PolyMerge: a quick and easy approach to hex-dominant mesh generationC. Antonio Sanchez and Sidney Fels

Human Communication Technologies, University of British Columbia, Vancouver, Canada

Introduction

I Hexahedral meshes are often preferred andsometimes necessary in engineering/simulations. higher convergence rates. avoid volumetric/shear locking artifacts

(a) Tet (b) Hex (c) Locking

I Well-conditioned hex meshes are difficult to generate. require estimation of a grid-like structure. cannot resolve sharp features

I Hex-dominant meshes allow best of both worlds

We present a novel technique for automatichex-dominant mesh generation. It is fast, easy toimplement, and allows fine control of hex quality.

Contributions

I Finite-element-based PolyCube deformationframework for estimating structure

I Efficient tetrahedral-merging algorithm for convertingtetrahedral meshes to hex-dominant ones

Related Work

Estimating grid structureI Solid-packing [4]; Cross-fields [1]; Centroidal Voronoi

tesselation [2]; PolyCube deformation [3]Hex-dominant mesh generationI Direct hex construction based on structure [3]I Tetrahedral merging, using Yamakawa and

Shimada’s graph-search algorithm [4]

Conclusions

PolyMerge: FEM-based PolyCube deformation→tetrahedralization→ new efficient tet-mergingalgorithm for generating hex-dominant meshes.I automatic, involving two linear elasticity parameters

and an element quality thresholdI fast, generating meshes of 100k+ elements in under

a minute, can easily be parallelizedI simple, involving quadratic minimization and basic

pattern identificationI control of hex quality, leverages tetrahedralization

algorithms, only forms hexes if quality criteriasatisfied

References

[1] BAUDOUIN, T. C., ET AL. A frontal approach tohex-dominant mesh generation. Adv. Model. andSim. in Eng. Sci. (AMSES) 1, 1 (2014), 1–30.

[2] LEVY, B., AND LIU, Y. Lp centroidal voronoi tessellation andits applications. ACM Trans. Graph. 29, 4 (July 2010),119:1–119:11.

[3] LIVESU, M., ET AL. Polycut: Monotone graph-cuts forpolycube base-complex construction. ACM Trans. Graph. 32, 6(Nov. 2013), 171:1–171:12.

[4] YAMAKAWA, S., AND SHIMADA, K. Fully-automatedhex-dominant mesh generation with directionality control viapacking rectangular solid cells. Int. J. for Num. Meth. inEng. (IJNME) 57, 15 (2003), 2099–2129.

Algorithm Overview

(a) Initial Surface (b) PolyCube Deformation (c) Node Distribution (d) Tetrahedralization (e) Hex Dominant Mesh

Algorithm Summary: An initial triangulated surface (a) is deformed to a PolyCube (b) to estimate internal structure. This is used to distribute nodes throughout the volume (c). The surface and nodes aretetrahedralized (d), and then processed using a greedy tetrahedral-merging algorithm to form a hex-dominant mesh (e).

PolyCube Deformation

Quadratic minimization frameworkI Penalty term to align surface faces with nearest

coordinate plane:

QS(Ω) =∑

f

Af

∑vi∈f

‖(v ′i − c′f )− Rf (vi − cf )‖2

vi

cf

Rf

I Finite element strain energy to regularize deformation:

QV (Ω) = xT K x− xT f + d

I Minimizing sum QS + QV yields linear system:

(L + K) x = f + b

I Iterated to solve for node positions x, updating optimal face rotations Rf

Advantages:I Minimum volumetric distortion between original shape and PolyCubeI Sparse symmetric linear system, constant system matrix (linear model)

Tetrahedral Merging

3-diag.(×2) 4-diag. (×8)

4-diag. (×4) 5-diag. (×10) 6-diag. (×2)

Central (×1) Slivered (×9)

Algorithm:I For each edge in tetrahedral mesh. Count # tets around edge, n. Given n, check for all k possible hexes

I For each tet in mesh. Check for “central tet” possibility. Check for slivers separating wedges

I If valid hex found, add to priority queue sortedby quality (e.g. minimum scaled Jacobian)

I Loop through queue, add hex if. all its tets still exist in mesh. all its faces conform to another hex or two tets

Advantages:I 40× faster than Yamakawa and Shimada’s

graph search algorithm [4]I Upper bound on number of potential hex

elements: 12× # edges + 10× # tets

Results

Quality and Timings

Results of PolyMerge algorithm

Output Hex fraction Timings (s)Model Nodes Hexes Tets # Elements Volume PolyCube Tet MergeFertility 16268 10161 17738 36.4% 81.9% 19.4 1.52

56179 37987 59657 38.9% 85.0% - 6.75Kitten 7242 6471 14254 31.2% 83.4% 17.8 0.900

36720 55604 75968 42.3% 85.7 % - 11.0Cow 8486 5169 11131 31.7% 81.8% 34.2 0.817

43540 34700 43190 44.5% 86.7% - 4.57Dragon 8993 4621 12483 27.0% 79.3% 29.6 0.385

138697 107023 172684 38.2% 83.2% - 13.6

Acknowledgements

Models of the fertility statue and kitten were providedcourtesy of UU by the AIM@SHAPE ShapeRepository. The dragon model was obtained courtesyof the Stanford Computer Graphics Laboratory.

This work is partially funded by NSERC, by theGRAND NCE, Autodesk Research through theParametric Human Project, and by UBC.

Video

Electrical and Computer Engineering - University of British Columbia - Vancouver, Canada EMAIL: antonios@ece.ubc.ca WWW: http://www.artisynth.org