cubit mesh generation project technical history of hex mesh generation scott a. mitchell sandia...

CUBIT Mesh Generation Project

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.



Outline

Introduction Relationship Map

» Sweep-like techniques» 3d techniques» Synthesis

Where are we now?» Hopes



Introduction



Introduction

What I will cover What I won’t cover Why is hex meshing hard



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



What I won’t cover

Tools, packaging, ease of use… Nodal positioning for good quality

» Bias and sizing control» Except as approach to connectivity

problem I’ll get some names right, but not all,

and leave some people out. Sorry, no offense intended.



What I won’t 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



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



Why is Hex Meshing Hard?

Quads harder than tris (i.e. even 2d is harder)» Can’t 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

? ?

q q

bdyeeq int24Euler:



3D Approaches

Assume: assembly problem» Fixed boundary mesh

Geometry first» constrained by topology

Topology first» unconstrained

– fix-up afterwards

» constrained by geometry



Relationship Map

Will refer to these map for context throughout talk



Sweep Relationship Map

Sweeping Node Placement Strategies

2D2.

5D2.

75D

Good Source SurfaceMesh Strategies

1-1 Sweeping / Projection / Rotation (lots)

M-1 Sweeping (several)

M-M Sweeping

Multi-axisCooper Tool Grafting

Cooper Tool

Inside-out3d



3D Relationship MapPaving/Plastering

Whisker Weavingstate list

STC dual structurerediscovered

Whisker Weavingcurve contraction

STC folds

Existence Proofs

Dual cycleElimination& Shelling

Recursive bisection

Linear-size

Local reconnection and smoothing

Flipping, (planar faces)

Prepare the boundary mesh:splicing

Refinement andcoarsening

Schneider’s open problem

Geode

Hex-to-void

clea

n-up

Geode betweenshells

geometry first topology firstTHEXcons

truc

tion

disc

over

y

Sweep-liketechniques

THEX

samitch: be careful to imply conceptual relationship, not cause and effect of efforts or people

samitch: be careful to imply conceptual relationship, not cause and effect of efforts or people

? Hex dominant?

Dual-ParticleMethods

Inside-outtechniques

Slicing doughnuts







STC folds

Existence Proofs


Recursive bisection

Linear-size






Geode

Hex-to-void

clea

n-up

Geode betweenshells


truc

tion

disc

over

y


THEX

? Hex dominant?



Slicing doughnuts



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



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)



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




Paving



Form new row and check for

overlap



Paving



Insert “Wedge

”



Paving



Seams



Paving



Close Loops and smooth



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

?

?



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



Plastering

Kind of worked» Could fill up most of space



Plastering

Difficulties» Advancing fronts rarely seemed to close

– Seams and Wedges (knives) Blacker et al.

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

Drive or collapse later

Remove from front for now







STC folds

Existence Proofs


Recursive bisection

Linear-size






Geode

Hex-to-void

clea

n-up

Geode betweenshells


truc

tion

disc

over

y


THEX

? Hex dominant?



Slicing doughnuts



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

We’ll 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







STC folds

Existence Proofs


Recursive bisection

Linear-size






Geode

Hex-to-void

clea

n-up

Geode betweenshells


truc

tion

disc

over

y


THEX

? Hex dominant?



Slicing doughnuts



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



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 2+1 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



Plastering’s downfall

» Seaming and connecting together based on geometry won’t create this nice structure.







STC folds

Existence Proofs


Recursive bisection

Linear-size






Geode

Hex-to-void

clea

n-up

Geode betweenshells


truc

tion

disc

over

y


THEX

? Hex dominant?



Slicing doughnuts



Murdoch’s 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



Murdoch’s 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 hadn’t been done yet): ½ time spent on surface isomorphism!



Murdoch’s 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– Ostenson’s idea– Thuston’s minimal energy surface (soap bubble) idea to define

geometry– Get a good surface mesh – visit later

Shepherd’s splicing Mattias Muller-Hanneman WW surface mesh fixup WW add geode templates

» Surface mesh first, then volume, turned out to be hard, too!







STC folds

Existence Proofs


Recursive bisection

Linear-size






Geode

Hex-to-void

clea

n-up

Geode betweenshells


truc

tion

disc

over

y


THEX

? Hex dominant?



Slicing doughnuts



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



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

green red blue purple




» 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 not…although it is easier to understand and visualize than free-floating primal edges.

crossing = weaving




Now it’s 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, knives=wedges sometimes show up at the end of chords where loops self-intersect…try 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



WW complicated sheets

edge w/ 10 hexes

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







STC folds

Existence Proofs


Recursive bisection

Linear-size






Geode

Hex-to-void

clea

n-up

Geode betweenshells


truc

tion

disc

over

y


THEX

? Hex dominant?



