1 Introduction
Tsplines were originally introduced in Computer Aided Design (CAD) as a superior alternative to NURBS [1] and have since emerged as an important technology across several disciplines including industrial, architectural, and engineering design, manufacturing, and engineering analysis. Tsplines can model complicated designs as a single, watertight geometry and can be locally refined [2, 3]. These basic properties make it possible to merge multiple NURBS patches into a single Tspline [4, 1] and any trimmed NURBS model can be represented as a watertight Tspline [5].
The use of Tsplines as a basis for isogeometric analysis has gained widespread attention [6, 7, 3, 8, 9, 10, 11, 12, 13]. Isogeometric analysis was introduced in [14] and described in detail in [15]. The isogeometric paradigm is simple: use the smooth spline basis that defines the geometry as the basis for analysis. Traditional designthroughanalysis procedures such as geometry cleanup, defeaturing, and mesh generation are simplified or eliminated entirely. Additionally, the higherorder smoothness provides substantial gains to analysis in terms of accuracy and robustness of finite element solutions [16, 17, 18].
An important development in the evolution of isogeometric analysis was the advent of Analysissuitable Tsplines (ASTS). ASTS are a mildly restricted subset of Tsplines which are optimized to simultaneously meet the needs of design and analysis [19, 3]. Linear independence of analysissuitable Tspline blending functions was established in [19]. An efficient local refinement algorithm for ASTS was developed in [3]
. Later, it was shown that a dual basis, constructed as in the tensor product settting, could be generalized to ASTS
[20]. This characteristic of ASTS is called dual compatibility. These results were then generalized to ASTS surfaces of arbitrary degree in [21].In this paper we continue to develop the theory of ASTS spaces. Specifically, we provide a rigorous characterization of ASTS and show that the space of smooth parametric bicubic polynomials, defined over the extended Tmesh of an ASTS, is contained in the corresponding ASTS space. To accomplish this, the theory of perturbed ASTS spaces is developed and an ASTS dimension formula is established in terms of the topology of the extended Tmesh. We note that, unlike existing approaches, our dimension formula does not require that the Tmesh have any particular nesting structure. We then show that this characterization, coupled with the dual compatibility of ASTS, can be used to prove that ASTS spaces possess the same optimal approximation properties as tensor product Bspline spaces [22]. Next, we prove under what conditions two ASTS spaces are nested. This provides the theoretical justification for the analysissuitable local refinement algorithm in [3] and provides a foundation upon which adaptive isogeometric analysis procedures may be developed in the future.
This paper is organized as follows. Section 2 describes the Tmesh in index space, Tjunctions, and the extended Tmesh. Tsplines in the parametric domain, blending functions, and Tspline spaces are defined in Section 3. Section 4 describes the conditions under which a Tspline is analysissuitable. The theory of smoothly perturbed ASTS is developed in Section 5. Section 6 proves the conditions under which two ASTS spaces are nested. Using the characterization of ASTS spaces and dual compatibility several basic approximation results are proven in Section 7. Finally, Section 8 proves the dimension of ASTS spaces.
2 The Tmesh
An important object underlying Tspline spaces is the Tmesh. A Tmesh is used to determine Tspline basis functions and how they are arranged with respect to one another. In other words, the mesh topology of the Tmesh determines the functional properties of the resulting space. In an attempt to adhere to a single notation and to reduce confusion, we define a Tmesh following much of the notation given in [19, 20]. For quick reference, A lists the most important notational conventions used throughout the text and where they are defined.
2.1 Definition
A Tmesh is a rectangular partition of the index domain , , where all rectangle corners (or vertices) have integer coordinates and all rectangles are open sets. Each vertex in is a singleton subset of . We denote all vertices of by . An edge of is a segment between vertices of that does not intersect any rectangle of . We note that edges do not contain vertices and they are open at their endpoints. We denote all edges of by . Figure 1 shows an example of a Tmesh. The notation will indicate that can be created by adding vertices and edges to .
The valence of a vertex is the number of edges such that is an endpoint. We only allow valence three (called Tjunctions) or four vertices. Note that valence two vertices, other than the four corners, are eliminated from the definition.
