1. Simplification and Improvement of Tetrahedral Models for Simulation
Barbara Cutler, MIT
Julie Dorsey, Yale
Leonard McMillan, UNC - Chapel Hill

2. Motivation
Physical simulations are now in widespread use in computer graphics
Interactive deformation & fracture simulations
However, preparing appropriate models is challenging

3. [Cutler et al. 02] & [Müller et al. 02]
Motivation
1,200 bone tetras
800 skin tetras
300 bone tetras
850 skin tetras
(head only)
[Cutler et al. 02] & [Müller et al. 02]

4. Contributions
Simplification and shape improvement to meet interactive simulation requirements
Element quality metric
Models from high-resolution scanned meshes with interior boundaries
Robust & efficient implementation

5. Overview Previous Meshing Research Goals and Requirements Algorithm
Mesh Generation
Mesh Simplification
Mesh Improvement
Mesh Refinement
Goals and Requirements
Algorithm
Results
Conclusions & Future Work

6. Mesh Generation Given some boundary, fill the interior with elements
Advancing Front / Advancing Layers [Lohner 88, Pirzadeh 96]
Delaunay Triangulation [Baker 89, Shewchuk 97, Cavalcanti & Mello 99, Persson & Strang 04]
Structured/Octree Tetrahedralization [Yerry & Shepard 84, Nielson & Sung 97]

7. Mesh Simplification Reduce the overall number of elements
Progressive Mesh - edge collapses only
2D [Hoppe 96]
3D [Staadt 98, Cignoni et al. 00, Chiang & Lu 03, Natarajan & Edelsbrunner 04]
Complex transformations – more difficult to implement, allows topology change
2D [Shroeder et al. 92, Turk 92]
3D [Trotts et al. 98, Chopra & Meyer 02]

8. Mesh Improvement Improve the quality/shape of elements in the mesh
Local transformations to improve shape [Frey & Field 91, Hoppe et al. 93, Joe 95]
Sliver removal from Delaunay Triangulations [Cheng et al. 99, Edelsbrunner & Guoy 02]
Combination of transformations more effective than a single type [Freitag & Ollivier-Gooch 97]

9. Mesh Refinement Increase the local resolution of the mesh
(while maintaining element quality)
Regular Subdivision [Bank et al. 83, Bey 95, Edelsbrunner & Grayson 00]
Edge Bisection [Alder 83, Rivara & Levin 92, Liu & Joe 95, Maubach 95, Arnold et al. 00]

10. Overview Previous Meshing Research Goals and Requirements Algorithm
Element Quality Metric
Algorithm
Results
Conclusions & Future Work

11. Goals and Requirements
Reduce the overall number of elements
Maintain the material boundaries
Improve the shape of each element
Reasonable distribution of elements
Robustness
Scalability

12. Why is Element Shape Important?
Very small dihedral angles → the stiffness matrix is constrained [Babuska & Aziz 76]
Very large dihedral angles → errors in FEM increase [Krizek 92]
All elements must meet minimum shape requirements

13. Element Quality Metric
Geometric mean of 3 components:
Shape
minimum solid angle (equilateral ≈ 0.55 steradians)
Volume
ideal volume =
Edge Length
ideal edge length = 3√ ideal volume
total volume target tetra count

14. Overview Previous Meshing Research Goals and Requirements Algorithm
Local mesh transformations
Block iteration
Results
Conclusions & Future Work

15. Local Mesh Transformations
Tetrahedral Swaps
Edge Collapse
Vertex Smoothing
Vertex Addition

16. Local Mesh Transformations
Tetrahedral Swaps
Choose the configuration with the best local element shape
Edge Collapse
Vertex Smoothing
Vertex Addition

17. Local Mesh Transformations
Tetrahedral Swaps
Edge Collapse
Delete a vertex & the elements around the edge
Vertex Smoothing
Vertex Addition
Before
After

18. Prioritizing Edge Collapses
Spanning: never collapse
Boundary: check error
Preserve topology
Thin layers should not pinch together
Collapse weight
Edge length + boundary error
No negative volumes
Local element quality does not significantly worsen
Boundary-Touching: one-way collapse
Interior: ok to collapse

