Technical History of Hex Mesh Generation

1
Technical History of Hex Mesh Generation

• Scott A. Mitchell
• Sandia National Labs

Sandia is a multiprogram laboratory operated by
Sandia Corporation, a Lockheed Martin
Company,for the United States Department of
Energy under contract DE-AC04-94AL85000.
2
Outline

• Introduction
• Relationship Map
• Sweep-like techniques
• 3d techniques
• Synthesis
• Where are we now?
• Hopes

3
Introduction
4
Introduction

• What I will cover
• What I wont cover
• Why is hex meshing hard

5
What I will cover

• Asked to give technical talk
• Unstructed hex meshing algorithms and ideas
• Relationships between algorithms
• Technical threads from one technique to another
• Focus on assembly meshing problems
• Multiple interlocking parts
• Focus on connectivity problem
• 3d Algorithms
• A little sweeping

6
What I wont cover

• Tools, packaging, ease of use
• Nodal positioning for good quality
• Bias and sizing control
• Except as approach to connectivity problem
• Ill get some names right, but not all, and leave
  some people out. Sorry, no offense intended.

7
What I wont cover

• Techniques involving
• Mixed elements
• Hex-dominant methods
• Non-conforming meshes
• Overset grids
• Primitives and Special algorithms for restricted
  classes of geometries
• Sphere-like assemblies (Hex3d at LANL)
• Air around airplanes, cars,
• CFD block primitives
• Medial-axis and Midpoint Subdivision

8
Why is Hex Meshing Hard?

• Connectivity of tet meshing is easy
• Delaunay triangulations
• Given set of points, can connect them up to form
  triangles, tetrahedra, d-simplices
• Empty-sphere property shows you how
• Caveats
• Good quality is difficult in 3 dimensions
• Boundary constraints is difficult in 3 dimensions
• Octree (subdivision) techniques
• Keep dividing polyhedron until you get simplices
• No new points on the boundary are necessary
• Advancing front techniques
• Remaining space always dividable into tetrahedra

9
Why is Hex Meshing Hard?

• Quads harder than tris (i.e. even 2d is harder)
• Cant just divide polygons into quads
• New points on the boundary may be necessary

• In practice, an even number of boundary edges can
  be divided into quads (with new interior points).
• Necessary and sufficient

10
3D Approaches

• Assume assembly problem
• Fixed boundary mesh
• Geometry first
• constrained by topology
• Topology first
• unconstrained
• fix-up afterwards
• constrained by geometry

11
Relationship Map

• Will refer to these map for context throughout
  talk

12
Sweep Relationship Map
2D
1-1 Sweeping / Projection / Rotation (lots)
Sweeping Node Placement Strategies
M-1 Sweeping (several)
2.5D
Good Source Surface Mesh Strategies
M-M Sweeping
Cooper Tool
Inside-out 3d
Multi-axis Cooper Tool
2.75D
Grafting
13
3D Relationship Map
samitch be careful to imply conceptual
relationship, not cause and effect of efforts or
people
Paving/Plastering
THEX
STC dual structure rediscovered
discovery
Hex-to-void
Schneiders open problem
Existence Proofs
Slicing doughnuts
Whisker Weavingcurve contraction
Dual cycle Elimination Shelling
Whisker Weavingstate list
construction
Dual- Particle Methods
STC folds
Recursive bisection
Linear-size
geometry first
topology first
THEX
Sweep-like techniques
Geode
? Hex dominant?
Local reconnection and smoothing
Flipping, (planar faces)
Refinement andcoarsening
Inside-out techniques
clean-up
Geode betweenshells
Prepare the boundary meshsplicing
14
3D Relationship Map
Paving/Plastering
THEX
STC dual structure rediscovered
discovery
Hex-to-void
Schneiders open problem
Existence Proofs
Slicing doughnuts
Whisker Weavingcurve contraction
Dual cycle Elimination Shelling
Whisker Weavingstate list
construction
Dual- Particle Methods
STC folds
Recursive bisection
Linear-size
geometry first
topology first
THEX
Sweep-like techniques
Geode
? Hex dominant?
Local reconnection and smoothing
Flipping, (planar faces)
Refinement andcoarsening
Inside-out techniques
clean-up
Geode betweenshells
Prepare the boundary meshsplicing
15
Paving/Plastering

• 12 years ago
• Blacker, Stephenson, Cass, Benzley
• Paving
• 2d advancing front
• Advance based on geometric criteria
• Merge connectivity when fronts collide
• Theme geometry first, connectivity second

16
Paving
Paving

• Advancing Front Begins with front at boundary
• Forms rows or single of elements based on front
  angles
• Must have even number of intervals for all-quad
  mesh

(Blacker,92)(Cass,96)
17
Paving
Paving

• Advancing Front Begins with front at boundary
• Forms rows of elements based on front angles
• Must have even number of intervals for all-quad
  mesh

