# 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 Inradius L^1 -inf to inf None None 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 5 Timestep Seconds 0 to inf None 4 Equivolume Skew Fluent Tet Squish Fluent

## Tetrahedral Quality Definitions

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

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

Inradius: Radius of the smallest sphere that can be fully contained in a linear tet.

Jacobian:  Minimum pointwise volume at any corner. Trelis also supports Jacobian calculations for tetra15 elements.

For tetra15 elements, all 15 nodes are included for the Jacobian calculation. For all other tet types, only the corner nodes are considered.

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.  Cubit also supports Distortion calculations for tetra10 elements.

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.

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 tets, the tets 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.

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. 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.
5. SDRC/IDEAS Simulation: Finite Element Modeling - User's Guide