A meshless local natural neighbour interpolation method for stress analysis of solids

Abstract

A meshless local natural neighbour interpolation method (MLNNI) is presented for the stress analysis of two-dimensional solids. The discrete model of the domain Ω consists of a set of distinct nodes, and a polygonal description of the boundary. The whole interpolation is constructed with respect to the natural neighbour nodes and Voronoi tessellation of the given point. A local weak form over the local Delaunay triangular sub-domain is used to obtain the discretized system of equilibrium equations. Compared with the natural element method using a standard Galerkin procedure which needs three point quadrature rule, the numerical integral can be calculated at the center of the background triangular quadrature meshes in the MLNNI method. Since the shape functions possess the Kronecker delta function property, the essential boundary conditions can be directly implemented with ease as in the conventional finite element method (FEM). The proposed method is a truly meshless method for software users, since the properties of the natural neighbour interpolation are meshless and all the numerical procedures are automatically accomplished by the computer. Application of the method to various problems in solid mechanics, which include the patch test, cantilever beam and gradient problem, are presented, and excellent agreement with exact solutions is acheived. Numerical results show that the accuracy of the proposed method is as good as that of the quadrangular FEM, and the time cost is less than that with the quadrangular FEM.