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Journal Articles SIAM Journal on Applied Mathematics Year : 2018

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

Abstract

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.
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Dates and versions

hal-01742396 , version 1 (24-03-2018)
hal-01742396 , version 2 (28-03-2018)
hal-01742396 , version 3 (09-08-2018)

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Marc Bonnet, Rémi Cornaggia, Bojan B Guzina. Microstructural topological sensitivities of the second-order macroscopic model for waves in periodic media. SIAM Journal on Applied Mathematics, 2018, 78 (4), pp.2057-2082. ⟨10.1137/17M1149018⟩. ⟨hal-01742396v3⟩
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