • the initial state vector, i.e., initial displacements and velocities, has been set to be zero, as well as the initial error covariance matrix P0 ; • the noise covariance matrix R has been computed as a diagonal matrix with elements equal to the noise covariance value used in the construction of the “measured” quatities; • the process noise covariance matrix Qa has been built assuming that the uncertainty related to the unknown inputs is higher that the model uncertainty. Q has been therefore set equal to zero, while Qu has been built as a diagonal matrix, the elements of which have resulted from the following formula: Qu = ( t · ωu · a u )2

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From Figs. 11.4 and 11.5, it can be concluded that a good input and response estimation can be potentially obtained using AKF for environmental tests applications. The following section will therefore discuss the results obtained for a real measured data set.

11.5 Input-State Estimation: Measured Data This section treats the state-input estimation for a test data set acquired during a measurement campaign carried out at Siemens Industry Software.

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Fig. 11.6 Test setup. (a) BARC mounted on a monoaxial electrodynamic shaker. (b) Detailed view of sensors and BARC’s attachment on the shaker

11.5.1 Measurement Campaign Figure 11.6a shows the complete test setup of the measurement campaign conducted on the BARC, while detailed focus on the sensors and BARC’s connection to the shaker is provided in Fig. 11.6b. The four M5 holes on the BARC base have been used to create a 4-points bolted connection with the “The Modal Shop” 2025E electrodynamic modal shaker. In addition, four PCB monoaxial force cells have been placed between the BARC and the shaker in order to measure the applied forces in Z direction. Sixteen strain gauges and four triaxial accelerometers have been attached to the system to both measure strain and acceleration data. Simcenter LMS SCADAS Mobile hardware and Simcenter Testlab software have been used to acquire data during the hereby described tests. Several excitation signals with different amplitude levels have been used during tests: continuous random (maximum level: 0.3, 0.4 and 0.5 V), pseudo random (RMS level: 0.1,0.15,0.2,0.25,0.3 V), sine (3.2, 20, 300, 650 Hz), linear sine sweep (sweep rate: 2, 3 Hz; voltage level: 0.1, 0.2 V), logarithmic sine sweep (sweep rate: 1, 2 oct/min; voltage level: 0.1, 0.2 V).

11.5.2 Input-State Estimation Using the AKF The AKF for joint input-state estimation has been applied to the data set acquired using a pseudo random excitation, i.e., a random multi-sine, with 0.3 V RMS level. Indeed, random signals are, amongst all the type of signals used during the test campaign, the most representative of the excitation mechanisms adopted in environmental tests. During this test, four forces are applied to the BARC through the bolted connections at the base. All the forces are considered to be unknown and they are going to be estimated using the AKF. Data acquired by the force cells will be used for comparison with the estimated quantities. The same BARC ROM described in Sect. 11.4 has been adopted. Figure 11.7 shows sensors locations and directions obtained by applying the OSP strategy described in Sect. 11.3.3. For this application, only strain sensors have been considered. In order to check for the validity of the algorithm in terms of response estimation, only a reduced subset of nine strains has been included in the vector of the measured quantities y, the remaining ones have been used for validation of the estimated responses. In particular, the measured locations reported in Fig. 11.7 have been selected among the total sixteen available strains according to the following requirements: • Measurements collocated with the applied forces should be included in order to allow for a more accurate reconstruction of the inputs [12]; • The application of the AKF in the framework of the BCC focuses on response estimation at the component level. Therefore, only one strain on the component has been included in the measured quantities, while the remaining ones have been used for comparison with the estimated responses.

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Fig. 11.7 Strain sensors map: measured (yellow) and unmeasured (magenta) locations

Observability requirements for the full sensors set are guaranteed by the OSP strategy. To ensure their fulfillment for the reduced sensors set, the same order of magnitude of the observability indicator O in the full sensors set case must be guaranteed. O results in a 10−12 order of magnitude for both the sets, thereby confirming system observability for the reduced set case. As carried out in Sect. 11.4, some parameters should be set in order to apply the AKF. In particular, the system initial displacement and velocity have been assumed to be equal to zero, as well as the initial error covariance matrix P0 . The covariance matrix R associated to the measurement noise is as a diagonal matrix, the elements of which are equal to the covariance of the noise associated to each of the measurements. These values have been retrieved by performing a test with the sole purpose of measuring noise on each channel. To what concerns the process noise covariance matrix Qa , Q has been set equal to zero as in Sect. 11.4, while Qu has been built as a diagonal matrix, the elements of which have resulted from a calibration procedure based on the use of the so-called L-curve [13]. In Fig. 11.8, the L-curve for each input involved in the estimation problem treated in this work is reported. This picture shows, for each input, the variation of the smoothing and error norms when the non-zero elements of Qu equally vary from 10 to 1012 . The uncertainty level, and therefore the process noise covariance, has been indeed assumed to be equivalent for all the inputs. The smoothing error is defined as: #N #N

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Figure 11.8 shows that the minimum of the error norm and a stable value of the smoothing norm are achieved when the diagonal elements of Qu are all equal to 107 , which is the value that has been set for the estimation. The adopted values of R and Qu are such that Q  R, i.e., the assumption that the model is less reliable than the measured data has been made. Figure 11.9 shows the input estimation results, offering a comparison between the estimated quantities and the forces measured through the force cells. The AKF fails in providing a precise estimation of the inputs, except for input 4. The inaccurate estimation can be explained by the evident differences in the measured forces PSDs shown in Fig. 11.9 (right),

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which should theoretically be equivalent. These differences are introduced by the presence of strong boundary conditions uncertainties that can be summarized by the following aspects: • Errors in reproducing the same connection conditions for all the four bolts during tests are present; • Reaction forces at the connection between the BARC and the shaker arise during tests; • The boundary conditions used in the model introduce a strong approximation of the test configuration. For the above mentioned reasons, different frequency components are estimated for each of the four forces. In this respect, a potential improvement could consists in additionally estimating torques that arise at the inputs application points. Furthermore, including the shaker in the model and modeling its interaction with the BARC represents another additional development. Figure 11.10 offers a comparison between measured, estimated and simulated responses at two of the measured locations shown in Fig. 11.7 (sensors 4 on the base and sensors 12 on the beam). Simulated time histories have been obtained by applying the measured inputs to the ROM. It is demonstrated that a good estimation is provided by the AKF, while evident mismatches arise between measured and simulated responses. These inaccuracies are indeed a clear representation of the previously mentioned model uncertainties, which affect the input estimation. Response estimation results at two unmeasured locations are reported in Fig. 11.11. This picture shows the comparison between estimated strains on the component (sensor 11 in y direction and sensor 10 in x direction) and their correspondent measured values. It is possible to notice that a good response estimation is provided, except for the poor reconstruction of high frequency components magnitude on the sensor measuring in y direction. This model gap is well compensated for measured signals in x direction, such as sensor 10, because x axis resulted to be more excited during the performed measurement campaign, i.e., lower measured response levels appeared for all the sensors in y direction. Moreover, strain measurements usually feature very low amplitude levels at high frequencies. The fusion of acceleration measurements, which are characterized by higher amplitudes at high frequencies, could lead to a better estimation of both inputs and responses.

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11.6 Conclusions The application of VS techniques to environmental tests on the BARC has been proposed in this work. The BARC is a simple hardware demonstrator used in the framework of the BCC, a challenge that focuses on improving operational environment replication at the component level during environmental tests. In particular, the potential of the AKF for both input and state estimation has been presented in this paper. A first simulated example, which shows good results in terms of input and component response reconstruction, has been reported. Furthermore, a test campaign has been conducted placing the BARC on a monoaxial electrodynamic shaker of comparable size. Results from the AKF application to the acquired data set have been discussed in this paper. It is shown that the AKF estimates different frequency components on each input and therefore fails in identifying the correct time histories. The main reason for the errors in the inputs’ estimation is attributed to the test boundary conditions uncertainties. A possible improvement could consist in adding torque estimation, or in including the shaker in the simulation model in order to take into account its interaction with the BARC. Satisfying results are obtained in terms of component response estimation, even if a low magnitude estimate for the high frequency components of the responses in y direction is provided. Adding acceleration data to the measurements vector represents a future step that can help to recover high frequency components in the estimated quantities. Future investigation aims at applying VS techniques to operational data to obtain the full-field service responses and compare them to the test ones in order to approach the BCC goal. Acknowledgments The author gratefully acknowledge the European Commission for its support of the Marie Sklodowska Curie program through the ITN DyVirt project (GA 764547).

References 1. Larsen, W., Blough, J.R., DeClerck, J.P., VanKarsen, C.D., Soine, D.E., Jones, R.: Initial modal results and operating data acquisition of shock/vibration fixture. In: Topics in Modal Analysis & Testing, vol. 9, pp. 363–370. Springer (2019) 2. Musella, U., Blanco, M.A., Mastrodicasa, D., Monco, G., Simone, M., Peeters, B., Mucchi, E., Guillaume, P., et al.: Combining test and simulation to tackle the challenges derived from boundary conditions mismatches in environmental testing. In: Sensors and Instrumentation, Aircraft/Aerospace, Energy Harvesting & Dynamic Environments Testing, vol. 7, pp. 259–269. Springer (2020) 3. Rohe, D.P., Smith, S., Brake, M.R.W., DeClerck, J., Blanco, M.A., Schoenherr, T.F., Skousen, T.J.: Testing summary for the box assembly with removable component structure. In: Sensors and Instrumentation, Aircraft/Aerospace, Energy Harvesting & Dynamic Environments Testing, vol. 7, pp. 167–177. Springer (2020) 4. Simon, D.: Optimal State Estimation: Kalman, H Infinity, and Nonlinear Approaches. Wiley (2006)

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5. Craig, R.R., Jr.: A review of time-domain and frequency-domain component mode synthesis method (1985) 6. Tamarozzi, T., Risaliti, E., Rottiers, W., Desmet, W., et al.: Noise, ill-conditioning and sensor placement analysis for force estimation through virtual sensing. In: In International Conference on Noise and Vibration Engineering (ISMA2016), pp. 1741–1756. Katholieke Univ Leuven, Dept Werktuigkunde (2016) 7. Cumbo, R., Tamarozzi, T., Janssens, K., Desmet, W.: Kalman-based load identification and full-field estimation analysis on industrial test case. Mech. Syst. Signal Process. 117, 771–785 (2019) 8. Naets, F., Croes, J., Desmet, W.: An online coupled state/input/parameter estimation approach for structural dynamics. Comput. Methods Appl. Mech. Eng. 283, 1167–1188 (2015) 9. Lourens, E., Reynders, E., De Roeck, G., Degrande, G., Lombaert, G.: An augmented kalman filter for force identification in structural dynamics. Mech. Syst. Signal Process. 27, 446–460 (2012) 10. Van Loan, C.: Computing integrals involving the matrix exponential. IEEE Trans. Autom. Control 23(3), 395–404 (1978) 11. Risaliti, E., Van Cauteren, J., Tamarozzi, T., Cornelis, B., Desmet, W.: Virtual sensing of wheel center forces by means of a linear state estimator. In: International Conference on Noise and Vibration Engineering (ISMA2016), Leuven (2016) 12. Maes, K., Van Nimmen, K., Lourens, E., Rezayat, A., Guillaume, P., De Roeck, G., Lombaert, G.: Verification of joint input-state estimation for force identification by means of in situ measurements on a footbridge. Mech. Syst. Signal Process. 75, 245–260 (2016) 13. Hansen, P.C., O’Leary, D.P.: The use of the l-curve in the regularization of discrete ill-posed problems. SIAM J. Sci. Comput. 14(6), 1487–1503 (1993) Silvia Vettori has graduated in Mechanical Engineering at University of Rome “La Sapienza” in 2018. Since September 2018, she is involved in the Marie Curie DyVirt PhD program with Siemens Industry Software NV in Leuven (Belgium) and ETH Zurich. Her research topic consists in employing Virtual Sensing techniques for structural dynamics applications.

Chapter 12

Virtual Sensing of Wheel Position in Ground-Steering Systems for Aircraft Using Digital Twins Mattia Dal Borgo, Stephen J. Elliott, Maryam Ghandchi Tehrani, and Ian M. Stothers

Abstract The ground-steering system is a part of the nose landing gear, which is fundamental to an aircraft’s safety. A sensing mechanism estimates the wheel direction, which is then fed back to the controller in order to calculate the error between the desired steering angle and the actual steering angle. As in many safety-critical control systems, the sensing mechanism for the nose wheel direction requires the use of multiple redundant sensors to estimate the same controlled signal. A virtual sensing technique is commonly employed, which estimates the steering angle using the measurements of multiple remote displacement sensors. The wheel position is then calculated on the basis of the nonlinear alignment of these sensors. In practice, however, each sensor is subject to uncertainty, minor and major faults and there is also ambiguity associated with the estimate of the steering angle because of the geometric nonlinearity. The redundant sensor outputs are thus different from each other, and it is important to reliably estimate the controlled signal under these conditions. This paper presents the development of a digital twin of the ground-steering system, in which the effect of uncertainties and faults can be systematically analysed. A number of state estimation algorithms are investigated under several scenarios of uncertainty and sensor faults. Two of these algorithms are based on a least squares estimation approach, the other algorithm, instead, calculates the steering angle estimate using a soft-computing approach. It is shown that the soft-computing estimation algorithm is more robust than the least squares based methods in the presence of uncertainties and sensor faults. The propagation of an uncertainty interval from the sensor outputs to the steering angle estimate is also investigated, in order to calculate the error bounds on the estimated controlled signal. The optimal arrangement of the sensors is also investigated using a parametric study of the uncertainty propagation, in which the optimal model parameters are the ones that generates the smallest uncertainty interval for the estimate. Keywords Digital twin · Virtual sensing · Aircraft ground-steering system · Soft-computing · Redundant sensors

12.1 Introduction The ground-steering system is a part of the nose landing gear, which is fundamental to an aircraft’s safety. A ground-steering system commonly consists of two linear actuators (electric or hydraulic), a power supply, a control valve, a follow-up device between the gear and the steering collar, a sensing mechanism, a control valve, a pilot control (switch) and a pilot-operated steering wheel, which can be either a bar or the rudder pedals or a combination of them [1]. The feedback signal of the ground-steering control loop is the steering angle of the nose wheel, which is then used to calculate the error between the desired steering angle and the actual steering angle. Although the wheel direction cannot usually be measured directly with an angular sensor, a virtual sensing technique is adopted, which estimates the steering angle using redundant measurements remote from the position of interest. Multiple linear displacement sensors are arranged on a mechanism that is similar to the dual push-pull steering mechanism [2] and

M. Dal Borgo () · S. J. Elliott · M. Ghandchi Tehrani Institute of Sound and Vibration Research, University of Southampton, Southampton, UK e-mail: [email protected]; [email protected]; [email protected] I. M. Stothers Ultra Electronics Precision Control Systems, St Johns Innovation Park, Cambridge, UK e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2020 Z. Mao (ed.), Model Validation and Uncertainty Quantification, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47638-0_12

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the nonlinear geometry of the sensor arrangement is used to calculate the wheel angular position. If the sensors were to operate perfectly, only a subset of them would need to be used for the estimation. In practice, however, the sensors are subject to uncertainty, minor or major faults and their operation may be nonlinear. Therefore, it is crucial to reliably estimate the controlled signal under these conditions, assessing the degree of confidence with which each sensor should be treated. This paper presents the development of a digital twin of the ground-steering system, in which the effect of uncertainties and faults can be systematically analysed. Three state estimation algorithms are discussed that calculate the steering angle estimate under several scenarios including model uncertainty, measurement noise and sensor faults. The paper is organized as follows. The development of the digital twin is first presented in Sect. 12.2. An interval type of uncertainty in the sensor outputs is then introduced in Sect. 12.3 in order to calculate the propagated uncertainty interval in the steering angle estimation. Three state estimation algorithms are investigated in Sect. 12.4, two of which are least squares based approaches and the remaining one is a soft-computing method. The results of the comparison among the estimation algorithms in terms of estimation accuracy when the digital twin is affected by model uncertainty, sensor noise and potential sensor faults, are presented in Sect. 12.5. The conclusions are summarized in Sect. 12.6.

12.2 Digital Twin of the Ground-Steering System The steering angle is estimated using a virtual sensing technique, which consists of multiple linear variable differential transformer (LVDT) sensors remotely located from the position of interest. The wheel direction is then calculated based on the nonlinear geometry of the sensor alignment. A model of the ground-steering sensing mechanism is shown in Fig. 12.1, in which four LVDTs are used, two on the left arm and two on the right arm. The model parameters, which are given in Table 12.1, have been selected according to the study in [2]. According to the schematic in Fig. 12.1, the controlled signal α is estimated from the measurements of the left (dL1 , dL2 ) and right (dR1 , dR2 ) LVDTs, respectively. In practice, however, the model parameters of the sensing mechanism are subject to uncertainty, as well as the sensor outputs. The LVDTs can also be affected by minor or major faults and there is ambiguity

Fig. 12.1 Schematic of the ground-steering sensing mechanism for the steering angle estimation, which includes the 4 LVDT sensors

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associated with all of the sensors due to the nonlinearity. The digital twin of the ground-steering system consists of the sensing mechanism model together with the uncertainties associated with its parameters, the noise in the sensor outputs and the potential faults in the LVDTs. The nonlinear relationship between the wheel direction and the outputs from the LVDTs can be studied through a kinematic analysis of position in which it is assumed that the model parameters are exact and the output from the sensors are ideal, hence without any noise of faults. The derivation of the relationship between the steering angle and the sensor outputs can be found in [3] and is omitted in this paper. However, it results in the following set of equations, 

2 = l 2 + a 2 + b2 − 2l [a cos (α + γ ) + b sin (α + γ )] dLi , 2 dRi = l 2 + a 2 + b2 − 2l [a cos (α − γ ) − b sin (α − γ )]

(12.1)

in which α is the steering angle, (dLi , dRi ) are the sensor outputs and l, a, b, γ are the model parameters. The relationship between the outputs from the LVDTs and the steering angle defined by Eq. (12.1) is also shown by the solid red line in Fig. 12.2(a). Since it was assumed that the sensor were ideal, hence without being affected by noise and faults, both right LVDTs give the same output and both left LVDTs give the same output. In this case only a subset of the four sensors would be needed to calculate the steering angle. A couple of one left and one right sensor measurements will determine a unique value of the steering angle, as shown in Fig. 12.2(a). The red line in Fig. 12.2(a) can be projected onto the x-y plane, which results on the black solid line shown in Fig. 12.2(b). This curve relates a subset of left and right sensor measurements to its unique steering angle value. However, if only a left or a right sensor output was used, there would be ambiguity on the value of the actual steering angle, because the nonlinear relationship between the steering angle and each sensor output generates double solutions. When both left and right steering angle are used, only two out of the four possible solutions are recurring and can be considered as the actual steering angle. The set of Eq. (12.1) can be rewritten in an explicit form in order to solve the inverse problem of determining the unknown steering angle from the known sensor outputs. Hence, Eq. (12.1) becomes, ⎧ ⎪ ⎨ sin (α) [b cos (γ ) − a sin (γ )] + cos (α) [a cos (γ ) + b sin (γ )] − ⎪ ⎩

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(12.2)

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Given a pair of measurements (dL , dR ), the steering angle estimate can be found by solving Eq. (12.2) by iteration using for example Newton-Raphson method. In this ideal scenario without uncertainties and faults, the solution of Eq. (12.2) generates the same solution of Eq. (12.1), which is shown with the red line in Fig. 12.2(a).

12.3 Interval Uncertainty Propagation In practice, the outputs from the sensors deviate from the ideal values because they are subject to measurement noise and faults. In this section we assume that none of the sensors is faulty, but all of them are subject to an interval type of uncertainty around the true measurement values. The output intervals from the sensors can then be written as,   [dLi ] = dLi dLi ,

  [dRi ] = dRi dRi ,

(12.3)

where d = μd − d is the lower bound and d = μd + d is the upper bound of the uncertainty interval, whereas [d] denotes the interval of possible values with an unspecified probability density function [4]. The interval [d] has a mean value μd and

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an uncertainty amplitude d . According to Eqs. (12.1 and 12.2), an uncertainty on the sensor outputs causes an uncertainty on the steering angle, which can be written as,   [α] = α α ,

(12.4)

where α = μα − α,l and α = μα + α,u are the lower and upper bounds of the steering angle interval [α]. The mean value of the steering angle interval is denoted as μα , whereas, α, l and α, u are the lower and upper bounds of the propagated uncertainty, respectively. Figure 12.3 shows, with the thick black lines, the left and right LVDT outputs with respect to the steering angle, according to Eq. (12.1). The thinner horizontal black lines represent interval uncertainties around an admissible measurement pair (dL , dR ), which is shown with green lines. The uncertainty bounds shown in Fig. 12.3 have been set to d = ± 10mm for each sensor for visual clarit. Each uncertainty interval associated with each sensor output generates two uncertainty intervals on the steering angle. All the propagated uncertainty intervals can be combined to calculate the smallest uncertainty interval around the true value of the steering angle. The upper and lower bounds of the propagated uncertainty interval around the true steering angle are shown in Fig. 12.4 for all steering angles between −90◦ and +90◦ . The upper and lower error bounds on α scale approximately linearly on the magnitude of d , up to d = ± 20mm. The geometric parameters presented in Table 12.1 can be varied in a parametric study to investigate how they affect the propagated uncertainty. Figure 12.5(a–d) show the influence of the parameters e, η, l, γ on the propagated uncertainty, respectively. The contour plots of Fig. 12.5 show only the upper bound of the uncertainty interval, as this is symmetric with respect to the true value of the steering angle. For each design parameter, an optimal value can be determined, which generates the smallest interval of propagated uncertainty. The optimal values of the parameters are indicated with the red lines in Fig. 12.5(a–d). The ground-steering sensing mechanism can then be redesigned using the newly determined values of the sensor arrangements. This guarantees a smaller propagation of uncertainties from the sensors to the estimated steering angle. It should be noticed that the parameters e, η and γ indicated in Table 12.1 are very close to the optimal parameters shown in Fig. 12.5(a, b and d). The optimal parameter l, instead, is larger than the one appearing in the table, but the values of the propagated uncertainty are very close to each other, making this difference almost negligible.

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12.4 State Estimation Algorithms The previous section dealt with the measurement uncertainty propagation, however, the true value of the steering angle was assumed to be known. In practice, the steering angle is the unknown variable and the only variables that are known are the outputs from the LVDTs and the model parameters. Both the model parameters and the sensor outputs, however, are affected by uncertainties on their values, and the sensors are subject to potential faults. Hence, it is crucial to reliably

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Fig. 12.5 Parametric analysis of the propagated uncertainty interval with respect to the values of the model parameters of Table 12.1. Maximum error in the steering angle estimation for a measurement uncertainty d = ± 1mm: (a) varying e; (b) varying η; (c) varying l; (d) varying γ . The red dashed lines represent the optimal values of the parameters that generate the smallest uncertainty interval on the steering angle

estimate the controlled signal α and to calculate the level of uncertainty associated with this estimation. This section presents a comparison among three different algorithms for the estimation of the steering angle. The first algorithm is based on an ordinary least-squares (OLS) approach that is commonly implemented in Global Positioning Systems (GPS), where the position of a GPS receiver is calculated using the signals sent by a set of redundant satellites [5, 6]. For the OLS estimation algorithm, the set of Eqs. (12.2) can be rewritten in matrix form as follows, Aˆx = d − vˆ ,

(12.5)

which is also known as observation equation, where ⎡

(b cos γ − a sin γ ) (a cos γ ⎢ ⎢ (b cos γ − a sin γ ) (a cos γ A=⎢ ⎣ (a sin γ − b cos γ ) (a cos γ (a sin γ − b cos γ ) (a cos γ is the matrix of model parameters,

⎤ + b sin γ ) ⎥ + b sin γ ) ⎥ ⎥, + b sin γ ) ⎦ + b sin γ )

(12.6)

12 Virtual Sensing of Wheel Position in Ground-Steering Systems for Aircraft Using Digital Twins



 sin αˆ , cos αˆ

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(12.7)

is the vector of estimated states, ⎧ 2 2 ⎫ 2 2 ⎪ ⎪ ⎪ l 2 + a 2 + b2 − dL1 ⎪ ⎨ 2 ⎬ 1 l + a + b − dL2 , d= 2 2 2 2 2l ⎪ ⎪ l + a + b − dR1 ⎪ ⎪ ⎩ ⎭ 2 l 2 + a 2 + b2 − dR2

(12.8)

is the vector containing the measurements, also known as observation vector, and vˆ is the error vector between the estimated and the measured observations. The OLS estimate is computed by minimizing the sum of the squared errors, which results in, −1 xˆ = AT A AT d,

(12.9)

And the steering angle estimation is derived from Eq. (12.9) as follows, sin αˆ cos αˆ

αˆ = atan

! (12.10)

.

The OLS estimator is computationally inexpensive and provides accurate enough estimates with noisy measurements, however, it lacks of robustness in presence of larger deviations or faults in the sensors. This motivates to look for estimation algorithms that are more robust against outliers. Given the four measurements from the left and right LVDTs, eight possible steering angles are calculated by solving the inverse kinematic problem given in Eq. (12.2). The solutions given by the left sensors can be written as, ⎛ αˆ Li,1,2 = 2atan ⎝

b±



a 2 + b2 − L2i a + Li

⎞ ⎠ − γ,

(12.11)

and for the right sensors as, ⎛ αˆ Ri,1,2 = 2atan ⎝

−b ±



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a + Ri

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(12.12)

where Li and Ri have been defined as, Li 

2 l 2 + a 2 + b2 − dLi , 2l

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2 l 2 + a 2 + b2 − dRi . 2l

(12.13)

The eight possible steering angle estimates cast the observation vector as follows,   αˆ = αˆ L1,1 αˆ L1,2 αˆ L2,1 αˆ L2,2 αˆ R1,1 αˆ R1,2 αˆ R2,1 αˆ R2,2 .

(12.14)

One of the most common methods of robust estimation are M-estimators, which can be regarded as a generalization of the OLS with a higher tolerance for outliers. In this case, instead of using the sum of the squared errors, the cost function can be written as [7], n  i=1

ρ (ei ) ,

(12.15)

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where ρ(e) is a general function of the error. Solving Eq. (12.15) is equivalent to solve a weighted least squares problem. However, in this case, the weights depend upon the errors, the errors depend upon the estimates, and the estimates depend upon the weights. Hence, an iterative approach is required, which is called iteratively reweighted least squares (IRLS) [7]. The IRLS algorithm works as follow, • Select an initial estimate, such as the OLS estimate, or the median estimate; • At each iteration calculate the error residuals ei and associate weights from the previous iteration; • Solve for the new weighted least squares estimate, −1 ˆ αˆ = HT WH HT Wα,

(12.16)

where αˆ is given by Eq. (12.14), H ∈ R8x1 with all elements of H equal to 1/8 , and W is a diagonal matrix containing the weights; • The previous two steps are repeated until the estimate converges. Different ρ(e) can be used for the IRLS algorithm, the one used in this paper is the Huber objective function [7], for which the weight become,  w (ei ) =

|ei | ≤ k 1 , k/ |ei | |ei | > k

(12.17)

where k is a tuning constant, which has been chosen to be twice the median absolute deviation (MAD) in this study. The estimator is more robust against outliers for smaller values of k. Although the IRLS is more robust against outliers with respect to the OLS, it is not always able to deal with sensor faults, because in certain circumstances more than half of the elements of Eq. (12.14) are incorrect and far from the true value of the steering angle. A different approach for the estimation of the steering angle is the soft-computing (S-C), or soft-voting method. The S-C approach is a sensor management technique that is similar to the majority voting approach, but in this case the observations are assigned weights that are based on fuzzy membership functions and the final estimate is computed as a weighted average of all valid observations. The S-C algorithm can be summarized as follows [8, 9]. Each element of the observation vector given by Eq. (12.14) is assigned a weight and the consolidated steering angle estimate is the weighted average of all valid observations, αˆ =

n 

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(12.18)

i=1

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,

(12.19)

where μi is the membership degree of α i and is bounded between zero and one. The membership degree of each signal is computed as shown in Fig. 12.6, where the current value of each observation (red vertical line) forms the centre of its corresponding membership function, which is shown with the black solid line. The membership degree of the signal is the largest membership degree of the remaining valid observations (green vertical lines) according to,   μi = maxμi αˆ j . i=j

(12.20)

There are various shapes for the membership functions, such as triangular, trapezoidal, sigmoidal or polynomial. A ◦ polynomial membership function has been used in this paper, which has a flat top between ±2 and is null for observation ◦ differences above 7 . This membership function is shown in Fig. 12.6.

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Fig. 12.7 Schematic of the soft-computing algorithm

The implementation of the S-C algorithm is illustrated in Fig. 12.7. The vector of observations is split into the eight observation signals that are then sorted. The membership degrees of the signals are computed according to Eq. (12.20), reorganized in the original order and combined in a vector. Finally, the consolidated steering angle estimation is computed according to Eqs. (12.18 and 12.19).

12.5 Results The methods of OLS, IRLS and S-C are compared in this section for the following scenarios, • First, the ideal scenario in which both the model parameters and the sensor outputs are assumed to be known perfectly. The results of implementing the OLS, IRLS and S-C approaches are shown in Fig. 12.8(a and b) with the dash-dotted red, dashed green and solid blue line, respectively. The black solid line represents the true value of the steering angle. • The second scenario includes an uncertainty of = ± 2mm for the model parameters given in Table 12.1, and measurement noise of d = ± 2mm for the LVDT sensors, however, faults have not been taken into account in this scenario. The results are shown in terms of estimated steering angle and estimation accuracy in Fig. 12.9(a and b).

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• The third and final scenario includes parameter uncertainties ( = ± 2mm), measurement noise ( d = ± 2mm), and sensor faults. In this case, the faulty sensor id one of the left LVDTs, dL1 , which is locked at 120 mm. The results and comparison amongst the different estimation algorithms are shown in terms of estimated steering angle and estimation accuracy in Fig. 12.10(a and b). The results presented in Figs. 12.8 and 12.9 show that the least squares based estimators can have a slightly better accuracy than the S-C algorithm when none of the sensors is faulty, regardless of the uncertainty. Figure 12.10, however, show that the S-C approach is more robust to sensor faults when compared to the OLS and the IRLS. In fact, it is the only one of the three proposed methods to give a consistent estimation accuracy across all the scenarios that have been studied.

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12.6 Conclusion This paper presented the estimation of wheel direction in ground-steering systems for aircraft using the measurements of remote and redundant LVDT sensors subject to uncertainties and faults. Firstly, the digital twin of the ground steering system has been introduced, which consists of a kinematic model of the sensing mechanism together with the model uncertainties, the measurement noise and the potential sensor faults. The nonlinear relationship between the steering angle and the sensor outputs was then discussed. Secondly, an interval type of uncertainty was introduced for each LVDT output and was propagated through the model to calculate the maximum error in the steering angle estimation. The variation of such error with the wheel position was also presented. A parametric study of the uncertainty propagation was carried out in order to calculate the optimal arrangement of the sensors that generates the smallest error in the estimation of the steering angle. Finally, three estimation algorithms were presented for the estimation of the wheel direction in presence of model uncertainties, measurement noise and sensors faults. Two least squared based approaches, namely ordinary least squares (OLS) and iteratively reweighted least squares (IRLS), and a soft-computing (S-C) method were introduced. A comparison among these three methods under different scenarios of uncertainties and faults was carried out in terms of accuracy of the ◦ ◦ estimation for a steering angle ranging from −90 to +90 . It is shown that the S-C estimation algorithm is more robust than the least squares based methods in presence of uncertainties and sensor faults. Future work will relate to the definition of a metric that calculates the degree of confidence in the estimate and the monitoring of the sensor conditions. Acknowledgements The authors gratefully acknowledge the support of the UK Engineering and Physical Sciences Research Council (EPSRC) through the DigiTwin project (grant EP/R006768/1).

References 1. Currey, N.S.: Aircraft Landing Gear design: Principles and Practices. American Institute of Aeronautics and Astronautics, Washington, DC (1988) 2. Zhang, M., Jiang, R.M., Nie, H.: Design and test of dual actuator nose wheel steering system for large civil aircraft. Int. J. Aerospace Eng., no. Article ID 1626015. 14 (2016) 3. Dal Borgo, M., Elliott, S. J., Ghandchi Tehrani, M., Stothers, I. M.: Robust estimation of wheel direction in ground-steering systems for aircraft. In Proceedings of the 26th International Congress on Sound and Vibration (ICSV26), Montreal, Canada, pp. 1–8 (2019) 4. Jaulin, L., Kieffer, M., Didrit, O., Walter, E.: Applied Interval Analysis: with Examples in Parameter and State Estimation, Robust Control and Robotics. Springer, London (2001) 5. Blewitt, G.: Basics of the GPS technique: observation equations. In: Geodetic Applications of GPS, pp. 10–54. Swedish Land Survey (1997)

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6. Langley, R.B.: The mathematics of GPS. GPS world. 2(7), 45–50 (1991) 7. Huber, P.J., Ronchetti, E.M.: Robust Statistics. Wiley, Hoboken (2009) 8. Oosterom, M., Babuska, R., Verbruggen, H.B.: Soft computing applications in aircraft sensor management and flight control law reconfiguration. IEEE Trans. Syst. Man Cybern. Part C Appl. Rev. 32(2), 125–139 (2002) 9. Berdjag, D., Zolghadri, A., Cieslak, J., Goupil, P.: Fault detection and isolation for redundant aircraft sensors. In: 2010 Conference on Control and Fault-Tolerant Systems (SysTol), Nice, France, pp. 137–142 (2010) Mattia Dal Borgo graduated in Mechanical Engineering in 2014. He then joined the ISVR, where he received his PhD in active control of nonlinear systems. Mattia’s research interests are in digital twins of safety-critical control systems, particularly for aerospace applications.

