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Cover
Title Page
Acknowledgements
Nomenclature
1 Introduction to Finite Element Model Updating
1. 1.1 Introduction
2. 1.2 Finite Element Modelling
3. 1.3 Vibration Analysis
4. 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. 1.7 Bayesian Approach versus Maximum Likelihood Method
8. 1.8 Outline of the Book
9. References
2 Model Selection in Finite Element Model Updating
1. 2.1 Introduction
2. 2.2 Model Selection in Finite Element Modelling
3. 2.3 Simulated Annealing
4. 2.4 Asymmetrical H‐Shaped Structure
5. 2.5 Conclusion
6. References
3 Bayesian Statistics in Structural Dynamics
1. 3.1 Introduction
2. 3.2 Bayes’ Rule
3. 3.3 Maximum Likelihood Method
4. 3.4 Maximum a Posteriori Parameter Estimates
5. 3.5 Laplace’s Method
6. 3.6 Prior, Likelihood and Posterior Function of a Simple Dynamic Example
7. 3.7 The Posterior Approximation
8. 3.8 Sampling Approaches for Estimating Posterior Distribution
9. 3.9 Comparison between Approaches
10. 3.10 Conclusions
11. References
4 Metropolis–Hastings and Slice Sampling for Finite Element Updating
1. 4.1 Introduction
2. 4.2 Likelihood, Prior and the Posterior Functions
3. 4.3 The Metropolis–Hastings Algorithm
4. 4.4 The Slice Sampling Algorithm
5. 4.5 Statistical Measures
6. 4.6 Application 1: Cantilevered Beam
7. 4.7 Application 2: Asymmetrical H‐Shaped Structure
8. 4.8 Conclusions
9. References
5 Dynamically Weighted Importance Sampling for Finite Element Updating
1. 5.1 Introduction
2. 5.2 Bayesian Modelling Approach
3. 5.3 Metropolis–Hastings (M‐H) Algorithm
4. 5.4 Importance Sampling
5. 5.5 Dynamically Weighted Importance Sampling
6. 5.6 Application 1: Cantilevered Beam
7. 5.7 Application 2: H‐Shaped Structure
8. 5.8 Conclusions
9. References
6 Adaptive Metropolis–Hastings for Finite Element Updating
1. 6.1 Introduction
2. 6.2 Adaptive Metropolis–Hastings Algorithm
3. 6.3 Application 1: Cantilevered Beam
4. 6.4 Application 2: Asymmetrical H‐Shaped Beam
5. 6.5 Application 3: Aircraft GARTEUR Structure
6. 6.6 Conclusion
7. References
7 Hybrid Monte Carlo Technique for Finite Element Model Updating
1. 7.1 Introduction
2. 7.2 Hybrid Monte Carlo Method
3. 7.3 Properties of the HMC Method
4. 7.4 The Molecular Dynamics Algorithm
5. 7.5 Improving the HMC
6. 7.6 Application 1: Cantilever Beam
7. 7.7 Application 2: Asymmetrical H‐Shaped Structure
8. 7.8 Conclusion
9. References
8 Shadow Hybrid Monte Carlo Technique for Finite Element Model Updating
1. 8.1 Introduction
2. 8.2 Effect of Time Step in the Hybrid Monte Carlo Method
3. 8.3 The Shadow Hybrid Monte Carlo Method
4. 8.4 The Shadow Hamiltonian
5. 8.5 Application: GARTEUR SM‐AG19 Structure
6. 8.6 Conclusion
7. References
9 Separable Shadow Hybrid Monte Carlo in Finite Element Updating
1. 9.1 Introduction
2. 9.2 Separable Shadow Hybrid Monte Carlo
3. 9.3 Theoretical Justifications of the S2HMC Method
4. 9.4 Application 1: Asymmetrical H‐Shaped Structure
5. 9.5 Application 2: GARTEUR SM‐AG19 Structure
6. 9.6 Conclusions
7. References
10 Evolutionary Approach to Finite Element Model Updating
1. 10.1 Introduction
2. 10.2 The Bayesian Formulation
3. 10.3 The Evolutionary MCMC Algorithm
4. 10.4 Metropolis–Hastings Method
5. 10.5 Application: Asymmetrical H‐Shaped Structure
6. 10.6 Conclusion
7. References
11 Adaptive Markov Chain Monte Carlo Method for Finite Element Model Updating
1. 11.1 Introduction
2. 11.2 Bayesian Theory
3. 11.3 Adaptive Hybrid Monte Carlo
4. 11.4 Application 1: A Linear System with Three Degrees of Freedom
5. 11.5 Application 2: Asymmetrical H‐Shaped Structure
6. 11.6 Conclusion
7. References
12 Conclusions and Further Work
1. 12.1 Introduction
2. 12.2 Further Work
3. References
Appendix A: Experimental Examples
1. A.1 Cantilevered Beam
