An efficient method for the dynamic responses of periodic structures based on the physical features of the structure and group theory

Abstract

This paper proposes an efficient numerical method for solving the dynamic responses of one-dimensional and two-dimensional periodic structures. Efficiently solving a system of linear equations is a crucial issue for computation of the dynamic responses of large-scale periodic structures. By using condensation technology and based on the periodic properties of a structure, the scale of the system of linear equations that corresponds to the entire structure is reduced. Based on the properties of the periodic structure, it is strictly proved in mathematics that, within a time step, the external force applied on a local unit cell can cause only the dynamic responses of a limited number of adjacent unit cells. By using this conclusion, the dynamic responses of periodic structures can be converted into the response analysis of some small-scale structures. Then, the dynamic responses of the small-scale structures are converted into the response analysis of the cyclic periodic structures, which can be efficiently solved by using the group theory. Outstanding efficiency and low computational resources are achieved because the dynamic responses for the entire structure are converted into that of some small-scale structures. Numerical examples are presented to illustrate the high efficiency and low computational resources of the proposed method.