(Blacker,92)(Cass,96)
18
Paving
Form new row and check for overlap
Paving

• Advancing Front Begins with front at boundary
• Forms rows of elements based on front angles
• Must have even number of intervals for all-quad
  mesh

(Blacker,92)(Cass,96)
19
Paving
Insert Wedge
Paving

• Advancing Front Begins with front at boundary
• Forms rows of elements based on front angles
• Must have even number of intervals for all-quad
  mesh

(Blacker,92)(Cass,96)
20
Paving
Seams
Paving

• Advancing Front Begins with front at boundary
• Forms rows of elements based on front angles
• Must have even number of intervals for all-quad
  mesh

(Blacker,92)(Cass,96)
21
Paving
Close Loops and smooth
Paving

• Advancing Front Begins with front at boundary
• Forms rows of elements based on front angles
• Must have even number of intervals for all-quad
  mesh

(Blacker,92)(Cass,96)
22
Paving

• Interesting / relevant details for later
• Single loop if seaming, keep two resulting loops
  even

?

• Multiple loops?
• Finesse by having each loop even at all stage

?
23
Plastering

• 10 years ago, Blacker, Canann, Stephenson
• Paving worked so well, why not try paving in 3d?
  Plastering
• Start with fixed surface mesh
• Add proto-hexes (faces) one by one, conforming to
  bdy
• Seam together nearby points, edges, faces
• Build hexes

24
Plastering

• Kind of worked
• Could fill up most of space

25
Plastering

• Difficulties
• Advancing fronts rarely seemed to close
• Seams and Wedges (knives) Blacker et al.

Remove from front for now
Drive or collapse later

• Ted Blacker to Pete Murdoch
• Find some way of closing up remaining region

26
3D Relationship Map
Paving/Plastering
THEX
STC dual structure rediscovered
discovery
Hex-to-void
Schneiders open problem
Existence Proofs
Slicing doughnuts
Whisker Weavingcurve contraction
Dual cycle Elimination Shelling
Whisker Weavingstate list
construction
Dual- Particle Methods
STC folds
Recursive bisection
Linear-size
geometry first
topology first
THEX
Sweep-like techniques
Geode
? Hex dominant?
Local reconnection and smoothing
Flipping, (planar faces)
Refinement andcoarsening
Inside-out techniques
clean-up
Geode betweenshells
Prepare the boundary meshsplicing
27
Plastering as hex dominant

• Years later
• Knives are ok finite elements, as are pyramids
• Leave knives in?
• vaporware
• Hex-tet interface, fill remaining void with
  pyramids and tets
• Geode transition template
• Developed but not used
• Well leave this thread and come back to it later
• Ted Blacker to Pete Murdoch (Meyers, Stephenson,
  Benzley)
• Find some way of closing up remaining region

28
3D Relationship Map
Paving/Plastering
THEX
STC dual structure rediscovered
discovery
Hex-to-void
Schneiders open problem
Existence Proofs
Slicing doughnuts
Whisker Weavingcurve contraction
Dual cycle Elimination Shelling
Whisker Weavingstate list
construction
Dual- Particle Methods
STC folds
Recursive bisection
Linear-size
geometry first
topology first
THEX
Sweep-like techniques
Geode
? Hex dominant?
Local reconnection and smoothing
Flipping, (planar faces)
Refinement andcoarsening
Inside-out techniques
clean-up
Geode betweenshells
Prepare the boundary meshsplicing
29
STC Dual Structure Rediscovered for Engr.

• Dual of tri / tet mesh
• is just the dual!

• Dual of quad / hex mesh
• is structure of arrangement of curves or surfaces
• Murdoch, 16 September 1993

30
STC Dual Structure Rediscovered

• Rediscovered a couple of times
• Physicists divide high-dimensional space into
  cubes to study relativity
• A.B. Zamalodchikov Tetrahedron equations and the
  relativistic S-matrix of straight strings in 21
  dimensions, Comm. Math. Phys. 79, 489-505 (1981)
• Mathematics arrangements
• B. Grünbaum Arrangements and Spreads, Reg. Conf.
  Ser. in Math. no 10, Amer. Math Soc. (1972)
• Topology geometric
• I. R. Aitchison, J. H. Rubinstein An
  introduction to polyhedral metrics of nonpositive
  curvature on 3-manifolds, Geometry of manifolds,
  ed S Donaldson and C Thomas, Cambridge Univ.
  Press 1990

31
Plasterings downfall

• Seaming and connecting together based on geometry
  wont create this nice structure.