Chapter 13

Assessing Model Form Uncertainty in Fracture Models Using Digital Image Correlation Robin Callens, Matthias Faes, and David Moens

Abstract Today, in aerospace and automotive industries, structural components are more and more designed up to their functional limits, pursuing weight minimization without compromising the mechanical integrity. Especially in the aforementioned domains, the fracture behavior is of utmost importance in this respect. Yet, due to the complexity of the underlying physical phenomena and corresponding models, taking fracture into account in a virtual design process still proves to be highly challenging. Nevertheless, in recent years, there has been a strong development in various numerical simulation techniques that can simulate crack initiation and growth, i.e., techniques such as the Extended Finite Element Method (XFEM), meshless methods as the Element Free Galerkin Method (EFG) or the Phase Field Fracture (PFF) modeling approach. In this paper, the objective is to examine to what extend these approaches are applicable in a design context by assessing their relative value regarding validity and uncertainty. Also, some numerical aspects regarding efficiency and stability are discussed. The work is based on the study of a specific case. First, standard compact tension experiments are performed. These results are used to identify the necessary material properties and associated parametric uncertainty levels for each of the considered fracture modelling approaches. Next, a validation study is performed in which stereo Digital Image Correlation (DIC) is applied during the actual crack propagation phase in the same compact tension test. The model form uncertainty is assessed by comparing the set of DIC full field displacement and strain measurements with the range of the numerical predictions including uncertainty of the different modelling strategies. Keywords Digital image correlation · Model form uncertainty · Extended finite element method · Phase field fracture · Compact tension test

13.1 Introduction Crack propagation is one of the most important failure causes in engineering application. This is especially true in aerospace and automotive industries, where structural components are designed up to their functional limits, pursuing weight minimization without compromising the mechanical integrity. Therefore, in these domains, the inclusion of crack propagation analysis in numerical simulations for early design assessment is rapidly gaining importance. Today, there are several techniques available for this. The most commonly applied ones include the Finite Element Method (FEM) [7], the Extended Finite Element Method (XFEM) [11], the Element Free Galerkin Method (EFG) [4] and the Phase Field Fracture (PFF) modeling approach [10]. While describing the same phenomenon, all these methods rely on very diverse fundamental numerical techniques to represent the crack and its propagation behaviour in the model. Looking at the evolution in these approaches, the first attempts to use the classical finite element modelling scheme for simulating crack propagation soon revealed two major disadvantages. On the one hand, as the crack propagation is by nature linked to local geometric effects, the results of the FE analyses prove to be highly mesh dependent. Furthermore, the strain concentration can only be estimated accurately when sufficient element resolution is present at the crack tip [11]. Consequently, a very fine mesh is necessary already in the initial simulation. Then, when the crack starts to propagate, a fine mesh in the crack propagation direction is needed, for which often remeshing is used. For complex geometries, this remeshing involves a significant computational burden [4, 11]. To resolve both these issues, XFEM was introduced in 1999 by Moës et al. [11]. XFEM adds the ability to describe the crack inside the elements, so a much coarser mesh can be used.

R. Callens · M. Faes · D. Moens () Department of Mechanical Engineering, LMSD, KU Leuven, Sint-Katelijne-Waver, Belgium e-mail: [email protected]; [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2020 Z. Mao (ed.), Model Validation and Uncertainty Quantification, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47638-0_13

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However, the implementation in commercial software still remains a major issue [15]. Another disadvantage is that XFEM assumes no damage around the crack itself. To cope with this issue, the Phase Field Fracture (PFF) modeling approach is proposed [12]. In this approach, the crack is described as being diffuse over the geometry, and consequently, damage is spread over the part. With this technique, there is no issue with mesh dependency and remeshing. However the PFF technique requires a fine mesh of typically order 0.05 mm around the crack [5, 10]. With XFEM and PFF as fundamentally different yet competing approaches available for crack simulation, this paper aims to quantify the model form uncertainty of both. This is done based on the standard compact tension tests according to ASTM D5045-14 [3]. The model form uncertainty is characterized based on a validation study. As the exact composition of the used material is unknown, first, a parametric model updating is performed. In a second step, XFEM and PFF are compared with respect to calculation time and stability. In the experimental part of this study, an innovative approach is followed, based on Digital Image Correlation (DIC). Whereas in literature, crack simulations are typically validated using force displacement curves or images from before and after crack propagation [1, 9], the DIC approach enables a totally new insight in the crack behaviour. By looking at local strain evolutions in the geometry during crack propagation [8], important observations can be made regarding the accuracy of the studied approaches. Still, this DIC approach poses challenges regarding image capturing, as the crack propagation speed can be very high. For instance, recently we measured propagation speeds of 256 m/s in ABS [17]. Therefore, in this work, DIC is used based on images from high speed cameras. The paper is structured as follows. The next Sect. 13.2 gives an overview of XFEM and PFF. Section 13.3 then introduces the compact tension model, and compares both modeling approaches for this specific case. Section 13.4 focuses on the parametric model updating for XFEM and PFF. Finally, in Sect. 13.5 the models are compared with the experiments to identify the model form uncertainty. The last Sect. 13.6 gives the conclusions of this paper.

13.2 Crack Simulations 13.2.1 Finite Element Method The Finite Element Method (FEM) is a widely available and used technique to approximate the solution of partial differential equations. As already mentioned in the introduction, when it comes to crack simulation, FEM has two major disadvantages. The first is related to the fact that when in an FE model a crack is growing, it is limited to do that over the element edges, introducing a significant mesh dependency in the solution. Therefore, the solution is biased with an error depending on the element size, which can only be remedied using a dense element mesh around the moving crack tip. A problem that arises at this point is that the crack path is not know a priori, so selecting a restricted region where a fine mesh is needed, is not possible. On the other hand, meshing the entire geometry with fine elements is computationally intractable. To overcome this issue, remeshing approaches ‘following’ the moving crack tip have been introduced in literature. The disadvantage here is that the outcome of these models is strongly depending on the applied remeshing approach [13].

13.2.2 Extended Finite Element Method The Extended Finite Element Method (XFEM) is FEM with two added functions to overcome the two major disadvantages of FEM addressed above, i.e., mesh dependent results and remeshing. A Heaviside function H (x) is added to give the crack the possibility to propagate trough an element, which is given for  ⊂ Rdr with dr the number of Cartesian spatial dimensions under consideration. The Heaviside function in a cracked element has two values: H (x) = −1 and H (x) = 1. For example, for  ⊂ R1 , in Fig. 13.1a H (x) = 1 from node 1 until the crack and H (x) = −1 from the crack until node 2. This way the elements are split in two. A second function is needed such that the crack can stop inside an element: the asymptotic crack tip function. For example, for  ⊂ R2 , the asymptotic displacement field around the crack tip in function of (r, θ ) can be described as [11]: Fl (r, θ ) =

! ! ! !   √ √ √ √ θ θ θ θ , r cos , r sin sin(θ ), r cos sin(θ ) r sin 2 2 2 2

(13.1)

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Fig. 13.1 Comparison of XFEM with a Heaviside function for a cracked element to the Phantom Node Method for e ⊂  in  ⊂ R1 (a) XFEM and (b) Phantom Node Method [15]

where Fl is the asymptotic crack tip function r and θ are the polar coordinates with origin at the crack tip. The approximated displacement field for  ⊂ R2 with XFEM then becomes: u = h



ui Ni +

i∈I



bj Nj H (x) +

j ∈J

 k∈K

Nk

/ 4 

0 ckl Fl (x)

(13.2)

l=1

# where Ni,j,k are basis functions, (ui , bj , ckl ) are degrees of freedom, i∈I ui Ni is the FEM displacement approximation for # # # all nodes I , j ∈J bj Nj H (x) is the Heaviside enrichment for the nodes J of cracked elements and k∈K Nk ( 4l=1 ckl Fl (x)) is the asymptotic enrichment function for the nodes K of elements where the crack tip is present. Now it is clear that XFEM uses 2 types of enrichments. What should not be neglected is that each enrichment comes with an extra degree of freedom in the stiffness matrix. So an open question arises “which nodes needs what kind of enrichment?” so that the stiffness matrix and the calculation time do not grow too large. When this is not known, the full geometry must be enriched, leading to a higher calculation time due to the increased number of degrees of freedom. To account for this, the level-set method is used for selecting the nodes to enrich before and during crack propagation. This approach uses 2 functions to describe where the crack is situated. The first function gives the description of the crack path, the other function describes the crack tip [16]. The implementation of XFEM in commercial software is a difficult task due to the enrichment functions. To reduce this implementation complexity, the Phantom Nodes Method is proposed bij Song et al. in 2006 [15]. They use only the Heaviside function such that the crack can only grow trough one or more elements at the time. Using this method, it is not possible for the crack to stop in the middle of an element. The implication to have one enrichment function less than XFEM is not only in implementation difficulty. The approximation for XFEM can furthermore be formulated in such a way that two overlapping elements are assumed, each representing a part of the cracked element. Phantom nodes therefore eliminate the need for the Heaviside function of XFEM, which makes the implementation easier [15]. Figure 13.1 visualizes the difference between XFEM (Fig. 13.1a) and the Phantom Nodes Method (Fig. 13.1b) in  ⊂ R1 . Using XFEM there is only one element e ⊂  with a Heaviside function. When using the phantom nodes, there are two overlapping elements with one normal and one phantom node.

13.2.3 Phase Field Fracture Method Opposed to XFEM, Phase Field Fracture (PFF) methods consider damage introduced in the geometry around the crack by means of a Phase Field function. This damage typically consists of micro voids or micro cracks before the crack is visually detectable. Phase Field Fracture methods have the major advantage that they simulate crack propagation without additional enrichments, mesh dependent solutions and remeshing. The typical Phase Field function for  ⊂ Rd1 problems equals [12]: d(x) = e−

|x| l

(13.3)

where d(x) is the damage, x is the distance from the crack and l gives the length scale factor which is described later in this paragraph. The damage function is defined such that the damage is spread over the part geometry in an exponentially decreasing way, the scale of which is defined by the length scale factor l. Amor et al. [2] determined that the length scale factor in  ⊂ Rd1 for uni-axial stress is function of several material properties: l=

27EGC 512σf2

(13.4)

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where σf is the fracture strength, E the Young’s modulus and GC the fracture toughness. Even though equation 13.4 is only valid for uni-axial stress, it also gives a good indication of the length scale factor for multi-axial stress. The disadvantage of PFF is the need for small element sizes to approximate the steep gradient of the exponential function. The minimal required element size is l/2 near the crack [5, 10]. Some papers [2] report that the simulation starts to converge only when the element size is l/10.

13.3 Application to Compact Tension Test 13.3.1 Compact Tension Model In this work, XFEM and PFF are applied to study crack propagation behaviour in a standard compact tension test according to standard AST M D5045-14. To ensure plain strain throughout the part, a thickness of 20 mm is selected. From the width the other dimensions can be defined according to the standard. As material, a PVC is chosen, the composition if which is not known. Therefore, the material properties are initially estimated based on the properties of standard PVC from a material database [6], summarized in Table 13.1 for XFEM and PFF. With the PFF technique, the length scale factor is calculated with equation 13.4. Before performing the model form uncertainty study, the material properties are updated to reduce the parametric uncertainty. An indicative 2D shell mesh for both XFEM and PFF is shown in Fig. 13.2a and b. The used elements are smaller than the indicative elements to ensure numerical convergence of the crack propagation prediction. Table 13.1 Initial material properties of PVC from a material properties database for XFEM and PFF [6], the length scale factor is calculated with equation 13.3

Material properties Young’s modulus Poisson-coefficient Energy release rate Fracture strength length scale factor

XFEM 2890 0,4 4,94 − −

PFF 2890 0,4 4,94 47,05 0,34

Units (SI) MP a − √ MP a m MP a mm

Fig. 13.2 Indicative mesh of the models for XFEM and PFF, with real used converging element size and number of elements. (a) Extended Finite Element Method fictive shell mesh, (1) 2D tetra elements and (2) 2D quad elements around the crack zone, the real mesh size is h = 0, 1 mm for the quads (9735 elements). (b) Phase Field Fracture Method fictive shell mesh, (1) 2D tetra elements and (2) 2D quad elements around the crack zone, the real mesh size is h = 0, 03 mm for the quads (20,236 elements)

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h = 2 mm h = 1 mm h = 0; 5 mm h = 0; 3 mm h = 0; 1 mm h = 0; 05 mm

2000 1800

Force (N)

1600 1400 1200 1000 0; 7 0; 8 0; 9 1; 0 1; 1 1; 2 1; 3

Displacement (mm) Fig. 13.3 Convergence study for XFEM when reducing the element size Table 13.2 XFEM convergence study when reducing the element size around the crack with number of elements and calculation time for 2 cores of an Intel Xeon CPU E5-2695 v3 @ 2.30GHz

Element Number size (mm) of elements 2 1513 1 2501 0,5 3329 0,3 3454 0,1 9735 0,05 22628

Calculation time (h:min) 0 : 04 0 : 08 0 : 16 0 : 27 2 : 24 6 : 53

Maximal force (N) 1362 1467 1693 1735 1935 2021

13.3.2 XFEM Convergence Study A convergence study is performed to select the appropriate element size for XFEM. Convergence is assessed in terms of the maximal force on the force-displacement curve. The force-displacement curves for different element sizes are visualized in Fig. 13.3. From Table 13.2, it can be noticed that the calculation time increases substantially when using smaller element sizes. It should be noted that for small elements not always a solution is obtained. It is know that XFEM has stability problems when the crack reaches element nodes, causing the solution of XFEM to be perturbed by round-off errors as a result of an ill-conditioned or even singular system matrix [14, 18]. Furthermore when using small elements, the distance between crack and nodes is small, causing instability as well. For the compact tension case, the stability is a major issue when the elements are smaller then 0, 1 mm. Therefore, in this paper, an element size of 0, 1 mm is used. The error on the maximal force is 4, 4%, so the simulation is considered to be converged.

13.3.3 PPF Convergence Study For PFF a convergence study is performed to define the element size and time step such that the simulation output is converged. Opposed to the automatic time stepping that is available for XFEM, such methods are not readily available for PFF. For the first part of the convergence study the element size is fixed to h = 0, 3 mm and the time step is varied. The results of this convergence study are shown in Fig. 13.4a and Table 13.3. From these results, it is clear that a time step of 0, 0001 s is sufficient, as further reduction of the time step only reduces the maximal force with 0, 5%.

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3800 3400 3000

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2600 2200 1; 5

1

2

2; 5 0; 01 sec 0; 005 sec 0; 001 sec 0; 0005 sec 0; 0001 sec 0; 00008 sec 0; 00006 sec

Displacement (mm) (a)

h = 0; 5 mm h = 0; 3 mm

2400 2300

h = 0; 1 mm h = 0; 05 mm h = 0; 03 mm

2200 2100 Force (N)

2000 1900 1800 1; 1

1; 2

1; 3

1; 4

1; 5

Displacement (mm) (b)

Fig. 13.4 PFF convergence study with reduction of the time step and element size. (a) PFF convergence study with reduction of the time step. (b) PFF convergence study for PFF with reduction of the element size

After the time step convergence, the convergence for the element size is checked with fixed time step of 0, 0001 s. The convergence study is visualized in Fig. 13.4b and Table 13.4. When the element size is h = 0, 05 mm, the maximal force increases with 1.3%. From these results, it is clear that an element size of h = 0, 03 mm is sufficient for convergence. This is in agreement with literature stating that an element size of h = l/10 should be sufficient [12].

13 Assessing Model Form Uncertainty in Fracture Models Using Digital Image Correlation Table 13.3 PFF convergence study when reducing the time step and calculation time for one core of an Intel Xeon CPU E5-2695 v3 @ 2,30GHz

Time step (s) 0, 01 0, 005 0, 001 0, 0005 0, 0001 0, 00008 0, 00006

125 Calculation time (h:min) 0 : 02 0 : 04 0 : 22 0 : 34 2 : 22 2 : 45 3 : 30

Maximal force (N) 6318 5128 3493 3121 2791 2771 2750

Table 13.4 PFF convergence study when reducing the element size with indication of total number of elements, the calculation time and the number of cores of one Intel Xeon CPU E5-2695 v3 @ 2,30GHz Element size (mm) 0, 5 0, 3 0, 1 0, 05 0, 03

Number of elements 1927 3562 17035 42435 20236

Calculation time (h:min) 2 : 22 1 : 25 4 : 22 2 : 48 4 : 30

Table 13.5 Quantified results from the mean curve from Fig. 13.5

Maximal force (N) 2779 2791 2293 2260 2231

Number of cores 1 2 2 6 2

Young’s modulus Maximal force Displacement at maximal force

1200 2850 3, 64

MP a N mm

13.3.4 Comparison of XFEM and PFF When comparing XFEM with PFF it is concluded that XFEM is not sufficiently stable for small elements h < 0, 1 mm. On the other hand, PFF gives always a result. Therefore, it is concluded that PFF is a more stable simulation technique. For PFF, the element size must be smaller than for XFEM: 0, 03 mm for PFF and 0, 1 mm for XFEM. As a consequence, the computation time for PFF (4 h 30 min) is found to be larger than for XFEM (2 h 24 min).

13.4 Parametric Model Updating The exact material composition of the used PVC is not know in advance, and so are the material properties. Therefore, first, an experimental campaign is set-up in which force displacement curves are quantified for 21 compact tension test. Consequently, the mean, the maximal en minimal curves are calculated. This is visualized in Fig. 13.5. From the mean curve, the maximal force, displacement at maximal force and Young’s modulus are measured and summarized in Table 13.5. When comparing the Young’s modulus of 2890 Mpa from typical material databases with the results in Table 13.5, there is a difference of more than 50%. This large discrepancy between justifies the parametric update prior to the model form uncertainty assessment. The model update is performed with the following minimization problem:  θˆ = arg min α||Fs − Fe ||22 + β||Xs − Xe ||22

(13.5)

θ∈

where θˆ = [E, GC ] ∈  for XFEM and θˆ = [E, l, GC ] ∈  for PFF, with  a set of admissible parameters. The parameters are searched such that the model-experiment discrepancy is minimized. ||Fs −Fe ||22 and ||Xs −Xe ||22 are the squared L2 -norm of the difference between the simulated and experimental values of maximal forces (F) and displacements at maximal forces

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Table 13.6 Results of the optimization problem

Simulation model XFEM PFF

Surrogate 0, 004 −0, 023

Full simulation 0, 005 0, 00075

Table 13.7 Original and updated material properties for XFEM and PFF with corresponding difference in the optimum Parameter − Fracture toughness (J/mm2 ) Young’s modulus (Mpa) Length scale factor (mm) 2 (-) eˆtot

XFEM init. 4, 94 2890 − 0, 294

opt. 19, 76 1195, 48 − 0, 004

PFF init. 4, 94 2890 0, 34 0, 919

opt. 18, 34 1298, 78 0, 26 −0, 023

Difference − −7, 19% +8, 64% − −

(X), where subscripts ( s ) and ( e ) indicate respectively the simulated and experimental values. Weights α and β are defined such that the initial value at the beginning of the optimization is 1/2 for each of both terms in the summation. This is done to achieve an equal weighting for both contributions. Consequently, the total goal function variation can also be expected to range between 1 (initialization) and 0 (full convergence). When solving the minimization problem, a large number of simulations is required, resulting in a large computation time (see Tables 13.2 and 13.4). Therefore, in this paper, the minimization is done using surrogate models for the goal function as defined in equation 13.5. For the training of the surrogate model, a supervised learning approach is applied that actively selects the most informative additional training samples, as well as the most appropriate surrogate modelling strategies. In this case, batches of 8 samples are used per iteration. At each iteration, the active learning algorithm selects between several response surfaces with varying basis. New batches of samples are generated until convergence of one of the surrogate models is sufficient. Both resulting surrogate models for the XFEM and PFF goal functions are cubic surfaces, with residuals in all the sample points smaller than 1 · 10−12 . The minimization is then performed on the surrogate models, reaching convergence at small goal function values, as indicated in the first column of Table 13.6. These small values show that the models reach good correspondence with the experimental data. Still, it can be noted that for PFF, there is a negative value, caused by approximation errors of the surrogate in the optimum. In order to estimate this error, full simulations are performed at the final optimization step. These results, reported in the second column of Table 13.6, show that the goal functions remain small. It can be concluded that the surrogate models produced a useful optimum, that can be used as starting point for the validation in the next section. The initial and optimized material properties are summarized in Table 13.7. From this table, based on relative difference of less than 10%, it is concluded that the material properties agree well for both simulation methods.

13.5 Model Validation After the parametric model updating, the model validation is performed to assess the residual model form error in both modelling approaches. For the validation, two criteria are used: the force displacement curve and the full field DIC strains in the loading direction eyy . First the force displacement curves from both simulations are compared to the experiments. This is visualized in Fig. 13.5. The force displacement curve can be cut in two parts, before and after the maximal force. In the first part there is linear elastic behavior and in the second part the crack is propagating. When comparing the simulations with the experiments some conclusions are made. In the first part, the simulations are similar to the experiments, with a small model form uncertainty. The maximal force is almost the same, this due to the model updating sequence. In the second part, during the crack propagation, the experiments give higher forces then the simulations. In this part, the model form uncertainty is comparably large for both simulations. When comparing both simulation models, they have almost the same model form

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PFF XFEM Min Exp Max Exp

Force (N)

Mean Exp

Displacement (mm)

Fig. 13.5 Validation of XFEM en PFF with the experimental force displacement curves

uncertainty. Next, the full field DIC strains give some interesting additional insights in this remaining uncertainty. When looking at the full field DIC strains on Fig. 13.6a–b before and after crack propagation, it can be seen that the strains eyy before crack propagation are almost the same to the simulations, Fig. 13.6c, e. During crack propagation, the strains eyy show remaining strains near the cracked surfaces. Those remaining strains are not visible in the simulations on Fig 13.6d, f. The conclusion here is that the remaining strains at the cracked surfaces are due to plastic deformation. As both simulations do not have the possibility to simulate plastic deformation, a large model form uncertainty for both simulation techniques is present, as also clearly depicted in the second part of the force displacement curve.

13.6 Conclusions The aim of this paper is to reduce the parametric model uncertainty and to assess the model form uncertainty for different crack simulation approaches. This is done using a compact tension test. Also a comparison between XFEM and PFF is performed with respect to calculation time and stability. When comparing both simulation techniques XFEM and PFF, it can be concluded that XFEM has the disadvantages that the simulation is not stable for small element sizes. PFF gives always a result, this simulation technique is thus more stable. When comparing the computational cost, XFEM is less demanding then PFF. With PFF, the element size needed for convergence is smaller then with XFEM. This work also confirms that the element size for PFF must be h = l/10 to get converged results, as claimed in literature. The parametric model uncertainty is successfully reduced for both models, with results reducing model-experiment discrepancy to less than 5%. These good results where obtained with an efficient parametric model updating sequence on a surrogate model. Finally, when looking at the model form uncertainty, both models are capable to simulate the linear elastic behavior, but not capable to simulate any plastic behavior that the experiments have. The implementation of techniques that can simulate plastic deformation is thus required to reduce the model form uncertainty.

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Fig. 13.6 eyy before and during crack propagation. (a) eyy before crack propagation with DIC. (b) eyy during crack propagation with DIC. (c) eyy before crack propagation with XFEM. (d) eyy during crack propagation with XFEM. (e) eyy before crack propagation with PFF. (f) eyy during crack propagation with PFF

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Acknowledgments The authors like to thank the Research Foundation Flanders (FWO) for funding the research with the project G0C2218N. Mathias Faes is a post-doctoral researcher of the Research Foundation Flanders (FWO) under grant 12P3519N.

References 1. Ambati, M., Kruse, R., De Lorenzis, L.: A phase-field model for ductile fracture at finite strains and its experimental verification. Comput. Mech. 57(1), 149–167 (2016) 2. Amor, H., Marigo, J.-J., Maurini, C.: Regularized formulation of the variational brittle fracture with unilateral contact: numerical experiments. J. Mech. Phys. Solids 57(8), 1209–1229 (2009) 3. ASTM International: ASTM D 5054 Standard Test Methods for Plane-Strain Fracture Toughness and Strain Energy Release Rate of Plastic Materials. ASTM Book of Standards 99, Reapproved 2007, 1–9 (2013) 4. Belytschko, T., Black, T.: Elastic crack growth in finite elements with minimal remeshing. Int. J. Numer. Methods Eng. 45(5), 601–620 (1999) 5. Borden, M.J., Verhoosel, C.V., Scott, M.A., Hughes, T.J., Landis, C.M.: A phase-field description of dynamic brittle fracture. Comput. Methods Appl. Mech. Eng. 217–220, 77–95 (2012) 6. Edupack, E.: ECS Edupack 2018 (2018) 7. Formica, G., Milicchio, F.: Crack growth propagation using standard FEM. Eng. Fract. Mech. 165, 1–18 (2016) 8. Kasvayee, K.A., Salomonsson, K., Ghassemali, E., Jarfors, A.E.: Microstructural strain distribution in ductile iron; comparison between finite element simulation and digital image correlation measurements. Mater. Sci. Eng.: A 655, 27–35 (2016) 9. Marco, M., Belda, R., Miguélez, M.H., Giner, E.: A heterogeneous orientation criterion for crack modelling in cortical bone using a phantomnode approach. Finite Elem. Anal. Des. 146, 107–117 (2018) 10. Miehe, C., Welschinger, F., Hofacker, M.: Thermodynamically consistent phase-field models of fracture: variational principles and multi-field FE implementations. Int. J.. Numer. Methods Eng. 83(10), 1273–1311 (2010) 11. Moës, N., Dolbow, J., Belytschko, T.: A finite element method for crack growth without remeshing. Int. J. Numer. Methods Eng. 46(1), 131–150 (1999) 12. Molnár, G., Gravouil, A.: 2D and 3D Abaqus implementation of a robust staggered phase-field solution for modeling brittle fracture. Finite Elem. Anal. Des. 130, 27–38 (2017) 13. Nguyen, V.P., Rabczuk, T., Bordas, S., Duflot, M.: Meshless methods: a review and computer implementation aspects. Math. Comput. Simul. 79(3), 763–813 (2008) 14. Sillem, A., Simone, A., Sluys, L.J.: The orthonormalized generalized finite element method-OGFEM: efficient and stable reduction of approximation errors through multiple orthonormalized enriched basis functions. Comput. Methods Appl. Mech. Eng. 287, 112–149 (2015) 15. Song, J.H., Areias, P.M., Belytschko, T.: A method for dynamic crack and shear band propagation with phantom nodes. Int. J. Numer. Methods Eng. 67(6), 868–893 (2006) 16. Stolarska, M., Chopp, D.L., Mos, N., Belytschko, T.: Modelling crack growth by level sets in the extended finite element method. Int. J. Numer. Methods Eng. 51(8), 943–960 (2001) 17. Van Eekert, B.: Karakterisatie van kunststoffen voor 3D-geprinte matrijzen. Master’s thesis, KU Leuven (2018) 18. Wu, J.-Y., Li, F.-B.: An improved stable XFEM (Is-XFEM) with a novel enrichment function for the computational modeling of cohesive cracks. Comput. Methods Appl. Mech. Eng. 295, 77–107 (2015) Robin Callens, a PhD student at the Reliable and Robust Design group of KU Leuven. My PhD project is about spatial uncertainty modeling for mechanical reliability. I graduated in Engineering Technology at KU Leuven with option Mechanical and Electrical Engineering in June 2019.

Chapter 14

Identification of Lack of Knowledge Using Analytical Redundancy Applied to Structural Dynamic Systems Jakob Hartig, Florian Hoppe, Daniel Martin, Georg Staudter, Tugrul Öztürk, Reiner Anderl, Peter Groche, Peter F. Pelz, and Matthias Weigold

Abstract Reliability of sensor information in today’s highly automated systems is crucial. Neglected and not quantifiable uncertainties lead to lack of knowledge which results in erroneous interpretation of sensor data. Physical redundancy is an often-used approach to reduce the impact of lack of knowledge but in many cases is infeasible and gives no absolute certainty about which sensors and models to trust. However, structural models can link spatially distributed sensors to create analytical redundancy. By using existing sensor data and models, analytical redundancy comes with the benefits of unchanged structural behavior and cost efficiency. The detection of conflicting data using analytical redundancy reveals lack of knowledge, e.g. in sensors or models, and supports the inference from conflict to cause. We present an approach to enforce analytical redundancy by using an information model of the technical system formalizing sensors, physical models and the corresponding uncertainty in a unified framework. This allows for continuous validation of models and the verification of sensor data. This approach is applied to a structural dynamic system with various sensors based on an aircraft landing gear system. Keywords Interpretation of sensor data · Data-induced conflicts · Analytical redundancy · Lack of knowledge · Sensor error

14.1 Introduction On 14 March 2017, the ExoMars Schiaparelli Mars probe crashed, resulting in a total loss of the descent stage [1]. According to ESA, the cause was a defective orientation sensor that provided invalid values. Although conflicts with other data were detected, they were ignored. In the following two years, two fully manned Boeing 737 Max crashed, which in both cases led to the death of all passengers and the crews. According to the National Transportation Safety Committee, the cause was a malfunction of one of the angle of attack sensors which caused the control system to push the nose of the airplane down [2]. A difference between left and right angle of attack sensor of 20◦ was detected until the end of record. Highly automated systems rely on an increasing amount of information derived from data. However, the incidents show that the origin and context of data has not yet been sufficiently taken into account. Often data are assumed to be the true values and uncertainty or lack of knowledge is often assumed to be non-existent or is ignored in their generation process. If, on the other hand, data from several sources are interpreted simultaneously, they can induce conflicts. These in turn reveal

J. Hartig · P. F. Pelz Chair of Fluid Systems, Technical University of Darmstadt, Darmstadt, Germany e-mail: [email protected]; [email protected] F. Hoppe () · D. Martin · P. Groche Institute for Production Engineering and Forming Machines, Technical University of Darmstadt, Darmstadt, Germany e-mail: [email protected]; [email protected]; [email protected] G. Staudter · R. Anderl Institute of Computer Integrated Design, Technical University of Darmstadt, Darmstadt, Germany e-mail: [email protected]; [email protected] T. Öztürk · M. Weigold Institute of Production Management, Technology and Machine Tools, Technical University of Darmstadt, Darmstadt, Germany e-mail: [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2020 Z. Mao (ed.), Model Validation and Uncertainty Quantification, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47638-0_14

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lack of knowledge in the interpretation of the data and at the same time allow continuous validation of the data and of the models used in their creation. The generation of sensor data often involves a multitude of models and parameters in addition to the physical sensor structure. We summarize these models and parameters under the term metadata. The data, the metadata and the physical sensor are a data source. Often, different data sources share part of the metadata, e.g. strain gauges, which are applied to the same technical structure and therefore share the same modulus of elasticity. Thus there is a chance not only to conclude from the conflicting data that there is lack of knowledge, but also to infer the cause by means of patterns in the metadata used. This requires methods for uncovering and exploiting the content and context of metadata in data sources. We propose a method to link sensor and model metadata with data induced conflicts to reveal lack of knowledge.

14.2 Background Different methods have been developed to deal with conflicting data sources. On the one hand conflicts between data sources can be seen as part of erroneous system behavior. Thus different methods use conflicting data for fault detection and fault isolation [3, 4]. On the other hand, conflicts can also be seen as part of the systems normal behavior. Then data from multiple sources can be used to reduce uncertainty and to improve the overall level of data quality. Simple methods for data reconciliation of conflicting sensor data are voting systems [5]. More elaborate fusion methods are the Bayes method [6, 7], Dempster-Shafer method [8, 9], and heuristic methods [10, 11]. In the process industry for the estimation of process states data reconciliation methods are implemented. The goal is to fuse the conflicting data, i.e. reconcile the state of the system with the conservation laws of mass and energy. For this the conservation laws and the measured values have to be an overdetermined equation system. With a quadratic minimization method, the system states are changed until the values satisfy the conservation laws [3]. Data sources consist of sensors and models. Models only represent a certain part of the relevant reality, see Fig. 14.1. At each step of concretion, from the relevant reality to the model and from the model to its data, uncertainty due to simplifications and assumptions occurs, so uncertainty is illustrated by the model not covering the relevant reality completely [12]. Model uncertainty results from the fact that the functional relationships are suspected, unknown, incomplete or ignored. Ignorance can lead to data-induced conflicts that arise when the interpretation and use of uncertain data from more than one source leads to contradictory statements. Contradicting values of different redundant data sources are in conflict, when their confidence intervals do not overlap. Data-induced conflicts can therefore be attributed to the model (model uncertainty), to the parameters of the model and to sensor errors. If uncertainty is not sufficiently taken into account, if too few or uncertain data sources are considered, these conflicts remain unnoticed. Once a sufficient amount of information is collected and combined from multiple sources, data-induced conflicts may become visible. The methods mentioned above for dealing with conflicting data sources, fail to differentiate between sensor and model and do not take the uncertainty into account. Therefore, we propose a method to support interpretation and decision in case of data induced conflicts with redundant data sources. Besides physical redundancy of sensor systems, analytical redundancy in the form of mathematical process models is used [13]. For the method the following two points have to be addressed: (i)

Fig. 14.1 Visualization of model uncertainty and lack of knowledge [12]

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Conflicts emerge when the confidence intervals do not overlap. Hence, the uncertainty in an interconnected system has to be propagated. How can this be done efficiently in an environment with many sensors and models? (ii) The different data sources, i.e. sensors and models are linked with the help of analytical redundancy. How can the dependency of different sensors and models on each other be modeled in an information model and used in the solving process?

