Metrics for Tetrahedral Elements

The metrics used for tetrahedral elements in CUBIT 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 Node Distance L^1 -inf to inf None 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 High Order Metrics Distortion L^0 -1 to 1 0.6 to 1 Normalized Inradius L^0 -1 to 1 0.15 to 1 Mass Increase Ratio L^0 1 to inf None Timestep Seconds 0 to inf None 4

Tetrahedral Quality Definitions

With a few exceptions, as noted below, Cubit supports quality metric calculations for linear tetrahedral elements only.  When calculating quality metrics, that only support linear elements, for a higher order tetrahedral element, Cubit 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

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

Inradius: For all tets but tetra10s, the radius of the smallest, fully contained sphere of the linear tet. For tetra10s, the mid-edge nodes are used to subdivide the tet into 12 linear sub-tets. The inradius is the smallest inradius of the 12 linear sub-tets * 2.3.

Jacobian: Minimum pointwise volume at any corner. Cubit 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.

Node Distance: Minimum distance between any two adjacent corner nodes.

Scaled Jacobian: For linear elements the 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

High Order Elements

The preceding metrics will measure quality based only on the 4 corner nodes of the tetrahedron. The following metrics also take into account the mid nodes.

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.

Element Volume: For linear tets, (1/6) * Jacobian at corner node. For higher order tets, the tet is subdivided into sub-tets, the volumes of which are summed.

Normalized Inradius: Ratio of minimum subtet inner radius to tet outer radius (circumsphere). Subtets are defined by subdividing the tet into 12 smaller tets by using a common point at the centroid of the tet and the 6 mid-edge nodes as shown in Figure 1. The minimum in-radius of any of these 12 tets normalized by its parent outer-radius and a constant is used to determine this metric. The Normalized Inradius metric is also valid for linear elements, except that all mid-edge nodes are defined as the midpoint of their corner nodes. Figure 1. Subtet subdivision used for determining Normalized Inradius quality metric

Mass Increase Ratio: This metric stems from the global target time step and the element time step. The density required to fulfill the target time step (via mass scaling) divided by the block density is termed the mass increase ratio. Because the density within each element is constant, a ratio in the element density is equivalent to a ratio in the element mass. This metric calculates the requisite density for each element to attain the prescribed target time step. If that density is greater than the defined density, the metric yields a value greater than one. This desired global time step is set by the user with the command:

[Set] Target Timestep <value>

As stated, this metric computes the element based timestep metric and consequently element blocks must be defined with material properties of Young’s modulus, Poisson’s ratio, and a target timestep must be set.

If this metric is computed in the context of a block ('quality block 1 mass increase ratio') an accompanying printout of the mass increase per block is given.

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

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