A characterization of supersmoothness of multivariate splines

06/19/2019 ∙ by Michael S. Floater, et al. ∙ University of Minnesota UNIVERSITETET I OSLO 0

We consider spline functions over simplicial meshes in R^n. We assume that the spline pieces join together with some finite order of smoothness but the pieces themselves are infinitely smooth. Such splines can have extra orders of smoothness at a vertex, a property known as supersmoothness, which plays a role in the construction of multivariate splines and in the finite element method. In this paper we characterize supersmoothness in terms of the degeneracy of spaces of polynomial splines over the cell of simplices sharing the vertex, and use it to determine the maximal order of supersmoothness of various cell configurations.

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1 Introduction

Polynomial splines over a simplicial partition of a domain in (a triangular mesh in 2D, a tetrahedral mesh in 3D, and so on) are functions whose pieces are polynomials up to a certain degree and which join together with some order of continuity . Such spline functions may have extra orders of smoothness at a vertex of the mesh, a property known as supersmoothness as suggested by Sorokina [13]. For example, the Clough-Tocher macroelement, which is piecewise cubic, is twice differentiable at the refinement point, as first observed by Farin [5], and so this element can be said to have supersmoothness of order 2 at that point.

For the construction of splines or finite elements with higher orders of continuity, it is important to recognize and make use of supersmoothness. For example, it plays a role in many of the macroelement constructions surveyed by Lai and Schumaker[8]

, where applications of splines to approximation theory and computer-aided geometric design are discussed. The concept of supersmoothness is also relevant to the finite element method. Motivated by structure-preserving or compatible discretizations there has recently been an increased interest in investigating the use of splines for vector fields and differential complexes 

[2, 3, 4, 7]. The de Rham complex reveals a connection between smooth, e.g., , finite elements and the Stokes problem in fluid mechanics. In a discrete de Rham complex, the spline spaces for the velocity field may inherit the supersmoothness of the scalar field, [2, 4, 7, 11]. Thus, supersmoothness is also of importance in the study of these problems.

Since Farin’s observation about the Clough-Tocher element, Sorokina, in [13] and [14] has derived further supersmoothness properties of polynomial splines, and in particular higher order supersmoothness in a cell in 2D; see equation (4). More recently, Shekhtman and Sorokina [12] observed that supersmoothness is a phenomenon of more general spline functions, not only piecewise polynomials. Their results imply that at the vertex of a triangulation with incoming edges all having different slopes, any spline with has supersmoothness, i.e., the spline has supersmoothness of order at least . This holds as long as the spline pieces themselves have continuity.

The results of [12] were the motivation for this paper. If we simplify the framework of [12] and assume that all the spline pieces are smooth, which is the case for polynomials and many other functions of interest, can we extend the results to higher orders of supersmoothness and also to higher Euclidean space dimensions? Our solution is to simplify the problem by deriving a characterization of supersmoothness in terms of the degeneracy of polynomial spline spaces over the cell (in Theorem 1). Using this, the maximal order of supersmoothness at a vertex can be determined once a general formula for the dimensions of the polynomial spline spaces over the cell is known. At the end of the paper we apply these results to various cell configurations.

2 Cells and supersmoothness

We start with some definitions.

2.1 Cells

Suppose is an interior vertex of a simplicial partition of , . We call the collection of -simplices in the mesh that share the common vertex a cell and we let . For example, in 2D a cell is a sequence of triangles , , that form a star-shaped polygon , as in Figure 1.

Figure 1: Cells in 2D.

In the special case that , is known as a Clough-Tocher split since it can also be constructed by refinement. We could start with any triangle in the plane (the outer triangle in the figure), then let be any point inside and connect the three edges of to , thus creating three sub-triangles of .

In 3D a cell is a collection of tetrahedra. A simple example is the Alfeld split, constructed by choosing a tetrahedron , then any point inside and connecting  to the four triangular faces of . The resulting cell has four tetrahedra, as in Figure 2.

Figure 2: Alfeld split in 3D.

2.2 Splines

We want to study functions that are defined piecewise over the simplices of . Let be an integer, . Then we let

and we will call a function a spline. We are assuming here that for each there is some open set containing and a function such that for . The pieces could, for example, be polynomials of any degree (in which case we can take ), or rational functions, trigonometric functions, and so on.

