Time response analysis of periodic structures via wave-based absorbing boundary conditions

A finite element procedure is proposed to compute the dynamic response of infinite periodic structures subject to localized time-dependent excitations. Straight periodic structures which are made up of cells/substructures of arbitrary shapes (e.g., 2D substructures) are analyzed. The proposed approach involves considering a periodic structure of finite length with excitation sources and absorbing boundary conditions which are expressed in the time domain. The absorbing boundary conditions are first described in the frequency domain by means of impedance matrices using a wave approach. Afterwards, they are switched to the time domain by decomposing the impedance matrices via rational functions, and expressing these rational functions in terms of polynomials of the frequency up to order 2. The related matrix system involves the usual vectors of displacements, velocities and accelerations, as well as vectors of supplementary variables. As such, it can be simply and quickly convert to the time domain yielding a classical second-order time differential equation which can be integrated with the Newmark algorithm. Numerical experiments are proposed which highlight the relevance of the approach.