Generalized spline spaces over T-meshes: Dimension formula and locally refined generalized B-splines

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Abstract

Univariate generalized splines are smooth piecewise functions with sections in certain extended Tchebycheff spaces. They are a natural extension of univariate (algebraic) polynomial splines, and enjoy the same structural properties as their polynomial counterparts. In this paper, we consider generalized spline spaces over planar T-meshes, and we deepen their parallelism with polynomial spline spaces over the same partitions. First, we extend the homological approach from polynomial to generalized splines. This provides some new insights into the dimension problem of a generalized spline space defined on a prescribed T-mesh for a given degree and smoothness. Second, we extend the construction of LR-splines to the generalized spline context.

Introduction

Generalized splines are smooth piecewise functions with sections in spaces of the form (see [8]) PpU,V:=1,t,,tp2,U(t),V(t),t[a,b],2pN.Classical polynomial splines are obtained by taking the functions U, V equal to tp1,tp. In such a case, the space PpU,V is the space of algebraic polynomials of degree p, denoted by Pp. Other interesting examples are trigonometric or exponential generalized splines for which U, V are taken as cos (αt), sin (αt), or cosh (αt), sinh (αt), respectively.

Under suitable conditions on U, V, the space (1) has the same structural properties as Pp. Similarly, generalized splines possess all the desirable properties of polynomial splines. In particular, they admit a representation in terms of basis functions that are a natural extension of the polynomial B-splines. Moreover, classical algorithms (like degree elevation, knot insertion, differentiation formulas, etc.) can be explicitly rephrased for them. Such basis functions are referred to as generalized B-splines (GB-splines).

Generalized splines are popular tools in the computer aided geometric design (CAGD) community. Besides their theoretical interest, generalized spline spaces offer the possibility of controlling the shape of their elements by means of some shape parameters (the value α in the case of trigonometric and exponential generalized splines mentioned above), see [7], [8], [18], [19], [29]. Moreover, they are an interesting alternative to non-uniform rational B-splines (NURBS), see [6], [23], [37], [38] and references therein. In particular, trigonometric and exponential generalized splines allow for an exact representation of conic sections as well as some transcendental curves (helix, cycloid, etc.) and are attractive from the geometrical point of view. Indeed, in contrast with NURBS, they are able to provide parameterizations of conic sections with respect to the arc length so that equally spaced points in the parameter domain correspond to equally spaced points on the described curve. It is also worth mentioning that, contrarily to NURBS, trigonometric and exponential generalized B-splines behave completely similarly to polynomial B-splines with respect to differentiation and integration.

Thanks to the above properties, tensor-products of generalized B-splines are also an interesting problem-dependent alternative to tensor-product polynomial B-splines and NURBS in isogeometric analysis (IgA), see [9], [24], [25], [26]. Introduced nearly a decade ago in a seminal paper by Hughes et al. [16], IgA is nowadays a well-established paradigm for the analysis of problems governed by partial differential equations (PDEs), see, e.g., [10] and references therein. It aims at improving the connection between numerical simulation and computer aided design (CAD) systems. The main idea of IgA is to use the functions adopted in CAD systems not only to describe the domain geometry, but also to represent the numerical solution of the differential problem, within an isoparametric framework.

Adaptive local refinement is fundamental in geometric modeling and is a crucial ingredient for obtaining, in an efficient way, an accurate numerical solution of PDEs. However, any tensor-product structure lacks adequate local refinement. The introduction and the success of the IgA paradigm triggered the interest in alternative structures that support local refinements. Confining the discussion to local tensor-product structures, we mention T-splines [21], [32], [33], hierarchical splines [14], [15], [36], and locally refined (LR-) splines [13], [17].

T-splines, hierarchical splines and LR-splines can be seen as a special case of splines over T-meshes [11], [12], [30], [31]. A complete understanding of these spline spaces requires the knowledge of the dimension of the spline space defined on a prescribed T-mesh for a given degree and smoothness, see [11], [20], [28], [30] and references therein. Among the various techniques to tackle this difficult problem, one can use the homological approach proposed in [28], where the technique presented in [1] for splines on triangulations has been fine-tuned for splines on planar T-meshes. The resulting dimension formula is a key ingredient in the analysis of the properties of LR-splines, see [13].

As mentioned above, generalized splines enjoy the fundamental properties of polynomial splines, including the behavior with respect to local refinement. In particular, GB-splines support (locally refined) hierarchical structures in the same way as polynomial B-splines, see [26] (and also [15], [35]). T-spline structures based on GB-splines have been addressed in [3], [4]. Results on the dimension of generalized spline spaces over T-meshes have been provided in [5] by extending the approach based on so-called minimal determining sets [30].

In this paper, we deepen the parallelism between polynomial splines and generalized splines over planar T-meshes. More precisely,

  • we extend the homological approach in [28] to generalized splines, in order to address the problem of determining the dimension of a generalized spline space on a prescribed T-mesh for a given degree and smoothness;

  • we extend the construction of LR-splines presented in [13] to generalized splines.

The remainder of the paper is divided into four sections. In Section 2 we give the definition of generalized spline spaces over T-meshes. Section 3 is devoted to the study of the dimension of such spaces by means of the homological approach. Then, we describe generalized LR-splines in Section 4. Finally, we end in Section 5 with some concluding remarks.

Section snippets

Generalized spline spaces over T-meshes

In this section we formulate the definitions of the meshes and of the spaces we are dealing with. We consider a region ΩR2 which is a finite union of closed axis-aligned rectangles, called cells, with pairwise disjoint interiors. We assume that Ω is simply connected and its interior Ωo is connected; see Fig. 1 for an illustration. The smallest rectangle containing Ω is denoted by [ah, bh] × [av, bv].

We start by defining a T-mesh on Ω using the notation and definition given in [28].

Definition 1 T-mesh

A T-meshT:=(

Dimension of the generalized spline space

In this section we study the dimension of SpU,V,r(T). Our arguments are based on homological techniques similar to the ones used in [28] for investigating the dimension of the space Spr(T).

Locally refined generalized splines

In the context of algebraic polynomial splines, the classical knot insertion process of tensor-product B-splines gives rise to LR B-splines [13]. In this section we extend this construction to the generalized spline setting.

Conclusions

In this paper we have considered generalized spline spaces over planar T-meshes, and we have shown that they share several structural properties with polynomial spline spaces over the same partitions. First, we have provided some new insights into the problem of determining the dimension of a generalized spline space defined on a prescribed T-mesh for a given degree and smoothness, by extending the homological approach in [28] to generalized splines. Second, we have shown that the construction

Acknowledgments

This work was partially supported by INdAM-GNCS Gruppo Nazionale per il Calcolo Scientifico, by the MIUR ‘Futuro in Ricerca 2013’ Programme through the project DREAMS, and by the ‘Uncovering Excellence’ Programme of the University of Rome ‘Tor Vergata’ through the project DEXTEROUS.

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