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Mesh refinement:sequential, parallel,

and dynamic

Benoît Hudson, CMU

Joint work with Umut Acar, TTI-CGary Miller and Todd Phillips, CMU

Papers available athttp://www.cs.cmu.edu/~bhudson

2

Benoît Hudson, CMU

Joint work with Umut Acar, TTI-CGary Miller and Todd Phillips, CMU

Papers available athttp://www.cs.cmu.edu/~bhudson

Mesh refinement:sequential, parallel,

and dynamic

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4

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Visualize

Finite Element Simulation Overview

Model

Partial Diff. Eqs.Mesh

Solve

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Our results

• First optimal-time sequential mesher– Fast in implementation

• First provably fast parallel mesher

• First optimal-time dynamic mesher

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Outline

1. Precise problem description

2. Prior solutions

3. Sequential

4. Parallel

5. Dynamic

6. Many open problems

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Outline

1. Precise problem description2. Prior solutions

3. Sequential

4. Parallel

5. Dynamic

6. Many open problems

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Input

Points

Segments

Polygons

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Input

Points

Segments

Polygons

‘P’ courtesy of Shewchuk

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Output

‘P’ courtesy of Shewchuk

ConformingAll features

appear (subdivided)

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Big angles are bad

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No small angle ) no big angle

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Need Steiner Points

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Sizing

Big featuresBig triangles

No-small-angle )Medium size betweensmall, large features

Small featureSmall triangles

Size-optimalityOutput O(mopt) points

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Formal problem

• Input: – Points 2 Rd, Segments, Polygons, …– Quality bound: angle

• Output:– Conforms: All features appear – Quality: No angle smaller than – Size-optimal: O(mopt) vertices

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Outline

1. Precise problem description2. Prior solutions

3. Sequential

4. Parallel

5. Dynamic

6. Many open problems

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Outline

1. Precise problem description

2. Prior solutions3. Sequential

4. Parallel

5. Dynamic

6. Many open problems

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Delaunay Triangulation

Maximizes minimum

angle

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Delaunay Triangulation

Maximizes minimum

angle

May not begood enough

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Ruppert (1992)

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Ruppert: Identify skinny triangle

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Ruppert: Find circumcenter

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Ruppert: Snap to segment

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Ruppert: Insert, retriangulate

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Ruppert: Repeat until done

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Evaluation criteriaF

eatu

re h

andl

ing

Runtime

2d points

3d points

segments

polygons

n2 n lg nn polylog(n)

Miller04Ruppert92

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Ruppert 3D: a bad example

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Ruppert 3D: a bad example

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Ruppert 3D: (n2)

n/2 pointsaround circle

n/2 pointsalong line

Delaunay hasn2/4 tets

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Evaluation criteriaF

eatu

re h

andl

ing

Runtime

2d points

3d points

segments

polygons

n2 n lg nn polylog(n)

Mil04

Shewchuk98

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Quadtree: Bern et al (1990)

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Quadtree: Bern et al (1990)

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Quadtree: Bern et al (1990)

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Quadtree: Bern et al (1990)

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Evaluation criteriaF

eatu

re h

andl

ing

Runtime

2d points

3d points

segments

polygons

n2 n lg nn polylog(n)

Mil04

She98

BEG90

MV92

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Compare and contrast

55 triangles, 30 86 triangles, 17

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Outline

1. Precise problem description

2. Prior solutions3. Sequential

4. Parallel

5. Dynamic

6. Many open problems

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Outline

1. Precise problem description

2. Prior solutions

3. Sequential4. Parallel

5. Dynamic

6. Many open problems

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Fast sequential meshing

Hudson, Miller, Phillips 2006Sparse Voronoi Refinement

15th International Meshing Roundtable

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The intuitionQuadtree’s fast runtime: top-down.Intermediate meshes are good quality.

Ruppert’s small size: bottom-up,good feature recovery.

SVR: always good quality, find features quickly

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Sparse Delaunay Refinement

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Add a bounding box

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Triangulate just the box!

