Probabilistic Finite Element Model Updating Using Bayesian Statistics
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Content
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
6.2 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
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
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
1
Introduction to Finite Element Model Updating
1.1 Introduction
Finite element model updating methods are intended to correct and improve a numerical model to match the dynamic behaviour of real structures (Marwala, 2010). Modern computers, with their ability to process large matrices at high speed, have facilitated the formulation of many large and complicated numerical models, including the boundary element method, the finite difference method and the finite element models. This book deals with the finite element model that was first applied in solving complex elasticity and structural analysis problems in aeronautical, mechanical and civil engineering. Finite element modelling was proposed by Hrennikoff (1941) and Courant and Robbins (1941). Courant applied the Ritz technique and variational calculus to solve vibration problems in structures (Hastings et al., 1985). Despite the fact that the approaches used by these researchers were different from conventional formulations, some important lessons are still relevant. These differences include mesh discretisation into elements (Babuska et al., 2004).
The Cooley-Turkey algorithms, which are used to speedily obtain Fourier transformations, have facilitated the development of complex techniques in vibration and experimental modal analysis. Conversely, the finite element model ordinarily predicts results that are different from the results obtained from experimental investigation. Among reasons for the discrepancy between finite element model prediction and experimentally measured data are as the following (Friswell and Mottershead, 1995; Marwala, 2010; Dhandole and Modak, 2011):
- model structure errors resulting from the difficulty in modelling damping and complex shapes such as joints, welds and edges;
- model order errors resulting from the difficulty in modelling non-linearity and often assuming linearity;
- model parameter errors resulting in difficulty in identifying the correct material properties;
- errors in measurements and signal processing.
In finite element model updating, it is assumed that the measurements are correct within certain limits of uncertainty and, for that reason, a finite element model under consideration will need to be updated to better reflect the measured data. Additionally, finite element model updating assumes that the difficulty in modelling joints and other complicated boundary conditions can be compensated for by adjusting the material properties of the relevant elements. In this book, it is also assumed that a finite element model is linear and that damping is sufficiently low not to warrant complex modelling (Mottershead and Friswell, 1993; Friswell and Mottershead, 1995). Using data from experimental measurements, the initial finite element model is updated by correcting uncertain parameters so that the model is close to the measured data. Alternatively, finite element model updating is an inverse problem and the goal is to identify the system that generated the measured data (Brincker et al., 2001; Dhandole and Modak, 2010; Zhang et al., 2011; Boulkaibet, 2014; Fuellekrug et al., 2008; Cheung and Beck, 2009; Mottershead et al., 2000).
There are two main approaches to finite element model updating, namely, maximum likelihood and Bayesian methods (Marwala, 2010; Mottershead et al., 2011). In this book, we apply a Bayesian approach to finite element model updating.
1.2 Finite Element Modelling
Finite element models have been applied to aerospace, electrical, civil and mechanical engineering in designing and developing products such as aircraft wings and turbo-machinery. Some of the applications of finite element modelling are (Marwala, 2010): thermal problems, electromagnetic problems, fluid problems and structural modelling. Finite element modelling typically entails choosing elements and basis functions (Chandrupatla and Belegudu, 2002; Marwala, 2010). Generally, there are two types of finite element analysis that are used: two-dimensional and three-dimensional modelling (Solin et al., 2004; Marwala, 2010).
Two-dimensional modelling is simple and computationally efficient. Three-dimensional modelling, on the other hand, is more accurate, though computationally expensive. Finite element analysis can be formulated in a linear or non-linear fashion. Linear formulation is simple and usually does not consider plastic deformation, which non-linear formulation does consider. This book only deals with linear finite element modelling, in the form of a second-order ordinary differential equation of relations between mass, damping and stiffness matrices. A finite element model has nodes, with a grid called a mesh, as shown in Figure 1.1 (Marwala, 2001). The mesh has material and structural properties with particular loading and boundary conditions. Figure 1.1 shows the dynamics of a cylinder, and the mode shape of the first natural frequency occurring at 433 Hz.
Figure 1.1 A finite element model of a cylindrical shell
These loaded nodes are assigned a specific density all over the material, in accordance with the expected stress levels of that area (Baran, 1988). Sections which undergo more stress will then have a higher node density than those which experience less or no stress. Points of stress concentration may have fracture points of previously tested materials, joints, welds and high-stress areas. The mesh may be imagined as a spider's web so that, from each node, a mesh element extends to each of the neighbouring nodes. This web of vectors has the material properties of the object, resulting in a study of many elements.
On implementing finite element modelling, a choice of elements needs to be made and these include beam, plate, shell elements or solid elements. A question that needs to be answered when applying finite element analysis is whether the material is isotropic (identical throughout the material), orthotropic (only identical at 90°) or anisotropic (different throughout the material) (Irons and Shrive, 1983; Zienkiewicz, 1986; Marwala, 2010).
Finite element analysis has been applied to model the following problems (Zienkiewicz, 1986; Marwala, 2010):
- vibration analysis for testing a structure for random vibrations, impact and shock;
- fatigue analysis to approximate the life cycle of a material or a structure due to cyclical loading;
- heat transfer analysis to model conductivity or thermal fluid dynamics of the material or structure.
Hlilou et al. (2009) successfully applied finite element analysis in softening material behaviour, while Zhang and Teo (2008) successfully applied it in the treatment of a lumbar degenerative disc disease. White et al. (2008) successfully applied finite element analysis for shallow-water modelling, while Pepper and Wang (2007) successfully applied it in wind energy assessment of renewable energy in Nevada. Miao et al. (2009) successfully applied a three-dimensional finite element analysis model in the simulation of shot peening. Bürg and Nazarov (2015) successfully applied goal-oriented adaptive finite element methods in elliptic problems, while Amini et al. (2015) successfully applied finite element modelling in functionally graded piezoelectric harvesters. Haldar et al. (2015) successfully applied finite element modelling in the study of the flexural behaviour of singly curved sandwich composite structures, while Millar and Mora (2015) successfully applied finite element methods to study the buckling in simply supported Kirchhoff plates. Jung et al. (2015) successfully used finite element models and computed tomography to estimate cross-sectional constants of composite blades, while Evans and Miller (2015) successfully applied a finite element model to predict the failure of pressure vessels. Other successful applications of finite element analysis are in the areas of metal powder compaction processing (Rahman et al., 2009), ferroelectric materials (Schrade et al., 2007), rock mechanics (Chen et al., 2009), orthopaedics (Easley et al., 2007), carbon nanotubes (Zuberi and Esat, 2015), nuclear reactors (Wadsworth et al., 2015) and elastic wave propagation (Gao et al., 2015; Gravenkamp et al., 2015).
1.3 Vibration Analysis
An important aspect to consider when implementing finite element analysis is the kind of data that the model is supposed to predict. It can predict data in many domains, such as the time, modal, frequency and time-frequency domains (Marwala, 2001, 2010). This book is concerned with constructing finite element models to predict measured data more accurately. Ideally, a finite element model is supposed to predict measured data irrespective of the domain in which the data are presented. However, this is not necessarily the case because models updated in the time domain will not necessarily predict data in the modal domain as accurately as they will for data in the time domain. To deal with this issue, Marwala and Heyns (1998) used data in the modal and frequency domains simultaneously to update the finite element model in a multi-criteria optimisation fashion. Again, whichever domain is used, the updated model performs less well on data in a different domain than those used in the updating process. In this book, we use data in the modal domain. Raw data are measured in the time domain and Fourier analysis techniques...
