Checking the Skewness

Skewness is one of the primary quality measures for a mesh. Skewness determines how close to ideal (that is, equilateral or equiangular) a face or cell is (see Figure 1).

Fig. 1 Ideal and Skewed Triangles and Quadrilaterals

The following table lists the range of skewness values and the corresponding cell quality for equilateral-volume-based skewness and skewness-normalized equiangular skewness.

Value of Skewness	Cell Quality
0	degenerate
< 0.02	bad (sliver)
0.02 - 0.25	poor
0.25 - 0.5	fair
0.5 - 0.75	good
0.75 - 1	excellent
1	equilateral

According to the definition of skewness, a value of 1 indicates an equilateral cell (best) and a value of 0 indicates a completely degenerate cell (worst). Degenerate cells (slivers) are characterized by nodes that are nearly coplanar (collinear in 2D).

Highly skewed faces and cells are unacceptable because the equations being solved assume that the cells are relatively equilateral/equiangular.

Two methods for measuring skewness are:

Based on the equilateral volume (applies only to triangles).
Based on the deviation from a normalized equilateral angle. This method applies to all cell and face shapes, for example, pyramids and prisms.

The skewness method is equiangular for every type.

Equilateral-Volume-Based Skewness

In the equilateral volume deviation method, skewness is defined as

where the optimal cell size is the size of an equilateral cell with the same circumradius.

The presence of cells that are fair or worse indicates poor boundary node placement. You should try to improve your boundary mesh as much as possible, because the quality of the overall mesh can be no better than that of the boundary mesh.

In 3D, most cells should be good or better, but a small percentage will generally be in the fair range and there are usually even a few poor cells.

Skewness-Normalized Equiangular

In the normalized angle deviation method, skewness is defined (in general) as

where

= largest angle in the face or cell

= smallest angle in the face or cell

= angle for an equiangular face or cell (for example, 60 for a triangle, 90 for a square)

Check the Skewness

To check skewness for the currently loaded mesh:

From the Icepak>Fields menu, select Mesh Viewer.
Click on the Quality tab to show the mesh diagnostic tools.
Select the Skewness option. Icepak displays a histogram of the skewness, as shown in Figure 2.
Note:
By default, All is selected, which displays values for mesh within the entire global mesh region. Select Geometry selection to display values for geometry selected in the 3D Modeler window or history tree, including non-model boxes representing mesh regions.

Fig. 2 Skewness Histogram

To modify the range of skewnesses viewed, enter a new value in the Min or Max field and then press the Enter key on your keyboard or click Apply to update the histogram. To modify the number of bars in the histogram, enter a new value in the Bar field and click Apply.Click Reset to rest to the range.
To view the elements of the mesh within a particular range of skewnesses, click on a bar in the histogram or a range in the Curve Info legend. Icepak displays the elements in the selected range in the 3D Modeler window. Select the Shaded option if you want to view these elements with solid shading. Click Color and select a desired color if needed.