Speaker: Remi Cornaggia (Sorbonne University)

Title: Modeling and controlling dispersive waves in architected materials : second-order homogenization and topological optimization

We are interested in waves in two-phase periodic materials, whose phase distribution is to be optimized to obtain specific dispersive properties (typically, to maximize the dispersion in given directions of wave propagation).

The two-scale asymptotic homogenization procedure will first be recalled. In particular, the second-order asymptotic expansion enables to model the low-frequency dispersive behavior of waves in these media. Illustratrations will be given for bilaminates in 1D, for which we designed correctors for boundary and transmission conditions, that complement the wave model to obtain an overall second-order approximation in bounded domains [1].

The topological optimization algorithm [2] will then be presented for scalar waves in 2D media (e.g. acoustic or antiplane shear waves). First, simple dispersion indicators are extracted from the homogenized model. Cost functionals to be minimized to achieve certain goals are then defined using these indicators. The minimization is then performed thanks to an iterative algorithm, which relies on the concept of topological derivative (TD) of the cost functional. The TD quantifies the sensitivity of the functional to a localized phase change in the unit cell, and therefore indicates optimal locations where to perform these phase changes. The TD of the cost functionnal can be computed from the TDs of the coefficients of the homogenized model, whose expressions were determined in a previous work [3]. At each step, the cell problems underlying the homogenized model, whose solutions are needed to compute the TDs, are solved thanks to FFT-accelerated solvers [4].

Two applications of the method will be presented: maximizing the dispersion in given directions, and determining the microstructure of an architected material from phase velocity measurements.

[1] Second-order homogenization of boundary and transmission conditions for one-dimensional waves in periodic media,

Remi Cornaggia and Bojan B. Guzina, International Journal of Solids and Structures, 2020

[2] Tuning effective dynamical properties of periodic media by FFT-accelerated topological optimization,

Rémi Cornaggia, Cédric Bellis, International Journal for Numerical Methods in Engineering, 2020

[3] Microstructural topological sensitivities of the second-order macroscopic model for waves in periodic media.

Marc Bonnet, Rémi Cornaggia and Bojan B. Guzina, SIAM Journal on Applied Mathematics, 2018

[4] A numerical method for computing the overall response of nonlinear composites with complex microstructure,

Hervé Moulinec et Pierre Suquet, Computer Methods in Applied Mechanics and Engineering, 1998

(files of [1,2,3] available on HAL : **https://cv.archives-ouvertes.fr/remi-cornaggia **)