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Optimaltime Dynamic Mesh Refinement

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Optimal-time Dynamic Mesh Refinement. Beno t Hudson, CMU. Joint work with Umut Acar, TTI-C ... [MPW02]: Miller, Pav, Walkington 'Fully incremental ...', 2002 ... – PowerPoint PPT presentation

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Title: Optimaltime Dynamic Mesh Refinement


1
Optimal-time Dynamic Mesh Refinement
  • Benoît Hudson, CMU
  • Joint work with Umut Acar, TTI-C
  • www.cs.cmu.edu/bhudson

2
credit SCEC
3
The mesh
credit CMU quake project
4
  • Fine elementsto resolvefine features
  • topography
  • surface effects
  • Coarse elementsfor coarse features
  • deep underground
  • hard rock

5
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

6
Mesh size near Riversidehow big?
7
Adaptive Mesh Refinement
  • Adapt the mesh during the simulation
  • Refine steep gradients
  • Coarsen flat gradients
  • Use dynamic mesh refinement!

8
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

9
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

10
Outline
  • Precise problem description
  • Static meshing using quadtrees
  • Dynamic
  • Applications
  • Generating tiny meshes in 2-d

11
Quadtree refinement
BEG90, MV92, BET93
Þ
12
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.
13
Quadtree Build Overlay
14
Quadtree Warp
15
Quadtree Triangulate
16
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

17
(No Transcript)
18
Optimal spacing No overly small elements
Define closest distance s Smallest cell size
s / 4 ) timesteps not too small.
19
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
  • ... ?

20
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

21
Runtime crowded cells
  • Crowded cellsblamed on p if
  • Cell contains p
  • Cell neighbours p

p
  • Max 9 neighbours of each size.

22
Runtime unbalanced cells
  • Unbalanced cellsblamed on p if
  • Cell unbalanced by cell blamed on p

p
How to count these?
23
Runtime unbalanced cells
  • Unbalanced cell blamed on p with size k ?
  • ) Distance 4 k

24
Packing Lemma
O(1) objects of size k fit in distance
O(k) Worst case 42
4 k
25
Runtime O(n lg L/s)
  • For each point p,
  • For each size i
  • 16 unbalanced splits
  • 9 crowded splits

26
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

27
Outline
  • Precise problem description
  • Static meshing using quadtrees
  • Dynamic meshing
  • Applications
  • Generating tiny meshes in 2-d

28
Dynamic Trace
main
29
Dynamic Trace
main
init
split
split
warp
split
30
Dynamic Trace
main
init
split
split
warp
split
warp
31
Dynamic Trace
main
init
split
split
warp
triang
split
warp
triang
32
Comparing traces
main
main
Trace stability How much red?
33
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

34
Outline
  • Precise problem description
  • Static meshing using quadtrees
  • Dynamic meshing
  • Applications
  • Generating tiny meshes in 2-d

35
(1) Dynamic Simulation
Partial Diff. Eqs.
Model
36
(1) Dynamic Simulation
Mesh
Partial Diff. Eqs.
Model
Visualize
Solve
37
(1) Dynamic Simulation
Mesh
Partial Diff. Eqs.
Model
Visualize
Solve
38
(1) Dynamic Simulation
  • New requirements
  • Dynamic matrix assembly
  • Dynamic linear solver
  • Dynamic visualizer
  • Dynamic AMR

Mesh
Partial Diff. Eqs.
Model
Visualize
Solve
39
(2) Dynamic with Features
  • Dynamic algorithmdoes not handle segments,
    polygons, ...
  • Dynamic SVR?
  • Coming soon

40
(3) Out of core refinement
  • Engineers want billions of elements
  • Doesnt fit in memory
  • Dynamic refinementallows partial meshing

41
Outline
  • Precise problem description
  • Static meshing using quadtrees
  • Dynamic meshing
  • Applications
  • Generating tiny meshes in 2-d

42
Delaunay Refinement
  • Quadtree criticism generates very (constant
    factor) large meshes.
  • Har-Peled, Üngör 05
  • Dont insert every point in the quadtree

43
Üngör off-centers
Any third point in circle forms a quality triangle
44
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
45
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

46
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

47
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

48
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
49
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

50
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
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