A Discrete Sine‐Cosine Transforms Galerkin Method for the Conductivity of Heterogeneous Materials With Mixed Dirichlet/Neumann Boundary Conditions

Abstract

This work aims at developing a numerical method for conductivity problems in heterogeneous media subjected to mixed Dirichlet/Neumann boundary conditions. The method relies on a fixed‐point iterative solution of an auxiliary problem obtained by a Galerkin discretization using an approximation space spanned by mixed cosine‐sine series. The solution field is written as a known term verifying the boundary conditions and an unknown term described by cosine‐sine series, having no contribution on the boundary. Discrete sine‐cosine transforms, of Type I and III depending on the boundary conditions, are used to approximate the elementary integrals involved in the Galerkin formulation, which makes the method relying on the numerical complexity of fast Fourier transforms. The method is finally assessed in a problem of a composite material.