An introduction to mesh generation Part II : Delaunay …perso.uclouvain.be/.../teaching/documents/meca2170-jfr-cours2.pdf · An introduction to mesh generation Part II : Delaunay-based

Delaunay-based meshing Algorithmic issues

An introduction to mesh generationPart II : Delaunay-based mesh generation

techniques

Jean-François Remacle

Department of Civil Engineering,Université catholique de Louvain, Belgium

Jean-François Remacle Mesh Generation


Mediator

Let S1 and S2 be two points of R2. We note S1S2 the line going from S1 toS2. The mediator M(S1,S2) is the locus of all the points which areequidistant to S1 and S2.

M(S1,S2) = {P ∈ R2,d(P,S1) = d(P,S2)}

where d(., .) is the euclidian distance between two points of R2.Geometrically, it is the orthogonal bissector of the segment of line betweenthe two points.

S1 S2 R2 S1S2 S1

S2 M(S1, S2) S1 S2

M(S1, S2) = {P ! R2 | d(P, S1) = d(P, S2)},d(·, ·)

S1

d(P, S1) P

d(P, S2)

S2

M

S1 S2

S1 S2

"



Partition of the plane

The mediator separates the plane into two regions. The first region containsall the points that are closer to S1, the second one contains the ones that arecloser to S2. Any point of R2M are therefore associated to one of those twopoints.Note that

this relation depends on the way distances are computed,this relation can be defined in Rd.



The Voronoï diagram

Let us consider a set of N points S = {S1, . . . ,SN }.S = {Si}i=1,...,N N

S2

S7

S10

S8

S4S9

S1

S11

S3

S12S5

S6

S2

S7

S10

S8

S4S9

S1

S11

S3

S12S5

S6

!

S2

S7

S10

S8

S4S9

S1

S11

S3

S12S5

S6

C(Si) Si

Si

C(Si) = {P " R2 | d(P, Si) # d(P, Sj), $j %= i}.

!




We assume that there exists no triplet of points in S that are colinear

S = {Si}i=1,...,N N

S2

S7

S10

S8

S4S9

S1

S11

S3

S12S5

S6

S2

S7

S10

S8

S4S9

S1

S11

S3

S12S5

S6

!

S2

S7

S10

S8

S4S9

S1

S11

S3

S12S5

S6

C(Si) Si

Si


!




We assume that there exists no quadruplet of points in S that are cocircularS = {Si}i=1,...,N N

S2

S7

S10

S8

S4S9

S1

S11

S3

S12S5

S6

S2

S7

S10

S8

S4S9

S1

S11

S3

S12S5

S6

!

S2

S7

S10

S8

S4S9

S1

S11

S3

S12S5

S6

C(Si) Si

Si


!




The Voronoï cell C(Si) associated to point Si is the locus of points of R2

that are closer to Si than any other point Sj, j = 1, . . . ,N, i 6= j.

S = {Si}i=1,...,N N

S2

S7

S10

S8

S4S9

S1

S11

S3

S12S5

S6

S2

S7

S10

S8

S4S9

S1

S11

S3

S12S5

S6

!

S2

S7

S10

S8

S4S9

S1

S11

S3

S12S5

S6

C(Si) Si

Si


!



Construction of the Voronoï diagram

The Voronoï cell is constructed as follow. Consider all mediators M(Si,Sj).Each of those mediators define a half plane. The Voronoï diagram is the partof the space that is always closer to Si, i.e. that always associate Si to thepoint P, for all possible mediators.

M(Si, Sj) Si

Sj

!

•N

N

••

•

•

•

!



Construction of the Voronoï diagram

The set of all Voronoï cells is called the Voronoï diagram of S.

M(Si, Sj) Si

Sj

!

•N

N

••

•

•

•

!



Properties of the Voronoï diagram

Voronoï cells are polygons in 2D,polyhedra in 3D and d-polytopes in dD,Voronoï cells are convex,The vertices of the Voronoï are alwayslocated at intersection of 3 mediators,Voronoï cells are either closed or open.They can only be open for points that arelocated on the convex hull of the 2Ddomain.Any points is loacted at the centroïd ofits Voronoï cell.

M(Si, Sj) Si

Sj

!

•N

N

••

•

•

•

!Jean-François Remacle Mesh Generation




M(Si, Sj) Si

Sj

!

•N

N

••

•

•

•





M(Si, Sj) Si

Sj

!

•N

N

••

•

•

•





M(Si, Sj) Si

Sj

!

•N

N

••

•

•

•





M(Si, Sj) Si

Sj

!

•N

N

••

•

•

•



Triangulation

The set of points S can be triangulated. This means that we can build a setof triangles with the Si’s as vertices. Note that

All triangulations do not cover the same subset of R2 (this is not thecase in our figure),Even with the same domain to cover, several triangulations arepossible.

M(Si, Sj) Si

Sj

!

•N

N

••

•

•

•

!



Delaunay triangulation

Among all possible triangulations, there exists one triangulation based onthe Voronoï diagram. This triangulation is build up as follows:

Each centroid of a Voronoï cell is a point of the triangulation,Each of these centroid is linked with all its neighbors in the Voronoïdiagram,The resulting figure is a triangulation because Voronoï points arealways the meeting points of 3 mediators.

