A fast numerical method for the conductivity of heterogeneous media with Dirichlet boundary conditions based on discrete sine–cosine transforms
Léo Morin, Joseph Paux
Abstract
The aim of this work is to develop a fast numerical method for solving conductivity problems in heterogeneous media subjected to Dirichlet boundary conditions. The method is based on a fixed-point iterative solution to an integral Lippmann–Schwinger type equation that is obtained by a Galerkin discretization of the cell problem using an approximation space spanned by sine series; the solution field is split between a known term verifying the Dirichlet boundary conditions and an unknown term described by sine series, which is null on the boundary by construction. With a suitable numerical integration scheme of the elementary integrals involved in the Galerkin formulation, based on discrete sine–cosine transforms, the method relies on the numerical complexity of fast Fourier transforms. The method is assessed in several problems including composite materials and fibrous networks.