# An introduction to mesh generation Part III : Finite ...perso. Finite Element Mesh Generation...

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Finite Element Mesh Generation

An introduction to mesh generationPart III : Finite Element Mesh Generation

Jean-Franois Remacle

Department of Civil Engineering,Universit catholique de Louvain, Belgium

Jean-Franois Remacle Mesh Generation

Finite Element Mesh Generation

Euler-Poincar

A mesh M is a geometrical discretization of a domain that consists ofA collection of mesh entities Mdi 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.

Jean-Franois Remacle Mesh Generation

Finite Element Mesh Generation

Euler-Poincar

A mesh M is a geometrical discretization of a domain that consists ofA collection of mesh entities Mdi 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.

Jean-Franois Remacle Mesh Generation

Finite Element Mesh Generation

Euler-Poincar

A mesh M is a geometrical discretization of a domain that consists ofA collection of mesh entities Mdi 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.

Jean-Franois Remacle Mesh Generation

Finite Element Mesh Generation

Euler-Poincar

Jean-Franois Remacle Mesh Generation

Finite Element Mesh Generation

Euler-Poincar

Jean-Franois Remacle Mesh Generation

Finite Element Mesh Generation

Euler-Poincar (3D)

The Euler-Poincar 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-Poincarformula, 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.

Jean-Franois Remacle Mesh Generation

Finite Element Mesh Generation

Euler-Poincar (3D)

The Euler-Poincar formula is

#V #E +#F (L#F)2(SG) = 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(SG) = 812+6(66)2(10) = 0.

Jean-Franois Remacle Mesh Generation

Finite Element Mesh Generation

Euler-Poincar

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(SG) = 1624+10(1210)2(11) = 0.

Jean-Franois Remacle Mesh Generation

Finite Element Mesh Generation

Euler-Poincar (2D)

Consider M, a 2D mesh of a domain .#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-Poincar relation gives the following relation between thosequantities:

#V #E +#F () = 0

where () is the Euler-Poincar characteristic of the surface.

Jean-Franois Remacle Mesh Generation

Finite Element Mesh Generation

Euler-Poincar (2D)

The genus of different surfaces is given

for the sphere, = 2 (#V #E +#F = 24+4),for the torus, = 0 (#V #E +#F = 48+4),for the disk, = 1,for the Klein bootle, = 1.

Jean-Franois Remacle Mesh Generation

Finite Element Mesh Generation

Euler-Poincar (2D)

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

= #V #E +#F = 22g+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.

Jean-Franois Remacle Mesh Generation

Finite Element Mesh Generation

Euler-Poincar (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.

Jean-Franois Remacle Mesh Generation

Finite Element Mesh Generation

Euler-Poincar (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.

Jean-Franois Remacle Mesh Generation

Finite Element Mesh Generation

Euler-Poincar (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.

Jean-Franois Remacle Mesh Generation

Finite Element Mesh Generation

Euler-Poincar (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

Jean-Franois Remacle Mesh Generation

Finite Element Mesh Generation

Euler-Poincar (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 tri

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