Hierarchical analysissuitable Tsplines: Formulation, Bézier extraction, and application as an adaptive basis for isogeometric analysis
Abstract
In this paper hierarchical analysissuitable Tsplines (HASTS) are developed. The resulting spaces are a superset of both analysissuitable Tsplines and hierarchical Bsplines. The additional flexibility provided by the hierarchy of Tspline spaces results in simple, highly localized refinement algorithms which can be utilized in a design or analysis context. A detailed theoretical formulation is presented including a proof of local linear independence for analysissuitable Tsplines, a requisite theoretical ingredient for HASTS. Bézier extraction is extended to HASTS simplifying the implementation of HASTS in existing finite element codes. The behavior of a simple HASTS refinement algorithm is compared to the local refinement algorithm for analysissuitable Tsplines demonstrating the superior efficiency and locality of the HASTS algorithm. Finally, HASTS are utilized as a basis for adaptive isogeometric analysis.
keywords:
isogeometric analysis, hierarchical splines, adaptive mesh refinement, Tsplines1 Introduction
In this work, a hierarchical extension of analysissuitable Tsplines is developed and utilized in the context of isogeometric design and analysis. We call this new spline description hierarchical analysissuitable Tsplines (HASTS). The class of HASTS is a strict superset of both analysissuitable Tsplines LiZhSeHuSc10 (); ScLiSeHu10 (); BeBuChSa12 (); BeBuSaVa12 (); LiScSe12 () and hierarchical Bsplines FoBa88 (); SchDeScEvBoRaHu12 (); VuGiJuSi11 (); ScThEv13 (); GiJu13 (); GiJuSp12 ().
Tsplines, introduced in the CAD community SeZhBaNa03 (), are a generalization of nonuniform rational Bsplines (NURBS) which address fundamental limitations in NURBSbased design. For example, a Tspline can model a complicated design as a single, watertight geometry and are also locally refinable SeCaFiNoZhLy04 (); ScLiSeHu10 (). Since their advent they have emerged as an important technology across multiple disciplines and can be found in several major commercial CAD products TSManual12 (); Autodesk360 ().
Isogeometric analysis was introduced in HuCoBa04 () and described in detail in Cottrell:2009rp (). The isogeometric paradigm is simple: use the smooth spline basis that defines the geometry as the basis for analysis. As a result, exact geometry is introduced into the analysis, the smooth basis can be leveraged by the analysis EvBaBaHu09 (); HuEvRe13 (); CoHuRe07 (), and new innovative approaches to model design WaZhScHu11 (); LiZhHuScSe14 (), analysis SchDeScEvBoRaHu12 (); ScSiEvLiBoHuSe12 (); ScWuBl12 (); BeBaDeHsScHuBe09 (), optimization Wall08 (), and adaptivity Bazilevs2009 (); DoJuSi09 (); ScThEv13 (); ScThEv13 () are made possible. The use of Tsplines as a basis for isogeometric analysis (IGA) has gained widespread attention across a number of application areas Bazilevs2009 (); ScBoHu10 (); ScLiSeHu10 (); Verhoosel:2010vn (); Verhoosel:2010ly (); BoScLaHuVe11 (); BeBaDeHsScHuBe09 (); SchDeScEvBoRaHu12 (); ScSiEvLiBoHuSe12 (); SiScTaThLi14 (); DiLoScWrTaZa13 (); HoReVeBo14 (); BaHsSc12 (); BuSaVa12 (); GiKoPoKaBeGeScHu14 (). Particular focus has been placed on the use of Tspline local refinement in an analysis context ScLiSeHu10 (); BoScLaHuVe11 (); Verhoosel:2010ly (); Verhoosel:2010vn ().
In the context of CAD, where a designer interacts directly with the geometry, Tspline local refinement is most useful if confined to a single level. In other words, all local refinement is done on one control mesh and all control points have similar influence on the shape of the surface. In this way, the geometric behavior of the surface is easily controlled through the manipulation of control points before and after refinement. In the context of analysis, however, where not all control points need to have a geometric interpretation, the single level restriction can be relaxed. This hierarchical point of view has important advantages:

Hierarchical local refinement remains completely local. Single level Tspline local refinement always entails a degree of nonlocal control point propagation ScLiSeHu10 ().

