A stable and accurate X-FFT solver for linear elastic homogenization problems in 3D

Abstract: Although FFT-based methods are renowned for their numerical efficiency and stability, traditional discretizations fail to capture material interfaces that are not aligned with the grid, resulting in suboptimal accuracy. To address this issue, the work at hand introduces a novel FFT-based solver that achieves interface-conforming accuracy for three-dimensional mechanical problems. More precisely, we integrate the extended finite element (X-FEM) discretization into the FFT-based framework, leveraging its ability to resolve discontinuities via additional shape functions. We employ the modified abs(olute) enrichment and develop a preconditioner based on the concept of strongly stable GFEM, which mitigates the conditioning issues observed in traditional X-FEM implementations. Our computational studies demonstrate that the developed X-FFT solver achieves interface-conforming accuracy, numerical efficiency, and stability when solving three-dimensional linear elastic homogenization problems with smooth material interfaces.