用点刚度矩阵的简单Galerkin无网格法--脆性点法（FPM）求解复杂区域中的二维线弹性裂纹扩展问题
A Simple Galerkin Meshless Method, the Fragile Points Method (FPM) Using Point Stiffness Matrices, for 2D Linear Elastic Problems in Complex Domains with Crack and Rupture Propagation
摘要：
脆弱点法（FPM）是一种非常简单的伽辽金无网格法，它只使用基于点的间断试函数和检验函数，而不使用基于单元的试函数和检验函数。在本研究中，借由探讨点刚度矩阵与数值通量修正的概念，详细地给出线弹性问题的有限体积法的演算公式。文中还详细讨论了有限单元法在模拟复结构的变形、模拟复杂裂纹扩展和断裂发展等方面的优越性。基准问题的变形和应力分析的数值例子，以及具有复杂几何形状的现实结构，表明所提出的FPM的准确性，效率和鲁棒性。在这项研究中也给出了裂纹萌生和扩展的模拟，证明了本FPM在模拟复杂的断裂和断裂现象的优势。FPM中的裂纹和断裂扩展建模无需像标准、扩展或广义FEM中那样重新划分网格或增加测试函数。FPM在碰撞、侵彻等极端问题的模拟将在我们以后的论文中讨论。
英文摘要：
The Fragile Points Method (FPM) is an elementarily simple Galerkin meshless method, employing Point-based discontinuous trial and test functions only, without using element-based trial and test functions. In this study, the algorithmic formulations of FPM for linear elasticity are given in detail, by exploring the concepts of point stiffness matrices and numerical flux corrections. Advantages of FPM for simulating the deformations of complex structures, and for simulating complex crack propagations and rupture developments, are also thoroughly discussed. Numerical examples of deformation and stress analyses of benchmark problems, as well as of realistic structures with complex geometries, demonstrate the accuracy, efficiency and robustness of the proposed FPM. Simulations of crack initiation and propagations are also given in this study, demonstrating the advantages of the present FPM in modeling complex rupture and fracture phenomena. The crack and rupture propagation modeling in FPM is achieved without remeshing or augmenting the trial functions as in standard, extended or generalized FEM. The simulation of impact, penetration and other extreme problems by FPM will be discussed in our future papers.
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