Dislocation transport using a time-explicit Runge–Kutta discontinuous Galerkin finite element approach
Dislocation transport using a time-explicit Runge–Kutta discontinuous Galerkin finite element approach
复制标题
使用时间显式龙格-库塔不连续伽辽金有限元方法进行位错传输
DOI：
10.1088/1361-651x/ac44a7
复制
发表时间：
2021
影响因子：
1.8
通讯作者：
J. Bleyer
中科院分区：
文献类型：
--
作者：
M. Upadhyay;J. Bleyer
提出了一种时间显式龙格-库塔不连续伽辽金（RKDG）有限元格式来求解三维位错输运初始边值问题。作为该问题核心的位错密度输运方程是一阶非定常平流-反应型双曲偏微分方程；DG方法非常适合求解缺乏任何扩散项的方程。RKDG方案的开发遵循线法方法。首先，利用上旋DG方法进行空间半离散化，得到一个常微分方程组。然后，采用显式RK格式进行时间离散化求解。采用强稳定性保持方法对一阶（正演欧拉）、二阶和三阶RK方法进行了RKDG方案的三维数值实现。这些实现为光滑解提供了（准）最优收敛速率。斜率限制器用于防止由粗解的高阶空间近似（多项式度大于或等于1）引起的虚假吉布斯振荡。为了了解RKDG方案的关键参数对螺旋位错输运模拟中预测溶液稳定性的影响，进行了参数化研究。然后，模拟了两个相反符号螺位错的湮灭和一个多边形位错环的扩展。RKDG格式能够解决位错湮灭过程中产生的激波而没有任何伪振荡，并且能够以非常低的数值扩散预测棱柱形环的扩展。结果表明，与基于连续伽辽金有限元法或快速傅立叶变换法的现有方法相比，该方法具有更强的鲁棒性和准确性。
A time-explicit Runge–Kutta discontinuous Galerkin (RKDG) finite element scheme is proposed to solve the dislocation transport initial boundary value problem in 3D. The dislocation density transport equation, which lies at the core of this problem, is a first-order unsteady-state advection–reaction-type hyperbolic partial differential equation; the DG approach is well suited to solve such equations that lack any diffusion terms. The development of the RKDG scheme follows the method of lines approach. First, a space semi-discretization is performed using the DG approach with upwinding to obtain a system of ordinary differential equations in time. Then, time discretization is performed using explicit RK schemes to solve this system. The 3D numerical implementation of the RKDG scheme is performed for the first-order (forward Euler), second-order and third-order RK methods using the strong stability preserving approach. These implementations provide (quasi-)optimal convergence rates for smooth solutions. A slope limiter is used to prevent spurious Gibbs oscillations arising from high-order space approximations (polynomial degree ⩾ 1) of rough solutions. A parametric study is performed to understand the influence of key parameters of the RKDG scheme on the stability of the solution predicted during a screw dislocation transport simulation. Then, annihilation of two oppositely signed screw dislocations and the expansion of a polygonal dislocation loop are simulated. The RKDG scheme is able to resolve the shock generated during dislocation annihilation without any spurious oscillations and predict the prismatic loop expansion with very low numerical diffusion. These results indicate that the proposed scheme is more robust and accurate in comparison to existing approaches based on the continuous Galerkin finite element method or the fast Fourier transform method.