Abstract Summary
One-dimensional periodic media have been extensively studied in the past, on the one hand to determine the dispersion curves linking the frequencies to the wavenumbers and on the other hand for the calculation of bounded periodic structures. In the latter case, effective methods have been obtained, in particular based on methods such as the Wave Finite Element (WFE) and allow the rapid determination of the response of a structure to a given excitation from computations made mainly on one substructure. For two-dimensional periodic media, the situation is much less favourable. Dispersion curves can be obtained as in the one-dimensional case and can account for certain properties of these media. For the calculation of structures and obtaining the response of two-dimensional periodic structures to an excitation, the results mainly concern infinite media. For finite media, few results and effective methods are available. This communication aims to treat this last case of response of finite two-dimensional periodic structures to an excitation. One will be limited to structures having symmetries compared to two orthogonal planes parallel to the edges of a substructure. One bases oneself on the WFE by modeling a rectangular substructure by a finite element model. By imposing the wavenumber in one direction, we can numerically calculate the wavenumbers and mode shapes associated with propagation in the perpendicular direction. We can thus build a set of waves propagating respectively along the two directions parallel to the sides of the rectangle which will serve as the basis on which we will decompose the solution. By taking appropriately chosen solutions, we can decouple the waves in the two directions parallel to the sides of the rectangle. The solution of each of these two problems is obtained by a fast Fourier transformation which gives the amplitudes associated with the waves. We can then calculate the global solution at selected points by summing the contributions of all these waves. We thus obtain the global solution for a two-dimensional periodic medium with a low cost in computing time. Examples are given for the case of a two-dimensional membrane.
