B.1 Neumann Boundary for the Rotor-Rotor-Operator

The Neumann boundary integral from (5.74) is modified to read

$$\begin{split}\int_{\mathcal{C}_{N1}}\vec{n}\cdot \left[ \vec{... ...mathrm{d}s = & = \left[D\right]\left\{c\right\}. \end{split}$$ (B.1)

Three different cases for each of the three edges of every triangular element must be considered. $\gamma$ is assumed to be a constant scalar in each element.

For the computation of the element matrix

$$\begin{split}D_{ij}^e & = \int_{\mathcal{C}^e_{k}}\vec{n}_k\c... ... \right), i\in[1;3], j\in[1;3], k\in[1;3] \end{split}$$ (B.2)

the following expressions must be calculated

$$\left(l_{23}\vec{n}_1\right)\times\vec{\nabla}\lambda^e_1 = -\left(l_{23}\vec{n}_1\right)\times\frac{l_{23}\vec{n}_1}{2F_e} = \vec{0}$$ (B.3)

$$\begin{split}\left(l_{23}\vec{n}_1\right)\times\vec{\nabla}\l... ...{31}\times\vec{r}_{23}\right)}_{-2F_e} = -\vec{e}_z \end{split}$$ (B.4)

$$\left(l_{23}\vec{n}_1\right)\times\vec{\nabla}\lambda^e_3 = \frac... ...\vec{e}_{z}\cdot\left(\vec{r}_{12}\times\vec{r}_{23}\right)}_{2F_e} = \vec{e}_z$$ (B.5)

$$\left(l_{31}\vec{n}_2\right)\times\vec{\nabla}\lambda^e_1 = \vec{... ... \left(l_{31}\vec{n}_2\right)\times\vec{\nabla}\lambda^e_3 = -\vec{e}_z$$ (B.6)

$$\left(l_{12}\vec{n}_3\right)\times\vec{\nabla}\lambda^e_1 = -\vec... ...z \left(l_{12}\vec{n}_3\right)\times\vec{\nabla}\lambda^e_3 = \vec{0}$$ (B.7)

$$\begin{split}\vec{\nabla}\times\vec{N}^e_1 & = 2l_{12} \left(... ...right)\right]}_{2F_e} = \frac{l_{12}}{F_e}\vec{e}_z \end{split}$$ (B.8)

$$\vec{\nabla}\times\vec{N}^e_2 = \frac{l_{23}}{F_e}\vec{e}_z$$ (B.9)

$$\vec{\nabla}\times\vec{N}^e_3 = \frac{l_{31}}{F_e}\vec{e}_z.$$ (B.10)

Along Edge $12$ :

$$\left. \begin{array}{l} \lambda^e_3 = 0 \lambda^e_2 = 1 - \lam... ...{r}_1 - \vec{r}_2)\lambda^e_1 + \vec{r}_2 = \vec{r}_{21}\lambda^e_1 + \vec{r}_2$$ (B.11)

$$s = \vec{r}\cdot\vec{e}_{21} \Rightarrow \mathrm{d}s = \vec{r}_{21}\cdot\vec{e}_{21} d\lambda^e_1 = l_{12} d\lambda^e_1$$ (B.12)

or for $\lambda^e_2$

$$\left. \begin{array}{l} \lambda^e_3 = 0 \lambda^e_1 = 1 - \lam... ...{r}_2 - \vec{r}_1)\lambda^e_2 + \vec{r}_1 = \vec{r}_{12}\lambda^e_2 + \vec{r}_2$$ (B.13)

$$s = \vec{r}\cdot\vec{e}_{12} \Rightarrow \mathrm{d}s = \vec{r}_{12}\cdot\vec{e}_{12} d\lambda^e_2 = l_{12} d\lambda^e_2$$ (B.14)

$$\begin{split}D_{ij}^e & = \frac{1}{\gamma}\int_{\mathcal{C}^e... ...mes\vec{N}^e_j \right), i\in[1;3], j\in[1;3] \end{split}$$ (B.15)

$$\begin{split}\int_{\mathcal{C}^e_{12}}\left(\vec{n}_3\times\v... ...^e_2 d\lambda^e_2\right\} = & = l_{12}\vec{e}_z \end{split}$$ (B.16)

$$\begin{split}\int_{\mathcal{C}^e_{12}}\left(\vec{n}_3\times\v... ...mbda^e_3 \mathrm{d}s}_{0}\right\} = & = \vec{0} \end{split}$$ (B.17)

$$\begin{split}\int_{\mathcal{C}^e_{12}}\left(\vec{n}_3\times\v... ...1\lambda^e_1 d\lambda^e_1\right\} = & = \vec{0} \end{split}$$ (B.18)

$$\left[D\right]^e = \frac{1}{\gamma{} F_e} \left[ \begin{array}{c... ...2} & l_{12}l_{23} & l_{12}l_{31} 0 & 0 & 0 0 & 0 & 0 \end{array} \right].$$ (B.19)

Analogously the element matrices for the remaining edges are obtained.

Along Edge $23$ :

$$\left[D\right]^e = \frac{1}{\gamma{} F_e} \left[ \begin{array}{c... ... l_{23}l_{12} & l_{23}l_{23} & l_{23}l_{31} 0 & 0 & 0 \end{array} \right].$$ (B.20)

Along Edge $31$ :

$$\left[D\right]^e = \frac{1}{\gamma{} F_e} \left[ \begin{array}{c... ...& 0 & 0 l_{31}l_{12} & l_{31}l_{23} & l_{31}l_{31} \par \end{array} \right].$$ (B.21)