Probabilistic Finite Element Model Updating Using Bayesian Statistics E-Book

Tshilidzi Marwala

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 430

Veröffentlichungsjahr: 2016

Beschreibung

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.

Leseprobe

Table of Contents

Cover

Title Page

Acknowledgements

Nomenclature

1 Introduction to Finite Element Model Updating

1.1 Introduction

1.2 Finite Element Modelling

1.3 Vibration Analysis

1.4 Finite Element Model Updating

1.5 Finite Element Model Updating and Bounded Rationality

1.6 Finite Element Model Updating Methods

1.7 Bayesian Approach

versus

Maximum Likelihood Method

1.8 Outline of the Book

References

2 Model Selection in Finite Element Model Updating

2.1 Introduction

2.2 Model Selection in Finite Element Modelling

2.3 Simulated Annealing

2.4 Asymmetrical H‐Shaped Structure

2.5 Conclusion

References

3 Bayesian Statistics in Structural Dynamics

3.1 Introduction

3.2 Bayes’ Rule

3.3 Maximum Likelihood Method

3.4 Maximum a Posteriori Parameter Estimates

3.5 Laplace’s Method

3.6 Prior, Likelihood and Posterior Function of a Simple Dynamic Example

3.7 The Posterior Approximation

3.8 Sampling Approaches for Estimating Posterior Distribution

3.9 Comparison between Approaches

3.10 Conclusions

References

4 Metropolis–Hastings and Slice Sampling for Finite Element Updating

4.1 Introduction

4.2 Likelihood, Prior and the Posterior Functions

4.3 The Metropolis–Hastings Algorithm

4.4 The Slice Sampling Algorithm

4.5 Statistical Measures

4.6 Application 1: Cantilevered Beam

4.7 Application 2: Asymmetrical H‐Shaped Structure

4.8 Conclusions

References

5 Dynamically Weighted Importance Sampling for Finite Element Updating

5.1 Introduction

5.2 Bayesian Modelling Approach

5.3 Metropolis–Hastings (M‐H) Algorithm

5.4 Importance Sampling

5.5 Dynamically Weighted Importance Sampling

5.6 Application 1: Cantilevered Beam

5.7 Application 2: H‐Shaped Structure

5.8 Conclusions

References

6 Adaptive Metropolis–Hastings for Finite Element Updating

6.1 Introduction

6.2 Adaptive Metropolis–Hastings Algorithm

6.3 Application 1: Cantilevered Beam

6.4 Application 2: Asymmetrical H‐Shaped Beam

6.5 Application 3: Aircraft GARTEUR Structure

6.6 Conclusion

References

7 Hybrid Monte Carlo Technique for Finite Element Model Updating

7.1 Introduction

7.2 Hybrid Monte Carlo Method

7.3 Properties of the HMC Method

7.4 The Molecular Dynamics Algorithm

7.5 Improving the HMC

7.6 Application 1: Cantilever Beam

7.7 Application 2: Asymmetrical H‐Shaped Structure

7.8 Conclusion

References

8 Shadow Hybrid Monte Carlo Technique for Finite Element Model Updating

8.1 Introduction

8.2 Effect of Time Step in the Hybrid Monte Carlo Method

8.3 The Shadow Hybrid Monte Carlo Method

8.4 The Shadow Hamiltonian

8.5 Application: GARTEUR SM‐AG19 Structure

8.6 Conclusion

References

9 Separable Shadow Hybrid Monte Carlo in Finite Element Updating

9.1 Introduction

9.2 Separable Shadow Hybrid Monte Carlo

9.3 Theoretical Justifications of the S2HMC Method

9.4 Application 1: Asymmetrical H‐Shaped Structure

9.5 Application 2: GARTEUR SM‐AG19 Structure

9.6 Conclusions

References

10 Evolutionary Approach to Finite Element Model Updating

10.1 Introduction

10.2 The Bayesian Formulation

10.3 The Evolutionary MCMC Algorithm

10.4 Metropolis–Hastings Method

10.5 Application: Asymmetrical H‐Shaped Structure

10.6 Conclusion

References

11 Adaptive Markov Chain Monte Carlo Method for Finite Element Model Updating

11.1 Introduction

11.2 Bayesian Theory

11.3 Adaptive Hybrid Monte Carlo

11.4 Application 1: A Linear System with Three Degrees of Freedom

11.5 Application 2: Asymmetrical H‐Shaped Structure

11.6 Conclusion

References

12 Conclusions and Further Work

12.1 Introduction

12.2 Further Work

References

Appendix A: Experimental Examples

A.1 Cantilevered Beam

A.2 H‐Shaped Structure Simulation

A.3 GARTEUR SM‐AG19 Structure

References

Appendix B: Markov Chain Monte Carlo

B.1 Introduction

B.2 Basic Definition of the Markov Chain

B.3 Invariant Distribution

B.4 Reversibility and Ergodicity

References

Appendix C: Gaussian Distribution

C.1 Introduction

C.2 Gaussian Distribution

C.3 Properties of the Gaussian Distribution

References

Index

End User License Agreement

List of Tables

Chapter 01

Table 1.1 Comparison of finite element model and real measurements

Chapter 02

Table 2.1 Finite element model updating of asymmetrical H‐shaped structure using simulated annealing and regularised objective function

