Microstructural topological sensitivities of the second-order macroscopic model for waves in periodic media

Marc Bonnet (1) , Rémi Cornaggia (2) , Bojan B Guzina (3)

1 POEMS - Propagation des Ondes : Étude Mathématique et Simulation
2 IRMAR - Institut de Recherche Mathématique de Rennes
3 Department of Civil Engineering [Minneapolis]

We consider scalar waves in periodic media through the lens of a second-order effective i.e. macroscopic description, and we aim to compute the sensitivities of the germane effective parameters due to topological perturbations of a microscopic unit cell. Specifically, our analysis focuses on the tensorial coefficients in the governing mean-field equation – including both the leading order (i.e. quasi-static) terms, and their second-order companions bearing the effects of incipient wave dispersion. The results demonstrate that the sought sensitivities are computable in terms of (i) three unit-cell solutions used to formulate the unperturbed macroscopic model; (ii) two adoint-field solutions driven by the mass density variation inside the unperturbed unit cell; and (iii) the usual polarization tensor, appearing in the related studies of non-periodic media, that synthesizes the geometric and constitutive features of a point-like perturbation. The proposed developments may be useful toward (a) the design of periodic media to manipulate macroscopic waves via the microstructure-generated effects of dispersion and anisotropy, and (b) sub-wavelength sensing of periodic defects or perturbations.

Mots clés: Topological sensitivity; Second-order homogenization; Waves in periodic media

Domaines: Equations aux dérivées partielles [math.AP]; Mécanique des solides [physics.class-ph]

Marc Bonnet, Rémi Cornaggia, Bojan B Guzina. Microstructural topological sensitivities of the second-order macroscopic model for waves in periodic media. 2018. ⟨hal-01742396v1⟩