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

Covers the probabilistic finite element model based on Bayesian statistics with applications to aeronautical and mechanical engineering

Finite element models are used widely to model the dynamic behaviour of many systems including in electrical, aerospace and mechanical engineering.

The book covers probabilistic finite element model updating, achieved using Bayesian statistics. The Bayesian framework is employed to estimate the probabilistic finite element models which take into account of the uncertainties in the measurements and the modelling procedure. The Bayesian formulation achieves this by formulating the finite element model as the posterior distribution of the model given the measured data within the context of computational statistics and applies these in aeronautical and mechanical engineering.

Probabilistic Finite Element Model Updating Using Bayesian Statistics contains simple explanations of computational statistical techniques such as Metropolis-Hastings Algorithm, Slice sampling, Markov Chain Monte Carlo method, hybrid Monte Carlo as well as Shadow Hybrid Monte Carlo and their relevance in engineering.

Key features:

Contains several contributions in the area of model updating using Bayesian techniques which are useful for graduate students.Explains in detail the use of Bayesian techniques to quantify uncertainties in mechanical structures as well as the use of Markov Chain Monte Carlo techniques to evaluate the Bayesian formulations.

The book is essential reading for researchers, practitioners and students in mechanical and aerospace engineering.

Tshilidzi Marwala is a Professor of Mechanical and Electrical Engineering as well as Deputy Vice-Chancellor at the University of Johannesburg. He holds a Bachelor of Science in Mechanical Engineering from Case Western Reserve University, a Master of Mechanical Engineering from the University of Pretoria, a PhD in Engineering from Cambridge University and was a post-doctoral researcher at Imperial College (London). He is a Fellow of TWAS and a distinguished member of the ACM. His research interests are multi-disciplinary and include the applications of computational intelligence to engineering, computer science, finance, social science and medicine. He has supervised 45 Masters and 19 PhD students and has published 8 books and over 260 papers. He is an associate editor of the International Journal of Systems Science.

Dr. Ilyes Boulkaibet is currently a researcher at the University of Johannesburg. He received a PhD from the University of Johannesburg, a second MSc from Stellenbosch University, an MSc from the University of Constantine 1 Algeria, and a Bachelor of Engineering from University of Constantine 1 Algeria. Dr. Ilyes Boulkaibet has published papers in international journals and has participated in numerous conferences including the International Modal Analysis Conference. Dr. Boulkaibets research areas are multidisciplinary in nature and include uncertainty quantification in computational mechanics, dynamics of complex systems, inverse problems for linear and non-linear dynamics and control systems.Professor Adhikari is the chair of Aerospace Engineering in the College of Engineering of Swansea University. He received his MSc from the Indian Institute of Science and a PhD from the University of Cambridge. He was a lecturer at the Bristol University and a Junior Research Fellow in Fitzwilliam College, Cambridge. He has been a visiting Professor at the University of Johannesburg, Carleton University and the Los Alamos National Laboratory . Professor Adhikari's research areas are multidisciplinary in nature and include uncertainty quantification in computational mechanics, bio- and nano-mechanics (nanotubes, graphene, cell mechanics, nano-bio sensors), dynamics of complex systems, inverse problems for linear and non-linear dynamics and vibration energy harvesting.

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

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 Laplaces 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 MetropolisHastings and Slice Sampling for Finite Element Updating 65

4.1 Introduction 65

4.2 Likelihood, Prior and the Posterior Functions 66

4.3 The MetropolisHastings Algorithm 69

4.4 The Slice Sampling Algorithm 71

4.5 Statistical Measures 72

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 MetropolisHastings (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 MetropolisHastings for Finite Element Updating 104

6.1 Introduction 104

6.2 Adaptive MetropolisHastings 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

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 MetropolisHastings 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

12.2.2 Multiple-Try MetropolisHastings 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