Hexahedral Metrics

automesh metrics hex --help
Quality metrics for an all-hexahedral finite element mesh

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

Options:
  -i, --input <FILE>   Mesh input file (exo | inp | stl)
  -o, --output <FILE>  Quality metrics output file (csv | npy)
  -q, --quiet          Pass to quiet the terminal output
  -h, --help           Print help

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

Maximum edge ratio $ER_{m a x}$
Minium scaled Jacobian $SJ_{m i n}$
Maximum skew
Element volume

A brief description of each metric follows.

Maximum Edge Ratio

$ER_{m a x}$ 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

$SJ_{m i n}$ 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. Illustration of minimum scaled Jacobian² with acceptable range [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 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].

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	$ER_{m a x}$	$SJ_{m i n}$	$ske w_{m a x}$	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 (aspect ratio): 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 (aspect ratio): 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. Cubit uses the term Aspect Ratio for Edge Ratio.

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

Local Numbering Scheme

Nodes

The local numbering scheme for nodes of a hexadedral element:

      7---------6
     /|        /|
    / |       / |
   4---------5  |
   |  3------|--2
   | /       | /
   |/        |/
   0---------1

node	connected nodes
0	1, 3, 4
1	0, 2, 5
2	1, 3, 6
3	0, 2, 7
4	0, 5, 7
5	1, 4, 6
6	2, 5, 7
7	3, 4, 6

Faces

The local numbering scheme for faces of a hexadedral element:

             7---------6
             |         |
             |    5    |
             |         |
   7---------4---------5---------6---------7
   |         |         |         |         |
   |    3    |    0    |    1    |    2    |
   |         |         |         |         |
   3---------0---------1---------2---------3
             |         |
             |    4    |
             |         |
             3---------2

face	nodes
0	0, 1, 5, 4
1	1, 2, 6, 5
2	2, 3, 7, 6
3	3, 0, 4, 7
4	3, 2, 1, 0
5	4, 5, 6, 7

References

Knupp PM, Ernst CD, Thompson DC, Stimpson CJ, Pebay PP. The verdict geometric quality library. SAND2007-1751. Sandia National Laboratories (SNL), Albuquerque, NM, and Livermore, CA (United States); 2006 Mar 1. link ↩ ↩2 ↩3 ↩4
Hovey CB. Naval Force Health Protection Program Review 2023 Presentation Slides. SAND2023-05198PE. Sandia National Lab.(SNL-NM), Albuquerque, NM (United States); 2023 Jun 26. link ↩
Livesu M, Pitzalis L, Cherchi G. Optimal dual schemes for adaptive grid based hexmeshing. ACM Transactions on Graphics (TOG). 2021 Dec 6;41(2):1-4. link ↩
Bracci M, Tarini M, Pietroni N, Livesu M, Cignoni P. HexaLab.net: An online viewer for hexahedral meshes. Computer-Aided Design. 2019 May 1;110:24-36. link ↩

Keyboard shortcuts

automesh