Keywords: power flow, Krylov subspace, model reduction, parallel implementation, adaptive interpolation.

This paper focuses on power flow prediction in vibrating systems over a frequency band of interest. An efficient method is proposed to predict power flow from finite element models based on the rational Krylov subspace projection [1]. It belongs to a family of well-established moment-matching techniques, which have been successfully used in circuit simulation [2], electromagnetic problems [3] and acoustic field computation [4,5]. In this paper, general formulation of power flow is derived as multiple-input multiple-output transfer functions of linear time-invariant systems. Instead of constructing the reduced-order models (ROMs) by directly projecting the original systems onto the rational Krylov subspaces, a matrix-free algorithm is developed to construct the ROMs from the interpolation values of the transfer functions and their higher-order derivatives [6]. The ROMs match the moments of the transfer functions of the original systems at interpolation frequencies to a specified order, which guarantees the preservation of power flow and its higher-order derivatives up to the same order. The matrix-free algorithm reduces both the computational cost and the memory requirement in constructing the ROMs.

The accuracy of the ROMs is mainly determined by the choice of interpolation frequencies. This paper investigates how to improve the accuracy of the ROMs by adaptive interpolation. After an initial ROM is constructed, its accuracy is improved by adding new interpolation points at the frequencies corresponding to large approximation error. Unfortunately, the exact error is not known without solving the original system. The results presented in this paper show that the residual norms provide a good way to identify the frequencies with large approximation error. Since the residual norms are obtained directly from the ROM, this approach minimizes the approximation error at a reasonable cost. The performance of the matrix-free algorithm can be enhanced by adapting it for simultaneous execution in multiple processes. The main process adaptively chooses interpolation frequencies according to the current error estimate, evenly distributes them among all processes to achieve load balancing, gathers computation results from the rest processes, and constructs the ROM according to the matrix-free algorithm. Inter-process communication is implemented by using the message passing interface [7]. Significant parallel speedup is achieved because the major computational task in each process is independent.

Case studies are conducted on the testbeds from Harwell-Boeing collection [8]. Four numerical examples are presented to illustrate the efficiency of adaptive interpolation, the effect of matching-moment order, and the effect of different combinations of initial and updating sizes. For the ROMs of the same size, the one using more interpolation frequencies is usually more accurate. Initial and updating sizes also affect the accuracy of the resulting ROMs. It is observed that starting from a small-sized ROM and iterating many times with small updating size does not necessarily guarantee an accurate ROM. It is preferable to use a large initial size and update only one or two times in order to construct an accurate ROM efficiently.

1

E.J. Grimme, "Krylov Projection Methods for Model Reduction", Ph.D. dissertation, University of Illinois at Urbana-Champaign, 1997.

2

R.W. Freund, "Krylov-subspace methods for reduced-order modeling in circuit simulation", Journal of Computational and Applied Mathematics, 123(1-2):395-421, 2000. doi:10.1016/S0377-0427(00)00396-4

3

M. Kuzuoglu, R. Mittra, "Finite element solution of electromagnetic problems over a wide frequency range via the Padé approximation", Computer Methods in Applied Mechanics and Engineering, 169(3-4):263-277, 1999. doi:10.1016/S0045-7825(98)00157-1

4

M. Malhotra, P.M. Pinsky. "Efficient computation of multi-frequency far-field solutions of the Helmholtz equation using Padé approximation", Journal of Computational Acoustics, 8(1):223-240, 2000. doi:10.1016/S0218-396X(00)00014-5

5

M.M. Wagner, P.M. Pinsky, M. Malhotra, "Application of Padé via Lanczos approximations for efficient multifrequency solution of Helmholtz problems", Journal of the Acoustical Society of America, 113(1):313-319, 2003. doi:10.1121/1.1514932

6

X. Li, "Power Flow Prediction in Vibrating Systems via Model Reduction", Ph.D. dissertation, Boston University, 2004.

7

W. Gropp, E. Lusk, A. Skjellum, "Using MPI: Protable Parallel Programming with the Message-Passing Interface, Edition", The MIT Press, Cambridge, 1999.

8

I.S. Duff, R.G. Grimes, J.G. Lewis. "Users' guide for the Harwell-Boeing sparse matrix collections (release I)", Technical Report TR/PA/92/86, Research and Technology Division, Boeing Computer Services, 1992.

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