Problèmes d'interface en présence de métamatériaux : modélisation, analyse et simulations

Valentin Vinoles 1, 2
1 POEMS - Propagation des Ondes : Étude Mathématique et Simulation
Inria Saclay - Ile de France, UMA - Unité de Mathématiques Appliquées, CNRS - Centre National de la Recherche Scientifique : UMR7231
Abstract : We are interested in transmission problems between dielectrics and metamaterials, that is to say media with unusual electromagnetic properties such as negative constants at some frequencies. These media are often made of periodic assemblies of resonant micro-structures and in this case the homogenization theory can justify mathematically these effective properties. A preliminary part deals with these problems in the harmonic domain and in geometry with separation of variables.Analytical computations are done and reveal in the so-called critical cases some mathematical diffculties: the solutions do not have the standard regularity and the problem can even be ill-posed.The first part examines these transmission problems in the time domain for which metamaterials are modelled by dispersive models (Drude model or Lorentz model for instance). The diffculties reside in the choice of a discretization scheme but especially in the construction of absorbing conditions. The method used here is the use of Perfectly Matched Layers (PMLs). Since classical PMLs are unstable for these models due to the presence of backward waves, we propose a new class of PMLs for which we conduct a stability analysis. The latter allows us to build stable PMLs. They are then used to simulate the long-time behaviour of a transmission problem; we illustrate the fact that the limit amplitude principle can be faulted because of interface resonances.The second part aims to overcome these phenomena by coming back to the classical homogenization in the harmonic domain, for dissipative media. For transmission problems, it is known that models resulting from this method lose accuracy due to the presence of boundary layers at the interface. We propose an enriched model at the interface: by combining the method of two-scale homogenization and that of matched asymptotic expansions, we build non-standard transmission conditions involving tangential derivatives along the interface (Laplace-Beltrami operators). This requires to solve cell problems and non-standardproblems in infinite periodic bands. An error analysis confirms the improvement of the accuracy of the model and numerical simulations show the effectiveness of these new conditions. Finally, this approach is formally reproduced in the case of high contrast materials which behave like metamaterials. We show that these new conditions regularise the transmission problem in the critical cases.
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Valentin Vinoles. Problèmes d'interface en présence de métamatériaux : modélisation, analyse et simulations. Modélisation et simulation. Université Paris-Saclay, 2016. Français. ⟨NNT : 2016SACLY009⟩. ⟨tel-01426226⟩

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