Optimal FFT-accelerated Finite Element Solver for Homogenization

We propose a matrix-free finite element (FE) homogenization scheme that is considerably more efficient than generic FE implementations. The efficiency of our scheme follows from a preconditioned well-scaled reformulation allowing for the use of the conjugate gradient or similar iterative solvers. The geometrically-optimal preconditioner -- a discretized Green's function of a periodic homogeneous reference problem -- has a block-diagonal structure in the Fourier space which permits its efficient inversion using the fast Fourier transform (FFT) techniques for generic regular meshes. This implies that the scheme scales as $\mathcal{O}(n \log(n))$ like FFT, rendering it equivalent to spectral solvers in terms of computational efficiency. However, in contrast to classical spectral solvers, the proposed scheme works with FE shape functions with local supports and is free of the Fourier ringing phenomenon. We showcase that the scheme achieves the number of iterations that are almost independent of spatial discretisation and scales mildly with the phase contrast. Additionally, we discuss the equivalence between our displacement-based scheme and the recently proposed strain-based homogenization technique with finite-element projection.