14.3 Method To increase the availability of information about the desired value it is intended to observe redundant sources composed of physical sensors and a variety of models. Each source is associated with a certain amount of uncertainty due to precision, accuracy and model uncertainty, which are propagated to the target quantity. For efficient propagation of uncertainty, automatic differentiation is used for the derivatives [14]. When models change, the derivatives do not have to be implemented again. The implementation was done by operator overloading [15]. In a best case scenario, the signals of the redundant data sources can then be merged using mathematical methods of data fusion (c.f. BACKGROUND). In case of a data-induced conflict, where any subset of confidence intervals of the redundant values does not overlap, each source must first be checked for plausibility. Necessary to that end is to have sufficient knowledge (metadata) about the underlying technical system along with its sensors and known uncertainties to derive boundary conditions in order to validate if the signals are within certain limits. The next step is to investigate if different (conflicting) sources make use of common models and/or sensors. To gain the maximum amount of information it is mandatory to expand the methodology by providing a global view of the connections between data, models and sensors throughout the entire technical system. The observation is no longer limited to a single value from redundant sources, but provides information on whether the sensors/models used for their generation are also used in the redundant observation of other values and, if so, whether inconsistencies occur there. In case of a data-induced conflict, this information can then be used to restrict whether an error is caused by a sensor or due to model uncertainty and thus, reveal lack of knowledge in the interpretation of sensor data. In case of a data-induced conflict the consideration of only two redundant sources is not sufficient to draw conclusions on which source to trust. Therefore, it is mandatory to reify the investigation on the sensors and models used within the sources and to collect information whether those sensors/models are also used in the redundant observation of other values and, if so, whether inconsistencies occur there. There are two possible causes for data-induced conflicts: (i) sensor error and (ii) model uncertainty. (i) A data-induced conflict caused by a sensor error can then be identified in two ways. The first logical step is to check the sensor signal for plausibility using metadata to derive boundary conditions in order to check if the signals fulfill/meet certain characteristics or limits. If the sources pass the plausibility check further investigations must be done by examining if the same models with different sensors are used for the redundant observation of a value of the same physical variable at another location of the technical system. If this scenario reveals no inconsistencies, the cause can be restricted to a sensor error. Sensor errors can be categorized into technical failure and applications errors. A technical failure is any behavior departing from the sensors principal functionality, e.g. lack of power. An application error is any disorder caused by misconductions, such as dislocation during assembly, invalid or erroneous calibration, contaminations etc. (ii) If a conflict is caused by model uncertainty, it can be revealed by using different models but the same sensors. If this scenario reveals no inconsistencies, the cause can be restricted to model uncertainty. In technical systems possible model errors can be caused by either insufficient simplifications, parameters or assumptions, which can also differ as the system behavior changes unexpectedly, e.g. due to the wear, deformations, failure of components.

14.4 Validation There are a variety of possible sensor errors that can occur in a technical system. One example, which was detected during the investigation for this paper, is a wrong local sensor coordinate system due to swapped signal cables of a 3-ax piezoelectric force sensor. This is a sensor error difficult to detect, since the sensor signals are below its respective limit values. Lack of knowledge comes into effect by means of not knowing which source to trust. The following section shows the application of the proposed approach on an experiment series conducted at the modular active spring-damper system (MASDS) revealing data-induced conflicts caused by the introduced application error which was detected in the upper load-bearing structure.

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Fig. 14.2 Modular Active Spring-Damper System (MASDS)

The MASDS developed at the Collaborative Research Center (CRC) 805 “Control of Uncertainties in Load-Carrying Structures in Mechanical Engineering” serves as an application example for the proposed approach. The MASDS is a structural dynamic system based on an aircraft landing gear, see Fig. 14.2. The MASDS consists of two load-bearing structures of beams, connected via a joint module and a semi-active spring-damper. To quantify and evaluate the system response to drop impact tests in different test scenarios the MASDS is equipped with a variety of sensors to measure force, displacement, strain and acceleration. A 3-axial strain gauge force sensor is used to measure the forces that are introduced via the elastic foot. Above and below the spring-damper module 1-axial strain gauge force sensors (SG) are installed. The forces in the support housing of the upper structure are measured by 3-axial piezoelectric force sensors (PFS). Accelerometers are used to determine the acceleration of the upper structure, the demonstrator elastic foot and the spring-damper module. Displacement sensors, attached at the upper and lower gears of the supporting structure provide the measurement of the spring-damper module compression. Furthermore, tensile, pressure and bending loads of a multitude of bars are detected by strain gauges. By utilization of symmetries and different physical models redundant data is generated deliberately. The upper truss structure of the MASDS consists of symmetrical tetrahedral elements, which represent a rod system. The assumption of a perfectly perpendicular drop impact to the ground results in three symmetrical force paths passing the three support points (1–3) and the outermost rods. The force in each support points (1–3) is measured by 3-ax. piezoelectric force sensors. Strain gauges located in the surrounded rods are used to measure normal and bending strains, see Fig. 14.4. Under the assumption of a perpendicular drop impact, the resulting quantities can be investigated with 9 sensors used in redundant data sources of four models for this subassembly of the test rig. (i) Due to symmetry, the sum of the forces in x- and y-direction of all three piezoelectric force sensors must be zero. (ii) The force equilibrium at each support point has to be fulfilled in each direction. As shown in in the proposed approach (c.f. Fig. 14.3), the plausibility of each data sources needs to be verified by checking whether the signals are within certain limits derived from the characteristics of the examined system like mass, dimensions, sensor and mechanical properties. In the next step the conflict detection for the data sources takes place. Data induced conflicts emerge when confidence intervals of the sources do not overlap. The uncertainty of measured data is specified by confidence intervals. The assumption is made that the values show a gaussian distribution. The level of confidence is 95%. The uncertainty is propagated through the different models with gaussian error propagation. The propagation was implemented by using automatic differentiation (AD). Applied to an algorithm AD automatically gives the value of the algorithm as well as its derivative. AD is implemented by overloading the operators (e.g. addition, multiplication, trigonometric functions) of the used programming language. This makes the treatment of uncertain values straightforward and flexible. The following example deals with an application error of the 3-ax piezoelectric force sensor at the housing support 1, see Fig. 14.4. The wiring was done in such a way that the components in y- and z- direction are interchanged. In addition to that, the eventuality of an unexpected impact angel during a drop test is regarded by setting an inclination angle of 6◦ so that

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Fig. 14.3 Methodology for the identification of lack of knowledge

Fig. 14.4 Upper truss structure with sensor placements

the assumption of the MASDS being perpendicular to the ground is incorrect. This results in data-induced conflicts between different data sources. The redundant sources for the symmetry model are represented by the two graphs in Fig. 14.5, in which the force components in x- and y-direction are plotted over time. The force in support 1 is compared with the sum of the forces in support 2 and 3. The shading represents the confidence interval. A conflict in the force in y-direction can be seen. For the force equilibrium models, the results can be found in Figs. 14.6 and 14.7. The three force components are plotted over time. For FE1 a data induced conflict emerges for y- and z-direction. For FE2 no conflict appears. The results of the experiment are summarized in Table 14.1 below. Two data-induced conflicts were detected resulting in two consensus sets respectively with the models “Symmetry” and “FE1 ”. Within the model for symmetry one consensus subset contains only PFS1 , while PFS2 and PFS3 form the other subset. A second conflict is detected in the force equilibrium (FE1 ) at the housing support 1. The sensor PFS1 is the binding link which appears in both conflicting scenarios. Consequently, it is recommended to check PFS1 for a variety of sensor errors.

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Fig. 14.5 Source values for symmetry model under erroneous signal processing of PFS1

Fig. 14.6 Source values for FE1 under erroneous signal processing of PFS1

source values for joint FE2y

time in s

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joint force z in N

joint force x in N

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Fig. 14.7 Source values for FE2 under erroneous signal processing of PFS1

The verification of the force equilibrium model for housing support 1 shows that the y- and z- component of the force sensor PFS1 are interchanged (c.f. Fig. 14.6). After this error is found and fixed, the results can be checked again (c.f. Table 14.2). The error in force equilibrium 1 vanished. However, there is still a conflict in the symmetry model. This allows the conclusion that there is a modelling error as a consequence of the unexpected impact angle.

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Table 14.1 Result-matrix with two data induced-conflicts Symmetry

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Table 14.2 Result-matrix with one data induced-conflict Symmetry

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14.5 Conclusion It was shown that lack of knowledge in the interpretation of conflicting sensor data can be identified by differentiating data sources into models and sensors and spanning the investigation from the redundant observation of a single redundant value to the interconnection between models and sensors throughout the system. The proposed approach was applied to a structural dynamic system - the modular active spring-damper system (MASDS) developed at the Collaborative Research Center (CRC) 805 “Control of Uncertainties in Load-Carrying Structures in Mechanical Engineering”. Therefore, an information model was built containing all relevant metadata of the underlying sensor system, such as known uncertainties and calibration data. Methods for the propagation and calculation of the resulting uncertainties were implemented, resulting in a software-tool for the automatic detection of data-induced conflicts. A method for the visualization of data-induced conflicts as well as for the interconnections between models and sensors was developed and implemented in form of a matrix. This kind of representation provides information on whether sensors/models resulting in data-induced conflicts are also used in the redundant observation of other values and, if so, whether inconsistencies occur there. In case of a data-induced conflict, this information allows to restrict whether an error is caused by a sensor or due to model uncertainty and thus, reveal lack of knowledge in the interpretation of sensor data.

14.6 Outlook In future works, the approach should be expanded by implementing a solver for dynamic models with the help of automatic differentiation and the possibility to propagate uncertainties. This would lead to more redundant data sources. Furthermore, the different redundancy scenarios should be interconnected in a comprehensive network, so that sensor validity can be checked throughout the entire system covering all values of interest. Acknowledgements The authors would like to thank the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) for funding this research – Projektnummer 57157498 – SFB 805.

References 1. ESA, EXOMARS 2016 – Schiaparelli Anomaly Inquiry, Report DG-I/2017/546/TTN, 18 May 2017 2. KNKT, Aircraft Accident Investigation Report: Boeing 737-8 (MAX), PK-LQP, Report KNKT.18.10.35.04, 29 October 2018 3. Sbárbaro, D., Del Villar, R.: Advanced control and supervision of mineral processing plants. In: Advances in Industrial Control, London. Springer (2006) 4. Isermann: Fault-Diagnosis Systems. In: An Introduction from Fault Detection to Fault Tolerance. Springer, Berlin/Heidelberg (2006) 5. Fischer, M.J.: The consensus problem in unrealiable distributed systems (a brief survey). In: Foundations of Computation Theory, pp. 127–140. Springer, Berlin/Heidelberg (1983) 6. Castanedo, F.: A review of data fusion techniques. Sci. World J. (2013) 7. Kumar, M., Garg, D.P., Zachery, R.A.: A generalized approach for inconsistency detection in data fusion from multiple sensors. In: American Control Conference, 2006: 14–16 June 2006, [Minneapolis, MN]. IEEE Operations Center, Piscataway (2006) 8. Khaleghi, B., et al.: Multisensor data fusion: a review of the state-of-the-art. Inf. Fusion. 14, 28–44 (2013) 9. Yager, R.R.: On the dempster-Shafer framework and new combination rules. Inf. Sci. 41, 93–137 (1987) 10. Kreß, R., et al.: Fault detection and diagnosis for electrohydraulic actuators. IFAC Proceed. 33, 983–988 (2000) 11. Steinhorst, W.: Sicherheitstechnische Systeme. Vieweg+Teubner Verlag, Wiesbaden (1999) 12. Pelz, P. F., Hedrich, P.: Unsicherheitsklassifizierung anhand einer Unsicherheitskarte (Internal Report, Chair of Fluid Systems). Darmstadt (2015) 13. Isermann, R., Ballé, P.: Trends in the application of model-based fault detection and diagnosis of technical processes. Control. Eng. Pract. 5, 709–719 (1997) 14. Baydin, A.G., Pearlmutter, B.A., Radul, A.A., Siskind, J.M.: Automatic differentiation in machine learning: a survey. J. Mach. Learn. Res. 18(153), 1–43 (2018) 15. Walther, A., Griewank, A.: Getting started with ADOL-C. Version: 2012. In: Naumann, U., Schenk, O. (eds.) Combinatorial Scientific Computing Bd. 20121684. CRC Press, Boca Raton (2012) Jakob Hartig, 04/2018-Today: PhD candidate at chair of fluid systems, Technical University of Darmstadt. Research in uncertainty in engineering applications, hydraulic systems, modelling of urban structures. 01/2018: Master of Science in Mechanical and Process engineering. Florian Hoppe, 02/2015-06/2020: PhD candidate at institute for production engineering and forming machines, Technical University of Darmstadt. Research on digitization and control of process chains and servo presses. 12/2014: Master of Science in Electrical Engineering and Information Technology.

Chapter 15

A Structural Fatigue Monitoring Concept for Wind Turbines by Means of Digital Twins János Zierath, Sven-Erik Rosenow, Johannes Luthe, Andreas Schulze, Christiane Saalbach, Manuela Sander, and Christoph Woernle

Abstract Wind turbines are complex mechatronic systems designed to withstand high dynamic loads over a considerable period due to environmental influences. Following the guidelines from DNVGL or IEC, the design process of modern wind turbines is governed by a rather rough classification of annual mean wind speed and turbulence intensity resulting in very conservative design loads compared to the actual load conditions at the specific erection site of the wind turbine. Thus, structural reserves are very likely at the end of the turbine’s approved lifetime. Driven by this, a research collaboration between different institutions from wind industry and research facilities was initiated with the aim to exploit these structural reserves by means of continuous, model-based fatigue monitoring of individual wind turbine structures. Within this contribution, the authors present the fundamental process chain, arising challenges and first solutions employing the broad expertise of the individual project partners. Generally, the following key requirements are set for the monitoring concept: Precise estimation of the endured fatigue loads at critical spots by means of a digital twin; low sensor installation effort on the operating turbine comprising a minimal set of long-term reliable measurement hardware at accessible positions; online data processing on the turbine control system unit (SCADA). The research project comprises the design of state estimators for tower and blade structure, the definition of an optimal sensor network, the development of damage models for materials of blades and tower, concept studies on a scaled testbed of the turbine and the tests on the real wind turbine. Keywords Fatigue monitoring · Lifetime extension · Digital twin · Wind turbines

15.1 Introduction Today’s wind turbines operate with respect to a permitted lifetime of 20–25 years [1]. After that time span wind turbines are typical dismantled and replaced without taking into account that the technical lifetime might possibly not have been reached. In this contribution a structural fatigue monitoring concept is proposed that determines and records the actual loads and stresses acting on a wind turbine to estimate its remaining useful lifetime. The superior objective is to operate wind turbines until their technical lifetime has been reached. This solution should help to reduce the energy consumption for the wind turbine production and thus reducing the economical and ecological costs for the transition to sustainable energy sources. The idea to this contribution arises from two former research projects of W2E Wind to Energy in cooperation with the Institute for Automatic Control of the RWTH Aachen University (koRola, grant number 01IS17015A/B) and the Chair of Technical Dynamics of the University of Rostock (DynAWind, grant number 0325228C/D). In the research project koRola

J. Zierath () · S.-E. Rosenow W2E Wind to Energy, Rostock, Germany e-mail: [email protected]; [email protected] J. Luthe · A. Schulze · C. Woernle Department of Mechanical Engineering and Marine Technology, Chair of Technical Dynamics, University of Rostock, Rostock, Germany e-mail: [email protected]; [email protected]; [email protected] C. Saalbach WINDnovation Engineering Solutions, Berlin, Germany e-mail: [email protected] M. Sander Department of Mechanical Engineering and Marine Technology, Chair of Structural Mechanics, University of Rostock, Rostock, Germany e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2020 Z. Mao (ed.), Model Validation and Uncertainty Quantification, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47638-0_15

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the use of model predictive control for wind turbines is investigated. The reduced process model used within the controller scheme represents a digital twin of the wind turbines. The control variables are the result of an optimisation process with a process model within its core. In the DynAWind project the dynamical properties of wind turbines are experimentally und numerically investigated. One of the main lessons learned is how to observe the dynamic behaviour of a wind turbine structure by a minimum number of sensors supported by a numerical model.

15.2 Background Based on the experiences of the former research projects, W2E started in 2019 cooperation with the blade designer WINDnovation (WN) and the chairs for structural mechanics (STM) and technical dynamics (LTMD) of the University of Rostock the DynAWind2 project (grant number 0325228E/F/G). The aim of the current research project is to develop methods for a structural fatigue monitoring concept to operate wind turbines towards their technical life time. The structural fatigue monitoring concept comprises three main tasks: 1. Within a dynamic process model supported by a minimal sensor concept the displacement fields of the flexible components such as blades and tower are estimated. 2. The displacement fields of the flexible components are used to obtain the stresses at critical fatigue points within the structures or at their connections. 3. An online fatigue calculation is carried out to estimate the remaining useful lifetime by comparison to the design values. The main advantage of such a fatigue monitoring concept compared to the static concept of guidelines or rules is the possibility to obtain the actual fatigue values of a wind turbine. The typical guideline concept bases on a very rough site classification with respect to its wind conditions. Typically the environmental conditions lie below the guideline classification so that a remaining useful lifetime could be expected.

15.3 Analysis The following description provides an overview about the process chain for the fatigue monitoring (Fig. 15.1). The structure of an industrial wind turbine mainly consists of the tower, three rotor blades, the hub and the nacelle comprising the drive train and electrical components. From the mechanical point of view blades and tower are slender structures behaving like a beam. While the tower is regarded as fixed mounted with respect to the foundation, the motion of the blades can be divided

Stress Estimation

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into large rigid body motions overlaid by small deformations. With respect to the overall wind turbine dynamics the hub and the nacelle are regarded to be rigid. Based on these assumptions a mechanical model of the wind turbine is built up [2, 3]. The mechanical model is supported by a minimum set of measurements realised by Inertial Measurement Units (IMU). A IMU placed in the nacelle nearby the yaw bearing enables to measure the absolute position of the tower top. The methods used to obtain the stable position information are already known, e.g., from quadrocopter technology [4]. Furthermore one IMU is placed in each blade. From the known tower top position the undeformed reference configurations of the IMU positions in the blades are obtained by relative kinematics using the measured rotor position and rigid body kinematics. The difference of the measured IMU signal and the undeformed reference configuration provides the deformation vector of the blade at the sensor position. In a second step the mechanical wind turbine model and the deformation vectors at the sensor positions are used to reconstruct the displacement fields of the flexible structures. The estimated displacement field is the basis for the stress calculation at critical fatigue points, e.g., the bolt connections at tower bottom. The stress calculation for tower components and bolt connections, respectively, is obtained directly from beam mechanics. More difficult is the stress calculation of the rotor blades. Rotor blades are made from fibre-reinforced composites leading to more complex stress states. Due to the fact that a lot of steps within the working process of a rotor blade are handmade, the quality of the blades varies strongly. In first step it is assumed that the blades are ideally manufactured. The basis for the stress calculations at critical fatigue points is a detailed finite element model with the original design properties. In the last step the fatigue values are estimated. Here, also two different fatigue calculations are applied. The fatigue of the steel structures and bolt connections are estimated using the well-known, linear Palmgren-Miner schemes. More difficult to estimate is the fatigue behaviour of the rotor blade structures. The fatigue behaviour of fibre-reinforced composites is nonlinear. That means effects of the sequence of load cycles have to be taken into account. To obtain a more realistic fatigue model various laboratory tests with fibre-reinforced materials are carried out. The proposed structural fatigue monitoring concept bases on the record of 10 min time series and subsequent application of the fatigue calculations.

15.4 Conclusion The proposed structural fatigue monitoring concept provides the estimation of the remaining useful lifetime for wind turbines based on the actual loads and stresses. The main advantage of this concept compared to theoretical calculations as proposed in the common rules and guidelines is the record of actual events seen by the wind turbine. This includes especially extreme events such as storms, grid failures and unsymmetrical aerodynamic loads e.g. in wind parks. Thus the structural fatigue monitoring concept provides a more realistic lifetime for wind turbines. A minimal sensor concept based on micromechanical Inertial Measurement Units is introduced. The sensors are very robust and widely used in automotive and computer industry. A digital twin of the wind turbine is created to estimate the loads and stresses of the wind turbine where no sensors could be placed. For steel structures well-known fatigue models are applied. A fatigue model for fibre-reinforced composites will be created based on material tests. The calculated useful remaining lifetime has some uncertainties caused by known mistakes in the tool chain. An uncertainty analysis has to be carried out to provide a confidence interval of remaining lifetime. The proposed structural fatigue monitoring concept provides a possibility to reduce the ecological and economical costs of wind turbine for a sensible exit from nuclear and fossil-fuel energy. Acknowledgements This research is part of the DYNAWIND2 project funded by the German Federal Ministry for Economic Affairs and Energy under grant number 0325228E/F/G.

References 1. Lloyd, G.: Guideline for the Certification of Wind Turbines, Edition 2010. Germanischer Lloyd, Hamburg (2010) 2. Luthe, J.; Schulze, A.; Rachholz, R.; Zierath, J.; Woernle, C.: Acceleration-based strain estimation in a beam-like structure. In: Proceedings of the IMSD 2018 – The 5th Joint International Conference on Multibody System Dynamics, Lisboa, Portugal, 2018 3. Luthe, J., Schulze, A., Rachholz, R., Zierath, J., Woernle, C.: State observation in beam-like structures under unknown excitation. In: Kecskeméthy, A., Geu, F.F. (eds.) Multibody Dynamics 2019. Computational Methods in Applied Sciences, vol. 53. Springer, Cham (2020) 4. Konrad, T., Gehrt, J.-J., Lin, J., Zweigel, R., Abel, D.: Advanced state estimation for navigation of automated vehicles. Annu. Rev. Control. 46, 181–195 (2018). https://doi.org/10.1016/j.arcontrol.2018.09.002

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János Zierath, Since 2011 – Senior R&D engineer at W2E Wind to Energy. Since 2011 – Lecturer on Flexible Multibody Dynamics in Rostock. 2016/17 – visiting lecturer on Mechanics 3 and Structural Dynamics at Duisburg-Essen. 2015 – Associate Professor at University of Rostock. 2014 – Postdoctoral lecture qualification. 2011 – PhD

Chapter 16

Damage Identification of Structures Through Machine Learning Techniques with Updated Finite Element Models and Experimental Validations Panagiotis Seventekidis, Dimitrios Giagopoulos, Alexandros Arailopoulos, and Olga Markogiannaki

Abstract Structural Health Monitoring (SHM) Techniques have recently started to draw significant attention in engineering applications due to the need of maintenance cost reductions and catastrophic failures prevention. Most of the current research on SHM focuses on developing either purely experimental models or stays on purely numerical data without real application validation. The potential of SHM methods however could be unlocked, having accurate enough numerical models and classifiers that not only recognize but also locate or quantify the structural damage. The present study focuses on the implementation of a methodology to bridge the gap between SHM models with numerically generated data and correspondence with measurements from the real structure to provide reliable damage predictions. The methodology is applied in a composite carbon fiber tube truss structure which is constructed, using aluminum elements and steel bolts for the connections. The composite cylindrical parts are produced on a spinning axis by winded carbon fibers, cascaded on specified number of plies, in various angles and directions. 3D FE models of the examined cylindrical parts are developed in robust finite element analysis software simulating each carbon fiber ply and resin matrix and analyzed against static and dynamic loading to investigate their linear and nonlinear response. In addition, experimental tests on composite cylindrical parts are conducted based on the corresponding analysis tests. The potential damage to the structure is set as loose bolts defining a multiclass damage identification problem. The SHM procedure starts with optimal modeling of the structure using an updated Finite Element (FE) model scheme, for the extraction of the most accurate geometrical and physical numerical model. To develop a high-fidelity FE model for reliable damage prediction, modal residuals and mode shapes are combined with response residuals and time-histories of strains and accelerations by using the appropriate updating algorithm. Next, the potential multiclass damage is simulated with the optimal model through a series of stochastic FE load cases for different excitation characteristics. The acceleration time series obtained through a network of optimally placed sensors are defined as the feature vectors of each load case, which are to be fed in a supervised Neural Network (NN) classifier. The necessary data processing, feature learning and limitations of the NN are discussed. Finally, the learned NN is tested against the real structure for different damage cases identification. Keywords Structural health monitoring · Optimal modeling · Modal identification · Neural networks · Damage identification

16.1 Introduction Carbon fiber reinforced polymer (CFRP) composites have gained much attention in recent years through their industrialized implementation and use, as a structural material for static and dynamic load bearing as well as resistance to accidental excitations and actions. Due to its low-density, low thermal expansion and high strength, stiffness and corrosion resistance, applications from aerospace and automotive industry to building reinforcement and retrofit, as well as cryogenic fuel storage tanks are emerging rapidly [1, 2]. CFRP composites are manufactured on a spinning axis of various radii, by compressing multiple cascaded plies of pre-tensed carbon fibers, which are winded in certain volume fractions and patterns of angles and directions, against a liquid resin polymer matrix. The final product is obtained after leaving the composite material in a

P. Seventekidis · D. Giagopoulos () · A. Arailopoulos · O. Markogiannaki Department of Mechanical Engineering, University of Western Macedonia, Kozani, Greece e-mail: [email protected]; [email protected]; [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2020 Z. Mao (ed.), Model Validation and Uncertainty Quantification, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47638-0_16

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furnace for specific duration in order to achieve full strength and hardening characteristics [3, 4]. Being inherently sensitive to manufacturing treatment and due to its material variability, CFRPs strongly require certification results through numerical validation and hybrid (numerical - experimental) verification [5]. The most popular carbon fiber-reinforced composites, which have been extensively investigated by researchers, are the plain-woven CFRPs. Their popularity is attributed mainly to the low production cost combined to their effectiveness and efficiency under in-plane loading conditions. Presenting tension-compression asymmetric characteristics and orthotropic or even strong anisotropic mechanical behavior, due to varying fiber patterns, plain woven CFRPs are categorized to matrixdominant presenting low strength and to fiber-dominant presenting high strength [6]. Thus, it is of high importance to fully understand and grow high confidence about the mechanical behavior and in-plane loading capacity of each CFRP made structure. Moreover, as most engineering applications require multi-axial loading strength, their behavior in such loading conditions need also to be examined. Combined experimental measurements, conducted in and out of laboratory, to numerical Finite Element (FE) model simulations are employed in order to investigate in the macroscopic mechanical characteristics and material properties of CFRP structures [5, 7]. In this work, the material properties of a specific woven CFRP structure are classified and tuned reconciling experimental data to equivalent numerical (FE) model computations. This is achieved through combining modal residuals, that include the lowest identified modal frequencies and mode shapes, with response residuals, that include shape and amplitude correlation coefficients considering measured and analytical frequency response functions and time-histories of strains and accelerations [8–10]. Single objective structural identification strategies without the need of sub-structuring methods, are used for estimating the parameters of the finite element model. A state-of-the-art optimization algorithm, namely, covariance matrix adaptation evolution strategy (CMA-ES) [11, 12], is applied in parallel computing, to solve the single-objective optimization problem, arising from combining the above residuals [13, 14]. The applicability and effectiveness of the methods applied, is explored by updating the finite element model of a lightweight small-scale CFRP pin-joined structure. Issues related to estimating unidentifiable solutions [15, 16] arising in FE model updating formulations are also addressed. A systematic study is carried out to demonstrate the effect of model error, finite element model parameterization, number of measured modes and number of mode shape components on the optimal models and their variability. Evaluated as accurate enough, FE generated data may afterwards be used as input for SHM computations. Vibration based SHM methods is an emerging field along different engineering applications with promising results for reliable damage identification tools. Experimental mostly in its current form, SHM requires series of measurements in a structure that are afterwards fed in statistical classifiers that try to recognize the status of the structure [17, 18]. Numerical methods on the other hand may seem attractive due to their ability to simulate arbitrary damage cases in a structure, therefore creating large numbers of data as inputs to supervised learning classifiers. However, uncertainties in the modeling procedures, as well as in the experimental measurements that may try to validate the numerically derived results, lead to unreliable data derived from standard nominal numerical models. FE nominal models may be used for simpler tasks such as sensitivity analysis of the structure [19] for optimal sensor placement. A potential use of the model updating techniques in this work therefore rises, as substitute of the real experimental measurements to be used for training data in SHM classifiers. Provided that FE models are accurate enough, advantages of this approach in comparison to pure experimental approaches include simulation of any damage type and replacement of the time-consuming experimental measurements with computer generated data. The generated data can afterwards be used as input to supervised classifiers that may identify but also locate and quantify the anomalies in a structure, something that is usually hard to achieve with pure experimental approaches. In the present context the optimal FE model of a truss structure consisting of CFRP tubes with aluminum connections and steel bolts is used as a benchmark case, investigating the capabilities of a machine learning SHM framework through the information carried in the acceleration histories. The current work is organized as follows. The theoretical formulation of finite element model updating based on modal characteristics, frequency response functions is briefly presented in Sect. 16.2. Section 16.3 presents the adopted residual in time domain. Section 16.4 presents the experimental application, the development of the FE model of small-scale cantilever CFRP beam, its modal identification along with the FE model updating parameterization and results for orthotropic material characterization. In Sect. 16.5 the updated orthotropic material properties are verified on a small-scale pin-joined CFRP frame under dynamic excitation comparing experimental data and numerical results. Section 16.6 presents the SHM framework along with the Machine Learning FE training and experimental testing/validation procedure. Conclusions are shown in Sect. 16.7.

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16.2 Frequency and Time Domain Response Residuals Let a parameterized class of linear structural models used to model the dynamic behavior of the structure and let θ ∈ R Nθ be the set of free structural model parameters to be identified using the measured modal data. The overall measure of fit of the linear model, between the measured and the model predicted characteristics is formed in the following expression, combining modal and frequency response residuals [20, 21]:           J θ ; w = w1 J1 θ + w2 J2 θ + w3 J3 θ + w4 J4 θ

(16.1)

using equally weighting factorswi ≥ 0, i = 1, 2,  3, 4, with w1 + w2 + w3 + w4 = 1. For the first group, the measure of fit J1 θ is selected to represent the difference   between the measured and the model predicted frequencies for all modes. For the second group, the measure of fit J2 θ is selected to represent the difference between the measured and the model predicted mode shape components for all modes, given by: m m           J1 θ = εω2 r θ and J2 θ = εφ2 θ r=1

r=1

r

(16.2)

2   3   where the modal data are used ωr θ , φ r θ ∈ R N0 , r = 1, · · · , m to formulate the following residuals:

εωr

   



    ωr2 θ − ωˆ r2   βr θ φ r θ − φˆ r



θ = and εφ θ = r

ˆ

ωˆ r2

φ

(16.3)

r

and for the second group the measure of fit J3 (θ ) and J4 (θ ) represent the frequency response measures of fit as follows: m  m      2      2  1 − xs ωˆ r , θ and J4 θ = 1 − xa ωˆ r , θ J3 θ = r=1

(16.4)

r=1

where " " "{HX (ωk )}H {HA (ωk )}"2 xs (ωk ) =    {HX (ωk )}H {HX (ωk )} {HA (ωk )}H {HA (ωk )}

(16.5)

" " 2 "{HX (ωk )}H {HA (ωk )}" xa (ωk ) =     {HX (ωk )}H {HX (ωk )} + {HA (ωk )}H {HA (ωk )}

(16.6)

and

constitute the global and amplitude correlation coefficients [22], where {HX (ωk )} and {HA (ωk )} are the experimental (measured) and the numerical (predicted) response vectors at matching excitation – response locations, for any measured frequency point, ωk . Additionally, parameter estimation of nonlinear model is based on response time history measurements such as acceleration and displacements. This formulation has the advantage of applicability over both linear and non-linear systems; it compares the measured raw data of the experimental arrangement to the equivalent predictions of the numerical model. In this way, all available information is preserved and systematic errors of the identification procedure are alleviated. The measure of fit is given by: n   1  J θ; M = m i=1

#m

j =1



2   g ij θ m |M − yˆij #m   2 j =1 yˆij

(16.7)

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  where g ij θ m |M is the numerical time-history of the introduced FE model and yˆij is the respective experimental signal. Subscripts i correspond to the sensor (accelerometer) location and measurement direction, and j corresponds to the timestep instant. n is the total number of measured sensor locations and directions, whereas m is the total number of measured time-steps (number of observations).

