Metrics for Hexahedral Elements

The metrics used for hexahedral 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 0.4
1
Element Volume
L^3
-inf to inf
None
1
Stretch
L^0
0 to 1
0.25 to 1
2
Diagonal Ratio
L^0
0 to 1
0.65 to 1
3
Dimension
L^1
0 to inf
None
1
Condition No.
L^0
1 to inf
1 to 8
5
Jacobian
L^3
-inf to inf
None
5
Scaled Jacobian
L^0
-1 to +1
0.5 to 1
5
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
0.5 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^0
0 to 1
0.6 to 1
6

Hexahedral Quality Definitions

Unless otherwise noted, Trelis supports calculations for linear hexahedral elements. Metric calculations involving higher-order hexahedral elements will only use the corner nodes of the element.

Aspect Ratio: Maximum edge length ratios at hex center.

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

Taper: Maximum ratio of lengths derived from opposite edges.

Element Volume: Jacobian at hex center.

Stretch: Sqrt(3) * minimum edge length / maximum diagonal length.

Diagonal Ratio: Minimum diagonal length / maximum diagonal length.

Dimension: Pronto-specific characteristic length for stable time step calculation. Char_length = Volume / 2 grad Volume.

Condition No. Maximum condition number of the Jacobian matrix at 8 corners.

Jacobian: Minimum pointwise volume of local map at 8 corners & center of hex.

Scaled Jacobian: Minimum Jacobian divided by the lengths of the 3 edge vectors.

Shear: 3/Mean Ratio of Jacobian Skew Matrix

Shape: 3/Mean Ratio of weighted Jacobian Matrix

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

Shear & Size: Product of Shear and Size Metrics

Shape & Size: Product of Shape and Size Metrics

Distortion: {min(|J|)/actual volume}*parent volume, parent volume = 8 for hex

References for Hexahedral Quality Measures

  1. (Taylor, 89)
  2. FIMESH code
  3. Unknown
  4. (Knupp, 00)
  5. P. Knupp, Algebraic Mesh Quality Metrics for Unstructured
    Initial Meshes, to appear in Finite Elements for Design
    and Analysis.
  6. SDRC/IDEAS Simulation: Finite Element Modeling - User's Guide