An efficient and quadratic accurate linear-gradient smoothing integration scheme for meshfree Galerkin methods

Abstract

To address domain integration of meshfree Galerkin methods with quadratic base, we propose an efficient and accurate linear-gradient smoothing integration (LGSI) scheme in this study. In our scheme, the smoothed gradient is expressed as the linear polynomial form with respect to the center of the smoothing domain by means of Taylor's expansion. The unknown coefficients can be uniquely determined in terms of the smoothed gradient technique, which is low-cost because it transforms the complex domain integration into its boundary integration. The LGSI is also simple because there is no need to correct test functions and introduce additional computational costs. The integration error of the LGSI scheme for the stiffness matrix in the 2D case is proved. Consequently, the LGSI is exact with respect to the quadratic meshfree Galerkin method. Numerical examples demonstrated the performance of the LGSI scheme for solving the 2D anisotropy potential and elasticity problems as well as the 3D potential problem.