Group Theory and Applications in Structural Mechanics

Abstract

Group theory is known as the mathematical language of the symmetry, and the representation theory is the powerful means of group theory in analysis of physical problems. Methods are available for studying symmetry in science and engineering using the theory of groups, Stiefel and Fässler [1] and Boardman et al. [2]. Methods for symmetry analysis of physical systems are developed in crystallogy and quantum mechanics for decomposing the problems of complicated symmetric systems. Although in structural mechanics, such techniques have not been introduced as much as in other fields of science, the method has been successfully utilised in different cases. Zingoni [3, 4] has studied the application of group theory in bifurcation problems, dynamic problems and in finite element method. Application of the group theory in the force method of structural analysis can be found in the joint work of Zingoni and Zlokovic [5]. Healy and Treacy [6] have developed innovative methods by combination of symmetry properties of structures with repeated substructures and the group theoretic method. Kaveh and Nikbakht have implemented methods based on group theory for decomposition of the problems of topological graphs [7], vibration analysis of simple dynamic systems [8] and the stability analysis of symmetric frames with simple forms of symmetry [9, 10].