Macro Elements on the Clough-Tocher Split

The Clough-Tocher Macro Panel lets you impose symmetric super smoothness conditions on the Clough-Tocher split and then check if a macro element can be built with those conditions.

To obtain the macro panel select the configuration "Clough-Tocher", and click on the button "macro elements" in the control panel. The buttons on the macro panel have the following effects:

Hide causes the macro panel to disappear. To make it reappear click again on "macro elements".
Draw causes the current smoothness conditions to be drawn initially, or if the linear algebra in the drawing window has been initialized. (Otherwise the smoothness conditions are drawn automatically whenever they are changed.)
Init initializes the linear algebra (and exits super selection mode) in the drawing window.
Set initializes the linear algebra if necessary and attempts to set the natural data that go with the current set of smoothness conditions and the current polynomial degree.
Clear removes all super smoothness conditions.
The status window gives information about the current analysis in some situations.

Super smoothness conditions can be imposed using the text fields in the second row of the macro panel, and the buttons to either side of those text fields. The text fields lists the additional degree of smoothness. So if all the text fields show zero then the spline is the ordinary space without any super smoothness conditions at all.

Groups of super smoothness conditions are indicated by representative points, edges, or faces. Recall that the boundary vertices are labeled , , , and , and the centroid is

$\displaystyle V_4 = \frac{V_0+V_1+V_2+V_3}{4}.$

: extra smoothness at the boundary vertices.
: extra smoothness at the centroid.
$E_{01}$ : extra smoothness at the boundary edges.
$E_{04}$ : extra smoothness at the interior edges.
$F_{014}$ : extra smoothness across the interior faces. For the Clough-Tocher split there is just one kind of interior face, and so increasing the smoothness across interior faces is equivalent to increasing .

Natural Data

Let and be as usual. Let denote the total degree of smoothness at the boundary vertices and let denote the total degree of smoothness along the boundary edges. Then in terms of the domain points the following are the natural data:

The balls around the vertices, requiring $4{R+3 \choose 3}$ imposable points.
The globs around the boundary edges requiring

$\displaystyle 6\sum_{i=0}^E (i+1)(d-2R-1+i)_+$
imposable points. The index runs from the edge outwards through the shells surrounding the edge. In going from shell to , the number of coefficients in an -th derivative not defined by the vertex data increases by 1 over the same number for an -th derivative, and the factor accounts for the number of mixed partial derivatives of order .
The remaining points in the layers zero through r for each boundary face, requiring a total of

$\displaystyle 4 \sum_{i=0}^r {{d-i+2-3(E-i+1)\choose 2 }}_{+} - 3{R+2-i-2(E+1-i)\choose 2}_+$
imposable points. Of course, these expressions can be simplified, but in this form they show their origin. The first term gives the number of points on the -th layer, working inwards, that are not contained in the edge globs. The second term accounts for the possibility of the vertex balls intersecting with that set.

The code attempts to impose the right number of conditions and informs you of whether or not this is possible.

The following examples will illustrate the ideas:

A Quintic Element

Let and , and require smoothness at the boundary vertices. The dimension of the spline space is 68. The natural data comprise 64 conditions that can all be imposed. The four remaining degrees of freedom may be used, for example, to interpolate to function and gradient values at the centroid of the tetrahedron.

A nonic polynomial element.

Let and , and require smoothness around the centroid. The resulting space is actually polynomial of dimension 220. Imposing 4-balls around the boundary vertices and 2-globs along the boundary edges, in addition to the natural face data, gives rise to the well known polynomial element. The 216 natural data can be imposed, and the remaining four degrees of freedom can be used, for example, to interpolate to function and gradient values at the centroid of the tetrahedron.