C.2.2 For the Vector Function

For the vector functions $\vec{H}_1$ the Neumann boundary term (5.31) is given by

$$\int_{\mathcal{A}_{N2}}\lambda_i\vec{n}\cdot(\utilde{\mu}\cdot\ve... ...2}}\lambda_i \vec{n}\cdot(\utilde{\mu}\cdot\vec{N}_j) \mathrm{d}A = [D]\{c\}.$$ (C.35)

Assuming that $\mu$ is a constant scalar in each element the corresponding element matrix is given by

$$D_{ij}^e = \mu\int_{\mathcal{A}^e_k}\lambda^e_i \vec{n}_k\cdot\vec{N}^e_j \mathrm{d}A, k\in[1;4], i\in[1;4], j\in[1;6].$$ (C.36)

For the face $\mathcal{A}^e_1$ opposite to the node $1$ ($k = 1$ ):

The element function $\lambda^e_1$ is 0 on the element face $\mathcal{A}^e_1$ . Thus the first row of the element matrix is zero.

$$D_{1j}^e = \mu\int_{\mathcal{A}^e_1}\lambda^e_1 \vec{n}_1\cdot\vec{N}^e_j \mathrm{d}A = 0.$$ (C.37)

The remaining entries can be calculated as follows

$$\begin{split}D_{21}^e & = \mu\int_{\mathcal{A}^e_1}\lambda^e_... ...\nabla}\lambda^e_1 = \frac{\mu{}l_1}{72V_e} f_{11} \end{split}$$ (C.38)

$$\begin{split}D_{22}^e & = \mu\int_{\mathcal{A}^e_1}\lambda^e_... ...nabla}\lambda^e_1 = \frac{\mu{}l_2}{144V_e} f_{11} \end{split}$$ (C.39)

$$D_{23}^e = \frac{\mu{}l_3}{144V_e} f_{11}$$ (C.40)

$$\begin{split}D^e_{24} & = \mu\int_{\mathcal{A}^e_1}\lambda^e_... ...\right] = \frac{\mu{}l_4}{144V_e}(f_{12} - 2f_{13}) \end{split}$$ (C.41)

$$D^e_{25} = \frac{\mu{}l_5}{144V_e}(2f_{14} - f_{12})$$ (C.42)

$$D^e_{26} = \frac{\mu{}l_6}{144V_e}(f_{13} - f_{14})$$ (C.43)

$$\begin{split}D^e_{31} & = \frac{\mu{}l_1}{144V_e}f_{11} ... ...^e_{36} = \frac{\mu{}l_6}{144V_e}(f_{13} - 2f_{14}) \end{split}$$ (C.44)

$$\begin{split}D^e_{41} & = \frac{\mu{}l_1}{144V_e}f_{11} ... ...e_{46} = \frac{\mu{}l_6}{144V_e}(2f_{13} - f_{14}). \end{split}$$ (C.45)

For $\mathcal{A}^e_2$ :

$$\begin{split}D^e_{11} & = -\frac{\mu{}l_1}{72V_e}f_{22} ... ...D^e_{16} = \frac{\mu{}l_6}{144V_e}(f_{23} - f_{24}) \end{split}$$ (C.46)

$$D^e_{2j} = 0, \j\in[1;6]$$ (C.47)

$$\begin{split}D^e_{31} & = -\frac{\mu{}l_1}{144V_e}f_{22} \... ...^e_{36} = \frac{\mu{}l_6}{144V_e}(f_{23} - 2f_{24}) \end{split}$$ (C.48)

$$\begin{split}D^e_{41} & = -\frac{\mu{}l_1}{144V_e}f_{22} \... ...e_{46} = \frac{\mu{}l_6}{144V_e}(2f_{23} - f_{24}). \end{split}$$ (C.49)

For $\mathcal{A}^e_3$ :

$$\begin{split}D^e_{11} & = \frac{\mu{}l_1}{144V_e}(f_{31} - 2f... ...{32}) D^e_{16} = \frac{\mu{}l_5}{144V_e}f_{33} \end{split}$$ (C.50)

$$\begin{split}D^e_{21} & = \frac{\mu{}l_1}{144V_e}(2f_{31} - f... ...{32}) D^e_{26} = \frac{\mu{}l_5}{144V_e}f_{33} \end{split}$$ (C.51)

$$D^e_{3j} = 0, \j\in[1;6]$$ (C.52)

$$\begin{split}D^e_{41} & = \frac{\mu{}l_1}{144V_e}(f_{31} - f_... ...{32}) D^e_{46} = \frac{\mu{}l_5}{72V_e}f_{33}. \end{split}$$ (C.53)

For $\mathcal{A}^e_4$ :

$$\begin{split}D^e_{11} & = \frac{\mu{}l_1}{144V_e}(f_{41} - 2f... ... D^e_{16} = -\frac{\mu{}l_6}{144V_e}f_{44} \end{split}$$ (C.54)

$$\begin{split}D^e_{21} & = \frac{\mu{}l_1}{144V_e}(2f_{41} - f... ... D^e_{26} = -\frac{\mu{}l_6}{144V_e}f_{44} \end{split}$$ (C.55)

$$\begin{split}D^e_{31} & = \frac{\mu{}l_1}{144V_e}(f_{41} - f_... ... D^e_{36} = -\frac{\mu{}l_6}{72V_e}f_{44} \end{split}$$ (C.56)

$$D^e_{4j} = 0, j\in[1;6].$$ (C.57)