New results on mesh refinement

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New results onmesh refinement

• Benoît Hudson, CMU
• Joint work with Gary Miller and Todd Phillips

Papers available at http//www.cs.cmu.edu/bhudson
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CAD model
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Will it work?
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Equations
Navier-Stokes
Heat transfer
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Run simulations!
Partial Diff. Eqs.
CAD model
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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
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Simpler Input
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Two possible meshes
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Which is better?
Example f(x,y) 1-x2
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This is a bad mesh
f(x,y) 1-x2

• At mesh vertices
• read the value
• At other points p
• Find t that contains p
• Weighted average of points of t

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This is a better mesh
0
0
1
0
0
1
0
0
1
0
0
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Good aspect ratio Þ good interpolation
R
Good aspect ratio if
r
Equivalent all angles ³ a
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Aspect ratio
Bad aspect ratio if
R
r
Equivalent some angle lt a
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This is a better mesh
0
0
1
This mesh is Delaunay best possible for those
points
0
0
1
0
0
1
0
0
17
This is a good mesh
Add Steiner points
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This is a good mesh
0
0
0
0
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Formal problem

• Input
• Piecewise Linear Complex (PLC)
• points, segments, polygons,
• quality bound angle ³ a
• Output
• quality triangulation (angle ³ a)
• all features appear (perhaps subdivided)
• O(mopt) vertices

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Grading
Good aspect ratio Þ good grading
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Outline

• Two older methods
• quadtree refinement
• Delaunay refinement
• Our algorithm SVR as a hybrid
• Parallel SVR

Dynamic meshing too much for this talk. See me
afterwards.
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Older Solutions

• 1950-now construct mesh by hand!
• Takes days to months.
• Automatic methods ca. 1980-now
• no quality, size, or time bounds
• BEG90 quality and size in O(n lg n m)
• (buggy proof, fixed in BET93)
• Rup92 smaller in practice, no time bound.

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Simpler input
Just points for now
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Quadtree refinement
BEG90, MV92, BET93
Þ
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Quadtree rules

• Unbalanced
• Neighbor is small

• Crowded
• two points in cell, or
• one point in cell, one in neighbour

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Quadtree
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Quadtree
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Quadtree
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Quadtree in a word

• Quality mesh (17)
• O(mopt) output size
• O(n lg n m) runtime
• General approach top-down

(think binary search tree)
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Delaunay Refinement
Rup92, She97
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Add a bounding box
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Triangulate (use Delaunay)
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Identify skinny triangle
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Find circumcenter
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Insert, retriangulate
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Find more skinny triangles
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Find more skinny triangles
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Find more skinny triangles
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Find more skinny triangles
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Find more skinny triangles
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Find more skinny triangles
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Rupperts algorithm in a word

• Quality mesh (any a 20.7)
• O(mopt) output size
• O(n2 m) runtime worst case
• works fast in practice
• General approach bottom-up

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Compare and contrast
69 triangles, 17
35 triangles, 30
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Rupperts algorithm in a word

• Quality mesh
• O(mopt) output size
• smaller than quadtree in practice
• O(n2 m) runtime worst case
• works fast in practice
• General approach bottom-up

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Restrictions

• Rupperts algorithm handles segments only at 60
  angles to each other.
• 3d extensions She97, MPW02

segment segment face
segment face face
60 70 90
Smaller angles may cause infinite loops
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Meshing well, quickly

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

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The Goal
Rupperts small meshes
Quadtrees fast runtime
In dimension 3!
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The intuition
Quadtrees fast time slowly zero in on
small features, keep mesh good quality always.
Large size warp to input too late.
SVR always good quality, resolve when necessary
Rupperts small size immediately resolve
all features. Rupperts bad timing allow
horrid mesh quality.
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SVR
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Add a bounding box
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Triangulate just the box!
Triangles keep track of points
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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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General flavour
Like quadtree, refine top-down Like Ruppert, use
input points and circumcenters
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SVR in a word
L largest distance s smallest distance L/s º
spread Normally L/s Î poly(n) lg L/s lg n

• Quality mesh
• O(mopt) output size
• O(n lg L/s m) runtime worst case
• General approach hybrid

