Uncertainty Quantification in Dynamic Problems …...Dedication To my parents Babasaheb and Shamshad for their love and support; to my all mentors and teachers for developing my scientiﬁc

Uncertainty Quantification in Dynamic ProblemsWith Large Uncertainties

Sameer B. Mulani

Dissertation submitted to the Faculty of the

Virginia Polytechnic Institute and State University

in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

in

Aerospace Engineering

Rakesh K. Kapania (Committee Chair)

Robert W. Walters (Committee Co-Chair)

Mahendra P. Singh

Mayuresh Patil

July 17, 2006

Blacksburg, Virginia

Keywords: Karhunen-Loeve expansion, metamodeling, polynomial chaos, probabilistic

sound power sensitivity, Monte-Carlo simulation, stochastic eigenvalue problem,

random variable, random process, uncertainty quantification.

Copyright c©2006, Sameer B. Mulani

Uncertainty Quantification in Dynamic Problems With Large

Uncertainties

Sameer B. Mulani

(ABSTRACT)

This dissertation investigates uncertainty quantification in dynamic problems. The Ad-

vanced Mean Value (AMV) method is used to calculate probabilistic sound power and the

sensitivity of elastically supported panels with small uncertainty (coefficient of variation).

Sound power calculations are done using Finite Element Method (FEM) and Boundary

Element Method (BEM). The sensitivities of the sound power are calculated through

direct differentiation of the FEM/BEM/AMV equations. The results are compared with

Monte Carlo simulation (MCS). An improved method is developed using AMV, meta-

model, and MCS. This new technique is applied to calculate sound power of a composite

panel using FEM and Rayleigh Integral. The proposed methodology shows considerable

improvement both in terms of accuracy and computational efficiency.

In systems with large uncertainties, the above approach does not work. Two Spectral

Stochastic Finite Element Method (SSFEM) algorithms are developed to solve stochastic

eigenvalue problems using Polynomial chaos. Presently, the approaches are restricted

to problems with real and distinct eigenvalues. In both the approaches, the system

uncertainties are modeled by Wiener-Askey orthogonal polynomial functions. Galerkin

projection is applied in the probability space to minimize the weighted residual of the

error of the governing equation. First algorithm is based on inverse iteration method.

A modification is suggested to calculate higher eigenvalues and eigenvectors. The above

algorithm is applied to both discrete and continuous systems. In continuous systems, the

uncertainties are modeled as Gaussian processes using Karhunen-Loeve (KL) expansion.

Second algorithm is based on implicit polynomial iteration method. This algorithm is

found to be more efficient when applied to discrete systems. However, the application

of the algorithm to continuous systems results in ill-conditioned system matrices, which

seriously limit its application.

Lastly, an algorithm to find the basis random variables of KL expansion for non-Gaussian

processes, is developed. The basis random variables are obtained via nonlinear transfor-

mation of marginal cumulative distribution function using standard deviation. Results

are obtained for three known skewed distributions, Log-Normal, Beta, and Exponential.

In all the cases, it is found that the proposed algorithm matches very well with the known

solutions and can be applied to solve non-Gaussian process using SSFEM.

iii

Dedication

To my parents Babasaheb and Shamshad for their love and support; to my

all mentors and teachers for developing my scientific mind and to all

mathematicians, a constant source of inspiration.

iv

Acknowledgments

I am indebted to all my “GURUS” for imparting knowledge and the quality which thrives

for the truth about everything in each field. My first “GURU” is my mother, Mrs.

Shamshad B. Mulani, who built a discipline in my life and ethics about life at an early

stage. Always, I have observed that lack of guidance has created downfall in my studies

so it will be inadequate to say “Thank You” to my all “GURUS” for their invaluable

contribution in my life. My “GURUS”’ list is ever-increasing.

I am grateful to my advisor, Dr. Rakesh K. Kapania for giving me an opportunity to

work with him on different projects. Importantly, he agreed to become my advisor during

the second phase of my research. During this phase, he gave constant encouragement

and guidance whenever there were problems in my research fields and personal matters.

I am thankful to Almighty God who gave me an opportunity to work with my co-advisor,

Dr. Robert W. Walters who has the greatest scientific mind. Dr. Walters always created

interest in stochastic mechanics using polynomial chaos and gave valuable inputs during

my research. Dr. Michael J. Allen during his stay at Virginia Tech., developed my interest

in stochastic mechanics and helped me to understand the underlying physics. From him,

I learnt that all physical phenomenon are governed by differential equations, and solution

to these equations are obtained using available different numerical methods. This helped

me to conduct my research in different fields efficiently. I learnt probability and reliability

fundamentals from Dr. Mahendra P. Singh’s classes and personal discussions. I am

thankful to Dr. Mayuresh Patil for serving on my advisory committee.

Particularly, I would like to express my deepest thanks to my friend, Urmila Maitra

who encouraged me to pursue higher studies and provided me with moral and emotional

support. Thanks are also due to my friends and colleagues in our department for their

valuable company and encouragement, especially to Shereef Sadek, Dhaval Makhecha

v

and Omprakash Seresta. My friend, Sachin Patil in my home-town always took care of

problems in India.

I also want to acknowledge the help of Dr. Naira Hovakiyam for letting me teach her class,

“Computational Methods”, and the project funded from NASA Langley and National

Institute of Aerospace (NIA) for funding the project with Dr. David Peake (NIA), and

Ms. Karen Taminger (NASA) as the grant monitors.

I would like to express my sincere thanks to the AOE computing staff; especially Luke

Scharf and David Koh were always there to solve my weird problems. I thank the entire

administrative staff of the Aerospace and Ocean Engineering Department, especially Ms.

Betty Williams, Wanda Foushee and Gail Coe, for taking care of all the paper work.

Finally and uttermost, I thank the Almighty God for giving me this opportunity to work

in this field successfully at this university with the company of my brothers from Muslim

Student Association during difficult as well as happy times and took care of me during

the holy month of Ramdan. Last but not the least, I would like to thank my family for

always being there.

vi

Contents

Title Page i

Abstract ii

Dedication iv

Acknowledgments v

Table of Contents vii

List of Figures xi

List of Tables xviii

Nomenclature xxi

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Classification of Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Uncertainty Quantification Methods . . . . . . . . . . . . . . . . . . . . . 6

1.3.1 Probabilistic Methods . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3.2 Possibilistic Methods . . . . . . . . . . . . . . . . . . . . . . . . . 8

vii

1.4 Objectives of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . . 11

1.5 Outline of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . 13

2 Literature Survey 16

2.1 Possibilistic Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1.1 Interval Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.1.2 Convex Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.1.3 Fuzzy Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.1.4 Evidence Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2 Probabilistic Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2.1 Asymptotic Reliability Methods . . . . . . . . . . . . . . . . . . . 28

2.2.2 Perturbation Stochastic Finite Element Method (PSFEM) . . . . 35

2.2.3 Spectral Stochastic Finite Element Method (SSFEM) . . . . . . . 38

2.2.4 Sampling Techniques . . . . . . . . . . . . . . . . . . . . . . . . . 46

3 Probabilistic Sound Power and its Sensitivity 49

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.2 Theoretical Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.2.1 Sound Power Calculations . . . . . . . . . . . . . . . . . . . . . . 54

3.2.2 Probabilistic Structural Acoustic Analysis . . . . . . . . . . . . . 56

3.2.3 Probabilistic Sound Power Sensitivity . . . . . . . . . . . . . . . . 61

3.2.4 Non-Monotonic Response and Associated Sensitivity . . . . . . . 63

3.3 Application and Validation . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.3.1 Piston in an Infinite Baffle . . . . . . . . . . . . . . . . . . . . . . 65

3.3.2 Elastically-Supported Plate . . . . . . . . . . . . . . . . . . . . . 69

viii

3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4 Probabilistic Metamodeling 80

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.2 Hybrid Metamodel for Nonmonotonic, Nonlinear Response Function Anal-

ysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.3 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.3.1 Analytical Functions . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.3.2 Composite Panel Sound power . . . . . . . . . . . . . . . . . . . . 88

4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5 Fundamental Eigenvalue using Polynomial Chaos 98

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.2 Spectral Stochastic Finite Element Analysis . . . . . . . . . . . . . . . . 100

5.3 Stochastic Eigenvalue Problem . . . . . . . . . . . . . . . . . . . . . . . . 101

5.4 Eigenvalue Extraction Algorithms . . . . . . . . . . . . . . . . . . . . . . 104

5.5 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.5.1 Two Degrees-of-Freedom System . . . . . . . . . . . . . . . . . . 107

5.5.2 Continuous System . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6 Higher Eigenvalues using Polynomial Chaos 131

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131


6.3 Eigenvalue Extraction Algorithms . . . . . . . . . . . . . . . . . . . . . . 136


ix

6.4.1 Three Degree-of-Freedom System . . . . . . . . . . . . . . . . . . 139

6.4.2 Continuous System . . . . . . . . . . . . . . . . . . . . . . . . . . 146

6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

7 A New Algorithm for Eigenvalue Analysis using Polynomial Chaos 162

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162


7.3 Eigenvalues Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

7.4 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

8 Karhunen-Loeve Expansion of Non-Gaussian Random Process 173

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

8.2 Nonlinear Transformation Method . . . . . . . . . . . . . . . . . . . . . . 176


8.3.1 Log-Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . 178

8.3.2 Beta Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

8.3.3 Exponential Distribution . . . . . . . . . . . . . . . . . . . . . . . 184

8.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

9 Future Research in Uncertain Dynamic Problems 192

Bibliography 194

Vita 212

x

List of Figures

1.1 Uncertainty Quantification Methods . . . . . . . . . . . . . . . . . . . . . 5

2.1 General Membership Function of Input Fuzzy Variable . . . . . . . . . . 22

2.2 Belief and Plausibility Measures . . . . . . . . . . . . . . . . . . . . . . . 25

3.1 Baffled Circular Piston Configuration . . . . . . . . . . . . . . . . . . . . 66

3.2 Deterministic and 98% Probabilistic Radiated Sound Power for Baffled

Circular Piston Configuration . . . . . . . . . . . . . . . . . . . . . . . . 68

3.3 Deterministic and Random Design Parameter Configuration of Flexible

Panel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.4 Finite Element and Boundary Element Models of the Elastically Supported

Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.5 Deterministic and 98% Probabilistic Radiated Sound Power for Flexible

Panel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.1 MCS using Hybrid Metamodel . . . . . . . . . . . . . . . . . . . . . . . . 85

4.2 Response CDFs for Function 1 with MCS and AMV . . . . . . . . . . . . 87

4.3 Response CDFs for Function 1 with AMVMC, LHS, MCS, and AMV . . 87

4.4 Percent Error of Monotonic CDFs with respect to MCS for Function 1 . 88

4.5 Monotonic Response CDFs for Function 2 . . . . . . . . . . . . . . . . . 89

4.6 Percent Error of Monotonic CDFs with respect to MCS for Function 2 . 89

xi

4.7 Geometry of the Baffled Panel and Coordinate System . . . . . . . . . . 91

4.8 Sound power CDFs for the Composite Panel at 188 Hz . . . . . . . . . . 93

4.9 Percent Error in the Sound power CDFs with respect to MCS at 188 Hz 94





4.14 Deterministic and 95% Probabilistic Radiated Sound power . . . . . . . . 96

5.1 Two Degrees-of-Freedom Spring-Mass Model . . . . . . . . . . . . . . . . 107

5.2 PDFs of Fundamental Eigenvalue using First and Second Order Chaos for

the 2-DOF System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.3 PDFs of Fundamental Eigenvalue using Third and Fourth Order Chaos

for the 2-DOF System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.4 First-Order Chaos Eigenvalue Convergence for the 2-DOF System . . . . 110

5.5 Second-Order Chaos Eigenvalue Convergence for the 2-DOF System . . . 111

5.6 Third-Order Chaos Eigenvalue Convergence for the 2-DOF System . . . . 111

5.7 Fourth-Order Chaos Eigenvalue Convergence for the 2-DOF System . . . 112

5.8 Probabilistic Eigenvector for the 2-DOF System at Different Probability

Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.9 PDFs of the Fundamental Eigenvalue for Different Order Chaos with Dif-

ferent Probability Spaces for Mass and Stiffness for the 2-DOF System . 116

5.10 Fundamental Eigenvector at Different Probability Levels with 1− 3 order

chaos with for the 2-DOF System . . . . . . . . . . . . . . . . . . . . . . 116

5.11 First-Order Eigenvalue Coefficients Convergence for Different Probability

Spaces for Mass and Stiffness for the 2-DOF System . . . . . . . . . . . . 119

xii

5.12 Second-Order Eigenvalue Coefficients Convergence for Different Probabil-

ity Spaces for Mass and Stiffness for the 2-DOF System . . . . . . . . . . 120

5.13 Third-Order Eigenvalue Coefficients Convergence for Different Probability

Spaces for Mass and Stiffness for the 2-DOF System . . . . . . . . . . . . 121

5.14 Cantilever Beam as a Continuous Structure . . . . . . . . . . . . . . . . 122

5.15 PDFs of the Fundamental Eigenvalue using First and Second-Order Chaos

with Same Probability Space for Mass and the Bending Rigidity . . . . . 124

5.16 Fundamental Eigenvector using First-Order Chaos at 95% Probability


5.17 Fundamental Eigenvector using Second-Order Chaos at 95% Probability


5.18 Fundamental Eigenvector’s Mid-point Displacement using Second-Order

Chaos with Same Probability Space for Mass and the Bending Rigidity . 126

5.19 PDFs of the Fundamental Eigenvalue of the Cantilever Beam with Differ-

ent Probability Spaces for Mass and the Bending Rigidity . . . . . . . . . 128

5.20 Fundamental Eigenvector of the Cantilever Beam at Different Probabilities

using First-Order Chaos with the Different Probability Spaces for Mass

and the Bending Rigidity . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5.21 Fundamental Eigenvector of the Cantilever Beam at Different Probabilities

using Second-Order Chaos with the Different Probability Spaces for Mass

and the Bending Rigidity . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

6.1 Three Degrees-of-Freedom Spring-Mass Model . . . . . . . . . . . . . . . 139

6.2 PDFs of Fundamental Eigenvalue using Fourth Order Chaos for Perfectly

Correlated Masses and Stiffness for the 3-DOF System . . . . . . . . . . 142

6.3 PDFs of Second Eigenvalue using Fourth Order Chaos for Perfectly Cor-

related Masses and Stiffness for the 3-DOF System . . . . . . . . . . . . 142

6.4 PDFs of Third Eigenvalue using Fourth Order Chaos for Perfectly Corre-

lated Masses and Stiffness for the 3-DOF System . . . . . . . . . . . . . 143

xiii

6.5 Convergence of λ for Perfectly Correlated Masses and Stiffness for the

3-DOF System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.6 Fundamental Eigenvector for Perfectly Correlated Masses and Stiffness for

the 3-DOF System at Different Probabilities . . . . . . . . . . . . . . . . 144

6.7 Second Eigenvector for Perfectly Correlated Masses and Stiffness for the

3-DOF System at Different Probabilities . . . . . . . . . . . . . . . . . . 144

6.8 Third Eigenvector for Perfectly Correlated Masses and Stiffness for the


6.9 PDFs of Fundamental Eigenvalue using Third Order Chaos for Uncorre-

lated Masses and Stiffness for the 3-DOF System . . . . . . . . . . . . . 147

6.10 PDFs of Second Eigenvalue using Third Order Chaos for Uncorrelated

Masses and Stiffness for the 3-DOF System . . . . . . . . . . . . . . . . . 147

6.11 PDFs of Third Eigenvalue using Third Order Chaos for Uncorrelated

Masses and Stiffness for the 3-DOF System . . . . . . . . . . . . . . . . . 148

6.12 Convergence of λ for Uncorrelated Masses and Stiffness for the 3-DOF

System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

6.13 Fundamental Eigenvector for Uncorrelated Masses and Stiffness for the


6.14 Second Eigenvector for Uncorrelated Masses and Stiffness for the 3-DOF

System at Different Probabilities . . . . . . . . . . . . . . . . . . . . . . 149

6.15 Third Eigenvector for Uncorrelated Masses and Stiffness for the 3-DOF

System at Different Probabilities . . . . . . . . . . . . . . . . . . . . . . 150

6.16 Simply-Supported Beam as a Continuous Structure . . . . . . . . . . . . 150

6.17 PDFs of Fundamental Eigenvalue using Fourth Order Chaos for Fully Cor-

related Masses and Stiffness for Simply-Supported Beam . . . . . . . . . 153

6.18 PDFs of Second Eigenvalue using Fourth Order Chaos for Fully Correlated

Masses and Stiffness for Simply-Supported Beam . . . . . . . . . . . . . 153

xiv

6.19 PDFs of Third Eigenvalue using Fourth Order Chaos for Fully Correlated


6.20 Convergence of λ for Fully Correlated Masses and Stiffness for Simply-

Supported Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

6.21 Fundamental Eigenvector for Fully Correlated Masses and Stiffness for

Simply-Supported Beam at Different Probabilities . . . . . . . . . . . . . 155

6.22 Second Eigenvector for Fully Correlated Masses and Stiffness for Simply-

Supported Beam at Different Probabilities . . . . . . . . . . . . . . . . . 155

6.23 Third Eigenvector for Fully Correlated Masses and Stiffness for Simply-


6.24 PDFs of Fundamental Eigenvalue using Second Order Chaos for Uncorre-

lated Masses and Stiffness for Simply-Supported Beam . . . . . . . . . . 157

6.25 PDFs of Second Eigenvalue using Second Order Chaos for Uncorrelated


6.26 PDFs of Third Eigenvalue using Second Order Chaos for Uncorrelated


6.27 Convergence of λ for Uncorrelated Masses and Stiffness for Simply-Supported

Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

6.28 Fundamental Eigenvector for Uncorrelated Masses and Stiffness for Simply-


6.29 Second Eigenvector for Uncorrelated Masses and Stiffness for Simply-


6.30 Third Eigenvector for Uncorrelated Masses and Stiffness for Simply-Supported

Beam at Different Probabilities . . . . . . . . . . . . . . . . . . . . . . . 160

7.1 Three Degrees-of-Freedom Spring-Mass Model for Eigenvalue Analysis . . 166

7.2 PDFs of Fundamental Eigenvalue for Perfectly Correlated Masses and

Stiffness for the 3-DOF System using Fourth-Order Polynomial Chaos Ex-

pansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

xv

7.3 PDFs of Second Eigenvalue for Perfectly Correlated Masses and Stiffness

for the 3-DOF System using Fourth-Order Polynomial Chaos Expansion . 168

7.4 PDFs of Third Eigenvalue for Perfectly Correlated Masses and Stiffness

for the 3-DOF System using Fourth-Order Polynomial Chaos Expansion . 169

7.5 PDFs of Fundamental Eigenvalue for Uncorrelated Masses and Stiffness

for the 3-DOF System using Third-Order Polynomial Chaos Expansion . 170

7.6 PDFs of Second Eigenvalue for Perfectly Correlated Masses and Stiffness


7.7 PDFs of Third Eigenvalue for Perfectly Correlated Masses and Stiffness


8.1 CDF and PDF of Non-Gaussian Marginal Log-Normal Distribution . . . 180

8.2 CDFs of Marginal Log-Normal Distribution and Transformed Distribution 181

8.3 PDFs of Marginal Log-Normal Distribution and KL Expansion Basis Ran-

dom Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

8.4 Log-Normal CDFs of Analytical and Numerical KL Expansion Basis Ran-

dom Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

8.5 Scatter Plots of Log-Normal KL Expansion Basis Random Variables . . . 182

8.6 CDF and PDF of Non-Gaussian Marginal Beta Distribution . . . . . . . 184

8.7 CDFs of Marginal Beta Distribution and Transformed Distribution . . . 185

8.8 PDFs of Marginal Beta Distribution and KL Expansion Basis Random

Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

8.9 Beta CDFs of Analytical and Numerical KL Expansion Basis Random

Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

8.10 Scatter Plots of Beta KL Expansion Basis Random Variables . . . . . . . 186

8.11 CDF and PDF of Non-Gaussian Marginal Exponential Distribution . . . 188

8.12 CDFs of Marginal Exponential Distribution and Transformed Distribution 189

xvi

8.13 PDFs of Marginal Exponential Distribution and KL Expansion Basis Ran-

dom Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

8.14 Exponential CDFs of Analytical and Numerical KL Expansion Basis Ran-

dom Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

8.15 Scatter Plots of Exponential KL Expansion Basis Random Variables . . . 190

xvii

List of Tables

3.1 Characteristics of the Baffled Circular Piston . . . . . . . . . . . . . . . . 66

3.2 Radiated Sound Power and Sound Power Sensitivity Values for Circular

Piston . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.3 Predicted and Actual 98% Probabilistic PW Values due to 2% Change in

Masses, m1, m2,and Dampers, b1, b2 . . . . . . . . . . . . . . . . . . . . . 71

3.4 Predicted and Actual 98% Probabilistic PW Values due to 1% Change in

Masses, m1, m2,and Dampers, b1, b2 . . . . . . . . . . . . . . . . . . . . . 71

3.5 Characteristics of the Flexible Panel . . . . . . . . . . . . . . . . . . . . 73

3.6 Characteristics of the Elastic Panel Support . . . . . . . . . . . . . . . . 73

3.7 Radiated Sound Power and Sound Power Sensitivity Values for Both Fre-

quency Range for Flexible Panel . . . . . . . . . . . . . . . . . . . . . . . 77

3.8 Predicted and Actual 98% Probabilistic PW Values due to 3% Independent

Changes in t1 and t2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.9 Predicted and Actual 98% Probabilistic PW Values due to a 2% Change

in t1 and t3 simultaneously for the first frequency range . . . . . . . . . . 77

3.10 Predicted and Actual 98% Probabilistic PW Values due to a 2% Change

in t2 and t3 simultaneously for the second frequency range . . . . . . . . 78

3.11 Predicted and Actual 98% Probabilistic PW Values due a to 2% Change

in t1 and t2 simultaneously for the second frequency range . . . . . . . . 78

4.1 Composite Panel Properties . . . . . . . . . . . . . . . . . . . . . . . . . 91

xviii

5.1 Fundamental Eigenvalue Coefficients with Same Probability Space for Mass

and Stiffness for the 2-DOF System . . . . . . . . . . . . . . . . . . . . . 108

5.2 Fundamental Eigenvalue Coefficients with Different Probability Spaces for

Mass and Stiffness for the 2-DOF System . . . . . . . . . . . . . . . . . . 115

5.3 Fundamental Eigenvalue Coefficients for Cantilever Beam with Same Prob-

ability Space for Mass and Stiffness . . . . . . . . . . . . . . . . . . . . . 124

5.4 Fundamental Eigenvalue Coefficients for Cantilever Beam with Different

Probability Space for Mass and Stiffness . . . . . . . . . . . . . . . . . . 127

6.1 Eigenvalue Coefficients for Perfectly Correlated Masses and Stiffness for

the 3-DOF System using Fourth-Order Chaos . . . . . . . . . . . . . . . 141

6.2 Mean and Standard Deviation for Perfectly Correlated Masses and Stiff-

ness for the 3-DOF System using LHS . . . . . . . . . . . . . . . . . . . 141

6.3 Eigenvalue Coefficients for Uncorrelated Masses and Stiffness for the 3-

DOF System using Third-Order Chaos . . . . . . . . . . . . . . . . . . . 146

6.4 Mean and Standard Deviation for Uncorrelated Masses and Stiffness for

the 3-DOF System using LHS . . . . . . . . . . . . . . . . . . . . . . . . 146

6.5 Mean and Standard Deviation of Eigenvalues for Fully Correlated Masses

and Stiffness for the Simply-Supported Beam using Fourth-Order Chaos

and LHS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

6.6 Mean and Standard Deviation of Eigenvalues for Uncorrelated Masses and

Stiffness for the Simply-Supported Beam using Second-Order Chaos and

LHS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

7.1 Eigenvalue Coefficients for Perfectly Correlated Masses and Stiffness for

the 3-DOF System using Fourth-Order Chaos . . . . . . . . . . . . . . . 167

7.2 Eigenvalues Mean and Standard Deviation for Perfectly Correlated Masses

and Stiffness for the 3-DOF System using LHS . . . . . . . . . . . . . . . 167

7.3 Eigenvalue Coefficients for Uncorrelated Masses and Stiffness for the 3-

DOF System using Third-Order Chaos . . . . . . . . . . . . . . . . . . . 169

xix

7.4 Eigenvalues Mean and Standard Deviation for Uncorrelated Masses and

Stiffness for the 3-DOF System using LHS . . . . . . . . . . . . . . . . . 170

8.1 Independency of KL Expansion Basis Random Variables for Log-Normal

Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

8.2 Independency of KL Expansion Basis Random Variables for Beta Distri-

bution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

8.3 Independency of KL Expansion Basis Random Variables for Exponential

Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

xx

Nomenclature

FEM Finite Element Method

CFD Computational Fluid Dynamics

BEM Boundary Element Method

SFEM Stochastic Finite Element Method

PSFEM Perturbation Stochastic Finite Element Method

SSFEM Spectral Stochastic Finite Element Method

PDF Probability Density Function

CDF Cumulative Distribution Function

FOSM First Order Second Moment method

SOSM Second Order Second Moment method

AMV Advanced Mean Value method

MCS Monte Carlo Simulation

NN Neural Network

LHS Latin Hypercube Sampling

ODE Ordinary Differential Equation

PDE Partial Differential Equation

L Differential operator associated with differential equation

Bi Linear homogeneous differential operators associated with boundary conditions

Ci Linear homogeneous differential operators associated with initial conditions

xi Upper bound of uncertain variable, x

xi Lower bound of uncertain variable, x

∀ for all; for any; for each

∈ set membership

: such that ...

⊆ is a subset of

xxi

⇒ implies; if .. then⋂intersected with; intersect⋃the union of ... and ...; union

∅ the empty set

sup supremum

BBA Basic Belief Assignment

f : x → y mapping

Bel Belief

Pl Plausibility∑sum over ... from ... to ... of

Θ Sample space

θ Trial realisation

R Set of Real Numbers

σ Standard deviation

ρ (x, x′) Autocorrelation coefficient

fX (x) Joint PDF of the variables

β Safety factor

MPPL Most Probable Point Locus

Φ (.) Standard normal CDF

DOE Design of Experiment

K Stiffness matrix

U Displacement vector

F Load vector

ξi (θ) Uncorrelated random variables

Rhh (x,y) Autocorrelation function

Γn (ξ1, . . . , ξn) the Hermite polynomial of order n

δij Kronecker delta

〈〉 Expectation operator

M Mass matrix

C Damping matrix

U Acceleration vector

U Velocity vector

Vn Normal velocity vector

ω Excitation frequency

xxii

K Acoustic wave number

A and B Acoustic system matrices

Ps Surface pressure vector

PW Sound power

g (.) Limit state equation

J1 First order Bessel function of first kind

λ Eigenvalue of the system

det(.) Matrix determinant

xxiii

Chapter 1

Introduction

1.1 Motivation

Most of the physical systems are modeled using differential equations in which coefficients

and/or inhomogeneous parts are uncertain because those are found through experimen-

tations. The definition of “uncertainty” is given as “A potential deficiency in any phase

or activity of the modeling process that is due to lack of knowledge” [1]. The progress

in dealing with uncertainty as theoretical probability has started a long ago. The first

monograph on the probability was published by Pierre Simon, Marquis de Laplace [2].

The first application of probability was applied to the problems in Physics by J. W.

Gibbs [3] in 1903. Einstein and Smoluchowski derived the probability density function of

the particle displacement of Brownian motion in (1905− 1906) [3]. Mathematicians and

1

1.1 Motivation

physicists took a lead in the development of random processes, time series [4]. Heisen-

berg in 1927 proposed his “Uncertainty Principle” in Quantum Mechanics for locating

electron’s position or momentum (www.aip.org/history/heisenberg/p08.htm), but be-

cause of difficulty of understanding this principle at that time, Schrodinger put forth

his “Wave Equation” which describes probability functions for electrons’ orbits about

nuclei (www.online.redwoods.cc.ca.us). The excellent interpretation of Heisenberg’s Un-

certainty Principle in natural systems is given in Vanmarcke [5] as “true patterns of

point-to-point variation can not be known: there is a basic trade-off between the accu-

racy of a measurement and the (time or distance) interval within which the measurements

are made”. Simultaneously, engineers started the development of reliability in mechanical

and civil engineering based on probability theory.

Historical development of structural reliability is very well discussed by Madsen et al [6].

The development can be divided into three eras. In the first era (1920 − 1960), the

reliability approach was initiated by Mayer (1926) and carried on by Weibull (1939),

Freudenthal (1947), Plum (1950), and Basler (1960) independently, this era was the be-

ginning of the development of reliability fields with smaller steps of progress. Then in

the second era (1960− 1980), the reliability field made rapid progress because of efforts

of Cornell, Ferry-Borges, Castanheta, Bolotin, Ditlevsen, Lind, Rackwitz, Hasofer and

Veneziano. Still these methods’ application was limited to analytical or to semi-analytical

problems. Then in third era , 1984−till now, the development of numerical methods of

finite element method (FEM), computational fluid dynamics (CFD), boundary element

2

http://www.aip.org/history/heisenberg/p08.htm

http://online.redwoods.cc.ca.us/DEPTS/science/chem/storage/Schrod/page3.htm

1.2 Classification of Uncertainty

method (BEM), and other numerical methods dealing with differential equations as well

as digital simulation using high speed computers evolved rapidly so these reliability meth-

ods were tried to apply to more complex problems using numerical approaches. In 1983,

Vanmarcke and Grigoriu [7] presented a finite element analysis of a simple shear beam

with random rigidity. This research initiated the development to Perturbation Stochas-

tic Finite Element Method (PSFEM) to deal with random inputs. There are two very

good monographs written on PSFEM [8, 9] by Kleiber et al and Haldar at al. To refine

response of the stochastic systems for large coefficient of variation, Ghanem and Spanos

proposed Spectral Stochastic Finite Element Method (SSFEM) [10]. But still SSFEM

is in primary phase, there is a lot of scope of improvements in the method as well as

its applications to different problems and developments of numerical algorithms. This

particular need of applying SSFEM and PSFEM is addressed in the paper by Oden et

al [11].


Oberkampf et al has extensively discussed the sources of uncertainty and the methods

dealing with the uncertainty. The authors classified uncertainties into different classes [12]

1. Aleatoric uncertainty (Inherent uncertainty) :

The uncertainty associated with the observed phenomenon which can not be de-

scribed by deterministic description. This uncertainty is because of inherent ran-

3


domness of the phenomenon.

2. Epistemic uncertainty:

This type of uncertainty can be subclassified as

(a) Physical Modeling:

This occurs because of limited knowledge of the phenomenon being observed,

so mathematical model (for example, formulas, equations, algorithms) of the

phenomenon is imperfect. This associates randomness in the material proper-

ties, geometric properties, boundary conditions, and initial conditions.

(b) Discretization errors:

Complex systems solutions are obtained using numerical methods like FEM,

BEM, and CFD in which spatial and temporal discretizations are carried. This

is approximate modeling of our analytical (mathematical) modeling which will

have truncation errors. For non-linear problems, equilibrium is set up using

iterative methods which further adds errors to the calculated response of a

system.

(c) Computer round-off errors:

Numerical solutions are obtained using digital simulation which has finite pre-

cision like 32 or 64 bit processors so this simulation chops off infinite repre-

sentation of decimal numbers into binary numbers and carries mathematical

operations.

4

1.3 Uncertainty Quantification Methods

Uncertainty Methods

Possibilistic Methods Probabilistic Methods

1. Interval Analysis2. Convex Modeling3. Fuzzy Set Theory4. Evidence Theory

1. Asymptotic Methods2. Sampling Techniques3. Perturbation Stochastic

Finite Element Method (PSFEM)

4. Spectral Stochastic Finite Element Method (SSFEM)

Fuzzy Random Variable Approach

Figure 1.1: Uncertainty Quantification Methods

5



In Fig. 1.1, different methods dealing with uncertainties are classified according to the in-

put data available. These methods can be combined and modified according to our needs

like efficiency and accuracy. These methods can be classified as Probabilistic Methods,

Section 1.3.1 and Possibilistic Methods, Section 1.3.2. In probabilistic methods, infor-

mation of random variables and/or random processes is available, so response variable

will be a random variable. When the information of an input random variable is not

complete or can not be defined exactly as a random variable, deterministic methods are

often used to calculate the variability in the response.