2. A.2 H‐Shaped Structure Simulation
3. A.3 GARTEUR SM‐AG19 Structure
4. References
Appendix B: Markov Chain Monte Carlo
1. B.1 Introduction
2. B.2 Basic Definition of the Markov Chain
3. B.3 Invariant Distribution
4. B.4 Reversibility and Ergodicity
5. References
Appendix C: Gaussian Distribution
1. C.1 Introduction
2. C.2 Gaussian Distribution
3. C.3 Properties of the Gaussian Distribution
4. References
Index
End User License Agreement

List of Tables

Chapter 01
1. Table 1.1 Comparison of finite element model and real measurements
Chapter 02
1. Table 2.1 Finite element model updating of asymmetrical H‐shaped structure using simulated annealing and regularised objective function
2. Table 2.2 Finite element model updating of asymmetrical H‐shaped structure using simulated annealing and cross‐validation
3. Table 2.3 Bayes factors for asymmetrical H‐shaped structure
Chapter 03
1. Table 3.1 Types of uncertainty problems (Adhikari, 2015)
2. Table 3.2 Final remarks regarding the overall study
3. Table 3.3 The updated value of the spring stiffness using maximum likelihood, Gaussian approximation and simulated annealing techniques
4. Table 3.4 Frequencies and errors when maximum likelihood, Gaussian approximation and simulated annealing techniques used to update spring stiffness
Chapter 04
1. Table 4.1 The updated vector of Young’s modulus using the M‐H and SS techniques
2. Table 4.2 Frequencies and errors when the M‐H and SS techniques used to update Young’s modulus
3. Table 4.3 Initial and updated parameters using the SS and M‐H algorithms
4. Table 4.4 Natural frequencies and errors when the SS and M‐H algorithms were used to update the parameters
Chapter 05
1. Table 5.1 MCDWIS algorithm parameters
2. Table 5.2 Variable parameter vector results
3. Table 5.3 Cantilever frequency results
4. Table 5.4 MCDWIS algorithm parameters
5. Table 5.5 Updating vector
6. Table 5.6 Natural frequencies
Chapter 06
1. Table 6.1 Cantilever frequency results
2. Table 6.2 Cantilevered beam frequency results
3. Table 6.3 H‐beam variable parameter vector results
4. Table 6.4 Beam frequency results
5. Table 6.5 Updating vector results
6. Table 6.6 Natural frequency results
Chapter 07
1. Table 7.1 The updated vector of Young’s modulus using HMC technique
2. Table 7.2 Results when the HMC technique is used to update Young’s modulus
3. Table 7.3 Initial and updated parameters using the HMC algorithm
4. Table 7.4 Results when the HMC algorithm was used to update the parameters
Chapter 08
1. Table 8.1 The initial values of the updating parameters for the GARTEUR example
2. Table 8.2 The bounds of the updating parameters for the GARTEUR example
3. Table 8.3 Initial and updated parameter values for HMC and SHMC^4,8 algorithms at
4. Table 8.4 Initial and updated parameter values for HMC and SHMC^4,8 algorithms at the
5. Table 8.5 Modal results for the HMC and SHMC^4,8 algorithms at
6. Table 8.6 Modal results and errors for HMC and SHMC^4,8 algorithms for a time step of 4.8 ms
Chapter 09
1. Table 9.1 The momentum, MD and reweighting steps for HMC, SHMC and S2HMC algorithms
2. Table 9.2 Initial and updated parameters using HMC, SHMC and S2HMC
3. Table 9.3 Natural frequencies and errors when HMC, SHMC and S2HMC are used to update the parameters
4. Table 9.4 Initial and updated parameter values for the HMC, SHMC and S2HMC algorithms at two different time steps
5. Table 9.5 Modal results and errors for HMC, SHMC and S2HMC at two different time steps
Chapter 10
1. Table 10.1 Initial and updated parameters using EMCMC and M‐H algorithms
2. Table 10.2 Natural frequencies when using EMCMC and M‐H algorithms
3. Table 10.3 Errors when using EMCMC and M‐H algorithms
Chapter 11
1. Table 11.1 The updated vector of stiffness parameters using the AHMC technique
