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:

1. Definitions from graph theory
2. Graph symmetry for matrices of special patterns
3. Graph representation of the structural members of frames for dynamic analysis
4. Linear stability analysis
5. Concluding remarks

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.

1

Biggs, N.L. "Algebraic Graph Theory", Cambridge University Press, 2nd edition, Cambridge, 1993.

2

Cvetkovic, D.M., Dobb, M. and Sachs, H., "Spectra of Graphs", Academic Press, New York 1980.

3

Kaveh, A. "Optimal Structural Analysis", John Wiley, 2nd edition, UK, 2006.

4

Kaveh, A. and Rahimi Bondarabady, H.A., "A multi-level finite element nodal ordering using algebraic graph theory", Finite Elements in Analysis and Design, 2002; 38, pp. 245-261. doi:10.1016/S0168-874X(01)00062-2

5

Kaveh, A., and Rahimi Bandarabady, H.A., "Finite element mesh decompositions using complementary Laplacian matrix", Communications in Numerical Methods in Engineering, 2000; 16, pp. 379-389. doi:10.1002/1099-0887(200006)16:6<379::AID-CNM332>3.0.CO;2-G

6

Kaveh, A. and Rahami, H., "An efficient method for decomposition of regular structures using graph products", International Journal Numerical Methods in Engineering, 2004; 61, pp. 1797-1808. doi:10.1002/nme.1126

7

Kaveh, A and Sayarinejad, MA, Graph symmetry in dynamic systems, Computers and Structures, 2004; 82, pp. 2229-2240. doi:10.1016/j.compstruc.2004.03.066

8

Kaveh, A and Slimbahrami, B., "Eigensolution of symmetric frames using graph factorization", Communications in Numerical Methods in Engineering, 2004; 20, pp. 889-910. doi:10.1002/cnm.711

9

Kaveh, A. and Sayarinejad, MA, "Eigensolutions for matrices of special patterns, Communications in Numerical Methods in Engineering", 2003; 19, pp. 125-136. doi:10.1002/cnm.576

10

Kaveh, A and Sayarinejad, MA, "Eigensolutions for factorable matrices of special patterns", Communications in Numerical Methods in Engineering, 2004; 20, pp. 133-146. doi:10.1002/cnm.656

11

Kaveh, A. "Structural Mechanics: Graph and Matrix Methods", Research Studies Press, Somerset, UK, 3rd edition, 2004.

12

Kaveh, A and Syarinejad, MA. "Eigenvalues of factorable matrices with form IV symmetry", Communications in Numerical Methods in Engineering, No. 6, 2005; 21, pp. 269-278. doi:10.1002/cnm.744

13

Kaveh, A and Sayarinejad, MA. "Augmented canonical forms and factorization of graphs", Asian Journal of Civil Engineering, No. 6, 2005;6, pp. 495-509.

14

Kaveh, A and Sayarinejad, MA, "Eigenvalues of factorable matrices with form IV symmetry". Communications in Numerical Methods in Engineering, No. 6, 2005; 21, pp. 269-287. doi:10.1002/cnm.744

15

Kaveh, A and Slimbahrami, B., "Buckling Load of Frames Using Graph Symmetry", Proc. of the Fourth Intl Conference on Engineering Computational Technology, Civil-Comp Press, UK, September 2004.

purchase the full-text of this chapter (price £20)

go to the previous chapter
go to the next chapter
return to the table of contents
return to the book description
purchase this book (price £90 +P&P)