19. Local Mesh Transformations
Tetrahedral Swaps
Edge Collapse
Vertex Smoothing
Move a vertex to the centroid of its neighbors
Convex or concave, but avoid negative-volume elements
Vertex Addition
Before
After

20. Local Mesh Transformations
Tetrahedral Swaps
Edge Collapse
Vertex Smoothing
Vertex Addition
At the center of a tetra, face, or edge
Useful when mesh is simplified, but needs further element shape improvement

21. Ensuring Consistency Prevent mesh degeneracies
Examine the neighbors sharing each face, edge and vertex
(see paper for list)
Implementation must be tolerant of negative- and zero-volume elements
May be present in input models or at intermediate stages of deformation

22. Block Iteration Algorithm
while (tetra count > target tetra count)
T = a subset of all elements
randomly reorder T
foreach t T, try:
tetrahedral swaps
edge collapse
move vertex
add vertex
Look for an action that improves or removes this element

23. Block Iteration Algorithm
E is the allowable boundary error
E = ideal edge length
while (tetra count > target tetra count)
T = a subset of all elements
randomly reorder T
foreach t T, try:
tetrahedral swaps
edge collapse
move vertex
add vertex
E *=
As ∆E → 0, the Block Iteration Algorithm is equivalent to a Progressive Mesh √
tetra count
target tetra count 3

24. Block Iteration Algorithm
E = ideal edge length
percent = 10%
while (tetra count > target tetra count)
T = the poorest percent of all elements
randomly reorder T
foreach t T, try:
tetrahedral swaps
edge collapse
move vertex
add vertex
E *=
percent += 10%
The Block Iteration Algorithm is a partial order Not all of the edge weights must be recomputed before the next transformation √
tetra count
target tetra count 3

25. Computing Edge Collapse Weight
Expensive to determine legality of collapse, especially in 3D
On average 100 edge weights are invalidated when an edge is collapsed
Progressive Mesh maintains a priority queue of all collapse weights (total order)
Before
After

26. Edge Collapse Weight Recomputation
Average number of edge weight
re-computations before an edge is collapsed
Block Iteration
Progressive Mesh
ratio
461K → 2K
28.2
240.5
8.5
461K → 10K
40.3
240.2
6.0
461K → 50K
67.7
238.9
3.5
Edge collapse weight re-computation dominates the running time (~80%)

28. Overview Previous Meshing Research Goals and Requirements Algorithm
Results
Meshes, Performance, Quality
Comparison to Previous Work
Conclusions & Future Work

29. Results: Hand original (100K faces) 10K tetras (7K faces) 100K tetras

30. Results: Dragon original (100K faces) 100K tetras (48K faces)

31. Performance
Block Iteration
(all transformations)
mm:ss
461K → 2K
9:20
461K → 10K
12:12
461K → 50K
27:45
Extreme simplification is faster because E, the allowable error, is larger
(optimizing over fewer elements)

32. Performance
Block Iteration
(all transformations)
mm:ss
(edge collapse only)
Progressive Mesh
ratio
461K → 2K
9:20
6:38
1:02:08
9.4
461K → 10K
12:12
7:25
1:01:35
8.3
461K → 50K
27:45
13:24
57:15
4.3
Edge collapse weight re-computation dominates the running time (~80%)

33. Visualization of Element Quality
good angle, but small-volume
1,050K tetras (133K faces)
zero-angle & zero-volume
near-equilateral & ideal-volume

34. Visualization of Element Quality
Octree or Adaptive Distance Field (ADF)
461K tetras (108K faces)

35. Visualization of Element Quality
After Simplification & Mesh Improvement
10K tetras (3K faces)

36. Variety of Transformations
All Transformations
Edge Collapse Only
More likely to contain poor quality elements or require large boundary error, E

37. Overview Previous Meshing Research Goals and Requirements Algorithm
Results
Conclusions & Future Work

38. Conclusions Element quality metric Robust & efficient implementation
Ensures removal of poorest quality elements to meet FEM requirements
Encourages iterative modeling

39. Future Work
Switch to a total-order mesh optimization (e.g. Progressive Mesh) at end to improve performance
Order of transformations attempted
Multiple transformation look-ahead
Topological simplification
Online local re-meshing & refinement

40. Thank You
Matthias Müller, Rob Jagnow, Justin Legakis, Derek Bruening, Frédo Durand
MIT Computer Graphics Group