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Cooperation with: W. Heinrich, T. Tischler, H. Zscheile (FerdinandBraunInstitut für Höchstfrequenztechnik (FBH) Berlin)
Description: Fieldoriented methods which describe the physical properties of microwave circuits and optical structures are an indispensable tool to avoid costly and timeconsuming redesign cycles. Commonly the electromagnetic characteristics of the structures are described by their scattering matrix which is extracted from the orthogonal decomposition of the electric field at a pair of neighboring crosssectional planes on each waveguide, [2]. The electric field is the solution of a twodimensional eigenvalue and a threedimensional boundary value problem for Maxwell's equations in the frequency domain, [7]. The computational domain is truncated by electric or magnetic walls; open structures are treated using the Perfectly Matched Layer (PML) ([11]) absorbing boundary condition.
The subject under investigation are threedimensional structures of arbitrary geometry which are connected to the remaining circuit by transmission lines. Ports are defined at the transmission lines' outer terminations. In order to characterize their electrical behavior the transmission lines are assumed to be infinitely long and longitudinally homogeneous. Short parts of the transmission lines and the passive structure (discontinuity) form the structure under investigation, [7].
The equations are discretized with orthogonal grids using the Finite Integration Technique (FIT), [1, 4, 16]. Maxwellian grid equations are formulated for staggered nonequidistant rectangular grids and for tetrahedral nets with corresponding dual Voronoi cells.
A threedimensional 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:
^{ . }d  =  j[]^{ . }d,  ([])^{ . }d  =  0, 
^{ . }d  =   j[]^{ . }d,  ([])^{ . }d  =  0, 
(1) Eigenmode problem ([2]): The interesting modes of smallest attenuation are found solving a sequence of eigenvalue problems of modified matrices with the aid of the invert mode of the Arnoldi iteration using shifts implemented in the package ARPACK, [9]. To reduce the execution time for highdimensional problems, a coarse and a fine grid are used. The use of the linear sparse solver PARDISO ([13]) and two levels of parallelization results in an additional speedup of computation time. The eigenvalue problem for rectangular grids is described in [58]. The mode fields at the ports of a transmission line, which is discretized by means of tetrahedral grids, are computed interpolating the results of the rectangular discretization. The PML influences the mode spectrum. Modes that are related to the PML boundary can be detected using the power part criterion [15].
(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. Independent set orderings, Jacobi and SSOR preconditioning techniques, [10], and a block quasiminimal residual algorithm, [3], are applied to solve the systems of the linear algebraic equations. Details are given in [7] and [14]. In comparison to the simple lossy case, the number of iterations of Krylov subspace methods increases significantly in the presence of PML. Moreover, overlapping PML conditions at the corner regions of the computational domain lead to an increase of the magnitude of the corresponding offdiagonal elements in comparison to the diagonal ones of the coefficient matrix. This downgrades the properties of the matrix, [7].
Using rectangular grids, a mesh refinement in one point results in an accumulation of small elementary cells in all coordinate directions. In addition, rectangular grids are not well suited for the treatment of curved and nonrectangular structures. Thus, tetrahedral nets with corresponding Voronoi cells are used for the threedimensional boundary value problem. The primary grid is formed by tetrahedra and the dual grid by the corresponding Voronoi cells, which are polytopes, [12]. The gradient of the electric field divergence at an internal point is obtained considering the partial volumes of the appropriate Voronoi cell.
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