摘要
基于自然邻结点近似位移函数提出了一种用于求解弹性力学平面问题的无网格局部Petrov-Galerkin方法.这种方法在结构的求解域Ω内任意布置离散的结点,并且利用需求结点的自然邻结点和Voronoi结构来构造整体求解的近似位移函数.对于构造好的近似位移函数,在局部的Delaunay三角形子域上采用局部Petrov-Galerkin方法建立整体求解的平衡控制方程,这样平衡方程的积分可在背景三角形积分网格的形心上解析计算得到,而采用标准Galerkin方法的自然单元法需要三个数值积分点.该方法能够准确地施加边界条件,得到的系统矩阵是带状稀疏矩阵,对软件用户来说,它还是一种完全的、真正的无网格方法.所得计算结果表明,该方法的计算精度与有限元法四边形单元相当,但计算和形成系统平衡方程的时间比有限元法四边形单元提高了将近一倍,是一种理想的数值求解方法.
The meshless local Petrov-Galerkin method, which is based on the Voronoi cells and natural neighbour interpolations, is presented for solving the planar elasticity problem in this paper. 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 nature neighbour nodes and Voronoi tessellation of the gived point. The natural neighbour interpolants are strictly linear between adjacent nodes on the boundary of the convex hull, which facilitates imposition of essential boundary conditions with ease as it is in the conventional finite element method. The triangular FEM shape functions and the natural neighbour interpolations are differently chosen as the trial and test functions, and a local weak form is used to form the system discrete equation for two-dimensional solids. Compared with natural element method using standard Galerkin procedure which need three point quadrature rule, the numerical integral can be analytical calculated at the center of the backgroud triangular quadrature meshes in this method, and hence the difficulty in the numerical integral due to complexity of shape functions from meshless approximations can be avoided. The method is also a truly meshless method for software users, because the properties of natural neighbour interpolations are meshless and all the numerical procedures are automatically accomplished by the computer. The method is applied to various problems in solid mechanics, which include, the patch test, the cantilever beam and gradient problems is, and excellent agreement with exact solution is obtained. Numerical results also show that the accuracy of the present method is almost equal to the quadrangular finite element method, and the time cost is less than the quadrangular finite element method.
出处
《力学学报》
EI
CSCD
北大核心
2003年第2期187-193,共7页
Chinese Journal of Theoretical and Applied Mechanics
