An introduction to mesh generation Part III : Finite...

Finite Element Mesh Generation

An introduction to mesh generationPart III : Finite Element Mesh Generation

Jean-François Remacle

Department of Civil Engineering,Université catholique de Louvain, Belgium

Jean-François Remacle Mesh Generation


Euler-Poincaré

A mesh M is a geometrical discretization of a domain Ω that consists ofA collection of mesh entities Md

i of controlled size and distribution;Topological relationships or adjacencies forming the graph of themesh.In unstructured meshes, those relationships are explicit, i.e. they haveto be given explicitely.In other words, the relations between the number of mesh vertices,edges, faces and regions is unknown a prioriYet, topology provides some general relations.. explicitely.



Euler-Poincaré (3D)

The Euler-Poincaré formula describes the relationship of the number ofvertices, the number of edges and the number of faces of the cellulardecomposition (a mesh) of a manifold. It has been generalized to includepotholes and holes that penetrate the solid. To state the Euler-Poincaréformula, we need the following definitions:

#V is the number of vertices in the mesh,#E is the number of edges in the mesh,#F is the number of faces in the mesh.G is the number of holes that penetrate the solid, usually referred to asgenus in topologyS is the number of shells. A shell is an internal void of a solid. A shellis bounded by a 2-manifold surface, which can have its own genusvalue. Note that the solid itself is counted as a shell. Therefore, thevalue for #S is at least 1.L is the number of loops. All outer and inner loops of faces arecounted.




The Euler-Poincaré formula is

#V −#E +#F −(L−#F)−2(S−G) = 0.

A cube has eight vertices (#V = 8), 12 edges (#E = 12) and six faces(#F = 6), no holes and one shell (S = 1); but #L = #F since each facehas only one outer loop. Therefore, we have

#V −#E+#F−(L−#F)−2(S−G) = 8−12+6−(6−6)−2(1−0) = 0.



Euler-Poincaré

The following solid has 16 vertices, 24 edges, 10 faces, 1 hole (i.e.,genus is 1), 1 shell and 12 loops (10 faces + 2 inner loops on top andbottom faces). Therefore,

#V −#E+#F−(L−#F)−2(S−G) = 16−24+10−(12−10)−2(1−1) = 0.




Consider M, a 2D mesh of a domain Ω.#V is the number of vertices in the mesh,#E is the number of edges in the mesh,#F is the number of faces in the mesh.

The Euler-Poincaré relation gives the following relation between thosequantities:

#V −#E +#F −χ(Ω) = 0

where χ(Ω) is the Euler-Poincaré characteristic of the surface.




The genus of different surfaces is given

for the sphere, χ = 2 (#V −#E +#F = 2−4+4),for the torus, χ = 0 (#V −#E +#F = 4−8+4),for the disk, χ = 1,for the Klein bootle, χ = 1.




Another form of the relation, more useful for general domains:

χ = #V −#E +#F = 2−2g+b

whereb is the number of boundaries (1 for the plane or 0 for a torus or asphere),g is the genus of the surface. The genus is the largest number ofnonintersecting simple closed curves that can be drawn on the surfacewithout separating it. Roughly speaking, it is the number of holes in asurface.



Euler-Poincaré (triangular meshes)

We consider a mesh of a domain that is isomorph to a disk. Then, thefollowing relation holds

#F +2(#V −1)−#Vb = 0

where #Vb is the number of vertices on b.This is an important relation that gives a relation between the number oftriangles and the number of vertices in a triangular mesh.

Demonstration :The relation is true for one only triangle.All triangulations with #N given are equivalent. Edge swaps allow totransform a given triangulation to any other.An edge swap does not modify neither #V nor#F.Inserting a point inside a triangle adds one vertex, 2 triangles and 3edge, so the relation is recurrent.Inserting a point on the boundary adds one vertex, one triangle, 2edges and one boundary point.



Euler-Poincaré (triangular meshes)

Asymptotically#F =' 2#V , there are about 2 times more triangles than nodes.#V −#E +#F ' 0, which gives 3#V ' #E : the number of edges isabout 3 times the number of vertices.



