应变能最小的有理样条插值曲线

发布时间：2018-02-03 22:21

本文关键词： 有理样条插值保单调保正保凸应变能最优化　出处：《安徽理工大学》2017年硕士论文　论文类型：学位论文

【摘要】：插值是构造简单的连续函数,使得所构造的连续函数曲线能够通过全部给定的离散数据点。插值法是数值逼近中最基本的方法,包括多项式插值、有理插值、埃尔米特插值、样条插值和有理样条插值等。其中多项式插值的结构简单,便于进行计算和理论分析,所以被广泛用于函数逼近、数值微分和数值积分等问题。但是高次多项式插值,特别是等距节点的高次多项式插值容易出现Runge现象,这使得高次多项式插值的逼近效果不佳。有理插值比多项式插值的逼近效果好,在节点处近似导数的求取问题上引起许多学者的研究兴趣。但是有理插值方法,如连分式插值方法会出现极点、不可达点以及逆差商不存在等问题。有理样条插值有很好的逼近效果,不仅能避免出现极点、不可达点等,而且可以通过选择适当的参数来保持插值数据的单调性、凹凸性等。本文引入了曲线的应变能。插值曲线的应变能越小,曲线则越光顺。因此要使有理样条插值曲线满足保形性要求,可以用最优化理论,建立优化模型来解最优形状参数和节点导数值。文章是以形状参数和插值函数在节点处的导数为决策变量,以插值曲线应变能最小为目标函数,以插值函数保形以及形状控制参数和节点处的导数大于零作为约束条件,建立优化模型,求解获得在曲线应变能最小的情况下的最优形状参数。由于给定的插值数据可能具有单调性、凹凸性等性质,所以就有不同决策变量,目标函数和约束条件,构造出不同性质的有理样条插值曲线,通过计算选择出适当的参数来保持插值数据的单调性、凹凸性等。给出的数值例子表明新方法能获得光顺的插值曲线。
[Abstract]:Interpolation is the construction of simple continuous function, so that the constructed continuous function curve can pass through all the given discrete data points. Interpolation is the most basic method in numerical approximation, including polynomial interpolation, rational interpolation. Hermite interpolation, spline interpolation and rational spline interpolation, among which polynomial interpolation is simple, easy to calculate and theoretical analysis, so it is widely used in function approximation. But the higher degree polynomial interpolation, especially the higher order polynomial interpolation of equidistant nodes, is easy to appear Runge phenomenon. This makes the approximation effect of higher degree polynomial interpolation not good. Rational interpolation is better than polynomial interpolation, and many scholars are interested in finding approximate derivatives at nodes. But rational interpolation methods. Such as continuous fraction interpolation method will appear pole, unreachable point and deficit quotient does not exist. Rational spline interpolation has a good approximation effect, not only can avoid the emergence of pole, non-reachable point and so on. Moreover, the monotonicity and convexity of interpolation data can be maintained by selecting appropriate parameters. In this paper, the strain energy of the curve is introduced. The smaller the strain energy of the interpolation curve is. The curve is more smooth, so to make the rational spline interpolation curve meet the requirements of shape preservation, we can use the optimization theory. In this paper, the derivative of shape parameter and interpolation function at the node is taken as decision variable, and the minimum strain energy of interpolation curve is taken as objective function. The optimization model is established based on the shape preserving of interpolation function and the fact that the shape control parameter and the derivative at the node are larger than zero. The optimal shape parameters are obtained when the strain energy of the curve is minimum. Because the given interpolation data may have monotonicity, concave convexity and other properties, there are different decision variables, objective functions and constraints. The rational spline interpolation curves with different properties are constructed and the proper parameters are selected by calculation to maintain the monotonicity and convexity of the interpolation data. Numerical examples are given to show that the new method can obtain fairing interpolation curves.
【学位授予单位】：安徽理工大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.3

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