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Uncertainty Quantification in Dynamic Problems With 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. c Copyright 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 Advanced 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, metamodel, 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 transformation 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 2.1

2.2

16

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

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

Application and Validation . . . . . . . . . . . . . . . . . . . . . . . . . .

64

3.3.1

Piston in an Infinite Baffle . . . . . . . . . . . . . . . . . . . . . .

65

3.3.2

Elastically-Supported Plate . . . . . . . . . . . . . . . . . . . . .

69

3.3

viii

3.4

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 Probabilistic Metamodeling

80

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2

Hybrid Metamodel for Nonmonotonic, Nonlinear Response Function Anal-

4.3

4.4

78

80

ysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

4.3.1

Analytical Functions . . . . . . . . . . . . . . . . . . . . . . . . .

84

4.3.2

Composite Panel Sound power . . . . . . . . . . . . . . . . . . . .

88

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

5 Fundamental Eigenvalue using Polynomial Chaos

98

5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

5.6

98

5.5.1

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

5.5.2

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

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6 Higher Eigenvalues using Polynomial Chaos

131

6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

6.2


6.3

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

6.4


ix

6.5

6.4.1

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

6.4.2

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

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

7 A New Algorithm for Eigenvalue Analysis using Polynomial Chaos

162

7.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

7.2


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


8.4

8.3.1

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

8.3.2

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

8.3.3

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

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 . . . . . . . . . . . . . . . . . . . . . . . .

3.3

Deterministic and Random Design Parameter Configuration of Flexible Panel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.4

74

Finite Element and Boundary Element Models of the Elastically Supported Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.5

68

75

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

4.10 Sound power CDFs for the Composite Panel at 364 Hz . . . . . . . . . .

94

4.11 Percent Error in the Sound power CDFs with respect to MCS at 364 Hz

95

4.12 Sound power CDFs for the Composite Panel at 406 Hz . . . . . . . . . .

95

4.13 Percent Error in the Sound power CDFs with respect to MCS at 406 Hz

96

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 Different 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 Probability 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 with Same Probability Space for Mass and the Bending Rigidity . . . . . 125 5.17 Fundamental Eigenvector using Second-Order Chaos at 95% Probability with Same Probability Space for Mass and the Bending Rigidity . . . . . 125 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 Different 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 Correlated Masses and Stiffness for the 3-DOF System . . . . . . . . . . . . 142

6.4

PDFs of Third Eigenvalue using Fourth Order Chaos for Perfectly Correlated Masses and Stiffness for the 3-DOF System . . . . . . . . . . . . . 143 xiii

6.5

ˆ for Perfectly Correlated Masses and Stiffness for the Convergence of λ 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 3-DOF System at Different Probabilities . . . . . . . . . . . . . . . . . . 145

6.9

PDFs of Fundamental Eigenvalue using Third Order Chaos for Uncorrelated 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 ˆ for Uncorrelated Masses and Stiffness for the 3-DOF 6.12 Convergence of λ System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 6.13 Fundamental Eigenvector for Uncorrelated Masses and Stiffness for the 3-DOF System at Different Probabilities . . . . . . . . . . . . . . . . . . 149 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 Correlated 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 Masses and Stiffness for Simply-Supported Beam . . . . . . . . . . . . . 154 ˆ for Fully Correlated Masses and Stiffness for Simply6.20 Convergence of λ 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 SimplySupported Beam at Different Probabilities . . . . . . . . . . . . . . . . . 155 6.23 Third Eigenvector for Fully Correlated Masses and Stiffness for SimplySupported Beam at Different Probabilities . . . . . . . . . . . . . . . . . 156 6.24 PDFs of Fundamental Eigenvalue using Second Order Chaos for Uncorrelated Masses and Stiffness for Simply-Supported Beam . . . . . . . . . . 157 6.25 PDFs of Second Eigenvalue using Second Order Chaos for Uncorrelated Masses and Stiffness for Simply-Supported Beam . . . . . . . . . . . . . 157 6.26 PDFs of Third Eigenvalue using Second Order Chaos for Uncorrelated Masses and Stiffness for Simply-Supported Beam . . . . . . . . . . . . . 158 ˆ for Uncorrelated Masses and Stiffness for Simply-Supported 6.27 Convergence of λ Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 6.28 Fundamental Eigenvector for Uncorrelated Masses and Stiffness for SimplySupported Beam at Different Probabilities . . . . . . . . . . . . . . . . . 159 6.29 Second Eigenvector for Uncorrelated Masses and Stiffness for SimplySupported Beam at Different Probabilities . . . . . . . . . . . . . . . . . 159 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 Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 for the 3-DOF System using Third-Order Polynomial Chaos Expansion . 171

