singularity-constrained octahedral fields for hexahedral ...ˆ˜衡.pdf · singularity graph...
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Heng Liu, RWTH Aachen UniversityPaul Zhang, MITEdward Chien, MITJustin Solomon, MIT David Bommes, RWTH Aachen University
Singularity-Constrained Octahedral Fields for Hexahedral Meshing
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Mesh Generation and Optimization Group | Heng LiuMesh Generation and Optimization Group | Heng Liu
Why Hexahedral Meshes?
Automotive [CM2 MeshTools]
Aerospace [MeshGems]
Biomechanics [TrueGrid]
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Mesh Generation and Optimization Group | Heng LiuMesh Generation and Optimization Group | Heng Liu
Motivation
VS
• Advantage:superior numerical properties, e.g. performance, accuracy• Problem:difficult to generate good quality hex meshes
hexahedral mesh tetrahedral mesh
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Mesh Generation and Optimization Group | Heng LiuMesh Generation and Optimization Group | Heng Liu
Motivation
hexahedral mesh
• Problem:difficult to generate good quality hex meshes.
• Good quality:• approximation➡ faithful & boundary aligned
• regularity➡few singularities/irregularities
• element quality ➡scaled Jacobians close to 1
…
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Mesh Generation and Optimization Group | Heng LiuMesh Generation and Optimization Group | Heng Liu
Motivation
• Hex Meshing• automation• robustness• high quality
automatic robust
high quality
No robust & automatic algorithm!
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Mesh Generation and Optimization Group | Heng Liu
high quality
robustautomatic
Mesh Generation and Optimization Group | Heng Liu
• Hex Meshing• Advancing fronts
[Kremer, et al. 2013]
State-of-the-art
automatic
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Mesh Generation and Optimization Group | Heng Liu
high quality
robustautomatic
Mesh Generation and Optimization Group | Heng Liu
• Hex Meshing• Grid based methods
State-of-the-art
[Maréchal 2009]
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Mesh Generation and Optimization Group | Heng LiuMesh Generation and Optimization Group | Heng Liu
• Hex Meshing• Polycube based: Gregson et al. 2011, Livesu et al. 2013, Huang et al.
2014, Fu et al. 2016, Fang et al. 2016, Xu et al. 2017…
[Gregson, et al. 2011]
State-of-the-art
robust
high quality
automatic
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Mesh Generation and Optimization Group | Heng LiuMesh Generation and Optimization Group | Heng Liu
• Hex Meshing• Octahedral field based
[Nieser, et al. 2011]
State-of-the-art
high quality
robust
high quality
automatic
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Mesh Generation and Optimization Group | Heng Liu
high quality
robustautomatic
Mesh Generation and Optimization Group | Heng Liu
• Hex Meshing• Octahedral field based
[Jiang, et al. 2013]
[Li, et al. 2012]
State-of-the-art
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David Bommes – TITANE TeamMesh Generation and Optimization Group | Heng Liu
Octahedral Field Based Hex Meshing
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Mesh Generation and Optimization Group | Heng Liu
input tetrahedral mesh
boundary aligned octahedral field
[Huang et al. 2011][Li et al. 2012]
[Jiang et al. 2014][Ray et al. 2016]
[Vaxman et al. 2017]
Mesh Generation and Optimization Group | Heng Liu
[Nieser et al. 2011][Lyon et al. 2016]
hex meshsingularities
octahedral-fieldsingularities
invalid singularities➡ integer-grid map
degenerates
Octahedral Field Based Hex Meshing
correctedsingularity graph
constrainedoctahedral field
outputhexahedral mesh
outputhexahedral meshsingularity graph
problematic step
Modified Algorithm
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Mesh Generation and Optimization Group | Heng LiuMesh Generation and Optimization Group | Heng Liu
Contribution 1
corrected singularity graphsingularity graph
• local configurations• global necessary condition
guidesingularity graph correctioncurrent: manual & future: automatic
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Mesh Generation and Optimization Group | Heng LiuMesh Generation and Optimization Group | Heng Liu
Modified Algorithm
input tetrahedral mesh
boundary aligned octahedral field
correctedsingularity graph
constrainedoctahedral field
outputhexahedral mesh
singularity graph
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Mesh Generation and Optimization Group | Heng Liu
• algorithm to generate octahedral fields with prescribed hex-meshable singularity graphs
Mesh Generation and Optimization Group | Heng Liu
Contribution 2
corrected singularity graphconstrained octahedral field
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Mesh Generation and Optimization Group | Heng LiuMesh Generation and Optimization Group | Heng Liu
Modified Algorithm
input tetrahedral mesh
boundary aligned octahedral-field
correctedsingularity graph
constrainedoctahedral-field
outputhexahedral mesh
singularity graph
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David Bommes – TITANE TeamMesh Generation and Optimization Group | Heng Liu
What are hex mesh singularities?
