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

• Published in: European journal of mechanics, A, Solids Vol. 91; p. 104418
• Main Authors: Duhamel, D., Mencik, J.-M.
• Format: Journal Article
• Language: English
• Published: Berlin Elsevier Masson SAS 01.01.2022; Elsevier BV; Elsevier
• Subjects: Absorbing boundary conditions; Algorithms; Boundary conditions; Differential equations; Dynamic response; Engineering Sciences; Excitation; Impedance; Periodic structures; Polynomials; Rational functions; Time dependence; Time domain analysis; Time response; Two dimensional analysis; Wave finite element method; Wave finite element method; Time response; Absorbing boundary conditions; Periodic structures
• ISSN: 0997-7538; 1873-7285
• DOI: 10.1016/j.euromechsol.2021.104418

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 iω 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. •The wave finite element method is extended to time analysis of periodic structures.•Absorbing boundary conditions are formulated in the time domain.•Rational approximation of boundary impedances leads to classical temporal formulation.•Numerical simulations via Newmark algorithm yield accurate results.