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
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
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
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 (
To get a well-posed problem, Myers boundary condition, which just requires the normal displacement to belong to
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
As it is the case to calculate the vorticity
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
Recently, we have been able to provide sharp conditions for the
Last (with L. Chesnel and X. Claeys), we investigated the case of a 2D corner which can be ill-posed (in the classical
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
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
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.