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.

A local basis algorithm for searching natural neighbours in Natural Element Method (NEM) is presented for solving the elasticity problems in this paper. Comparison with the global sweep algorithm used in natural element method or Natural Neighbour Method (NNM) for searching natural neighbours, the proposed algorithm is more expedient and convenient in the constructions and computation of natural neighbour interpolations. In the proposed NEM based on local search, the Laplace (non-sibson) interpolations are constructed with respect to the natural neighbour nodes of the given point which have been locally defined. The shape functions from the Laplace approximations have the delta function property and the Laplace interpolants are strictly linear between adjacent nodes, which facilitate imposition of essential boundary conditions and treatment of material discontinuity with ease as it is in the conventional finite element method. The Laplace interpolants derived from the local algorithm and the global algorithm in NEM are identical because of the uniqueness of the Voronoi diagram. Numerical results and convergence studies also show that the present NEM based on local search algorithm possesses the same accuracy and rate of convergence as they are in previous NEM.