Peter Alfeld Department of Mathematics College of Science University of Utah The Bernstein Bézier Form Home Page Examples Spline Spaces and Minimal Determining Sets User's Guide Residual Arithmetic Triangulations How does it work? Bibliography

## Designing Finite Elements

To design triangular finite elements click on the red button labeled fin. El. next to the triangulation menu or choose finite elements on that menu.

A control panel will pop up. It lets you choose m which is the number of interior edges attached to each vertex of the macro triangle. m=2k if m is even, and m = 2k-1 if m is odd. You can increment m in steps of 1, or k in steps of 1, which means incrementing m in steps of 2. As you change m and k the split of the macrotriangle changes accordingly. Choose d and r suitably using the control panel or the drawing window.

To impose supersmoothness conditions press SUPER or SPECIAL on the finite element control panel. Additional rows will appear on that panel. They can be used to impose supersmoothness conditions in appropriate groups, to save time and to make it easier for situations where m, d, and r are large. Whether your press SUPER or SPECIAL affects whether in the drawing window you can impose ordinary vertex and edge superspline conditions, or special conditions as described in the User's Guide. In either case, in the finite element control panel you can impose both kinds of conditions.

The colors of the textfields indicate the kind of superspline conditions that can be imposed, as follows:

• Cyan: Vertex Conditions
• Blue: Edge Conditions
• Gray: Special Conditions

The best way to familiarize yourself with the controls is probably to click on them and see what they do. However, here is a description:

• The first text field in the second row controls supersmoothness at the vertices of the macro triangle.
• The remainder of that row controls the imposition of individual smoothness conditions. You can choose one of these the types:
• BV-i: Edge from a boundary vertex to an interior vertex, the greater i the more that vertex is removed from the boundary.
• interior i: Edges along the line from the outermost Clough-Tocher point towards the centroid, the greater i the farther you are in the interior of the triangle.
• inner T: In the case that m is even, the three edges of the innermost triangle.
Order is the order of the smoothness condition, greater than r and less than or equal to d. Location is the location of the smoothness condition along the edge. The Submit button actually imposes the condition after you have selected it.
• The (cyan) vertex textfield in each row controls the super smoothness at the interior vertices. As you progress through the rows you proceed towards the center.
• The first (blue) edge field in each row controls smoothness conditions across edges emanating from the boundary vertices, again moving inwards as you approach the center.
• The second edge field controls smoothness conditions across edges that do not have a boundary vertex.
To exit the setup mode exit Superspline mode as usual or click on Super in the finite element control panel.

Once everything is set up, clicking on the "set It" button will add points the MDS that correspond to the usual boundary data on a finite element.

The "Try again" button reenters superspline mode and lets you modify the supersmoothness conditions.

[14-Feb-2000]