Convergence of Damped Polarization Schemes for the FFT-Based Computational Homogenization of Inelastic Media With Pores

Porous microstructures represent a challenge for the convergence of FFT-based computational homogenization methods. In this contribution, we show that the damped Eyre–Milton iteration is linearly convergent for a class of nonlinear composites with a regular set of pores, provided the damping factor is chosen between zero and unity. First, we show that an abstract fixed-point method with non-expansive fixed-point operator and non-trivial damping converges linearly, provided the associated residual mapping satisfies a monotonicity condition on a closed subspace. Then, we transfer this result to the framework of polarization schemes and conclude the linear convergence of the damped Eyre–Milton scheme for porous materials. We present general arguments which apply to a class of nonlinear composites and mixed stress-strain loadings, as well. We show that the best contraction estimate is reached for a damping factor of $1/2$, that is, for the polarization scheme of Michel–Moulinec–Suquet, and derive the corresponding optimal reference material. Our results generalize the recent work of Sab and co-workers who showed that an adaptively damped Eyre–Milton scheme leads to linear convergence for a class of linear composites with pores. Finally, we report on computational experiments which support our findings.