Keywords: group theory, symmetry groups, representation theory, structural mechanics, solid mechanics, eigenvalue problems, bifurcation analysis, vibration analysis, finite-element formulation.

This paper is a review of applications of symmetry groups and associated representation theory in the analysis and study of problems involving symmetry within the fields of solid and structural mechanics. Such techniques are very well-established in various branches of physics and chemistry [1], but the need for a more systematic and thorough exploitation of symmetry in tackling problems within solid and structural mechanics has provided the impetus over the past 30 years for the development of group-theoretic methods. This paper traces the advances made in the exploitation of group theory in areas such as bifurcation analysis, vibration analysis and finite-element analysis, and outlines the various schemes of implementation currently available. In all cases, it is shown that through the characteristic vector-space decomposition, group-theoretic methods afford considerable simplifications and reductions in computational effort in comparison with conventional methods, and render the computations amenable to the use of parallel processors.

Of course, and as is well known, it is possible to exploit symmetry in structural analysis without resorting to the mathematical tools of group theory and associated representation theory. However, conventional methods cannot take into account the types of symmetry for which they are not suited, and in the case of structural configurations possessing complex symmetry properties, only a portion (or even none) of the total symmetry can be exploited. On the other hand, the group-theoretic approach, based on the theory of symmetry groups and associated representation theory, has the capability of taking into account all the symmetry properties of a configuration in a systematic manner, leading to often dramatic reductions in computational effort, regardless of the complexity of the symmetry.

In 1979, Sattinger [2] demonstrated the applicability of group-theoretic methods to local bifurcation problems, and since then a number of investigators have shown that similar techniques can be used to exploit symmetry in global bifurcation analysis [3,4]. It is recognised that symmetric structures have a much more complex bifurcation behaviour than non-symmetric structures. Multiple critical points, where more than one eigenvalue simultaneously vanishes, are inherent in such structures owing to symmetry [4]. Fortunately for structural configurations of this kind, the group-theoretic approach may be used to simplify the analysis. Through the process, the tangent-stiffness matrix may be brought into block-diagonal form, eliminating problems of numerical ill-conditioning usually associated with the bifurcation analysis of thin shells, or facilitating tests for singularity of the tangent-stiffness matrix.

In vibration analysis, the group-theoretic approach consists in decomposing the -dimensional vector space of a problem with degrees of freedom into a number of independent subspaces each of dimension (where ), and then solving for the eigenvalues associated with a given subspace independently of those of the other subspaces [5]. A feature of the group-theoretic approach is its prediction of the number of normal modes of vibration of a given symmetry type, and of the occurrence of degenerate normal modes (i.e. normal modes of the same frequency of vibration). The latter are always associated with irreducible representations of dimension greater than . The number of linearly independent normal modes of a particular frequency is simply given by the dimension of the corresponding irreducible representation.

In finite-element problems, the variables of interest are the degrees of freedom associated with a given finite element, but what requires to be determined at the elemental level are usually the element matrices such as stiffness and mass matrices. The numerical integration associated with the evaluation of these matrices can involve the expenditure of huge computational effort, particularly in the case of new finite-element proposals with large numbers of nodes and degrees of freedom. This is where a group-theoretic formulation can be most advantageous in reducing computational effort [6], when deliberate choice is made of symmetric finite elements.

1

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

2

Sattinger D.H., "Group Theoretic Methods in Bifurcation Theory (Lecture Notes in Mathematics No. 762)", Springer-Verlag, Berlin, 1979.

3

Healey T.J., "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

Ikeda K., Murota K., "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

Zingoni A., "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

6

Zlokovic G.M., "Group Supermatrices in Finite Element Analysis", Ellis Horwood, Chichester (UK), 1992.

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