32
3D Relationship Map
Paving/Plastering
THEX
STC dual structure rediscovered
discovery
Hex-to-void
Schneiders open problem
Existence Proofs
Slicing doughnuts
Whisker Weavingcurve contraction
Dual cycle Elimination Shelling
Whisker Weavingstate list
construction
Dual- Particle Methods
STC folds
Recursive bisection
Linear-size
geometry first
topology first
THEX
Sweep-like techniques
Geode
? Hex dominant?
Local reconnection and smoothing
Flipping, (planar faces)
Refinement andcoarsening
Inside-out techniques
clean-up
Geode betweenshells
Prepare the boundary meshsplicing
33
Murdochs thesis (1995)

• The Spatial Twist Continuum (S.T.C.) is a dual
  representation of the hexahedral mesh. The S.T.C.
  method of representing a mesh inherently captures
  the global connectivity constraint of all
  hexahedral meshes and thereby allows the
  construction of the mesh in a robust ordered or
  random manner
• Ideas
• Geometric curvature and continuity of twist
  planes
• Envision mostly regular arrangement, with careful
  attention paid to few places of irregular
  connectivity, and how to combine them together

34
Murdochs vaporware STC fold algorithm

• a valid hexahedral mesh will have a valid
  S.T.C., the question is then whether to create
  the S.T.C. first or use it as a tool to guide the
  internal mesh connectivity
• Suggested creating surface and volume mesh
  simultaneously by inserting geometric twist
  planes one by one, guided by desired mesh density
  functions
• Seemed hard
• Could you do that for an assembly? Or just single
  volume?
• How to define geometry of twist-planes?
• How to ensure dual of surface mesh captured the
  topology of the model? Compare to octree, inside
  out method by Schneiders (which hadnt been done
  yet) ½ time spent on surface isomorphism!

35
Murdochs vaporware STC fold algorithm

• To my knowledge, no serious investment in
  simultaneous surface and volume approach. They
  all create the surface mesh, then try to get the
  volume mesh
• If it worked for fixed surface mesh, it would
  solve the assembly problem
• Refinement and coarsening, mesh improvement,
  flipping, sheet pushing after initial
  arrangement, mostly topology
• Ostensons idea
• Thustons minimal energy surface (soap bubble)
  idea to define geometry
• Get a good surface mesh visit later
• Shepherds splicing
• Mattias Muller-Hanneman
• WW surface mesh fixup
• WW add geode templates
• Surface mesh first, then volume, turned out to be
  hard, too!

36
3D Relationship Map
Paving/Plastering
THEX
STC dual structure rediscovered
discovery
Hex-to-void
Schneiders open problem
Existence Proofs
Slicing doughnuts
Whisker Weavingcurve contraction
Dual cycle Elimination Shelling
Whisker Weavingstate list
construction
Dual- Particle Methods
STC folds
Recursive bisection
Linear-size
geometry first
topology first
THEX
Sweep-like techniques
Geode
? Hex dominant?
Local reconnection and smoothing
Flipping, (planar faces)
Refinement andcoarsening
Inside-out techniques
clean-up
Geode betweenshells
Prepare the boundary meshsplicing
37
Whisker Weaving State List

• Instead of Plastering hexes,
• With so much structure capturing these implicit
  constraints, why not start with the STC, then
  dualize to get hexes?
• Blacker, Tautges, Mitchell

38
Whisker Weaving State List Ideas

• Main ideas of state list Whisker Weaving
• Start with fixed surface mesh
• This defines boundary loops of the sheets
  twist-planes

purple
green
red
blue
39
Whisker Weaving State List Ideas

• Add dual of hex (STC vertex) from bdy inward, one
  at a time
• Compare to plastering
• Each sheet is 2d did we reduce this 3d problem
  to a set of 2d problems? I think notalthough it
  is easier to understand and visualize than
  free-floating primal edges.

crossing weaving
40
Whisker Weaving State List Ideas

• Now its different than Plastering
• Try to get good local connectivity
• Connectivity Rules about where to add, when to
  seam,
• Easily identify seaming joining cases
• Rarely join two loops together
• Keep sheets as disk
• Hmm, kniveswedges sometimes show up at the end
  of chords where loops self-intersecttry tracing
  a knife
• Geometric rules consider geometric dihedral
  angles of faces near surface mesh, no flat hexes
• Heroic effort by Tautges to build rules for good
  all-hexes and close the weave
• In practice, often unclosed region left-over,
  like Plastering
• In practice, very complicated sheets

41
WW complicated sheets

• One or two sheets going through
• most of the surface quads
• Related to expected connectivity
• of random graphs on lattice?
• Red knives (wedges)

42
3D Relationship Map
Paving/Plastering
THEX
STC dual structure rediscovered
discovery
Hex-to-void
Schneiders open problem
Existence Proofs
Slicing doughnuts
Whisker Weavingcurve contraction
Dual cycle Elimination Shelling
Whisker Weavingstate list
construction
Dual- Particle Methods
STC folds
Recursive bisection
Linear-size
geometry first
topology first
THEX
Sweep-like techniques
Geode
? Hex dominant?
Local reconnection and smoothing
Flipping, (planar faces)
Refinement andcoarsening
Inside-out techniques
clean-up
Geode betweenshells
Prepare the boundary meshsplicing
43
Schneiders Open Problem

• Can anyone create an all-hex mesh conforming to
  the following surface mesh?
• http//www-users.informatik.rwth-aachen.de/robert
  s/open.html
• He didnt really say why youd want to. But, if
  you could, then for any model, you could Plaster
  a while, add these, fill with tets, then divide
  into all-hex mesh.
• Are there bad surface meshes that arent
  fillable?

