A FFT-based numerical scheme for the transient conductivity of heterogeneous materials with non-periodic boundary conditions

The aim of this work is to develop FFT-based solvers for transient diffusion in heterogeneous materials subjected to non-periodic (Dirichlet/Neumann) boundary conditions. We focus on a problem of thermal conductivity and derive a theta-method which includes an implicit solver for transient thermal conductivity in heterogeneous materials. The method is based on a fixed-point iterative solution of an auxiliary problem obtained by a Galerkin discretization using an approximation space based on mixed sine–cosine series. The solution field is decomposed as a known term verifying the boundary conditions and a fluctuation (unknown) term described by appropriate sine–cosine series. The solution of the auxiliary problem involves discrete sine–cosine transforms, of type I and III, which makes the solver rely on the computational complexity of fast Fourier transforms. The method is applied to the prediction of transient thermal fields in a composite material subjected to non periodic boundary conditions.