## Section: New Results

### Time-harmonic diffraction problems

#### Numerical computation of variational integral equation methods

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.

The formulas have been extended to double layer potentials and, for self influence coefficients, to affine basis functions. Their efficiency for the solution of Maxwell equations has been proved in the framework of a collaboration with CERFACS. The redaction of a paper devoted to the case of parallel panels has been completed and submitted to SIAM J. Sci. Comp. Some preliminary work on the numerical integration of the regular part of the integrand has been undertaken in the context of an internship.

#### Fast Multipole Method for Viscoelastodynamics

Participants : Marc Bonnet, Stéphanie Chaillat.

This work is done in collaboration with Eva Grasso (LMS, Ecole Polytechnique) and Jean-François Semblat (IFSTTAR). We have extended the single- and multi-domain time-harmonic elastodynamic multi-level fast multipole BEM (Boundary Element Method) formulations to the case of weakly dissipative viscoelastic media [21] . The underlying boundary integral equation and fast multipole formulations are formally identical to that of elastodynamics, except that the wavenumbers are complex-valued due to attenuation. Attention was focused on evaluating the multipole decomposition of the visco-elastodynamic fundamental solution, involving complex-valued wavenumbers. As a result, a damping-dependent modification of the selection rule for the multipole truncation parameter was proposed and assessed on 3D single-region and multi-region visco-elastodynamic examples involving up to about $3\phantom{\rule{0.166667em}{0ex}}{10}^{5}$ boundary nodal unknowns.

#### Formulation and Fast Evaluation of the Multipole Expansions of the Elastic Half-Space Fundamental Solutions

Participants : Marc Bonnet, Stéphanie Chaillat.

This ongoing work is concerned with a formulation and computation algorithm for the elastodynamic Green's tensor for the traction-free half-space allowing its use within a Fast Multipole Boundary Element Method (FM-BEM). Due to the implicit satisfaction of the traction-free boundary condition achieved by the Green's tensor, discretization of (parts of) the free surface is no longer required. Unlike the full-space fundamental solution, the elastodynamic half-space Green's tensor cannot be expressed in terms of usual kernels such as ${e}^{\mathrm{i}kr}/r$ or $1/r$. Its multipole expansion thus cannot be deduced from known expansions, and is formulated in this work using a spatial two-dimensional Fourier transform approach. The latter achieves the separation of variables which is required by the FMM. A key numerical issue, upon which current work is focused, is concerned with the definition of an efficient numerical quadrature for the evaluation of the inverse Fourier transform, whose integrand is both singular and oscillatory, as classical Gaussian quadratures would perform poorly, fail or require unacceptably large number of quadrature points.

#### Multiple scattering by small scatterers

Participants : Maxence Cassier, Christophe Hazard.

We consider the scattering of an acoustic time-harmonic wave by an arbitrary number of sound-soft obstacles located in a homogeneous medium. When the size of the obstacles is small compared with the wavelength, the numerical simulation of such a problem by classical methods (e.g., integral equation techniques or methods based on a Dirichlet-to Neumann map) can become highly time-consuming, particularly when the number of scatterers is large. In this case, the use of an asymptotic model may reduce considerably the numerical cost. Such a model was introduced by Foldy and Lax in the middle of the last century to study multiple isotropic scattering in a medium which contains randomly distributed small scatterers. Their asymptotic model is based on the fact that the scattered wave can be approximated by a wave emitted by point sources placed at the centers of the scatterers; the amplitudes of the sources are calculated by solving a linear system which represents the interactions between the scatterers. Nowadays, the Foldy–Lax model is still used in numerous physical and numerical applications to approximate the scattered wave in a deterministic media. But to the best of our knowledge, there was no mathematical justification of this asymptotic model. We have proposed such a justification which provides local error estimates for the two-dimensional problem in the case of circular obstacles. An article on this subject has been recently submitted to Wave Motion.

#### Harmonic wave propagation in locally perturbed infinite periodic media

Participants : Julien Coatléven, Sonia Fliss, Patrick Joly.

