An accurate and robust FFT-based solver for transient diffusion in heterogeneous materials

The purpose of the present letter is to propose an efficient, accurate and robust FFT-based solver for transient diffusion in heterogeneous materials with “realistic” BC, taking advantage of two recent advances in terms of boundary conditions and finite difference schemes to overcome their actual limitations (periodic BC and spurious oscillations). It is an essential step towards couplings between mechanics and other physics (such as the diffusion of species) through FFT-based solvers. Discrete Trigonometric Transform are used to implement non-periodic boundary conditions, and a finite difference (FD) scheme recently proposed by Finel is advantageously compared to the common hexahedral FD scheme. “Accurate” refers to two properties: accurate in term of locality with a small size of Finite Difference pencil to capture fluctuations around material heterogeneities, and accurate in term of precision with the absence of spurious spatial oscillations (at least in the reported cases with well-separated inclusions). The “robustness” is here associated to the stability of the solver, especially associated to the implicit time integration method. The description of the method focuses on thermal diffusion but applies to any other similar diffusion process (with the same type of parabolic equation). As a by-product, the FD scheme proposed by Finel is introduced in a more general (for mixing finite different schemes) and simple way (no introduction of FCC subgrids), extending its usage to any type of grid parity (not only even grids).