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Section: New Results

Asymptotic models

Modeling and numerical methods for the study underground storage of waste

Participant : Grégoire Allaire.

On the topic "Numerical multi-scale simulations" I work in collaboration with Philippe Montarnal and Thomas Abballe from CEA. We work on the extension of a multiscale finite element method to a finite volume framework with an application to the diffusion of chemical species in concrete. We rely on one of my previous works with Robert Brizzi at Ecole Polytechnique, where we developed a new multiscale finite element method that allows to perform a numerical homogenization. The principle is to build a basis of finite elements that contains information on the heterogeneity of the underlying medium. We can then make precise calculations without having to use of very fine mesh size (smaller than the characteristic size of the heterogeneities). We successfully extended this approach to finite volume method (more adapted to diffusivity high contrasts) in 2D and 3D within the framework of TRIO-U code. An article, based on the presentation of Thomas Abballe in NTMC'O9 Congress (New Trends in Model Coupling), has been published on this subject [2] .

The second topic on the hydrodynamic dispersion, is a collaboration with Robert Brizzi (CMAP), Andro Mikelic (Lyon 1) and Andrey Piatnitski (Narvik). It consists in finding macroscopic models for transport and dispersion of chemical species in flows within porous media and in presence of chemical reactions. Given the fluid speed field, the microscopic equations are of the convection-diffusion-reaction with two possible fields: the concentration in the fluid and the one on the solid surface of the pores. Using a homogenization method (more precisely two-scale convergence with drift) we obtain a new effective model where the homogenized coefficients obtained from microscopic cell problems combine the three aspects (convection, diffusion, chemical reactions) in a strongly coupled form. One can then study the dependence of the effective dispersion tensor with respect to the Peclet and the Damkohler dimensionless numbers . Numerical simulations indicate that the asymptotic behavior of the dispersion for large Peclet numbers are radically different in the presence or in the absence of chemical reactions. Our work is published in [6] , cite63.

Moreover, with Andrey Piatnitski we showed in [9] that a reaction-diffusion model in porous media with periodically time oscillating coefficients (for example, as a result of a cyclical external forcing) can lead at the macroscopic level to a significant convection effect. This surprising effect, also studied by other authors, is one of the possible explanations for the mechanism of bio-engines. In the context of waste storage in porous media, it could be responsible for transportation of radionuclide larger than one would predict from microscopic flow velocities.

Finally, and more recently with the same people we are interested in the homogenization of microscopic models of clays. More generally, this work is part of a collaboration with the GDR PARIS and the team of chemists around Pierre Turq. In a first step we considered the mechanism of electrophoresis in porous media for which there are already many homogenized models in the literature of physics or mechanics. We mathematically justified this homogenization process in a linearized case (a usual assumption in previous works) and we formally derived a new homogenized model in the nonlinear case. We published an article on the subject [7] .

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 [18] .

Jointly with B. Delourme and P. Joly we investigated 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, 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 [38] . The case of 3-D Maxwell's equations is under preparation.

Generalized Impedance Boundary Conditions: the inverse problem

Participants : Nicolas Chaulet, Houssem Haddar.

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 $ \upper_gamma$: = $ \partial$D . More precisely we consider the cases where a boundary condition of the form: $ \partial$u/$ \partial$$ \nu$ + Zu = 0 on $ \upper_gamma$ is satisfied, where Z is a boundary operator and $ \nu$ denotes the outward normal field on $ \upper_gamma$ .

The exact impedance operator Z corresponds to the so-called Dirichlet-to-Neumann (DtN) map, i.e. Im2 ${f\#8614 -\#8706 u/\#8706 \#957 |}_\#915 $ where u solves the Hemholtz equation inside D and satisfies u = f on $ \upper_gamma$ . 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 $ \delta$ then for TM polarization, the approximate boundary conditions of order 1 corresponds to Z = 1/$ \delta$ while for the TE polarization it corresponds to Z = $ \delta$($ \partial$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 collaboration with L. Bourgeois we continued the investigation of the problem of unique identification and stability of the reconstruction of Im3 ${Z=\#956 \#916 _\#915 +\#955 }$ from the knowledge of the far fields created by one or several incident plane waves at a fixed frequency. We first derived a new type of stability estimate for the identification of $ \lambda$ and $ \mu$ from the far field when inexact knowledge of the boundary is assumed. We then introduced an optimization method to identify $ \lambda$ and $ \mu$ , using in particular a H1 -type regularization of the gradient. We also conducted some numerical validation in two dimensions, including a study of the impact of some various parameters, and by assuming either an exact knowledge of the shape of the obstacle or an approximate one [33] . We are currently finalizing a work on simultaneous reconstruction of the GIBC and the shape of the obstacle from the knowledge of the farfield operator using a nonlinear optimization method.


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