2. Table 11.2 Frequencies and errors when the AHMC technique is implemented to update the stiffness parameters
3. Table 11.3 Initial and updated parameters using the AHMC algorithm
4. Table 11.4 Natural frequencies and errors when the AHMC algorithm is implemented to update the structure

List of Illustrations

Chapter 01
1. Figure 1.1 A finite element model of a cylindrical shell
2. Figure 1.2 Finite element model updating procedure
Chapter 02
1. Figure 2.1 The multifold cross‐validation technique applied where the model is updating K times, each time leaving out the data indicated by the shaded area and not initialising the finite element model parameter when moving to the next updating stage
Chapter 03
1. Figure 3.1 How to handle uncertainty (Adhikari, 2015)
2. Figure 3.2 A single‐degree‐of‐freedom system
3. Figure 3.3 A two‐degrees‐of‐freedom system
Chapter 04
1. Figure 4.1 Target distribution and histogram of the M‐H samples with 100 iterations
2. Figure 4.2 Target distribution and histogram of the M‐H samples with 1000 iterations
3. Figure 4.3 Slice sampling approach
4. Figure 4.4 Scatter plot of θ₁ versus θ₂with marginal histograms using the M‐H method
5. Figure 4.5 Scatter plot of θ₃ versus θ₄ with marginal histograms using the M‐H method
6. Figure 4.6 Scatter plot of θ₁ versus θ₂ with marginal histograms using the SS method
7. Figure 4.7 Scatter plot of θ₃ versus θ₄ with marginal histograms using the SS method
8. Figure 4.8 The correlation between the updated parameters using the M‐H algorithm
9. Figure 4.9 The correlation between the updated parameters using the SS algorithm
10. Figure 4.10 The correlation between the updated parameters using the SS algorithm
11. Figure 4.11 The correlation between the updated parameters using the M‐H algorithm
Chapter 05
1. Figure 5.1 Sample distribution
2. Figure 5.2 Population size adaptation
3. Figure 5.3 Log weight of sixth state
4. Figure 5.4 Log weight of 10th state
5. Figure 5.5 W_up adaptation
6. Figure 5.6 Correlation of samples
7. Figure 5.7 Multidimensional array with each page a matrix of samples for each variable parameter
8. Figure 5.8 Log weight of the sixth state
9. Figure 5.9 Log weight of the 10th state
10. Figure 5.10 W_up adaptation
11. Figure 5.11 Population size
12. Figure 5.12 Correlation between samples
Chapter 06
1. Figure 6.1 Adaptation of scaling parameter
2. Figure 6.2 Adaptation of the covariance matrix
3. Figure 6.3 Parameter correlation
4. Figure 6.4 Adaptation of (a) first updating parameter and (b) probability density estimation
5. Figure 6.5 Adaption of scaling factor
6. Figure 6.6 Adaption of covariance matrix
7. Figure 6.7 Adaption of (a) first parameter and (b) the kernel density estimate
8. Figure 6.8 3D bar graph of the parameter correlation
9. Figure 6.9 Adaptation of the scaling factor
10. Figure 6.10 Adaptation of the covariance matrix
11. Figure 6.11 Adaptation of (a) second updated parameter and (b) the kernel density
12. Figure 6.12 Correlation between different parameters
13. Figure 6.13 Adaptation of population size
14. Figure 6.14 The sixth state of the log weight
15. Figure 6.15 The 10th state of the log weight
Chapter 07
1. Figure 7.1 Scatter plots with marginal histograms using the HMC method. Scatter plot of θ₁ versus θ₂
2. Figure 7.2 Scatter plots with marginal histograms using the HMC method. Scatter plot of θ₃ versus θ₄
3. Figure 7.3 The correlation between the updated parameters (HMC algorithm)
4. Figure 7.4 Scatter plots with marginal histograms using the HMC method. Scatter plot of θ₄ versus θ₂
5. Figure 7.5 Scatter plots with marginal histograms using the HMC method. Scatter plot of θ₅ versus θ₆
6. Figure 7.6 The correlation between the updated parameters (HMC algorithm)
Chapter 08
