An inherently consistent reproducing kernel gradient smoothing framework toward efficient Galerkin meshfree formulation with explicit quadrature

Abstract

A reproducing kernel gradient smoothing framework with explicit quadrature rules is proposed for efficient Galerkin meshfree formulation. In this framework, the meshfree smoothed gradients are formulated via a reproducing kernel representation of the standard gradients of field variables. It is interesting to find that this reproducing kernel construction of smoothed gradients of meshfree shape functions enables an identical satisfaction of the integration constraint derived from Galerkin meshfree formulation. In other words, the integration consistency is an inherent property of the proposed reproducing kernel gradient smoothing methodology regardless of integration schemes. Consequently, without violation of the integration constraint, the conventional normal low order Gauss quadrature rules in finite element analysis now can be used to properly integrate the meshfree stiffness matrix simply through replacing the standard meshfree gradients with the reproducing kernel smoothed gradients at Gauss points. In order to efficiently compute the reproducing kernel smoothed gradients, a set of explicit quadrature rules referring to 2D triangular and 3D tetrahedral integration cells are systematically developed. The total number of sample points associated with these quadrature rules is minimized from a global point of view through introducing an equivalent number of sample points, which implies as many as sample points are shared by neighboring integration cells. The proposed methodology recovers the stabilized conforming nodal integration when linear basis function is employed, while the focus of the present work is multidimensional higher order basis functions such as quadratic and cubic ones. Superior convergence, accuracy as well as efficiency performances of the proposed reproducing kernel gradient smoothing framework are thoroughly demonstrated by a series of numerical examples.