On degree elevation of T-splines☆
Introduction
T-splines (Sederberg et al., 2003, Sederberg et al., 2004) address many limitations inherent in the NURBS representation, such as local refinement (Sederberg et al., 2004, Scott et al., 2012), watertightness via merging (Ipson, 2005, Sederberg et al., 2003) and trimmed NURBS conversion (Sederberg et al., 2008). T-splines have proved to be an important technology across several disciplines including industrial, architectural and engineering design, manufacturing and engineering analysis. Knot insertion and degree elevation algorithms are two fundamental algorithms which are used to enrich a spline space (Farin, 2002). Degree elevation is the process of raising the degree of a curve or a surface while keeping the shape unchanged. For NURBS, these issues have been well studied (Farin, 2002, Wang and Deng, 2007, Huang et al., 2005). For T-splines, the local refinement algorithm has also been well studied (Sederberg et al., 2004, Scott et al., 2012, Morgenstern and Peterseim, 2015). However, no previous articles address degree elevation for T-splines.
Another motivation for T-spline degree elevation is from the analysis community. T-splines are attractive not only in geometric modeling but also in iso-geometric analysis (IGA), which uses the smooth spline basis that defines the geometry as the basis for analysis. IGA is introduced in Hughes et al. (2005) and described in detail in Cottrell et al. (2009). The use of T-splines as a basis for IGA has gained widespread attention (Bazilevs et al., 2010, Scott et al., 2012, Scott et al., 2013, Borden et al., 2012, Benson et al., 2010, Veiga et al., 2011, Buffa et al., 2012, Dimitri et al., 2014, Liu et al., 2014, Schillinger et al., 2014). While the whole class of T-splines are not suitable as a basis for IGA because of possible linear dependence (Buffa et al., 2010), a mildly topological restricted subset of T-splines, analysis-suitable T-splines (AS T-splines), are optimized to meet the needs both for design and analysis (Li et al., 2012, Scott et al., 2012, Veiga et al., 2012, Veiga et al., 2013, Li and Scott, 2014). B-spline based IGA uses the operations of h-refinement (knot insertion), p-refinement (degree elevation) and k-refinement (both h and p-refinement are preformed) (Cottrell et al., 2007, da Veiga et al., 2011). The k-refinement provides smoother functions and increases the accuracy over the classical -continuous p-refinement for the problems of structural vibrations (Cottrell et al., 2007). Thus, the development of k-refinement or even local k-refinement for T-splines is important both for IGA and geometric modeling.
Our goal is to develop algorithms for T-spline degree elevation. Given a bi-degree T-spline space defined on a T-mesh , the algorithm finds a new T-mesh such that . If is a tensor-product mesh, then the new mesh is also a tensor-product mesh by increasing the knot multiplicity, which is the mesh defined in Section 4.3. However, this is not true for general T-splines because the relationship between and is unknown. Thus, we first provide a recursive algorithm in Section 4 based on degree elevation of each blending function. But in the process of the algorithm, we need to insert some additional vertices and edges such that the blending functions after degree elevation correspond to a valid T-mesh. If we restrict the resulting T-spline to be analysis-suitable, then the relation between T-mesh and is simplified (Theorem 5.3), which enables us to develop two optimized algorithms in Section 5. If the original T-mesh is also analysis-suitable, we can explicitly give the new T-mesh (Remark 5.4).
The paper is structured as follows. Section 2 provides the background on T-splines and AS T-splines. Section 3 recalls B-splines degree elevation. Section 4 presents a degree elevation algorithm for generic T-splines. Section 5 gives two optimized degree elevation algorithms. The last section is discussion.
Section snippets
Index T-mesh
An index T-mesh (Bazilevs et al., 2010) for a bi-degree T-spline is a collection of all the elements of a rectangular partition of the index domain , where all rectangle corners (or vertices) have integer coordinates. Denote the active region as a rectangle region , here is the maximal integers equal to or less than d. The active region carries the anchors that will be associated with the blending functions while the other indices
Degree elevation of B-splines curves
This section recalls degree elevation for a B-spline curve and a tensor-product B-spline surface.
Given a knot vector , , , a set of degree p B-spline basis functions can be defined in terms of the local knot vector . If we rewrite the knot vector by getting rid of the multiplicities as , here is the multiplicities of knots . Then the B-spline space can also be defined as
Degree elevation of T-splines
In this section, we provide a recursive degree elevation for general T-splines.
Analysis suitable degree elevation
Because AS T-splines possess many desirable good properties for geometric modeling and iso-geometric analysis, in this section, we develop two algorithms such that the T-spline after degree elevation is analysis-suitable.
Results and conclusion
We conclude by presenting some numerical experimentations about these three degree elevation algorithms for T-splines. The key issue to compare the algorithms is the number of resulting anchors. Thus all the examples are shown with the index T-meshes in this section.
The first example is a T-spline defined on a T-mesh refined from a tensor-product T-mesh along the diagonal faces. Using the general T-spline degree elevation (Fig. 12b), just similar as the behavior of local refinement algorithm in
Acknowledgements
The authors are supported by the NSFC (No. 11031007, No. 60903148, No. 11371341), a NKBRPC (2011CB302400), the Chinese Universities Scientific Fund, SRF for ROCS SE, and the Youth Innovation Promotion Association CAS.
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This paper has been recommended for acceptance by Thomas Sederberg.