An efficient nesting sub-domain gradient smoothing integration algorithm with quadratic exactness for Galerkin meshfree methods

Abstract

Abstract An efficient nesting sub-domain gradient smoothing integration algorithm is proposed for Galerkin meshfree methods with particular reference to the quadratic exactness. This approach is consistently built upon the smoothed gradients of meshfree shape functions defined on two-level nesting triangular sub-domains, where each integration cell consists of four equal-area nesting sub-domains. First, a rational measure is designed to evaluate the error of the gradient smoothing integration for the stiffness matrix. Thereafter through a detailed analysis of the gradient smoothing integration errors associated with the two-level nesting triangular sub-domains, a quadratically exact algorithm for the stiffness matrix integration is established through optimally combining the contributions from the two-level nesting sub-domains. Meanwhile, the integration of force terms consistent with the stiffness integration is presented in order to ensure exact quadratic solutions within the Galerkin formulation. It is noted that the proposed approach with quadratic exactness shares the same foundation as the well-established stabilized conforming nodal integration method with linear exactness, i.e., the smoothed derivatives of meshfree shape functions are directly built upon the values of meshfree shape functions on the boundary of the integration cells and the time consuming derivative computations are completely avoided. Moreover, the present formulation has even less integration sampling points than the four point Gauss integration. Numerical examples show very favorable performance regarding the accuracy and efficiency for the proposed approach.