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.

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

Participants : Marc Bonnet, Stéphanie Chaillat.

The use of the elastodynamic half-space Green's tensor in the FM-BEM is a very promising avenue for enhancing the computational performances of 3D BEM applied to analyses arising from e.g. soil-structure interaction or seismology. 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. To address the critical need of an efficient quadrature for the 2D Fourier integral, whose singular and oscillatory character precludes using usual (e.g. Gaussian) rules, generalized Gaussian quadrature rules have been used instead. The latter were generated by tailoring for the present needs the methodology of Rokhlin's group. Numerical tests have been conducted to demonstrate the accuracy and numerical efficiency of the proposed FMM. In particular, a complexity significantly lower than that of the non-multipole version was shown to be achieved. A full FM-BEM based on the proposed acceleration method for the half-space Green's tensor is currently under way.

Domain decomposition methods for time harmonic wave propagation

Participants : Francis Collino, Patrick Joly, Mathieu Lecouvez.

This work is motivated by a collaboration with the CEA-CESTA (B. Stupfel) through the PhD thesis of M. Lecouvez that has started at the beginning of the year.

We are interested in the diffraction of time harmonic electromagnetic waves by perfectly conducting objects covered by multi-layered (possibly thin) dielectric coatings. This problem is computationally hard when the size of the object is large (typically 100 times larger) with respect to the incident wavelength. In such a situation is to use a domain decomposition method in which each layer would contitute a subdomain. More precisely, we want to use a non overlaping iterative domain decomposition method based on the use of Robin type transmission conditions, a subject to which people at Poems gave substantial contributions in the 90's through the works of Collino, Desprès, and Joly.

The novelty of our approach consists in using new transmission conditions using some specific impedance operators in order to improve the convergence properties of the method (with respect to more standard Robin conditions). Provided that such operators have appropriate functional analytic properties, the theory shows that one achieves geometric convergence (in opposition the the slow algebraic convergence obtained with standard methods). These properties prevent the use of local impedance operator, a choice that was commonly done for the quest of optimized transmission conditions (following for instance the works of Gander, Japhet, Nataf). We propose a solution that uses nonlocal integral operators using appropriate Riesz potentials. To overcome the disadvantage of dealing with completely nonlocal operators, we suggest to work with truncated kernels, i.e. with operators of the form ($\Gamma$ represents one interface)

where $K\left(\right|x\left|\right)$ is an appropriate singlar kernel (typically $K\left(\right|x\left|\right)={\left|x\right|}^{-\gamma }$) and $\chi \left(\rho \right)$ an adequate smooth cut-off function. Playing with a few parameters such as the size of the support of $\chi$, we expect to achieve an optimal compromise between the reduction of the number of iterations of the method and the cost of each iteration.

Time harmonic aeroacoustics

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

We are still working on the numerical simulation of the acoustic radiation and scattering 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. Let us recall that our method combines, a Finite Element resolution of the augmented Galbrun equation and of the coupled vorticity transport equation, and the use of Perfectly Matched Layers (PML) to bound the computational domain. The main recent improvements concern the test of the method in presence of unstable modes.

When determining the aeroacoustics modes propagating in a flow, unstable modes exist for certain types of flows: when an inflection point exists in the velocity profile and when the shear in this point is strong enough. Such modes grow exponentially in space. Up to recently, our numerical simulations have been performed for stable flows. We have tested the behavior of PML in the presence of unstable modes, which usually convert a propagating field in a decaying field. Therefore we do not have a theoretical framework to characterize the behavior of PML in the presence of spatially growing modes but the various conducted numerical tests have shown that our numerical method is still able to select the outgoing solution, even in the presence of instabilities, if the attenuation in the PML is strong enough.

Multiple scattering in a duct

Participant : Jean-François Mercier.

This topis is developed in collaboration with Agnès Maurel (Langevin Institute ESPCI).

The objective of this work, part of the ANR Procomedia, is to develop analytical methods to describe the propagation of acoustic waves in 2D waveguides containing penetrable inclusions. Scatterers of arbitrary shape with a contrast in both density and sound speed are considered. A modal approach is adopted, in which the wave equation is projected onto the transverse modes of the homogeneous guide. For each mode a 1D wave equation is obtained with a source term which characterizes the scatterers and couples modes together. In weak scattering regime (small scatterers or low contrasts or low frequency), the Born approximation is used to solve analytically this family of coupled ODE. This gives an explicit prediction for the scatterered field, in particular the reflection and transmission coefficients are obtained in two cases of interest: periodically or randomly distributed scatterers. In both cases, expressions similar to those in free space (available only for low frequencies) are obtained without frequency limit, thanks to the presence of a shape factor sensitive to the geometry of the scatterers at high frequencies.

Recently the obtained analytical expressions have been exploited to develop a very simple imaging method in a heterogeneous waveguide. Measurements of low-frequency reflection and transmission allow to find the position of the object while the higher frequency measurements give access to the shape and to the physical characteristics of the scatterers. The results are good in the case of low contrast and small scatterers, for which the Born approximation is perfectly valid.

Localization in perturbed periodic metamaterials

Participant : Jean-François Mercier.

This topis is developed in collaboration with Agnès Maurel, Abdelwaheb Ourir (Langevin Institute ESPCI) and Vincent Pagneux (LAUM).

The aim of this work, part of the ANR Procomedia, is to study the propagation of electromagnetic waves through 1D perturbed periodic media. The attenuation length in a medium consisting of alternating materials of optical indices $n_1>0$ and $n_2<0$ (metamaterials) is determined. When such medium is randomly disturbed, the localization properties differ significantly from those obtained in a classical disturbed medium: in the homogeneous case $n_1=n_2$, a random perturbation of the indices induces the Anderson localization with a strong field attenuation. In contrast, in the case $n_1=-n_2$, it was recently shown that the introduction of disorder on the permittivities ${ϵ}_1$ and ${ϵ}_2$ gave rise to an "anomaly", the suppression of the Anderson localization. This anomaly results in a significant increase of the attenuation length $l_N$ for large sample sizes $N$.

We have made two improvements to existing works: simple analytical expressions of the attenuation length have been determined, valid over a wide range of frequencies and of number of layers. In addition we considered realistic metamaterials by taking into account disorder in both the permittivity and the permeability $\mu$. When only the permeability is disturbed (or only the permittivity), our analytical expression can explain the transition to the abnormal behavior when the number of layers increases. Furthermore we show that the anomaly is strongly affected when disturbances in permeability and permittivity are jointly considered: the coupling of the two effects is capable of reseting the usual localization.

Modeling of meta-materials in electromagnetism

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

This topis is developed in collaboration with Eric Chung (Chinese Univ. of Hong Kong) and Xavier Claeys (Paris VI).

Meta-materials can be seen as particular media whose dielectric and/or magnetic constant are negative, at least for a certain range of frequencies. 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 year 2012 saw the completion of Lucas Chesnel PhD thesis. We present below the main results obtained these last three years. 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, 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. 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).

As a second topic, 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 (with X. Claeys). 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). We proposed a numerical approach to approximate the solution which consists in adding some PMLs in the neighbourhood of the corner.

Last, we studied the transmission problem in a purely 3D electromagnetic setting from a theoretical point of view. We proved that the Maxwell problem is well-posed if and only if the two associated scalar problems (with Dirichlet and Neumann boundary conditions) are well-posed. Of course, these scalar problems involves sign-changing coefficients but they can be studied using simple scalar T -coercivity approach.

C. Carvalho started her PhD thesis this fall in the continuation of these works.

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.