Transcript All-Hex Meshing using Singularity-Restricted Field Yufei Li1, Yang Liu2, Weiwei Xu2, Wenping Wang1, Baining Guo2 1.

All-Hex Meshing using
Singularity-Restricted Field
Yufei Li1, Yang Liu2, Weiwei Xu2, Wenping Wang1, Baining Guo2
1. The University of Hong Kong
2. Microsoft Research Asia
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Motivation
• All-hex mesh
– A 3D volume tessellated entirely by hexahedron elements.
All-hex mesh
Tetrahedral mesh
• Why alll-hex mesh?
– Reduced number of elements.
– Improved speed and accuracy of physical simulations [Shimada
2006; Shepherd and Johnson 2008].
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Motivation
• Issues
– Highly constrained connectivity.
– Require much user interaction.
Semi-automatic
User interaction
• Industrial practice
– Multiple sweeping [Shepherd et al. 2000];
– Paving and plastering [Staten et al. 2005];
– …
ANSYS software
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Motivation
• Quality criteria for all-hex mesh
– Boundary conformity
– Feature alignment
– Low distortion
Feature
Alignment
Boundary
Conformity
Low
Distortion
All-hex mesh
Goal: automatically generate all-hex meshes with high-quality
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Existing methods: all-hex meshing based on
volume 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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Existing methods: all-hex meshing based on
volume parameterization guided by 3D frame field
Input volume
(tetrahedral mesh)
3D frame field
(inside the volume)
[Huang et al. 2011]
Volume parameterization
(guided by 3D frame field)
All-hex mesh
Hex-dominant mesh
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Existing methods: all-hex meshing based on
volume 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
Manually
designed
meta-mesh
CubeCover
[Nieser et al. 2011]
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Our approach: all-hex meshing framework
based on singularity-restricted field (SRF).
Key condition
Input volume
(tetrahedral mesh)
3D frame field
SRF
3D frame field
(singularity-restricted field)
Volume parameterization
(guided by 3D frame field)
All-hex mesh
Major contribution
Automatic SRF conversion
SRF
Hex-dominant mesh
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Basics of 3D frame field
• Discrete setting [Nieser et al. 2011]
– 3D frame:
24 permutations.
Chiral Cubical
Symmetry Group
(24 matrices)
– Discrete 3D frame field for input tet mesh: a constant 3D frame
for each tet.
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Basics of 3D frame field
• A pair of arbitrary frames
– Difference: a general rotation.
– Matching: the permutation that best matches the two frames (24
choices).
Fs
Matching
Ft
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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).
Proposition: Any singular edge
does not end inside the volume.
Singular graph
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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 subset
of rotations:
3D frame field
(24 edge types)
SRF
(10 edge types)
– Ru, Rv, Rw represent the 90 degree rotations around u-, v-, wcoordinate axes, respectively.
SRF is necessary for inducing a valid all-hex structure
[Nieser et al. 2011]
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Converting general 3D frame field to
singularity-restricted field (SRF)
3D frame field
(24 edge types
)
SRF
(10 edge types
Eliminate the improper
singular edges (14 types)
• Operations for SRF conversion:
– Matching adjustment: tentatively adjust the matching
for any triangular face, and check if improper singular
edges could be eliminated.
Geometric
– Improper singular edge collapse.
operation
Necessary for
all-hex meshing
)
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Converting general 3D frame field to
singularity-restricted field (SRF)
• Improper singular edge collapse (topological operation)
– Collapse improper singular edges without introducing new ones;
Key
– Preserve the validity of mesh topology during the collapsing process.
s1
s2
Our algorithm could eliminate all the
improper singular edges, except two extreme
t e
cases that do not happen in practice.
(See proof in the paper)
Collapse improper singular edge e
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SRF Conversion
Input frame field
Matching adjustment could
also smooth the singular graph.
Output SRF
Improper singular edges (in red)
are collapsed.