16.3 Machine Learning with Neural Networks Artificial Neural Networks (ANN) comprise a relatively old idea [23] that mimics the biological function of neurons output and input. Mathematically a NN in its simplest form, called perceptron, computes an output based on a weighted sum of previous input as follows: Yi = f (Xi ) and Xi = bi + Wi yi−1

(16.8)

Where Yi is the output of neuron i computed with the transfer function f and Xi is weighted sum of its inputs yi − 1 which are weighted over with an array Wi and bias b. The goal in a trivial machine learning problem like the above is to correctly choose the weight matrix W and this process is called NN training, which nowadays is usually performed through stochastic gradient optimization methods using the Back-Propagation (BP) algorithm [24]. The simple perceptron however would not be suitable or adequate in a SHM problem where raw signal is intended to be fed in the training process due to structured way it expects data and also due to the inability to extract training features, requiring high user pre-processing. The above difficulties however are handled conveniently by a class of ANN called Convolutional Neural Networks (CNN) that by applying learned filters to the input data, they can extract training features in an unstructured manner. The convolution is filtering process that this time the neurons produce as output and an acceleration signal a may be filtered at a convolutional layer as follows: X = b + conv1D (k, a)

(16.9)

Where k would be the learned network filter, b the neuron bias and conv1D represents the filtering process of the initial signal. The learning of the filters adds in as an extra task for network BP training, however filtered signals now contain enhanced information about the status of structure. Moreover, additional layers isolate and extract the important signal characteristics, which end up in a simple perceptron classifier. This time however, the necessary feature extraction and structuring has already been performed by the convolutional and the intermediate layers, accounting for the wide success and use of this type of networks [25].

16.4 Experimental Application In order to examine the complexity and orthotropic material mechanical behavior of the used CFRP, dynamically induced excitation tests were conducted at a cantilever CFRP tubular beam as presented in Fig. 16.1. Specifically, Fig. 16.1 presents the cantilever CFRP small-radius tube along with two (2) tri-axial accelerometers, a strain gauge sensor and a load cell at the free end of the cantilever beam, where an electromagnetic shaker device is mounted. Both arrangements were introduced in order to acquire knowledge of the mechanical behavior of the CFRP material and thus characterize its orthotropic behavior. The CFRP is consisted of a stack of nine (9) plies with equal thickness and orientation angles apart from one ply. ◦ ◦ Specifically, plies 1 to 6 and 8 to 9 have a thickness of t = 0.175 mm at θ = 55 and θ = − 55 orientation angles ◦ consecutively. Ply 7 has a thickness of t = 0.16 mm at θ = 86 . The nominal material parameters of the 2D orthotropic material used to model the CFRP was E1 = 146, 45GPa and E2 = 7.73GPa for the modulus of elasticity in X and Y direction respectively, vxy = vyx = 0.12 is the Poisson’s ration for in-plane bi-axial loading, and G12 = 3.54GPa, Gxz = 3.95GPa and Gyz = 2.80GPa are the in-plane, transverse for shear in XZ plane and transverse for shear in YZ plane shear moduli and ρ = 1600 kgr/m3 is the density.

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Fig. 16.1 Experimental setup of cantilever CFRP tube under dynamic load excitation

Fig. 16.2 FE model of cantilever CFRP tube along with aluminum drop-outs

Fig. 16.3 Typical eigenmodes predicted by the nominal FE model

The geometry of the cantilever CFRP beam is discretized with composite shell elements and tetrahedral solid elements for the aluminum ends using appropriate pre-processing commercial software [26]. The total number of DOFs was 1,500,000 [27]. The detailed FE model is presented in Fig. 16.2. Indicative mode shapes of the predicted by the nominal FE model are presented in Fig. 16.3 colored by spectrum colors of the normalized deformations. After developing the nominal finite element model, an experimental modal analysis procedure of the CFRP cantilever beam was performed in order to quantify the dynamic characteristics of the examined structure. First, all the necessary elements of the FRF matrix, required for determining the response of the structure were determined by imposing impulsive loading [8–10, 28, 29]. The measured frequency range of the test was 0–600 Hz. An initial investigation indicated that the beam has seven (7) natural frequencies in this frequency range.

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Fig. 16.4 Typical FRFs for modal identification

Figure 16.4 presents typical Frequency Response Functions (FRFs) at three components X, Y and Z for two specific measuring points under a specific impulse location and direction. Moreover, the top diagram of Fig. 16.5 presents a stabilization diagram of a detailed FRF for modal identification, whereas the lower diagram is the detailed view of the FRF. The parameterization of the finite element model is introduced in order to facilitate the applicability of the updating framework. The parameterized model is consisted of five (5) parts, as shown in Fig. 16.6. Part 1 is modeled with composite shell elements and orthotropic material properties while parts 2 to 5 are modeled with solid elements and isotropic material properties. Specifically, part 2 is a steel base, parts 3 and 4 represent the aluminum dropouts of the beam and part 5 is the glue between the CFRP and the aluminum end. All orthotropic material properties along with the nine ply thicknesses t and orientation angles θ were used as design variables of part 1. Additionally, Young’s moduli and the material densities of isotropic material parts were also used as design variables. Apart from material properties parameters, the Rayleigh modal damping ratios are used as design variables. Specifically, modal damping ratios ζ 1 to ζ 7 pertaining to the first seven (7) eigenmodes are included in the design variables, so as to enhance fitting of compared time histories and FRFs, using nominal damping ratio of 3%, as the most common for a composite and steel structures. The total number of design variables for the FE model is thirty-six (36). The CMA-ES framework is applied at ±10% from the nominal values as design bounds, in order to update the developed FE model using the objective function of Eq. (16.1) in combination to Eq. (16.7), combining modal residuals that include the lowest identified modal frequencies with mode shapes and response residuals that include shape and amplitude correlation coefficients considering measured and numerical frequency response functions including components at all sensor locations, along with time domain acceleration time-histories. Finally, the results of the FE model-updating framework are presented in Table 16.1. A comparison between identified, nominal and updated FE predicted modal frequencies are also presented.

16.5 Analysis of a Small-Scale Pin-Jointed CFRP Structure Analysis Finally, the experimental arrangement presented in Fig. 16.7 was set up in order to verify the updated material parameters of the CFRP. Four (4) tri-axial accelerometers were placed on the pin-joined CFRP frame structure, which was anchored on flat plate parallel to the ground, on a vertical concrete column. An electromagnetic shaker was mounted on a free end of the frame where a load cell sensor was placed to record imposed forces under dynamic excitation load.

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Stabilization Diagram

10–4 Stable in frequency Stable in frequency and damping Not stable in frequency Averaged response function

Model Order

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0.1

0.15

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0.35

0.4

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Fig. 16.5 Detailed FRF and stabilization diagram for modal identification

Additionally, a detailed FE model of a small-scale CFRP pin-joined structure was also developed. The geometry of the structure is discretized with composite shell and solid elements as presented in Fig. 16.8. The same figure also presents a detailed view of the FE model at a pin-joint and two indicative mode shapes of the FE model, using the updated orthotropic material parameters, colored by spectrum colors of the normalized deformations. Finally, a comparison between experimental and numerical acceleration time histories at matching locations and excitation loading is presented in Fig. 16.9. Specifically, time-histories of acceleration at the measured components X, Y and Z of the experimental arrangement under harmonic excitation is presented for two measured locations with black continuous line, whereas the numerically predicted equivalent response of the FE model using the updated parameters is presented with red continuous line. The experimentally obtained acceleration time histories, result very close to those numerically computed, concluding in a high fidelity FE model that could be used for damage identification of the composite cylindrical parts of the structure.

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Fig. 16.6 Parts of the parameterized FE model. Detail of CFRP tube and aluminum drop-out Table 16.1 Comparison between identified, nominal and updated FE predicted modal frequencies Mode 1 2 3 4 5 6 7

Identified Frequency (Hz) 18.16 18.18 149.45 167.51 414.32 436.14 531.36

Damping (%) 0.85 0.63 0.82 0.25 0.71 0.91 1.2

Numerical (before updating) Frequency (Hz) Error (%) 15.54 16.86 16.62 11.79 137.23 8.90 144.22 16.15 408.12 1.52 428.12 1.87 472.35 12.49

Numerical (after updating) Frequency (Hz) Error (%) 18.48 1.73 19.12 2.82 147.53 1.30 168.8 0.76 415.34 0.25 435.54 0.14 524.87 1.24

16.6 Damage Identification Labeled (healthy or damaged) FE acceleration histories may afterwards be used for a Machine Learning based SHM methodology. Damage is defined as loose bolts in the pin-joined structure and with the high-fidelity FE model it is possible to simulate damage at different extends. The fed in data is used to train a supervised 1D Convolutional NN (CNN) architecture, which is by nature a powerful and flexible classifier [30] that requires minimal data preprocessing. The NN is synthesized, trained and tuned using the open source python library Keras [31]. Training accuracy is used as the indicative parameter of the network’s capability to capture and identify the correct features in the fed-in acceleration signal. The trained network may afterwards be validated against the real experimental measurements (Fig. 16.10) of each damage type and the accuracy limits of the proposed framework are explored. Dependence of the accuracy on the number and type of FE time histories is found to govern the problem, however the methodology is proven to be reliable for identifying structural damage in the structure.

Fig. 16.7 Experimental setup of small-scale CFRP pin-joined structure

>19.5394 17.5855 15.6315 13.6776 11.7236 9.7697 7.81576 5.86182 3.90788 1.95394 17.8105 16.0295 14.2484 12.4674 10.6863 8.90527 7.12422 5.34316 3.56211 1.78105 d,

where knl = 900 N/mm represents the stiffness of the bumper, and d = 0.5 mm the gap width. The base excitation yG was a version of the El Centro earthquake scaled in time in order to excite the first three bending modes of the structure. Its peak amplitude was varied in the simulations to identify the level that brings the bumper into contact and activates the nonlinear response of the building. For a qualitative assessment of the effect of an imperfect control system on the results, an amplitude error aerr = ±0.1 was deliberately introduced at the interface with the actuator so that the physical displacement would differ from the numerical one (y2p = (1 + aerr ) y2n ).

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Excitation amplitude (mm) Fig. 18.10 Identification of the threshold of nonlinear behaviour and dependence on the accuracy of the control and actuation system

The simulation results are shown in Fig. 18.10. With perfect control, the bumper nonlinearity would be activated at a peak excitation amplitude of 1.05 mm. It can also be seen that, with errors higher than +7% in the actuation amplitude, this hybrid testing set-up would yield a nonlinear response at all excitation levels, thereby completely misrepresenting the dynamics of the real assembly. However, below this threshold, even relatively large errors in the actuator amplitude would yield an acceptable estimation of the critical excitation amplitude. This simple analysis illustrates how hybrid testing can provide experimental data representative of the whole physical twin without having to modify the physical structure and in controlled conditions, thanks to the real-time interaction between the component of interest and the digital twin of the rest of assembly. It can also be used to selectively target the physics of complex components in the context of the larger assembly, as the effects of the harsh nonlinearity introduced by the bumper mechanism can be measured and then propagated to the numerical substructure without modelling approximations.

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At the same time, this analysis highlights the need for active research in order to address the mutual influence between the uncertainties introduced by the hybrid test control and those associated with the digital twin itself.

18.6 Impact on Control A key feature that distinguishes a digital twin from a simple model is the connection to its physical twin through an exchange of data in real time. The physical twin sends the data that the sensors have gathered to the digital twin, which in turn manipulates the data and sends back some scheduling signals. This arrangement can be investigated as a feedback control system, as shown in Fig. 18.11. In general, the feedback loop wants to follow a reference r, which generates an error e that the controller manipulates to create a control signal u. The control signal is then used to drive the actuators that exert a control force x on the structure, which responds with an output y. Sensors are then used to measure the output and observers can be used to estimate the other states of the system, which are then fed back to close the loop. In practice, as it is shown in Fig. 18.11, the actuator, the structure and the sensors are subject to potential faults, their model parameters may be uncertain and there can be disturbances that are not known beforehand, or that are not measured. All these uncertainties, or differences, between the digital and physical twin can have an impact on the performance, but also on the stability of the control loop. Changes in the dynamics of the structure can be addressed by the data-augmented modelling, as discussed in Sect. 18.4, which can then be used to update, or to redesign, the controller. Hence, the controller would need to adapt itself to the new plant dynamics. Although this could lead to a better performance, it remains necessary for the controller to be robust against any other uncertainties and to be tolerant of failures in each of the components of the control loop. The objectives of the controller design, for this case study, are the disturbance rejection, which is given by the primary force Fs (see Fig. 18.1), and the robustness of the controller against uncertainties. The metric used to judge the performance of a controller is the time-averaged kinetic energy of the three storey structure [12], which is defined as, 1 t→+∞ t

*

T = lim

0

t

1 mi y˙i2 (t  )dt  . 2 3

(18.4)

i=1

The control action is given by a secondary force Fa on the third floor of the three storey structure and it is assumed that the accelerations of each of the three storeys are measured. Different control architectures are investigated, one that is model based and one that is non-model based. A direct velocity feedback (VFC) [13] on the third floor of the structure has been chosen as the non-model based control method. In this case, the velocity of the structure is measured at the control location and is fed back to the controller, which amplifies the signal and then applies an equal but opposite force on the structure at the same location, damping out the vibration. This control strategy does not need the knowledge of the plant dynamics, hence it is potentially robust against changes in the dynamic behaviour of the structure, however, it may give a non optimal performance since it reduces the velocity at a single location instead of the entire kinetic energy of the structure.

faults Δ a r +

e

controller

u

da

actuator

faults Δ x

p

dp

plant

y

-

sensor (observer)

faults

ds

Fig. 18.11 General block diagram of a feedback control system, in which the actuators, the structure and the sensors are affected by potential faults, parameter uncertainties and un-modelled disturbances d

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20 10 0

T [dB]

-10 -20 -30 -40 KE passive KE VFC 10.77dB reduction KE LQR 12.77dB reduction

-50 -60 10-1

100

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f [Hz] Fig. 18.12 Time-averaged kinetic energy of the 3-storey building with and without a control action on the third floor. The control gains of the Velocity Feedback Controller and the Linear Quadratic Regulator are designed in terms of equal control effort. The kinetic energy reduction is calculated as the integral of the frequency dependent kinetic energy

A linear quadratic regulator (LQR) [14] has been chosen as a full state control method that is model based. The LQR solves the linear quadratic Gaussian problem that minimises a cost function that is a trade off between the performance of the controller and the control effort required to achieve that performance. The control action, in this case, is still applied only on the third floor of the structure, but the matrix of feedback gains is calculated to achieve a global performance of vibration reduction, hence considering all the states of the system. This method can give a better performance than the previous one, however, if the dynamics of the structure changes during the operation, the control performance could degrade, or worse, the controller could become unstable. Figure 18.12 shows a comparison among these control strategies for the data set one (bumper not in contact) in terms of kinetic energy for the same control effort requirements. The LQR strategy performs slightly better than the VFC in this scenario, giving a better reduction of the kinetic energy. If the plant dynamics changes, however, the LQR feedback gains are not designed to adapt to these changes and could lead to a worse than expected performance. The design parameters of the LQR controller, are computed for the nominal system and usually stay the same throughout the asset operation even though the plant dynamics could change. Instead of using a LQR controller, one could use another model based control method, such as model predictive control (MPC). MPC would still be liable to the same problems faced by the LQR, however, the internal model of MPC can be made adaptive and it could be updated by the data-augmented model. Adaptive MPC can be implemented to achieve an optimal performance, however, its robustness against uncertainties and potential faults needs to be compared against the robustness and performance of the non-model based control methods, such as VFC.

18.7 Decisions in a Digital Twin Decision-making is a key process required for digital twins. In order to remain a useful tool for the end-user, digital twins must continuously adapt so that they are representative of the physical twins. In the three-storey building structure case study, data-augmented modelling allows the digital twin to identify when a decision may be required as an inflation in the prediction variance is observed. Upon seeing this trigger caused by the previously unseen nonlinear condition, the digital twin should decide whether to perform one of the following actions:

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Update the model parameters. Learn an improved data-augmented model. Learn missing physics online from observed data. Advise to learn missing physics offline with hybrid testing.

Decision theory provides a formal framework for comparing courses of action [15]. In order to make a decision, one must evaluate the expected utilities associated with each action. In the context of the three storey building structure, each possible action has a different computational and financial cost whilst also offering varying improvements in predictive performance and thus value of information. In general, the value of information provided by the improvement in predictive performance is dependent on the utility gain achieved in the high-level decision-making for which the digital twin is implemented. Retaining the uncertainties associated with the models is important as it allows robust decisions to be made; if the uncertainty is high conservative actions will be preferred, however, with lower uncertainties cost-optimal decisions can be made. Furthermore, the cost of poor predictive performance is significant if a model based control design is utilised. In this context, the digital twin is required to predictive accurately to avoid instabilities in the controller and sub-optimal control. As a result, the uncertainty associated with any prediction must be used in deciding how the model interacts with the controller, potentially leading to more robust control.

18.8 Conclusions Developing a digital twin poses several challenges which require optimal decision making. This paper has explored the required decision making by a digital twin by investigating a three storey building structure. This case study has motivated different approaches to overcoming poor predictive performance, such as model updating, data-augmented a physics-based model and improvement of the model offline through hybrid testing. Due to the coupled nature of a digital twin and controller, any model based controller design with be significantly impacted by any poor predictive performance, unless mitigated for in the design stage. Realising a digital twin will require each of these stages to be formally combined using decision theory, based on cost optimal decisions making that will improve predictive performance and controllability. This is a valuable area of further research, and will help develop a workflow required for developing more general digital twins. Acknowledgments The authors would like to acknowledge the support of the UK Engineering and Physical Sciences Research Council via grants EP/R006768/1.

References 1. Grieves, M., Vickers, J.: Digital twin: mitigating unpredictable, undesirable emergent behavior in complex systems. In: Transdisciplinary Perspectives on Complex Systems: New Findings and Approaches (2016) 2. Sharma, P., Hamedifar, H., Brown, A., Green, R.: The dawn of the new age of the industrial internet and how it can radically transform the offshore oil and gas industry. In: Proceedings of the Annual Offshore Technology Conference (2017) 3. Smith, R.C.: Uncertainty Quantification: Theory, Implementation, and Applications (2014) 4. Mottershead, J.E., Friswell, M.I.: Model updating in structural dynamics: a survey. J. Sound Vib. 167(2), 347–375 (1993) 5. Au, S.K., Ni, Y.C.: Fast Bayesian modal identification of structures using known single input forced vibration data. Struct. Control Health Monit. 21(3), 318–402 (2014) 6. Kennedy, M.C., O’Hagan, A.: Bayesian calibration of computer models. J. R. Stat. Soc.: Ser. B (Stat. Methodol.) 63(3), 425–464 (2001) [Online]. Available: https://doi.org/10.1111/1467-9868.00294 7. Berger, J.O., Cavendish, J., Paulo, R., Lin, C.-H., Cafeo, J.A., Sacks, J., Bayarri, M.J., Tu, J.: A framework for validation of computer models. Technometrics 49(2), 138–154 (2007) 8. Higdon, D., Gattiker, J., Williams, B., Rightley, M.: Computer model calibration using high-dimensional output. J. Am. Stat. Assoc. 103(482), 570–583 (2008) 9. Rasmussen, C.E., Williams, C.K.I.: Gaussian processes for machine learning. MIT Press, Cambridge, Massachusetts (2006) 10. Wagg, D., Neild, S., Gawthrop, P.: Real-time testing with dynamic substructuring. In: Bursi, O., Wagg, D. (eds.) Modern Testing Techniques for Structural Systems. CISM International Centre for Mechanical Sciences, vol. 502. Springer, Wien (2008) 11. Horiuchi, T., Inoue, M., Konno, T., Namita, Y.: Real-time hybrid experimental system with actuator delay compensation and its application to a piping system with energy absorber. Earthq. Eng. Struct. Dyn. 28(10), 1121–1141 (1999) 12. Fahy, F., Gardonio, P.: Sound and structural vibration: radiation, transmission and response, 2nd edn. Elsevier Science, Oxford, UK (2007)

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13. Baumann, O.N., Elliott, S.J.: The stability of decentralized multichannel velocity feedback controllers using inertial actuators. J. Acoust. Soc. Am. 121(1), 188–196 (2007) 14. Zhang, J., He, L., Wang, E., Gao, R.: A lqr controller design for active vibration control of flexible structures. In: 2008 IEEE Pacific-Asia Workshop on Computational Intelligence and Industrial Application, vol. 1, pp. 127–132 (2008) 15. Parmigiani, G., Lurdes, I.: Decision Theory: Principles and Approaches, 1st edn. Wiley, Hoboken, NJ (2009)

Chapter 19

An Improved Optimal Sensor Placement Strategy for Kalman-Based Multiple-Input Estimation Lorenzo Mazzanti, Roberta Cumbo, Wim Desmet, Frank Naets, and Tommaso Tamarozzi

Abstract The knowledge of the dynamic behavior of a mechanical system in a certain operating scenario is essential in many industrial applications. In particular, nowadays, the accurate and concurrent identification of the response fields and external loads represents a challenging target. Several experimental techniques, some exploiting a coupling with simulated solutions based on predictive methodologies, have recently been proposed and are in current use. However, in practice there is a common issue in the selection of the optimal types of sensors and their measurement location selection in order to reconstruct the desired quantities (e.g., loads, displacement or acceleration field) for a desired accuracy and dynamic range. This paper focuses on a Kalman filter approach for multiple input/state estimation, combining operational measurement and numerical model data. In the presented framework, an existing Optimal Sensor Placement (OSP) strategy for load identification is discussed and an improvement of this sensor selection is proposed. The reference OSP approach, previously proposed by the authors, is mainly focused on system observability, which is only a minimum requirement to obtain a stable estimator. For this reason it does not necessarily lead to the most accurate estimator or the highest dynamic range. In this work, we propose two alternative metrics based respectively on estimator covariance convergence and closed-loop estimator bandwidth with respect to the available set of measurements. The existing OSP is compared with the proposed metrics for multiple input/state estimation, showing improved accuracy of estimated quantities when these new metrics are accounted for in the sensor selection. Keywords Kalman filter · Input/state estimation · Optimal sensor placement · Observability · Augmented filter

19.1 Introduction The identification of the forces acting on a mechanical component is a relevant research topic in many engineering applications. In this framework, the usage of Kalman filtering [1, 2] techniques in the augmented formulation [3, 4] have been explored combining numerical and experimental data. The Augmented Kalman Filter (AKF) aims to identify the behavior of a mechanical system in terms of external forces and full-field responses (e.g. strain, acceleration, displacement), starting from the knowledge of the response in some measured points (output) along the structure. Analytical and numerical studies [5, 6] proved that the nature of the output influences the accuracy of the estimation: using acceleration only measurements leads indeed to an unstable filter [4, 5]. In order to obtain a stable estimation, a number of position-level output equal or greater

L. Mazzanti Siemens Digital Industries Software, Leuven, Belgium Engineering Department, University of Ferrara, Ferrara, Italy e-mail: [email protected] R. Cumbo () · T. Tamarozzi Siemens Digital Industries Software, Leuven, Belgium Katholieke Universiteit Leuven, Heverlee, Belgium e-mail: [email protected]; [email protected] W. Desmet · F. Naets Katholieke Universiteit Leuven, Heverlee, Belgium DMMS Core Lab, Flanders Make, Belgium e-mail: [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2020 Z. Mao (ed.), Model Validation and Uncertainty Quantification, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47638-0_19

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to the number of forces to be identified is required [4]. Tamarozzi et al. [7] propose an Optimal Sensor Placement (OSP) strategy to identify the optimal type (position/acceleration/strain), number and location of the sensors needed to identify a certain set of external forces. This strategy evaluates the performance of a subset of sensors on the estimation by the means of an observability metric PBH. The latter is evaluated at 0 Hz, which is the frequency associated to the adopted 0th-order input model [5]. Starting from the augmented state-space formulation of the dynamic equations of a mechanical system: x˙∗ = A∗ x ∗ , y = H x ∗   1 sI − A PBH(s) = , OPBH = #ni H cond(PBH(0)) i i=1

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Where A∗ and x ∗ are the augmented state matrix and vector respectively and ni is the number of loads to be estimated. The metric OPBH selects sequentially the sensors for which the removal implies a significant decrease in the condition number of the observability matrix [7, 8]. The aim of this work is to improve and advance current procedure, defining an Advanced Optimal Sensor Placement (AOSP) criterion based upon different and more suitable metrics.

19.2 Virtual Measurement Uncertainty and Bandwidth Prediction for Optimal Sensor Placement The OSP is focused on system observability, which is the minimum requirement to obtain a stable estimation through Kalman filtering techniques. However, this kind of metric can lead to conclusions that are not consistent from a physical point of view, as shown in Fig. 19.1 for a simple mass-spring-damper system with 30 degrees of freedom. The OPBH decreasing trend is shown for a single input estimation, using sensors expressed in meters [m] and in millimeters [mm]. The two curves clearly depend on the units used to express the measurements. If the user selects a certain percentage threshold value, a different number of optimal sensors will be chosen. Two new metrics are proposed in this paper in order to define a more reliable OSP. The step of observability check [7, 8] is replaced by: • the steady-state value of the estimation error covariance matrix P which can be evaluated before the estimation and computed from the continuous algebraic Riccati equation associated to the observation problem of the AKF: A∗ P + P A∗T + Q∗ − (P H T R −1 )R(P H T R −1 )T = 0

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where A∗ is the augmented system transition matrix, H the augmented measurement matrix, Q∗ the model covariance matrix, R the sensor covariance matrix. The AOSP criterion defined by using this matrix is called Input Error Covariance criterion (IEC) and the selection metric is linked to the minimization of the input estimation error by considering the diagonal terms of P related to the forces to be identified. • the estimator transfer function Tuu ˆ , evaluated as the product of the input-output (physical system) transfer function Tyu and output-estimated input (result of the AKF) transfer function Tuy ˆ . A block diagram representation of both functions is shown in Fig. 19.2, where G = P H T R −1 is the Kalman gain matrix and Huˆ is a boolean selection matrix. The full equation of the transfer function is:  ∗ −1 −1 (s) = H ((I s − A + GH ) G) C(I s − A) B + D Tuu ˆ uˆ

(19.4)

For a single input identification, a good estimation shows a transfer function value close to 1 over the full frequency range of interest. The AOSP criterion defined by using Tuu ˆ is called Transfer Function Identity index criterion (TFI). The selection metric for multiple-input estimation is linked to how much the transfer function is similar to an identity matrix across the frequency bandwidth defined by the user. The main advantages of the proposed metrics are that they both do not show any units-dependency and they consider the matrix Q∗ and R, which are related to the model and sensors uncertainties respectively.

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19.3 Results A numerical experiment on a Finite Element Model (FEM) of an automotive rear twistbeam suspension, described in [7, 8], was performed in order to evaluate the performance of the two proposed selection criteria IEC and TFI. The objective is the estimation of 6 sinusoidal input (3 forces and 3 torques) applied in the center of the left wheel along x, y, z directions. The minimum number of sensors, i.e. 6 position-level (strain), are used to perform the 6-input estimation. The force Fz and the torque Mz are shown in Fig. 19.3, comparing all the three criteria. A significant improvement in the quality of the estimation (especially for torque) has been achieved through IEC and TFI sensor selection. Further studies will be conducted in order to evaluate the effect of the combination of strain and acceleration data on the input estimation.

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References 1. Kalman, R.: A new approach to linear filtering and prediction problems. J. Basic Eng. 82(1), 35–45 (1960) 2. Simon, D.: Optimal State Estimation: Kalman, H inf and Nonlinear Approaches. Wiley, New York (2006) 3. Lourens, E., Reynders, E., De Roeck, G., Degrande, G., Lombaert, G.: An augmented Kalman filter for force identification in structural dynamics. Mech. Syst. Signal Process. 27, 446–460 (2012) 4. Maes, K.: Filtering techniques for force identification and response estimation in structural dynamics. Ph.D. thesis (2016) 5. Naets, F., Cuadrado, J., Desmet, W.: Stable force identification in structural dynamics using Kalman filtering and dummy-measurements. Mech. Syst. Signal Process. 50–51, 235–248 (2015) 6. Azam, S.E., Chatzi, E., Papadimitriou, C.: A dual Kalman filter approach for state estimation via output-only acceleration measurements. Mech. Syst. Signal Process. 60, 866–886 (2015) 7. Tamarozzi, T., Risaliti, E., Rottiers, W., Janssens, K., Desmet, W.: Noise, ill-conditioning and sensor placement analysis for force estimation through virtual sensing. In: Leuven (ed.) International Conference on Noise and Vibration Engineering (ISMA), Department Werktuigkunde Katholieke University (2016) 8. Cumbo, R., Tamarozzi, T., Janssens, K., Desmet, W.: Kalman-based load identification and full-filed estimation analysis on industrial test case. Mech. Syst. Signal Process. 117, 771–785 (2019)

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Roberta Cumbo studied Aeronautical Engineering at La Sapienza University of Rome (Italy) and obtained her Master degree in 2017. She is currently employed at Siemens PLM Software in Leuven (Belgium) as fellowship holder of a Baekeland PhD Mandate (VLAIO, Belgium). Her research activity converned with Kalman-based Virtual Sensing strategy for input-state estimation for helicopter main rotor systems.

Chapter 20

Towards Population-Based Structural Health Monitoring, Part IV: Heterogeneous Populations, Transfer and Mapping Paul Gardner, Lawerence A. Bull, Julian Gosliga, Nikolaos Dervilis, and Keith Worden

Abstract Population-based structural health monitoring (PBSHM) involves utilising knowledge from one set of structures in a population and applying it to a different set, such that predictions about the health states of each member in the population can be performed and improved. Central ideas behind PBSHM are those of knowledge transfer and mapping. In the context of PBSHM, knowledge transfer involves using information from a structure, defined as a source domain, where labels are known for a given feature, and mapping these onto the unlabelled feature space of a different, target domain structure. If the mapping is successful, a machine learning classifier trained on the transformed source domain data will generalise to the unlabelled target domain data; i.e. a classifier built on one structure will generalise to another, making Structural Heath Monitoring (SHM) cost-effective and applicable to a wide range of challenging industrial scenarios. This process of mapping features and labels across source and target domains is defined as domain adaptation, a subcategory of transfer learning. However, a key assumption in conventional domain adaptation methods is that there is consistency between the feature and label spaces. This means that the features measured from one structure must be the same dimension as the other (i.e. the same number of spectral lines of a transmissibility), and that labels associated with damage locations, classification and assessment, exist on both structures. These consistency constraints can be restrictive, limiting to which types of population domain adaptation can be applied. This paper, therefore, provides a mathematical underpinning for when domain adaptation is possible in a structural dynamics context, with reference to topology of a graphical representation of structures. By defining when conventional domain adaptation is applicable in a structural dynamics setting, approaches are discussed that could overcome these consistency restrictions. This approach provides a general means for performing transfer learning within a PBSHM context for structural dynamics-based features. Keywords Population-based structural health monitoring · Transfer learning · Domain adaptation

20.1 Introduction The process of utilising information across a population of structures in order to perform and improve inferences that generalise for the complete population is known as Population-based Structural Health Monitoring (PBSHM) [1]. This approach to Structural Health Monitoring (SHM) clearly provides significant benefits, as any knowledge, whether about the behaviour of features, or any damage-labelled data, that may have been obtained from one member of the population, aids predictions across the whole population. For example, in the case of an aeroplane fleet, an assortment of damagelabelled data may be available for different members of the fleet, all under differing operational conditions. The concept of PBSHM is to incorporate the feature and label data from each aeroplane to generate a machine learning-based approach that generalises across the complete fleet for all damage scenarios, especially when many members of the fleet have no labelled data associated with them. This work is Part IV of a series of papers on PBSHM [2–6], where for an overview of PBSHM and further motivation the reader is referred to earlier parts [2–4]. Central concepts for performing PBSHM are those of knowledge transfer and mapping. These two processes are crucial for realising PBSHM for various reasons. Firstly, conventional approaches to data-driven SHM, whether supervised, unsupervised or semi-supervised machine learning methods, assume that the training and test data are drawn from the same distribution. This assumption breaks down in PBSHM, as each member of the population will have differences in their

P. Gardner () · L. A. Bull · J. Gosliga · N. Dervilis · K. Worden Dynamics Research Group, Department of Mechanical Engineering, The University of Sheffield, Sheffield, UK e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2020 Z. Mao (ed.), Model Validation and Uncertainty Quantification, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47638-0_20

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data distributions, whether due to environmental variations, manufacturing and assembly differences, operational conditions etc. As a result, conventional machine learning approaches fail to generalise; for example, a classifier trained on data from one aeroplane in the fleet will generally fail to classify for a different aeroplane from the same population. Mapping data from one member of the population to another is therefore required such that a general classifier can be generated. Another issue is that damage-labelled data, for all possible damage states of interest, are often not obtainable for each individual in the population. In fact, it is generally infeasible to obtain a complete label set for all possible damage states, even for one structure. However, it may be possible to obtain a relatively complete damage-label set by combining labels from all structures in a population, or from a historic database from other SHM campaigns. Leveraging this label knowledge from across a population and mapping it onto a consistent space means that knowledge transfer is possible, aiding the generation of a general machine learning method for the complete population. By utilising these two processes, SHM methods can be generated that are cost-effective and applicable across a wider variety of challenging industrial applications. Transfer learning is one approach that aims to improve the performance of a learner by transferring knowledge between different domains. Within this branch of machine learning, various methods exist with differing assumptions about consistency between domains and what knowledge is being transferred. This paper outlines the mapping and knowledge transfer problems within PBSHM, with a particular focus on modal-based features. Specifically, the work presented here focuses on recent successes in the application of domain adaption, a sub-category of transfer learning, in PBSHM. Domain adaption assumes labelled data are available in a source domain and that this can be used to aid classification in an unlabelled (or partially-labelled) target domain, by mapping the two domains onto a common subspace. However, conventional approaches to domain adaptation assume consistency between the feature and label space for each domain. This means that the features from one structure must be the same dimension as the other structure (i.e. the same number of natural frequencies), and that any damage labels for detection, location, classification or assessment can exist for both structures. These constraints can be limiting in the context of PBSHM. With reference to the topology of a graphical representation of a structure (discussed in Parts II and III of the paper series [3, 4]), this paper presents a mathematical underpinning for when domain adaption is possible within the context of PBSHM. The outline of this paper is as follows. Section 20.2 outlines the PBSHM contexts where transfer learning is applicable, providing key definitions and an overview of the mapping and knowledge transfer problems. Following the mapping problem descriptions, Sect. 20.3 demonstrates the contexts in which domain adaptation is applicable, referencing graphical representations of structures. Finally, conclusions are presented highlighting where technological advances are required in order to produce general learning tools for PBSHM.

