![\begin{align}
\oint_{\partial \Omega} \vec H
\cdot d\vec s & = \int_{\Omega}
j\o...
... \oint_{\cup \Omega} ([\mu] \vec H) \cdot d \vec
\Omega & = 0 \notag,\end{align}](../2001/img234.gif)
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Cooperation with: W. Heinrich, M. Kunze, T. Tischler, H. Zscheile (Ferdinand-Braun-Institut für Höchstfrequenztechnik, Berlin (FBH))
Supported by: DFG: ``Finite-Integrations-Methode mit Tetraedergitter zur elektromagnetischen Simulation von Mikrowellenschaltungen'' (Electromagnetic simulation of microwave circuits using finite integration techniques with tetrahedral meshes), FBH
Description: The design of microwave and optoelectronic devices requires efficient CAD tools in order to avoid costly and time-consuming redesign cycles. The fields of application are mobile communications, radio links, automobile radar systems, optical communications, and material processing. The commercial applications of microwave circuits cover the frequency range between 1 GHz and about 100 GHz, special applications in radioastronomy use even higher frequencies up to 1 THz. For optoelectronic devices, frequencies about several hundred THz are common.
The structures under investigation can be described as an interconnection of infinitely-long transmission lines, which have to be longitudinally homogeneous. Ports are defined on the transmission lines. The junction, the so-called discontinuity, may have an arbitrary structure. The whole structure may be surrounded by an enclosure.
For numerical treatment, the computational domain has to be truncated by electric or magnetic walls or by so-called Absorbing Boundary Conditions (ABC) simulating radiation effects. Among the ABCs, the Perfectly Matched Layer (PML) technique represents the most powerful formulation.
The scattering matrix describes the structure in terms of wave modes at the ports, which
can be computed from the electromagnetic field.
A three-dimensional boundary value problem can be formulated using the integral
form of Maxwell's equations in the frequency domain in order to compute
the electromagnetic field and subsequently the scattering matrix:
![\begin{displaymath}
\vec D = [\epsilon] \vec E, \enspace \vec B = [\mu] \vec H,
...
...[\mu] = \mbox{diag} \left ( {\mu}_x, {\mu}_y, {\mu}_z \right ).\end{displaymath}](../2001/img235.gif){kind=small}
In the period under report, studies were made in the following areas:
(A) Eigenmode problem ([1]): We are interested only in a few modes having the smallest attenuation. These are the modes with the smallest magnitudes of the imaginary part, but possibly with a large real part of associated propagation constants. First the method to find all eigen modes within a certain region of the complex plane was developed for microwave structures ([3], [4]). The modes are found solving a sequence of eigenvalue problems of modified matrices with the aid of the invert mode of the Arnoldi iteration using shifts. The number of the corresponding propagation constants increases using PML. Due to the short wavelength, the dimension and the number of eigenvalue problems grows strongly for applications in optoelectronics (self-aligned stripe (SAS) laser) ([7]) in comparison to microwave circuits. All propagation constants which are located in the long strip of the complex plane covered by a sequence of Cassini curves (see Fig. 1) must be found. New strategies to handle these problems in a feasible time were developed. To reduce the execution times, in a first step, the problem is solved using a coarse grid in order to find approximately the locations of the propagation constants of interest. The accurate modes are calculated in a second step for an essentially reduced region, using a fine grid. Examining the eigenfunctions, in an additional step, nonphysical PML modes are eliminated.
(B) Boundary value problem ([2]): Using a finite volume approach with staggered nonequidistant rectangular grids, Maxwell's equations are transformed into a set of Maxwellian grid equations. The corresponding high-dimensional systems of linear-algebraic equations are solved using Krylov subspace methods with preconditioning techniques ([6]). Depending on the PML, the number of iterations increases strongly in some cases. To get feasible execution times, techniques how to choose the PML and the PML cell sizes have been developed.
(C) Tetrahedral grids ([9]):
For the above-mentioned methods, the geometry to be analyzed is subdivided into elementary rectangular cells
using three-dimensional nonequidistant rectangular cells. Due to the
high spatial resolution, CPU time and storage requirements are very
high. Supported by DFG and FBH, a finite volume method using
unstructured meshes (tetrahedrons) was developed for the boundary
value problem, with the aim to reduce the
number of elementary cells by improved possibilities of local grid refinement and to improve the treatment
of curved boundaries. Using a Delaunay triangulation
(grid generator COG
(see page ![[*]](http://www.wias-berlin.de/misc/icons/cross_ref_motif.gif){kind=icon}
) and
lbg
) and corresponding
Voronoi cells, Maxwell's equations are transformed into a set of Maxwellian grid equations. The circumcenter
of a generated tetrahedron can be located outside the
tetrahedron. This is taken into account when generating
the matrix. Substituting the components of the magnetic flux density,
the number of unknowns is reduced by a factor of two.
The new research results have been published in [4] - [9].
References:
, On the computation of eigen modes for lossy microwave
transmission lines including perfectly matched layer boundary conditions,
Internat. J. Comput. Math.
Electrical Electronic Engng., 20 (2001), pp. 948-964.
, Perfectly matched layers in microwave transmission lines, in:
ENUMATH 2001, European Conference on Numerical Mathematics and Advanced
Applications, Abstracts, July 23-28, 2001, Tech. Rep. no. TR-01-9, Luglio
2001, Ischia Porto, Italy, July 2001, Centro di Ricerche per il Calcolo
Parallelo e i Supercalcolatori del Consiglio Nazionale delle Ricerche,
pp. 107-108.
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