1.3.1 Probabilistic Methods

Research performed over last several years has led to the development of probabilistic

methods to account for the uncertainties in material and geometric properties of the

systems. Particularly, to account for uncertainty in forcing functions in dynamic systems,

a number of methods are available and have been successfully applied in studying the

resulting random vibrations of these systems [13, 14, 15]. These methods can be divided

into two main categories: methods having (a) an implicit definition and (b) those having

an explicit definition of the system response.

The methods having an implicit definition of the system response can be subdivided into

(a) moment methods [6, 16] and (b) sampling methods [17]. Moment methods only re-

6


quire approximate first and second moment information of the response and the implicit

representation of the response function in the form of function evaluations at a point.

This can be used to get satisfactory information. Moment methods represent the re-

sponse function implicitly as a low order (1 or 2 degree) polynomial that can be used for

computationally efficient calculation of the cumulative distribution function (CDF) char-

acterizing the system response. The two most well known and utilized moment methods,

the first order second moment (FOSM) method and the second order second moment

(SOSM) method, have laid the foundation for the development of more sophiscated

methods that account for higher order effects [18, 19, 20, 21] and efficiently represent

a complex response surface over the random variable domain of interest [22, 23, 24]. The

accuracy of these methods however still suffers when the coefficient of variation, δ, of ran-

dom variables is greater than 0.1. Sampling methods can be used to calculate accurately

probabilistic response characteristics, but these methods require very large sample set of

realizations of random variables. Variance reduction [25, 26] and stratified sampling [27]

methods have been developed to decrease the size of sample set. Still sampling methods

lacks computational efficiency when used with numerical techniques like FEM, BEM and

CFD.

A second category of probabilistic methods has been developed that utilizes explicit rep-

resentation of the response function. The methods are referred to as the stochastic finite

element methods (SFEM), or random field methods. In these methods, the uncertain

characteristics of the response are related to an explicit representation of the uncer-

7


tainty in the structural parameters, and loads. The two most popular SFEMs are the

perturbation technique described by Kleiber and Hein [8, 9] and the Karhunen-Loeve

expansion scheme used along with the FEM and was initially proposed by Ghanem and

Spanos [28, 29, 30] and further developed by Ghanem [31, 32]. In the former method, first

or second-order perturbation expansions of all random quantities are taken about their

mean values via a Taylor series expansion. These expansions are then used to recursively

solve the stochastical moments of the system response (mean and correlation function).

In the latter, the Karhunen-Loeve expansion is performed on the structural properties

exhibiting uncertainty and is subsequently combined with a truncated polynomial chaos

representation of the response. The resulting system of linear algebraic equations ob-

tained when considering the system’s discretized governing differential equations can be

solved for the unknown coefficient in the polynomial chaos expansion. Once these coef-

ficients are known, the statistics of the response can be readily obtained. In the initial

development of polynomial chaos, Hermite orthogonal polynomials were used for repre-

senting second order Gaussian random process. To account for non-Gaussian processes,

generalized polynomial chaos was proposed [33].

1.3.2 Possibilistic Methods

When the information of input random variables can not be defined in terms of joint

probability density function (PDF) but the range of input random variables is known,

these possibilistic methods (deterministic methods) are used to find the uncertainty in

8


response variable. The new possibilistic methods are (a) interval analysis [34]; (b) convex

modeling [35]; (c) fuzzy set theory [36]; and (d) evidence theory [37]. In 1958, Ramon

E. Moore [38] initiated the development of interval analysis. In interval analysis, the

uncertain variables are described by upper and lower bounds that forms a hypercube

for the input uncertain variables. The aim of the interval analysis is to find upper and

lower bound of any of the response variables. In interval analysis, interval algebra is used

to process all algebraic calculations. An extensive literature on Interval Analysis can

be found at the internet site www.interval-comp.com. With convex modeling, uncertain

variables lie within hyper convex region as opposed to hypercube in interval analysis.

The shape of this hyper convex region can be adjusted easily by changing the definition

of the input uncertain variables. Instead of representing the uncertainty using intervals,

input uncertainty is represented as some function (like PDF) in fuzzy theory.

The application of fuzzy set theory to engineering systems is described in the book by

Kaufmann et al [39]. But the application of fuzzy set theory in uncertainty quantification

is limited. Recently a monograph on the application of fuzzy set theory in uncertainty

quantification has been published [40]. In fuzzy set theory, input uncertain variables are

defined as fuzzy numbers via a membership function. The confidence in the uncertain

variables are represented by α-cuts, if α = 1, the random variable becomes deterministic

and α = 0 denotes that the uncertain variable can take any value between the whole

range of random variable.

Interval analysis, convex modeling, and fuzzy set theory application in uncertainty quan-

9

http://www.cs.utep.edu/interval-comp/main.html


tification becomes computationally intensive for complex problems. The range of re-

sponse becomes prohibitively large as the algebraic calculations on the random variables

become large which overestimates the uncertainty in the response from the actual uncer-

tainty. In reality, all complex systems will have aleatory as well as epistemic uncertainty.

So the application of evidence theory would be the best. Evidence theory can be viewed

as combination of probability theory and possibility theory, but there are no PDFs or

interval of input variables or membership function as in the fuzzy set theory, uncertain

variables are represented in terms of evidences. Shafer [41] extended Dempster’s original

work and the theory is now generally called Dempster-Shafer theory.

If the input random variables are uncertain, it means that their definition is fuzzy, some

efforts has been made to solve such problems using probability theory. Thacker et al [42]

have tried to use Bayesian estimation techniques to predict the reliability of the system.

Even fuzzy set theory is combined with the probability theory which is called as “Fuzzy

Random Variable Approach” [40]. Evidence theory narrows the range of the response

variable as compared to other possibilistic methods [43]. Recently a comparison has been

made between the evidence theory and Bayesian theory and it is suggested that if the

difference between the minimum and the maximum probabilities of the response due to

impreciseness in input parameters is large, then Bayesian analysis should be used [44].

10

1.4 Objectives of the Dissertation


The overall aim of this thesis can be classified into the following main objectives:

• Develop an algorithm to calculate probabilistic sound power and its sen-

sitivity with respect to deterministic design variables.

Lot of research has been done to deal with the uncertainties in structural mechan-

ics and reliability fields using numerical methods like FEM, and BEM, but limited

work has been done to apply probabilistic techniques to acoustics or vibro-acoustics

problems. The concept of probabilistic structural acoustic sensitivity with respect

to deterministic design variables has yet to be addressed. An algorithm is de-

veloped which calculates the probabilistic sound power at different probabilities

and its sensitivities with respect to deterministic design variables using numerical

methods such as FEM, BEM, and Advanced Mean Value (AMV) method. Using

this algorithm, vibro-acoustic system can be optimized to have desirable structural

acoustic characteristics.

• Develop a new metamodeling technique to calculate probabilistic acous-

tic power of composite panels.

Use of probabilistic techniques for calculating vibro-acoustic response of composites

is limited. In this work, a new technique is presented that better represents the

complex dynamic response of a composite structure during implicit probabilistic

11


calculations. A metamodeling technique is presented which fits the response sur-

face between FOSM and AMV at different probabilities. Neural Network is used

to capture the nonlinear relationship between FOSM and AMV response.

• Develop an algorithm which calculates stochastic eigenvalues and eigen-

vectors of the systems with large uncertainties.

In vibrations and acoustics, eigenvalues characterize resonance of the systems.

Stochastic eigenvalues of the systems with small coefficient of variation of the ma-

terial properties can be obtained using Perturbation Stochastic Finite Element

Method (PSFEM). Since, the assumption of PSFEM is that the underlying ran-

dom variables are Gaussian, so response of the system becomes Gaussian. PSFEM

fails to capture the complete distribution of the response for the systems with large

uncertainties. Stochastic eigenvalue problems were solved using Stochastic Spectral

Finite Element Method (SSFEM) for the systems in which the material properties

are defined as random fields.

• Develop a method which calculates stochastic eigenvalues without cal-

culating eigenvectors of the systems with large uncertainties.

Many times, we are interested in stochastic eigenvalues alone and not necessarily

in stochastic eigenvectors. An algorithm is developed which calculates stochastic

eigenvalues without calculating eigenvectors for the system with large uncertainties

12

1.5 Outline of the Dissertation

using polynomial iteration method.

• Find basis random variables of Karhunen-Loeve (KL) expansion for non-

Gaussian random process.

For Gaussian random process, basis random variables of KL expansion are uncor-

related standard normal variables. Basis random variables of KL expansion should

be identically distributed random variables with zero mean and unit variance and

should be independent. Few efforts have been made to obtain these basis vari-

ables for non-Gaussian random process, but the question of independence of these

variables remains unanswered. The method to obtain independent basis random

variables of KL expansion for non-Gaussian is presented using nonlinear transfor-

mation of the CDF of the marginal distribution function of the random processes.


Uncertainty quantification methods are extensively discussed in Chapter 2. In Chapter 2,

both probabilistic and possibilistic methods, introduced in Section 1.3, are reviewed.

Advantages, disadvantages, and shortcomings are discussed in Chapter 2.

In Chapter 3, probabilistic sensitivity analysis for sound power is presented which can be

implemented easily for other dynamic systems. In this analysis FEM, BEM, and AMV

are combined to define probabilistic sensitivity. FEM, BEM, and AMV are introduced

13


in Chapter 3. This algorithm is applied to an analytical model of a baffled circular

piston as well as to numerical model of an elastically-supported plate. The sensitivity of

sound power at high probability level to changes in deterministic structural parameters

is calculated through direct differentiation of the FEM/BEM/AMV procedure. The

probabilistic sound power computations are validated through comparison with the data

obtained from a Monte Carlo simulation and the probabilistic sound power sensitivities,

are validated through comparison with data computed by performing re-analysis.

An efficient, new probabilistic metamodel is presented in Chapter 4 which calculates the

probabilistic vibro-acoustic response of a composite structure. FEM and Rayleigh Inte-

gral are used to calculate the vibration response and the radiated sound pressure in the

far-field, respectively. Using this far-field pressure values, sound power of vibrating struc-

ture is calculated. The new probabilistic technique combines the AMV, metamodeling

and simple Monte Carlo Sampling (MCS). The new technique is applied to a compos-

ite panel with geometric and structural uncertainty. Neural networks (NN) are used to

construct the proposed metamodel.

Most accurate, polynomial chaos is applied to obtain fundamental and higher eigenvalues

in Chapters 5 and 6, respectively. These algorithms are developed which will be useful in

probabilistic aeroelastic analysis. Important points like Karhunen-Loeve expansion and

Galerkin projection are introduced in Chapter 5. These algorithms are intrusive because

uncertainties are propagated using Galerkin projection.

In Chapter 7, new efficient and accurate intrusive method is developed and applied to

14


3 degree-of-freedom system and all three eigenvalues are calculated and compared to

Monte Carlo Simulation (MCS) using Latin Hypercube Sampling (LHS). Ill-conditioned

system matrices obtained for continuous systems limits the application of this algorithm

to complex systems.

To find basis random variables of Karhunen-Loeve expansion for non-Gaussian random

process, a non-iterative algorithm is presented in Chapter 8 for analytical non-Gaussian

random variables with input marginal density function. This algorithm involves the

transformation of CDF of marginal density function using standard deviation. This

algorithm is very accurate as compared to previous algorithm. Future research directions

in uncertainty quantification are discussed in Chapter 9.

15

Chapter 2

Literature Survey

During last two centuries, research has been carried out to deal with the uncertainty in

physical systems. As described in Chapter 1, Section 1.2, uncertainties are classified into

epistemic ( inherent ) and aleatoric (lack of knowledge) uncertainty [12]. Most of the

time, all physical systems are governed by differential equations, those equations may

be ordinary differential Equation (ODE), partial differential equation (PDE), or simple

algebraic equations. These equations may be time invariant or time varying or only

functions of spatial dimensions. This can be very well explained using the following set

of equations,

Lu (x, t) = f (x, t) (2.1)

Bi (x) u (x) = 0, i = 1, 2, . . . , p (2.2)

Ci (t) u (x) = 0, i = 1, 2, . . . , n (2.3)

16

2.1 Possibilistic Methods

where L is differential operator of order 2 × p in space and 2 × n in time, Bi and Ciare

linear homogeneous differential operators associated with boundary conditions and initial

conditions respectively, f (x, t) is the source term and u (x, t) is the response of the

differential equation. This differential equation is well posed with appropriate number of

boundary and initial conditions.

If the uncertainty is in f (x, t) like the force spectra, analysis tools for such systems are

well developed [13, 14, 15]. If the uncertainty is in L, Bi and Bi are not yet fully developed

to obtain solutions that have a certain level of accuracy. Coefficients of these operators

become random processes and random variables depending upon the differential equation.

If the knowledge of these random processes and random variables are known in terms of

the joint PDF, then numerical solution for the response can be obtained using PSFEM

or SSFEM. If these random variable definitions are not available, then methods based on

Possibility theory should be used. In the next sections, all these methods are described.


When the information about uncertain variables is available in terms of their range or

can not be defined in terms of joint PDF, these methods should be used to get the range

of response variable. These methods are further classified according to the information

available and analysis tools that are used.

17


2.1.1 Interval Analysis

In interval analysis, input uncertain variables are defined in terms of their upper, xi and

lower, xi bounds respectively so that

xi ≤ xi ≤ xi ; i = 1, 2, . . . , N (2.4)

and fundamental mathematical operations for input uncertain x = [x, x] and y = [y, y]

variables are defined in following ways

x + y =[x + y, x + y

], (2.5)

x − y =[x − y, x − y

], (2.6)

x × y =[min

xy, xy, xy, xy

, max

xy, xy, xy, xy

], (2.7)

1

x= [1/x, 1/x ] if x > 0 or x < 0 (2.8)

x ÷ y = x × 1/y (2.9)

In computational mechanics, discretized equations are represented as

A y = b (2.10)

where A is a matrix whose elements are representative of input parameters, b is a forcing

function and y is the unknown vector. Matrix A and vector, b are uncertain and are

18


represented by intervals. The aim of the analysis is to find the range of y. Interval

arithmetic, combinatorial approach can be used to get the response. In combinatorial

approach, the response, yr can be written as

yr = y(xi

1, xj2, . . . , xk

n

);

i, j, . . . , k = 1, 2; r = 1, 2, . . . , 2n

y =[y, y

]=

[min

r = 1,2,..., 2nyr, max

r = 1,2,..., 2nyr

](2.11)

This combinatorial approach can be applied to small problems because it generates so

many combinations for the response sample space for large problems so it limits the ap-

plication of this method. So Elishakoff puts forth the concept of “antioptimization” in

which sequentially all the scalars or any element of y can be obtained using an optimiza-

tion procedure. But this method becomes computationally inefficient if we want to find

all the elements of y for large problems; it can thus only be applied to small problems.

Otherwise, interval arithmetic can be used but it overestimates the response. First, this

was applied in structural reliability by Rao et al [34]. This is the simplest of all the

methods but is applicable to only problems with small dimensions and furthermore it

overestimates the response by a large amount. So research continued so as to improve

this method and Convex modeling and Fuzzy Set theory came into picture.

19


2.1.2 Convex Modeling

In interval analysis, uncertain variables form a hypercube; convex modeling modifies

this hypercube with choice of convex hyper region. Convex modeling is discussed in the

monograph [35]. This monograph particularly deals with the uncertainty associated with

loading conditions. The other definition of the convex region can be obtained by the

following function

xt Ω x ≤ a (2.12)

where x is a vector of uncertain variables, Ω is a positive definite matrix and a is a

positive constant. By changing the definition of Ω and a, the shape of convex region

can be changed easily. Once these variables are defined then maximum and minimum

value of the response is obtained using the methods discussed in Section 2.1.1 or using

optimization. Still this method is computationally inefficient and prohibitive for a large

degree of freedom system.

2.1.3 Fuzzy Set Theory

Fuzzy set theory was initiated by Lotfi Zadeh. Initially, it was applied in artificial in-

telligence, image processing, communication systems and control systems in electronic

devices to deal with the uncertainty in these systems. Its application in structural analy-

sis started in the 1980′s and Rao et al [45] proposed a “Fuzzy Finite Element Approach”.

20


In this theory, uncertain variables are represented by membership functions which have

following properties:

∀x ∈ S : µ (x) ∈ 0, 1 (2.13)

∀ A, B ∈ S , A ⊆ B ⇒ µ (A) ≤ µ (B) (2.14)

∀ Ai, i ∈ I,

I⋂i=1

Ai = ∅ ⇒ µ

(I⋃

i=1

Ai

)= max

i∈I(µ (Ai)) (2.15)

where S is the domain of the uncertain variable, µ is the membership function.

Typical membership function is shown in Fig. 2.1. In this membership function, typical α-

cut is also shown. Mathematical operations between multiple fuzzy numbers are carried

out using Zadeh’s Extension Principle. If y = f(x1, x2, . . . , xn) and µx1 , µx2 , . . . , µxn

are associate membership functions for input uncertain variables, then the associated

membership with y is given as:

µY (y) = supy=f(x1,x2,...,xn)

min [µX1 (x1) , µX2 (x2) , . . . , µXn (xn)] (2.16)

The above operation is possible for explicit expressions. It is implemented in commercial

software, MATLABR© in “Fuzzy Logic Toolbox” [46]. In MATLAB R©, uncertain variables

are defined in terms of their membership function using available library of functions, one

can also write one’s own functions. This process is called “Fuzzification”. Mathematical

operations are carried on input variables to get the response variable, then it is defuzzified

to obtain the solution. This whole procedure is explained in [46]. For implicit functions,

21


α-cuts are made at different levels and for each α level, interval analysis is carried. This α

represents confidence in the uncertain numbers, if α = 1, it is called as “crisp value” which

means it is deterministic and if α = 0, the uncertain variable covers the feasible range.

As the interval analysis is used at different α levels, computations become prohibitive

and inefficient. So the Vertex Method was proposed which is efficient [47]. Its application

in studying reliability of structures is increasing [44, 48, 49, 50, 51, 52].

0 2 4 6 8 10

0

0.2

0.4

0.6

0.8

1

µ(x)

x

α

Figure 2.1: General Membership Function of Input Fuzzy Variable

22


2.1.4 Evidence Theory

In physical systems, both epistemic and aleatory uncertainties will be present. Proba-

bility theory can not deal with epistemic uncertainties, so only possibilistic theories can

be used for such uncertainties. Probability theory deals with aleatory type uncertainty.

But both theories can not be used at the same time because of their different set of rules.

In such cases, “Evidence Theory” [53] seems to be best possible solution. Shafer fur-

ther developed Dempster’s [54] work and came with this theory, called Dempster-Shafer

theory [41]. Evidence theory is a generalization of the classical probability and possibil-

ity theories in terms of evidences and their measures. Major technical terms and their

definitions are discussed in [53, 55], these are briefly discussed in the following sections.

2.1.4.1 Frame of Discernment

Mutually exclusive elementary propositions from the universal set, X, become the Frame

of Discernment. These elementary propositions may be overlapping each other or may

be nested in one another. According to their overlap and nesting, those are classified as

Consonant, Consistent, Arbitrary and Disjoint frame of discernment. These elementary

propositions are combined to form a power set, 2X, which represents all the available

combinations including the null set, ∅ and frame of discernment, X.

23


2.1.4.2 Basic Belief Assignment

In evidence theory, for each member of frame of discernment, X, basic belief assign-

ment (BBA), m is assigned which represents the confidence in individual members of X.

Following rules, while assigning BBAs to the propositions must be satisfied,

m : 2X → [0, 1] (2.17)

m (∅) = 0 (2.18)∑A∈2X

m (A) = 1 (2.19)

While assigning BBAs, evidences may not be there so those elements’ BBA should be

assigned 0. Evidences’ information is not passed from one element to other elements of

the power set, 2X , so following relations are valid.

m (x1) + m (x2) 6= m (x1, x2) (2.20)

m (x1) ≥ m (x1, x2) (2.21)

m (X) ≤ 1 (2.22)

where x1 and x2 are elementary propositions of frame of discernment, X. m (x1, x2)

means that there is a confidence in either x1 or x2 but not in both at the same time.

24


2.1.4.3 Belief and Plausibility Functions

From the BBA assignment, the lower bound, Belief, and upper bound, Plausibility of any

proposition, A can be obtained using following equations.

Bel (A) =∑

B|B⊆A

m (B) (2.23)

Pl (A) =∑

B|B∩A6=∅

m (B) (2.24)

These Bel (A) and Pl (A) are represented in the Fig. 2.2. Following properties can be

obtained.

Bel (A) + Bel(A)≤ 1 (2.25)

Pl (A) = 1 − Bel(A)

(2.26)

Pl (A) + Pl(A)≥ 1 (2.27)

where A represents the classical compliment of A.

Bel (A) Bel¡A¢

Pl (A)

Uncertainty

Figure 2.2: Belief and Plausibility Measures

25


2.1.4.4 Rules for the Combination of Evidences

When the evidences come from different sources, these evidences are combined to ob-

tain BBA of a element, A, using different rules as described in [53]. Dempster’s rule

is discussed here, for other rules can be found in [53]. Dempster’s rule assumes that

these sources of evidences are independent. The Dempster rule of combination is purely

a conjunctive operation (AND). Specifically, the combination (called the joint m12) is

calculated from the aggregation of two BBA m1 and m2 in the following manner:

m12 (A) =

∑B∩C=A

m1 (B) m2 (C)

1 − K, A 6= ∅ (2.28)

K =∑

B∩C=∅

m1 (B) m2 (C) (2.29)

K represents basic probability mass associated with conflict. This is determined by the

summing the products of the BBAs of all sets where the intersection is null. Basic

assumption of Dempster’s rule is that the evidences come from consistent resources. If

there is lot of conflict for evidences, a numerical instability occurs. In such cases other

rules of combining should be used [53].

This theory has been applied in structural reliability [37, 43, 55]. This theory seems

to be promising in applied mechanics fields to deal with epistemic as well as aleatory

uncertainties.

26

2.2 Probabilistic Methods


When the definition of uncertainty is available interms of random process or random vari-

ables, probabilistic methods should be preferred to possibilistic methods. Probabilistic

methods in current form are mathematical elegant, give very accurate results and pro-

duce unique results. Realisation of a random phenomenon is called a trial. All possible

outcomes of the phenomenon forms a sample space, Θ, and elements of this set, outcomes

are denoted by θ. For each θ, probability (confidence), P , is assigned in terms of a number

such that P ∈ [0, 1]. The collection of possible events having well-defined probabilities

is called the σ-algebra associated with Θ, and is denoted by F . The probability space is

defined by (Θ,F ,P). A real random variable X is a mapping X : (Θ, F , P) → R.

All definitions like PDF, CDF, their correlation coefficients, ρ, expectations, covariances,

and their functions are extensively discussed in the book by Papoulis [56].

The vectorial space of real random variables with finite second moment (〈X2〉 < ∞) is

denoted by L2 (Θ, F , P). Here 〈〉 is the expectation of the given quantity. A random

field w (x, θ) can be defined as a curve in L2 (Θ, F , P), that is a collection of random

variables as a function of x. A random field is said to be multivariate or univariate

depending upon the physical dimensions of the process. If the mean, µ (x), the variance,

σ2 (x) are constant and autocorrelation coefficient, ρ (x, x′) is a function of the difference

x − x′ only, the random field is called homogeneous. Details of random processes can

be found in [5]. All the probabilistic methods are well discussed in the report [57] which

27


will be described in the following sections. This is the first document that tried to draw

a comparison between different methods, advantages and disadvantages of individual

methods.

2.2.1 Asymptotic Reliability Methods

In this section, we will review existing implicit probabilistic methods, their limitations,

and state how they have been combined with metamodels. For systems in which uncer-

tainty can be represented as a set of discrete random design variables, the probability

that the system response, Z (X), will be less than or equal to a particular value, Z0, can

be expressed as

P (Z ≤ Z0) =

∫Ω

fX (x) dΩ = p (2.30)

where fX (x) is the joint PDF of the random variables and Ω is the domain of integration

defined by the limit state equation, g (X). The limit state equation, g (X) = Z (X)−Z0 ≤

0, is simply the difference between the response function and a particular value that

produces a negative or zero resultant. In structural reliability, when the response function

represents the difference between resistance and load and Z0 = 0, the resultant domain

of integration is referred to as the failure region. Generally, the joint PDF of the input

random variables is not available and, if it is, evaluation of Eq. (2.30) is typically very

difficult. As a result, analytical approximations for evaluating Eq. (2.30) have been

developed.

28


2.2.1.1 Moment Based Methods

In these methods, the mean and covariance of the input random variables are used to

define the mean (first moment) and standard deviation (second moment) of the response.

The probability of failure is obtained as the function of the minimum distance from the

origin to the limit state surface in a reduced standard normal space. This distance is called

the safety factor, β. The collection of random variable values that define the location

of β on the limit state surface for different probability levels is referred to as the most

probable point locus (MPPL). Moment based methods assume that the input random

variables are normal and uncorrelated. Often this is not the case and a transformation

must be employed. The Rosenblatt transformation [6, 58] uses the marginal PDFs and the

covariances of all the random variables to convert non-normal correlated random variables

into a set of independent normal variables. Information about joint and conditional

PDFs may not always be available for the calculation of marginal PDFs. In this case,

the Rackwitz-Fiessler algorithm can be used to approximate the mean and standard

deviation of equivalent normal variables at points along the response function [6].

First Order Second Moment Method(FOSM)

In this method, the response function is linearized about the mean values, µi, of the

random variables,

Z (X) ≈ Z (µ) +n∑

i=1

(∂Z

∂Xi

)µ

(Xi − µi) (2.31)

29


where n is the total number of random variables. The limit state equation is then:

g (X) ≈ a0 +n∑

i=1

aiXi − Z0 (2.32)

where ai are obtained from Eq. (2.31). The probability that g (X) ≤ 0 is calculated as,

p = Φ (−β) (2.33)

where Φ is standard normal CDF and β is computed from

β =µZ

σZ

(2.34)

=

a0 +n∑

i=1

aiµi√n∑

i=1

a2i σ

2i

(2.35)

By varying Z0 ,CDF or PDF can be constructed for the system response. This method is

efficient and accurate when the response function is linear or mildly nonlinear. When the

random variables are non-normal, in the standard normal space response, the resultant

equation may become highly nonlinear. To account for this nonlinearity, Second Order

Second Moment Method (SOSM), and the Advanced Mean Value Method (AMV) have

been developed.

Second Order Second Moment Method

In the SOSM method, the response function is represented by a second order Taylor

30


series expansion about the MPP, X∗, defined by Eq. (2.36) corresponding to a particular

probability level.

Z (X) ∼= Z (X∗)+n∑

i=1

(∂Z

∂Xi

)X∗

(Xi −X∗i )+

n∑i=1

n∑j=1

(∂2Z

∂Xi∂Xj

)X∗

(Xi −X∗i )(Xj −X∗

j

)(2.36)

For large β, the probability of g (X) ≤ 0 is calculated as [17],

p = Φ (−β)n−1∏i=1

(1 + β κi)1/2 (2.37)

where β is the safety index calculated in the FOSM method and κi are the principal

curvatures of the limit state defined by Eq. (2.36). The SOSM method gives accurate

probabilities for second order response functions. When the nonlinearity of the response

surface is greater than second order, the AMV method may be used to account for the

higher order terms neglected in Eq. (2.36).

Advanced Mean Value Method

In the AMV method [18], as in the case FOSM method, the response function Z (X) is

expanded using a Taylor series about the mean values of the random variables

Z (X) = Z (µ)+n∑

i=1

(∂Z

∂Xi

)µ

(Xi − µi)+H (X) (2.38)

31


Z (X) = a0 +n∑

i=1

aiXi+H (X) (2.39)

Z (X) = Z1 (X)+H (X) (2.40)

where Z1 (X) represents the first order response given by Eq. (2.31) and H (X) represents

higher order terms. The first step in the AMV method is to conduct an FOSM analysis

using Z1 (X). Once this is done, the Z1 (X) values in the first order response CDF

corresponding to each probability (β) level are replaced with the ZAMV values shown

below

ZAMV = Z1+H (Z1) (2.41)

by simply revaluating Eq. (2.38) at the MPPL.

The AMV method as described above gives accurate CDF curves when the most probable

point locus calculated using Z1 is close to the exact MPPL. The AMV method can be

improved by iteratively expanding Eq. (2.38) about the Z1 MPPL to obtain an updated

Z∗1 and MPPL∗ until a specified convergence criteria is met. This will give a MPPL equal

to the exact MPPL when there is only one minimum in the response function. If there

is more than one local minimum in the response function, the AMV method may not

converge. For highly nonlinear, nonmonotonic response functions, the AMV method will

produce nonmonotonic CDFs. A correction scheme based on the theory of one random

variable has been proposed to convert the nonmonotonic CDF to an equivalent monotonic

CDF [18]. The total number of response evaluations is n + m + 1, where n represents the

32


total number of random variables and m is the total number of probability levels used to

define the CDF.

Metamodeling And Moment Based Methods

When advanced numerical methods like the finite element or the boundary element

method are used to obtain the response function evaluations and sensitivities for defining

Eqs. (2.31), (2.36), and (2.38), these equations become metamodels. More specifically

Eqs. (2.31), (2.36), and (2.38) represent the numerical response of the model (FEM or

BEM) used to model the original process. This idea of creating approximate models of

models, or metamodels, can be formulated in the following three steps [59].

1. Choosing an experimental design for generating data.

2. Choosing a model to represent the data.

3. Fitting the model to the observed data.

The first step is commonly referred to as “Design of Experiment” (DOE). The essence

of this step is to select a limited number of input variable values that when used in

numerical simulation produce response values that adequately define the response over

the range of interest. These DOE methods include Random Selection Designs, Factorial

Designs, Space Filling Designs, and Orthogonal Array Design, to name a few. Once a

DOE method (set of input values) is selected and a set of response values is generated, an

analytical model is selected to represent the data. The FOSM, SOSM, and AMV methods

33


described above, carry out metamodeling by generating an MPPL based on approximate

first and second order response moments and then representing the response with the

same first order Taylor series expansion as used to generate the MPPL or with a second

order Taylor series expansion. The AMV method adds a correction step to the first order

response representation. The third step in the metamodeling procedure is not needed as

there are no unknowns in the response model that need to be determined.

In probabilistic mechanics, a methodology has been developed that conforms more closely

to the three steps for creating a metamodel. This method, known as the Response

Surface Method, uses low order polynomial response function representation and least

square regression for model fitting. The resultant analytical expression can then be used

as opposed to a numerical procedure (FEM or BEM) in probability calculations. In

this method, the implicitly defined response functions are transformed to a closed form,

analytical equations which can be expressed as:

Z = f (X) + ε (2.42)

where ε is a normally distributed error with zero mean and standard deviation σε, f (X)

is an unknown function that is approximated for slightly nonlinear response functions as:

f (X) = b0 +n∑

i=1

biXi (2.43)

34


or for nonlinear surfaces,

f (X) = b0 +n∑

i=1

biXi +n∑

i=1

n∑j=1

bijXiXj (2.44)

where Xi are the input variables and bi are unknown coefficients to be determined using

regression by the least squares method. When constructing the response surface, the

DOE typically defines sampling points that are at µi ± fiσi; where µi is the mean of

the i’th random variable, σi is the standard deviation and fi is the scaling factor [22].

Various modifications of the above method have been developed to better approximate

the response surface around the MPPL. These methods have been applied to stiffened

plate reliability analysis [23] and for the reliability of clamped-clamped end beams [60].

2.2.2 Perturbation Stochastic Finite Element Method (PSFEM)

This method is based on the Taylor series expansion of the response and system matrices

using input random variables. Application of Taylor series has been implemented since

1970 in many different fields. In PSFEM, random variables are represented as the sum

of mean, µi and variation about the mean, αi. Response as well as input variables are

expanded using Taylor series upto second order expansion about the mean values of

random variables. So this method only results in mean and covariance matrix of the

response vector. This method was applied by many researchers [61, 62, 63, 64, 7] in

different fields. Recently two monographs are published on this subject [8, 9].

