Collocation Scheme for Computing the Coefficients of the PolynomialChaos Expansion in Stochastic Finite Element Analysis

摘要

The Stochastic Finite Element Method developed by Ghanem et al [1] allows to solve stochasticboundary value problems involving spatially randomly varying materials usually described asGaussian or lognormal random fields. The method is based on the discretization of the input randomfields and the expansion of the mechanical response onto the so-called polynomial chaos. A similarprocedure allowing to model random material properties and loading by means of any number ofrandom variables of any type has been recently proposed [2-3].In both cases, the coefficients of the response expansion are computed using a Galerkin procedure,which leads to a linear system whose size is equal to the number of degrees of freedom of the systemmultiplied by the number of coefficients retained in the response expansion. This approach presentsthree main drawbacks, namely 1) the programming of ad-hoc software to assemble and solve thissystem, 2) the total computational cost, 3) the difficulty to address non linear problems (although anattempt can be found in [4]).In this paper, a collocation scheme is proposed to compute the response coefficients [5]. Thiscalculation reduces to solving selected deterministic finite element problems. This task may becarried out with any finite element code at hand without additional implementation. The threedrawbacks mentioned above are solved ab initio. The approach is illustrated by the analysis of acrack in a pipe weld in the context of fracture mechanics.