Quadratically consistent one-point (QC1) quadrature for meshfree Galerkin methods

摘要

A robust and efficient integration method, named quadratically consistent one-point (QC1) scheme, which evaluates the Galerkin weak form only at the centers of background triangle elements (cells) is proposed for meshfree methods using quadratic basis. The strain at the evaluation points is approximated by corrected (smoothed) nodal derivatives which are determined by a discrete form of the divergence theorem between nodal shape functions and their derivatives in Taylor's expansion. We prove that such smoothed nodal derivatives also meet the differentiation of the approximation consistency (DAC). The same Taylor's expansion is applied to the weak form and the smoothed nodal derivatives are used to compute the stiffness matrix. The proposed QC1 scheme can pass both the linear and the quadratic patch tests exactly in a numerical sense. Several examples are provided to demonstrate its better numerical performance in terms of convergence, accuracy, efficiency and stability over other one-point integration methods in the meshfree literature, especially its superiority over the existing linearly consistent one-point (LC1) quadratures.