The horizontal (resp., vertical) skeleton of a Tmesh is denoted by (resp., ), and is the union of all horizontal (resp., vertical) edges and all vertices. Finally, we denote the skeleton to be the union . For a given vertex we define and . We assume that these two sets are ordered.
We split the index domain into an active region and a frame region such that and , and . Note that both and are closed.
A symbolic Tmesh [19] is created from a Tmesh by assigning a symbol in Table 1 to each vertex in a tensor product mesh formed from the index coordinates, . The symbol is chosen to match the mesh topology of . The symbolic Tmesh corresponding to the Tmesh in Figure 1 is shown in Figure 2.
Symbol  Correspondence with 

Valence 4 vertex, corner vertex, or valence 3 boundary vertex in  
, , ,  Oriented valence three vertex in 
Vertical edge in  
Horizontal edge in  
No corresponding vertex or edge in 
2.2 Admissible Tmeshes
We say that a Tmesh is admissible if it satisfies three basic conditions. First, we require that contains the vertical segments for and the horizontal segments for . These horizontal and vertical lines are for basis function definition near the boundary. Second, we require that contains the vertical segments for and the horizontal segments for . Third, we require that for any two vertices in , such that for some , if (resp., ), then (resp., ). From a practical point of view these are minor restrictions. The Tmesh in Figure 1 is admissible.
We note that for convenience and simplicity, we often refer to only the active region of an admissible Tmesh when speaking of a Tmesh. In all cases, we assume that the frame region has an admissible topology.
2.3 Anchors and Tjunctions
We define the anchors . We denote the total number of anchors in by . We define to be the set of all valence three vertices. These are called Tjunctions. The symbols , , , indicate the four possible orientations of a Tjunction in a symbolic Tmesh. A Tjunction (resp., ) of type and (resp., , ) and their extensions are called horizontal (resp., vertical). The solid white and red circles in Figure 1 are anchors and the red circles are Tjunctions.
2.4 Segments
We define a segment to be a closed line segment of contiguous vertices and edges whose beginning and ending vertices are Tjunctions (interior or boundary). Given two horizontal (resp., vertical) segments defined over the intervals and we say that if . We denote by (resp., ) the collection of all horizontal (resp., vertical) segments, and by the collection of all segments. We define and . We assume these two sets are ordered. We denote the total number of segments in by . We denote the total number of horizontal (resp., vertical) segments in by (resp., ). We denote the number of line segments in (resp., ) by (resp., ).
2.5 The extended Tmesh
Tjunction extensions can be associated with each Tjunction. For example, given a Tjunction of type we extract from four consecutive indices such that . We call the face extension, the edge extension for such kind of Tjunction. Similarly, we can define the face and edge extensions for the other kinds of Tjunctions , , which are illustrated in Figure 3.
We denote the extension of Tjunction and the union of all horizontal (resp., vertical) face extensions by (resp., ), the union of all face extensions by , and the union of all extensions (face and edge) by . We define the extended Tmesh, , as the Tmesh created by adding to all the Tjunction extensions. In other words, . We denote the total number of vertices in by . The extended Tmesh corresponding to the Tmesh in Figure 1 is shown in Figure 4.
Adding Tjunction extensions to may introduce three additional collections of vertices. The first, called crossing vertices and denoted by , is created from the intersection of crossing face extensions. In other words,
We denote the number of crossing vertices in by . In Figure 4 the crossing vertices are denoted by red stars.
The second, called overlap vertices and denoted by , is created from the intersection of overlapping face extensions with . In other words,
We denote the number of overlap vertices in by . In Figure 4 the overlap vertices are denoted by green triangles.
The third, called extended vertices and denoted by , is created from the intersection of face extensions and
while removing those vertices which already correspond to overlap vertices. Additionally, all nonanchor vertices are classified as extended vertices. In other words,
We denote the number of extended vertices in by . In Figure 4 the extended vertices are denoted by black squares.
