Simplified methods and a posteriori error estimation for the homogenization of representative volume elements (RVE)

Abstract

Homogenization techniques usually rely on solving a boundary value problem on the representative element volume (RVE). This problem is generally complex to solve when the micro-structure is realistic, especially in three dimensions. In this paper, we develop two simplified methods providing approximate micro-fields over the RVE. These fields yield upper and lower bounds to the exact homogenized property. The a posteriori estimation of the modeling error introduced by the simplified methods is thus straightforward. Both simplified methods are based on a two-scale strategy. The RVE is decomposed into subdomains over which the solution is sought as a smooth part (meso-scale) plus a correction (micro-scale). The correction is expressed in terms of smooth part through a prolongation operator. This operation is performed independently on each subdomain and is thus readily parallelizable. Then, the smooth part of the solution is obtained by solving a ‘meso’ problem involving all the subdomains. In the numerical experiments, we consider 2-D linear scalar diffusion problems with periodic boundary conditions on the RVE. The RVE is made of a two-phase material consisting of a matrix in which circular or elliptical inclusions are distributed randomly. Numerical examples are computed in a parallel computation done on a cluster of 16 Intel P.C.s.