Isogeometric interpolation by generalized splines

Abstract

It is well known that polynomial splines generally do not retain the geometric properties of the initial data. This paper defines a class of functions I(V) which has shape-retaining properties ('isogeometry') and is determined by a given set of points V-{P. = (jc. JC) €/? 2 : X Q <x^ < ... <X N }. The necessary and sufficient conditions for I(V) to be nonempty are given. A local algorithm for convex and monotone interpolation by C generalized splines is obtained. Its application enables us to give a complete solution to the isogeometric interpolation problem for data of arbitrary form and isolate the sections of linearity, the angles, and the breaks.One of the most widespread approximation methods for mesh functions is the interpolation by C 2 cubic splines. These splines combine the smoothness properties suitable for many applications with the simplicity of their computer realization and high accuracy of results obtained. But in a number of cases the behaviour of cubic splines does not agree with the qualitative characteristics of the initial data. Visually it makes itself evident in the presence of jumps, oscillations, various deviations not characteristic of a given set of points. These features may be expressed mathematically as nonmonotonicity and presence of inflection points in the data on monotonicity and convexity sections.Attempts to improve the shape properties of cubic splines were made long before. For this purpose various generalizations of cubic splines were introduced. The paper by Schweikert [19] is supposed to be one of the first papers devoted to cubic splines in tension. The papers [2][3][4]16,20] are also worth mentioning. The authors of all these investigations introduced several parameters into the structure of the spline to control the qualitative behaviour of the curve obtained. They did not offer, however, any procedure for automatic choice of these parameters, they gave only recommendations as to their choice in the dialog mode. Naturally, it was due to insufficient formalization of the cubic spline shape behaviour notion and with the difficulties in the development of algorithms for free parameters automatic choice.The problem of spline construction with given geometric characteristics seems to be formalized for the first time in the papers by A. I. Grebennikov [9,10], where it was referred to as the problem of isogeometric approximation. In particular, based on the local B-spline approximation methods, it was shown that on a sufficiently detailed mesh the cubic spline retains the geometric properties of the approximated function.In [7] Fritsch and Carlson gave an effective partial solution to the isogeometric interpolation problem, using the Hermite cubic spline. The algorithm constructed by the authors is local and produces the C 1 monotone curve. However, it does not retain the initial data convexity. More recently this result was improved in [6].

Design of Shell Conjunction Zones in Pressure Vessels

“…The resulting system of equations has a diagonal form with respect to the unknown second-order derivatives of the spline function Mi [5].…”