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.

针对任意阶的Galerkin无网格方法，提出了一种高效且内在一致的积分方法，称为投影积分（PI）。与传统的数值积分方法相比，PI 定义了一个基于加权范数的投影算子，以在适当且一致的函数空间中系统地逼近无网格形函数。然后以投影形函数为基础建立替代函数空间，用投影形函数代替原始无网格形函数近似计算Galerkin弱形式。由于投影算子的投影一致性，投影后的形函数仍能满足一致的条件。为了方便实现，三角有限元选择空间来近似无网格形状函数，并仔细选择加权范数中的权重函数以简化本文中的投影算子。利用具有低阶高斯求积规则的相应阶三角有限元形函数可以很容易地满足任意阶积分约束条件，这意味着可以获得最优收敛速度。提出了一系列数值例子来证明所提出的方法的效率和优越的收敛性能。