7.7

PDFs of Third Eigenvalue for Perfectly Correlated Masses and Stiffness for the 3-DOF System using Third-Order Polynomial Chaos Expansion . 171

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 Random Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

8.4

Log-Normal CDFs of Analytical and Numerical KL Expansion Basis Random 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 Random Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 8.14 Exponential CDFs of Analytical and Numerical KL Expansion Basis Random 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 . . . . . . . . . . . . . . . .

3.2

Radiated Sound Power and Sound Power Sensitivity Values for Circular Piston . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3

70

Predicted and Actual 98% Probabilistic PW Values due to 2% Change in Masses, m1 , m2 ,and Dampers, b1 , b2 . . . . . . . . . . . . . . . . . . . . .

3.4

66

71

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 Frequency Range for Flexible Panel . . . . . . . . . . . . . . . . . . . . . . .

3.8

Predicted and Actual 98% Probabilistic PW Values due to 3% Independent Changes in t1 and t2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.9

77

77

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

4.1

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

78

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 Probability 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 Stiffness for the 3-DOF System using LHS . . . . . . . . . . . . . . . . . . . 141

6.3

Eigenvalue Coefficients for Uncorrelated Masses and Stiffness for the 3DOF 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 3DOF 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 Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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

⇒ T

implies; if .. then

S

the union of ... and ...; union

∅

the empty set

sup

supremum

BBA

Basic Belief Assignment

f :x→y

mapping

Bel

Belief

Pl P

Plausibility

Θ

Sample space

θ

Trial realisation

R

Set of Real Numbers

intersected with; intersect

sum over ... from ... to ... of

σ

Standard deviation 0

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

hi

Expectation operator

M

Mass matrix

C ¨ U

Damping matrix 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 experimentations. 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]. Heisenberg in 1927 proposed his “Uncertainty Principle” in Quantum Mechanics for locating electron’s position or momentum (www.aip.org/history/heisenberg/p08.htm), but because of difficulty of understanding this principle at that time, Schr¨odinger 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 Uncertainty 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 accuracy 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 beginning 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

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 methods 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 Stochastic 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].

1.2

Classification of Uncertainty

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 described by deterministic description. This uncertainty is because of inherent ran3

1.2 Classification of Uncertainty

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 properties, 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 precision like 32 or 64 bit processors so this simulation chops off infinite representation of decimal numbers into binary numbers and carries mathematical operations.

4

1.3 Uncertainty Quantification Methods

Uncertainty Methods

Probabilistic Methods

Possibilistic Methods

1. 2. 3. 4.

1. 2. 3.

Interval Analysis Convex Modeling Fuzzy Set Theory Evidence Theory

4.

Asymptotic Methods Sampling Techniques Perturbation Stochastic Finite Element Method (PSFEM) Spectral Stochastic Finite Element Method (SSFEM)

Fuzzy Random Variable Approach

Figure 1.1: Uncertainty Quantification Methods

5


1.3

Uncertainty Quantification Methods

In Fig. 1.1, different methods dealing with uncertainties are classified according to the input 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, information 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


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 re6


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 response 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) characterizing 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 random 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 representation 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 obtained when considering the system’s discretized governing differential equations can be solved for the unknown coefficient in the polynomial chaos expansion. Once these coefficients are known, the statistics of the response can be readily obtained. In the initial development of polynomial chaos, Hermite orthogonal polynomials were used for representing second order Gaussian random process. To account for non-Gaussian processes, generalized polynomial chaos was proposed [33].

