Optimal-time Dynamic Mesh Refinement

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Optimal-time Dynamic Mesh Refinement

• Benoît Hudson, CMU
• Joint work with Umut Acar, TTI-C
• www.cs.cmu.edu/bhudson

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credit SCEC
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The mesh
credit CMU quake project
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• Fine elementsto resolvefine features
• topography
• surface effects
• Coarse elementsfor coarse features
• deep underground
• hard rock

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Formal problem Meshing
Drives Simulation Accuracy
Drives Simulation Runtime

• Input
• Point set 2 Rd
• Fixed quality bound angle ³ a
• Output Triangulation that is
• Conforming All input points appear
• Quality No angle smaller than a
• Optimal size Not too many elements

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Mesh size near Riversidehow big?
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Adaptive Mesh Refinement

• Adapt the mesh during the simulation
• Refine steep gradients
• Coarsen flat gradients
• Use dynamic mesh refinement!

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Formal problem Meshing

• Input
• Point set 2 Rd
• Fixed quality bound angle ³ a
• Output Triangulation that is
• Conforming All input points appear
• Quality No angle smaller than a
• Optimal size Not too many elements

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Formal problem Dynamic Meshing
AdditionsDeletions

• Input
• Dynamic point set 2 Rd
• Fixed quality bound angle ³ a
• Maintain Triangulation that is
• Conforming All input points appear
• Quality No angle smaller than a
• Optimal size Not too many elements

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Outline

1. Precise problem description
2. Static meshing using quadtrees
3. Dynamic
4. Applications
5. Generating tiny meshes in 2-d

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Quadtree refinement
BEG90, MV92, BET93
Þ
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Quadtree BEG90 rules

• Unbalanced
• Neighbor is small

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

My rule Order the work largest first.
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Quadtree Build Overlay
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Quadtree Warp
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Quadtree Triangulate
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Guarantees

• Conforming All input points appear
• Quality No angle smaller than a
• Optimal size Not too many elements
• Fast O(n lg L/s) time
• in any fixed dimension d

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(No Transcript)
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Optimal spacing No overly small elements
Define closest distance s Smallest cell size
s / 4 ) timesteps not too small.
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Runtime O(n lg L/s)

• Want to show O(n lg L/s) splits performed.
• Namely, O(lg L/s) splits per input point
• ... ?

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Runtime O(n lg L/s)

• Want to show O(n lg L/s) splits performed.
• Namely, O(lg L/s) splits per input point
• Account separately for splits due to
• Crowding
• Balance

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Runtime crowded cells

• Crowded cellsblamed on p if
• Cell contains p
• Cell neighbours p

p

• Max 9 neighbours of each size.

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Runtime unbalanced cells

• Unbalanced cellsblamed on p if
• Cell unbalanced by cell blamed on p

p
How to count these?
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Runtime unbalanced cells

• Unbalanced cell blamed on p with size k ?
• ) Distance 4 k

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Packing Lemma
O(1) objects of size k fit in distance
O(k) Worst case 42
4 k
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Runtime O(n lg L/s)

• For each point p,
• For each size i
• 16 unbalanced splits
• 9 crowded splits

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Guarantees BEG90

• Conforming All input points appear
• Quality No angle smaller than a
• Optimal size Not too many elements
• Fast O(n lg L/s) time
• in any fixed dimension d

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Outline

1. Precise problem description
2. Static meshing using quadtrees
3. Dynamic meshing
4. Applications
5. Generating tiny meshes in 2-d

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Dynamic Trace
main
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Dynamic Trace
main
init
split
split
warp
split
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Dynamic Trace
main
init
split
split
warp
split
warp
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Dynamic Trace
main
init
split
split
warp
triang
split
warp
triang
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Comparing traces
main
main
Trace stability How much red?
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Self-adjusting computation Acar et al, 2006

• Update speed O( stability)
• ) Quadtree update is O(log L/s)
• History-independent
• Retain guarantees from static algorithm
• Insert / delete are exactly symmetric
• Implementation in SML

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Outline

1. Precise problem description
2. Static meshing using quadtrees
3. Dynamic meshing
4. Applications
5. Generating tiny meshes in 2-d

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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?
• 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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Outline

1. Precise problem description
2. Static meshing using quadtrees
3. Dynamic meshing
4. Applications
5. Generating tiny meshes in 2-d

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Delaunay Refinement

• Quadtree criticism generates very (constant
  factor) large meshes.

• Har-Peled, Üngör 05
• Dont insert every point in the quadtree

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Üngör off-centers
Any third point in circle forms a quality triangle
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Har-Peled, Üngör 2005

• Goal O(n lg L/s) runtime
• Question how to do point location?

Refinement algorithm For all pairs, If 9 p 2
circle Choose p Otherwise Create p at
off-center
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Har-Peled, Üngör 2005

• Use quad-tree overlay
• For all pairs (a, b) with a in smallest cell, b
  in neighbouring cell
• If 9 p 2 circle
• Choose p
• Otherwise
• Create p at off-center

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Har-Peled, Üngör 2005

• Use quad-tree overlay
• For all pairs (a, b) with a in smallest cell, b
  in neighbouring cell
• If 9 p 2 circle
• Choose p
• Otherwise
• Create p at off-center

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Har-Peled, Üngör 2005

• Use quad-tree overlay
• For all pairs (a, b) with a in 2nd - smallest
  cell, b in neighbouring cell
• If 9 p 2 circle
• Choose p
• Otherwise
• Create p at off-center

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Dynamic stability of HPÜ

• Use quad-tree overlay
• O(lg L/s)-stable
• Off-center blames (a,b)
• Blame is transitive
• Same analysis as balance
• O(lg L/s)-stable

HPÜ is O(lg L/s)-stable
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Conclusions

• Dynamic Meshing in O(lg L/s) per update
• Simple algorithm, optimal time
• Implementation near-trivial
• Output guarantees same mesh as static
• In 2-d maintain mesh as small as known
• To-do
• features
• dynamize the rest of the world

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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