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

Homogenization techniques usually rely on solving a boundary value problem on the representative element volume (RVE). This problem is generally compl ex to solve when the micro-structure is realistic, especially in three dime nsions. In this paper, we develop two simplified methods providing approxim ate micro-fields over the RVE. These fields yield upper and lower bounds to the exact homogenized property. The a posteriori estimation of the modelin g 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-s cale) plus a correction (micro-scale). The correction is expressed in terms of smooth part through a prolongation operator. This operation is performe d independently on each subdomain and is thus readily parallelizable. Then, the smooth part of the solution is obtained by solving a 'meso' problem in volving all the subdomains. In the numerical experiments, we consider 2-D l inear scalar diffusion problems with periodic boundary conditions on the RV E. The RVE is made of a two-phase material consisting of a matrix in which circular or elliptical inclusions are distributed randomly. Numerical examp les are computed in a parallel computation done on a cluster of 16 Intel P. C.s. (C) 1999 Elsevier Science S.A. All rights reserved.