,PROBABILISTIC FINITE
ELEMENT MODEL
UPDATING USING
BAYESIAN STATISTICS
,PROBABILISTIC FINITE
ELEMENT MODEL
UPDATING USING
BAYESIAN STATISTICS
APPLICATIONS TO
AERONAUTICAL AND
MECHANICAL ENGINEERING
Tshilidzi Marwala and Ilyes Boulkaibet
University of Johannesburg, South Africa
Sondipon Adhikari
Swansea University, UK
,This edition first published 2017
© 2017 John Wiley & Sons, Ltd
Registered Office
John Wiley & Sons, Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom
For details of our global editorial offices, for customer services and for information about how to apply for
permission to reuse the copyright material in this book please see our website at www.wiley.com.
The right of the author to be identified as the author of this work has been asserted in accordance with the
Copyright, Designs and Patents Act 1988.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted,
in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted
by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher.
Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be
available in electronic books.
Designations used by companies to distinguish their products are often claimed as trademarks. All brand names
and product names used in this book are trade names, service marks, trademarks or registered trademarks of their
respective owners. The publisher is not associated with any product or vendor mentioned in this book.
Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing
this book, they make no representations or warranties with respect to the accuracy or completeness of the contents
of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose.
It is sold on the understanding that the publisher is not engaged in rendering professional services and neither the
publisher nor the author shall be liable for damages arising herefrom. If professional advice or other expert
assistance is required, the services of a competent professional should be sought.
Library of Congress Cataloging-in-Publication Data
Names: Marwala, Tshilidzi, 1971– author. | Boulkaibet, Ilyes, 1981– author. | Adhikari, Sondipon, author.
Title: Probabilistic finite element model updating using Bayesian statistics: applications to aeronautical and mechanical
engineering / Tshilidzi Marwala, Ilyes Boulkaibet and Sondipon Adhikari.
Description: Chichester, UK ; Hoboken, NJ : John Wiley & Sons, 2017. | Includes bibliographical references and index.
Identifiers: LCCN 2016019278| ISBN 9781119153030 (cloth) | ISBN 9781119153016 (epub)
Subjects: LCSH: Finite element method. | Bayesian statistical decision theory. | Engineering–Mathematical models.
Classification: LCC TA347.F5 M3823 2016 | DDC 620.001/51825–dc23
LC record available at https://lccn.loc.gov/2016019278
A catalogue record for this book is available from the British Library.
Cover image: Godruma/Gettyimages
Set in 10 /12pt Times by SPi Global, Pondicherry, India
10 9 8 7 6 5 4 3 2 1
,Contents
Acknowledgements x
Nomenclature xi
1 Introduction to Finite Element Model Updating 1
1.1 Introduction 1
1.2 Finite Element Modelling 2
1.3 Vibration Analysis 4
1.3.1 Modal Domain Data 4
1.3.2 Frequency Domain Data 5
1.4 Finite Element Model Updating 5
1.5 Finite Element Model Updating and Bounded Rationality 6
1.6 Finite Element Model Updating Methods 7
1.6.1 Direct Methods 8
1.6.2 Iterative Methods 10
1.6.3 Artificial Intelligence Methods 11
1.6.4 Uncertainty Quantification Methods 11
1.7 Bayesian Approach versus Maximum Likelihood Method 14
1.8 Outline of the Book 15
References 17
2 Model Selection in Finite Element Model Updating 24
2.1 Introduction 24
2.2 Model Selection in Finite Element Modelling 25
2.2.1 Akaike Information Criterion 25
2.2.2 Bayesian Information Criterion 25
2.2.3 Bayes Factor 26
,vi Contents
2.2.4 Deviance Information Criterion 26
2.2.5 Particle Swarm Optimisation for Model Selection 27
