Stochastic Finite Elements: A Spectral Approach

ISBN-13: 9781461277958 / Angielski / Miękka / 2011 / 214 str.

Roger G. Ghanem; Pol D. Spanos

Opis

This monograph considers engineering systems with random parame ters. Its context, format, and timing are correlated with the intention of accelerating the evolution of the challenging field of Stochastic Finite Elements. The random system parameters are modeled as second order stochastic processes defined by their mean and covari ance functions. Relying on the spectral properties of the covariance function, the Karhunen-Loeve expansion is used' to represent these processes in terms of a countable set of un correlated random vari ables. Thus, the problem is cast in a finite dimensional setting. Then, various spectral approximations for the stochastic response of the system are obtained based on different criteria. Implementing the concept of Generalized Inverse as defined by the Neumann Ex pansion, leads to an explicit expression for the response process as a multivariate polynomial functional of a set of un correlated random variables. Alternatively, the solution process is treated as an element in the Hilbert space of random functions, in which a spectral repre sentation in terms of the Polynomial Chaoses is identified. In this context, the solution process is approximated by its projection onto a finite subspace spanned by these polynomials."

Spis treści

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 Stochastic Finite Element Method: 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 Stochastic Finite Elements: 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.