The support integration scheme in the least-squares mesh-free method

The support integration in the least-squares mesh-free method (LSMFM) is presented to avoid any use of extrinsic cells in mesh-free framework and some aspects of the stabilized conforming nodal integration (SCNI) in Galerkin mesh-free method are investigated. The SCNI employs Voronoi cells for integration and enforces the linear exactness. Therefore, it works well with linear basis in mesh-free approximation and it is computationally less intensive than the integration with element-like cells. Despite the high computational efficiency of SCNI, it is shown through numerical tests that with higher-degree basis functions SCNI fails to reproduce the basis functions and degrades the rate of convergence. For LSMFM, the concept of support integration, where integration points are generated in each nodal support, is introduced. Its effectiveness results from the robustness of LSMFM to integration errors, which has been shown in the authors’ previous works. The numerical examples demonstrate that with only a few integration points in each nodal support LSMFM reproduces the basis functions of any degree and its solution accuracies are comparable to those with accurate integration.