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Full Description
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.
Contents
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.- 2.4 Homogeneous Chaos.- 3 Stochastic Finite Element Method: Response Representation.- 3.1 Preliminary Remarks.- 3.2 Deterministic Finite Elements.- 3.3 Stochastic Finite Elements.- 4 Stochastic Finite Elements: Response Statistics.- 4.1 Reliability Theory Background.- 4.2 Statistical Moments.- 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.- 5.3 Two Dimensional Static Problem.- 5.4 One Dimensional Dynamic Problem.- 6 Summary and Concluding Remarks.
