Computation of the dynamic response of large two-dimensional elastic periodic structures: case of mixed boundary conditions

The analysis of two-dimensional periodic elastic media is addressed using an extension of the Wave Finite Element (WFE) method. This approach determines propagation constants from the finite element model of a substructure, based on its mass and stiffness matrices obtained with any standard FE code. For each substructure, two sets of waves are computed by determining propagation constants and wave modes in one direction for prescribed propagation constants in the transverse direction. The general solution is then expressed as a combination of these waves, leading to a linear system whose unknowns are the wave amplitudes. We consider here the special case of mixed boundary conditions where the displacement is imposed in one direction and the force in the complementary direction. This allows for a very efficient solution using FFT, as in the scalar case considered previously. The proposed formulation enables the study of large but finite two-dimensional periodic structures, for which classical finite element approaches, even with model reduction techniques such as Craig–Bampton, become computationally prohibitive. Several examples illustrate the methodology, involving substructures with various material compositions and an increasing number of unit cells. Special attention is given to configurations with inclusions embedded in a matrix, analyzed over a wide frequency range and for large periodic assemblies.