This study presents a numerical modelling framework based on complex variable meshless methods, which can accurately and efficiently track arbitrary crack paths in two-dimensional linear elastic solids. The key novelty of this work is that the proposed meshless modelling scheme enables a direct element-free approximation for the solutions of linear elastic fracture mechanics problems. The complex variable moving least-squares approximation with a group of simple complex polynomial basis is applied to implement this meshless model, with which the fracture problems with both stationary or progressive cracks are considered and studied. The effects of choosing different definitions of weighted complex variable error norm and different forms of complex polynomial basis on the computational accuracy of crack tip fields and crack paths prediction are analysed and discussed. Five benchmark numerical examples were studied to demonstrate the superiority of the present complex variable meshless framework over a standard element-free Galerkin method in tracking arbitrary crack paths in two-dimensional elastic solids.

本研究提出了一种基于复变量无网格方法的数值建模框架，可以准确有效地跟踪二维线弹性实体中的任意裂纹路径。这项工作的关键新颖之处在于，所提出的无网格建模方案能够为线性弹性断裂力学问题的解决方案提供直接的无单元近似。应用具有一组简单复多项式基的复变量移动最小二乘逼近来实现该无网格模型，该模型考虑和研究了具有静止或渐进裂纹的断裂问题。分析讨论了选择不同的加权复变量误差范数和不同形式的复多项式基对裂纹尖端场计算精度和裂纹路径预测的影响。研究了五个基准数值示例，以证明本复杂变量无网格框架在跟踪二维弹性固体中任意裂纹路径方面优于标准无单元 Galerkin 方法。