Section: New Results

Asymptotic models

Inverse scattering problem for coated obstacles

Participants : Nicolas Chaulet, Houssem Haddar.

In collaboration with L. Bourgeois, we considered the inverse scattering problem consisting in the identification of both an obstacle and its “equivalent impedance” from farfield measurements at a fixed frequency. The first specificity of this work is to consider the cases where this impedance is not a scalar function but a second order surface operator. The latter can be seen as a more general model for effective impedances and is for instance widely used for scattering from thin coatings. The second specificity of this work is to characterize the derivative of a least square cost functional with respect to this complex configuration. We provide in particular an extension of the notion of shape derivative to the cases where the impedance parameters cannot be considered as the traces of given functions. For instance, the obtained derivative does not vanish (in general) for tangential perturbations. The efficiency of considering this type of derivative is illustrated by some 2D numerical experiments based on a (classical) steepest descent method. The feasibility of retrieving both the obstacle and the impedance functionals is discussed in further numerical experiments [33] .

Interface conditions for thin dielectrics

Participant : Houssem Haddar.

Jointly with B. Delourme and P. Joly we established transmission conditions modelling thin interfaces that has (periodic) rapid variations along tangential coordinates. Motivated by non destructive testing experiments, 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 [15] . The analysis of the approximate problem for Maxwell's equations is the subject of a forthcoming publication.

Homogenization

Participant : Grégoire Allaire.

With I. Pankratova and A. Piatnitski we considered the homogenization of a non-stationary convection-diffusion equation posed in a bounded periodic heterogeneous domain with homogeneous Dirichlet boundary conditions. Assuming that the convection term is large, we give the asymptotic profile of the solution and determine its rate of decay. In particular, it allows us to characterize the “hot spot”, i.e., the precise asymptotic location of the solution maximum which lies close to the domain boundary and is also the point of concentration. Due to the competition between convection and diffusion, the position of the “hot spot” is not always intuitive as exemplified in some numerical tests.

With Z. Habibi we studied the homogenization of heat transfer in periodic porous media where the fluid part is made of long thin parallel cylinders, the diameter of which is of the same order than the period. The heat is transported by conduction in the solid part of the domain and by conduction, convection and radiative transfer in the fluid part (the cylinders). A non-local boundary condition models the radiative heat transfer on the cylinder walls. To obtain the homogenized problem we first use a formal two-scale asymptotic expansion method. The resulting effective model is a convection-diffusion equation posed in a homogeneous domain with homogenized coefficients evaluated by solving so-called cell problems where radiative transfer is taken into account. In a second step we rigorously justify the homogenization process by using the notion of two-scale convergence. One feature of this work is that it combines homogenization with a 3D to 2D asymptotic analysis since the radiative transfer in the limit cell problem is purely two-dimensional. Eventually, we provide some 3D numerical results in order to show the convergence and the advantages of our homogenization method.

Modelling and simulation for underground nuclear waste storage.

Participants : Grégoire Allaire, Harsha Hutridurga.

In the framework of the GDR MOMAS (Groupement de Recherches du CNRS sur les MOdélisations MAthématiques et Simulations numériques liées aux problèmes de gestion des déchets nucléaires) I am working with R. Brizzi, H. Hutridurga, A. Mikelic and A. Piatnitski on upscaling of microscopic models by homogenization (i.e. finding macroscopic models and effective coefficients).

We studied the Taylor dispersion of a contaminant in a porous medium. The originality of the model is that it takes into account surface diffusion and convection on the pores boundaries. We rigorously obtained the homogenized equation and studied the behavior of the effective dispersion tensor when varying various parameters.

In collaboration with a team of chemists (around J.-F. Dufrêche from the GNR Paris), we have undertaken the rigorous homogenization of a system of PDEs describing the transport of a N-component electrolyte in a dilute Newtonian solvent through a rigid porous medium. The motion is governed by a small static electric field and a small hydrodynamic force, which allowed us to use O'Brien's linearized equations as the starting model. Convergence of the homogenization procedure was established and the homogenized equations were discussed. Based on the rigorous study of the underlying equations, it was proved that the effective tensor satisfies Onsager properties, namely is symmetric positive definite. This result justified the approach of many authors who used Onsager theory as a starting point.

A new membrane/plate modeling

Participant : Olivier Pantz.

Using a formal asymptotic expansion, we have proved with K. Trabelsi, that non-isotropic thin-structure could behave (when the thickness is small) like a shell combining both membrane and bending effects. It is the first time to our knowledge that such a model is derived. An article on this project is in preparation.

A new Liouville type Rigidity Theorem

Participant : Olivier Pantz.

We have recently developed a new Liouville type Rigidity Theorem. Considering a cylindrical shaped solid, we prove that if the local area of the cross sections is preserved together with the length of the fibers, then the deformation is a combination of a planar deformation and a rigid motion. The results currently obtained are limited to regular deformations and we are currently working with B. Merlet to extend them. Nevertheless, we mainly focus on the case where the conditions imposed to the local area of the cross sections and the length of the fibers are only "almost" fulfilled. This will enable us to derive rigorously new non linear shell models combining both membranar and flexural effects that we have obtained using a formal approach.

Lattices

Participant : Olivier Pantz.

With A. Raoult and N. Meunier (Université Paris Descartes), we have compute the asymptotic limit of a square lattice with three-points interactions. An article currently under review has been submitted on this work.

Homogenization of axon Bundles

Participant : Olivier Pantz.

With E. Mandonnet (Lariboisière Hospital), we have developed a new modeling for bundles of axons using homogenization technique. Previous works only focus (even if not explicitly) in the low density case: That is when the axon density is small. The aim is to determine which kind of electrical stimulation could trigger a signal into the axon. Under the low density assumption, the external electric field is independent of the membrane potential of the axon. If not, both are strongly coupled. Moreover, we have performed numerical simulations to determine what is the best position of the electrodes to enable the activation of the axons. This work has lead to the publications of an article [20] and a technical report [29] . Finally, we have begin to investigate more realistic modelings of the ionic flux based on the works of FitzHuch-Nagumo with a student, Xinxin Cheng, who spend three months at the CMAP.