Keywords: automated multi-level substructuring, AMLS, nonlinear eigenproblem, eigenvalue, eigenvector, sparse matrix, iterative projection method, Arnoldi method.

In this paper we consider the nonlinear eigenvalue problem

where is a family of large and sparse matrices depending on a parameter . As in the linear case a parameter is called an eigenvalue of if problem (28) has a nontrivial solution which is called a corresponding eigenvector.

Problems of this type arise in damped vibrations of structures, vibrations of rotating structures, stability of linear systems with retarded argument, lateral buckling problems, vibrations of fluid-solid structures, or numerical simulation of quantum dot heterostructures, to name just a few.

Over the last few years, a new method for huge linear eigenvalue problems

where and are Hermitian and positive definite, known as Automated Multi-Level Substructuring (AMLS), has been developed by Bennighof and co-authors, and has been applied to frequency response analysis of complex structures [1]. Here the large finite element model is recursively divided into very many substructures on several levels based on the sparsity structure of the system matrices. Assuming that the interior degrees of freedom of substructures depend quasistatically on the interface degrees of freedom, and modelling the deviation from quasistatic dependence in terms of a small number of selected substructure eigenmodes the size of the finite element model is reduced substantially yet yielding satisfactory accuracy over a wide frequency range of interest. Recent studies [2] 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 AMLS is considerably faster than Lanczos type approaches.

From a mathematical point of view, AMLS is nothing else but a projection method where the large problem under consideration is projected to a subspace spanned by a small number of eigenmodes of undamped clamped substructures on several levels. With respect to this basis the projection of the stiffness matrix becomes diagonal, and the mass matrix is projected to a generalized arrowhead form.

In this presentation we discuss the generalization of AMLS to the nonlinear eigenvalue problem (28). To this end we identify an essential linear part of the nonlinear eigenproblem, i.e. we rewrite problem (28) as

where and are Hermitean and positive definite matrices, and is small. We construct the ansatz space for the projection by the AMLS approach using the matrices and only, and apply all transformations and projections to the nonlinear part simultaneously. Thus, we obtain a nonlinear eigenproblem of the same type as (28) but of much smaller dimension which can be solved by any appropriate method, i.e. by a dense solver if the projected problem is small, by linearization if the underlying problem is a polynomial eigenproblem, or by an iterative projection method of Arnoldi [3] or Jacobi-Davidson type [4].

The efficiency of the method is demonstrated by three classes of problems, a gyroscopic eigenproblem modelling a rotating tire, a rational eigenvalue problem

modelling the damped vibrations of a structure with nonproportional damping where a viscoelastic constitutive relation is assumed to describe the behaviour of the material, and a rational eigenvalue problem governing free vibrations of fluid-solid structure.

1

J. Bennighof. Vibroacoustic frequency sweep analysis using automated multi-level substructuring. In Proceedings of the AIAA 40th SDM Conference, St. Louis, Missouri, 1999.

2

A. Kropp and D. Heiserer. Efficient broadband vibro-acoustic analysis of passenger car bodies using an FE-based component mode synthesis approach. J.Comput.Acoustics, 11:139 - 157, 2003. doi:10.1142/S0218396X03001870

3

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

4

H. Voss. A Jacobi-Davidson method for nonlinear and nonsymmetric eigenproblems. Technical report, Section of Mathematics, Hamburg University of Technology, 2004. Submitted to Computers & Structures.

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