35


The application of PSFEM is shown in the context of static problem. The basic principles

of PSFEM remains same for all type of problems. The static equilibrium using FEM is

written as:

K U = F (2.45)

where K, U, and F are stiffness matrix, displacement vector, and applied load vector

respectively. K, U, and F will be random because of randomness in geometric as well as

material properties. So K, U, and F are expanded using Taylor series about the mean

values of random variables which are shown below.

K = K0 +N∑

i=1

KIi αi +

1

2

N∑i=1

N∑j=1

KIIij αiαj + O(‖α‖2) (2.46)

U = U0 +N∑

i=1

UIi αi +

1

2

N∑i=1

N∑j=1

UIIij αiαj + O(‖α‖2) (2.47)

F = F0 +N∑

i=1

FIi αi +

1

2

N∑i=1

N∑j=1

FIIij αiαj + O(‖α‖2) (2.48)

where K0, U0, and F0 are the mean values of respective tensors. ()Ii and ()I

i I represents

the first and second order derivatives evaluated at α = 0, e. g. :

KIi =

∂K∂αi

∣∣∣∣α=0

(2.49)

KIIij =

∂2K∂αi∂αj

∣∣∣∣α=0

(2.50)

36


After substituting Eqs.( 2.46-2.48) in Eq. (2.45) and collecting the similar order of terms,

following equations are obtained

U0 = K−10 F0 (2.51)

UIi = K−1

0

(FI

i − KIi U0

)(2.52)

UIIij = K−1

0

(FII

ij − KIi UI

j − KIj UI

i − KIIij U0

)(2.53)

From these mean and covariance matrix of the response vector, U can be obtained as

〈U〉 ≈ U0 +1

2

N∑i=1

N∑j=1

UIIij Cov [αi, αj] (2.54)

Cov [U, U] ≈N∑

i=1

N∑j=1

UIi

(UI

j

)TCov [αi, αj] (2.55)

After Cov [αi, αj] is substituted in terms of correlation coefficients ρijin Eq. 2.55, final

expression for Cov [U, U] is obtained as

Cov [U, U] ≈N∑

i=1

N∑j=1

∂U∂αi

∣∣∣∣α=0

∂UT

∂αj

∣∣∣∣α=0

ρij σαiσαj

(2.56)

For applying PSFEM, random processes representing material properties, geometric prop-

erties are discretized using different discretizations which are given below.

• Midpoint method

37


• Shape function method

• Integration point method

• Spatial Average method

• Weighted integral method

Further information of these discretization methods can be found in [57] and related

references. As the Taylor series expansion is used in this approach, its applicability is

limited to the problems where coefficient of variation, δ, of input random variables are

small. As the number of input random variables becomes large, this method becomes

time consuming and inefficient.

2.2.3 Spectral Stochastic Finite Element Method (SSFEM)

This method was initially proposed by Ghanem and Spanos [28, 29, 30] using Karhunen-

Loeve expansion and further developed [31, 32] to account for higher coefficient of vari-

ation, δ of input random variables. Galerkin procedure is employed in random (proba-

bility) space which is exponentially convergent. All the tools needed for this method are

discussed in subsequent sections.

38


2.2.3.1 Karhunen-Loeve Expansion

Using Karhunen-Loeve expansion, a random process, w (x, θ), can be written like Fourier

decomposition in terms of eigenfunction and eigenvalues of the correlation function [65]

w (x, θ) = w (x)+∞∑i=1

√κiφi (x) ξi (θ) (2.57)

where w (x) is the mean value of the random process, ξi (θ) are uncorrelated random vari-

ables and Ω is the domain over which the random process w (x, θ) is defined. κi and φi (x)

are the eigenvalues and eigenfunctions of the autocorrelation function Rhh (x,y) which

is positive semi-definite and symmetric (i.e. Rhh (x,y) = Rhh (y,x)). The eigenvalues

and eigenfunctions of the correlation function are obtained by solving a homogeneous

Fredholm integral equation of the second kind

∫Ω

Rhh (x,y) φi (y) dy = κiφi (x) (2.58)

Only for very few correlation functions, analytical solution is available for Eq. (2.58).

To solve Eq. (2.58) for any arbitrary correlation function, Galerkin procedure, explained

in Section 2.2.3.3 is employed. The random process is expanded using finite terms in

Karhunen-Loeve expansion. The number of terms used in the expansion depends upon

the eigenvalue’s magnitude and the expansion converges in the mean square sense to the

correlation function.

39


2.2.3.2 Generalized Polynomial Chaos

Since the correlation function of the response of an uncertain system to an uncertain input

is not available in an analytical form, the Karhunen-Loeve expansion, in its original form

can not be used to represent the correlation function. So the response should be written

in terms of a nonlinear function of the random variables that are the basis functions

of the input correlation function. This is known as polynomial chaos. Wiener [4] first

introduced homogeneous chaos to represent a second order Gaussian random process. It

was first used by Ghanem and Spanos to solve structural mechanics problem using the

FEM [28]. In Wiener Polynomial Chaos theory, a random process G (θ,x) is expressed

as

G (θ,x) = a0 (x) Γ0 +∞∑

i1=1

ai1 (x) Γ1 (ξi1 (θ)) +

∞∑i1=1

i1∑i2=1

ai1i2 (x) Γ2 (ξi1 (θ) , ξi2 (θ)) + · · · , (2.59)

where Γn (ξ1, . . . , ξn) denotes the Hermite polynomial of order n, an n dimensional poly-

nomial function of ξi, i = 1, 2, . . . , n; ξi are uncorrelated standard normal variables, and

θ is a realization of these variables, and ain are deterministic coefficients. If G (θ) is a

random variable then ain are constants. These Hermite polynomial functions are given

as,

Γn (ξ1, . . . , ξn) = e12ξT ξ (−1)n ∂n

∂ξi1 · · · ∂ξin

e−12ξT ξ (2.60)

40


Equation (2.59) can be written in a simple form as

G (θ) =P∑

j=1

GjΓj (ξ) (2.61)

The orthogonality property of polynomials used in polynomial chaos can be written as

〈ΓiΓj〉 =⟨Γ2

i

⟩δij (2.62)

where δij is the Kronecker delta and 〈., .〉 represents the expectation of the weighted inner

product of these polynomials in variable ξ, i.e.

〈f (ξ) g (ξ)〉 =

∫Σ

f (ξ) g (ξ) W (ξ) dξ (2.63)

where W (ξ) is the weight function and Σ is the domain of the random variable. For

Hermite chaos, W (ξ) is a multidimensional standard orthonormal joint probability den-

sity function. Hermite polynomial chaos solution approaches in the mean square sense

to exact response of the system if the input random process is Gaussian. Mean square

sense convergence of the response implies that the error in the mean and the variance

of the polynomial chaos solution approaches exponentially to exact mean and variance

of the response as number of terms in the Karhunen-Loeve expansion are increased.

The ‘mean-square’ error of the numerical solution from the chaos expansion up (x, θ) is

41


computed

e2 (x) =(E [up (x, θ)− ue (x, θ)]2

)1/2(2.64)

where E denotes the ‘expectation’ operator and p is the order of the chaos expansion.

The ‘mean-square’ convergence (L2 convergence in random space) of the L∞ norm (in

physical space) of e2 (x) as p increases. To converge in the mean square sense for other

processes, Wiener-Askey polynomials should be used as explained in [33]. For these

processes, the formulation will still be same as given in Eq. (2.59).

2.2.3.3 Galerkin Procedure

After the expansion of the input random process using the Karhunen-Loeve theorem,

the response of a system is written as polynomial chaos. For discrete inputs, the input

random variables are expanded using polynomial chaos. The inputs and outputs in terms

of polynomial chaos are substituted in the governing stochastic differential equation. A

solution is obtained using Galerkin procedure, a well known approach for solving ordinary

and partial differential equations in complex spatial domains. A tremendous amount of

literature is available on using Galerkin method for solving problems in structural me-

chanics, fluid mechanics, and applied mathematics. In polynomial chaos, the Galerkin

method is used to minimize the weighted residual of the differential equation by mul-

tiplying Γj (ξ), polynomial function from polynomial chaos expansion terms. Here, the

42


error criteria is defined in two ways, the error in the mean and variance of the polynomial

chaos solution are compared to exact mean and variance of the output. As number of

terms in the Karhunen-Loeve expansion and in polynomial chaos increases, these error

norms decrease exponentially. The whole procedure is summarized as follows,

• Expand the input random process using the Karhunen-Loeve theorem, Eq. (2.57),

and the output using polynomial chaos, Eq. (2.61), as a function of the appropriate

random variables.

• Substitute input random process expansion and response expansion in the governing

differential equation,

Lu (x, t, θ) = f (x, t, θ) (2.65)

Bi (x, θ) u (x, θ) = 0, i = 1, 2, . . . , k (2.66)

u (x, t, θ) =P∑

j=0

uj (x, t) Γj (ξ (θ)) (2.67)

f (x, t, θ) =P∑

j=0

fj (x, t) Γj (ξ (θ)) (2.68)

43


P∑j=0

Biuj (x, t) Γj (ξ (θ)) = 0, i = 1, 2, . . . , k (2.69)

P∑i=0

√κiφiξi L

(P∑

j=0

uj (x, t) Γj (ξ (θ))

)=

P∑j=0

fjΓj (2.70)

where L is differential operator, Bi are stochastic linear homogeneous differential

operators associated with boundary conditions, f (x, t, θ) is the source term and

u (x, t, θ) is the response of the differential equation. This differential equation is

well posed with appropriate number of boundary and initial conditions. Here, the

uncertainties can be in the boundary, and/or initial conditions, material, and/or

source terms. The variable, P indicates the chaos order; uj and fj are polynomial

chaos coefficients of u (x, t, θ) and f (x, t, θ) respectively. Bi operate on u (x, t, θ)

as given in Eq. (2.69).

• Error in the mean and the variance of the response is minimized by multiplying

Eq. (2.70) by Γk and taking the expectation of Eq. (2.70) results in Eq. (2.71).

⟨P∑

i=0

√κiφiξi L

(P∑

j=0

uj (x, t) Γj (ξ (θ))

), Γk

⟩= fk

⟨Γ2

k

⟩,

k = 0, 1, . . . , P. (2.71)

The orthogonality property of polynomials will be used in these calculations. Equa-

tion (2.71) is a set of multidimensional algebraic equations or equations in multi-

dimensional tensors. By solving this multidimensional system, Eq. (2.71), the de-

terministic coefficients uj and probabilistic characteristics of the response, u, will

44


be found.

The same type of procedure should be used to find the eigenvalues and the corresponding

eigenfunctions of the correlation function numerically. In this case, the eigenfunction is

written as∑

dkNk, dk are deterministic coefficients and Nk are the shape functions. After

substituting∑

dkNk in Eq. (2.58), this equation is multiplied by Nq, the shape function,

and the matrices are formulated and the resulting general eigenvalue problem is solved

to get the eigenvalues and eigenfunctions of the correlation function.

The discussed SSFEM is called as “Intrusive Method”, the requirement of this method is

that FEM code is developed from the scratch. So Choi et al [66] proposed “Non-intrusive

method” in which deterministic FEM code is used to get the probabilistic response sub-

jected to input random variable uncertainties. In this method, Latin Hypercube Sampling

(LHS) is used to get response polynomial chaos coefficients. If the uncertainties are ran-

dom process, those are decomposed using the Karhunen-Loeve expansion and evaluated

at gauss points during FEM calculation and stochastic response is obtained. This method

can be called as “Semi-Intrusive Method”, as this method requires the decomposition of

input random processes and evaluations at gauss points. This method is still computa-

tionally expensive for dynamical response. Therefore in this work, a new algorithm for

finding fundamental eigenvalue of linear stochastic differential equation is presented.

45


2.2.4 Sampling Techniques

For highly nonlinear, nonmonotonic response functions moment based methods may not

give accurate results. This is because moment based methods utilize linear approxima-

tions to locate the MPPL and hence may converge to a local minimum, not to the true

MPPL represented by the global minimum. As such sampling methods may have to be

employed to produce accurate results. Three of the most popular sampling methods are

described in the following.

2.2.4.1 Standard Monte Carlo Sampling (MCS)

In this technique, sampling points defined as vectors of random variable values are ran-

domly generated using the definition of the input variables. The response of the system

is evaluated for each vector of input variable values. The CDF or PDF of the response is

constructed using the ratio of values less than a particular response to the total number

of responses. The accuracy of this CDF or PDF can be examined using the Coefficient of

Variation, δ of each probability or the associated confidence interval [17]. This method

is very simple and efficient when used with analytical response functions but can become

computationally inefficient when numerical methods are used to calculate the system

response. Therefore sampling techniques have been developed that employ a reduced

number of sample points.

46


2.2.4.2 Latin Hypercube Sampling (LHS)

One method that uses fewer sampling points than MCS was proposed by McKay et al [27].

In this method, the range of each random variable is divided into N , non-overlapping

intervals of 1/N probability. For each input variable, one value is randomly selected

from each probability interval to form a data set for that variable. If n is the number

of random variables, this gives n data sets of N values. The data set of first random

variable is then combined randomly with the data set of the second random variable to

produce a N × 2 matrix. This matrix is combined randomly with the data set of the

third random variable and so on until an N ×n matrix is obtained [67]. The rows of this

matrix give N sample points that can be used to produce the response ratios in MCS

that define the CDF of the response.

2.2.4.3 Importance Based Sampling

Sampling is carried over the whole input variable domain in MCS (randomly) and LHS

(stratified). In Importance Based Sampling, sampling is conducted only in the region

where g (X) ≤ 0 [68]. Given this condition Eq. (2.30) can be written as:

p =

∫I [g (X) ≤ 0]

fX (x)

fS (x)fS (x) dx (2.72)

where I [g (X)] is indicator function that is equal to 1 when g (X) ≤ 0 and equal to 0

when g (X) > 0, fS (x) is the density function around the most probable point and s is

47


the domain around the most probable point. Generally fS (s) is chosen as a normally

distributed density function with mean at the most probable point and standard deviation

equal to that of the original density function. This procedure reduces the covariance of

the probability calculation and therefore requires less number of sample points than MCS.

2.2.4.4 Combined Use With Metamodels

When using sampling techniques with numerical methods computational efficiency is ei-

ther realized by significantly reducing the number of required samples or by replacing

the numerically generated response function with an equivalent analytical expression.

The latter has become the most popular as the numerical procedures used to describe

the response of complex systems can themselves be computationally intensive. In these

instances sampling methods typically take the role of a DOE and provide the framework

for evaluating the probabilities of interest. Response surfaces, as described in Eqs. (2.43)

and (2.44), can be constructed using samples from MCS or LHS and then used in sub-

sequent FOSM or SOSM probability calculations (e. g. [69]). Alternatively, a neural

network [70] can be created and trained using sampling points to replace the system

response functions for performing the probabilistic calculations [71].

48

Chapter 3

Probabilistic Sound Power and its

Sensitivity

3.1 Introduction

The goal of probabilistic modeling in the design and analysis of structural-acoustic sys-

tems is to adequately account for uncertainty when predicting vibro-acoustic perfor-

mance. In this work, a methodology is presented for calculating the CDF, characterizing

the radiated sound power of a vibrating structure comprised of deterministic and random

structural parameters. A sensitivity algorithm is also presented that predicts the change

in the sound power CDF due to a change in deterministic structural parameters.

49

3.1 Introduction

Bernhard and Kompella [72, 73] performed pioneering work that documents the exis-

tence of uncertainty in the interior noise of automotive vehicles. Their work showed large

statistical variation in the measured acoustic frequency response functions of nominally

identical, post-production vehicles. They speculated that such variation was induced

during manufacturing and assembly. In part as a result of this work, the automotive

industry has recognized the need to develop efficient acoustic prediction methods that

account for inherent uncertainty when generating the acoustic frequency response func-

tions that characterize a vehicle [74].

FEM and BEM are the main numerical techniques used to predict and analyze the

response of structural acoustic systems that display distinct model characteristics. When

predicting the frequency response of structures that are uncoupled from the surrounding

acoustic medium, a harmonic excitation is applied to the structure and the FEM is used

to calculate resultant vibration [75, 76, 77]. This vibration forms the boundary conditions

in the BEM used to calculate the acoustic response on the surface of the structure and/or

at a point in the acoustic domain [78, 79, 80, 81]. Once the initial system response has

been predicted sensitivity analysis can be performed to indicate desirable design changes.

Many sensitivity analysis techniques utilize FEM and BEM to predict the change in vibro-

acoustic response due to a change in a structural parameter. In these techniques, FEM

is used to predict the change in structural vibration due to a change in a structural

parameter, or the structural sensitivity [82, 83]. The BEM is used to calculate the

sensitivity of the acoustic response due to a change in the prescribed vibration, or the

50

3.1 Introduction

acoustic sensitivity [84, 85]. Once the structural and acoustic sensitivities are known they

can be combined to predict the change in the acoustic response of a vibrating structure

due a change in its structural parameters [86] in hopes of obtaining a more favorable

acoustic response.

Although there has been abundance of work that combines probabilistic methods with

FEM and BEM to treat uncertainty in traditional areas of mechanics and reliability,

limited work has been done to apply these techniques to acoustic or structural acous-

tic systems and the concept of probabilistic structural acoustic sensitivity with respect

to deterministic design variables has yet to be addressed. This definition excludes the

well known inverse techniques developed by Soize that account for the influence of struc-

tural uncertainty in system response where the uncertainty has been defined as random

impedances with partially defined statistics (mean and standard deviation) [87]. To the

authors’ knowledge the only work done before the turn of the century involving proba-

bilistic analysis of an acoustic system occurred in 1990 by Ettouney and Daddazio [88].

These researchers combined BEM with a perturbation approach to recursively solve for

second order expressions of the unknown surface pressure and normal velocity written in

terms of uncertain parameters. The uncertain parameters were taken to be the character-

istics of the acoustic medium and were described as random variables with known PDFs.

Using the second order pressure and velocity equations statistical properties of surface

impedance were calculated. This method follows the PSFEM as defined by Kleiber and

Hein [8].

51

3.1 Introduction

The AMV method has been shown to work efficiently with implicitly defined, non-

monotonic response functions and an implementation is readily available [18]. In 2002,

Allen and Vlahopoulos combined FEM and BEM with the AMV method to calculate

the cumulative distribution function characterizing interior sound pressure for structural

enclosures with uncertainty in some of the structural design parameters [89]. Uncer-

tainty was described in terms of random variables with known PDFs. Sub-structuring

techniques and stored vibration invariant information from the boundary element anal-

ysis were employed to ensure efficiency in the multiple structural acoustic computations

required in the probabilistic method. Large variation between deterministic and acoustic

response at high probability levels were observed. This variation was shown to be caused

by the interaction between structural and acoustic modes. Specifically, structural uncer-

tainty allows the location of structural modes to vary therefore increasing coupling with

acoustic modes located nearby.

In this work, the FEM/BEM/AMV methodology mentioned above is extended to calcu-

late probabilistic radiated sound power. The radiated sound power at a high probability

level over a frequency range is taken to represent an acoustic performance envelope for

the structure. A sensitivity algorithm is also presented that predicts the change in the

radiated sound power at a given probability level due to change in the deterministic

structural parameters. To illustrate the concept of an acoustic performance envelope

and its sensitivity the probabilistic method and the probabilistic sound power sensitivity

algorithm are applied to a simple two-degree-of-freedom system whose acoustic response

52

3.2 Theoretical Derivation

can be solved analytically. The FEM/BEM/AMV method and the sensitivity algorithm

are then applied to a complex structural acoustic system representative of an automo-

tive windshield. The windshield is modeled as an elastically supported plate subject to

a deterministic load. In both applications the structure is considered to be comprised

of deterministic structural parameters and parameters with inherent uncertainty. Prob-

abilistic sound power computations are validated through comparison with data from

Monte Carlo simulation and probabilistic sound power sensitivities are validated through

comparison with data computed through re-analysis.


The numerical algorithms for calculating the probabilistic sound power of a vibrating

structure and its sensitivity are presented in this section. A brief review of how FEM

and BEM are employed in deterministic sound power calculations is provided. For proba-

bilistic sound power calculations the structure is considered to be comprised of uncertain

and deterministic parameters and subject to a deterministic excitation. The uncertain

structural parameters are described as random variables with known PDFs. The AMV

method is used to evaluate the joint PDFs describing the system over the performance

surface defined by the sound power computations, thus producing the CDF describing

the acoustic response. Direct differentiation of the FEM/BEM/AMV procedure is em-

ployed for calculating the change in the radiated sound power associated with a particular

53


probability level due to a change in a deterministic structural design parameter. It has

been noted in earlier work that certain performance functions can exhibit non-monotonic

behavior [18, 89]. Such behavior has been accounted for in this formulation.

3.2.1 Sound Power Calculations

Following the standard Galerkin finite element method, the governing differential equa-

tion describing the motion of an arbitrarily shaped structure subject to a harmonic

excitation can be written in matrix form as [75, 77, 82]:

M U + C U + K U = F (3.1)

where U =nodal displacement vector, M =mass matrix, C =damping matrix, K =stiffness

matrix, and F =the nodal forcing vector. Solving Eq. (3.1) for U and multiplying each

side of the equation by a transformation matrix, T1 produces the normal velocity com-

ponents on the surface of the structure. The normal velocity, Vn, is written as:

Vn = T1 U = T1 S−1t F (3.2)

where the structural matrix, St, is complex and is equal to −ω2M + iωC + K. The

transformation matrix, T1 represents a conversion of structural displacement to structural

vibration and a projection of that vibration onto the acoustic boundary element model.

54


In this formulation structural vibration is considered to be independent of the effects of

the surrounding acoustic medium. Such an assumption is valid when the structure is

immersed in light fluids such as air. Although a direct matrix inversion technique has

been presented here, solution via modal superposition is equally valid. It is recommended

that for large structures, sub-structuring techniques be employed to isolate components

with random structural parameters, see [89].

Once the structural vibration response is known, a collocation procedure can be followed

to solve the time invariant wave equation for the resultant acoustic response. Specifically,

the direct [79, 80, 81] boundary element method is utilized for computing the acoustic

surface pressure, Ps, generated due to the velocity, Vn, boundary condition. In this

method the Surface Helmholtz integral equation, given by Seybert et al [90] as

1 +

∫S

∂

∂n

(1

4πR

)PS (r) dS =

∫S

(Ps (r0)

∂

∂n

(e−iKR

4πR

)− iωρ

(e−iKR

4πR

)vn (r0)

)dS (3.3)

is discretized into a set of nodes and elements. In Eq. (3.3), S represents the vibrating

surface, K is the acoustic wave number, ∂/∂n implies partial differentiation with respect

to the surface normal, r and r0 represent locations on the vibrating surface and R is the

magnitude of the distance between r and r0. After properly accounting for singularities

encountered during integration [91, 92], and non-unique solutions associated with interior

characteristic frequencies [81], Eq. (3.3) can be written in matrix form as:

A Ps = B Vn (3.4)

55


where A and B are the acoustic system matrices which are a function of frequency and

acoustic medium. After Eq. (3.4) has been solved for the unknown nodal pressure on the

surface of the boundary element model, Ps, the radiated sound power can be calculated

as [91]:

PW =1

2Re(PT

selAr V∗

nel

)(3.5)

where vectors Pseland Vnel

represent elemental surface pressure and normal velocity,

respectively. Ar is a diagonal matrix of elemental areas. The elemental surface pressure

and normal velocity relate to nodal quantities via transformation matrix T2 as

Psel= T2 Ps, V∗

nel= T2 V∗

n (3.6)

Transformation matrix, T2 averages nodal values over the surface of the element to obtain

elemental quantities.

3.2.2 Probabilistic Structural Acoustic Analysis

In order to account for the presence of structural uncertainty the procedure outlined

above is combined with an asymptotic reliability method. Physical parameters of the

structure that produce randomness in the acoustic response are taken to be random

design variables with known PDFs. The AMV method is employed to integrate the joint

56


PDF associated with the random variables up to the limit state surface defined by the

sound power computations. Thus the CDF characterizing the radiated sound power is

produced.

Radiated sound power constitutes the performance function for our system and can be

written in terms of deterministic and random structural variables as:

PW = PW (H, X) (3.7)

where X is the vector of random variables and H is a vector of deterministic structural

design parameters. Due to the presence of X, it follows that the acoustic response is also

a random variable and the probability that the radiated sound power will be less than

some value, PW0, can be expressed as [18]:

P (PW < PW0) =

∫Ω

fX (x) dx (3.8)

where fX (x) is the joint PDF of the random design variables and Ω is the region where

the performance function, i.e. sound power, is less than the particular value PW0. In

standard reliability analysis, Ω is referred to as the failure domain. It should be noted

that the region of interest in this work corresponds to what is called the safety domain

in standard reliability analysis as we are concerned with the probabilistic performance of

our system and not its reliability.

57


The AMV method is a mean value, first order reliability technique combined with a cor-

rection procedure. As with most moment based reliability methods the AMV method

evaluates Eq. (3.8) using the standard normal function. This implies that all the random

variables must be uncorrelated and Gaussian. In theory, the Rosenblatt Transforma-

tion [16, 58] can be employed to ensure this condition. Once this is done the location

at which to evaluate the standard normal must be determined. This point is referred to

as the reliability index and represents the minimum distance between the origin and the

limit state surface in the normalized random variable domain. Due to the complexity

of Eq. (3.5), the limit state surface is taken to be defined implicitly through a linear

response surface. This is done by expanding the radiated sound power in a Taylor series

expansion about the means of the random variables and neglecting higher order terms.

The limit state surface defines the boundary of (PW < PW0) and is written as:

g (H, X) ≈ PW (H, µ) +n∑

j=1

(∂PW

∂Xj

)(Xj − µj) − PW0 (3.9)

where g (H, X) is known as the limit state equation, n = the total number of random

design variables, ∂PW/∂Xj is the structural acoustic sensitivity of the radiated sound

power with respect to design variable, Xj evaluated at the mean value, µj, of design

variable Xj. The sound power sensitivity is obtained by differentiating Eq. (3.5) with

58


respect to the j’th random design variable to give:

PW

∂Xj

=1

2Re

(PT

selAr

(∂V∗

n

∂Xj

)el

+

(∂Ps

∂Xj

)T

el

Ar V∗nel

)(3.10)

In differentiating Eq. (3.5), it was assumed that changes in the random design variables

do not affect the shape of the boundary element model. This limits the design variables

to be either material or sizing variables. Both the sensitivity of the surface pressure and

normal velocity to the j’th random variable, in Eq. (3.10), are defined in terms of nodal

displacement sensitivity using Eqs. (3.2), (3.4), and (3.6);

(∂Ps

∂Xj

)el

= T2 A−1 B T1∂U∂Xj(

∂V∗n

∂Xj

)el

= T2 T1∂U∂Xj

(3.11)

The sensitivity of the nodal displacements is calculated as [83]:

∂U∂Xj

= −(−ω2 M + i ω C + K

)−1(−ω2 ∂M

∂Xj

+ i ω∂C∂Xj

+∂K∂Xj

)U (3.12)

Having developed an approximation for the limit state surface, the reliability index can

be determined. For random variables with normal distribution, the reliability index, β,

59


is equal to:

β =µg

σg

=PW (H, µ) − PW0√

n∑j=1

(∂PW∂Xj

σj

)2(3.13)

where σj represents the standard deviation of random variable j. Once the reliability

index is known the probability is evaluated as

P (PW < PW0) = Φ (−β) (3.14)

where Φ denotes the standard normal function.

Unlike other first order reliability methods the AMV method updates the CDF defined by

Eq. (3.14) with a corrective procedure that accounts for the higher order terms originally

neglected in Eq. (3.9). Given the reliability index, β, and the limit state equation the

design point in the standard normal space can be obtained. In the standard normal

design space the coordinates for the design point are expressed as:

λi = β(∇g)i

|∇g|(3.15)

where λi represents the normal coordinate of the i’th random variable, and (∇g)i/|∇g|

is the i’th component of the unit normal to the limit state surface evaluated at µ. The

normal coordinates can be translated into the X design space through the following

60


relationship:

X∗i = λi σi + µi (3.16)

The design point, X∗ is employed for correcting the CDF defined by Eq. (3.14). If one

assumes that the design point for the linear response surface defined in Eq. (3.16) is close

to the design point for the actual limit state surface then the CDF can be corrected by

simply evaluating the sound power at the design point. Retaining the probability defined

by Eq. (3.14) the corresponding acoustic pressure of interest becomes

Φ (−β) = P (PW < PW (H, X∗)) (3.17)

Equation (3.17) defines the CDF generated by the AMV method. Sound power values as-

sociated with high probability levels are considered to represent an acoustic performance

envelope for the system.

3.2.3 Probabilistic Sound Power Sensitivity

Equation (3.17) implies that the sound power associated with a certain probability level

is simply the sound power evaluated at the design point associated with that probability

level. This is denoted by the following expression.

PWβ = PW (H, X∗) (3.18)

61


The change in the sound power at a given probability level due to a change in the m’th

deterministic design parameter is obtained by differentiating Eq. (3.18) as follows:

∂PWβ

∂Hm

=∂PW

∂Hm

∣∣∣∣X

+N∑

j=1

∂PW

∂X∗j

∂X∗j

∂Hm

(3.19)

The first term in Eq. (3.19) and ∂PW/∂X∗j represent the sensitivity of the sound power

with respect to the m’th deterministic design parameter and the j’th random variable

evaluated at the design point respectively. These sensitivity values are computed us-

ing Eqs. (3.10), (3.11), and (3.12). ∂X∗j /∂Hm is the sensitivity of the design point to

the m’th deterministic parameter. The sensitivity of the design point is calculated by

differentiating Eqs. (3.16) and (3.15):

∂X∗j

∂Hm

= σj

∂λ∗j∂Hm

= σj β∂

∂Hm

((∇g)j

|∇g|

)(3.20)

where the sensitivity of the unit normal is given by:

∂

∂Hm

((∇g)j

|∇g|

)= σj

∂PW

∂Xj

−

n∑k=1

∂PW∂Xk

σ2k

∂2PW∂Hm∂Xk(

n∑k=1

(∂PW∂Xk

)2

σ2k

)1.5

+

∂2PW

∂Hm∂Xj(n∑

k=1

(∂PW∂Xk

)2

σ2k

)0.5

(3.21)

In the numerical implementation of this sensitivity algorithm the second partials in

Eq. (3.21) are obtained by finite difference analysis where the perturbation of the m’th

deterministic design variable is taken to be 0.1% of its original value.

62


3.2.4 Non-Monotonic Response and Associated Sensitivity

In earlier work [89], it was shown that structural acoustic systems can exhibit a non-

monotonic CDF response at certain frequencies. These frequencies correspond to in-

stances where multiple design points on the acoustic response surface produce the same

sound power. For structural acoustic systems non-monotonic probabilistic response is

often encountered around resonance.

For instances where Eq. (3.5) is concave or convex over the locus of design points that

correspond to Eq. (3.17) there exist two design points that produce the same sound power,

yet correspond to different probability levels. Using a previously developed correction

scheme [18], an accurate probability value can be obtained at these sound power values.

The corrected CDF, CDFc, associated with non-monotonic performance functions is given

by the following approximations [18]:

CDFc = 1 − CDF1 + CDF2 (3.22)

or

CDFc = CDF1 − CDF2 (3.23)

where CDF1 and CDF2 are the two original probability values calculated using Eq. (3.18)

that correspond to the same sound power value, ordered such that CDF1 > CDF2.

Eq. (3.22) represents the relationship between corrected CDF and the non-monotonic

63

3.3 Application and Validation

CDF for the case where Eq. (3.5) is concave over the locus of design points and a maxi-

mum sound power value has been identified. Eq. (3.23) is used when Eq. (3.5) is convex

over the locus of design points and a minimum sound power value has been identified.

