Probabilistic Finite Element Model Updating Using Bayesian Statistics

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. Boulkaibet’s 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 xi1 Introduction to Finite Element Model Updating 11.1 Introduction 11.2 Finite Element Modelling 21.3 Vibration Analysis 41.3.1 Modal Domain Data 41.3.2 Frequency Domain Data 51.4 Finite Element Model Updating 51.5 Finite Element Model Updating and Bounded Rationality 61.6 Finite Element Model Updating Methods 71.6.1 Direct Methods 81.6.2 Iterative Methods 101.6.3 Artificial Intelligence Methods 111.6.4 Uncertainty Quantification Methods 111.7 Bayesian Approach versus Maximum Likelihood Method 141.8 Outline of the Book 15References 172 Model Selection in Finite Element Model Updating 242.1 Introduction 242.2 Model Selection in Finite Element Modelling 252.2.1 Akaike Information Criterion 252.2.2 Bayesian Information Criterion 252.2.3 Bayes Factor 262.2.4 Deviance Information Criterion 262.2.5 Particle Swarm Optimisation for Model Selection 272.2.6 Regularisation 282.2.7 Cross-Validation 282.2.8 Nested Sampling for Model Selection 302.3 Simulated Annealing 322.4 Asymmetrical H-Shaped Structure 352.4.1 Regularisation 352.4.2 Cross-Validation 362.4.3 Bayes Factor and Nested Sampling 362.5 Conclusion 37References 373 Bayesian Statistics in Structural Dynamics 423.1 Introduction 423.2 Bayes’ Rule 453.3 Maximum Likelihood Method 463.4 Maximum a Posteriori Parameter Estimates 463.5 Laplace’s Method 473.6 Prior, Likelihood and Posterior Function of a Simple Dynamic Example 473.6.1 Likelihood Function 493.6.2 Prior Function 493.6.3 Posterior Function 503.6.4 Gaussian Approximation 503.7 The Posterior Approximation 523.7.1 Objective Function 523.7.2 Optimisation Approach 523.7.3 Case Example 553.8 Sampling Approaches for Estimating Posterior Distribution 553.8.1 Monte Carlo Method 553.8.2 Markov Chain Monte Carlo Method 563.8.3 Simulated Annealing 573.8.4 Gibbs Sampling 583.9 Comparison between Approaches 583.9.1 Numerical Example 583.10 Conclusions 60References 614 Metropolis–Hastings and Slice Sampling for Finite Element Updating 654.1 Introduction 654.2 Likelihood, Prior and the Posterior Functions 664.3 The Metropolis–Hastings Algorithm 694.4 The Slice Sampling Algorithm 714.5 Statistical Measures 724.6 Application 1: Cantilevered Beam 744.7 Application 2: Asymmetrical H-Shaped Structure 784.8 Conclusions 81References 815 Dynamically Weighted Importance Sampling for Finite Element Updating 845.1 Introduction 845.2 Bayesian Modelling Approach 855.3 Metropolis–Hastings (M-H) Algorithm 875.4 Importance Sampling 885.5 Dynamically Weighted Importance Sampling 895.5.1 Markov Chain 905.5.2 Adaptive Pruned-Enriched Population Control Scheme 905.5.3 Monte Carlo Dynamically Weighted Importance Sampling 925.6 Application 1: Cantilevered Beam 935.7 Application 2: H-Shaped Structure 975.8 Conclusions 101References 1016 Adaptive Metropolis–Hastings for Finite Element Updating 1046.1 Introduction 1046.2 Adaptive Metropolis–Hastings Algorithm 1056.3 Application 1: Cantilevered Beam 1086.4 Application 2: Asymmetrical H-Shaped Beam 1116.5 Application 3: Aircraft GARTEUR Structure 1136.6 Conclusion 119References 1197 Hybrid Monte Carlo Technique for Finite Element Model Updating 1227.1 Introduction 1227.2 Hybrid Monte Carlo Method 1237.3 Properties of the HMC Method 1247.3.1 Time Reversibility 1247.3.2 Volume Preservation 1247.3.3 Energy Conservation 1257.4 The Molecular Dynamics Algorithm 1257.5 Improving the HMC 1277.5.1 Choosing an Efficient Time Step 1277.5.2 Suppressing the Random Walk in the Momentum 1287.5.3 Gradient Computation 1287.6 Application 1: Cantilever Beam 1297.7 Application 2: Asymmetrical H-Shaped Structure 1327.8 Conclusion 135References 1358 Shadow Hybrid Monte Carlo Technique for Finite Element Model Updating 1388.1 Introduction 1388.2 Effect of Time Step in the Hybrid Monte Carlo Method 1398.3 The Shadow Hybrid Monte Carlo Method 1398.4 The Shadow Hamiltonian 1428.5 Application: GARTEUR SM-AG19 Structure 1438.6 Conclusion 152References 1539 Separable Shadow Hybrid Monte Carlo in Finite Element Updating 1559.1 Introduction 1559.2 Separable Shadow Hybrid Monte Carlo 1559.3 Theoretical Justifications of the S2HMC Method 1589.4 Application 1: Asymmetrical H-Shaped Structure 1609.5 Application 2: GARTEUR SM-AG19 Structure 1659.6 Conclusions 171References 17210 Evolutionary Approach to Finite Element Model Updating 17410.1 Introduction 17410.2 The Bayesian Formulation 17510.3 The Evolutionary MCMC Algorithm 17710.3.1 Mutation 17810.3.2 Crossover 17910.3.3 Exchange 18110.4 Metropolis–Hastings Method 18110.5 Application: Asymmetrical H-Shaped Structure 18210.6 Conclusion 185References 18611 Adaptive Markov Chain Monte Carlo Method for Finite Element Model Updating 18911.1 Introduction 18911.2 Bayesian Theory 19111.3 Adaptive Hybrid Monte Carlo 19211.4 Application 1: A Linear System with Three Degrees of Freedom 19511.4.1 Updating the Stiffness Parameters 19611.5 Application 2: Asymmetrical H-Shaped Structure 19811.5.1 H-Shaped Structure Simulation 19811.6 Conclusion 202References 20312 Conclusions and Further Work 20612.1 Introduction 20612.2 Further Work 20812.2.1 Reversible Jump Monte Carlo 20812.2.2 Multiple-Try Metropolis–Hastings 20812.2.3 Dynamic Programming 20912.2.4 Sequential Monte Carlo 209References 209Appendix A: Experimental Examples 211Appendix B: Markov Chain Monte Carlo 219Appendix C: Gaussian Distribution 222Index 226