5.3.2 Assembling

The two-dimensioanl case leads to a linear equation system similar to (5.37). The sub-matrix $[A]$ is written in the form (5.67). With $\gamma$ constant in each element $[S]^e$ is given by

$$\begin{split}S_{ij}^e & = \frac{1}{\gamma}\int_{\mathcal{A}_e... ...times\vec{N}_j^e\right)F_e, i\in[1;3], j\in[1;3], \end{split}$$ (5.82)

where the rotor terms are calculated in the following manner

$$\vec{\nabla}\times\vec{N}_1^e = 2l_{12}\left(\vec{\nabla}\lambda_... ..._e}\times\frac{\vec{r}_{31}\times\vec{e}_z}{2F_e} = \frac{l_{12}}{F_e}\vec{e}_z$$ (5.83)

$$\vec{\nabla}\times\vec{N}_2^e = \frac{l_{23}}{F_e}\vec{e}_z \vec{\nabla}\times\vec{N}_3^e = \frac{l_{31}}{F_e}\vec{e}_z.$$ (5.84)

Thus $[S]^e$ becomes

$$S_{11}^e = \frac{1}{\gamma}\left(\vec{\nabla}\times\vec{N}_1^e\right) \cdot\left(\vec{\nabla}\times\vec{N}_1^e\right)F_e = \frac{l_{12}^2}{\gamma F_e}$$ (5.85)

$$S_{21}^e = S_{12}^e = \frac{l_{12}l_{23}}{\gamma F_e}$$ (5.86)

$$S_{31}^e = S_{13}^e = \frac{l_{12}l_{31}}{\gamma F_e}$$ (5.87)

$$S_{22}^e = \frac{l_{23}^2}{\gamma F_e}$$ (5.88)

$$S_{32}^e = S_{23}^e = \frac{l_{23}l_{31}}{\gamma F_e}$$ (5.89)

$$S_{33}^e = \frac{l_{33}^2}{\gamma F_e}.$$ (5.90)

For element-wise constant $\mu$ the entries from $[M]^e$ are obtained from

$$M_{ij}^e = \mu\int_{\mathcal{A}_e}\vec{N}_i^e\cdot\vec{N}_j^e \mathrm{d}A, i\in[1;3], j\in[1;3].$$ (5.91)

One entry calculation is demonstrated in detail

$$\begin{split}M_{11}^e & = \mu\int_{\mathcal{A}_e}\vec{N}_1^e\... ..._e}\left(\lambda_2^e\right)^2 \mathrm{d}A \right]. \end{split}$$

The arrising integrals are solved as follows

$$\begin{split}\int_{\mathcal{A}_e}\left(\lambda_1^e\right)^2 ... ...(1-\lambda_1^e\right) d\lambda_1^e = \frac{F_e}{6} \end{split}$$ (5.92)

$$\begin{split}\int_{\mathcal{A}_e}\lambda_1^e\lambda_2^e \mat... ...da_1^e\right)^2}{2} d\lambda_1^e = \frac{F_e}{12}, \end{split}$$ (5.93)

using the integral domain transformation Appendix A. Generally it can be written

$$\int_{\mathcal{A}_e}\lambda_i^e\lambda_j^e \mathrm{d}A = \left\{... ...}{12} & \mathrm{for} i \neq j \end{array} \right. j\in[1;3], j\in[1;3].$$ (5.94)

In a similar manner all entries of $[M]^e$ are given

$$M_{11}^e = \frac{\mu l_{12}^2}{24 F_e}\left(\vec{r}_{23}\cdot\vec{r}_{23} - \vec{r}_{23}\cdot\vec{r}_{31} + \vec{r}_{31}\cdot\vec{r}_{31}\right)$$ (5.95)

$$M_{21}^e = M_{12}^e = \frac{\mu l_{12} l_{23}}{48 F_e}\left(\vec{r}_{31}\cdot\vec{... ..._{31} - 2 \vec{r}_{23}\cdot\vec{r}_{12} + \vec{r}_{23}\cdot\vec{r}_{31}\right)$$ (5.96)

$$M_{31}^e = M_{13}^e = \frac{\mu l_{12} l_{31}}{48 F_e}\left(\vec{r}_{23}\cdot\vec{... ...{r}_{12} - \vec{r}_{23}\cdot\vec{r}_{23} + \vec{r}_{23}\cdot\vec{r}_{12}\right)$$ (5.97)

$$M_{22}^e = \frac{\mu l_{23}^2}{24 F_e}\left(\vec{r}_{12}\cdot\vec{r}_{12} - \vec{r}_{12}\cdot\vec{r}_{31} + \vec{r}_{31}\cdot\vec{r}_{31}\right)$$ (5.98)

$$M_{32}^e = M_{23}^e = \frac{\mu l_{23} l_{31}}{48 F_e}\left(\vec{r}_{23}\cdot\vec{... ..._{12} - 2 \vec{r}_{23}\cdot\vec{r}_{31} + \vec{r}_{31}\cdot\vec{r}_{12}\right)$$ (5.99)

$$M_{33}^e = \frac{\mu l_{31}^2}{24 F_e}\left(\vec{r}_{12}\cdot\vec{r}_{12} - \vec{r}_{12}\cdot\vec{r}_{23} + \vec{r}_{23}\cdot\vec{r}_{23}\right).$$ (5.100)

