Improved Bounds on the Elastic Properties of Porous Media with the Variational Equivalent Inclusion Method (VEIM)

Abstract

The equivalent inclusion method (EIM) was initially proposed by Moschovidis and Mura [12] to find an approximate solution to the problem of two ellipsoidal inhomogeneities embedded in an infinite medium. More recently, it has been applied to large assemblies of inhomogeneities [7]. Although less accurate than traditional (e.g. finite elements) numerical techniques, the equivalent inclusion method is an attractive solution for numerical homogenization problems. Indeed, it leads to relatively small linear systems, and does not require time-consuming preliminaries (such as mesh creation). However, the standard EIM has some shortcomings which are mainly due to the fact that point collocation techniques are used to discretize the underlying integral equation [6, 12]. In contrast, the Variational Equivalent Inclusion Method (VEIM) introduced here uses Galerkin techniques to carry out this discretization. In this paper, we briefly recall the fundamentals of the standard EIM. We then present the VEIM and show that it can lead to rigorous bounds on the macroscopic properties. Finally, some illustrations of the method are presented.