Table 2.2 Finite element model updating of asymmetrical H‐shaped structure using simulated annealing and cross‐validation

Table 2.3 Bayes factors for asymmetrical H‐shaped structure

Chapter 03

Table 3.1 Types of uncertainty problems (Adhikari, 2015)

Table 3.2 Final remarks regarding the overall study

Table 3.3 The updated value of the spring stiffness using maximum likelihood, Gaussian approximation and simulated annealing techniques

Table 3.4 Frequencies and errors when maximum likelihood, Gaussian approximation and simulated annealing techniques used to update spring stiffness

Chapter 04

Table 4.1 The updated vector of Young’s modulus using the M‐H and SS techniques

Table 4.2 Frequencies and errors when the M‐H and SS techniques used to update Young’s modulus

Table 4.3 Initial and updated parameters using the SS and M‐H algorithms

Table 4.4 Natural frequencies and errors when the SS and M‐H algorithms were used to update the parameters

Chapter 05

Table 5.1 MCDWIS algorithm parameters

Table 5.2 Variable parameter vector results

Table 5.3 Cantilever frequency results

Table 5.4 MCDWIS algorithm parameters

Table 5.5 Updating vector

Table 5.6 Natural frequencies

Chapter 06

Table 6.1 Cantilever frequency results

Table 6.2 Cantilevered beam frequency results

Table 6.3 H‐beam variable parameter vector results

Table 6.4 Beam frequency results

Table 6.5 Updating vector results

Table 6.6 Natural frequency results

Chapter 07

Table 7.1 The updated vector of Young’s modulus using HMC technique

Table 7.2 Results when the HMC technique is used to update Young’s modulus

Table 7.3 Initial and updated parameters using the HMC algorithm

Table 7.4 Results when the HMC algorithm was used to update the parameters

Chapter 08

Table 8.1 The initial values of the updating parameters for the GARTEUR example

Table 8.2 The bounds of the updating parameters for the GARTEUR example

Table 8.3 Initial and updated parameter values for HMC and SHMC

4,8

algorithms at

Table 8.4 Initial and updated parameter values for HMC and SHMC

4,8

algorithms at the

Table 8.5 Modal results for the HMC and SHMC

4,8

algorithms at

Table 8.6 Modal results and errors for HMC and SHMC

4,8

algorithms for a time step of 4.8 ms

Chapter 09

Table 9.1 The momentum, MD and reweighting steps for HMC, SHMC and S2HMC algorithms

Table 9.2 Initial and updated parameters using HMC, SHMC and S2HMC

Table 9.3 Natural frequencies and errors when HMC, SHMC and S2HMC are used to update the parameters

Table 9.4 Initial and updated parameter values for the HMC, SHMC and S2HMC algorithms at two different time steps

Table 9.5 Modal results and errors for HMC, SHMC and S2HMC at two different time steps

Chapter 10

Table 10.1 Initial and updated parameters using EMCMC and M‐H algorithms

Table 10.2 Natural frequencies when using EMCMC and M‐H algorithms

Table 10.3 Errors when using EMCMC and M‐H algorithms

Chapter 11

Table 11.1 The updated vector of stiffness parameters using the AHMC technique

Table 11.2 Frequencies and errors when the AHMC technique is implemented to update the stiffness parameters

Table 11.3 Initial and updated parameters using the AHMC algorithm

Table 11.4 Natural frequencies and errors when the AHMC algorithm is implemented to update the structure

List of Illustrations

Chapter 01

Figure 1.1 A finite element model of a cylindrical shell

Figure 1.2 Finite element model updating procedure

Chapter 02

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

Figure 3.1 How to handle uncertainty (Adhikari, 2015)

Figure 3.2 A single‐degree‐of‐freedom system

Figure 3.3 A two‐degrees‐of‐freedom system

Chapter 04

Figure 4.1 Target distribution and histogram of the M‐H samples with 100 iterations

Figure 4.2 Target distribution and histogram of the M‐H samples with 1000 iterations

Figure 4.3 Slice sampling approach

Figure 4.4 Scatter plot of

θ

1

versus

θ

2

with marginal histograms using the M‐H method

Figure 4.5 Scatter plot of

θ

3

versus

θ

4

with marginal histograms using the M‐H method

Figure 4.6 Scatter plot of

θ

1

versus

θ

2

with marginal histograms using the SS method

Figure 4.7 Scatter plot of

θ

3

versus

θ

4

with marginal histograms using the SS method

Figure 4.8 The correlation between the updated parameters using the M‐H algorithm

