A consistent projection integration for Galerkin meshfree methods

An efficient and inherently consistent integration method, named projection integration(PI), is proposed for Galerkin meshfree methods with arbitrary order. In contrast to traditional numerical integration methods, PI defines a projection operator based on a weighted norm to systematically approximate meshfree shape functions in an appropriate and consistent function space. Then a substitute function space is established by utilizing projected shape functions as a basis, and Galerkin weak form is approximately calculated by substituting the projected shape functions for the original meshfree shape functions. Owing to the projection consistency of the projection operator, the projected shape functions can still meet consistent conditions. For convenient implementations, the triangular finite element space is selected to approximate meshfree shape functions and the weight function in the weighted norm is carefully chosen to simplify the projection operator in this paper. Arbitrary order integration constraint conditions can be easily met by utilizing the corresponding order triangular finite element shape functions with low order Gauss quadrature rules, which means the optimal convergence rate can be obtained. A series of numerical examples are presented to demonstrate the proposed method’s efficiency and superior convergence performance.