The sensitivity of the sound power associated with the corrected probability level defined

by Eqs. (3.22) and (3.23) is calculated using a simple average. The sound power associated

with the probability values denoted by CDFc, CDF1, and CDF2 in Eqs. (3.22) and (3.23)

are by definition identical. By differentiating Eqs. (3.22) and (3.23) and averaging on

two branches, we can write the sensitivities for corrected monotonic CDFs at a given

probability level,

(∂PWβ

∂Hm

)c

=∓((

∂PWβ

∂Hm

)1

+(

∂PWβ

∂Hm

)2

)2

(3.24)

The sign of the sensitivity in Eq. (3.24) is dependent upon the concavity of Eq. (3.5), if

the non-monotonic CDF is concave, the sign is negative and it is positive, if the CDF is

convex. Eq. (3.24) is directly employ for calculating probabilistic sensitivities for non-

monotonic results.


In this section the probabilistic sound power and probabilistic sensitivity analysis for two

vibrating structures are presented. In each case, the probabilistic response is validated

64


through comparison with results from standard Monte Carlo Simulation and the sensi-

tivity values are validated through comparison with results from a re-analysis. The first

structural acoustic system is used to illustrate the concept of an acoustic performance

envelope and its sensitivity without the use of FEM or BEM computations. This simple

system represents a two degrees-of-freedom piston placed in an infinite baffle. Uncertainty

is considered in the system stiffness and probabilistic sensitivities are calculated with re-

spect to the deterministic system mass and damping. The second system employs the

FEM/BEM/AMV algorithms outlined in Section 3.2 and represents a deterministically

excited automotive windshield. The windshield is modeled as an elastically supported

plate with model characteristics taken from literature [93]. Uncertainty is considered

in the stiffness of the elastic support and probabilistic sensitivities are calculated with

respect to thickness sizing variables.

3.3.1 Piston in an Infinite Baffle

The two-degree-of-freedom piston in an infinite baffle is illustrated in Fig. 3.1 and system

characteristics are provided in Table 3.1. Analytical expressions for sound power and

sound power sensitivity can be readily obtained for this system. Given that the system

is excited by a time-harmonic point load, the radiated sound power can be determined

as

65


b1

k1

m1

b2

k2

m2

F

Figure 3.1: Baffled Circular Piston Configuration

Mass, m1, m2 8kgDamping, b1, b2 1.3E02N − s/m

Diameter, d 0.1mForce amplitude, F 200N

Table 3.1: Characteristics of the Baffled Circular Piston

66


PW =1

2ρ0 c S

(1 −

2 c J(

ω dc

)ω d

) (a′2 + b′2

a2 + b2

)a′ = −ω2 F (b1 + b2)

b′ = −F ω(k1 + k2 −m1 ω2

)a = k1 k2 − ω2

(b1 b2 + k2 m1 + k1 m2 + k2 m2 − m1 m2 ω2

)b = −ω (b2 k1 + b1 k2) + ω3 (b2 m1 + b1 m2 + b2 m2) (3.25)

where ρ0 =density of the air, c =speed of sound in the air, S =cross-sectional area of the

piston, d =piston diameter, F =magnitude of the harmonic load, and ω =frequency in

radians. Equation (3.25) is derived on the assumption that frequency of excitation is in

low frequency regime to satisfy K d/2 << 1 and diameter of the piston is chosen accord-

ing to this condition. Uncertainty is considered in the spring constants of the system.

Spring stiffness values, k1 and k2, are taken to be independent, normally distributed ran-

dom variables with a mean value of 4.39E05N/m and standard deviation equal to 15%

of the mean value.

The radiated sound power is calculated for the deterministic configuration where the

random variables are considered equal to their mean value, Fig. 3.2. The sound power

associated with a 98% probability level is also calculated and plotted in Fig. 3.2 using

Eq. (3.25) where the FEM/BEM computations have been replaced with analytical ex-

pressions. Given that the radiated sound power at a particular frequency will be less

than or equal to the corresponding value on the probabilistic response curve 98% of the

67


time, this curve is taken to represent an acoustic performance envelope. Monte Carlo

simulation for this structural acoustic configuration was conducted using 104 samples.

Results from the Monte Carlo Simulation are also plotted in Fig. 3.2 and confirm the

accuracy of the AMV method.

20 40 60 80 10050

55

60

65

70

75

80

85

90

95

Frequency (Hz)

Sou

nd p

ower

(dB

)

Deterministic98% probabilisticMonte−carlo

Figure 3.2: Deterministic and 98% Probabilistic Radiated Sound Power for Baffled CircularPiston Configuration

The sensitivity of the acoustic performance envelope to changes in the deterministic

structural parameters of the system can be calculated using Eqs. (3.19) and (3.24). For

this configuration the system masses and dampers are taken to be deterministic design

variables. Sensitivity values associated with these variables are given in Table 3.2 along

with the corresponding sound power value. From the sensitivity values it can be seen

that the mass is going to affect the sound power as compared to damping, but for overall

68


reduction of sound power, damping needs to be changed as compared to mass because

mass change will shift the curve horizontally only. Perturbation in damping creates

shifting of the sound power envelope vertically and mass perturbation moves sound power

curve horizontally. So to have a general case of perturbation of sound power envelope;

mass, m1 and m2 and dampers b1 and b2 are perturbed by +2% simultaneously.

To validate the sensitivity algorithm away from resonance, mass, m1 and m2 and dampers,

b1 and b2 are perturbed by +2% simultaneously; new 98% probabilistic sound powers are

calculated and compared with the predicted 98% sound powers using sensitivities through

finite difference which are given in Table 3.3. The error between the predicted sound

power and actual sound power at 98% probability is less than 4%. Around resonance,

sensitivities become very high. To predict 98% probabilistic sound power for perturbed

system, a +1% perturbations in both masses m1 and m2 and same for dampers, b1 and b2

respectively are used to obtain the sensitivities. Predicted sound power calculated using

probabilistic sensitivities obtained through this algorithm and actual 98% probabilistic

sound power obtained through MCS are given in Table 3.4. The error between these

sound powers is less than 4%.

3.3.2 Elastically-Supported Plate

To check the robustness of this algorithm, a complex structural/acoustic system is used

so that FEM and BEM, numerical techniques will have to be used. A flexible, elastically

69


Freq Sound Sound Power Sensitivity,(Hz) Power N/s

N −m/s m1 m2 b1 b2

15.0 8.655E − 06 6.929E − 07 1.929E − 06 −6.666E − 10 −2.025E − 1020.0 1.509E − 03 −1.044E − 04 1.173E − 05 −1.335E − 05 −3.467E − 0621.5 2.349E − 03 4.762E − 04 1.395E − 03 −2.784E − 06 −8.685E − 0624.5 3.355E − 03 −9.949E − 04 −2.563E − 03 −3.492E − 05 −1.670E − 0526.0 2.222E − 03 −1.951E − 03 −5.469E − 03 −1.157E − 05 −6.598E − 0630.0 7.475E − 05 −1.936E − 05 −4.876E − 05 −9.715E − 06 −5.925E − 0935.0 2.476E − 05 −4.652E − 06 −9.345E − 06 −3.558E − 10 −6.881E − 1037.0 1.912E − 05 −3.610E − 06 −6.341E − 06 1.118E − 10 −4.483E − 1042.0 1.115E − 05 −2.731E − 06 −2.973E − 06 9.145E − 10 −4.238E − 1145.0 8.693E − 06 −2.646E − 06 −1.942E − 06 1.599E − 09 4.343E − 1047.0 7.075E − 06 −2.634E − 06 −1.423E − 06 2.289E − 09 1.015E − 0949.0 5.611E − 06 −2.563e− 06 −9.991E − 07 3.296E − 09 1.947E − 0952.0 4.657E − 06 9.493E − 05 2.179E − 06 1.675E − 07 1.348E − 0754.0 5.589E − 05 2.572E − 05 −1.986E − 05 −1.768E − 07 −4.992E − 0755.0 6.057E − 05 2.970E − 05 −2.005E − 05 −1.989E − 07 −5.836E − 0757.0 6.785E − 05 2.124E − 05 −2.536E − 05 −2.065E − 07 −6.158E − 0758.0 7.132E − 05 3.422E − 05 −2.328E − 05 −2.279E − 07 −6.859E − 0763.0 9.313E − 05 3.320E − 05 −3.510E − 05 −2.962E − 07 −8.979E − 0765.0 1.036E − 04 5.143E − 05 −3.416E − 05 −3.430E − 07 −1.053E − 0667.5 1.154E − 04 5.194E − 05 −4.031E − 05 −3.785E − 07 −1.173E − 0670.0 1.004E − 04 −2.519E − 05 −6.065E − 05 −1.966E − 07 −6.142E − 0775.0 5.004E − 05 −1.074E − 05 −2.592E − 05 2.383E − 08 −8.384E − 0880.0 3.335E − 05 −3.859E − 06 −1.445E − 05 −5.781E − 09 −2.347E − 0885.0 2.611E − 05 −1.777E − 06 −1.008E − 05 −2.130E − 09 −9.966E − 0990.0 2.219E − 05 −9.624E − 07 −7.914E − 06 −9.932E − 10 −5.322E − 0995.0 1.976E − 05 −5.802E − 07 −6.658E − 06 −5.369E − 10 −3.269E − 09100.0 1.811E − 05 −3.770E − 07 −5.849E − 06 −3.208E − 10 −2.204E − 09

Table 3.2: Radiated Sound Power and Sound Power Sensitivity Values for Circular Piston

70


Freq PW (N-m/s) % Error % Change(Hz) Predicted Actual15.0 9.074E − 06 9.091E − 06 −0.178 5.03426.0 1.035E − 03 1.320E − 03 −21.60 −40.59830.0 6.385E − 05 6.492E − 05 −1.652 −13.14535.0 2.252E − 05 2.266E − 05 −0.620 −8.47737.0 1.753E − 05 1.761E − 05 −0.471 −7.89342.0 1.062E − 05 1.067E − 05 0.389 −4.40145.0 7.959E − 06 7.990E − 06 −0.393 −8.08447.0 6.426E − 06 6.453E − 06 −0.429 −8.78249.0 5.041E − 06 5.067E − 06 −0.530 −9.68170.0 8.669E − 05 8.722E − 05 −0.611 −13.14675.0 4.418E − 05 4.468E − 05 −1.117 −10.72580.0 3.042E − 05 3.062E − 05 −0.643 −8.19685.0 2.421E − 05 2.431E − 05 −0.434 −6.86090.0 2.077E − 05 2.083E − 05 −0.331 −6.09095.0 1.860E − 05 1.865E − 05 −0.274 −5.604100.0 1.712E − 05 1.716E − 05 −0.239 −5.273

Table 3.3: Predicted and Actual 98% Probabilistic PW Values due to 2% Change in Masses,m1, m2,and Dampers, b1, b2

Freq PW (N-m/s) % Error % Change(Hz) Predicted Actual20.0 1.528E − 03 1.534E − 03 −0.399 1.63754.0 4.860E − 05 4.971E − 05 −2.226 −11.06655.0 5.261E − 05 5.379E − 05 −2.180 −11.20557.0 6.040E − 05 6.035E − 05 −0.088 −11.06558.0 6.212E − 05 6.351E − 05 −2.200 −10.94063.0 8.220E − 05 8.235E − 05 −0.180 −11.57765.0 8.987E − 05 9.143E − 05 −1.705 −11.71867.5 1.006E − 04 1.025E − 04 −1.817 −11.802

Table 3.4: Predicted and Actual 98% Probabilistic PW Values due to 1% Change in Masses,m1, m2,and Dampers, b1, b2

71


supported plate subjected to deterministic multipoint excitation is used to illustrate the

new probabilistic sound power and probabilistic sound power sensitivity algorithms. The

structural configuration is taken from Allen et al [93] and represents an automotive side

windshield and windshield seal. Three thickness regions are defined for the plate and the

thickness values are taken as deterministic sizing parameters. Stiffness properties of the

plate’s elastic support are considered as random variables.

Physical and geometric characteristics of the flexible plate are provided in Table 3.5.

The plate is divided into three concentric rectangular regions denoted t1, t2, and t3 in

Fig. 3.3. The thickness of each region constitutes a deterministic sizing parameter with

initial value equal to 0.0032m. The elastic support is provided along the inner perimeter,

which consists of not only stiffness but also viscous damping where damping is expressed

in terms of elastic stiffness as,

c = η k/ ω (3.26)

where c =actual viscous damping value entered in the finite element analysis in Ns/m,

η =structural damping, k =seal stiffness in N/m, and ω =frequency in radians. The

elastic support is divided into two sections denoted by k1 and k2 in Fig. 3.3. This is

done to represent a typical seal division appearing in an automotive side windshield.

The stiffness values of these two seal sections are taken to be independent, normally

distributed random variables. Values for the mean stiffness and frequency dependent

72


damping are given in Table 3.6. The standard deviation of the stiffness random variables

is taken to be 15% of the mean value. Four harmonic point forces are placed at off center

and off diagonal locations as shown in Fig. 3.3. All four forces have a magnitude of 1N

and different magnitude of phases.

Finite element and boundary element models are shown in Fig. 3.4 which are used to

calculate the radiated sound power. Radiated sound power for the deterministic config-

uration is calculated as described in Section 3.2 with input variables equal to their mean

values and plotted in Fig. 3.5. Probabilistic sound power response at 98% probability

level is also calculated and plotted in Fig. 3.5. As shown in this figure the sound power

at 98% probability ranges from 0 to 5db higher than the deterministic response through

the frequency range. 98% probabilistic sound power is confirmed through Monte Carlo

simulation and Monte Carlo sound power points are plotted in the Fig. 3.5. In the Monte

Carlo simulation only 100 points were used.

Density 2.7× 103kg/m3

Poisson′sRatio 0.33Y oung′sModulus 7.3× 1010N/m2

Length(outer/innerperimeter) 0.475m/0.450mWidth(outer/innerperimeter) 0.375m/0.321m

StructuralDamping 1.0%

Table 3.5: Characteristics of the Flexible Panel

Freq.Range MeanStiffness, (N/m)/m Damping20 to 76Hz 1.2× 105 5%76 to 145Hz 1.2× 105 14%

Table 3.6: Characteristics of the Elastic Panel Support

73


F1

F4

F3 F2

spring

damper

t1

t2

t3

k1

k2

Figure 3.3: Deterministic and Random Design Parameter Configuration of Flexible Panel

74


Boundary ElementFinite Element

Figure 3.4: Finite Element and Boundary Element Models of the Elastically Supported Plate

40 60 80 100 120 140

60

65

70

75

80

85

90

95

Frequency (Hz)

Sou

nd p

ower

(dB

)

Deterministic98% probabilisticMonte−carlo

Figure 3.5: Deterministic and 98% Probabilistic Radiated Sound Power for Flexible Panel

75


Sensitivity values associated with the 98% probabilistic sound power response are calcu-

lated with respect to design variables t1, t2, and t3. These sensitivity values are provided

in Table 3.7 for both the frequency ranges. As expected, across the frequency range, the

sensitivity values imply a reduction in the probabilistic response due to an increase in

thickness. In the first frequency range, 20 − 76 Hz, thickness variables t3, t2 and t1 are

influential in decreasing order. In the second frequency range, 76 − 145 Hz, there is no

definite order of influence of thickness variables on 98% probabilistic sound power.

In order to validate the sensitivity algorithm presented above, finite difference analyses

were conducted. For the first range of frequencies, +3% changes in thickness t1 and t2

independently, is used for re-analysis and it is compared with finite difference results

using sensitivity information, which are shown in Table 3.8. As shown in Table 3.8, there

is a less then 3% error between predicted and actual results. For the same frequency

range, a +2% variation in t1 and t3 is imposed simultaneously. Both the predicted and

the actual sound power are presented in Table 3.9. Error between the predicted and the

actual results for this analysis does not exceed 4%. For the second frequency range, two

analyses were performed. In the first case, a +2% variations in both thicknesses t2 and

t3 is applied simultaneously and results for predicted and actual 98% probabilistic sound

powers are compared, see Table 3.10, and the error between them is less than 4%. For

the same frequency range, a variation of +2% in t1 and t2 is employed simultaneously and

it is found that there is a less than 1% error for actual and predicted 98% probabilistic

sound powers, see Table 3.11.

76


Freq Sound Sound Power Sensitivity,(Hz) Power N/s

N −m/s t1 t2 t335.0 5.141E − 5 −2.984E − 3 −9.002E − 4 6.310E − 346.5 4.343E − 3 −2.838E − 1 −7.090E − 1 −1.851E − 051.5 4.096E − 3 −1.592E − 1 −6.043E − 1 −1.752E − 058.0 2.771E − 4 −3.929E − 2 −1.184E − 1 −4.158E − 169.0 1.622E − 4 −8.845E − 3 −1.893E − 3 6.748E − 272.0 1.809E − 4 1.668E − 2 −8.043E − 3 5.027E − 275.0 1.781E − 4 1.206E − 1 −1.849E − 1 −1.583E − 080.0 8.916E − 5 1.536E − 2 −3.644E − 2 −1.908E − 187.0 5.214E − 5 1.247E − 3 −9.724E − 3 −1.605E − 2115.0 1.233E − 4 5.655E − 2 4.004E − 2 −5.189E − 2125.0 4.640E − 5 1.848E − 2 1.044E − 2 −1.950E − 2135.0 2.378E − 5 1.532E − 2 1.519E − 2 4.125E − 3

Table 3.7: Radiated Sound Power and Sound Power Sensitivity Values for Both FrequencyRange for Flexible Panel

Freq PW (N-m/s) for t1 PW (N-m/s) for t2(Hz) Predicted Actual Predicted Actual35.0 5.114E − 5 5.125E − 5 5.133E − 5 5.154E − 546.5 4.318E − 3 4.216E − 3 4.279E − 3 4.197E − 358.0 2.736E − 4 2.713E − 4 2.664E − 4 2.635E − 469.0 1.614E − 4 1.612E − 4 1.624E − 4 1.580E − 472.0 1.801E − 4 1.755E − 4 1.854E − 4 1.802E − 4

Table 3.8: Predicted and Actual 98% Probabilistic PW Values due to 3% Independent Changesin t1 and t2

Freq PW (N-m/s)(Hz) Predicted Actual35.0 5.174E − 5 5.226E − 546.5 4.189E − 3 4.109E − 358.0 2.451E − 4 2.361E − 469.0 1.618E − 4 1.683E − 572.0 1.834E − 4 1.885E − 4

Table 3.9: Predicted and Actual 98% Probabilistic PW Values due to a 2% Change in t1 andt3 simultaneously for the first frequency range

77

3.4 Conclusions

Freq PW (N-m/s)(Hz) Predicted Actual87.0 5.059E − 5 5.100E − 5115.0 1.226E − 4 1.180E − 4125.0 4.585E − 5 4.537E − 5

Table 3.10: Predicted and Actual 98% Probabilistic PW Values due to a 2% Change in t2and t3 simultaneously for the second frequency range

Freq PW (N-m/s)(Hz) Predicted Actual80.0 8.789E − 5 8.806E − 587.0 5.163E − 5 5.153E − 5125.0 4.813E − 5 4.808E − 5

Table 3.11: Predicted and Actual 98% Probabilistic PW Values due a to 2% Change in t1and t2 simultaneously for the second frequency range

3.4 Conclusions

New algorithms for calculating both the sound power and its sensitivity are presented for

structural acoustic systems in the presence of structural uncertainty in various structural

parameters. The probabilistic radiated sound power algorithm is developed by following

a previously validated technique using finite element/boundary element analysis and an

advanced mean value method. The sensitivity algorithm is obtained by differentiating the

new probabilistic sound power algorithm with respect to deterministic structural design

parameters. A circular piston in infinite baffle and an elastically supported flexible plate

are used to illustrate the new algorithms. The validity of this algorithm and robustness is

shown through these examples. In piston problem, stiffness of supporting frame is taken

as random and mass and damping are taken as deterministic. For second problem, the

78

3.4 Conclusions

panel is taken to be comprised of deterministic design parameters while the elastically

support possesses random stiffness. The sensitivity algorithm is valid as evidenced by

comparison with results obtained from finite difference analysis.

79

Chapter 4

Probabilistic Metamodeling

4.1 Introduction

The use of laminated composites in aerospace, automotive and naval structures has be-

come more due to the high strength to weight ratio and high stiffness to weight ratios

they exhibit. In addition, vibro-acoustic simulation has become an important issue in

the design of these structure [94, 95, 96, 97, 98, 99, 100]. The magnitude of structure

borne noise strongly influences the habitability, comfort, and perceived quality of these

vehicles. And in many cases, these vehicles must meet strict vibro-acoustic standards for

operation. For composite materials, fiber orientation, percentage fiber volume, ply thick-

nesses and fiber packing determine the material and geometric properties of components

within these vehicles. Due to structural as well as manufacturing complexities, there is

80

4.1 Introduction

often a large amount of uncertainty in these properties. Statistics of these properties are

obtained experimentally for different types of composites [101] and can influence vibro-

acoustic response. As such, there is a need to adequately address uncertainty in the

vibro-acoustic performance of these systems.

In recent years, many of the probabilistic methods requiring an implicit response eval-

uations have been applied to the static analysis of composite structures for reliabil-

ity [102, 103], buckling failure analysis [104], and ultimate strength [105]. However ap-

plications of these methods for the dynamic analysis of composites is limited. To date,

the calculation of natural frequencies [106, 107] and supersonic flutter of composite pan-

els [108] has been addressed. The application of composites in vibro-acoustic simulation

has also been addressed [109, 110]. To the author’s knowledge, however, uncertainty

of composite characteristics in vibro-acoustic simulation is yet to be addressed. In this

work, a new technique will be presented that better represents the complex dynamic

response of a composite structure during implicit probabilistic calculations.

The new probabilistic technique combines existing implicit probabilistic methods with

the concept of metamodeling. Put simply, a metamodel is model of a model. The concept

of a metamodel is inherent to the probabilistic methods that utilize implicit representa-

tion of the system response. In Chapter 2 and Section 2.2.1.1, available Metamodeling

in probabilistic techniques from previous research work are discussed. Three examples

are used to illustrate the accuracy of the new technique. In the two examples, the tech-

nique is applied to analytical expressions of random variables exhibiting highly nonlinear

81

4.2 Hybrid Metamodel for Nonmonotonic, Nonlinear Response FunctionAnalysis

and nonmonotonic characteristics. In the third example, the sound power of a baffled

composite panel is calculated using the FEM and Rayleigh’s Integral. Uncertainty is

considered in material as well as geometric properties of the panel. CDFs characterizing

the radiated sound power are generated at specific frequencies and, in all of the examples

the calculated CDFs are compared to those generated using the AMV, LHS, and MCS.

4.2 Hybrid Metamodel for Nonmonotonic, Nonlin-

ear Response Function Analysis

As described before moment based methods accurately account for monotonic, linear,

and nonlinear response functions while sampling methods can account for nonmonotonic,

nonlinear response. However, when numerical response functions are employed even the

reduced number of samples given by LHS can prove computationally prohibitive. As

such, new methods are needed that exhibit the computational efficiency of the moment

based methods and the accuracy of the sampling methods.

In this work, a new technique is presented for the analysis of highly nonlinear and non-

monotonic response functions. Instead of using a single response surface or neural network

to replace a numerical response function in probabilistic calculations we use both in the

context of the AMV analysis. In the AMV method [18], the response function Z (X) is

82

4.2 Hybrid Metamodel for Nonmonotonic, Nonlinear Response FunctionAnalysis

expanded using a Taylor series about the mean values of the random variables

Z (X) = Z (µ)+n∑

i=1

(∂Z

∂Xi

)µ

(Xi − µi)+H (X) (4.1)

Z (X) = Z1 (X)+H (X) (4.2)

where Z1 (X) represents the first order response given by Eq. (2.31) and H (X) represents

higher order terms. The first step in the AMV method is to conduct a FOSM analysis

using Z1 (X). Once this is done, the Z1 (X) values in the first order response CDF

corresponding to each probability (β) level are replaced with the ZAMV values shown

below

ZAMV = Z1+H (Z1) (4.3)

by simply revaluating Eq. (4.1) at the MPPL.

The AMV analysis itself uses a first order response surface (metamodel) and a correction

based on the MPPL to construct the response CDF. In the new method, this first order

response surface (metamodel 1, Fig. 4.1) is combined with a metamodel representing the

AMV correction (metamodel 2, Fig. 4.1) to produce a “hybrid” metamodel that is used

in Monte Carlo simulation. Use of the hybrid metamodel with standard MCS is identified

by the bold arrows in Fig. 4.1. This new technique is referred to as the AMVMC method

in the plots that follow.

The metamodel representing the AMV correction is a multilayered neural network trained

with information obtained in the original AMV analysis. As stated earlier the AMV

83

4.3 Application

method conducts a FOSM analysis to identify the MPPL. The MPPL is then used to

update the response associated with the first order probability values implying the rela-

tionship given in Eq. (4.3). The Z1 values associated with each point in the MPPL are

given as input and the corrected AMV values, ZAMV , are given as output when training

the neural network to represent the single random variable function in Eq. (4.3).

4.3 Application

To investigate the ability of the new technique to accurately capture nonlinear, nonmono-

tonic vibro-acoustic response; it is applied to two analytical functions. These analytical

examples represent simple nonlinear, nonmonotonic expressions that may arise in acous-

tic analysis. This new technique is then applied to a vibro-acoustic system where FEM

and Rayleigh’s Integral are employed to calculate radiated sound power of a composite

panel in an infinite baffle excited by point load.

4.3.1 Analytical Functions

For these expressions, all xi are taken to be uncorrelated, normally distributed random

variables. In the first expression, the random variables have a mean of 10.0 and a standard

deviation of 1.5. In the second expression, the random variables have a mean of 10.0

and a standard deviation of 0.15. The new probabilistic technique is applied to both

expression where the sensitivities for the initial AMV analysis are obtained through a

84

4.3 Application

Metamodel II: Neural Network

betweenFirst Order

Response and AMV Response

Calculation ofMost Probable

Point using First Order Response

Metamodel I:First Order Response Function

Expansion using Taylor Series

Calculation of AMV Response

AMV Analysis

Generation of Random

Realizations of variables

Construction of

CDF

Figure 4.1: MCS using Hybrid Metamodel

85

4.3 Application

1% perturbation of the random variables. The two analytical functions are given below.

4.3.1.1 Function 1:

y = 10.0 + sin (x1 + x2) + sin (1.5 (x1 + x2)) (4.4)

4.3.1.2 Function 2:

y =

[2 J1 (0.5 x3 sin (x1))

0.5 x3 sin (x1)

]2 [sin (1.5 x2 sin (x1))

1.5 x2 sin (x1)

]2

(4.5)

where J1 is the first-order Bessel function of the first kind. The CDFs for the first function

obtained in the AMV analysis at 100 probability levels are plotted in the Fig. 4.2 along

with the CDFs generated using MCS with 105 sample points. Note that in this plot, the

non-monotonocity of the response function creates a nonmonotonic response CDF. This

CDF is corrected using the theory of one random variable, mentioned earlier, to give

an equivalent monotonic CDF, denoted AMVcor. The bottom plot in Fig. 4.3 displays

the monotonic CDFs along with the CDF produced by a LHS with 104 sample points.

The % error of the monotonic CDFs compared to the Monte Carlo results are given in

Fig. 4.4. Note that although the errors are small, the new technique gives more accurate

results over the entire probability range as compared to the corrected AMV response.

The CDFs for the second function are given in Fig. 4.5. Once again the AMV response

is calculated and corrected at 100 probability levels, the MCS is conducted with 105

samples, and the LHS is conducted with 104 samples. The % error of the monotonic

86

4.3 Application

8 8.5 9 9.5 10 10.5 11 11.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y

Pro

babi

lity

Monte CarloAMV

incorAMV

cor

Figure 4.2: Response CDFs for Function 1 with MCS and AMV

8 8.5 9 9.5 10 10.5 11 11.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y

Pro

babi

lity

AMVMCMonte CarloAMV

corLHS

Figure 4.3: Response CDFs for Function 1 with AMVMC, LHS, MCS, and AMV

87

4.3 Application

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−4

−3

−2

−1

0

1

2

3

4

Probability

% E

rror

AMVMCAMV

corLHS

Figure 4.4: Percent Error of Monotonic CDFs with respect to MCS for Function 1

CDFs with respect to MCS are given in Fig. 4.6. As before, results for the new technique

show greater accuracy over the entire probability range. It should be noted that error in

the corrected AMV response exceeded a factor of 2 in the probability ranges 0.01 to 0.18

and 0.28 to 0.6 and was not plotted. It should be further noted that the LHS with 104

samples gives more accurate results than the new technique over the entire probability

range for both function 1 and 2.

4.3.2 Composite Panel Sound power

In this example, the radiated acoustic power of a composite panel in an infinite baf-

fle is analyzed. The panel is excited by a harmonic point load of magnitude 1Nat

88

4.3 Application

0 0.005 0.01 0.015 0.02 0.025 0.03

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y

Pro

babi

lity

AMVMCAMV

corMonte CarloLHS

Figure 4.5: Monotonic Response CDFs for Function 2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−100

−80

−60

−40

−20

0

20

40

60

80

100

Probability

% E

rror

AMVMCAMV

corLHS

Figure 4.6: Percent Error of Monotonic CDFs with respect to MCS for Function 2

89

4.3 Application

(−0.04, 0.06, 0) m as referenced to the cartesian coordinate system in Fig. 4.7. The panel

dimensions are 0.4m long by 0.3m wide and the stacking sequence of the 4-ply lami-

nate is [0/90/90/0]. Material properties for the panel are taken from Oh et al [106] and

represent a graphite/epoxy constituent material, see Table 4.1. The properties listed

in Table 4.1 and the fiber orientation angles are considered to be uncorrelated normal

random variables. The mean values of the random variables are listed in Table 4.1 and

the standard deviation of the first eleven variables is taken to be 10% of the mean values.

The standard deviation for the fiber angles is taken to be 2.5 deg, as is common in the

literature [105, 106].

The dynamic structural response of the plate is calculated using finite element analysis

with 441 nodes and 800 elements based on first-order shear deformation theory. The finite

element analysis is conducted using the commercial code, MSC Nastran. The modal

damping in the material is taken to be 0.05 as reported in [111]. Although damping

typical varies with frequency, for simplicity, it is taken as a constant. It is assumed that

the plate does not experience delamination, matrix cracking, or fiber breakage during

vibration.

Once the vibratory response has been determined, the pressure at a data recovery point

can be calculated using Rayleigh’s Integral [112] as:

p (r, θ, φ) = −ikρceikr

2πr

∫ b/2

−b/2

∫ a/2

−a/2

vn (x, y) exp

[−i(αx

a

)− i

(βy

b

)]dxdy (4.6)

90

4.3 Application

y

x

z

a

b

p (r,θ,φ)

θ

r

φ

Figure 4.7: Geometry of the Baffled Panel and Coordinate System

Variable Mean ValueElastic Modulus, E11 1.2755E + 11 N/m2

Elastic Modulus, E22 1.1032E + 10 N/m2

Shear Modulus, G12 5.736E + 09 N/m2



Poission Ratio, ν12 0.35Density, ρ 1552.84 Kg/m3

Ply thickness,(4 plys), h 5.0E − 04 mStacking Sequence [0/90/90/0]

Table 4.1: Composite Panel Properties

91

4.3 Application

where

α = k a sin θcos φ

β = k b sin θsin φ, (4.7)

k is the wave number, vn is the complex surface velocity at the (x, y), r is the distance

to the data recovery point, ρ is the density of air, and c is the speed of sound in air.

Equations (4.6), and (4.7) refer to the geometry and coordinate system shown in the

Fig. 4.7. Using Eqs. (4.6), and (4.7), the average acoustic power radiated from one side

of the panel can be written as,

Π =

∫ 2π

0

∫ π/2

0

|p|2

ρ cr2 sin θ dθ dφ (4.8)

Equations (4.6) through (4.8) along with the associated structural finite element analysis

are used as an implicit definition of the systems vibro-acoustic response. The new proba-

bilistic technique was conducted using this response to calculate a sound power CDFs at

particular frequencies. The required sensitivities were calculated using 1% perturbation

of the random variables.

