An FFT-based method for computing weighted minimal surfaces in microstructures with applications to the computational homogenization of brittle fracture

Cell formulae for the effective crack resistance of a heterogeneous medium obeying Francfort-Marigo's formulation of linear elastic fracture mechanics have been proved recently, both in the context of periodic and stochastic homogenization. This work proposes a numerical strategy for computing the effective, possibly anisotropic, crack resistance of voxelized microstructures using the fast Fourier transform (FFT). Based on Strang's continuous minimum cut-maximum flow duality, we explore a primal-dual hybrid gradient method for computing the effective crack resistance, which may be readily integrated into an existing FFT-based code for homogenizing thermal conductivity. We close with demonstrative numerical experiments.