Mesh Generation

Mesh Generation

Jean-Daniel Boissonnat DataShape, INRIA

http://www-sop.inria.fr/datashape

Meshing surfaces and 3D domains

visualization and graphics applications CAD and reverse engineering

geometric modelling in medecine, geology, biology etc.

autonomous exploration and mapping (SLAM) scientific computing : meshes for FEM

P d

dO

Ω

O o

Mesh generation : from art to science

Grid methods

Lorensen & Cline [87] : marching cube Lopez & Brodlie [03] : topological consistency

Plantiga & Vegter [04] : certified topology using interval arithmetic

Morse theory

Stander & Hart [97]

B., Cohen-Steiner & Vegter [04] : certified topology

Delaunay refinement

Hermeline [84]

Ruppert [95]

Shewchuk [02]

Chew [93]

B. & Oudot [03,04]

Cheng et al. [04]

Main issues

Sampling

How do we choose points in the domain ?

What information do we need to know/measure about the domain ?

Meshing

How do we connect the points ?

Under what sampling conditions can we compute a good approximation of the domain ?

Scale and dimension

Sampling conditions

Medial axis ofM: axis(M)

set of points with at least two closest points onM

Local feature size and reach

∀x∈M, lfs(x)=d(x,axis(M)) rch(M)=infx∈Mlfs(x)

(,η)-net of¯ M

1.P ⊂M,∀x∈M: d(x,P)≤lfs(x)

2.∀p,q∈ P, kp−qk ≥ηε¯ min(lfs(p),lfs(q))

Sampling conditions

Medial axis ofM: axis(M)

set of points with at least two closest points onM

Local feature size and reach

∀x∈M, lfs(x)=d(x,axis(M)) rch(M)=infx∈Mlfs(x)

(,η)-net of¯ M

1.P ⊂M,∀x∈M: d(x,P)≤lfs(x)

2.∀p,q∈ P, kp−qk ≥ηε¯ min(lfs(p),lfs(q))

Restricted Delaunay triangulation

Definition

Therestricted Delaunay triangulationDelX(P)to X⊂R^d is the nerve of Vor(P)∩X

If P is anε-sample, any ball centered onXthat circumscribes a facetfof DelX(P)has a radius≤εlfs(cf)

Restricted Delaunay triangulation

Definition

Therestricted Delaunay triangulationDelX(P)to X⊂R^d is the nerve of Vor(P)∩X

If P is anε-sample, any ball centered onXthat circumscribes a facetfof DelX(P)has a radius≤εlfs(cf)

Restricted Delaunay triangulation

Definition

Therestricted Delaunay triangulationDelX(P)to X⊂R^d is the nerve of Vor(P)∩X

If P is anε-sample, any ball centered onXthat circumscribes a facetfof DelX(P)has a radius≤εlfs(cf)

Restricted Delaunay triangulation

Definition

Therestricted Delaunay triangulationDelX(P)to X⊂R^d is the nerve of Vor(P)∩X

Restricted Delaunay triang. of ( ε, η ¯ ) -nets

[Amenta et al. 1998-], [B. & Oudot 2005]

If

S ⊂R³ is a compact surface of positive reach without boundary

P is an(ε,η)-net with¯ η¯≥cstandεsmall enough

then

Del_|S(S)provides good estimates of normals

There exists ahomeomorphism φ:Del_|S(P)→ S

sup_x(kφ(x)−xk) =O(ε²)

Surface mesh generation by Delaunay refinement

[Chew 1993, B. & Oudot 2003]

φ:S→R= Lipschitz function

∀x∈S, 0< φ₀= ¯η₀εrch(S)≤φ(x)<

εlfs(x)

ORACLE: For a facetf ofDel|S(P), returncf,rf andφ(cf) A facetf isbadifrf > φ(cf)

Context and Motivation

Delaunay refinement meshing engine

The algorithms refines:

Bad facets: f ∈Del_|S(P) – oversized (sizing field)

– badly shaped (min angle bound) – inaccurate (distance bound) Bad Tetrahedra : t∈Del_|O(P) – oversized (sizing field)

– badly-shaped (radius-egde ratio) Required oracle on domain to be meshed

• point location in domain and subdomains

• intersection detection/computation between boundary surfaces and segments (Delaunay edges)

PA-MY (INRIA Geometrica) CGALmesh ARC-ADT-AE-2010 9 / 36

Algorithm

INIT compute an initial (small) sampleP0⊂S REPEAT IFf is a bad facet

insert in Del3D(cf), updatePandDel|S(P) UNTIL no bad facet remains

Surface mesh generation by Delaunay refinement

[Chew 1993, B. & Oudot 2003]

φ:S→R= Lipschitz function

∀x∈S, 0< φ₀= ¯η₀εrch(S)≤φ(x)<

εlfs(x)

ORACLE: For a facetf ofDel|S(P), returncf,rf andφ(cf) A facetf isbadifrf > φ(cf)

Context and Motivation

Delaunay refinement meshing engine

The algorithms refines:

Bad facets: f ∈Del_|S(P) – oversized (sizing field)

– badly shaped (min angle bound) – inaccurate (distance bound) Bad Tetrahedra : t∈Del_|O(P) – oversized (sizing field)

– badly-shaped (radius-egde ratio) Required oracle on domain to be meshed

• point location in domain and subdomains

• intersection detection/computation between boundary surfaces and segments (Delaunay edges)

PA-MY (INRIA Geometrica) CGALmesh ARC-ADT-AE-2010 9 / 36

Algorithm

INIT compute an initial (small) sampleP0⊂S REPEAT IFf is a bad facet

insert in Del3D(cf), updatePandDel|S(P) UNTIL no bad facet remains

The meshing algorithm in action

The algorithm outputs a loose net of S

Separation

∀p∈ P,d(p,P \ {p}) =kp−qk

≥min(φ(p), φ(q))

