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
Timestep

Seconds

0 to inf

None

6

Distortion
L^0
0 to 1
0.6 to 1
7

Hexahedral Quality Definitions

With a few exceptions, as noted below, Trelis supports quality metric calculations for linear hexahedral elements only. When calculating quality metrics (that only support linear elements) for a higher-order hexahedral element only the corner nodes will be used.

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. Trelis also supports Jacobian calculations for hex27 elements.

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

Timestep: The approximate maximum timestep that can be used with this element in explicit transient dynamics analysis. This critical timestep is a function of both element geometry and material properties. To compute this metric on hexes, the hexes must be contained in a element block that has a material associated to it, where the material has poisson's ratio, elastic modulus, and density defined.

Distortion: {min(|J|)/actual volume}*parent volume, parent volume = 8 for hex. Trelis also supports Distortion calculations for hex20 elements.

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. Flanagan, D.P. and Belytschko, T., 1984, "Eigenvalues and Stable Time Steps for the Uniform Hexahedron and Quadrilateral", Journal of Applied Mechanics, Vol. 51, pp.35-40.
  7. SDRC/IDEAS Simulation: Finite Element Modeling - User's Guide