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Mesh Generation and Optimization Group | Heng Liu
regularvalence 4
singularvalence 3
singularvalence 5
Mesh Generation and Optimization Group | Heng Liu
Hex Mesh Singularities
• Local Conditions - Edge types(interior)
singularvalence 6
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Mesh Generation and Optimization Group | Heng LiuMesh Generation and Optimization Group | Heng Liu
Hex Mesh Singularities
singularvalence 6
split
twice valence 5
• Local Conditions - Edge types(interior)
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Mesh Generation and Optimization Group | Heng Liu
regularvalence 4
Mesh Generation and Optimization Group | Heng Liu
Hex Mesh Singularities
singularvalence 3
singularvalence 5
singularvalence 6
• Local Conditions - Edge types(interior)
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Mesh Generation and Optimization Group | Heng LiuMesh Generation and Optimization Group | Heng Liu
• Singularity graph
singular arcs
singular nodes
Hex Mesh Singularities
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Mesh Generation and Optimization Group | Heng LiuMesh Generation and Optimization Group | Heng Liu
Hex Mesh Singularities
• Vertex typeshex mesh vertices are isomorphic to triangulations of the sphere
• How many different?➡ infinitely many➡ but only 11 when restricted to edge
valence 3,4,5
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Mesh Generation and Optimization Group | Heng LiuMesh Generation and Optimization Group | Heng Liu
• How many sphere triangulations exists with vertex valences restricted to 3, 4, 5?➡Answer: only 11
• Assume triangulation has #V vertices with signature (i, j, k)
#valence 3
#valence 4
#valence 5
Euler formula#V - #E + #F = 2
3i + 2j + k = 12with
i+j+k = #V
consequences:1. minimal #V = 4 (i=4)2. maximal #V = 12 (k=12)
Hex Mesh Singularities
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Mesh Generation and Optimization Group | Heng LiuMesh Generation and Optimization Group | Heng Liu
Hex Mesh Singularities• #V=4
(4,0,0)• #V=5
(3,1,1), (2,3,0)• #V=6
(3,0,3), (2,2,2), (1,4,1), (0,6,0)• #V=7
(0,5,2), (1,3,3), (2,1,4)• #V=8
(0,4,4), (1,2,5), (2,0,6)• #V=9
(0,3,6), (1,1,7)• #V=10
(0,2,8), (1,0,9)• #V=11
(0,1,10)• #V=12
(0,0,12)
3i + 2j + k = 12with
i+j+k = #Vvalence 5 with 5 vertices requires self-connection
[Schmeichel and Hakimi 1977]“On Planar Graphical Degree Sequences”
[Mishra and Sarvate 2007]“A note on Non-Regular Planar Graphs”
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Mesh Generation and Optimization Group | Heng LiuMesh Generation and Optimization Group | Heng Liu
• #V=4(4,0,0)
• #V=5(2,3,0)
• #V=6(2,2,2), (0,6,0)
• #V=7(0,5,2), (1,3,3)
• #V=8(0,4,4), (2,0,6)
• #V=9(0,3,6)
• #V=10(0,2,8)
• #V=12(0,0,12)
Hex Mesh Singularities
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Mesh Generation and Optimization Group | Heng Liu
• #V=4(4,0,0)
• #V=5(2,3,0)
• #V=6(2,2,2), (0,6,0)
• #V=7(0,5,2), (1,3,3)
• #V=8(0,4,4), (2,0,6)
• #V=9(0,3,6)
• #V=10(0,2,8)
• #V=12(0,0,12)
Mesh Generation and Optimization Group | Heng Liu
Hex Mesh Singularities
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Mesh Generation and Optimization Group | Heng Liu
• #V=4(4,0,0)
• #V=5(2,3,0)
• #V=6(2,2,2), (0,6,0)
• #V=7(0,5,2), (1,3,3)
• #V=8(0,4,4), (2,0,6)