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Apply splitting rules

1. If a triangle is skinny,split it.

2. If a triangle containsinput, split it.

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Apply splitting rules

1. If a triangle is skinny,split it.

2. If a triangle containsinput, split it.

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Split

Split(t)

2. Shrink by k3. Choose a point4. Insert it, retriangulate

1. If a triangle is skinny,split it.

2. If a triangle containsinput, split it.

1. Draw circle

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Apply splitting rules

1. If a triangle is skinny,split it.

2. If a triangle containsinput, split it.

Split(t)

2. Shrink by k3. Choose a point4. Insert it, retriangulate

1. Draw circle

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Apply splitting rules

1. If a triangle is skinny,split it.

2. If a triangle containsinput, split it.

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Apply splitting rules

1. If a triangle is skinny,split it.

2. If a triangle containsinput, split it.

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Apply splitting rules

1. If a triangle is skinny,split it.

2. If a triangle containsinput, split it.

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Apply splitting rules

1. If a triangle is skinny,split it.

2. If a triangle containsinput, split it.

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General flavour

Like quadtree, refinetop-down

Like Ruppert, use inputpoints, circumcenters

Warp to input whenpossible.

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Runtime proof(1) Quality always “good”

– Never warp close tomesh vertex

– No angle smaller than ®’

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Runtime proof(1) Quality always good

(2) Quality ) bounded degree– 360 degrees– degree 360/®’

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Runtime proof(1) Quality always good

(2) Quality ) bounded degree

(3) Bounded degree ) O(1) operations– insertions– range queries

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Runtime proof(1) Quality always good

(2) Quality ) bounded degree

(3) Bounded degree ) O(1) operations

(4) Quality ) divide & conquer– !!!

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Quality ) divide & conquer

p

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Quality ) divide & conquer

p

e0(p)

e1(p)

e1(p)e0(p)

·

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Quality ) divide & conquer

p

e0(p)

e2(p)

e2(p)e0(p)

· 2

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Quality ) divide & conquer

p

e0(p)

ei(p)

ei (p)e0(p)

· i

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Quality ) divide & conquer

p

e0(p)

ei(p)

farthest(p)nearest(p)

· k

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Quality ) divide & conquer

p

farthest(p)nearest(q)

· 2kq

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Packing Lemma

p

At most O(1) qi

before farthest(p)falls by half.

q2

q1q3

q5

q4

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s

Packing Lemma

At most O(1) qi

before farthest(p)falls by half.

farthest(p) falls by half at mostlog (L/s) times

L

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Runtime proof(1) Quality always good(2) Quality ) bounded degree(3) Bounded degree )

O(1) operations(4) Quality ) divide & conquer

O(m + n log L/s)

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Evaluation criteriaF

eatu

re h

andl

ing

Runtime

2d points

3d points

segments

polygons

n2 n lg nn polylog(n)

Mil04

She98

BEG90

MV92

SVR(HMP06)

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In practice: point clouds

N points around ring

N points on line

Preparata (N=103, 104) Stanford Bunny (34834 points)

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In practice: N=1,000

N pointsaround circle

N pointsalong line

AlgorithmSVR

Pyramid

Max #tets70K

1.014M

Max degree70

1,010

Output #tets70K87K

Runtime2s21s

1 GHz Pentium III1 GB RAM

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(Calculate: (104)2 = 108 tets in Delaunay,each tet is 8 pointers ~ 3.2 GB)

In practice: N=10,000

N pointsaround circle

N pointsalong line

SVR: outputs 722K tets in 22sPyramid: thrashes, ^c after 4 hours

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In practice: the bunnyAlgorithm

SVRPyramid

Output #tets430K383K

Runtime12s25s

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Outline

1. Precise problem description

2. Prior solutions

3. Sequential4. Parallel

5. Dynamic

6. Many open problems

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Outline

1. Precise problem description

2. Prior solutions

3. Sequential

4. Parallel5. Dynamic

6. Many open problems

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Parallel SVR

Hudson, Miller, Phillips 2007Sparse Parallel Delaunay Refinement

To appear, SPAA.

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Why parallel?

