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

Over the last few years, a new method for large-scale linear eigenvalue problems 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 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.

On each level of the hierarchical substructuring AMLS consists of two steps. First for every substructure of the current level a congruence transformation based on block-Gaussian elimination is applied to the matrix pencil to decouple in the stiffness matrix the substructure from the degrees of freedom of higher levels. Secondly, the dimension of the problem is reduced by modal truncation of the corresponding diagonal blocks discarding eigenmodes according to eigenfrequencies which exceed a predetermined cut-off frequency.

Hence, AMLS is nothing else but a projection method where the large problem under consideration is projected to a subspace spanned by a smaller number of eigenmodes of undamped clamped substructures on several levels.

Dynamic analysis of structures frequently involves finite element discretisations with millions of unknowns, and the frequency range needed for the analysis often is so large that the number of required eigenpairs can easily reach into the thousands. In this situation it may happen that the dimension of the reduced eigenproblem is still so high that the CPU time required for solving it is much larger than the CPU time for reducing the original problem to its condensed form.

We discuss a way how to further reduce the dimension of the projected problem on-the-fly by intermediate reduction steps applying modal condensation to aggregated substructures which consist of the union of already reduced substructures corresponding to a subtree of the partitioning graph. This yields smaller although more populated condensed eigenproblems. We thus ballance the cost of the two subtasks, reducing the dimension of the original problem and solving the reduced problem, and thereby we decrease the overall time and effort. We demonstrate its efficiency solving a huge gyroscopic eigenvalue problem that models the dynamic behaviour of a rotating tyre [2] where the total CPU time is reduced by approximately 25%.

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

K. Elssel, H. Voss, "Reducing huge gyroscopic eigenproblem by Automated Multi-Level Substructuring", Arch. Appl. Mech., 76, 171 - 179, 2006. doi:10.1007/s00419-006-0013-0

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