• #V=9(0,3,6)
• #V=10(0,2,8)
• #V=12(0,0,12)
Mesh Generation and Optimization Group | Heng Liu
Hex Mesh Singularities
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Mesh Generation and Optimization Group | Heng LiuMesh Generation and Optimization Group | Heng Liu
Hex Mesh Singularities• #V=4
(4,0,0)• #V=5
(2,3,0)• #V=6
(2,2,2), (0,6,0)• #V=7
(0,5,2), (1,3,3)
• #V=8(0,4,4), (2,0,6)
• #V=9(0,3,6)
• #V=10(0,2,8)
• #V=12(0,0,12)
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Mesh Generation and Optimization Group | Heng LiuMesh Generation and Optimization Group | Heng Liu
Hex Mesh Singularities• #V=4
(4,0,0)• #V=5
(2,3,0)• #V=6
(2,2,2), (0,6,0)• #V=7
(0,5,2), (1,3,3)
• #V=8(0,4,4), (2,0,6)
• #V=9(0,3,6)
• #V=10(0,2,8)
• #V=12(0,0,12)
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Mesh Generation and Optimization Group | Heng LiuMesh Generation and Optimization Group | Heng Liu
Hex Mesh Singularities• #V=4
(4,0,0)• #V=5
(2,3,0)• #V=6
(2,2,2), (0,6,0)• #V=7
(0,5,2), (1,3,3)• #V=8
(0,4,4), (2,0,6)
• #V=9(0,3,6)
• #V=10(0,2,8)
• #V=12(0,0,12)
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Mesh Generation and Optimization Group | Heng LiuMesh Generation and Optimization Group | Heng Liu
Hex Mesh Singularities• #V=4
(4,0,0)• #V=5
(2,3,0)• #V=6
(2,2,2), (0,6,0)• #V=7
(0,5,2), (1,3,3)
• #V=8(0,4,4), (2,0,6)
• #V=9(0,3,6)
• #V=10(0,2,8)
• #V=12(0,0,12)
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Mesh Generation and Optimization Group | Heng LiuMesh Generation and Optimization Group | Heng Liu
Hex Mesh Singularities• #V=4
(4,0,0)• #V=5
(2,3,0)• #V=6
(2,2,2), (0,6,0)• #V=7
(0,5,2), (1,3,3)
• #V=8(0,4,4), (2,0,6)
• #V=9(0,3,6)
• #V=10(0,2,8)
• #V=12(0,0,12)
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Mesh Generation and Optimization Group | Heng LiuMesh Generation and Optimization Group | Heng Liu
Hex Mesh Singularities• #V=4
(4,0,0)• #V=5
(2,3,0)• #V=6
(2,2,2), (0,6,0)• #V=7
(0,5,2), (1,3,3)
• #V=8(0,4,4), (2,0,6)
• #V=9(0,3,6)
• #V=10(0,2,8)
• #V=12(0,0,12)
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Mesh Generation and Optimization Group | Heng LiuMesh Generation and Optimization Group | Heng Liu
Hex Mesh Singularities• #V=4
(4,0,0)• #V=5
(2,3,0)• #V=6
(2,2,2), (0,6,0)• #V=7
(0,5,2), (1,3,3)
• #V=8(0,4,4), (2,0,6)
• #V=9(0,3,6)
• #V=10(0,2,8)
• #V=12(0,0,12)
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Mesh Generation and Optimization Group | Heng LiuMesh Generation and Optimization Group | Heng Liu
Hex Mesh Singularities• #V=4
(4,0,0)• #V=5
(2,3,0)• #V=6
(2,2,2), (0,6,0)• #V=7
(0,5,2), (1,3,3)
• #V=8(0,4,4), (2,0,6)
• #V=9(0,3,6)
• #V=10(0,2,8)
• #V=12(0,0,12)
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Mesh Generation and Optimization Group | Heng LiuMesh Generation and Optimization Group | Heng Liu
Hex Meshable Singularity Graphs (valence 3/4/5)
• Local Conditions - Vertex types(interior)each singular node is one of the 11 configurations
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Mesh Generation and Optimization Group | Heng Liu
• Global Necessary Condition
Mesh Generation and Optimization Group | Heng Liu
Hex Meshable Singularity Graphs (valence 3/4/5)
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David Bommes – TITANE TeamMesh Generation and Optimization Group | Heng Liu
What are Octahedral Field singularities?
