Anisotropic Voronoi diagrams and Delaunay Meshes Mariette Yvinec - - PowerPoint PPT Presentation

Anisotropic Voronoi diagrams and Delaunay Meshes

Mariette Yvinec

INRIA Sophia Antipolis

Geometric Computing Workshop Heraklion, January 2013

MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 1 / 49

Introduction Motivation and Definitions

The need for anisotropy

What is an anisotropic simplicial mesh ?

• a mesh with simplicial elements
• elongated according to prescribed directions.
• Required anisotropy is described

at each point using an anisotropic metric

• with spatial variations: metric field

Why anisotropic meshes ?

◮ reduce interpolation error:

anisotropy according to Hessian

◮ adaptative solving of PDE

for anisotropic phenomenon

◮ accurate surface discretisation

anisotropy according to curvature tensor

Images from Adrien Loiselle s Phd MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 2 / 49

Introduction Motivation and Definitions

How to prescribe anisotropy

Metric field each point z of the domain − → M(z) a positive definite quadratic matrix M-distance between two points a and b: dM(a, b) =

• (a − b)tM(a − b)

◮ Fonction approximation

Metric field related to the Hessian of the function

◮ Approximation of surfaces

Metric field related to the normal field and to the second fundamental form (principal curvatures)

◮ Adaptative FEM for PDE

Metric field related to the error estimation on previous iteration solution

MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 3 / 49

Introduction Previous work

Previous Work

Many heuristics for anisotropic simplicial 2D or 3D meshes

• Ellipses packing

[ Li et al. 99 ] , [ Yamakawa Shimada 03 ]

• Anisotropic Delaunay refinement [ Borouchaki et al. 97 ]

[ Frey Alauzet 04 ] , [ Dobrzynski Frey 08 ]

• Continuous mesh [ Loseille Alauzet 09 ]
• Anisotropic mesh optimization

myciteBossen Heckbert 96, [ Li et al. 05 ] More heuristics for surface meshes

• Alliez et al (03), based on prinicipal curvature lines
• Jiao et al (06), anisotropic adaptation

using Garland Heckbert quadratic error

• Azernikov Fischer 05 , grid based

MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 4 / 49

Introduction Previous work

Previous Work (cont’d)

Voronoi approaches

• Voronoi diagram on Riemannian manifold

[ Leibon Letscher 00 ] , [ Bouglueux et al. 08 ]

• Anisotropic Voronoi diagram

[ Labelle Shewchuk 03 ] , [ Boissonnat et al. 05 ] [ Du Wang 05 ] [ Cheng at al. 06 ] surface meshes based on 3D AVD Delaunay approach

• Anisotropic Delaunay meshes

The locally uniform anisotropic Delaunay meshes approach [ Boissonnat et al. 10 ] , [ Boissonnat et al. (to appear) ]

MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 5 / 49

Outline

Introduction Motivation and Definitions Previous work Anisotropic Voronoi diagrams Meshing Algorithm I Meshing Algorithm II Anisotropic Delaunay meshes Meshing Algorithm Proof of the Algorithm Termination Anisotropic surface meshes

MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 6 / 49

Anisotropic Voronoi diagrams

Definition of Anisotropic Voronoi Diagrams

[Labelle and Shewchuk 03]

Let D ⊂ Rd be a domain with a metric field defined on D : ∀p ∈ D − → Mp. dp(x, y) =

• (x − y)tMp(x − y)

Anisotropic Voronoi diagram P a set of sites in D ∀p ∈ P, Voronoi cell V (p) V (p) = {x ∈ Rd : dp(p, x) ≤ dq(q, x), ∀q ∈ P, q = p} Bisector of {p,q} is a conic: (x − p)tMp(x − p) = (x − q)tMq(x − q) The Du et Wang variant V (p) = {x ∈ Rd : dx(p, x) ≤ dx(q, x), ∀q ∈ P, q = p}

MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 7 / 49

Anisotropic Voronoi diagrams

Definition of Anisotropic Voronoi Diagrams

[Labelle and Shewchuk 03]

◮ Each site is within its cells ◮ Cells may be not connected. ◮ The dual need not be a triangulation.

MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 8 / 49

Anisotropic Voronoi diagrams Meshing Algorithm I

Meshing Algorithm from AVD

[ Labelle and Shewchuk 03]

Summary of the algorithm

◮ Anisotropic Voronoi diagram computed as a lower enveloppe:

{fi(x) = (x − pi)tMpi(x − pi)}

◮ refine the Voronoi diagram by adding new sites on bisectors ◮ until bisectors are straigth enough ,

so that Voronoi cells are connected and the dual is a triangulation. Remarks

◮ Terminates and works only in 2D ◮ No need to maintain the exact anisotropic Voronoi diagram :

a loose version is enough, where only the main connected component of each new site is computed.

MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 9 / 49

Anisotropic Voronoi diagrams Meshing Algorithm II

Anisotropic Voronoi diagrams from power diagrams

Boissonnat et al.

Linearization of the distance function x(x, y) − → ˆ x, ˆ xt = (x, y, x2, xy, y 2) site(pi, Mi) − → ˆ pi, ˆ pt

i = (Mipi, −Mxx i , −1/2Mxy i , −Myy i )

dpi(x, pi)2 = xtMix − 2pt

i Mix + pt i Mipi

= −2 ˆ pi

tˆ

x + pt

i Mipi

Anisotropic Voronoi diagram from a power diagram

◮ Compute the power diagram, in dimension d + d(d + 1)/2 = 5,

• f Σ = {σi( ˆ

pi,

• ˆ

pi2 − pt

i Mipi), i = 1 · · · n}

◮ Intersect it with the 2 manifold M = {x, y, x2, xy, y 2} ◮ Project the result in R2

MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 10 / 49

Anisotropic Voronoi diagrams Meshing Algorithm II

Meshing Algorithm from AVD II

Boissonnat et al.

Algorithms

◮ Maintain the power diagram V (Σ) of Σ in dimension 5, ◮ Maintain the set of vertices of the intersection V (Σ) M ◮ Refine inserting new sites at Voronoi vertices

until the set of dual triangles form a triangulation and are well shaped (according to local metrics) Remarks

◮ Terminates and works in 2D ◮ Problems for termination in higher dimension

MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 11 / 49

Anisotropic Delaunay meshes

Anisotropic Metric and Space Transform

Metric definition

An anisotropic metric in Rd is defined in some basis by a symmetric positive definite d × d matrix M M-distance between two points a and b dM(a, b) =

• (a − b)tM(a − b)

Associated space transform

∃ matrix FM such that det(FM) > 0 and F t

MFM = M.

dM(a, b) =

• (a − b)tF t

MFM(a − b)

= FM(a − b) FM

↓

MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 12 / 49

Anisotropic Delaunay meshes

Delaunay triangulation in a uniform anisotropic metric

metric M M-distance, dM(a, b)=

• (a − b)tM(a − b)

= FM(a − b) M-Volume M-sphere: CM(c, r) = {x : dM(c, x) = r} M-ball: BM(c, r) = {x : dM(c, x) ≤ r} M-balls are ellipses with axes along eigenvectors of M M-circumball of a k-simplex τ: among the M-ball circumscribing τ the one with smallest radius. Delaunay triangulation DelM(V ): the M-circumball of each d-simplex is empty Del(FM(V ))

F −1

M

− → DelM(V ) FM − →

MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 13 / 49

Anisotropic Delaunay meshes

Simplex Quality in Anisotropic Metric

A k-simplex τ in a metric M CM(τ)(cM(τ), rM(τ)) the M-circumsphere of τ eM(τ) the shortest edge ρM(τ) = rM(τ)

eM(τ) the radius-edge ratio

σM(τ) =

• VolM(τ)

ek

M(τ)

1

k the sliverity ratio.

