Asymptotics and numerical efficiency of the Allen-Cahn model for phase interfaces with low energy in solids

TL;DR

This paper derives an asymptotic expansion for the interface speed in the Allen-Cahn model for phase transformations in solids, revealing how low interface energy affects simulation accuracy and computational effort.

Contribution

It provides a second-order asymptotic expansion of the interface speed and analyzes the model error dependence on interface parameters.

Findings

Model error proportional to interface width divided by interface energy

Low interface energy requires very small interface width for accurate simulations

Adaptive mesh refinement is necessary for efficient simulations

Abstract

We study how the propagation speed of interfaces in the Allen-Cahn phase field model for phase transformations in solids consisting of the elasticity equations and the Allen-Cahn equation depends on two parameters of the model. The two parameters control the interface energy and the interface width but change also the interface speed. To this end we derive an asymptotic expansion of second order for the interface speed, called the kinetic relation, and prove that it is uniformly valid in both parameters. As a consequence we show that the model error is proportional to the interface width divided by the interface energy. We conclude that simulations of interfaces with low interface energy based on this model require a very small interface width, implying a large numerical effort. Effective simulations thus need adaptive mesh refinement or other advanced techniques. This version of the paper contains the proofs of Theorem 4.5 and Lemma 5.8, which are omitted in the version published in Continuum Mechanics and Thermodynamics.