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Figure 2: Finite element meshes .
T-Splines Correct the Deficiencies of NURBS
3
Background
The concept of B-splines was introduced in [1] and have found many applications in Finite Element analysis. A more general class of splines called T-splines was introduced in [2]. A NURBS (Non Uniform Rational B-spline) based T-splines was introduced in [3]. This work provides a comparison between NURBS and T-splines on the basis of finite elements. The next step in this direction is the development of isogeometric analysis and isogeometric analysis T-splines. The direct use of isogeometric analysis T-splines has not been studied yet. However, in [4], the authors presented the first isogeometric analysis T-spline construction algorithm for a Trimmed NURBS approximation which only use three degrees of freedom per control point.
In [5], the authors develop isogeometric analysis of T-splines, and show that each knot interval has one degree of freedom. In [6], the authors show that the S-shape functions provide an embedding of the space of T-splines. Finally in [7], the authors use the S-shape functions to construct isogeometric analysis T-splines.
Objective
In this report, we will compare NURBS and T-splines on the basis of isogeometric analysis for triangle meshes. Two NURBS and three T-splines will be compared, and the differences and similarities between them will be presented.
Comparison between B-spline and T-spline basis functions
B-Splines
A B-spline basis function, denoted by w(x), is a polynomial spline [1]. The minimal number of B-splines required to represent a given function is the degree of B-spline or the order of the B-spline. The B-spline basis functions form a family of basis functions for the space of polynomial spline over the interval [0,1]. They can be expressed as the linear combination of sines and cosines of the given polynomial.
Figure 3: Two B-spline basis functions and S-shape functions .
T-Spline
A T-spline basis function, w(x), is a piece
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