Gradient reproducing kernel particle method

Abstract

This paper presents an innovative formulation of the RKPM (reproducing kernel particle method) pioneered by Liu. A major weakness of the conventional RKPM is in dealing with the derivative boundary conditions. The EFGM (element free Galerkin method) pioneered by Belytschko shares the same difficulty. The proposed RKPM referred to as GRKPM (gradient RKPM), incorporates the first gradients of the function in the reproducing equation. Therefore in three-dimensional space GRKPM consists of four independent types of shape functions. It is due to this feature that the corrected collocation method can be readily generalized and combined with GRKPM to enforce the EBCs (essential boundary conditions), involving both the field quantity and its first derivatives simultaneously. By considering several plate problems it is observed that GRKPM yields solutions of higher accuracy than those obtained using the conventional approach, while for a desired accuracy the number of particles needed in GRKPM is much less than in the traditional methodology.