Metrics for Tetrahedral Elements

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

Function Name
Dimension
Full Range
Acceptable Range
Reference
Aspect Ratio Beta
L^0
1 to inf
1 to 3
1
Aspect Ratio Gamma
L^0
1 to inf
1 to 3
1
Element Volume
L^3
-inf to inf
None
1
Condition No
L^0
1 to inf
1 to 3
2
Jacobian
L^3
-inf to inf
None
2
Scaled Jacobian
L^0
-1 to 1
0.2 to 1
2
Shape
L^0
0 to 1
0.2 to 1
3
Relative Size
L^0
0 to 1
0.2 to 1

3

Shape and Size
L^0
0 to 1
0.2 to 1
3
Distortion
L^0
-1 to 1
0.6 to 1
4
Equivolume Skew       Fluent
Tet Squish       Fluent

Tetrahedral Quality Definitions

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

Aspect Ratio Beta: CR / (3.0 * IR) where CR = circumsphere radius, IR = inscribed sphere radius

Aspect Ratio Gamma: Srms**3 / (8.479670*V) where Srms = sqrt(Sum(Si**2)/6), Si = edge length

Element Volume: (1/6) * Jacobian at corner node

Condition No.: Condition number of the Jacobian matrix at any corner

Jacobian: Minimum pointwise volume at any corner

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

Shape: 3/Mean Ratio of weighted Jacobian Matrix

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

Shape & Size: Product of Shape and Relative Size Metrics

Distortion: {min(|J|)/actual volume}*parent volume, parent volume = 1/6 for tet

Equivolume Skew: (metric used by Fluent)

Tet Squish: (metric used by Fluent)

For tetra10 elements, the distortion metric can be used in conjunction with the shape metric to determine whether the mid-edge nodes have caused negative Jacobians in the element. The shape metric only considers the linear (parent) element. If a tetra10 has a non-positive shape value then the element has areas of negative Jacobians. However, for elements with a positive shape metric value, if the distortion value is non-positive then the element contains negative Jacobians due to the mid-side node positions.

Note that, for tetrahedral elements, there are several definitions of the term "aspect ratio" used in literature and in software packages. Please be aware that the various definitions will not necessarily give the same or even comparable results.

References for Tetrahedral Quality Measures

  1. (Parthasarathy, 93)
  2. (Knupp, 00)
  3. P. Knupp, Algebraic Mesh Quality Metrics for Unstructured Initial Meshes, to appear in Finite Elements for Design
    and Analysis.
  4. SDRC/IDEAS Simulation: Finite Element Modeling - User's Guide