Hierarchical local coarsening is achieved by simply removing higher levels of refinement where needed ScThEv13 (). Local coarsening operations for single level Tspline descriptions are possible but their algorithmic complexity remains uncertain SeCaFiNoZhLy04 ().

Hierarchical refinement and coarsening operations use a fixed control mesh which simplifies algorithmic developments, especially for parallel computations. Single level local refinement requires expensive mesh manipulation and modification operations.

Hierarchies of finitedimensional subspaces are the natural setting for many optimized iterative solvers and preconditioning techniques for largescale linear systems.
Initial investigations employing hierarchical Bspline refinement in the context of IGA have demonstrated the promise of the hierarchical approach KuVeZeBr14 (); SchDeScEvBoRaHu12 (); SchEvReScHu13 (); GrKrSch02 (); ScRa11 ().
HASTS inherit the design strengths of Tsplines without the single level restriction. In this way, a complex Tspline design can be encapsulated in the first level of the hierarchy while higher levels can be leveraged to develop adaptive multiresolution schemes which are smooth, highly localized, geometrically exact, and appropriate for the analysis task at hand. We feel that this provides the appropriate mathematical foundation for the development of integrated isogeometric design and analysis methodologies for demanding applications in science and engineering. Note that, in this paper, we restrict our theoretical developments to HASTS defined over foursided domains. However, extending HASTS to domains of arbitrary topological genus should be straightforward in the context of the recently introduced spline forest ScThEv13 ().
We note that in addition to Tsplines, hierarchical Bsplines, and NURBS a number of alternative spline technologies have been proposed as a basis for IGA with varying strengths and weaknesses. Truncated hierarchical Bsplines (THBsplines) GiJuSp12 (); KiGiJu14 (); GiJuSp13 (); BeKiBrChMoOhKi14 () are a modification of hierarchical Bsplines FoBa88 (); VuGiJuSi11 (); SchEvReScHu13 () which possess a partition of unity and enhanced numerical conditioning. Bspline forests ScThEv13 () are a generalization of hierarchical Bsplines to surfaces and volumes of arbitrary topological genus. Polynomial splines over hierarchical Tmeshes (PHTsplines) htspline2 (); htspline3 (); htspline4 (); htspline5 (), modified Tsplines KaChDe13 (), and locally refined splines (LRsplines) DoLyPe13 (); Br13 () are closely related to Tsplines with varying levels of smoothness and approaches to local refinement. Generalized Bsplines MaPeSa11 (); CoMaPeSa10 () and Tsplines BrBeChOhKi14 () enhance a piecewise polynomial spline basis by including nonpolynomial functions, typically trigonometric or hyperbolic functions. Generalized splines permit the exact representation of conic sections without resorting to rational functions. Generalized splines can also be used to represent solution features with known nonpolynomial characteristics exactly in certain circumstances.
1.1 Structure and content of the paper
In Section 2 the Tmesh is described and appropriate notational conventions are introduced. Analysissuitable Tsplines are then described in Section 3. The local linear independence of analysissuitable Tsplines is established in Section 4. Hierarchical analysissuitable Tsplines are then defined in Section 5. In preparation for their use in design and analysis a Bézier extraction framework is introduced in Section 6. HASTS are then utilized as a basis for isogeometric analysis in Section 7. In Section 8 we draw conclusions. We note that the paper has been written so the proof of local linear independence in Section 4 is selfcontained and can be skipped if the reader is not interested in the detailed theory of analysissuitable Tsplines.
2 The Tmesh
The Tmesh is used to define the topological structure of the associated Tspline space. In other words, the Tmesh defines the basis functions and their relationship to one another. We closely follow the notational conventions introduced in LiScSe12 (); BeBuSaVa12 (); BeBuChSa12 ().
A Tmesh in two dimensions is a rectangular partition of such that all vertices have integer coordinates. All cells are rectangular, nonoverlapping, and open. An edge is a horizontal or vertical line segment between vertices which does not intersect any cell. The valence of a vertex is the number of edges coincident to that vertex. Since all cells are assumed rectangular, only valence three (i.e., Tjunction) or four is allowed for all vertices . The sets of horizontal and vertical coordinates in the Tmesh are denoted by and . The horizontal and vertical skeletons, and , of a Tmesh are the union of all horizontal and vertical edges, respectively, and associated vertices. The entire skeleton is denoted by .
We split into an active region and a frame region such that and , and where and are polynomial degrees. Note that both and are closed. Further, all Tmeshes considered in this work are admissible as described in LiScSe12 (), a mild restriction always adopted in practice. The notation will indicate that can be created by adding vertices and edges to . Fig. 1 shows an example Tmesh.
2.1 Analysissuitable Tmeshes
Analysissuitable Tsplines (ASTS) were introduced in LiZhSeHuSc10 (). The analysissuitability of a Tspline is dictated by the structure of the underlying Tmesh. We define face and edge extensions to be closed line segments that originate at Tjunctions. For example, to define a horizontal face extension we trace out a horizontal line by moving in the direction of the missing edge until vertical edges or vertices are intersected. To define an edge extension we trace out a horizontal line by moving in the direction opposite the face extension until vertical edges or vertices are intersected. A Tjunction extension includes both the face and edge extensions. Since extensions are defined as closed line segments they may intersect at their end points. An extended Tmesh, , is the Tmesh formed by adding the Tjunction extensions to . The collection of rectangular cells in is denoted by . We say a Tmesh is analysissuitable if no horizontal Tjunction extension intersects a vertical Tjunction extension. Face and edge extensions (along with analysissuitability) are illustrated in Figure 2.
2.2 Anchors
Anchors are used in the construction of Tspline blending functions. For an analysissuitable Tmesh the anchors are located only in the active region and are defined as follows:

