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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 (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 in-product has a simple geometrical meaning and the compliance (2.10) (2.21)

leads to an angle criterion for each edge of the mesh (2.22)

 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: Three-dimensional 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 .

## 2.5.1 Two-Dimensional Criterion

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. 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 well-known formulation for two dimensions. This is the same criterion as for Box Integration, given in (2.17). Therefore, the general two-dimensional 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 three-dimensional differences can be found in .    Next: 2.6 Grid Refinement Up: 2. Grid Types Previous: 2.4 Demands for the

J. Cervenka: Three-Dimensional Mesh Generation for Device and Process Simulation