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:

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 $ V_0$, $ V_1$, $ V_2$, and $ V_3$, and the centroid is

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

Natural Data

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

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 $ C^1$ Element

Let $ r=1 $ and $ d=5$, and require $ C^2$ 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 $ C^1$ element.

Let $ r=1 $ and $ d=9$, and require $ C^9$ 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 $ C^1$ 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.