Figure 4.9 The correlation between the updated parameters using the SS algorithm

Figure 4.10 The correlation between the updated parameters using the SS algorithm

Figure 4.11 The correlation between the updated parameters using the M‐H algorithm

Chapter 05

Figure 5.1 Sample distribution

Figure 5.2 Population size adaptation

Figure 5.3 Log weight of sixth state

Figure 5.4 Log weight of 10th state

Figure 5.5

W

up

adaptation

Figure 5.6 Correlation of samples

Figure 5.7 Multidimensional array with each page a matrix of samples for each variable parameter

Figure 5.8 Log weight of the sixth state

Figure 5.9 Log weight of the 10th state

Figure 5.10

W

up

adaptation

Figure 5.11 Population size

Figure 5.12 Correlation between samples

Chapter 06

Figure 6.1 Adaptation of scaling parameter

Figure 6.2 Adaptation of the covariance matrix

Figure 6.3 Parameter correlation

Figure 6.4 Adaptation of (a) first updating parameter and (b) probability density estimation

Figure 6.5 Adaption of scaling factor

Figure 6.6 Adaption of covariance matrix

Figure 6.7 Adaption of (a) first parameter and (b) the kernel density estimate

Figure 6.8 3D bar graph of the parameter correlation

Figure 6.9 Adaptation of the scaling factor

Figure 6.10 Adaptation of the covariance matrix

Figure 6.11 Adaptation of (a) second updated parameter and (b) the kernel density

Figure 6.12 Correlation between different parameters

Figure 6.13 Adaptation of population size

Figure 6.14 The sixth state of the log weight

Figure 6.15 The 10th state of the log weight

Chapter 07

Figure 7.1 Scatter plots with marginal histograms using the HMC method. Scatter plot of

θ

1

versus

θ

2

Figure 7.2 Scatter plots with marginal histograms using the HMC method. Scatter plot of

θ

3

versus

θ

4

Figure 7.3 The correlation between the updated parameters (HMC algorithm)

Figure 7.4 Scatter plots with marginal histograms using the HMC method. Scatter plot of

θ

4

versus

θ

2

Figure 7.5 Scatter plots with marginal histograms using the HMC method. Scatter plot of

θ

5

versus

θ

6

Figure 7.6 The correlation between the updated parameters (HMC algorithm)

Chapter 08

Figure 8.1 Normal probability plot for

ρ

(HMC)

Figure 8.2 Normal probability plot for

ρ

(SHMC

4

)

Figure 8.3 Normal probability plot for

ρ

(SHMC

8

)

Figure 8.4 A two‐dimensional histogram of

ρ

versus

LI

max

(HMC)

Figure 8.5 A two‐dimensional histogram of

ρ

versus

LI

max

(SHMC

4

)

Figure 8.6 A two‐dimensional histogram of

ρ

versus

LI

max

(SHMC

8

)

Figure 8.7 The correlation between the updated parameters (HMC)

Figure 8.8 The correlation between the updated parameters (SHMC

8

)

Figure 8.9 The total average error for HMC and SHMC

4,8

for different time steps

Figure 8.10 The acceptance rate obtained for HMC and SHMC

4,8

for different time steps

Chapter 09

Figure 9.1 Scatter plot of

A

x

1

versus

I

x

2

with marginal histograms using S2HMC methods

Figure 9.2 Scatter plot of

A

x

2

versus

A

x

3

with marginal histograms using S2HMC methods

Figure 9.3 The correlation between the updated parameters (S2HMC algorithm)

Figure 9.4 The total average error using the HMC, SHMC and S2HMC methods

Figure 9.5 The acceptance rate obtained for different time steps using the HMC, SHMC and S2HMC methods

Figure 9.6 Histograms of updating model parameters

ρ

using the S2HMC method. The normalisation constant

θ

1

0

is the initial (mean) values

Figure 9.7 Histograms of updating model parameters

using the S2HMC method

Figure 9.8 Histograms of updating model parameters

LI

min

using the S2HMC method

Figure 9.9 Histograms of updating model parameters

LI

max

using the S2HMC method

Figure 9.10 The correlation between the updated parameters (S2HMC algorithm)

Figure 9.11 The total average error using the HMC, SHMC and S2HMC methods

Figure 9.12 The acceptance rate obtained for different time steps using HMC, SHMC and S2HMC methods.