44
3D Relationship Map
Paving/Plastering
THEX
STC dual structure rediscovered
discovery
Hex-to-void
Schneiders open problem
Existence Proofs
Slicing doughnuts
Whisker Weavingcurve contraction
Dual cycle Elimination Shelling
Whisker Weavingstate list
construction
Dual- Particle Methods
STC folds
Recursive bisection
Linear-size
geometry first
topology first
THEX
Sweep-like techniques
Geode
? Hex dominant?
Local reconnection and smoothing
Flipping, (planar faces)
Refinement andcoarsening
Inside-out techniques
clean-up
Geode betweenshells
Prepare the boundary meshsplicing
45
Existence Proofs

• Parallel thread, unknown at the time to Murdoch,
  Blacker, Mitchell
• Mac Casel, of PDA, posting to sci-math
• Under what conditions can you hex-mesh a
  polyhedron without modifying its boundary?
• Bill Thurston (worlds best topologist)
• 25 Oct 1993, 39 days after Murdoch rediscovered
  the STC, knew about STC all along, answers in 1
  ½ pages
• If the dual curves (loops) intersect an even
  number of times, then there is a topological hex
  mesh
• As for the geometric question My first guess
  is it can be done geometrically if it can be done
  topologically, but it looks very tricky. Which
  question are you really trying to answer?

46
Existence Proofs

• Mitchells proof, very similar to Thurstons
  outline, a couple years later (mid 1995)
• Map surface mesh to a sphere (smooth)
• Form STC loops smooth closed curves
• Extend loops into closed surfaces
• Fix arrangement of surfaces to avoid degenerate
  elements
• Dualize to form hexes
• Map back to original object.

47
Necessary Conditions
48
Aside What about tet meshing?

• Tets have 4 (even) triangles, so all tet meshes
  are bounded by an even number of tets, too!
• But triangles have 3 (odd) edges, so any
  triangle surface mesh has same parity of
  triangles as bounding edges
• Any triangle mesh of a closed volume is even

1,3
2,4
Any triangle mesh of top has same parity as any
triangle mesh of bottom. Hence sum is even
49
Sufficient Conditions

• Every pair of loops on sphere intersect each
  other even number of times
• A single loop can self-intersect an even or odd
  number of times
• Recall trouble with knives and self-intersecting
  loops

50
Sufficient Conditions

• Topology theorems
• For any single curve with even
  self-intersections, can construct a surface
• For any pair of curves with odd
  self-intersections, can construct a surface
• (Proof is constructive rumor that
  non-constructive is false)

51
Sufficient Conditions
If you dualized a very coarse arrangement, youd
get degenerate hexes. Easy to fix with extra
balls.
52
Sufficient Conditions
Do similar buffering sheet next to surface mesh
53
Schneiders Open Problem Closed?

• Using these kind of techniques, a dozen people
  have constructed a mesh for the open problem.
• One self-intersecting curve, 8 times. 2 curves.
• Non-degenerate meshes have gt 100 hexes
• Poor quality
• But positive jacobians last IMR! 1.0E-4)

54
What about non-spheres?

• Map surface mesh to a sphere (smooth)
• David Eppstein told me about Thurstons proof.
• Bill Thurston and I were going to write joint
  paper.
• Thurston worked a couple days and decided
  rigorously extending to non-balls was harder than
  it looked, lost interest. So single-author paper ?

55
What about non-spheres?

• Necessary even surface mesh and some edge-chains
  is necessary. Volume property.
• Is it sufficient?
• Nate Folwell showed exponential number of hexes
  necessary for knotted holes

56
What about non-spheres?

• Algorithms
• V.A. Gasilov (Moscow) reduce topology of
  non-spheres to collection of spheres.
• slicing doughnuts
• Mesh with tets, grow a maximal tet collection
  that is a ball and doesnt self-touch, boundary
  is a cut.
• Jeff Ericksons recent paper Optimally cutting a
  surface into a disk may be relevant?

