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

机译：固体应力分析的无网格局部自然邻域插值方法

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摘要

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.
机译：提出了一种无网格局部自然邻域插值方法（MLNNI），用于二维实体的应力分析。区域Ω的离散模型由一组不同的节点和边界的多边形描述组成。相对于给定点的自然邻居节点和Voronoi镶嵌来构造整个插值。使用局部Delaunay三角子域上的局部弱形式来获得平衡方程的离散系统。与使用标准Galerkin程序（需要三点正交规则）的自然元素方法相比，可以在MLNNI方法中在背景三角形正交网格的中心计算数值积分。由于形状函数具有Kroneckerδ函数性质，因此可以像常规有限元方法（FEM）一样轻松地直接实现基本边界条件。所提出的方法对于软件用户来说是一种真正的无网格方法，因为自然邻居插值的属性是无网格的，并且所有数值过程都由计算机自动完成。提出了该方法在固体力学中各种问题的应用，包括斑块试验，悬臂梁和梯度问题，并与精确解具有良好的一致性。数值结果表明，所提方法的精度与四边形有限元法相当，且时间成本小于四边形有限元。

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