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Subsections
2.5 Demands for Finite Elements
Within Finite Elements the coefficients of the matrix
shown in (2.8) and (2.9) can be generally expressed as [46][54][75]

(2.19) 
respectively

(2.20) 
for the element volumes
. The symbols and denote the Ansatzfunctions of the points and .
If
is a constant scalar within the tetrahedrons,
this inproduct has a simple geometrical meaning and the compliance
(2.10)

(2.21) 
leads to an angle criterion for each edge of the mesh [75]
Sum of dihedral angles:
Let
be an edge with adjacent tetrahedra . For each tetrahedron two planes exist which do not contain
and which span a dihedral angle
. The two planes share an edge with length . The sum over
of the cotangent of
weighted by must be greater or equal than zero.

Figure 2.10:
Threedimensional grid criterion for Finite Elements.
Dihedral angle of the edge
of a tetrahedron

Dihedral angles around the edge
, arisen from the four tetrahedrons which share the edge


Within Figure 2.10 the criterion is clarified. In Figure 2.10(a) the dihedral angle of the edge
is shown. It is the angle between the faces which do not contain the edge
. Here, is the length of the edge opposing to
.
In Figure 2.10(b) all the tetrahedrons connected to
are shown. Each of the participating tetrahedrons spans its own dihedral angle to
.
In two dimensions the criterion (2.22) is also valid. The two triangles connected to
can be considered as tetrahedrons with the same (small) edge , which can be canceled out of the summation, and the dihedral angle of the edge
simplifies to the angle between the two edges of the triangle which do not contain this edge.
Figure 2.11:
Twodimensional mesh criterion for Finite Elements

The dihedral angle reduces to

(2.23) 
As shown in Figure 2.11 in two dimensions exactly two triangles are connected by each edge and (2.22) simplifies to

(2.24) 
With the following transformation and knowing that
or 
(2.25) 
we get

(2.26) 
which is equivalent to

(2.27) 
Therefore,

(2.28) 
is the wellknown formulation for two dimensions. This is the same criterion as for Box Integration, given in (2.17).
Therefore, the general twodimensional grid criterion for Finite Boxes and Finite Elements results in the same formulation which is satisfied if the tessellation of the grid points is a Delaunay tessellation. In three dimensions, the criterion for Finite Boxes and Finite Elements differ. Detailed information of the threedimensional differences can be found in [13][14][19][28].
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Up: 2. Grid Types
Previous: 2.4 Demands for the
J. Cervenka: ThreeDimensional Mesh Generation for Device and Process Simulation