【作者】 刘博；

【导师】 黄云清；

【作者基本信息】 湘潭大学 ， 计算数学， 2010， 硕士

【摘要】 自然邻接插值是一种广泛应用于多元数据拟合的插值方法.自然元方法就是以自然邻接插值(形函数)作为试探函数和检验函数应用在Galerkin过程的数值计算方法,其依赖于Voronoi图和Delaunay三角几何结构.自然邻接插值只与节点的分布有关,而与网格形状无关.因此在处理不规则区域的问题时,自然元方法具有天然的优势.形函数满足插值性质；在凸包边界的两相邻点之间是严格线性连续等性质.自然元方法既具有经典有限元方法、无网格方法的优点,又克服了两者的一些缺陷.本文讨论以Sibson插值为形函数的自然单元方法,介绍了自然元形函数的性质、构造以及自然单元方法的应用.我们首先给出了Sibson插值的定义和光滑性、插值性质等各项性质的证明,并构造了自然元方法Galerkin过程的核心内容一形函数；其次利用标准的Galerkin离散方法,介绍了自然元程序的设计思想和主要结构；最后将自然元方法用于求解平板问题和圆孔问题的数值试验,并在圆孔问题中比较了自然元方法和有限元方法的数值结果.由于自然元形函数在凸包边界上是线性的,在数值实验中可以很便利地强加本质边界条件.以上算例也说明了自然元方法应用于求解不规则区域的偏微分方程是非常有吸引力的.更多还原

【Abstract】 Natural neighbor interpolation is a widely used multivariate scattered data interpolation method.Natural element method(NEM),which depends on the Voronoi diagram and Delaunay triangulation,is a numerical methods that natural neighbor interpolation(shape function)is used as trial and test approximations in a Galerkin method.Natural neighbor interpolation is only related to the distribution of nodes,nothing with the grid shape. Therefore, for the issue of irregular region, NEM has a natural advantage. Furthermore the shape function meets the interpolation property; and on the boundary of convex hull it is strictly linear continuous between two neighbor points, etc. so NEM not only has the advantages of the classical finite element(FEM) method and meshless method, but also overcomes some shortcomings of both.In this paper,we discuss NEM in which Sibson interpolation is used as shape function, and describe the properties,construction of shape function and the ap-plication of the NEM.First,we illustrate the definition of Sibson interpolation and give out some proofs involving smoothness,interpolation property, etc.Mean-while, we construct the key of the Galerkin method-shape function.Second, using a standard Galerkin method,we introduce the design and main structure of the NEM.Finally, we use NEM to solve the plane problem and hole plate prob-lem.In the experiment of hole plate problem,we compare the numerical results of the NEM and FEM.Since shape function is linear on the boundary of con-vex hull,we can conveniently impose the essential conditions on the boundary in the numerical experiment.At the same time,these examples also illustrate that NEM is very attractive,which is applied to solve partial differential equations of the irregular region.更多还原