1.3.2


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 quan9


tification becomes computationally intensive for complex problems. The range of response becomes prohibitively large as the algebraic calculations on the random variables become large which overestimates the uncertainty in the response from the actual uncertainty. 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

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 sensitivity with respect to deterministic design variables. Lot of research has been done to deal with the uncertainties in structural mechanics 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 developed 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 acoustic 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

1.4 Objectives of the Dissertation

calculations. A metamodeling technique is presented which fits the response surface 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 eigenvectors 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 material properties can be obtained using Perturbation Stochastic Finite Element Method (PSFEM). Since, the assumption of PSFEM is that the underlying random 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 calculating 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 nonGaussian random process. For Gaussian random process, basis random variables of KL expansion are uncorrelated 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 variables 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 transformation of the CDF of the marginal distribution function of the random processes.

1.5

Outline of the Dissertation

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 Integral 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 structure is calculated. The new probabilistic technique combines the AMV, metamodeling and simple Monte Carlo Sampling (MCS). The new technique is applied to a composite 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 Ci are 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.

2.1


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 = [1/x, 1/x ] x

(2.8)

if x > 0 or x < 0

x ÷ y = x × 1/y

(2.9)

In computational mechanics, discretized equations are represented as

Ay = 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 xi1 , xj2 , . . . , xkn ; r = 1, 2, . . . , 2n

i, j, . . . , k = 1, 2; y

= =

y, y min

r = 1,2,..., 2n

yr ,

max

r = 1,2,..., 2n

yr

(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 application 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 optimization 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 intelligence, image processing, communication systems and control systems in electronic devices to deal with the uncertainty in these systems. Its application in structural analysis started in the 19800 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) ! I I \ [ ∀ Ai , i ∈ I, Ai = ∅ ⇒ µ Ai = max (µ (Ai ))

(2.14)

i=1

i=1

i∈I

(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 αcut 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) =

sup

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

(2.16)

y=f (x1 ,x2 ,...,xn )

The above operation is possible for explicit expressions. It is implemented in commercial R in “Fuzzy Logic Toolbox” [46]. In MATLAB R , uncertain variables software, MATLAB 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].

1

0.8

α

µ(x)

0.6

0.4

0.2

0 0

2

4

x

6

8

10

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. Probability 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 further 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 possibility 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 assignment (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)

X

(2.19)

m (A) = 1

A∈2X

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.

X

Bel (A) =

m (B)

(2.23)

B|B⊆A

X

P l (A) =

(2.24)

m (B)

B|B∩A6=∅

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

Bel (A) + Bel A¯ ≤ 1

(2.25)

P l (A) = 1 − Bel A¯

(2.26)

P l (A) + P l A¯ ≥ 1

(2.27)

where A¯ represents the classical compliment of A.

P l (A)

Bel (A)

U ncertainty Figure 2.2: Belief and Plausibility Measures

25

¡ ¢ Bel A¯


2.1.4.4

Rules for the Combination of Evidences

When the evidences come from different sources, these evidences are combined to obtain 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: P m12 (A) = K =

m1 (B) m2 (C)

B∩C=A

X

1 − K m1 (B) m2 (C)

,

A 6= ∅

(2.28) (2.29)

B∩C=∅

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

2.2


When the definition of uncertainty is available interms of random process or random variables, probabilistic methods should be preferred to possibilistic methods. Probabilistic methods in current form are mathematical elegant, give very accurate results and produce 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 (hX 2 i < ∞) is denoted by L2 (Θ, F, P). Here h i 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, σ 2 (x) are constant and autocorrelation coefficient, ρ (x, x0 ) is a function of the difference x − x0 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 uncertainty 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 Z P (Z ≤ Z0 ) =

fX (x) dΩ = p

(2.30)

Ω

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 X ∂Z i=1

29

∂Xi

(Xi − µi ) µ

(2.31)


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

g (X) ≈ a0 +

n X

ai Xi − Z0

(2.32)

i=1

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)

n P a0 + ai µ i = r ni=1 P 2 2 ai σi

(2.35)

i=1

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

2.2 Probabilistic Methods series expansion about the MPP, X∗ , defined by Eq. (2.36) corresponding to a particular probability level.

