Reformulation and influence of discretization in Fourier-series-based FFT methods for heterogeneous materials

The aim of this work is to revisit the formulation of FFT-based methods for heterogeneous materials in the periodic setting. These numerical methods are based on the iterative resolution of an auxiliary problem involving a reference homogeneous material and a polarization tensor. Assuming a description of the fields by Fourier series, we show the equivalence between three discretization approaches based either on the strong and the weak formulation of the problem. A special emphasis is put on the representation of the local fields described by Fourier series including the possibility of using non-uniform grid. Numerical experiments are performed on a model problem of conductivity with a checkerboard microstructure for which an analytical solution allows to assess the effect of Fourier modes together with the grid discretization. The occurrence of oscillations is finally addressed by studying (generalized) discrete Green operators (still in the context of Fourier series) and interface spreading approaches based on smoothing techniques.