if and are odd the anchors are vertices. It is written as or equivalently .

if is even and is odd the anchors are horizontal edges. It is written as .

if is odd and is even the anchors are vertical edges. It is written as .

if and are even the anchors are cells. It is written as .

The set of all anchors is denoted by . The set of anchors for varying values of and are shown in Figure 3.
3 Analysissuitable Tsplines
Analysissuitable Tsplines form a 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 LiZhSeHuSc10 ().

The basis constitutes a partition of unity LiScSe12 ().

Each basis function is nonnegative.

They can be generalized to arbitrary degree BeBuSaVa12 ().

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 Hug00 ()) are satisfied a priori.

They obey the convex hull property.

They can be locally refined SeZhBaNa03 (); ScLiSeHu10 (); LiScSe12 ().

A dual basis can be constructed BeBuChSa12 (); BeBuSaVa12 ().

Optimal approximation LiScSe12 ().
The important properties of ASTS emanate directly from the topological properties of the underlying analysissuitable Tmesh and resulting set of Tspline basis functions constructed from it.
3.1 Tspline basis functions, spaces, and geometry
Given a parametric domain we define global horizontal and vertical open knot vectors and , respectively. In other words,
and
As a result, every Tmesh vertex has the parametric representation . For reasons that will become apparent, we refer to a cell with positive parametric area as a Bézier element. The parametric domain of a Bézier element is denoted by . The set of all Bézier elements in a Tmesh is denoted by .
For each anchor we construct horizontal and vertical local index vectors and made up of increasing (but not necessarily consecutive) indices in and , respectively. Note that for odd, , and for even, . Similar relationships hold for . The procedure for determining local index vectors is shown in Fig. 4 for various polynomial degrees. To clarify this procedure we describe the anchors and associated local index vectors. In Figure 4a, and and thus the example anchor is the cell . The horizontal local index vector is and the vertical local index vector is We observe that subset of the vertical skeleton located at index does not span the entire height of the anchor cell, hence it is not included in the horizontal local index vector; similarly since subset of the horizontal skeleton located at index does not span the entire width of the cell it not included in the vertical local index vector. In Figure 4b, and thus the example anchor is the vertical edge . The horizontal local index vector is and the vertical local index vector is Similar to the prior example the subset of the vertical skeleton at index does not span the entire height of the anchor edge, hence it is not included in the horizontal local index vector. Figure 4c shows the case where and thus the example anchor is the horizontal edge . The horizontal local index vector is and the vertical local index vector is In the last case, shown in Figure 4d, and thus the example anchor is the vertex . The horizontal local index vector is and the vertical local index vector is


(a) 
(b) 