3 The parametric domain and Tspline spaces
Let and
be two global knot vectors defined on the interval
. Interior knots may have a multiplicity of three while end knots may have a multiplicity of four. The global knot vectors define a full parametric domain, , where and a reduced parametric domain, , where . The Tmesh in the parametric domain is defined as the collection of nonempty elements of the form where . We denote those elements where by . The extended Tmesh in the parametric domain as well as all element related concepts are defined similarly. Throughout this paper we use the index and parametric representation of a Tmesh interchangeably with the context making the use clear.For each anchor we define its horizontal (vertical) index vector (, respectively) as a subset of (, respectively) where contains five unique consecutive indices in with . The vertical index vector, denoted by , is constructed in an analogous manner. We then associate a Tspline blending function with anchor . The Tspline blending functions are given by
(1) 
where and are the cubic Bspline basis functions associated with the local knot vectors
(2)  
(3) 
and and .
Figure 5 illustrates the construction of a Tspline blending function corresponding to anchor . In this case, the local knot vectors are and .
A Tspline space is simply the span of the blending functions, , .
4 Analysissuitable Tsplines
Analysissuitable Tsplines form a practically useful subset of Tsplines. ASTS maintain the important mathematical properties of the NURBS basis while providing an efficient and highly localized refinement capability. Several important properties of ASTS have been proven:

The blending functions are linearly independent for any choice of knots [19].

The basis constitutes a partition of unity (see Corollary 6.7).

Each basis function is nonnegative.

They can be generalized to arbitrary degree [21].

An affine transformation of an analysissuitable Tspline is obtained by applying the transformation to the control points. We refer to this as affine covariance. This implies that all “patch tests” (see [23]) are satisfied a priori.

They obey the convex hull property.
Definition 4.1.
An analysissuitable Tspline is a Tspline whose Tmesh is analysissuitable [19]. A Tmesh is said to be analysissuitable if it is admissible and no horizontal Tjunction extension intersects a vertical Tjunction extension.
An analysissuitable Tmesh is shown in Figure 6a. The corresponding extended Tmesh is shown in Figure 6b. Notice that no horizontal extension intersects a vertical extension. The dual basis for an ASTS equips these spaces with a rich mathematical structure which we leverage in this paper [20].
Lemma 4.2.
For a bicubic ASTS, each dual basis function, corresponding to a Tspline basis function with local knot vectors and , is
where and are dual basis functions corresponding to univariate cubic Bsplines [24] whose knot vectors are and , respectively.
5 Perturbed Tsplines
From a theoretical point of view, developing a complete and rigorous characterization of Tspline spaces is complicated by the presence of zero knot intervals (especially near Tjunctions) and overlap vertices. However, allowing both is important when Tsplines are used as a tool in design and analysis.
To overcome this difficulty, we develop the theory of the perturbed Tmesh (and resulting perturbed Tspline space). A perturbed Tspline can be used to prove properties about the original Tspline. In other words, we will generate a perturbed Tmesh, establish the result in the perturbed setting, and then show that the result holds as the perturbation converges to the original Tspline.
5.1 Perturbed Tmeshes
A perturbed Tmesh is created by first generating perturbed global knot vectors, , , where is a vector of perturbation parameters. A perturbed global knot vector is written as
where takes the index of the knot in and the segment in and returns a unique index in the perturbed global knot vector. The knot values are initialized as . In other words, a knot index which corresponds to a which contains multiple segments in the Tmesh is repeated times. Notice that this operation induces an index map (resp., ) from the indices in the perturbed global knot vector onto the original global knot vector. The knot values are then perturbed using a small parameter as
where , if , and is equal to , otherwise. The constant, . This same procedure is applied to to form . The Tmesh, , is then modified to form the perturbed Tmesh, , by associating the vertices and edges contained in the segment of with knot . Notice that the number of anchors does not change when forming . A perturbed Tspline space is a Tspline space formed from perturbed global knot vectors and Tmesh. A strictly perturbed Tmesh or Tspline space is one where .