Chapter 10

Figure 10.1 The correlation between the updated parameters (EMCMC algorithm)

Figure 10.2 The correlation between the updated parameters (M‐H algorithm)

Chapter 11

Figure 11.1 A mass‐and‐spring linear system with three degrees of freeedom

Figure 11.2 Scatter plots with error ellipses using the AHMC method

Figure 11.3 The total average error using the AHMC algorithm

Figure 11.4 The kernel smoothing density estimation of updating model parameters using the AHMC method

Figure 11.5 The correlation between uncertain parameters

Figure 11.6 The total average error for the AHMC algorithm

Appendix A

Figure A.1 Cantilevered beam mode 1

Figure A.2 Cantilevered beam mode 2

Figure A.3 Cantilevered beam mode 3

Figure A.4 Cantilevered beam mode 4

Figure A.5 Cantilevered beam mode 5

Figure A.6 The H‐shaped aluminium structure

Figure A.7 H‐shaped aluminium structure mode 1

Figure A.8 H‐shaped aluminium structure mode 2

Figure A.9 H‐shaped aluminium structure mode 3

Figure A.10 H‐shaped aluminium structure mode 4

Figure A.11 H‐shaped aluminium structure mode 5

Figure A.12 Aeroplane structure mode 1

Figure A.13 Aeroplane structure mode 2

Figure A.14 Aeroplane structure mode 3

Figure A.15 Aeroplane structure mode 4

Figure A.16 Aeroplane structure mode 5

Figure A.17 Aeroplane structure mode 6

Figure A.18 Aeroplane structure mode 7

Figure A.19 Aeroplane structure mode 8

Figure A.20 Aeroplane structure mode 9

Figure A.21 Aeroplane structure mode 10

Appendix C

Figure C.1 Graph of a probability distribution function for a univariate Gaussian distribution function

Figure C.2 Graph of a probability distribution function graph for a bivariate Gaussian distribution function

Guide

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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

This edition first published 2017© 2017 John Wiley & Sons, Ltd

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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–dc23LC record available at https://lccn.loc.gov/2016019278

A catalogue record for this book is available from the British Library.

Cover image: Godruma/Gettyimages

Acknowledgements

We would like to thank the University of Johannesburg and the University of Swansea for contributing towards the writing of this book. We also would like to thank Michael Friswell, Linda Mthembu, Niel Joubert and Ishmael Msiza for contributing towards the writing of this book.

We dedicate this book to the schools that gave us the foundation to always seek excellence in everything we do: the University of Cambridge and the University of Johannesburg.

Tshilidzi Marwala, PhDJohannesburg1 February 2016

Ilyes Boulkaibet, PhD

Johannesburg

1 February 2016

Sondipon Adhikari, PhD

Swansea

1 February 2016

Nomenclature

AI

Artificial intelligence

AIC

Akaike information criterion

APEPCS

Adaptive pruned‐enriched population control scheme

AR

Acceptance rate

BFGS

Quasi‐Newton Broyden–Fletcher–Goldfarb–Shanno

BIC

Bayesian information criterion

CG

Conjugate gradient

c.o.v.

Coefficient of variation

DIC

Deviance information criterion

DOF

Degree of freedom

DWIS

Dynamically weighted importance sampling

FEM

Finite element model

FRF

Frequency response function

GA

Genetic algorithm

GS

Gibbs sampling

HMC

Hybrid Monte Carlo

MC

Markov chain

MCDWIS

Monte Carlo dynamically weighted importance sampling

MD

Molecular dynamics

MCMC

Markov chain Monte Carlo

M‐H

Metropolis–Hastings

ML

Maximum likelihood

MAP

Maximum a posteriori

NS

Nested sampling

PDF

Probability distribution function

PSO

Particle swarm optimisation

SA

Simulated annealing

SHMC

Shadow hybrid Monte Carlo

S2HMC

Separable shadow hybrid Monte Carlo

SS

Slice sampling

VV

Velocity verlet

N

Number of degrees of freedom

Z

X

Experimental data vector

Z

i

Analytical data vector

θ

Uncertain parameter vector

Dev

(

θ

)

Deviance of

θ

P

D

Posterior mean deviance parameter

S

Structure’s sensitivity matrix

J

Objective function

Z

Evidence

Acceleration

W

Weighting matrix

H

Hessian matrix

I

Unit matrix

η

Step size used by the conjugate gradient technique

V

Variance matrix

Ω

Diagonal matrix with diagonal elements of the natural frequencies

x

i

Chromosome vector or position vector

p

i

Best position

v

i

Velocity

d

Dimension of the updated vector

ℝ

One‐dimensional real domain

ℝ

n

n

‐dimensional real domain

‐dimensional real domain

T

Transformation matrix

Covariance matrix of the updated vector

θ

at the

j

th iteration

Covariance of the measured data

P

Probability function

Experimental model data

The posterior probability distribution function

Proposed probability distribution function

Transition matrix

Normal distribution with mean

μ

and variance

σ

Joint density

μ

f

Expectation value of the function

f

i

th measured natural frequency

i

th measured circular natural frequency

N

m

Number of measured modes

f

i

i

th analytical frequency obtained from the finite element model

j

Imaginary unit of a complex number

‖

Λ

‖

Euclidean norm of

Λ

λ

Lagrange multiplier

K

Bayes factor

E

i

Error vector

E

(·)