A part of the PhD of J. Coatléven consists in developing a method for solving harmonic wave problems with locally perturbed line defects in periodic media. For the treatment of these unbounded defects, which are structured apart from a local perturbation, a new approach has been developed, based on a perturbation principle. The solution is written as a the sum of a solution corresponding to the unperturbed line defect and a contribution of the local perturbation. This decomposition leads to a generalization of the so-called Lippmann-Schwinger equation, whose coefficients are computed through their Floquet-Bloch transform, which leads to solve wave-guide problems, these last problems being solved using the transparent boundary condition method developed during S. Fliss's PhD. The discretization of the inverse Floquet-Bloch transform is done using appropriate quadrature rules, whereas the space discretization requires classical finite elements. The theoretical basis as well as the numerical analysis of this method are well understood, and the method has been successfully tested numerically. In particular, the theoretical convergence estimates have been checked in practice, and the method has a satisfying behavior in limit cases not fully covered by the theory, such as the non-absorbing case.

Concerning the non absorbing case, the question of the limiting absorption principle has been treated for locally perturbed periodic media with particular assumptions. In this case, we are studying the behavior of the solution at large distance of the local perturbation.

#### Time harmonic aeroacoustics

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

We are still working on the numerical simulation of the acoustic scattering and radiation in presence of a mean flow. This is the object of the 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. The main recent improvements concern: the consideration of ducts with treated boundaries and the development of an alternative model to Galbrun's equation.

**Treated boundaries**

Our aim is to extend the time harmonic equation of Galbrun to take into account acoustically treated boundaries. Such boundaries are generally described by the Myers boundary condition. Since this condition is naturally expressed in terms of Galbrun's unknown, the displacement $\mathbf{u}$, Galbrun's equation easily extends to treated boundaries. However we face a difficulty: the original equation of Galbrun leads to a non coercive problem. For rigid boundaries, considering an augmented variational formulation leads to well-posedness. But this approach does not work anymore for treated boundaries.

We have improved our understanding of this difficulty. We are now convinced that the Augmented Galbrun's equation combined with Myers condition leads to an ill-posed problem. More precisely source terms for which the solution of Galbrun's equation does not belong to standard functional spaces exist. Such source terms are very particular: located on the treated boundary and singular. This is why during the numerical validations performed at Cerfacs, for "standard" source terms (${L}^{2}$ functions compactly supported in the fluid) we did not get any problem.

To get a well-posed problem, Myers boundary condition, which just requires the normal displacement to belong to ${L}^{2}$, must be regularized. It can be achieved by requiring the tangential derivative of $\mathbf{u}\xb7\mathbf{n}$ to belong to ${L}^{2}$ . We have also understood that less regularity is sufficient to get a well-posed problem, as it is the case if the fluid is in contact of an elastic medium (such interface cannot be described by a Myers condition since it is necessarily non-local). In particular in the case of a uniform flow well-posedness is proved for a sufficiently slow flow.

**Alternative to Galbrun's model**

We have kept on considering the model of Goldstein's equations, alternative to Galbrun's equation. The Goldstein's equations couples two unknowns: the velocity potential $\varphi $ and a vectorial unknown $\xi $. $\varphi $ satisfies a modified Helmholtz's equation with variable coefficients linked to the flow, in which $\xi $ is added as a source term. $\xi $ satisfies a transport equation coupled to the velocity potential. This new model facilitate the treatment of 3D problems since Galbrun's equation requires to introduce many unknowns. Moreover the vectorial unknown in Goldstein's formulation vanishes in the areas where the flow is potential which is interesting since realistic flows are mainly potential, the non-potential areas being located near the boundaries or behind obstacles.