20.2 Population-Based Structural Health Monitoring and Transfer Learning Population-Based Structural Health Monitoring (PBSHM) involves mapping data and labels from different structures within a population, such that a general learner can be inferred across the complete population. As a consequence, health monitoring can be performed online for any member of the population. This section seeks to define the mapping and knowledge transfer scenarios within PBSHM and demonstrate the applicable forms of learning for each scenario, with an emphasis on transfer learning.

20.2.1 Population Types Key definitions are required in order to outline the mapping problems within PBSHM. A population, in the context of PBSHM, is a group of structures (the smallest being a group of two structures) that provide information required for performing health monitoring. This general definition of a population can be further divided into two categories: homogenous and heterogeneous populations [3]; these groupings relate to the level of dissimilarity within a population, where both population types benefit from knowledge transfer. Colloquially, a homogeneous population is one in which each structure in the population can be deemed nominally identical for a given context [2, 3]. A heterogeneous population is thus a group of non-identical, and therefore different structures. Conceptually, differences will occur for a multitude of reasons, and structures will be deemed different due to various properties, based on specific contexts. This leads to a spectrum of different types of heterogeneous populations, which as similarities increase, will approach a homogenous population. One method of quantifying these differences, utilising Irreducible Element (IE) models and Attributed Graphs (AGs), has been discussed in

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Fig. 20.1 Categories of heterogeneous population within population-based SHM

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Part II of this series of papers [3]. This approach highlights three main sources of difference within a structure, relating to the topology of the graph and types of attribute stored within the nodes and edges: Geometry: relates to the shape, size and scaling of a structure in a population, e.g. a group of aluminium rectangular plates where each plate has a different length, width and thickness. Material: relates to different material classes, specific materials, and their properties for structures in a population, e.g. a pair of the same size rectangular plates where one is made from aluminium and another from steel. Topology: relates to different topologies for graphical representations of structures in a populations, e.g. a pair of aluminium beams where one is a cantilever and the other is éncastre, where there is a difference between the two lumpedmass representations due to the difference in boundary conditions. These classes of differences are summarised in Fig. 20.1, where each of these categories may overlap forming a different category of heterogeneous population. For example, consider a heterogeneous population of two beam-like structures, these may have geometric differences, e.g. structure one is a beam with a tapered rectangular geometry and structure two is a beam of the same length with a uniform I-beam cross-section. In addition to these geometric differences, aspects of the materials may be different, e.g. Structure One is constructed from a unidirectional carbon fibre layup and Structure Two is formed from the same carbon fibre but in a plain weave layup. This population is therefore a heterogeneous population due to geometric and material differences, which are properties stored in the attributed graph, as defined in Part II [3], meaning the population would be located in the Venn diagram between material and geometry. More formal definitions may also be attached to these types of population, as discussed in paper II [3]. A homogeneous population can be defined as a group of structures that are topologically homogeneous and where the geometric θ g and material θ m properties (collectively θ = {θ g , θ m }) in the nodes and edges of the attributed graph are defined as being random draws from a concentrated base distribution i.e. p(θ ). It is noted that topologically homogeneous, as defined in paper II [3], relates to a group of structures that can be considered pairwise topologically equivalent, with respect their graphical representation, e.g. two structures that could be modelled as five degree-of-freedom lumped mass models where the first mass is fixed to ground. The probability mass in the base distribution p(θ ) therefore defines the small differences between members in the population. A strongly-homogeneous population would have a unimodal concentrated base distribution for the geometric and material properties, where the strictest, perfect, form of homogeneous population is one in which the base distribution is a Dirac delta function, i.e. each member of the population is exactly the same, a scenario that never occurs in reality, but is the assumption in applying conventional machine learning methods trained on one structure, to another. Using these population definitions and categories of difference helps determine the difficulty of the mapping problem for PBSHM. It is also noted that differences observed in data may also occur outside the structural properties of individuals in a population, not captured in the attributes and topology discussed above. These differences will relate to how data acquisition and any further processing to obtain features were performed. For example, differences in sensor placement will lead to differences in data distributions, even if those sensors are placed in the ‘same’ location. Properties of the data acquisition may also affect the obtained feature for each member of the population, e.g. the transmissibilities from two members of a homogeneous population may be different due to differences in the raw data sample rate. These sources of difference related to data acquisition and processing will need to be considered in defining the mapping problems in PBSHM; however, this paper simplifies the scenarios such that only structural differences in a population are considered.

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20.2.2 Transfer Learning Transfer learning offers several techniques for dealing with scenarios where domains, tasks and distributions used within training and testing a learner are different [7]. Distinct from multi-task learning, where the objective is to learn multiple tasks across different domains [8], transfer learning leverages knowledge from source tasks in improving specific target tasks, i.e. the focus is on target tasks rather than all tasks (both source and target) equally [7]. This type of learning is applicable to PBSHM, even in the homogeneous population scenario, as variations in each structure within the population will lead to differences in the data distribution, meaning that a learner trained on one structure will not apply to another structure in the population. Before formally defining transfer learning, several other objects are required. Definition 1 A domain D = {X, p(X)} is an object that consists of a feature space X and a marginal probability distribution p(x) over the feature data x = {x i }N i=1 ∈ X, a finite sample from X. Definition 2 A task T = {Y, f (·)} is an object that consists of a label space Y and a predictive function f (·) (p(y | x) in a probabilistic setting) which can be inferred from training data {x i , yi }N i=1 where x i ∈ X and yi ∈ Y. s In the case of a single source and task domain, there will be a source domain data set Ds = {x s,i , ys,i }N i=1 where x s,i ∈ Xs t and ys,i ∈ Ys , and a target domain data set Dt = {x t,i , yt,i }N i=1 where x t,i ∈ Xt and yt,i ∈ Yt , where it is generally assumed that 0 ≤ Nt  Ns .

Definition 3 Transfer learning states that given a source domain Ds and task Ts , and a target domain Dt and task Tt , it is the process of improving the target predictive function ft (·) in Tt by using knowledge from Ds and Ts , whilst assuming Ds = Dt and/or Ts = Tt . Transfer learning methods then differ in their assumptions about whether X, p(x), Y or p(y | x) are consistent across the source and target. Another distinction is also made within transfer learning about the dimensions of the feature spaces. Definition 4 Homogeneous transfer learning assumes that the feature and label spaces represent the same attributes, Xs = Xt and Ys = Yt , and therefore that the dimensions of the feature space are equal, ds = dt . Definition 5 Heterogeneous transfer learning assumes that the features spaces are non-equivalent, Xs = Xt , and often that the source and target domains share no common features, meaning ds = dt . In addition, heterogeneous transfer learning can also assume that Ys = Yt . Clearly heterogeneous transfer learning is the more challenging, but more general, category of approach, with the mapping having to account for dimensionality transformations, a more complex mapping of the joint distributions, as well as the potential to handle inconsistent label spaces. In order to clarify terminology, it is noted that the categories of homogeneous and heterogeneous transfer learning do not directly relate to homogeneous and heterogeneous populations, i.e. homogeneous transfer learning is not the only way of performing knowledge transfer for a homogeneous population and heterogeneous transfer learning is not the only way of performing knowledge transfer for a heterogeneous population. This is clarified further in Sect. 20.2.3, in which different transfer learning approaches are linked to a variety of population types, and in Sect. 20.3 with two case studies in which homogeneous transfer learning is applied to both a homogeneous and heterogeneous population.

20.2.3 Mapping Scenarios Before defining mapping scenarios within a population-based view of SHM, it is first important to define the categories of problem established in conventional SHM. There are several types of challenge within SHM, categorised by Rytter into a hierarchy [9], where generally difficulty increases with each level in the hierarchy. • • • • •

Detection: identify the presence of damage. Location: identify the position of damage. Classification: identify the type of damage. Assessment: identify the extent of damage. Prognosis: identify the safety and residual life of the structure.

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Table 20.1 Feature and label consistencies for types of population when considering modal features and location resolution at the graphical representation level Homogeneous Feature consistency Detection label consistency Location label consistency Classification label consistency Assessment label consistency

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This hierarchy provides a framework for defining the mapping problems within a population-based SHM context, as each level describes the required label agreement. By grounding the mapping scenarios within PBSHM to the hierarchy of SHM problems, it shows that the degree of similarity required is problem dependent. Table 20.1 states the level of feature and label1 consistency for the categories of population outlined in Sect. 20.2.1 from a physical viewpoint. The table considers modal features and that the location label resolution is at the level of the graphical representation of the structure [3]. Each column in the heterogeneous category assumes that all other columns are equivalent, and that more complex heterogeneous populations are formed by combining columns, as indicated by Fig. 20.1. Two types of heterogeneous population in Table 20.1, material and topology, are stated as having scenarios where label consistency may or may not occur. Both of these categories have a hierarchy of additional attributes (discussed in Part II [3]) that will define whether these labels are consistent. For example, if populations are heterogeneous in terms of their material class the classification and assessment labels will be inconsistent, e.g. a metallic structure will not experience delamination, whereas there are delamination labels associated with a composite. Furthermore, even when the material class and specific materials are the same, the grade or batch properties will cause variation that could cause assessment label inconsistency. In a similar manner, classification and assessment label consistency can change within the topology category. If the topologies between a population are different, but all the joints within the graph are equivalent, then the classification and assessment labels will be consistent. In contrast, if the joints change between structures, e.g. structure one is assembled via bolted joints and the other structure welded, then the classification and assessment label spaces will be inconsistent. In scenarios where the label spaces are not consistent, some form of label space matching is required. This is particularly important in an unlabelled (or partially-labelled) target space where information on cluster relationships between the source and target domain are not defined in the training data and the label space mismatch means the problem is ill-defined. For example, in a SHM location problem, where some locations do not actually exist on both structures. Even when the damage label space between members of the population is inconsistent globally, a subset of these labels may be consistent, as shown in Fig. 20.2. This demonstrates the power of graph and attribute matching between members of a population (see Part III [4]), as by locating the common attributes/subsystems Sc between members of a population helps identify the level of information that can be exchanged through a mapping φ (where this mapping may be through a latent space [7], as in Joint Domain Adaptation (JDA) [11]). Within an IE and AG representation of structures, these common subsystems Sc become common subgraphs in the AGs, which can be determined by graph matching, as demonstrated in Part III [4]. There will be SHM scenarios in which only part of a population needs to be similar for mapping to be possible. For example, if the SHM problem is performing damage location on a five and three degree-of-freedom structure, then locally features and location labels can be considered consistent. This would occur if, for example, the first three natural frequencies are considered, and that damage labels are only used for the first three degrees-of-freedom for each structure. It is also noted that global feature consistency can be obtained, even for topologically heterogeneous populations, by selecting an appropriate feature. For example, if in the previous example the features were transmissibilities, with the same number of frequencies bins over the same frequency range, then the feature spaces are consistent. However, this leads to questions over whether the key damage sensitive information is contained within the feature for both structures, and may lead to a phenomenon known as negative transfer, discussed in Sect. 20.2.4. From Table 20.1 it is clear that the start of any population-based SHM campaign is considering what SHM scenario is required and then obtaining the level of consistency and similarities between members of the population. Once this has been obtained, there are then four mapping problems that occur: consistency in feature and label spaces, consistency in the feature space and inconsistency in the label space, inconsistency in the feature space and consistency in the label space, and inconsistency in feature and label spaces. Table 20.2 defines the transfer learning approach appropriate for each assumption.

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Label-space matching refers to the process of identifying and pairing equivalent label classes in the transfer learning mapping, where some classes will be left unmatched between the source and target domains. This process must be performed correctly to prevent negative transfer of class labels, and is challenging, as often the task domain is unlabelled (or minimally labelled with undamaged labels only) requiring some form of semi-supervised learning [12]. It is important to state that this risk of negative transfer increases when label inconsistency is assumed, for both homogenous and heterogeneous transfer learning, and so as far as possible, a PBSHM scenario should be posed as one that is consistent in the label space. At this point it is useful to compare transfer learning to other forms of learning that are also appropriate for a populationbased approach to SHM. As mentioned in the first paper of this series [2], a form is one approach to PBSHM for a homogeneous population, and will assume consistent feature and label spaces. This technique seeks to create a general representation of a population within the data-domain. A form can be inferred from a group of structures and used to transfer this knowledge to remaining members of a population. When used in this manner, it can be seen as a type of transfer learning. However, a form can also be inferred across a complete population and in this scenario aims to infer an improved general learner for the complete population, in which it is a type of multi-task learning. The other difference with a form is that the representation is usually in the data-domain, and therefore the variance of general learner will often be inflated, unlike transfer learning or multi-task learning that often project to a latent space, with the aim of removing uncertainty not related to the latent process. As stated previously, multi-task learning differs from transfer learning in that the aim is to infer a general learning over all domains rather than transfer from one group to another. Multi-task learning can be both consistent and inconsistent in the feature space but will be consistent in the label space. In a PBSHM context, multi-task learning may be useful in removing the effects of confounding influences [13] and improving a learner for a structure with minimal labels within a population.

20.2.4 Negative Transfer A major concern when performing transfer learning, is if information is incorrectly mapped from one domain to another and reduces the performance of a learner when compared to learning from the target domain alone [7]. This phenomenon is known as negative transfer, and often occurs when source domains are very dissimilar to the target domain [14]. The fundamental idea behind transfer learning is that the source domains contain useful and relevant information about a target domain. This can be hard to determine before transfer, and becomes especially problematic when label inconsistency is assumed, the target domain is unlabelled, or that an example for each label class is not available. A binary classification scenario where negative transfer occurs is when source domain classes are mapped on to the opposite classes on the target domain, due to the target domain being unlabelled, and the source classes being dissimilar to the target domain. Negative transfer highlights the question of when to transfer, and motivates the reasoning behind developing a measure of similarity between structures [4], as this provides an informative method for determining if a dataset will harm the learning process. Within the machine learning literature, several approaches have been proposed that seek to address negative transfer

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[15–17]. Graph-based methods that define the relationships between source domains by stating them graphically using a transferability metric have been developed [15]. This provides similar motivation for the Irreducible Element and Attributed Graph approach [3, 4]; however, in an SHM context these graphical representations can be formed from a physics-based viewpoint, aiding the strength of knowledge about similarity. Other approaches in avoiding negative transfer have sought to weight each source domain based on its relevance to the target domain; this is known as instance weighting [18]. Local cluster-based weighting has also been proposed, meaning that for each class on a source domain, an individual weighting is provided, stating that informativeness may not be globally shared in a particular source domain [16]. As a result, it is important to consider and account for the possibility of negative transfer when identifying what structures, and their corresponding datasets, to use in transfer learning, as well as developing algorithms to reduce or remove the possibility of negative transfer within PBSHM.

20.3 Domain Adaptation Domain adaptation is one form of transfer learning that seeks to transfer feature spaces between source and target domains, assuming that their marginal distributions over the finite sample set are not equal p(Xs ) = p(Xt ). These techniques are primarily designed for homogeneous transfer learning, where the feature space and label space are consistent [7, 11, 19, 20]; however, heterogeneous transfer learning forms of domain adaptation do exist [21–23]. This section outlines domain adaptation and its assumptions, before demonstrating its applicability to homogenous populations, and a heterogeneous population; that contain both geometric and material differences. These case studies motivate what aspects of PBSHM are currently achievable, highlighting the required areas of further research in making PBSHM applicable across the complete range of problems outlined in Sect. 20.2.3. Domain adaptation is formally defined (for a single source and target domain) as: Definition 6 Domain adaptation applies when a given inference is required for a target domain Dt and task Tt , and is the process of improving the target predictive function ft (·) in Tt given a source domain Ds and task Ts , whilst assuming Xs = Xt and Ys = Yt but that p(Xs ) = p(Xt ), and one can further assume p(Ys | Xs ) = p(Yt | Xt ). Various algorithms have been developed for this scenario [11, 19, 20], and several have been applied in a PBSHM context [24]. One approach is implemented in this section, Joint Domain Adaption (JDA) [11], in order to visually motivate the PBSHM problems in which a domain adaptation approach is applicable, although it is noted that other techniques may offer different levels of classification performance. Briefly, JDA is a technique that assumes the joint distributions between the source and target are different p(Ys , Xs ) = p(Yt , Xt ), and finds the optimal latent mapping in which p(φ(Xs )) ≈ p(φ(Xt )) and p(φ(Xs ) | Ys = c) ≈ p(φ(Xt ) | Yt = c) for each class c ∈ {1, . . . , C} in Y (the class conditionals are matched as the conditionals are often challenging and computationally expensive to compute). This match is performed by leveraging the empirical form of the Maximum Mean Discrepancy (MMD) distance [25]; the difference between two mean embeddings in a Reproducing Kernel Hilbert Space (RKHS), induced by a kernel K [26] (where a variety of kernels can be implemented). The summation of the marginal and class conditional distributions can be formed as,  Dist(p(φ(Xs )) , p(φ(Xt ))) + Dist(p(φ(Xs ) | Ys ) , p(φ(Xt ) | Yt )) ≈ tr W T KMc KW (20.1) where W are weights associated with the empirical MMD distance and Mc is the MMD matrix, defining the mean embedding; this summation can be minimised to find the optimal latent mapping. It is noted that JDA assumes an unlabelled target domain and utilises a simplistic form of semi-supervised learning, using a classifier trained on the source domain projection to predict pseudo-labels Yˆt for the projected target domain. The MMD matrix is defined as, ⎧ 1 ⎪ xi , xj ∈ D(c) ⎪ (c) (c) , s ⎪ Ns Ns ⎪ ⎪ ⎪ (c) 1 ⎪ ⎪ ⎨ Nt(c) Nt(c) , xi , xj ∈ Dt 4 (c) Mc (i, j ) = xi ∈ D(c) ⎪ s xj ∈ Dt −1 ⎪ , ⎪ (c) (c) (c) ⎪ Ns Nt ⎪ xi ∈ D(c) ⎪ s xj ∈ Dt ⎪ ⎪ ⎩ 0, otherwise

(20.2)

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where D(c) s = {x i : x i ∈ Ds ∧ y(x i ) = c} are the instances that belong in class c given the true source label y(x i ) of x i and (c) (c) Ns = |D(c) ˆ i ) = c} are the instances that belong in class c given the pseudo-target label s |; and Dt = {x i : x i ∈ Dt ∧ y(x (c) (c) y(x ˆ i ) of x i and Nt = |Dt | (where ∧ is the logical AND symbol). The optimisation problem is then formed (subject to the regularisation constraint, where μ governs the level of regularisation, and kernel Principle Component Analysis (PCA)) and becomes an eigenvalue problem where the optimal weights W are obtained from the eigenvectors corresponding to the k smallest eigenvalues from, / K

C 

0 Mc K + μI W = KH KW φ.

(20.3)

c=0

Due to the pseudo-labelling of the target features, [11] recommends running several iterations of the optimisation to find the optimal W . A k-dimensional transformed feature space can be calculated by Z = KW ∈ R(Ns +Nt )×k , and a classifier trained on the transformed source data can be applied to the transformed target data.

20.3.1 Homogeneous Populations A homogenous population is one in which the label spaces will be consistent, as stated in Table 20.1, and depending on the feature utilised, may also have consistent feature spaces. This makes homogenous populations an ideal scenario for applying JDA, and demonstrating the benefits of transfer learning. This section presents a homogenous population of two, five storey shear structures depicted in Fig. 20.3. For both structures the stiffness elements {ki }5i=1 are constructed from the summation of the tip stiffness of four rectangular cantilever beams i.e. 4kb , presented in Fig. 20.3b, where damage is applied to one beam of the four beams at a given degree-of-freedom, in the form of an open crack using the reduction in stiffness model from [27]. The geometric and material properties are nominally the same and are shown in Table 20.3. The two structures are a homogeneous population as they are topologically2 and structurally3 equivalent in their lumpedmass representation and the material and geometry parameters can be defined by a concentrated unimodal distribution. The SHM problem considered here is a location problem, where a crack that is 10% of the beam width is applied at a distance 15% of the beam length from the base of the beam. This damage scenario is simulated for each of the degrees-of-freedom meaning that there are six labelled scenarios: undamaged labelled ‘1’, damage at k1 labelled ‘2’, damage at k2 labelled ‘3’, damage at k3 labelled ‘4’, damage at k4 labelled ‘5’, and damage at k5 labelled ‘6’. The features considered in this case study, depicted in Fig. 20.4, are the five damped natural frequencies meaning Xs = RNs ×5 and Xt = RNt ×5 , where Ns = 1800 and Nt = 1200 and each class has an equal weighting of data points. In order to test the inferred mapping, a separate test data set Fig. 20.3 Schematic of the five degree-of-freedom shear structure: panel (a) is of the full system and panel (b) depicts the cantilever beam component where {ki }5i=1 = 4kb i.e. the stiffness coefficients in (a) are generated from four times the tip bending stiffness in (b)

m5 c5

k5 m4

c4

kb k4

m3 c3

k3 m2

c2

k2 m1

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lloc

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2 Topologically-equivalent

lcr

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structures are those with the same graphical representation. In terms of IE and AGs, two graphs are topologicallyequivalent if the underlying unattributed graphs are identical. 3 Structurally-equivalent structures are those with the same graphical representation with the same locations of ground nodes. In terms of IE and AGs, two graphs are structurally equivalent if they are topologically equivalent with ground nodes in the AGs occurring in the same place.

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Table 20.3 Properties of the source and target structures for the homogenous population case study Property Beam geometry, {lb , wb , tb } Mass geometry, {lm , wm , tm } Elastic modulus, E Density, ρ Damping coefficient, c

Unit mm mm GPa kg/m3 Ns/m

Source {5000.0, 350.0, 350.0} {12000.0, 12000.0, 350.0}   N 210.00, 1 × 10−9 N (8000.0, 50) G (8.0000, 0.8)

Target {4999.0, 351.2, 349.7} {12001.0, 11998.8, 351.6}   N 210.89, 1 × 10−9 N (8019.4, 10) G (7.9981, 0.8)

Fig. 20.4 Source and target domain features for the homogeneous population case study. The source domain data are denoted by (·), the targetdomain training and testing data are denoted by (+) and (◦) respectively

from the target data was obtained where Xtest = RNtest ×5 and Ntest = 1500, again where each class has an equal number of data points. JDA was implemented with ten iterations, a linear kernel, and a k-Nearest Neighbour (k-NN) classifier (five neighbours), such that the emphasis is on the domain adaptation mapping and not the classifier. Cross-validation (5-folds) was implemented in order to identify the regularisation parameter, μ = 1 × 10−3 , and the number of transfer components, k = 4. The inferred mapping is presented in Fig. 20.5, where it can be seen that the source and target datasets have been successfully mapped onto each other, where the histograms of the marginal distributions are shown on the diagonal. The classification results are given in Table 20.4, where it can be seen that compared to a k-NN without JDA, domain adaptation provides a substantial improvement in classification performance.

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Fig. 20.5 Transfer components for the source and target domains in the homogeneous population case study. The source domain transfer components are denoted by (·), the target-domain training and testing transfer components are denoted by (+) and (◦) respectively Table 20.4 Classification results for the homogenous population case study trained on the labelled source domain and applied to the unlabelled target domain Method Mapping training Mapping testing

Accuracy Accuracy

k-NN 83.8% 83.9%

JDA 100.0% 100.0%

20.3.2 Heterogeneous Populations The heterogeneous population case study considers a population with geometry and material differences. The two structures in the population are structurally equivalent and are both five-storey shear structures (Fig. 20.3). The material differences exist, as the first structure is steel and the other aluminium; the geometric dissimilarities occur due to differences in the dimensions of the structure, displayed in Table 20.5. The SHM problem is the same as in the homogeneous case study i.e. a six-class location problem where damage is introduced as open cracks located at each degree-of-freedom. The features are damped natural frequencies such that Xs = RNs ×5 and Xt = RNt ×5 , Xtest = RNtest ×5 ; Ns = 1800, Nt = 1200 and Ntest = 1500 all with equal weighting of data points in each class. The source and target domain features are presented in Fig. 20.6, the magnitudes of which are very different highlighting the need for transfer learning. In a similar manner to the homogeneous case study, JDA was implemented with ten iterations, a linear kernel and a k-NN classifier (five neighbours). Cross-validation (5-folds) identified the regularisation parameter, μ = 1 × 10−3 , and number of transfer components, k = 4, used to generate the transfer components in Fig. 20.7. Once more, domain adaptation in the form of JDA, shows vast improvements in classification performance when compared to training a classifier on the source domain and applying it to the target, which has a classification performance equivalent to random guessing, as demonstrated

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Table 20.5 Properties of the source and target structures for the heterogeneous population case study Property Beam geometry, {lb , wb , tb } Mass geometry, {lm , wm , tm } Elastic modulus, E Density, ρ Damping coefficient, c

Unit mm mm GPa kg/m3 Ns/m

Source {5000, 350, 350} {12000, 12000, 350}   N 210, 1 × 10−9 N (8000, 10) G (50, 0.1)

Target {4300, 500, 500} {10000, 10000, 500}   N 71, 75 × 10−7 N (2700, 2.5) G (8, 0.8)

Fig. 20.6 Source (left) and target (right) domain features for the heterogeneous population case study. The source domain data are denoted by (·), the target-domain training and testing data are denoted by (+) and (◦) respectively

in Table 20.6. This case study demonstrates that for a selection of heterogeneous problems, homogeneous transfer learning, in the form of domain adaptation, is still applicable.

20.4 Discussion and Conclusions Knowledge transfer and mapping are important processes in developing a population-based approach to SHM. The idea of mapping data from different structures within a population such that knowledge about health states are transferred can be achieved via transfer learning. It is important when applying transfer learning to determine what similarities exist between structures within a population, and what information should be transferred, such that negative transfer is avoided. With this aim in mind, it is helpful to categorise structures based on their differences. Here two main types of population within PBSHM have been considered: homogenous populations and heterogeneous populations. Further to these two main categories, three more subdivisions have been discussed for heterogeneous populations: geometry, material and topology. These sources of dissimilarity form part of an Irreducible Element and Attributed Graph representation of structures, [3, 4]. By starting from what type of SHM problem is required, it is possible to categorise the level of consistency that exists for each type of population. This helps define what assumptions are required for a given transfer learning method, and whether homogenous or heterogeneous transfer learning should be implemented. Furthermore, label inconsistency between members of the population will increase the likelihood of negative transfer. This issue shows the power of an Irreducible Element and Attributed Graph representation of structures, as this approach means that similarities between members of the population can be extracted, reducing the transfer learning problem to one in which labels are considered consistent. Transfer learning, in the form of domain adaptation has been demonstrated to be applicable for problems when both feature and label spaces are consistent. By defining the mapping problem within an SHM and population type context, several PBSHM scenarios can be considered to fulfil these requirements. This paper has demonstrated that domain adaptation is applicable for both a homogeneous population, and heterogeneous population context; where the geometric differences are changes in dimensions and two different metallic materials are used. This work has highlighted that heterogeneous

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Fig. 20.7 Transfer components for the source and target domains in the heterogeneous population case study. The source domain transfer components are denoted by (·), the target-domain training and testing transfer components are denoted by (+) and (◦) respectively Table 20.6 Classification results for the heterogeneous population case study trained on the labelled source domain and applied to the unlabelled target domain Method Mapping training Mapping testing

Accuracy Accuracy

k-NN 16.7% 16.7%

JDA 100.0% 100.0%

populations, where topology differences occur, represent a significant challenge. When label spaces overlap, and can be considered consistent, heterogeneous domain adaptation may provide a solution to SHM problems for this type of population, and this is emphasised as an area of further research. Acknowledgments The authors would like to acknowledge the support of the UK Engineering and Physical Sciences Research Council via grants EP/R006768/1 and EP/R003645/1.

References 1. Papatheou, E., Dervilis, N., Maguire, A.E., Antoniadou, I., Worden, K.: Population-based SHM: A case study on an offshore wind farm. In: Proceeding of the 10th International Workshop on Structural Health Monitoring (2015) 2. Bull, L.A., Gardner, P.A., Gosliga, J., Dervilis, N., Papatheou, E., Maguire, A.E., Campos, C., Rogers, T.J., Cross, E.J., Worden, K.: Towards a population-based structural health monitoring. Part I: Homogeneous populations and forms. In: Proceedings of IMAC XXXVIII – the 38th International Modal Analysis Conference, Houston (2020) 3. Gosliga, J., Gardner, P.A., Bull, L.A., Dervilis, N., Worden, K.: Towards a population-based structural health monitoring. Part II: Heterogeneous populations and structures as graphs. In: Proceedings of IMAC XXXVIII – the 38th International Modal Analysis Conference, Houston (2020)

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4. Gosliga, J., Gardner, P.A., Bull, L.A., Dervilis, N., Worden, K.: Towards a population-based structural health monitoring. Part III: Graphs, networks and communities. In: Proceedings of IMAC XXXVIII – the 38th International Modal Analysis Conference, Houston (2020) 5. Worden, K.: Towards a population-based structural health monitoring. Part VI: Structures as geometry. In: Proceedings of IMAC XXXVIII – the 38th International Modal Analysis Conference, Houston (2020) 6. Lin, W., Worden, K., Maguire, A.E., Cross, E.J.: Towards a population-based structural health monitoring. Part VII: EoV fields: Environmental mapping. In: Proceedings of IMAC XXXVIII – the 38th International Modal Analysis Conference, Houston (2020) 7. Pan, S.J., Yang, Q.: A survey on transfer learning. IEEE Trans. Knowl. Data Eng. 22, 1345–1359 (2010) 8. Zhang, Y., Yang, Q.: An overview of multi-task learning. Natl. Sci. Rev. 5, 30–43 (2018) 9. Rytter, A.: Vibrational based inspection of civil engineering structures. Ph.D. thesis, Aalborg University (1993) 10. Farrar, C.R., Lieven, N.A.J.: Damage prognosis: the future of structural health monitoring. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 365, 623–632 (2007) 11. Long, M., Wang, J., Ding, G., Sun, J., Yu, P.S.: Transfer feature learning with joint distribution adaptation. In: 2013 IEEE International Conference on Computer Vision, pp. 2200–2207 (2013) 12. Chapelle, O., Schölkopf, B., Zien, A.: Semi-Supervised Learning. MIT Press, Cambridge (2006) 13. Sohn, H.: Effects of environmental and operational variability on structural health monitoring. Phil. Trans. R. Soc. A 365, 539–560 (2007) 14. Rosenstein, M.T., Marx, Z., Kaelbling, L.P., Dietterich, T.G.: To transfer or not to transfer. In: NIPS 2005 Workshop on Transfer Learning (2005) 15. Eaton, E., DesJardins, M., Lane, T.: Machine learning and knowledge discovery in databases. Springer, Berlin Heidelberg (2008). https://link. springer.com/chapter/10.1007/978-3-540-87479-9_39 16. Ge, L., Gao, J., Ngo, H., Li, K., Zhang, A.: On handling negative transfer and imbalanced distributions in multiple source transfer learning. In: Proceedings of the 2013 SIAM International Conference on Data Mining, SDM (2013) 17. Wang, Z., Dai, Z., Póczos, B., Carbonell, J.: Characterizing and avoiding negative transfer. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 11293–11302 (2019) 18. Quiñonero-Candela, J., Sugiyama, M., Schwaighofer, A., Lawrence, N.D.: Dataset Shift in Machine Learning. Mit Press, Cambridge (2009) 19. Pan, S.J., Tsang, I.W., Kwok, J.T., Yang, Q.: Domain adaptation via transfer component analysis. IEEE Trans. Neural Netw. 22, 199–210 (2011) 20. Long, M., Wang, J., Ding, G., Pan, S.J., Yu, P.S.: Adaptation regularization: a general framework for transfer learning. IEEE Trans. Knowl. Data Eng. 26, 1076–1089 (2014) 21. Duan, L., Xu, D., Tsang, I.W.: Learning with augmented features for heterogeneous domain adaptation. In: Proceedings of the 29th International Conference on Machine Learning, ICML (2012) 22. Li, W., Duan, L., Xu, D., Tsang, I.W.: Learning with augmented features for supervised and semi-supervised heterogeneous domain adaptation. IEEE Trans. Pattern Anal. Mach. Intell. 36, 1134–1148 (2014) 23. Day, O., Khoshgoftaar, T.M.: A survey on heterogeneous transfer learning. J. Big Data 4, 29 (2017) 24. Gardner, P.A., Worden, K.: On the application of domain adaptation for aiding supervised SHM methods. In: Proceedings of the 12th International Workshop on Structural Health Monitoring, pp. 3347–3357 (2019) 25. Schölkopf, B., Smola, A., Müller, K-R.: Nonlinear component analysis as a kernel eigenvalue problem. Neural Comput. 10, 1299–1319 (1998) 26. Gretton, A., Borgwardt, K.M., Rasch, M.J., Schöolkopf, B., Smola, A.: A kernel two-sample test. J. Mach. Learn. Res. 13, 723–773 (2012) 27. Christides, S., Barr, A.D.S.: One-dimensional theory of cracked Bernoulli-Euler beams. Int. J. Mech. Sci. 26, 639–648 (1984)

Chapter 21

Feasibility Study of Using Low-Cost Measurement Devices for System Identification Using Bayesian Approaches Alejandro Duarte and Albert R. Ortiz

Abstract The development of low-cost measurement devices for dynamic identification of structures has gained considerable interest in recent times. The high cost of traditional systems has led several authors to evaluate the feasibility of using affordable acquisition systems. This article aims to quantify the uncertainties of the modal parameters of a flexible structure, instrumented with traditional and low-cost measurement devices. The evaluated low-cost acquisition systems were smartphones and Raspberry Pi microprocessors, while the conventional system consisted of highly sensitive piezoelectric accelerometers. The instrumented structure was a steel pedestrian bridge located in the city of Barranquilla. Bayesian methodologies were used to identify the modal parameters and their associated uncertainties in terms of probability distributions. The results obtained show the feasibility of using low-cost devices to determine the dynamic properties of a flexible structure. Keywords System identification · Bayesian inference · Low-cost measurement device · Uncertainties quantification · Structural health monitoring

21.1 Introduction In many applications of Structural Health Monitoring (SHM) it is essential to use acquisition systems to obtain information on different types of structures; however, in many cases, these systems represent a considerable investment due to the high cost in acquisition and maintenance. With the advancement of technology, new alternatives have been developed to address this problem, among the most innovative are the use of sensors built into smartphones. These devices have accelerometers, magnetometers, gyroscopes, memory, processors, among other sensors that convert these devices into potential acquisition systems [1]. Likewise, the boom in the microcontroller and microprocessor sector has generated the appearance of devices such as the Raspberry Pi (RBP) and Arduinos. These devices can acquire and even process data by adding different types of sensors, also, because they are portable, economical, and versatile, several authors have used them to control or acquire experimental data. In the context of system identification, it is desired to know the main modal parameters that govern the response of a structure. To measure the structure’s vibration response is used an acquisition system with a set of accelerometers. Several studies have been conducted to evaluate the feasibility of using smartphone accelerometers as a substitute for traditional acquisition systems [2–7]. However, these studies have focused on modal identification using deterministic techniques, and the uncertainty associated with the modal parameters identified with these low-cost devices has not been addressed. The identification process is subject to uncertainties generated by the sensitivity of the sensors. The modal parameters identified may be affected by the low signal-to-noise ratio characteristics of these devices; therefore, they should be treated differently in the identification process if compared to traditional systems [8].

