自然单元法研究进展

自然单元法是一种基于Voronoi图和Delaunay三角化几何结构,以自然邻点插值为试函数的一种新型数值方法.其既具有无网格方法和经典有限元方法的优点,又克服了两者的一些缺陷,是一种发展前景广阔的求解微分方程的数值方法.自然单元法的形函数满足插值性质,可以像有限元法一样直接施加本质边界条件,不存在基于移动最小二乘拟合的无网格方法不能直接施加本质边界条件的难题.由于自然单元法是无网格方法,可以方便处理有限元方法较难处理的一些问题,例如移动边界和大变形等问题.自然单元法与其他数值方法的最根本区别于其插值格式的不同.将自然邻点插值用于Galerkin过程,就得到基于Voronoi结构的自然单元Galerkin法.自然邻点插值有自然邻点Sibson插值和Laplace插值(非Sibson插值)两种.Laplace插值比Sibson插值在计算上要简单的多,并且不论对凸的或非凸的区域都能精确施加本质边界条件.以Laplace插值为试函数的自然单元法在数值实施上比以Sibson插值为试函数的自然单元法简单.本文对基于Voronoi结构的自然邻点插值和自然单元法的基本思想作了介绍,综述了国内外关于自然单元法的研究成果,总结了自然单元法的优点和尚需解决的问题.

ADVANCES IN NATURAL ELEMENT METHOD

The natural element method (NEM) is a new numerical computational method based on Voronoi diagram and Delaunay triangulation. It is a Galerkin-based meshless method that is built upon the notion of the natural neighbor interpolation. The natural element method has advantages of both finite element method and meshless method, and does not have their disadvantages. As its shape functions satisfy interpolating properties, the natural element method is similar to the finite element method and can exactly interpolate piece-wise linear boundary conditions. The meshless methods, based on moving least square approximation as trial and test functions, can always exactly reproduce essential boundary conditions. As a meshless method, the natural element method can easily treat some problems, such as moving boundary and large deformation problems, which finite element method is difficult to treat. The essential difference of the natural element method and other numerical methods is their trial and test functions. Using the natural neighbor interpolation in a Galerkin procedure, we obtain the natural element Galerkin method based on Voronoi Structure. There are two natural neighbor interpolants: natural neighbor-based Sibson interpolation and Laplace interpolation (non-Sibsonian interpolation). Laplace interpolation is easier than Sibson interpolation in computation. In its numerical implementation, the natural element method based on Laplace interpolation as trial and test functions is easier than that based on Sibson interpolation. In this paper, the basic ideas of natural neighbor-based interpolation and the natural element method based on Voronoi structure are presented. The recent advances in the natural neighbor interpolation and the natural element method are reviewed. Some problems that have to be solved for NEM in the future are discussed.