As it is the case to calculate the vorticity $\psi =\mathrm{curl}\mathbf{u}$, used to regularize Galbrun's equation, a Discontinuous Galerkin (DG) discretization is used to determine $\xi $ and numerical simulations have been performed at Cerfacs. We have also developed an alternative method allowing to solve Goldstein's equations with simple Lagrange Finite Element and this was the object of Jean-Emmanuel Lauzet's intership. We have developed a method combining the Streamline Upwind Petrov Galerkin (SUPG) scheme to discretize the Goldstein's model with the introduction of PML to bound the calculation domain. To test the efficiency of the method, a non-potential flow has been determined analytically. It consists of a lid-cavity flow connected to a uniform flow in a duct. The viscous cavity flow is solution of the Stoke's equation and is determined in a rectangular domain by a modal method.

#### Modeling of meta-materials in electromagnetism

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

A collaboration with Eric Chung (Chinese Univ. of Hong Kong) and Xavier Claeys (ISAE).

Meta-materials can be seen as particular media whose dielectric and/or magnetic constant are negative, at least for a certain range of frequency. This type of behavior 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 first topic we considered a few years ago was: when is the (simplified) scalar model well-posed in the classical ${H}^{1}$ framework? It turned out this issue could be solved with the help of the so-called $T$-coercivity framework. While numerically, we proved that the (simplified) scalar model could be solved efficiently by the most "naive" discretization, still using $T$-coercivity.

Recently, we have been able to provide sharp conditions for the $T$-coercivity to hold in general 2D and 3D geometries (with L. Chesnel), which involve explicit estimates in simplified geometries together with localization arguments. We then analyzed the discretization of the scalar problem with a classical, ${H}^{1}$ conforming, finite element method, and proved the convergence under the same sharp conditions (with L. Chesnel). We also showed that the problem can be solved with the help of a Discontinuous Galerkin discretization, which allows one to approximate both the field and its gradient (with E. Chung).

Last (with L. Chesnel and X. Claeys), we investigated the case of a 2D corner which can be ill-posed (in the classical ${H}^{1}$ framework). Using the Mellin transform, we showed that a radiation condition at the corner has to be imposed to restore well-posedness. Indeed there exists a wave which takes an infinite time to reach the corner: this "black hole" phenomenon is observed in other situations (elastic wedges for example).

As a second topic, we studied the transmission problem in a purely 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 ${L}^{2}$. For that, we had to assume that the interface is "sufficiently smooth", excluding in particular corners. With L. Chesnel, we have been able to remove this assumption, so that we can solve the problem around an interface with corners. It turns out the $T$-coercivity framework can be applied once more, under the same assumptions as for the scalar model. In the process, we recover more compact embedding results.

#### Numerical MicroLocal Analysis

Participants : Jean-David Benamou, Francis Collino, Simon Marmorat.

Numerical microlocal analysis of harmonic wavefields is based on a family of linear filters using Bessel functions and applied to wave data collected on a circle of fixed radius ${r}_{0}$ around the observation point ${x}_{0}$ where we want to estimate the Geometric Optics/ High Frequency components. The data can easily be reconstructed from more conventional line array or grid geometry. The output is an angular function presenting picks of amplitudes in the direction angles of rays.

The original `NMLA` algorithm relied on a local plane wave assumption for the data. For arbitrary waves, it meant linearization errors
and accuracy limitations. Also, only the directions of the (multiple) rays are recovered but
the traveltime and amplitudes are not reliably computed.
We recently introduced a new "impedant" observable
which allows to prove a stability theorem.
Numerical results confirm that the new `NMLA` filter is robust to random and correlated noise.

Using asymptotic expansion on `NMLA` filtered point sources data,
we designed a correction method for the angle which also estimates the wavefront curvature.
It can be used to correct the linearization errors mentioned above and provides a second order correction
in the Taylor approximation of the traveltime.

The parameters of the method (size of observation circle, discretization) are automatically optimized and
a posteriori quantitative error on angles and curvature are available.
Numerical studies validate the stability result and
confirm the superior accuracy of the curvature corrected `NMLA` version over image processing methods.

When some bandwith is available we can also compute the traveltime. The amplitude remains polluted by phase errors. Its determination is still open.