Lippmann-Schwinger solvers for the computational homogenization of materials with pores

We show that under suitable hypotheses on the nonporous material law and a geometric regularity condition on the pore space, Moulinec-Suquet's basic solution scheme converges linearly. We also discuss for which derived solvers a (super)linear convergence behavior may be obtained, and for which such results do not hold, in general. The key technical argument relies on a specific subspace on which the homogenization problem is nondegenerate, and which is preserved by iterations of the basic scheme. Our line of argument is based in the nondiscretized setting, and we draw conclusions on the convergence behavior for discretized solution schemes in FFT-based computational homogenization. Also, we see how the geometry of the pores' interface enters the convergence estimates. We provide computational experiments underlining our claims.