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.
