Discrete weighted cubic splines

This paper presents methods for shape preserving spline interpolation. These methods are based on discrete weighted cubic splines. The analysis results in two algorithms with automatic selection of the shape control parameters: one to preserve the data monotonicity and other to retain the data convexity. Discrete weighted cubic B-splines and control point approximation are also considered.     

Shape-preserving interpolation by cubic splines

We consider the problem of shape-preserving interpolation by cubic splines. We propose a unified approach to the derivation of sufficient conditions for the k-monotonicity of splines (the preservation of the sign of any derivative) in interpolation of k-monotone data for k = 0, …, 4.     

Interpolation and approximation by monotone cubic splines

We study the reconstruction of a function defined on the real line from given, possibly noisy, data values and given shape constraints. Based on two abstract minimization problems characterization results are given for interpolation and approximation (in the euclidean norm) under monotonicity constraints. We derive from these results Newton-type algorithms for the computation of the monotone spline approximant.     

Convex interval interpolation with cubic splines

A necessary and sufficient criterion is presented under which the problem of the convex interval interpolation with cubicC ¹-splines has at least one solution. The criterion is given as an algorithm which turns out to be effective.Dedicated to Professor Julius Albrecht on the occasion of his 60th birthday.     

Monotone and convex interpolation by weighted cubic splines

Algorithms for interpolating by weighted cubic splines are constructed with the aim of preserving the monotonicity and convexity of the original discrete data. The analysis performed in this paper makes it possible to develop two algorithms with the automatic choice of the shape-controlling parameters (weights). One of them preserves the monotonicity of the data, while the other preserves the convexity. Certain numerical results are presented.     

Modified nodal cubic spline collocation for biharmonic equations

We formulate a modified nodal cubic spline collocation scheme for the solution of the biharmonic Dirichlet problem on the unit square. We prove existence and uniqueness of a solution of the scheme and show how the scheme can be solved on an N × N uniform partition of the square at a cost O(N ² log₂ N + mN ²) using fast Fourier transforms and m iterations of the preconditioned conjugate gradient method. We demonstrate numerically that m proportional to log₂ N guarantees the desired convergence rates. Numerical results indicate the fourth order accuracy of the approximations in the global maximum norm and the fourth order accuracy of the approximations to the first order partial derivatives at the partition nodes.      

Deficient cubic splines with average slope matching

We obtain a deficient cubic spline function which matches the functions with certain area matching over a greater mesh intervals, and also provides a greater flexibility in replacing area matching as interpolation. We also study their convergence properties to the interpolating functions.     

Accurate modeling of contact using cubic splines

In this paper, we develop and implement a new method for the accurate representation of contact surfaces. This approach overcomes the difficulties arising from the use of traditional node-to-linear surface contact algorithms. In our proposed method, contact surfaces were modeled accurately using C¹-continuous cubic splines, which interpolate the finite element nodes. In this case, the unit normal vectors are defined uniquely at any point on the contact surfaces. These splines preserve the local deformation of the nodes on each flexible contact surface. Consequently, a consistent linearization of the kinematic contact constraints, based on the spline interpolation, was derived. Moreover, the gap between two contact surfaces was modeled accurately using an efficient surface-to-surface contact search algorithm. Since the continuity of the splines is not affected by the number of nodes, accurate stress distribution can be obtained with less finite elements at the contact surface than that using the traditional linear discretization of the contact surface. Two numerical examples are used to illustrate the advantages of the proposed representation. They show a significant improvement in accuracy compared to traditional piecewise element-based surface interpolation. This approach overcomes the problem of mismatch in a finite element mesh. This is very useful, since most realistic engineering problems involve contact areas that are not known a priori.     

A shape-preserving approximation by weighted cubic splines

This paper addresses new algorithms for constructing weighted cubic splines that are very effective in interpolation and approximation of sharply changing data. Such spline interpolations are a useful and efficient tool in computer-aided design when control of tension on intervals connecting interpolation points is needed. The error bounds for interpolating weighted splines are obtained. A method for automatic selection of the weights is presented that permits preservation of the monotonicity and convexity of the data. The weighted B-spline basis is also well suited for generation of freeform curves, in the same way as the usual B-splines. By using recurrence relations we derive weighted B-splines and give a three-point local approximation formula that is exact for first-degree polynomials. The resulting curves satisfy the convex hull property, they are piecewise cubics, and the curves can be locally controlled with interval tension in a computationally efficient manner.     

The local integro cubic splines and their approximation properties

We show the integro cubic splines proposed by Behforooz [1] can be constructed locally by using B-representation of splines. The approximation properties of the local splines are also considered.     

Convex interval interpolation with cubic splines,II

The problem of convex interval interpolation with cubicC ¹-splines has an infinite number of solutions, if it is solvable at all. For selecting one of the solutions a regularized mean curvature is minimized. The arising finite dimensional constrained program is solved numerically by means of a dualization approach.Dedicated to Professor Julius Albrecht on the occasion of his 65th birthday.     

Some approximation properties of periodic parametric cubic splines

The parametric cubic splines interpolating to such closed curves as the circle and ellipse are derived in a form where their parameters are given by simple algebraic expressions. The structure of these expressions enables the error in approximation of the given curves to be precisely determined and some additional features of the spline deduced.     

Numerical calculation of fourier integrals with cubic splines

The most used formula for calculation of Fourier integrals is Filon's formula which is based on approximation of the function by a quadratic in each double interval. In order to obtain a better approximation we use the cubic spline fit. The method is not restricted to equidistant points, but the final formulas are only derived in this case. Test computations show that the spline formula may be superior to Filon's formula.     

Range restricted interpolation using Gregory's rational cubic splines

The construction of range restricted univariate and bivariate interpolants to gridded data is considered. We apply Gregory's rational cubic C¹ splines as well as related rational quintic C² splines. Assume that the lower and upper obstacles are compatible with the data set. Then the tension parameters occurring in the mentioned spline classes can be always determined in such a way that range restricted interpolation is successful.     

On cubic formulas related to the mixed Hermite splines

A cubic formula containing partial integrals is considered on a class of functions of two variables. It is shown that the integral of a mixed Hermite spline gives the best cubic formula for the given class. The coincidence of cubic formulas, which are exact for odd and even mixed Hermite splines, is established.Translated from Ukrainskii Matematicheskii Zhurnal, Vol.45, No. 4, pp. 579–581, April, 1993.     

Fitting data using optimal Hermite type cubic interpolating splines

In this work we obtain a new optimal property for cubic interpolating splines of Hermite type applied to data-fitting problems. The existence and uniqueness of the Hermite type cubic spline with minimal quadratic oscillation in average are proved.