Euler-Poincaré (tet. meshes)

We consider a tet. mesh of a 3D domain that is isomorphic to a sphere. Thefollowing relation holds

#E −#R = #V +#Vb −3

where #Vb is the number of vertices on the boundary.

This relation states that, for any tetraedrization of the domain, the differencebetween the number of edges and elements is constant.

There exist no relation between the number of tets. and the number ofnodes, as it exists in 2D. For exmaple, it is possible to define a “face swap”transformation : 2 tets that have a face in common are transformed in 3 tetswithout changing the number of vertices. Yet one additional edge has to beadded to the mesh in order to verify the relation.



Euler-Poincaré (tet. meshes)

Asymptotically Euler-Poincaré gives:

#V −#E +#F −#R' 0.

This relation is true if #E = (n+1)#V , #F = 2n#V and #R = n#V for anypoistive integer n.

It seems that n = 6 is a good bound. For very regular fine 3D tet. meshes, agood bound is n = 5.6.

Tetrahedral Mesh T Hexahedral Mesh H#R(T) = 6#V(T)

#F(T) = 12#V(T)

#E(T) = 7#V(T)

#R(H) = #V(H)

#F(H) = 3#V(H)

#E(H) = 3#V(H)

Table: Relation between number of entities in a mesh.



Other statistics

A second interesting set of statistics concerns the average number of meshentities of dimension d adjacent to a mesh entity of dimension q. We callthis Nd(Mq). These statistics are represented in this table

Tetrahedral Mesh T Hexahedral Mesh Hd 3 2 1 0

N3(Md) 1 2 5 23N2(Md) 4 1 5 35N1(Md) 6 3 1 14N0(Md) 4 3 2 1

d 3 2 1 0N3(Md) 1 2 4 8N2(Md) 6 1 4 12N1(Md) 12 4 1 6N0(Md) 8 4 2 1

Table: Average number of adjacencies per entity



Data structures

Each mesh generation algorithms require an appropriate choice of thedata structures,The issue is principally the memory signature of the mesh generator,not the c.p.u.Any “good” 3D mesh generator is able to build more that a milliontets/minute.Large finite element computations can take days of computations onparallel computers. So, waiting 10 minutes for building a 3D mesh ofmore than a million nodes is not an issue.Most meshers cannot build meshes of 1 million nodes on oneprocessor without saturating the memory.Consequently, data structures should be as light as possible.



Data structures

If we consider that each adjacency has a unit storage, the total storage costfor a given representation can be computed as follow:

C(I,M) =

4∑d=0

4∑q=0

Nd(M)Id,qNd(Mq) (1)

A mesh representation is said to be complete if any adjacencies can beretrieved for any mesh entity without a global traversal of the mesh. In theother case, the representations are termed incomplete.

In a complete representation, any adjacency requires a number of operationsthat does not depend on the size of the mesh. In an incompleterepresentation, getting some adjacencies will require a complete traversal ofthe mesh containers (referred to as linear behavior). It is evident that wecannot afford this traversal each time we ask for an adjacency. Completerepresentations are then the only acceptable mesh representations if we areto work with a single I for all the mesh related algorithms.



Incidence matrices

I2 =

1 0 0 00 0 0 00 0 0 01 0 0 1

I3 =

1 0 0 01 1 0 01 1 1 01 1 1 1

I ′3 =

1 0 0 01 1 0 00 1 1 00 0 1 1

C(I2,T) = 31N0 C(I3,T) = 232N0 C(I ′

3,T) = 100N0

C(I2,H) = 10N0 C(I3,H) = 64N0 C(I ′3,H) = 32N0

I4 =

1 1 0 01 1 0 00 0 0 00 1 0 1

I5 =

1 0 0 10 0 0 00 0 0 01 0 0 1

I6 =

1 0 1 00 0 0 01 0 1 00 0 1 1

C(I4,T) = 103N0 C(I5,T) = 64N0 C(I6,T) = 164N0

C(I4,H) = 24N0 C(I5,H) = 18N0 C(I6,H) = 35N0

Table: Some classical mesh adjacencies requirements: I2 for Lagrangian finiteelements, I3 and I ′

3 for Hierarchical finite elements, I4 for Finite Volumes orDiscontinuous Galerkin solvers, I5 for mesh smoothing and I6 for edge swapping.