(c) 
(d) 
The Tspline blending function is given by
(1) 
where and are Bspline basis functions associated with the local knot vectors and .
We define to be the set of all basis functions associated with a Tmesh. Given a weight for each a rational Tspline basis function can be written as
(2)  
(3) 
where is called a weight function. For clarity we will often suppress the dependence on the polynomial degrees and write the basis function as . Figure 5 shows several Tspline basis functions plotted in the parametric domain .
An ASTS space, denoted by , is the span of the blending functions in constructed from an analysissuitable Tmesh. Given vector valued control points, , or , the geometry of a Tspline can be written as
(4) 
4 Local linear independence of analysissuitable Tsplines
The local linear independence of analysissuitable Tsplines is an important theoretical result in its own right and is critical for our definition of hierarchical analysissuitable Tsplines in Section 5. The (global) linear independence of analysissuitable Tsplines was first shown in LiZhSeHuSc10 (). Local linear independence is a stronger result than global linear independence and is the notion of linear independence enjoyed by standard finite element bases. Local linear independence implies that the finite element basis is linearly independent over every element domain. For smooth locallyrefined bases this elementlevel notion of linear independence may be lost.
4.1 Preliminaries
Given a knot vector and degree we can recursively define Bspline basis functions as follows:
(5) 
(6) 
The de Boor algorithm Farin99 () provides a standard method for evaluating a Bspline, although other possibilities exist Se10 (); Ra89 (). A Bspline basis function can also be denoted by . Following Schu93 (), there exist dual functionals such that where the Kronecker delta is zero when and one otherwise. We have the following two results:
Lemma 4.1.
Suppose . Then
Lemma 4.2.
For any function , if , then for we have that
As shown in BeBuSaVa12 (); BeBuChSa12 () the notion of a dual basis can be extended to analysissuitable Tsplines.
Theorem 4.3.
Given an analysissuitable Tmesh and associated basis functions the set of functionals form a dual basis. Specifically, we have that
where and are dual basis functions corresponding to univariate Bsplines Schu93 () with local knot vectors and , respectively.
4.2 Proof of local linear independence
Lemma 4.4.
Let be a cell from an analysissuitable extended Tmesh with vertices and . If the basis function anchored at with local index vectors and is nonzero over then there must either exist an integer , , such that and or an integer , , such that and .
Proof.
Suppose the lemma is false, then there exists at least one cell in that violates the condition. We denote this cell by be . Since violates the condition there exists at least one corner of that lies in where and . Without loss of generality we may assume that and . We have following three cases:

The corner is a vertex of the original Tmesh. This violates Lemma 3.2(a) in BeBuSaVa12 ().

The corner is the result of the intersection of two perpendicular Tjunction extensions. This violates the assumption that the Tmesh is analysissuitable.

The corner is the result of the intersection of one Tjunction extension and a Tmesh edge. Without loss of generality, we assume the edge is a horizontal edge and the Tjunction extension is vertical and is associated with a Tjunction . As the edge cannot intersect the vertical line , it must terminate in a Tjunction . Examining the extensions associated with and we find that they must intersect. This violates the assumption that the Tmesh is analysissuitable.
Hence, such cell cannot exist. ∎
Theorem 4.5.
The basis for an analysissuitable Tspline is locally linearly independent.
Proof.
Let an arbitrary Bézier element, , from an analysissuitable Tmesh be given. We denote the element vertices by and . Let be the set of anchors whose corresponding basis functions are nonzero over . Assume that for all . By Lemma 4.4, since is nonzero over , there exists either a , , such that and or an , , such that and By Lemma 4.3
Since for or for we have by Lemma 4.2 that
or
which completes the proof. ∎
5 Hierarchical analysissuitable Tsplines
A hierarchical Tspline space is constructed from a finite sequence of nested ASTS spaces, , , and bounded open index domains, , which define the nested domains for the hierarchy. Two important theoretical results for ASTS will be used in the construction of hierarchical analysissuitable Tsplines:
Theorem 5.1.
Given two analysissuitable Tmeshes with nonoverlapping Tjunction extensions, and , if , then .
Theorem 5.2.
Analysissuitable Tsplines are locally linear independent.
We note that to accommodate overlapping Tjunction extensions requires a minor generalization of Theorem 5.1 which is not reproduced here to maintain clarity of exposition. For a complete description of the underlying theory we refer the interested reader to LiScSe12 (). The local linear independence of ASTS is proven in Section 4.
5.1 Sequences of analysissuitable Tmeshes
We construct a sequence of analysissuitable Tmeshes such that , , as follows:

Create from by subdividing each cell in into four congruent cells.

