5.3 Two-Dimensional Vector Finite Element Method

The two-dimensional case is, of course, very similar to the three-dimensional one. Equations (5.35) and (5.36) correspond to

$\displaystyle \int_{\mathcal{A}}\left(\vec{\nabla}\times\vec{N}_i\right) \cdot\... ... - \jmath\omega\int_{\mathcal{A}}\mu\vec{N}_i\cdot\vec{\nabla}\psi \mathrm{d}A$	$\displaystyle = 0$	(5.72)
$\displaystyle \int_{\mathcal{A}}\vec{\nabla}\lambda_i\cdot\left(\mu\vec{H}_1\ri... ...hcal{A}}\vec{\nabla}\lambda_i\cdot\left(\mu\vec{\nabla}\psi\right) \mathrm{d}A$	$\displaystyle = 0.$	(5.73)

The boundary $\partial\mathcal{A}$ of the two-dimensional domain $\mathcal{A}$ is divided into a Dirichlet boundary $\mathcal{C}_{D1}$ and a Neumann boundary $\mathcal{C}_{N1}$ for (5.72) and into a Dirichlet boundary $\mathcal{C}_{D2}$ and a Neumann boundary $\mathcal{C}_{N2}$ for (5.73)

$\displaystyle \partial\mathcal{A} = \mathcal{C}_{D1} + \mathcal{C}_{N1} , \partial\mathcal{A} = \mathcal{C}_{D2} + \mathcal{C}_{N2}.$

The Dirichlet boundary conditions define values for $\psi$ on the nodes belonging to the Dirichlet boundary $\mathcal{C}_{D2}$ or for the tangential component of $\vec{H}_1$ on the edges belonging to the Dirichlet boundary $\mathcal{C}_{D1}$ . The finite element analysis is performed as for the three-dimensional case: The unknown functions in (5.72) and (5.73) are substituted by their approximations, the corresponding residua are weighted by vector and scalar trial functions, and the weak formulation (the law of Gauß) is applied. Thereby the boundary conditions on $\mathcal{C}_{N1}$ and $\mathcal{C}_{N2}$ arise analogously to (5.29) and (5.31) and read

$\displaystyle f_{N1} = \int_{\mathcal{C}_{N1}}\vec{n}\cdot\left[\vec{N}_i\times... ...}\vec{N}_i\cdot\left(\vec{E}\times\vec{n}\right) \mathrm{d}s \mathrm{and}$

(5.74)

$\displaystyle f_{N2} = \int_{\mathcal{C}_{N2}}\vec{n}\cdot\left\{\lambda_i\left... ...athrm{d}s = \int_{\mathcal{C}_{N2}}\lambda_i \vec{n}\cdot\vec{B} \mathrm{d}s.$

(5.75)

Often the shape and the dimensions of the simulation domain are chosen to assume the boundary conditions $f_{N1}$ and $f_{N2}$ are zero. For example this is the case, if the electric field $\vec{E}$ is normal to the Neumann boundary $\mathcal{C}_{N1}$ or if the magnetic flux $\vec{B}$ has no normal component to the Neumann boundary $\mathcal{C}_{N2}$ , or if the simulation domain is sufficiently large to assume that $\vec{E}\times\vec{n}$ is zero on $\mathcal{C}_{N1}$ and $\vec{n}\cdot\vec{B}$ is zero on $\mathcal{C}_{N2}$ . If the boundary conditions $f_{N1}$ and $f_{N2}$ have to be considered, for example to combine the finite element analysis with the boundary element method, the corresponding element matrices are calculated in the Appendix in Section B.1, Subsection B.2.1 and Subsection B.2.2.

The domain is discretized in triangular elements and the boundary in curves, respectively.