S3

S11S10

S2

S6

S1

S5

S7

S4

S12

S9

S8

S3

S11S10

S2

S6

S1

S5

S7

S4

S12

S9

S8

!

K N

T B N

N + 1 K

K

K

K

K

K

K

!



Delaunay triangulation

Because the Voronoï diagram is unique, the Delaunay triangulation is alsounique.

S3

S11S10

S2

S6

S1

S5

S7

S4

S12

S9

S8

S3

S11S10

S2

S6

S1

S5

S7

S4

S12

S9

S8

!

K N

T B N

N + 1 K

K

K

K

K

K

K



Properties of the Delaunay triangulation

By construction, the Delaunay triangulation has the following propertiesElements are simplices (triangles in 2D, tetraedra in 3D...),Elements cover the convex hull and do not overlap,Elements respect the max-min angle principle: the minimum angle forall corners of triangles is maximal in the Delaunay triangulation.

As corrolaries, three very important properties can be establishedElements of the Delaunay triangulation all satisfy the empty sphereproperty,Equilateral triangulation,Duality between Delaunay and Voronoï




























The empty sphere property

Let Ki be a simplex in dD of a triangulation T. Let Bi be the ball that passesthrough the d +1 points of Ki. This ball is called the circumsphere of Ki.A simplex Ki satisfies the empty sphere property if its circumscribed spheredoes not contain any other point of the triangulation.

S3

S11S10

S2

S6

S1

S5

S7

S4

S12

S9

S8

S3

S11S10

S2

S6

S1

S5

S7

S4

S12

S9

S8

!

K N

T B N

N + 1 K

K

K

K

K

K

K

!



The empty sphere property

Any simplex Ki of the Delaunay triangulation satisfiy the empty sphereproperty.If every simplex Ki of a triangulation satisfies the empty sphereproperty, then it is a Delaunay triangulation.

Delaunay ⇐⇒ Empty Sphere

K

K

K

K

K

K

K T

!

S3

S11S10

S2

S6

S1

S5

S7

S4

S12

S9

S8

•

•

•

•

•

!



Duality Delaunay-Voronoï

To each vertex of the Delaunaytriangulation is associated aVoronoï cell,To each edge of the Delaunaytriangulation is associated anedge (piece of a mediator) of theVoronoï diagram,To each triangle of the Delaunaytriangulation is associated avertex of the Voronoï diagram.



Duality Delaunay-Voronoï (in 3D)

To each vertex of the Delaunay tetraedrization is associated a Voronoïcell,To each edge of the Delaunay tetraedrization is associated a face (pieceof a mediator plane) of the Voronoï diagram,To each face of the Delaunay tetraedrization is associated an edge ofthe Voronoï diagram,To each tetraedra of the Delaunay tetraedrization is associated a vertexof the Voronoï diagram.



Maxmin property

The Delaunay triangulationmaximizes the minimum of angles formes by triangle edges,minimizes the maximal circumcircle radiuses of triangles in thetriangulation

This property is only valid in 2D. In 2D,

Lemma : if there exists a triangulation for which all angles are acute, it is aDelaunay triangulation.



Maxmin property

The Delaunay triangulationmaximizes the minimum of angles formes by triangle edges,minimizes the maximal circumcircle radiuses of triangles in thetriangulation

This property is only valid in 2D. In 2D,

Lemma : if there exists a triangulation for which all angles are acute, it is aDelaunay triangulation.



Uniqueness of the Delaunay triangulation (in dD)

The Delaunay triangulation of a set of points S is uniqueif no set of 4 points are co-cyclical in 2Dif no set of 5 points are co-spherical in 3D...

N

•

NT S

••

!

N

NT

S

T T



Minmax properties (in dD)

If all simplices of Delaunay triangulation contain their circumcenter, itis a Delaunay triangulation.This property is the generalization of the acute angle condition in dD.

In a Delaunay triangulation, the sum of the squares of edge lengthsdivided by the volume of elements adjacent to these edges is minimal.

A Delaunay triangulation is regular, in some sense. Yet, Delaunaytriangulations are not the mostly regular triangulations. In 3D,delaunay triangulation often contain slivers which are elements of verybad quality for computation.

22 J.F. REMACLE, X. LI, M.S. SHEPHARD AND J.E. FLAHERTY.

by the three squares) and a mesh vertex (indicated by the circle). Table II lists the promising localmesh modification operation(s) to be evaluated for each type. To be effective, mesh modifications areevaluated in three priority levels and local mesh modifications in the next priority level are evaluateduntil all mesh modification in previous level can not improve local mesh configuration. In case morethan one operation is possible, the one producing best local mesh configuration is applied.

two key edges

key face

key vertex

type I type II

Figure 12. Sliver tetrahedron types and associated key entities.

Priority Mesh modifications for type I Mesh modifications for type II1 swap either key mesh edge swap the key face or relocate the key vertex2 split either key edge and relocate the new vertex, split the face then split/relocate the new vertex,

split both edges and collapse the interior edge swap the three edges that bound the face3 relocate vertices of the tetrahedron relocate the three vertices that bound the face

Table II. Determination of local mesh modifications.