Extend Tjunctions in until it is analysissuitable and .
This algorithm is graphically demonstrated in Figure 6 for a particular Tmesh. For an efficient and general algorithm to produce nested analysissuitable Tspline spaces see ScLiSeHu10 ().


(b) Create from by subdividing Bézier elements. 
(c) Extend Tjunctions until is analysissuitable and . 
5.2 Hierarchical Tspline spaces
The hierarchical analysissuitable Tspline basis can be constructed recursively in a manner analogous to that used for hierarchical Bsplines VuGiJuSi11 ():

Initialize

Recursively construct from by setting
where
and

Set
We denote the number of functions in by . We call the space spanned by the functions in a hierarchical analysissuitable Tspline space and denote it by . To make the ideas concrete a univariate hierarchical spline space is shown in Figure 7.
The linear independence of the functions in follows immediately from the definition of hierarchical Tsplines and the local linear independence of ASTS (see Section 4).
Lemma 5.3.
The functions in the hierarchical basis are linearly independent.
Proof.
See Lemma 2 in VuGiJuSi11 () ∎
Lemma 5.4.
Given , a sequence of hierarchical analysissuitable Tspline bases, , .
Proof.
See Lemma 3 in VuGiJuSi11 () ∎
By construction, , thus the approximation properties of analysissuitable Tsplines are inherited by their hierarchical counterpart. In particular, constants are exactly represented and all patch tests are exactly satisfied LiScSe12 (); Hug00 ().
6 Bézier extraction of hierarchical analysissuitable Tsplines
The Bézier extraction framework ScBoHu10 (); Borden:2010nx (); ScThEv13 () can be extended to HASTS in a straightforward fashion. Using Bézier extraction, the spline hierarchy is collapsed onto a single level finite element mesh which can then be processed by standard finite element codes without any explicit knowledge of HASTS algorithms or data structures.
6.1 Bernstein basis functions
The univariate Bernstein basis functions are written as
(7) 
where and the binomial coefficient , . In CAGD, the Bernstein polynomials are usually defined over the unit interval , but in finite element analysis the biunit interval is preferred to take advantage of the usual domains for Gauss quadrature. The univariate Bernstein basis has the following properties:

Partition of unity.

Pointwise nonnegativity.

Endpoint interpolation.

Symmetry.
Figure 8 shows the Bernstein basis for polynomial degrees .
We construct a bivariate Bernstein basis function of degree by where , , and , as the tensor product of univariate basis functions
(8) 
with
(9) 
6.2 The geometry of a hierarchical representation
In a single level Tspline, basis functions and control points have a onetoone relationship and each control point influences the geometry in a similar manner. In a hierarchical context it is common to only associate control points with the functions in . This is the convention adopted in this paper. Note that by construction every blending function in can be written in terms of basis functions in (see Lemma 5.4). We call the functions in geometric blending functions. We use to denote the number of geometric blending functions.
Given vector valued control points, , or , and weights , the geometry of a hierarchical representation can be written as
(10)  
(11) 
where , is used to index the geometric blending functions, and is the weight function. The decoupling of geometry from the basis functions in is an additional complexity unique to hierarchical representations which is elegantly addressed via Bézier extraction.
6.3 Bézier Elements
The set of Bézier elements underlying a hierarchical Tspline are determined recursively in a manner similar to the basis. We denote the set of Bézier elements in a hierarchy by . We construct as follows:

