Elsevier

Computer Aided Geometric Design

Volume 39, November 2015, Pages 17-49
Computer Aided Geometric Design

Characterization of analysis-suitable T-splines

https://doi.org/10.1016/j.cagd.2015.06.007Get rights and content

Highlights

  • We study the analysis suitable T-spline spaces as a piecewise polynomials space.

  • We characterize the difference between two neighboring polynomial expressions.

  • We study when the space can be described by smoothness conditions.

Abstract

In this article we provide the characterization of analysis suitable T-spline spaces (Beirão da Veiga et al., 2013) as the space of piecewise polynomials with appropriate linear constrains on the subdomain interfaces. We describe AST-meshes for which the linear constrains are equivalent to smoothness conditions and provide examples showing that this is not always the case.

Introduction

IsoGeometric Analysis (IGA) is a recent numerical technique for solving partial differential equations (PDEs) proposed by T.J.R. Hughes et al. (2005). The key feature of IGA is that the PDE solution is approximated by B-splines or Non-Uniform Rational B-Splines (NURBS) that are commonly used to represent the domain geometry in Computer Aided Design (CAD). Many recent papers in the engineering literature deal with IGA in different applicative areas such as fluid dynamics, structural mechanics, and electromagnetics (see Cottrell et al., 2009 and the references therein). Smoothness of B-splines and NURBS shape functions is also beneficial for PDE approximation, compared to classical C0 continuity of standard finite elements, and it allows new methods enjoying properties which would be hard to obtain with finite elements (e.g., see Buffa et al., 2011 and Evans and Hughes, 2013a, Evans and Hughes, 2013b, Evans and Hughes, 2013c).

Tensor-product B-splines or NURBS, often adopted in CAD and IGA, limit flexibility in mesh generation and refinement. However, local refinement strategies in IGA are possible thanks to the non-tensor product extensions of B-splines, such as T-splines (Sederberg et al., 2004, Sederberg et al., 2003), LR-splines (Dokken et al., 2013), hierarchical splines (Vuong et al., 2011) and polynomial splines over T-meshes (Deng et al., 2008). The focus of this paper is on T-splines. T-spline shape functions are constructed from a T-mesh, they are individually tensor-product of univariate B-splines, but the T-mesh breaks the global tensor-product structure by allowing so-called T-junctions.

T-splines have been recognized as a promising tool for IGA in Bazilevs et al. (2010) and have been the object of growing interest in the literature: Beirão da Veiga et al. (2011), Buffa et al. (2010), Dörfel et al. (2010), Li and Scott (2011), Li et al. (2012), Scott et al. (2012), Wang et al. (2011). More recently, in Li et al. (2012) analysis-suitable (AS) T-splines have been defined: AST-splines are a sub-class of T-splines having fundamental mathematical properties needed in a PDE solver. Linear independence of AST-splines blending functions has been first shown in Li et al. (2012) for the bi-cubic case. In Beirão da Veiga et al. (2013), AST-splines are generalized to arbitrary degree and it is shown that the condition of being AS, which is mainly a condition on the connectivity of the T-mesh, is equivalent to the fact that the T-spline basis functions admit a dual basis that can be constructed as in the tensor-product setting. As a corollary, the T-spline space enjoys the optimal approximation properties of standard (tensor-product) B-splines spaces. Still, one key part of the theory of T-splines is incomplete: the characterization of the T-spline space is known only for the special case of T-meshes with simple lines (Li and Scott, 2014).

By characterization here we mean the identification of the T-spline space as the space of piecewise polynomial with respect to a mesh (the so-called polynomial-mesh) with some constraints at the mesh edges, for example smoothness constraints. Our goal in this paper is to prove this result.

T-spline characterization is an important mathematical target per se but it is also a useful tool for other results. For example, the isogeometric vector fields proposed in Buffa et al. (2014) for electromagnetic problems are based on AST-splines and their properties depends on the characterization. Furthermore, a complete theory of approximation properties of T-spline spaces in the context if isogeometric analysis will benefit from the characterization proposed here, see Beirão da Veiga et al. (2014).

For our T-spline characterization we adopt a general technique that is based on the previous works of Billera (1988) and Mourrain (2014) and Pettersen (2013). The main tool is homology, first used in Billera (1988) to study the dimension of spline spaces on triangular meshes, later generalized by Mourrain (2014) to rectangular elements and by Pettersen (2013) to higher dimension cubes. The results of Mourrain (2014) and Pettersen (2013) have been extensively used in Dokken et al. (2013) to study LR-splines. In this paper we further extend Mourrain (2014), Pettersen (2013) in order to represent more general smoothness conditions that arise with T-splines.