Complete representations

Iol =

1 1 0 01 1 1 00 1 1 10 0 1 1

Icirc =

1 0 0 11 1 0 00 1 1 00 0 1 1

C(Iol,T) = 163N0 C(Icirc,T) = 123N0

C(Iol,H) = 56N0 C(Icirc,H) = 33N0

Table: Some complete representations. The unidirectional representation is used inGmsh.



Mesh generation

Mesh generation often uses CAD modelsas input,A 3D model can be defined using itsBoundary Representation (BRep). Anyvolume (or region) is bounded by a set ofsurfaces and a surface is bounded by aserie of curves. Finally, a curve isbounded by two end points.Solid modelers usually provide an APIfor the creation, manipulation,interrogation and storage of 3D models.

C. Geuzaine and J.-F. Remacle · 5

Fig. 4. CAD model of a propeller (left) and its volume mesh (right)

There are various ways to define the mesh size field. In Gmsh, the user canprovide a mesh size field that is defined on another mesh (a background mesh) ofthe domain and/or provide mesh sizes associated with model entities or adapt themesh to the geometry of the model using curvatures. When two size fields !1(x, y, z)and !2(x, y, z) are provided, we use the minimum of both functions as size field.

3.2 1D mesh generation

Let us consider a curve "c(#) : [#1, #2] ! R3. The number of subdivisions N of thecurve is a function of the size field

N =

! !2

!1

1

!(x, y, z)"d!"c"d#.

The N +1 mesh points on the curve are located at coordinates #0, . . . , #N where#i is computed using the following rule

i =

! !i

!1

1

!(x, y, z)"d!"c"d#.

In Gmsh, we compute this primitive using a recursive integration rule.


Curved surface shapes designed by CAD system are usually defined by parametricsurfaces, for example, NURBS [?]. Let us consider a model face G2

i with its un-derlying geometry, in this case a surface S(u, v) # R3. The domain of definitionof parameters (u, v) is defined by a serie of boundary curves. Figure 5 shows oneof the 76 model faces of the propeller. Left part of the figure shows the surface inthe parametric space while the right one is for the real space. This surface con-

ACM Transactions on Mathematical Software, Vol. V, No. N, March 2007.



Geometry engineComputational Modeling Sciences Department

6

Geometry

vertices:

x,y,z location




7

Geometry

vertices:

x,y,z location

curves: bounded

by two vertices




8

Geometry

vertices:

x,y,z location

surfaces: closed

set of curvescurves: bounded

by two vertices




9

Geometry

vertices:

x,y,z location

surfaces: closed

set of curves

volumes: closed

set of surfaces

curves: bounded

by two vertices




11

Geometry

body: collection

of volumes

vertices:

x,y,z location

volumes: closed

set of surfaces

surfaces: closed

set of curves

loops: ordered

set of curves on

surface

curves: bounded

by two vertices




10

Geometry

body: collection

of volumes

vertices:

x,y,z location

surfaces: closed

set of curves

volumes: closed

set of surfaces

curves: bounded

by two vertices




12

Geometry

body: collection

of volumes

vertices:

x,y,z location

volumes: closed

set of surfaces

loops: ordered

set of curves on

surface

surfaces: closed

set of curves

(loops)coedges:

orientation of curve

w.r.t. loop

curves: bounded

by two vertices




13

Geometry

body: collection

of volumes

vertices:

x,y,z location

volumes: closed

set of surfaces

(shells)

surfaces: closed

set of curves

(loops)

loops: ordered

set of curves on

surface

coedges:


w.r.t. loop

shell:

oriented set

of surfaces

comprising

a volume

curves: bounded

by two vertices




15

Geometry

body: collection

of volumes

vertices:

x,y,z location

volumes: closed

set of surfaces

(shells)

surfaces: closed

set of curves

(loops)

loops: ordered

set of curves on

surface

coedges:


w.r.t. loop

shell:

oriented set

of surfaces

comprising

a volume

coface:

oriented

surface

w.r.t. shell

curves: bounded

by two vertices




16

Geometry

Volume 1

Surface 1 Surface 2 Surface 3 Surface 4 Surface 5 Surface 6

Volume 2

Surface 8 Surface 9 Surface 10 Surface 11

Surface 7

Volume 1

Volume 2

Surface 11

Surface 7

Manifold Geometry:

Each volume maintains

its own set of unique

surfaces




17

Geometry

Volume 1

Surface 1 Surface 2 Surface 3 Surface 4 Surface 5 Surface 6

Volume 2

Surface 8 Surface 9 Surface 10

Surface 7

Volume 1

Volume 2

Surface 7

Non-Manifold

Geometry: Volumes

share matching surfaces



Mesh generation

Mesh generation works as followsCurves are first discretized,Then surfaces are meshed using curve’sdiscretizations,Finallly, regions are meshed using surfacemeshes.

The mesh generation process’s aim is to insertpoints at the “right” locations i.e. a mesh thatrespects some criterion on element sizes.

C. Geuzaine and J.-F. Remacle · 5

Fig. 4. CAD model of a propeller (left) and its volume mesh (right)

There are various ways to define the mesh size field. In Gmsh, the user canprovide a mesh size field that is defined on another mesh (a background mesh) ofthe domain and/or provide mesh sizes associated with model entities or adapt themesh to the geometry of the model using curvatures. When two size fields !1(x, y, z)and !2(x, y, z) are provided, we use the minimum of both functions as size field.


Let us consider a curve "c(#) : [#1, #2] ! R3. The number of subdivisions N of thecurve is a function of the size field

N =

! !2

!1

1

!(x, y, z)"d!"c"d#.

The N +1 mesh points on the curve are located at coordinates #0, . . . , #N where#i is computed using the following rule

i =

! !i

!1

1

!(x, y, z)"d!"c"d#.



Curved surface shapes designed by CAD system are usually defined by parametricsurfaces, for example, NURBS [?]. Let us consider a model face G2

i with its un-derlying geometry, in this case a surface S(u, v) # R3. The domain of definitionof parameters (u, v) is defined by a serie of boundary curves. Figure 5 shows oneof the 76 model faces of the propeller. Left part of the figure shows the surface inthe parametric space while the right one is for the real space. This surface con-

ACM Transactions on Mathematical Software, Vol. V, No. N, March 2007.



Mesh generation

The appropriate means to ensure that a mesh-based numerical analysisprocedure produces the most effective solution results is to apply anadaptive solution strategy. Efforts on the development of adaptativestrategies applied to finite element computations have been underwayfor more that twenty years now. Such an adaptive strategy usuallyinvolve some kind of error estimation procedure together with a meshrefinement algorithm.We define the mesh size function δ(x,y,z) as a function the defines atevery point of the domain a target size for the elements at the point.The aim of the mesh generation procedure is to be able to buile a meshthat complies with the mesh size field.There are various ways to define the mesh size field. In Gmsh, the usercan provide a mesh size field that is defined on another mesh (abackground mesh) of the domain and/or provide mesh sizes associatedwith model entities or adapt the mesh to the geometry of the modelusing curvatures. When two size fields δ1(x,y,z) and δ2(x,y,z) areprovided, we use the minimum of both functions as size field.



Mesh Generation



Mesh generation

Let us consider a curve ~c(ξ) : [ξ1,ξ2]→ R3. The number of subdivisions Nof the curve is a function of the size field

N =

∫ξ2

ξ1

1δ(x,y,z)

‖dξ~c‖dξ.

