all-hex meshing using singularity-restricted field yufei li 1, yang liu 2, weiwei xu 2, wenping wang...
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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
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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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Input volume(tetrahedral mesh)
3D frame field(inside the volume)
Volume parameterization(guided by 3D frame field)
All-hex mesh Hex-dominant mesh
[Huang et al. 2011]
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Existing methods: all-hex meshing based onvolume parameterization guided by 3D frame field
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Input volume(tetrahedral mesh)
3D frame field(inside the volume)
Volume parameterization(guided by 3D frame field)
All-hex mesh
CubeCover[Nieser et al. 2011]
Manually designed
meta-mesh
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Existing methods: all-hex meshing based onvolume parameterization guided by 3D frame field
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Input volume(tetrahedral mesh)
3D frame field
Volume parameterization(guided by 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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Basics of 3D frame field
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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Basics of 3D frame field
• 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.
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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
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• 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
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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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SRF(singularity-restricted field)
Input volume(tetrahedral mesh)
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
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SRF(singularity-restricted field)
Input volume(tetrahedral mesh)
Volume parameterization(guided by SRF)
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
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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
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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
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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
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SRF(singularity-restricted field)
Input volume(tetrahedral mesh)
Volume parameterization(guided by SRF)
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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Limitations & Future work
• No theoretical guarantee that SRF always leads to a valid all-hex structure.
• Singularity mis-alignment: no global control of singularities.
Singularity mis-alignment
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Limitations & Future work
• No theoretical guarantee that SRF always leads to a valid all-hex structure.
• 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
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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)
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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.
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Frame field guiding
Small features, not enough tets
User intention
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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
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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
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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
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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
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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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Statistics