• My laptop: two cores today– Paterson: expect 100 cores in 10 years

• National Labs: 41,000 cores last week

• Understanding the dependencies:– Helps with constant factors in sequential case– Helps with out-of-core, distributed,

compression, dynamic case, …

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Parallel Algorithms Comparison

SVR [HMP07]Quadtree [BET93]Ruppert [STU02]Ruppert [STU02]

any d2d2d3d

O(lg (L/s) lg(m))O(lg n)O(polylog(L/s))O(polylog(L/s))

n: number of input points, segments, …m: number of output pointsL/s: spread of the input

O(n lg L/s + m)O(n lg n + m)O(m polylog(L/s))(n2)

Depth WorkDimensionAlgorithm

L: largest distances: smallest distance

L/s spreadNormally: L/s poly(n)

lg L/s lg n

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Outline

1. Precise problem description

2. Prior solutions

3. Sequential

4. Parallel5. Dynamic

6. Many open problems

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Outline

1. Precise problem description

2. Prior solutions

3. Sequential

4. Parallel

5. Dynamic6. Many open problems

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Dynamic meshing

Acar, Hudson, 2007 (in submission)Dynamic mesh refinement using

Quadtrees and Off-centers

81O’Brien, Hodgins 1999

QuickTime™ and a decompressor

are needed to see this picture.

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Stability

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Stability

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Stability

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Stability

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Stability

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Stability bound: O(log L/s)

Proof in paper:Newly split cells packaround new point

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Self-adjusting computation: Acar et al, 2006

• Update speed = O(# stability)

• Insert / delete are exactly symmetric

• Stability is O(log L/s)

) Quadtree update is O(log L/s)• Implementation in SML

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Outline

1. Precise problem description

2. Prior solutions

3. Sequential

4. Parallel

5. Dynamic6. Many open problems

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Outline

1. Precise problem description

2. Prior solutions

3. Sequential

4. Parallel

5. Dynamic

6. Many open problems

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Visualize

(1) Dynamic Simulation

Model

Partial Diff. Eqs.Mesh

Solve

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Visualize

(1) Dynamic Simulation

Model

Partial Diff. Eqs.Mesh

Solve

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Visualize

(1) Dynamic Simulation

Model

Partial Diff. Eqs.Mesh

Solve

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Visualize

(1) Dynamic Simulation

Model

Partial Diff. Eqs.Mesh

Solve

New requirements:(1)Dynamic matrix assembly(2)Dynamic linear solver(3)Dynamic visualizer(4)Dynamic AMR

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(2) Dynamic with Features

• Dynamic algorithmdoes not handle segments, polygons, ...

• Dynamic SVR?– Likely coming soon

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(3) Out of core refinement

• Engineers want billions of elements

• Doesn’t fit in memory• Dynamic refinement

allows partial meshing

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(4) Distributed refinement

• Engineers want billions of elements

• Engineers have supercomputers

• Need to address:– Mesh partitioning

– Load balancing

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(5) Surface reconstruction

Traditional approach:• Compute Delaunay• Skinny tet ) surface

Refinement:• Compute Delaunay• Skinny tet ) refine

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Summary

• Everyone needs a mesher

• Sequential: First optimal-time mesher– First sub-quadratic in 3d!– Implementation in progress

• Parallel: First optimal-work, shallow-depth

• Dynamic: First optimal-time mesher

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Bibliography

[Che89]: Chew “Guaranteed quality triangular meshes”, 1989[BEG90]: Bern, Eppstein, Gilbert “Provably good mesh generation”, 1994[MV92]: Mitchell, Vavasis “Quality mesh generation …”, 2000[Rup92]: Ruppert “A Delaunay refinement algorithm for …”, 1995[BET93]: Bern, Eppstein, Teng “Parallel construction …”, 1999[She97]: Shewchuk “Delaunay refinement mesh generation”, 1997[MPW02]: Miller, Pav, Walkington “Fully incremental …”, 2002[STU02]: Spielman, Teng, Ungor “Parallel Delaunay …”, 2002[Mil04]: Miller, “A time-efficient Delaunay Refinement …”, 2004[HPU05]: Har-Peled, Ungor, “A time-optimal Delaunay …”, 2005[HMP06]: Hudson, Miller, Phillips, “Sparse Voronoi Refinement”, 2006[HMP07]: ~, “Sparse Parallel Delaunay Refinement”, 2007[MPS07]: Miller, Phillips, Sheehy, “Size competitive …”, 2007[HA07]: Acar, Hudson, “Dynamic quad-tree mesh refinement ...”, submitted