The N +1 mesh points on the curve are located at coordinates ξ0, . . . ,ξN

where ξi is computed using the following rule

i =

∫ξi

ξ1

1δ(x,y,z)

‖dξ~c‖dξ.




Delaunay-based mesh generationComputational Modeling Sciences Department

36

Lawson Algorithm

•Locate triangle containing X

•Subdivide triangle

•Recursively check adjoining

triangles to ensure empty-

circle property. Swap

diagonal if needed

•(Lawson,77)

X

Given a Delaunay

Triangulation of n nodes,

How do I insert node n+1 ?

Delaunay




37

X

Lawson Algorithm

•Locate triangle containing X

•Subdivide triangle

•Recursively check adjoining

triangles to ensure empty-

circle property. Swap

diagonal if needed

•(Lawson,77)

Delaunay




38

Bowyer-Watson Algorithm

•Locate triangle that contains

the point

•Search for all triangles

whose circumcircle contain

the point (d<r)

•Delete the triangles (creating

a void in the mesh)

•Form new triangles from the

new point and the void

boundary

•(Watson,81)

X

r c

d

Given a Delaunay



Delaunay




39

X

Bowyer-Watson Algorithm

•Locate triangle that contains

the point

•Search for all triangles

whose circumcircle contain

the point (d<r)

•Delete the triangles (creating

a void in the mesh)

•Form new triangles from the

new point and the void

boundary

•(Watson,81)

Given a Delaunay



Delaunay




40

•Begin with Bounding Triangles (or Tetrahedra)

Delaunay




41

Delaunay

•Insert boundary nodes using Delaunay method

(Lawson or Bowyer-Watson)




42

Delaunay






43

Delaunay






44

Delaunay






45

Delaunay






46

Delaunay

•Recover boundary

•Delete outside triangles

•Insert internal nodes




47

Delaunay

Node Insertion

Grid Based

•Nodes introduced based on a regular lattice

•Lattice could be rectangular, triangular, quadtree, etc…

•Outside nodes ignored

h




48

Delaunay

Node Insertion

Grid Based

•Nodes introduced based on a regular lattice

•Lattice could be rectangular, triangular, quadtree, etc…

•Outside nodes ignored




49

Delaunay

Node Insertion

Centroid

•Nodes introduced at triangle centroids

•Continues until edge length, hl !




50

Delaunay

Node Insertion

Centroid

•Nodes introduced at triangle centroids

•Continues until edge length, hl !

l




51

Delaunay

Node Insertion

Circumcenter (“Guaranteed Quality”)

•Nodes introduced at triangle circumcenters

•Order of insertion based on minimum angle of any triangle

•Continues until minimum angle > predefined minimum

!

)30( o!"

(Chew,Ruppert,Shewchuk)




52

Delaunay

Circumcenter (“Guaranteed Quality”)

•Nodes introduced at triangle circumcenters

•Order of insertion based on minimum angle of any triangle

•Continues until minimum angle > predefined minimum )30( o!"

Node Insertion (Chew,Ruppert,Shewchuk)




53

Delaunay

Advancing Front

•“Front” structure maintained throughout

•Nodes introduced at ideal location from current front edge

Node Insertion

A B

C

(Marcum,95)




54

Delaunay

Advancing Front

•“Front” structure maintained throughout

•Nodes introduced at ideal location from current front edge

Node Insertion

(Marcum,95)




55

Delaunay

Voronoi-Segment

•Nodes introduced at midpoint of segment connecting the

circumcircle centers of two adjacent triangles

Node Insertion

(Rebay,93)




56

Delaunay

Voronoi-Segment

•Nodes introduced at midpoint of segment connecting the

circumcircle centers of two adjacent triangles

Node Insertion

(Rebay,93)




57

Delaunay

Edges

•Nodes introduced at along existing edges at l=h

•Check to ensure nodes on nearby edges are not too close

Node Insertion

h

h

h

(George,91)




58

Delaunay

Edges

•Nodes introduced at along existing edges at l=h

•Check to ensure nodes on nearby edges are not too close

Node Insertion(George,91)




