Stochastic Finite Elements A Spectral Approach, Revised Edition

Discrepancies frequently occur between a physical system's responses and predictions obtained from mathematical models. The Spectral Stochastic Finite Element Method (SSFEM) has proven successful at forecasting a variety of uncertainties in calculating system responses. This text analyzes a class of discrete mathematical models of engineering systems, identifying key issues and reviewing relevant theoretical concepts, with particular attention to a spectral approach.
Random system parameters are modeled as second-order stochastic processes, defined by their mean and covariance functions. Relying on the spectral properties of the covariance function, the Karhunen-Loeve expansion is employed to represent these processes in terms of a countable set of uncorrected random variables, casting the problem in a finite dimensional setting. Various spectral approximations for the stochastic response of the system are obtained. Implementing the concept of generalized inverse leads to an explicit expression for the response process as a multivariate polynomial functional of a set of uncorrelated random variables. Alternatively, the solution process is treated as an element in the Hilbert space of random functions, in which a spectral representation is identified in terms of polynomial chaos. In this context, the solution process is approximated by its projection onto a finite subspace spanned by these polynomials.

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
1.1 Motivation
1.2 Review of Available Techniques
1.3 The Mathematical Model
1.4 Outline
2 REPRESENTATION OF STOCHASTIC PROCESSES
2.1 Preliminary Remarks
2.2 Review of the Theory
2.3 Karhunen-Loeve Expansion
2.3.1 Derivation
2.3.2 Properties
2.3.3 Solution of the Integral Equation
2.4 Homogeneous Chaos
2.4.1 Preliminary Remarks
2.4.2 Definitions and Properties
2.4.3 Construction of the Polynomial Chaos
3 SFEM: Response Representation
3.1 Preliminary Remarks
3.2 Deterministic Finite Elements
3.2.1 Problem Definition
3.2.2 Variational Approach
3.2.3 Galerkin Approach
3.2.4 "p-Adaptive Methods, Spectral Methods and Hierarchical Finite Element Bases"
3.3 Stochastic Finite Elements
3.3.1 Preliminary Remarks
3.3.2 Monte Carlo Simulation (MCS)
3.3.3 Perturbation Method
3.3.4 Neumann Expansion Method
3.3.5 Improved Neumann Expansion
3.3.6 Projection on the Homogeneous Chaos
3.3.7 Geometrical and Variational Extensions
4 SFEM: Response Statistics
4.1 Reliability Theory Background
4.2 Statistical Moments
4.2.1 Moments and Cummulants Equations
4.2.2 Second Order Statistics
4.3 Approximation to the Probability Distribution
4.4 Reliability Index and Response Surface Simulation
5 NUMERICAL EXAMPLES
5.1 Preliminary Remarks
5.2 One Dimensional Static Problem
5.2.1 Formulation
5.2.2 Results
5.3 Two Dimensional Static Problem
5.3.1 Formulation
5.3.2 Results
5.4 One Dimensional Dynamic Problem
5.4.1 Description of the Problem
5.4.2 Implementation
5.4.3 Results
6 SUMMARY AND CONCLUDING REMARKS
6.1 SUMMARY AND CONCLUDING REMARKS
BIBLIOGRAPHY
ADDITIONAL REFERENCES
INDEX