A threshold dislocation dynamics method

TL;DR

This paper introduces a novel threshold dislocation dynamics method utilizing an anisotropic fractional Laplacian, accurately capturing both local and nonlocal forces affecting dislocation motion, validated through numerical simulations.

Contribution

It extends the threshold dynamics framework to dislocation motion with anisotropic fractional Laplacian, including correction techniques for efficient and precise simulations.

Findings

Accurately captures local and nonlocal dislocation forces

Generalizes threshold dynamics with fractional Laplacian

Validated through numerical simulations of dislocation interactions

Abstract

The Merriman-Bence-Osher threshold dynamics method is an efficient algorithm to simulate the motion by mean curvature. It has the advantages of being easy to implement and with high efficiency. In this paper, we propose a threshold dynamics method for dislocation dynamics in a slip plane, in which the spatial operator is essentially an anisotropic fractional Laplacian. We show that this threshold dislocation dynamics method is able to give { two correct leading orders} in dislocation velocity, including both the $O(\log\varepsilon)$ local curvature force and the $O(1)$ nonlocal force due to the long-range stress field generated by the dislocations as well as the force due to the applied stress, where $\varepsilon$ is the dislocation core size, { if the time step is set to be $\Delta t=\varepsilon$. This generalizes the available result of threshold dynamics with the corresponding fractional Laplacian, which is on the leading order $O(\log\Delta t)$ local curvature velocity under the isotropic kernel.} We also propose a numerical method based on spatial variable stretching to correct the mobility and to rescale the velocity for efficient and accurate simulations, which can be applied generally to any threshold dynamics method. We validate the proposed threshold dislocation dynamics method by numerical simulations of various motions and interaction of dislocations.