# Metrics for Quadrilateral Elements

The metrics used for quadrilateral elements in Trelis are summarized in the following table:

 Function Name Dimension Full Range Acceptable Range Reference Aspect Ratio L^0 1 to inf 1 to 4 1 Skew L^0 0 to 1 0 to 0.5 1 Taper L^0 0 to +inf 0 to 0.7 1 Warpage L^0 0 to 1 0.9 to 1.0 NEW Element Area L^2 -inf to inf None 1 Stretch L^0 0 to 1 0.25 to 1 2 Minimum Angle degrees 0 to 90 45 to 90 3 Maximum Angle degrees 90 to 360 90 to 135 3 Condition No. L^0 1 to inf 1 to 4 4 Jacobian L^2 -inf to inf None 4 Scaled Jacobian L^0 -1 to +1 0.5 to 1 4 Shear L^0 0 to 1 0.3 to 1 5 Shape L^0 0 to 1 0.3 to 1 5 Relative Size L^0 0 to 1 0.3 to 1 5 Shear & Size L^0 0 to 1 0.2 to 1 5 Shape & Size L^0 0 to 1 0.2 to 1 5 Distortion L^2 -1 to 1 0.6 to 1 6

## Quadrilateral Quality Definitions

Aspect Ratio: Maximum edge length ratios at quad center

Skew: Maximum |cos A| where A is the angle between edges at quad center

Taper: Maximum ratio of lengths derived from opposite edges

Warpage: Cosine of Minimum Dihedral Angle formed by Planes Intersecting in Diagonals

Element Area: Jacobian at quad center

Stretch: Sqrt(2) * minimum edge length / maximum diagonal length

Minimum Angle: Smallest included quad angle (degrees).

Maximum Angle: Largest included quad angle (degrees).

Condition No. Maximum condition number of the Jacobian matrix at 4 corners

Jacobian: Minimum pointwise volume of local map at 4 corners & center of quad

Scaled Jacobian: Minimum Jacobian divided by the lengths of the 2 edge vectors

Shear: 2/Condition number of Jacobian Skew matrix

Shape: 2/Condition number of weighted Jacobian matrix

Relative Size: Min( J, 1/J ), where J is determinant of weighted Jacobian matrix

Shear and Size: Product of Shear and Relative Size

Shape and Size: Product of Shape and Relative Size

Distortion: {min(|J|)/actual area}*parent area, parent area = 4 for quad

## Comments on Algebraic Quality Measures

Shape, Relative Size, Shape & Size, and Shear are algebraic quality metrics that apply to quadrilateral elements. Trelis encourages the use of these metrics since they have certain nice properties (see reference 5 below). The metrics are referenced to a square-shaped quadrilateral element, thus deviations from a square are measured in various ways.

Shape measures how far skew and aspect ratio in the element deviates from the reference element.

Relative size measures the size of the element vs. the size of reference element. If the element is twice or one-half the size of the reference element, the relative size is one-half. The reference element for the Relative Size metric is a square whose area is determined by the average area of all the quadrilaterals on the surface mesh under assessment

Shape and size metric measures how both the shape and relative size of the element deviate from that of the reference element.

The SHEAR metric is based on the condition number of the skew matrix. SHEAR is really just an algebraic skew metric but, since the word skew is already used in the list of quad quality metrics, Trelis has chosen to use the word 'shear.'

Shear = 1 if and only if quadrilateral is a rectangle.

The Robinson 'skew' metric equals the ideal (zero) if the quad is a rectangle. It also attains the ideal if the quad is a trapezoid, a kite, or even triangular!

## References for Quadrilateral Quality Measures

1. (Robinson, 87)
2. FIMESH code.
3. Unknown.
4. (Knupp, 00)
5. P. Knupp, Algebraic Mesh Quality Metrics for Unstructured Initial Meshes, submitted for publication.
6. 6. SDRC/IDEAS Simulation: Finite Element Modeling--User's Guide

## Details on Robinson Metrics for Quadrilaterals

The quadrilateral element quality metrics that are calculated are aspect ratio, skew, taper, element area, and stretch. The calculations are based on metrics described in (Robinson, 87). An illustration of the shape parameters is shown in Figure 1, below. The stretch metric is calculated by dividing the length of the shortest element edge divided by the length of the longest element diagonal.

Figure 1. Illustration of Quadrilateral Shape Parameters (Quality Metrics)