CAE Fidesys 9.0 Documentation

Metrics for Quadrilateral Elements

The metrics used to evaluate the quality of quadrilateral elements are presented in Table 1.

Table 1 - Quality Evaluation Metrics for Quadrilateral Elements

Quadrilateral Quality Definitions

Aspect ratio - the maximum ratio of the lengths of the sides, measured in the center of the quadrilateral.

Skew - the maximum cosine of the angle between the sides, measured in the center of the quadrilateral.

Taper - the maximum ratio of the lengths of the opposite sides of the quadrilateral.

Warpage - the cosine of the minimum dihedral angle formed by planes intersecting on the diagonals of the quadrilateral.

Area of the element - the value of the Jacobian in the center of the quadrilateral.

Stretch - 1.414×(ratio of minimum side length to the maximum diagonal length of the quadrilateral).

Minimum angle - the smallest internal angle of the quadrilateral.

Maximum angle - the largest inner angle of the quadrilateral.

Conditioning number - the maximum conditioning number of the Jacobi matrix obtained at the nodes of the quadrilateral.

Jacobian - the minimum of the point values of the Jacobian obtained at the nodes and in the center of the quadrilateral.

Scaled Jacobian - the minimum value of the Jacobian divided by the product of the lengths of the vectors corresponding to the two sides of the quadrilateral.

Shear - 2/the number of conditionality of a skew-symmetric matrix Jacobi.

Shape - 2/the number of conditionality of the weighted matrix Jacobi.

Relative size - min(J, 1/J), where J is the determinant the weighted Jacobi matrix.

Shear and size - the product of shear by relative size.

Shape and size - the product of shape by relative size.

Distortion - {min(|J|)/actual area}×("parent" area). "Parental&"square is the area of the "parent"; a quadrilateral characterized by rectilinear sides; it is assumed equal to 4.

Explanations

Shear, shape, relative size, shear and size, shape and size are algebraic quality metrics applicable to quadrilateral elements. These metrics are based on comparing the element's parameters with a square "reference" and characterize various deviations from this "reference."

In particular, the shape metric shows how much the skew and aspect ratio of an element differs from a square.

The relative size metric is based on comparing the element's dimensions with those of a square "reference." If an element is half the size of the "reference," then the metric is set to 0.5. The "reference" for the corresponding evaluation is a square whose area is equal to the average area of ​​all quadrangles in the considered surface mesh.

The shape and size metric is used to evaluate the extent to which both the shape and relative size of an element differ from the characteristics of the "reference."

The shear metric is based on the condition number of the skew-symmetric Jacobian matrix. Like the skew metric, it characterizes the slope of its sides. If the quadrilateral is a rectangle, then shear= 1. The Robinson skew metric in this case is equal to zero. It is also zero when the quadrilateral is a trapezoid or even a triangle.

Sources

• Robinson, 1987.

• FIMESH program.

• Confidential information.

• Knupp, 2000.

• P. Knupp, Algebraic Mesh Quality Metrics for Unstructured Initial Meshes, submitted for publication.

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

On Robinson's Metrics for Quadrilaterals

Robinson's metrics, which characterize the quality of quadrilateral elements, include: aspect ratio, skew, taper, and element area [Robinson, 1987]. These also include stretch. A graphical representation of three of these metrics is shown in Fig. 1. The stretch metric is defined (see above) as the ratio of the minimum side length of a quadrilateral to the maximum length of its diagonal.

Figure 1 - Graphical representation of Robinson metrics (quadrilateral shape parameters)