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Volume 35, Issue 2
A Threshold Dislocation Dynamics Method

Xiaoxue Qin, Alfonso H.W. Ngan & Yang Xiang

Commun. Comput. Phys., 35 (2024), pp. 273-312.

Published online: 2024-03

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  • 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 $\mathcal{O}(log ε)$ local curvature force and the $\mathcal{O}(1)$ nonlocal force due to the long-range stress field generated by the dislocations as well as the force due to the applied stress, where $ε$ is the dislocation core size, if the time step is set to be $∆t = ε.$ This generalizes the available result of threshold dynamics with the corresponding fractional Laplacian, which is on the leading order $\mathcal{O}(log \Delta t)$ 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.

  • AMS Subject Headings

65R20, 65N12, 74A50, 35R11

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-35-273, author = {Qin , XiaoxueNgan , Alfonso H.W. and Xiang , Yang}, title = {A Threshold Dislocation Dynamics Method}, journal = {Communications in Computational Physics}, year = {2024}, volume = {35}, number = {2}, pages = {273--312}, 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 $\mathcal{O}(log ε)$ local curvature force and the $\mathcal{O}(1)$ nonlocal force due to the long-range stress field generated by the dislocations as well as the force due to the applied stress, where $ε$ is the dislocation core size, if the time step is set to be $∆t = ε.$ This generalizes the available result of threshold dynamics with the corresponding fractional Laplacian, which is on the leading order $\mathcal{O}(log \Delta t)$ 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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2023-0188}, url = {http://global-sci.org/intro/article_detail/cicp/22942.html} }
TY - JOUR T1 - A Threshold Dislocation Dynamics Method AU - Qin , Xiaoxue AU - Ngan , Alfonso H.W. AU - Xiang , Yang JO - Communications in Computational Physics VL - 2 SP - 273 EP - 312 PY - 2024 DA - 2024/03 SN - 35 DO - http://doi.org/10.4208/cicp.OA-2023-0188 UR - https://global-sci.org/intro/article_detail/cicp/22942.html KW - Dislocation dynamics, threshold dynamics method, nonlocal velocity, anisotropic mobility, variable stretching. AB -

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 $\mathcal{O}(log ε)$ local curvature force and the $\mathcal{O}(1)$ nonlocal force due to the long-range stress field generated by the dislocations as well as the force due to the applied stress, where $ε$ is the dislocation core size, if the time step is set to be $∆t = ε.$ This generalizes the available result of threshold dynamics with the corresponding fractional Laplacian, which is on the leading order $\mathcal{O}(log \Delta t)$ 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.

Xiaoxue Qin, Alfonso H.W. Ngan & Yang Xiang. (2024). A Threshold Dislocation Dynamics Method. Communications in Computational Physics. 35 (2). 273-312. doi:10.4208/cicp.OA-2023-0188
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