Rational Spline Interpolation Curve Of Minimum Strain Energy

Interpolation is to construct a continuous function,which makes the continuous function through all the given discrete data points.Interpolation is the most basic method of numerical approximation,including polynomial interpolation,rational interpolation,Hermite interpolation,spline interpolation,rational spline interpolation.The structure of polynomial interpolation is simple,it is convenient to carry out the calculation and theoretical analysis,so it is widely used to deal with the problem of function approximation,numerical differentiation and numerical integration.However,the high-order polynomial interpolation,especially the equidistant nodes of high-order is prone to Runge phenomenon,which makes the polynomial interpolation approximation effect.Rational interpolation is better than polynomial interpolation,so many scholars have studied the problem of finding the derivative at nodes.However,the rational interpolation method,such as the continued fraction interpolation method,will appear the poles,the unreachable points and the deficit quotient.Rational spline interpolation has good approximation effect,which can not only avoid the poles and unattainable points,but also can adjust the appropriate parameters to ensure the monotonicity and concavity of the interpolation data.This paper introduces the strain energy of curves.The smaller the strain energy of the interpolation curve,the more smooth it is.Therefore,in order to make the rational spline interpolation curve can satisfy the shape preserving requirements,we can use optimization theory to establish optimization model to solve the most shape parameters and the derivative value of node.This article is based on the shape parameters and the interpolation function at the nodes being decision variables,the minimum strain energy of the interpolation curve being the objective function,and the interpolation function for shape as well as the shape control parameters and the derivative of the node greater than zero being the constraint conditions.The shape preserving rational spline interpolation curve with minimum strain energy is obtained based on the optimization model.Because the interpolation data may have given monotonicity and concavity properties,so they have different decision variables,objective function and constraint conditions,construct the different nature of the rational cubic spline curve,through calculation and choose appropriate parameters to keep the interpolation data monotonicity and concavity.A numerical example is given to show that the new method can obtain a smooth interpolation curve.The numerical examples are given to show that the new method can obtain fairing interpolation curve.