An Efficient RVE-EMC Approach for Multiscale Equations of Random Heterogeneous Materials

Resolving multiscale equations in heterogeneous materials presents significant computational challenges arising from rapid spatial oscillations and random variations in solutions induced by intricate microstructural configurations. This study proposes and analyzes a two-stage stochastic homogenization framework designed to efficiently compute homogenized solutions for multiscale diffusion equations. The methodology unfolds through two distinct phases. In the first stage, each realization of the random microstructure undergoes spatial homogenization through a representative volume element (RVE)-based approach, effectively replacing the original multiscale diffusion equation with a random counterpart featuring piecewise constant coefficients. Through ensemble-based Monte Carlo (EMC) averaging of diffusion coefficients, we reformulate the random diffusion equation into a deterministic diffusion problem with a random source term in the second stage. This critical reformulation enables the implementation of an efficient fixed-point iteration scheme for solving the resultant constant-coefficient diffusion equation. In addition, the convergence of the homogenized solution to the solution of multiscale diffusion equation is proved. Numerical examples are provided to demonstrate the ability and accuracy of the proposed method.