A general superelement generation strategy for piecewise periodic media

Structures composed of repetitions of multiple identical cells are common and have been the object of a large body of literature on waveguides, periodic media, and cyclic symmetry. Starting from a cell model, possibly with a large number of Degrees of Freedom both inside the cell and on its edges, the objective of this paper is to propose a Ritz-Galerkin reduction procedure retaining the standard second order model form and periodicity properties of the original model, while controlling accuracy in terms of model bandwidth and reproduction of the forced response to applied loads. The procedure is classically decomposed in two phases: subspace learning and basis generation. For the learning phase, Wave Finite Element (WFE) and periodic computations are presented as alternatives. The latter are then preferred for their easier control on the reduced model accuracy using classical modal synthesis and the simple choice of few target wavelengths. For the basis generation phase, constraints needed to generate a periodic superelement are defined and numerical procedures to generate a basis verifying those constraints are proposed. The validity of the reduction is demonstrated for the case of a cell with random elastic properties presenting bandgaps, local modes and wavemode crossing. Accurate predictions of modes and damped forced response are given for both the infinite and finite cases, using frequency and time simulations. The proposed analysis illustrates tracking of waveshapes, evaluation of significant waves by computation of the forced response in the frequency/wavenumber domain and interpretation of the relation between the infinite and finite forced responses. The case of a railway track with edges and transitions between multiple periodic zones is finally used to illustrate scalability issues