Slivers

Let ρ0 and σ0 be two constants. With respect to the metric M, the k-simplex τ is:

• well shaped if ρM(τ) ≤ ρ0 and σM(τ) ≥ σ0
• a sliver if ρM(τ) ≤ ρ0 and σM(τ) ≤ σ0
• a k-sliver if it is a sliver and all its (k − 1)-faces are well-shaped.

MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 14 / 49

Anisotropic Delaunay meshes

Locally Uniform Anisotropic Delaunay Meshes

A mesh such that: the star of each vertex is Delaunay and well shaped wrt the metric at that vertex. V set of vertices, v ∈ V : Mv metric at v Delv(V ) Delaunay triangulation of V computed with metric Mv Sv: the star of v in Delv(V )

MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 15 / 49

Anisotropic Delaunay meshes

Inconsistencies

w x y v Sw Sv

w(wxy) v(vwx)

Inconsistency : some simplex τ with vertices {v, w, . . .} appears in star Sv but not in star Sw.

MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 16 / 49

Anisotropic Delaunay meshes Meshing Algorithm

Overview of the meshing algorithm

◮ Maintain the set of stars S(V ) = {Sv : v ∈ V } ◮ Refine V until stars are well shaped and consistent. ◮ Consistent stars are stitched

MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 17 / 49

Anisotropic Delaunay meshes Meshing Algorithm

Meshing Algorithm

V current set of sites, S(V ) = {Sv : v ∈ V } the star set Domain Ω: each star Sv is restricted to Ω i.e. Sv = {τ ∈ Delv(V ) : v ∈ τ and cv(τ) ∈ Ω} Apply following refinement rules with priority order:

• 1. Sizing field - Distorsion

While ∃τ ∈ Sv s.t. rv(τ) ≥ α lf(cv(τ)), refine τ

• 2. Radius-edge ratio

While ∃τ ∈ Sv s.t. ρv(τ) ≥ ρ0, refine τ

• 3. Slivers

While ∃ a sliver τ ∈ Sv (ρv(τ) ≤ ρ0, σv(τ) ≤ σ0), refine τ

• 4. Inconsistencies

While ∃ an inconsistent simplex τ ∈ Sv, refine τ

MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 18 / 49

Anisotropic Delaunay meshes Meshing Algorithm

Meshing Algorithm

Refine τ

◮ Find a refinement point p for τ ◮ insert p in any star Sv such that p ∈ Bv(τ) for some τ ∈ Sv ◮ create a new star for p

The remaining of this talk

• 1. Discuss the sizing field (Rule 1)
• 2. Prove the termination of the algorithm.

Main problem comes from quasi-cospherical configurations. Avoid them through carefull choice of refinement points.

MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 19 / 49

Anisotropic Delaunay meshes Meshing Algorithm

Metric Field and Distorsion

Distorsion between two metrics (M, N): γ(M, N) = max{F −1

M FN, F −1 N FM}

with A = supx∈Rd Ax

x .

In the context of a metric field, Distorsion between two points (p, q) : γ(p, q) = γ(Mp, Mq) Distorsion bounds the difference between dp and dq: ∀x, y, 1 γ(p, q)dq(x, y) ≤ dp(x, y) ≤ γ(p, q) dq(x, y).

MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 20 / 49

Anisotropic Delaunay meshes Meshing Algorithm

The Bounded Distorsion Radius

Definition

Assume a metric field on a domain Ω. For each p ∈ Ω, the bounded distorsion radius bdr(p, γ0) is a function Ω × R → R defined as follows: bdr(p, γ0) = max{l ∈ R} such that dp(p, q) ≤ l dp(p, r) ≤ l

• =

⇒ γ(q, r) ≤ γo.

Bounded distorsion radius lemma

∀p, q ∈ Ω, 1 γ(p, q) [bdr(p, γ0) − dp(p, q)] ≤ bdr(q, γ0) ≤ γ(p, q) [bdr(p, γ0) + dp(p, q)] .