2.2.6 Regularisation 28
2.2.7 Cross-Validation 28
2.2.8 Nested Sampling for Model Selection 30
2.3 Simulated Annealing 32
2.4 Asymmetrical H-Shaped Structure 35
2.4.1 Regularisation 35
2.4.2 Cross-Validation 36
2.4.3 Bayes Factor and Nested Sampling 36
2.5 Conclusion 37
References 37
3 Bayesian Statistics in Structural Dynamics 42
3.1 Introduction 42
3.2 Bayes’ Rule 45
3.3 Maximum Likelihood Method 46
3.4 Maximum a Posteriori Parameter Estimates 46
3.5 Laplace’s Method 47
3.6 Prior, Likelihood and Posterior Function of a Simple Dynamic Example 47
3.6.1 Likelihood Function 49
3.6.2 Prior Function 49
3.6.3 Posterior Function 50
3.6.4 Gaussian Approximation 50
3.7 The Posterior Approximation 52
3.7.1 Objective Function 52
3.7.2 Optimisation Approach 52
3.7.3 Case Example 55
3.8 Sampling Approaches for Estimating Posterior Distribution 55
3.8.1 Monte Carlo Method 55
3.8.2 Markov Chain Monte Carlo Method 56
3.8.3 Simulated Annealing 57
3.8.4 Gibbs Sampling 58
3.9 Comparison between Approaches 58
3.9.1 Numerical Example 58
3.10 Conclusions 60
References 61
4 Metropolis–Hastings and Slice Sampling for Finite Element Updating 65
4.1 Introduction 65
4.2 Likelihood, Prior and the Posterior Functions 66
4.3 The Metropolis–Hastings Algorithm 69
4.4 The Slice Sampling Algorithm 71
4.5 Statistical Measures 72
,Contents vii
4.6 Application 1: Cantilevered Beam 74
4.7 Application 2: Asymmetrical H-Shaped Structure 78
4.8 Conclusions 81
References 81
5 Dynamically Weighted Importance Sampling for Finite Element Updating 84
5.1 Introduction 84
5.2 Bayesian Modelling Approach 85
5.3 Metropolis–Hastings (M-H) Algorithm 87
5.4 Importance Sampling 88
5.5 Dynamically Weighted Importance Sampling 89
5.5.1 Markov Chain 90
5.5.2 Adaptive Pruned-Enriched Population Control Scheme 90
5.5.3 Monte Carlo Dynamically Weighted Importance Sampling 92
5.6 Application 1: Cantilevered Beam 93
5.7 Application 2: H-Shaped Structure 97
5.8 Conclusions 101
References 101
6 Adaptive Metropolis–Hastings for Finite Element Updating 104
6.1 Introduction 104
6.2 Adaptive Metropolis–Hastings Algorithm 105
6.3 Application 1: Cantilevered Beam 108
6.4 Application 2: Asymmetrical H-Shaped Beam 111
6.5 Application 3: Aircraft GARTEUR Structure 113
6.6 Conclusion 119
References 119
7 Hybrid Monte Carlo Technique for Finite Element Model Updating 122
7.1 Introduction 122
7.2 Hybrid Monte Carlo Method 123
7.3 Properties of the HMC Method 124
7.3.1 Time Reversibility 124
7.3.2 Volume Preservation 124
7.3.3 Energy Conservation 125
7.4 The Molecular Dynamics Algorithm 125
7.5 Improving the HMC 127
7.5.1 Choosing an Efficient Time Step 127
7.5.2 Suppressing the Random Walk in the Momentum 128
7.5.3 Gradient Computation 128
7.6 Application 1: Cantilever Beam 129
7.7 Application 2: Asymmetrical H-Shaped Structure 132
7.8 Conclusion 135
References 135
,viii Contents
8 Shadow Hybrid Monte Carlo Technique for Finite Element Model Updating 138
8.1 Introduction 138
8.2 Effect of Time Step in the Hybrid Monte Carlo Method 139
8.3 The Shadow Hybrid Monte Carlo Method 139
8.4 The Shadow Hamiltonian 142
8.5 Application: GARTEUR SM-AG19 Structure 143
8.6 Conclusion 152
References 153
9 Separable Shadow Hybrid Monte Carlo in Finite Element Updating 155
9.1 Introduction 155
9.2 Separable Shadow Hybrid Monte Carlo 155
9.3 Theoretical Justifications of the S2HMC Method 158
9.4 Application 1: Asymmetrical H-Shaped Structure 160
9.5 Application 2: GARTEUR SM-AG19 Structure 165
9.6 Conclusions 171
References 172
10 Evolutionary Approach to Finite Element Model Updating 174
10.1 Introduction 174
10.2 The Bayesian Formulation 175
10.3 The Evolutionary MCMC Algorithm 177
10.3.1 Mutation 178
10.3.2 Crossover 179
10.3.3 Exchange 181
10.4 Metropolis–Hastings Method 181
10.5 Application: Asymmetrical H-Shaped Structure 182
10.6 Conclusion 185