2.3 Supersmoothness

Now we look at enhanced smoothness of splines at . We will say that a spline has supersmoothness of order at if all its pieces , , have common derivatives up to order at the point . In this case we will follow convention and write even though will not in general be in any neighbourhood of . To make this clearer let us consider the following two cases.

  • If a spline is in then has all derivatives of order at , but not in general in any neighbourhood of .

  • If a spline is in for some then will not in general even have derivatives of order at because such derivatives are only defined if has derivatives of order in a neighbourhood of . However, the restriction of  to any straight line in passing through will have smoothness of order .

3 Taylor approximations

Our aim is to characterize supersmoothness in terms of the degeneracy of polynomial splines. The first step in the derivation is to study Taylor approximations.

Let be a function in for some domain and let be a multi-index, with . We denote the corresponding partial derivative of by

Then, with respect to a point , we denote the Taylor approximation of of order by

where

We will make use of the following property of these Taylor approximations.

Lemma 1

Let be distinct points in and let be the line segment connecting them. Let be some domain containing . Suppose that and that . Then, for any , .

  • Proof. We can represent the line segment parametrically as

    where . Letting for some , we find that

    where

    Since and are equal on , we also have

    and so

We want to generalize this property to derivatives of and . To do this we first show

Lemma 2

Let and suppose for some domain containing . Then, for any integer and any multi-index with ,

  • Proof. From the definition of , for ,

From Lemmas 1 and 2 we obtain

Lemma 3

Let be as in Lemma 1. Suppose that and that for some ,

(1)

Then, for any ,

(2)
  • Proof. If , equation (2) trivially holds since both sides are equal to . If , by Lemma 2, equation (2) is equivalent to

    and by Lemma 1, this is implied by equation (1).

4 Characterization of supersmoothness

We are now approaching our characterization of supersmoothness.

4.1 Polynomial spline spaces and degeneracy

For integers and with let

where  is the linear space of polynomials in of degree at most . Thus is the usual space of polynomial splines on of smoothness and degree at most .

By definition, for any cell and any we have . Sometimes, however, depending on and , we might have . In this case contains no ‘true’ splines, only polynomials, and we view as being ‘degenerate’ in this sense.

Definition 1

We will say that is degenerate if .

As an example, the space is degenerate for any and any cell .

4.2 Piecewise Taylor approximations

Next recall the more general spline space , , and let . For any , we can make a piecewise Taylor approximation of by piecing together the individual Taylor approximations of order at of the pieces of . We will refer to this piecewise approximation as . Due to Lemma 3 we next show

Lemma 4

If for any then for any , .

  • Proof. We only need to show that . Let be two simplices that share a common -dimensional face . The face is the union of all the line segments that connect to the -dimensional face of opposite to . The pieces and have the same derivatives up to order on . Therefore, by Lemma 3, the two Taylor approximations and have the same derivatives up to order on . Therefore, they have the same derivatives up to order  on the whole face . Thus as claimed.

4.3 Characterization

Finally we arrive at the characterization. For , we will write if for all .

Theorem 1

For , if and only if is degenerate.

  • Proof. Suppose that is degenerate and let . Since , Lemma 4 implies that . It then follows that . For each , the two functions and have the same derivatives at  up to order . Therefore, also . This proves that .

    Conversely, suppose that . Then since , every polynomial spline is in and therefore . This shows that is degenerate.

4.4 Maximal order of supersmoothness

We can also consider the mos (maximal order of supersmoothness) of , i.e.,

To characterize this, observe that we have a nested sequence of spaces,

Therefore, if is non-degenerate for some , then is non-degenerate for all . Thus, for any cell and any , there is a unique highest degree such that is degenerate. From Theorem 1 we deduce

Corollary 1

.

5 Applications

We now apply the characterization theorem to some concrete examples. For a cell  in and smoothness the spline space , with , is degenerate if

(3)

For some cell configurations degeneracy is known for specific degrees . We then conclude from Theorem 1 that all splines in have supersmoothness of order , but we do not know whether is optimal. However, if we know the dimensions of all the spaces , , we obtain the maximal supersmoothness from Corollary 1 by finding the largest satisfying (3).

We note also that Alfeld [1] has computed the dimension of many spline spaces over various kinds of cell. These computational results also determine supersmoothness by Theorem 1 or Corollary 1.

