


[Contents]  [Index] 
Cooperation with: W. Heinrich, M. Kunze, T. Tischler, H. Zscheile (FerdinandBraunInstitut für Höchstfrequenztechnik, Berlin (FBH))
Supported by: FerdinandBraunInstitut für Höchstfrequenztechnik (FBH)
Description:
The electric properties of circuits are described in terms of their scattering matrix using a threedimensional boundary value problem for Maxwell's equations. The computational domain is truncated by electric or magnetic walls; open structures are treated using the Perfectly Matched Layer (PML) absorbing boundary condition.
The subject under investigation is about passive structures of arbitrary geometry. They are connected to the remaining circuit by transmission lines. Ports are defined on the waveguides. In order to characterize their electrical behavior the transmission lines are assumed to be infinitely long and longitudinally homogeneous.
The boundary value problem
Due to the fact that only small fractions of a circuit can be simulated, the pressure for larger problem sizes and shorter computing times is evident. Thus, in the period under report, studies were continued in the following areas:
(1) Eigenmode problem [2]: Only a few modes of smallest attenuation are able to propagate and have to be taken into consideration. Using a conformal mapping between the plane of propagation constants and the plane of eigenvalues, the task is to compute all eigenmodes in a region, bounded by two parabolas. The region is covered by a number of overlapping circles. The eigenmodes in these circles 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 method, developed initially for a reliable calculation of all complex eigenvalues from microwave structure computations [4], is expanded to meet the very special requirements of optoelectronic structure calculations. Relatively large cross sections and highest frequencies yield increased dimensions and numbers of eigenvalue problems. Using the results of a coarse grid calculation within the final fine grid calculation yields a remarkable reduced numerical effort. The use of the new linear sparse solver PARDISO [12] (see page ) rather than UMFPACK [3], and two levels of parallelization result in an additional speedup of computation time.
(2) Boundary value problem [1]: The electromagnetic fields are computed by the solution of largescale systems of linear equations with indefinite complex symmetric coefficient matrices. In general, these matrix problems have to be solved repeatedly for different righthand sides, but with the same coefficient matrix. The number of righthand sides depends on the number of ports and modes. The systems of linear equations are solved using a block Krylov subspace iterative method. Independent set orderings, Jacobi and SSOR preconditioning techniques are applied to reduce the dimension and the number of iterations [13]. In comparison to the simple lossy case the number of iterations of Krylov subspace methods increases significantly in the presence of Perfectly Matched Layers. This growth could be further reduced by applying suitable grids and PML. The speed of convergence depends on the relations of the edges in an elementary cell of the nonequidistant rectangular grid in this case. The best results can be obtained using nearly cubic cells grids. Moreover, overlapping PML conditions on the corners downgrade the properties of the coefficient matrix, and should be avoided.
(3) Consistent subgridding scheme: For the abovementioned methods the geometry to be analyzed is subdivided into elementary rectangular cells using threedimensional nonequidistant rectangular cells. Applying rectangular grids, a mesh refinement in one point results in an accumulation of elementary cells in all coordinate directions even though generally the refinement is needed only in certain regions. Due to the high spatial resolution required, CPU time and storage requirements are very high. A general approach to handle small structure details is to introduce local mesh refinement ([14]) with hanging nodes. Energy and divergence conservation have to be conserved throughout the discretization process for Maxwell's equations. Maxwell's grid equations have been derived for some kinds of submeshes.
(4) Tetrahedral grids [11]: Rectangular grids are not well suitable for the treatment of curved and nonrectangular boundaries. Thus, supported by FBH, the development of a finitevolume method was continued by using unstructured grids (tetrahedrals, see page ) for the boundary value problem.
The new research results have been published in [5]  [10].
References:



[Contents]  [Index] 