Section: New Results

Asymptotic methods and approximate models

Multiscale modelling in electromagnetism

Participants : Bérangère Delourme, Patrick Joly.

This topic is developed in collaboration with the CEA-Grenoble (LETI) and H.Haddar (INRIA-Saclay-DEFI) and is dedicated to the study of asymptotic models associated with the scattering of electromagnetic waves from a complex periodic structure. More precisely, this structure is made of a dielectric ring that contains two layers of wires winding around it (see figure 3 ). We are interested in situations where the thickness of the ring and the distance between two consecutive wires are very small compared to the wavelength of the incident wave and the diameter of the ring. One easily understands that in those cases, direct numerical computations of the solution would become prohibitive as the small scale (denoted by ) goes to 0, since the used mesh would need to accurately follow the geometry of the heterogeneities.

Figure 3. the periodic ring

In order to overcome this difficulty, we derive approximate models where the periodic ring is replaced by effective transmission conditions. The numerical discretization of approximate problems is expected to be much less expensive than the exact one, since the used mesh has no longer to be constrained by the small scale.

In a first part, we have studied a simplified 2D case: we have constructed a complete and explicit expansion of the solution with respect to the small parameter and we have derived approximate models. These models are theoretically and numerically validated. For one year, we have been interesting in the 3D Maxwell case (which is the interesting model for the application) which presents new difficulties. The first one is due to the finite length of the ring: we need to understand the behavior of the electromagnetic field in the vicinity of the two extremities of the ring. This work has been partially done in collaboration with X.Claeys: we have studied a simplified two dimensional case and proved the relative accuracy of the first order intuitive approximate model. However, the building of approximate models of higher orders seems to be difficult. To avoid this first difficulty, we now consider the 3D Maxwell case with periodic boundary conditions on the extremities of the ring. We have derived an asymptotic expansion of the solution and an approximate model. From both hand computations and functional analysis points of view, the study of Maxwell's equations is more difficult than the study of Helmholtz equation.

Quasi-singularities and electrowetting

Participants : Patrick Ciarlet, Thu Huyen Dao.

A collaboration with Claire Scheid (Nice Univ.).

This is a twofold work.

First, the Master internship of Thu Huyen Dao. Following the PhD thesis of Samir Kaddouri (2007), she studied quasi-singularities for the 2D-cartesian electrostatic model around rounded tips, using the electric field as the primary unknown, instead of the potential. To complement this work, we are investigating the computation of accurate maps of the values of the electric field, to model corona discharge phenomena around 2D and 2 D tips.

Handling very small amount of liquid on a solid surface is of great industrial interest. In this field, electrowetting process is now broadly used: one charges a droplet posed on a solid by applying a given voltage between this droplet and a counter-electrode placed beneath the insulator. This allows one to control precisely the wetting of the drop on the solid. For modeling purposes, one has to compute very accurately the shape of the drop near the counter-electrode by solving an electrostatic problem with a piecewise constant electric permittivity. We recently considered 3D configurations, based on the numerical approximation of the electric field, using a generalized Weighted Regularization Method (see §  6.2.1 ). A paper on this topic has been accepted for publication in M2AN.

Asymptotic models for junctions of thin slots

Participants : Katrin Boxberger, Patrick Joly, Adrien Semin.

We have almost finished the work started in 2007 for the acoustic case. We have considered the most general possible case (a finite number of slots and junctions, and the slots may have different width). We have studied the two differents aspects of this problem:

• the theoretical point of view: as for the case of two junctions and one slot, we completly justify the asymptotic expansion. We plan to publish (at least) one INRIA Research Report and one article in Asymptotic Analysis,

• the numerical point of view: with the fellowship of Katrin Boxberger, we developed a C++ oriented-object code named "Net Waves" (this code is available on the INRIA GForge web site at url http://gforge.inria.fr/projects/netwaves ). This code is particular in the sense that there's no code at our knowledge which solves acoustic wave equation on a general finite network, even with classical Kirchhoff conditions. This code boards graphical output and is still maintained in the project.

Wave propagation on infinite trees

Participants : Patrick Joly, Adrien Semin.

We have continued the work started in 2007, on two different ways.

• Firstly, we have implemented an approximation of transparent boundary conditions for the Helmoltz equation on a self-similar p -adic tree in the code "Net Waves" mentionned in §  6.5.3 . We are currently making some regressions tests to be able to test these conditions. The next step, to be able to do many computations, is to write a GPU version of this code.

• Secondly, we started a collaboration with Serge Nicaise from the University of Valenciennes since July 2009 to look the functionnal framework and the notion of trace at infinity on a general (not necessarily self-similar) p -adic tree.

Approximate models in aeroacoustics

Participants : Anne-Sophie Bonnet-Ben Dhia, Patrick Joly, Lauris Joubert, Ricardo Weder.

This is the subject of the PhD thesis of Lauris Joubert and the object of a collaboration with M. Duruflé.

Two aspects of the subject have been considered.

First we have completed our work on a simplified model for the propagation of acoustic waves in a duct in the presence of a laminated flow. The theoretical analysis of this model has been completed in two directions:

• The stability analysis of the model in function of the Mach profile has been achieved completely. In the unstable case, an analogy with the known results about Kelvin-Helmholtz instabilities for incompressible fluids (Rayleigh anf Fjorjtoft criteria) has been established. An article has been submitted for publication.

• We have developed a general method for obtaining a quasi-analytic representation of the solution that results into a priori estimates. This method in based on the use of the Fourier-Laplace transform and complex analysis methods. An article has been submitted.

The quasi-analytic representation of the solution has been exploited numerically (see for instance the result of figure in the case of a parabolic Mach profile). The comparison with results obtained by discretizing the full model (Galbrun's equations) is under way.

The second aspect we have first developed is the construction of effective boundary conditions for taking into account boundary layers in aeroacoustics. On the basis of the analysis of the thin duct problem, we have proposed a first effective condition whose stabilty has been proven. This condition has the practical disadvantage to be nonlocal with respect to the normal coordinate inside the boundary layer. One can obtain a local condition after approximating the exact Mach profile by a piecewise linear profile. The study and the implementation of this new condition will be the subject of our next contribution to the problem.

Impedance boundary conditions for the aero-acoustic wave equations in the presence of viscosity

Participants : Bérangère Delourme, Patrick Joly, Kersten Schmidt.

This is a joint work with Sébastien Tordeux (INSA Toulouse).

In compressible fluids the propagating sound can be described by linearised and perturbed Navier-Stokes equations. This project is dedicated to the case of viscous fluid without mean flow. By multiscale expansion and matched asymptotic expansion we are deriving impedance boundary conditions taking into account the viscosity of the fluid. The first case is a plain wall and the second a wall with periodic perforations where we apply surface homogenisation.