Keywords: unsymmetric eigenvalue problem, rational eigenproblem, fluid-solid interaction, variational characterisation of eigenvalues, iterative projection method, nonlinear Arnoldi method, Jacobi-Davidson method.

The free vibrations of a plate with elastically attached loads are governed by an unsymmetric eigenvalue problem which is a coupled system of a plate problem and a finite number of one dimensional oscillators [1,2,3]. It has similar properties to eigenproblems governing free vibrations of fluid solid structures [4]. Although symmetry is missing its eigenvalues are real and they are shown to satisfy a minmax characterization with respect to the Rayleigh functional. Discretising the plate problem using a finite element method one gets an unsymmetric, sparse matrix eigenvalue problem which preserves the structure of the continuous problem, i.e. its eigenvalues are also real and they allow for a variational characterisation.

Solving the discretised problem by an iterative projection method like shift-and-invert Arnoldi or a rational Krylov method the special structure that guarantees the realness of the spectrum and its variational characterisation is destroyed and even non-real approximations to eigenvalue may appear. In this paper we suggest a structure preserving projection approach based on the nonlinear Arnoldi method which was introduced in [5] and on the Cayley transformation. Taking advantage of the minmax characterization we are able to compute the eigenvalues one after the other in a safe way.

The efficiency of the method is evaluated by a finite element discretisation of a plate problem of dimension 18650. While our method required 19.8 seconds to determine the 56 smallest eigenvalues on an Intel Pentium D CPU at 3.2 GHZ with 3.5 GB RAM using MATLAB 7.7.0 (R2008b), the nonlinear Arnoldi method for an equivalent rational eigenvalue problem [3] needed 79.0 seconds, and the shift-and-invert Arnoldi method implemented in the MATLAB function sptarn required 106.1 seconds.

1

S.I. Solov'ëv, "Eigenvibrations of a plate with elastically attached loads", Preprint SFB393/03-06, Sonderforschungsbereich 393 an der Technischen Universitt Chemnitz, Germany, 2003.

2

L. Mazurenko, H. Voss, "Low rank rational perturbations of linear symmetric eigenproblems", Z. Angew. Math. Mech., 86, 606-616, 2006. doi:10.1002/zamm.200510267

3

H. Voss, "Eigenvibrations of a plate with elastically attached loads", in "Proceedings of the European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS 2004", P. Neittaanmäki, T. Rossi, S. Korotov, E. Onate, J. Periaux, D. Knörzer, (Editors), Jyväskylä, Finland, 2004.

4

M. Stammberger, H. Voss, "On an unsymmetric eigenvalue problem governing free vibrations of fluid-solid structures", Technical Report 128, Institute of Numerical Simulation, Hamburg University of Technology, 2009.

5

H. Voss, "An Arnoldi method for nonlinear eigenvalue problems", BIT Numerical Mathematics, 44, 387-401, 2004. doi:10.1023/B:BITN.0000039424.56697.8b

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