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.

为了解决具有二次基的无网格伽辽金方法的域集成，我们在本研究中提出了一种有效且准确的线性梯度平滑集成（LGSI）方案。在我们的方案中，平滑梯度通过泰勒展开表示为相对于平滑域中心的线性多项式形式。未知系数可以根据平滑梯度技术唯一确定，这是低成本的，因为它将复域积分转换为其边界积分。LGSI 也很简单，因为不需要修正测试函数和引入额外的计算成本。证明了 LGSI 方案在二维情况下刚度矩阵的积分误差。因此，LGSI 相对于二次无网格伽辽金方法是精确的。