1. Figure 8.1 Normal probability plot for ρ (HMC)
2. Figure 8.2 Normal probability plot for ρ (SHMC⁴)
3. Figure 8.3 Normal probability plot for ρ (SHMC⁸)
4. Figure 8.4 A two‐dimensional histogram of ρ versus LI_max (HMC)
5. Figure 8.5 A two‐dimensional histogram of ρ versus LI_max (SHMC⁴)
6. Figure 8.6 A two‐dimensional histogram of ρ versus LI_max (SHMC⁸)
7. Figure 8.7 The correlation between the updated parameters (HMC)
8. Figure 8.8 The correlation between the updated parameters (SHMC⁸)
9. Figure 8.9 The total average error for HMC and SHMC^4,8 for different time steps
10. Figure 8.10 The acceptance rate obtained for HMC and SHMC^4,8 for different time steps
Chapter 09
1. Figure 9.1 Scatter plot of A_x1 versus I_x2 with marginal histograms using S2HMC methods
2. Figure 9.2 Scatter plot of A_x2 versus A_x3 with marginal histograms using S2HMC methods
3. Figure 9.3 The correlation between the updated parameters (S2HMC algorithm)
4. Figure 9.4 The total average error using the HMC, SHMC and S2HMC methods
5. Figure 9.5 The acceptance rate obtained for different time steps using the HMC, SHMC and S2HMC methods
6. Figure 9.6 Histograms of updating model parameters ρ using the S2HMC method. The normalisation constant θ₁⁰ is the initial (mean) values
7. Figure 9.7 Histograms of updating model parameters using the S2HMC method
8. Figure 9.8 Histograms of updating model parameters LI_min using the S2HMC method
9. Figure 9.9 Histograms of updating model parameters LI_max using the S2HMC method
10. Figure 9.10 The correlation between the updated parameters (S2HMC algorithm)
11. Figure 9.11 The total average error using the HMC, SHMC and S2HMC methods
12. Figure 9.12 The acceptance rate obtained for different time steps using HMC, SHMC and S2HMC methods.
Chapter 10
1. Figure 10.1 The correlation between the updated parameters (EMCMC algorithm)
2. Figure 10.2 The correlation between the updated parameters (M‐H algorithm)
Chapter 11
1. Figure 11.1 A mass‐and‐spring linear system with three degrees of freeedom
2. Figure 11.2 Scatter plots with error ellipses using the AHMC method
3. Figure 11.3 The total average error using the AHMC algorithm
4. Figure 11.4 The kernel smoothing density estimation of updating model parameters using the AHMC method
5. Figure 11.5 The correlation between uncertain parameters
6. Figure 11.6 The total average error for the AHMC algorithm
Appendix A
1. Figure A.1 Cantilevered beam mode 1
2. Figure A.2 Cantilevered beam mode 2
3. Figure A.3 Cantilevered beam mode 3
4. Figure A.4 Cantilevered beam mode 4
5. Figure A.5 Cantilevered beam mode 5
6. Figure A.6 The H‐shaped aluminium structure
7. Figure A.7 H‐shaped aluminium structure mode 1
8. Figure A.8 H‐shaped aluminium structure mode 2
9. Figure A.9 H‐shaped aluminium structure mode 3
10. Figure A.10 H‐shaped aluminium structure mode 4
11. Figure A.11 H‐shaped aluminium structure mode 5
12. Figure A.12 Aeroplane structure mode 1
13. Figure A.13 Aeroplane structure mode 2
14. Figure A.14 Aeroplane structure mode 3
15. Figure A.15 Aeroplane structure mode 4
16. Figure A.16 Aeroplane structure mode 5
17. Figure A.17 Aeroplane structure mode 6
18. Figure A.18 Aeroplane structure mode 7
19. Figure A.19 Aeroplane structure mode 8
20. Figure A.20 Aeroplane structure mode 9
21. Figure A.21 Aeroplane structure mode 10
Appendix C
1. Figure C.1 Graph of a probability distribution function for a univariate Gaussian distribution function
2. Figure C.2 Graph of a probability distribution function graph for a bivariate Gaussian distribution function

This edition first published 2017
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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.

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