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Simulation of microwave and laser structures using rectangular grids and tetrahedral nets

Collaborator: G. Hebermehl, F.-K. Hübner, R. Schlundt

Cooperation with: W. Heinrich, T. Tischler, H. Zscheile (Ferdinand-Braun-Institut für Höchstfrequenztechnik (FBH) Berlin)

Description: Field-oriented methods which describe the physical properties of microwave circuits and optical structures are an indispensable tool to avoid costly and time-consuming 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 cross-sectional planes on each waveguide, [2]. The electric field is the solution of a two-dimensional eigenvalue and a three-dimensional 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 three-dimensional 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 non-equidistant rectangular grids and for tetrahedral nets with corresponding dual Voronoi cells.

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:

$$\oint_{{\partial \Omega}}^{} \vec{H} \vec{s} \int_{{\Omega}}^{} \omega \epsilon \vec{E} \vec{\Omega} \oint_{{\cup \Omega}}^{} \epsilon \vec{E} \vec{\Omega}$$ = = 0,

$$\oint_{{\partial \Omega}}^{} \vec{E} \vec{s} \int_{{\Omega}}^{} \omega \mu \vec{H} \vec{\Omega} \oint_{{\cup \Omega}}^{} \mu \vec{H} \vec{\Omega}$$ = = 0,

$$\vec{D}$$ = [$$\epsilon$$]$$\vec{E}$$,$$\vec{B}$$ = [$$\mu$$]$$\vec{H}$$,

with

[$$\epsilon$$] = $$\epsilon_{{0}}^{{}}$$diag$$\left(\vphantom{ {\epsilon}_x, {\epsilon}_y, {\epsilon}_z }\right.$$$$\epsilon_{{x}}^{{}}$$,$$\epsilon_{{y}}^{{}}$$,$$\epsilon_{{z}}^{{}}$$$$\left.\vphantom{ {\epsilon}_x, {\epsilon}_y, {\epsilon}_z }\right)$$,[$$\mu$$] = $$\mu_{{0}}^{{}}$$diag$$\left(\vphantom{ {\mu}_x, {\mu}_y, {\mu}_z }\right.$$$$\mu_{{x}}^{{}}$$,$$\mu_{{y}}^{{}}$$,$$\mu_{{z}}^{{}}$$$$\left.\vphantom{ {\mu}_x, {\mu}_y, {\mu}_z }\right)$$

for rectangular grids and

[$$\epsilon$$] = $$\epsilon_{{0}}^{{}}$$$$\epsilon_{{r}}^{{}}$$,[$$\mu$$] = $$\mu_{{0}}^{{}}$$$$\mu_{{r}}^{{}}$$

for tetrahedral grids. This results in a two-step procedure: an eigenvalue problem for complex matrices and the solution of large-scale systems of linear algebraic equations with indefinite symmetric complex matrices.

(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 high-dimensional 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 speed-up of computation time. The eigenvalue problem for rectangular grids is described in [5-8]. 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 large-scale systems of linear equations with indefinite complex symmetric coefficient matrices. In general, these matrix problems have to be solved repeatedly for different right-hand sides, but with the same coefficient matrix. The number of right-hand sides depends on the number of ports and modes. Independent set orderings, Jacobi and SSOR pre-conditioning techniques, [10], and a block quasi-minimal 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 off-diagonal 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 non-rectangular structures. Thus, tetrahedral nets with corresponding Voronoi cells are used for the three-dimensional 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.

References:

1. K. BEILENHOFF, W. HEINRICH, H.L. HARTNAGEL, Improved finite-difference formulation in frequency domain for three-dimensional scattering problems, IEEE Trans. Microwave Theory Techniques, 40 (1992), pp. 540-546.
2. A. CHRIST, H.L. HARTNAGEL, Three-dimensional finite-difference method for the analysis of microwave-device embedding, IEEE Trans. Microwave Theory Techniques, 35 (1987), pp. 688-696.
3. R.W. FREUND, W. MALHOTRA, A Block-QMR algorithm for non-Hermitian linear systems with multiple right-hand sides, Linear Algebra Applications, 254 (1997), pp. 119-157.
4. G. HEBERMEHL, R. SCHLUNDT, H. ZSCHEILE, W. HEINRICH, Improved numerical methods for the simulation of microwave circuits, Surv. Math. Ind., 9 (1999), pp. 117-129.
5. G. HEBERMEHL, F.-K. HÜBNER, R. SCHLUNDT, TH. TISCHLER, H. ZSCHEILE, W. HEINRICH, Numerical simulation of lossy microwave transmission lines including PML, in: Scientific Computing in Electrical Engineering, U. van Rienen, M. Günther, D. Hecht, eds., vol. 18 of Lecture Notes Comput. Sci. Eng., Springer, Berlin, 2001, pp. 267-275.
6.          , in: Perfectly matched layers in transmission lines, in: Numerical Mathematics and Advanced Applications, ENUMATH 2001, F. Brezzi, A. Buffa, S. Corsaro, A. Murli, eds., Springer, Italy, 2003, pp. 281-290.
7.          , Simulation of microwave and semiconductor laser structures including absorbing boundary conditions, in: Challenges in Scientific Computing -- CISC 2002, E. Bänsch, ed., vol. 35 of Lecture Notes Comput. Sci. Eng., Springer, Berlin, 2003, pp. 131-159.
8.          , Eigenmode computation of microwave and laser structures including PML, in: Scientific Computing in Electrical Engineering, W.H.A. Schilders, S.H.M.J. Houben, E.J.W. ter Maten, eds., Mathematics in Industry, Springer, Berlin, 2004, pp. 196-205.
9. R.B. LEHOUCQ, Analysis and implementation of an implicitly restarted Arnoldi iteration, Technical Report no. 13, Rice University, Department of Computational and Applied Mathematics, Houston, USA, 1995.
10. Y. SAAD, Iterative Methods for Sparse Linear Systems, PWS Publishing Company, Boston, Mass., 1996.
11. Z.S. SACKS, D.M. KINGSLAND, R. LEE, J.-F. LEE, A perfectly matched anisotropic absorber for use as an absorbing boundary condition, IEEE Trans. Antennas Propagation, 43 (1995), pp. 1460-1463.
12. J. SCHEFTER, Discretisation of Maxwell equations on tetrahedral grids, WIAS Technical Report no. 6, 2003.
13. O. SCHENK, K. GÄRTNER, W. FICHTNER, Efficient sparse LU factorization with left-right looking strategy on shared memory multiprocessors, BIT, 40 (2000), pp. 158-176.
14. R. SCHLUNDT, G. HEBERMEHL, F.-K. HÜBNER, W. HEINRICH, H. ZSCHEILE, Iterative solution of systems of linear equations in microwave circuits using a block quasi-minimal residual algorithm, in: Scientific Computing in Electrical Engineering, U. van Rienen, M. Günther, D. Hecht, eds., vol. 18 of Lecture Notes Comput. Sci. Eng., Springer, Berlin, 2001, pp. 325-333.
15. TH. TISCHLER, W. HEINRICH, The perfectly matched layer as lateral boundary in finite-difference transmission-line analysis, IEEE Trans. Microwave Theory Techniques, 48 (2000), pp. 2249-2253.
16. T. WEILAND, A discretization method for the solution of Maxwell's equations for six-component fields, Int. J. Electronics Communication (AEÜ), 31 (1977), pp. 116-120.

Fig. 1: Primary and dual grid; An eight-cell primary grid and its one interior dual cell; Voronoi cell and single tetrahedron. The electric field intensity components marked with red color are located at the centers of the edges, and the magnetic flux density components marked with black color are normal to the cell faces.

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