A numerical convergence study regarding homogenization assumptions in phase field modeling

Abstract

SummaryFrom a mathematical point of view, phase field theory can be understood as a smooth approximation of an underlying sharp interface problem. However, the smooth phase field approximation is not uniquely defined. Different phase field approximations are known to converge to the same sharp interface problem in the limiting case—if the thickness of the diffuse interface converges to zero. In this respect and focusing on numerics, a question that naturally arises is as follows: What are the convergence rates of the different phase field models? The paper deals precisely with this question for a certain family of phase field models. The focus is on an Allen–Cahn‐type phase field model coupled to continuum mechanics. This model is rewritten into a unified variational phase field framework that covers different homogenization assumptions in the diffuse interfaces: Voigt/Taylor, Reuss/Sachs and more sound homogenization approaches falling into the range of rank‐one convexification. It is shown by means of numerical experiments that the underlying phase field model—that is, the homogenization assumption in the diffuse interface—indeed influences the convergence rate. Copyright © 2017 John Wiley & Sons, Ltd.