In recent years, an alternative approach for huge linear eigenvalue problems known as Automated Multi-Level Substructuring (AMLS), has been developed by Bennighof and co-authors, and has been applied successfully to frequency response analysis of complex structures [1]. Recent studies in vibro-acoustic analysis of passenger car bodies, where very large FE models with more than six million degrees of freedom appear and several hundreds of eigenfrequencies and eigenmodes are needed, have shown that for this type of problems AMLS is considerably faster than Lanczos type approaches.

Free vibrations of fluid-solid structures are governed by nonsymmetric eigenproblems which have a particular structure: all eigenvalues are real, and right and left eigenvectors satisfy a special relation. Due to the missing symmetry the original AMLS method does not apply, but these types of problems are covered in the following way: one first solves the symmetric eigenproblems governing free vibrations of the fluid and of the structure independently by AMLS, thus determining approximations to eigenmodes of both media corresponding to the lower end of the spectra. Then the original nonsymmetric eigenvalue problem is projected to the space spanned by these eigenmodes. Hence, the coupling is not considered when constructing the search space, but only in the projected problem.

In a recent paper [2] we proposed an AMLS variant which incorporates the coupling already into the reduction process. Numerical experiments indicate that this improves the accuracy of AMLS. However, more research is necessary to make the method successful for realistic models.

In this paper we eliminate the degrees of freedom of one of the media thus getting a rational eigenvalue problem which is symmetric and the eigenvalues of which can be characterised by a minmax principle. Reducing its dimension by applying AMLS to its linear part and performing all transformations and projections for the rational part as well one arrives at a much smaller projected rational eigenvalue problem which preserves the structure of the original one and which can be solved easily. In this way one obtains approximations to eigenvalues at the lower end of the spectrum. The accuracy can be improved considerably if all degrees of freedom on the interface of the media are included into the substructuring on the coarsest level. Since this strategy inflates the dimension of the final projected eigenproblem significantly we discuss in this paper whether a small part of the interface degrees of freedom already suffices.

1

J.K. Bennighof, R.B. Lehoucq, "An automated multilevel substructuring method for the eigenspace computation in linear elastodynamics", SIAM J. Sci. Comput., 25, 2084-2106, 2004. doi:10.1137/S1064827502400650

2

M. Stammberger, H. Voss, "Automated multi-level substructuring for a fluid-solid vibration problem", Technical Report 117, Institute of Numerical Simulation, Hamburg University of Technology, 2007.

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