57
3D Relationship Map
Paving/Plastering
THEX
STC dual structure rediscovered
discovery
Hex-to-void
Schneiders open problem
Existence Proofs
Slicing doughnuts
Whisker Weavingcurve contraction
Dual cycle Elimination Shelling
Whisker Weavingstate list
construction
Dual- Particle Methods
STC folds
Recursive bisection
Linear-size
geometry first
topology first
THEX
Sweep-like techniques
Geode
? Hex dominant?
Local reconnection and smoothing
Flipping, (planar faces)
Refinement andcoarsening
Inside-out techniques
clean-up
Geode betweenshells
Prepare the boundary meshsplicing
58
Whisker Weave via Curve Contraction

• Implemented the existence proof for curves w/out
  self-intersections, sphere surfaces. (Nate
  Folwell, Mitchell)
• To get surface, take loop and extend
  topologically next to the remaining void
• Pre-process surface mesh to get rid of self
  intersections.
• Face collapse and refinement.
• Always solves the topology problem (existence
  proof is a proof, after all), but rarely the
  geometric problem
• Thurston As for the geometric question My
  first guess is it can be done geometrically if it
  can be done topologically, but it looks very
  tricky. Which question are you really trying to
  answer?

59
Whisker WeavingSelf-Intersection Removal

• Face collapse
• and pillowing

60
Whisker WeavingSelf-Intersection Removal

• Bad surface mesh quality in practice

61
Whisker Weaving via Curve Contraction

• Small, isolated regions of poor quality
• 30 hexes sharing one node
• Revisit later

62
3D Relationship Map
Paving/Plastering
THEX
STC dual structure rediscovered
discovery
Hex-to-void
Schneiders open problem
Existence Proofs
Slicing doughnuts
Whisker Weavingcurve contraction
Dual cycle Elimination Shelling
Whisker Weavingstate list
construction
Dual- Particle Methods
STC folds
Recursive bisection
Linear-size
geometry first
topology first
THEX
Sweep-like techniques
Geode
? Hex dominant?
Local reconnection and smoothing
Flipping, (planar faces)
Refinement andcoarsening
Inside-out techniques
clean-up
Geode betweenshells
Prepare the boundary meshsplicing
63
Dual Cycle Elimination

• Matthias Muller-Hannemann
• Similar to curve contraction algorithm
• Advantages
• No fix-up, instead prevents bad cases
• Sufficient conditions to eliminate a cycle
  without introducing degenerate or flat elements
• (but sometimes not possible to make progress)
• No self-intersecting curves for arrangements
• German CAD tool geometry decomposer into 4-sided
  surfaces, and well structured surface mesher

64
Dual Cycle Elimination

• Basic principle
• Cycles eliminated one by one
• As WW curve contraction

65
Dual Cycle Elimination

• Cycle elimination order must be a shelling
• With other algebraic rules
• sufficient to avoid degeneracies
• E.g. forbidden cycle
• Maybe a different order is a shelling

66
Dual Cycle Elimination

• Remove self-intersecting loops ahead of time
• Dice one level
• High element count
• Replace self-intersections with template
• Good quality compared to WW approach
• Works for assembly with shared surfaces
• Doesnt introduce new self-intersections

67
Dual Cycle Elimination

• Application success
• Human mandible
• Good quality hex mesh
• Had to do some manual work to identify good
  surface mesh.

68
3D Relationship Map
Paving/Plastering
THEX
STC dual structure rediscovered
discovery
Hex-to-void
Schneiders open problem
Existence Proofs
Slicing doughnuts
Whisker Weavingcurve contraction
Dual cycle Elimination Shelling
Whisker Weavingstate list
construction
Dual- Particle Methods
STC folds
Recursive bisection
Linear-size
geometry first
topology first
THEX
Sweep-like techniques
Geode
? Hex dominant?
Local reconnection and smoothing
Flipping, (planar faces)
Refinement andcoarsening
Inside-out techniques
clean-up
Geode betweenshells
Prepare the boundary meshsplicing
69
Recursive Bisection

• Calvo and Idelsohn in Argentina
• Anyone want to pay for their visit?
• Select a cycle for elimination insert a layer
  of hexes
• Instead of layer next to boundary (as WW or
  Shelling), it cuts the model in two.
• Layer has connectivity of one of the sides
• Maybe better geometrically?
• Two halves resolved independently
• Topological monsters (degeneracies) as WW

70
Bisection v.s. Cycle Elimination
Recursive bisection
Shelling / WW
71
3D Relationship Map
Paving/Plastering
THEX
STC dual structure rediscovered
discovery
Hex-to-void
Schneiders open problem
Existence Proofs
Slicing doughnuts
Whisker Weavingcurve contraction
Dual cycle Elimination Shelling
Whisker Weavingstate list
construction
Dual- Particle Methods
STC folds
Recursive bisection
Linear-size
geometry first
topology first
THEX
Sweep-like techniques
Geode
? Hex dominant?
Local reconnection and smoothing
Flipping, (planar faces)
Refinement andcoarsening
Inside-out techniques
clean-up
Geode betweenshells
Prepare the boundary meshsplicing
72
Dual Particle Methods