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A high-quality all-hex meshing framework
based on singularity-restricted field (SRF).
Improvement
Adaptive rounding
Input volume
(tetrahedral mesh)
SRF
(singularity-restricted field)
Volume parameterization
(guided by SRF)
Improved CubeCover [Nieser et al. 2011]
to solve this mixed-integer problem.
Gradient field
Given SRF
Input domain
Parameter domain
All-hex mesh
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Obstacle 1: degenerate element
Input domain
Parameter domain
Zero
volume
Element
Degeneration
Fail to trace iso-lines
Missing hex elements!
Singular edge combination
on triangular face.
b
a
Degenerate elements (in red)
Why
degenerate?
c
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Handling degenerate elements
See the paper
Input volume
(tetrahedral mesh)
SRF
(singularity-restricted field)
All degeneration cases
for any triangular face.
Topological operations
Preprocessing
Volume parameterization
(guided by SRF)
All-hex mesh
All the degenerate elements (in red) are removed
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Obstacle 2: flipped element
Erroneous topology
of iso-curve network
Fix the topology
Restore a complete
all-hex mesh
Negative
volume
Input domain
Parameter domain
Flipped Elements
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Comparison with [Huang et al. 2011]
Free of improper singular edges.
Red edges are
improper edges.
More smooth
SRF by our method
Frame field by [Huang et al. 2011]
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Optimized SRF with different frame field initializations
SRF
All-hex mesh
Singular
structure
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Comparison with CubeCover [Nieser et al. 2011]
Cube-like
elements
Distorted
elements
Our method: J_min = 0.609
J_min
CubeCover: J_min = 0.073
[-1,1]: the minimal scaled Jacobian of hexes, bigger is better.
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Comparison with PolyCube [Gregson et al. 2011]
Distortion
Boundary
conformity
Our method: J_min = 0.351
Poor quality due to
PolyCube nature
PolyCube: J_min = 0.196
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More results by our method
J_min = 0.185
Feature alignment
Cube-like elements
J_min = 0.729
J_min = 0.599
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A high-quality all-hex meshing framework
based on singularity-restricted field (SRF).
Input volume
(tetrahedral mesh)
SRF
(singularity-restricted field)
Volume parameterization
(guided by SRF)
All-hex mesh
Contributions
Effective smoothness of 3D frame fields
Effective operations
for SRF conversion
Key ingredient
Improved volume parameterization by
handling degenerate& flipped elements
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Limitations & Future work
• No theoretical guarantee that SRF always leads to a valid
all-hex structure.
Necessary condition
SRF
Singularity-restricted field
All-hex structure
“Almost” but NOT sufficient
Open problem: what is the sufficient
condition for all-hex structures?
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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)
Backup slides
Boundary-aligned 3D frame field generation
• Difference of two frames Fs and Ft
• Smoothness: closeness from
to
(24 types of permutations)
• Optimization
– Solved by the L-BFGS method [Liu and Nocedal 1989]
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
User intention
Small features, not
enough tets
Robustness of SRF conversion
• Test on a random frame field
– Initialization: principal-dominant cross-field on the boundary +
random frames inside.
– Without optimization.
•
•
•
•
32320 tets
6825 vertices
775 proper singular edges
61 improper singular
edges
SRF
conversion
•
•
•
•
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
• Computation
Gradient field
Integer variables:
– Boundary faces
– Vertices on the singular graph
– Adjacent face gaps
Given SRF
.
CANNOT guarantee degenerationfree 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.
b
a
c
The triangle might still
degenerate due to the integer
rounding on vertices a, b and c
SRF is not sufficient
What is the sufficient condition
for all-hex structures?
Triangular face
The topology of the singular graph prohibits the
existence of all-hex structures.
CANNOT guarantee flip-free
volume parameterization!
The singular graph consists
of two spiral and close curves
inside a torus volume.
Fail to retrieve an
all-hex mesh.
The tets mapped to negative
volumes in the parameterization
are rendered.
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