Z (X) ∼ = Z (X∗ )+

n X ∂Z ∂Xi

i=1 n X n X i=1 j=1

2

∂ Z ∂Xi ∂Xj

(Xi − Xi∗ )+ X∗

(Xi − Xi∗ ) Xj − Xj∗

(2.36)

X∗

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

p = Φ (−β)

n−1 Y

(1 + β κi )1/2

(2.37)

i=1

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 X ∂Z i=1

∂Xi

31

(Xi − µi )+H (X) µ

(2.38)


Z (X) = a0 +

n X

ai Xi +H (X)

(2.39)

i=1

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 ZAM V values shown below ZAM V = 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 Z1∗ and M P P L∗ 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) +

(2.42)

where is a normally distributed error with zero mean and standard deviation σ , f (X) is an unknown function that is approximated for slightly nonlinear response functions as:

f (X) = b0 +

n X i=1

34

bi Xi

(2.43)


or for nonlinear surfaces,

f (X) = b0 +

n X

bi Xi +

i=1

n X n X

bij Xi Xj

(2.44)

i=1 j=1

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 ± fi σi ; 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:

KU = 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 X

KIi αi

i=1

U = U0 +

N X

N N 1 X X II + Kij αi αj + O(kαk2 ) 2 i=1 j=1

(2.46)

N N 1 X X II U αi αj + O(kαk2 ) 2 i=1 j=1 ij

(2.47)

N N 1 X X II + Fij αi αj + O(kαk2 ) 2 i=1 j=1

(2.48)

UIi αi +

i=1

F = F0 +

N X i=1

FIi αi

where K0 , U0 , and F0 are the mean values of respective tensors. ()Ii and ()Ii I represents the first and second order derivatives evaluated at α = 0, e. g. :

∂K = ∂αi α=0 ∂ 2 K II Kij = ∂αi ∂αj α=0

KIi

36

(2.49) (2.50)


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

U0 = K−1 0 F0

(2.51)

FIi − KIi U0 UIi = K−1 0

(2.52)

−1 I I I I II UII FII ij = K0 ij − Ki Uj − Kj Ui − Kij U0

(2.53)

From these mean and covariance matrix of the response vector, U can be obtained as N N 1 X X II hUi ≈ U0 + U Cov [αi , αj ] 2 i=1 j=1 ij

Cov [U, U] ≈

N X N X

UIi UIj

T

Cov [αi , αj ]

(2.54)

(2.55)

i=1 j=1

After Cov [αi , αj ] is substituted in terms of correlation coefficients ρij in Eq. 2.55, final expression for Cov [U, U] is obtained as N X N X ∂UT ∂U Cov [U, U] ≈ ρij σαi σαj ∂α ∂α i j α=0 α=0 i=1 j=1

(2.56)

For applying PSFEM, random processes representing material properties, geometric properties 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 KarhunenLoeve expansion and further developed [31, 32] to account for higher coefficient of variation, δ of input random variables. Galerkin procedure is employed in random (probability) 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)+

∞ X √

κi φi (x) ξi (θ)

(2.57)

i=1

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

Z Rhh (x, y) φi (y) dy = κi φi (x)

(2.58)

Ω

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 +

∞ X

ai1 (x) Γ1 (ξi1 (θ)) +

i1 =1 i1 ∞ X X

ai1 i2 (x) Γ2 (ξi1 (θ) , ξi2 (θ)) + · · · ,

(2.59)

i1 =1 i2 =1

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

1 T ξ

Γn (ξ1 , . . . , ξn ) = e 2 ξ

(−1)n

40

1 T ∂n e− 2 ξ ξ ∂ξi1 · · · ∂ξin

(2.60)


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

P X

G (θ) =

Gj Γj (ξ)

(2.61)

j=1

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

hΓi Γj i =

Γ2i δij

(2.62)

where δij is the Kronecker delta and h., .i represents the expectation of the weighted inner product of these polynomials in variable ξ, i.e.

