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Meshing: A (Biased) Crash Course

Meshing: A (Biased) Crash Course

Discrete Differential Geometry

DDG Course 2012

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The Big PictureMeshing, an essential preprocessing step For surface representation

complex geometry

with a few basic geometric primitives

For simulation of physical phenomena realistic/accurate animation of fluid,

deformable solids, electromagnetism often modeled as PDE

domain discretization Then FEM/FVM to integrate PDE in space (& time)

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DDG Course 2012

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The Big PictureMeshing, an essential preprocessing step For surface representation

complex geometry

with a few basic geometric primitives

For simulation of physical phenomena realistic/accurate animation of fluid,

deformable solids, electromagnetism often modeled as PDE

domain discretization Then FEM/FVM to integrate PDE in space (& time)

DDG Course 2012

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Meshing, an essential preprocessing step For surface representation

complex geometry

with a few basic geometric primitives

For simulation of physical phenomena realistic/accurate animation of fluid,

deformable solids, electromagnetism often modeled as PDE

domain discretization Then FEM/FVM to integrate PDE in space (& time)

The Big Picture

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DDG Course 2012

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The Big PictureMeshing, an essential preprocessing step For surface representation

complex geometry

with a few basic geometric primitives

For simulation of physical phenomena realistic/accurate animation of fluid,

deformable solids, electromagnetism often modeled as PDE

domain discretization Then FEM/FVM to integrate PDE in space (& time)

DDG Course 2012

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What is Meshing?General Idea: breaking up a physical domain

2D domain in 2D, or in 3D, or 3D domain,…

into simpler subdomains—the elements simplices (triangles, tetrahedra)

or not (quads, polygons, hexahedra, polyhedra)

“flat” 2D “flat” 3D

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Brief Glossary

Node, element:

Structured/Unstructured Mesh: regular valence and degree

Isotropic/Anisotropic Mesh: without/with stretched elements

Graded Mesh: w/ elements varying in size (fct of

position)

DDG Course 2012

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Brief Glossary

Node, element:

Structured/Unstructured Mesh: regular valence and degree

Isotropic/Anisotropic Mesh: without/with stretched elements

Graded Mesh: w/ elements varying in size (fct of

position)

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Why (Un)Structured?Structure brings: simpler data structure

better compression

reuse of DSP algos

smoothing

wavelets

Unstructuredmeshing Structured

meshing

DDG Course 2012

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Brief Glossary

Node, element:

Structured/Unstructured Mesh: regular valence and degree

Isotropic/Anisotropic Mesh: without/with stretched elements

Graded Mesh: w/ elements varying in size (fct of

position)

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Why Anisotropy?

#V #FIsotropicmeshing

Anisotropicmeshing

#V #F

Anisotropicquad meshing

#V #F

DDG Course 2012

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Brief Glossary

Node, element:

Structured/Unstructured Mesh: regular valence and degree

Isotropic/Anisotropic Mesh: without/with stretched elements

Graded Mesh: w/ elements varying in size (fct of

position)

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Why Gradation?

Allows for

better capture

of details for same vertex budget

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Goals in Mesh Generation

Control over shape of elements mass matrix conditioning

shape often induced by PDE good bet: regular shape

Control over sizing often dictated by simulation … &

boundary

Control over total size low number of elements is preferred

Tetrahedrus Commonus

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How to Mesh a Domain?

Usually, "bottom-up" approach vertices are first placed (point sampling)

then connected by edges, triangles, etc…

But not always… remeshing thru simplification (top-down)

placement of edges first [Alliez et al. 2003]

as strokes along important directions

placement of elements first [Alliez et al. 2004]

to best approximate local shape, based on normal field

DDG Course 2012

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How to Mesh a Domain?

Usually, "bottom-up" approach vertices are first placed (point sampling)

then connected by edges, triangles, etc…

But not always… remeshing thru simplification (top-down)

placement of edges first [Alliez et al. 2003]

as strokes along important directions

placement of elements first [Alliez et al. 2004]

to best approximate local shape, based on normal field

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DDG Course 2012

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How to Mesh a Domain?