Mean value

E

(

zz

T

)

Variance matrix of

z

R

t

Normalisation constant ratio

X

m

The Fourier‐transformed displacement

F

m

Force matrix

W

Kinetic energy

V

Potential energy

Gradient of V

H

Hamiltonian function

H

[2

k

]

Shadow Hamiltonian function of order 2

k

p

Momentum vector

Gradient

K

B

Boltzmann constant

T

Temperature

δt

1Introduction 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 transform the data into the frequency domain. Modal analysis is applied to transform the data from the frequency domain to the modal domain. All of these domains include similar information, but each domain reveals different data representations.

1.3.1 Modal Domain Data

The modal domain expresses data as natural frequencies, damping ratios and mode shapes. The technique used for extracting the modal properties is a process called modal analysis (Ewins, 1995). Natural frequencies are basic characteristics of a system and can be extracted by exciting the structure and analysing the vibration response. Cawley and Adams (1979) used changes in the natural frequencies to identify damage in composite materials. Farrar et al. (1994) successfully used the shifts in natural frequencies to identify damage on an I‐40 bridge. Other successful applications of natural frequencies include damage detection in tabular steel offshore platforms (Messina et al., 1996, 1998), spot welding (Wang et al., 2008) and beam‐like structures (Zhong and Oyadiji, 2008; Zhong et al., 2008).

A mode shape represents the curvature of a system vibrating at a given mode and a particular natural frequency. West (1982) successfully applied the modal assurance criterion for damage on a Space Shuttle orbiter body flap, while Kim et al. (1992) successfully used the coordinate modal assurance criterion of Lieven and Ewins (1988) for damage detection in structures. Further applications of mode shapes include composite laminated plates (Araújo dos Santos et al., 2006; Qiao et al., 2007), linear structures (Fang and Perera, 2009), beam‐type structures (Qiao and Cao, 2008; Sazonov and Klinkhachorn, 2005), optical sensor configuration (Chang and Pakzad, 2015), multishell quantum dots (Vanmaekelbergh et al., 2015) and creep characterisation (Hao et al., 2015).

1.3.2 Frequency Domain Data

The measured excitation and response of the structure are converted into the frequency domain using Fourier transforms (Ewins, 1995; Maia and Silva, 1997), and from these the frequency response function is extracted. Frequency response functions have, in general, been used to identify faults (Sestieri and D’Ambrogio, 1989; Faverjon and Sinou, 2009). D’Ambrogio and Zobel (1994) used frequency response functions to identify the presence of faults in a truss structure, while Imregun et al. (1995) used frequency response functions for damage detection. Lyon (1995) and Schultz et al. (1996) used measured frequency response functions for structural diagnostics. Other direct applications of the frequency response functions include the work of Shone et al. (2009), Ni et al. (2006), X. Liu et al. (2009), White et al. (2009) and Todorovska and Trifunac (2008). Additional applications include missing‐data estimation (Ugryumova et al., 2015), identification of a non‐commensurable fractional transfer (Valério and Tejado, 2015), as well as damage detection (Link and Zimmerman, 2015).

1.4 Finite Element Model Updating

In real life, it turns out that the predictions of the finite element model are quite different from the measurements. As an example, for a finite element model of a simply suspended beam, the differences between the model‐predicticted natural frequencies and the measured frequencies are shown in Table 1.1 (Marwala and Sibisi, 2005; Marwala, 2010). These results are for a fairly easy structure to model, and they demonstrate that the finite element model’s data are different from the measured data. Finite element model updating has been studied quite extensively (Friswell and Mottershead, 1995; Mottershead and Friswell, 1993; Maia and Silva, 1997; Marwala, 2010). There are three approaches used in finite element model updating: direct methods, iterative deterministic and uncertainty quantification methods. Direct approaches are computationally inexpensive, but reproduce modal properties that are physically unrealistic.

