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Section: New Results

Time-harmonic diffraction problems

Robust computation of eigenmodes in electromagnetism

Participant : Patrick Ciarlet.

This is a twofold work.

First, a collaboration with Annalisa Buffa (CNR, Pavia, Italy), Grace Hechme (former PostDoc) and Erell Jamelot (former PhD student).

To overcome the traditional difficulty linked to the apparition of spurious modes of the Maxwell's operator, we have proposed, analyzed and implemented a method based on a saddle-point formulation of the eigenvalue problem. Convincing numerical experiments have been carried out. Two papers on this topic have been published: one in CMAME (Dec. 2008), and one in Numer. Math. (2009).

Second, a collaboration with François Lefèvre and Stephanie Lohrengel (Reims Univ.) and Serge Nicaise (Valenciennes Univ.).

We have extended this method to composite materials, with the help of a generalized Weighted Regularization Method. Numerical results have been obtained. A paper on this topic has been accepted for publication in M2AN.

Time harmonic aeroacoustics

Participants : Anne-Sophie Bonnet-Ben Dhia, Jean-François Mercier.

We are still working on the finite element approximation of the time harmonic Galbrun's equation for simulating the acoustic scattering and radiation in presence of a mean flow. This is now the object of an ANR project, AEROSON, in collaboration with Florence Millot and Sébastien Pernet at CERFACS, Nolwenn Balin at EADS and Vincent Pagneux at the Laboratoire d'Acoustique de l'Université du Maine. Remember that we use an augmented variational formulation to overcome the lack of coerciveness of the original model. The regularizing term requires the evaluation of the vorticity $ \psi$ (curl of the main unknown which is the displacement Im7 $\#119854 $ ) which solves a hydrodynamic equation. Finally the complete problem couples the time harmonic augmented Galbrun's equation with a second order time harmonic advection equation. Our recent results concern mainly the theoretical and numerical study of the following time harmonic advection equation:

Im18 ${-i\#969 \#968 +\#119829 ·\#968 =f~\mtext in~\#937 ~\mtext and~\#968 =0~\mtext on~\#915 ^-}$

where the flow Im19 $\#119829 $ is incompressible and $ \upper_gamma$- is the inflow boundary of $ \upper_omega$ . Available results in the literature concern the case $ \omega$ = ia with Im20 ${a\#8712 \#8477 }$ and the extension to the case Im21 ${\#969 \#8712 \#8477 }$ is not straightforward. We proved that:

Using these results, we have proved that the complete problem, restricted to a bounded computational domain by using Perfectly Matched Layers, is coercive + compact, if the flow varies slowly enough. The numerical method which couples continuous nodal finite element for Im7 $\#119854 $ with the DG scheme for $ \psi$ has been implemented at CERFACS and the validation is in progress.

Harmonic wave propagation in locally perturbed infinite periodic media

Participants : Julien Coatléven, Sonia Fliss, Patrick Joly.

Two main domains of applications are concerned :

There is a need for efficient numerical methods for computing the propagation of waves inside media that are not necessarily periodic but differ from periodic media only in bounded regions (small with respect to the total size of the propagation domain) containing scatterers (obstacles, cracks, inhomogeneities). For these problems, a major difficulty is the reduction of the pure numerical computations to bounded domains surrounded the perturbations. The key point is to take advantage of the periodic structure of the problem outside to construct artificial 9but exact) boundary conditions. That is why we have investigated then the generalization of the DtN approach to periodic media.

This was the subject of the PhD Thesis of Sonia Fliss who has defended her thesis in May 2009. In this work, we proposed a solution for construction DtN operator for the 2D Helmholtz equation in the case of a single perturbation. We show in particular that we can factorize the DtN operator as a product of two operators. The first operator corresponds to a halfspace DtN operator and is constructed with a method which combines analytical tools such as the Floquet Bloch Transform with the numerical solution of local cell problems. The second operator requires the numerical solution of non stantdard integral equations in the (x, k) plane, where k denotes the dual variable in the Floquet Bloch Transform. The method is well established, rigorously justified and successfully tested in the case of absorbing media.

The treatment of non absorbing media raises complicated and new questions and requires the definition of an appropriate numerical procedure that should correspond to the continuous limiting absorption principle. The difficulties concern the discretization of the (x, k) -compact set and the resolution of the non standard integral equation.

Moreover, for theoretical and numerical issues, we have extended the method to the construction of Robin-to-Robin operator : instead of relating the Dirichlet and the Neumann traces of the solution, we want to relate two different Robin traces of the solution. From a theoretical point of view, this approach makes sense for a non absorbing media. Indeed, the construction of the DtN operator needs the resolution of family of local cell problems with Dirichlet conditions. For non absorbing media, this introduces a set of frequencies corresponding to the eigenvalues of the laplace operator with periodic coefficients on periodicity cell. This set of forbidden frequencies, due to the construction of DtN operator, is artificial and disappears with the construction of RtR operator. From a numerical point of view, one of the interest of RtR operators is that, contrary to DtN operators for instance, they are bounded operators with bounded inverse and their discretization leads to well-conditionned matrices. This extension is the subject of a chapter that we have written for the E-Book "Wave Propagation in Periodic Media Analysis, Numerical Techniques and practical Applications".

