Rank-three laminates are good approximants of the optimal microstructures for the diffusion problem in dimension two

Abstract

In two dimensions there are optimal bounds for the effective conductivity of arbitrary mixtures of two heat conducting materials: one isotropic and the other anisotropic; used in fixed volume fractions and allowing for rotations. Some of those bounds involve a rank-two lamination, but others involve a microstructure of coated disks. We create a region of laminates of rank at most three, which gives a very good approximation of the optimal bound if the starting material has a moderate degree of anisotropy. We also study the stability under homogenization of this region, meaning that whenever one homogenizes a mixture of two materials belonging to it, the effective diffusion tensor also belongs to this region. This is done to show that the region we create cannot be easily enlarged. (C) 2003 Elsevier Ltd. All rights reserved.