Stochastic Finite Element Expansion for Random Media

A new method for the solution of problems involving material variability is proposed. The material property is modeled as a stochastic process. The method makes use of the Karhunen-Loeve expansion to represent the random material property. The expansion is a representation of the process in terms of a finite set of uncorrelated random variables. The resulting formulation is compatible with the finite element method. A Neumann expansion scheme is subsequently employed to obtain a convergent expansion of the response process. The response is thus obtained as a homogeneous multivariate polynomial in the uncorrelated random variables. From this representation various statistical quantities may be derived. The usefulness of the proposed method, in terms of accuracy and efficiency, is exemplified by considering a cantilever beam with random rigidity. The derived results pertaining to the second-order statistics of the response are found in good agreement with those obtained by a Monte Carlo simulation solution of the problem.