Slicing doughnuts



Schneiders Open Problem

Can anyone create an all-hex mesh conforming to the following surface mesh?

http://www-users.informatik.rwth-aachen.de/~roberts/open.html

» He didn’t really say why you’d 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 aren’t fillable?







STC folds

Existence Proofs


Recursive bisection

Linear-size






Geode

Hex-to-void

clea

n-up

Geode betweenshells


truc

tion

disc

over

y


THEX

? Hex dominant?



Slicing doughnuts



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 (world’s 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?”



Existence Proofs

Mitchell’s proof, very similar to Thurston’s 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.



Necessary Conditions



Aside: What about tet meshing?

Tet’s have 4 (even) triangles, so all tet meshes are bounded by an even number of tets, too!

But triangle’s 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 hassame parity as any triangle mesh of bottom.Hence sum is even



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…




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)




If you dualized a very coarse arrangement, you’d get degenerate hexes.Easy to fix with extra balls.




Do similar buffering sheet next to surface mesh



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 > 100 hexes» Poor quality

– But positive jacobians last IMR! 1.0E-4)



What about non-spheres?

“Map surface mesh to a sphere (smooth)” David Eppstein told me about Thurston’s

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




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




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 doesn’t self-touch, boundary is a cut.

» Jeff Erickson’s recent paper “Optimally cutting a surface into a disk” may be relevant?







STC folds

Existence Proofs


Recursive bisection

Linear-size






Geode

Hex-to-void

clea

n-up

Geode betweenshells


truc

tion

disc

over

y


THEX

? Hex dominant?



Slicing doughnuts



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?”



Whisker WeavingSelf-Intersection

Removal

Face collapse

and pillowing



Whisker WeavingSelf-Intersection

Removal

Bad surface mesh quality in practice



Whisker Weaving via Curve Contraction

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







STC folds

Existence Proofs


Recursive bisection

Linear-size






Geode

Hex-to-void

clea

n-up

Geode betweenshells


truc

tion

disc

over

y


THEX

? Hex dominant?



Slicing doughnuts



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




Basic principle» Cycles eliminated one by one

– As WW curve contraction




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




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– Doesn’t introduce new self-intersections




Application success» Human mandible

– Good quality hex mesh– Had to do some manual

work to identify good surface mesh.







STC folds

Existence Proofs


Recursive bisection

Linear-size






Geode

Hex-to-void

clea

n-up

Geode betweenshells


truc

tion

disc

over

y


THEX

? Hex dominant?



Slicing doughnuts



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



Bisection v.s. Cycle Elimination

Recursive bisection Shelling / WW







STC folds

Existence Proofs


Recursive bisection

Linear-size






Geode

Hex-to-void

clea

n-up

Geode betweenshells


truc

tion

disc

over

y


THEX

? Hex dominant?



Slicing doughnuts



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 Euler’s formula that would say how many

» Position them using optimization (minimum energy configuration)» Connect them up, dualize to get quads



Dual Particle Methods

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




Forces are local, didn’t always close up nicely» Heuristic local

reconnection» Particle add &

remove» Helped in 2d




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

random Laplace



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.




Geometric problems still- interpenetrations







STC folds

Existence Proofs


Recursive bisection

Linear-size






Geode

Hex-to-void

clea

n-up

Geode betweenshells


truc

tion

disc

over

y


THEX

? Hex dominant?



Slicing doughnuts



Linear-size Hex Meshes

Eppstein» Mitchell-Thurston algorithms produce too many

hexes» n = number of surface quads

O(n) on faceO(n) on each side

O(n) on edge

O(n1/2) on edge



Linear-Size Hex Meshes

Steps

Hex-shaped boundary tiles

THex interiorO(n) tets

Original shape



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 didn’t…like Schneiders open problem







STC folds

Existence Proofs


Recursive bisection

Linear-size






Geode

Hex-to-void

clea

n-up

Geode betweenshells


truc

tion

disc

over

y


THEX

? Hex dominant?



Slicing doughnuts



Geode - motivation

Recall linear-hex Recall Plastering, hex to void



Geode Motivation

3d problem

How to conformally interface in 3d?



Geode

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

Hexesdivided into 8

Tetsdivided into 4 hexes



Geode – 2d



Geode – 3d

Hex side

Tet side



Geode – 3d



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…

thexTHexgeode

geode

mesh with hexes

hex



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 don’t know how to make it easy.