Z hf (ξ) g (ξ)i =

f (ξ) g (ξ) W (ξ) dξ

(2.63)

Σ

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 density 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 mechanics, fluid mechanics, and applied mathematics. In polynomial chaos, the Galerkin method is used to minimize the weighted residual of the differential equation by multiplying 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 X

uj (x, t) Γj (ξ (θ))

(2.67)

fj (x, t) Γj (ξ (θ))

(2.68)

j=0

f (x, t, θ) =

P X j=0

43

2.2 Probabilistic Methods P X

Bi uj (x, t) Γj (ξ (θ)) = 0, i = 1, 2, . . . , k

(2.69)

j=0 P X √

κi φi ξi L

i= 0

P X

! uj (x, t) Γj (ξ (θ))

=

j=0

P X

fj Γj

(2.70)

j=0

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 X√ i= 0

κi φi ξi L

P X

! uj (x, t) Γj (ξ (θ)) , Γk

+

= fk Γ2k ,

j=0

k = 0, 1, . . . , P. (2.71)

The orthogonality property of polynomials will be used in these calculations. Equation (2.71) is a set of multidimensional algebraic equations or equations in multidimensional tensors. By solving this multidimensional system, Eq. (2.71), the deterministic 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

P

substituting

dk Nk , dk are deterministic coefficients and Nk are the shape functions. After

P

dk Nk 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 subjected to input random variable uncertainties. In this method, Latin Hypercube Sampling (LHS) is used to get response polynomial chaos coefficients. If the uncertainties are random 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 computationally 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 approximations 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 randomly 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:

Z p=

I [g (X) ≤ 0]

fX (x) fS (x) dx fS (x)

(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 either 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 subsequent 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 systems is to adequately account for uncertainty when predicting vibro-acoustic performance. 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 existence 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 functions 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 vibroacoustic 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 acoustic 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 structural 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 probabilistic 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 characteristics 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, nonmonotonic 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]. Uncertainty was described in terms of random variables with known PDFs. Sub-structuring techniques and stored vibration invariant information from the boundary element analysis 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 uncertainty 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 calculate 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 automotive 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. Probabilistic 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.

3.2

Theoretical Derivation

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 probabilistic 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 employed 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 equation describing the motion of an arbitrarily shaped structure subject to a harmonic excitation can be written in matrix form as [75, 77, 82]:

¨ + C U˙ + K U = F MU

(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 components on the surface of the structure. The normal velocity, Vn , is written as:

Vn = T1 U = T1 S−1 t F

(3.2)

where the structural matrix, St , is complex and is equal to −ω 2 M + 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

Z 1+ S

∂ ∂n

1 4πR

−iKR Z ∂ e−iKR e PS (r) dS = Ps (r0 ) − iωρ vn (r0 ) dS (3.3) ∂n 4πR 4πR S

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 V n 55

(3.4)


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 Re PTsel Ar V∗nel 2

(3.5)

where vectors Psel and 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:

P W = P W (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, P W0 , can be expressed as [18]:

Z P (P W < P W0 ) =

fX (x) dx

(3.8)

Ω

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 P W0 . 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 correction 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 Transformation [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 (P W < P W0 ) and is written as:

g (H, X) ≈ P W (H, µ) +

n X ∂P W ∂Xj

j=1

(Xj − µj ) − P W0

(3.9)

where g (H, X) is known as the limit state equation, n = the total number of random design variables, ∂P W/∂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 1 = Re PTsel Ar ∂Xj 2

∂V∗n

+

∂Xj

el

∂Ps ∂Xj

T

! Ar V∗nel

(3.10)

el

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 ∂U = T2 A−1 B T1 ∂Xj ∂Xj ∗ el ∂U ∂Vn = T2 T1 ∂Xj el ∂Xj

