A comparison study of the efficiency and accuracy of IEFG in solving elasticity problems using different nodal integration schemes

The interpolation property (Kronecker delta property) is a favorable feature for the shape functions of meshless methods. The meshless methods with interpolation property can directly impose the essential boundary conditions of problems, which saves computing resources. The interpolating element-free Galerkin method (IEFG) based on the improved interpolating moving least-squares (IMLS) method is an efficient meshless method with high accuracy, which changes the disadvantage that the traditional element-free Galerkin method (EFG) does not have interpolation property. To achieve higher efficiency, direct nodal integration (DNI) is introduced into IEFG in this paper. DNI is simple in execution and cheaper in computing resources, compared with Gauss integration (GI), but it will produce stability issues and sub-optimal convergence. To eliminate the instability of DNI, this paper adopts three improved direct nodal integration schemes, residual balance direct nodal integration (RDNI), naturally stabilized direct nodal integration (NDNI), and modified direct nodal integration (MDNI). Through three 2D elasticity numerical examples, the meshless method combining IEFG with improved direct nodal integration schemes is verified and proved to be efficient, and the effectiveness of several nodal integration schemes in improving integration efficiency and retaining integration stability is compared.