All-Hex Meshing using Singularity-Restricted Field Yufei Li 1, Yang Liu 2, Weiwei Xu 2, Wenping Wang 1, Baining Guo 2 1. The University of Hong Kong 2

All-Hex Meshing using Singularity-Restricted Field

Yufei Li1, Yang Liu2, Weiwei Xu2, Wenping Wang1, Baining Guo2

1. The University of Hong Kong2. Microsoft Research Asia

Motivation

• All-hex mesh– A 3D volume tessellated entirely by hexahedron elements.

• Why alll-hex mesh?– Reduced number of elements.– Improved speed and accuracy of physical simulations [Shimada

2006; Shepherd and Johnson 2008].

All-hex mesh Tetrahedral mesh

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Motivation

• Issues– Highly constrained connectivity.– Require much user interaction.

• Industrial practice– Multiple sweeping [Shepherd et al. 2000];– Paving and plastering [Staten et al. 2005];– …

Semi-automaticUser interaction

ANSYS software

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Motivation

• Quality criteria for all-hex mesh– Boundary conformity– Feature alignment– Low distortion

Goal: automatically generate all-hex meshes with high-quality

Feature Alignment

Low Distortion

All-hex mesh

Boundary Conformity

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Existing methods: all-hex meshing based onvolume parameterization guided by 3D frame field

Input volume(tetrahedral mesh)

3D frame field(inside the volume)

Volume parameterization(guided by 3D frame field)

All-hex mesh

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All-hex mesh Hex-dominant mesh

[Huang et al. 2011]

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All-hex mesh

CubeCover[Nieser et al. 2011]

Manually designed

meta-mesh

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3D frame field


All-hex mesh Hex-dominant mesh

Our approach: all-hex meshing frameworkbased on singularity-restricted field (SRF).

SRF(singularity-restricted field)

All-hex mesh

3D frame field

SRF

Major contributionAutomatic SRF conversion

Key condition

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Basics of 3D frame field

• Discrete setting [Nieser et al. 2011]– 3D frame:

– Discrete 3D frame field for input tet mesh: a constant 3D frame for each tet.

24 permutations.

Chiral Cubical Symmetry Group

(24 matrices)

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Fs

Ft

• A pair of arbitrary frames– Difference: a general rotation.– Matching: the permutation that best matches the two frames (24

choices).

Matching

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• An interior edge– How the frames rotate around it?

• Identity matrix: regular edge.• Non-identity matrix: singular edge (23 types).

Singular graph

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Proposition: Any singular edge does not end inside the volume.

Singularity-restricted field (SRF)

• Definition of SRF – A 3D frame field is an SRF if all of its

edge types fall into the following subsetof rotations:

– Ru, Rv, Rw represent the 90 degree rotations around u-, v-, w- coordinate axes, respectively.

SRF is necessary for inducing a valid all-hex structure

SRF(10 edge types)

3D frame field(24 edge types)

[Nieser et al. 2011]

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Converting general 3D frame field to singularity-restricted field (SRF)

• Operations for SRF conversion:– Matching adjustment: tentatively adjust the matching

for any triangular face, and check if improper singular edges could be eliminated.

– Improper singular edge collapse.

SRF(10 edge types )

3D frame field(24 edge types )

Necessary for all-hex meshing

Eliminate the improper singular edges (14 types)

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Geometric operation

• Improper singular edge collapse (topological operation)– Collapse improper singular edges without introducing new ones; – Preserve the validity of mesh topology during the collapsing process.

Converting general 3D frame field to singularity-restricted field (SRF)

Collapse improper singular edge e

et

s2

s1

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Our algorithm could eliminate all the improper singular edges, except two extreme

cases that do not happen in practice.(See proof in the paper)

Key

Improper singular edges (in red) are collapsed.

Matching adjustment could also smooth the singular

graph.

SRF Conversion

Input frame field Output SRF

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Volume parameterization(guided by SRF)

All-hex mesh

A high-quality all-hex meshing frameworkbased on singularity-restricted field (SRF).

Input domain Parameter domain

Gradient field Given SRF

Improved CubeCover [Nieser et al. 2011]to solve this mixed-integer problem.

ImprovementAdaptive rounding

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Obstacle 1: degenerate element

Element Degeneration

Input domain Parameter domain

Zero volume

Fail to trace iso-lines Missing hex elements!

