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INNOVATION IN ENGINEERING COMPUTATIONAL TECHNOLOGY
Edited by: B.H.V. Topping, G. Montero, R. Montenegro
Symmetry and Structures
Department of Civil Engineering, Iran University of Science and Technology, Narmak, Tehran, Iran
A. Kaveh, "Symmetry and Structures", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Innovation in Engineering Computational Technology", Saxe-Coburg Publications, Stirlingshire, UK, Chapter 19, pp 403-424, 2006. doi:10.4203/csets.15.19
Keywords: symmetry, graph theory, decomposition, Laplacian matrix, eigenvalues, canonical forms, free vibration, stability.
Many engineering problems require the calculation of eigenvalues and eigenvectors of matrices. As an example, eigenvalues correspond to natural frequencies in vibrating systems and buckling loads in the stability analysis of structures. Eigenvalues and eigenvectors of matrices associated with adjacency and Laplacian of graphs form the basis of the algebraic graph theory [1,2]. These eigensolutions have found many applications in sparse matrix technology, and are particularly employed in the ordering [3,4] and partitioning of graphs, and decomposition of large-scale finite element meshes for parallel computing [5,6]. Applications in structural mechanics can also be found in [7,8]. General methods are available in literature for such calculations, however, for matrices with special structures, it is beneficial to make use of their additional properties [9,10,11,12,13,14,15].
The mathematical models of many practical structures have various kinds of symmetry, which can be used in order to reduce the computational time for their analysis. In this paper, efficient methods are presented for eigenproblems involved in structural mechanics. Special canonical forms are presented which employ a decomposition process followed by special healings of the corresponding graph models. Four such forms are introduced in this paper and applied to eigensolutions that occur in the free vibration and stability analysis of frames. The proposed methods are illustrated by means of simple examples.
The present paper consists of the following sections:
A collection of applications of graph theory for the optimal analysis of structures is presented in this article. Such applications not only simplify the problems related to structural mechanics but also produces a power bridge between the development of graph theory on the one hand and structural mechanics on the other. Many structures and in particular, space structure, have different types of symmetry and using this property simplifies the calculation to a great extent.
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