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Metrics for Triangular Elements

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

Table 1 - Quality Evaluation Metrics for Triangular Elements

Metric Name

Dimension

Range of Change

Acceptable Range

Information Source

Element area

<length×length>

0 to ∞

Not defined

[1]

Minimum angle

degrees

0 to 60

30 to 60

[1]

Maximum angle

degrees

60 to 180

60 to 90

[1]

Condition number

-

from 1 to ∞

from 1 to 1.3

[2]

Scale Jacobian

-

from -1 to +1

from 0.2 to 1

[2]

Relative size

-

0 to 1

0.25 to 1

[3]

Shape

-

0 to 1

0.25 to 1

[3]

Shape & Size

-

0 to 1

0.25 to 1

[3]

Distortion

-

from -1 to +1

0.6 to 1

[4]

Metric Definitions

Element Area - 0.5 × the Jacobian value at a corner node of a triangle.

Minimum Angle - the smallest angle of a triangle.

Maximum Angle - the largest angle of a triangle.

Condition Number - the condition number of the Jacobian matrix.

Scale Jacobian - the minimum Jacobian value, divided by the product of the lengths of the vectors corresponding to the two sides of the triangle.

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

Shape - 2/condition number of the weighted Jacobian matrix.

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

Distortion - {min(|J|)/actual area}×("parent" area). "Parent" area is the area of ​​the "parent" triangle, characterized by straight sides; it is taken to be equal to 0.5.

Explanations

Relative size, shape, shape and size are algebraic quality metrics. The program gives them preference over any other metrics. The metrics used are based on comparing the element's parameters with the parameters of an equilateral triangle and characterize various deviations from this "standard."

In particular, the relative size metric is used to compare the dimensions of an element with the dimensions of a "reference." If the element is half the size of the "reference," then the metric is set to 0.5. The "reference" for this assessment is a triangle whose area is equal to the average area of ​​all triangular mesh elements under consideration.

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

Sources of information