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Civil-Comp Proceedings
ISSN 1759-3433 CCP: 81
PROCEEDINGS OF THE TENTH INTERNATIONAL CONFERENCE ON CIVIL, STRUCTURAL AND ENVIRONMENTAL ENGINEERING COMPUTING Edited by: B.H.V. Topping
Paper 231
Automated Multi-Level Substructuring for Nonlinear Eigenproblems K. Elssel and H. Voss
Section of Mathematics, Hamburg University of Technology, Germany Full Bibliographic Reference for this paper
K. Elssel, H. Voss, "Automated Multi-Level Substructuring for Nonlinear Eigenproblems", in B.H.V. Topping, (Editor), "Proceedings of the Tenth International Conference on Civil, Structural and Environmental Engineering Computing", Civil-Comp Press, Stirlingshire, UK, Paper 231, 2005. doi:10.4203/ccp.81.231
Keywords: automated multi-level substructuring, AMLS, nonlinear eigenproblem, eigenvalue, eigenvector, sparse matrix, iterative projection method, Arnoldi method.
Summary
In this paper we consider the nonlinear eigenvalue problem
where ![]() ![]() ![]() ![]() ![]() ![]() 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 ![]() ![]()
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 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 ![]() ![]() ![]() ![]() ![]() ![]() 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. References
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