Sizing field: lf(p) = bdr(p, γ0)

to adapt mesh density to spatial field variation.

MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 21 / 49

Anisotropic Delaunay meshes Meshing Algorithm

Sizing field and Distorsion

Distorsion for a simplex τ = p0p1 . . . pd. τ has a circumball Bi(τ) for the metrics Mi of each of vertex pi. distorsion γ(τ) = max γ(p, q) for (p, q) within the same circumball Bi(τ). 1.Sizing field - Distorsion While ∃τ ∈ Sv s.t. rv(τ) ≥ αlf (cv(τ)), refine τ. Rule 1. ensures that the distorsion of each simplex in the mesh is less than γ0 if α ≤ 1.

τ q Bb(τ) Ba(τ) Bc(τ) a b c p

MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 22 / 49

Anisotropic Delaunay meshes Meshing Algorithm

Quasi-cosphericities

Quasi-cospherical configurations

A subset U of d + 2 sites {p0, p1, . . . , pd+1} is a (γ0, M)-cospherical configuration if there exist two metrics N and N′ such that :

◮ γ(M, N) ≤ γ0, γ(M, N′) ≤ γ0 and γ(N, N′) ≤ γ0; ◮ DelN(U) and DelN′(U) are different.

When M and γ0 are omitted U is called a quasi-cospherical configuration.

Lemma

Configuration (τ, q) is (γ0, M)-cospherical, with witnesses metrics N and N′, iff: q ∈ BN(τ) and q ∈ BN′(τ)

τ q BN(τ) BN′(τ) v u w

MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 23 / 49

Anisotropic Delaunay meshes Meshing Algorithm

Quasi-cosphericities and Inconsistencies

τ q Bw(τ) Bv(τ) v w

Lemma

Let τ be an inconsistent simplex: τ ∈ Sv, τ ∈ Sw where v, w vertices of τ If γ(τ) < γ0, ∃q ∈ Sw such that the configuration (τ, q) is (γ0, Mv)-cospherical.

MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 24 / 49

Anisotropic Delaunay meshes Meshing Algorithm

Picking Region

◮ Using circumcenters as refinement points

may lead to cascading occurencees of quasi-cospherical configurations

◮ The refinement point is chosen in a small region

around the circumcenter, called the picking region

◮ Inside a picking region, the refinement point is chosen so as

to avoid formation of slivers and quasi-cospherical configurations

Picking region [Li 00]

τ a bad simplex in star Sv Bv(cv(τ), rv(τ)) the Mvcircumball Picking region of τ : Pv(τ) Pv(τ) = Bv(cv(τ), δrv(τ))

v Pv(τ) q r s p τ MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 25 / 49

Anisotropic Delaunay meshes Meshing Algorithm

Valid refinement point

Let τ be a simplex in star Sv, with Mv-circumradius rv(τ). A point p ∈ Pv(τ) is a valid refinement point for τ, if:

◮ There is no well shaped subset σ ⊂ V s.t.

τ ′ = (p, σ) is a sliver for a metric M s.t γ(M, Mp) ≤ γ0, with M-circumradius rM(τ ′) ≤ βrv(τ) .

◮ There is no well shaped subset of σ ⊂ V s.t.

U = (p, σ) is a (γ0, M)-cospherical configuration for a metric M s.t γ(M, Mp) ≤ γ0, with M-circumradius rM(U) ≤ βrv(τ). Note: Circumradius of a (γ0, M)-cospherical configuration U rM(U) = minτ ′⊂U,|τ ′|=d+1 rM(τ ′)

MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 26 / 49

Anisotropic Delaunay meshes Meshing Algorithm

Picking Lemma

Lemma

For any value of parameters α, β, δ, ρ0 it is possible to choose σ0 small enough and γ0 close enough to 1, so that a valid refinement point exists in the picking region Pv(τ) of any bad simplex τ ∈ Sv

v Pv(τ) q r s p τ

Proof

hitting sets of Pv(τ): subsets of V forming with some p ∈ Pv(τ) and for some metric M with γ(M, Mp) ≤ γ0 a small M-sliver or a small (γ0, M)-cospherical configuration.