References 186
11 Adaptive Markov Chain Monte Carlo Method for Finite Element
Model Updating 189
11.1 Introduction 189
11.2 Bayesian Theory 191
11.3 Adaptive Hybrid Monte Carlo 192
11.4 Application 1: A Linear System with Three Degrees of Freedom 195
11.4.1 Updating the Stiffness Parameters 196
11.5 Application 2: Asymmetrical H-Shaped Structure 198
11.5.1 H-Shaped Structure Simulation 198
11.6 Conclusion 202
References 203
12 Conclusions and Further Work 206
12.1 Introduction 206
12.2 Further Work 208
12.2.1 Reversible Jump Monte Carlo 208
,Contents ix
12.2.2 Multiple-Try Metropolis–Hastings 208
12.2.3 Dynamic Programming 209
12.2.4 Sequential Monte Carlo 209
References 209
Appendix A: Experimental Examples 211
Appendix B: Markov Chain Monte Carlo 219
Appendix C: Gaussian Distribution 222
Index 226
, Acknowledgements
We would like to thank the University of Johannesburg and the University of Swansea for con-
tributing towards the writing of this book. We also would like to thank Michael Friswell, Linda
Mthembu, Niel Joubert and Ishmael Msiza for contributing towards the writing of this book.
We dedicate this book to the schools that gave us the foundation to always seek excellence in
everything we do: the University of Cambridge and the University of Johannesburg.
Tshilidzi Marwala, PhD
Johannesburg
1 February 2016
Ilyes Boulkaibet, PhD
Johannesburg
1 February 2016
Sondipon Adhikari, PhD
Swansea
1 February 2016
ELEMENT MODEL
UPDATING USING
BAYESIAN STATISTICS
,PROBABILISTIC FINITE
ELEMENT MODEL
UPDATING USING
BAYESIAN STATISTICS
APPLICATIONS TO
AERONAUTICAL AND
MECHANICAL ENGINEERING
Tshilidzi Marwala and Ilyes Boulkaibet
University of Johannesburg, South Africa
Sondipon Adhikari
Swansea University, UK
,This edition first published 2017
© 2017 John Wiley & Sons, Ltd
Registered Office
John Wiley & Sons, Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom
For details of our global editorial offices, for customer services and for information about how to apply for
permission to reuse the copyright material in this book please see our website at www.wiley.com.
The right of the author to be identified as the author of this work has been asserted in accordance with the
Copyright, Designs and Patents Act 1988.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted,
in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted
by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher.
Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be
available in electronic books.
Designations used by companies to distinguish their products are often claimed as trademarks. All brand names
and product names used in this book are trade names, service marks, trademarks or registered trademarks of their
respective owners. The publisher is not associated with any product or vendor mentioned in this book.
Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing
this book, they make no representations or warranties with respect to the accuracy or completeness of the contents
of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose.
It is sold on the understanding that the publisher is not engaged in rendering professional services and neither the
publisher nor the author shall be liable for damages arising herefrom. If professional advice or other expert
assistance is required, the services of a competent professional should be sought.
Library of Congress Cataloging-in-Publication Data
Names: Marwala, Tshilidzi, 1971– author. | Boulkaibet, Ilyes, 1981– author. | Adhikari, Sondipon, author.
Title: Probabilistic finite element model updating using Bayesian statistics: applications to aeronautical and mechanical
engineering / Tshilidzi Marwala, Ilyes Boulkaibet and Sondipon Adhikari.
Description: Chichester, UK ; Hoboken, NJ : John Wiley & Sons, 2017. | Includes bibliographical references and index.