The entries of $[B]$ are expressed as

$$B_{ij}^e = - \jmath\omega\mu\int_{\mathcal{A}_e}\vec{N}_i^e\cdot\vec{\nabla}\lambda_j^e \mathrm{d}A, i\in[1;3], j\in[1;3].$$ (5.101)

The entry $B_{11}^e$ is calculated in detail as

$$\begin{split}B_{11}^e & = - \jmath\omega\mu\int_{\mathcal{A}_... ...nt_{\mathcal{A}_e}\lambda_2^e \mathrm{d}A \right). \end{split}$$ (5.102)

For the integral terms the integral domain transformation from Appendix A is used again to obtain

$$\int_{\mathcal{A}_e}\lambda_i^e \mathrm{d}A = \frac{F_e}{3}, i\in[1;3].$$ (5.103)

Thus $[B]^e$ is given by

$$B_{11}^e = - \jmath\omega\mu{}l_{12} \left(\frac{\vec{r}_{23}\times\vec{e}... ...e}_z}{2F_e}\cdot\frac{\vec{r}_{23}\times\vec{e}_z}{2F_e} \right)\frac{F_e}{3} =$$ (5.104)

$$= - \jmath\omega\frac{\mu{}l_{12}}{12F_e} \left( \vec{r}_{23}\cdot\vec{r}_{31} - \vec{r}_{23}\cdot\vec{r}_{23} \right)$$

$$B_{12}^e = - \jmath\omega\frac{\mu{}l_{12}}{12F_e} \left( \vec{r}_{31}\cdot\vec{r}_{31} - \vec{r}_{31}\cdot\vec{r}_{23} \right)$$ (5.105)

$$B_{13}^e = - \jmath\omega\frac{\mu{}l_{12}}{12F_e} \left( \vec{r}_{12}\cdot\vec{r}_{31} - \vec{r}_{12}\cdot\vec{r}_{23} \right)$$ (5.106)

$$B_{21}^e = - \jmath\omega\frac{\mu{}l_{23}}{12F_e} \left( \vec{r}_{23}\cdot\vec{r}_{12} - \vec{r}_{23}\cdot\vec{r}_{31} \right)$$ (5.107)

$$B_{22}^e = - \jmath\omega\frac{\mu{}l_{23}}{12F_e} \left( \vec{r}_{31}\cdot\vec{r}_{12} - \vec{r}_{31}\cdot\vec{r}_{31} \right)$$ (5.108)

$$B_{23}^e = - \jmath\omega\frac{\mu{}l_{23}}{12F_e} \left( \vec{r}_{12}\cdot\vec{r}_{12} - \vec{r}_{12}\cdot\vec{r}_{31} \right)$$ (5.109)

$$B_{31}^e = - \jmath\omega\frac{\mu{}l_{31}}{12F_e} \left( \vec{r}_{23}\cdot\vec{r}_{23} - \vec{r}_{23}\cdot\vec{r}_{12} \right)$$ (5.110)

$$B_{32}^e = - \jmath\omega\frac{\mu{}l_{31}}{12F_e} \left( \vec{r}_{31}\cdot\vec{r}_{23} - \vec{r}_{31}\cdot\vec{r}_{12} \right)$$ (5.111)

$$B_{33}^e = - \jmath\omega\frac{\mu{}l_{31}}{12F_e} \left( \vec{r}_{12}\cdot\vec{r}_{23} - \vec{r}_{12}\cdot\vec{r}_{12} \right).$$ (5.112)

Analogously to (4.49) for the element matrix $\left[C\right]^e$ one obtains:

$$C_{11}^e = -\jmath\omega\frac{\mu}{4F_e} \vec{r}_{23}\cdot\vec{r}_{23}$$ (5.113)

$$C_{12}^e = C_{21}^e = -\jmath\mu\frac{\mu}{4F_e} \vec{r}_{23}\cdot\vec{r}_{31}$$ (5.114)

$$C_{13}^e = C_{31}^e = -\jmath\mu\frac{\mu}{4F_e} \vec{r}_{23}\cdot\vec{r}_{12}$$ (5.115)

$$C_{22}^e = -\jmath\omega\frac{\mu}{4F_e} \vec{r}_{31}\cdot\vec{r}_{31}$$ (5.116)

$$C_{23}^e = C_{32}^e = -\jmath\omega\frac{\epsilon}{4F_e} \vec{r}_{31}\cdot\vec{r}_{12}$$ (5.117)

$$C_{33}^e = -\jmath\omega\frac{\mu}{4F_e} \vec{r}_{12}\cdot\vec{r}_{12}.$$ (5.118)