• Analyst group at SNL tired of waiting for WW-like
  approaches, start with 2d
• Jung, Dohrmann, Witkowski, Wolfenbarger, Gerstle,
  Mitchell, Panthaki, Segalman
• Wanted something more mathematical
• Solve geometric placement and connectivity
  simultaneously
• Throw dual particles into a domain
• Thought there would be an Eulers formula that
  would say how many
• Position them using optimization (minimum energy
  configuration)
• Connect them up, dualize to get quads

73
Dual Particle Methods

• Physics forces between particles
• Attraction, repulsion, damping, torque

74
Dual Particle Methods

• Forces are local, didnt always close up nicely
• Heuristic local reconnection
• Particle add remove
• Helped in 2d

75
Dual Particle Methods

• Initial position dual particles based on solution
  to Laplace equation.
• Almost as good as paving, when it worked

Laplace
random
76
Dual Particle Methods 3D

• 3D - Leung
• Position particles as before
• Write down all the constraints from existence
  proof
• Solve constraints via discrete optimization
• Add dual geometric constraints as they are
  violated (no cells interpenetrate)
• Huge running time
• Mostly solved topology of hex mesh. Geometric
  quality not good enough.

77
Dual Particle Methods 3D
78
Dual Particle Methods 3D
Geometric problems still - interpenetrations
79
3D Relationship Map
Paving/Plastering
THEX
STC dual structure rediscovered
discovery
Hex-to-void
Schneiders open problem
Existence Proofs
Slicing doughnuts
Whisker Weavingcurve contraction
Dual cycle Elimination Shelling
Whisker Weavingstate list
construction
Dual- Particle Methods
STC folds
Recursive bisection
Linear-size
geometry first
topology first
THEX
Sweep-like techniques
Geode
? Hex dominant?
Local reconnection and smoothing
Flipping, (planar faces)
Refinement andcoarsening
Inside-out techniques
clean-up
Geode betweenshells
Prepare the boundary meshsplicing
80
Linear-size Hex Meshes

• Eppstein
• Mitchell-Thurston algorithms produce too many
  hexes
• n number of surface quads

O(n) on edge
O(n1/2) on edge
O(n) on face
O(n) on each side
81
Linear-Size Hex Meshes

• Steps

Hex-shaped boundary tiles
Original shape
THex interior O(n) tets
82
Linear-Size Hex Meshing

• For each tile
• Fill with hexes
• Matching to create a couple of types of tiles,
  all satisfy sufficient conditions of existence
  proofs

• Solves topology problem, not geometry problem
• Anybody figured out exactly how to fill?I
  didntlike Schneiders open problem

83
3D Relationship Map
Paving/Plastering
THEX
STC dual structure rediscovered
discovery
Hex-to-void
Schneiders open problem
Existence Proofs
Slicing doughnuts
Whisker Weavingcurve contraction
Dual cycle Elimination Shelling
Whisker Weavingstate list
construction
Dual- Particle Methods
STC folds
Recursive bisection
Linear-size
geometry first
topology first
THEX
Sweep-like techniques
Geode
? Hex dominant?
Local reconnection and smoothing
Flipping, (planar faces)
Refinement andcoarsening
Inside-out techniques
clean-up
Geode betweenshells
Prepare the boundary meshsplicing
84
Geode - motivation

• Recall linear-hex
• Recall Plastering, hex to void

85
Geode Motivation

• 3d problem
• How to conformally interface in 3d?

86
Geode

• There is a transition template that someone (me)
  has filled with ok-quality hexes
• Based on constructing sheets, interleaving

Tets divided into 4 hexes
Hexes divided into 8
87
Geode 2d
88
Geode 3d

• Hex side

• Tet side

89
Geode 3d
90
Geode between shells

• Ideal template has scaled jacobian 0.26
• Comparable to THex mesh
• In practice, almost never get well-shaped layer
• Good for single layer on planar surface

91
Geode

• Why geode and not other templates?
• Self-intersections? Then why not 4-triangle
  geode
• We know a few things that make a surface mesh
  hard to fill with hexes, but we still dont know
  how to make it easy.

92
Geode Eliminates Self-Intersecting Loops!

• Unpublished
• Recall Dual Cycle Elimination
• 1. Dice
• 2. Replace self-int with template
• Instead,
• 2. Add one geode template (not layer)
• Remaining region has no self-intersection
• Conjecture after dicing, every surface mesh of
  any volume admits a compatible, linear-size,
  positive jacobian hexahedral mesh.

