Analysis of in-plane crack problems using the localized method of fundamental solutions

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 inplane 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.