5.1 Clough-Tocher split

In , when has three triangles it is a Clough-Tocher split, , and, using the theory of Bernstein-Bézier polynomials, Farin showed in [5, Theorem 7] that is degenerate for any . He then concluded in  [5, Corollary 8] that for any .

We can now apply Theorem 1 to conclude more generally that for . However, this is not the optimal supersmoothness for general .

5.2 An arbitrary cell in 2D

Sorokina made a substantial generalization of Farin’s result. She showed in [13, Theorem 3.1] that if has triangles, and the interior edges have different slopes, then for ,

(4)

The proof was based on comparing the dimension of with those of superspline spaces. Since in (4) is independent of the degree , one might expect a more general result. This was also suggested by the work of Shekhtman and Sorokina [12]. From (4) it follows that there is at least one order of supersmoothness when for any degree . Shekhtman and Sorokina showed that this is also true for more general splines, in other words, in our notation, when . Their proof was based on expressing partial derivatives as linear combinations of directional derivatives along the edges meeting at . Using Corollary 1, we can now improve this result to match that of the polynomial case, in other words, we can remove the ‘d’ in (4). To do this, we first transform the dimension formula of Lai and Schumaker [8] into a more suitable form.

Lemma 5

Suppose has triangles and suppose there are different slopes among the interior edges of . For ,

(5)

where , and if and otherwise.

  • Proof. The dimension of was derived in [8, Theorem 9.3] in the form

    (6)

    Using the fact that

    we can rewrite (6) as

    Then, using the fact that , the result follows.

Theorem 2

Suppose has triangles and suppose there are different slopes among the interior edges of . Then for ,

(7)
  • Proof. By Corollary 1, it is sufficient to determine the highest degree such that is degenerate, i.e., such that . To do this we use Lemma 5. Suppose . If , the second term in (5) is strictly positive and so is non-degenerate. Therefore is degenerate if and only if . Otherwise, . Then considering the third term in (5), is degenerate if and only if for all , or equivalently , which is equivalent to

As an example, for the Clough-Tocher split we have and so

(8)

5.3 The Alfeld split in

The dimensions of the spaces are not currently known for a general cell in  for . However, they are known in special cases. One of these is the Alfeld split in . In , , the split is constructed by choosing any -dimensional simplex  and splitting it into smaller simplices by choosing an arbitrary interior point in and connecting it to each of the faces (of dimension ) of . We denote this split by . The 3D case is shown in Figure 2.

Using the theory of Bernstein-Bézier polynomials, Worsey and Farin showed in [15, Lemma 3.1] that is degenerate. From this, Theorem 1 implies that . But we can make a further generalization by invoking the recently derived dimension formula of Foucart and Sorokina [6] and Schenck [9]. Let us define, for and ,

Theorem 3

The maximal order of supersmoothness of the Alfeld split is

  • Proof. The dimensions of the polynomial spline spaces on the Alfeld split were generated and conjectured by Foucart and Sorokina [6] and proved by Schenck [9]: for ,

    where

    Therefore, is degenerate if and only if , or equivalently if

    By Corollary 1, the maximal order of supersmoothness is the largest such , i.e.,

    or equivalently, .

For example, , and in particular, , which shows that the macro-element on the Alfeld split described in [8, Section 18.3] has supersmoothness.

We note that Theorem 3 in the case agrees with the supersmoothness of the Clough-Tocher split in equation (8).

5.4 The split

Worsey and Farin [15] proposed an alternative generalization of the Clough-Tocher split to , using recursion through the Euclidean dimensions. To split an -simplex , they first split the faces of of dimension 2 (triangles) by making a Clough-Tocher split. They next split each 3-face (a tetrahedron) of by choosing any point in the relative interior of and connecting it to the twelve triangles on the boundary of constructed in the previous step. They continue in a similar way, next splitting faces of of dimension 4 and so on. Part of a Worsey-Farin split in 3D is shown in Figure 3, viewed as a refinement of an Alfeld split. One of the subsimplices of the Alfeld split has been split into three.

Figure 3: Part of a Worsey-Farin split in 3D.

