Abstract. The Automated Multi-Level Substructuring (AMLS) method has been developed to reduce the computational demands of frequency response analysis and has recently been proposed as an alternative to iterative projection methods like Lanczos or Jacobi-Davidson for computing a large number of eigenvalues for matrices of very large dimension. Based on Schur complements and modal approximations of submatrices on several levels AMLS constructs a projected eigenproblem which yields good approximations of eigenvalues at the lower end of the spectrum. Rewriting the original problem as a rational eigenproblem of the same dimension as the projected problem, and taking advantage of a minmax characterization for the rational eigenproblem we derive an a priori bound for the AMLS approximation of eigenvalues. 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 modeling 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 ([16], [14], e.g.) in vibroacoustic analysis of passenger car bodies where very large FE models with more than one 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.We stress the fact that substructuring does not mean that it is obtained by a domain decomposition of a real structure, but it is understood in a purely algebraic sense, i.e. the dissection of the matrices can be derived by applying a graph partitioner like CHACO [11] or METIS [15] to the matrix under consideration. However, because of its pictographic nomenclature we will use terms like substructure or eigenmode from frequency response problems when introducing the AMLS method.From a mathematical point of view AMLS is a projection method where the ansatz space is constructed exploiting Schur complements of submatrices and truncation of spectral representations of subproblems. In this paper we will take advantage of the facts that the original eigenproblem is equivalent to a rational eigenvalue problem of the same dimension as the projected problem in AMLS, which can be interpreted as exact condensation of the original eigenproblem with respect to an appropriate basis. Its eigenvalues at the lower end of the spectrum can be characterized as minmax values of a Rayleigh functional of this rational eigenproblem. Hence, comparing the Rayleigh quotient of the projected problem and the Rayleigh functional of the rational

Simulating numerically the sound radiation of a rolling tire requires the solution of a very large and sparse gyroscopic eigenvalue problem. Taking advantage of the automated multi-level substructuring (AMLS) method it can be projected to a much smaller gyroscopic problem, the solution of which however is still quite costly since the eigenmodes are non-real and complex arithmetic is necessary. This paper discusses the application of AMLS to huge gyroscopic problems and the numerical solution of the AMLS reduction. A numerical example demonstrates the efficiency of AMLS.

A parallelization concept for the adaptive multi-level substructuring (AMLS) method is presented, the idea of which is to hierarchically substructure the system under consideration, and at the same time to use a truncated eigenvalue decomposition on the interfaces to reduce the excessive number of interface degrees of freedom in component mode methods.

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