4.2.3 Assembling

The assembling of the global matrix

and the right hand side vector $\{d\}$ is done similarly to the two-dimensional method. The entire region is processed element wise. In each tetrahedron the element matrix

is calculated. The local matrix entries $K_{ij}^e$ are added to the global entries $K_{ij}$ or are multiplied by the known coefficient

(obtained from the Dirichlet boundary condition) and added to the right hand side vector element

depending on the global index

, by which it is distinguished between Dirichlet or unknown node coefficients. The superscript

indicates that the indexes

are local element indexes, which are converted to global

indexes (without superscript

) by the connectivity array. In the three-dimensional case the connectivity array has four row entries for the four element nodes.

For the assembling of the element matrix it is assumed that the permittivity $\utilde{\epsilon}$ is a scalar and element wise constant. This assumption is not an essential restriction. It allows $\epsilon$ to change from one element to the next. Thus inhomogeneous materials (with respect to $\epsilon$ ) can be easily simulated. The regions, in which $\epsilon$ changes substantially, must be discretized in appropriately small elements. Using the expressions for the gradient of the barycentric coordinates (4.71) to (4.74) the element matrix for the Laplace term is given by

$\displaystyle K_{11}^e$	$\displaystyle = \int_{\mathcal{V}_e}\vec{\nabla}\lambda_1^e\cdot\utilde{\epsilo... ...thrm{d}V = \epsilon{}V_e \vec{\nabla}\lambda_1^e\cdot\vec{\nabla}\lambda_1^e =$
	$\displaystyle = \epsilon{}V_e \frac{\vec{r}_4\times\vec{r}_5}{6V_e}\cdot\frac{... ...(\vec{r}_5\cdot\vec{r}_5) - (\vec{r}_4\cdot\vec{r}_5)(\vec{r}_4\cdot\vec{r}_5)]$	(4.80)
$\displaystyle K_{12}^e = K_{21}^e$	$\displaystyle = \frac{\epsilon}{36V_e}[(\vec{r}_2\cdot\vec{r}_6)(\vec{r}_4\cdot\vec{r}_6) - (\vec{r}_2\cdot\vec{r}_4)(\vec{r}_6\cdot\vec{r}_6)]$	(4.81)
$\displaystyle K_{13}^e = K_{31}^e$	$\displaystyle = \frac{\epsilon}{36V_e}[(\vec{r}_3\cdot\vec{r}_4)(\vec{r}_5\cdot\vec{r}_5) - (\vec{r}_3\cdot\vec{r}_5)(\vec{r}_4\cdot\vec{r}_5)]$	(4.82)
$\displaystyle K_{14}^e = K_{41}^e$	$\displaystyle = \frac{\epsilon}{36V_e}[(\vec{r}_1\cdot\vec{r}_6)(\vec{r}_4\cdot\vec{r}_4) - (\vec{r}_1\cdot\vec{r}_4)(\vec{r}_4\cdot\vec{r}_6)]$	(4.83)
$\displaystyle K_{22}^e$	$\displaystyle = \frac{\epsilon}{36V_e}[(\vec{r}_2\cdot\vec{r}_2)(\vec{r}_3\cdot\vec{r}_3) - (\vec{r}_2\cdot\vec{r}_3)(\vec{r}_2\cdot\vec{r}_3)]$	(4.84)
$\displaystyle K_{23}^e = K_{32}^e$	$\displaystyle = \frac{\epsilon}{36V_e}[(\vec{r}_1\cdot\vec{r}_6)(\vec{r}_3\cdot\vec{r}_3) - (\vec{r}_1\cdot\vec{r}_3)(\vec{r}_3\cdot\vec{r}_6)]$	(4.85)
$\displaystyle K_{24}^e = K_{42}^e$	$\displaystyle = \frac{\epsilon}{36V_e}[(\vec{r}_1\cdot\vec{r}_3)(\vec{r}_4\cdot\vec{r}_6) - (\vec{r}_1\cdot\vec{r}_6)(\vec{r}_3\cdot\vec{r}_4)]$	(4.86)
$\displaystyle K_{33}^e$	$\displaystyle = \frac{\epsilon}{36V_e}[(\vec{r}_1\cdot\vec{r}_1)(\vec{r}_3\cdot\vec{r}_3) - (\vec{r}_1\cdot\vec{r}_3)(\vec{r}_1\cdot\vec{r}_3)]$	(4.87)
$\displaystyle K_{34}^e = K_{43}^e$	$\displaystyle = \frac{\epsilon}{36V_e}[(\vec{r}_1\cdot\vec{r}_3)(\vec{r}_1\cdot\vec{r}_4) - (\vec{r}_1\cdot\vec{r}_1)(\vec{r}_3\cdot\vec{r}_4)]$	(4.88)
$\displaystyle K_{44}^e$	$\displaystyle = \frac{\epsilon}{36V_e}[(\vec{r}_1\cdot\vec{r}_1)(\vec{r}_4\cdot\vec{r}_4) - (\vec{r}_1\cdot\vec{r}_4)(\vec{r}_1\cdot\vec{r}_4)].$	(4.89)

If inhomogeneous Neumann boundary conditions are used, the corresponding element matrix is depicted in Section C.2.1, which is also used for the quasi magnetostatic case. Thus the magnetic scalar potential $\psi$ and the permeability $\utilde{\mu}$ are used instead $\varphi$ and $\utilde{\epsilon}$ . Additionally it is assumed that the permeability is a scalar and element wise constant ( $\utilde{\mu} \rightarrow \mu$ ).