A simplified threshold dynamics algorithm for isotropic surface energies

TL;DR

This paper introduces a simplified threshold dynamics algorithm for modeling isotropic surface energies, removing the need for retardation functions while maintaining efficiency and allowing for complex surface tension and mobility configurations.

Contribution

The authors develop a more straightforward threshold dynamics algorithm that uses linear combinations of Gaussians, simplifying implementation without sacrificing generality or efficiency.

Findings

The new algorithm maintains efficiency with only two Gaussian convolutions.

It allows specifying distinct surface tensions and mobilities in multi-phase networks.

Counterexamples show cases where convergence may fail.

Abstract

We present a simplified version of the threshold dynamics algorithm given in the work of Esedoglu and Otto (2015). The new version still allows specifying N-choose-2 possibly distinct surface tensions and N-choose-2 possibly distinct mobilities for a network with N phases, but achieves this level of generality without the use of retardation functions. Instead, it employs linear combinations of Gaussians in the convolution step of the algorithm. Convolutions with only two distinct Gaussians is enough for the entire network, maintaining the efficiency of the original thresholding scheme. We discuss stability and convergence of the new algorithm, including some counterexamples in which convergence fails. The apparently convergent cases include unequal surface tensions given by the Read \& Shockley model and its three dimensional extensions, along with equal mobilities, that are a very common choice in computational materials science.