Table 1.1 Comparison of finite element model and real measurements

Mode number

Finite element frequencies (Hz)

Measured frequencies (Hz)

1

42.30

41.50

2

117.0

114.5

3

227.3

224.5

4

376.9

371.6

Although the finite element model can predict measured quantities, the updated model is limited in that it loses the connectivity of nodes, results in populated matrices and in loss of matrix symmetry. All these factors are physically unrealistic. Iterative techniques use changes in physical parameters to update the finite element models and, thereby, generate models that are physically realistic (Marwala, 2010). However, since they are based on optimisation techniques, the problems of global versus local optimum solution and over‐fitting the measured data, these methods still produce unrealistic results. Esfandiari et al. (2009) used the sensitivity approach, frequency response functions and natural frequencies for model updating in structures, while Wang et al. (2009) used the Zernike moment descriptor for finite element model updating. Yuan and Dai (2009) updated a finite element model of damped gyroscopic systems, while Kozak et al. (2009) used a miscorrelation index for model updating. Arora et al. (2009) proposed a finite element model updating approach that used damping matrices, while Schlune et al. (2009) implemented finite element model updating to improve bridge evaluation. Yang et al. (2009) investigated several objective functions for finite element model updating in structures, while Bayraktar et al. (2009) applied modal properties for updating a finite element model of a bridge. Li and Du (2009) used the most sensitive design variable for finite element model updating of stay‐cables, while Steenackers et al. (2007) successfully applied transmissibility for finite element model updating. Xu Yuan and Ping Yu (2015) proposed finite element model updating of damped structures, while Shabbir and Omenzetter (2015) applied particle swarm optimisation for finite element model updating. The uncertainty quantification techniques, however, include the uncertainties related to the modelled structure (or systems) during the updating procedure. The uncertainty quantification approaches that treat uncertain parameters as random parameters with joint distribution functions are called the probabilistic techniques and these comprise Bayesian and perturbation methods, whereas the non‐probabilistic (possibilistic) approaches use the interval method or membership functions (fuzzy technique) to define uncertain parameters. In this book, only the Bayesian approach is used to update structures.

Other successful implementations of finite element model updating methods include applications in bridges (Huang and Zhu, 2008; Jaishi et al., 2007; Niu et al., 2015), composite structures (Pavic et al., 2007), helicopters (Shahverdi et al., 2006), atomic force microscopes (Chen, 2006), footbridges (Živanović et al., 2007), estimating constituent‐level elastic parameters of plates (Mishra and Chakraborty, 2015) and identifying temperature‐dependent thermal‐structural properties (Sun et al., 2015). The process of finite element updating is illustrated in Figure 1.2 (Boulkaibet, 2014).

Figure 1.2 Finite element model updating procedure

1.5 Finite Element Model Updating and Bounded Rationality

As illustrated in Figure 1.2, optimisation involves minimising the distance between measurements and the model output, whichever way the model is defined, whether deterministically or probabilistically. The minimisation process gives either a local optimum solution or a global one, and one is never sure whether the solution is global or local, particularly for complex problems. Furthermore, the data to be used should be universally represented, meaning that all the domain representations must be used, and this is not possible. A definition of ‘rational solution’ implies that the solution is optimised, all information is used and an optimal objective function for optimisation is used. In finite element model updating, this is not possible.

In rational theory, the limited availability of information required in making a rational decision, and the limitations of devices for making sense of incomplete decisions, are covered by the theory of bounded rationality, and it was proposed by Herbert Simon (Simon, 1957, 1990, 1991; Tisdell, 1996). The theory of bounded rationality has been used in modelling by researchers such as Lee (2013), Gama (2013), Jiang et al. (2013), Stanciu‐Viziteu (2012), Aviad and Roy (2012) and Murata et al. (2012). Herbert Simon coined the term satisficing, by combining the terms ‘satisfying’ and ‘sufficing’, which is the concept of making optimised decisions under the limitations that the data used in making such decisions are imperfect and incomplete, while the model used to make such decisions is inconsistent and imperfect. In the same vein, the finite element model updating problem is a satisficing problem, not a process of seeking the correct model.

1.6 Finite Element Model Updating Methods

This section reviews methods that have been used for finite element model updating. They are grouped into classes, and more details on these may be found in Marwala (2010). There are three categories of finite element model updating techniques: direct methods; iterative methods; and uncertainty quantification methods.

1.6.1 Direct Methods

Direct methods (Friswell and Mottershead, 1995; Marwala, 2010) are one of the earliest strategies used for finite element model updating. They possess the ability to reproduce the exact experimental data and without using iterations, which makes these algorithms computationally efficient. These methods are still used for finite element model updating, and modern instruments and sensors that have lately been used in experiments allow these methods to overcome some of their disadvantages, such as lack of node connectivity and the need for a large amount of data to reproduce the exact experimental matrices. In this subsection, several direct methods – the matrix update methods, the Lagrange multiplier method, the optimal matrix methods and the eigenstructure assignment method – are briefly described.