5. RESULTS

5.1. Cannon Blast Problem

Consider the problem of a 2D perforated tube of diameter 155 mm (a section of a cannon). The diameterof the perforated holes inside the barrel (the muzzle break) are d = 28.6 mm.

The initial conditions for the problem are the one of a shock tube. We consider a virtual interfaceinside the barrel (see mesh refinement for t = 0 at Figure 13). The initial pressure for the gas insideof the tube are P = 57, 273, 627.96 Pa i.e. more than 500 times the external atmospheric pressure ofPatm = 101, 300 Pa. The initial temperature of the air inside the tube is T = 2, 111.5 K and its initialvelocity along x direction is 0.

The final time of the computation was tend = 5 ! 10!4 sec. A second order Runge-Kutta timeintegration scheme was used. The time steps were computed adaptively with a CFL limit of 1.0.Starting time steps were about 5 ! 10!8 seconds and the final time steps were about 1.5 ! 10!8

seconds. The mesh was refined every 10!6 seconds so that the total number of mesh refinements is501, including the initial refinement that enables the correct capture of the initial discontinuous state

Copyright c! 2000 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2000; 00:1–6Prepared using nmeauth.cls









two key edges

key face

key vertex

type I type II





5. RESULTS















two key edges

key face

key vertex

type I type II





5. RESULTS















two key edges

key face

key vertex

type I type II





5. RESULTS









Interior Cavity

We consider a triangulation T and a point P inside T. The cavity CPassociated to P consist in all the simplices for which their circumspherecontains P.

•

N

•

y

z

x S2

S4

S1

S3

!

u

erreur = "u # !u" $ CHK

!K

!!"2u!!

• HK K

• !K K

HK

!K

CP P TT

P

!



Exterior Cavity

We consider a triangulation T and a point P outside T. The cavity CPassociated to P consist in all the simplices for which their circumspherecontains P completed by all elements that can be formed by joining all theedges (faces in 3D) of T visible by P.

CP P TT P

P T P

P

P

BP P

T P

P P

!

Ti

i S = {Sj}j=1,...,N

Ti+1 Ti P = Si+1

Ti+1 = Ti " CP + BP .

Ti i

S Ti+1

P (i + 1)

Ti P "# CP

CP

P

PP

!



The Bowl

The cavity BP associated to a point P of a triangulation T consist in all thesimplices that are formes with P. The number of elements in the bowl isundertermined.

CP P TT P

P T P

P

P

BP P

T P

P P

!

Ti

i S = {Sj}j=1,...,N

Ti+1 Ti P = Si+1

Ti+1 = Ti " CP + BP .

Ti i

S Ti+1

P (i + 1)

Ti P "# CP

CP

P

PP

!



The Delaunay Kernel

Let us assume that we have build the delaunay triangulation Ti with the firsti points of the set S = S1, . . . ,SN .It is possible to build iteratively the delaunay triangulation Ti+1, i.e. usingTi and P = Si+1. We define the Delaunay kernel as:

Ti+1 = Ti −CP +BP

Theorem : if Ti is the Delaunay triangulation of the convex hull of the i first

points of a set S, then Ti+1, the triangulation constructed using theDelaunay kernel is a Delaunay triangulation Note : the Delaunay kernel is

usually localized in the neighborhood of p. The algorithmic complexityrelated to its computation can be considered as constant.



Insertion of an interior point P

From Ti and P = Si+1, we find CP,The cavity CP includes all elements for which their circumsphereincludes P,

CP P TT P

P T P

P

P

BP P

T P

P P

!

Ti

i S = {Sj}j=1,...,N

Ti+1 Ti P = Si+1

Ti+1 = Ti " CP + BP .

Ti i

S Ti+1

P (i + 1)

Ti P "# CP

CP

P

PP





CP P TT P

P T P

P

P

BP P

T P

P P

!

Ti

i S = {Sj}j=1,...,N

Ti+1 Ti P = Si+1

Ti+1 = Ti " CP + BP .

Ti i

S Ti+1

P (i + 1)

Ti P "# CP

CP

P

PP




The cavity CP is removed from the current triangulation,The bowl BP is added to the triangulation.Ti+1 = Ti −CP +BP

