In the past, the study of periodic media mainly focused on one-dimensional periodic structures (meaning periodic along one direction), on the one hand to determine the dispersion curves linking the frequencies to the wavenumbers and on the other hand to obtain the response of a structure to an external excitation, both for bounded or unbounded structures. In the latter case, effective approaches have been obtained, based on methods such as the Wave Finite Element (WFE). Two-dimensional periodic media are more complex to analyse but dispersion curves can be obtained rather easily as in the one-dimensional case. Obtaining the steady state response of two-dimensional periodic structures to time-harmonic excitations is much more difficult than for one-dimensional media and the results mainly concern infinite media. This work is about this last case of the steady state response of finite two-dimensional periodic structures to time-harmonic excitations by limiting oneself to structures described by a scalar variable (acoustic, thermal, membrane behaviour) and having symmetries compared to two orthogonal planes parallel to the edges of a substructure. Using the WFE for a rectangular substructure and imposing the wavenumber in one direction, we can numerically calculate the wavenumbers and mode shapes associated with propagation in the perpendicular direction. By building solutions with null forces on parallel boundaries, 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 giving the amplitudes associated with the waves. By summing the contributions of all these waves we obtain the global solution for a two-dimensional periodic medium with a large number of substructures and a low computing time. Examples are given for the case of a two-dimensional membrane with many substructures and different types of heterogeneities.

过去，周期性介质的研究主要集中在一维周期性结构（即沿一个方向的周期性），一方面确定频率与波数之间的色散曲线，另一方面获得外部激励的结构，无论是有界结构还是无界结构。在后一种情况下，基于波有限元（WFE）等方法，已经获得了有效的方法。二维周期性介质的分析比较复杂，但与一维情况一样可以很容易地获得色散曲线。获得二维周期结构对时谐激励的稳态响应比一维介质困难得多，并且结果主要涉及无限介质。这项工作是关于有限二维周期性结构对时谐激励的稳态响应的最后一种情况，通过将自身限制为由标量变量（声学、热学、膜行为）描述的结构，并且与两个正交平面相比具有对称性平行于下部结构的边缘。使用矩形子结构的 WFE 并在一个方向上施加波数，我们可以数值计算与垂直方向上的传播相关的波数和模态形状。通过在平行边界上构建零力解，我们可以解耦平行于矩形边的两个方向上的波。这两个问题中的每一个的解决方案都是通过快速傅里叶变换获得的，给出与波相关的振幅。通过对所有这些波的贡献求和，我们获得了具有大量子结构和低计算时间的二维周期性介质的全局解。给出了具有许多子结构和不同类型异质性的二维膜的例子。