SUMMARYThe use of group theory in simplifying the study of problems involving symmetry is a well-established approach in various branches of physics and chemistry, and major applications in these areas date back more than 70 years. Within the engineering disciplines, the search for more systematic and more efficient strategies for exploiting symmetry in the computational problems of solid and structural mechanics has led to the development of group-theoretic methods over the past 40 years. This paper reviews the advances made in the application of group theory in areas such as bifurcation analysis, vibration analysis and finite element analysis, and summarizes the various implementation procedures currently available. Illustrative examples of typical solution procedures are drawn from recent work of the author. It is shown how the group-theoretic approach, through the characteristic vector-space decomposition, enables considerable simplifications and reductions in computational effort to be achieved. In many cases, group-theoretic considerations also allow valuable insights on the behaviour or properties of a system to be gained, before any actual calculations are carried out.

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Where a finite element possesses symmetry properties, derivation of fundamental element matrices can be achieved more efficiently by decomposing the general displacement field into subspaces of the symmetry group describing the configuration of the element. In this paper, the procedure is illustrated by reference to the simple truss and beam elements, whose well-known consistent-mass matrices are obtained via the proposed method. However, the procedure is applicable to all one-, two- and three-dimensional finite elements, as long as the shape and node configuration of the element can be described by a specific symmetry group.