The Reproducing Singularity and Polynomial Particle Shape Functions for Meshless Methods

无网格方法如再生核粒子法（RKPM）和单元划分有限元法（PUFEM）比传统有限元法更有效。然而，这些方法存在以下局限性：(1)重构核粒子（RKP）形状函数的构造比较复杂。单位配分函数在PUFEM中是必不可少的，常用的Shepard单位配分函数一般是复杂的有理函数。因此，无网格方法需要较长的计算时间才能达到合理的精度；(2) RKPM和PUFEM在处理基本边界条件方面存在困难。为了弥补这些限制，PI构造了高度规则的分段多项式再现多项式粒子（RPP）形状函数，该函数满足Kronecker特性。此外，他还构造了简单的分段多项式PU函数。然而，如果解包含奇点（如裂纹奇点），则RPP形状函数不太实用。为了在无网格方法框架下处理奇异性问题，引入了用于二维奇异性问题的再现奇异粒子（RSP）形状函数。现在，PI提出将他的二维RPP和RSP形状函数扩展到三维奇异问题的三维情况。在美国，每年因骨折造成的损失是一个天文数字。金属疲劳被认为是最近几起航空事故和桥梁倒塌的可能原因。此外，老化的桥梁和飞机的安全是一个全国性的问题。材料失效是其他工程结构（核电站、水电站等）的主要问题。因此，为了进行有效的检查和预防性维护计划，需要进行准确的断裂分析。本研究旨在为裂纹材料提供精确的应力分析。这些分析对于精确估计材料中的三维裂纹扩展是必不可少的。实际上，所提出的研究结果将适用于老化桥梁、海上石油平台和许多其他需要密切监测结构完整性的工程应用的有效维护。最终，这项研究将对公众安全和环境产生直接影响。

Meshless methods such as Reproducing Kernel Particle method (RKPM) and Partition of Unity Finite Element Method (PUFEM), are much more effective than the conventional FEM. However, these methods have the following limitations: (1) the constructions of Reproducing Kernel Particle (RKP) Shape functions are complicate. The partition of unity (PU) functions is essential in PUFEM and the popular Shepard PU functions are generally complicated rational functions. Thus, Meshless methods require a lengthy computing time for reasonable accuracy; (2) RKPM and PUFEM have difficulties in dealing with essential boundary conditions. To compensate for these limitations, the PI constructed the highly regular piecewise polynomial Reproducing Polynomial Particle (RPP) shape functions that satisfy the Kronecker delta property. Furthermore, he also constructed simple piecewise polynomial PU functions for arbitrary partitioned patches. Nevertheless, the RPP shape functions are not very practical if the solution contains singularities (such as crack singularity). To deal with singularities in the framework of Meshless Methods, the PI introduced the Reproducing Singularity Particle (RSP) shape functions for two dimensional singularity problems. Now, the PI proposes to extend his two dimensional RPP and RSP shape functions to the three dimensional cases for three dimensional singularity problems. The annual cost of fracture-related damage in the United States is an astronomical amount. Metal fatigue has been cited as the probable cause of several recent airline accidents and the bridge collapses. Moreover, the safety of aging bridges and airliners is a national concern. Material failures are a major concern for other engineering structures (nuclear power plants, hydroelectric dams, etc). Thus, for effective inspection and preventive maintenance programs, an accurate fracture analysis is needed. The proposed research is to provide accurate stress analysis of cracked materials. These analyses are essential for precise estimates of three dimensional crack propagation in materials. Practically, the results of the proposed research will be applicable to the effective maintenance of aging bridges, off-shore oil platforms, and numerous other engineering applications where structural integrity should be closely monitored. Ultimately, this research will have direct impacts on the safety of the public and the environment.