Removing sliver triangles

For a meaningful table lookup when using a Delaunay method, you typically want to use the tabular data that lies in closest proximity to the lookup point in the interpolation calculation. However, if two legs of a triangle in the Delaunay triangularization are distinctly longer than the third, there is a likelihood that the three tabular data points closest to the lookup point are not those used in the interpolation calculation. A triangle that has such a shape is referred to as a sliver triangle. To improve the likelihood of a meaningful table lookup, you can remove sliver triangles from the Delaunay triangularization.

Sliver triangles typically arise when the tabular data defines a region that contains a convex shape. For example, consider the following tabular data that defines pressure as a function of (x,y) coordinates:

Data point	X-coordinate	Y-coordinate	Pressure
1	0	0	6
2	1	0.25	7
3	2	0.4	8
4	3	0.25	9
5	4	0	10
6	0	1	8
7	1	1.25	9
8	2	1.4	10
9	3	1.25	11
10	4	1	12
11	0	2	10
12	1	2.25	11
13	2	2.4	12
14	3	2.25	13
15	4	2	14

When the tabular data is plotted, you can see that connecting the first five data points reveals a convex shape.

The following figure shows the Delaunay triangularization for the tabular data. Unlike FE meshing algorithms that use part geometry to define the region to be meshed, the Delaunay triangularization mesh is based solely on the tabular data. Thus, the convex region is filled by three triangles. In the figure, these triangles are shaded.

How you proceed depends on whether you want to use interpolation or extrapolation to determine the table lookup values for points within the convex region. To use extrapolation, you must:

• Remove the triangles in the convex region, but otherwise leave the Delaunay triangularization unaltered.Using this approach, the final Delaunay triangularization might look as follows.

For information on the extrapolation options available when a Delaunay method is selected, see Table field interpolation and extrapolation options.

To use interpolation, you have two options:

• Accept the Delaunay triangularization in its current form.

• Remove the triangles in the convex region, redefine the coordinates of the tabular data along the convex feature so as to eliminate the convex feature, and remesh the Delaunay triangularization.Using this approach, the final Delaunay triangularization might look as follows.

For both of these options, the software obtains the table lookup value for a point within the convex region by interpolating the values at the vertices of the triangle within which the point lies.

Identifying sliver triangles

A sliver triangle is a triangle with one leg much shorter than the other two. To make a quantitative determination whether a triangle is a sliver triangle, you use one of the following criteria:

• Edge length ratio criterionA triangle is considered a sliver if:1 - \frac{{\overline {AC} }}{{\left( {\overline {AB} + \overline {BC} } \right)}} < \varepsilon where ε is a value that you specify, lengths AC, AB, and BC are the lengths of the legs, and edge AC is the free edge.

• Aspect ratio criterionA triangle is considered a sliver if:\frac{{\overline {BD} }}{{\overline {AC} }} < \alpha where α is a value that you specify, lengths BD and AC are the lengths of legs, and edge AC is the free edge.

Free edges are the edges that define the perimeter of the Delaunay triangularization.

You select the sliver detection criterion and specify the values for α and ε in the Table Field dialog box, on the Table Options page.

Note:

For spatial independent domains, the lengths are physical distances. For non-spatial independent domains, the lengths are not physical distances.

For example, suppose the independent variables are time, t, and temperature, T. The values for time and temperature at a tabular data point represent a set of coordinates in time-temperature space. Thus, in the edge length ratio and aspect ratio formulas, the software calculates the length between points i and j as follows:

\overline {IJ} = \sqrt {{{\left( {{t_j} - {t_i}} \right)}^2} + {{\left( {{T_j} - {T_i}} \right)}^2}}

Removing sliver triangles

The sliver triangle removal algorithm is a recursive procedure that works as follows:

1. The software applies the sliver detection criterion to each triangle along the perimeter of the Delaunay triangularization. Triangles that are determined to be slivers are removed.For example, suppose that you specify the sliver detection criterion such that the triangle in the convex region is determined to be a sliver and the other triangles with free edges are not. The software eliminates the sliver and the new Delaunay triangularization is as follows:

2. Because the software identified and removed a sliver triangle, it automatically repeats the previous step for the new Delaunay triangularization. Triangles that are determined to be slivers are removed.Assuming that a second sliver triangle is removed from the convex region, the new Delaunay triangularization is as follows:

3. The process continues until the software no longer identifies and removes a sliver triangle.Assuming that a third sliver triangle is removed from the convex region, the new Delaunay triangularization is as follows:

To remove sliver triangles, in the Table Field dialog box, on the Table Options page, select the Remove 2D Delaunay Slivers check box.

Removing sliver triangles and relocating data points

An algorithm removes the sliver triangles in the convex region, redefines the coordinates of the tabular data along the convex feature so as to eliminate the convex feature, and remeshes the Delaunay triangularization. This algorithm works as follows:

1. The software applies the sliver detection criterion to each triangle along the perimeter of the Delaunay triangularization. Triangles that are determined to be slivers are removed. For each removed sliver, the software projects the vertex opposite the free edge onto the free edge. Then the software remeshes the Delaunay triangularization.For information on how the software projects vertices, see Projecting vertices.For example, suppose that you specify the sliver detection criterion such that only the triangle in the convex region is determined to be a sliver. The software eliminates the sliver and projects the vertex onto the free edge of the removed sliver.Delaunay triangularization after sliver removal****Delaunay triangularization after vertex relocation

2. Because the software identified and removed a sliver triangle, it repeats the previous step for the new Delaunay triangularization. Triangles that are determined to be slivers are removed and vertices are projected.Assuming that a second sliver triangle is removed from the convex region, the Delaunay triangularization is as follows:Delaunay triangularization after sliver removal****Delaunay triangularization after vertex relocation

3. The process continues until the software no longer identifies and removes a sliver triangle.Assuming that a third sliver triangle is removed from the convex region, the Delaunay triangularization is as follows:Delaunay triangularization after sliver removal****Delaunay triangularization after vertex relocation

To remove sliver triangles and relocate data points, in the Table Field dialog box, on the Table Options page, select the Remove 2D Delaunay Slivers and Snap Remaining 2D Vertices check boxes.

Projecting vertices

Suppose triangle ABC is determined to be a sliver triangle when AC is the free edge. The software then projects vertex B along the normal to edge AC as indicated to position D.

Where do I find it?

Application	Pre/Post
Prerequisites	A table field that has a Delaunay interpolation method and the independent domains consists of two variables or consists of three spatial variables that either degenerate to a planar surface or are mapped to a surface
Simulation Navigator	Right-click the field node→Edit
Location in dialog box	Table Options page

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