(3.11)

The sensitivity of the nodal displacements is calculated as [83]:

−1 ∂U = − −ω 2 M + i ω C + K ∂Xj

∂M ∂C ∂K −ω + iω + ∂Xj ∂Xj ∂Xj 2

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 =

P W (H, µ) − P W0 µg = s 2 n σg P ∂P W σ j ∂Xj

(3.13)

j=1

where σj represents the standard deviation of random variable j. Once the reliability index is known the probability is evaluated as

P (P W < P W0 ) = 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:

Xi∗ = 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 (P W < P W (H, X∗ ))

(3.17)

Equation (3.17) defines the CDF generated by the AMV method. Sound power values associated 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.

P Wβ = P W (H, X∗ ) 61

(3.18)


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:

N X ∂P W ∂P Wβ ∂P W ∂Xj∗ = + ∂Hm ∂Hm X ∂Xj∗ ∂Hm j=1

(3.19)

The first term in Eq. (3.19) and ∂P W/∂Xj∗ 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 using Eqs. (3.10), (3.11), and (3.12). ∂Xj∗ /∂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):

∂Xj∗ ∂λ∗j ∂ = σj = σj β ∂Hm ∂Hm ∂Hm

(∇g)j |∇g|

(3.20)

where the sensitivity of the unit normal is given by:  ∂ ∂Hm

(∇g)j |∇g|

∂P W = σj ∂Xj

n P

∂2P W







∂P W 2 σk ∂2P W −    ∂Hm ∂Xj  k=1 ∂Xk ∂Hm ∂Xk    +    (3.21) 1.5 0.5 n n 2 2  P    P ∂P W ∂P W 2 2 σ σ k k ∂Xk ∂Xk k=1

k=1

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 nonmonotonic CDF response at certain frequencies. These frequencies correspond to instances 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)

CDFc = CDF1 − CDF2

(3.23)

or

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 maximum 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,

∂P Wβ ∂Hm

∓

=

∂P Wβ ∂Hm

+

1

2

c

∂P Wβ ∂Hm

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 nonmonotonic results.

3.3

Application and Validation

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 sensitivity 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 respect 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


k1

k2

F

b1

b2 m1

m2

Figure 3.1: Baffled Circular Piston Configuration

M ass, m1 , m2 Damping, b1 , b2 Diameter, d F orce amplitude, F

8kg 1.3E02N − s/m 0.1m 200N

Table 3.1: Characteristics of the Baffled Circular Piston

66


1 PW = ρ0 c S 2

2 c J ωcd 1 − ωd

!

a02 + b02 a2 + b 2

a0 = −ω 2 F (b1 + b2 ) b0 = −F ω k1 + k2 − m1 ω 2

a = k1 k 2 − ω 2 b1 b2 + k2 m 1 + k1 m 2 + k 2 m 2 − m 1 m 2 ω 2 b = −ω (b2 k1 + b1 k2 ) + ω 3 (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 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


8.3 Numerical Examples

1 0.8 0.6 0.4 Original CDF Transformed CDF

0.2

2

4 6 8 Log-Normal Variable Domain

10

12

Figure 8.2: CDFs of Marginal Log-Normal Distribution and Transformed Distribution


1.2 1 0.8

KL Basis PDF Original PDF

0.6 0.4 0.2

0

2 4 Log-Normal Variable Domain

6

8

Figure 8.3: PDFs of Marginal Log-Normal Distribution and KL Expansion Basis Random Variable

181



1 0.8 0.6 0.4 Analytical KL Basis CDF Numerical KL Basis CDF

0.2

0

1 2 3 Log-Normal Variable Domain

4

5

Figure 8.4: Log-Normal CDFs of Analytical and Numerical KL Expansion Basis Random Variables

0.5 0.25 0

-0.25 -0.5 -0.75 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 Log-Normal Variable, Hx1L

