Subsections

• 2.5.1 Two-Dimensional Criterion

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]

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

respectively

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

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

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

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

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

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

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

With the following transformation and knowing that

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

we get

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

which is equivalent to

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

Therefore,

$$\Theta_{k}+\Theta_{l} \leq 180^\circ$$ (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].