Initialize

Recursively construct from by setting
where
and

Set
We denote the number of Bézier elements in by .
6.4 Element localization
Using standard techniques ScBoHu10 (); Borden:2010nx () it is possible to determine the set of functions in which are nonzero over any element in . This gives rise to a standard element connectivity map which, given an element index and local function index , returns a global function index . In other words . The reader is referred to Hug00 () for additional details on common approaches to finite element localization and the array. Note that can indicate an anchor or a global function index.
We write a rational hierarchical Tspline basis function, restricted to element , as
(12) 
where and is the element weight function restricted to element . The element geometric map is the restriction of to element .
6.5 Bézier extraction
To present the basic ideas, Bézier extraction for a Bspline curve is shown graphically in Figure 9. Bézier extraction constructs a linear transformation defined by a matrix referred to as the extraction operator. The extraction operator maps a Bernstein polynomial basis defined on Bézier elements to the global spline basis. The transpose of the extraction operator maps the control points of the spline to the Bézier control points.
Each hierarchical basis function supported by element can be written in Bézier form as
(13)  
(14) 
where the dependence of the Bernstein polynomial on the polynomial degrees and has been suppressed for clarity. The overbar will be used to denote a quantity written in terms of the Bernstein basis defined over the element domain . The Bézier coefficients are computed using standard knot insertion techniques ScBoHu10 (). We denote the vector of hierarchical basis functions supported by element by and the vector of Bernstein basis functions by . We then have that
(15)  
(16) 
where is the element extraction operator (see ScThEv13 ()). In other words, the element extraction operator is composed of the Bézier coefficients .
We write the element weight function as
where is the number of geometric basis functions which are nonzero over element . We may also write the rational hierarchical basis functions as
Finally, the element geometric map can be written as
The implementation of a finite element framework based on Bézier extraction is described in detail in ScBoHu10 (); Borden:2010nx ().
7 Computational Results
We illustrate the use of hierarchical Tsplines in the context of isogeometric analysis. We consider problems that highlight the unique attributes of both hierarchical refinement and Tsplines. The examples used are inspired by those found in ScLiSeHu10 (); VuGiJuSi11 (); ScThEv13 ().
7.1 A comparison between ASTS and HASTS local refinement
We compare local refinement of ASTS to local refinement of HASTS. When working with ASTS all refinement is performed on a single level whereas when working with HASTS this constraint is relaxed. For additional algorithmic details on local refinement of ASTS see ScLiSeHu10 (). We locally refine the Tspline ship hull design shown in Fig. 10 using both methods. The geometry is constructed using the Autodesk Tspline plugin for Rhino3d tspline_rhino (). Tsplines are popular in ship hull design because an entire hull can be modeled by a single watertight surface with a minimal number of control points SeSe10 (). Tjunctions can be used to efficiently model local features. Note that the initial Tspline of the hull contains just control points and Bézier elements.
We restrict the refinement region to the locations detailed in Fig. 11. It is assumed that the original design is too coarse to be used as a basis for analysis and additional resolution is required in the rectangular region followed by highly localized refinements along the region corresponding to the curve.
The HASTS refinement algorithm is based on the algorithm presented in ScThEv13 () for spline forests. The algorithm is elementbased, meaning refinement is driven by the subdivision of Bézier elements. The hierarchical basis is then reextracted into the new hierarchical Tmesh topology to generate the new set of Bézier elements. A detailed description of the underlying algorithms, in the context of HASTS, will be postponed to a future publication. Figure 12 shows three HASTS local refinements along the curve shown in Figure 11. The elements are colored according to their level, , in the hierarchy. Note that no nonlocal propagation of local refinement occurs for HASTS. Only those elements specified for refinement are subdivided. This is possible due to the relaxation of the single level constraint inherent in ASTS. The refinements form a nested sequence of continuous hierarchical analysissuitable Tspline spaces. The geometry of the hull is exactly preserved during refinement. The final HASTS is composed of Bézier elements and basis functions. However, only geometric blending functions and control points are used to define the hull geometry.
As a comparison, Figure 13 shows the results of ASTS local refinement using the algorithm from ScLiSeHu10 (). The top figure shows the control points added during local refinement (black dots) along the curve. The region selected for refinement is shown in red. Observe the propagation of the control points away from the selected refinement region. The bottom figure shows the resulting Bézier elements after refinement. Superfluous control points and elements are added just to satisfy the single level constraint inherent in the definition of ASTS.
7.2 HASTS as an adaptive basis
We now consider HASTS as an adaptive basis for isogeometric analysis. We choose as a benchmark the advection skew to the mesh problem shown in Figure 14. This problem is advection dominated, with diffusivity of . Along the external boundary, the boundary conditions are selected such that sharp interior and boundary layers are present in the solution. In this case, degrees.
7.2.1 Problem Statement
Let be a bounded region in and assume has a piecewise smooth boundary . Let denote a general point in , and let the temperature at a point be denoted by . Given Dirichlet boundary data, , the steadystate advectiondiffusion boundary value problem consists of finding the temperature such that
(17)  
where and are the spatially varying solenoidal velocity vector and symmetric, positivedefinite, diffusivity tensor, respectively. Note that in this paper we define where is a positive constant called the diffusivity coefficient. We employ SUPG BroHug82 () with a standard definition for the element stabilization parameter, .
7.2.2 A residual based error estimator
To estimate the error we employ a simple residualbased explicit estimator based on the variational multiscale theory for fluids Hug95 (); LaMa05 (); HaDoMi06 (); HaDoFu12 (). It is given by
(18) 
Note that this error estimator underestimates the error for diffusiondominated flows but is adequate for the advectiondominated benchmark presented in this paper. Using standard techniques Hug00 () we use the element scaling
(19) 
where are the mesh size distributions for and , respectively, and is the order of convergence of the method. The element size, , is the square root of the element area. We flag elements for refinement if . The adaptive process is repeated until a specified convergence tolerance is attained or a maximum number of hierarchical levels are introduced.
7.3 Results
We solve the problem with biquadratic and bicubic hierarchical Tsplines. The initial Tmesh for both cases is shown in Figure 15. Note that the initial Tmesh is locally refined to accommodate the presence of sharp boundary layers in the solution. Note that this refinement is not hierarchical. We have found that judiciously performing local refinement of the first level of a Tspline hierarchy to accommodate geometric features or boundary conditions leads to smaller hierarchies and more efficient solution procedures.
During each adaptive step the error is assessed as described in Section 7.2.2, elements are flagged for refinement and subdivided, and a new hierarchical basis is then extracted into the new hierarchical Tmesh topology. This generates a refined set of Bézier elements. The sequence of Bézier mesh refinements is shown for both the biquadratic and bicubic case in Figures 16–17. The sequence of biquadratic refinements form a nested sequence of continuous HASTS spaces, whereas the sequence of bicubic refinements form a nested sequence of continuous HASTS spaces. Note that fewer elements are required for convergence as the smoothness and order of the basis increases ScThEv13 ().
To illustrate the structure and distribution of the hierarchical basis the Greville abscissae ScSiEvLiBoHuSe12 () are plotted in Figures 18–20. Note that a linear parameterization was employed for all meshes and the level zero control points and blending functions define the geometry. The dots are scaled according to their level in the hierarchy; a larger dot denotes a lower level. The sequence of solutions are shown in Figures 21–22.
(a) Initial biquadratic Bézier mesh 
(b) Initial bicubic Bézier mesh 
(c) Second biquadratic Bézier mesh 
(d) Second bicubic Bézier mesh 


