# SMS:ARR Mesh Quality Assessment Plot

The Angle Representation Region (ARR) plot is used to assess the overall quality of a triangular mesh such as those used by ADH, ADCIRC and other numerical engines. When the plot wizard is selected, this option appears if the mesh module is enabled. Clicking finish in the Plot Wizard results in an ARR plot for the current unstructured mesh, loaded in SMS.

The plot includes the ARR region (defined below), a point for each element in the mesh, and three contour lines (0.3 in red, 0.45 in yellow and 0.6 in green) of the currently selected element quality measure (also defined below). As a general rule, elements with quality lower than 0.3 should be reviewed and improved (mesh editing) if possible.

Click on any point in the plot to see the element ID associated with that point and the six quality measure values for that element.

Once the mesh is edited in any way, update the ARR plot by right-clicking in the plot and selecting Refresh. Until this is done, the plot will continue to reflect the mesh that existed when it was generated (or most recently refreshed).

This plot is based on a paper entitled "Numerical representation of the quality measures of triangles and triangular meshes".[1] Several of the figures below are derived from this paper.

To assess the quality of a triangular mesh, such as those used by ADH or ADCIRC, the quality of each element is represented as a point, based on the interior angles of that element. These interior angles are labeled α, β, and γ as shown:

Plot these three angles into an equilateral triangle.

If ordering the three angles so that α > β > γ as shown below, all of the points will fall into the shaded portion of the equilateral triangle. This is referred to as the ARR region.

 Angels ordered so that α > β > γ The ARR region

The quality of the elements is further assessed by computing a quality measure from attributes of the triangle. These attributes include:

• The minimum interior angle αmin(γ from previous figure).
• The lengths of edges.
• The triangle area.
• The inner and outer radius.
• The minimum distance through the triangle (hmin).

These measures vary from 0.0 at the edges of the equilateral triangle to 1.0 at the center. The measures supported by SMS include:

${\displaystyle q_{x_{min}}={\frac {3x_{min}}{\pi }},\ q_{Ll}={\frac {l_{min}}{l_{max}}},\ q_{ALS}={\cfrac {4{\sqrt {3}}A}{l{\frac {2}{1}}+1{\frac {2}{2}}+1{\frac {2}{3}}}}}$

${\displaystyle q_{Rr}={\frac {2r}{R}},\ q_{Lr}={\frac {2{\sqrt {3}}r}{l_{max}}},\ q_{Lh}={\frac {2h_{min}}{{\sqrt {3}}l_{max}}}}$

The following figures show how each of these quality measures cover the ARR

## References

1. ^ Sarrate, J. , Palau, J. and Huerta, A. (2003), Numerical representation of the quality measures of triangles and triangular meshes. Communications in Numerical Methods in Engineering., 19: 551-561.[1]