Metrics for Triangular Elements

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

Function Name
Dimension
Full Range
Acceptable Range
Reference
Maximum Angle
degrees
60 to 180
60 to 90
1
Minimum Angle
degrees
0 to 60
30 to 60
1
Aspect Ratio
L^0
1 to inf
1 to 3
Aspect Ratio Alpha
L^0
1 to inf
1 to 3
Condition No
L^0
1 to inf
1 to 1.3
2
Deviation
L^2
0 to inf
None
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
High Order Metrics
Distortion
L^2
-1 to 1
0.6 to 1
4
Element Area
L^2
0 to inf
None
1
Normalized Inradius
L^2
0 to 1
None

Approximate Triangular Quality Definitions:

Maximum Angle: Maximum included angle in triangle

Minimum Angle: Minimum included angle in triangle

Aspect Ratio: Ratio of circumcirle to inradius

Aspect Ratio Alpha: Ratio of longest edge length to inradius

Condition No: Condition number of the Jacobian matrix

Deviation: Absolute distance from triangle centroid to associated surface

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

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

Normalized Inradius: 4.0 * minimum_subtri_inradius / radius of circle containing three corner nodes

Comments on Algebraic Quality Measures

Relative Size, Shape, and Shape & Size are algebraic metrics, which have well behaved properties. Cubit 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