(e) Third biquadratic Bézier mesh 
(f) Third bicubic Bézier mesh 
(a) Fourth biquadratic Bézier mesh 
(b) Fourth bicubic Bézier mesh 
(c) Fifth biquadratic Bézier mesh 
(d) Fifth bicubic Bézier mesh 
(e) Sixth biquadratic Bézier mesh 
(f) Sixth bicubic Bézier mesh 
(a) Initial quadratic Greville abscissae 
(b) Initial bicubic Greville abscissae 
(c) Greville abscissae for second biquadratic mesh 
(d) Greville abscissae for second bicubic mesh 


(a) Greville abscissae for third biquadratic mesh 
(b) Greville abscissae for third bicubic mesh 
(c) Greville abscissae for fourth biquadratic mesh 
(d) Greville abscissae for fourth bicubic mesh 
(a) Greville abscissae for fifth biquadratic mesh 
(b) Greville abscissae for fifth bicubic mesh 
(c) Greville abscissae for sixth biquadratic mesh 
(d) Greville abscissae for sixth bicubic mesh 


(f) Solution 5 


(f) Solution 5 
8 Conclusion
We have presented hierarchical analysissuitable Tsplines which is a superset of both analysissuitable Tsplines and hierarchical Bsplines. We have also developed the necessary theoretical formulation of HASTS including a proof of the local linear independence of analysissuitable Tsplines. We presented a simple algorithm for the creation of nested Tspline spaces and also extended Bézier extraction to HASTS. We then demonstrated the potential of HASTS by comparing HASTS to a local refinement algorithm for Tsplines which demonstrated the improved efficiency and locality of using a hierarchical approach. We also demonstrated the use of HASTS in the context of isogeometric analysis by solving the benchmark static skew advection problem.
In future work we will provide a detailed description of the underlying algorithms to perform hierarchical refinement in the context of HASTS. We will also consider hierarchical and refinements of Tsplines. Finally, we intend to extend the definition of spline forests in ScThEv13 () to the Tspline regime. This will allow us to accommodate smooth interfaces and also interface directly with commercial Tspline products.
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