A perturbation of an analysissuitable Tmesh is shown in Figure 7. The analysissuitable Tmesh is shown in Figure 7a and the perturbed Tmesh is shown in Figure 7b. Knot intervals are shown instead of knots for simplicity. Recall that a knot interval is simply the difference between adjacent knots in a global knot vector. Notice that the horizontal and vertical zero knot intervals have been replaced by nonzero knot intervals and . The vertical segments with Tjunctions are perturbed resulting in a new knot interval .
Proposition 5.1.
If is analysissuitable then is analysissuitable.
Proof.
Suppose the extensions of two Tjunctions ( or ) and ( or ) in intersect. This implies that and . According to the construction of , we don’t change the order of the indices, so and , i.e., there are intersecting extensions in . ∎
Lemma 5.1.
Let be an analysissuitable Tmesh. For every anchor, , and horizontal index vector, , in the perturbed Tmesh, , where . This also holds for the vertical index vectors.
Proof.
If the topological symbols corresponding to the indices where , , are not or then the result immediately follows since the indices are contained in a single horizontal segment. Thus, we only need to prove that the result holds when the topological symbol corresponding to is or . Without loss of generality we assume it to be . Since, according to Lemma 5.1, the Tmesh, , is analysissuitable the symbols for the first two indices in can only be or . Thus, . ∎
Theorem 5.2.
If is analysissuitable and is an offset perturbation then as , where .
6 Refineability and nestedness
We now explore the refineability and nesting behavior of analysissuitable Tspline spaces. In other words, given two analysissuitable Tsplines spaces, and , we establish the conditions under which . We first establish basic refineability properties when the analysissuitable Tmesh does not have any knot multiplicities or overlap vertices. Using the theory of perturbed Tsplines, we then extend those results to encompass Tmeshes which do have zero knot intervals and overlap vertices.
Definition 6.1.
The notation denotes a perturbed Tmesh where and is created by removing those edges and vertices from the strictly perturbed Tmesh which correspond to unperturbed edges and vertices in . By inspection, it is clear that , constructed in this way, is a perturbed Tmesh which satisfies Proposition 5.1, Lemma 5.1, and Theorem 5.2 and that .
The construction of is depicted in Figure 8. Two analysissuitable Tmeshes are shown in Figure 8a and Figure 8b. Notice that . The perturbed Tmesh (shown in Figure 8c) is formed by removing the dotted lines (shown in Figure 8c) from (shown in Figure 8d).
Definition 6.2.
Given a Tmesh, , with no knot multiplicities, and the corresponding extended Tmesh, , the homogeneous extended spline space is defined as
(4) 
where is the space of bivariate functions which are continuous in and over all of . is the space of bicubic polynomials. The extended spline space is defined to be .
Proposition 6.1.
Proof.
We first prove that the dimension of is not less than the dimension of . Notice that for any function , . We now show that the dimension of is not less than the dimension of . This is equivalent to showing that there is only one function in which is zero over . It is easy to see that the only function which is zero over must be zero over all of since the minimum support of a cubic spline function is four intervals. ∎
Lemma 6.3.
If the extended Tmesh, , of an analysissuitable Tmesh, , has no knot multiplicities or overlap vertices, then . In other words, the analysissuitable Tspline space, , and the extended spline space, , are the same space.
Proof.
We have that (see [20], Lemma 4.3), so the dimension of is less than that of , which, according to Theorem 8.5, Proposition 6.1, and Theorem 8.9 is the number of active vertices. Since the blending functions for anaysissuitable Tsplines are linearly independent the dimension of is also the number of active vertices. Thus, the two spline spaces are identical. ∎
Lemma 6.4.
Given two analysissuitable Tmeshes, and , neither of which has knot multiplicities or overlap vertices, if , then .
Proof.
Obviously, . Since and are analysissuitable, according to Lemma 6.3, . ∎
Lemma 6.5.
Given two analysissuitable Tspline spaces, and , if , then .
Proof.
Suppose the perturbed Tspline space, , is spanned by the basis functions, , and the perturbed Tspline space, , is spanned by the basis functions, . We have that
where , are the dual functionals for the analysissuitable Tspline basis as described in [20]. According to Theorem 5.2,
According to Theorem 4.41 in [24], is bounded, so
i.e.,
∎
Theorem 6.6.
