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.

翻译：Merriman-Bence-Osher阈值动力学方法是一种高效模拟平均曲率运动的算法，具有易于实现且效率高的优点。本文针对滑移面内的位错动力学，提出了一种阈值动力学方法，其空间算子本质上是各向异性的分数阶拉普拉斯算子。我们证明，当时间步长设为$\Delta t=\varepsilon$时，该阈值位错动力学方法能够正确给出位错速度的前两阶主导项，包括$O(\log\varepsilon)$的局部曲率力、由位错产生的长程应力场所引起的$O(1)$非局部力，以及外加应力导致的力，其中$\varepsilon$为位错核心尺寸。这一结果推广了现有关于相应分数阶拉普拉斯算子的阈值动力学结论——后者在各向同性核函数下仅给出$O(\log\Delta t)$量级的局部曲率速度主导项。我们还提出了一种基于空间变量拉伸的数值方法，用于修正迁移率并重新标定速度，以实现高效精确的模拟，该方法可普遍适用于任何阈值动力学方法。通过位错多种运动模式及相互作用的数值模拟，我们验证了所提阈值位错动力学方法的有效性。