Ti+1 = Ti ! CP + BP

PP

Ti P !" CP

P

P

P

P

#

Ti+1 = Ti ! CP + BP

P

P

P

P

S = {Sj}j=1,...,N

T0

S

Sj j=1,...,N

S

#





Ti+1 = Ti ! CP + BP

PP

Ti P !" CP

P

P

P

P

#

Ti+1 = Ti ! CP + BP

P

P

P

P

S = {Sj}j=1,...,N

T0

S

Sj j=1,...,N

S

#





Ti+1 = Ti ! CP + BP

PP

Ti P !" CP

P

P

P

P

#

Ti+1 = Ti ! CP + BP

P

P

P

P

S = {Sj}j=1,...,N

T0

S

Sj j=1,...,N

S

#



Insertion of an exterior point P


Ti+1 = Ti ! CP + BP

PP

Ti P !" CP

P

P

P

P

#

Ti+1 = Ti ! CP + BP

P

P

P

P

S = {Sj}j=1,...,N

T0

S

Sj j=1,...,N

S

#





Ti+1 = Ti ! CP + BP

PP

Ti P !" CP

P

P

P

P

#

Ti+1 = Ti ! CP + BP

P

P

P

P

S = {Sj}j=1,...,N

T0

S

Sj j=1,...,N

S

#





Ti+1 = Ti ! CP + BP

PP

Ti P !" CP

P

P

P

P

#

Ti+1 = Ti ! CP + BP

P

P

P

P

S = {Sj}j=1,...,N

T0

S

Sj j=1,...,N

S

#





Ti+1 = Ti ! CP + BP

PP

Ti P !" CP

P

P

P

P

#

Ti+1 = Ti ! CP + BP

P

P

P

P

S = {Sj}j=1,...,N

T0

S

Sj j=1,...,N

S

#





Ti+1 = Ti ! CP + BP

PP

Ti P !" CP

P

P

P

P

#

Ti+1 = Ti ! CP + BP

P

P

P

P

S = {Sj}j=1,...,N

T0

S

Sj j=1,...,N

S

#



A Delaunay mesher, how it works!

4 additional points are added that encompass the convex hull of S. Twotriangles are first inserted in the triangulation. The advantage is that onlyinterior points will be inserted!

T0

S = {Sj}j=1,...,N

P CP

!

BP P

Ti+1




The Delaunay Kernel is used sequentially for all points of S.T0

S = {Sj}j=1,...,N

P CP

!

BP P

Ti+1

!

T0

S = {Sj}j=1,...,N

P CP

!

BP P

Ti+1

!




At the end, the elements that contain the four points of the box areremoved. The resulting mesh does not contain necessary the edges of theconvex hull of S.

P

P

P

!

CP

P

Ti " CP

P

P

P

Ti+1 = Ti " CP + BP

P

P

!



Computing the Delaunay Kernel ?

If the Delaunay Kernel is used for producing a mesh, the crucial pointis to efficiently find the Delaunay cavity CP of a point P,We’ll see later that a 2D triangulation of N points contains O(2N)

triangles.If all the O(2i) triangles of triangulation Ti have to be checked (point Pis inside or ouside the circumsphere), the algorithmic complexity ofthe algorithm is N2.Using the “4 points box trick” and inserting interior points one by oneusing the Delaunay kernel is perhaps not the best idea for triangulatinga known set of points.Yet, for most finite element applications, the set S is not known apriori.























Divide and Conquer

There exists a “Divide and Conquer” type of algorithm fortriangulating a known set of points.It allows a O(N logN) complexityDivide and conquer (DC) is an important algorithm design paradigm.It works by recursively breaking down a problem into two or moresub-problems of the same (or related) type, until these become simpleenough to be solved directly. The solutions to the sub-problems arethen combined to give a solution to the original problem.Gmsh provides this kind of algorithm (it is used at some point formesh generation, we’ll see later).



Divide and Conquer




Divide and Conquer




Divide and Conquer




Divide and Conquer

The Delaunay Divide and Conquer algorithm starts by sorting the N pointsof S lexicographically.



Divide and Conquer

Divide and Conquer algorithm for triangulations in two dimensions isdue to Lee and Schachter which was improved by Guibas and Stolfiand later by Dwyer. In this algorithm, one recursively draws a line tosplit the vertices into two sets. The Delaunay triangulation is computedfor each set, and then the two sets are merged along the splitting line.



Divide and Conquer

When the recursion comes to a point when 2 or 3 points are present atthe given level, trivial triangulation is performed. The convex hull oftriangulation parts have to be saved at each stage.



Divide and Conquer

Using some clever tricks, the merge operation can be done in timeO(N), so the total running time is O(N logN).



Divide and Conquer in Gmsh

The source code can be found in Mesh/DivideAndConquer.cc

The source codeint DelaunayAndVoronoi(DocPeek doc)

{

pPointArray = doc->points;

if(doc->numPoints < 2)

return 1;

qsort(doc->points, doc->numPoints, sizeof(PointRecord),

comparePoints);

recur_trig(0, doc->numPoints - 1);

return 1;

}




An example with 200 points

!" !"# $%"&'()# *%# *+,'-.# "'.,'# $%+# &"# $%"&'()# *+# &/'01&)0+.("('2.3# 452',# 6&",,+,# *+752.(#

!"#$%&'!$()*)+'# 8(5+# *)9'.'+,# :*/"%(5+,# 6&",,+,# 12%552.(# 8(5+# *)9'.'+,# +.# 2%(5+# *+#

6+&&+,# &;<3#=.+#6&",,+#12%5#&+,#12'.(,>#%.+#6&",,+#12%5#&+,#(5'".-&+,#+(#%.+#6&",,+#12%5#

&+,#*2..)+,3#!"#6&",,+#?#*2..)+#@#,+5"#%.+#6&",,+#,$('-)##)./-()#*2.(#*+752.(#*)5'7+5#