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Overall Comparison
SVR HMP06 Quadtree BEG90 Ruppert
Rup92 Octtree MV92 Shewchuk She97
any d 2d 2d 3d 3d
O(n lg L/s m) O(n lg n m) O(mn) O(mn) W(n2)
, O(m2n)
hybrid top-down bottom-up top-down bottom-up
n number of input points, segments, m number
of output points L/s spread of the input lg n
assumes L/s Î poly(n)
Miller04 O((n lg G m) lg m)
HPU05O(n lg L/s m) on point sets
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Overall Comparison
O(n lg n m) on point sets
SVR HMP06 Quadtree BEG90 Ruppert
Rup92 Octtree MV92 Shewchuk She97
any d 2d 2d 3d 3d
O(n lg L/s m) O(n lg n m) O(mn) O(mn) W(n2)
, O(m2n)
hybrid top-down bottom-up top-down bottom-up
W(n2) even on point sets
n number of input points, segments, m number
of output points L/s spread of the input lg n
assumes L/s Î poly(n)
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In practice
n/2 points along line
n/2 points around circle
Delaunay has n2/4 tets
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In practice n2000
Algorithm SVR Shewchuk
Max tets 81K 1.014M
Max degree 39 1,000
Output tets 81K 87K
n/2 points along line
n/2 points around circle
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In practice n20000
(Calculate Pyramid needs 108 tets, each tet is 8
pointers Þ 3.2 GB)
SVR outputs 774K tets in 8 minutes Shewchuk
thrashes, c after 4 hours
n/2 points along line
n/2 points around circle
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Parallel SVR

• Hudson, Miller, Phillips 2007Sparse Parallel
  Delaunay RefinementTo appear, SPAA.

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Why parallel?

• Low-end desktops have dual-core CPUs.
• Understanding the dependencies
• Helps with constant factors in sequential case
• Helps with out-of-core, distributed, compression,
  dynamic case,

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Parallel Comparison
Depth
Work
SVR HMP07 Quadtree BET93 Ruppert STU02
any d 2d 2d, 3d
O(lg (L/s) lg(m)) O(lg n) O(polylog(L/s))
O(n lg L/s m) O(n lg n m) O( ??? )
n number of input points, segments, m number
of output points L/s spread of the input lg n
assumes L/s Î poly(n)
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A sample work set
What can we do in parallel?
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Conflict
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Block
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Moot
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Independent events
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The algorithm parallel

• Build a bounding box
• While workset not empty
• Compute conflicts, blocking, mooting
• Defer blocked moves to later
• Colour the conflict graph in colours 1..d
• for 1..d
• perform all the moves in parallel
• remove mooted moves
• collect new moves for next round

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Why does it work?

• No move is blocked more than O(1) rounds
• either it happens
• or it gets mooted
• Proof
• If a break move blocks, there is a clean move
  nearby.
• Clean moves dont block.
• By packing, only O(1) clean moves can be nearby.
• QED.

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Summary

• No move is blocked long
• Break moves are discovered quickly
• Thus the time bound of O(lg L/s) rounds
• And O(lg m) per round overhead
• In practice we can use any number of
  shared-memory processors

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Conclusions SVR

• O(n lg (L/s) m) time
• Matches best 2d bound
• First sub-quadratic 3d time bound
• First 4d mesh refinement algorithm
• Practical
• Parallelizes!

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Open problems
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(1) Constrained Delaunay Refinement

• SVR meshes tiny gap.
• Gap is exterior how can we ignore it?
• Traditional method Constrained Delaunay
  Triangulation.
• Hope CDT/SVR

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(2) Slivers

• Points almost coplanar
• Good radius/edge ratio
• Bad solid anglesÞ numerically bad
• Best proofs guarantee 0.001
• In practice get 10

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(3) Small(ish) angles

• SVR needs ³ 90 angles between faces
• 80? 60? 30?

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(4) Small angles

• What if angle is 1?
• Whats a legal output?
• MPS07 guarantee no large angles in 2d
• Extends to 3d?

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(5) Curves

• CAD models use curves, curved surfaces
• No good theoretical bounds on refining curved
  surfaces

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(6) Laundry list

• Dynamic meshing
• (my thesis topic)
• Moving meshes
• Streaming
• Distributed-memory
• CAD cleaning

• Handling large spread
• Non-Euclidian metrics
• Better constant bounds

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Bibliography

• 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 , submitted
• HA07 Hudson, Acar, Dynamic quad-tree mesh
  refinement, submitted