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Making Meshesand Putting Them To Good Use

Making Meshesand Putting Them To Good Use

Mathieu DesbrunCaltech

CS176

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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…

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Scrutinizing the Beast

Four (well-spaced) vertices near the equatorial plane of their circumsphere

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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

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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

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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

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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

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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

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(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

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Under/Overlaid Approximant

CVT PWL approximant compact Voronoi cells isotropic sampling

ODT [Chen 04] PWL interpolant compact simplices isotropic meshing

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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

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Results of Optimization in 2D

Note: needs proper boundary conditions…

Demo

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Optimization

Distribution of radius ratios

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Optimization

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Optimization

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Optimization

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Optimization

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Optimization

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The Visible Human

Meshing of MRI datasets

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Other Big Meshes

Dihedral angles ≥ 15.0

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Fandisk (Mechanical Part)

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Comparisons

Dihedral angles ≥ 3°4,189 vertices

TetGen

Dihedral angles ≥ 0.1°15,157 vertices

DelPSCNODT

Dihedral angles ≥ 12°4,050 vertices

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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]

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Meshes with power diagrams

Can find a dual diagram by tweaking the weights

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Adaptive Sampling (who needs meshing :)

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Architecture via power diagrams

Discrete Force field within the walls forms cells

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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

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Smoking Bunny

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Smoke without Aliasing

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Liquid Shape

Fluid with free surface (using foliation processing)

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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

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Meshes that Deform

Elastic deformations of gargoyles and armadillos

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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!