Bounds on the Dimensions of Trivariate Spline Spaces

The Tables referenced below give the dimensions, and bounds on those dimensions for various configurations, and a range of values of r and d. The bounds can be computed with the 3DMDS package. Three bounds are given. Details of these bounds will be described elsewhere, but here is an outline.
1. l.b. A lower bound obtained by counting domain points and smoothness conditions while building the configuration by adding one tetrahedron at a time, joining it at 1, 2, or 3 faces. The lower bound may be less than the dimension of the polynomial space. In that case of course it can be easily augmented by setting it to the polynomial dimension. However, the smaller value is shown since this is a better indication of the quality of the bounds. Lower bounds below the polynomial dimension are marked in Gray.
2. c.o.b. A complex upper bound obtained as follows: Pick a plane P that is not parallel to any (interior) face of any of the tetrahedra. Intersect each face with that plane and translate the intersections (without changing their slopes) so that they contain the origin. Pick another line L in P that is not parallel to any of the given lines and that does not contain the origin. Compute the intersections of L with the intersections. On the resulting line analyze the univariate spline space obtained by associating a univariate polynomial with each tetrahedra and making those polynomials join smoothly across the points on L corresponding to the interior faces of those tetrahedra. Use those dimensions of univariate spaces to obtain the upper bound on the trivariate space.
3. s.o.b. A simple upper bound obtained by counting domain points and smoothness conditions when building the triangulation.
In the tables, background colors indicate the following circumstances:
• Green (Green) The spline space is actually polynomial.
• Yellow (Yellow) The dimension is greater than the polynomial dimension.
• Red (Red) The bound gives the true dimension.
• Gray (Gray) The lower bound is lower than the polynomial dimension.
• Cyan (Cyan) The upper bound overestimates the dimension but it is better than the other upper bound.
• Light Blue (Light Blue) Some of the domain points do not enter any smoothness conditions.

Tables are available for the following configurations:

1. Two tetrahedra sharing a common face.
2. a 3-orange , three tetrahedra sharing an interior edge.
3. a split of a cube into six tetrahedra about a space diagonal, an analog of the two dimensional type-I triangulation.
4. The 3D Clough-Tocher Split.
5. The 3D Morgan-Scott split.
6. The regular octahedron .
7. The Worsey-Farin split. The CT split is applied to the overall tetrahedron, and to each of its faces. This gives a Powell-Sabin type split of a Tetrahedron.
8. The double Clough-Tocher Split. The CT split is applied to the overall tetrahedron, and then to each of the four subtetrahedra.
9. An analog of the 2D type-II triangulation, a tessellation of a cubic by 24 tetrahedra sharing an interior point. This is sometimes called a "type IV" split.
10. A generic version of the Morgan-Scott split .
11. A generic Octahedron.
12. A generic version of the Worsey-Farin split.
13. A generic version of the double Clough-Tocher split.
14. A generic type IV split .
15. A slight modification of Rudin's example . This is a configuration consisting of 41 tetrahedra and 14 points. It is the simplest example of an unshellable triangulation in the sense that no unshellable triangulation with fewer tetrahedra exists. "unshellable" means that the configuration cannot be built by adding one tetrahedron at a time, maintaining at each stage a configuration. (Removing any one tetrahedron from the configuration leaves a set of tetrahedra whose union is homeomorphic to a ball.) For details see Mary E. Rudin, An unshellable triangulation of a tetrahedron, Bulletin of the AMS, 64 (1958), 90--91. Many arguments concerning configurations are based on shelling, and this example may serve as a counter example. Note that the configuration has no interior points. In this example the angle of one degree used by Rudin is replaced by an angle of approximately 12 degrees. The angle can be increased up to 19 degrees without some of the tetrahedra beginning to overlap.)
16. The Morgan-Scott Face Split , a tetrahedron where each face has been split by the symmetric Morgan-Scott Split and every vertex on the boundary of the tetrahedron connected to the centroid. Meant for Macro design.
17. The same as before, with the Wang version of the Morgan-Scott Split on the faces.
18. The double Clough-Tocher Face Split , similar to the previous splits and the Worsey-Farin Element, except that each face is split by the double Clough-Tocher split. A macro control panel is available for this split.
19. An 8-cell . Following bivariate notions, a cell is a configuration with one interior vertex such that all boundary vertices are connected to the interior vertex by an edge. The are two topologically different cells consisting of 8 tetrahedra. One is the octahedron, and this is the other one.
20. aligned Powell-Sabin . Two Worsey-Farin Splits next to each other such that the two edges meeting at the centroid of the common face are collinear. Note that V4-V8-V9 are collinear. V3-V4-V8 are also aligned.
21. unaligned Powell-Sabin . A version of the previous configuration. Two Worsey-Farin Splits next to each other such that the two edges meeting at the centroid of the common face are not collinear. V4-V8-V9 are collinear, but V3-V4-V8 are not.
22. generic Powell-Sabin . A generic version of the previous configuration. Two Worsey-Farin Splits next to each other such that the two edges meeting at the centroid of the common face are not collinear. V4-V8-V9 are not collinear.
23. inverted . A partition of a tetrahedron into 11 subtetrahedra: 4 vertex tetrahedra, 6 edge tetrahedra, and one inverted tetrahedron.
24. T60 A partition of a tetrahedron into 60 tetrahedra. A geometrically unconstrained C1 cubic macro element can be built on this split.
25. T504. A partition of a tetrahedron into 504 tetrahedra. A geometrically unconstrained C1 quadratic macro element can be built on this split.
26. symmetric Worsey-Piper . Each face of a tetrahedron is split into 6 triangles, using the centroid of the face, and the midpoints of the edges. Thus the Powell-Sabin 6-split is applied to each face. The tetrahedron is then split into 24 subtetrahedra all sharing the barycenter of the tetrahedron.
27. generic Worsey-Piper . A less symmetric version of the previous split. The union of the 24 tetrahedra is still a tetrahedron, but the split points are no longer in the centers of the edges, faces, and the overall tetrahedron.
28. symmetric MS cone . The planar Morgan-Scott split coned to three dimensions. Thus the spline space decouples into d+1 bivariate spline spaces. This configuration can be used to illustrate the fact that in order to obtain the dimension in three variables for any sufficiently large value of d one has to get the dimension of bivariate spaces for all values of d.
29. generic MS cone . A similar configuration built on the generic planar Morgan-Scott split.
The columns in the tables contain the following information:
• r is the degree of smoothness. (No supersplines are illustrated at present.)
• d is the polynomial degree.
• pol the dimension of the polynomial space, listed for reference and comparison.
• l.b. is the above mentioned lower bound.
• dim is the computed dimension.
• c.u.b. is the complex upper bound.
• s.u.b. is the simple upper bound.
• M is the initial number of equations.
• N is the initial number of unknowns.
• rank is the rank of the initial linear system.
• MDS is the number of active points (excluding those who do not enter any smoothness conditions at all) which need to be in a minimal determining set. This is also the number of rows in the reduced linear system.
• delta is the density of the reduced system, measured in percent.
• mods is the number of times a residual was computed in the generation of this particular line.