In this paper, the localized method of fundamental solutions (LMFS), a recently developed meshless collocation method, is applied to the numerical solution of problems with cracks in linear elastic fracture mechanics. The main idea of the LMFS is to divide the entire computational domain into a set of overlapping sub-domains, and in each sub-domain, the classical MFS formulation and the moving least squares (MLS) technique are applied to form the corresponding local system of equations. The LMFS will finally generate a banded and sparse matrix system which makes the method very attractive for large-scale engineering applications. To deal with in-plane crack problems, an enriched LMFS approach is proposed by combining the LMFS formulation for linear elasticity problems and a set of enrichment functions which take into account the asymptotic behavior of the near-tip displacement and stress fields. The enriched LMFS can significantly improve the computational accuracy of the calculation of stress intensity factor (SIF) of the cracked materials, even with a very coarse LMFS node distribution. Several benchmark numerical examples are presented to illustrate the accuracy and efficiency of the proposed method.

在本文中，基本解的局部化方法（LMFS）是一种最近发展起来的无网格配置方法，被应用于线弹性断裂力学中裂纹问题的数值求解。LMFS 的主要思想是将整个计算域划分为一组重叠的子域，并在每个子域中应用经典的 MFS 公式和移动最小二乘法 (MLS) 技术形成相应的局部系统的方程。LMFS 最终将生成带状稀疏矩阵系统，这使得该方法对大规模工程应用非常有吸引力。为了处理面内裂纹问题，通过将线性弹性问题的 LMFS 公式和考虑近尖端位移和应力场的渐近行为的一组富集函数相结合，提出了一种富集 LMFS 方法。富集的 LMFS 可以显着提高裂纹材料应力强度因子 (SIF) 计算的计算精度，即使 LMFS 节点分布非常粗糙。给出了几个基准数值例子来说明所提出方法的准确性和效率。