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Section: New Results
Asymptotic Analysis
Effective boundary conditions for thin periodic coatings
Participants : Mathieu Chamaillard, Houssem Haddar.
This topic is the object of a collaboration with Patrick Joly and is a continuation of our earlier work on interface conditions done in the framework of the PhD thesis of Berangère Delourme. Th goal here is to derive effective conditions that model scattering from thin periodic coatings where the thickness and the periodicity are of the same length but very small compared to the wavelength. The originality of our work, compared to abundant literature is to consider the case of arbitrary geometry (2-D or 3-D) and to consider higher order approximate models. We formally derived third order effective conditions after exhibiting the full asymptotic expansion of the solution in terms of the periodicity length.
Homogenization of electrokinetic models in porous media
Participant : Grégoire Allaire.
With R. Brizzi, J.-F. Dufrêche, A. Mikelic and A. Piatnitski, we are interested in the homogenization (or upscaling) of a system of partial differential equations describing the non-ideal transport of a N-component electrolyte in a dilute Newtonian solvent through a rigid porous medium. Our work can be divided in two different contributions. First, in the case of an ideal model (for which the homogenized system was already known) we consider the various limits which can be obtained in the effective parameters when the ratio between the characteristic pore length and the Debye length is either small or large. Second, we studied the homogenization process in the non-ideal case, namely when considering the so-called mean spherical approximation (MSA) model which takes into account finite size ions and screening effects.
A new shell modeling 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 is currently under review.
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 membrane and flexural effects that we have obtained using a formal approach.