1

Log-Normal Variable, Hx3L


1 0.75

1

0


Log-Normal Variable, H x4L


0.5 0.25

0.5 0.25 0


1

1

1 0.75

0.75

1

0.75 0.5 0.25 0


1

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 (x − a)α−1 (b − x)β−1 B (α, β) (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 independency 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 cumulants calculations in Table 8.2. Scatter plots of these variables are shown in Fig. 8.10

183


3.5 3 2.5 CDF PDF

2 1.5 1 0.5 0.2

0.4 0.6 0.8 Beta Variable Domain

1

Figure 8.6: CDF and PDF of Non-Gaussian Marginal Beta Distribution n 1 2 3 4

κn (ξ1 + ξ2 ) κn (ξ1 ) + κn (ξ2 ) −5.26490E − 15 −5.26495E − 15 1.99904E + 00 1.99889E + 00 5.25263E − 03 2.05754E − 03 −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



1 0.8 0.6 0.4 Original CDF Transformed CDF

0.2

0.5

1 1.5 2 Beta Variable Domain

2.5

Figure 8.7: CDFs of Marginal Beta Distribution and Transformed Distribution


2.5

2

KL Basis PDF Original PDF 1.5

1

0.5

-1

-0.5 0 0.5 Beta Variable Domain

1

Figure 8.8: PDFs of Marginal Beta Distribution and KL Expansion Basis Random Variable

185



1 0.8 0.6 0.4 0.2

Analytical KL Basis CDF Numerical KL Basis CDF

-1

-0.5 0 0.5 Beta Variable Domain

1

Figure 8.9: Beta CDFs of Analytical and Numerical KL Expansion Basis Random Variables

1

Beta Variable, H x3 L

Beta Variable, H x2L

1 0.5 0

-0.5

0

-0.5 -1

-1 -1

-0.5 0 0.5 Beta Variable, Hx1L

1

1

-1


1

-1


1

1

Beta Variable, H x4 L

Beta Variable, H x4L

0.5

0.5 0

-0.5 -1

0.5 0

-0.5 -1

-1


1

Figure 8.10: Scatter Plots of Beta KL Expansion Basis Random Variables

186


Nonlinear transformation using the standard deviation and the definition of CDF produces 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 random 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 independency 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 cumulants calculations in Table 8.3. Scatter plots of these variables are shown in Fig. 8.15

187

8.4 Conclusions

1.6 1.4 1.2 1 0.8

CDF PDF CDF PDF

0.6 0.4 0.2 1

2 3 4 5 Exponential Variable Domain

6

Figure 8.11: CDF and PDF of Non-Gaussian Marginal Exponential Distribution n 1 2 3 4

κn (ξ1 + ξ2 ) κn 7.45081E − 15 2.00689E + 00 4.04189E + 00 1.22226E + 01

(ξ1 ) + κn (ξ2 ) 7.45095E − 15 2.00317E + 00 4.01956E + 00 1.20717E + 01

Table 8.3: Independency of KL Expansion Basis Random Variables for Exponential Distribution

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 process, KL expansion should have independent random variables with zero mean and unit variance. This particular requirement is satisfied. In this method, nonlinear transformation is applied to the marginal distribution function of given random process using

188


8.4 Conclusions

1 0.8 0.6 Original CDF Transformed CDF

0.4 0.2

1

2 3 4 Exponential Variable Domain

5

6

Figure 8.12: CDFs of Marginal Exponential Distribution and Transformed Distribution

1.6


1.4

KL Basis PDF Original PDF

1.2 1 0.8 0.6 0.4 0.2 0


4

5

Figure 8.13: PDFs of Marginal Exponential Distribution and KL Expansion Basis Random Variable

189

8.4 Conclusions


1 0.8 0.6 0.4

Analytical KL Basis CDF CDF Analytical KL Basis Numerical KL Basis CDF CDF Numerical KL Basis

0.2

0


4

5

Figure 8.14: Exponential CDFs of Analytical and Numerical KL Expansion Basis Random Variables

