Abstract: According to the ideal of Natural Neighbor Interpolation, based on the geometric characteristics of Voronoi cells, this paper describes a C^∞ interpolation basis function, which is smooth and continuously differentiable everywhere except at its defining node. The parametric expression of first derivatives of the interpolation basis function is also given. The new theory for function and derivative interpolation has application in CFD to develop a novel Finite Element Method (FEM) which is called Natural Element Method (NEM). In addition, it demonstrates that the C^∞ interpolation basis function can be the promising role in a range of visualization problems of CFD.

Key words: interpolation, Voronoi cells, natural neighors, C^∞ interpolation basis function, CFD

摘要： 根据自然邻点插值(NNI)方法的思想，基于Voronoi cells的几何特性，从自然邻点(Natural Neighbors)的概念出发，对C^∞插值基函数N_i(x)的数学性质进行了研究，给出了N_i(x)的一阶导数的一种数学表达式及其数学性质。将Voronoi cells和C^∞插值基函数应用于流体力学有限元方法(即自然元方法)，通过对二维Burgers方程的数值算例说明了该方法在计算流体力学中的良好应用前景。结合实例讨论了该基函数的插值效果，同时说明了插值方法可很好地应用于计算流体力学的可视化(Visualization)处理。

关键词: 插值, Voronoicells, 自然邻点, C^∞插值基函数, 计算流体力学