Metrics

automesh metrics --help
Quality metrics for an existing finite element mesh

Usage: automesh metrics [OPTIONS] --input <FILE> --output <FILE>

Options:
  -i, --input <FILE>   Name of the mesh input file
  -o, --output <FILE>  Name of the quality metrics output file
  -q, --quiet          Pass to quiet the terminal output
  -h, --help           Print help

# example:
automesh metrics -i single_valence_10.inp -o single_valence_10.csv

Input/Output File Types

inp -> csv

Description

automesh implements the following element quality metrics defined in the Verdict report.¹

• Maximum edge ratio ${\mathrm{ER}}_{\max }$
• Minium scaled Jacobian ${\mathrm{SJ}}_{\min }$
• Maximum skew
• Element volume

A brief description of each metric follows.

Maximum Edge Ratio

• ${\mathrm{ER}}_{\max }$ measures the ratio of the longest edge to the shortest edge in a mesh element.
• A ratio of 1.0 indicates perfect element quality, whereas a very large ratio indicates bad element quality.
• Knupp et al.¹ (page 87) indicate an acceptable range of [1.0, 1.3].

Minimum Scaled Jacobian

• ${\mathrm{SJ}}_{\min }$ evaluates the determinant of the Jacobian matrix at each of the corners nodes, normalized by the corresponding edge lengths, and returns the minimum value of those evaluations.
• Knupp et al.¹ (page 92) indicate an acceptable range of [0.5, 1.0], though in practice, minimum values as low as 0.2 and 0.3 are often used.

Figure. Illustrate of minimum scaled Jacobian² with acceptable range for quality occurring in [0.3, 1.0].

Maximum Skew

• Skew measures how much an element deviates from being a regular shape (e.g., in 3D a cube; in 2D a square or equilateral triangle). A maximum skew value of 0 indicates a perfectly regular shape, while higher values indicate increasing levels of distortion.
• Knupp et al.¹ (page 97) indicate an acceptable range of [0.0, 0.5].

Element Volume

• Measures the volume of the element.

Unit Tests

Inspired by Figure 2 of Livesu et al.³ reproduced here below

we examine several unit test singleton elements and their metrics.

valence	singleton	${\mathrm{ER}}_{\max }$	${\mathrm{SJ}}_{\min }$	${\mathrm{skew}}_{\max }$	volume
3		1.000000e0 (1.000)	8.660253e-1 (0.866)	5.000002e-1 (0.500)	8.660250e-1 (0.866)
3' (noised)		1.292260e0 (2.325) ** Cubit: 1.292	1.917367e-1 (0.192)	6.797483e-1 (0.680)	1.247800e0 (1.248)
4		1.000000e0 (1.000)	1.000000e0 (1.000)	0.000000e0 (0.000)	1.000000e0 (1.000)
4' (noised)		1.167884e0 (1.727) ** Cubit: 1.168	3.743932e-1 (0.374)	4.864936e-1 (0.486)	9.844008e-1 (0.984)
5		1.000000e0 (1.000)	9.510566e-1 (0.951)	3.090169e-1 (0.309)	9.510570e-1 (0.951)
6		1.000000e0 (1.000)	8.660253e-1 (0.866)	5.000002e-1 (0.500)	8.660250e-1 (0.866)
...	...	...	...	...	...
10		1.000000e0 (1.000)	5.877851e-1 (0.588)	8.090171e-1 (0.809)	5.877850e-1 (0.588)

Figure: Maximum edge ratio, minimum scaled Jacobian, maximum skew, and volume. Leading values are from automesh. Values in parenthesis are results from HexaLab.⁴ Items with ** indicate where automesh and Cubit agree, but HexaLab disagrees.

The connectivity for all elements:

1,    2,    4,    3,    5,    6,    8,    7

with prototype:

The element coordinates follow:

# 3
    1,      0.000000e0,      0.000000e0,      0.000000e0
    2,      1.000000e0,      0.000000e0,      0.000000e0
    3,     -0.500000e0,      0.866025e0,      0.000000e0
    4,      0.500000e0,      0.866025e0,      0.000000e0
    5,      0.000000e0,      0.000000e0,      1.000000e0
    6,      1.000000e0,      0.000000e0,      1.000000e0
    7,     -0.500000e0,      0.866025e0,      1.000000e0
    8,      0.500000e0,      0.866025e0,      1.000000e0

# 3'
    1,      0.110000e0,      0.120000e0,     -0.130000e0
    2,      1.200000e0,     -0.200000e0,      0.000000e0
    3,     -0.500000e0,      1.866025e0,     -0.200000e0
    4,      0.500000e0,      0.866025e0,     -0.400000e0
    5,      0.000000e0,      0.000000e0,      1.000000e0
    6,      1.000000e0,      0.000000e0,      1.000000e0
    7,     -0.500000e0,      0.600000e0,      1.400000e0
    8,      0.500000e0,      0.866025e0,      1.200000e0

# 4
    1,      0.000000e0,      0.000000e0,      0.000000e0
    2,      1.000000e0,      0.000000e0,      0.000000e0
    3,      0.000000e0,      1.000000e0,      0.000000e0
    4,      1.000000e0,      1.000000e0,      0.000000e0
    5,      0.000000e0,      0.000000e0,      1.000000e0
    6,      1.000000e0,      0.000000e0,      1.000000e0
    7,      0.000000e0,      1.000000e0,      1.000000e0
    8,      1.000000e0,      1.000000e0,      1.000000e0

# 4'
    1,      0.100000e0,      0.200000e0,      0.300000e0
    2,      1.200000e0,      0.300000e0,      0.400000e0
    3,     -0.200000e0,      1.200000e0,     -0.100000e0
    4,      1.030000e0,      1.102000e0,     -0.250000e0
    5,     -0.001000e0,     -0.021000e0,      1.002000e0
    6,      1.200000e0,     -0.100000e0,      1.100000e0
    7,      0.000000e0,      1.000000e0,      1.000000e0
    8,      1.010000e0,      1.020000e0,      1.030000e0

# 5
    1,      0.000000e0,      0.000000e0,      0.000000e0
    2,      1.000000e0,      0.000000e0,      0.000000e0
    3,      0.309017e0,      0.951057e0,      0.000000e0
    4,      1.309017e0,      0.951057e0,      0.000000e0
    5,      0.000000e0,      0.000000e0,      1.000000e0
    6,      1.000000e0,      0.000000e0,      1.000000e0
    7,      0.309017e0,      0.951057e0,      1.000000e0
    8,      1.309017e0,      0.951057e0,      1.000000e0

# 6
    1,      0.000000e0,      0.000000e0,      0.000000e0
    2,      1.000000e0,      0.000000e0,      0.000000e0
    3,      0.500000e0,      0.866025e0,      0.000000e0
    4,      1.500000e0,      0.866025e0,      0.000000e0
    5,      0.000000e0,      0.000000e0,      1.000000e0
    6,      1.000000e0,      0.000000e0,      1.000000e0
    7,      0.500000e0,      0.866025e0,      1.000000e0
    8,      1.500000e0,      0.866025e0,      1.000000e0

# 10
    1,      0.000000e0,      0.000000e0,      0.000000e0
    2,      1.000000e0,      0.000000e0,      0.000000e0
    3,      0.809017e0,      0.587785e0,      0.000000e0
    4,      1.809017e0,      0.587785e0,      0.000000e0
    5,      0.000000e0,      0.000000e0,      1.000000e0
    6,      1.000000e0,      0.000000e0,      1.000000e0
    7,      0.809017e0,      0.587785e0,      1.000000e0
    8,      1.809017e0,      0.587785e0,      1.000000e0

References