Let us consider a more general split. We choose any Euclidean dimension , . We then initialize the splitting by splitting each -face of by choosing any point in the relative interior of and connecting it to the -faces of . Then, for in sequence, we split each -face of by choosing any point in the relative interior of and connecting it to the

simplices of dimension on the boundary of constructed in the previous step. The resulting split of is a cell around the point in the interior of  chosen at the last step (). It has sub-simplices and we denote it by .

For example, in 2D, is a Clough-Tocher split and is a Powell-Sabin 6-split. In 3D, is an Alfeld split, is a Worsey-Farin split and is a Worsey-Piper split.

By construction, each of the -faces of is itself split into a split. A split , , can also be viewed as a refinement of a split .

It was shown by Worsey and Farin [15] that is degenerate for any . Based on this observation, they concluded, as ‘an interesting aside’, that their piecewise-cubic element has supersmoothness at . Theorem 1 implies more generally that . Using now degeneracy over the Alfeld split in we obtain a more general result.

Theorem 4

The maximal order of supersmoothness of a split, , is bounded as follows:

  • Proof. First let . We will show that is degenerate. The proof of this is similar to that of [15, Theorem 3.2] and is by induction on . Consider first . Since is a -dimensional Alfeld split it follows from Lemma 5 that is degenerate. Now suppose and let . Let be one of the -faces of . Let be the point in the relative interior of  used to make the -dimensional split of in the construction of . Let

    be any point on the line segment and let be the -simplex

    which passes through and is parallel to . The split induces an analogous split . By the induction hypothesis, is degenerate and so all the pieces of meeting at have common derivatives within up to order at . Since all these pieces join continuously along , they also have common derivatives along . Therefore all these pieces are the same polynomial and thus belongs to . Since , it follows, as in the proof of Theorem 3, that .

    This proves the lower bound on . To prove the upper bound we just need to observe that is a refinement of an Alfeld split , which implies that

    for any . Thus if is non-degenerate, so is is non-degenerate,

5.5 The split

Consider the special case of the split, which has subsimplices. It can be constructed by first making an Alfeld split () of an -simplex using some interior point . We then choose an interior point of each boundary face (an -simplex) of and use it to split into subsimplices and then connect them to .

Let us say that is aligned if, for every face , the splitting point chosen for is the unique point in that is collinear with and the vertex of opposite . This is what Schenck and Sorokina [10] called a facet split.

Theorem 5

The maximal order of supersmoothness of an aligned split is

  • Proof. The dimensions of the polynomial spline spaces for an aligned split were derived by Schenck and Sorokina [10]. For ,

    (9)

    where is as in Theorem 3 and

    Therefore, is degenerate if and only if . Since when , this is equivalent to the condition that , which holds when

    The largest possible in both cases gives the result by Corollary 1.

It is remarked in [10, Remark 4.3] that for , the dimension formula (9) also holds even without the collinearity condition, from which we conclude that for an arbitrary split,

For example, in 3D, for an arbitrary Worsey-Farin split we have

5.6 -cells

Finally, we consider a slightly different kind of cell, constructed as follows. Let be an -dimensional simplex and choose an interior point of and connect it to just one -face of , forming a simplex contained in . We now let be the the closure of . The two elements and form what we will call a 2-cell, . Of course it is not a cell of simplices because is not a simplex. Figure 4 shows a 2-cell in .

Figure 4: A 2-cell in 2D.

Now we can consider the supersmoothness of splines in . Even though 2-cells do not occur in simplicial meshes, the local configuration of edges emanating from could occur in a polytopal mesh if we allowed non-convex polytopes. Shektman and Sorokina [12] studied this kind of configuration in 2D and showed that the order of supersmoothness is at least for any . We can now extend this result using our characterization. Even though a 2-cell contains the non-simplicial element , the intersection of and is the union of faces (of dimension ) and so our characterization of supersmoothness at also holds for a -cell, i.e., we can apply Theorem 1 and Corollary 1 to a -cell . To use these results we need the dimensions of the spline spaces , .

Lemma 6

For any ,

  • Proof. We have

    where

    Letting be the -dimensional faces common to and , we have

    Let be any equation for the face , . Then we can express any uniquely in the form

    where if and if .

Theorem 6

For , .

  • Proof. By Lemma 6, is degenerate if and only if , or equivalently that . Thus, from Corollary 1, the maximal order of supersmoothness is

For example, in , and in 3D, .

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