◮ the number of hitting subsets is bounded by a constant K = K(α, β, δ) ◮ the forbidden region induced by each hitting subset

has a volume that − → 0 when σ0 − → 0 and γ0 − → 1 in such way that γ0−1

σd

− → 0.

MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 27 / 49

Anisotropic Delaunay meshes Meshing Algorithm

Anisotropic Meshing Algorithm

Pick valid(τ, Mv)

• Pick a random point p ∈ Pv(τ)
• While p is not a valid refinement point

Pick a random point p ∈ Pv(τ)

Meshing Algorithm

Apply following refinement rules with priority order:

• 1. Sizing field - Distorsion

While ∃τ ∈ Sv s.t. rv(τ) ≥ α lf(cv(τ)), Insert(cv(τ)).

• 2. Radius-edge ratio

While ∃τ ∈ Sv s.t. ρv(τ) ≥ ρ0, Insert( Pick valid(τ, Mv)).

• 3. Slivers

While ∃ a sliver, Insert( Pick valid(τ, Mv)).

• 4. Inconsistencies

While ∃ an inconsistent simplex τ ∈ Sv, Insert( Pick valid(τ, Mv)).

MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 28 / 49

Anisotropic Delaunay meshes Proof of the Algorithm Termination

Termination of the Algorithm

◮ Based on a volume argument. ◮ Shortest intersites distance to p ∈ V ,

l(p) = min

q∈V d(p, q)

with d(p, q) = min(dp(p, q), dq(p, q)) . We prove a lower bound for l(p) as a function of lf(p).

◮ Lower bound on l(p) is based on a lower bound

• n the insertion radius r(p) of each vertex

r(p) = min

q∈V d(p, q), at the time p is inserted.

MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 29 / 49

Anisotropic Delaunay meshes Proof of the Algorithm Termination

Termination of the Algorithm

Insertion radius lemma

If p is the refinement point of simplex τ within star Sv

• r(p) ≥ rv(τ)

Γ

if Rule 1 applies

• r(p) ≥ 1−δ

Γ rv(τ) if Rule 2,3 or 4.

where Γ = supx,y∈Ω γ(x, y)

Insertion radius- Rule 1

When Rule 1 is applied, the insertion radius of the new site is such that r(p) ≥ C1 lf(p) with C1 = α Γ .

Insertion radius - Rule 2,3-4

. . .

MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 30 / 49

Anisotropic Delaunay meshes Proof of the Algorithm Termination

Termination of the Algorithm

◮ Insertion radius : provided constants ρ0, γ0, β, δ satisfy

(1 − δ)ρ0 Γγ2 ≥ 2 and (1 − δ)β Γγ2

0(1 + δ) ≥ 2

there is a constant C, such that, for any vertex in the mesh r(p) ≥ C lf(p).

◮ Then, for any mesh vertex p,

min

q∈V d(p, q) ≥

C (1 + C)Γ lf(p)

◮ n: number of mesh vertices

n ≤ Γ2d u(d)

• 1 +

C (1+C)2Γ2 C (1+C)2Γ2

d IΩ ΦΩ where u(d) is the volume of the unit Euclidean d-dim ball, IΩ =

• Ω

dpp lf(p)d ,

and ΦΩ is another domain characteristic.

MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 31 / 49

Anisotropic Delaunay meshes Proof of the Picking Lemma

Circumspheres Lemma

Let Mv and Mw be two metrics such that γ(Mv, Mw) ≤ γ0, τ a k-simplex, Mv- well shaped i.e. ρv(τ) ≤ ρ0 and σv(τ) ≥ σ0. cv and rv the Mv-circumcenter and Mv-circumradius of τ cw and rw its Mw-circumcenter and Mw-circumradius of τ.

