## Section: New Results

### Waveguides, resonances, and scattering theory

#### Elastic waveguides

Participants : Vahan Baronian, Anne-Sophie Bonnet-Ben Dhia, Sonia Fliss, Éric Lunéville.

In partnership with the CEA, we are developing numerical tools to simulate ultrasonic non-destructive testing in elastic waveguides.

During the PhD of Vahan Baronian (which has been defended in November), a finite element method has been developed to compute the scattering by an arbitrary local perturbation of an isotropic 2D or 3D guide. By using modal expansions, specific boundary conditions are written on the artificial boundaries which are perfectly transparent, allowing the FE computation zone to be reduced as small as possible.

During this year, Vahan Baronian has extended the method to the more complicated configuration of a junction of several possibly different waveguides. From a theoretical point of view, he also proved that the method works in an orthotropic waveguide, as soon as one symmetry axis of the material is parallel to the axis of the waveguide.

We are now investigating the case of a 3D fully anisotropic plate, in partnership with CEA and EADS. We first consider the diffraction by a 2D obstacle (a stiffener for instance) at oblique incidence, which leads to a 2D problem.

#### Acoustic propagation in a lined waveguide

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

We have continued our work in collaboration with Emmanuel Redon of University of Dijon on the acoustic radiation in an infinite 2D lined guide (absorbing boundary condition on a wall) with a uniform mean flow. The aim was, in order to bound the domain, to build transparent boundary conditions, introduced by means of a Dirichlet to Neumann (DtN) operator based on a modal decomposition. Such decomposition is easy to carry out in a hard-walled guide. With absorbing lining and in presence of a mean flow, many difficulties occur. Even without flow, the eigenvalue problem is no longer selfadjoint but a relation for which the modes are orthogonal is found. In the flow case is only found a generalized orthogonality relation asymptotically satisfied by the high-order modes, which allows to define transparent boundary conditions.

We focus now on improving the mathematical framework of this problem and in characterizing the modes : clarifying their asymptotic behavior and proving their completness. In the no flow case, the modes behave asymptotically like the modes of a hard-walled guide. Then, following the approach developed in the literature for water waves, completness of the modes by comparison with the rigid modes and well-posedness of the radiation problem (defining a coercive+compact problem) can be proved. In the flow case, the asymptotic behavior of the modes is unexpected : they correspond to a soft boundary, even for a slow flow. Our objective is now to extend the completness proof of the no flow case.

#### Propagation in non uniform open waveguide

Participants : Anne-Sophie Bonnet-Ben Dhia, Benjamin Goursaud, Christophe Hazard.

This topic follows the work in collaboration with Lahcene Chorfi (University
of Annaba, Algeria) and Ghania Dakhia (University of Biskra, Algeria) about
the problem of the scattering of a time-harmonic acoustic wave by a defect
located in a two-dimensional uniform open waveguide. The theoretical analysis
of this problem using a generalized Fourier transform has been presented in
the previous activity reports, and has led to an article published in
2009. The natural continuation of this work concerns the propagation in a non
uniform open waveguide, more precisely the question of the junction of two
different uniform open waveguides. The question we consider concerns the
existence and the uniqueness of the solution to the propagation equations
together with two *modal radiation conditions* imposed to the scattered
field on both sides of the junction. The direction we have followed consists
in a combination of the technique used in the above mentioned work and the
idea developed a few year ago in the lab by Anne-Sophie Bonnet-Ben Dhia and
Axel Tillequin in the case of an abrupt junction (along a line perpendicular
to the direction of propagation). We have made a lot of progress this year,
but some difficulties concerning the question of uniqueness remain.

#### Multiple scattering in a duct

Participants : Éric Lunéville, Jean-François Mercier.

We study the multiple scattering of acoustic waves by rigid obstacles randomly placed in a duct. Multiple scattering regime corresponds to the propagation of a field with a wavelength comparable to the scatterers size. The aim is to find the characteristic parameters of a homogeneous medium modelizing this heterogeneous medium. The usual approach to find such equivalent medium is to send a plane wave and to calculate the mean field by averaging on many configurations of scatterers. We have developped a fast numerical method to compute such field. It is based on the coupling of finite element in a neighborhood of the scatterers (to reduce the mesh size) with an integral representation of the scattered field. A difficulty is that this last part requires the computation of the Green's function of the guide, which expresses as an expansion converging slowly. To enhance the reduction of computation time, perturbed periodic arrangements of scatterers are considered : they are placed on a periodic lattice and then randomly locally moved. Then it is possible to parallelize the computations : the scattering area is splited in slices and scattering matrices of each slice is computed. We prove numerically that only the plane wave has a coherent part in the heterogeneous medium : all the amplitudes of the other guided modes vanish after averaging. Moreover we obtain that the transmission through the averaged medium follows the law predicted by analytical models originally developed in the free space (not in a duct) excepting at the band gap frequencies of the periodic lattice, where the transmission vanishes as in a periodic medium.

#### Modelling of TE and TM Modes in Photonic Crystal Wave-Guides

Participant : Kersten Schmidt.

This is a joint project with Roman Kappeler (ETH Zürich).

In this project we study the band structure in photonic crystal wave-guides
of finite width and infinite periodicity in the other direction. The TE and
TM modes for a given frequency are determined by a quadratic
eigenvalue problem in the quasi-momentum k in the unit cell. The unit cell
– an infinite strip – is truncated and the decay conditions are replaced by
approximative boundary conditions. We perform numerical experiments for
realistic wave-guides with the *p* -version of the finite element method
on meshes with curved cells.