*+,# 6&",,+,# *+# *2..)+# ,6"&"'5+># 7+6(25'+&&+# +(# (+.,25'+&&+3# A2(+B# $%+# 72%,# "&&+B# 8(5+#

"0+.),#;#0".'1%&+5#*+,#62.(+.+%5,#:&',(+,># ("C&+"%D<#*+#("'&&+#7"5'"C&+#:&+#.20C5+#*+#

(5'".-&+,# "%-0+.(+# +(# 6+5("'.,# (5'".-&+,# ,+52.(# )&'0'.),# *+# &"# &',(+# "%# 62%5,# *+#

&/"&-25'(E0+<3# !/%('&',"('2.# *+# &"# F(".*"5*# 4+01&"(+# !'C5"5G# :F4!>#

HHH3,-'3620I(+6EI,(&<#12%55"'(#72%,#9"6'&'(+5#-5".*+0+.(#&"#(J6E+3#

!"#$%&'()*

!+# *)7+&211+0+.(# *%# !"#$"%&&'# ,+# 9+5"# 2C&'-"(2'5+0+.(# *".,# &/+.7'52..+0+.(# !KA=L3#

M2%5# 6+%D# $%'# %('&',+5"'+.(# &+%5# 25*'."(+%5# 1+5,2..+&# ,2%,#N'.*2H,># 72%,# "7+B# &"# 12,,'C'&'()#

*/'.,("&&+5# %.# +.7'52..+0+.(# *+# (G1+# !KA=L# :-5"(%'(<# +.# ()&)6E"5-+".(# &/+.7'52..+0+.(#

6G-H'.# :HHH36G-H'.3620<3# F'# 72%,# *+7'+B# '.,("&&+5# 6G-H'.># '.,("&&+B# &/+.,+0C&+# *+,#

62012,".(,>#G#62015',#&+,#2%('&,#*+#*)7+&211+0+.(3#K&#+D',(+#%.+#7+5,'2.#0'.'0"&+#*+#6G-H'.#

,%5#E((1OIIHHH30'.-H325-I#$%+#.2%,#./"72.,#1",#75"'0+.(# (+,()+#0"',#$%'# P2%'(#*/%.+#C2..+#

5)1%("('2.3# K&#./+,(#)7'*+00+.(#1",# '.(+5*'(#*+#*)7+&211+5#72(5+#62*+#+.#%('&',".(#"%(5+#6E2,+#

0"',# '&# 9"%(# ,"72'5# $%+# 6+5("'.,# &2-'6'+&,# 6200+56'"%D# C'+.# 62..%,# .+# 92%5.',,+.(# 1",# %.#

6201'&"(+%5# QRR# $%'# 5+,1+6(+# ,65%1%&+%,+0+.(# &"# .250+# SAFK3# T2(5+# 152-5"00+# 12%55"'(#

*2.6#(5U,#C'+.#6201'&+5#,%5#72(5+#1&"(+9250+#0"',#MSF#,2%,#!KA=L3##

!+# &2-'6'+&# -0,E# +,(# ()&)6E"5-+"C&+# +.# 7+5,'2.# +D)6%("C&+# 12%5# &"# 1&%1"5(# *+,# 1&"(+9250+,#

,(".*"5*# :HHH3-+%B325-I-0,E<3# K&# +,(# '.,("&&)# +(# %('&',"C&+# ,%5# &+,# 0"6E'.+,# *%# 5),+"%#

'.9250"('$%+#9"6%&("'5+3#!/+D)6%("C&+#,+#(52%7+#*".,#($')*+,-./*).$&,012T2%,#12%7+B#

"P2%(+5#&"#&'-.+#,%'7".(+#

'3!#"42 56789:5678;($')*+,-./*)2

*".,#&+#9'6E'+5#1/%,0"<#$%'#,+#(52%7+#;#&"#5"6'.+#*+#72(5+#6201(+#%('&',"(+%53#Q+#9"',".(>#72%,#

12%55+B#("1+5#,'01&+0+.(#-0,E#;#1"5('5#*+#./'0125(+#$%+&#5)1+5(2'5+3#

+,-./%-*

T2'6'#%.#+D+01&+#,'01&+#$%'#02.(5+#&+#(G1+#*+#5),%&("(,#"((+.*%,#:&+,#*2..)+,#,2.(#*',12.'C&+,#

,%5#&+#,'(+#N+C#*%#62%5,<3#A2%,#"72.,#-).)5)#;#&/"'*+#*+#VS4!SW#(52',#9'6E'+5,#XD+01&+Y3'.>#

XD+01&+Z3'.# +(#XD+01&+[3'.# $%'# 62.('+..+.(# 5+,1+6('7+0+.(#Z\\>#Z\\\#+(# Z\\\\#12'.(,# +.(5+#

*".,# &+,# '.(+57"&&+,# L#]^Z$# +(# Z$_# +(# `#]^Z$# +(# Z$_3# !+,# 12'.(,# ,2.(# 6E2',',# "%# E","5*#

:92.6('2.#5".*<#+(>#+.#6E"$%+#12'.(#L#`#:5>"%<>#2.#"#6"&6%&)#&"#92.6('2.#7:5>"%<abY:5<,'.:c%<3#!+,#

5),%&("(,#,2.(#15),+.(),#*".,#&+,#d'-%5+,#[>e#+(#c3#

## #

d'-%5+#[#O#f),%&("(,#*+#&"#(5'".-%&"('2.#12%5#&/+D+01&+#XD+01&+Y3'.#"7+6#Z\\#12'.(,3#




An example with 2000 points!

