Tetrahedron Meshes. Linh Ha

Tetrahedron

Meshes

Overview

 Fundamental Tetrahedron meshing

Meshing polyhedra Tetrahedral shape

Delaunay refinements Removing silver

Meshing Polyhedra

 Is that easy ?

 A Polyhedron is union of convex polyhedra

i I

i

H

Facets and Segments

 Neighborhood of x

b : open unit ball

 Face figure of x

 Face of P: closure of maximal collection of point with identical face figures

(

x b

)

P N_ε = + ε . _

( )

(

N

x

x

)

What does it look like ?

 1- Faces

Face figures a line Polyhedra segment

 2- Faces

Face figure a plane Polydedra facets

 24 verties, 30 segments, 11s facets

Tetrahedrization

 Tetrahedrization of P is a simplicial complex

K whose underlying space is P: |K|=P

Only bounded polyhedra have tetrahedrization Every vertex of P is vertex of K

Is that easy ?

 Prim – easy : 3 tetrahedrons

 Ruppert and Seidel: the problem of deciding if a polyhedron can be tetrahedrized without inserting extra vertices is NP-hard and problem of deciding if a polyhedron can be tetrahedrized with only k additional vertices is NP-hard.

8 tetrahedrons , 7verties, 12 edges, 8 triangles

A C B a b c M

Fencing off - Simple methods

Fencing algorithms

 Step1: Erect the fence of each segment.  Step2:Triangulate the bottom facet of

every cylinder and erect fences from the new segments

 Step3:Decompose each wall into triangles and finally tetrahedrize each cylinder by

constructing cones from an interior point to the boundary.

Limits

 Upper bound: 28m2 - number of

tetrahedrons for m-segments Polyhedra

 Lower bound: (n+1)2 with n : number of cut

Tetraheral Shape

 What is good and bad tetrahedra ?  Bad tetrahedra

Bad tetrahedral classification

 Skinny: vertices are close to a line

 Flat : vertices are close to a plane

Quality Metrics

 2D : minimum angle in the triangulation  3D : Radius-edge ratio Aspect ratio Volume ratio R: outer radius L : shortest egde r: inner radius V: volume

Ratio properties

 A mesh of tetrahedra has ratio property if for all tetrahedra

 For all triangles

 Characteristics:

Claim A: if abc is a triangle in K then

 _{Proof: geometry}  Ratio properties 0 ρ ρ ≤ 0 ρ ρ ≤ b a c a b a − ≤ − ≤ ₀ − 0 2 2 / 1 ρ ρ R b a− ≤ 2 0 /ρ R b a− ≥ 0 2 / ρ a c R R ≤ − ≤

Ratio property

 Denote

 Claim B: if angle between ab and ap less than then

(

1/ 4

)

2 arctan 2 0 0 0 = ρ − ρ − η 0 η b a p a b a − ≤ − ≤ 2 − 2 / 1 Proof: ( ) a b b a R R b a b a R R v a − − − ≥ − − + − = − − − = − . 4 / 1 4 / 4 / 4 / 0 0 2 2 2 ρ ρ ( )

(

)

0 2 0 1/4 2 arctan / 2 arctan η ρ ρ = − − ≥ − − = ∠ o b a v x bax

Length Variation and Constant

degree

 Length variation: if ab, ap are edges in K

Lengths of edges with common endpoint is

bound

 Constant degree : Every vertex a in K belong to at most

However this constraint is too loose

( ) (1 cos / 4 ) / 2 ₀ 0 = − η m 1 0 1 2 0 2 0 0 − − = m m v ρ b a v p a v b a − / ₀ ≤ − ≤ ₀ −

(

₂

)

3 0 0 = 2v + 1 δ

Delaunay Triangulation

Properties of Delaunay triangulation

 maximizing the minimum angle in the triangulation:

 Better lighting performance

 Improvements on the accuracy

 Even stronger: lexicographically maximized sequence of

angles

 Every triangle of a Delaunay triangulation has an

Delaunay Tetrahedration

 _{Every tetrahedra of a Delaunay} Tetrahedration

has an empty circumsphere. However

max-min angle optimality in 2D dont’s

maintain

Left: Delaunay tetrahedralization: have arbitrarily thin tetrahedron known as a sliver

Edge flip

Incremental algorithms in 2D: edge flips

Problems

 Segment that not cover by the edges

 Faces are not covered by triangles of D

Have to add new points and update Delaunary tetrahedration

 well-spaced points generate only round or _{well-spaced points generate only round or} sliver Delaunay tetrahedra

Delaunary refinement

 Construct Delaunary tetrahedration has ratio property  Definition  encroached upon sub – segment  encroached upon sub – facet V a b V A B C

Refinement algorithms

 Rule1: If a subsegment is encroached upon, we

split it by adding the mid- point as a new vertex to the Delaunay tetrahedrization

 Rule2 : If a subfacet is encroached upon, we

split it by adding the circum-centre x as a new vertex to the Delaunay tetrahedrization

 Rule3: If a tetrahedron inside P has

then split the tetrahedron by adding the

circumcentre x as a new vertex to the Delaunay tetrahedrization

 Proprity : Rule1 > Rule2 > Rule3

0

/ L > ρ R

Local density

 _{Local feature size :f R}3 – R, f(x) f varies slowly with x

 Insertion radius : r_x length of the shortest Delaunay edge

Radii and Parrents

 Parent vertex: p(v) is the vertex that is “responsible” for the insertion of v

Radius claims

 Let x be a vertex of D and p is its parents, if it exits. Then or where

if x has type 1 or 2 and

( )

x f r_x ≥ _{rx ≥ c.rp} 2 / 1 = c c = ρ 0

Graded meshes

 Ratio claim: Let x be a Delaunay vertex with

parent and Then

 Invariant: if x is a type I vertex in the Delaunay

tetrahedrization, for i=1..3 then

 Conclusion: p x c r r ≥ .

