Section: New Results

Asymptotic methods and approximate models

Effective boundary conditions for thin periodic coatings

Participants : Mathieu Chamaillard, Patrick Joly.

This topic is the object of a collaboration with Houssem Haddar. We are interested in the construction of "equivalent" boundary condition for the diffraction of waves by an obstacle with smooth boundary $\Gamma$ covered with a thin coating of width $\delta$ whose physical characteristics vary "periodically along $\Gamma$ with a period proportional to the small parameter $\delta$. For a general boundary $\Gamma$, the notion of periodicity is ambiguous: we have chosen to define the coating as the image, or the deformation, by a smooth mapping of a flat layer of width delta (the reference configuration) that preserves the normals, which appears consistent with a manufactoring process. The electromagnetic parameters in the coating are then defined as the images through ${\Phi }_{\Gamma }$ of periodic functions in the reference configuration.

We have first considered the case of the scalar wave equation. Using an asymptotic analysis in $\delta$, which combines homogenization and matched asymptotic expansions, we have been able to establish a second order boundary condition of the form

where $B_{\Gamma }^1$ and $B_{\Gamma }^2$ are second order tangential differential operators along $\Gamma$ whose coefficients depend on both the geometrical characteristics of $\Gamma$ (through the curvature tensor) and the material properties of the coating, through the resolution of particular cell problems in the flat reference configuration. When the coating is homogeneous, we have checked that one recovers the well known second order thin layer condition. This new condition is expected to provide $O\left({\delta }^3\right)$ accuracy. Its implementation and its rigorous analysis (error estimates) are ongoing.

Thin Layers in Isotropic Elastodynamics

Participants : Marc Bonnet, Aliénor Burel, Patrick Joly.

This research is developed in the framework the numerical modeling of non-destructive testing experiments using ultrasonic waves. Most realistis propagation media involves thin layers of resin (typically for gluing together different homogeneous media), which are, until now, difficult to take into account numerically, the principal issue being the very small space step needed for meshing such a thin layer. An idea to get rid of this complication is to use asymptotic analysis in order to establish effective transmission conditions. We have studied the simple model problem in two dimensions, with an infinite flat layer of thickness $\epsilon$. Using a formal approach based on a scaling inside the layer and an power series expansion in $\epsilon$ solution as a polynomial in $\epsilon$, we have established first and second order conditions. Energy techniques parmit to guaranty the stability of our approximation.

Homogenization and metamaterials

Participants : Sonia Fliss, Patrick Joly, Valentin Vinoles.

This topic is developped in collaboration with Xavier Claeys (LJLL, Paris VI).

The mathematical modeling of electromagnetic metamaterials and the homogenization theory are intimately related because metamaterials are precisely constructed by a periodic assembly of small microstructures involving dielectric materials presenting a high contrast with respect to a reference medium. As a consequence, each microstructure behaves as a resonator which induces surprising properties to the effective or homogenized material such as negative permittivity and / or permeability at certain frequencies. The relevant theoretical approach to this question is the non standard (or high contrast) homogenization theory developed in particular in France by G. Bouchitté.

In the framework of the ANR Metamath, we wish to deepen this question by looking carefully at the treatment of boundaries and interfaces that are generally poorly taken into account by the first order homogenization. This is developed in collaboration with X. Claeys (Paris VI).

This question is already relevant for standard homogenization for which taking into account the presence of a boundary induces a loss of accuracy due to the inadequation of the standard homogenization approach to take into account the boundary layers induced by the boundary. Our objective is to construct approximate effective boundary conditions that would restore the desired accuracy.

With the PhD thesis of V. Vinoles, we aim at extending the previous approach to the treatment of metamaterials via high contrast homogenization. In particular, we intend to treat the challenging question of interfaces between metamaterials and standard materials (see also sections).

Asymptotic analysis and negative materials

Participants : Lucas Chesnel, Sergei Nazarov.

This topic is developped in collaboration with Xavier Claeys (LJLL, Paris VI) and S.A. Nazarov (IPME RAS, St Petersburg, Russia).

