Section: New Results

Imaging and inverse problems

Sampling methods in waveguides

Participants : Laurent Bourgeois, Anne-Claire Egloffe, Sonia Fliss, Mathieu Guenel, Eric Lunéville.

First, we have adapted the modal formulation of sampling methods (Linear Sampling Method and Factorization Method) to the case of a periodic waveguide in the acoustic case. This study is based on the analysis of the far field of scattering solutions in cylindrical waveguides, in particular for the fundamental solution, which enables us to obtain a far field formulation of sampling methods, and then a modal formulation of such methods. The aim of the inverse problem is to retrieve a defect from the scattered fields which correspond to the incident fields formed by the Floquet modes. The corresponding numerical implementation was the subject of the Master internship of Mathieu Guenel who obtained some first promising results.

Secondly, going back to the homogeneous waveguide in the acoustic case, we have started a study of the sampling methods in the time domain. This will be the subject of Anne-Claire Egloffe's post-doc. The aim is to use the modal formulation of the sampling methods at all frequencies and recompose the best possible image of the defect.

The exterior approach to retrieve obstacles

Participant : Laurent Bourgeois.

This theme is a collaboration with Jérémi Dardé from IMT (Toulouse).

We have adapted the exterior approach developped for the Laplace equation to the Stokes system. The aim is to find a fixed Dirichlet obstacle in a fluid which is governed by the Stokes system with the help of boundary measurements. The exterior approach consists in defining a decreasing sequence of domains that converge in some sense to the obstacle. More precisely, such iterative approach is based on a combination of a quasi-reversibility method to update the solution of the ill-posed Cauchy problem outside the obstacle obtained at previous iteration and of a level set method to update the obstacle with the help of the solution obtained at previous iteration. In particular, we have introduced two different mixed formulations of quasi-reversibility for the ill-posed Stokes systems in order to use standard Lagrange finite elements.

Inverse scattering with generalized impedance boundary conditions

Participants : Laurent Bourgeois, Mathieu Chamaillard, Nicolas Chaulet.

This work is a collaboration between POEMS and DEFI projects (more precisely Houssem Haddar) and constitutes the subject of the PhD thesis of N. Chaulet, which was defended on the 27/11/2012. We are concerned with the identification of some obstacle and some Generalized Impedance Boundary Conditions (GIBC) on the boundary of such obstacle from far field measurements generated by the scattering of harmonic incident waves. The GIBCs are approximate models for thin coatings, corrugated surfaces, rough surfaces or imperfectly conducting media.

During this last year, we complemented our previous work in two directions. First, we justified the use of the Factorization method to solve the inverse obstacle problem in the presence of GIBCs. This method gives a uniqueness proof as well as a fast algorithm to reconstruct the obstacle from the knowledge of the far field produced by incident plane waves for all the directions of incidence at a given frequency. We also provided some numerical reconstructions of obstacles for several impedance operators.

Meanwhile, we studied the application of non linear optimization techniques to solve the inverse problem for the $3D$ Maxwell's equations. The main advantage of this type of method is that they can be applied with much less data than the Factorization method. Nevertheless, we had to compute the partial derivatives of the electromagnetic field with respect to the parameters we want to reconstruct. In our case, these parameters are the coefficients that define the impedance operator and the shape of the obstacle. We characterized these derivatives in the case where the GIBC is defined by a second order surface operator. The applicability of such methods has been illustrated by some numerical experiments in dimension 3 in which we reconstructed the shape of the scatterer as well as the coefficients that characterize the impedance operator. As demonstrated in the two dimensional case, we think that the GIBCs could be efficiently used to identify the shape of coated objects as well as the parameters of the coating in the $3D$ Maxwell case.

Linear sampling methods in the time domain

Participant : Simon Marmorat.

This work is developed in collaboration with H. Haddar (DEFI, Inria Saclay) and A. Lechleiter (Bremen University). We are concerned with the inverse problem of reconstructing obstacles from the knowledge of scattered acoustic waves in the time domain. We tackle this problem using a linear sampling method that directly acts on time domain data: this imaging technique yields a picture of the scatterer by solving a linear operator equation involving the measured data for many right-hand sides given by singular solutions to the wave equation. We have illustrated the method on numerical examples and have shown a good behaviour with respect to aperture (the quality of reconstruction is better than in the frequency case in the case of limited aperture) and the ability of simultaneously reconstructing obstacles with different boundary conditions among the Dirichlet, Neumann and Robin-Fourier ones.

Space-time focusing on unknown scatterers

Participants : Maxence Cassier, Patrick Joly, Christophe Hazard.

This topic concerns the studies started two years ago about time-reversal in the context of Maxence Cassier's thesis. The main question is to generate a time-dependent wave that focuses on one given scatterer not only in space, but also in time. Our recent works concern two items. On one hand, we have proposed a way to construct such a focusing wave which does not require an a priori knowledge of the location of the obstacle. This wave is represented by a suitable superposition of the eigenvectors of the so-called time-reversal operator in the frequency domain. Numerical results show the focusing properties of such a wave. On the other hand, we try to understand how to translate the physical idea of ìfocusingî into mathematical terms. We proposed and and implemented energy criterion which can be used in numerical experiments in order to evaluate the quality of the focus.