!!

!

"#$%&'!(!)!*+,%-./.,!0'!-/!.&#/1$%-/.#21!32%&!-4'5'63-'!75'63-'89#1!/:';!8<<<!32#1.,9!

!

!!

!

"#$%&'!=!)!*+,%-./.,!0'!-/!.&#/1$%-/.#21!32%&!-4'5'63-'!75'63-'>9#1!/:';!8<<<<!32#1.,9!

?2.'@!A%'B!32%&!-',!8<<<<!32#1.,B!-4/-$2&#.C6'!%.#-#,+!3/&!12,!,2#1,!/!1+;',,#.+!D'/%;2%3!62#1,!

04%1'!,';210'!32%&!'EE';.%'&!,21!.&/:/#-!FG%,.'!32%&!:2%,!+1'&:'&!%1!3'%H9!I'-%#!A%#!3/&:#'10&/B!

'1!2%.&'B!J!&+3210&'!J!-/!A%',.#21!,%#:/1.'!D+1+E#;#'&/!0'!12.&'!6/1,%+.%0'!)!

!" 75#,.'K.K#-!%1'!&'-/.#21!'1.&'!-'!126D&'!0'!.&#/1$-',!'.!-'!126D&'!0'!32#1.,!F%1'!E2&6%-'!

A%#!,'&/#.!#10+3'10/1.'!0'!-/!.&#/1$%-/.#21H!L!

M21!.&/:/#-9!




An example with 20000 points

!

!!

!

"#$%&'!(!)!*+,%-./.,!0'!-/!.&#/1$%-/.#21!32%&!-4'5'63-'!75'63-'89#1!/:';!8<<<!32#1.,9!

!

!!

!

"#$%&'!=!)!*+,%-./.,!0'!-/!.&#/1$%-/.#21!32%&!-4'5'63-'!75'63-'>9#1!/:';!8<<<<!32#1.,9!

?2.'@!A%'B!32%&!-',!8<<<<!32#1.,B!-4/-$2&#.C6'!%.#-#,+!3/&!12,!,2#1,!/!1+;',,#.+!D'/%;2%3!62#1,!

04%1'!,';210'!32%&!'EE';.%'&!,21!.&/:/#-!FG%,.'!32%&!:2%,!+1'&:'&!%1!3'%H9!I'-%#!A%#!3/&:#'10&/B!

'1!2%.&'B!J!&+3210&'!J!-/!A%',.#21!,%#:/1.'!D+1+E#;#'&/!0'!12.&'!6/1,%+.%0'!)!

!" 75#,.'K.K#-!%1'!&'-/.#21!'1.&'!-'!126D&'!0'!.&#/1$-',!'.!-'!126D&'!0'!32#1.,!F%1'!E2&6%-'!

A%#!,'&/#.!#10+3'10/1.'!0'!-/!.&#/1$%-/.#21H!L!

M21!.&/:/#-9!



Geometrical Predicates

The exactness of the Divide and Conquer algorithm heavily relies onthe robustness of one Geometrical Predicate.A predicate is an operator or function which returns a boolean value,true or false.We already know that the computation of the Delaunay kernel alsorelies on the robustness of another Geometrical Predicate (inside oroutside the circumsphere).In general, a predicate is an operator or function which returns aboolean value, true or false.Many other computational geometry applications use numerical testsknown as the orientation and incircle tests.




















The orientation test determines whether a point lies to the left of, to theright of, or on a line or plane defined by other points.The incircle test determines whether a point lies inside, outside, or on acircle defined by other points.Each of these tests is performed by evaluating the sign of adeterminant. The determinant is expressed in terms of the coordinatesof the points.If these coordinates are expressed as single or double precisionfloating-point numbers, roundoff error may lead to an incorrect resultwhen the true determinant is near zero.In turn, this misinformation can lead an application to fail or produceincorrect results.

























Lawson’s Algorithm

Lawson’s algorithm allow to insert a new point on a given edge orinside a given triangle of an existing triangulationThe number of operations for the insertion is constant. In consequense,the mesh generation process can be performed with a linearcomplexity, i.e. a O(N) complexity.Lawson’s algorithm is well adapted to finite element mesh generation.in this case, the points of S are not known in advance.











The edge swap mesh modification operator

In 2D, we consider two triangles that share the same edge,This edge is one of the diagonals of a quadrilateral that is made of therecombination of the two triangles,The edge swapping operator consist in changing the diagonal in thequadrilateral

S2

S4

a4T2a1S1

T4T3

a2 a3T1

T5T6

a5

S3

S2

S4

a4

a1S1

T4T3

a2 a3

T5T6

a5

S3T1 T2

E E ! S

!!

"





S2

S4

a4T2a1S1

T4T3

a2 a3T1

T5T6

a5

S3

S2

S4

a4

a1S1

T4T3

a2 a3

T5T6

a5

S3T1 T2

E E ! S

!!

