On non‐stationary polarization methods in FFT‐based computational micromechanics

Abstract

Polarization-type methods are among the fastest solution methods for FFT-based computational micromechanics. However, their performance depends critically on the choice of the reference material. Only for finitely contrasted materials, optimum-selection strategies are known. This work focuses on adaptive strategies for choosing the reference material, details their efficient implementation, and investigates the computational performance. The case of porous materials is explicitly included. As a byproduct, we introduce a suitable convergence criterion that permits a fair comparison to strain-based FFT solvers and Eyre-Milton type implementations.

Computing the effective crack energy of heterogeneous and anisotropic microstructures via anisotropic minimal surfaces

“…If not mentioned otherwise, the governing equations were solved for a prescribed tolerance tol = 10 −4 and the convergence criterion [66]…”

An FFT-based method for computing the effective crack energy of a heterogeneous material on a combinatorially consistent grid

“…As a byproduct, we arrive at an expression for the minimum-cut problem that is much simpler than in Couprie et al [58], see Section 3.1. We discuss the alternating direction method of multipliers (ADMM) in Section 3.2, enriched by various adaptive parameter-selection strategies, recently studied in Schneider [60]. Finally, we demonstrate the capabilities of our approach in applications of industrial size, see Section 4.…”

“…A recent study [60] highlighted the importance of utilizing a damping factor and choosing the penalty factor ρ adaptively. For this purpose, we consider the modified scheme…”

“…with damping δ ∈ (0, 1) and adaptive penalty parameter ρ k . In general, the over-relaxation δ = 1/4 is recommended [60,68,69]. Simple choices for the parameter ρ k are based on the Lorenz-Tran-Dinh scaling [70]…”

“…The algorithm (3.18) was integrated into an existing FFT-based computational homogenization code for thermal conductivity [74], written in Python with Cython extensions (and OpenMP). In the context of small-strain inelasticity, the implementation of the ADMM (3.18) and the memory-efficient computation of the penalty factor is discussed in Schneider [60]. In the same paper, the convergence criterion…”