hex meshsingularities
octahedral-fieldsingularities
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Mesh Generation and Optimization Group | Heng Liu
• Octahedral-Fields
Mesh Generation and Optimization Group | Heng Liu
Octahedral Field Singularities
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Mesh Generation and Optimization Group | Heng LiuMesh Generation and Optimization Group | Heng Liu
Octahedral Field Singularities
• Octahedral-Fields• one frame per tet (unit quaternion)
• one matching quaternion per face• 24 classes of matchings
singularities form a graph
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Mesh Generation and Optimization Group | Heng LiuMesh Generation and Optimization Group | Heng Liu
Octahedral-Field Singularities
: valance 3
: valance 5
: valid type: invalid type
• Defects
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Mesh Generation and Optimization Group | Heng LiuMesh Generation and Optimization Group | Heng Liu
Octahedral-Field Singularities
: valance 3
: valance 5
: valid type: invalid type
• Corrected Singularity Graph
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Mesh Generation and Optimization Group | Heng Liu
input tetrahedral mesh
boundary aligned octahedral-field
correctedsingularity graph
constrainedoctahedral-field
outputhexahedral mesh
singularity graph
Mesh Generation and Optimization Group | Heng Liu
Modified Algorithm
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David Bommes – TITANE TeamMesh Generation and Optimization Group | Heng Liu
Singularity Constrained Hexable Octahedral-Fields
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Mesh Generation and Optimization Group | Heng LiuMesh Generation and Optimization Group | Heng Liu
• Input:• Singularity graph , satisfying➡global necessary condition➡local conditions
• Output: • an octahedral field O that is boundary-
aligned and matches the singularity graph
Challenge
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Mesh Generation and Optimization Group | Heng LiuMesh Generation and Optimization Group | Heng Liu
• Input:• alignment constraints As• edge type constraints Es• vertex type constraints Vs
• Output: • topological octahedral field➡matchings and field alignment
Challenge
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Mesh Generation and Optimization Group | Heng LiuMesh Generation and Optimization Group | Heng Liu
• Input:• alignment constraints As• edge type constraints Es• vertex type constraints Vs
Singularity Graph Constraints
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Mesh Generation and Optimization Group | Heng LiuMesh Generation and Optimization Group | Heng Liu
Singularity Graph Constraints
singularvalence 3
• Input:• alignment constraints As• edge type constraints Es• vertex type constraints Vs
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Mesh Generation and Optimization Group | Heng LiuMesh Generation and Optimization Group | Heng Liu
Singularity Graph Constraints
• 11 cases
• Input:• alignment constraints As• edge type constraints Es• vertex type constraints Vs
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Mesh Generation and Optimization Group | Heng LiuMesh Generation and Optimization Group | Heng Liu
• Vertex type constraints
Singularity Graph Constraints
1. “axis-to-axis”➡4 possibilities
2. “3-axes-corner”➡4 possibilities
3. “2-axes-to-2-axes”➡unique
“right-handed” frame
Singularity graph constraints involve matchings and field alignment
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Mesh Generation and Optimization Group | Heng LiuMesh Generation and Optimization Group | Heng Liu
• Field alignment topologically matters for• singular edges• surface normals
• Idea:• fix field alignment (arbitrarily) and then
derive consistent matchings
Singularity Graph Constraints
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Mesh Generation and Optimization Group | Heng LiuMesh Generation and Optimization Group | Heng Liu
Simplified Algebraic System
• Input:• axis alignment constraints• one edge type constraint for each tetmesh edge• vertex type constraints between adjacent singularities
• Task: • find consistent matchings
• Idea:• proceed from where local constraint information is available1. locally unique matchings (“2-axes-to-2-axes”) 2. partially constrained matchings(“axis-to-axis”)
} difficult algebraic system
