Mesh refinement: sequential, parallel, and dynamic

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Mesh refinementsequential, parallel,and dynamic

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

Papers available at http//www.cs.cmu.edu/bhudson
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Mesh refinementsequential, parallel,and dynamic

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

Papers available at http//www.cs.cmu.edu/bhudson
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Finite Element Simulation Overview
Partial Diff. Eqs.
Model
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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 description
2. 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
ConformingAll features appear (subdivided)
P courtesy of Shewchuk
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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
Size-optimality Output O(mopt) points
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Formal problem

• Input
• Points 2 Rd, Segments, Polygons,
• Quality bound angle ³ a
• Output
• Conforms All features appear
• Quality No angle smaller than a
• Size-optimal O(mopt) vertices

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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 description
2. Prior solutions
3. Sequential
4. Parallel
5. Dynamic
6. Many open problems

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Delaunay Triangulation
Maximizes minimumangle
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Delaunay Triangulation
Maximizes minimumangle
May not be good 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 criteria
polygons
segments
Miller04
Ruppert92
Feature handling
3d points
2d points
n2
n lg n
n polylog(n)
Runtime
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Ruppert 3D a bad example
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Ruppert 3D a bad example

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Ruppert 3D W(n2)
n/2 points along line
n/2 points around circle
Delaunay has n2/4 tets
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Evaluation criteria
polygons
Shewchuk98
segments
Mil04
Feature handling
3d points
2d points
n2
n lg n
n polylog(n)
Runtime
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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 criteria
She98
polygons
MV92
segments
Mil04
BEG90
Feature handling
3d points
2d points
n2
n lg n
n polylog(n)
Runtime
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Compare and contrast
86 triangles, 17
55 triangles, 30
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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 description
2. Prior solutions
3. Sequential
4. Parallel
5. Dynamic
6. Many open problems

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Fast sequential meshing

• Hudson, Miller, Phillips 2006Sparse Voronoi
  Refinement15th International Meshing Roundtable

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The intuition
Quadtrees fast runtime top-down. Intermediate
meshes are good quality.
SVR always good quality, find features quickly
Rupperts small size bottom-up,good feature
recovery.
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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

1. If a triangle is skinny,split it.
2. If a triangle containsinput, split it.

Split(t)
1. Draw circle
2. Shrink by k
3. Choose a point
4. Insert it, retriangulate
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Apply splitting rules

1. If a triangle is skinny,split it.
2. If a triangle containsinput, split it.

Split(t)
1. Draw circle
2. Shrink by k
3. Choose a point
4. Insert it, retriangulate
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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, refine top-down Like Ruppert, use
input points, 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
e0(p)
e1(p)
p
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Quality ) divide conquer
e0(p)
p
e2(p)
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Quality ) divide conquer
e0(p)
p
ei(p)
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Quality ) divide conquer
farthest(p)
?k
e0(p)
nearest(p)
p
ei(p)
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Quality ) divide conquer
farthest(p)
q
?2k
nearest(q)
p
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Packing Lemma
At most O(1) qibefore farthest(p)falls by half.
p
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Packing Lemma
At most O(1) qibefore farthest(p)falls by half.
farthest(p) falls by half at most log (L/s) times
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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 criteria
SVR (HMP06)
She98
polygons
MV92
segments
Mil04
BEG90
Feature handling
3d points
2d points
n2
n lg n
n polylog(n)
Runtime
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In practice point clouds
Stanford Bunny (34834 points)
Preparata (N103, 104)
N points around ring
N points on line
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In practice N1,000
N points along line
N points around circle
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In practice N10,000
(Calculate (104)2 108 tets in Delaunay, each
tet is 8 pointers Þ 3.2 GB)
SVR outputs 722K tets in 22s Pyramid thrashes,
c after 4 hours
N points along line
N points around circle
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In practice the bunny
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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 description
2. Prior solutions
3. Sequential
4. Parallel
5. Dynamic
6. Many open problems

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Parallel SVR

• Hudson, Miller, Phillips 2007Sparse Parallel
  Delaunay RefinementTo 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
L largest distance s smallest distance L/s º
spread Normally L/s Î poly(n) lg L/s lg n
Depth
Work
Dimension
Algorithm
SVR HMP07 Quadtree BET93 Ruppert
STU02 Ruppert STU02
any d 2d 2d 3d
O(lg (L/s) lg(m)) O(lg n) O(polylog(L/s)) O(polylo
g(L/s))
O(n lg L/s m) O(n lg n m) O(m
polylog(L/s)) W(n2)
n number of input points, segments, m number
of output points L/s spread of the input
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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 description
2. Prior solutions
3. Sequential
4. Parallel
5. Dynamic
6. Many open problems

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Dynamic meshing

• Acar, Hudson, 2007 (in submission)Dynamic mesh
  refinement using Quadtrees and Off-centers

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OBrien, Hodgins 1999
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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 pack around 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. Dynamic
6. 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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(1) Dynamic Simulation
Partial Diff. Eqs.
Model
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(1) Dynamic Simulation
Mesh
Partial Diff. Eqs.
Model
Visualize
Solve
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(1) Dynamic Simulation
Mesh
Partial Diff. Eqs.
Model
Visualize
Solve
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(1) Dynamic Simulation

• New requirements
• Dynamic matrix assembly
• Dynamic linear solver
• Dynamic visualizer
• Dynamic AMR

Mesh
Partial Diff. Eqs.
Model
Visualize
Solve
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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
• Doesnt fit in memory
• Dynamic refinementallows 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