Dislocation transport using an explicit Galerkin/least-squares formulation
Dislocation transport using an explicit Galerkin/least-squares formulation
复制标题
使用显式伽辽金/最小二乘公式进行位错传输
DOI：
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发表时间：
2006
期刊：
影响因子：
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通讯作者：
C. Fressengeas
中科院分区：
文献类型：
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作者：
S. Varadhan;A. Beaudoin;Amit Acharya;C. Fressengeas
引入了场位错力学（FDM）中准线性输运方程的显式Galerkin/最小二乘公式，并将其应用于位错湮灭、多边形位错环展开和Frank-Read源模拟等物理环境下位错密度演化的运动学研究。对相应的线性一维（1D）情况进行了稳定性分析。当Galerkin项和最小二乘项采用等权且形状函数为线性时，该公式可简化为一维方程的Lax-Wendroff有限差分格式。这种条件稳定的方法可以得到对称的常系数良条件方程组，对大规模问题很有吸引力。结果表明，在上述情况下，输运方程可简化为控制几何光学和水平集方法的哈密顿-雅可比方程。这些方程的弱解不是唯一的，通过适当的算法修改，数值方法可以捕获激波和折射波对应的解。
An explicit Galerkin/least-squares formulation is introduced for a quasilinear transport equation in field dislocation mechanics (FDM) and applied to the study of the kinematics of dislocation density evolution in the following physical contexts: annihilation of dislocations, expansion of a polygonal dislocation loop and simulation of a Frank–Read source. Stability analysis is carried out for the corresponding linear one-dimensional (1D) case. The formulation reduces to the Lax–Wendroff finite difference scheme for the 1D equation when equal weighting is used for the Galerkin and least-squares terms and the shape functions are linear. This conditionally stable method leads to a symmetric well-conditioned system of equations with constant coefficients, making it attractive for large-scale problems. It is shown that the transport equation, in the contexts mentioned above simplifies to the Hamilton–Jacobi equations governing geometrical optics and level-set methods. The weak solutions to these equations are not unique, and the numerical method is able to capture solutions corresponding to shock as well as rarefraction waves by appropriate algorithmic modifications.