One of the applications of negative materials (metals at optical frequencies or negative metamaterials) is the construction of subwavelength cavities. In this kind of application, the idea is to use the following result: an inclusion of a negative material in a positive material changes radically the spectrum of the Maxwell's operators. We demonstrated this result for the scalar operator in a configuration where a positive material contains a small negative inclusion whose size tends to zero. As a second topic, we proved an instability result for a configuration where the interface between the positive and the negative material has a rounded corner. It appears that the solution depends critically on the value of the rounding parameter and does not converge when the rounded corner tends to the actual corner. We also studied the spectrum of the scalar operator in this configuration. This spectrum does not converge but seems (for the moment, the proof is not complete) to oscillate like $ln\delta$ where $\delta \to 0$ is the rounding parameter.

Modelling of non-homogeneous lossy coaxial cable for time domain simulation.

Participants : Geoffrey Beck, Sébastien Imperiale, Patrick Joly, Martina Novelinkova.

This topic, initiated at the end of the PhD thesis of S. Imperiale, has been the subject of the internship of M. Novelinkova and is the subject of the PhD thesis of G. Bech which started in October.

We investigate the question of the electromagnetic propagation in thin electric cables from a mathematical point of view via an asymptotic analysis with respect to the (small) transverse dimension of the cable: as it has been done in the past in mechanics for the beam theory from 3D elasticity, we use such an approach for deriving simplified effective 1D models from 3D Maxwell's equations. Doing so, we have been able to derive a generalized telegraphist's equation, a 1D wave equation with additional time convolution terms that results from the conjugated effect of electromagnetic losses and heterogeneity of the cross section. This new model has been fully justified through error estimates. We are currently working on a higher order generalized telegraphist's equation that would include dispersive effects through nonlocal capacity and inductance operators.

From the pratical point of view, a code that computes the coefficients (including the convolution kernel) of the effective model and solves the generalized telegraphist's equation has been implemented. It has been exploited to measure the presence of localized defects on the propagation of electromagnetic waves. This application has been motivated by the ANR project SODDA, in collaboration with CEA-LETI, about the non destructive trsting of networks of electric cables (a subject that we are investigating in collaboration with M. Sorine from Inria Rocquencourt).

Elastic wave propagation in strongly heterogeneous media

Participants : Patrick Joly, Simon Marmorat.

This subject enters our long term collaboration with CEA-LIST on the development on numerical methods for time-domain non destructive testing experiments using ultra-sounds. This is also the subject of the PhD thesis of Simon Marmorat. Our objective is to develop an efficient numerical approach for the propagation of elastic waves in a medium which is made of many small inclusions / heterogeneities embedded in a smooth (or piecewise smooth) background medium, without any particular assumption (such as periodicity) on the spatial distribution of these heterogeneities. Our idea is to exploit the smallness of the inclusions (with respect to the wavelength in the background medium) to derive a simplified approximate model in which each inclusion would be described by very few parameters (functions of time) coupled to the displacement field in background medium for which we could use a computational mesh that ignores the presence of the heterogeneities. For deriving such a model. we intend to use and adapt the asymptotic methods previously developed at Poems (such as matched asymptotic expansions).

Multiple scattering by small scatterers

Participants : Maxence Cassier, Christophe Hazard.

We consider the scattering of an acoustic time-harmonic wave by an arbitrary number of sound-soft obstacles located in a homogeneous medium. When the size of the obstacles is small compared with the wavelength, the numerical simulation of such a problem by classical methods (e.g., integral equation techniques or methods based on a Dirichlet-to Neumann map) can become highly time-consuming, particularly when the number of scatterers is large. In this case, the use of an asymptotic model may reduce considerably the numerical cost. Such a model was introduced by Foldy and Lax in the middle of the last century to study multiple isotropic scattering in a medium which contains randomly distributed small scatterers. Their asymptotic model is based on the fact that the scattered wave can be approximated by a wave emitted by point sources placed at the centers of the scatterers; the amplitudes of the sources are calculated by solving a linear system which represents the interactions between the scatterers. Nowadays, the FoldyñLax model is still used in numerous physical and numerical applications to approximate the scattered wave in a deterministic media. But to the best of our knowledge, there was no mathematical justification of this asymptotic model. We have proposed such a justification which provides local error estimates for the two-dimensional problem in the case of circular obstacles. An article on this subject has been accepted and will be published in Wave Motion in January 2013.