## Section: New Results

### Imaging and inverse problems

#### Sampling methods in waveguides

Participants : Laurent Bourgeois, Eric Lunéville, Alexandre Routier.

We have derived a modal formulation of sampling methods (both the Linear Sampling Method and the Factorization Method) in an acoustic waveguide when the obstacle to recover is a set of cracks. This was the subject of the Master internship of Alexandre Routier. For such particular obstacle, we have analyzed the importance of the test function we introduce in the sampling method if we a priori know the type of boundary condition we have on the lips of the crack, both from the theoretical and the numerical point of view. Besides we have proved that in our modal formulation, the Factorization Method is applicable by using the same data as those used in the Linear Sampling Method, which is a novelty as concerns sampling methods in waveguides. The Linear Sampling Method has been extended to the elastic case, for which the usual obstacle is a set of traction free cracks. This makes the choice of the test function crucial, and we have emphasized such fact on numerical examples.

#### Inverse scattering with generalized impedance boundary conditions

Participants : Laurent Bourgeois, 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. 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. The GIBCs are approximate models for thin coatings or corrugated surfaces. During this last year, we have completed the computation of the partial derivatives of the far field with respect to the unknowns, among which is the boundary of the obstacle, and we have implemented many numerical experiments. In particular, we have shown the efficiency of the method consisting in approximating a perfect conductor which is coated with a thin dielectric layer of variable width by a second order GIBC in order to retrieve the obstacle, as well as the refraction index and the width of the layer.

#### Detection of targets using time-reversal

Participants : Maxence Cassier, Patrick Joly, Christophe Hazard.

This topic concerns the studies started last year 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 “focusing” into mathematical terms. We have proposed an energy criterion which can be used in numerical experiments in order to evaluate the quality of the focus. The question is to relate such a criterion with the construction of the above mentioned focusing wave. Works on this topic are in progress.

#### Interior transmission problem

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

This work is a collaboration with Houssem Haddar from the DEFI project. 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 say 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 trivial 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. 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.

#### Flaw identification using elastodynamic topological derivative

Participant : Marc Bonnet.

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). This approach, which allows a qualitative reconstruction of cracks in terms of their location but also their orientation (utilizing the fact that the polarization tensor depends on the normal to the trial small crack), has been implemented within a conventional FEM platform. A standard Newmark unconditionally-stable time-marching scheme is used for simulating data, and for computing the free and adjoint solutions used in the evaluation of the TD field. Extensive 3D time-domain numerical experiments for the detection of cracks buried either in a homogeneous pipe-like structure or on the interface between two sandwiched plates highlight its usefulness and performance. The application of TD to flaw identification has thus far rested upon a heuristic basis. Its justification in limiting situations such as the Born approximation is currently being investigated.

#### Topological derivative in anisotropic elasticity

Participant : Marc Bonnet.

In collaboration with Gabriel Delgado (CMAP, Ecole Polytechnique). This work addresses the current lack of a comprehensive treatment of the topological derivative for anisotropic elasticity, by addressing the case where both the background material and the trial small inhomogeneity have arbitrary anisotropic elastic properties. Accordingly, a formula for the topological derivative of any cost functional defined in terms of regular volume or surface densities depending on the displacement is established, by combining small-inhomogeneity asymptotics and the adjoint solution approach. The latter feature makes the proposed result simple to implement and computationally efficient. Both three-dimensional and plane-strain settings are covered; they differ mostly on details pertaining to the elastic moment tensor. This result achieves a direct generalization to the fully anisotropic case of previously-known formulations for isotropic elasticity. Moreover, the main properties of the EMT, a critical feature of any elastic topological derivative formula, are studied for the fully anisotropic case, generalizing available results on the isotropic case. Finally, further generality is achieved by also deriving the topological derivative of strain energy-based cost functionals, which depend on the displacement gradient. This case, seldom addressed so far, requires a specific, and separate, treatment. Applications of these results include topology optimization of composite structures (a topic currently pursued by G. Delgado) or flaw identification using experimental data from nondestructive testing.

#### 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.

#### Accelerated boundary element method for diffuse optical imaging

Participant : Marc Bonnet.

In collaboration with Simon Arridge and Josias Elisee (University College London, UK). Numerical methods for calculating forward models of light propagation in tissue are extensively used in diffuse optical imaging (DOI). DOI involves a Helmholtz-type PDE with a complex-valued wavenumber. It requires modelling large optical regions whose parameters are known and piecewise constant. The boundary element method (BEM) answers this need and avoids the detailed interior meshing of these regions. The single-level Fast Multipole Method has been applied for solving the DOI governing equation, allowing substantial reduction of computational costs. The enhanced practicability of the BEM in DOI was demonstrated through test examples on single-layer problems, where two-digit reduction factors on solution time are achieved, and on a high-resolution version of a three-layered neonate’s head.

#### High-Velocity Estimates and Inverse Scattering for Quantum N-Body Systems with Stark Effect

Participants : Ricardo Weder, Gerardo Daniel Valencia.

In an $N$–body quantum system with a constant electric field, by inverse scattering, we uniquely reconstruct pair potentials, belonging to the optimal class of short-range potentials and long-range potentials, from the high-velocity limit of the Dollard scattering operator. We give a reconstruction formula with an error term.

#### Small-Energy Analysis for the Matrix Schrödinger Operator on the Half-Line

Participant : Ricardo Weder.

In collaboration with Tuncay Aktosun (University of Texas Arlington) and Martin Klaus (Virginia Tech). The matrix Schrödinger equation with a selfadjoint matrix potential is considered on the half line with the most general selfadjoint boundary condition at the origin. When the matrix potential is integrable and has a first moment, it is shown that the corresponding scattering matrix is continuous at zero energy. An explicit formula is provided for the scattering matrix at zero energy. The small-energy asymptotics are established also for the related Jost matrix, its inverse, and various other quantities relevant to the corresponding direct and inverse scattering problems.