◮ dv(cv, cw) ≤ fk(ρ0, σ0, γ0)rv, where

fk(ρ0, σ0, γ0) =

• 1 + 2k

k γ2

0ρk

σk γ2

0 − 1

• =

O(γ0 − 1).

◮ rw ∈

• h−

k (γ0)rv, h+ k (γ0)rv

• , with

h−

k (γ0)

= 1 γ0 (1 − fk(γo)) = 1 − O(γ0 − 1) h+

k (γ0)

= γ0 (1 + fk(γo)) = 1 + O(γ0 − 1)

cw cv Cw Cv

MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 32 / 49

Anisotropic Delaunay meshes Proof of the Picking Lemma

Slivers forbidden regions

Sliver Lemma

Let τ be a k-sliver for metric M. τ(v) the face of τ opposite to vertex v. C(v) the M-circumsphere of τ(v). C′(v) = C(v) ∩ Aff(τ(v)) dM(v, C′(v)) ≤ 4πρ0σ0r(v)

Forbidden region

If τ ′ is hitting set of P(τ) likely to form a k′-sliver with some p ∈ P(τ), the Mv volume of the forbidden region is: Volv(Yv(τ ′)) ≤ µk′(ρ0, σ0, γ0)βdr d

v ,

where µk′(ρ0, σ0, γ0) → 0 when σ0 → 0 and γ0 → 1 with γ0−1

σk′

→ 0.

MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 33 / 49

Anisotropic Delaunay meshes Proof of the Picking Lemma

Quasi-cospherical forbidden regions

Cospherical Lemma

If τ is a d-simplex well shaped wrt to metric M, and p is s.t. (τ, p) is (γ0, M)-cospherical, then p ∈ BM(c, g +

d (γ0) r) \ BM(c, g − d (γ0) r),

where C(c, r) is the M-circumsphere of τ, and g +

d (γ0) =

• γ2

0 + (1 + γ2 0)fd(γ0)

• ,

g −

d (γ0) =

• 1

γ2

0 − (1 + 1

γ2

0 )fd(γ0)

• .

Forbidden region

If τ ′ is a d-simplex hitting P(τ) the Mv volume of the forbidden region is: Volv(Wv(τ ′)) ≤ ω(ρ0, σ0, γ0)βdr d

v ,

where ω(ρ0, σ0, γ0) → 0 when σ0 → 0 and γ0 → 1 with γ0−1

σd

→ 0.

τ p

τ p

MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 34 / 49

Anisotropic Delaunay meshes Proof of the Picking Lemma

Proof of Picking Lemma

Number of forbidden regions

If αγ0

• δ + 2βγ2

0)

• ≤ 1,

when a refinement point is searched in Pv(τ), the number of hitting set is Kv(τ) ≤ K(α, β, δ, γ0) where K(α, β, δ, γ0) is bounded when γ0 → 1.

Proof

q ∈ τ ′, p ∈ P(τ ′) the refinement point

• 1. Because configuration (τ ′, p) is small, dM(p, q) ≤ 2βrv,

dv(cv, q) ≤

• δ + 2γ2

0β

• rv ≤ α
• δ + 2γ2

0β

• lf(cv)
• 2. The intersites distance remains locally bounded : ∀q′ ∈ V , = q,

d(q, q′) ≥ C (1 + C)Γ lf(q), dv(q, q′) ≥ l2 lf(cv), with l2 = C (1 + C)Γγ2

• 1 − αγ0(δ + 2γ2

0β)

• MY (INRIA Sophia Antipolis)

Anisotropic Voronoi-Delaunay GC2013 35 / 49

Anisotropic Delaunay meshes Experimental Results

Anisotropic 3D Meshes

MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 36 / 49

Anisotropic Delaunay meshes Experimental Results

Anisotropic 3D Meshes

MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 37 / 49

Anisotropic Delaunay meshes Experimental Results

Anisotropic 3D Meshes

E(x, y, z) = tanh 1 λ(2x − sin(5y)) +y 3 + x3 Mesh elements elongated

• rthogonally to the gradient.