Inside of volume
93
Geode and Linear-Size

• Whats so special about dicing?
• Is quality going to be that good?
• Doubt it would be that practicle
• Revisit loops that are self-nearby later

94
3D Relationship Map
Paving/Plastering
THEX
STC dual structure rediscovered
discovery
Hex-to-void
Schneiders open problem
Existence Proofs
Slicing doughnuts
Whisker Weavingcurve contraction
Dual cycle Elimination Shelling
Whisker Weavingstate list
construction
Dual- Particle Methods
STC folds
Recursive bisection
Linear-size
geometry first
topology first
THEX
Sweep-like techniques
Geode
? Hex dominant?
Local reconnection and smoothing
Flipping, (planar faces)
Refinement andcoarsening
Inside-out techniques
clean-up
Geode betweenshells
Prepare the boundary meshsplicing
95
Prepare the Boundary Mesh

• Recall WW, Shelling, and Geode
• approaches to preparing the surface mesh to admit
  a (good quality?) hex mesh

96
Algorithms - splicing

• Mesh Editing and Splicing

Remove self intersecting chord Leave boundary
nodes fixed
Initial Grid
97
Algorithms - splicing

• Mesh Editing and Splicing

Intermediate mesh needs new chord to
reconcile boundary
New non self intersecting chord inserted to
reconcile boundary nodes
98
Surface Mesh Modification - Splicing
The procedure
Undesirable mesh interval
Completed Mesh
Mesh from W76 AFF
Remove unwanted quads
99
Fixing Surface Mesh for WW

• No self-intersections, but pretty convoluted
• Remove it!

100
Fixing Surface Meshes

• In progress
• Automation
• How general of a method will it be?
• Whats properties are really sufficient for the
  surface mesh?

101
3D Relationship Map
Paving/Plastering
THEX
STC dual structure rediscovered
discovery
Hex-to-void
Schneiders open problem
Existence Proofs
Slicing doughnuts
Whisker Weavingcurve contraction
Dual cycle Elimination Shelling
Whisker Weavingstate list
construction
Dual- Particle Methods
STC folds
Recursive bisection
Linear-size
geometry first
topology first
THEX
Sweep-like techniques
Geode
? Hex dominant?
Local reconnection and smoothing
Flipping, (planar faces)
Refinement andcoarsening
Inside-out techniques
clean-up
Geode betweenshells
Prepare the boundary meshsplicing
102
Where are we today?

• Summary
• Nothing yet is
• Boundary mesh conforming
• Automatic
• Good quality
• Applicable to all geometries
• Matthias very good for mandible-like problems

103
Best hopes

• Suspect that only certain surface meshes admit a
  good-quality mesh
• Combined geom topology techniques
• Difficult to get tight necessary and sufficient
  conditions -gt sufficient might be doable

104
Current Activity

• Show why hexes are hard
• Flipping
• Fix existing mesh in unstructured way
• Past efforts not promising
• Semi-structured sweep-like techniques
• Grafting, Coopering
• Improve existing mesh in semi-structured way
• Looks promising
• Surface and volume all-at-once looking better
• Mesh cutting

105
3D Relationship Map
Paving/Plastering
THEX
STC dual structure rediscovered
discovery
Hex-to-void
Schneiders open problem
Existence Proofs
Slicing doughnuts
Whisker Weavingcurve contraction
Dual cycle Elimination Shelling
Whisker Weavingstate list
construction
Dual- Particle Methods
STC folds
Recursive bisection
Linear-size
geometry first
topology first
THEX
Sweep-like techniques
Geode
? Hex dominant?
Local reconnection and smoothing
Flipping, (planar faces)
Refinement andcoarsening
Inside-out techniques
clean-up
Geode betweenshells
Prepare the boundary meshsplicing
106
Flipping

• Bern and Eppstein
• Goals
• Find set of operations sufficient for all desired
  topological changes
• Relate local improvement to problems of mesh
  existence
• Computational Geometry contributions to meshing,
  welcome complement to engineering approaches

107
Flipping Triangle Meshes

• Triangle meshes
• Two types of flips
• Equivalent to gluing tetrahedron on top of
  existing triangle mesh, take non-touching tris
• Flip graph is connected

108
Flipping Quads
no

• Flip graph is connected
• Up to parity
• For disk surfaces

no
109
Flipping Tetrahedra

• Flip graph connected for
• Topological tets, unknown for geometric tets
• Swap top and bottom of 4d simplex

110
Flipping Hexestop and bottom of 4D cuboid
111
Flipping Hexes

• Some flips preserve flatness of faces
• Does warped boundary mesh imply warped hexes?
• Beyond flatness, can we someday determine the
  maximum quality of a hex mesh matching a given
  surface mesh?