Identifiers: LCCN 2016019278| ISBN 9781119153030 (cloth) | ISBN 9781119153016 (epub)
Subjects: LCSH: Finite element method. | Bayesian statistical decision theory. | Engineering–Mathematical models.
Classification: LCC TA347.F5 M3823 2016 | DDC 620.001/51825–dc23
LC record available at https://lccn.loc.gov/2016019278
A catalogue record for this book is available from the British Library.
Cover image: Godruma/Gettyimages
Set in 10 /12pt Times by SPi Global, Pondicherry, India
10 9 8 7 6 5 4 3 2 1
,Contents
Acknowledgements x
Nomenclature xi
1 Introduction to Finite Element Model Updating 1
1.1 Introduction 1
1.2 Finite Element Modelling 2
1.3 Vibration Analysis 4
1.3.1 Modal Domain Data 4
1.3.2 Frequency Domain Data 5
1.4 Finite Element Model Updating 5
1.5 Finite Element Model Updating and Bounded Rationality 6
1.6 Finite Element Model Updating Methods 7
1.6.1 Direct Methods 8
1.6.2 Iterative Methods 10
1.6.3 Artificial Intelligence Methods 11
1.6.4 Uncertainty Quantification Methods 11
1.7 Bayesian Approach versus Maximum Likelihood Method 14
1.8 Outline of the Book 15
References 17
2 Model Selection in Finite Element Model Updating 24
2.1 Introduction 24
2.2 Model Selection in Finite Element Modelling 25
2.2.1 Akaike Information Criterion 25
2.2.2 Bayesian Information Criterion 25
2.2.3 Bayes Factor 26
,vi Contents
2.2.4 Deviance Information Criterion 26
2.2.5 Particle Swarm Optimisation for Model Selection 27
2.2.6 Regularisation 28
2.2.7 Cross-Validation 28
2.2.8 Nested Sampling for Model Selection 30
2.3 Simulated Annealing 32
2.4 Asymmetrical H-Shaped Structure 35
2.4.1 Regularisation 35
2.4.2 Cross-Validation 36
2.4.3 Bayes Factor and Nested Sampling 36
2.5 Conclusion 37
References 37
3 Bayesian Statistics in Structural Dynamics 42
3.1 Introduction 42
3.2 Bayes’ Rule 45
3.3 Maximum Likelihood Method 46
3.4 Maximum a Posteriori Parameter Estimates 46
3.5 Laplace’s Method 47
3.6 Prior, Likelihood and Posterior Function of a Simple Dynamic Example 47
3.6.1 Likelihood Function 49
3.6.2 Prior Function 49
3.6.3 Posterior Function 50
3.6.4 Gaussian Approximation 50
3.7 The Posterior Approximation 52
3.7.1 Objective Function 52
3.7.2 Optimisation Approach 52
3.7.3 Case Example 55
3.8 Sampling Approaches for Estimating Posterior Distribution 55
3.8.1 Monte Carlo Method 55
3.8.2 Markov Chain Monte Carlo Method 56
3.8.3 Simulated Annealing 57
3.8.4 Gibbs Sampling 58
3.9 Comparison between Approaches 58
3.9.1 Numerical Example 58
3.10 Conclusions 60
References 61
4 Metropolis–Hastings and Slice Sampling for Finite Element Updating 65
4.1 Introduction 65
4.2 Likelihood, Prior and the Posterior Functions 66
4.3 The Metropolis–Hastings Algorithm 69
4.4 The Slice Sampling Algorithm 71
4.5 Statistical Measures 72
,Contents vii
4.6 Application 1: Cantilevered Beam 74
4.7 Application 2: Asymmetrical H-Shaped Structure 78
4.8 Conclusions 81
References 81
5 Dynamically Weighted Importance Sampling for Finite Element Updating 84
5.1 Introduction 84
5.2 Bayesian Modelling Approach 85
5.3 Metropolis–Hastings (M-H) Algorithm 87
5.4 Importance Sampling 88
5.5 Dynamically Weighted Importance Sampling 89
5.5.1 Markov Chain 90
5.5.2 Adaptive Pruned-Enriched Population Control Scheme 90
5.5.3 Monte Carlo Dynamically Weighted Importance Sampling 92
5.6 Application 1: Cantilevered Beam 93
5.7 Application 2: H-Shaped Structure 97
