Section: New Results

Asymptotic models

Long time asymptotic models for the wave equation in periodic media

Participant : Grégoire Allaire.

In a joint work with M. Palombaro and J. Rauch, we studied the homogenization and singular perturbation of the wave equation in a periodic media for long times of the order of the inverse of the period . We consider initial data that are Bloch wave packets, i.e., that are the product of a fast oscillating Bloch wave and of a smooth envelope function. We prove that the solution is approximately equal to two waves propagating in opposite directions at a high group velocity with envelope functions which obey a Schrödinger type equation. Our analysis extends the usual WKB approximation by adding a dispersive, or diffractive, effect due to the non uniformity of the group velocity which yields the dispersion tensor of the homogenized Schrödinger equation [6] , [32] .

Jointly with L. Friz, we extended these previous results in the case of a locally periodic media. In such a case, on top of homogenization appears another effect, called localization (similar to the so-called Anderson localization for the Schrödinger equation in quantum mechanics). We consider initial data that are localized Bloch wave packets, i.e., that are the product of a fast oscillating Bloch wave at a given frequency and of a smooth envelope function whose support is concentrated at a point x with length scale . We assume that (, x) is a stationary point in the phase space of the Hamiltonian (, x) , i.e., of the corresponding Bloch eigenvalue. Upon rescaling at size we prove that the solution of the wave equation is approximately the sum of two terms with opposite phases which are the product of the oscillating Bloch wave and of two limit envelope functions which are the solution of two Schrödinger type equations with quadratic potential. Furthermore, if the full Hessian of the Hamiltonian (, x) is positive definite, then localization takes place in the sense that the spectrum of each homogenized Schrödinger equation is made of a countable sequence of finite multiplicity eigenvalues with exponentially decaying eigenfunctions [4] .

Interface conditions for thin dielectrics

Participant : Houssem Haddar.

In a first work, in collaboration with S. Chun and J. Hesthaven from Brown University, we established transmission conditions modelling thin anisotropic media in time dependent electromagnetic diffraction problems. The derived interface conditions turn out to be well suited for Discontinuous Galerkin methods since the latter implicitly support discontinuities between elements. The interface conditions only results into a modification of the numerical flux used in DG methods. These conditions has been successfully tested in the 1-D case up the fourth order where stabilization in time has been applied to the fourth order condition. It is also worth noticing that the expression of these conditions in the anisotropic case cannot be simply deduced from the isotropic one by just replacing constant coefficients with their matrix equivalent. We extended the 1-D case to the 2-D and 3-D ones, where stable conditions are designed for curved geometries up tor order 3 and for flat ones up to order 4. These conditions are numerically validated in the 2-D case [34] .

Jointly with B. Delourme and P. Joly we are investigating the extension of this work to the cases where the thin interface has (periodic) rapid variations along tangential coordinates. Motivated by non destructive testing experiments of tires, we considered the case of cylindrical geometries and time harmonic waves. We already obtained a full asymptotic description of the solution in terms of the thickness in the scalar case using so called matched asymptotic expansions. This asymptotic expansion is then used to derive generalized interface conditions and establish error estimates for obtained approximate models. The case of 3-D Maxwell's equations is under study.

Generalized Impedance Boundary Conditions: the forward problem

Participants : Houssem Haddar, Armin Lechleiter.

We studied so-called Generalized Impedance Boundary Conditions (gibc ) in the context of time-harmonic rough surface and rough layer scattering. In such problems one considers scattering objects like an unbounded hypersurface or an inhomogeneous infinitely extended layer. For a variety of interface and boundary conditions including the GBICs, we showed existence and uniqueness of solution for scattering of acoustic or TE/TM polarized electromagnetic waves from such structures. This result is achieved by Rellich identities yielding explicit a-priori bounds on the solution - those also allow to transfer the asymptotic analysis of GBICs for bounded obstacles to the rough surface setting [36] . Currently under investigation is whether these results can be extended to the full electromagnetic rough layer scattering problem.

In collaboration with B. Aslanyurek, who was visiting our group for 9 months in 2009, we derived Generalized Impedance Boundary Conditions that model thin dielectric coatings with variable width. We treated the 2-D electromagnetic problem for both TM and TE polarizations. The expressions of the gibc s are derived up to the third order (with respect to the coating width). The order of convergence is numerically validated through various numerical examples. A particular attention is given to the cases where the inner boundary has corner singularities [29] .

Generalized Impedance Boundary Conditions: the inverse problem

Participants : Houssem Haddar, Nicolas Chaulet.

We are interested here in the identification of a medium impedance from the knowledge of far measurements of a scattered wave at a given frequency. Assuming that the unknown medium occupies a domain D , the medium impedance is understood as a “local” operator that links the Cauchy data of the field u on the medium boundary : = D . More precisely we consider the cases where a boundary condition of the form: u/ + Zu = 0 on is satisfied, where Z is a boundary operator and denotes the outward normal field on .

The exact impedance operator Z corresponds to the so-called Dirichlet-to-Neumann (DtN) map, i.e. where u solves the Hemholtz equation inside D and satisfies u = f on . Consequently determining this map is “equivalent” to identify the physical properties inside D , which is in general a severely ill-posed problem that requires more than a finite number of measurements.

We are interested here in situations where the operator Z is an approximation of the exact DtN map. In general these approximations correspond to asymptotic models associated with configurations that involve a small parameter. These cases include small amplitude roughness, thin coatings, periodic gratings, highly absorbing media, ...

The simplest form is the case where Z is a scalar function, which corresponds in general to the lowest order (non trivial) approximations, for instance in the case of very rough surfaces of highly absorbing media (the Leontovich condition). However, for higher order approximations or in other cases the operator Z may involve boundary differential operators. For instance when the medium contains a perfect conductor coated with a thin layer of width then for TM polarization, the approximate boundary conditions of order 1 corresponds to Z = 1/ while for the TE polarization it corresponds to Z = (_ss + k²n) where s denotes the curvilinear abscissa, k the wave number and n is the mean value of the thin coating index with respect to the normal coordinate. Higher order approximations would include curvature terms or even higher order derivatives. This type of conditions will be referred to as Generalized Impedance Boundary Conditions gibc . One easily sees, from the given example, how the identification of the impedance would provide information on some effective properties of the medium (for instance, the thickness of the coating and the normal mean value of its index). Determining these effective properties would be less demanding in terms of measurements than solving the inverse problem with the exact DtN map (the unknown parameters have one dimension less) and we also expect that the inherent ill-posedness to be less severe.

In a first work with L. Bourgeois and motivated by the example above we addressed the question of unique identification and stability of the reconstruction of from the knowledge of one scattered wave. After pointing out that uniqueness does not hold in the general case, we propose some additional assumptions for which uniqueness can be restored. We also considered the question of stability when uniqueness holds. We prove in particular Lipschitz stability when the impedance parameters belong to a compact set. We also extend local stability results to the case of back-scattering data [10] .

The general goal of the PhD thesis of Nicolas Chaulet (started in October 2009) is to extend this work to more complex expressions of the impedance operator and validate theoretical results through numerical experiments. Some 2-D numerical results are obtained in this direction as well as stability results with respect to error on the boundary location. We also would like to investigate the problem where the boundary is also unknown and analyze whether a gibc induces cloaking.