Computational stochastic homogenization of random media elliptic problems using Fourier Galerkin method

Abstract

In mechanics, research topics on probabilistic effects and combination of atomistic, statistical and continuum approaches are being identified as a future research direction. Challenges of complex multiscale interactions and limits of available tools provide an opportunity for probability theory and stochastic processes, so far remained in the background, being brought to the frontier. As far as real problems characterized with non-periodic and random processes are concerned, stochastic homogenization has been mostly tackled with pure mathematical formulations without giving a practical computational recipe. To provide a numerical stochastic homogenization procedure, a recent attempt has been made by Xu and Graham-Brady [A stochastic computation method for evaluation of global and local behavior of random elastic media, Comput. Methods Appl. Mech. Eng. 194(42–44) (2005) 4362–4385] proposing a concept of stochastic representative volume element (SRVE). In this work, the SRVE concept is applied to general divergence-type stochastic partial differential equation (PDE), which is numerically solved with a numerical Fourier Galerkin recipe and the stochastic Galerkin method. This technique provides not only a means of global homogenization but also solution for statistical descriptors (such as variance) of the local solutions to such PDEs. A convergence study is conducted for the computing algorithm of Gaussian random media problems.