Tuning effective dynamical properties of periodic media by FFT-accelerated topological optimization

International audience; This works concerns the propagation of waves in periodic media, whose microstruc-ture is optimized to obtain specific dynamical properties (typically, to maximize the dispersion in given directions). The present study, focusing on scalar waves in two dimensions, e.g. antiplane shear waves, aims at setting a generic optimization framework. The proposed optimization procedure relies on a number of mathematical and numerical tools. First, the two-scale asymptotic homogenization method is deployed up to second-order to provide an effective dispersive model, valid for low and medium frequencies. Simple dispersion indicators and cost functionals are then considered on the basis of this model. Then, the minimization of these functionals is performed thanks to an algorithm that relies on the concept of topological derivative to iteratively perform phase changes in the unit cell characterizing the material. Finally, FFT-accelerated solvers are extensively used to solve the cell problems underlying the homogenized model. To illustrate the proposed approach, the resulting procedure is applied to the design of anisotropic media with maximal dispersion in specific directions, and to the reconstruction of unknown microstructures from effective phase velocity data. Several extensions are discussed, notably to address microstructured interfaces, elastic waves, and higher frequency regimes.