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Computational Science, Engineering & Technology Series
ISSN 1759-3158
CSETS: 28
CIVIL AND STRUCTURAL ENGINEERING COMPUTATIONAL TECHNOLOGY
Edited by: B.H.V. Topping and Y. Tsompanakis
Chapter 6

Applications of Graph Products and Canonical Forms in Structural Mechanics: A Review

A. Kaveh<sup>1</sup> and H. Rahami<sup>2</sup>

1School of Civil Engineering, Iran University of Science and Technology, Tehran, Iran
2School of Engineering Science, College of Engineering, University of Tehran, Iran

Full Bibliographic Reference for this chapter
A. Kaveh, H. Rahami, "Applications of Graph Products and Canonical Forms in Structural Mechanics: A Review", in B.H.V. Topping and Y. Tsompanakis, (Editor), "Civil and Structural Engineering Computational Technology", Saxe-Coburg Publications, Stirlingshire, UK, Chapter 6, pp 157-186, 2011. doi:10.4203/csets.28.6
Keywords: canonical forms, graph products, eigensolution, diagonalization, decomposition, circulant.

Summary
In spite of considerable advances in the computational capability of computers in recent years, efficient methods for more time saving solutions of structures are of great interest. Large problems arise in many scientific and engineering problems. While the basic mathematical ideas are independent of the size of the matrices, the numerical determination of the displacement and internal forces become more complicated as the dimensions of matrices increase and their sparsity decreases. The use of prefabrication in industrialized building construction, often results in structures with regular patterns of elements exhibiting symmetry of various types, and special methods are beneficial for efficient solution of such problems.

In this chapter different canonical forms which have applications in engineering problems are studied. These matrices are in block forms and the arrangement of blocks lead to different canonical forms changing the solution of the problem.

Here first we use Schur's method and Form I and Form II are introduced as special cases. Then Form II is generalized to Forms F and G. In all cases the aim is to calculate the eigenvalues and eigenvectors followed by finding their inverse. Then the graph products are introduced. Then Form III is introduced and it is transformed to Form II. Parallel to this the transformation of a special case of F to Gis performed. Finally the circulant matrices as a special class of block matrices are discussed.

In this chapter different canonical forms are studied. Depending on the layout of the constituting sub-matrices of the matrix different methods should be used for calculating the eigenvalues and eigenvectors, and finding their inverse. Such matrices arise in the adjacency and Laplacian matrices of the product graphs. The stiffness matrices of repetitive structures in block form may also be formed in canonical forms using appropriate coordinate systems. Here conditions required for the formation of such matrices are studied and methods are presented for their decomposition into smaller sub-matrices. In certain cases some row and column operations alter the layout of the block matrices and make them suitable for decomposition and block diagonalization. In certain matrices the circulant nature helps to achieve the decomposition by introducing a matrix function to calculate the eigenvalues and eigenvectors. For each case examples are presented for clarification.

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