Stochastic Finite Elements A Spectral

Designed for those involved in the analysis and design of random systems, this graduate-level text analyzes a class of discrete mathematical models of engineering systems. It clearly identifies key issues and offers an instructive review of relevant theoretical concepts, with particular attention to a spectral approach. Contents: 1. Introduction. 2. Representation of Stochastic Processes. 3. Stochastic Finite Element Method: Response Representation. 4. Stochastic Finite Elements: Response Statistics. 5. Numerical Examples. 6. Summary and Concluding Remarks. Bibliography. Index. Unabridged republication of the edition published by Springer-Verlag, New York, 1991. 93 Figures. 7 Tables.

Roger G. Ghanem is a Professor at University of Southern California's Department of Aerospace and Mechanical Engineering in Los Angeles.
Pol D. Spanos is the L. B. Ryon Chair in Engineering in the Department of Mechanical Engineering and Materials Science at Rice University.

1 INTRODUCTION

1

(12)

1.1 Motivation

1

(2)

1.2 Review of Available Techniques

3

(6)

1.3 The Mathematical Model

9

(3)

1.4 Outline

12

(1)

2 REPRESENTATION OF STOCHASTIC PROCESSES

13

(182)

2.1 Preliminary Remarks

13

(1)

2.2 Review of the Theory

14

(3)

2.3 Karhunen-Loeve Expansion

17

(25)

2.3.1 Derivation

17

(4)

2.3.2 Properties

21

(3)

2.3.3 Solution of the Integral Equation

24

(18)

2.4 Homogeneous Chaos

42

(21)

2.4.1 Preliminary Remarks

42

(2)

2.4.2 Definitions and Properties

44

(3)

2.4.3 Construction of the Polynomial Chaos

47

(16)

3 SFEM: Response Representation

63

(32)

3.1 Preliminary Remarks

63

(2)

3.2 Deterministic Finite Elements

65

(5)

3.2.1 Problem Definition

65

(1)

3.2.2 Variational Approach

66

(2)

3.2.3 Galerkin Approach

68

(1)

3.2.4 p-Adaptive Methods, Spectral Methods and Hierarchical Finite Element Bases

69

(1)

3.3 Stochastic Finite Elements

70

(25)

3.3.1 Preliminary Remarks

70

(1)

3.3.2 Monte Carlo Simulation (MCS)

71

(1)

3.3.3 Perturbation Method

72

(3)

3.3.4 Neumann Expansion Method

75

(2)

3.3.5 Improved Neumann Expansion

77

(4)

3.3.6 Projection on the Homogeneous Chaos

81

(11)

3.3.7 Geometrical and Variational Extensions

92

(3)

4 SFEM: Response Statistics

95

(18)

4.1 Reliability Theory Background

95

(3)

4.2 Statistical Moments

98

(11)

4.2.1 Moments and Cummulants Equations

98

(9)

4.2.2 Second Order Statistics

107

(2)

4.3 Approximation to the Probability Distribution

109

(3)

4.4 Reliability Index and Response Surface Simulation

112

(1)

5 NUMERICAL EXAMPLES

113

(76)

5.1 Preliminary Remarks

113

(1)

5.2 One Dimensional Static Problem

114

(28)

5.2.1 Formulation

114

(5)

5.2.2 Results

119

(23)

5.3 Two Dimensional Static Problem

142

(37)

5.3.1 Formulation

142

(10)

5.3.2 Results

152

(27)

5.4 One Dimensional Dynamic Problem

179

(10)

5.4.1 Description of the Problem

179

(2)

5.4.2 Implementation

181

(4)

5.4.3 Results

185

(4)

6 SUMMARY AND CONCLUDING REMARKS

189

(6)

6.1 SUMMARY AND CONCLUDING REMARKS

189

(6)

BIBLIOGRAPHY

195

(16)

ADDITIONAL REFERENCES

211

(8)

INDEX

219

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Stochastic Finite Elements A Spectral Approach, Revised Edition

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Table of Contents

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