Keywords: model order reduction, multivariate Krylov subspace methods, parameterized systems, finite integration technique.

In this paper we consider large linear time invariant systems which depend on several parameters s₁,s₂,...,s_n. These may be the frequency, material or geometric parameters. It is presented how they arise from the FIT-discretized [1,2] Maxwell equations. In order to solve such systems in a reasonable time, model order reduction is used which aims to reduce the n dimensional system to an m<<n dimensional system which preserves some qualities of the original system. Especially model order reduction via projection on Krylov-subspaces has proven to be very effective and its characteristics for one-parameter linear systems are briefly highlighted here.

In the single parameter case [3,4] the order reduction is carried out via projection on a subspace with dimension m<<n. The solution vector x of the original system is restricted on a space V spanned by orthonormal trial vectors v₁...v₂. By projection with a test matrix W in C^l*m the dimension of the system matrix is reduced from n*n to m*m. The matrices V and W are chosen so that the resulting reduced order model is a partial realization, Padé approximation or rational interpolant of the transfer function of the original system.

Details concerning the relation of the approximations and the projection matrices are found in [5] as well as in [3,4]. It is a common characteristic of all these cases, that colsp(V) and colsp(W) must contain unions of Krylov subspaces of the system matrices and vectors [5]. An analogous method exists for multivariate linear systems (see also [6,7,8]) and is presented also in this paper. Unfortunately it cannot be used for FIT-systems directly as they are nonlinear with respect to line length. This work shows a linearization method. We consider the mesh being stretched according to the length variation, so that the mesh topology is maintained and expresses all matrices according to this mesh variation. The procedure presented here is demonstrated on a numerical example.

1

T. Weiland, "Eine Methode zur Lösung der Maxwellschen Gleichungen für sechskomponentige Felder auf diskreter Basis", Electronics and Communictions (AEÜ), Vol31(3), pp. 116-120, March 1977.

2

T. Weiland, "Time Domain Electromagnetic Field Computation with Finite Difference Methods", Int.J.Num.Modelling, 9:295-319, 1996. doi:10.1002/(SICI)1099-1204(199607)9:4<295::AID-JNM240>3.0.CO;2-8

3

T. Wittig, "Zur Reduzierung der Modellordnung in elektromagnetischen Feldsimulationen", PhD Dissertation, Cuvillier Verlag Göttingen, 2003.

4

T. Wittig, R. Schuhmann, T. Weiland, "Model Order Reduction for Large Systems in Computational Electromagnetics", LAA 2006, Vol.415(2), pp. 499-530, May 2006.

5

E.J. Grimme, "Krylov projection methods for model reduction", Ph.D.Thesis, University of Illinois at Urbana Champaign, 1997.

6

O. Farle, V. Hill, P. Ingelström, R. Dyczij-Edlinger, "Model Order Reduction of Linear Finite Element Models Parameterized by Polynomials in Several Variables", Automatisierungstechnik, vol 54, pp. 161-169, 2006. doi:10.1524/auto.2006.54.4.161

7

R. Dyczij-Edlinger, O. Farle, V. Hill, P. Ingelström, "Multi-level and parametric Finite Element Techniques for Time-Harmonic Fields", EMB 07, Lund University, pp. 185-192, 2007.

8

D.S. Weile, E. Michielsen, E. Grimme, K. Gallivan, "A method for generating rational interpolant reduced order models of two parameter linear systems", Applied Mathematics Letters, Vol. 12, pp. 93-102, July 1999. doi:10.1016/S0893-9659(99)00063-4

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