# Metrics for Triangular Elements

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

 Function Name Dimension Full Range Acceptable Range Reference Element Area L^2 0 to inf None 1 Maximum Angle degrees 60 to 180 60 to 90 1 Minimum Angle degrees 0 to 60 30 to 60 1 Condition No L^0 1 to inf 1 to 1.3 2 Scaled Jacobian L^0 -1 to 1 0.2 to 1 2 Relative Size L^0 0 to 1 0.25 to 1 3 Shape L^0 0 to 1 0.25 to 1 3 Shape and Size L^0 0 to 1 0.25 to 1 3 Distortion L^2 -1 to 1 0.6 to 1 4

## Approximate Triangular Quality Definitions:

Element Area: (1/2) * Jacobian at corner node

Maximum Angle: Maximum included angle in triangle

Minimum Angle: Minimum included angle in triangle

Condition No. Condition number of the Jacobian matrix

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

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

Shape: 2/Condition number of weighted Jacobian matrix

Shape & Size: Product of Shape and Relative Size

Distortion: {min(|J|)/actual area}*parent area, parent area = 1/2 for triangular element

## Comments on Algebraic Quality Measures

Relative Size, Shape, and Shape & Size are algebraic metrics, which have well behaved properties. Trelis encourages the use of these metrics over other metrics. These metrics are referenced to an ideal element which, in the case of triangular elements, is an equilateral triangle. Thus deviations from an equilateral triangle are measured in various ways by the algebraic metrics.
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. By default, the size of the reference element is the average size of all the elements that the quality command is currently evaluating.

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

## References for Triangular Quality Measures

1. Traditional.
2. Knupp, 2000.
3. P. Knupp, Algebraic Mesh Quality Metrics for Unstructured Initial Meshes, submitted for publication.
4. SDRC/IDEAS Simulation: Finite Element Modeling--User's Guide