1.6.1.1 Matrix Update Methods

Matrix update methods operate by modifying structural model matrices, that is, the mass, stiffness and damping matrices (Baruch, 1978). These are obtained by minimising the distance between analytical and measured matrices as follows (Friswell and Mottershead, 1995; Marwala, 2010):

(1.1)

where M is the mass matrix, C is the damping matrix, K is the stiffness matrix of the structure, Ei is the error vector (also known as the residual force), , ωi is the ith natural frequency and ϕi is the ith mode shape. The residual force is the harmonic force with which the unupdated model will have to be excited at a frequency of ωi so that the structure will respond with the mode shape ϕi. The Euclidean norm of Ei is minimised by updating physical parameters in the model (Ewins, 1995; Marwala, 2010) and choosing an optimisation routine. These techniques are classified as iterative since they are employed by iteratively changing the relevant parameters until the error is minimised. Ojalvo and Pilon (1988) minimised the Euclidean norm of the residual force for the ith mode of the structure by using the modal properties. The residual force in the equation of motion in the frequency domain may be minimised as (Friswell and Mottershead, 1995):

(1.2)

where Xm and Fm are the Fourier‐transformed displacement and force matrices, respectively. Each column of the matrix corresponds to a measured frequency point. The Euclidean norm of the error matrix E is minimised by updating physical parameters in the model. The methods described in this subsection are computationally expensive. In addition, it is challenging to identify a global minimum because of multiple stationary points, which are caused by the non‐unique nature of inverse problems (Janter and Sas, 1990; Mares and Surace, 1996; Friswell et al., 1994; Dunn, 1998).

1.6.1.2 Lagrange Multiplier Method

The Lagrange multiplier method is an optimisation technique that deals with the objective function and constraints of an optimisation equation (Rad, 1997). It is implemented by minimising a constrained objective function, where the constraints are imposed by Lagrange multipliers (Marwala, 2010; Minas and Inman, 1988; Heylen and Sas, 1987).

1.6.1.3 Optimal Matrix Methods

Optimal matrix methods employ analytical rather than numerical solutions to obtain matrices from the damaged systems. They are formulated using Lagrange multipliers and perturbation matrices, and the optimisation problem is posed to minimise (Friswell and Mottershead, 1995)

(1.3)

where E is the objective function, λ is the Lagrange multiplier, R is the constraint of the equation and Δ denotes the perturbation of the system matrices. Different permutations of perturbations are tried until the difference between the finite element model results and the measured results is minimised. Baruch and Bar Itzhack (1978), Berman and Nagy (1983) and Kabe (1985) formulated Equation 1.3 by minimising the Frobenius norm of the error, while maintaining the symmetry of the matrices. McGowan et al. (1990) introduced an extra constraint that maintained the connectivity of the structure and used measured mode shapes to update the stiffness matrix to locate structural damage. Zimmerman et al. (1995) used a partitioning method for matrix perturbations as sums of element or substructural perturbation matrices to reduce the rank of unknown perturbation matrices. Carvalho et al. (2007) successfully applied a direct method for model updating with incomplete measured modal data. A limitation of these approaches is that the updated model is physically unrealistic.

1.6.1.4 Eigenstructure Assignment Methods

Eigenstructure assignment approaches are based on control system theory, and the system under consideration is made to respond in a predetermined configuration. An updated finite element model is that with eigenstructure which is obtained from measured data. Zimmerman and Kaouk (1992) applied this approach successfully to update a finite element model of a cantilevered beam based on modal properties, while Schultz et al. (1996) updated a finite element model using the measured frequency response functions. The limitation of this technique that the number of sensor locations is less than the degrees of freedom in the finite element model. To deal with this limitation, either the mode shapes and frequency response functions are expanded to the size of the finite element model or the mass and stiffness matrices of the finite element model are reduced to the size of the measured data. The reduction/expansion approaches that are applied are static reduction (Guyan, 1965; Gysin, 1990; Imregun and Ewins, 1993), dynamic reduction (Paz, 1984), improved reduced systems (O’Callahan, 1989) and the system‐equivalent reduction process (O’Callahan et al., 1989).

1.6.2 Iterative Methods

Iterative methods (Friswell and Mottershead, 1995; Marwala, 2010) were developed to overcome the weakness of the direct methods and to update finite element models of complex systems. These methods use non‐linear equations to deal with the non‐convex optimisation problem which arises when a complex system is updated. In these methods, a set of parameters are iteratively adjusted to minimise an objective function (also called a penalty function), where most of the objective functions used in model updating contain only modal and/or response functions data. In this subsection, two popular iterative methods are briefly discussed.

1.6.2.1 Sensitivity Methods

Sensitivity approaches work on the premise that experimentally measured data are perturbations of design data around a finite element model. Therefore, experimentally measured data ought to be approximately equal to data predicted by the finite element model for this approach to work. These approaches uses the derivatives of either the modal properties or the frequency response functions as a basis for finite element model updating. Many procedures have been developed to calculate the derivative of the modal properties and frequency response functions, including Fox and Kapoor (1968), Norris and Meirovitch (1989), Haug and Choi (1984), Chen and Garba (1980) and Adhikari and Friswell (2001). Ben‐Haim and Prells (1993) used frequency response function sensitivity to update a finite element model, while Lin et al. (1995) used modal sensitivity for finite element model updating and Hemez (1993) used element sensitivity for finite element updating. Alvin (1997) improved the convergence rate by using statistical confidence measurements in finite element model updating.

