The Reproducing Singularity and Polynomial Particle Shape Functions for Meshless Methods

无网格法如再生核粒子法(RKPM)和单位划分有限元法(PUFE)比传统的有限元方法更有效。但这些方法都存在以下局限性：(1)再生核粒子(RKP)形状函数构造复杂。单位分解(PU)函数是PUFE中的基本问题，而目前流行的Shepard单位分解函数一般都是复杂的有理函数。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PUFE在处理本质边界条件方面存在困难。为了弥补这些局限性，PI构造了满足Kronecker Delta性质的高度正则分段多项式再生多项式粒子(RPP)形函数。此外，他还构造了任意分块曲面的简单分段多项式PU函数。然而，如果解包含奇点(如裂纹奇点)，则RPP形函数不是很实用。为了在无网格法框架内处理奇异性，PI引入了二维奇异性问题的再生奇异性粒子(RSP)形函数。现在，PI建议将他的二维RPP和RSP形函数推广到三维情形，用于三维奇性问题。在美国，每年与骨折相关的损害成本是一个天文数字。金属疲劳被认为是最近几起航空事故和大桥坍塌的可能原因。此外，老化桥梁和客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，为了有效的检查和预防性维护计划，需要进行准确的断裂分析。这项研究的目的是为裂纹材料提供准确的应力分析。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，拟议的研究结果将适用于老化桥梁、近海石油平台的有效维护，以及应密切监测结构完整性的许多其他工程应用。最终，这项研究将对公众和环境的安全产生直接影响。

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.