The outline of the paper is as follows. After some preliminary notation and definitions of Section 2, we introduce the concept of T-mesh, AST-mesh, T-splines in Section 3. In Section 4 we define a space of piecewise polynomial functions with constraints at the mesh lines. These constraints may represent smoothness properties or more general conditions. Then we express the dimension of such a space in terms of the dimension of a homology space, and then we are able to show that, for the kind of constraints that are of interests when dealing with T-spline spaces, we can prove that the dimension of the homology space is indeed zero. Therefore the formula simplifies and the dimension of the piecewise polynomial space is related to the T-mesh topology. In Section 5 we show that the dimension of the T-spline space is the same, and finally that the T-spline space coincides with the piecewise polynomial space. Section 6 contains some example in which the proposed characterization is computed explicitly. In Section 7 we discuss some cases where the characterization can be simplified to smoothness conditions on the inter-element edges. The details are in Theorem 7.1.

Unluckily our result is technical and the proof is decomposed in many small lemmas. In order to facilitate the comprehension of the main ideas of the proof we include a map in Fig. 1.

Section snippets

Preliminaries

In this section we cover the background material on meshes and B-splines. It is also a review of the notation used afterwards. In this paper we study spaces of bivariate piecewise polynomials. For our purposes the degree of the polynomials can be considered fixed at the beginning. We denote by Pp=P(px,py) the space of polynomials in x,y of degree px with respect of x and py with respect of y. We will use Pp to denote the space of univariate polynomials of degree p.

In order to avoid writing

T-splines

T-spline spaces are usually described in the physical domain. This is not appropriate for our purposes and thus we restrict our description to the index-T-mesh. Index T-meshes are box-meshes whose skeleton is contained in Z×Z. They are instrumental in the description of a T-spline space because the generators of the space are associated to topological features of the index-T-mesh: the anchors. More details can be found in Beirão da Veiga et al. (2013).

Definition 3.1

An index-T-mesh is a pair MT=(Q,Ξ) where

  • 1.

    Q

Piecewise polynomials

In this section we will introduce a general framework for the description of piecewise polynomial spaces with linear constrains at the mesh edges. In particular we define constraints that characterize AST-spline spaces. The characterization will be proved in Section 5.

Definition 4.1

Let MP be a box-mesh on Ω and J a function from the set of internal edges (i.e. F1I(MP)) to the set of subspaces of Pp. Then we defineS(MP,J)={f:ΩR,eF2(MP),f|e is a polynomial in Pp,e1,e2F2(MP),aF1I(MP):a=(e1e2)f|e1f|e2J(a

Characterization

In this section we prove the main result of the paper. Namely the following characterization theorem.

Theorem 5.1

For all AST-meshes MT it holdsT(MT)=S(MP,J), where J is defined in Definition 4.4.

We begin by proving the following inclusion

Lemma 5.2

Let MT be an AST-mesh, thenT(MT)S(MP,J).

Proof

Consider an element Bη of the basis of T(MT), where η is the corresponding anchor. Let Θx=[θx,1,,θx,px+2] be the horizontal local-knot-vector of Bη and Θy=[θy,1,,θy,py+2] be its vertical local-knot-vector. A vertical line i

Examples

In this section we collect some examples that illustrate the polynomial spaces associated to edges of the polynomial-mesh. Since the definition of J uses some information that is contained in the AST-mesh we will use a representation that mimics the polynomial-mesh while preserving the information of the index-T-mesh.

We decided to limit the examples to the case of bi-cubic AST-splines as this is the most commonly used degree. The examples are ordered by complexity. Example 1 is a simple

Equivalence with smoothness conditions

The characterization of the AST-spline space provided in Section 5 uses complex gluing conditions on the inter-element boundaries. In some cases the same piecewise polynomial space can be equivalently defined in terms of required smoothness on the internal edges. This is possible also when the two conditions are not equivalent on edge by edge basis. In this section we describe a subset of AST-meshes for which T is characterized by smoothness condition. We then apply the content of this section

Acknowledgements

The authors were partially supported by the European Research Council through the Starting Grant GeoPDEs (ERC-2007-StG 205004) and through the Consolidator Grant HIgeoM (ERC-2013-CoG 616563) and by the Italian MIUR through the FIRB “Futuro in Ricerca” Grant RBFR08CZ0S. This support is gratefully acknowledged.

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  • Cited by (0)

    This paper has been recommended for acceptance by Thomas Sederberg.

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