Modified T-splines☆
Introduction
T-splines were introduced by Sederberg et al., 2003, Sederberg et al., 2004 and have been studied extensively in the last ten years. T-splines are a generalization of NURBS surfaces, the control meshes of which permit T-junctions. Unlike NURBS, T-junctions allow T-splines to be locally refinable without propagating entire columns or rows. This property makes T-splines an ideal technology for removing superfluous control points in NURBS surfaces and for adaptive isogeometric analysis (Hughes et al., 2005, Cottrell et al., 2009). Initial investigations using T-splines as a basis for isogeometric analysis demonstrate that T-splines possess similar convergence properties as NURBS with far fewer degrees of freedom (Dörfel et al., 2009, Bazilevs et al., 2010). However, the blending functions of T-splines are not always linearly independent. Buffa et al. (2010) gave an example of a T-spline with linearly dependent blending functions. This causes concerns about the linear independence of T-splines. A solution to this problem is the so-called analysis-suitable T-splines (AST-splines for short) (Li et al., 2012, Scott et al., 2012). AST-splines are a subset of T-splines defined over a restricted T-mesh whose T-junction extensions do not intersect, and the blending functions are always linearly independent and thus are suitable for isogeometric analysis. However, the topology of the meshes of AST-splines is relatively restrict. For example, the mesh for common local refinement in isogeometric analysis as shown in Fig. 1 is not an AST-mesh. Algorithm exists for modifying a non-AST-mesh into an AST-mesh (Scott et al., 2012).
In Deng et al. (2008), the authors introduced the concept of splines over T-meshes, and specifically PHT-splines were proposed. PHT-splines are polynomial splines defined over a hierarchical T-mesh, and the basis functions of PHT-splines are linearly independent, form a partition of unity and have compact supports. The local refinement algorithm of PHT-splines is local and very simple. Furthermore, since a PHT-spline is a polynomial (instead of a piecewise polynomial) over each cell of the T-mesh, it holds a good approximation property. These properties make PHT-splines an ideal tool for isogeometric analysis (Nguyen-Thanh et al., 2011a, Nguyen-Thanh et al., 2011b). However, PHT-splines are only continuous, which is a disadvantage for geometric modeling.
Another type of local refinement splines is LR-splines introduced by Dokken et al. (2013). LR-splines are defined on a μ-extended LR-mesh which is constructed by inserting line segments starting from a tensor product mesh according to certain rules. LR-spline also forms a non-negative partition of unity and spans the complete piecewise polynomial space on the mesh when the mesh construction follows certain rules. Different strategies can be employed to construct linearly independent LR B-splines by mesh modification. However, unlike T-splines and tensor product B-splines, there is no one-to-one correspondence between the 3D control mesh and the LR B-spline functions.
Hierarchical B-splines were introduced by Forsey and Bartels (1988), which have been recently further elaborated by Giannelli et al. (2012). The idea is to suitably truncate hierarchical B-spline functions according to finer levels in the hierarchy, which are called THB-splines. A THB-spline is a linear combination of B-splines, and THB-splines form a partition of unity and are linearly independent.
In this paper, we introduce a new type of local refinement splines called Modified T-splines, which inherit some good properties of the above splines while preventing some undesirable properties. The intuitive idea is to construct a set of basis functions which have good properties, such as non-negativity, partition of unity, linear independence and compact support. With the help of an auxiliary T-mesh (defined in Lemma 3.2), the basis functions are constructed as a linear combination of T-splines defined over . In this sense, Modified T-splines have some similarity with THB-splines.
The remainder of the current paper is organized as follows. In Section 2, we recall some preliminary knowledge about knot deletion of B-splines and T-splines. In Section 3, the construction details of Modified T-splines are described, and some properties, especially approximation property of Modified T-splines are discussed. Section 4 demonstrates some applications of Modified T-splines in surface fitting. Section 5 concludes the paper with a summary and future work.
Section snippets
Preliminaries
In this section, we recall some preliminary knowledge about knot deletion of B-splines in one dimension which is useful in the construction of Modified T-splines. Then the basic concepts about T-meshes and T-splines are reviewed.
Construction of modified T-splines
In this section, we are going to construct a new type of local refinement splines over T-meshes which inherit the major properties of current local refinement splines. We call such splines Modified T-splines. We start with an outline of the construction process. Then the detailed construction approach is described, and the properties of Modified T-splines are presented.
Conversion of Modified T-splines to TP-splines
Since Modified T-spline basis functions are linear combinations of the T-spline functions , and by knot insertion, the T-spline functions are linear combinations of the TP B-spline basis functions , so are linear combinations of . In matrix form, Thus
Suppose we have a Modified T-spline surface then it can
Conclusions and future work
This paper proposes a new local refinement splines called Modified T-splines. The basic idea is to construct a set of basis functions over a T-mesh which have good properties such as non-negativity, partition of unity and compact support. Due to the properties of the basis functions, Modified T-splines inherit many good properties of current local refinement splines, and thus should be useful both in geometric modeling and isogeometric analysis.
There are a few problems worthy of further
Acknowledgements
The authors thank the reviewers for providing useful comments and suggestion. The work is supported by 973 Program 2011CB302400, the NSF of China No. 11031007.
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This paper has been recommended for acceptance by B. Juettler.