The Merriman-Bence-Osher threshold dynamics method is an efficient algorithm to simulate the motion by mean curvature, in which two steps of convolution with diffusion kernel and thresholding alternate. It has the advantages of being easy to implement and with high efficiency. In this paper, we propose an efficient threshold dynamics method for dislocation dynamics in a slip plane. We show that this proposed threshold dislocation dynamics method is able to give correct two 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, where $\varepsilon$ is the dislocation core size. This is different from the available threshold dynamics methods in the literature which only give the leading order local velocities associated with mean curvature or its anisotropic generalizations of the moving fronts. We also propose a numerical method based on spatial variable stretching to overcome the numerical limitations brought by physical settings in this threshold dislocation dynamics method. Specifically, this variable stretching method is able to correct the mobility and to rescale the velocity, 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通过扩散核卷积与阈值交替实现平均曲率运动的高效模拟算法，具有易于实现和计算效率高的优势。本文针对滑移面内的位错动力学问题，提出了一种高效的阈值动力学方法。研究表明，该阈值位错动力学方法能够正确给出位错速度的两个主导阶次，包括由位错产生的长程应力场所致的$O(\log \varepsilon)$局部曲率力和$O(1)$非局部力（其中$\varepsilon$为位错芯尺寸）。这与现有阈值动力学方法不同——现有方法仅能给出与平均曲率或各向异性推广相关的移动界面主导阶次局部速度。针对该阈值位错动力学方法中物理设置带来的数值限制，我们提出了一种基于空间变量拉伸的数值方法。具体而言，该变量拉伸方法能修正迁移率并重新标度速度，可普遍适用于各类阈值动力学方法。最后，通过多种位错运动与相互作用的数值模拟，验证了所提阈值位错动力学方法的有效性。