112
3D Relationship Map
Paving/Plastering
THEX
STC dual structure rediscovered
discovery
Hex-to-void
Schneiders open problem
Existence Proofs
Slicing doughnuts
Whisker Weavingcurve contraction
Dual cycle Elimination Shelling
Whisker Weavingstate list
construction
Dual- Particle Methods
STC folds
Recursive bisection
Linear-size
geometry first
topology first
THEX
Sweep-like techniques
Geode
? Hex dominant?
Local reconnection and smoothing
Flipping, (planar faces)
Refinement andcoarsening
Inside-out techniques
clean-up
Geode betweenshells
Prepare the boundary meshsplicing
113
Local Reconnection and Smoothing

• Knupps untangle smoothing
• Is able to get positive jacobians in interior of
  meshes that we know admit them
• Was unable to untangle WW meshes
• Issues
• Provable?
• Perfect implementation?
• Idea
• Combine reconnection and smoothing for
  unstructured 3d hex meshes
• As done in every other class of unstructured
  meshers

114
Local Reconnection and Smoothing

• Find local area of poor quality
• Classify type of problem, possible improvements
• Find best improvement
• Swap local connectivity via primitives
• Locally smooth
• Assess quality
• Undo
• Apply best improvement, if any

115
Local Reconnection and Smoothing

• Flips, but dual interpretation
• Might not be flips (non-parity preserving) for
  non-ball volumes

Moving sheets to untangle
Also sheet pushing ideas for local element size
control
116
Local Reconnection and Smoothing
Inserting sheets
Connecting sheets
117
Local Reconnection and Smoothing

• Most-untangled sheets not always the best quality
• Wasnt able to improve specific small examples,
  by any manual means
• Just push problem around
• Process didnt work
• Search space very large
• Progress possible but not monotonic? (unlike
  quad, tri, tet)
• Space of valid hex meshes too coarse, not well
  connected?
• Progress not possible?

118
3D Relationship Map
Paving/Plastering
THEX
STC dual structure rediscovered
discovery
Hex-to-void
Schneiders open problem
Existence Proofs
Slicing doughnuts
Whisker Weavingcurve contraction
Dual cycle Elimination Shelling
Whisker Weavingstate list
construction
Dual- Particle Methods
STC folds
Recursive bisection
Linear-size
geometry first
topology first
THEX
Sweep-like techniques
Geode
? Hex dominant?
Local reconnection and smoothing
Flipping, (planar faces)
Refinement andcoarsening
Inside-out techniques
clean-up
Geode betweenshells
Prepare the boundary meshsplicing
119
Sweep Relationship Map
2D
1-1 Sweeping / Projection / Rotation (lots)
Sweeping Node Placement Strategies
M-1 Sweeping (several)
2.5D
Good Source Surface Mesh Strategies
M-M Sweeping
Cooper Tool
Inside-out 3d
Multi-axis Cooper Tool
2.75D
Grafting
120
Sweeping Types
One-to-One Sweep
Many-to-Many Multisweep Coopering
Many-to-One Sweep
121
MultiSweep

• Create virtual surfaces w/ imprints
• Surface mesh the combination
• Sweep the underlying volumes

122
MultiSweep
Image of target curves partitions source surfaces

• Image created using
• matrix (affine transformation)
• or weighted winslow smoothing
• one mesh layer at a time
• The image is a mesh image
• depends on resolution, premesh

123
Multisweep in progress
124
Cooper Tool

• Like multisweep
• Nice language of caps, barrels
• Different projection methods
• All layers at once
• Intersection of projections different

125
Multi-Axis Coopering

• Blacker and Miyoshi
• Break model into tree

126
Multi-Axis Coopering

• Cut section to sweep in new direction
• Cuts can intersect and overlap - tricky

127
Mesh grafting
128
Mesh grafting processConforming Only Affects 1
Layer
129
3D Relationship Map
Paving/Plastering
THEX
STC dual structure rediscovered
discovery
Hex-to-void
Schneiders open problem
Existence Proofs
Slicing doughnuts
Whisker Weavingcurve contraction
Dual cycle Elimination Shelling
Whisker Weavingstate list
construction
Dual- Particle Methods
STC folds
Recursive bisection
Linear-size
geometry first
topology first
THEX
Sweep-like techniques
Geode
? Hex dominant?
Local reconnection and smoothing
Flipping, (planar faces)
Refinement andcoarsening
Inside-out techniques
clean-up
Geode betweenshells
Prepare the boundary meshsplicing
130
Refinement and Coarsening Shepherd, Borden,
Boundary Refinement
Sheet Refinement Cleaving
Mesh Coarsening
131
Sheet Insertion

• Modify mesh to capture geometry, through STC
  sheets

132
Mesh Cutting Sheet Insertion
The problem is to fit the mesh on the left to the
volume on the right. The rest of the volumes in
the model can then be swept away from this volume.
133
Step 1 Capturing the first imprint.
134
Step 2 Capturing the other two imprints
135
Step 3 Capturing the Cutouts
136
Mesh Cutting

• Questions
• Can we do this for assemblies?
• How wild of shapes can it handle?
• Compare vs. Schneiders thesis
• Compare vs. Murdoch thesis

137
Conclusions

• Technical history of unstructured meshing ideas
• Common themes
• Lots left to do and explore
• Enough background ideas to start research
• Probably not much low-hanging fruit left
• Hope of practical tools before hex meshing is
  obsolete
• Gap between theory (understanding) and practice
• closing a bit