1.6.2.2 Optimisation Methods

Huang and Zhu (2008) applied optimisation methods for the finite element model updating of bridge structures. The optimisation method was augmented by a sensitivity analysis. Schwarz et al. (2007) updated a finite element model which minimised the difference between the modes of a finite element model and those from the experiment. Bakir et al. (2007) applied sensitivity approaches for finite element model updating. They used a constrained optimisation method to minimise the differences between the natural frequencies and mode shape.

Jaishi and Ren (2007) applied a multi‐objective optimisation approach for finite element model updating. Their multi‐objective cost function was based on the differences between eigenvalues and strain energy. Liu et al. (2006) updated a finite element model of a 14‐bay beam with semi‐rigid joints and a boundary using a hybrid optimisation method. Zhang and Huang (2008) applied a gradient descent optimisation method for the finite element model updating of bridge structures. The objective function was formulated as the summation of the frequency difference and modal shapes. Parameter alteration was guided by engineering judgement.

1.6.3 Artificial Intelligence Methods

Finite element modelling updating can be achieved through the use of artificial intelligence techniques. Artificial intelligence techniques are computational tools that are inspired by the way nature and biological systems work. Within the context of finite element model updating, some of the techniques that have been applied are genetic algorithms, particle swarm optimisation, fuzzy logic, neural networks, and support vector machines. A genetic algorithm simulates natural evolution, where the law of the survival of the fittest is applied to a population of individuals. This natural optimisation method is used for optimising a function (Mitchell, 1998). Particle swarm optimisation is an evolutionary optimisation method that was developed by Kennedy and Eberhart (1995), inspired by algorithms that model the flocking behaviour seen in birds. The response surface method is a procedure that functions by generating a response for a given input and then constructs an approximation to a complicated model such as a finite element model (Kamrani et al., 2009).

Finite element models are computationally expensive methods. To manage the computational load, some form of emulator to approximate the finite element model can be implemented. Y. Liu et al. (2009) used fuzzy theory, while Jung and Kim (2009) employed the hybrid genetic algorithm for finite element model updating. Tan et al. (2009) used support vector machines and wavelet data for finite element model updating in structures, while Zapico et al. (2008) applied neural networks. Further successful applications of artificial intelligence methods to finite element model updating include Tu and Lu (2008) and Yan et al. (2007), as well as Fei et al. (2006) who applied genetic algorithms. Feng et al. (2006) applied a hybrid of a genetic algorithm and simulated annealing, and He et al. (2008) applied a hybrid of a genetic algorithm and neural networks.

Marwala (2010) used the particle swarm optimisation technique for finite element model updating, and the results were compared to those obtained from the genetic algorithm. Furthermore, simulated annealing was also introduced and applied to finite element model updating, and the results were compared to those from particle swarm optimisation. To deal with the issue of computational efficiency, a response surface method that combines the multi‐layer perceptron and particle swarm optimisation was introduced and applied to finite element model updating. The results were compared to those from the genetic algorithm, particle swarm optimisation and simulated annealing.

1.6.4 Uncertainty Quantification Methods

Due to the numerical and experimental uncertainties associated with the updated models, formulating the updating problems as iterative optimisation with constraints may not produce stable and accurate results. Modelling uncertainties are caused by predictions used to model the systems, especially when the physical components used to model the systems are complex and not sufficiently well understood. On the other hand, experimental uncertainties are caused by noise resulting from the measurements or by the variability of the system parameters (Der Kiureghian and Ditlevsen, 2009; Soize, 2010; Walker et al., 2003). In this subsection, the perturbation method, minimum variance method and Bayesian approach are briefly described.

1.6.4.1 Perturbation Method

The perturbation technique uses a Taylor series to extend the terms in model updating equations around a predefined point and then to estimate the mean and variance of the updated parameters (Khodaparast, 2010; Hua et al., 2008; Khodaparast et al., 2008). One type of perturbation technique uses the least‐squares method for stochastic finite element model updating, by assuming that the measured data and updating parameters are statistically independent. Another perturbation technique was developed by Hua et al. (2008) and assumes that the measured vector ZX can be obtained by adding a random component (ΔZX) to a deterministic component (mean value) as follows (Khodaparast, 2010; Hua et al., 2008; Boulkaibet, 2014):

(1.4)

where the perturbation vector (ΔZX) has zero mean and represents the uncertainty in the measured data. The structural parameters θ, the sensitivity matrix S and the predictions Z are defined around the mean value of these vectors and/or matrices as follows (Friswell and Mottershead, 1995; Khodaparast, 2010; Boulkaibet, 2014):

(1.5)

(1.6)

(1.7)