The first part of the PhD of J. Coatléven has been devoted to devicing a unified vision for multi-scattering problems, showing that for any situation where one could identify a reference medium and several compact perturbations, the DtN map on the boundary of the union of the scatterers can be expressed directly from the resolution of songle-scattering problems, thus allowing to treat multiple-scattering in a periodic medium (see the activity report of 2008). This work has been concretized by a talk during the 9th International Conference on Mathematical and Numerical Aspects of Waves (WAVES'09) in Pau, another during the Progress In Electromagnetics Research Symposium (PIERS 2009) in Moscow, and an article has been submitted. Recent work has also showed that the theoretical framework developped readily extends to the situation of a perturbed interface problem (seen as a multiple scattering problem), and thus will allow quasi-realistic non-destructive testing simulations in near future.

Moreover, we intend to treat more general pertubations and in particular line defects (i.e. the perturbation is infinite in one dimension). In optics, such defects are created to construct an (open) waveguide to concentrate light. The existence and the computation of the eigenmodes is a crucial issue. This is related to a seladjoint eigenvalue problem associated to a PDE in an unbounded domain (namely in the directions orthogonal to the line defect), which makes both the analysis and the computation hard. We believe that, by adapting the DtN approach developed for scattering problems, we shall offer a rigorously justified alternative to existing methods and an improvement of the computational time since we can reduce the numerical computation to a small neighborhood of the defect. However, there is a price to be paid : the reduction of the problem leads to a nonlinear eigenvalue problem, of a fixed point nature.

This research offers a number of interesting perspectives and developpments from theoretical, numerical and practical point of view. Applied to photonic crystals modelization, this work enters in the framework of the collaboration with the laboratory of Fundamental Electronics of Orsay University. For now, our collaboration has focused on the modelling of the light refraction at the surface of a photonic crystal and has been concretized by a co-signed article submitted to Physical Review B. Some perspectives of the collaboration with the electronic and optic communities is to generalize this study to a 3D problem, to analyse the extension of this method to electromagnetism and to study more general defects. These perspectives encourage us to ask for an ARC project with the BACCHUS team project of Inria Bordeaux for their high precision numerical expertise and with the laboratory of Fundamental Electronics for the modelling and the comparison with realistic experimentations.

Modeling of meta-materials in electromagnetism

Participants : Anne-Sophie Bonnet-Ben Dhia, Patrick Ciarlet, Lucas Chesnel.

Meta-materials can be seen as particular dielectric media whose dielectric and/or magnetic constant are negative, at least for a certain range of frequency. This type of behaviour can be obtained, for instance, with particular periodic structures. Of special interest is the transmission of an electromagnetic wave between two media with opposite sign dielectric and/or magnetic constants. As a matter of fact, applied mathematicians have to address challenging issues, both from the theoretical and the discretization points of view.

The (simplified) scalar model can be solved efficiently by the most "naive" discretization. It turns out that the convergence of the numerical approximation can be proved via a uniform stability estimate. Interestingly, it can also be solved numerically by introducing some dissipation in the model (first topic addressed by L. Chesnel, during his Master intership).

Last year, we considered the study of the transmission problem in a 3D electromagnetic setting from a theoretical point of view: to achieve well-posedness of this problem, we had to proceed in several steps, proving in particular that the space of electric fields is compactly embedded in L2 . For that, we assumed some regularity results on the interface. The second topic now studied by L. Chesnel is how to remove this assumption, and allow for instance to solve the problem around an interface with corners.

Finally, we are currently investigating configurations in which the problem is ill-posed. The aim is to define a suitable functional framework to recover well-posedness.

A Multiscale Finite Element Method for 2D Photonic Crystals

Participant : Kersten Schmidt.

This is a joint project with Christoph Schwab and Holger Brandsmeier (ETH Zürich).

Photonic crystals (PhC) structures are dielectric materials with a periodic fine structure. Light injected into the PhC is diffracted and refracted by the many dielectric scatterers arranged in the periodic arrays. Some properties of photonic crystal structures can predicted by the model of the infinite crystal with the same pattern which leads to the well-known band structure. For each wave-length comparable to the size of the periodicity there exist a set of functions – the Bloch modes – propagating in the infinite crystal. We propose a multiscale finite element method for finite structures of photonic crystals with basis functions of two scales. The small scale is represented by Bloch modes where the large scale functions are polynomials. We derived a fast quadrature formula leading to computational costs independent of the number of crystal periods inside a macroscopic cell.

At a first model we investigated the TE and TM modes in a photonic crystal band with finite extend in one and infinite in the other. We hold the number of Bloch modes and increase the macroscopic polynomial degree. We observed the relative error decaying rapidly when increasing hte polynbomial degree is independent of the number of crystal periods inside a macroscopic cell. The next step are true two-dimensional PhC structures.

Accurate computation of influence coefficients

Participants : Marc Lenoir, Nicolas Salles.

The dramatic increase of the efficiency of the variational integral equation methods for the solution of scattering problems must not hide the difficulties remaining for an accurate numerical computation of some influence coefficients, especially when the panels are close and almost parallel.

A complete set of explicit formulas has been derived for the case of constant basis functions and plane triangular panels, when the singularity of the kernel is homogeneous of order -1. The object of the thesis of Nicolas Salles is to test the accuracy and efficiency of these formulas, to extend their range of use to affine basis functions and other degrees of homogeneity and to implement them in a finite elements computer software.


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