Usually, "bottom-up" approach vertices are first placed (point sampling)

then connected by edges, triangles, etc…

But not always… remeshing thru simplification (top-down)

placement of edges first [Alliez et al. 2003]

as strokes along important directions

placement of elements first [Alliez et al. 2004]

to best approximate local shape, based on normal field

DDG Course 2012

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What We Won’t Mention [Much]Meshing of curved manifolds often, can be flattened thru

parameterizationsee STAR report aim@shape, Alliez et al. 2004

Meshing of non manifold objects

Polygonal Meshing

Surface Approximation vertices can be off the original surface

see [Nadler ‘86, Simpson ’94, Heckbert-Garland ‘99]

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So How do I Mesh a Domain?Seemingly, vertex placement is crucial we must “sample” the domain appropriately

well-spaced point sets? spread nodes in the (flat or curved) domain

spread them evenly

using an attraction/repulsion simulation for instance

and we are done, right? most papers in graphics use this approach

good news: sufficient in 2D

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So How do I Mesh a Domain?

An example from the sea…

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Intro to Delaunay Triang.‘Canonical’ triangulation of a point set with numerous optimal properties

see: Eppstein, Meshing Roundtable 2001Shewchuck, his UC Berkeley’s websiteLots of robust, existing codes (Triangle, CGAL, …)

“territories” = Voronoi diagram

“dual” = Delaunay triangulation (DT)

In 3DIn 2D

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Moving on to 3DEverything breaks… the regular tet does NOT tile space

perfectly shaped tets everywhere not attainable

DT based on well-spaced points have slivers

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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

often, non-linear and slow

no guarantee of success—in fact, may be worse

locally altering the Delaunay triangulation called “sliver exudation”

no longer Delaunay, obviously

guarantees, but no practical bounds

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Popular 3D Meshing MethodsDelaunay-based refinement

sphere packing

specific subdivision octree

regular lattice

advancing front

spring energy Laplacian (bad!)

non-zero rest length

aspect / radius ratios

dihedral / solid angles

max-min/min-max volumes, lengths

sphere radii

sliver exudation

Local Node OptimizationsGlobal Node Placement

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An Interesting Alternative Emergence of variational approaches Centroidal Voronoi Tesselation

Optimal Delaunay Triangulation

Idea: position of vertices also optimal in some way

often based on lifting transformation

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Lifting Transformation (nD)Lift points (x, y) to paraboloid (x, y, x2 + y2) placement of origin irrelevant

Delaunay triang. = convex hull of 3D points

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Lifting Transformation (nD)Lift points (x, y) to paraboloid (x, y, x2 + y2) placement of origin irrelevant

Voronoi Diagram = proj. of upper enveloppe

Before…

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Pushing This Analogy FurtherIdea: move vertices to perfect the approximation of the paraboloid roots in function approximation

Hessian(x2) = Id

studied quite a bit in 2D

Lloyd algo: for given points,

DT gives optimal L1 approx

for given conn., find optimal vertex positions

volume btw approximant and paraboloid

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(after Lloyd relaxation)

Lloyd Algorithm in 3D

Breaks too, unfortunately…

why? dual cells well-shaped

but not sufficient for well-shaped tets!

use of a dual approach, maybe?

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Under/Overlaid Approximant

CVT approximant

compact Voronoi cells

isotropic sampling

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Under/Overlaid Approximant

CVT approximant compact Voronoi cells

isotropic sampling

ODT [Chen 04]

PWL interpolant compact simplices

isotropic meshing

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Results of Optimization in 2D

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Optimization: init

Distribution of radius ratios

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Optimization: step 1

Distribution of radius ratios

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Optimization: step 2

Distribution of radius ratios

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Optimization

Distribution of radius ratios

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Optimization

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Optimization

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Input mesh

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Hand: Sizing

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Hand Remeshed

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Stanford Bunny

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Stanford Bunny (uniform)

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The Visible Human

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Fandisk

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Conclusions

Meshing: a fascinating research field needs theoretical guarantees

needs practical results

Please, consider using Delaunay not slow (unlike what you’ve heard)