The meshless numerical manifold method (MNMM) inherits two covers of the numerical manifold method. A mathematical cover is composed of nodes' influence domains and a physical cover consists of physical patches, which are produced through cutting mathematical cover by physical boundaries. Because two covers are adopted, MNMM can naturally and uniquely solve both the continuous and discontinuous problems under Galerkin's variational framework. However, Galerkin's meshless method needs background integration grids to realize solving, which often does not match the nodes' influence domains, so the accuracy of numerical integration is reduced. Consider that MNMM allows even distribution of nodes and the physical cover contains the characteristics of the boundary and nodes' influence domains, the study presents a new numerical integration strategy to ensure that the background integration grids match the nodes' influence domains. The method can be applied to continuous and discontinuous problems, and is proved to be equivalent to the influence domain integration. At the same time, a reasonable arrangement of mathematical nodes is made to assure the background integration grid that it is accordant and straightforward. In this way, the number of physical patches of each integral point is the same, which improves the accuracy of interpolation calculation. The effectiveness of the proposed method is verified by numerical examples of both continuous and discontinuous problems.

无网格数值流形方法（MNMM）继承了数值流形方法的两个覆盖。数学覆盖由节点的影响域组成，物理覆盖由物理补丁组成，物理补丁是通过物理边界切割数学覆盖产生的。由于采用了两个覆盖，MNMM可以自然而独特地解决Galerkin变分框架下的连续和不连续问题。然而，Galerkin 的无网格方法需要背景积分网格来实现求解，这往往与节点的影响域不匹配，因此降低了数值积分的精度。考虑到 MNMM 允许节点均匀分布，物理覆盖包含边界和节点影响域的特征，该研究提出了一种新的数值积分策略，以确保背景积分网格与节点的影响域相匹配。该方法可以应用于连续和不连续问题，并被证明等效于影响域集成。同时，合理安排数学节点，保证背景积分网格一致、直观。这样，每个积分点的物理块数相同，提高了插值计算的精度。通过连续和不连续问题的数值算例验证了所提出方法的有效性。该方法可以应用于连续和不连续问题，并被证明等效于影响域集成。同时，合理安排数学节点，保证背景积分网格一致、直观。这样，每个积分点的物理块数相同，提高了插值计算的精度。通过连续和不连续问题的数值算例验证了所提出方法的有效性。该方法可以应用于连续和不连续问题，并被证明等效于影响域集成。同时，合理安排数学节点，保证背景积分网格一致、直观。这样，每个积分点的物理块数相同，提高了插值计算的精度。通过连续和不连续问题的数值算例验证了所提出方法的有效性。每个积分点的物理块数相同，提高了插值计算的精度。通过连续和不连续问题的数值算例验证了所提出方法的有效性。每个积分点的物理块数相同，提高了插值计算的精度。通过连续和不连续问题的数值算例验证了所提出方法的有效性。