Some studies of the reproducing kernel particle method

Interests in meshfree (or meshless) methods have grown rapidly in the recent years in solving boundary value problems arising in mechanics, especially in dealing with difficult problems involving large deformation, moving discontinuities, etc. Rigorous error estimates of a meshfree method, the reproducing kernel particle method (RKPM), have been theoretically derived and experimentally tested in [13,14]. In this paper, we provide some further studies of the meshfree method. First, improved local meshfree interpolation error estimates are derived. Second, a new and efficient technique is proposed to implement Dirichlet boundary conditions. Numerical experiments indicate that optimal convergence orders are maintained for Dirichlet problems over higher dimensional domains. Finally, the meshfree method is applied to solve 4th-order equations. Since the smoothness of meshfree functions is the same as that of the window function, the meshfree method is a natural choice for conforming approximation of higher-order differential equations.