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Mesh Generation and Optimization Group | Heng Liu
m
Mesh Generation and Optimization Group | Heng Liu
• Algorithm1. MergeSingularArcCharts2. MergeBoundaryCharts3. MergeVolumeCharts
Algorithm: Chart-Merging
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Mesh Generation and Optimization Group | Heng LiuMesh Generation and Optimization Group | Heng Liu
Algorithm: Chart-Merging
• Algorithm1. MergeSingularArcCharts2. MergeBoundaryCharts3. MergeVolumeCharts
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Mesh Generation and Optimization Group | Heng LiuMesh Generation and Optimization Group | Heng Liu
Algorithm: Chart-Merging
• Algorithm1. MergeSingularArcCharts2. MergeBoundaryCharts3. MergeVolumeCharts
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Mesh Generation and Optimization Group | Heng LiuMesh Generation and Optimization Group | Heng Liu
Algorithm: Chart-Merging
• Algorithm1. MergeSingularArcCharts2. MergeBoundaryCharts3. MergeVolumeCharts
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Mesh Generation and Optimization Group | Heng LiuMesh Generation and Optimization Group | Heng Liu
Algorithm: Chart-Merging
“2-axes-to-2-axes”➡unique
• Algorithm1. MergeSingularArcCharts2. MergeBoundaryCharts3. MergeVolumeCharts
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Mesh Generation and Optimization Group | Heng LiuMesh Generation and Optimization Group | Heng Liu
• Input:• Axes alignment• Matchings
Algorithm: Topological Field to Geometrical Field
minimize
subject to:
minimize
subject to:
Relaxation
Eigenvalue Problem
topological field geometrical field
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David Bommes – TITANE TeamMesh Generation and Optimization Group | Heng Liu
Results
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Mesh Generation and Optimization Group | Heng LiuMesh Generation and Optimization Group | Heng Liu
Results-Bone[Tets: 71k Time: 0.9s/54.0s]
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Mesh Generation and Optimization Group | Heng LiuMesh Generation and Optimization Group | Heng Liu
Results-Hand[Tets: 113k Time: 1.8s/34.9s]
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Mesh Generation and Optimization Group | Heng LiuMesh Generation and Optimization Group | Heng Liu
Results-Bunny[Tets: 130k Time: 2.7s/53.2s]
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Mesh Generation and Optimization Group | Heng LiuMesh Generation and Optimization Group | Heng Liu
Results-Kitten[Tets: 55k Time: 1.8s/28.3s]
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Mesh Generation and Optimization Group | Heng LiuMesh Generation and Optimization Group | Heng Liu
Results-Joint[Tets: 44k Time: 0.5s/13.6s]
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Mesh Generation and Optimization Group | Heng LiuMesh Generation and Optimization Group | Heng Liu
Results-Rockerarm[Tets: 122k Time: 3.1s/67.5s]
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Mesh Generation and Optimization Group | Heng LiuMesh Generation and Optimization Group | Heng Liu
Results-Knot[Tets: 131k Time: 1.8s/47.3s]
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Mesh Generation and Optimization Group | Heng LiuMesh Generation and Optimization Group | Heng Liu
Results-Armadillo[Tets: 206k Time: 6.0s/83.4s]
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Mesh Generation and Optimization Group | Heng LiuMesh Generation and Optimization Group | Heng Liu
Results-Elephant[Tets: 301k Time: 9.9s/88s]
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David Bommes – TITANE TeamMesh Generation and Optimization Group | Heng Liu
Summary & Outlook
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Mesh Generation and Optimization Group | Heng LiuMesh Generation and Optimization Group | Heng Liu
Summary & Outlook
• Contribution• Necessary conditions: hex mesh singularities with valence 3, 4, 5 (and some more)• Algorithm: octahedral-fields with fixed topology
• Future Challenges• Sufficient conditions: local consistency & global consistency • Automatic correction of invalid singularity graph• Robust integer-quantization & mapping automatic high-quality hex meshing
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David Bommes – TITANE TeamMesh Generation and Optimization Group | Heng Liu
Thank You!
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