A. Duarte Department of Civil and Environmental Engineering, Universidad del Norte, Barranquilla, Colombia e-mail: [email protected] A. R. Ortiz () School of Civil Engineering and Geomatics, Universidad del Valle, Cali, Colombia e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2020 Z. Mao (ed.), Model Validation and Uncertainty Quantification, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47638-0_21

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Bayesian methodologies for modal identification have gained significant attention in recent times. In SID, these methodologies allow obtaining the optimal modal parameters and their associated uncertainties in terms of probability distributions, which means that it is possible to calculate the probability of frequencies, modal shapes, and damping ratios using response data [9, 10]. Therefore, this probabilistic approach allows for estimating the reliability of the acquisition system. This article seeks to identify the main modal parameters of a steel pedestrian bridge located in the city of Barranquilla. Three different data acquisition systems are used: MEMS on smartphones, MEMS sensors on RBP, and high-resolution piezoelectric accelerometers. A comparative evaluation will be carried out between each acquisition system based on the uncertainties found in the identification process. The document has four main sections. Section 2 presents background about the uncertainty quantification in system identification. Section 3 describes the experimental setup. The probability density function of the parameters identified for each acquisition system is then described in Sect. 4. Conclusions and future work expectative are presented in Sect. 5.

21.2 Background 21.2.1 Bayesian Spectral Density Approach for System Identification (BSDA) This article uses a Bayesian approach to identify the modal parameters of the bridge. The method is based on modal updating techniques to determine the probability density functions (PDF) of modal parameters. Bayesian methodologies have attracted a lot of attention because they allow the uncertainty quantification of the parameters updated as a function of the data. Therefore, it means the identification of modal frequencies of small amplitude, damping ratios, and modal shapes of the system using ambient response data. The BSDA estimates the most likely power spectral density (PSD) function given ambient data records. Equation 21.1   ∗ ˆ ˆ shows the expected value of the PSD, E Fk Fk |θ . This model depends on modal parameters, θ = [f, ζ , S, Se ]. Therefore, the parameters to be identified are the natural frequency, f, damping ratio, ζ , the spectral density of the modal excitation, S, and the modeling error or PSD of the specular noise, Se [11].     E Fˆk Fˆk∗ |θ = E Sˆk (θ ) = SDk + Se

−1  2 f 2 2 , βk = Dk = βk − 1 + (2ζβk ) fk

(21.1)

(21.2)

The experimental part of this methodology is introduced in equation 3. This expression represents the spectral density of ambient acceleration records. For a single degree of freedom (SDOF), the PSD is obtained with the square of the absolute value of the Fourier transform. " "2 " " Sˆk = "Fˆk "

: Fˆk =

N 2Δt  ˆ −2π i(j −1)(k−1) N x¨ j e N

(21.3)

j =1

Bayes’ theorem, as shown in equation 21.4, is used for the update process and to obtain the subsequent posterior distribution, p θ |Sˆk . It represents the conditional probability by having the parameters θ given a set of experimental data Sˆk . This expression is calculated p(θ ), which represents the prior knowledge of as the multiplication of the prior distribution,  ˆ ˆ the parameters θ , times p Sk |θ , a term known as likelihood. p Sk |θ follows the distribution of Eq. 21.5 and it represents the likelihood between the model given a set of parameters θ and the experimental data, Sˆk [12, 13].   p θ |Sˆk = cp (θ ) p Sˆk |θ

(21.4)

21 Feasibility Study of Using Low-Cost Measurement Devices for System Identification Using Bayesian Approaches

 p Sˆk |θ =

⎛ 1

Sˆk

203

⎞

  exp ⎝−  ⎠ E Sˆk (θ ) E Sˆk (θ )

(21.5)

21.3 Experimental Setup 21.3.1 Structure Description The study structure was a pedestrian bridge located in the city of Barranquilla. The bridge is one of the primary access to the Universidad del Norte campus. The bridge has a span length of 22.4 m, and it consists of a series of metal drawer profiles shaping a truss, as shown in Fig. 21.1. Due to the shape as it was built, the deck can be considered as a simple supported structure, and consequently, the predominant vibration modes are expected to be vertical and transversal, independently of ramps.

21.3.2 Structure Instrumentation and Tests Description The response of the structure was measured using three different types of sensors: Piezoelectric, MEMS on RBP, and MEMS on Smartphones. The identification focused on the first vertical vibration mode of the structure, and therefore, all sensors were measuring in a vertical direction as indicated in Table 21.1. The first type of sensors consists of piezoelectric accelerometers connected to a National Instruments NI-9234 module. Accelerometers are piezoelectric sensors PCB reference 333B50 with a sensitivity of 100 mV/g. The DAQ system was connected to a laptop where data was stored and preprocessed. The second family of devices is smartphones reference iPhone 6. Each smartphone six has a Bosch MEMS accelerometer with a sensitivity of 4096 LSB/g. Data were acquired using the Vibsensor application stored in the smartphone memory. The third type of sensors were accelerometers MEMS ADXL355 connected to RBP 3B+ microprocessors. The accelerometers have a sensitivity of 400 mV/g. They were at the same location as smartphones. The acquired data in RBP was sent to a server through an internet network.

Fig. 21.1 Schema of the bridge. The deck is a tridimensional truss as it is simple supported by ramps (left). Position of sensors are shown as numbers 1 to 5 and are detailed in Table 21.1 (right) Table 21.1 Sensor location along the span of the bridge ID 1 2 3 4 5

Position from the south side 5.36 m 8.17 m 11.25 m 13.60 m 16.75 m

PCB accelerometer Uniaxial Uniaxial Triaxial Uniaxial Triaxial

Bosch MEMS on Smartphones Triaxial – Triaxial – Triaxial

ADXL355 on Rasberry Pi platform Triaxial – Triaxial – Triaxial

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Ambient vibration tests were performed in early mornings during holidays to guarantee the continuity of the data without pedestrians crossing the bridge. All devices acquired simultaneously for 10 min with a frequency sampling of 100 Hz for Smartphones and RBP, and 2000 Hz for the NI DAQ system.

21.4 Results Ambient vibration data is processed using the Bayesian methodology introduced in Sect. 2. Several 10-minute data are obtained with each acquisition system. Figure 21.2 shows the PSD spectrums of each acquisition system.

21.4.1 Probabilistic SID in Low-Cost Devices The structure was identified by starting the BSDA methodology in its variant for a single degree of freedom system (SDOF). A Markov Chain Monte Carlo was used for sampling the posterior distribution and finding the parameters θ that  (MCMC) maximize the likelihood, p Sˆk |θ . To avoid bias in the Bayesian update, the prior term, p(θ ), was taken as a constant for each

a) RBP

c) N I Fig. 21.2 PSD functions for each acquisition system (a-c) and combined PSD plot (d)

b) Iphone6

d) NiDaq, RBP, Iphone6

21 Feasibility Study of Using Low-Cost Measurement Devices for System Identification Using Bayesian Approaches

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MEMS sensor in a RBP Fig. 21.3 PSD of the experimental data recorded by each system and the Expected value found by the maximization of the likelihood

of the measuring devices. The acceleration measurements of the RBP, smartphones, and piezoelectric accelerometers located in the center of the bridge were used for the identification of the modal properties associated with the first vertical vibration mode. For the MCMC, 30,000 samples were obtained, and the first 10,000 were burned to guarantee the convergence of the parameters. Figure 21.2 shows the estimate of the expected PSD value after the sampling process. Figure 21.3 shows the PSD of each system and the maximum a posteriori value.

21.4.2 Marginal Distributions and Correlations The samples generated in each Markov chain are used to estimate the marginal distributions of the parameters. Figure 21.4 shows the marginal distributions (diagonal) and the correlation plots of samples generated for all acquisition systems. Results in Fig. 21.4 show an evident correlation between the damping ratio, the PSD of the modal force and the PSD of the modeling error or spectral noise. These results are consistent with those obtained by Yan & Katafygiotis [14]. As stated in the previous section, there are several correlations between the parameters in the identification process. If the uncertainties, represented in the marginal distributions, are compared with the optimal parameters, estimated as the first moment of each distribution, it is possible to evaluate how the characteristics of the acquisition system affect the uncertainties and parameters identified. Figure 21.5 shows the marginal distributions of each of the systems evaluated.

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f (Hz)

8.76

8.74

0.8

0.4 0.2 0.52

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From the results obtained in Fig. 21.5c, it is evident that the marginal distribution of spectral noise in the iPhone 6 has very little dispersion. This indicates that the optimal value of the parameter has a great influence on the identification process. It differs from the marginal distribution of the National Instruments (NiDaq) acquisition system, which can even approximate a uniform distribution due to its wide dispersion. Therefore, if the modal parameters of the structure are estimated without including the spectral noise parameter with the NiDaq system, the results would not vary significantly. In contrast, if the modal parameters of the structure are estimated with a smartphone or RBP without including the term Se , the identified parameters would be influenced by not modeling the noise. Finally, in the identification process, no significant correlation was found between the natural frequency and the remaining parameters [ζ , S, Se ]. The characteristics of the acquisition systems do not significantly influence the identification of this parameter. The moments of the subsequent distributions are shown in Table 21.2. The moments serve as the basis for comparing each acquisition system. The Raspberri Pi and the iPhone 6 have very similar standard deviations, except in the spectral noise. As expected, data obtained by the DAQ with piezoelectric sensors (NI DAQ) got the lowest deviations in frequency, damping, and modal excitation. Although both the iPhone 6 and the Raspberry Pi could not correctly estimate the damping deterministically, confidence intervals of the damping values obtained by the NI DAQ system are contained in the 95% confidence interval of the iPhone 6 and the Rasberry Pi.

21 Feasibility Study of Using Low-Cost Measurement Devices for System Identification Using Bayesian Approaches

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DAQ fˆ = 8.7358σ f = 0.0052 ζˆ = 0.3539σ f = 0.0617  log Sˆ = −10.10σ ˆ  = 0.0274 log S  ˆ = −10.17σ  = 0.515 log Se ˆ log Se

RBP fˆ = 8.7354σ f = 0.0071 ζˆ = 0.4566σ f = 0.0804  log Sˆ = −9.93σ ˆ  = 0.0392 log S  ˆ = −9.19σ  = 0.152 log Se ˆ log Se

iPhone 6 fˆ = 8.7355σ f = 0.0069 ζˆ = 0.4579σ f = 0.0843  log Sˆ = −10.00σ ˆ  = 0.0476 log S  ˆ = −8.125σ  = 0.0322 log Se ˆ log Se

21.5 Conclusions Within the high technology available in smartphones and other low-cost microprocessors such as the RBP, the use of SHM techniques in real scale structures is becoming common. Traditionally techniques for processing the data and obtaining dynamic properties are being implemented on data acquired with this technology, which could increase uncertainties in the identification process given the differences in resolution. This paper identified the uncertainty in the modal parameters of a full-scale structure using three different devices. The devices are MEMS accelerometers sensors attached to RBP and smartphones (considered low cost-devices), and piezoelectric sensors in a robust data acquisition system. The identified structure consists of a 22 m pedestrian bridge. The processing methodology was the Bayesian Spectral Density Approach for System Identification found in the literature. Results show that the identification process of the natural frequency of the structure has similar values in all three devices, with small differences in mean values (less than 0.01%) and standard deviations (%). This means that low-cost measuring devices successfully identified the natural frequency of the structure, giving results like those obtained with traditional acquisition systems. The independence between the natural

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frequency and the other parameters implies that the identification of the frequency is not affected by the characteristics of the acquisition system for these types of structures. Other parameters identified, such as damping ratio, noise identification, and modal excitation, vary considerably. The damping ratio has differences up to 30%, which could affect predictions in the response of the structure. PDF shows strong correlations between the parameters ζ , S, Se . Consequently, the uncertainties of these propagate affecting the identification of the optimal parameters. Correlations are related to spectral noise and modal excitation. This means that modeling the PSD of spectral noise in the identification process is necessary for low-cost measuring devices. The omission of this directly affects the identification of modal excitation and the damping ratio if low-cost devices are used. However, for traditional acquisition systems, this parameter does not significantly affect the identification process. The use of a low-cost acquisition system is a viable methodology to identify the modal parameters of flexible structures. However, it is advisable to use Bayesian methodologies that allow calculating the uncertainties of modal parameters in the identification process.

References 1. Feng, M., Fukuda, Y., Mizuta, M., Ozer, E.: Citizen sensors for SHM: use of accelerometer data from smartphones. Sensors. 15(2), 2980–2998 (2015) 2. Alavi, A.H., Buttlar, W.G.: An overview of smartphone technology for citizen-centered, real-time and scalable civil infrastructure monitoring. Future Gener. Comput. Syst. 93, 651–672 (2019) 3. Castellanos-Toro, S., Marmolejo, M., Marulanda, J., Cruz, A., Thomson, P.: Frequencies and damping ratios of bridges through operational modal analysis using smartphones. Constr. Build. Mater. 188, 490–504 (2018) 4. Ervasti, M., Dashti, S., Reilly, J., Bray, J.D., Bayen, A., Glaser, S.: iShake: mobile phones as seismic sensors–user study findings. In: Presented at the Proceedings of the 10th International Conference on Mobile and Ubiquitous Multimedia, pp. 43–52 (2011) 5. Feldbusch, A., Sadegh-Azar, H., Agne, P.: Vibration analysis using mobile devices (smartphones or tablets). Procedia Eng. 199, 2790–2795 (2017) 6. Mourcou, Q., Fleury, A., Franco, C., Klopcic, F., Vuillerme, N.: Performance evaluation of smartphone inertial sensors measurement for range of motion. Sensors. 15(9), 23168–23187 (2015) 7. Ozer, E., Feng, M.Q.: Direction-sensitive smart monitoring of structures using heterogeneous smartphone sensor data and coordinate system transformation. Smart Mater. Struct. 26(4), 045026 (2017) 8. Dorvash, S., Pakzad, S.: Effects of measurement noise on modal parameter identification. Smart Mater. Struct. 21(6), 065008 (2012) 9. Yuen, K.-V., Kuok, S.-C.: Bayesian methods for updating dynamic models. Appl. Mech. Rev. 64(1), 010802 (2011) 10. Yuen, K.-V., Katafygiotis, L.S.: Bayesian time-domain approach for modal updating using ambient data. Probabilistic Eng. Mech. 16(3), 219–231 (2001) 11. Katafygiotis, L.S., Yuen, K.: Bayesian spectral density approach for modal updating using ambient data. Earthq. Eng. Struct. Dyn. 30(8), 1103–1123 (2001) 12. Yuen, K.-V., Katafygiotis, L.S., Beck, J.L.: Spectral density estimation of stochastic vector processes. Probabilistic Eng. Mech. 17(3), 265–272 (2002) 13. Au, S.-K.: Insights on the Bayesian spectral density method for operational modal analysis. Mech. Syst. Signal Process. 66, 1–12 (2016) 14. Yan, W.-J., Katafygiotis, L.S.: An analytical investigation into the propagation properties of uncertainty in a two-stage fast Bayesian spectral density approach for ambient modal analysis. Mech. Syst. Signal Process. 118, 503–533 (2019)

Chapter 22

Kernelised Bayesian Transfer Learning for Population-Based Structural Health Monitoring Paul Gardner, Lawrence A. Bull, Nikolaos Dervilis, and Keith Worden

Abstract Population-based structural health monitoring is the process of utilising information from a group of structures in order to perform and improve inferences that generalise to the complete population. A significant challenge in inferring a general representation for structures is that feature spaces will be inconsistent for a wide variety of populations and datasets. This scenario, where the dimensions of the feature spaces for each structure are different, occurs for a variety of reasons. Firstly, the group of structures themselves may be a heterogeneous population, where differences occur due to topology, leading to inconsistency in modal-based features. Secondly, feature spaces may be inconsistent across the population due to differences in the raw data (i.e. different sample frequencies etc.) and feature extraction processing. In this context, where feature spaces are inconsistent between different structure in a population, a general model that describes their behaviours becomes challenging to infer. This issue is because dimensionality reduction must be performed such that each domain’s feature set projects to a consistent shared latent space where a model can be inferred. This paper introduces a technique, kernelised Bayesian transfer learning, that seeks to learn a projection matrix and kernel embedding that map to a latent space where a discriminative classifier can be inferred in a Bayesian manner, using variational inference. This algorithm allows a general discriminative classifier to be inferred across a population where the feature spaces for each structure are inconsistent. A numerical case study is presented, demonstrating the effectiveness of this approach and for providing a discussion of its implications for population-based structural health monitoring. Keywords Population-based structural health monitoring · Transfer learning · Multi-task learning

22.1 Introduction Performing health monitoring collectively for a group of structures is known as PBSHM [1]. A requirement of performing PBSHM is inferring a general representation of a population of structures that applies to the complete population [2, 3]. By inferring a general machine learning model, label knowledge can be exchanged between structures in the population, improving the overall accuracy of any health monitoring approach for the population. Several benefits are provided by a population-view of SHM, such as improved knowledge of feature and label relationships, a removal of confounding influences across the population in a latent space representation, and an increase in classification performance for each member of the population. Another aim of PBSHM is to overcome a lack of available damage state data for each member of the population by pooling and transferring label knowledge between the population, as demonstrated in [3]. This paper is concerned with the scenario in PBSHM in which the aim is to generate an improved general representation of the population of structures, related to a form [2]. Specifically, the context considered is when the feature spaces are inconsistent, which arises for several reasons. Firstly, the population may be heterogeneous [3, 4], based on differences in topology (i.e. differences in physical connections between parts of the structure) [4], leading to feature spaces that are different dimensions for each structure, especially when features are modal-based. The second scenario is when the raw data and feature extraction process lead to inconsistent feature spaces. This may occur due to different signal processing and data acquisition processes for each member of the population. As a result, any general model must include a projection into a consistent shared feature space, such that label information can be shared. Here kernelised Bayesian transfer learning is introduced as a multi-task learning algorithm [5]. This technique seeks to use information from multiple sources with inconsistent feature spaces, such that an improved discriminative classifier can be inferred for the complete population. This P. Gardner () · L.A. Bull · N. Dervilis · K. Worden Dynamics Research Group, Department of Mechanical Engineering, The University of Sheffield, Sheffield, UK e-mail: [email protected]; [email protected]; [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2020 Z. Mao (ed.), Model Validation and Uncertainty Quantification, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47638-0_22

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technique is beneficial when several structures have large overlap between clusters in their feature space, and when limited label knowledge is known for each structure. Health monitoring for wind farms provides an illustration of this approach. Typically, an asset manager of a wind farm is required to monitor the complete fleet, where each turbine has a distinct individual behaviour due to unique environmental conditions and manufacturing tolerances. This results in the finite datasets from each turbine having a different data distribution and will cause a machine learner trained on one member of the population to fail to generalise to another member of the population. In fact, the company will often have access to a variety of wind farms, each of which might have a different model of wind turbine in them. Within all these data sources (from members of the population) a variety of labelled health state data will often be available. Obtaining a complete label set for each turbine in the population is clearly not feasible; however, the asset manger wishes to use their label knowledge from their complete population of turbines from multiple wind farms in health monitoring, as labelled data is a valuable asset. However, the feature spaces may not be the same dimension for each member of the population, due to different data acquisition processes and feature extraction. In this context, kernelised Bayesian transfer learning is applicable in inferring a model in a consistent latent feature space in which all members of the population can be mapped, and their label information utilised. The outline of this paper is as follows. Firstly, kernelised Bayesian transfer learning is introduced in Sect. 22.2 within the context of transfer learning and multi-task learning. Section 22.3 demonstrates kernelised Bayesian transfer learning on a numerical example in which different sets of shear buildings with varying degrees-of-freedom are utilised in creating a general classification model of the population. Finally, conclusions are presented highlighting areas of further research.

22.2 Kernel-Based Transfer Learning Kernelised Bayesian transfer learning is a supervised multi-task learning algorithm for inconsistent feature domains [5], i.e. the dimension d of the feature space X for each domain is not equal e.g. d1 = d2 for a two-domain problem. Before formally introducing multi-task learning, the definition of two objects are required: A domain D = {X, p(X)} is an object that consists of a feature space X and a marginal probability distribution p(X) over a finite sample of feature data X = {x i }N i=1 ∈ X from X. A task T = {Y, f (·)} is the combination of a label space Y and a predictive function f (·). Multi-task learning is a branch of transfer learning in which the aim is to use knowledge from multiple domains to improve a particular task [6, 7]. The main difference to other transfer learning methods is that multi-task learning weights knowledge transfer equally for all domains. The goal therefore, is to generate an improved prediction function f (·) for one consistent label space Y using the feature data from several source domains. Kernelised Bayesian transfer learning is a particular form of multi-task learning that can be defined as heterogeneous transfer learning i.e. at least one feature space Xj for a domain Dj is not the same dimension as another feature space Xk in the domain set such that dj = dk . The assumptions are that there is a relationship between the feature space for each domain and the label space and that each domain provides informative knowledge that will improve the predictive function f (·). Several other heterogeneous transfer learning methods exist [8, 9]; however, kernelised Bayesian transfer learning is a multi-task learning that performs two tasks: (1) finding a shared latent subspace for each domain and (2) inferring a discriminative classifier in the shared latent subspace, in a Bayesian manner, are visualised in Fig. 22.1. Kernelised Bayesian transfer learning assumes T domains {Dj }Tj=1 with consistent label spaces Yj = Yk ∀ j ∈ T and ∀ k ∈ T but with features spaces that are not consistent. For each domain, an associated finite label set is available y j = {yj,i ∈ {−1, +1}}i∈Ij where Ij is the index set for each data point in domain j . For each domain, the data are embedded

in a kernel matrix {Kj ∈Nj ×Nj }Tj=1 using a particular kernel function, where Nj is the number of data point in domain j . An optimal projection matrix {Aj ∈Nk ×R }Tj=1 is constructed for each domain and projects the kernel embedding into a shared latent subspace {Hj = ATj Kj }Tj=1 . In this subspace a discriminative functional classifier {f j = HjT w + 1b}Tj=1 is inferred that predicts the outputs in the shared latent subspace with shared classification parameters {b ∈, w ∈R }. The quantities are inferred in a Bayesian manner, utilising variational techniques and conjugate analysis. In keeping with the brevity of the current paper the reader is referred to [5] for mathematical details.

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Mass geometry {lm , wm , tm } m {12, 12, 0.35} {10, 10, 0.50} {11, 11, 0.40} {11, 11, 0.45} {12, 12, 0.46}

Elastic modulus E GPa   N 210, 1.0 × 10−9   N 200, 2.0 × 10−9   N 205, 1.5 × 10−9   N 208, 1.0 × 10−9   N 201, 2.0 × 10−9

Density ρ kg/m3 N (8000, 10) N (7800, 50) N (7950, 25) N (7900, 15) N (7850, 20)

Damping coefficient c Ns/m G (50, 0.1) G (8, 0.8) G (25, 0.2) G (20, 0.1) G (50, 0.1)

22.3 Shear Building Case Study In order to demonstrate the applicability of kernelised Bayesian transfer learning for PBSHM, a numerical case study is demonstrated. The SHM problem is a two-class location problem (which can be seen as binary damage detection as there are only two classes). The population of structures is composed of five different shear building structures, schematically depicted in Fig. 22.2 as lumped-mass models in bending, forming a heterogeneous population [3, 4]. Each structure is represented by n mass {mi }ni=1 , stiffness {ki }ni=1 and damping {ci }ni=1 coefficients. The masses are specified by a length lm , width wm , thickness tm and density ρ. The stiffness elements are calculated from four cantilever beams in bending 4kb = 4(3EI / lb3 ), where E is the elastic modulus, I the second moment of area and lb the length of the beam. The damping coefficients are directly specified and are not derived from a physical model. Damage is introduced to the structure via an open crack, using a reduction in EI (using the model in [10]) across one of the beams in k1 , i.e. k1 = 3(3EI / lb3 ) + kd , where kd is the tip stiffness of a cantilever beam subject to an open crack of length lcr at location lloc along the length of the beam. Each observation, for a particular structure, is composed from random draws from a base distribution for E, ρ and c. The properties of the five structures in the population are shown in Table 22.1. The features in this case study are damped natural frequencies and damping ratios i.e. X = {ωi , ζi }ni=1 . The first two principal components for each domain are depicted in Fig. 22.3, indicating the level of overlap in the classes, where the

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22 Kernelised Bayesian Transfer Learning for Population-Based Structural Health Monitoring Table 22.2 Number of data points in each class for each domain

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number of training and testing data point for each domain are presented in Table 22.2. Each domain has a different degree of class imbalance in the training dataset (Table 22.2), reflecting scenarios where certain structures may provide more informative data about a particular damage class. Figure 22.4 presents the classification function and margin (set as zero) for the test data (top panel) and the probability of the data point belong to the +1 class (bottom panel) compared to the true label. These plots show the value of a probabilistic approach, as a soft margin reduces the number of misclassified points. It can also be seen, that due to imbalance in the class (weighted towards −1 the undamaged class for each structure), there is greater uncertainty in the damaged class (+1). The figures also depict more confident predictions for domains 1 and 4, which can be expected due to their relatively large number of data points and separability. The classification result are presented in Table 22.3. Accuracies and F1 macro scores are based on data points belonging to class +1 if p(y∗,j = +1)f∗,j ≥ 0.5. Considering classification performance, kernelised Bayesian transfer learning has accurately inferred a general model for the five domains. Although there is a large performance decrease for domain 2, this is expected due to the few data points used to train the method, compared with the other domains, and a relatively large degree of overlap in the classes. Performance remains high for the two separable domains (1 and 4), meaning that the method has not negatively impacted classification in these domains by learning a multi-task model.

22.4 Conclusions Population-based SHM requires machine learning techniques that utilise knowledge from a range of sources in order to construct improved general models of the population. One challenge is creating general models when feature spaces are inconsistent. This paper demonstrates one approach to multi-task learning, kernelised Bayesian transfer learning, where each domain has an inconsistent feature space. This approach is applicable to heterogeneous populations (with topological differences) and when data processing has led to feature inconsistencies. Kernelised Bayesian transfer learning, has been demonstrated to provide a good level of classification performance for a numerical case study involving a heterogeneous population of shear structures with a different number of storeys. The approach aids classification in scenarios of high class overlap, and in scenarios with a large class imbalance. Further research is required in investigating the multi-class problem, and in extending the method to scenarios in which one domain has no labels for a particular class. This will extend the approach from a multi-task learner to a full transfer learning method.

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Acknowledgments The authors would like to acknowledge the support of the UK Engineering and Physical Sciences Research Council via grants EP/R006768/1 and EP/R003645/1.

References 1. Papatheou, E., Dervilis, N., Maguire, E.A., Antoniadou, I., Worden, K.: Population-based SHM: a case study on an offshore wind farm. In: Proceeding of the 10th International Workshop on Structural Health Monitoring (2015) 2. Bull, L.A., Gardner, P., Gosliga, J., Rogers, T.J., Haywood-Alexander, M., Dervilis, N., Cross, E.J., Worden, K.: Towards a population-based structural health monitoring. Part I: homogeneous populations and forms. In: Proceedings of IMAC XXXVIII – the 38th International Modal Analysis Conference, Houston (2020) 3. Gardner, P., Bull, L.A., Gosliga, J., Dervilis, N., Worden, K.: Towards population-based structural health monitoring, Part IV: heterogeneous populations, transfer and mapping. In: Proceedings of IMAC XXXVIII – the 38th International Modal Analysis Conference, Houston (2020) 4. Gosliga, J., Gardner, P., Bull, L.A., Dervilis, N., Worden, K.: Towards a population-based structural health monitoring. Part II: heterogeneous populations and structures as graphs. In: Proceedings of IMAC XXXVIII – the 38th International Modal Analysis Conference, Houston (2020) 5. Gönen, M., Margolin, A.A.: Kernelized Bayesian transfer learning. In: Proceedings of the National Conference on Artificial Intelligence (2014) 6. Pan, S.J., Yang, Q.: A survey on transfer learning. IEEE Trans. Knowl. Data Eng. 22(10), 1345–1359 (2010) 7. Zhang, Y., Yang, Q.: An overview of multi-task learning. Natl. Sci. Rev. 5(1), 30–43 (2018) 8. Duan, L., Xu, D., Tsang, I.W.: Learning with augmented features for heterogeneous domain adaptation. In: Proceedings of the 29th International Conference on Machine Learning, ICML (2012) 9. Day, O., Khoshgoftaar, T.M.: A survey on heterogeneous transfer learning. J. Big Data 4(1), 29 (2017) 10. Christides, S., Barr, A.D.: One-dimensional theory of cracked Bernoulli-Euler beams. Int. J. Mech. Sci. 26, 639–648 (1984)

Chapter 23

Predicting System Response at Unmeasured Locations Using a Laboratory Pre-Test Randy Mayes, Luke Ankers, and Phil Daborn

Abstract One can estimate unmeasured acceleration spectral density responses of a structure utilizing measured responses from a relatively small number of accelerometers and the active mode shapes provided from a finite element model. The objective in this paper is to demonstrate a similar concept, but purely based on information from a laboratory pre-test. Response predictions can only be calculated at degrees of freedom that have been instrumented in the experimental pretest, but greater accuracy may be possible than with a finite element-based expansion. A multi-reference set of frequency response functions is gathered in the laboratory pre-test of the field hardware. Two response instrumentation sets are included in the pre-test. One set corresponds to the measurements that will be taken in the field environment. The second set is the field responses that are of great interest but will not be measured in the field environment due to logistical constraints. For example, the second set would provide definition of the component field environment. A set of basis vectors is extracted from the pre-test experimental data in each of multiple frequency bands. Then the field environment is applied to the hardware and the data gathered from the field accelerometers. The basis vectors are then used to expand the response from the field accelerations to the other locations of interest. The proof of concept is provided with an acoustic test environment on the Modal Analysis Test Vehicle. Predicted acceleration spectral density simulations at 14 degrees of freedom (known as “truth responses”) are compared against truth acceleration measurements collected for this work from the acoustic environment. Due to the segregated bandwidth analysis, the required number of field accelerometers to provide the simulation is much smaller than the number of modes in the entire frequency bandwidth. Keywords Flight environments · Flight accelerations · Expansion of field response

23.1 Motivation and Approach The dynamic qualification of components requires a specification for testing the component. Often the specification is an acceleration spectral density that has undergone an “enveloping” and sanitization process such that the resulting flat profiled curve is not representative of the dynamics measured on the component in the system environment. The acceleration measurements from the system environment are often made at different locations than the component connection point, resulting in a great deal of uncertainty. The specification is generally considered conservative, without understanding how conservative, or if it is truly conservative. Examples have been shown when traditional specified single degree of freedom acceleration spectral densities were not conservative compared to the multi-degree of freedom field component response [1, 2]. Generating a complete dynamic description of a component specification for a certain system environment is logistically unfeasible with traditional methods. In many systems, the required number of sensors cannot be mounted at the location

Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia LLC, a wholly owned subsidiary of Honeywell International Inc. for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525. R. Mayes () Structural Dynamics Department, Sandia National Laboratories, Albuquerque, NM, USA e-mail: [email protected] L. Ankers · P. Daborn Structural Dynamics Team, AWE Plc, Aldermaston, UK e-mail: [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2020 Z. Mao (ed.), Model Validation and Uncertainty Quantification, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47638-0_23

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connecting the component to the system for specification of base input. Often there are no sensors mounted on the component or subsystem in traditional field tests. This generates the motivation. Is there a way to obtain a good estimate of the component motion and, thus, to remove the uncertainty that traditionally exists in the system environment for that component motion? A finite element-based method was presented previously [3]. This work demonstrates proof of concept of an experimentbased approach to estimate the true acceleration spectral densities at enough component locations of interest to establish the component’s environment in the system. The approach is somewhat analogous to the SEREP method [4] by O’Callahan et al. Consider the approximation of acceleration motion as     x¨ m Um {p} ¨ (23.1) ≈ x¨ u Uu where x is the vector of physical response, U is a shape matrix of orthogonal basis vectors derived from an experimental laboratory pre-test for a certain frequency band and p is the vector of generalized DoF responses for these basis vectors. Subscript m is for measured accelerations during a system test. Subscript u is for unmeasured system test accelerations at locations where we wish to produce an estimated acceleration spectral density for use in specifications. The U vectors are determined from a laboratory pre-test. Only the subscript m accelerations are recorded in the typical field test. If Eq. (23.1) holds, the upper partition can be used with measured accelerations from the system environment to estimate p with a spatial filter as {p} ¨ = U+ xm } m {¨

(23.2)

where the superscript + represents the pseudo-inverse. The pseudo-inverse mitigates some experimental error when there are more accelerometers than generalized dof. A well-conditioned Um matrix provides more accurate results, which comes from proper placement of the measurement accelerometers. Once p has been extracted from the system environment, the unmeasured dof characterized in the laboratory pre-test, can be estimated using the lower partition of Eq. (23.1). This experiment-based approach was implemented using research hardware known as the Modal Analysis Test Vehicle (MATV) developed at the Atomic Weapons Establishment (AWE). 67 accelerometers were mounted at various locations in MATV. Some of the accelerometers were used as measurement dof, and others were placed at locations of interest and were dubbed the truth accelerometers. The truth accelerometers correspond to the unmeasured u accelerations in eq. (23.1). The proof of concept attempts to simulate the truth measurements, in the form of acceleration spectral densities, for u dof using the laboratory measured U vectors and p derived from the measured m accelerations. The system environment was provided for MATV by suspending it from bungee cords in an acoustic chamber and exciting it with a random pressure loading from an acoustic horn. The Institute of Sound and Vibration Research (ISVR) at University of Southampton provided the environment and facility. The experimental approach described here was an afterthought to a previous work [3]. A multi-shaker modal test had been performed by AWE on the MATV for finite element modal validation. After the finite element-based approach was tried, the authors posed the question, could basis vectors be extracted from the frequency response functions gathered for the modal test for use in predicting responses as described in Eqs. (23.1 and 23.2). This extended abstract provides the overview of the work.