5

Exponential Variable, Hx3L

Exponential Variable, H x2 L

5 4 3 2 1 0 0

1 2 3 4 Exponential Variable, Hx1L

3 2 1 0

5

0


5

0


5

5

Exponential Variable, Hx4L

5

Exponential Variable, Hx4 L

4

4 3 2 1 0 0


5

4 3 2 1 0

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. Inverse 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, moments 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 Applications, 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, Dordrecht/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, Cambridge, 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 Perturbation Technique and Computer Implementation, John Wiley & Sons, New York, 1992. [9] Haldar, A., and Mahadevan, S., Reliability Assessment Using Stochastic Finite Element 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 Directions 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 Publishing 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., “Capabilities 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 Uncertainties in Probabilistic Engineering Analysis”, 42nd AIAA/ASME/ASCE/ASC Structures, Structural Dynamics, and Materials Conference, Seattle, WA, AIAA Paper, 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 Random 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 Elements,” 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 Stochastic 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 Mechanics, 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 Uncertainty”, Transactions of the ASME, Vol. 126, 2004, pp. 386-394. [37] Bae, H.-R., Grandhi, R. V., and Canfield, R. A., “Epistemic Uncertainty Quantification 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 Materials 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., “Comparison of Evidence Theory and Bayesian Theory for Uncertainty Modeling”, Reliability 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, Structural Dynamics, and Materials Conference, Reston, VA, AIAA Paper 1996-1610, 15-17 April 1996. R , Users Guide Version 2.2.2, C The [46] Fuzzy Logic Toolbox for Use with MATLAB 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 Composite 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 Mapping”, 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 Reliability, 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 Mathematical 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 Statistical 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 Structural Safety and Reliability”, Proceedings of ICOSSAR-81, the 3rd International Conference on Structural Safety and Reliability, 1981, pp. 395-408. [63] Liu, W.-K., Belytschko, T., and Mani, A., “ Random Field Finite Elements”, International 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 Settlements”, Journal of Geotechnical Engineering, ASCE, Vol. 116, No. 11, 1990, pp. 1717-1735. [65] Lo` eve, M., Probability Theory, 4th ed., Springer-Verlag, 1977. [66] Choi, S-K., Grandhi, R. V., and Canfield, R. A., “Structural Reliability under NonGaussian 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 Vibration 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#1999-011791, 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 Acoustical 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 MacNeal - 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 System 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 Reliability in Design, Vol. 112, No. 4, 1990, pp. 556-560. [89] Allen, M. J., and Vlahopoulos, N., “Numerical Probabilistic Analysis of Structural/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. 271335. [95] Junger, M. C., “Shipboard Noise: Sources, Transmission, and Control”, Noise Control 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 Transmission 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 Element 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 Distribution 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 Ultimate Strength”, AIAA Journal, Vol. 40, No. 7, 2002, pp. 1408-1414. [106] Oh, D. H., and Librescu, L., “Free Vibration and Reliability of Composite Cantilevers 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 Composite 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 LineExcited 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, 19961411, 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 Systems 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 Analysis,” 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 Problem 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 Stochastic Partial Differential Equations (Stochastic Finite Elements)”, Report No. 200306, Department of Mathematics and Computer Science, Technical University Braunschweig, 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 Expansion for Non-Gaussian Random Fields,” Engineering Mechanics Conference, Baltimore, MD, 13-16 June 1999. [123] Sakamoto, S., and Ghanem, R., “Polynomial Chaos Decomposition for the Simulation 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 NonGaussian Processes using Karhunen-Loeve Expansion”, Probabilistic Engineering Mechanics, 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 Stochastic 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 Uncertainty 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 completing his high school education at His Highness Raja Chintamanrao Patwardhan High School, Sangli, India in 1990, he entered Wilingdon College of Sangli and passed successfully Maharashtra State Board of Higher Secondary Education examination in 1992. He pursued his Bachelor’s degree in Civil Engineering from Walchand College of Engineering, 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 completing 10 months of Master’s thesis work at the Institut f¨ ur Statik und Dynamik der Luft- und Raumfahrtkonstruktionen (ISD), Universit¨at 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.

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