All-Hex Meshing using Singularity-Restricted Field Yufei Li 1, Yang Liu 2, Weiwei Xu 2, Wenping Wang...
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Transcript of All-Hex Meshing using Singularity-Restricted Field Yufei Li 1, Yang Liu 2, Weiwei Xu 2, Wenping Wang...
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
1/28
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
3/28
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
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
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.
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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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
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
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
18/28
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
20/28
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
23/28
Feature alignment
Cube-like elements
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
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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