Keywords: substructure, model reduction, modal reduction, Dirichlet-to-Neumann map, vibration, linear dynamic systems, finite elements.

In structural dynamics, one often wishes to approximate a given discrete model of a structure by another model which involves a much smaller number of degrees of freedom. Such a reduction is very important in cases where the computational effort associated with the direct analysis of the given system is prohibitive. There is a very large volume of literature on the subject, often called "model order reduction;" . For example the review papers [1,2,3,4]. Some of this work is related to robust control, where there is a need for the repetitive real-time solution of large structural dynamic problems [].

Modal reduction is a special case of model reduction. The original linear system is first decomposed into its eigenmodes. Then a small number of these eigenmodes is retained to represent the system, whereas all the other modes are discarded. The following question then arises: "Which of the modes should be retained?" In structural dynamics it is traditional to retain those modes associated with the lowest frequencies [6]. In control theory, a common procedure is the balanced realisation method proposed by Moore [7], where a special mode truncation is used to obtain a reduced system with equal amounts of controllability and observability. Both these approaches are simple and easy to code, but they are not based on any optimality criterion. Hence, although in many cases they produce very good approximations, they are not guaranteed to do so.

In this paper, we consider a linear substructure "attached" to a main structure. Such a substructure may represent a piece of equipment such as an antenna, connected to the main structure. On the other hand, if an engineer is interested in the dynamics of different antenna designs, the "substructure" may be most of the structure, while the antenna may be treated as the "main" structure. We wish to reduce the substructure alone, without modifying the main structure. When doing this, we are not interested in the accurate representation of the dynamics of the substructure itself, but in accurately representing the effect this substructure has on the dynamic behaviour of the main structure. Here we ask a question similar to the one mentioned above: In the reduction process, which of the substructure's modes should be retained? In contrast to the low-frequency rule dominating structural dynamics, we shall obtain a new criterion for "modal importance." We shall show that the most important modes of the substructure are those whose coupling matrices, to be defined in a particular way, have the highest norm. This will lead to a simple and effective algorithm for optimal modal reduction.

Recently [8,9] we have developed such a modal reduction scheme for substructures with linear damping-free time-dependent behaviour. In the present paper we first review the general subject of structure reduction. Then we extend the scheme mentioned above to substructures which experience Rayleigh damping; this is an important extension which turns out to be quite involved technically. The approach we take here involves the Dirichlet-to-Neumann map as the main analysis tool. We show the resulting scheme to be approximately optimal in a certain sense. We compare the scheme via a numerical experiment to standard modal reduction.

1

T. Mukherjee, G.K. Fedder, D. Ramaswamy and J. White, "Emerging Simulation Approaches for Micromachined Devices," IEEE Trans. Computer-Aided Design of Integrated Circuits and Systems, 19, 1572-1589, 2000. doi:10.1109/43.898833

2

A.C. Cangellaris and L. Zhao, "Model Order Reduction Techniques for Electromagnetic Macromodelling Based on Finite Methods," Int. J. Numer. Modell. Electron. Networks Devices Fields, 13, 181-197, 2000. doi:10.1002/(SICI)1099-1204(200003/06)13:2/3<181::AID-JNM355>3.0.CO;2-8

3

E. Livne and G.D. Blando, "Reduced-Order Design-Oriented Stress Analysis Using Combined Direct and Adjoint Solutions," AIAA J., 38, 898-909, 2000. doi:10.2514/2.1045

4

C.B. Allen, N.V. Taylor, C.L. Fenwick, A.L. Gaitonde and D.P. Jones, "A Comparison of Full Non-linear and Reduced Order Aerodynamic Models in Control Law Design Using a Two-dimensional Aerofoil Model," Int. J. Numerical Methods in Engineering, 64, 1628-1648, 2005. doi:10.1002/nme.1421

5

M. Morari and E. Zafiriou, "Robust Process Control", Prentice Hall, Englewood Cliffs, NJ, 1989.

6

L. Meirovitch, "Computational Methods in Structural Dynamics", Sijthoff & Noordhoff, Alphen aan den Rijn, The Netherlands, 1980.

7

B.C. Moore, "Principal Component Analysis in Linear Systems: Controllability, Observability and Model Reduction," IEEE Trans. Automatic Control, 26, 17-32, 1981. doi:10.1109/TAC.1981.1102568

8

P. Barbone, D. Givoli and I. Patlashenko, "Optimal Modal Reduction of Vibrating Substructures," Int. J. Numerical Methods in Engineering, 57, 341-369, 2003. doi:10.1002/nme.680

9

D. Givoli, P. Barbone and I. Patlashenko, "Which are the Important Modes of a Subsystem?" Int. J. Numerical Methods in Engineering, 59, 1657-1678, 2004. doi:10.1002/nme.935

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