Asymptotic analysis of the interior transmission eigenvalues related to coated obstacles

Participant : Nicolas Chaulet.

This work is a collaboration with Fioralba Cakoni from the University of Delaware (USA) and Houssem Haddar from the DEFI project. The interior transmission eigenvalues play an important role in the area of inverse scattering problems. These eigenvalues can actually be determined by multi-static far field data. Thus, they could be used for non destructive testing. We focused on the case where the obstacle is a perfectly conducting body coated by some thin dielectric material. We derived and justified the asymptotic expansion of the first interior transmission eigenvalue with respect to the thickness of the coating for the $TM$ electromagnetic polarization. This expansion provided interesting qualitative information about the behavior of these eigenvalues and also gave an explicit formula to compute the thickness of the coating.

Interior transmission problem

Participants : Anne-Sophie Bonnet-Ben Dhia, Lucas Chesnel, Jérémi Firozaly.

This work is a collaboration with F. Cakoni from the University of Delaware (U.S.) and H. Haddar from the DEFI project at Inria Saclay. The interior transmission problem plays an important role in the inverse scattering theory for inhomogeneous media. In particular, it arises when one is interested in the reconstruction of an inclusion embedded in a background medium from multi-static measurements of diffracted fields at a given frequency. Physically, it is important to prove that, for a given frequency, there are no waves which do not scatter. Mathematically, this last property boils down to state that the frequency is not a transmission eigenvalue, that is, an eigenvalue of the interior transmission problem. An important issue is to prove that transmission eigenvalues form at most a discrete set with infinity as the only accumulation point. This is not straightforward because the operator associated with this problem exhibits a sign changing in its principal part and its study is not standard. Using the T-coercivity approach, we proved the discreteness under relatively weak assumptions both for the scalar and Maxwell cases. In particular, the simple technique we proposed allows to treat cases, which were not covered by existing methods, where the difference between the inclusion index and the background index changes sign. Now, we are trying to understand the fundamental links which exist between this problem and the transmission problem between a positive and a negative material. In some configurations, the study of the interior transmission problems leads to consider the operator $\Delta (\sigma \Delta ·):H_0^2\left(\Omega \right)\to H^{-2}\left(\Omega \right)$ where $\Omega$ is the domain and $\sigma$ is a coefficient which changes sign on $\Omega$. During the internship of Jérémy Firozaly, we proved that this operator exhibits properties very different from the operator $\text{div}(\sigma \nabla ·):H_0^1\left(\Omega \right)\to H^{-1}\left(\Omega \right)$.

Flaw identification using elastodynamic topological derivative

Participants : Marc Bonnet, Rémi Cornaggia.

In collaboration with Cédric Bellis (Columbia Univ. USA), Bojan Guzina (Univ. of Minnesota, USA).

The concept of topological derivative (TD) quantifies the perturbation induced to a given cost functional by the nucleation of an infinitesimal flaw in a reference defect-free body, and may serve as a flaw indicator function. In this work, the TD is derived for three-dimensional crack identification exploiting over-determined transient elastodynamic boundary data. This entails in particular the derivation of the relevant polarization tensor, here given for infinitesimal trial cracks in homogeneous or bi-material elastic bodies. Simple and efficient adjoint-state based formulations are used for computational efficiency, allowing to compute the TD field for arbitrarily shaped elastic solids. The latter is then used as an indicator function for the spatial location of the sought crack(s). Current investigations focus on justifying the heuristic underpinning TD-based identification, which consists in deeming regions where the TD is most negative as the likeliest locations of actual flaws and on formulating higher-order topological expansions in the elastodynamic case.

Topological derivative in anisotropic elasticity

Participant : Marc Bonnet.

In collaboration with Gabriel Delgado (CMAP, Ecole Polytechnique).

Following up on previous work on the topological derivative (TD) of displacement-based cost functionals in anisotropic elasticity, a TD formula has been derived for general cost functionals that involve strains (or displacement gradients) rather than displacements. The small-inclusion asymptotics of such cost functionals are quite different than in the previous case, due to the fact that the strain perturbation inside an elastic inclusion remains finite no matter how small the inclusion size. Cost functionals of practical interest having this format include von Mises equivalent stress (often used in plasticity or failure criteria) and energy-norm error functionals for coefficient-identification inverse problems.

Energy functionals for elastic medium reconstruction using transient data

Participant : Marc Bonnet.

In collaboration with Wilkins Aquino (Cornell Univ., USA).

Energy-based misfit cost functionals, known in mechanics as error in constitutive relation (ECR) functionals, are known since a long time to be well suited to (electrostatic, elastic,...) medium reconstruction. In this ongoing work, a transient elastodynamic version of this methodology is developed, with emphasis on its applicability to large time-domain finite element modeling of the forward problem. The formulation involves coupled transient forward and adjoint solutions, a fact which greatly hinders large-scale computations. A computational approach combining an iterative treatment of the coupled problem and the adjoint to the discrete Newmark time-stepping scheme is found to perform well on large FE models, making the time-domain ECR functional a worthwhile tool for medium identification.