Sound power CDFs are plotted in Figs. 4.8, 4.10, and 4.12 for 188 Hz, 364 Hz, and 406

Hz respectively. The CDFs for the new method have been created using results at 100

probability levels. CDFs created using the corrected AMV response, MCS (104 sampling

points), and LHS (2× 103 sampling points) have also been plotted in these figures. The

92

4.3 Application

response at 188 Hz is monotonic over the majority of probability values. Hence the

corrected AMV response gives results close in accuracy to the new technique. However

the response at 364 Hz and 406 Hz is monotonic over smaller probability intervals. At

these frequencies the CDF created using the new technique more closely match the shape

of the CDFs created using MCS and LHS as compared to the corrected AMV results in

Fig. 4.10 and 4.12. In Figs. 4.9, 4.11 and 4.13, % error of these CDFs are plotted

with respect to MCS. Sound power CDFs were constructed at 1 Hz intervals over the

frequency range 50 to 1000 Hz using new technique. Sound power values associated with

95% certainity were extracted from these CDFs and plotted in Fig. 4.14 along with the

deterministic response calculated using the mean values of the random variables.

0 0.005 0.01 0.015 0.02 0.0250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Pro

babi

lity

Sound power (dB) (Ref. 1E−12)

AMVMCAMV

corMonte CarloLHS

Figure 4.8: Sound power CDFs for the Composite Panel at 188 Hz

93

4.3 Application

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−15

−10

−5

0

5

10

15

Probability

% E

rror

AMVMCAMV

corLHS

Figure 4.9: Percent Error in the Sound power CDFs with respect to MCS at 188 Hz

0 1 2 3 4 5 6 7 8 9 10

x 10−4

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Pro

babi

lity

AMVMCAMV

corMonte CarloLHS


94

4.3 Application

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−40

−30

−20

−10

0

10

20

30

40

Probability

% E

rror

AMVMCAMV

corLHS


1 2 3 4 5 6 7 8

x 10−4

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Pro

babi

lity

AMVMCAMV

corMonte CarloLHS


95

4.3 Application

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−80

−60

−40

−20

0

20

40

60

80

Probability

% E

rror

AMVMCAMV

corLHS


100 200 300 400 500 600 700 800 900 1000

65

70

75

80

85

90

95

100

105

Frequency (Hz)

Sou

nd

Pow

er (

dB

) (R

ef. 1

E−

12)

Deterministic95% Probabilistic

Figure 4.14: Deterministic and 95% Probabilistic Radiated Sound power

96

4.4 Conclusions

4.4 Conclusions

In this work, a new probabilistic technique has been presented for calculating the CDF of

a nonlinear, nonmonotonic response function. The motivation behind this effort was the

desire to resolve probabilistic information associated with the vibro-acoustic response

of composite structures with uncertain material and geometric sizing properties. The

new technique combines AMV analysis, MCS, and metamodeling concepts. The new

technique combines the efficiency of the moment based methods with the improved ac-

curacy of the sampling methods. Although additional function evaluations are required

in the new technique beyond those of the initial AMV analysis, these additional func-

tions are considered to have a negligible impact on computational efficiency because they

are evaluated using an analytical expression (neural network metamodel). As shown in

Figs. 4.4, 4.6, 4.11, and 4.13 the new technique gives greater accuracy than the corrected

AMV results. This is because the MCS employed in the new technique is not subject

to the local minima problem that appears in the moment based methods. However, as

in the AMV method, the new method’s accuracy depends on the proximity of the ap-

proximated MPPL to the actual MPPL. Although, like the corrected AMV response,the

new technique is less accurate than the LHS response. It is felt that this inaccuracy is

compensated by an improved computational efficiency.

97

Chapter 5

Fundamental Eigenvalue using

Polynomial Chaos

5.1 Introduction

For linear dynamic systems, eigenvalues are important properties. These eigenvalues rep-

resent resonance frequencies of the systems. The steady state response of the system can

be constructed using modal superposition. Because of uncertainties in the material as

well as geometric properties, there will be variation in eigenvalues of these systems, hence

also in the response. A considerable effort has been made to obtain the probabilistic char-

acteristics of these eigenvalues [113, 114, 106, 107, 115], mostly based on the perturbation

based FEM [8]. This particular method breaks down if coefficient of variation of input

98

5.1 Introduction

random variables is greater than 0.1 [114]. To account for higher coefficient of variation,

generalized SSFEM should be used. SSFEM was initially developed by Ghanem and

Spanos [28, 29, 30] and further developed by Ghanem [31, 32]. Efforts also have been

made to study dynamical response of a system having uncertain properties [116], but in

that research, time integration is used to get response in the time domain, an approach

that for linear systems is generally considered to be inefficient as compared to modal

superposition. The other approach to use polynomial chaos for dynamic systems is the

so called non-intrusive formulation [66]. In this method, the Karhunen-Loeve expansion

is used with LHS to get response polynomial chaos coefficients and probabilistic charac-

teristics of the response. This method is still computationally expensive for dynamical

response. Therefore in this work, a new algorithm for finding fundamental eigenvalue of

linear stochastic differential equation is presented.

In this new algorithm, material properties are written as either the Karhunen-Loeve

expansion for a random process or as polynomial chaos for the random variable case.

Even eigenvalues and eigenvectors are expressed as polynomial chaos. Using proposed

algorithm, undetermined coefficients of eigenvalue and eigenvector of respective polyno-

mial chaos are obtained. After obtaining these coefficients, probabilistic characteristics

can be derived. Two examples are used to illustrate the accuracy of the new algorithm.

In the first example, the fundamental eigenvalue of a two degrees-of-freedom model is

calculated. In this example, masses and stiffnesses are assumed to belong to the same

probability space in the first case and to different probability spaces in the second case. In

99

5.2 Spectral Stochastic Finite Element Analysis

the second example, free vibration response of a cantilever beam where both the bending

rigidity and mass per unit length are assumed to be random processes with exponential

autocorrelation functions is studied. The fundamental eigenvalue is calculated when mass

and rigidity are from the same probability space as well as from two different probability

spaces. The results of both examples are compared with those obtained using LHS and

probabilistic characteristics of fundamental eigenvector are also given.

5.2 Spectral Stochastic Finite Element Analysis

In Chapter 2 and Section 2.2.3, SSFEM is explained in depth. So the procedure is given

here.

• Karhunen-Loeve Expansion

Using the definition of autocorrelation function of the material or geometric prop-

erties, eigenvalues and eigenvectors are obtained for the second kind of Fred-

holm Integral equation. Random process is represented as sum of infinite terms

which are functions of autocorrelation function’s eigenvalues and eigenvectors us-

ing Karhunen-Loeve Expansion.

• Generalized Polynomial Chaos

As the autocorrelation function of response is unknown, the response is written

sum of nonlinear functions of the random variables, these random variables are the

basis of input autocorrelation function. This was proposed by Wiener [4].

100

5.3 Stochastic Eigenvalue Problem

• Galerkin Method in Random Space

Once input random properties and response are developed using Karhunen-Loeve

expansion and polynomial chaos respectively, Galerkin method is used to minimize

the error in the mean and that in the standard deviation of the random response.

Galerkin method is applied in random (probability) space meaning the stochas-

tic differential equation is multiplied by appropriate polynomials from polynomial

chaos expansion and one tries to reduce the weighted residual (integral of the resid-

ual multiplied by suitably chosen weight functions) of these resultant equations.


For linear dynamic problems, eigenvalues of the dynamic systems is an important prop-

erty of the phenomenon governing the behavior of these systems. Response of these

systems can be constructed using linear combination of eigenvectors, the methodology is

known as modal superposition approach in structural dynamics. Most of the time, the

direct time integration algorithms are computationally inefficient as compared to modal

superposition, so dynamic response of linear problems is very often obtained using modal

superposition. In vibrations and acoustics, eigenvalues characterize resonance of the sys-

tems. For deterministic systems, there are number of algorithms to solve eigenproblem

and many of these are implemented in commercial finite element method softwares. But

for stochastic systems, solution to eigenproblems using polynomial chaos is not as de-

101


veloped. Solving stochastic eigenvalue problem by developing appropriate algorithms for

such problems is the key objective of this research. Stochastic differential eigenproblem

can be defined as:

K (x, θ) u (x, θ) = λ (x, θ)M (x, θ) u (x, θ) (5.1)

where K and M are stochastic linear homogeneous differential operators of order 2p and

2q, respectively such that p ≥ q. There are p boundary conditions associated with the

system governed by differential equations given in Eq. (5.1).

Bi (x, θ) u (x, θ) = 0, i = 1, 2, . . . , k (5.2)

Bi (x, θ) u (x, θ) = λ (x, θ) Ci (x, θ) u (x, θ) , i = k + 1, k + 2, . . . , p (5.3)

Here Bi and Ci are stochastic linear homogeneous differential operators of maximum order

2p − 1 and 2q − 1 respectively. Application of Galerkin procedure to Eq. (5.1) results

in a discrete system of algebraic equations. For simplicity, K and M are assumed to

be matrices. This assumption leads to a discrete problem, hence the governing equation

becomes

Kx = λMx (5.4)

102


Matrices K and M are written as polynomial chaos,

K =P∑

i=0

KiΓi, M =P∑

i=0

MiΓi, λ =P∑

i=0

λiΓi, x =P∑

i=0

xiΓi (5.5)

Here, λi and xi are unknowns. Therefore, governing Eq. (5.4) becomes:

P∑i=0

P∑j=0

KixjΓiΓj =P∑

k=0

P∑i=0

P∑j=0

λkMixjΓkΓiΓj (5.6)

To minimize the weighted residual of the response, multiply Eq. (5.6) by Γm and take

expectation

P∑i=0

P∑j=0

Kixj 〈ΓiΓjΓm〉 =P∑

k=0

P∑i=0

P∑j=0

λkMixj 〈ΓiΓjΓkΓm〉 , m = 0, . . . , P (5.7)

Above equation can be written in a simple form as:

K X = λM X (5.8)

where K, λM and X are matrices of the order n (P + 1)×n (P + 1), n (P + 1)×n (P + 1),

and n (P + 1)× 1 respectively. n× n is the order of individual Ki or Mi in polynomial

103

5.4 Eigenvalue Extraction Algorithms

chaos expansion. Generic elements of K, and λM are given as:

Kl,m =P∑

i=0

Ki 〈ΓiΓlΓm〉 , m, l = 0, . . . , P (5.9)

λMl,m =P∑

i=0

P∑k=0

Miλk 〈ΓiΓkΓlΓm〉 (5.10)

Xl = xl (5.11)

Equation (5.8) can be written in an alternative way as,

K X = MX Λ (5.12)

MXl,m =P∑

i=0

P∑j=0

Mixj 〈ΓiΓjΓlΓm〉 (5.13)

where XMl,m is generic element of this matrix.


For eigenvalue analysis, Eq. (5.8) needs to be solved iteratively. Different algorithms

are used to accomplish this. Differences in algorithms are only in the manner in which

the normalization of eigenvectors generated during iterative procedure is done. Equa-

tion (5.4) is multiplied by xT . The substitutions as given in Eq. (5.5) results in the

104


following equation

P∑i=0

P∑j=0

P∑k=0

xTi KjxkΓiΓjΓk =

P∑i=0

P∑j=0

P∑k=0

P∑l=0

λlxTi MjxkΓiΓjΓkΓl (5.14)

Multiplying above equation by Γm and taking expectation to minimize the weighted

residual of the solution produces following system of equations.

P∑i=0

P∑j=0

P∑k=0

xTi Kjxk 〈ΓiΓjΓkΓm〉 =

P∑i=0

P∑j=0

P∑k=0

P∑l=0

λlxTi Mjxk 〈ΓiΓjΓkΓlΓm〉 ,

m = 0, . . . , P (5.15)

The above equation can be written in simple form as:

K = M λ (5.16)

The generic elements of the matrices from Eq. (5.16) are given as:

Kl =P∑

i=0

P∑j=0

P∑k=0

xTi Kjxk 〈ΓiΓjΓkΓl〉 , l = 0, . . . , P (5.17)

Ml,m =P∑

i=0

P∑j=0

P∑k=0

xTi Mjxk 〈ΓiΓjΓkΓlΓm〉 , l, m = 0, . . . , P (5.18)

The following iterative procedure is used to get the eigenvalues

• Start with some initial xi, i = 0, . . . , P , generally it will be a vector of unit elements.

105

5.5 Numerical Examples

• Substitute these xi into Eq. (5.16) and find λi, i = 0, . . . , P .

• Substitute these λi into Eq. (5.8) to get new xi.

• Normalize these xi using Eq. (5.19)

xi =xi

||x0||∞(5.19)

where ||x0||∞ is the L∞ norm of the vector x0, mean eigenvector or 0th polynomial

chaos component of eigenvector.


The aforementioned algorithm is applied to a discrete system, a 2 degrees-of freedom

model, and to a continuous system, a cantilever beam. The fundamental eigenvalue and

fundamental eigenvector are obtained for both the discrete and the continuous systems.

These systems are described in subsequent subsections. These results obtained using

polynomial chaos are compared with Monte Carlo simulation using LHS. In LHS, the

range of each random variable is divided into N , non overlapping intervals of 1/N proba-

bility. From each such interval, one value is selected randomly for that random variable.

This N -value vector of the first random variable is combined with a corresponding vector

of second random variable realization. This resulting N × 2 matrix is combined with the

vector of third random variable to form an N × 3 matrix. This procedure is carried till

106


all n random variables are covered and an N × n sample space has been generated [67].

5.5.1 Two Degrees-of-Freedom System

m1 m2

k1k2

x1 x2

Figure 5.1: Two Degrees-of-Freedom Spring-Mass Model

For the discrete model shown in Fig. 5.1, equations of motion for free vibration analysis

become:

Mx+Kx = 0 (5.20)

m1 0

0 m2

x1

x2

+

k1 + k2 −k2

−k2 k2

x1

x2

=

0

0

(5.21)

For the system shown in Fig. 5.1; m1 = 1kg, m2 = 2kg and, k1 = 0.5N/m, k2 = 2N/m,

so mean mass matrix, M0 and mean stiffness matrix K0 become:

M0 =

1.0 0.0

0.0 2.0

K0 =

2.5 −2.0

−2.0 2.0

(5.22)

107


Here, it is assumed that the standard deviation in the mass matrix is 0.2M0 and standard

deviation in the stiffness matrix is 0.3K0. Fundamental eigenvalue is obtained for the

above system using algorithm as given in Section 5.4. If the uncertainties in the masses

and stiffnesses are from the same probability space, following eigenvalue polynomial co-

efficients, λi for different order of chaos are obtained as shown in the Table 5.1, where µ

and σ are the mean and standard deviation of eigenvalue for different order chaos.

First-Order PC Second-Order PC Third-Order PC Fourth-Order PCλ0 0.146137E + 00 0.145849E + 00 0.145819E + 00 0.145796E + 00λ1 0.155663E − 01 0.169852E − 01 0.172385E − 01 0.172610E − 01λ2 −0.337013E − 02 −0.382851E − 02 −0.396281E − 02λ3 0.877125E − 03 0.100671E − 02λ4 −0.169555E − 03µ 0.146137E + 00 0.145849E + 00 0.145819E + 00 0.145796E + 00σ 0.155663E − 01 0.176647E − 01 0.184245E − 01 0.184143E − 01

Table 5.1: Fundamental Eigenvalue Coefficients with Same Probability Space for Mass andStiffness for the 2-DOF System

When a Monte Carlo simulation using LHS is carried out with 1 million samples, the

mean, µ of the eigenvalue, 0.145762E +00, and its standard deviation, σ, 0.185367E−01

are obtained. In Figs. 5.2 and 5.3, the PDF of the fundamental eigenvalue are plotted.

The PDF obtained from LHS is also plotted to ascertain the accuracy of the solution

obtained using polynomial chaos. Figures 5.2 and 5.3 prove that the polynomial chaos

response converges to LHS PDF from below, also the first-order chaos PDF is normally

distributed PDF. From the PDF graphs, it can be seen that the third and the fourth-order

chaos results are close to the LHS results, but the validity of this statement, in general

will depend upon the coefficient of variation, δ, of the input parameters. In Figs. 5.4, 5.5,

108


5.6, and 5.7, the eigenvalue polynomial chaos coefficients are plotted as a function of the

iteration number along with the Log(|λ|2) of the first eigenvalue for different order chaos.

Note that |λ|2 is the L2 norm of the eigenvalue polynomial coefficients vector. From

Figs. 5.4, 5.5, 5.6, and 5.7, it can be seen that all eigenvalue polynomial chaos coefficients

for all order of chaos converge asymptotically to true value within 200 iterations. The

probabilistic fundamental eigenvector normalized with respect to maximum ordinate are

plotted in Fig. 5.8 for different probability ranges. For all order of chaos, if the mass and

stiffness are from the same probability space, the fundamental eigenvector at different

probabilities is same as the deterministic fundamental eigenvector. The probabilistic

eigenvector is normalized with respect to maximum ordinate.

0 0.05 0.1 0.15 0.2 0.250

5

10

15

20

25

30

35

Eigenvalue, λ (rad2/s2)

Prob

abili

ty D

ensi

ty F

unct

ion

Monte Carlo1st PC2nd PC

Figure 5.2: PDFs of Fundamental Eigenvalue using First and Second Order Chaos for the2-DOF System

109


0 0.05 0.1 0.15 0.2 0.250

5

10

15

20

25

30

35


Prob

abili

ty D

ensi

ty F

unct

ion

Monte Carlo3rd PC4th PC

Figure 5.3: PDFs of Fundamental Eigenvalue using Third and Fourth Order Chaos for the2-DOF System

100 200 300 400 500 600 700 800 900 1000

0.145

0.15

0.155

0.16

λ 0

100 200 300 400 500 600 700 800 900 1000

0.01

0.015

0.02

λ 1

100 200 300 400 500 600 700 800 900 1000−2

−1.9

−1.8

−1.7

−1.6

Iteration Number

Log(

||λ|| 2)

Figure 5.4: First-Order Chaos Eigenvalue Convergence for the 2-DOF System

110


200 400 600 800 1000

0.145

0.15

0.155

0.16

λ 0

200 400 600 800 10000.01

0.015

0.02

0.025

0.03

Iteration Number

λ 1

200 400 600 800 1000−0.01

−0.005

0

0.005

0.01

λ 2

200 400 600 800 1000−0.9

−0.85

−0.8

−0.75

−0.7

Iteration Number

Log

(||λ|

| 2)

Figure 5.5: Second-Order Chaos Eigenvalue Convergence for the 2-DOF System

200 400 600 800 1000

0.145

0.15

0.155

0.16

λ 0

200 400 600 8001000−0.01

0

0.01

λ 2

200 400 600 800 10000.01

0.02

0.03

λ 1

200 400 600 800 10000

0.01

0.02

λ 3

200 400 600 800 1000−0.9

−0.8

−0.7

Iteration Number

Log(

||λ|| 2)

Figure 5.6: Third-Order Chaos Eigenvalue Convergence for the 2-DOF System

111


200 400 600 800 1000

0.15

0.16

λ 0

200 400 600 800 10000.01

0.02

0.03

λ 1

200 400 600 800 1000−0.01

0

0.01

Iteration Number

λ 2

500 10000

0.005

0.01

λ 3

200 400 600 800 1000−1

0

1x 10

−3

λ 4

500 1000−0.9

−0.8

−0.7

Iteration Number

Log(

||λ|| 2)

Figure 5.7: Fourth-Order Chaos Eigenvalue Convergence for the 2-DOF System

1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Node

Dis

plac

emen

t

95% Prob Range80% Prob RangeMeanDeterministic

Figure 5.8: Probabilistic Eigenvector for the 2-DOF System at Different Probability Levels

112


The first order polynomial chaos eigenvector coefficient vectors for the two degrees-of-

freedom model using Eq. (5.19), and the mass and stiffness being from the same proba-

bility space, are obtained as:

x0 =

0.00000E + 00

5.15499E − 01

6.05913E − 01

, x1 =

0.00000E + 00

3.33618E − 12

3.92108E − 12

For the same system, second order polynomial chaos eigenvector coefficient vectors are

obtained as:

x0 =

0.00000E + 00

5.15398E − 01

6.05793E − 01

, x1 =

0.00000E + 00

5.09199E − 04

5.98414E − 04

, x2 =

0.00000E + 00

2.09756E − 05

2.46545E − 05

The third order polynomial chaos, eigenvector coefficient vectors are given as:

x0 =

0.00000E + 00

5.15285E − 01

6.05661E − 01

, x1 =

0.00000E + 00

1.06857E − 03

1.25599E − 03

,

x2 =

0.00000E + 00

1.67062E − 05

1.96363E − 05

, x3 =

0.00000E + 00

−4.39396E − 06

−5.16462E − 06

113


The fourth order polynomial chaos, eigenvector coefficient vectors for the two degree-of

freedom model using Eq. (5.19) are obtained as:

x0 =

0.00000E + 00

5.15316E − 01

6.05698E − 01

, x1 =

0.00000E + 00

9.10427E − 04

1.07011E − 03

, x2 =

0.00000E + 00

1.10034E − 05

1.29333E − 05

x3 =

0.00000E + 00

−2.82568E − 06

−3.32128E − 06

, x4 =

0.00000E + 00

6.02056E − 07

7.07651E − 07

where xi, i = 0, 1, . . . , P are polynomial chaos eigenvector components of probabilistic

eigenvector, x when developed as polynomial chaos expansion and P is the number of

chaos terms.

If the mass and the stiffness are from different probability spaces i.e. mass and stiffnesses

are uncorrelated to each other, then the fundamental eigenvalue polynomial coefficients

are given in Table 5.2. The mean, µ and the standard deviation, σ of the fundamental

eigenvalue for different order chaos are given in Table 5.2. The results of this analysis

are compared with the Monte Carlo simulation using LHS, 1E + 04 samples. This yields

µ = 0.155894E + 00 and σ = 0.607573E − 01 as the mean and standard deviation of

the fundamental eigenvalue, respectively. PDFs of stochastic fundamental eigenvalue

114


upto three order polynomial chaos are compared with LHS in Fig. 5.9. Second-order and

third-order chaos’ PDFs are close to the PDF obtained using LHS.

First Order PC Second Order PC Third Order PCλ0 0.154755E + 00 0.156272E + 00 0.156401E + 00λ1 −0.276268E − 01 −0.350419E − 01 −0.348062E − 01λ2 0.447680E − 01 0.467170E − 01 0.468764E − 01λ3 0.922787E − 02 0.885926E − 02λ4 −0.930116E − 02 −0.102245E − 01λ5 −0.201138E − 04 0.194770E − 08λ6 −0.222439E − 02λ7 0.209005E − 02λ8 −0.109885E − 07λ9 0.526129E − 09µ 0.154755E + 00 0.156272E + 00 0.156401E + 00σ 0.526062E − 01 0.605577E − 01 0.610394E − 01

Table 5.2: Fundamental Eigenvalue Coefficients with Different Probability Spaces for Massand Stiffness for the 2-DOF System

Probabilistic eigenvectors for different probability ranges are same as the determinis-

tic eigenvector. For all order polynomial chaos, probabilistic eigenvectors are shown in

Fig. 5.10, and polynomial chaos eigenvector coefficients are given in Eqs. (5.23), (5.24),

and (5.25) for first-order, second-order and third-order chaos respectively. Probabilis-

tic eigenvectors are normalized with respect to maximum ordinate. The convergence of

eigenvalue polynomial coefficients for different chaos orders are shown in Figs. 5.11, 5.12,

and 5.13.

115


0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

1

2

3

4

5

6

7

8


Prob

abili

ty D

ensi

ty F

unct

ion

Monte Carlo1st PC2nd PC3rd PC

Figure 5.9: PDFs of the Fundamental Eigenvalue for Different Order Chaos with DifferentProbability Spaces for Mass and Stiffness for the 2-DOF System

1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Node

Dis

plac

emen

t


Figure 5.10: Fundamental Eigenvector at Different Probability Levels with 1− 3 order chaoswith for the 2-DOF System

116


x0 =

0.00000E + 00

4.77312E − 01

5.61028E − 01

, x1 =

0.00000E + 00

9.39688E − 02

1.10450E − 01

,

x2 =

0.00000E + 00

−9.72281E − 02

−1.14281E − 01

(5.23)

x0 =

0.00000E + 00

5.10030E − 01

5.99485E − 01

, x1 =

0.00000E + 00

2.25668E − 02

2.65248E − 02

,

x2 =

0.00000E + 00

6.64541E − 03

7.81095E − 03

, x3 =

0.00000E + 00

−1.02569E − 03

−1.20559E − 03

,

x4 =

0.00000E + 00

1.94342E − 02

2.28428E − 02

, x5 =

0.00000E + 00

1.73580E − 05

2.04025E − 05

(5.24)

117


x0 =

0.00000E + 00

5.14279E − 01

6.04479E − 01

, x1 =

0.00000E + 00

3.98038E − 02

4.67850E − 02

,

x2 =

0.00000E + 00

2.04312E − 02

2.40146E − 02

, x3 =

0.00000E + 00

3.30034E − 05

3.87919E − 05

,

x4 =

0.00000E + 00

5.68721E − 06

6.68469E − 06

, x5 =

0.00000E + 00

−4.68793E − 07

−5.51015E − 07

,

x6 =

0.00000E + 00

−1.42692E − 05

−1.67719E − 05

, x7 =

0.00000E + 00

−9.00057E − 05

−1.05792E − 04

,

x8 =

0.00000E + 00

4.62186E − 06

5.43249E − 06

, x9 =

0.00000E + 00

3.97740E − 08

4.67499E − 08

(5.25)

where xi, i = 0, 1, . . . , P are polynomial chaos eigenvector coefficients vectors and P is

the number of terms in the polynomial chaos.

118


0 100 2000.15

0.154

0.158

0.16λ 0

0 100 2000.04

0.042

0.044

λ 2

0 100 200

−0.032

−0.028

−0.022

Iteration Number

λ 1

0 100 200−0.8

−0.79

−0.78

Iteration Number

Log

(||λ|

| 2)

Figure 5.11: First-Order Eigenvalue Coefficients Convergence for Different Probability Spacesfor Mass and Stiffness for the 2-DOF System

5.5.2 Continuous System

Here a simple cantilever beam of unit length and having Gaussian random process to

represent the bending rigidity, EI; the mass per unit length, ρA, and exponential covari-

ance, C (x1, x2), is taken to study for its free vibration response. Correlation length, b

is 1 unit. For this beam, the mean value, 〈EI〉 and the standard deviation, σEI of the

bending rigidity are 30 units and 9 units, respectively. Similarly the mean of the mass

per unit length, 〈ρA〉, and the standard deviation are 10 and 3 units, respectively. The

119


100 200 300 400 5000.154

0.155

0.156

0.157

0.158

λ 0

100 200 300 400 5004

5

6

7

8x 10

−3

λ 3

100 200 300 400 500−0.038

−0.036

−0.034

−0.032

−0.03

λ 1

100 200 300 400 500−10

−9

−8

−7

−6x 10

−3

λ 4

100 200 300 400 5000.045

0.046

0.047

0.048

λ 2

100 200 300 400 500−2

−1

0x 10

−4

λ 5

100 200 300 400 500−0.74

−0.73

−0.72

−0.71

−0.7

Iteration Number

Log

(||λ|

| 2)

Figure 5.12: Second-Order Eigenvalue Coefficients Convergence for Different ProbabilitySpaces for Mass and Stiffness for the 2-DOF System

120


20 40 60 80 100 120 1400.155

0.156

0.157

0.158

λ 0

20 40 60 80 100 120 140-4

-2

0

2x 10

-4

λ 5

20 40 60 80 100 120 140-0.04

-0.035

-0.03

λ 1

20 40 60 80 100 120 140-2.5

-2

-1.5

-1x 10

-3

λ 6

20 40 60 80 100 120 1400.045

0.05

0.055

λ 2

20 40 60 80 100 120 1401

1.5

2

2.5x 10

-3

λ 7

20 40 60 80 100 120 1406

8

10

12x 10

-3

λ 3

20 40 60 80 100 120 140-2

0

2

4

x 10-4

λ 8

20 40 60 80 100 120 140

-0.014-0.012

-0.01-0.008-0.006

λ 4

20 40 60 80 100 120 140-2

-1

0

1x 10

-4

λ 9

20 40 60 80 100 120 140-0.8

-0.78

-0.76

-0.74

Iteration Number

Log

(||λ|

| 2)

Figure 5.13: Third-Order Eigenvalue Coefficients Convergence for Different Probability Spacesfor Mass and Stiffness for the 2-DOF System

121


-0.5 0.50.0

x

Figure 5.14: Cantilever Beam as a Continuous Structure

governing differential equation for this cantilever beam is given as:

ρA∂2w

∂t2+

∂2

∂x2

(EI

∂2w

∂x2

)= 0, (5.26)

w(−0.5) = 0,∂w

∂x

∣∣∣∣x=−0.5

= 0 (5.27)

Governing differential eigenvalue problem is written as,

d2

dx2

(EI

d2W (x)

dx2

)= λρAW (x) (5.28)

The processes for the bending rigidity and the mass per unit length are written as the

Karhunen-Loeve expansion using analytical eigenvalues and eigenfunctions [28]. So all

mass and stiffness global matrices are obtained using FEM. Particularly for this beam,

10 elements are used with Hermite shape functions for the elements. In the first case, the

bending rigidity, EI and the mass per unit length, ρA, are assumed to belong to the same

122


probability space. The fundamental eigenvalue is obtained as described in Section 5.4.

The mean and standard deviation of the fundamental eigenvalue are obtained using

Monte Carlo simulation with LHS 20, 000 samples as µ = 3.795252E + 01 and σ =

1.042406E + 01, respectively. Here for LHS, the Karhunen-Loeve expansion is carried

out to represent input material random process. These expansion terms subsequently are

simulated using uncorrelated random variables and substituted into FEM representation

of the governing differential equation. This particular method is known as the non-

intrusive stochastic method [66]. For the first and second-order chaos with 4 dimensional

Karhunen-Loeve terms, the fundamental eigenvalue polynomial chaos coefficients are

given in Table 5.5. The PDFs of the fundamental eigenvalue using first and second

order chaos with 4 terms in the Karhunen-Loeve expansion are shown in Fig. 5.15. The

fundamental eigenvector’s 95% probability range is plotted for first-order and second-

order chaos as shown in Figs. 5.16 and 5.17 respectively. Second-order chaos eigenvector

result is confirmed with Monte-Carlo results by plotting PDF of eigenvector displacemet

of midpoint of beam as shown in the Fig. 5.18. While plotting the eigenvector, the

eigenvector is normalized with respect to maximum ordiante. Figure 5.18 shows that

the mean of the displacement of the mid-point of the eigenvector has converged. As

polynomial chaos is increased, the standard deviation of the displacement of the mid-

point of the eigenvector will converge to Monte Carlo results.

If the mass distribution and the bending rigidity are from the different probability spaces

then the fundamental eigenvalue chaos coefficients for the first-order and second-order

123


First-Order PC Second-Order PCλ0 3.629568E + 01 3.664825E + 01λ1 1.076528E + 00 −1.134500E + 00λ2 −6.796990E + 00 −1.081117E + 01λ3 −2.322211E − 01 3.440391E − 01λ4 5.960499E − 01 9.043374E − 01λ5 3.652826E − 01λ6 2.904858E + 00λ7 −2.540959E − 01λ8 −2.201586E − 01λ9 9.038602E − 01

λ10 −5.171474E − 01λ11 −1.545075E − 01λ12 −1.437138E − 01λ13 1.297258E − 01λ14 −4.856162E − 02

µ 3.629568E + 01 3.664825E + 01σ 6.911382E + 00 1.139785E + 01

Table 5.3: Fundamental Eigenvalue Coefficients for Cantilever Beam with Same ProbabilitySpace for Mass and Stiffness

0 20 40 60 80 100 120 1400

0.01

0.02

0.03

0.04

0.05

0.06


Prob

abili

ty D

ensi

ty F

unct

ion


Figure 5.15: PDFs of the Fundamental Eigenvalue using First and Second-Order Chaos withSame Probability Space for Mass and the Bending Rigidity

124


−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Node Position

Dis

plac

emen

t

Mean95% Prob Upper Limit95% Prob Lower Limit

Figure 5.16: Fundamental Eigenvector using First-Order Chaos at 95% Probability with SameProbability Space for Mass and the Bending Rigidity

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Node Position

Dis

plac

emen

t


Figure 5.17: Fundamental Eigenvector using Second-Order Chaos at 95% Probability withSame Probability Space for Mass and the Bending Rigidity

125


0.3 0.35 0.4 0.450

5

10

15

20

25

30

0.3 0.35 0.4 0.450

5

10

15

20

25

30

Prob

abili

ty D

ensi

ty F

unct

ion

Prob

abili

ty D

ensi

ty F

unct

ion

Eigenvector Mid-point Displacement

2nd PC Monte Carlo

Eigenvector Mid-point Displacement

Figure 5.18: Fundamental Eigenvector’s Mid-point Displacement using Second-Order Chaoswith Same Probability Space for Mass and the Bending Rigidity

chaos along with the mean and standard deviation are given in Table 5.6. The PDFs

for both the chaos order of fundamental eigenvalue are given in Fig. 5.19 along with the

Monte Carlo simulation using 20000 LHS samples. The mean and standard deviation of

the fundamental eigenvalue are obtained as µ = 4.009882E +01 and σ = 1.809414E +01

using Monte Carlo simulation. The mean and standard deviation from the second-order

chaos expansion are µ = 3.917371E + 01 and σ = 1.769511E + 01, respectively. The

fundamental eigenvector’s 95%probability range is given in Figs. 5.20 and 5.21 for first-

order and second-order chaos respectively.

