Computational & Technology Resources
an online resource for computational,
engineering & technology publications 

CivilComp Proceedings
ISSN 17593433 CCP: 83
PROCEEDINGS OF THE EIGHTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY Edited by: B.H.V. Topping, G. Montero and R. Montenegro
Paper 288
Improved GroupTheoretical Method for Eigenvalue Problems of Special Symmetric Structures Using Graph Theory A. Kaveh and M. Nikbakht
Department of Civil Engineering, Iran University of Science and Technology, Tehran, Iran A. Kaveh, M. Nikbakht, "Improved GroupTheoretical Method for Eigenvalue Problems of Special Symmetric Structures Using Graph Theory", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Proceedings of the Eighth International Conference on Computational Structures Technology", CivilComp Press, Stirlingshire, UK, Paper 288, 2006. doi:10.4203/ccp.83.288
Keywords: decomposition, symmetry, eigenvalues, group theory, representation theory, stiffness graph, mass graph, Laplacian matrix, natural frequency.
Summary
Grouptheoretical methods for decomposition of eigenvalue problems of skeletal
structures with symmetry employ the symmetry group of the structures and
blockdiagonalize their matrices. Such techniques have been well established by Zingoni [1,2,3]
and Healy and Treacy [4] among some others. In these methods,
n x n matrices
associated with a system having n degrees of freedom are changed into similar
block factored matrices under similarity transformation which is found using the
representation theory of symmetry groups. Therefore, at the end of the procedure,
the output is a number of matrices of smaller dimensions. However, it should be
noted that such submatrices are associated with physical subsystems and knowing
them can help us in some aspects. In some special cases, such decompositions can
further be continued. This particularly happens when submatrices resulted from
the decomposition process, correspond to substructures with new symmetrical
properties which are not among the properties of the original structure. Thus, a
grouptheoretical method is not able to recognize such additional symmetry from
the original problem. In this paper, a combinatorial approach based on grouptheoretical methods of decomposition of generalized eigenvalue problems for
symmetrical systems is presented in order to investigate the possibility of further
decomposition of the problems decoupled by group theory. The graph theory is
used in order to produce a physical interpretation for the results of block
diagonalization arising from the group theory. This algorithm identifies the cases
where the structure has the potential of being further decomposed, and also finds
the symmetry group, and subsequently the transformation which can further
decompose the system. It is also possible to find out when the blockdiagonalization is complete and no further decomposition is possible.
For this purpose, associated with an arbitrary structure, some graphs are introduced and these graphs are decomposed based on what has been presented before, for decomposition of a symmetric graph by the means of conventional grouptheoretical methods [5]. Then, the resulting subgraphs are explored in order to find new symmetrical properties. If such properties which can introduce a symmetry group are found in any of the subsystems, then the properties of that symmetry group and its representation can be implemented to form a transformation matrix by which further decomposition of the associated submatrices is possible. The focus of this paper is on dynamic analysis of rigid frames, in which the mass and stiffness matrices are involved. Therefore, corresponding to each frame, two graphs are introduced: the mass graph and the stiffness graph. Formation of such graphs has been introduced in detail, in reference [6], and here the grouptheoretical method is used to decompose them as mentioned above. Though the method is only applicable to special symmetric systems, such as problems in which the decomposed subproblems exhibit new symmetrical properties, the technique is very efficient and inexpensive using a computer, and if the method is found appropriate for a given problem, it results in considerable saving in computational time and effort. On the other hand, this method creates a physical sense for grouptheoretical methods of symmetry analysis and the resulted outputs, which can be very helpful in improving the conventional methods and analyzing the results. Although the focus of this paper is restricted to dynamic analysis of rigid frames, the method is general and is applicable to other types of structures. The application of the present method can easily be extended to stability analysis. References
purchase the fulltext of this paper (price £20)
go to the previous paper 
