5.2.2 Linear Vector Shape Functions on Tetrahedral Elements

Refer to the tetrahedron from Fig. <4.7>. The vector field in the tetrahedral element is interpolated by means of the Whitney $1$ -form basis function $W_i^e$ associated to the element edge $i$ [100]. For example along Edge $1$ between Node $1$ and Node $2$ this function is given by

$$\vec{W}_1^e = \vec{W}_{12}^e = \lambda_{1}^e\vec{\nabla}\lambda_2^e - \lambda_{2}^e\vec{\nabla}\lambda_1^e.$$

This approach was introduced first by Whitney [101]. The functions $\lambda^e_i$ are the barycentric coordinates from Subsection 4.2.2. If only one subscript index is used for the vector basis functions (for example $\vec{W}_1^e$ ), this index means an edge number (for example Edge 1). If two subscript indexes are used (for example $\vec{W}_{12}^e$ ), these indexes comply with the corresponding node numbers (for example Edge 1 belongs to the node pair Node $1$ and Node $2$ ). The edge indexes are associated to node pairs in Table 5.1.

Furthermore the following notation is used

$$\vec{r}_1 = \vec{r}_{12}, \vec{r}_2 = \vec{r}_{13}, \vec{r}_3 =... ... \vec{r}_4 = \vec{r}_{23}, \vec{r}_5 = \vec{r}_{42}, \vec{r}_6 = \vec{r}_{34}$$

and

$$l_1 = \vert\vec{r}_1\vert, l_2 = \vert\vec{r}_2\vert, l_3 = \ve... ... = \vert\vec{r}_4\vert, l_5 = \vert\vec{r}_5\vert, l_6 = \vert\vec{r}_6\vert.$$

Since $\lambda^e_1$ vanishes on the facet defined by $(2,3,4)$ and $\vec{\nabla}\lambda^e_1$ is perpendicular to that facet, $\vec{W}^e_1$ has no tangential component on this facet. Analogously it can be stated that $\vec{W}^e_1$ has no tangential component on the facet defined by $(1,3,4)$ . Thus the edge function $\vec{W}^e_1$ has no tangential component along the edges different from the Edge $1$ and it contributes only to their normal components. This is also valid for the Whitney vector basis functions $\vec{W}_i^e$ on the remaining element edges.

For the divergence and for the rotor of the Whitney function $\vec{W}_i^e$ it is written

$$\vec{\nabla}\cdot\vec{W}_1^e = \vec{\nabla}\lambda_{1}^e\cdot\vec{\nabla}\lambda_2^e - \vec{\nabla}\lambda_{2}^e\cdot\vec{\nabla}\lambda_1^e = 0$$

$$\vec{\nabla}\times\vec{W}_1^e = \vec{\nabla}\lambda_{1}^e\times\v... ...c{\nabla}\lambda_1^e = 2\vec{\nabla}\lambda_{1}^e\times\vec{\nabla}\lambda_2^e.$$

Along the direction of Edge $i$ the Whitney function $\vec{W}_i^e$ has a constant tangential component. This will be demonstrated for $\vec{W}_1^e$ .

$$\vec{W}_1\cdot\frac{\vec{r}_1}{l_1} = \frac{1}{l_1} \left( \lambd... ...V^e} - \lambda_2\frac{-6V^e}{6V^e} \right) = \frac{\lambda_1 + \lambda_2}{l_1}.$$

In direction parallel to Edge $1$ $\lambda_3$ and $\lambda_4$ are constant. On Edge $1$ $\lambda_3$ and $\lambda_4$ are zero. With (4.76) $\lambda_1 + \lambda_2$ are also constant parallel to the edge $1$ . On Edge $1$ $\lambda_1 + \lambda_2 = 1$ . Consequently parallel to Edge $1$

$$\vec{W}_1\cdot\frac{\vec{r}_1}{l_1} = \frac{\lambda_1 + \lambda_2}{l_1} = const$$ (5.43)

and on Edge $1$

$$\vec{W}_1\cdot\frac{\vec{r}_1}{l_1} = \frac{1}{l_1}.$$ (5.44)

Thus, the continuity of the tangential component along the element edge across the elements is guaranteed. Otherwise the normal component must not be continuous. Exactly this behavior is required for the fields $\vec{E}$ and $\vec{H}$ (if no surface current is available) and for the magnetic vector potential $\vec{A}$ . Thus, the element vector basis functions for the field associated with the edges are given by the very well suited $1$ -form Whitney functions $\vec{W}_i^e$

$$\vec{N}^e_1 = l_1\vec{W}_1^e = l_1(\lambda^e_1\vec{\nabla}\lambda^e_2 - \lambda^e_2\vec{\nabla}\lambda^e_1)$$ (5.45)

$$\vec{N}^e_2 = l_2\vec{W}_2^e = l_2(\lambda^e_1\vec{\nabla}\lambda^e_3 - \lambda^e_3\vec{\nabla}\lambda^e_1)$$ (5.46)

$$\vec{N}^e_3 = l_3\vec{W}_3^e = l_3(\lambda^e_1\vec{\nabla}\lambda^e_4 - \lambda^e_4\vec{\nabla}\lambda^e_1)$$ (5.47)

$$\vec{N}^e_4 = l_4\vec{W}_4^e = l_4(\lambda^e_2\vec{\nabla}\lambda^e_3 - \lambda^e_3\vec{\nabla}\lambda^e_2)$$ (5.48)

$$\vec{N}^e_5 = l_5\vec{W}_5^e = l_5(\lambda^e_4\vec{\nabla}\lambda^e_2 - \lambda^e_2\vec{\nabla}\lambda^e_4)$$ (5.49)

$$\vec{N}^e_6 = l_6\vec{W}_6^e = l_6(\lambda^e_3\vec{\nabla}\lambda^e_4 - \lambda^e_4\vec{\nabla}\lambda^e_2).$$ (5.50)

The performed multiplication with $l_i$ delivers the normalized and dimensionless edge functions $N_i$ .

Table 5.1: Tetrahedral Element Edge Definition.

Edge	Node 1	Node 2
1	1	2
2	1	3
3	1	4
4	2	3
5	4	2
6	3	4

The edge functions $\vec{N}^e_{1}$ and $\vec{N}^e_{6}$ are visualized in Fig. <5.5> and Fig. <5.6>, respectively.

Figure 5.5: The shape function along Edge $12$ .

Figure 5.6: The shape function along Edge $34$ .

If the field in the tetrahedral element (for instance the magnetic field $\vec{H}^e$ ) is given by

$$\vec{H}^e = \sum_{i=1}^6c_i\vec{N}_i^e,$$ (5.51)

the coefficient $c_i$ corresponds to the tangential component of the field $\vec{H}^e$ on the Edge $i$ . An example of the field in the element is depicted in Fig. <5.7> where all coefficients are set to $1$ .

Figure 5.7: The sum of all shape functions in the tetrahedron.