Innovation in Engineering Computational Technology
Edited by: B.H.V. Topping, G. Montero and R. Montenegro
A. Kaveh
Department of Civil Engineering, Iran University of Science and Technology, Narmak, Tehran, Iran
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:
- Definitions from graph theory
- Graph symmetry for matrices of special patterns
- Graph representation of the structural members of frames for dynamic analysis
- Linear stability analysis
- 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.
- 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.
- 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.
- 7
- Kaveh, A and Sayarinejad, MA, Graph symmetry in dynamic systems,
Computers and Structures, 2004; 82, pp. 2229-2240.
- 8
- Kaveh, A and Slimbahrami, B., "Eigensolution of symmetric frames using
graph factorization", Communications in Numerical Methods in Engineering,
2004; 20, pp. 889-910.
- 9
- Kaveh, A. and Sayarinejad, MA, "Eigensolutions for matrices of special
patterns, Communications in Numerical Methods in Engineering", 2003; 19,
pp. 125-136.
- 10
- Kaveh, A and Sayarinejad, MA, "Eigensolutions for factorable matrices of
special patterns", Communications in Numerical Methods in Engineering,
2004; 20, pp. 133-146.
- 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.
- 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.
- 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.
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