23.2 MATV Hardware, Instrumentation and Testing A cutaway of the MATV solid model as well as a picture of the MATV suspended in the acoustic chamber are given in Fig. 23.1. The MATV hardware is about a meter long and weighs about 47 kg. It has an external composite conical shell mounted on an aluminum substrate, aluminum large end cover plate, aluminum internal flat component plate, a steel pipe bolted to the internal flat component plate (representing a large internal component) and a bracket called the removable component (RC) bolted to the internal flat component plate (representing a small component). 67 accelerometer channels were available for instrumenting MATV. These were used in a laboratory pre-test, and the same accelerometers were also recorded in the system level acoustic test to provide the random vibration environment of interest. For the acoustic test, there were 14 channels selected as u dof as described in Eq. (23.1). These were the “truth” accelerometers that would normally be unmeasured, u DoF, in a system environment. Here they were measured to provide acceleration spectral densities against which the predicted responses could be compared. Nine u channels came from three triaxial accelerometers mounted at potential locations for a

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Fig. 23.1 MATV solid model cutaway (Left) and acoustic test setup (Right) Fig. 23.2 MATV modal test setup for pre-test

component, one on the cone wall near the large end, and two at locations on the flat component mounting plate. In addition, 4 channels on the RC cross beam were “truth” accelerometer (u) dof. This left 53 candidate accelerometer channels for possible measurement (m) dof. None of the measurement dof were a repeat of any of the unmeasured dof. Based on engineering judgment the m dof were at various locations on the cone, pipe, component plate, cover plate and RC. A multiple input multiple output (MIMO) shaker modal test with three shakers operating simultaneously was conducted to 2000 Hz using burst random excitation. The shaker arrangement for the test is shown in Fig. 23.2.

Fig. 23.3 Estimated (red) and Measured (blue) “Truth” Acceleration Spectral Densities

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23.3 Predictions of Acoustic Test Truth Responses The 2000 Hz frequency bandwidth was divided into 10 equal bands. Each band was analyzed separately. The real part of the FRF matrix and the imaginary part of the FRF matrix from the three reference shakers were stacked side by side. A singular value decomposition of these real numbers was calculated, and the first fifteen orthonormal shape vectors were saved as the U vectors for each band. Before the estimation process, an algorithm was developed to choose accelerometers to minimize the sum of the condition number for the ten Um matrices. This algorithm reduced the number of m accelerometers to 30 from the initial available 53. Eqs. (23.1) and (23.2) were manipulated to calculate cross spectra in the frequency domain. Fig. 23.3 shows the comparison of the estimated response (red) and the measured acceleration spectral densities (blue) for the 14 “truth” gages.

23.4 Conclusions From previous model validation work [3] it was known that there are at least 70 modes in the 2000 Hz bandwidth. By dividing into bands and using 15 shapes per band, a reasonable estimate of the 14 so called “unmeasured” acoustic test responses was obtained using an optimized set of 30 field measurement gages. Since this approach was an afterthought to the previous work, it is likely that the pre-test could have provided even better basis vectors by attaching more shakers to ensure exciting all possible modes. The advantage of this approach over the finite element approach is that there is great confidence in establishing the basis vectors applicable in each band by the singular value decomposition of the FRF matrix of the asbuilt system. The disadvantage of the experimental-based approach is that only responses instrumented in the pre-test may be predicted, as opposed to the finite element approach, which can at least attempt to predict unmeasured response at any DoF of the finite element model. Notice: This manuscript has been authored by National Technology and Engineering Solutions of Sandia, LLC. under Contract No. DE-NA0003525 with the U.S. Department of Energy/National Nuclear Security Administration. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes.

References 1. Hall, T.M.: Analytically investigating impedance-matching test fixtures. In: Sensors and Instrumentation, Aircraft/Aerospace, Energy Harvesting & Dynamic Environments Testing, Vol. 7, Springer (2019) 2. French, R.M.; Handy, R.; Cooper, H.L.: A comparison of simultaneous and sequential single-axis durability testing. In: Experimental Techniques, pp. 32–37, September/October 2006 (2006) 3. Mayes, R., Ankers, L., Daborn, P.: Predicting system response at unmeasured locations. In: Proceedings of the 37th International Modal Analysis Conference, Orlando, FL, January 2019, paper 4185 (2019) 4. O’Callahan, J., Avitabile, P., Reimer, R.: System Equivalent Reduction Expansion Process (SEREP). In: Proceedings of the 7th International Modal Analysis Conference, Las Vegas, NV, January 1989, pp. 29–37 1989 Randy Mayes has been in the modal testing group at Sandia for 30 years and had some previous experience in finite element modeling.

Chapter 24

Robust Estimation of Truncation-Induced Numerical Uncertainty François Hemez

Abstract Truncation error is ubiquitous in computational sciences, yet, rarely quantified and often ignored altogether. By definition, it is the difference between the exact-but-unknown solution of continuous equations that one wishes to solve, such as conservation laws, and what the computational software (finite elements, finite volumes, etc.) calculates. We contend that the commonly-accepted representation of truncation error as a single-term power-law (i.e., ε( x) = β · xp where x is the level of mesh resolution in the calculation and p is the accuracy of the numerical method) is inadequate and can lead to an erroneous quantification. The remedy proposed is to model this error as a series expansion of integer-valued powers (i.e., ε( x) = β 1 · x + β 2 · x2 + . . . + β N · xN where the expansion order N is unknown and potentially infinite). This representation is consistent with the theoretical form of truncation error derived from Modified Equation Analysis. Because N and the regression coefficients β k are not known, we further propose to use an info-gap model to numerically derive bounds of truncation error. These bounds, yExact − y( x) ≤ U( x), would express the worst-case error between what is calculated at resolution x and what is exact but unknown. Reporting such bounds is essential to assess the quality of a numerical simulation, much like an experimental uncertainty should accompany a measurement. The discussion proposed is, for the most part, conceptual and future efforts will focus on the numerical implementation of these ideas. Keywords Mesh refinement · Truncation error · Solution uncertainty · Robust bounds

24.1 Introduction Computational science and engineering codes implement models and algorithms to solve equations-of-motion relevant to a wide range of applications. Examples are finite element solvers of the mass, momentum and energy conservation laws for structural mechanics; and fluid dynamics solvers of the Navier-Stokes equations [1–3]. These numerical solutions are approximations that can deviate from the exact solutions of the continuous equations by unknown amounts. The difference between the exact solution, notionally referred to as yExact , and the solution provided by the computational software is, by definition, the numerical error written as yExact − y( x). The numerical solution is herein labelled y( x) to emphasize its dependence on discretization variables x which represent spatial discretization sizes, time steps, model orders, thresholds for iterative convergence, etc. This work discusses the quantification of uncertainty that originates from truncation effects in the numerical solution. Truncation originates from approximating and solving the continuous equations-of-motion or conservation laws on a spatial grid (or mesh) [4]. Figure 24.1 is a notional illustration where the dashed blue curve denotes the exact-but-unknown solution yExact and the solid red lines indicate a piecewise-constant approximation y( x) obtained with spatial resolution x. Truncation error is the difference between the exact and approximate solutions. It is defined either point-wise or in the sense of a global norm · consistent with the formulation of the numerical solver (for example, the H1 and H2 norms are commonly-encountered choices for the finite element method [5, 6]). Although many types of numerical error can pollute a calculation (time integration error, lack of convergence of a nonlinear solver, round-off, etc.) [7], it is assumed that they remain negligible compared to truncation effects. This assumption is reasonable because, first, these other errors can usually be controlled more easily than truncation effects and, second, F. Hemez () Lawrence Livermore National Laboratory, Livermore, CA, USA Design Physics (DP) Division, Livermore, CA, USA e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2020 Z. Mao (ed.), Model Validation and Uncertainty Quantification, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47638-0_24

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Approximate Solution, Exact-but-unknown Solution,

Fig. 24.1 Truncation error is the difference between the exact solution yExact (dashed blue) and the approximation y( x) (solid red) provided by the computational software. The figure suggests a numerical solver that generates a piecewise-constant solution within each interval (n − 1) · x ≤ x ≤ n · x

Solution Error, log10(|| y Exact ─ y(D x)||)

0.5 0.4 0.3 0.2 0.1 0 –0.1 p=1

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Mesh Size, log10(D x) Fig. 24.2 Fitting a linear model, log10 (yExact − y( x)) = log10 (β) + p · log10 ( x) (dashed red line), to three solution errors (blue cross symbols) for the simulation of a shock tube problem provides a rate-of-convergence of p = 1.03. IThis result verifies that the numerical solver is first-order accurate

many of these errors are “slaved” to a choice of mesh size. Explicit time integration solvers, for example, enforce a stability condition to select an appropriate time step depending on the level of spatial resolution. We focus on truncation because it often dominates the overall production of numerical error in the solution. Truncation error is usually studied by postulating that it behaves as a power-law function of the resolution used in the calculation, that is, yExact − y( x) = β · xp [4, 7, 8]. A rate-of-convergence of p = 2, for example, should be observed when refining a finite element mesh that uses quadratic shape functions to solve the equations-of-motion. Figure 24.2 depicts the results of mesh refinement applied to fluid dynamics simulations of a shock tube problem [9, 10]. The numerical solver is a second-order accurate finite volume method. The theory states that the accuracy should deteriorate to first-order when the solver approximates discontinuous solutions such as those of the transition of density, pressure and velocity states across shock or contact boundaries. The figure shows the log-values of solution error, log10 (yExact − y( x)), on the vertical axis as a function of log10 ( x) on the horizontal axis. The slope of the dashed red line, therefore, suggests the accuracy since this representation is simply a linear equation. It can be observed that the solution errors (blue crosses) follow this overall trend and an accuracy of p = 1.03. While representing truncation error as yExact − y( x) = β · xp is well-accepted in the scientific community, and usually appropriate (as anecdotally evidenced in Fig. 24.2), it ignores the fact that this error actually behaves as a potentially infinite series of integer powers:

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Exact − y (Δx) = β1 · Δx + β2 · Δx 2 + β3 · Δx 3 + β4 · Δx 4 + · · · = βk · Δx k .

y

(24.1)

1≤k≤N

The series expansion in Eq. (24.1) comes from Modified Equation Analysis [11, 12], an analytical procedure that derives the continuous equations solved by the numerical method for a given resolution, x. Modified equations differ from the equations-of-motion or conservation laws that one wishes to solve. These differences depend on the aggregate behavior of terms β k · xk in Eq. (24.1). A generic result is that the powers of x are integer-valued (i.e., k = 1, 2, . . . N) and the coefficients of the expansion, β k , are partial derivatives of the exact solution: βk =

∂ m y Exact , ∂x m

(24.2)

where the order of the derivative, m, does not necessarily equate the index of the coefficient (i.e., k = m). While the contributions of higher-order terms β k · xk to the truncation error can rapidly become negligible compared to that of the leading-order term, it nevertheless remains that reducing Eq. (24.1) to a single term, as it is commonly practiced, is an over-simplification that could be severely misleading. In fact, analyzing a mesh refinement study often results in non-integer convergence rates (for example, p = 1.75) when the theory in Eq. (24.1) states that truncation error behaves as a superposition of polynomial terms. Considering a large number of terms (up to, say, N = 5) rapidly becomes impractical from a computational pointof-view since it would require the execution of, at a minimum, (N + 1) mesh resolutions to best-fit the unknowns (yExact ; β 1 ; β 2 ; . . . β N ) of Eq. (24.1). Another issue is that the order itself, N, is unknown. This publication proposes a theoretic discussion of what it means to perform a mesh refinement study when the behavior of truncation error is represented by a potentially inappropriate formula. We contend that it is more constructive to displace the conversation from truncation error to truncation-induced uncertainty. The concept of error, defined in the Merriam-Webster dictionnary as “the difference between an observed or calculated value and a true value,” breaks down when “truth” (here, the exact solution yExact of the continuous equations) is unknown. This challenge is further aggravated by postulating an incorrect model-form, yExact − y( x) = β · xp , to express how truncation error behaves. Given these fundamental sources of ignorance, what seems to make more sense is to formulate a bound of uncertainty, which can be written as:





Exact − y (Δx) ≤ U (Δx) ,

y

(24.3)

and that expresses the extent to which a computed solution can deviate from the exact-but-unknown solution. Quantifying and reporting these bounds is essential to assess the quality of numerical simulations, much like estimates of experimental uncertainty or variability should always accompany physical measurements. We propose to analyze truncation-induced uncertainty using a truncated series expansion, such as shown in Eq. (24.1), where the exact solution yExact , regression coefficients (β 1 ; β 2 ; . . . β N ) and order-of-expansion, N, are unknown. The goal is to estimate bounds of uncertainty for specific levels of mesh resolution, i.e., U( x) in Eq. (24.3). Deriving these bounds in the general case, that is, while assuming as-little-as-possible about the types of equations solved and properties of numerical methods used, is an area of open research in the scientific community. We propose no solution. One promising path-forward, nevertheless, is to adopt the formalism of info-gap robustness [13, 14] to represent and propagate the aforementioned sources of uncertainty in the quantification of truncation effects. The discussion is, for the most part, conceptual and future efforts will focus on the implementation of these ideas.

24.2 The Truncation Error Actually Behaves According to the Modified Equation of the Numerical Method When a computational software is employed to solve equations-of-motion or conservation laws, one generally deals with, not one, but two equations. The first one is the continuous equation, such as Eq. (24.4) below, whose exact solution yExact is sought. The second one is the modified equation that represents the actual equation solved by the numerical method and whose solution is herein labelled yˆ Exact . These two equations are different as long as the level of resolution used by the numerical solver is finite (i.e., x = 0). P. Lax’s equivalence theorem [15] defines the conditions for consistency between

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these two equations and convergence of their solutions (i.e., yˆ Exact → y Exact as x → 0). Recognizing the difference between the original and modified equations is essential since it is where truncation error comes from. Numerical methods, such as the finite element and finite volume methods, solve equations-of-motion and conservation laws [1] that, without loss of generality, can be notionally written in one-dimensional geometry as:   ∂F y Exact ∂y Exact + = S, ∂t ∂x

(24.4)

where F(·) denotes the flux operator and S is the source or forcing function that drives the dynamics of the system. The continuity equation (or conservation of mass), for example, uses F(ρ) = u · ∂ρ/∂x and S = 0 where u is the flow velocity and ρ is the mass density. For momentum conservation (or Newton’s law), the flux is F(u) = u · ∇ u. When a numerical method is implemented to solve Eq. (24.4), one seeks the best-possible approximation of the continuous, exact-but-unknown solution, yExact . The approximation generally involves the spatial and temporal discretization of Eq. (24.4) [16]. For simplicity, this discretization is identified by the symbol x. The numerical solution, therefore, depends on x. A fact that is often left unrecognized is that the numerical solution y( x) does not explicitly approximate the exact solution yExact . It estimates, instead, the solution yˆ Exact of the modified equation that can generically be written as:   ∂F yˆ Exact ∂ yˆ Exact + =S+ ∂t ∂x

∂ yˆ Exact ∂ 2 yˆ Exact ∂ 2 yˆ Exact · Δt + · Δx 2 + . . . · ∂t ∂t 2 ∂x 2

! ,

(24.5)

It is emphasized that Eq. (24.5) is notional. The form of a modified equation actually depends on properties of the original Eq. (24.4) and those of the numerical method implemented for its resolution. The main point emphasized here is that the solutions yˆ Exact and yExact remain different as long as the resolution ( x, t, etc.) is finite. The terms between parentheses in the right-hand side of the modified Eq. (24.5) represent a potentially infinite series expansion that characterizes the truncation error introduced by the numerical method. It is what explains the difference between the continuous solutions yˆ Exact and yExact , which translates into differences between y( x) and yExact . These terms also indicate the accuracy of the numerical method. In Eq. (24.5), the leading-order term of spatial truncation, for example, is shown proportional to x2 , which means that this particular solver would be second-order accurate. Refining the mesh should yield a truncation error between y( x) and yExact that converges at a quadratic rate. Likewise, the solver is shown to be first-order accurate in time since the leading-order term of temporal truncation is proportional to t. The technique known as Modified Equation Analysis (MEA) [11] can be used to analytically derive the form of Eq. (24.5) for a particular numerical method. Time-integrating, for example, the linear harmonic oscillator equation with the Euler algorithm produces a modified equation that is different than the one obtained when the equation is solved with a central difference approximation. The derivations of MEA, however, are possible only for simple equations (e.g., linear oscillator, Euler equations in one dimension, advection and diffusion in one dimension, etc.). Applying this methodology is generally out-of-reach, even with the help of symbolic solver software, which leaves analysts to postulate the form of truncation error using a representation such as Eq. (24.1). This brief summary justifies our claim that, while it is broadly practiced, using a single-term power-law such as yExact − y( x) = β · xp to represent how truncation error behaves might not be appropriate in most situations.

24.3 Numerical Predictions Should Always be Accompanied by Their Bounds of Truncation-Induced Uncertainty Not knowing the mathematical form of truncation effects is further aggravated by the fact that the exact solution yExact of the continuous equations is usually unknown for real-world applications. The difference between what is computed with finite resolution and this unknown “truth” should, therefore, be thought of as an uncertainty. Consequently, it makes sense to derive lower and upper bounds of this uncertainty, which Eq. (24.3) suggests. An upper bound of truncation-induced uncertainty is proposed in Reference [7] for scalar-valued quantities-of-interest, that is, y (Δx) ∈ R. These quantities can be, for example, the resonant frequencies of a structure, average stress or temperature over a region of the computational mesh, coefficients of lift and drag of an airfoil, or the maximum pressure over time at a given location. For scalars, the bound of truncation-induced uncertainty is written as |yExact − y( x)| ≤ U( x) and indicates how a numerical solution provided by the computational software at resolution x (which is y( x)) might differ

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from the unknown “truth” (which is yExact ). References [7, 17] were amongst the first ones to derive this formalism and apply it to general-purpose fluid dynamics codes. In References [18, 19], the Minkowski inequality is used to generalize this approach and arrive at similar bounds free of the empirical factors introduced by earlier contributors. These results are briefly summarized to illustrate the discussion. The power of these results is that no assumption is formulated regarding the types of equations solved or properties of the numerical solvers. It means that the upper bound U( x) can be applied to a mesh refinement study irrespective of the types of equations solved. For completeness, it is also noted that few theories are available, if any, to derive bounds of uncertainty for multi-dimensional fields (i.e., y (Δx) ∈ Rm ) as opposed to scalars. One exception is Reference [20] that proposes a theory based on functional data analysis in an attempt to fill this gap. For a scalar quantity-of-interest, the upper bound of truncation-induced uncertainty is defined as: " " " " Exact − y (Δx)" ≤ U (Δx) , "y

(24.6)

where U( x) is obtained from two simulations performed with different levels of resolution denoted as x (the nominal resolution at which the bound of uncertainty is estimated) and xC = R · x (a coarser level of resolution of the mesh refinement study with, by definition, R > 1): U (Δx) =

|y (Δx) − y (ΔxC )| . R pˆ − 1

(24.7)

In Eq. (24.7), the rate-of-convergence pˆ is estimated using one of the techniques proposed in References [7, 21]. Eq. (24.7) is obtained by applying the Minkowski inequality to the modified equation that combines the (original) continuous equation and truncation terms generated by the numerical method. A proof is given in Reference [18]. A small value of the upper bound U( x), such as 10% or less relative to the computed solution y( x), suggests that predictions of the computational software are converging to the exact solution of the continuous equations. Figure 24.3 is an illustration with four simulations of the one-dimensional Burgers equation [22]. The quantity analyzed is the magnitude of the solution discontinuity that develops when the equation is integrated over a sufficiently long time-period. The figure shows that code predictions (green square symbols) converge as the mesh is refined (i.e., x → 0). The bounds U( x) are one-sided because convergence is monotonic. It is observed that these bounds correctly include the extrapolated solution yˆ that the solutions y( x) converge to when x → 0, as they should. We contend that, instead of estimating the exact-but-unknown solution yExact using an extrapolation yˆ as it is commonly practiced, and fail to quantify the uncertainty of this estimate, it would be more appropriate to report the prediction obtained at a given level of resolution as a double-sided interval y( x) ± U( x) or, if convergence is monotonic as shown in Fig. 24.3, as a single-sided interval [y( x); y( x) + U( x)] or [y( x) − U( x); y( x)]. Quantifying this uncertainty matters greatly, for example, for comparisons to physical observations. Just like a measurement should always be understood in the context of its overall experimental variability, an estimation of truncation-induced uncertainty should always accompany a code prediction.

24.4 The Proposed Path-Forward Is to Render Solution Uncertainty Robust to the Commonly-Practiced Assumptions One advantage of recognizing that truncation error behaves as a polynomial, such as Eq. (24.1), as opposed to the commonlyemployed formula yExact − y( x) = β · xp , is that the former is ameanable to linear algebra. The discussion starts by overviewing how this can be exploited to provide a solution for Eq. (24.1). Next, a path-forward is proposed to remedy the aforementioned short-comings of truncation error estimation. Our proposal is to arrive at robust bounds of truncation-induced uncertainty. A robust bound is one that guarantees to encompass the exact-but-unknown solution yExact sought, even if some of the assumptions made to estimate it are incorrect.

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0 .1 8

Density Jump, log10([y]S) (gm.cm–3)

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Fig. 24.3 The vertical red intervals are ranges y( x) ≤ yExact ≤ y( x) + U( x) of truncation-induced uncertainty for numerical solutions of the one-dimensional Burgers equation. The quantity analyzed, y( x), is the amplitude of the solution discontinuity. The horizontal black line indicates the extrapolated solution yˆ that the numerical solutions are converging to as x → 0

24.4.1 A Description of how Truncation Error Behaves Can Be Obtained through Simple Linear Algebra When Eq. (24.1) is specialized to scalars (i.e., a resonant frequency, average pressure, peak stress, etc.), the representation of truncation error originating from MEA can be written in a generic manner as: yˆ − y (Δx) = β1 · Δx + β2 · Δx 2 + . . . βN · Δx N ,

(24.8)

where yˆ is a to-be-determined estimate of the exact-but-unknown solution yExact . The potentially infinite series expansion is, here, truncated to the first N terms, an assumption that" is duly noted " and addressed below. Equation (24.8) further assumes monotonic convergence to convert the absolute value "yˆ − y (Δx)" in the left-hand side into a simple difference, which is not a significant loss of generality and which is adopted for simplicity. The equation can be written compactly as: ψ(Δx)T · β = −y (Δx) ,

(24.9)

where ψ( x) and β are single-column vectors that respectively collect the (known) basis functions and (unknown) extrapolated solution and regression coefficients: ⎛ ⎜ ⎜ ψ (Δx) = ⎜ ⎝

−1 Δx .. . Δx N

⎞

⎛

yˆ ⎜ β1 ⎟ ⎜ ⎟ ⎟, β = ⎜ . ⎝ .. ⎠

⎞ ⎟ ⎟ ⎟. ⎠

(24.10)

βN

Equation (24.9) is an easily-resolved linear system of equations. Assuming that m runs of the computational software are available at successively finer levels of resolution x1 , x2 . . . xm , the m instantiations of Eq. (24.9) can be collected in matrix form and written as  T · β = − Y, and where:

24 Robust Estimation of Truncation-Induced Numerical Uncertainty

⎡

ψ(Δx1 )T ⎢ ψ(Δx2 )T ⎢ ⎢ .. ⎣ . &

⎤

⎛

⎜ ⎥ ⎜ ⎥ ⎥ · β = −⎜ ⎝ ⎦

ψ(Δxm )T '( ) ΨT

229

&

⎞ y (Δx1 ) y (Δx2 ) ⎟ ⎟ ⎟. .. ⎠ .

(24.11)

y (Δxm ) '( ) Y

Many procedures are available to solve Eq. (24.11). If the system is over-determined, that is, m ≥ (N + 1), the least-squares solution can be computed as β = − ( ·  T )−1 · ( · Y). In the case of an under-determined or rank-deficient system, one can resort to a pseudo-inverse obtained, for example, from singular value decomposition [23]. Numerical optimization also provides versatile methods capable of reaching a solution that depends on how the problem is formulated (objective function, constraints, type of search algorithm, etc.) for implementation.

24.4.2 The Bounds of Truncation-Induced Uncertainty Can Be Made Robust to Assumptions Present in the Analysis The procedure described in the previous section to arrive at an equation, such as Eq. (24.8), that describes the behavior of truncation error for a given calculation is plagued with, sometimes difficult to justify, assumptions. The same observation is true for the state-of-the practice, represented by Eqs. (24.6 and 24.7), to estimate bounds of truncation-induced uncertainty, which is our ultimate objective. These assumptions are briefly summarized next. First and foremost, the mathematical representations assumed for the behavior of truncation error, and those formulated to derive the bounds U( x), might be incorrect. In the case of Eq. (24.8), the order N of the expansion is generally unknown " " and can rarely be guided by analytical derivations such as MEA. For the commonly adopted representation "yˆ − y (Δx)" = β · Δx p , the values obtained for the triplet of unknowns y; ˆ β; p are questionable since this model-form is inconsistent with the theoretic behavior of truncation error in Eq. (24.5) or Eq. (24.8). Another issue is that the regression coefficients β k of these equations are usually considered to be constant when, in fact, they are defined from partial derivatives of the exact solution yExact , such as suggested in Eq. (24.2) and Eq. (24.5). Their actual values could deviate significantly from the baseline values obtained by solving Eq. (24.11). Consider, for example, a solution yExact that represents a flow velocity in fluid dynamics. Spatial and temporal derivatives of this velocity are likely to be small in regions of the computational domain where the flow is regular and laminar, hence, yielding small magnitudes for the coefficients β k . Around an obstacle or airfoil, however, large gradients and even shock discontinuities might develop, which would increase the magnitudes of β k . Similar comments can be made for a solution yExact that would represent, for example, a stress field in solid mechanics. Large gradients might develop in a region of stress concentration or at the tip of a fracture, while they would be smaller elsewhere. Assuming constant-valued coefficients β k , therefore, might not be inappropriate in all cases. Finally, it is usually challenging to analyze a sufficient number of mesh resolutions. For example, halving a threedimensional mesh up to four times (i.e., running the problem with a resolution of x/16 instead of x) increases the number of elements, roughly speaking, by a factor of 163 = 4096. Assuming that the cost of the numerical method is proportional to the square of the number of unknowns (or number of elements), which is typical of implicit time integration or nonlinear solvers, the x/16-mesh-size solution would come at a cost more than 16 Million times that of the x-mesh-size solution. It grows to over a Billion times more expensive with five levels of refinement. In practice, therefore, the computational cost of mesh refinement seriously limits the number of discretization levels that can reasonably be analyzed. Not analyzing a sufficient number of refinement levels might deteriorate the quality of the extrapolated solution yˆ and regression coefficients β k . The least-squares solution to Eq. (24.11), for example, requires a full-rank matrix, which necessitates that m ≥ (N + 1), hence, limiting the value of N to two or three. It is likely that the values of yˆ and β k best-fitted with, say, three or four computational meshes would somewhat vary if one additional level of mesh resolution could be analyzed. For these reasons, it is reasonable to consider that the baseline solution of Eq. (24.11), denoted by β 0 for simplicity, might deviate from the “truth” that would be obtained if MEA could be analytically carried out. The implication that we are concerned about is that the corresponding bounds U( x) of truncation-induced uncertainty might not be appropriate (i.e., they might not bracket the exact-but-unknown solution yExact , as they should). The proposed path-forward is to produce bounds U( x) that are robust to the potential inappropriateness of these assumptions and the variations of β 0 -values that might result. Achieving robust bounds means that yExact would be guaranteed

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to be within y( x) ± U( x) even if some of the aforementioned assumptions are incorrect. While this topic is open research to a great extent and no solution is proposed herein, info-gap robustness offers an attractive theoretic [13] and computational [14] framework. It is briefly summarized below. An info-gap model can represent the uncertainty in the nominal solution β 0 of Eq. (24.11) as: S (β0 ; α) = {β such that β − β0  ≤ α} .

(24.12)

Equation (24.12) means that the values of the extrapolated solution and regression coefficients stored in β are allowed to deviate from the nominal values β 0 by up to a given amount α. This distance is defined in the sense of an appropriate norm ·. While this distance is written in the simplest possible manner for clarity, it can nevertheless accommodate a broad diversity of cases, such as adding information about the correlation between coefficients β k . Likewise, the order-of-expansion N can be included in the definition of S(β 0 ; α) to account for the fact that the number of terms of Eq. (24.8) is usually not known with absolute certainty. Robustness can be formulated by searching for the maximum-possible uncertainty in the extrapolated solution yˆ and  ∼ regression coefficients β k , which is denoted by α εF itting in Eq. (24.13) below, such that Eq. (24.8) is fitted with no more than a given level of error, εFitting . This condition can be expressed mathematically as: " " ⎞ ⎛ " "  " "   ∼ α εF itting = max ⎝ max ""yˆ − y (Δxk ) − βk · Δx k "" ≤ εF itting for all β ∈ S (β0 ; α)⎠ . {α>0} {1≤k≤m} " " 1≤k≤N

(24.13)

Finding the solution of Eq. (24.13) involves solving a set of nested optimization problems as explained in Reference [14] for an application to mechanical design. While doing so might require a non-trivial computational resource, this expense would surely remain negligible compared to the mesh refinement needed to compute the solutions y( xk ). The value of  ∼ α εF itting represents the maximum-tolerable uncertainty in yˆ and β k that guarantees that the model of " truncation "error accommodates the computed solutions with no more than εFitting error. The important distinction between "yˆ − y (Δxk )" and εFitting is emphasized. While the former estimates the truncation error of a computed solution y( xk ), the latter represents the goodness-of-fit of the model (i.e., Eq. (24.8)) itself. A user could require, for example, that the analysis be performed with no more than εFitting = 10% fitting error. The formulation proposed in Eq. (24.13) avoids the confusion between these two types of error.  ∼ The relationship between εFitting and α εF itting represents the robustness function of the truncation error model, which can always be constructed numerically. In some cases, a closed-form solution for the robustness function can be derived analytically [13]. It depends, for example, on the definition of the innermost optimization in Eq. (24.13). A direction of  ∼ future research is to explore the feasibility of obtaining a closed-form solution for α εF itting , which would be greatly beneficial from a computational point-of-view. The next and final step is to obtain robust bounds U( x) of truncation-induced uncertainty. The definition given in Eqs. (24.6 and 24.7) is conditioned on representing truncation as |yExact − y( x)| = β · xp . An avenue of research is to investigate if another closed-form formula for U( x) can be rigorously derived from a representation such as Eq. (24.8) instead. One area of concern is to define an appropriate rate-of-convergence for pˆ in Eq. (24.7) since it does not have an equivalent in Eq. (24.8) where the powers of x are integer-valued. In any case, more work is needed to define U( x) consistently with  ∼ Eq. (24.8). Robust bounds can then be obtained by exploring the uncertainty space S β0 ; α εF itting to search for the  ∼ minimum and maximum values of U( x) for all variables y, ˆ βk ∈ S β0 ; α εF itting . This last step would likely involve more numerical " It would, however, be facilitated by the fact that executing the representations of truncation " optimization. error, such as "yˆ − y (Δx)" = β · Δx p or Eq. (24.8), is trivial.