Degenerate elements (in red)Why

degenerate?

Singular edge combination on triangular face.

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c

ba




All-hex mesh

All degeneration cases for any triangular face.

Handling degenerate elements

Preprocessing

Topological operations

All the degenerate elements (in red) are removed

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See the paper

Obstacle 2: flipped element

Input domain Parameter domain Flipped Elements

Negative volume

Erroneous topology

of iso-curve networkFix the topology

Restore a complete all-hex mesh

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Comparison with [Huang et al. 2011]

SRF by our method Frame field by [Huang et al. 2011]

Red edges are improper edges.

More smooth

Free of improper singular edges.

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Optimized SRF with different frame field initializations

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Singular structure

All-hex mesh

SRF

Comparison with CubeCover [Nieser et al. 2011]

J_min [-1,1]: the minimal scaled Jacobian of hexes, bigger is better.

Our method: J_min = 0.609 CubeCover: J_min = 0.073

Cube-likeelements

Distorted elements

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Comparison with PolyCube [Gregson et al. 2011]

Our method: J_min = 0.351 PolyCube: J_min = 0.196

Poor quality due to PolyCube nature

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Distortion

Boundary conformity

More results by our method

J_min = 0.729

J_min = 0.185

J_min = 0.599

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Feature alignment

Cube-like elements




All-hex mesh

A high-quality all-hex meshing frameworkbased on singularity-restricted field (SRF).

Effective smoothness of 3D frame fields

Effective operations for SRF conversion

Improved volume parameterization by handling degenerate& flipped elements

Contributions

Key ingredient

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Limitations & Future work

• No theoretical guarantee that SRF always leads to a valid all-hex structure.

Open problem: what is the sufficient condition for all-hex structures?

SRFSingularity-restricted field

All-hex structure

Necessary condition

“Almost” but NOT sufficient

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• Singularity mis-alignment: no global control of singularities.

Singularity mis-alignment

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• Singularity mis-alignment: no global control of singularities.• CANNOT guarantee a degeneracy-free or flip-free

volume parameterization. Shortcoming shared by CubeCover [Nieser et al. 2011].

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Acknowledgements• Reviewers for constructive comments.• Ulrich Reitebuch, Jin Huang for providing comparison data.• Funding agencies: The National Basic Research Program of China

(2011CB302400), the Research Grant Council of Hong Kong (718209, 718010, and 718311)

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Backup slides

Boundary-aligned 3D frame field generation

• Difference of two frames andFs Ft

• Optimization– Solved by the L-BFGS method [Liu and Nocedal 1989]

• Smoothness: closeness from to (24 types of permutations)

Frame field initialization

• Boundary tets– Smooth boundary cross field + surface normals

• Interior tets– Propagation from boundary tets.– Assigned to be the same as the one of its nearest boundary tet.

Frame field guiding

Small features, not enough tets

User intention

Robustness of SRF conversion

• Test on a random frame field– Initialization: principal-dominant cross-field on the boundary +

random frames inside.– Without optimization.

SRF conversion

• 32320 tets• 6825 vertices• 775 proper singular edges• 61 improper singular

edges

• 31930 tets• 6766 vertices• 753 proper singular edges• 0 improper singular edges

SRF-Guided Volume Parametrization

• Definition

– The integer grids in induce a hex tessellation of the input volume .

Gradient field Given SRF

• Computation

Integer variables:– Boundary faces– Vertices on the singular graph– Adjacent face gaps

CANNOT guarantee degeneration-free volume parameterization

• The triangle has three regular edges (does not belong to the degeneration case in our analysis).

• Vertices a, b and c are on the singular graph.

c

b

a

The triangle might still degenerate due to the integer

rounding on vertices a, b and c

SRF is not sufficient

The topology of the singular graph prohibits the existence of all-hex structures.

What is the sufficient condition for all-hex structures?

Triangular face

The singular graph consists of two spiral and close curves

inside a torus volume.

The tets mapped to negative volumes in the parameterization

are rendered.

Fail to retrieve an all-hex mesh.

CANNOT guarantee flip-free volume parameterization!

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Documents

All-Hex Meshing using Singularity-Restricted Field Yufei Li 1, Yang Liu 2, Weiwei Xu 2, Wenping Wang 1, Baining Guo 2 1. The University of Hong Kong 2