An interpolating boundary element-free method (IBEFM) for elasticity problems

Abstract

The paper begins by discussing the interpolating moving least-squares (IMLS) method. Then the formulae of the IMLS method obtained by Lancaster are revised. On the basis of the boundary element-free method (BEFM), combining the boundary integral equation method with the IMLS method improved in this paper, the interpolating boundary element-free method (IBEFM) for two-dimensional elasticity problems is presented, and the corresponding formulae of the IBEFM for two-dimensional elasticity problems are obtained. In the IMLS method in this paper, the shape function satisfies the property of Kronecker δ function, and then in the IBEFM the boundary conditions can be applied directly and easily. The IBEFM is a direct meshless boundary integral equation method in which the basic unknown quantity is the real solution to the nodal variables. Thus it gives a greater computational precision. Numerical examples are presented to demonstrate the method. moving least-squares (MLS) approximation, interpolating moving least-squares (IMLS) method, boundary integral equation, meshless method, boundary element-free method (BEFM), interpolating boundary element-free method (IBEFM), elasticity problem PACS: 02.60.Ed, 02.70.-c, 02.70.Pt, 46.15.-x In recent years, more and more attention has been paid to research on the meshless (or meshfree) method [1,2]. The meshless method has some advantages over the traditional computational methods, such as finite element method (FEM) and boundary element method (BEM). The meshless boundary integral equation method points to one of the important research directions of the meshless method. Combining different approximation functions in the meshless method with the boundary integral equation method, researchers have developed meshless boundary integral equation methods, such as the boundary node method (BNM) [3-5], the local boundary integral equation (LBIE) method [6-8], and the boundary element-free method (BEFM) [9-19]. The moving least-squares (MLS) approximation is one of the bases of the meshless method [1,2,20]. The meshless method based on the MLS approximation can generate a solution possessing great precision. The meshless boundary integral equation methods are developed by combining the MLS approximation with boundary integral equation methods [3-19]. The boundary node method (BNM) is one of the meshless boundary integral equation methods, and Mukherjee et al. [3-5] used it to solve potential problems and linear elasticity problems. Another equally important method of the meshless boundary integral equation methods is the local boundary integral equation (LBIE) method, which Atluri et al. [6-8] present to solve linear and nonlinear boundary value problems.In the BNM, the basic unknown quantities are approximations of the nodal variables. However, as they are not the real nodal variables, the boundary conditions cannot be directly applied. In the LBIE method, the traction term is not included in the local boundary integral equations.

Natural element method for axisymmetric elastoplasticproblems

“…时依然保持较高的计算精度, 因此受到了国际计算力 学界的高度重视. 目前发展的无网格法主要有无单元 伽辽金法 [4,5] 、边界无单元法 [6,7] 、重构核粒子法 [8,9] 和 自然单元法 [10] 等. 与大多数无网格法不同 .…”

A proper orthogonal decomposition variational multiscale meshless interpolating element‐free Galerkin method for incompressible magnetohydrodynamics flow

“…An improved element‐free Galerkin (IEFG) method based on the interpolating moving least‐squares (IMLS) method is proposed in Reference for solving nonlinear elastic large deformation problems. The interpolating moving least squares approximation is combined with the boundary element‐free method in Reference to introduce the interpolating boundary element‐free method (IBEFM) for solving two‐dimensional elasticity problems. The main aim of Reference is to combine the improved interpolating moving least‐squares approximation with boundary element‐free method to solve two‐dimensional potential problems.…”

Error estimates for the interpolating moving least-squares method in n -dimensional space

“…Netuzhylov used the meshfree collocation method based on the IMLS method to solve boundary value problems [49]. Ren et al presented the interpolating boundary element-free (IBEF) method and interpolating element-free Galerkin (IEFG) method for two-dimensional potential and elasticity problems [57][58][59][60]. Based on the IMLS method, Ren et al presented the complex variable interpolating moving least-squares (CVIMLS) method [55,56], and Wang et al presented the improved interpolating moving least-squares (IIMLS) method with the nonsingular weight function [62,63,68,69].…”