126

Fir

st-O

rder

PC

Seco

nd-O

rder

PC

λ0

=4.

0166

54E

+01

λ0

=3.

9173

71e

+01

λ15

=2.

4900

66E−

01λ

30

=2.

1946

27E−

03λ

1=−

1.00

1466

E+

01λ

1=−

1.16

3847

E+

01λ

16

=−

7.70

7725

E−

02λ

31

=7.

7452

43E−

02λ

2=−

4.63

7340

E+

00λ

2=−

5.38

9872

E+

00λ

17

=4.

3442

78E−

01λ

32

=−

5.40

2073

E−

02λ

3=

1.30

6371

E+

00λ

3=

1.51

9605

E+

00λ

18

=−

2.39

8511

E−

01λ

33

=−

2.28

9357

E−

02λ

4=

3.91

1854

E−

01λ

4=

4.55

8093

E−

01λ

19

=−

7.11

1791

E−

02λ

34

=5.

1196

86E−

03λ

5=

9.25

0299

E+

00λ

5=

1.01

5173

E+

01λ

20

=−

9.60

2896

E−

01λ

35

=−

1.28

2781

E−

01λ

6=−

4.26

4305

E+

00λ

6=−

4.65

9577

E+

00λ

21

=4.

7459

71E−

01λ

36

=4.

4191

69E−

02λ

7=−

1.19

1993

E+

00λ

7=−

1.29

5053

E+

00λ

22

=1.

5655

41E−

01λ

37

=−

4.98

3524

E−

02λ

8=

3.60

1936

E−

01λ

8=

3.97

7313

E−

01λ

23

=−

5.32

2089

E−

02λ

38

=3.

5301

84E−

02λ

9=

2.04

0404

E+

00λ

24

=3.

0866

46E−

02λ

39

=−

1.14

6199

E−

01λ

10

=1.

8924

01E

+00

λ25

=1.

7278

02E−

02λ

40

=−

1.90

5374

E−

01λ

11

=−

5.35

5312

E−

01λ

26

=2.

6180

61E−

01λ

41

=5.

8464

86E−

02λ

12

=−

1.61

1618

E−

01λ

27

=−

1.61

2740

E−

01λ

42

=−

1.22

9168

E−

01λ

13

=−

2.10

3538

E+

00λ

28

=−

6.98

3114

E−

02λ

43

=1.

3454

49E−

01λ

14

=9.

4292

98E−

01λ

29

=2.

2644

51E−

02λ

44

=−

6.65

0745

E−

02µ

=4.

0166

54E

+01

µ=

3.91

7371

e+

01σ

=1.

5131

45e

+01

σ=

1.76

9511

E+

01

Tab

le5.

4:Fun

dam

enta

lEig

enva

lue

Coeffi

cien

tsfo

rCan

tile

ver

Bea

mw

ith

Diff

eren

tPro

babi

lity

Spa

cefo

rM

ass

and

Stiffne

ss


0 20 40 60 80 100 120 140 160 1800

0.005

0.01

0.015

0.02

0.025

0.03

0.035


Pro

bab

ilit

y D

ensi

ty F

un

ctio

n


Figure 5.19: PDFs of the Fundamental Eigenvalue of the Cantilever Beam with DifferentProbability Spaces for Mass and the Bending Rigidity

128


−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Node Position

Dis

plac

emen

t


Figure 5.20: Fundamental Eigenvector of the Cantilever Beam at Different Probabilities usingFirst-Order Chaos with the Different Probability Spaces for Mass and the Bending Rigidity

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Node Position

Dis

plac

emen

t


Figure 5.21: Fundamental Eigenvector of the Cantilever Beam at Different Probabilities usingSecond-Order Chaos with the Different Probability Spaces for Mass and the Bending Rigidity

129

5.6 Conclusions

5.6 Conclusions

In this research, a non statistical algorithm is developed which calculates the fundamental

eigenvalue and the eigenvector for stochastic differential eigenvalue problem for a contin-

uous system as well as for discrete systems. An iterative procedure like vector iteration

is used to obtain the fundamental eigenvalue. In this approach, the Karhunen-Loeve the-

orem is used to expand input spatial random material properties and polynomial chaos

for input random variables as well as for eigenvalue and the corresponding eigenvec-

tor. After substituting these inputs and responses into governing differential eigenvalue

equation, the Galerkin projection is applied in random space to minimize the weighted

residual of the response. Here, the uncertainties are considered in both the mass dis-

tribution and stiffness (or the bending rigidity). This algorithm is applied to find the

fundamental eigenvalue of a cantilever beam with exponential covariance for both the

bending rigidity and the mass distribution. In this problem, material uncertainties are

assumed as Gaussian random process so original Wiener chaos is applied for both the

problems. But the same algorithm can be readily applied to non-Gaussian random pro-

cesses or non-Gaussian random variables using Wiener-Askey Polynomial chaos. The

fundamental eigenvalues obtained for both the systems are compared with Monte Carlo

simulation using LHS. In the next Chapter 6, the higher eigenvalues and eigenvectors

using polynomial chaos with efficient algorithms are discussed.

130

Chapter 6

Higher Eigenvalues using

Polynomial Chaos

6.1 Introduction

In Chapter 5, an intrusive algorithm is discussed to calculate the fundamental stochastic

eigenvalue using Spectral Stochastic Finite Element Method (SSFEM). Generally, we are

interested in not only the fundamental eigenvalue but also in higher eigenvalues. The

other approach to use polynomial chaos for dynamic systems is the so-called non-intrusive

formulation [66]. In this method, the Karhunen-Loeve expansion is used with Latin Hy-

percube Sampling (LHS) to get response polynomial chaos coefficients and probabilistic

characteristics of the response. This method is still computationally expensive for dy-

131

6.1 Introduction

namical response. An algorithm for calculating stochastic fundamental eigenvalue using

polynomial chaos is presented in Chapter 5. In this work, an extension of that work, an

algorithm for finding any eigenvalue (not just the fundamental one) of linear stochastic

differential equation is presented.

In this new algorithm, material properties are written as either the Karhunen-Loeve ex-

pansion for a random process or as polynomial chaos for the random variable case. Even

eigenvalues, shifts in eigenvalues, and eigenvectors are expressed as polynomial chaos.

Using the proposed algorithm, undetermined coefficients of fundamental eigenvalue and

eigenvector of respective polynomial chaos are obtained using zero shift in eigenvalues.

The initial eigenvalue shift vector for other eigenvalues is predicted using eigenvalues ob-

tained from mean values of input properties and the stochastic fundamental eigenvalue

polynomial coefficient vector. After obtaining these coefficients, probabilistic character-

istics can be derived.

Two examples are used to illustrate the accuracy of this algorithm. In the first example,

all three eigenvalues of a three degrees-of-freedom model are calculated. For this example,

two cases are considered. The system masses and stiffnesses are assumed to belong

to the same probability space in the first case and to different probability spaces in

the second case. In the second example, free vibration response of a simply supported

beam where both the bending rigidity and the mass per unit length are assumed to be

random processes with exponential autocorrelation functions is studied. The first three

eigenvalues are calculated when the mass and the rigidity are from the same probability

132


space as well as from two different probability spaces. The results of both examples are

compared with those obtained using LHS and probabilistic characteristics of fundamental

eigenvector are also given.

In Chapter 2 and Section 2.2.3, all required fundamentals and tools of SSFEM were

discussed so those are not being repeated here again.


For linear dynamic problems, the eigenvalues of the dynamic systems is an important

property of the phenomenon governing the behavior of these systems. In vibration

and acoustics, eigenvalues characterize resonances of a system. For deterministic sys-

tems, a number of algorithms are available to solve the eigenproblem and many of

these are implemented in commercial softwares and also free available software from

NETLIB(http://www.netlib.org/). But for stochastic systems, solution to eigenprob-

lems using polynomial chaos is not as developed as for deterministic systems. Solving

stochastic eigenvalue problem by developing appropriate algorithms for such problems is

the key objective of this research. Stochastic differential eigenproblem can be defined as:

K (x, θ) u (x, θ) = λ (θ)M (x, θ) u (x, θ) (6.1)

133

http://www.netlib.org/



2q, respectively and p ≥ q. There are p boundary conditions associated with the system

governed by differential equation given in Eq. (6.1).

Bi (x, θ) u (x, θ) = 0, i = 1, 2, . . . , k (6.2)

Bi (x, θ) u (x, θ) = λ (θ) Ci (x, θ) u (x, θ) , i = k + 1, k + 2, . . . , p (6.3)

Here Bi and Ci are stochastic linear homogeneous differential operators of maximum order

2p− 1 and 2q − 1 respectively.

Application of Galerkin procedure to Eq. (6.1) in probability space as well as spatial

dimensions results in a discrete system of algebraic equations, this is very well ex-

plained [28, 29, 30]. This results into K and M as matrices. This assumption leads

to a discrete problem, hence the governing equation becomes:

Kx = λMx (6.4)

Equation (6.4) by using with eigenvalue shifting theorem [117] can be written as:

(K− ηM)x = λMx (6.5)

λ = λ− η (6.6)

where λ, λ, and η are shifted eigenvalue, original eigenvaule of Eq. (6.4), and shift in the

134


interested eigenvalue respectively. Matrices K and M are written as polynomial chaos,

K =P∑

i=0

KiΓi, M =P∑

i=0

MiΓi, η =P∑

i=0

ηiΓi, λ =P∑

i=0

λiΓi, x =P∑

i=0

xiΓi (6.7)

Here, scalars, λi and vectors, xi are unknowns.

Therefore, Eq. (6.5) becomes:

P∑i=0

P∑j=0

KixjΓiΓj −P∑

k=0

P∑i=0

P∑j=0

ηkMixjΓkΓiΓj =P∑

k=0

P∑i=0

P∑j=0

λkMixjΓkΓiΓj (6.8)

To minimize the weighted residual of the error, multiply Eq. (6.8) by Γm and taking

expectation, we get

P∑i=0

P∑j=0

Kixj 〈ΓiΓjΓm〉 −P∑

k=0

P∑i=0

P∑j=0

ηkMixj 〈ΓiΓjΓkΓm〉 =

P∑k=0

P∑i=0

P∑j=0

λkMixj 〈ΓiΓjΓkΓm〉 , m = 0, . . . , P (6.9)

The above equation can be written in a simple form as:

(K−Kr) X = λM X (6.10)

where K, Kr, λM and X are tensors of the order n (P + 1) × n (P + 1), n (P + 1) ×

n (P + 1), n (P + 1) × n (P + 1), and n (P + 1) × 1 respectively. n × n is the order of

individual Ki or Mi in polynomial chaos expansion. Elements of K, Kr and λM are given

135


as

Kl,m =P∑

i=0

Ki 〈ΓiΓlΓm〉 , m, l = 0, . . . , P (6.11)

Krl,m=

P∑i=0

P∑k=0

Miηk 〈ΓiΓkΓlΓm〉 (6.12)

λMl,m =P∑

i=0

P∑k=0

Miλk 〈ΓiΓkΓlΓm〉 (6.13)

Xl = xl (6.14)

Equation (6.10) can be written in another way as:

(K−Kr) X = MX λ (6.15)

MXl,m =P∑

i=0

P∑j=0

Mixj 〈ΓiΓjΓlΓm〉 (6.16)

where XMl,m is a typical l,mth element of this tensor.


Equation (6.10) is similar to the deterministic eigenvalue problem. To find eigenvalue

polynomial coefficients, Eq. (6.10) needs to be solved iteratively using such methods as

the vector iteration method. Equation (6.5) is multiplied by xT and the substitutions

136


are done as given in Eq. (6.7) which results in the following equation

P∑i=0

P∑j=0

P∑k=0

xTi KjxkΓiΓjΓk −

P∑i=0

P∑j=0

P∑k=0

P∑l=0

ηlxTi MjxkΓiΓjΓkΓl =

P∑i=0

P∑j=0

P∑k=0

P∑l=0

λlxTi MjxkΓiΓjΓkΓl (6.17)

Multiplying above equation by Γm and taking expectation to minimize the weighted

residual of the error produces following system of equations.

P∑i=0

P∑j=0

P∑k=0

xTi Kjxk 〈ΓiΓjΓkΓm〉 −

P∑i=0

P∑j=0

P∑k=0

P∑l=0

ηlxTi Mjxk 〈ΓiΓjΓkΓlΓm〉

=P∑

i=0

P∑j=0

P∑k=0

P∑l=0

λlxTi Mjxk 〈ΓiΓjΓkΓlΓm〉 , m = 0, . . . , P (6.18)

The above equation can be written in simple form as:

K = M λ (6.19)

The generic elements of the matrices from Eq. (6.19) are given as:

Kl =P∑

i=0

P∑j=0

P∑k=0

xTi Kjxk 〈ΓiΓjΓkΓl〉 −

P∑i=0

P∑j=0

P∑k=0

P∑s=0

ηsxTi Mjxk 〈ΓiΓjΓkΓlΓl〉 , l = 0, . . . , P (6.20)

Ml,m =P∑

i=0

P∑j=0

P∑k=0

xTi Mjxk 〈ΓiΓjΓkΓlΓm〉 , l, m = 0, . . . , P (6.21)

137


The following iterative procedure is used to get the eigenvalues

• For the fundamental eigenvalue, assume ηi = 0, i = 0, . . . , P ; for higher eigenvalues,

estimate ηi using fundamental eigenvalue polynomial coefficient vector, λf and

higher eigenvalues obtained using mean values of input properties such that

η =

(Λ (1) +

1

3(Λ (n)− Λ (1))

)λf

λf1

C (6.22)

where Λ (1) and Λ (n) are fundamental and higher eigenvalue obtained from mean

values of input properties, λf1 is the mean value of fundamental eigenvalue and C is

constant that imparts stability to the above algorithm and it is discussed throughly

elsewhere [118]. Here C ≈ 1.25 to 1.50 is taken.

• Start with some initial xi, i = 0, . . . , P , generally it will be a vector of unit elements.

• Substitute these xi into Eq. (6.19) and find λi, i = 0, . . . , P .

• Substitute these λi into Eq. (6.10) to get new xi.

• Normalize these xi with respect to mass matrix such that

xi =xi

||x0||∞(6.23)

where ||x0||∞ is the L∞ norm of the vector x0, mean eigenvector or 0th polynomial

chaos component of eigenvector.

138


• Once λi are converged, calculate eigenvalue coefficient vector as

λ = λ + η (6.24)


The aforementioned algorithm is applied to a discrete system, a 3 degree-of-freedom

model, and to a continuous system, a simply supported beam. The first three eigenvalues

and fundamental eigenvector are obtained for the discrete and the continuous systems.

These systems are described in subsequent subsections. These results obtained using

polynomial chaoses are compared with Monte Carlo simulation using LHS [67].

6.4.1 Three Degree-of-Freedom System

m1 m2

k1 k2x1 x2

m3

x3k3 k4

Figure 6.1: Three Degrees-of-Freedom Spring-Mass Model

139


For the discrete model shown in Fig. 6.1, equations of motion for free vibration analysis

become,

Mx+Kx = 0 (6.25)

m1 0 0

0 m2 0

0 0 m3

x1

x2

x3

+

k1 + k2 −k2 0

−k2 k2 + k3 −k3

0 −k3 k3 + k4

x1

x2

x3

=

0

0

0

(6.26)

For the system shown in Fig. 6.1; m1 = 2kg, m2 = 3kg, m3 = 1kg and, k1 = 3N/m,

k2 = 3N/m, k3 = 2N/m, k4 = 2N/m, so mean mass matrix, M0 and mean stiffness

matrix K0 become

M0 =

2.0 0.0 0.0

0.0 3.0 0.0

0.0 0.0 1.0

K0 =

6.0 −3.0 0.0

−3.0 5.0 −2.0

0.0 −2.0 4.0

(6.27)

Here, it is assumed that the standard deviation in the mass matrix is 0.2M0 and standard

deviation in the stiffness matrix is 0.3K0. Three eigenvalues are obtained for the above

system using algorithm as given Section 6.3. If the masses and stiffnesses are perfectly

correlated, eigenvalue polynomial coefficients, λi for fourth-order of chaos are obtained

as shown in the Table 6.1, where µ and σ are the mean and standard deviation of

140


eigenvalue for these eigenvalues. When a Monte Carlo simulation using LHS is carried

out with 10, 000 samples, obtained means, µ and the standard deviations, σ, of the three

eigenvalues are shown in the Table 6.2. In Figs. 6.2, 6.3, and 6.4, the PDFs of these

eigenvalues are plotted. The PDFs obtained from LHS are also plotted to ascertain the

accuracy of the solution obtained using polynomial chaos. As shifts in the eigenvalues

are constant, Log(|λ|2) are same as Log(∣∣∣λ∣∣∣

2), so Log(

∣∣∣λ∣∣∣2) are plotted as a function

of the iteration number in Fig. 6.5. Note that | |2 is L2 norm of the particular vector.

The probabilistic eigenvectors for three eigenvalues which are normalized with respect to

maximum ordinate, are plotted in Figs. 6.6, 6.7, and 6.8. The probabilistic eigenvectors

are the same as the deterministic eigenvectors.

First Eigenvalue Second Eigenvalue Third Eigenvalueλ0 6.211807E − 01 3.307638E + 00 4.537975E + 00λ1 7.335855E − 02 3.902001E − 01 5.358672E − 01λ2 −1.709919E − 02 −9.026097E − 02 −1.248484E − 01λ3 4.045195E − 03 2.066325E − 02 2.949488E − 02λ4 −7.865526E − 04 −3.664059E − 03 −5.735625E − 03µ 6.211807E − 01 3.307638E + 00 4.537975E + 00σ 7.796968E − 02 4.140461E − 01 5.695060E − 01

Table 6.1: Eigenvalue Coefficients for Perfectly Correlated Masses and Stiffness for the 3-DOFSystem using Fourth-Order Chaos

First Eigenvalue Second Eigenvalue Third Eigenvalueµ 6.216377E − 01 3.308550E + 00 4.540788E + 00σ 7.627693E − 02 4.064229E − 01 5.581128E − 01

Table 6.2: Mean and Standard Deviation for Perfectly Correlated Masses and Stiffness forthe 3-DOF System using LHS

If mass and stiffnesses are uncorrelated to each other, then the eigenvalue polynomial

141


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

1

2

3

4

5

6

7

First Eigenvalue, λ (rad2/s2)

Prob

abili

ty D

ensi

ty F

unct

ion

Monte Carlo

4th PC

Figure 6.2: PDFs of Fundamental Eigenvalue using Fourth Order Chaos for Perfectly Corre-lated Masses and Stiffness for the 3-DOF System

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

0.2

0.4

0.6

0.8

1

1.2

1.4

Second Eigenvalue, λ (rad2/s2)

Prob

abili

ty D

ensi

ty F

unct

ion

Monte Carlo4th PC

Figure 6.3: PDFs of Second Eigenvalue using Fourth Order Chaos for Perfectly CorrelatedMasses and Stiffness for the 3-DOF System

142


0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Third Eigenvalue, λ (rad2/s2)

Prob

abili

ty D

ensi

ty F

unct

ion

Monte Carlo

4th PC

Figure 6.4: PDFs of Third Eigenvalue using Fourth Order Chaos for Perfectly CorrelatedMasses and Stiffness for the 3-DOF System

20 40 60 80 100 120 140−3

−2

−1

20 40 60 80 100 120 140−5

0

5

20 40 60 80 100 120 140−2

0

2

Iteration Number

λλ

Log

(||λ

|| 2)

First Eigenvalue

Second Eigenvalue

Third Eigenvalue

Figure 6.5: Convergence of λ for Perfectly Correlated Masses and Stiffness for the 3-DOFSystem

143


1 2 3 4 5−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

Node

Dis

plac

emen

t


Figure 6.6: Fundamental Eigenvector for Perfectly Correlated Masses and Stiffness for the3-DOF System at Different Probabilities

1 2 3 4 5−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Node

Dis

plac

emen

t


Figure 6.7: Second Eigenvector for Perfectly Correlated Masses and Stiffness for the 3-DOFSystem at Different Probabilities

144


1 2 3 4 5−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Node

Dis

plac

emen

t


Figure 6.8: Third Eigenvector for Perfectly Correlated Masses and Stiffness for the 3-DOFSystem at Different Probabilities

coefficients for third-order chaos are given in Table 6.3. In the same Table 6.3, their mean

values, µ and the standard deviations, σ of these eigenvalues are given. The results of

this analysis are compared with the Monte Carlo simulation using LHS, 10, 000 samples.

The mean values, µ and the standard deviations, σ of LHS of these eigenvalues are

reported in Table 6.4. In Figs. 6.9, 6.10, and 6.11, the PDFs of the these eigenvalues

are plotted along with the PDF obtained from LHS. The excellent agreement is observed

between Polynomial Chaos and LHS results. In Fig. 6.12, Log(∣∣∣λ∣∣∣

2

)are plotted as a

function of the iteration number for these eigenvalues. The probabilistic eigenvectors for

three eigenvalues which are normalized with respect to maximum ordinate, are plotted

in Figs. 6.13, 6.14, and 6.15. Probabilistic eigenvectors range, normalized with respect

145


to maximum ordinate are same as the deterministic eigenvectors.

First Eigenvalue Second Eigenvalue Third Eigenvalueλ0 6.651182E − 01 3.542152E + 00 4.858313E + 00λ1 −1.463308E − 01 −7.774304E − 01 −1.071756E + 00λ2 1.994111E − 01 1.057139E + 00 1.456019E + 00λ3 3.328264E − 02 1.646364E − 01 2.466950E − 01λ4 −4.327536E − 02 −2.179527E − 01 −3.095934E − 01λ5 −1.771800E − 06 −2.203023E − 05 −4.004512E − 04λ6 −6.682629E − 03 −2.206436E − 02 −4.318611E − 02λ7 8.492407E − 03 3.298059E − 02 4.980368E − 02λ8 7.176791E − 06 −2.190436E − 04 1.724688E − 03λ9 7.504976E − 09 −7.440530E − 05 −2.836150E − 04µ 6.651182E − 01 3.542152E + 00 4.858313E + 00σ 2.562770E − 01 1.352317E + 00 1.871463E + 00

Table 6.3: Eigenvalue Coefficients for Uncorrelated Masses and Stiffness for the 3-DOF Sys-tem using Third-Order Chaos

First Eigenvalue Second Eigenvalue Third Eigenvalueµ 6.650723E − 01 3.541251E + 00 4.858611E + 00σ 2.561809E − 01 1.364063E + 00 1.871501E + 00

Table 6.4: Mean and Standard Deviation for Uncorrelated Masses and Stiffness for the 3-DOFSystem using LHS

6.4.2 Continuous System

Here a simply-supported beam of unit length and having Gaussian random process to

represent the bending rigidity, EI; the mass per unit length, ρA, and exponential covari-

ance, C (x1, x2), is taken to study for its free vibration response. Correlation length, b

is 1 unit. For this beam, the mean value, 〈EI〉 and the standard deviation, σEI of the

bending rigidity are 30 units and 9 units, respectively. Similarly the mean of the mass

146


0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8


Prob

abili

ty D

ensi

ty F

unct

ion

Monte Carlo3rd PC

Figure 6.9: PDFs of Fundamental Eigenvalue using Third Order Chaos for UncorrelatedMasses and Stiffness for the 3-DOF System

0 2 4 6 8 10 12 140

0.05

0.1

0.15

0.2

0.25

0.3

0.35


Prob

abili

ty D

ensi

ty F

unct

ion

Monte Carlo3rd PC

Figure 6.10: PDFs of Second Eigenvalue using Third Order Chaos for Uncorrelated Massesand Stiffness for the 3-DOF System

147


0 5 10 15 20 250

0.05

0.1

0.15

0.2

0.25


Prob

abili

ty D

ensi

ty F

unct

ion

Monte Carlo3rd PC

Figure 6.11: PDFs of Third Eigenvalue using Third Order Chaos for Uncorrelated Masses andStiffness for the 3-DOF System

10 20 30 40 50 60 70 80 90 100−0.4

−0.2

0

10 20 30 40 50 60 70 80 90 100−2

0

2

Log

(||λ

|| 2)

10 20 30 40 50 60 70 80 90 100−2

0

2

Iteration Number

First Eigenvalue

Second Eigenvalue

Third Eigenvalue

Figure 6.12: Convergence of λ for Uncorrelated Masses and Stiffness for the 3-DOF System

148


1 2 3 4 5−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

Node

Dis

plac

emen

t


Figure 6.13: Fundamental Eigenvector for Uncorrelated Masses and Stiffness for the 3-DOFSystem at Different Probabilities

1 2 3 4 5−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Node

Dis

plac

emen

t


Figure 6.14: Second Eigenvector for Uncorrelated Masses and Stiffness for the 3-DOF Systemat Different Probabilities

149


1 2 3 4 5−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Node

Dis

plac

emen

t


Figure 6.15: Third Eigenvector for Uncorrelated Masses and Stiffness for the 3-DOF Systemat Different Probabilities

-0.5 0.50.0

x

Figure 6.16: Simply-Supported Beam as a Continuous Structure

150


per unit length, 〈ρA〉, and the standard deviation are 10 and 3 units, respectively. The

governing differential equation for this cantilever beam is given as,

ρA∂2w

∂t2+

∂2

∂x2

(EI

∂2w

∂x2

)= 0, (6.28)

w(−0.5) = 0, w(0.5) = 0 (6.29)

Governing differential eigenvalue problem is written as,

d2

dx2

(EI

d2W (x)

dx2

)= λρAW (x) (6.30)

The processes for the bending rigidity and the mass per unit length are written as the

Karhunen-Loeve expansion using analytical eigenvalues and eigenfunctions [28]. So all

mass and stiffness global matrices are obtained using FEM. Particularly for this beam,

10 elements are used with Hermite shape functions for the elements. In the first case, the

bending rigidity, EI as well as the mass per unit length, ρA, are assumed to belong to the

same probability space. Their mean values, µPC and standard deviations, σPC obtained

using fourth-order chaos are given in Table 6.5. To compare the effectiveness of the

proposed algorithm, a Monte Carlo using 5, 000 LHS is carried out for these eigenvalues

and their mean values, µLHS and standard deviations, σLHS are given in Table 6.5. Here

for LHS, the Karhunen-Loeve expansion is carried out to represent input material random

process. These expansion terms subsequently are simulated using uncorrelated random

variables and substituted into FEM representation of the governing differential equation.

151


This particular method is known as the non-intrusive stochastic method [66]. The PDFs

for these eigenvalues along with the PDF obtained using LHS, are plotted in Figs. 6.17,

6.18, and 6.19. The convergence of eigenvalues are shown in Fig. 6.20 in which Log(∣∣∣λ∣∣∣

2

)are plotted as a function of the iteration number. The probabilistic eigenvectors for

three eigenvalues which are normalized with respect to maximum ordinate, are plotted

in Figs. 6.21, 6.22, and 6.23.

First Eigenvalue Second Eigenvalue Third EigenvalueµPC 2.723826E + 02 4.702985E + 03 2.202525E + 04

µLHS 2.897179E + 02 4.677086E + 03 2.372747E + 04σPC 3.484409E + 01 3.637537E + 02 1.081263E + 03

σLHS 3.502319E + 01 3.537238E + 02 1.046769E + 03

Table 6.5: Mean and Standard Deviation of Eigenvalues for Fully Correlated Masses andStiffness for the Simply-Supported Beam using Fourth-Order Chaos and LHS

If the mass distribution and the bending rigidity are from the different probability spaces

then their mean values, µPC and standard deviations, σPC of the first three eigenvalue’s

using the second-order chaos are given in Table 6.6. The eigenvalue polynomial coeffi-

cients for second-order chaos are not given here due to their size. When a Monte Carlo

simulation using LHS is carried out with 5, 000 samples, obtained mean values, µLHS

and the standard deviations, σLHS of the three eigenvalues are shown in the Table 6.6.

In Figs. 6.24, 6.25, and 6.26, the PDFs of the these eigenvalues are plotted along with

the PDF obtained from LHS. The excellent agreement is observed between Polynomial

Chaos and LHS results. In Fig. 6.27, Log(∣∣∣λ∣∣∣

2

)are plotted as a function of the iteration

number for these eigenvalues. The probabilistic eigenvectors for three eigenvalues which

152


200 220 240 260 280 3000

0.02

0.04

0.06

0.08

0.1

0.12


Prob

abili

ty D

ensi

ty F

unct

ion

Monte Carlo4th PC

Figure 6.17: PDFs of Fundamental Eigenvalue using Fourth Order Chaos for Fully CorrelatedMasses and Stiffness for Simply-Supported Beam

4200 4400 4600 4800 5000 52000

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016


Prob

abili

ty D

ensi

ty F

unct

ion

Monte Carlo4th PC

Figure 6.18: PDFs of Second Eigenvalue using Fourth Order Chaos for Fully Correlated Massesand Stiffness for Simply-Supported Beam

153


2.3 2.35 2.4 2.45 2.5

x 104

0

1

2

3

4

5

6

7

8

9

x 10−3


Prob

abili

ty D

ensi

ty F

unct

ion

Monte Carlo4th PC

Figure 6.19: PDFs of Third Eigenvalue using Fourth Order Chaos for Fully Correlated Massesand Stiffness for Simply-Supported Beam

10 20 30 40 50 60 70 80 90 1005.5

6

6.5

10 20 30 40 50 60 70 80 90 1007.5

8

8.5

10 20 30 40 50 60 70 80 90 100

8

10

12

Iteration Number

Log

(||λ

|| 2)

First Eigenvalue

Second Eigenvalue

Third Eigenvalue

Figure 6.20: Convergence of λ for Fully Correlated Masses and Stiffness for Simply-SupportedBeam

154


−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

Node Position

Dis

plac

emen

t


Figure 6.21: Fundamental Eigenvector for Fully Correlated Masses and Stiffness for Simply-Supported Beam at Different Probabilities

−0.5 0 0.5

−1

−0.5

0

0.5

1

Node Position

Dis

plac

emen

t


Figure 6.22: Second Eigenvector for Fully Correlated Masses and Stiffness for Simply-Supported Beam at Different Probabilities

155


−0.5 0 0.5

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Node Position

Dis

plac

emen

t


Figure 6.23: Third Eigenvector for Fully Correlated Masses and Stiffness for Simply-SupportedBeam at Different Probabilities

are normalized with respect to maximum ordinate, are plotted in Figs. 6.28, 6.29, and

6.30.

