View
1
Download
0
Category
Preview:
Citation preview
1
Making Meshesand Putting Them To Good Use
Making Meshesand Putting Them To Good Use
Mathieu DesbrunCaltech
CS176
31
So What Is Wrong in 3D?
Regular tets do NOT tile space perfectly shaped tets everywhere? dream on DT of well-spaced points can have (lotta) slivers
assuming "well-spaced" is well defined…
32
Scrutinizing the Beast
Four (well-spaced) vertices near the equatorial plane of their circumsphere
33
Taming the Beast?
Recipes to get rid of slivers local jiggling of faulty vertices
through optimization of shape’s “quality” for example, penalize small dihedral angles
no guarantee of success—in fact, may become worse
locally altering the Delaunay triangulation called “sliver exudation” no longer Delaunay, obviously
but weighted Delaunay—local tweak of the metric» will be discussed later
guarantees, but no practical bounds let’s go one dimension higher
34
Lifting Transformation (nD)
Lift points (x, y) to paraboloid (x, y, x2 + y2) lift points up; compute convex hull what is the induced connectivity?
Delaunay triang. = convex hull of 3D points
35
Lifting Transformation (nD)
Lift points (x, y) to paraboloid (x, y, x2 + y2) what about Voronoi diagrams?
Voronoi diagram = proj. of upper envelope
Before…
2
36
Pushing This Analogy Further
Idea: move vertices to perfect the approximation of the paraboloid roots in function approximation
Hessian(x2/2) = Id studied quite a bit in 2D Lloyd algo:
for given points,VD gives optimal L1 approx
for given conn., find optimal vertex positions always move a vertex to the centroid of its Voronoi region Live Demo
volume btw approximant and paraboloid
37
Lloyd Algorithm for Meshingread input boundary setup data structure & preprocessinggenerate initial sites inside do
- Delaunay triangulation of {xi}- move sites to optimal locations {xi*}
until convergence or stopping criterionextract interior mesh
38
(after Lloyd relaxation)
Breaks down, unfortunately…
why? Voronoi cells seem well-shaped but plenty of poorly-shaped tets, as mentioned… use of a dual approach, maybe?
Lloyd Algorithm in 3D
39
Under/Overlaid Approximant
CVT PWL approximant compact Voronoi cells isotropic sampling
ODT [Chen 04] PWL interpolant compact simplices isotropic meshing
40
Optimization of Vertices
Energy minimum for given connectivity iff:
Massage the equation and you get:
ij ikjk xxxkij
ii xxx
,
2* ||||)(||||2
1
j xi
xk
ij
jji
i cx
||||
1*
circumcenter of j
41
Results of Optimization in 2D
Note: needs proper boundary conditions…
Demo
3
42
0
Optimization
Distribution of radius ratios
43
Optimization
1
44
Optimization
2
45
Optimization
46
Optimization
47
Optimization
4
48
The Visible Human
Meshing of MRI datasets
49
Other Big Meshes
Dihedral angles ≥ 15.0
50
Fandisk (Mechanical Part)
51
Comparisons
Dihedral angles ≥ 3°4,189 vertices
TetGen
Dihedral angles ≥ 0.1°15,157 vertices
DelPSCNODT
Dihedral angles ≥ 12°4,050 vertices
52
Power Diagrams
Offseting squared distances by weights
Voronoi: all points treated equal
now “weight” wi per vertex
weights alter the L2 distance
covers set of orthogonal duals (V-1 DOFs)Simple and efficient generalization surprising applications as well
0.4
[Aurenhammer et al. ‘98]
53
Meshes with power diagrams
Can find a dual diagram by tweaking the weights
5
54
Adaptive Sampling (who needs meshing :)
55
Architecture via power diagrams
Discrete Force field within the walls forms cells
56
Putting Meshes to Work
Once meshing done, computations can happen Eulerian approaches
static mesh, motion encoded through local cell exchanges common for fluid simulations, level sets
57
Smoking Bunny
58
Smoke without Aliasing
59
Liquid Shape
Fluid with free surface (using foliation processing)
6
60
Putting Meshes to Work
Once meshing done, computations can happen Eulerian approaches
static mesh, motion encoded through local cell exchanges common for fluid simulations, level sets
Lagrangian approaches mesh moves (initial mesh = rest position) common for elastic and plastic deformable bodies
61
Meshes that Deform
Elastic deformations of gargoyles and armadillos
62
Conclusions
Meshing: fascinating research field needs theoretical guarantees
with practical bounds
needs practical results with theoretical foundations
can be extremely complex think quad/hexa meshes... plenty of remaining challenges
small angles, accelerating convergence, handling of anisotropy
Now it’s your turn!
Recommended