Minisymposia Abstracts

Speaker: Laurent Bourgeois
ENSTA, Paris, France
Title: Imaging an elastic waveguide with the linear sampling method by using the Lamb modes
Abstract: In the context of elasticity in the harmonic regime, we consider the problem of identication of some obstacles in a 2D or 3D waveguide by using the far elds produced by the propagating guided modes (called Lamb modes in 2D). The main issue is that contrary to the acoustic case, it is not possible to expand the displacement eld with respect to the transverse modes as it may be done for the acoustic eld. To cope with this issue, we use Fraser's biorthogonality relationship between two hybrid variables that mix displacement and stress components, which enables us expand those hybrid variables on the transverse modes. We develop the formalism of the Linear Sampling Method in such framework and show through 2D numerical experiments that the identication of Dirichlet obstacles and cracks is ecient with only a few Lamb modes.

Speaker: Bojan Guzina
University of Minnesota, USA
Title: Multi-frequency reconstruction of scatters via the linear sampling method
Abstract: This paper investigates the multi-frequency reconstruction of soundsoft and penetrable obstacles via the linear sampling method involving either far-field or near-field observations of the scattered field. On establishing a suitable approximate solution to the linear sampling equation and making an assumption of continuous frequency sweep, two possible choices for a cumulative multi-frequency indicator function of the scatterer's support are examined. The first alternative, termed the "serial" indicator, is taken as a natural extension of its monochromatic companion in the sense that its computation entails space-frequency (as opposed to space) L2-norm of a solution to the linear sampling equation. Under a set of assumptions that include experimental observations down to zero frequency and compact frequency support of the wavelet used to illuminate the obstacle, this indicator function is further related to its time domain counterpart. As a second possibility, the so-called "parallel" indicator is alternatively proposed as an L2-norm, in the frequency domain, of the monochromatic indicator function. On the basis of a perturbation analysis which demonstrates that the monochromatic solution of the linear sampling equation behaves as O(|k2 − k2*|−m), m ≤ 1 in the neighborhood of an isolated eigenvalue, k2*, of the associated interior (Dirichlet or transmission) problem, it is found that the "serial" indicator is unable to distinguish the interior from the exterior of a scatterer in situations when the prescribed frequency band traverses at least one such eigenvalue. In contrast the "parallel" indicator is, due to its particular structure, shown to be insensitive to the presence of pertinent interior eigenvalues (unknown beforehand and typically belonging to a countable set), and thus to be robust in a generic scattering configuration. A set of numerical results, including both "fine" and "coarse" frequency sampling, is included to illustrate the performance of the competing (multi-frequency) indicator functions, demonstrating behavior that is consistent with the theoretical results. The utility of multi-frequency obstacle illumination is further examined in the context of limited-aperture and noise-polluted observations.

Speaker: Marc Bonnet
Ecole Polytechnique, Paris, France
Title: 3D time-domain topological sensitivity for wave-based crack imaging in elastic solids
Abstract: The concept of topological sensitivity (TS) 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 communication, the TS 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 TS 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), is implemented within a conventional FEM platform (the Cast3m general purpose code developed by the French Atomic Energy Commission (CEA)). 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 TS 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. Tentative links with other qualitative identification approaches are discussed.

Speaker: Michele Piana University of Genova, Italy
Title: The inhomogeneous Lippmann–Schwinger equation and breast cancer detection using microwaves
Abstract: The problem of detecting breast cancer using microwaves is described by means of the inhomogeneous Lippmann–Schwinger equation and an hybrid reconstruction method is formulated for quantitatively evaluating the refractive index of the cancerous tissue.

Speaker: Drossos Gintides
Technical University of Athens, Greece
Title: The inverse transmission eigenvalue problem for spherically symmetric index of refraction
Abstract: In this talk we will present some new results about the inverse transmission eigenvalue problem for a spherically symmetric index of refraction ρ > 0. We will show that if ρ satisfies some integral bound, then it can be uniquely recovered from the transmission eigenvalues for which the corresponding eigenfunctions are spherically symmetric improving existing results. We also consider the associated problem for the Schrödinger operator. Finally, we will present some estimates concerning the existence of complex eigenvalues.

Speaker: Roland Griesmaier
University of Mainz, Germany
Title: Iterative and non-iterative methods for the reconstruction of wires and thin tubes
Abstract: We consider the inverse problem of reconstructing thin wires and tubular conductivity inhomogeneities inside some three-dimensional body from measurements of electrostatic currents and potentials on its boundary. An asymptotic analysis of the forward problem gives an asymptotic representation formula for the perturbation of the electrostatic potential on the boundary of the body caused by such inclusions. This representation is used to derive an iterative Newton-type reconstruction method to recover the position and the shape of the wires or tubes using one measurement of the boundary potential corresponding to some appropriate boundary current. We discuss the performance of this algorithm, present numerical examples, and compare those to results obtained by a non-iterative qualitative reconstruction method developed earlier for this inverse problem.

Speaker: Anne Cossonniere
Ecole Polytechnique, Paris, France
Title: The interior transmission eigenvalue problem: A surface integral equation approach for piecewise constant index
Abstract: We consider the so-called transmission eigenvalue problem in the scalar case for media with piecewise constant index n. The problem is formulated using a surface integral equation approach and Fredholm properties of the obtained problem is analysed in suitable boundary Sobolev spaces for the trace of the solution and the trace of its normal derivative, namely H−1/2 × H−3/2. We shall discuss in particular the cases where n − 1 changes sign. We also show how this formulation can be used to numerically compute the transmission eigenvalues, the main difficulty being the compactness of the underlying boundary operator in usual trace spaces H1/2 × H−1/2. The numerical part is a joint work with F. Collino and M'B Fares. The proposed method also applies to 3D Maxwell. 6

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