"





S2

S4

a4T2a1S1

T4T3

a2 a3T1

T5T6

a5

S3

S2

S4

a4

a1S1

T4T3

a2 a3

T5T6

a5

S3T1 T2

E E ! S

!!

"




The edge swapping operator is a purely topological operator. It doesnot modify the position of the vertices but only changes theconnectivity of the mesh

S2

S4

a4T2a1S1

T4T3

a2 a3T1

T5T6

a5

S3

S2

S4

a4

a1S1

T4T3

a2 a3

T5T6

a5

S3T1 T2

E E ! S

!!

"




Edge swap cannot be applied ifThe edge has only one neighboring triangle (if the edge is a boundaryedge of the domain)The quadrilaterai is concave.

TDel

S3 S2 S1 S4

S3

T2

T1S2

S1 S4

S3

S2

T1 T2

S1 S4

T2

S2

S1 S4

S3

T1

S1 S4

T1 T2

!

T1

S TDel

T1 TDel

T1

S T2

S

!




Edge swap cannot be applied ifThe edge has only one neighboring triangle (if the edge is a boundaryedge of the domain)The quadrilaterai is concave.

TDel

S3 S2 S1 S4

S3

T2

T1S2

S1 S4

S3

S2

T1 T2

S1 S4

T2

S2

S1 S4

S3

T1

S1 S4

T1 T2

!

T1

S TDel

T1 TDel

T1

S T2

S

!



Multiple edge swaps

Theorem : All triangulations of a given convex hull are equivalent withrespect to a finite number of edge swaps

Corollary : The unique Delaunay triangulation of a convex hull can beconstructed starting from any triangulation, thanks to a finite number ofswaps.

TDel

S3 S2 S1 S4

S3

T2

T1S2

S1 S4

S3

S2

T1 T2

S1 S4

T2

S2

S1 S4

S3

T1

S1 S4

T1 T2

!

T1

S TDel

T1 TDel

T1

S T2

S

!



Edge swap vs. Delaunay

If two adjacent triangles violate the empty sphere criterion, then we swapthe edge and the empty sphere criterion is satisfied for those two triangles.

S1

S3

S2

S4

T2T1

!

maxmin

maxmin

" "

!3

!2

!6

!4

!5!1!4 !6

!3

!5

!2!1

T

!




If the quadrilateral is concave, no edge swap is possible but ...If the quadrilateral is concave, it satisfies the empty sphere criterion.

S1

S3

S2

S4

T2T1

!

maxmin

maxmin

" "

!3

!2

!6

!4

!5!1!4 !6

!3

!5

!2!1

T

!




If the quadrilateral is concave, no edge swap is possible but ...If the quadrilateral is concave, it satisfies the empty sphere criterion.

S1

S3

S2

S4

T2T1

!

maxmin

maxmin

" "

!3

!2

!6

!4

!5!1!4 !6

!3

!5

!2!1

T

!



Method of Lawson

Start with any triangulation,Swap all the edges for which the quadrilateral formed by the twoneighboring triangles violate the empty sphere criterion,Repeat that until every edge is ok.The algorithm usually converges in 4-5 iterations.



Method of Lawson




Method of Lawson




Method of Lawson




Mesh generation

We aim at intserting P in a triangulation,

Ti i

S Ti+1

P (i + 1)

P Ti

P

K

K Ti

P

!

P K K

P

Ti Ti+1

P

K

P

Ti+1 P

i + 1

!



Mesh generation

Find the element that includes P and split the element in 3sub-elements

Ti i

S Ti+1

P (i + 1)

P Ti

P

K

K Ti

P

!

P K K

P

Ti Ti+1

P

K

P

Ti+1 P

i + 1

!



Mesh generation

Use Lawson’s algorithm to improve the connexions,

Ti i

S Ti+1

P (i + 1)

P Ti

P

K

K Ti

P

!

P K K

P

Ti Ti+1

P

K

P

Ti+1 P

i + 1

!



Bowyer and Watson Algorithm

Another algorithm to build a Delaunay Triangulation.Theorem : The Delaunay cavity CP is star shaped and simplyconnected.

Let us consider Ti,Find a triangle that is too small (we’ll define later what this means),Insert the point P at the center of the circumcircle of this triangle thathas not the right size,Eliminate locally all triangle that violate the incircle property, thecavity CP is composed of all those eliminated triangles. For that, use adata structure for which every triangle knows about its neighbors,Triangulate the cavity.

































Bowyer and Watson in Gmsh

The complete header can be found inMesh/meshGFaceDelaunayInsertion.h

The source codestruct MTri3

{

MVertex *v[3];

MTri3 *neigh[3];

};




Illustration:

!"#$%&'()*+,*+-$./*%+*'+01'2$3+

!"# $%&"'()*"$&+'#,-#.-*")'"/#0+112,-#3-%$"&'-1#0%+0%&4$41# &'$4%-11"'$-15#6+)1#-'#3&$-%+'1#)'-#

7)&# 8"# '+)1# 1-%8&%# 0+)%# *"# 3%4"$&+'# ,9)'# $%&"'()*"$-)%# ")$+:"$&7)-# ,-# .-*")'"/5# ;+&$# )'#

