# Probabilistic Finite Element Model Updating Using Bayesian Statistics: Applications to Aeronautical and Mechanical Engineering

ISBN: 978-1-119-15303-0

Nov 2016

248 pages

Select type: Hardcover

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\$130.00

## Description

Probabilistic Finite Element Model Updating Using Bayesian Statistics: Applications to Aeronautical and Mechanical Engineering

Tshilidzi Marwala and Ilyes Boulkaibet, University of Johannesburg, South Africa

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.

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 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

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

6.3 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

7.6 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

8.5 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

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