An efficient algorithm based on group theory and the Woodbury formula for the dynamic responses of periodic structures

An efficient numerical method for computing the dynamic responses of periodic structures is proposed. Efficiently solving a system of linear equations is a key issue for computing the dynamic response of a large-scale structure. Based on the periodic properties of a structure and using condensation technology, the system of linear equations that corresponds to the periodic structure is reduced to a small-scale system of linear equations. Based on the Woodbury formula, the solution for the small-scale system of linear equations is obtained by solving a new system of linear equations whose coefficient matrix corresponds to a cycle periodic structure. Using group theory, the new system of linear equations is efficiently solved. Superior efficiency is achieved because the scale of the system of linear equations for the entire structure is significantly reduced and the coefficient matrix for the new system of linear equations can be converted into a block-diagonal matrix using group theory. Numerical examples are presented to illustrate the high efficiency of the proposed method.