-'1-:<*-# ,-# 0+&'$1# =# ># ?!@A555A!6BA# !!&!C!D# -1$# )'# $%&"'(*-# ,-# .-*")'"/# ,-# .EF=G# 1&# -$#

1-)*-:-'$# 1&# 1+'# 3-%3*-# 3&%3+'13%&$# '-# 3+'$&-'$# ")3)'# ")$%-# 0+&'$# 7)-# !&A# !C# -$# !D5# H9-1$# *-#

$I4+%2:-#,)#3-%3*-5#

6+$+'1#7)J&*#-K&1$-#,-#'+:<%-)K#"*(+%&$I:-1#0-%:-$$"'$#,-#$%&"'()*-%#)'#')"(-#,-#0+&'$1#1"'1#

0"11-%#0"%#*"#(4'4%"$&+'#,)#,&"(%"::-#,-#=+%+'+L5#

!9"*(+%&$I:-#,-#M+N/-%#-1$#)'#"*(+%&$I:-#&$4%"$&O5#P*#)$&*&1-#*-#$I4+%2:-#,)#3-%3*-5#H+'1&,4%+'1#

*"#$%&"'()*"$&+'#,-#.-*")'"/#.EF=6G#"8-3#=6#>#?!@A555A!6B5#;+&$#)'#0+&'$#!6Q@#"#RF=6G#FS&()%-#

TG5#!9"*(+%&$I:-#1)&8"'$#0-%:-$#,-#(4'4%-%#*"#$%&"'()*"$&+'#,-#.-*")'"/#.EF=6Q@G#"8-3#=6Q@#>#

?"!@A555A!6A#!6Q@#B5#

$# U'# 3+::-'3-# 0"%# 4*&:&'-%# $+)1# *-1# $%&"'(*-1# ,-# .EF=6G# ,+'$# *-# 3-%3*-# 3&%3+'13%&$#

3+'$&-'$#!6Q@A#-$#3-*"#-'#"33+%,#"8-3#*-#$I4+%2:-#,)#3-%3*-5#P*#-1$#0+11&<*-#,-#,4:+'$%-%#

7)-#*-#0+*/(+'-#VF=6A"!6Q@G#3+'1$%)&$#W#0"%$&%#,-1#$%&"'(*-1#4*&:&'41#-1$#3+'8-K-#-$#,9)'#

1-)*#$-'"'$#FS&()%-#TG5#

$# U'#%-3+'1$%)&$#*"#$%&"'()*"$&+'#-'#%-*&"'$#*-#0+&'$#!6Q@#")K#0+&'$1#,-#RFVG5#U'#+<$&-'$#

"&'1&#)'-#'+)8-**-#$%&"'()*"$&+'#,-#.-*")'"/#FS&()%-#TG5##

# ##

S&()%-#T#X#Y*(+%&$I:-#,-#M+N/-%#-$#Z"$1+'5#

41(&*%+,*2+5(1%#*2+

[-3+'1$%)&%-# )'# +<C-$# W# 0"%$&%# ,-# ,+''4-1# 0"%$&-**-1# -1$# )'# 0%+<*2:-# &:0+%$"'$# -$# "/"'$# ,-#

'+:<%-)1-1#"00*&3"$&+'1#,"'1#,-#'+:<%-)K#,+:"&'-15#V'#:4,-3&'-#0"%#-K-:0*-A#+'#3I-%3I-#W#

%-3+'1$%)&%-# ,-1# :+,2*-1# $%&\,&:-'1&+''-*1# ,-1# +%("'-1# W# 0"%$&%# ,-# 3+)0-1# 13"''-%# ]# -'#

(4+*+(&-A#*"#1$%)3$)%-#,)#1+)1\1+*#W#0"%$&%#,-#:-1)%-1#1&1:&7)-1A#-'#&'(4'&-%&-A#)'#:+,2*-#HYU#

W#0"%$&%#,-#0+&'$1#:-1)%41#1)%#)'-#:"7)-$$-5##

!"#$%&"'()*"$&+'#,-#.-*")'"/#-$#*-#,&"(%"::-#,-#=+%+'+L#,4O&'&11-'$#)'-#$-3I'&7)-#0-%:-$$"'$#

,-#%-3+'1$%)&%-#)'-#1)%O"3-#T.#F+)#)'#8+*):-#1&#+'#$%"8"&**-#-'#^.GA#1&#*9+'#,&10+1-#)'&7)-:-'$#

,-#0+&'$1#1)%#3-$$-#1)%O"3-#F+)#1)%#3-#8+*):-G5#

!-# <)$# ,-# 3-# 0%+C-$# -1$# ,-# 3+'1$%)&%-# )'-# $%&"'()*"$&+'# ,-# .-*")'"/# W# 0"%$&%# ,J)'-# *&1$-# ,-#

,+''4-15#H-1#,+''4-1#1+'$#O+)%'&-1#1+)1#*"#O+%:-#,J)'#O&3I&-%#+%("'&14#,-#*"#O"_+'#1)&8"'$-#X#


Documents

An introduction to mesh generation Part II : Delaunay …perso.uclouvain.be/.../teaching/documents/meca2170-jfr-cours2.pdf · An introduction to mesh generation Part II : Delaunay-based