FM = P  

1 (1+φ∇E)

1 1   Pt

MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 38 / 49

Anisotropic Delaunay meshes Anisotropic surface meshes

How to get Anisotropic Surface Meshes

MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 39 / 49

Anisotropic Delaunay meshes Anisotropic surface meshes

Restricted Delaunay triangulation

V(P) S

sampling P, surface S

• Del(P)

Delaunay triangulation

• Vor(P)

its dual Voronoi diagram

• The Delaunay triangulation restricted to S

Del|S(P) subcomplex of Del(P) formed by faces whose dual intersect S

MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 40 / 49

Anisotropic Delaunay meshes Anisotropic surface meshes

Restricted Delaunay triangulation

V(P) S

sampling P, surface S

• Del(P)

Delaunay triangulation

• Vor(P)

its dual Voronoi diagram

• The Delaunay triangulation restricted to S

Del|S(P) subcomplex of Del(P) formed by faces whose dual intersect S [Edelsbrunner and Shah 97] [Amenta and Bern 98] [Boissonnat and Oudot 05] If P is dense enough on S

◮ Del|S(P) is homeomorphic to S ◮ Del|S(P) is a good approximation of S

in term of Hausdorff distance normals, area and curvature estimation

MY (INRIA Sophia Antipolis) Anisotropic Voronoi-Delaunay GC2013 40 / 49

Anisotropic Delaunay meshes Anisotropic surface meshes

Anisotropic Surface Mesh Algorithm

Star set

Maintain T(V ) = {Tv : v ∈ V } and S(V ) = {Sv : v ∈ V } Tv the star of v in Delv(V ) Sv the restriction of Tv to the surface S

Meshing Algorithm

Apply following refinement rules with priority order:

• 1. Sizing field - Distortion

While ∃τ ∈ Sv s.t. rv(τ ≥ α lf(cv(τ)), Insert(cv(τ)).

• 2. Radius-edge ratio

While ∃τ ∈ Sv s.t. ρv(τ) ≥ ρ0, Insert( Pick valid(τ, Mv)).

• 3. Inconsistency

While ∃ an inconsistent simplex τ ∈ Sv, Insert( Pick valid(τ, Mv)). cv(τ) the center of the Surface Delaunay ball

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Anisotropic Delaunay meshes Experimental results

Anisotropic Surface Meshes

Metric according to the curvature tensor

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Anisotropic Delaunay meshes Experimental results

Anisotropic Surface Mesh

Estimated curvature tensor on polyhedral surfaces

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Anisotropic Delaunay meshes Experimental results

Anisotropic Surface Mesh

Estimated curvature tensor onpolyhedral surfaces

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Anisotropic Delaunay meshes Experimental results

Anisotropic Surface Meshes

Metric according to a shock wave function

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Anisotropic Delaunay meshes Experimental results

Timings

CPU time to generate a curvature adapted mesh of a torus

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Anisotropic Delaunay meshes Experimental results

Approximation error

Anisotropic Vertices: 2314 Facets: 4622 Vertices: 1217 Facets: 2423 Vertices: 898 Facets: 1792 Vertices: 6310 Facets: 12616 Vertices: 6120 Facets: 12236 Vertices: 6040 Facets: 12076 κ = 36.7% R=10 Isotropic κ = 19.9% R=50 κ = 14.9% R=200

number of vertices needed to mesh pieces of torus with the same accuray

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Conclusion and perspectives

An anisotropic mesh generator

◮ provably correct. ◮ conceptually simple. It relies on Delaunay triangulation. ◮ works in any dimension

Ongoing and future work

◮ much room to improve the 3D and surface mesh prototype ◮ handle 3D domains with boundaries ◮ handle sharp edges ◮ tackle discontinuous metric field ◮ apply the trick of picking regions to anisotropic Voronoi diagrams

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Thank you

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