Multi-parameter polynomial order reduction of linear finite element models

In this paper we present a numerically stable method for the model order reduction of finite element (FE) approximations to passive microwave structures parameterized by polynomials in several variables. The proposed method is a projection-based approach using Krylov subspaces and extends the works of Gunupudi etal. (P. Gunupudi, R. Khazaka and M. Nakhla, Analysis of transmission line circuits using multidimensional model reduction techniques, IEEE Trans. Adv. Packaging 25 (2002), pp. 174–180) and Slone etal. (R.D. Slone, R. Lee and J.-F. Lee, Broadband model order reduction of polynomial matrix equations using single-point well-conditioned asymptotic waveform evaluation: derivations and theory, Int. J. Numer. Meth. Eng. 58 (2003), pp. 2325–2342). First, we present the multivariate Krylov space of higher order associated with a parameter-dependent right-hand-side vector and derive a general recursion for generating its basis. Next, we propose an advanced algorithm to compute such basis in a numerically stable way. Finally, we apply the Krylov basis to construct a reduced order model of the moment-matching type. The resulting single-point method requires one matrix factorization only. Numerical examples demonstrate the efficiency and reliability of our approach.

[1]  Juan Zapata,et al.  Full-wave analysis of cavity-backed and probe-fed microstrip patch arrays by a hybrid mode-matching generalized scattering matrix and finite-element method , 1998 .

[2]  Eric James Grimme,et al.  Krylov Projection Methods for Model Reduction , 1997 .

[3]  Roland W. Freund,et al.  Efficient linear circuit analysis by Pade´ approximation via the Lanczos process , 1994, EURO-DAC '94.

[4]  M. Nakhla,et al.  Analysis of transmission line circuits using multi-dimensional model reduction techniques , 2001, IEEE 10th Topical Meeting on Electrical Performance of Electronic Packaging (Cat. No. 01TH8565).

[5]  R. Craig,et al.  Model reduction and control of flexible structures using Krylov vectors , 1991 .

[6]  P. Dooren,et al.  Asymptotic Waveform Evaluation via a Lanczos Method , 1994 .

[7]  Daniel Boley.  Krylov space methods on state-space control models , 1994 .

[8]  B. Lohmann,et al.  Order reduction of large scale second-order systems using Krylov subspace methods , 2006 .

[9]  Alain Bossavit,et al.  Edge-elements for scattering problems , 1989 .

[10]  Michel Nakhla,et al.  Transient waveform estimation of high-speed MCM networks using complex frequency hopping , 1993, Proceedings 1993 IEEE Multi-Chip Module Conference MCMC-93.

[11]  Abdullah Atalar,et al.  Pole-zero computation in microwave circuits using multipoint Pade approximation , 1995 .

[12]  Jacob K. White,et al.  A multiparameter moment-matching model-reduction approach for generating geometrically parameterized interconnect performance models , 2004, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems.

[13]  Zhaojun Bai,et al.  Dimension Reduction of Large-Scale Second-Order Dynamical Systems via a Second-Order Arnoldi Method , 2005, SIAM J. Sci. Comput..

[14]  Lawrence T. Pileggi,et al.  PRIMA: passive reduced-order interconnect macromodeling algorithm , 1998, 1997 Proceedings of IEEE International Conference on Computer Aided Design (ICCAD).

[15]  Eric Michielssen,et al.  Analysis of frequency selective surfaces using two-parameter generalized rational Krylov model-order reduction , 2001 .

[16]  R. W. Freund,et al.  Passive reduced-order modeling via Krylov-subspace methods , 2000, CACSD. Conference Proceedings. IEEE International Symposium on Computer-Aided Control System Design (Cat. No.00TH8537).

[17]  Rodney D. Slone,et al.  Broadband model order reduction of polynomial matrix equations using single‐point well‐conditioned asymptotic waveform evaluation: derivations and theory , 2003 .

[18]  Lawrence T. Pileggi,et al.  Asymptotic waveform evaluation for timing analysis , 1990, IEEE Trans. Comput. Aided Des. Integr. Circuits Syst..

[19]  L. Codecasa,et al.  A novel approach for generating boundary condition independent compact dynamic thermal networks of packages , 2005, IEEE Transactions on Components and Packaging Technologies.

[20]  Axel Ruhe.  Rational Krylov sequence methods for eigenvalue computation , 1984 .

[21]  Kyle A. Gallivan,et al.  A method for generating rational interpolant reduced order models of two-parameter linear systems , 1999 .

[22]  Robert J. Lee,et al.  Applying Padé via Lanczos to the finite element method for electromagnetic radiation problems , 2000 .

[23]  Jin-Fa Lee,et al.  The transfinite element method for modeling MMIC devices , 1988, 1988., IEEE MTT-S International Microwave Symposium Digest.