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



Geode and Linear-Size

What’s so special about dicing? Is quality going to be that good?

» Doubt it would be that practicle» Revisit loops that are self-nearby later







STC folds

Existence Proofs


Recursive bisection

Linear-size






Geode

Hex-to-void

clea

n-up

Geode betweenshells


truc

tion

disc

over

y


THEX

? Hex dominant?



Slicing doughnuts



Prepare the Boundary Mesh

Recall WW, Shelling, and Geode » approaches to preparing the surface mesh to

admit a (good quality?) hex mesh



Algorithms - splicing

Mesh Editing and Splicing

Initial GridRemove self intersecting chord –

Leave boundary nodes fixed



Algorithms - splicing

Mesh Editing and Splicing

Intermediate mesh – needs new chord to reconcile boundary

New non self intersecting chord inserted to reconcile boundary nodes



Surface Mesh Modification - Splicing

Undesirable mesh interval

Remove unwanted quads Completed MeshMesh from W76 AFF

The procedure:



Fixing Surface Mesh for WW

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



Fixing Surface Meshes

In progress» Automation

How general of a method will it be?» What’s properties are really sufficient for

the surface mesh?







STC folds

Existence Proofs


Recursive bisection

Linear-size






Geode

Hex-to-void

clea

n-up

Geode betweenshells


truc

tion

disc

over

y


THEX

? Hex dominant?



Slicing doughnuts



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



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 -> sufficient might be doable



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







STC folds

Existence Proofs


Recursive bisection

Linear-size






Geode

Hex-to-void

clea

n-up

Geode betweenshells


truc

tion

disc

over

y


THEX

? Hex dominant?



Slicing doughnuts



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



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



Flipping Quads

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

no

no



Flipping Tetrahedra

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



Flipping Hexestop and bottom of 4D

cuboid



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?







STC folds

Existence Proofs


Recursive bisection

Linear-size






Geode

Hex-to-void

clea

n-up

Geode betweenshells


truc

tion

disc

over

y


THEX

? Hex dominant?



Slicing doughnuts



Local Reconnection and Smoothing

Knupp’s 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…




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




Flips, but dual interpretation» Might not be flips (non-parity preserving)

for non-ball volumes

Moving sheetsto untangle

Also sheet pushing ideas for local element size control




Connecting sheets

Inserting sheets




Most-untangled sheets not always the best quality Wasn’t able to improve specific small examples, by

any manual means» Just push problem around

Process didn’t 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?







STC folds

Existence Proofs


Recursive bisection

Linear-size






Geode

Hex-to-void

clea

n-up

Geode betweenshells


truc

tion

disc

over

y


THEX

? Hex dominant?



Slicing doughnuts



Sweep Relationship Map

Sweeping Node Placement Strategies

2D2.

5D2.

75D

Good Source SurfaceMesh Strategies

1-1 Sweeping / Projection / Rotation (lots)

M-1 Sweeping (several)

M-M Sweeping

Multi-axisCooper Tool Grafting

Cooper Tool

Inside-out3d



Sweeping Types

One-to-One SweepMany-to-Many

Multisweep & Coopering

Many-to-One Sweep



MultiSweep

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



MultiSweep

Image of target curvespartitions source surfaces

Image created using • matrix (affine transformation)• or weighted winslow smoothingone mesh layer at a time

The image is a mesh image depends on resolution, premesh



Multisweep – in progress



Cooper Tool

Like multisweep» Nice language of caps, barrels» Different projection methods

– All layers at once

» Intersection of projections different– …



Multi-Axis Coopering

Blacker and Miyoshi Break model into tree



Multi-Axis Coopering

“Cut” section to sweep in new direction» Cuts can intersect and overlap - tricky



Mesh grafting







STC folds

Existence Proofs


Recursive bisection

Linear-size






Geode

Hex-to-void

clea

n-up

Geode betweenshells


truc

tion

disc

over

y


THEX

? Hex dominant?



Slicing doughnuts



Refinement and Coarsening

Shepherd, Borden,…

Boundary Refinement

Mesh Coarsening

Sheet Refinement& Cleaving



Sheet Insertion

Modify mesh to capture geometry, through STC sheets



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.

Mesh Cutting & Sheet Insertion



Step 1: Capturing the first imprint.



Step 2: Capturing the other two imprints



Step 3: Capturing the Cutouts



Mesh Cutting

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



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

cubit mesh generation project technical history of hex mesh generation scott a. mitchell sandia...

Documents

hex meshing hard slide

technical talk

geometry slide

tetrahedra slide

algorithms technical

midpoint subdivision

d algorithms

little sweeping slide