First Eigenvalue Second Eigenvalue Third EigenvalueµPC 3.069019E + 02 5.178844E + 03 2.111937E + 04

µLHS 3.130792E + 02 4.999777E + 03 2.224798E + 04σPC 1.364490E + 02 2.170968E + 03 9.247072E + 03

σLHS 1.368285E + 02 2.156818E + 03 9.493314E + 03

Table 6.6: Mean and Standard Deviation of Eigenvalues for Uncorrelated Masses and Stiffnessfor the Simply-Supported Beam using Second-Order Chaos and LHS

156


0 200 400 600 800 1000 12000

0.5

1

1.5

2

2.5

3

3.5

4x 10

−3


Prob

abili

ty D

ensi

ty F

unct

ion

Monte Carlo2nd PC

Figure 6.24: PDFs of Fundamental Eigenvalue using Second Order Chaos for UncorrelatedMasses and Stiffness for Simply-Supported Beam

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 104

0

0.5

1

1.5

2

2.5x 10

−4


Prob

abili

ty D

ensi

ty F

unct

ion

Monte Carlo2nd PC

Figure 6.25: PDFs of Second Eigenvalue using Second Order Chaos for Uncorrelated Massesand Stiffness for Simply-Supported Beam

157


0 2 4 6 8 10 12

x 104

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

−5


Prob

abili

ty D

ensi

ty F

unct

ion

Monte Carlo2nd PC

Figure 6.26: PDFs of Third Eigenvalue using Second Order Chaos for Uncorrelated Massesand Stiffness for Simply-Supported Beam

20 40 60 80 100 120 140 160 180 2005.5

6

6.5

0 50 100 150 2008

8.5

9

Log

(||λ

|| 2)

20 40 60 80 100 120 140 160 180 2007.5

8

8.5

Iteration Number

First Eigenvalue

Second Eigenvalue

Third Eigenvalue

Figure 6.27: Convergence of λ for Uncorrelated Masses and Stiffness for Simply-SupportedBeam

158


−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Node Position

Dis

plac

emen

t


Figure 6.28: Fundamental Eigenvector for Uncorrelated Masses and Stiffness for Simply-Supported Beam at Different Probabilities

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5

−1

−0.5

0

0.5

1

Node Position

Dis

plac

emen

t


Figure 6.29: Second Eigenvector for Uncorrelated Masses and Stiffness for Simply-SupportedBeam at Different Probabilities

159

6.5 Conclusions

−0.5 0 0.5

−1

−0.5

0

0.5

1

Node Position

Disp

lace

men

t


Figure 6.30: Third Eigenvector for Uncorrelated Masses and Stiffness for Simply-SupportedBeam at Different Probabilities

6.5 Conclusions

An algorithm is developed which calculates not only fundamental eigenvalue and eigen-

vector but any higher eigenvalue and eigenvector for stochastic differential eigenvalue

problem for continuous system as well as for discrete systems. This algorithm which is

similar to deterministic vector iteration uses the eigenvalues obtained using mean value

properties and eigenvalue shift theorem effectively. Input spatial random material prop-

erties are expanded using the Karhunen-Loeve theorem and input random variables as

well as eigenvalues, eigenvalues’ shifts and the eigenvectors as polynomial chaos. After

substituting these inputs and responses into governing differential eigenvalue equation,

Galerkin projection is applied in random space to minimize the weighted residual of

the error. Here, uncertainties are considered in the material properties in terms of the

160

6.5 Conclusions

mass distribution and stiffness or the bending rigidity. This algorithm is applied to a

three degrees-of freedom system in which masses and stiffnesses are assumed as Gaussian

random variables. While applying this algorithm to a continuous problem, a simply-

supported beam, elastic rigidity and mass distribution are assumed as Gaussian random

process with exponential covariance so original Wiener chaos is applied. But the same

algorithm can be readily applied to non-Gaussian random processes or non-Gaussian

random variables using Wiener-Askey Polynomial chaos. First three eigenvalues ob-

tained for both the systems are compared with Monte Carlo simulation using LHS. Mass

normalized probabilistic eigenvectors show variation with respect to probability while

probabilistic eigenvectors normalized with respect to maximum ordinate are same as

deterministic eigenvector.While calculating eigenvalues, input properties are assumed ei-

ther fully correlated or uncorrelated. In the case of fully correlated input properties,

obtained coefficient variation of eigenvalues are very small and this becomes very large

for the case of uncorrelated input properties, these two cases form two extremes. For the

most part, these extreme cases may not occur, so it is essential to develop polynomial

chaos for partially correlated variables. At design stage, many times, our interest lies

in the eigenvalues only and not necessarily in the eigenvectors, so efficient algorithms

are needed to calculate eigenvalues which can be more efficient than the vector iteration

method proposed here. An algorithm is developed in Chapter 7 which calculates most

accurate eigenvalues without calculating eigenvector as compared to interactive methods

developed in this Chapter 6 as well as Chapter 5.

161

Chapter 7

A New Algorithm for Eigenvalue

Analysis using Polynomial Chaos

7.1 Introduction

In Chapters 5 and 6, iterative algorithms were developed to calculate eigenvalues and

eigenvectors of discrete and continuous systems using polynomial chaos. But many times,

we are interested in eigenvalues alone and not necessarily in eigenvectors. So an algorithm

which calculates only eigenvalues using polynomial iteration is developed.

In this new algorithm, Karhunen-Loeve expansion is used to write structural properties

into summation of orthogonal components and the response is written in terms of poly-

nomial chaos. Eigenvalues are obtained using an implicit iteration along with the Secant

162


method to solve eigenvalue polynomial equation. This algorithm is applied to a three

degree-of-freedom model. While applying this algorithm, two cases are considered: 1)

system masses and stiffnesses are correlated, 2) masses and stiffnesses are uncorrelated.

The results for both cases are confirmed using LHS.


When the system properties are random, resulting eigenvalues would be random and

hence the knowledge of eigenvalues would be essential in terms of PDF or bounds. Us-

ing vector iteration method, probabilistic eigenvalues are obtained in previous chapters.

Governing stochastic differential eigenproblem can be defined as:

K (x, θ) u (x, θ) = λ (θ)M (x, θ) u (x, θ) (7.1)


2q, respectively and p ≥ q. There are p boundary conditions associated with the system

governed by differential equation given in Eq. (7.1). Application of Galerkin procedure

to Eq. (7.1) results in a discrete system of algebraic equations. For simplicity, K and

M are assumed to be matrices. This assumption leads to a discrete problem, hence the

governing equation becomes:

Kx = λMx (7.2)

163

7.3 Eigenvalues Extraction

Matrices K and M are written as polynomial chaos,

K =P∑

i=0

KiΓi, M =P∑

i=0

MiΓi, λ =P∑

i=0

λiΓi, (7.3)

Here, λi are unknowns and x is not expanded as polynomial chaos as our interest is not

to find x. Therefore, the governing equation, Eq. (7.2) becomes:

(P∑

i=0

KiΓi −P∑

j=0

P∑i=0

λjMiΓiΓj

)x = 0 (7.4)

To minimize the weighted residual of the response, multiply Eq. (7.4) by Γm and take

expectation

P∑i=0

Ki 〈ΓiΓm〉 −P∑

j=0

P∑i=0

λjMi 〈ΓiΓjΓm〉 = 0, m = 0, . . . , P (7.5)


For nontrivial solution of Eq. 7.5,

det

(P∑

i=0

Ki 〈Γi Γm〉 −P∑

j=0

P∑i=0

λj Mi 〈Γi Γj Γm〉

)= 0, m = 0, . . . , P (7.6)

164


Equation 7.6 means that there will be P + 1 simultaneous equations which satisfy eigen-

value polynomial coefficient vector which are given as:

det

(P∑

i=0

Ki 〈Γi Γ0〉 −P∑

j=0

P∑i=0

λj Mi 〈Γi Γ0 Γj〉

)= 0,

det

(P∑

i=0

Ki 〈Γi Γ1〉 −P∑

j=0

P∑i=0

λj Mi 〈Γi Γ1 Γj〉

)= 0,

...

det

(P∑

i=0

Ki 〈Γi ΓP 〉 −P∑

j=0

P∑i=0

λj Mi 〈Γi ΓP Γk〉

)= 0

(7.7)

Eigenvalue polynomial coefficient vector is obtained as a solution of these simultaneous

equations using implicit secant iteration method which are presented in the following

equation.

J λi+1 = J λi − F (7.8)

where J is Jacobian evaluated at λ and obtained using through finite difference calcu-

lations. F is function vector of governing Eqs. 7.6 evaluated at λ and λ is eigenvalue

polynomial coefficient vector. While choosing initial guess, λ0 is chosen as deterministic

eigenvalue and rest of the coefficients are some fraction of this deterministic eigenvalue.

165

7.4 Numerical Example


This algorithm is applied to a discrete system, a 3 degrees-of-freedom model which is

shown in Fig. 7.1. In this model, all masses, mi = 0.5Kg and stiffnesses, ki = 400N/m

are assumed. Equation of motion for free vibration analysis become

m1 0 0

0 m2 0

0 0 m3

x1

x2

x3

+

k1 + k2 −k2 0

−k2 k2 + k3 −k3

0 −k3 k3

x1

x2

x3

=

0

0

0

(7.9)

m1 m2

k1 k2

x1 x2

m3

x3

k3

Figure 7.1: Three Degrees-of-Freedom Spring-Mass Model for Eigenvalue Analysis

It is assumed that the coefficients of variation for the mass matrix and stiffness matrix are

0.2 and 0.3 respectively. All three eigenvalues are obtained using all available equations.

When the masses and stiffness are perfectly correlated to each other, eigenvalue polyno-

mial coefficients are shown in Table 7.1 for 4th order using all 5 polynomial equations.

166


These eigenvalues are obtained using implicit polynomial iteration with 1E6 iterations.

In these Tables, µ, σ, and Λ are respectively, the mean, the standard deviation and the

norm of governing Eqs. 7.6. The obtained mean, µm and the standard deviation, σm

using Monte Carlo simulation with LHS with 10, 000 samples are given in Table 7.2. The

PDF of the first, the second, and the third eigenvalues are plotted in Figs. 7.2, 7.3, and

7.4 for the three eigenvalues using all 5 equations along with the LHS-PDF.

When the mass and stiffness matrices are uncorrelated to each other, the eigenvalue

polynomial coefficients are given in Table 7.3 for the third order chaos. In Table 7.4, the

mean, µm and the standard deviation, σm of these eigenvalues, obtained using LHS are

given. In Fig. 7.5, 7.6, and 7.7, PDFs of the first, the second, and the third eigenvalues

are plotted along with the PDF obtained using LHS (10, 000 samples)respectively.

First Eigenvalue Second Eigenvalue Third Eigenvalueλ0 1.547933E + 02 1.215259E + 03 2.537639E + 03λ1 1.828259E + 01 1.435355E + 02 2.997232E + 02λ2 −4.265764E + 00 −3.349351E + 01 −6.993967E + 01λ3 1.015672E + 00 7.975018E + 00 1.665338E + 01λ4 −2.031273E − 01 −1.594946E + 00 −3.330796E + 00µ 1.547933E + 02 1.215259E + 03 2.537639E + 03σ 1.943776E + 01 1.526064E + 02 3.186650E + 02Λ 2.538120E − 08 1.488407E − 07 1.217002E − 05

Table 7.1: Eigenvalue Coefficients for Perfectly Correlated Masses and Stiffness for the3-DOF System using Fourth-Order Chaos

First Eigenvalue Second Eigenvalue Third Eigenvalueµm 1.548558E + 02 1.215314E + 03 2.537753E + 03σm 1.909155E + 01 1.531909E + 02 3.198850E + 02

Table 7.2: Eigenvalues Mean and Standard Deviation for Perfectly Correlated Masses andStiffness for the 3-DOF System using LHS

167


0 20 40 60 80 100 120 140 160 180 2000

0.005

0.01

0.015

0.02

0.025

0.03

Fundamental Eigenvalue, λ (rad2/s2)

Pro

babi

lity

Den

sity

Fun

ctio

n

Monte Carlo4th PC

Figure 7.2: PDFs of Fundamental Eigenvalue for Perfectly Correlated Masses and Stiffnessfor the 3-DOF System using Fourth-Order Polynomial Chaos Expansion

0 200 400 600 800 1000 1200 1400 16000

0.5

1

1.5

2

2.5

3

3.5x 10

−3


Prob

abili

ty D

ensit

y Fu

nctio

n

Monte Carlo4th PC

Figure 7.3: PDFs of Second Eigenvalue for Perfectly Correlated Masses and Stiffness for the3-DOF System using Fourth-Order Polynomial Chaos Expansion

168


0 500 1000 1500 2000 2500 3000 35000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8x 10

−3


Pro

babi

lity

Den

sity

Fun

ctio

n

Monte Carlo4th PC

Figure 7.4: PDFs of Third Eigenvalue for Perfectly Correlated Masses and Stiffness for the3-DOF System using Fourth-Order Polynomial Chaos Expansion

First Eigenvalue Second Eigenvalue Third Eigenvalueλ0 1.657428E + 02 1.301222E + 03 2.717144E + 03λ1 −3.646515E + 01 −2.862797E + 02 −5.978011E + 02λ2 4.969570E + 01 3.901539E + 02 4.969570E + 01λ3 8.287889E + 00 6.506207E + 01 1.358715E + 02λ4 −1.080380E + 01 −8.481988E + 01 −1.080380E + 01λ5 0.000000E + 00 0.000000E + 00 0.000000E + 00λ6 −1.657519E + 00 −1.301288E + 01 −2.717528E + 01λ7 2.160683E + 00 1.696458E + 01 2.160683E + 00λ8 0.000000E + 00 0.000000E + 00 0.000000E + 00λ9 0.000000E + 00 0.000000E + 00 0.000000E + 00µ 1.657428E + 02 1.301222E + 03 2.717144E + 03σ 6.386933E + 01 5.014263E + 02 1.047049E + 03Λ 6.745344E − 07 1.948552E − 06 1.556611E − 04

Table 7.3: Eigenvalue Coefficients for Uncorrelated Masses and Stiffness for the 3-DOF Sys-tem using Third-Order Chaos

169


First Eigenvalue Second Eigenvalue Third Eigenvalueµm 1.657623E + 02 1.301412E + 03 2.717143E + 03σm 6.368426E + 01 5.014944E + 02 1.050691E + 03

Table 7.4: Eigenvalues Mean and Standard Deviation for Uncorrelated Masses and Stiffnessfor the 3-DOF System using LHS

0 100 200 300 400 500 600 7000

1

2

3

4

5

6

7

8x 10

−3


Prob

abili

ty D

ensi

ty F

unct

ion

Monte Carlo3rd PC

Figure 7.5: PDFs of Fundamental Eigenvalue for Uncorrelated Masses and Stiffness for the3-DOF System using Third-Order Polynomial Chaos Expansion

170


0 1000 2000 3000 4000 5000 6000 7000 80000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1x 10

−3


Prob

abili

ty D

ensi

ty F

unct

ion

Monte Carlo3rd PC

Figure 7.6: PDFs of Second Eigenvalue for Perfectly Correlated Masses and Stiffness for the3-DOF System using Third-Order Polynomial Chaos Expansion

0 2000 4000 6000 8000 10000 12000 140000

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

−4


Prob

abili

ty D

ensi

ty F

unct

ion

Monte Carlo3rd PC

Figure 7.7: PDFs of Third Eigenvalue for Perfectly Correlated Masses and Stiffness for the3-DOF System using Third-Order Polynomial Chaos Expansion

171

7.5 Conclusions

7.5 Conclusions

An algorithm is developed which calculates stochastic eigenvalues of the given system

using implicit secant iteration method. Here the information of the eigenvectors are not

needed. It is applied to a discrete system of a 3 degree-of-freedom model. In this example,

the mass and stiffness matrices are assumed to be Gaussian random variables. This

algorithm can be applied to other random variables easily as well as to random processes

representing structural properties of continuous systems. Eigenvalues of a 3 degree-of-

freedom model obtained using this algorithm are confirmed with LHS results. When this

algorithm is applied to continuous system problems, matrix conditioning problem occurs

even with nondimensionalization of the system. To find eigenvalue polynomial coefficient

vector, optimization techniques were also tried, these techniques also failed to converge.

172

Chapter 8

Karhunen-Loeve Expansion of

Non-Gaussian Random Process

8.1 Introduction

In Chapters 5 and 6, eigenvalues of continuous structures are obtained for exponential

auto-covariance using Karhunen-Loeve (KL) expansion of input Gaussian random pro-

cesses. In practical problems, most of the random processes are Non-Gaussian meaning

that their marginal PDFs have positive real domain which might be finitely bounded.

KL expansion of Gaussian random process is similar to Fourier decomposition in random

space (probability space), it decomposes random process into random variables (Basis

random variables) and deterministic orthogonal functions in real space. Basis random

173

8.1 Introduction

variables of KL expansion [65] of Gaussian random processes are uncorrelated standard

normal variables. Basis random variables when using KL expansion should be indepen-

dent identically distributed random variables with zero mean and unit variance. Uncor-

related standard normal variables are independent, so the question of independency does

not arise in the KL expansion of Gaussian random process. In the case of non-Gaussian

random variables, uncorrelatedness and independence are not equivalent [120].

To obtain KL expansion basis random variables for Non-Gaussian process, marginal den-

sity function’s information should be available. Based upon the available information of

marginal density functions, following thumb-rules should be used to obtain KL expansion

basis random variables [120],

• If the bounds of random variables are known, A random variable with Uniform

distribution should be used.

• If the mean and the variance are available, then a Gaussian distribution should be

chosen.

• If the information about higher moments along with the mean and the variance

is available, then the principle of the maximum entropy should be used to obtain

basis random variables [121].

Recently, efforts has been made to simulate non-Gaussian process using KL expan-

sion [122, 123, 124]. Poirion et al obtained basis random variables of KL expansion

174

8.1 Introduction

using Monte-Carlo simulation. Homogeneous chaos expansion was used to simulate non-

Gaussian random process using the definition of KL expansion of zero mean and unit

variance Gaussian random process [123]. This method requires calculation of expected

value of product of underlying Gaussian random process at two points and this calcula-

tion is done using FEM, see [123]. Phoon et al [124] used the definition of cumulative

distribution function, the correlation matrix as an Identity matrix, and various sampling

techniques to obtain KL expansion basis random variables for Non-Gaussian random pro-

cess. All these methods are either iterative or sampling based techniques, iterations are

carried to match the marginal distribution function. Apart from this, still the question

of independence of these variables remains unanswered. Here, an algorithm is presented

which calculates the KL expansion basis random variables for a Non-Gaussian random

process that are independent. The proposed method carries nonlinear transformation of

the marginal distribution CDF using the standard deviation of the marginal distribution

function which may produce nonzero mean. Another transformation on the obtained

CDF gives our target CDF. This algorithm is applied to available standard cases of ran-

dom variables like Log-Normal, Exponential and Beta distributions. The obtained basis

random variables are compared with analytical solutions defined in terms of either CDF

or PDF which have the mean and the variance as 0 and 1, respectively. This algorithm

is non-iterative and uses available information of marginal distribution function.

175

8.2 Nonlinear Transformation Method


In Nonlinear Transformation method, a nonlinear transformation is applied to obtain

a random variable with unit variance using the definition of standard deviation and

the CDF of the marginal distribution function. Then transformation is carried on this

random variable to render the mean of this random variable to be zero. Using inverse

CDF technique [9], required samples can be generated of these target variables. Following

steps should be carried to obtain KL expansion basis random variable:

• Calculate the standard deviation and the CDF of given random process’ marginal

distribution.

• Calculate new CDF such that

Fnew

(Xori

σ

)= Fori (Xori) (8.1)

xoriσ∫

−∞

fnew

(xori

σ

)dx =

xori∫−∞

fori (xori) dx (8.2)

where Fori and Fnew are CDF of the input parameter’s marginal CDF and normal-

ized marginal CDF using the standard deviation, σ of fori respectively. fori and

fnew are the marginal PDF of input density function and the normalized PDF. This

nonlinear transformation creates random variable with the unit variance and the

mean of this random variable may be nonzero.

176


• Now generate uncorrelated standard normal variables samples and transform these

into independent Uniform distribution samples using the definition of CDF of stan-

dard normal variables.

• Generate independent samples of fnew using CDF inverse technique. Now calculate

the mean, µ of fnew. If µ is nonzero, subtract µ from generated samples and plot

the CDF and the PDF of f ∗new. f ∗new is our basis random variable for K-L expansion.

One can check the independency of these variables here using Eq. 8.3 [125] or look

at the scatter plots of these variables. If a larger/smaller value of one variable is

associated with the larger/smaller value of another variable, then these variables

are dependent, otherwise they are independent; this can be seen easily in the scatter

plot of these variables.

κn (ξ1 + ξ2) = κn (ξ1) + κn (ξ2) (8.3)

where ξ are KL expansion basis random variables and κn is n’th cumulant of these

variables. These cumulants are invariants of the random variable and they are

related to the Characteristic Function as:

φ (u) =∞∫

−∞exp [i u x] fx (x) dx (8.4)

= exp

[∞∑

n=1

κn

n!(i u)n

](8.5)

177


where φ and fx (x) are the characteristic function of the PDF and the PDF of the

random variable, respectively.

This method requires the complete definition of marginal distribution function. These

basis random variables’ are independent and with zero mean and unit variance. As

this method uses the inverse CDF technique, this method is similar to the Rosenblatt

Transformation to obtain independent normal variables from non-normal variables.


To check the accuracy, this algorithm is applied to standard random variables like Log-

Normal, Beta, and Gamma variables. These standard cases have analytical solutions for

marginal distributions with zero mean and unit variance.

8.3.1 Log-Normal Distribution

fx (x | µ, σ) =1

x σ√

2 πexp

(−(ln (x)− µ)2

2 σ2

), σ ≥ 0,−∞ ≤ µ ≤ ∞, x ∈ [0,∞) (8.6)

Equation 8.6 represents the PDF of Log-Normal distribution with input parameters µ

and σ. For this case study, µ and σ are chosen as 0 and 1, respectively. This random vari-

able has the mean and the standard deviation equal to 1.64872 and 2.1612, respectively.

Figure 8.1 represents the CDF and the PDF of this distribution. Nonlinear transforma-

178


tion using the standard deviation and the definition of CDF produces a new CDF which

is shown in Fig. 8.2 along with the original marginal CDF. Using Uniform independent

random numbers and the inverse CDF technique, random numbers are generated to cal-

culate the mean of the transformed CDFs. Since mean of these independent samples are

nonzero, the transformed CDF is shifted using these mean values. Figure 8.3 shows the

PDFs of original distribution and KL expansion basis random variables. To ascertain

the accuracy, the transformed and the shifted CDF is compared with the analytical CDF

having the definition

FY (y | µ, σ, δ) = Φ

(ln (y − δ ) − µ

σ

)(8.7)

where FY is CDF of LogNormal with µ = −0.7707, σ = 1.0, and δ = −0.7628 and Φ is

standard Normal CDF. The comparison of the analytical CDF and the numerical CDF

is shown in Fig. 8.4.

In this case study, 4 KL expansion basis random variables are generated and their inde-

pendency is checked using Eq. 8.3. Two arbitrarily KL basis random variables are chosen

and results of the application of Eq. 8.3 are shown for first 4 cumulants calculations in

Table 8.1. Scatter plots of these variables are shown in Fig. 8.5.

179


2 4 6 8 10 12Log-Normal Variable Domain

0.2

0.4

0.6

0.8

1

CDFPDF

Figure 8.1: CDF and PDF of Non-Gaussian Marginal Log-Normal Distribution

n κn (ξ1 + ξ2) κn (ξ1) + κn (ξ2)1 −8.06340E − 15 −8.06361E − 152 1.99178E + 00 1.99949E + 003 1.04783E + 01 1.06117E + 014 1.08926E + 02 1.11864E + 02

Table 8.1: Independency of KL Expansion Basis Random Variables for Log-Normal Distribu-tion

8.3.2 Beta Distribution

The PDF of standard Beta distribution is given as

fx (x |α, β) =1

B (α, β)xα−1 (1− x)β−1 , α > 0, β > 0, x ∈ [0, 1] (8.8)

Here, both α and β are chosen as 0.5 and B ( ) is the Beta function. The mean and

the standard deviation of this distribution are 0.5 and 0.353553, respectively. Figure 8.6

180



0.2

0.4

0.6

0.8

1evitalu

muCnoitubirtsiD

noitcnuF

Original CDFTransformed CDF

Figure 8.2: CDFs of Marginal Log-Normal Distribution and Transformed Distribution

0 2 4 6 8Log-Normal Variable Domain

0.2

0.4

0.6

0.8

1

1.2

ytilibaborPytisneD

noitcnuF

KL Basis PDFOriginal PDF

Figure 8.3: PDFs of Marginal Log-Normal Distribution and KL Expansion Basis RandomVariable

181



0.2

0.4

0.6

0.8

1evitalu

muCnoitubirtsiD

noitcnuF

Analytical KL Basis CDFNumerical KL Basis CDF

Figure 8.4: Log-Normal CDFs of Analytical and Numerical KL Expansion Basis RandomVariables

-0.75 -0.5 -0.25 0 0.25 0.5 0.75 1Log-Normal Variable, Hx1L

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

goL-

lamroN

elbairaV,Hx

2L


-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

goL-

lamroN

elbairaV,Hx

3L


-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

goL-

lamroN

elbairaV,Hx

4L


-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

goL-

lamroN

elbairaV,Hx 4

L

Figure 8.5: Scatter Plots of Log-Normal KL Expansion Basis Random Variables

182


represents the CDF and the PDF of this distribution. Nonlinear transformation using

the standard deviation and the definition of CDF produces a new CDF which is shown

in Fig. 8.7 along with the original marginal CDF. Using Uniform independent random

numbers and the inverse CDF technique, random numbers are generated to calculate

the mean of transformed CDFs. Since mean of these independent samples are nonzero,

the transformed CDF is shifted using these mean values. Figure 8.8 shows the PDFs of

original distribution and KL expansion basis random variables. To ascertain the accuracy,

the transformed and shifted CDF is compared with the analytical CDF having the PDF

definition as

fx (x |α, β, a, b) =1

B (α, β)

(x− a)α−1 (b− x)β−1

(b− a)α+β−1(8.9)

Equation 8.9 represents the generalized Beta function with domain as [ a, b]. Again, α, β

and the domain are chosen 0.5, 0.5 and [−1.42 , 1.42] respectively, which gives the mean

and the variance as 0 and 1, respectively. The comparison of the analytical CDF and the

numerical CDF is shown in Fig. 8.9.

In this case study, 4 KL expansion basis random variables are generated and their in-

dependency is checked using Eq. 8.3. Two KL expansion basis random variables are

chosen arbitrarily and results of the application of Eq. 8.3 are shown for first 4 cumu-

lants calculations in Table 8.2. Scatter plots of these variables are shown in Fig. 8.10

183


0.2 0.4 0.6 0.8 1Beta Variable Domain

0.5

1

1.5

2

2.5

3

3.5

CDFPDF

Figure 8.6: CDF and PDF of Non-Gaussian Marginal Beta Distribution

n κn (ξ1 + ξ2) κn (ξ1) + κn (ξ2)1 −5.26490E − 15 −5.26495E − 152 1.99904E + 00 1.99889E + 003 5.25263E − 03 2.05754E − 034 −2.99033E + 00 −2.99544E + 00

Table 8.2: Independency of KL Expansion Basis Random Variables for Beta Distribution

8.3.3 Exponential Distribution

The PDF of Exponential distribution is given as

fx (x |λ) = λ exp (−λ x) , x ∈ [0, ∞) λ > 0 (8.10)

Here, λ is chosen as 1.5. The mean and the standard deviation of this distribution are

2/3 and 2/3 respectively. Figure 8.11 shows the CDF and the PDF of this distribution.

184


0.5 1 1.5 2 2.5Beta Variable Domain

0.2

0.4

0.6

0.8

1evitalu

muCnoitubirtsiD

noitcnuF


Figure 8.7: CDFs of Marginal Beta Distribution and Transformed Distribution

-1 -0.5 0 0.5 1Beta Variable Domain

0.5

1

1.5

2

2.5

ytilibaborPytisneD

noitcnuF


Figure 8.8: PDFs of Marginal Beta Distribution and KL Expansion Basis Random Variable

185


-1 -0.5 0 0.5 1Beta Variable Domain

0.2

0.4

0.6

0.8

1evitalu

muCnoitubirtsiD

noitcnuF


Figure 8.9: Beta CDFs of Analytical and Numerical KL Expansion Basis Random Variables

-1 -0.5 0 0.5 1BetaVariable, Hx1L

-1

-0.5

0

0.5

1

ateBelbairaV,Hx

2L


-1

-0.5

0

0.5

1

ateBelbairaV,Hx

3L


-1

-0.5

0

0.5

1

ateBelbairaV,Hx

4L


-1

-0.5

0

0.5

1

ateBelbairaV,Hx

4L

Figure 8.10: Scatter Plots of Beta KL Expansion Basis Random Variables

186


Nonlinear transformation using the standard deviation and the definition of CDF pro-

duces a new CDF which is shown in Fig. 8.12 along with the original marginal CDF.

Using Uniform independent random numbers and the inverse CDF technique, random

numbers are generated to calculate the mean of the transformed CDFs. Since mean of

these independent samples are nonzero, the transformed CDF is shifted using these mean

values. Figure 8.13 shows the PDFs of original distribution and KL expansion basis ran-

dom variables. To ascertain the accuracy, the transformed and shifted CDF is compared

with the analytical CDF having the PDF definition as

F (y | µ, λ ) = 1− exp (−λ (y − µ)) , λ = 1, µ = −1 (8.11)

Equation 8.11 represents the shifted Exponential function with domain as [−1, ∞]. This

distribution has zero mean and unit variance. The comparison of the analytical CDF

and the numerical CDF is shown in Fig. 8.14.

In this case study, 4 KL expansion basis random variables are generated and their in-

dependency is checked using Eq. 8.3. Two KL expansion basis random variables are

arbitrarily chosen and results of the application of Eq. 8.3 are shown for first 4 cumu-

lants calculations in Table 8.3. Scatter plots of these variables are shown in Fig. 8.15

187

8.4 Conclusions

CDFPDF

1 2 3 4 5 6Exponential Variable Domain

0.20.40.60.8

11.21.41.6

CDFPDF

Figure 8.11: CDF and PDF of Non-Gaussian Marginal Exponential Distribution

n κn (ξ1 + ξ2) κn (ξ1) + κn (ξ2)1 7.45081E − 15 7.45095E − 152 2.00689E + 00 2.00317E + 003 4.04189E + 00 4.01956E + 004 1.22226E + 01 1.20717E + 01

Table 8.3: Independency of KL Expansion Basis Random Variables for Exponential Distribu-tion

8.4 Conclusions

In this work, a non-iterative method is presented which calculates KL expansion basis

random variables for Non-Gaussian random processes. For Non-Gaussian random pro-

cess, KL expansion should have independent random variables with zero mean and unit

variance. This particular requirement is satisfied. In this method, nonlinear transfor-

mation is applied to the marginal distribution function of given random process using

188

8.4 Conclusions


0.2

0.4

0.6

0.8

1evitalu

muCnoitubirtsiD

noitcnuF


Figure 8.12: CDFs of Marginal Exponential Distribution and Transformed Distribution


0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

ytilibaborPytisneD

noitcnuF


Figure 8.13: PDFs of Marginal Exponential Distribution and KL Expansion Basis RandomVariable

189

8.4 Conclusions



0.2

0.4

0.6

0.8

1evitalu

muCnoitubirtsiD

noitcnuF


Figure 8.14: Exponential CDFs of Analytical and Numerical KL Expansion Basis RandomVariables

0 1 2 3 4 5ExponentialVariable, Hx1L

0

1

2

3

4

5

laitnenopxEelbairaV,Hx

2L

0 1 2 3 4 5Exponential Variable, Hx2L

0

1

2

3

4

5


3L


0

1

2

3

4

5


4L


0

1

2

3

4

5


4L

Figure 8.15: Scatter Plots of Exponential KL Expansion Basis Random Variables

190

8.4 Conclusions

the CDF definition and the standard deviation of the marginal distribution. Independent

Uniform random numbers are generated from uncorrelated standard Normal samples. In-

verse CDF technique is used to generate KL expansion basis random variables with zero

mean and unit variance. This algorithm is applied to standard cases of distributions such

as Log-Normal, Beta, and Exponential. These distributions have analytical solutions for

KL expansion variables. KL expansion basis variables obtained through this algorithm

is compared to these analytical solutions and it is found that this numerical method’s

solutions agree well with analytical solutions. This method requires the definition of

marginal distribution function in terms of CDF and the standard deviation, so it can be

applied to any non-Gaussian random process where the marginal distribution function

definition is not available in terms of analytical formula. As, Nonlinear Transformation

method is non-iterative and samples are generated only once during this algorithm, so it

more efficient as compared to other methods. It can be applied to both the intrusive and

the non-intrusive polynomial chaos methods. In the case of the intrusive method, mo-

ments of samples are required which can be calculated easily. The explained procedure to

generate independent random numbers can be used in the non-intrusive polynomial case

where more than one non-Gaussian variables are used. In future research, this algorithm

can be applied to SSFEM where input random processes are non-Gaussian.