59

Delaunay

Boundary Constrained

Boundary Intersection

•Nodes and edges introduced where Delaunay edges

intersect boundary




60

Delaunay


Boundary Intersection

•Nodes and edges introduced where Delaunay edges

intersect boundary




61

Delaunay


Local Swapping

•Edges swapped between adjacent pairs of triangles until

boundary is maintained




62

Delaunay


Local Swapping






63

Delaunay


Local Swapping






64

Delaunay


Local Swapping






65

Delaunay


Local Swapping


boundary is maintained(George,91)(Owen,99)



Delaunay-based mesh generation

Computational Modeling Sciences Department

66

D C

VS

Delaunay

Local Swapping Example

•Recover edge CD at vector Vs

Boundary ConstrainedJean-François Remacle Mesh Generation




67

D C

E1

E2

E3

E4E5

E6

E7

E8


•Make a list (queue) of all edges Ei, that intersect Vs

Delaunay





68

D CE1

E2

E3

E4E5

E6

E7

E8

Delaunay


•Swap the diagonal of adjacent triangle pairs for each

edge in the list





69

D C

E2

E3

E4E5

E6

E7

E8

Delaunay


•Check that resulting swaps do not cause overlapping

triangles. I they do, then place edge at the back of the

queue and try again later





70

D C

E3

E4E5

E6

E7

E8

Delaunay


•Check that resulting swaps do not cause overlapping

triangles. If they do, then place edge at the back of the

queue and try again later





71

D C

E6

Delaunay


•Final swap will recover the desired edge.

•Resulting triangle quality may be poor if multiple swaps

were necessary

•Does not maintain Delaunay criterion!Jean-François Remacle Mesh Generation



72

Delaunay

A

C

D E

B


3D Local Swapping

•Requires both boundary edge recovery and boundary

face recovery

Edge Recovery

•Force edges into triangulation by

performing 2-3 swap transformation

ABC = non-conforming face

DE = edge to be recovered

(George,91;99)(Owen,00)




73

Delaunay

A

B

C

D E


3D Local Swapping


face recovery

Edge Recovery



ABC = non-conforming face

DE = edge to be recovered

ABCE

ACBD

2-3 Swap





74

Delaunay

A

B

C

D E


3D Local Swapping


face recovery

Edge Recovery



ABCE

ACBD

2-3 Swap

BAED

CBED

ACED

DE = edge recovered





75

Delaunay

A

C

D E

B


3D Local Swapping


face recovery

Edge Recovery


performing 2-3 swap transformationDE = edge recovered

ABCE

ACBD

2-3 Swap

BAED

CBED

ECED





76

Delaunay

A B

A BS

S

3D Edge Recovery

•Form queue of faces through which edge AB will pass

•Perform 2-3 swap transformations on all faces in the list

•If overlapping tets result, place back on queue and try again later

•If still cannot recover edge, then insert “steiner” point

Edge AB to be recovered

Exploded view of

tets intersected by

AB




77

a

b

c

d e

a

b

c

d e

a

bc

d

e

a

bc

d

e

a

b

c

de

a

b

c

def

n2

a

b

n3

n4

n5

n1

a

b

n2

n3

n4

n5

n1

c

a

b

n1

n2

n3

n4

n5

a

b

n3,i

1

23

n3,j

n3,k

n2,i

n1,i

n1,j

n1,kn2,j

n2,k

Delaunay

2-3 Swap 2-2 Swap Face Split Edge Split Edge Suppress

abce, acbd

abde, bcde, cade

aceb, adcb

adeb, edcb

abce, acbd

abfe, bcfe, cafe

bafd, cbfd, acfd

abnini+1 i=1…N

abnini+1,

cbnini+1 i=1…N

abnini+1 i=1…N

nm,knm.jnm.ia,

nm,knm.jnm.ib m=1…M

N = no. adj. tets at edge ab

M = no. unique trias in

polygon P=n1,n2,n3


An introduction to mesh generation Part III : Finite...

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