Simplification and Improvement of Tetrahedral Models for Simulation

Barbara Cutler, MITJulie Dorsey, Yale

Leonard McMillan, UNC - Chapel Hill

Motivation

• Physical simulations are now in widespread use in computer graphics

• Interactive deformation & fracture simulations

• However, preparing appropriate models is challenging

Motivation

300 bone tetras

850 skin tetras

(head only)

1,200 bone tetras

800 skin tetras

[Cutler et al. 02] & [Müller et al. 02]

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

Overview

• Previous Meshing Research– Mesh Generation– Mesh Simplification– Mesh Improvement– Mesh Refinement

• Goals and Requirements• Algorithm• Results• Conclusions & Future Work

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]

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]

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]

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]

Overview

• Previous Meshing Research• Goals and Requirements

– Element Quality Metric

• Algorithm• Results• Conclusions & Future Work

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

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

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 volumetarget tetra

count

Overview

• Previous Meshing Research• Goals and Requirements• Algorithm

– Local mesh transformations– Block iteration

• Results• Conclusions & Future Work

Local Mesh Transformations

• Tetrahedral Swaps• Edge Collapse• Vertex Smoothing• Vertex Addition

Local Mesh Transformations

• Tetrahedral Swaps– Choose the

configuration with the best local element shape

• Edge Collapse• Vertex Smoothing• Vertex Addition

Local Mesh Transformations

• Tetrahedral Swaps• Edge Collapse

– Delete a vertex & the elements around the edge

• Vertex Smoothing• Vertex Addition

Before

After

Prioritizing Edge Collapses

• Preserve topology– Thin layers should not

pinch together

• Collapse weight– Edge length +

boundary error

• No negative volumes• Local element quality

does not significantly worsen

Interior: ok to collapse

Boundary:

check error

Spanning: never collapse

Boundary-Touching:one-way collapse

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

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

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

Block Iteration Algorithm

while (tetra count > target tetra count)

T = a subset of all elementsrandomly reorder Tforeach t T, try:

• tetrahedral swaps• edge collapse• move vertex• add vertex

Look for an action that improves or removes this element

Block Iteration Algorithm

E = ideal edge length while (tetra count > target tetra count)

T = a subset of all elementsrandomly reorder Tforeach t T, try:

• tetrahedral swaps• edge collapse• move vertex• add vertex

E *=

tetra count

target tetra count√3

As ∆E → 0, the Block Iteration Algorithm is equivalent to a Progressive Mesh

E is the allowable boundary error

Block Iteration Algorithm

E = ideal edge lengthpercent = 10%while (tetra count > target tetra count)

T = the poorest percent of all elementsrandomly reorder Tforeach t T, try:

• tetrahedral swaps• edge collapse• move vertex• add vertex

E *=

percent += 10%

tetra count

target tetra count√3

The Block Iteration Algorithm is a partial order

Not all of the edge weights must be recomputed before the next transformation

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

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 3.5Edge collapse weight re-

computation dominates the running time (~80%)

Overview

• Previous Meshing Research• Goals and Requirements• Algorithm• Results

– Meshes, Performance, Quality– Comparison to Previous Work

• Conclusions & Future Work

Results: Handoriginal

(100K faces)

100K tetras(57K faces) 30K tetras

(19K faces)

10K tetras(7K faces)

Results: Dragonoriginal

(100K faces)

5K tetras(3K faces)

100K tetras(48K faces)

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)

Performance

Block Iteration(all

transformations)

mm:ss

Block Iteration(edge collapse

only)

mm:ss

Progressive Mesh

(edge collapse only)

mm:ss 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%)

Visualization of Element Quality

1,050K tetras(133K faces)

zero-angle &

zero-volume

good angle, but small-

volume

near-equilateral

& ideal-volume

Visualization of Element Quality

Octree or Adaptive Distance Field (ADF)

461K tetras(108K faces)

Visualization of Element Quality

After Simplification& Mesh

Improvement10K tetras(3K faces)

Variety of Transformations

More likely to contain poor quality elements or require

large boundary error, E

All Transformations Edge Collapse Only

Overview

• Previous Meshing Research• Goals and Requirements• Algorithm• Results• Conclusions & Future Work

Conclusions

• Element quality metric• Robust & efficient implementation• Ensures removal of poorest quality

elements to meet FEM requirements• Encourages iterative modeling

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

Thank You

• Matthias Müller, Rob Jagnow, Justin Legakis, Derek Bruening, Frédo Durand

• MIT Computer Graphics Group