A Normalizing Field Flow Induced Two-Stage Stochastic Homogenization Method for Random Composite Materials

Abstract

The traditional stochastic homogenization method can obtain homogenized solutions of elliptic problems with stationary random coefficients. However, many random composite materials in scientific and engineering computing do not satisfy the stationary assumption. To overcome the difficulty, we propose a normalizing field flow induced two-stage stochastic homogenization method to efficiently solve the random elliptic problem with non-stationary coefficients. By applying the two-stage stochastic homogenization method, the original elliptic equation with random and fast oscillatory coefficients is approximated as an equivalent elliptic equation, where the equivalent coefficients are obtained by solving a set of cell problems. Without the stationary assumption, the number of cell problems is large and the corresponding computational cost is high. To improve the efficiency, we apply the normalizing field flow model to learn a reference Gaussian field for the random equivalent coefficients based on a small amount of data, which is obtained by solving the cell problems with the finite element method. Numerical results demonstrate that the newly proposed method is efficient and accurate in tackling high dimensional partial differential equations in composite materials with complex random microstructures.