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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 93
PROCEEDINGS OF THE TENTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY
Edited by: B.H.V. Topping, J.M. Adam, F.J. Pallarés, R. Bru and M.L. Romero
Paper 261

Eigensolution of Locally Modified Regular Structures using the Shifted Inverse Iteration Method

A. Kaveh and H. Fazli

Centre of Excellence for Fundamental Studies in Structural Engineering, Iran University of Science and Technology, Tehran, Iran

Full Bibliographic Reference for this paper
A. Kaveh, H. Fazli, "Eigensolution of Locally Modified Regular Structures using the Shifted Inverse Iteration Method", in B.H.V. Topping, J.M. Adam, F.J. Pallarés, R. Bru, M.L. Romero, (Editors), "Proceedings of the Tenth International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 261, 2010. doi:10.4203/ccp.93.261
Keywords: regular structures, product graphs, modified structure, eigensolution, shifted inverse iteration.

Summary
There are well-established formulations for eigensolution of regular structures [1,2]. A structure is called regular if its model can be considered as a product graph. Matrices associated with a product graph possess canonical forms feasible for block-diagonalization. This property admits the decomposition of the original model into sub-models of smaller dimensions. The structural models commonly encountered in practice, however, do not exactly conform to product graphs, unless some local modifications are made. Using the information obtained from a decomposable model, also called the base model, it is the aim of this paper to propose a numerical method based on single vector iterations, for finding a few eigenvalues and eigenvectors of the modified system. The proposed method makes the best use of regularity properties of the modified structures to transform the eigenproblem associated with their free vibration into a simpler problem which can be handled more efficiently through vector iterations such as the shifted inverse iteration method.

The efficiency of the proposed method in terms of the number of iterations required, speed and storage requirements, is demonstrated through examples and a comparison is made with the standard vector iteration method applied to the entire structural model.

References
1
A. Kaveh, H. Rahami, "Block diagonalization of adjacency and Laplacian matrices for graph products; Applications in structural mechanics", International Journal for Numerical Methods in Engineering, 68(1), 33-63, 2006. doi:10.1002/nme.1696
2
A. Kaveh, H. Rahami, "Factorization for efficient solution of eigenproblems of adjacency and Laplacian matrices for graph products", International Journal for Numerical Methods in Engineering, 75(1), 58-82, 2008. doi:10.1002/nme.2245

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