24.5 Conclusion Even though it is always present when discretizing ordinary or partial differential equations using a computational mesh, truncation error is rarely quantified and reported with the numerical predictions. While no rigorous experimentalist would present or publish results of a testing campaign without estimating the overall experimental variability of measurements, truncation effects are most often ignored by numerical analysts. This publication offers a discussion to, first, raise awareness

24 Robust Estimation of Truncation-Induced Numerical Uncertainty

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of this issue; second, overview techniques that have been developed to assess how truncation error behaves; and, third, propose changes to address short-comings with how the quantification of truncation error is currently practiced. By definition, truncation refers to the difference between the exact-but-unknown solution of equations-of-motion or conservation laws that one wishes to solve and what the computational software calculates. Its behavior is usually modeled using a single-term power-law equation. We contend that this representation is inconsistent with the theoretic behavior of truncation error that can, in simple cases, be analytically derived through Modified Equation Analysis. Adopting an incorrect model-form to represent truncation effects can lead to erroneous estimates of the rate at which the numerical method converges its solutions when the mesh is refined. This, in turn, deteriorates the estimation of numerical uncertainty. Our proposal to advance the state-of-practice is twofold. We suggest, first, to consider adopting a mathematical representation that is more consistent with the theoretic behavior of truncation error. This would be a series expansion of integer-valued powers of the mesh size, ε( x) = β 1 · x + β 2 · x2 + . . . + β N · xN , where ε( x) is the truncation error at mesh size x. We show how this representation is ameanable to linear algebra, which is simpler to implement than resorting to numerical optimization when a power-law of the form ε( x) = β · xp is (incorrectly) postulated. The second part of our proposal is to rely on the previously obtained representation to arrive at bounds of truncationinduced uncertainty, yExact − y( x) ≤ U( x), to bracket the difference between the exact-but-unknown solution of the continuous equations solved and what the computational software calculates at mesh resolution x. While techniques are available to estimate these bounds, we seek to make them robust to potentially incorrect assumptions. Robust bounds means that the difference between the calculation y( x) and unknown “truth” yExact is guaranteed not to be surpassed even if some of the assumptions present in the analysis are incorrect. These assumptions include the ability to analyze a sufficient number of mesh resolutions, adopting a suitable expansion order, N; and relying on an appropriate mathematical model to estimate the bounds U( x). Info-gap robustness is proposed to represent these sources of lack-ofknowledge and propagate them to the bounds U( x) of truncation-induced uncertainty. Our discussion is mostly conceptual and future efforts will focus on the implementation of these ideas.

References 1. Lax, P.D., Wendroff, B.: Systems of conservation laws. Commun. Pure Appl. Math. 13(2), 217–237 (1960) 2. Leveque, R.J.: Finite Volume Methods for Hyperbolic Problems Cambridge texts in applied mathematics. Cambridge University Press, New York (2002) 3. Grinstein, F.F., Margolin, L.G., Rider, W.J.: Implicit Large Eddy Simulation. Cambridge University Press, New York (2007) 4. Rider, W., Witkowski, W., Kamm, J.R., Wildey, T.: Robust verification analysis. J. Comput. Phys.,. Elsevier. 307, 146–163 (2016) 5. Ciarlet, P.G.: Basic error estimates for elliptic problems. In: Finite Element Methods: Handbook of Numerical Analysis. North-Holland, New York (1991) 6. Babuška, I.: Error-bounds for the finite element method. Numer. Math.,. Springer-Verlag, Berlin, Germany. 16, 322–333 (1971) 7. Roache, P.J.: Verification and Validation in Computational Science and Engineering. Hermosa Publishers, Albuquerque (1998) 8. Roy, C.J.: Review of code and solution verification procedures for computational simulations. J. Comput. Phys.,. Elsevier. 205, 131–156 (2005) 9. Sod, G.A.: A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. J. Comput. Phys.,. Elsevier. 27, 1–31 (1978) 10. Hemez, F.M., Brock, J.S., Kamm, J.R.: Nonlinear error Ansatz models in space and time for solution verification. In: 47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference. American Institute of Aeronautics and Astronautics, Newport, RI (May 2006) 11. Warming, R.F., Hyett, B.J.: The modified equation approach to the stability and accuracy analysis of finite difference methods. J. Comput. Phys.,. Elsevier. 14, 159–179 (1974) 12. Hirt, C.W.: Heuristic stability theory for finite difference equations. J. Comput. Phys.,. Elsevier. 2, 339–355 (1968) 13. Ben-Haim, Y.: Info-Gap Decision Theory: Decisions under Severe Uncertainty, 2nd edn. Academic Press Publisher, Oxford (2006) 14. Hemez, F.M., Van Buren, K.L.: Info-gap robust design of a mechanical latch. In: Marchau, V., Walker, W., Bloemen, P., Popper, S. (eds.) Decision-Making under Deep Uncertainty: from Theory to Practice, pp. 201–222. Springer, Cham (2019) 15. Lax, P.D., Richtmyer, R.D.: Survey of the stability of linear finite difference equations. Commun. Pure Appl. Math. 9(2), 267–293 (1956) 16. Leveque, R.J.: Numerical Methods for Conservation Laws. Birkhauser-Verlag Publishers, Bassel (1990) 17. Roache, P.J.: Perspective: a method for uniform reporting of grid refinement studies. ASME J. Fluids. Eng. 116, 405–413 (1994) 18. Mollineaux, M.G., Van Buren, K.L., Hemez, F.M., Atamturktur, S.: Simulating the dynamics of wind turbine blades, part 1: model development and verification. Wind Energy. 16, 694–710 (2013) 19. Van Buren, K.L., Mollineaux, M.G., Hemez, F.M., Atamturktur, S.: Simulating the dynamics of wind turbine blades, part 2: model validation and uncertainty quantification. Wind Energy. 16, 741–758 (2013) 20. Hemez, F.M., Marcilhac, M.: A fresh look at mesh refinement in computational physics and engineering. In: 49th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference. American Institute of Aeronautics and Astronautics, Schaumburg (April 2008) 21. Hemez, F.M., Kamm, J.R.: A brief overview of the state-of-the-practice and current challenges of solution verification. In: Graziani, F. (ed.) Computational Methods in Transport: Verification and Validation, pp. 229–250. Springer, Berlin (2008)

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22. Burgers, J.M.: A mathematical model illustrating the theory of turbulence. Adv. Appl. Mech.,. Elsevier. 1, 171–199 (1948) 23. Golub, G.H., Kahan, W.: Calculating the singular values and pseudo-inverse of a matrix. J. Numer. Anal.,. Society for Industrial and Applied Mathematics. 2, 205–224 (1965) François Hemez is a scientist at the Lawrence Livermore National Laboratory, where he supports global security programs of the U.S. Department of Energy and other governement agencies. His research interests revolve around large-scale numerical simulations, data science, uncertainty quantification and model validation.

Chapter 25

Fatigue Crack Growth Diagnosis and Prognosis for Damage-Adaptive Operation of Mechanical Systems Pranav M. Karve, Yulin Guo, Berkcan Kapusuzoglu, Sankaran Mahadevan, and Mulugeta A. Haile

Abstract The digital twin paradigm that aims to integrate the information obtained from sensor data, physics models, operational data and inspection/maintenance/repair history of the system or component of interest, can potentially be used to optimize operational parameters that achieve a desired performance or reliability goal. In this paper, we discuss such a methodology for intelligent operation planning in mechanical systems. The proposed approach discusses two key components of the problem: damage diagnosis and damage prognosis. We consider the problem of diagnosis and prognosis of fatigue crack growth in a metal component, as an example. We discuss a probabilistic, Lamb-wave-scattering-based crack diagnosis framework that incorporates both aleatory and epistemic uncertainties in the diagnosis process. We build a Bayesian network for the Lamb-wave pitch-catch NDE using a low-fidelity physics-based model of the same. We perform global sensitivity analysis to quantify the contribution of various parameters to the variance of the damage-sensitive output signal feature(s) using this model. We retain the parameters with higher contribution, and build a medium-fidelity, oneway coupled, multi-physics model to simulate the piezoelectric effect and Lamb wave propagation. We perform Bayesian diagnosis of crack growth using the medium-fidelity model, considering data corrupted by measurement noise, and fuse the information from multiple sensors. We build a finite-element-based high-fidelity model for crack growth under uniaxial cyclic loading, and calibrate a phenomenological (low-fidelity) fatigue crack growth model using the high-fidelity model output. We use the resulting multi-fidelity model in a probabilistic crack growth prognosis framework; thus achieving both accuracy and computational efficiency. We integrate the probabilistic diagnosis and prognosis engines to estimate the damage state using both sensor data as well as model prediction. Keywords Information fusion · Lamb-wave pitch catch · Fatigue crack growth · Digital twin

25.1 Extended Abstract Mechanical systems need to cope with degradation and failures of the physical components due to aging, operational stress, environmental conditions, etc. Strategies for extending the maintenance-free operation window, thus increasing the resilience of the system or components become important. These include reconfiguration of the system, such as changing the operational regime to reduce or redistribute the stress and thus slowing down the damage progression. The optimum choice of operational regime depends both on the health state and its diagnosis, and the prediction of how the damage will grow given the task to be performed. The digital twin paradigm [1, 2] is well-suited for achieving resilience-enhancing system operations. In this work, we discuss two key aspects of this problem: damage diagnosis and damage prognosis. We test the proposed methodology by conducting numerical and laboratory experiments on an Aluminum 7075 plate with seeded damage. The plate is subjected to uniaxial cyclic loading that represents a series of tasks, where each task consists of cyclic loading with three different minimum and maximum amplitudes (or three sub-tasks). The damage diagnosis method estimates the current crack length given the results of ultrasonic pith-catch tests. The damage prognosis methodology estimates the future crack

P. M. Karve · Y. Guo · B. Kapusuzoglu · S. Mahadevan () Vanderbilt University, Nashville, TN, USA e-mail: [email protected]; [email protected]; [email protected]; [email protected] M. A. Haile U.S. Army Research Laboratory, Aberdeen, MD, USA e-mail: [email protected] © The Society for Experimental Mechanics, Inc. 2020 Z. Mao (ed.), Model Validation and Uncertainty Quantification, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47638-0_25

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length using a fatigue crack growth model and the loading pattern. The salient features of these methodologies are briefly described below. We employ Lamb wave scattering for performing probabilistic damage diagnosis. This has been shown to be a useful method for monitoring fatigue-induced crack growth in thin metallic structures (e.g., aircraft skin, metal pipes, etc.) [3– 5]. The damage detection methodology uses a pitch-catch scheme. Typically, the diagnostic algorithm compares the signal detected in the (damaged) structure with the signal detected in the damage-free state to infer the presence and the severity of damage. A relationship between the selected damage sensitive feature (ydata ) of the sensed signal and the severity of the damage (crack length, Lc ) can be obtained by performing baseline experiments and/or numerical modeling of the governing wave physics. Here, we use a probabilistic, Lamb-wave-scattering-based crack diagnosis framework for an actuator-sensor network (shown in Fig. 25.1a). We first perform global sensitivity analysis using a low-fidelity Lamb-wave pitch catch model [6] to compute Sobol’ indices [7]. The sensitivity analysis helps ascertain the relative importance of various parameters. We retain the important parameters and generate training data using a high-fidelity finite element model of Lamb-wave pitch catch. Using the training data, we construct a Gaussian process (GP) surrogate model that predicts the ydata for each actuator-sensor path given the crack length. We perform Bayesian diagnosis of crack growth using the low-fidelity model, considering data accompanied by measurement noise, and fuse the information from multiple actuator-sensor paths. In a Bayesian context, we express the uncertainty in our knowledge of the crack length by means of a probability distribution function. We assume a prior distribution (pprior ) of the crack length based on intuition, experience, etc.; and update the knowledge using the data by computing the likelihood function (plikelihood ) as pposterior (Lc |ydata ) ∝ plikelihood (ydata |Lc ) ∗ pprior (Lc ) .

(a)

(b) 0.3 Prior A2S2 A2S3 A2S1 A3S2 A1S2 A3S1 A1S3

atrue = 4.9

AL 7075 plate 0.25

actuators

A3

A2

15 in ~= 0.38 m

crack

A1

D ~= 1.8mm

Crack Length S3

S2

S1

sensors

Probability density function

25.4 mm 0.2

0.15

0.1

0.05

y x 0 6 in ~= 0.15 m

0

5

15 10 Crack length (Lc) [mm]

20

Fig. 25.1 Probabilistic diagnosis of fatigue crack growth (a) AL 7075 plate and actuator-sensor network, (b) Bayesian information fusion for fatigue crack diagnosis

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Each actuator-sensor path provides a different value of the damage metric, and hence, some additional information about the state of the damage (crack length) (Fig. 25.1a). If the estimation is performed for a path (say) A2-S2, then the updated posterior for the crack length can be used as a prior for the Bayesian estimation for the next path (say A2-S3). The result at the end of the second Bayesian update is the result of the fusion of information contained in the data obtained from the two actuator-sensor paths. The results of this sequential Bayesian information fusion procedure are shown in Fig. 25.1b. We perform the probabilistic fatigue crack growth prognosis using an empirical fatigue crack growth model, namely, Foreman’s law. Thus, for cyclic loading, the crack growth rate is given by da C( K)m = , dN (1 − R) ∗ Kc − ΔK where C and m are Foreman’s law parameters, K is the change in stress intensity factor due to the cyclic loading, Kc is the fracture toughness and R is the stress ratio. We build (high-fidelity) finite element models to compute the stress intensity factors for different load levels and crack lengths for the aluminum plate shown in Fig. 25.1a. We build a GP surrogate model that estimates the stress intensity factor for a given crack length in the plate. We perform pilot cyclic loading tests to obtain crack growth data for known loading history, and perform Bayesian calibration of the Foreman’s law parameter C using Markov chain Monte Carlo method and the GP surrogate model trained above. The probabilistic damage prognosis can now be performed given a candidate load profile and calibrated Foreman’s law parameters. Figure 25.2 shows the how the methodology can be used to design load profiles (task intensities) for four tasks. Each task starts with the diagnosis of the current state of the damage (crack length), which includes the diagnosis uncertainty. The estimate is fed to a load profile optimizer engine which attains a predefined system performance metric (not discussed here for brevity) while minimizing the damage growth (estimated using the damage prognosis model discussed above). The process is repeated for four tasks (as an example) to demonstrate how the required system performance is achieved while ensuring that the damage growth remains below a predefined threshold (acrit ).

Fig. 25.2 Probabilistic damage diagnosis and prognosis for resilience-enhancing system operations

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References 1. Glaessgen, E., Stargel, D.: The digital twin paradigm for future NASA and US Air Force vehicles. In: 53rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference (2012) 2. Li, C., Mahadevan, S., Ling, Y., Choze, S., Wang, L.: Dynamic Bayesian network for aircraft wing health monitoring digital twin. AIAA J. 55(3), 930–941 (2017) 3. Alleyne, D.N., Cawley, P.: The interaction of Lamb waves with defects. IEEE Trans. Ultrason. Ferroelectr. Freq. Control. 39(3), 381–397 (May 1992) 4. Ihn, J.-B., Chang, F.-K.: Pitch-catch active sensing methods in structural health monitoring for aircraft structures. Struct. Heal. Monit. 7(1), 5–19 (2008) 5. Janapati, V., Kopsaftopoulos, F.P., Li, F., Lee, S.J., Chang, F.-K.: Damage detection sensitivity characterization of acousto-ultrasound-based structural health monitoring techniques. Struct. Heal. Monit. 15(2), 143–161 (2016) 6. di Scalea, F., Salamone, S.: Temperature effects in ultrasonic Lamb wave structural health monitoring systems. J. Acoust. Soc. Am. 124(1), 161–174 (2008) 7. Sobol, I.M.: Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates. Math. Comput. Simul. 55(1), 271– 280 (2001) Sankaran Mahadevanis Professor of Civil and Environmental Engineering at Vanderbilt University, Nashville, Tennessee, where he has served since 1988. His research interests are in reliability and uncertainty analysis methods, material degradation, structural health monitoring, design optimization, and model uncertainty.

Chapter 26

An Evolutionary Approach to Learning Neural Networks for Structural Health Monitoring Tharuka Devendra, Nikolaos Dervilis, Keith Worden, George Tsialiamanis, Elizabeth J. Cross, and Timothy J. Rogers

Abstract For widespread adoption of Structural Health Monitoring (SHM) strategies, a key challenge is to produce tools which automate the creation and learning of effective algorithms with minimal input from expert practitioners. Classification of damage in a structure is one of the key steps in constructing a useful SHM system. The artificial neural network has been shown, for a number of years, to be a powerful tool for building such a classifier. While the learning of the parameters in the network has been widely addressed in the literature, the hyperparameters of the model (i.e. the topology) remain a challenge. This paper investigates the use of an evolutionary algorithm tailored to this learning problem, namely, NeuroEvolution of Augmenting Topologies. The effectiveness of this approach is considered with regard to classification accuracy, user input, and generalisation. A benchmark structure a representative three-storey building is used to demonstrate the use of this methodology. Keywords Structural health monitoring · Neural networks · Topology

26.1 Introduction-SHM Motivation This paper focuses on the implementation of Neural Networks (NN) when are utilised for damage detection/classification in the context of Structural Health Monitoring (SHM). NNs present some issues such as the model complexity which can lead to the search for the optimal topology of a NN. Here some ideas from natural selection and evolution are incorporated to allow different topologies to evolve and establish the optimal solution. Data from an experimental SHM set up was used to demonstrate the effectiveness of the algorithm for classification of damage states. The methodology that is investigated here aims to find the simplest network required for data classification and optimal prediction. The use of neural networks introduces added complexity and configuration with many parameters to be tuned and an incorrect setup of the NN model could lead to inaccurate predictions that would directly affect the health assessment of the structure by failing to generalise beyond the training data. This paper explores the use of a tool built on natural selection and evolutionary theories for topology learning.

26.2 Neural Networks (NN) in SHM Briefly, NN is a type of machine learning (ML) algorithm which aims to uncover a pattern that exists within data [1]. For example, in SHM a NN can use features in data collected from a structure (such as mode shapes and natural frequencies [2, 3]) to detect damage [4] or find the location of damage, amongst other applications. The heavy use of NNs in SHM is highlighted throughout much of the literature in the field [2, 5, 6]. One of the defining features of a NN is the use of many interconnected, non-linear functions/nodes [7] to learn patterns. The number of these functions and how they are connected, which will be referred to as the topology throughout this paper, are largely dependent on the complexity of the problem and commonly are specified a priori by the user. Bilbao et al. [8]

T. Devendra · N. Dervilis · K. Worden · G. Tsialiamanis · E. J. Cross · T. J. Rogers () Dynamics Research Group, Department of Mechanical Engineering, The University of Sheffield, Sheffield, UK e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]; [email protected] © The Society for Experimental Mechanics, Inc. 2020 Z. Mao (ed.), Model Validation and Uncertainty Quantification, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series, https://doi.org/10.1007/978-3-030-47638-0_26

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address the makeup of NN architecture, stating that the determination of network topology is one that is not very clear and requires empirical knowledge. The makeup of a NN is one described as an ‘arduous task’ [8]. Moody and Bilbao [8, 9] highlight the importance of topology selection in problems with limited data as the lack of data (and hence training data) can lead to a bias/variance (or underfitting/overfitting) trade-off. Here the terms underfitting and overfitting refer to how well a model is able to generalise the pattern within the data. When overfitting occurs, the model focuses too heavily on the pattern within the training data. As a result, it is not able to generalise well when tested on separate data. In contrast, when underfitting occurs the pattern learnt is overly simplistic. This can be especially important in SHM when very complex problems have insufficient amounts of data to fully develop a robust relationship for prediction, a phenomena often referred to as the “curse-of-dimensionality” [1]. The issue of computational complexity is also very important for online SHM applications. Rojas [10] states that the time required for the learning process of a NN can increase exponentially with the size of the network. Although methods do exist for reducing this (for example parallel computing [11] or new hardware [10]) the root of the problem remains with the complexity of the model.

26.3 Neuroevolution of Augmenting Topologies The algorithm proposed in this paper builds strongly upon concepts from Neuroevolution of Augmenting Topologies (NEAT) proposed by Stanley and Miikkulainen [12]. This neuroevolution (NE) method makes use of genetic algorithms (GAs) to optimise both NN topology and weights. NE can be defined as a set of methods that uses evolutionary algorithms to learn NNs [13]. More specifically, evolutionary algorithms use a population based search [14] to define the optimal parameters and topology for the NN. One of the most popular evolutionary algorithms is the genetic algorithm [15]. Genetic algorithms are computational models which are inspired by evolution in nature. The implementation follows closely the idea of natural selection and survival of the fittest [16]. It begins with a population of random chromosomes [12], which, within NE, equate to the parameters and/or hyperparameters of NNs. The population is evaluated with each individual being assigned a fitness value based on a defined fitness function. The fitness function is defined such that it increases monotonically with “improvement” in the quality of the model as defined by the user, e.g. in a regression problem this may be the negative mean-squared-error. The algorithm then implements a number of genetic operations and carries over the highest performing members of the population and the cycle repeats until a solution is found. In this context, the genetic operators would be the changing of topology and connection weights for NNs. Stanley [12] builds on this definition and introduces an algorithm named Neuro-Evolution of Augmenting Topologies (NEAT). This algorithm has been shown to outperform traditional NE methods. It is stated that the increased efficiency of the method was due to the incorporation of ideas such as speciation, crossover of topologies and mutating topology from a minimal architecture. Here, the NEAT algorithm is adopted and combined with backpropagation tools. In doing so, connection weights are now optimised using gradient descent instead of a GA for each proposed topology, the aim being to decrease learning time.

26.4 Topology Optimisation Algorithm As mentioned before, the method in this short work follows closely the outline given in [12], an implementation of which can be found at [17]. However, modifications were made to incorporate backpropagation. A summary of the algorithm used is shown in Fig. 26.1. Its foundation relies on encoding NN topologies in such a way that allows for the flexibility regarding their architecture. NNs are “grown” from the simplest structure possible through the application of mutations. These mutations allow for the addition and removal of connections and nodes, which is how new topologies are explored. Networks are optimised using backpropagation in by implementing each generation. They are then evaluated on their performance of prediction accuracy on the training data. A fitness score is assigned for each network, dependent on its performance, here the negative cross-entropy. Networks are encouraged to score as highly as possible with this metric to allow them to carry over to the next generation and be able to create new structures. The ‘optimal’ solution is found when a defined fitness threshold is met or when the algorithm has run for a set number of generations.

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26.5 Neural Network Encoding The genetic encoding scheme in [12] is used to represent NNs. Here the concept of a genome is introduced. Genomes are designed to easily represent NN connectivity, in a manner that can be manipulated via the GA approach. Within the genomes, individual genes encode information about the network such as connections and weights. Two gene types are specified, connection genes and node genes. Connection genes simply hold information about two node genes between which a connection exists, this includes values for the weight along that connection and if the connection is activated. Node genes keep track of the node type (input, hidden or output) and their associated bias value.

Fig. 26.1 An outline of the steps within the algorithm to converge to a solution

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26.5.1 Population Initialisation Stanley and Miikkulainen [12] have given comprehensive justification as to why it is beneficial to start the entire population with the simplest network possible. The initialisation used in this paper follows this. The simplest network is defined as one that only has connections between each of the input features and the output node(s).

26.5.2 Genome Evaluation and Fitness Function To calculate the fitness of a genome, the prediction error is calculated using the cross entropy loss function (L).   L = −y log yˆ − (1 − y) log 1 − yˆ

(26.1)

where yˆ is defined as the prediction and y as the true label in the classification problem. Once the loss is calculated, the negative value is taken as the fitness (−L). This is done to reward a lower cost and therefore, a higher fitness.

26.5.3 Tracking Innovation of Topology Innovation tracking refers to the tracking of new structural additions to a network. When a new connection is formed, an associated innovation number is attributed to the pair of nodes between which the connection exists. If the connection has occurred before, it simply takes on the innovation number of the first instance of it occurring.

26.5.4 Genetic Operators Mutations occur when the population reproduces to create the next generation. It is through these mutations that the population moves towards the “optimal” solution. Each mutation operation has an associated percentage chance to occur. The weight mutation involves the perturbation of a connection weight by a random value, chosen from a standard Gaussian distribution. The mechanism of a mutation involves the chance to reset the connection weight or simply reset all the connection weights within the genome in order to prevent against local minima in the backpropagation learning. The add connection mutation simply creates a connection between two already existing nodes. The connection is initialised with a random weight. Connections that can also bypass multiple layers can be created. Simply, the addition of a node involves splitting an existing connection by inserting a node in its place. The old connection is disabled and is replaced by two new connections. Removal of nodes and connections are done with the limitation that at least one path from an input feature to the output must exist. This is to ensure forward and backpropagation is possible. Crossover combines two parents genomes to create a new one. The two genomes are lined up and compared using the innovation numbers. Individual genes can then be identified as matching, disjoint or excess, with respect to their innovation number [12]. The child genome is created by inheriting genes depending on how each gene is classified.

26.5.5 Speciation Speciation of the population is a concept introduced to protect innovation in the model [12]. This property is especially important, as often, networks need time to adjust to new nonlinearities introduced by topological structural additions; this can lead to poor initial fitness of the genome. A compatibility function is defined that can be used as a measure for similarity between two genomes: δ=

c2 D c1 E + + c3 W¯ N N

(26.2)

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E and D represent the number of excess and disjoint genes between the genomes, and W¯ represents the average weight difference of matching connection genes [12]. If the compatibility distance between two genomes is within some threshold (set in the configuration) then they are considered to be in the same species. Weights are given different aspects of the compatibility function (c1 , c2 , c3 ), this must be a user choice which requires tuning. Speciation of the population involves two processes, finding species representatives and then finding new members for those corresponding species. New representatives are found by comparing each member of the population to current ones. The individual that is most similar, in terms of compatibility distance, is assigned to be the new species representative. Once each species has found a representative, a species is found for each member of the remaining population. Comparisons are again made between individuals and species representatives. The species with the closest compatibility distance are chosen for each genome. If no suitable species are found then an entirely new species are created with this specific genome as the representative.

26.5.6 Reproduction Reproduction occurs by allowing some number of the next population to be created by each species. This amount is determined by first calculating the sum of all species fitness values. The amount a species can reproduce is then dependent on its contribution to this total value. For example, if there are two species with fitness values of 0.4 and 0.6, they are allowed to create 40% and 60% of the new population respectively. Species that contribute very little to the total value ( 0 is the initial precompression force, ls is the length of the deformed spring, and lp is the length of the undeformed spring. As can be seen, the ANSD force is a nonlinear function of the displacement representing a geometric-type nonlinearity. The main uncertain parameters of the ANSD are Ks and Pin ; these parameters will be estimated using a nonlinear system identification approach. The top chevron of the ANSD is connected to the bottom of the device through two nonlinear gap spring assemblies (GSA). The purpose of the GSA is to generate a ‘dead-zone’ where the ANSD has no net effect on the structure. The GSA consists of a pre-compressed soft spring and a stiff spring connected in series. The assembly behaves as a bilinear spring whose yield displacement and post-yield stiffness can be tuned with the parameters Ps1 , Ks1 and Ks2 . The GSA force Fg is given by

Fg (u) =

⎧ ⎪ ⎨ Ks2 u ⎪ ⎩

|u| ≤

Ps1 Ks2

|u| >

Ps1 Ks2

(47.11) Ks2 Ks1 +Ks2 Ps1

+

Ks1 Ks2 Ks1 +Ks2 u

For a more comprehensive discussion of the ANSD the interested reader can consult Ref. [3].

47.4.2 Identification Results To obtain the experimental data the structure was subjected to a record of the 1995 Kobe earthquake. The parameters with greater uncertainty and thus to be identified are " = [k1 k2 k3 ξ Ks Pin ], where k1 , k2 , k3 denote the floor stiffnesses, ξ is the damping ratio of all modes, Ks is the ANSD stiffness and Pin is the ANSD pre-compression force. The prior distributions were selected as independent Gaussian distribution, with parameters selected based on the design specifications for the primary structure and the ANS system. The parameters prior mean is " = [ k 1 k 2 k 3 ξ K s P in C p F c ] = [ 20 40 20 0.02 5 − 7 0.8 0.8 ] and a coefficient of variation of 20% (units of force are in kips and units of length in inches). The identification results are presented next. The output consists of acceleration response measurements of floors one and three. The identified primary structure parameters and their uncertainty are depicted in Fig. 47.2, while the estimates of the parameters of the ANSD system and their uncertainty are shown in Fig. 47.3. As can be seen all the parameters show convergence. The estimate of the first floor displacement is shown in Fig. 47.4, while Table 47.1 summarizes the peak dynamic response estimates. As can be seen the estimates are in well agreement with the measurements, with a maximum error of around 4%. Finally, Fig. 47.5 shows the estimate of the NSD force-displacement hysteresis loops. It can observed that the estimates are in agreement with the measurements obtained using string posts, potentiometers and load cells.

47.5 Conclusion In this paper an output-only approach for Bayesian identification of stochastic nonlinear systems subjected to non-stationary inputs was presented. The approach is based on a re-parameterization of the parameters joint posterior distribution to

Fig. 47.2 Primary structure parameters estimates. Units: Force – Kip; Length – inch

Fig. 47.3 ANSD system parameters estimates. Units: force – kip; length – inch

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Fig. 47.4 First floor displacement response estimate Table 47.1 Maximum (peak) response estimates Response quantity Displacement (in) – 1st Floor Displacement (in) – 3rd Floor Column force (Kip) – 1st Floor Base shear (Kip) NSD force (Kip) Damper force (Kip)

Measurement 0.54 0.57 5.42 4.55 4.49 2.92

Estimate 0.54 0.57 5.62 4.65 4.50 2.95

NSD Force

a)

Error (%) 0 0 3.6 2.2 0.2 1.0

Base Shear

b)

Fig. 47.5 (a) NSD force-displacement loops estimate; (b) Base shear force-displacement loops estimate

marginalize and recursively estimate the state predictive distribution in a state estimation step using an unscented Kalman filter, bypassing the state augmentation required by existing on-line methods. This chiefly reduces the ill-posedness of the output-only nonlinear system identification inverse problem allowing to handle higher dimensions of the parameter space than current methods. To compute expectations of functions of the parameters the posterior distribution is sampled using Markov chain Monte Carlo. The proposed approach was experimentally validated in the context of a structure equipped with a seismic protection device for structures. It was shown that the approach is robust to modeling errors in the description of the structure and the forcing input, providing consistent estimates of the parameters and accurate prediction of the dynamic response of the structure. Acknowledgments The second author was partially supported by the Instituto Tecnológico de Santo Domingo (INTEC) and the Ministerio de Educación Superior Ciencia y Tecnología (MESCYT) through the FONDOCYT program. The support is gratefully acknowledged.

References 1. Erazo, K., Nagarajaiah, S.: An offline approach for output-only Bayesian identification of stochastic nonlinear systems using unscented Kalman filtering. J. Sound Vib. 397(9), 222–240 (2017)

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2. Julier, S., Uhlmann, J.: A new extension of the Kalman filter to nonlinear systems. The Robotic Research Group Report. The University of Oxford 3. Pasala, D.T.R., Sarlis, A.A., Nagarajaiah, S., Reinhorn, A.M., Constantinou, M.C., Taylor, D.: Adaptive negative stiffness: new structural modification approach for seismic protection. J. Struct. Eng. 139(7), 1112–1123 (2013) 4. Erazo, K., Nagarajaiah, S.: Bayesian structural identification of a hysteretic negative stiffness earthquake protection system using unscented Kalman filtering. Struct. Control. Health Monit. 25(9), 25:e2203 (2018). https://doi.org/10.1002/stc.2203 5. Erazo, K., Moaveni, B., Nagarajaiah, S.: Bayesian seismic strong-motion response and damage estimation with application to a full-scale seven story shear wall structure. Eng. Struct. 186(1), 146–160 (2019)