( )

( )

p x f p c r r x f ( ) / ≤ 1+ / . ( ) i x f x C r ≥ /

( ) (

x / 1 C1

)

f y x − ≥ +

Silver exudation

 Silver exits frequently between well shaped

Variational Tetrahedral Meshing

Pierre Alliez Pierre Alliez David Cohen-Steiner David Cohen-Steiner Mariette Yvinec Mariette Yvinec Mathieu Desbrun Mathieu Desbrun

Goal

 Tetrahedral mesh generation  Focus on:

quality: shape of elements control over sizing

 dictated by simulation  constrained by boundary  low number of elements

Quality Metrics

 3D :

Radius-edge ratio Aspect ratio

Volume ratio

 In the paper use Radius ratio = Aspect

ratio – “fair” to measure silver

R: outer radius L : shortest egde

r: inner radius

Popular Meshing Approaches Optimization: • spring energy • aspect ratios • dihedral angles • solid angles • volumes • edge lengths

• containing sphere radii • etc.

[Freitag et al.; Amenta et al.]

[Freitag et al.; Amenta et al.]

 Advancing front  Specific subdivision  _octree  _{crystalline lattice}  Delaunay  _refinement  _{sphere packing}

Delaunay Triangulation

Optimizing Vertex Placement

Improve compactness of Voronoi cells by minimizing

∑ ∫

₌ ∈ − = k j _{x R} j j dx x x x E .. 1 2 || || ) ( ρ

Necessary condition for optimality:

Centroidal Voronoi Tessellation (CVT) site

Optimizing Vertex Placement

 CVT [Du-Wang 03]

 _{best PL approximant}  _{compact Voronoi cells}  _{isotropic sampling} optimized _dual Best L1 approximation of paraboloid Vi:local cell

Alas, Harder in 3D...

• _{well-spaced points generate only round or well-spaced points generate only round or} sliver Delaunay tetrahedra

Alas, Harder in 3D...

• well-spaced points generate only round or _{well-spaced points generate only round or}

sliver Delaunay tetrahedra

sliver Delaunay tetrahedra

• _{regular tetrahedron does not tile space regular tetrahedron does not tile space} (360

(360°° / 70.53 / 70.53°° = 5.1) = 5.1)

dihedral angle of the

dihedral angle of the

regular tetrahedron regular tetrahedron = = arcos(1/3) arcos(1/3) ∼_∼ 70.53° 70.53°

Idea: Underlaid

vs

Overlaid

 CVT

 _{best PL approximant}  _{compact Voronoi cells}  _{isotropic sampling}

 ODT [Chen 04]

 _{best PL interpolant}

 _{compact simplices – not}

dual

 _{isotropic meshing}

Best L1 approximation

Rationale Behind ODT

 Approximation theory:

linear interpolation: optimal shape of an

element related to the Hessian of f

[Shewchuk]

 Hessian(||x||2) = Id

 Which PL interpolating mesh best approximates the paraboloid?

for fixed vertex locations

 Delaunay triangulation is the optimal connectivity

for fixed connectivity

 min of quadratic energy leads to the optimal vertex

locations

Optimizing Connectivity

 Delaunay triangulation is the optimal connectivity

Optimizing Vertex Position

Simple derivation in x_i lead to optimal position in its 1-ring

Optimal Update Rule

circumcenter x x_ii

∑

Ω ∈

Ω

=

i j j j i i

c

x

τ

τ

|

|

|

|

1

*

Optimization

Alternate updates of

connectivity (Delaunay triangulation) vertex locations

Optimization: Init

distribution of radius ratios

distribution of radius ratios good

Optimization: Step 1

good

Optimization: Step 2

distribution of radius ratios

distribution of radius ratios good

good

bad

Optimization: Step 50

distribution of radius ratios

distribution of radius ratios good

Sizing Field

Goal reminder:

 _{minimize number of elements}  _{better approximate the boundary}

 _{while preserving good shape of elements}

Those are not independent!

Automatic Sizing Field

Properties:

 size ≤ lfs (local feature size) on boundary  sizing field = maximal K-Lipschitz

[

||

||

(

)

]

inf

)

(

x

K

x

y

lfs

y

y

−

+

=

Ω ∂ ∈

µ

parameter

Sizing Field: Example

K=1K=1 K=1

3D Example

local feature size

local feature size sizing fieldsizing field

∑

Ω ∈

Ω

=

i j j j i i

c

x

τ

τ

|

|

|

|

1

*

Hand

local feature size

Torso

Conclusion

 Generate high quality isotropic tetrahedral meshes (improved aspect ratios)

 Simple alternated optimizations

 connectivity: Delaunay

 vertex positions: weighted circumcenters

 Good in practice

 Limit

 Approximate input boudary intead of comforming