Synopses & Reviews
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 identifiedand#160;in terms ofand#160;polynomial chaos. In this context, the solution process is approximated by its projection onto a finite subspace spanned by these polynomials.
Synopsis
This text analyzes a class of discrete mathematical models of engineering systems,and#160;identifying key issues andand#160;reviewingand#160;relevant theoretical concepts, with particular attention to a spectral approach. 1991 edition.
Synopsis
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. 1991 edition.
Synopsis
Designed for those involved in analysis and design of random systems, this 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. 93 figures. 7 tables.
1991 edition.
Table of Contents
1 INTRODUCTION
and#160; 1.1 Motivation
and#160; 1.2 Review of Available Techniques
and#160; 1.3 The Mathematical Model
and#160; 1.4 Outline
2 REPRESENTATION OF STOCHASTIC PROCESSES
and#160; 2.1 Preliminary Remarks
and#160; 2.2 Review of the Theory
and#160; 2.3 Karhunen-Loeve Expansion
and#160;and#160;and#160; 2.3.1 Derivation
and#160;and#160;and#160; 2.3.2 Properties
and#160;and#160;and#160; 2.3.3 Solution of the Integral Equation
and#160; 2.4 Homogeneous Chaos
and#160;and#160;and#160; 2.4.1 Preliminary Remarks
and#160;and#160;and#160; 2.4.2 Definitions and Properties
and#160;and#160;and#160; 2.4.3 Construction of the Polynomial Chaos
3 SFEM: Response Representation
and#160; 3.1 Preliminary Remarks
and#160; 3.2 Deterministic Finite Elements
and#160;and#160;and#160; 3.2.1 Problem Definition
and#160;and#160;and#160; 3.2.2 Variational Approach
and#160;and#160;and#160; 3.2.3 Galerkin Approach
and#160;and#160;and#160; 3.2.4 "p-Adaptive Methods, Spectral Methods and Hierarchical Finite Element Bases"
and#160; 3.3 Stochastic Finite Elements
and#160;and#160;and#160; 3.3.1 Preliminary Remarks
and#160;and#160;and#160; 3.3.2 Monte Carlo Simulation (MCS)
and#160;and#160;and#160; 3.3.3 Perturbation Method
and#160;and#160;and#160; 3.3.4 Neumann Expansion Method
and#160;and#160;and#160; 3.3.5 Improved Neumann Expansion
and#160;and#160;and#160; 3.3.6 Projection on the Homogeneous Chaos
and#160;and#160;and#160; 3.3.7 Geometrical and Variational Extensions
4 SFEM: Response Statistics
and#160; 4.1 Reliability Theory Background
and#160; 4.2 Statistical Moments
and#160;and#160;and#160; 4.2.1 Moments and Cummulants Equations
and#160;and#160;and#160; 4.2.2 Second Order Statistics
and#160; 4.3 Approximation to the Probability Distribution
and#160; 4.4 Reliability Index and Response Surface Simulation
5 NUMERICAL EXAMPLES
and#160; 5.1 Preliminary Remarks
and#160; 5.2 One Dimensional Static Problem
and#160;and#160;and#160; 5.2.1 Formulation
and#160;and#160;and#160; 5.2.2 Results
and#160; 5.3 Two Dimensional Static Problem
and#160;and#160;and#160; 5.3.1 Formulation
and#160;and#160;and#160; 5.3.2 Results
and#160; 5.4 One Dimensional Dynamic Problem
and#160;and#160;and#160; 5.4.1 Description of the Problem
and#160;and#160;and#160; 5.4.2 Implementation
and#160;and#160;and#160; 5.4.3 Results
6 SUMMARY AND CONCLUDING REMARKS
and#160; 6.1 SUMMARY AND CONCLUDING REMARKS
and#160; BIBLIOGRAPHY
and#160; ADDITIONAL REFERENCES
and#160; INDEX