Given two analysissuitable Tmeshes, and , if , then .
Proof.
Proposition 6.2.
Every analysissuitable Tspline space contains the space of bicubic polynomials.
Proof.
Corollary 6.7.
Every analysissuitable Tspline space forms a partition of unity. In other words, , .
Proof.
This immediately follows from Proposition 6.2 and the fact that . ∎
7 Approximation
As described in [20, 21] approximation properties of analysissuitable Tsplines are directly linked to Proposition 6.2. In other words, having the bicubic polynomials in the Tspline space is the minimal requirement to obtain an convergence rate in the mesh size.
Following the approach in [22, 20, 21], the dual basis for an analysissuitable Tspline space, , can be used to construct a projection operator, , where
We denote the open support of a Tspline basis function by , and the extended support of an element by , where
We will denote by the smallest rectangle in containing and .
Proposition 7.1.
Given an analysissuitable Tspline space, , the projection operator is (locally) h–uniformly continuous in the norm. In other words, there exists a constant independent of such that
Note that the constant may depend on the polynomial degree.
Proposition 7.2.
Given an analysissuitable Tspline space, , there exists a constant independent of such that for
where denotes the diameter of . Note that the constant may depend on the polynomial degree.
8 Dimension
In this section, we develop a dimension formula for polynomial spline spaces defined over the extended Tmesh in the parametric domain of a Tspline and establish the connection between this dimension formula and analysissuitable Tspline spaces. The dimension formula, written only in terms of topological quantities of the original Tmesh, is an essential ingredient in establishing the refineability properties in Section 6 and the approximation results in Section 7 for analysissuitable Tsplines. The essential results are proven in Theorems 8.5 and 8.9.
Unlike existing approaches, our dimension formula does not require that the Tmesh have any nesting structure. Of critical importance is how this dimension formula can be directly related to analysissuitable Tspline spaces which can then be used to construct a simple set of basis functions for the spline space which are compatible with commercial CAD and analysis frameworks.
8.1 Smoothing cofactorconformality method
We use the smoothing cofactorconformality method [25, 24] to transform the smoothness properties of into a linear constraint matrix, . This constraint matrix is then analyzed to determine the dimension of . We recall that the spline space, , is defined using the extended Tmesh, , corresponding to a Tmesh, , which does not have any knot multiplicities.
8.1.1 Vertex and edge cofactors
As shown in Figure 9, for any vertex, , the surrounding bicubic polynomial patches are labeled, , . If the vertex, , is a Tjunction, then for some . Since and are continuous there exists a cubic polynomial , called the edge cofactor, such that
(5) 
8.1.2 Assembling the constraint matrix,
Referring to Figure 10, consider a horizontal segment with vertices and edge cofactors. Using (10) we have that
(11)  
(12)  
(13) 
and
(14) 
Summing (11)  (13) and using (14) results in the linear system
(15) 
We call the solution space, denoted by , for this linear system the edge conformality space. Similarly, for a vertical segment we have that
(16) 
where the solution space is denoted by . By (15) and (16), one immediately has that
Lemma 8.1.
If each and are different, then the dimension of and are and respectively.
The linear systems (15) and (16), associated with the horizontal and vertical segments in , can be assembled into the global system
(17) 
where is a column vector of all vertex cofactors in and is a real matrix. Each edge conformality condition corresponds to a submatrix consisting of rows of and each vertex cofactor corresponds to a column of .
Lemma 8.2.
The dimension of is the nullity of , i.e., the dimension is minus the rank of .
8.2 Simplifying the constraint matrix, , and
It is possible to simplify the constraint matrix, , and the topology of the extended Tmesh in the parametric domain, , such that the null space of is undisturbed. To remove a vertex from means we delete the corresponding column from and to remove a segment from means we delete the appropriate submatrix from . We form the reduced constraint matrix by removing the eight segments and contained vertices , , , ,
Comments
There are no comments yet.