5.8 Conclusions 101
References 101
6 Adaptive Metropolis–Hastings for Finite Element Updating 104
6.1 Introduction 104
6.2 Adaptive Metropolis–Hastings Algorithm 105
6.3 Application 1: Cantilevered Beam 108
6.4 Application 2: Asymmetrical H-Shaped Beam 111
6.5 Application 3: Aircraft GARTEUR Structure 113
6.6 Conclusion 119
References 119
7 Hybrid Monte Carlo Technique for Finite Element Model Updating 122
7.1 Introduction 122
7.2 Hybrid Monte Carlo Method 123
7.3 Properties of the HMC Method 124
7.3.1 Time Reversibility 124
7.3.2 Volume Preservation 124
7.3.3 Energy Conservation 125
7.4 The Molecular Dynamics Algorithm 125
7.5 Improving the HMC 127
7.5.1 Choosing an Efficient Time Step 127
7.5.2 Suppressing the Random Walk in the Momentum 128
7.5.3 Gradient Computation 128
7.6 Application 1: Cantilever Beam 129
7.7 Application 2: Asymmetrical H-Shaped Structure 132
7.8 Conclusion 135
References 135
,viii Contents
8 Shadow Hybrid Monte Carlo Technique for Finite Element Model Updating 138
8.1 Introduction 138
8.2 Effect of Time Step in the Hybrid Monte Carlo Method 139
8.3 The Shadow Hybrid Monte Carlo Method 139
8.4 The Shadow Hamiltonian 142
8.5 Application: GARTEUR SM-AG19 Structure 143
8.6 Conclusion 152
References 153
9 Separable Shadow Hybrid Monte Carlo in Finite Element Updating 155
9.1 Introduction 155
9.2 Separable Shadow Hybrid Monte Carlo 155
9.3 Theoretical Justifications of the S2HMC Method 158
9.4 Application 1: Asymmetrical H-Shaped Structure 160
9.5 Application 2: GARTEUR SM-AG19 Structure 165
9.6 Conclusions 171
References 172
10 Evolutionary Approach to Finite Element Model Updating 174
10.1 Introduction 174
10.2 The Bayesian Formulation 175
10.3 The Evolutionary MCMC Algorithm 177
10.3.1 Mutation 178
10.3.2 Crossover 179
10.3.3 Exchange 181
10.4 Metropolis–Hastings Method 181
10.5 Application: Asymmetrical H-Shaped Structure 182
10.6 Conclusion 185
References 186
11 Adaptive Markov Chain Monte Carlo Method for Finite Element
Model Updating 189
11.1 Introduction 189
11.2 Bayesian Theory 191
11.3 Adaptive Hybrid Monte Carlo 192
11.4 Application 1: A Linear System with Three Degrees of Freedom 195
11.4.1 Updating the Stiffness Parameters 196
11.5 Application 2: Asymmetrical H-Shaped Structure 198
11.5.1 H-Shaped Structure Simulation 198
11.6 Conclusion 202
References 203
12 Conclusions and Further Work 206
12.1 Introduction 206
12.2 Further Work 208
12.2.1 Reversible Jump Monte Carlo 208
,Contents ix
12.2.2 Multiple-Try Metropolis–Hastings 208
12.2.3 Dynamic Programming 209
12.2.4 Sequential Monte Carlo 209
References 209
Appendix A: Experimental Examples 211
Appendix B: Markov Chain Monte Carlo 219
Appendix C: Gaussian Distribution 222
Index 226
, Acknowledgements
We would like to thank the University of Johannesburg and the University of Swansea for con-
tributing towards the writing of this book. We also would like to thank Michael Friswell, Linda
Mthembu, Niel Joubert and Ishmael Msiza for contributing towards the writing of this book.
We dedicate this book to the schools that gave us the foundation to always seek excellence in
everything we do: the University of Cambridge and the University of Johannesburg.
Tshilidzi Marwala, PhD
Johannesburg
1 February 2016
Ilyes Boulkaibet, PhD
Johannesburg
1 February 2016
Sondipon Adhikari, PhD
Swansea
1 February 2016
