Keywords: symmetric structure, structural analysis, symmetry recognition, symmetry search, symmetry operation, symmetry group, group theory.

The systematic study of symmetry has been made possible through group theory and associated representation theory. Applications of group theory to problems involving symmetry in physics and chemistry are well-known [1]. Within engineering mechanics, group theory has been applied to simplify problems of the vibration [2] and bifurcation [3,4] of structures, among others. The identification of the symmetry group of a configuration is the first step in any group-theoretic computational procedure, so it is important to be able to recognise all the symmetry properties of a system. Most group-theoretic formulations to date have relied on manual or human recognition of symmetry. However, for complex systems with a large number of structural members, nodes or joints, or for complex objects often encountered in manufacture, the symmetry properties may not at all be obvious to the human eye. It is therefore desirable to incorporate automatic recognition of symmetry within the general framework of any group-theoretic computational procedure.

From the literature, it appears that many of the methods that have been proposed for searching for symmetry in the technological areas of electronic and manufacturing engineering require the user to specify the type of symmetry to be searched for [5]. Also, the algorithms developed for those purposes seem to be suited for the detection of symmetry based on the shape of objects, or the positions of boundary points, and therefore cannot detect the symmetries of structural configurations with interior points (e.g. a solid finite element with not only face, edge and corner nodes, but also interior nodes).

In this paper, a new procedure is described for the systematic search and identification of the symmetries of a group of points in two and three-dimensional space, without any assumptions as to the type of symmetry the configuration might have, and taking into account all points of a system without the need for first defining the boundaries of the system. Such a group of points may represent the nodes of a complex finite element (in finite element analysis), the connection points of a space truss with a large number of members, the point masses of a multi degree-of-freedom vibrating system, etc. The points may occur as mere spatial positions of the components of a system, or they may be associated with force and/or displacement vectors. Algorithms for the search and identification of symmetry in both two-dimensional and three-dimensional spaces are presented. The procedure is intended for incorporation into a standard group-theoretic computational framework, where the first step is always the identification of the symmetry properties (and hence the symmetry group) of a physical system.

1

M. Hamermesh, "Group Theory and its Application to Physical Problems", Pergamon Press, Oxford, 1962.

2

A. Zingoni, "An efficient computational scheme for the vibration analysis of high-tension cable nets", Journal of Sound and Vibration, 189(1), 55-79, 1996. doi:10.1006/jsvi.1996.0005

3

T.J. Healey, "A group-theoretic approach to computational bifurcation problems with symmetry", Computer Methods in Applied Mechanics and Engineering, 67, 257-295, 1988. doi:10.1016/0045-7825(88)90049-7

4

K. Ikeda, K. Murota, "Bifurcation analysis of symmetric structures using block-diagonalisation", Computer Methods in Applied Mechanics and Engineering, 86, 215-243, 1991. doi:10.1016/0045-7825(91)90128-S

5

S.J. Tate, G.E.M. Jared, "Recognising symmetry in solid models", Computer-Aided Design, 35, 673-692, 2003. doi:10.1016/S0010-4485(02)00093-3

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