delaunay meshing for piecewise smooth complexes

Delaunay Meshing for Piecewise Smooth Complexes

Tamal K. DeyThe Ohio State U.

Joint work: Siu-Wing Cheng, Joshua Levine, Edgar A. Ramos

2/22Department of Computer Science and Engineering

Piecewise Smooth Complexes

Sharp Edges Non-manifold


Piecewise Smooth Complexes

• D is a piecewise smooth complex (PSC) if• Each k-dimensional element is a manifold and

compact subset of a smooth (C2) k-manifold, 0≤k≤2.

• The k-th stratum, Dk : set of k-dim elements of D.

• D0 – vertices, D1 – 1-faces, D2 – 2-faces.

• D≤k = D0 … Dk.• D satisfies usual reqs for being a complex.

• Interiors of elements are disjoint and for σ D, bd σ D.• For any σ, D, either σ = or σ D .


Delaunay refinement : History

• Chew89, Ruppert92, Shewchuk98 (Linear domains with no small angle)

• Cohen-Steiner-Verdiere-Yvinec02, Cheng-Dey-Ramos-Ray04 (polyhedral domains with small angle)

• Chew93 (surface without guarantees)• Cheng-Dey-Edelsbrunner-Sullivan01 (skin surfaces)• Boissonnat-Oudot03 and Cheng-Dey-Ramos-Ray04

(smooth surface)• Boissonnat-Oudot06 (Lipschitz surfaces)• Oudot-Rineau-Yvinec06 (Volumes)


Basics of Delaunay Refinement

• Chew 89, Ruppert 92, Shewchuk 98• Maintain a Delaunay triangulation of

the current set of vertices.• If some property is not satisfied by the

current triangulation, insert a new point which is locally farthest.

• Burden is on showing that the algorithm terminates (shown by packing argument).


Challenges for PSC• Topology

• Polyhedral case (input conformity,topology trivial).

• Curved elements (topology is an issue).• Topological Ball Property (TBP) was used for

smooth manifolds [BO03,CDRR04].• We need extended TBP for nonmanifolds.

• Nonsmoothness• Lipschitz surfaces [BO06], Remeshing [DLR05].

• Small angles• Delaunay refinement is hard [CP03, CDRR05,

PW04].


Topological Ball Property• For a weighted point

set S, let Vor S and Del S denote the weighted Voronoi and Delaunay diagrams.

• S has the TBP for σDi if σ intersects any k-face in Vor S either in emptyset or in a closed topological (i+k-3)-ball.


CW-Complexes

• A CW-complex R is a collection of closed (topological) balls whose interiors are pairwise disjoint and whose boundaries are the union of other closed balls in R.

• Our algorithm builds a CW-complex, Vor S||D|, to satisfy an extended TBP[ES97].


Extended TBP• S |D| has the extended TBP

(eTBP) for D if there is a CW-complex R with |R| = |D| s.t.

• (C1) The restricted Voronoi face F |D| is the underlying space of a CW-complex R’ R.

• (C2) The closed balls in R’ are incident to a unique closed ball bF R.

• (C3) If bF is a j-ball then bF bd F is a (j-1)-sphere.

• (C4) Each k-ball in R’, except bF, intersects bd F in a (k-1)-ball.


Extended TBP• For a 1- or 2-face σ, let Del S|σ

denote the Delaunay subcomplex restricted to σ.• Del S||Di|

= σDi Del S|σ.

• Del S||D| = σD Del S|σ.

• Theorem. If S has the eTBP for D then the underlying space of Del S||D| is homeomorphic to |D| [ES97].


Feature Size• For analysis, we require a

feature size which is 1-Lipschitz and non-zero.

• For any x |D|, let f(x) = min{m(x), g(x)}.

• For any σ D, f() is 1-Lipschitz over int σ.

• For δ (0,1] and x |D|, • if x D0, lfsδ(x) = δf(x).• if x int |Di|, for i ≥ 1,

lfsδ(x) = max{δf(x), maxybd|Di|

{lfsδ(y)-||x-y||}}.


Protecting D1 1. Any 2 adjacent balls on a 1-face

must overlap significantly without containing each others centers

2. No 3 balls have a common intersection

3. For a point p σ D1, if we enlarge any protecting ball Bp by a factor c ≤ 8, forming B’:

1. B’ intersects σ in a single curve, and intersects all D2 adjacent to σ in a topological disk.

2. For any q in B’ σ, the tangent variation between p and q is bounded.

3. For any q in B’ ( D2 adjacent to σ), the normal variation between p and q is bounded.


Admissible Point Sets

• Protecting balls are turned into weighted points

• We call a point set S admissible if• S contains all weighted points placed on D1.

• Other points in S are unweighted and they lie outside of the protecting balls (the weighted points).

• We maintain an admissible point set at each step of the algorithm.


D1 conformation

• Lemma. Let S is an admissible point set. For a 1-face σ, if p and q are adjacent weighted vertices spanning segment σpq on σ then Vpq is the only Voronoi facet which intersects σpq and it does so exactly once.


Meshing PSCs

• Meshing algorithm uses four tests to detect eTBP violations.

• Upon violation, we insert points outside of protected balls of weighted vertices.


Test 1: Multi-Intersection(q,σ)

• For a point q S on a 2-face σ, find a triangle t Del S|σ incident to q s.t. Vt intersects σ multiple times.

