# Metrics for Hexahedral Elements

The metrics used for hexahedral elements in CUBIT 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 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 Node Distance L^1 -inf to inf None 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 High Order Metrics Distortion L^0 0 to 1 0.6 to 1 7 Element Volume L^3 -inf to inf None 1 Mass Increase Ratio L^0 1 to inf None Timestep Seconds 0 to inf None 6

## Hexahedral Quality Definitions

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

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

Diagonal Ratio: Minimum diagonal length / maximum diagonal length.

Dimension: Pronto-specific characteristic length for stable timestep 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 at center of hex. Cubit also supports Jacobian calculations for hex27 elements.

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

Scaled Jacobian: For linear elements the 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

### High Order Elements

The preceding metrics will measure quality based only on the 8 corner nodes of the hexahedron. The following metrics also take into account the mid nodes.

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

## References for Hexahedral Quality Measures

Element Volume: For linear hexes, the jacobian at hex center. For higher-order hexes, the hex is subdivided into sub-tets, the volumes of which are summed.

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 timestep is a function of both element geometry and material properties. To compute this metric on hexes, the hexes 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.

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