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2.5 Demands for Finite Elements

Within Finite Elements the coefficients of the matrix $ {\mathrm{\bf S}}$ shown in (2.8) and (2.9) can be generally expressed as [46][54][75]

$\displaystyle s_{ij}=\sum_{\mathrm{elements}} \;\int\limits_{V_{\mathrm{element...
...i}) \;\makebox{\boldmath$\underline\varepsilon$}\;(\nabla N_{j}) \;\mathrm{d}v,$ (2.19)


$\displaystyle s_{ij}=\sum_{\mathrm{elements}} \;\int\limits_{V_{\mathrm{element}}} (\nabla N_{i}) \cdot (\nabla N_{j}) \;\mathrm{d}v$ (2.20)

for the element volumes $ V_{\mathrm{element}}$. The symbols $ N_{i}$ and $ N_{j}$ denote the Ansatzfunctions of the points $ p_i$ and $ p_j$. If $ \makebox{\boldmath $\underline\varepsilon$}$ is a constant scalar within the tetrahedrons, this in-product has a simple geometrical meaning and the compliance (2.10)

$\displaystyle s_{ij}\leq 0$ (2.21)

leads to an angle criterion for each edge of the mesh [75]

$\displaystyle \sum_{k=1}^{n} l_{k} \; \cot \Theta_{k} \geq 0.$ (2.22)

Sum of dihedral angles:
Let $ < p_ip_j >$ be an edge with $ n$ adjacent tetrahedra $ t_{k}$. For each tetrahedron $ t_{k}$ two planes exist which do not contain $ < p_ip_j >$ and which span a dihedral angle $ \Theta_{k}$. The two planes share an edge with length $ l_{k}$. The sum over $ k=1 \dots n$ of the cotangent of $ \Theta_{k}$ weighted by $ l_{k}$ must be greater or equal than zero.

Figure 2.10: Three-dimensional grid criterion for Finite Elements.
Dihedral angle $ \Theta$ of the edge $ < p_ip_j >$ of a tetrahedron
Dihedral angles around the edge $ < p_ip_j >$, arisen from the four tetrahedrons which share the edge $ < p_ip_j >$

Within Figure 2.10 the criterion is clarified. In Figure 2.10(a) the dihedral angle of the edge $ < p_ip_j >$ is shown. It is the angle between the faces which do not contain the edge $ < p_ip_j >$. Here, $ l$ is the length of the edge opposing to $ < p_ip_j >$. In Figure 2.10(b) all the tetrahedrons connected to $ < p_ip_j >$ are shown. Each of the participating tetrahedrons $ t_k$ spans its own dihedral angle $ \Theta_k$ to $ < p_ip_j >$.

2.5.1 Two-Dimensional Criterion

In two dimensions the criterion (2.22) is also valid. The two triangles connected to $ < p_ip_j >$ can be considered as tetrahedrons with the same (small) edge $ l_k$, which can be canceled out of the summation, and the dihedral angle of the edge $ < p_ip_j >$ simplifies to the angle between the two edges of the triangle which do not contain this edge.

Figure 2.11: Two-dimensional mesh criterion for Finite Elements

The dihedral angle reduces to

$\displaystyle \Theta_{k}= \measuredangle (< p_ip_k >,< p_kp_j >).$ (2.23)

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

$\displaystyle \cot \Theta_{k} + \cot \Theta_{l} \geq 0.$ (2.24)

With the following transformation and knowing that

$\displaystyle 0 < \Theta \leq 180^\circ$   or$\displaystyle \qquad\sin \Theta > 0$ (2.25)

we get

$\displaystyle \sin\Theta_{l} \;\cos\Theta_{k} + \sin\Theta_{k} \;\cos\Theta_{l} \geq 0,$ (2.26)

which is equivalent to

$\displaystyle \sin (\Theta_{k}+\Theta_{l}) \geq 0.$ (2.27)


$\displaystyle \Theta_{k}+\Theta_{l} \leq 180^\circ$ (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 [13][14][19][28].
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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