## 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. The 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 the transmission eigenvalue problem with applications to inverse problems

Participant : Houssem Haddar.

In a joint work with F. Cakoni and I. Harris, we consider the interior transmission problem associated with the scattering by an inhomogeneous (possibly anisotropic) highly oscillating peri-odic media. We show that, under appropriate assumptions, the solution of the interior transmission problem converges to the solution of a homogenized problem as the period goes to zero. Furthermore, we prove that the associ-ated real transmission eigenvalues converge to transmission eigenvalues of the homogenized problem. Finally we show how to use the first transmission ei-genvalue of the period media, which is measurable from the scattering data, to obtain information about constant effective material properties of the periodic media. The obtained convergence results are not optimal. Such results with rate of convergence involve the analysis of the boundary correction and will be subject of a forthcoming paper.

#### Homogenization of electrokinetic models in porous media

Participant : Grégoire Allaire.

With R. Brizzi, J.-F. Dufrêche, A. Mikelic and A. Piatnitski, 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 recent work has focused on the so-called non-ideal case. Namely we consider the mean spherical approximation (MSA) model which takes into account finite size ions and screening effects. On the one hand we established a rigorous homogenized transport model starting from this microscopic model. On the other hand we did numerical simulations to ompute the corresponding effective parameters and make systematic comparisons between the idea model and the MSA model.

#### Modeling and Simulation of the Mechanical behavior of Vesicles and Red Blood Cells

Participant : Olivier Pantz.

##### Highly anisotropic thin shells

With K. Trabelsi (IPSA), we have proposed a new justification of various non linear highly anisotropic elastic shell models. Among others, we do derive the so called Helfrich functional, that describe the behavior of the lipid bilayer of the vesicle and red blood cells. Our results will soon be published in MEMOCS (Mathematics and Mechanics Complex Systems).

##### Minimization of the Helfrich functional

Our work with K. Trabelsi established that the mechanical behavior of vesicles and red blood cells can be approximated by thin non linear anisotropic elastic shells. Minimizing directly the Helfrich functional is not an easy task from the numerical point of view. Most methods require the use of high order finite elements and stabilization techniques so to prevent mesh degeneration. Instead, we propose to approximate the two dimensional membrane of a vesicle (or red blood cell) by a three dimensional non linear elastic body of small thickness. Firstly, this enable us to use standard finite elements and discretization (basically Lagrange of degree 2). Secondly, the discretized formulation is intrinsically stable, so no stabilization is needed. Finally, even if it leads us to solve a three dimensional problem (instead of the two dimensional initial one), it is no more costly than a direct two dimensional approach as the scale of the mesh can be chosen to be of the same order than the "thickness" of the shell. We have already obtained encouraging results for vesicles. We plan to extend them to the case of vesicles with spontaneous curvature and to red blood cells. Moreover, we are considering different strategies to minimize the computational cost (that is already quite satisfying compared with some other methods).

#### Modeling of Damage and Fracture

Participant : Olivier Pantz.

##### Fracture as limit of Damage

With Leila Azem (PhD Student), we use a model introduced by G. Allaire, F. Jouve and N. Van Goethem in a previous work to simulate the propagation of fracture. The main idea is to approximate the fracture as a damage material and to compute the evolution of the path of the crack using a shape gradient analysis. Our main contribution consists to propose to use a material derivative approach to compute the shape gradient. The advantage is that it drastically simplify the evaluation of the shape gradient, as no regularization is needed and no jump terms as to be computed on the interface bewteen the healthy and damaged areras. An article is currently in preparation to present our results.

##### Fracture with penalization of the jump

With Leila Azem, we propose to approximate a model of fracture with penalization of the jump of the displacement as a limit of a damage model. This is achieved by a specific choice of the softness of the damage material with respect to the cost to turn material from a healthy to a damaged state. We have carried out a formal analysis to justify our approach and have already obtained several numerical results.