In this paper, based on the improved interpolating moving least-squares (IMLS) method and the dimension splitting method, the interpolating dimension splitting element-free Galerkin (IDSEFG) method for three-dimensional (3D) potential problems is proposed. The key of the IDSEFG method is to split a 3D problem domain into many related two-dimensional (2D) subdomains. The shape function is constructed by the improved IMLS method on the 2D subdomains, and the Galerkin weak form based on the dimension splitting method is used to obtain the discretized equations. The discrete equations on these 2D subdomains are coupled by the finite difference method. Take the improved element-free Galerkin (IEFG) method as a comparison, the advantage of the IDSEFG method is that the essential boundary conditions can be enforced directly. The effects of the number of nodes, the direction of dimension splitting, and the parameters of the influence domain on the calculation accuracy are studied through four numerical examples, the numerical solutions of the IDSEFG method are compared with the numerical solutions of the IEFG method and the analytical solutions. It is verified that the numerical solutions of the IDSEFG method are highly consistent with the analytical solution, and the calculation efficiency of this method is significantly higher than that of the IEFG method.

本文基于改进的插值移动最小二乘（IMLS）方法和维数分割方法，提出了一种用于三维（3D）潜在问题的无插值维数分割元素的Galerkin（IDSEFG）方法。IDSEFG方法的关键是将3D问题域划分为许多相关的二维（2D）子域。通过改进的IMLS方法在2D子域上构造形状函数，并使用基于维数分裂方法的Galerkin弱形式获得离散方程。这些2D子域上的离散方程通过有限差分法耦合。以改进的无元素伽勒金（IEFG）方法作为比较，IDSEFG方法的优点是可以直接强制执行基本边界条件。通过四个数值例子研究了节点数量，维数分裂方向和影响域参数对计算精度的影响，将IDSEFG方法的数值解与IEFG方法的数值解进行了比较，分析解决方案。验证了IDSEFG方法的数值解与解析解高度一致，并且该方法的计算效率明显高于IEFG方法。