Keywords: decomposition, symmetry, eigenvalues, group theory, representation theory, stiffness graph, mass graph, Laplacian matrix, natural frequency.

Group-theoretical methods for decomposition of eigenvalue problems of skeletal structures with symmetry employ the symmetry group of the structures and block-diagonalize 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 group-theoretical method is not able to recognize such additional symmetry from the original problem. In this paper, a combinatorial approach based on group-theoretical 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 block-diagonalization 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 group-theoretical 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 group-theoretical 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 group-theoretical 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.

1

A. Zingoni, Group-theoretical applications in solid and structural mechanics: a review, Chapter 12 in Computational Structures Technology, Edited by BHV Topping and Z. Bittnar, Saxe-Coburg Publications, UK, 2002. doi:10.4203/csets.7.12

2

A. Zingoni, "Truss and beam finite elements revisited: A derivation based on displacement-field decomposition", International Journal of Space Structures, 1996; 11(4): 371-380.

3

A. Zingoni, M.N. Pavlovic, and G.M. Zlokovic, A symmetry-adapted flexibility approach for multi-storey space frames: General outline and symmetry-adapted redundants, Structural Engineering Review, 1995; 7:107-119.

4

T.J. Healy and J.A. Treacy, "Exact block diagonalisation of large eigenvalue problems for structures with symmetry", International Journal for Numerical Methods in Engineering, 1991; 31:265-285. doi:10.1002/nme.1620310205

5

A. Kaveh and M. Nikbakht, Groups, Graphs and Linear Algebra for Symmetry; Block Diagonalization of Laplacian matrices, submitted for publication, 2006.

6

A. Kaveh and B. Dadfar, Eigensolution for free vibration of structures by graph symmetry, submitted for publication, 2006.

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