≥φ(p)− kp−qk ^(φis Lipschitz)

⇒ kp−qk ≥ ¹₂φ(p)≥ ¹₂φ₀ >0 _⇒the algorithm terminates

Density

I Each facet has aS-radiusrf ≤φ(cf)< εlfs(cf)

I If correctly initialized,Del_|S(P)has a vertex on each cc ofS

Size of the sample = Θ R

S dx

φ²(x)

Upper bound

Bp=B(p,^φ(p)₂ ),p∈ P R

S dx φ²(x) ≥P

p

R

(Bp∩S) dx

φ²(x) (theBpare disjoint)

≥⁴₉P

parea(Bp∩S)

φ²(p) φ(x)≤φ(p) +kp−xk

≤ ³₂ φ(p))

≥P

pC = C|P|

reach(S)

p φ(p)2

Lower bound

Use a covering instead of a packing

Size of the sample = Θ R

S dx

φ²(x)

Upper bound

Bp=B(p,^φ(p)₂ ),p∈ P R

S dx φ²(x) ≥P

p

R

(Bp∩S) dx

φ²(x) (theBpare disjoint)

≥⁴₉P

parea(Bp∩S)

φ²(p) φ(x)≤φ(p) +kp−xk

≤ ³₂ φ(p))

≥P

pC = C|P|

reach(S)

p φ(p)2

Lower bound

Use a covering instead of a packing

Chord lemma

Letxandybe two points ofM. We have

1 sin∠(xy,Tx)≤ _{2 rch(}^kx−y_M^k₎;

2 the distance fromytoTx is at most _{2 rch(M)}^kx−yk2 .

p D q

q⁰⁰ θ

Tp M

2θ

q⁰

kx−yk ≥ kx−y⁰⁰k

=2 rch(M)sin∠(xy,Tx) ky−y⁰k=kx−yk sin∠(xy,Tx)

≤ _{2 rch}^kx−yk(M²)

Tangent space approximation

Lemma [Whitney 1957]

Ifσis a j-simplex whose vertices all lie within a distancehfrom a hyperplaneH⊂R^d, then

sin∠(aff(σ),H)≤ 2j h

D(σ) = 2h Θ(σ) ∆(σ)

Corollary

Ifσis a j-simplex,j≤k, vert(σ)⊂M, ∆(σ)≤2εrch(M)

∀p∈σ, sin∠(aff(σ),Tp)≤ 2ε Θ(σ)

(h≤ _{2 rch(M)}^∆(σ)2 by the Chord Lemma)

Schwarz lantern

Thickness and angle bounds for Thickness and angle bounds for triangles triangles

Good sampling implies angle and thickness bounds for triangles

a b

c

h R

h=ab⇥sinb= ^{ab ac}_2R

⇥(abc) = ₂^h _4R^h ₈²2 = ^¯₈² sinb ₂ = ^¯₂

h=ab × sinb= ^ab_2R^×ac Θ(abc) = _2∆^h ≥ _4R^h ≥ _32R^φ20 ≥ _32ε^φ_rch(⁰² _S)

sinb≥ _2εrch(^φ0_S)

Triangulated pseudo-surface

Forεsmall enough,Del_|S(P)is atriangulated pseudo-surface

v

v

Any edgeeofDel_|S(P)has exactly two incident facets The adjacency graph of the facets is connected

Triangulated pseudo-surface

ean edge ofM,ˆ f any face incident one

1 V(p)bounded ⇒ |V(e)∩S|is even

2 V(f)intersectsSat most once O ^V(e) V(e)

Proof of 2 by contradiction: c,c⁰∈V(f)∩S

a.kc−c⁰k ≤2εrch(S)⇒sin∠(cc⁰,Tc)≤ε (Chord L.) b. cc⁰ ⊥f (dual face)

c.Tc≈aff(f) (Whitney’s L.)

Triangulated pseudo-surface

ean edge ofM,ˆ f any face incident one

1 V(p)bounded ⇒ |V(e)∩S|is even

2 V(f)intersectsSat most once O ^V(e) V(e)

Proof of 2 by contradiction: c,c⁰∈V(f)∩S

a.kc−c⁰k ≤2εrch(S)⇒sin∠(cc⁰,Tc)≤ε (Chord L.) b. cc⁰ ⊥f (dual face)

c.Tc≈aff(f) (Whitney’s L.)

The main result

The Delaunay refinement algorithm produces

anε- netP ofS

a triangulated surfaceˆS

I homeomorphic toS

I close toS (Hausdorff/Fr´echet distanceO(ε²), approximation of normalsO(ε))

Applications

Implicit surfacesf(x,y,z) =0

Isosurfaces in a 3d image (Medical images) Triangulated surfaces (Remeshing)

Point sets (Surface reconstruction) see cgal.org, CGALmesh project

Results on smooth implicit surfaces

CGALmesh Achievements

Meshing 3D domains

Input from segmented 3D medical images

[INSERM] [SIEMENS]

Comparison with the Marching Cube algorithm

Delaunay refinement Marching cube

CGALmesh Achievements

Meshing with sharp features

A polyhedral example

PA-MY (INRIA Geometrica) CGALmesh ARC-ADT-AE-2010 30 / 36

CGALmesh Achievements

Meshing 3D multi-domains

Input from segmented 3D medical images [IRCAD]

PA-MY (INRIA Geometrica) CGALmesh ARC-ADT-AE-2010 21 / 36

Surface reconstruction from unorganized point sets