Generalized reproducing kernel particle method

A generalized reproducing kernel particle method (GRKPM) is formulated based on the reproducing kernel (RK) approximation to improve the efficiency and accuracy of the traditional meshfree method. This is based on the coupling of a primitive function and an enrichment function. The primitive function introduces specified properties, while the enrichment function constitutes reproducing conditions. The meshfree formulation using GRKPM has been developed to: (1) enhance boundary condition treatment; and (2) deal with problems involving inhomogeneous materials. In the first case, the primitive function is designed to recover the nodal Kronecker delta property, and therefore obtain a RK interpolation function. To maintain the convergence properties of the original RK approximation, a mixed interpolation is introduced. In the second case, a function with discontinuous derivative normal to the interface while maintaining continuous derivative in other direction is designed to capture the discontinuous gradient in the solution.; Stabilized conforming nodal integration (SCNI) has been developed to enhance computational efficiency of the Galerkin meshfree methods. This research employs von Neumann analyses to investigate the spatial semi-discretization of Galerkin meshfree methods using SCNI. Two model problems were presented with respect to the normalized phase and group speeds for the wave equation, and normalized diffusivity for the heat equation. Both consistent and lumped mass (capacity) are considered in the study. The results show superior dispersion behavior in meshfree methods integrated by SCNI compared to the Gauss integration (GI) when consistent mass (capacity) matrix is employed in the discretization. For the lumped mass case, SCNI performance is comparable to that of GI with considerable reduction of computational time.; Meshfree discretization of a problem domain and the solution of a partial differential equation is accomplished by a set of particles. This feature makes meshfree method an ideal choice for adaptivity analysis. An effective and efficient error indicator for meshfree adaptivity analysis is formulated through an RK filter and a two-scale decomposition of the meshfree solution. It is observed that the error indicator precisely locates the region of larger error, and provides an indication of the convergence of the numerical solution.