• If no t exists, return null, otherwise return the furthest (weighted) intersection point from q.

q


Test 2: Normal-Deviation(q,σ,Θ)

• For a point q S on a 2-face σ, check nσ(p), nσ(q) < Θ for all points p Vq|σ.• 2ω ≤ Θ ≤ /6.

• If so return null.• Otherwise return a point p where nσ(p), nσ(q) = Θ .


Test 3: Infringement(q,σ)

• For q S σ, return null if q is not infringed, otherwise let pq be the infringing edge.

• If the boundary edges of Vpq intersect int σ, return any intersection point.

• Else, Vpq σ is a collection of closed curves, return a critical point of Vpq σ in a direction parallel to Vpq.

• We say q is infringed w.r.t. σ if• σ is a 2-face containing q s.t. pq

Del S|σ for some p σ.• σ is a 2-face and there is a 1-face

in bd σ containing q and a non-adjacent vertex p s.t. pq Del S|σ.


Test 4: No-Disk(q,σ)

• If the star of q in Del S|σ is a topological disk, return null.

• Otherwise, find the triangle t Del S|σ incident to q which has the furthest (weighted) intersection point in Vt|σ from q and return the intersection point.


Meshing Algorithm

1. Protect elements in D≤1 with weighted points. Insert a point in each element of D2 outside of protected regions. Let S be this point set.

2. For any σ D2 and point q S σ:• If Infringed(q,σ), Multi-Intersection(q,σ), Normal-

Deviation(q,σ,Θ), or No-Disk(q,σ) (checked in that order) return a point x, insert x into S.

3. Repeat 2. until no points are inserted.4. Return Del S|D.


Admissibility is Invariant

Lemma. The algorithm never attempts to insert a point in any protecting ball

• Since no 3 weighted points intersect,

• all surface points (intersections of dual Voronoi edges and D) lie outside of every protecting ball


Initialization

• The algorithm must initialize with a few points from each patch in D2

• Otherwise, components can be missed.


Termination

• Each point x inserted is Ω(lfsδ(x)) away from all other points.

• Standard packing argument follows.


Topology Preservation

• To satisfy C1-C4 of eTBP, we show each Voronoi k-face F = Vp1 … Vp(4-k)

has:

• (P1) If F σ ≠ , for σ Dj, the intersection is a (k+j-3)-ball

• (P2) There is a unique lowest dimensional σF s.t. p1, …, p(4-k)σF.

• (P3) F intersects σF and only incident elements of σF.

• Theorem. If S satisfies P1-P3 then S satisfies C1-C4 of eTBP.


Feature Preservation

• h:|D| |Del S|D| can be constructed which respects each Di [ES97].

• Thus hi:|Di| |Del S|Di| also a homeomorphism with

vertex restrictions, ensuring that the nonsmooth features are preserved.

Delaunay Refinement made practical for PSCs

S.-W. Cheng, Tamal K. Dey, Joshua Levine


Definitions

• For a patch σ Di,• When sampled with S• Del S|σ is the Delaunay

subcomplex restricted to σ• Skli S|σ is the i-dimensional

subcomplex of Del S|σ,

• Skli S|σ = closure { t |

t Del S|σ is an i-simplex}

• Skli S|Di = σ Di Skli S|σ


Disk Condition• For a point p on a 2-face σ,

• UmbD(p) is the set of triangles in Skl2 S|D2 incident to p.

• Umbσ(p) is the set of triangles in Skl2 S|σ incident to p.

• Disk_Condition(p) requires:i. UmbD(p) = σ, p σ Umbσ(p) ii. For each σ containing p,

Umbσ(p) is a 2-disk where p is in the interior iff p int σ


Meshing Algorithm

DelPSC(D, r)1. Protect elements of D≤1.2. Mesh2Complex – Repeatedly insert surface

points for triangles in Skl2 S|σ for some σ if either1. Disk_Condition(p) violated for p σ, or2. A triangle has orthoradius > r.

3. Mesh3Complex – Repeatedly insert orthocenters of tetrahedra in Skl3 S|σ for some σ if

1. A tetrahedra has orthoradius > r and its orthocenter does not encroach any surface triangle in Skl2 S|D2.

4. Return i Skli S|Di.


Termination Properties

1. Curve Preservation• For each σ D1, Skl1 S|σ σ. Two vertices are joined

by an edge in Skl1 S|σ iff they were adjacent in σ.

2. Manifold• For 0 ≤ i ≤ 2, and σ Di, Skli S|σ is a manifold with

vertices only in σ. Further, bd Skli S|σ = Skli-1 S|bd σ.• For i=3, the above holds when Skli S|σ is nonempty

after Mesh2Complex.

3. Strata Preservation• There exists some r > 0 so that the output of

DelPSC(D, r) is homeomorphic to D. • This homeomorphism respects stratification.


Voronoi Cells Intersect “Discly”

• Given a vertex p on a 2-face σ, if• Triangles incident to p in Skl2 S|σ are

small enough.• Then,

• Vp|σ is a topological disk,• Any edge of Vp|σ intersects σ at most

once, and • Any facet of Vp|σ which intersects σ does

so in an open curve.


TBP holds globally

• if• All triangles incident in Skl2 S|σ are

smaller than a bound for all 2-faces,• Then

• TBP holds globally• This leads to the proof of ETBP and

more…topic of a new unpublished paper.


Adjusting MaxRad Example


Examples


Sharp Example


Conclusions

• Delaunay meshing for PSC with guarantees.

• Feature preservation is an extra `feature’.

• Making computations easier, faster?• Analyzing size complexity?

delaunay meshing for piecewise smooth complexes

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