191

Chapter 9

Future Research in Uncertain

Dynamic Problems

This is the beginning of application of polynomial chaos in dynamical systems. Till now,

polynomial chaos has been applied to only isotropic materials. So its application to

composites will be interesting. Recently stochastic optimization using polynomial chaos

is developed for simple systems [126, 127], its application to complex (huge) systems is yet

to be determined. We can use non-intrusive methods can be used to get the idea about

the reliability of these systems till intrusive algorithms are developed. This author feels

that non-intrusive methods, although increasingly popular due to ease of use, will yield

inaccurate higher polynomial coefficients as compared to intrusive polynomial chaos. So

there is a need to develop non-intrusive methods that yield accurate higher polynomial

chaos coefficients.

192

• Comparison of intrusive polynomial chaos and non-intrusive polynomial chaos should

be done in terms of accuracy of higher polynomial chaos coefficients for static and

dynamic systems. Number of samples required for non-intrusive polynomial chaos

for certain amount of accuracy should be quantified for different sampling schemes.

• Application of polynomial chaos to both aeroelasticity and fluid-structure problems,

is imminent. Polynomial chaos application to such fields which have uncertainty in

aerodynamics and structural systems definitely will be interesting.

• An intrusive algorithm to find complex stochastic eigenvalues and eigenvectors will

be an extension of the algorithm to find real eigenvalues.

• Most of the time, polynomial chaos is applied to Gaussian random processes.

Some of the system processes are non-Gaussian which represents either positive

or bounded system parameters. Application of non-Gaussian processes may result

into uni-modal or multi-modal PDFs of response [122].

• For complex and huge systems, parallel programming or grid computing should

be used. For non-intrusive polynomial case, it seems to be easy as compared to

intrusive polynomial chaos.

193

Bibliography

[1] Walters, R. W., and Huyse, L., Uncertainty Analysis for Fluid Mechanics with Ap-

plications, ICASE Report 2002-1/NASA CR 2002-211449, 2002.

[2] Laplace, P. S. M., A Philosophical Essay on Probabilities [microform], translated from

the 6th French edition by F. W. Truscott and F. L. Emory, John Wiley, New York,

1902.

[3] Sobczyk, K., Stochastic Differential Equations, Kluwer Academic Publishers, Dor-

drecht/Boston/London, 1991, pp. 1-5.

[4] Wiener, N., “The Homogeneous Chaos,” American Journal of Mathematics, Vol. 60,

No. 4, 1938, pp. 897-936.

[5] Vanmarcke, E. H., Random Fields: Analysis and Synthesis, The MIT Press, Cam-

bridge, Massachusetts, London, 1983.

[6] Madsen, H. O., Krenk, S., and Lind, N. C., Methods of Structural Safety, Prentice

Hall Inc, Englewood Cliffs, New Jersey, 1986.

194

BIBLIOGRAPHY

[7] Vanmarcke, E. H., and Grigoriu, M. “Stochastic Finite Element Analysis of Simple

Beams,” Journal of the Engineering Mechanics Division, ASCE, Vol. 109, No.EM5,

1983, pp.1203-1214.

[8] Kleiber, M., and Hein, T. D., The Stochastic Finite Element Method, Basic Per-

turbation Technique and Computer Implementation, John Wiley & Sons, New York,

1992.

[9] Haldar, A., and Mahadevan, S., Reliability Assessment Using Stochastic Finite Ele-

ment Analysis, John Wiley & Sons, New York, 2000.

[10] Ghanem, R. G., and Spanos, P. D., Stochastic Finite Elements, A Spectral Approach,

Dover Publications Inc, New York, 2003.

[11] Oden, J. T., Belytschko, T., Babuska, I., and Hughes, T. J. R., “Research Direc-

tions in Computational Mechanics,” Computer Methods in Applied Mechanics and

Engineering, Vol. 192, No. 7-8, 2003, pp. 913-922.

[12] Oberkampf, W. L., Helton, J. C., and Sentz, K., “Mathematical Representation of

Uncertainty,” 42nd AIAA/ASME/ASCE/ASC Structures, Structural Dynamics, and

Materials Conference, Seattle, WA, AIAA Paper 2001-1645, 16-19 April 2001.

[13] Yang, T. Y., and Kapania, R. K., “Finite Element Random Response Analysis of

Cooling Tower,” Journal of Engineering Mechanics, ASCE, Vol. 110, No. EM4, 1984,

pp. 589-609.

195

BIBLIOGRAPHY

[14] Wirsching, P. H., Paez, T. L., and Ortiz, K., Random Vibrations : Theory and

Practice, John Wiley & Sons, New York, 1995.

[15] Lin, Y. K., Probabilistic Theory of Structural Dynamics, Robert E. Krieger Publish-

ing Company, New York, 1976.

[16] Ang, A. H. S., and Tang, W. H., Probability Concepts in Engineering Planning and

Design, Volume II Decision, Risk, and Reliability, John Wiley & Sons, Inc., New

York, 1984, 274-326.

[17] Haldar, A., and Mahadevan, S., Probability, Reliability and Statistical Methods in

Engineering Design, John Wiley & Sons, Inc., New York, 2000.

[18] Wu, Y.-T., Millwater, H. R., and Cruse, T. A., “Advanced Probabilistic Structural

Analysis Method for Implicit Performance Functions”, AIAA Journal, Vol. 28, No.

9, 1990, pp. 1663-1669.

[19] Riha, D., Millwater, H., and Thacker, B., “Probabilistic Structural Analysis using a

General Purpose Finite Element Program”, Finite Elements in Analysis and Design,

Vol. 11, 1992, pp. 201-211.

[20] Riha, D. S., Thacker, B. H., Hall, D. A., Auel, T. R., and Pritchard, S. D., “Capa-

bilities and Applications of Probabilistic Methods in Finite Element Analysis”, Fifth

ISSAT International Conference on Reliability and Quality in Design, Las Vegas,

Nevada, August 11-13, 1999.

196

BIBLIOGRAPHY

[21] Thacker, B. H., Riha, D. S., Millwater, H. R., and Enright, M. P., “Errors and Un-

certainties in Probabilistic Engineering Analysis”, 42nd AIAA/ASME/ASCE/ASC

Structures, Structural Dynamics, and Materials Conference, Seattle, WA, AIAA Pa-

per, 2001-1239, 16-19, April 2001.

[22] Rajashekhar, M. R., and Ellingwood, B. R., “A New Look at the Response Surface

Approach for Reliability Analysis”, Structural Safety, Vol. 12, 1993, pp. 205-220.

[23] Zheng, Y., and Das, P. K., “Improved Response Surface Method and its Application

to Stiffened Plate Reliability Analysis”, Engineering Structures, Vol.22, 2000, pp.

544-551.

[24] Kmiecik, M., and Soares, C. G., “Response Surface Approach to the Probability

Distribution of the Strength of Compressed Plates”, Marine Structures, Vol. 15, 2002,

pp. 139-156.

[25] Harbitz, A., “An Efficient Sampling Method for Probability of Failure Calculation”,

Structural Safety, Vol. 3, 1986, pp. 109-115.

[26] Bjerager, P., “Probability Integration by Direct Simulation”, Journal of Engineering

Mechanics, Vol. 114, No. 8, 1988, pp. 1285-302.

[27] McKay, M. D., Conover, W. J., and Beckman, R. J., “A comparison of three methods

for selecting values of input variables in the analysis of output from a computer code”,

Technometrics, Vol. 21, No. 2, 1979, pp. 239-245.

197

BIBLIOGRAPHY

[28] Spanos, P. D., and Ghanem, R. G., “Stochastic Finite Element Expansion for Ran-

dom Media,” Journal of Engineering Mechanics, ASCE, Vol. 115, No. 5, 1989, pp.

1035-1053.

[29] Spanos, P. D., and Ghanem, R. G., “Boundary Element Formulation for Random

Vibration Problems,” Journal of Engineering Mechanics, ASCE, Vol. 117, No. 2,

1991, pp. 409-423.

[30] Ghanem, R. G., and Spanos, P. D., “Polynomial Chaos in Stochastic Finite Ele-

ments,” Journal of Applied Mechanics, ASME, Vol. 57, No. 1, 1990, pp. 197-202.

[31] Ghanem, R. G., and Kruger, R. M., “Numerical Solution of Spectral Stochastic

Finite Element Systems,” Computer Methods in Applied Mechanics and Engineering,

Vol. 129, No. 3, 1996, pp. 289-303.

[32] Ghanem, R. G., “Ingredients for a General Purpose Stochastic Finite Elements

Implementation,” Computer Methods in Applied Mechanics and Engineering, Vol.

168, No. 1-4, 1999, pp. 19-34.

[33] Xiu, D., and Karniadakis, G. E., “The Wiener-Askey Polynomial Chaos for Stochas-

tic Differential Equations,” SIAM Journal on Scientific Computing, Vol. 24, No. 2,

2003, pp. 619-644.

[34] Rao, S. S., and Berke, L., “Analysis of Uncertain Structural Systems Using Interval

Analysis”, AIAA Journal, Vol. 35, No. 4, 1997, pp. 727-735.

198

BIBLIOGRAPHY

[35] Ben-Haim, Y., and Elishakoff, I., Convex Models of Uncertainty in Applied Mechan-

ics, Elsevier Science, Amsterdam, 1990.

[36] Nikolaidis, E., Chen, S., Cudney, H., Haftka, R. T., and Rosca, R., “Comparison

of Probability and Possibility for Design Against Catastrophic Failure Under Uncer-

tainty”, Transactions of the ASME, Vol. 126, 2004, pp. 386-394.

[37] Bae, H.-R., Grandhi, R. V., and Canfield, R. A., “Epistemic Uncertainty Quantifi-

cation Techniques including Evidence Theory for Large-Scale Structures”, Computers

and Structures, Vol. 82, 2004, pp. 1101-1112.

[38] Moore, R. E., Methods and Application of Interval Analysis, SIAM, Philadelphia,

1979.

[39] Kaufmann, A., and Gupta, M. M., Introduction to Fuzzy Arithmetic, Van Nostrand

Reinhold Company, New York, 1984.

[40] Ayyub, B. M., Uncertainty Modeling and Analysis in Civil Engineering, CRC Press,

Washington D. C., 1998.

[41] Shafer, G., A Mathematical Theory of Evidence, Princeton Univ. Press, Princeton,

New Jersey, 1976.

[42] Thacker, B., and Huyse, L., “Probabilistic Assessment on the Basis of Interval

Data”, 44th AIAA/ASME/ASCE/ASC Structures, Structural Dynamics, and Ma-

terials Conference, Norfolk, VA, AIAA Paper, 2003-1753, 1-12, April 2003.

199

BIBLIOGRAPHY

[43] Bae, H.-R., Grandhi, R. V., and Canfield, R. A., “Uncertainty Quantification of

Structural Response Using Evidence Theory”, AIAA Journal, Vol. 41, No. 10, 2003,

pp. 2062-2068.

[44] Soundappan, P., Nikolaidis, E., Haftka, R. T., Grandhi, R., and Canfield, R., “Com-

parison of Evidence Theory and Bayesian Theory for Uncertainty Modeling”, Relia-

bility Engineering and System Theory, Vol. 85, No. , 2004, pp. 295-311.

[45] Rao, S. S., and Sawyer, J. P., “Fuzzy Finite Element Approach for the Analysis

of Imprecisely-Defined Systems”, 37th AIAA/ASME/ASCE/ASC Structures, Struc-

tural Dynamics, and Materials Conference, Reston, VA, AIAA Paper 1996-1610, 15-17

April 1996.

[46] Fuzzy Logic Toolbox for Use with MATLABR©, Users Guide Version 2.2.2, C© The

MathWorks, Inc., September 2005.

[47] Ross, T. J., Fuzzy Logic with Engineering Applications, McGraw Hill, Inc., New

York, 1995.

[48] Muhanna, R. L., and Mullen, R. L., “Formulation of Fuzzy Finite-Element Methods

for Solid Mechanics Problems”, Computer-Aided Civil and Infrastructure Engineering,

Vol. 14, 1999, pp. 107-117.

[49] Savoia, M, “Structural Reliability Analysis through Fuzzy Number Approach, with

Application to Stability”, Computers and Structures, Vol. 80, 2002, pp. 1087-1102.

200

BIBLIOGRAPHY

[50] Mourelatos, Z. P., and Zhou, J., “Reliability Estimation and Design with Insufficient

Data Based on Possibility Theory”, 10th Multidisciplinary Analysis and Optimization

Conference, Albany, New York, AIAA Paper, 2004-4586, 1-16, 30 August-1 September

2004.

[51] Noor, A. K., Jr. Starnes, J. H., and, Peters, J. M., “Uncertainty Analysis of Com-

posite Structures”, Computer Methods in Applied Mechanics and Engineering, Vol.

185, 2000, pp. 413-432.

[52] Noor, A. K., Jr. Starnes, J. H., and, Peters, J. M., “Uncertainty Analysis of Stiffened

Composite Panels”, Composite Structures, Vol. 51, 2001, pp. 139-158.

[53] Sentz, K., and Ferson, S. “Combination of Evidence in Dempster-Shafer Theory”,

SANDIA Report, SAND2002-0835, 2002.

[54] Dempster, A. P., “Upper and Lower Probabilities Induced by a Multivalued Map-

ping”, The Annals of Statistics, Vol. 38, No. 2, 1967, pp. 325-339.

[55] Bae, H.-R., “Uncertainty Quantification and Optimization of Structural Response

using Evidence Theory”, Ph. D. Dissertation, Wright State University, 2004.

[56] Papoulis, A, Probability, Random Variables, and Stochastic Processes, McGraw-Hill

Inc., New York, Third Edition, 1991.

201

BIBLIOGRAPHY

[57] Sudret, B., and Der Kiureghian, A, “Stochastic Finite Element Methods and Reli-

ability, A State-of-the-Art Report”, Report No. UCB/SEMM-2000/08, Department

of Civil & Environmental Engineering, University of California, Berkeley, 2000.

[58] Rosenblatt, M., “Remarks on a Multivariate Transformation”, The Annals of Math-

ematical Statistics, Vol. 23, No. 3, 1952, pp. 470-472.

[59] Simpson, T. W., Peplinski, J. D., Koch, P. N, and Allen, J. K., “Metamodels for

Computer-based Engineering Design: Survey and Recommendations”, Engineering

with Computers, Vol. 17, 2001, pp. 129-150.

[60] Gomes, H. M., and Awruch, A. M., “Comparison of Response Method and Neural

Network with Other Methods for Structural Reliability Analysis”, Structural Safety,

Vol. 26, 2004, pp. 49-67.

[61] Handa, K, and Anderson, K.,“Application of Finite Element Methods in the Sta-

tistical Analysis of Structures”, Proceedings of ICOSSAR-81, the 3rd International

Conference on Structural Safety and Reliability, 1981, pp. 409-420.

[62] Hisada, T. and Nakagiri, S., “Stochastic Finite Element Method Developed for Struc-

tural Safety and Reliability”, Proceedings of ICOSSAR-81, the 3rd International Con-

ference on Structural Safety and Reliability, 1981, pp. 395-408.

[63] Liu, W.-K., Belytschko, T., and Mani, A., “ Random Field Finite Elements”, Inter-

national Journal for Numerical Methods in Engineering, Vol. 23, No. 10, 1986, pp.

1831-1845.

202

BIBLIOGRAPHY

[64] Phoon, K., Quek, S., Chow, Y., and Lee, S., “Reliability Analysis of Pile Settle-

ments”, Journal of Geotechnical Engineering, ASCE, Vol. 116, No. 11, 1990, pp.

1717-1735.

[65] Loeve, M., Probability Theory, 4th ed., Springer-Verlag, 1977.

[66] Choi, S-K., Grandhi, R. V., and Canfield, R. A., “Structural Reliability under Non-

Gaussian Stochastic Behavior,” Computers and Structures, Vol. 82, No. 13-14, 2004,

pp. 1113-1121.

[67] Olsson, A., Sandberg, G., and Dahlbom, O., “On Latin Hypercube Sampling for

Structural Reliability Analysis,” Structural Safety, Vol. 25, No. 1, 2003, pp. 47-68.

[68] Melchers, R. E., “Importance Sampling in Structural System”, Structural Safety,

Vol. 6, 1989, pp. 3-10.

[69] Haskin, F. E., Staple, B. D., and Ding, C., “Efficient Uncertainty Analyses using

Fast Probability Integration”, Nuclear Engineering and Design, Vol. 166, 1996, pp.

225-248.

[70] Haykin, S., Neural Networks, Prentice Hall Inc, Upper Saddle River, New Jersey,

1999.

[71] Deng, J.,Yue, Z. Q., Tham, L. G., and Zhu, H. H., “Pillar Design by Combining

Finite Element Methods, Neural Networks and Reliability: A Case Study of the

203

BIBLIOGRAPHY

Feng Huangshan Copper Mine, China”, International Journal of Rock Mechanics and

Mining Sciences, Vol. 40, 2003, pp. 585-599.

[72] Kompella M., and Bernhard R. J., “Measurement of the Statistical Variation of

Structural-Acoustic Characteristics of Automotive Vehicles”, SAE Noise and Vibra-

tion Conference, 931272, Traverse City, 1993.

[73] Kompella M., and Bernhard R. J. “Variation of Structural-Acoustic Characteristics

of Automotive Vehicles”, Noise Control Engineering Journal, Vol. 44, No. 2, 1996,

pp. 93-99.

[74] Moeller, M. J., and Lenk P., “NVH CAE Quality Metrics”, SAE Paper#1999-01-

1791, 1999, pp. 1077-1085.

[75] Bathe, K. J., Finite Element Procedures in Engineering Analysis, New Jersey,

Prentice-Hall, 1982.

[76] Huebner, K. H.,and Thornton, E. A. The Finite Element Method for Engineers, New

York, John Wiley & Sons, Second Edition, 1995.

[77] Cook, R. D., Malkus, D. S., and Plesha, M. E., Concepts and Applications of Finite

Element Analysis, New York, John Wiley & Sons, Third Edition, 1989.

[78] Burton, A. J., and Miller, G. F. “The Application of Integral Equation Methods to

the Numerical Solutions of Some Exterior Boundary Value Problems”, Proceedings

of Royal Society of London, Vol. 323, 1971, pp. 201-210.

204

BIBLIOGRAPHY

[79] Chertock, G., “Sound Radiation from Vibrating Surfaces”, The Journal of the Acous-

tical Society of America, Vol. 36, No. 7, 1964, pp. 1305-1313.

[80] Koopman, G. H., and Benner H., “Method for Computing the Sound Power of

Machines Based on the Helmholtz Integral”, The Journal of the Acoustical Society of

America, Vol. 71, No.1, 1982, pp. 77-89.

[81] Schenck, H. A., “Improved Integral Formulation for Acoustic Radiation Problems”,

The Journal of the Acoustical Society of , Vol. 44, 1968, pp. 41-58.

[82] Gockel, M. A., “MSC/NASTRAN Handbook for Dynamic Analysis”, Version 63,

The MacNeal-Schwendler Corporation, 1983.

[83] Seminar Notes, Design Sensitivity and Optimization in MSC/NASTRAN. The Mac-

Neal - Schwendler Corporation, 1993.

[84] Cunefare, K. A., and Koopmann, G. H. “Acoustic Design Sensitivity for Structural

Radiators”, Transactions of the ASME, Vol. 114, 1992, pp. 178-186.

[85] Lee, D. H., “Design Sensitivity Analysis and Optimization of an Engine Mount Sys-

tem using an FRF-Based Substructuring Method”, Journal of Sound and Vibration,

Vol. 255, No. 2, 2002, pp. 383-397.

[86] Allen, M. J., Sbragio, R., and Vlahopoulos, N., “Structural/Acoustic Sensitivity

Analysis of a Structure Subjected to Stochastic Excitation”, AIAA Journal, Vol. 39,

No. 7, 2001, pp. 1270-1279.

205

BIBLIOGRAPHY

[87] Ohayon, R., and Soize, C., Structural Acoustics and Vibration, Academic Press

Limited, San Diego, CA, 1998.

[88] Doddazio, R. and Ettouney, M., “Boundary Element Methods in Probabilistic

Acoustic Radiation Problems”, Journal of Vibration, Acoustics, Stress, and Relia-

bility in Design, Vol. 112, No. 4, 1990, pp. 556-560.

[89] Allen, M. J., and Vlahopoulos, N., “Numerical Probabilistic Analysis of Struc-

tural/Acoustic Systems”, Mechanics of Structures and Machines, Vol. 30, No. 3, 2002,

pp. 353-380.

[90] Seybert, A. F., Soenarko, B., Rizzo, F. J., and Shippy, D. J. “Application of the BIE

Method to Sound Radiation Problems Using an Isoparametric Element”, Journal of

Vibration, Acoustics, stress and Reliability in Design, Vol. 106, 1984, pp. 414-420

[91] Ciskowski, R. D., and Brebbia, C. A., Boundary Element Methods in Acoustics,

Computational Mechanics Publications, and Elsevier Applied Science, Southampton,

Boston, London, New York, 1991.

[92] Vlahopoulos, N., “A Numerical Structure-Borne Noise Prediction Scheme Based on

the Boundary Element Method with a New Formulation for the Singular Integrals”,

Computers & Structures, Vol. 50, No. 1, 1994, pp. 97-109.

[93] Allen, M. J., and Vlahopoulos, N.,“ Noise Generated from a Flexible and Elastically

Supported Structure Subject to Turbulent Boundary Layer Flow Excitation”, Finite

Elements in Analysis and Design, Vol. 37, 2001, pp. 687-712.

206

BIBLIOGRAPHY

[94] Hubbard, H. H., Aeroacoustic of Flight Vehicles, Theory and Practice, Volume 2:

Noise Control, Acoustical Society of America, Woodbury, New York, 1995, pp. 271-

335.

[95] Junger, M. C., “Shipboard Noise: Sources, Transmission, and Control”, Noise Con-

trol Engineering Journal, Vol. 34, No. 1, 1989, pp. 3-8.

[96] Zheng, H., Liu, G. R., Tao, J. S., and Lam, K. Y., “FEM/BEM Analysis of Diesel

Piston-Slap Induced Hull Vibration and Underwater Noise”, Applied Acoustics, Vol.

62, 2001, pp. 341-358.

[97] Blanchet, A., Chatel, G., and Paradis, A., “Study of Structure-Borne Noise Trans-

mission Inside Cabines by Sound-Intensity Measurements”, Proceedings of the 2nd

International Symposium on Shipboard Acoustics, 1986, Hague, Netherlands, 7-9

October, pp. 377-392.

[98] Vethecan, J. K., and Wood, L. A., “Modeling of Structure-Borne Noise in Vehicles

with Four-Cylinder Motors”, International Journal of Vehicle Design, Vol. 8, No. 4-6,

1987, pp. 439-454.

[99] Kim, S. H., Lee, M. J., and Sung, M. H., “Structural-Acoustic Modal Coupling

Analysis and Application to Noise Reduction in a Vehicle Passenger Compartment”,

Journal of Sound and Vibration, Vol. 225, No. 5, 1999, pp. 989-999.

207

BIBLIOGRAPHY

[100] Lim, T. C., “Automotive Panel Noise Contribution Modeling Based on Finite El-

ement and Measured Structural-Acoustic Spectra”, Applied Acoustics, Vol. 60, 2000,

pp. 505-519

[101] Philippidis, T. P., Lekou, D. J., and Aggelis, D. G., “Mechanical Property Distri-

bution of CFRP Filament Wound Composites”, Composite Structures, Vol. 45, 1999,

pp. 41-50.

[102] Cederbaum, G., Elishakoff, I., and Librescu, L., “Reliability of Laminated Plates

via the First-Order Second Moment Method”, Composite Structures, Vol. 15, 1990,

pp. 161-167.

[103] Jeong, H. K., and Shenoi, R. A., “Reliability Analysis of Mid-Plane Symmetric

Laminated Plates using Direct Simulation Method”, Composite Structures, Vol. 43,

1998, pp. 1-13.

[104] Lin, S. C., Buckling Failure Analysis of Random Composite Laminates subjected

to Random Loads”, International Journal of Solids and Structures, Vol. 37, 2000, pp.

7563-7576.

[105] Mahadevan, S., and Liu, X., “Probabilistic Analysis of Composite Structure Ulti-

mate Strength”, AIAA Journal, Vol. 40, No. 7, 2002, pp. 1408-1414.

[106] Oh, D. H., and Librescu, L., “Free Vibration and Reliability of Composite Can-

tilevers featuring Uncertain Properties”, Reliability Engineering and System Safety,

Vol. 56, 1997, pp. 265-272.

208

BIBLIOGRAPHY

[107] Shiao, M. C., and Chamis, C. C., “Probabilistic Evaluation of Fuselage-Type Com-

posite Structures”, Probabilistic Engineering Mechanics, Vol. 14, 1999, pp. 179-187.

[108] Liaw, D. G., and Yang, H. T. Y., “Reliability and Nonlinear Supersonic Flutter of

Uncertain Laminated plates”, AIAA Journal, Vol. 31, No. 12 , 1993, pp. 2304-2311.

[109] Hwang, Y. F., Kim, M., and Zoccola, P. J., “Acoustic Radiation by Point-or Line-

Excited Laminated Plates”, Journal of Vibration and Acoustics, Vol. 122, 2000, pp.

189-195.

[110] Ghinet, S., and Atalla, N., “Vibro-Acoustic Behavior of Multi-Layer Orthotropic

Panels”, Canadian Acoustics, Vol. 30, No. 3, 2002, pp. 72-73.

[111] Eslimy-Isfahany, S. H. R., and Banerjee, J. R., “Response of Composite Beams

to Deterministic and Random Loads”, 37th AIAA/ASME/ASCE/ASC Structures,

Structural Dynamics, and Materials Conference, Seattle, WA, AIAA Paper, 1996-

1411, 15-17, April 1996.

[112] Wallace, C. E., “Radiation Resistance of a Rectangular Panel”, The Journal of

Acoustical Society of America, Vol. 51, No. 3(2), 1972, pp. 946-952.

[113] Collins, J. D., and Thomson, W. T., “The Eigenvalue Problem for Structural Sys-

tems with Statistical Properties,” AIAA Journal, Vol. 7, No. 4, 1969, pp. 642-648.

[114] Shinozuka, M., and Astill, C. J., “Random Eigenvalue Problems in Structural Anal-

ysis,” AIAA Journal, Vol.10, No. 4, 1972, pp. 456-462.

209

BIBLIOGRAPHY

[115] Kapania, R. K., and Goyal, V. K., “Free Vibration of Unsymmetrically Laminated

Beams having Uncertain Ply Orientations,” AIAA Journal, Vol. 40, No. 11, 2002, pp.

2336-2344.

[116] Xiu, D., Lucor, D., Su, C-H., and Karniadakis, G. E., “Stochastic Modeling of

Flow-Structure Interactions Using Generalized Polynomial Chaos,” Journal of Fluids

Engineering-Transactions of the ASME, Vol. 124, No.1, 2002, pp. 51-59.

[117] Bathe, K. L. and Wilson, E. L., Numerical Methods in Finite Element Analysis,

Prentice-Hall, Englewood Cliffs, New Jersey, 1976, pp. 431-436.

[118] Bathe, K. L. and Ramaswamy, S., “ An Accelerated Subspace Iteration Method,”

Computer Methods in Applied Mechanics and Engineering, Vol. 23, No. 3, 1980, pp.

313-331.

[119] Mulani, S. B., Kapania, R. K., and Walters, R. W., “Stochastic Eigenvalue Prob-

lem with Polynomial Chaos,” 47th AIAA/ASME/ASCE/ASC Structures, Structural

Dynamics, and Materials Conference, Newport, RI, AIAA Paper 2006-2068, 1-4 May

2006.

[120] Keese, A., “A Review of Recent Developments in the Numerical Solution of Stochas-

tic Partial Differential Equations (Stochastic Finite Elements)”, Report No. 2003-

06, Department of Mathematics and Computer Science, Technical University Braun-

schweig, Brunswick, Germany, 2003.

210

BIBLIOGRAPHY

[121] Jaynes, E. T., Probability Theory: The Logic of Science, Cambridge University

Press, Cambridge, 2003.

[122] Poirion, F., and Soize, C., “Monte Carlo Construction of Karhunen Loeve Expan-

sion for Non-Gaussian Random Fields,” Engineering Mechanics Conference, Balti-

more, MD, 13-16 June 1999.

[123] Sakamoto, S., and Ghanem, R., “Polynomial Chaos Decomposition for the Sim-

ulation Non-Gaussian Nonstationary Stochastic Processes”, Journal of Engineering

Mechanics, Vol. 128, No. 2, 2002, pp. 190-201.

[124] Phoon, K. K., Huang, H. W., and Quek, S. T., “Simulation of Strongly Non-

Gaussian Processes using Karhunen-Loeve Expansion”, Probabilistic Engineering Me-

chanics, Vol. 20, No. 2, 2005, pp. 188-198.

[125] Haberman, S. J., Advanced Statistics, Volume I: Description of Populations,

Springer-Verlag, Springer Series in Statistics, Berlin, 1996.

[126] Choi, S. K., Grandhi, R. V., and Canefield, R. A., “Optimization of Stochas-

tic Mechanical Systems using Polynomial Chaos Expansion,” 10th AIAA/ISSMO

Multidisciplinary Analysis and Optimization Conference, Albany, NY, AIAA Paper

2004-4590, 30 August-1 September 2004.

[127] Kim, N. H., Wang, H., and Quiepo, N. V., “Effcient Shape Optimization under Un-

certainty using Polynomial Chaos Expansions and Local Sensitivities,”AIAA Journal,

Technical Note, Vol. 44, No. 5, 2006, pp. 1112-1115.

211

Vita

Sameer Babasaheb Mulani was born in Sangli, Maharashtra, India on March 26, 1975,

the eldest son of Babasaheb Abdul Mulani and Shamshad Babasaheb Mulani. After com-

pleting his high school education at His Highness Raja Chintamanrao Patwardhan High

School, Sangli, India in 1990, he entered Wilingdon College of Sangli and passed success-

fully Maharashtra State Board of Higher Secondary Education examination in 1992. He

pursued his Bachelor’s degree in Civil Engineering from Walchand College of Engineer-

ing, Shivaji University, Sangli. He entered the Department of Aerospace Engineering at

the Indian Institute of Technology Bombay (IIT Bombay), Mumbai, Maharashtra, India

in 1998. In July, 2000, he graduated with a Master of Technology degree after com-

pleting 10 months of Master’s thesis work at the Institut fur Statik und Dynamik der

Luft- und Raumfahrtkonstruktionen (ISD), Universitat Stuttgart, Germany. Till March

2001, he worked in IIT Bombay as Research Associate. Doctoral studies began in the

Fall of 2001 under the guidance of Dr. Michael J. Allen at the Aerospace and Ocean

Engineering Department at Virginia Tech in the area of Uncertainty Quantification in

Vibro-Acoustics systems. Since, Fall of 2004, he is working under the guidance of Dr.

212

Rakesh K. Kapania and Dr. Robert W. Walters to study the Uncertainty Quantification

of Dynamic Systems. While pursuing his PhD, he has taught the Experimental Methods

Lab and Computational Methods course.

213

Documents

Uncertainty Quantification in Dynamic Problems …...Dedication To my parents Babasaheb and Shamshad for their love and support; to my all mentors and teachers for developing my scientiﬁc