arXiv subjects

Lucas Chesnel

Publications and source records attributed to Lucas Chesnel.

At least 37 records · Page 2Linked to original sources

arXiv2019

Exact zero transmission during the Fano resonance phenomenon in non symmetric waveguides

We investigate a time-harmonic wave problem in a waveguide. We work at low frequency so that only one mode can propagate. It is known that the scattering matrix exhibits a rapid variation for real frequencies in a vicinity of a complex resonance located close to the real axis. This is the so-called Fano resonance phenomenon. And when the geometry presents certain properties of symmetry, there are two different real frequencies such that we have either $R=0$ or $T=0$, where $R$ and $T$ denote the reflection and transmission coefficients. In this work, we prove that without the assumption of symmetry of the geometry, quite surprisingly, there is always one real frequency for which we have $T=0$. In this situation, all the energy sent in the waveguide is backscattered. However in general, we do not have $R=0$ in the process. We provide numerical results to illustrate our theorems.

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arXiv2019

On quasi-reversibility solutions to the Cauchy problem for the Laplace equation: regularity and error estimates

We are interested in the classical ill-posed Cauchy problem for the Laplace equation. One method to approximate the solution associated with compatible data consists in considering a family of regularized well-posed problems depending on a small parameter $\varepsilon>0$. In this context, in order to prove convergence of finite elements methods, it is necessary to get regularity results of the solutions to these regularized problems which hold uniformly in $\varepsilon$. In the present work, we obtain these results in smooth domains and in 2D polygonal geometries. In presence of corners, due the particular structure of the regularized problems, classical techniques à la Grisvard do not work and instead, we apply the Kondratiev approach. We describe the procedure in detail to keep track of the dependence in $\varepsilon$ in all the estimates. The main originality of this study lies in the fact that the limit problem is ill-posed in any framework.

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arXiv2019

Inside-outside duality with artificial backgrounds

We use the inside-outside duality approach proposed by Kirsch-Lechleiter to identify transmission eigenvalues associated with artificial backgrounds. We prove that for well chosen artificial backgrounds, in particular for the ones with zero index of refraction at the inclusion location, one obtains a necessary and sufficient condition characterizing transmission eigenvalues via the spectrum of the modified far field operator. We also complement the existing literature with a convergence result for the invisible generalized incident field associated with the transmission eigenvalues. This work is based on several of the pioneering works of our dearest colleague and friend Armin Lechleiter and is dedicated to his memory.

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arXiv2018

From zero transmission to trapped modes in waveguides

We consider the time-harmonic scattering wave problem in a 2D waveguide at wavenumber $k$ such that one mode is propagating in the far field. In a first step, for a given $k$, playing with one scattering branch of finite length, we demonstrate how to construct geometries with zero transmission. The main novelty in this result is that the symmetry of the geometry is not needed: the proof relies on the unitary structure of the scattering matrix. Then, in a second step, from a waveguide with zero transmission, we show how to build geometries supporting trapped modes associated with eigenvalues embedded in the continuous spectrum. For this second construction, using the augmented scattering matrix and its unitarity, we play both with the geometry and the wavenumber. Finally, the mathematical analysis is supplemented by numerical illustrations of the results.

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arXiv2018

On well-posedness of time-harmonic problems in an unbounded strip for a thin plate model

We study the propagation of elastic waves in the time-harmonic regime in a waveguide which is unbounded in one direction and bounded in the two other (transverse) directions. We assume that the waveguide is thin in one of these transverse directions, which leads us to consider a Kirchhoff-Love plate model in a locally perturbed 2D strip. For time harmonic scattering problems in unbounded domains, well-posedness does not hold in a classical setting and it is necessary to prescribe the behaviour of the solution at infinity. This is challenging for the model that we consider and constitutes our main contribution. Two types of boundary conditions are considered: either the strip is simply supported or the strip is clamped. The two boundary conditions are treated with two different methods. For the simply supported problem, the analysis is based on a result of Hilbert basis in the transverse section. For the clamped problem, this property does not hold. Instead we adopt the Kondratiev's approach, based on the use of the Fourier transform in the unbounded direction, together with techniques of weighted Sobolev spaces with detached asymptotics. After introducing radiation conditions, the corresponding scattering problems are shown to be well-posed in the Fredholm sense. We also show that the solutions are the physical (outgoing) solutions in the sense of the limiting absorption principle.

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arXiv2018

Invisibility and perfect reflectivity in waveguides with finite length branches

We consider a time-harmonic wave problem, appearing for example in water-waves and in acoustics, in a setting such that the analysis reduces to the study of a 2D waveguide problem with a Neumann boundary condition. The geometry is symmetric with respect to an axis orthogonal to the direction of propagation of waves. Moreover, the waveguide contains one branch of finite length. We analyse the behaviour of the complex scattering coefficients $\mathcal{R}$, $\mathcal{T}$ as the length of the branch increases and we show how to design geometries where non reflectivity ($\mathcal{R}=0$, $|\mathcal{T}|=1$), perfect reflectivity ($|\mathcal{R}|=1$, $\mathcal{T}=0$) or perfect invisibility ($\mathcal{R}=0$, $\mathcal{T}=1$) hold. Numerical experiments illustrate the different results.

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arXiv2018

Simple examples of perfectly invisible and trapped modes in waveguides

We consider the propagation of waves in a waveguide with Neumann boundary conditions. We work at low wavenumber with only one propagating mode in the leads, all the other modes being evanescent. We assume that the waveguide is symmetric with respect to an axis orthogonal to the longitudinal direction and is endowed with a branch of height $L$ whose width coincides with the wavelength of the propagating modes. In this setting, tuning the parameter $L$, we prove the existence of simple geometries where the transmission coefficient is equal to one (perfect invisibility). We also show that these geometries, for possibly different values of $L$, support so called trapped modes (non zero solutions of finite energy of the homogeneous problem) associated with eigenvalues embedded in the continuous spectrum.

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arXiv2018

Trapped modes and reflectionless modes as eigenfunctions of the same spectral problem

We consider the reflection-transmission problem in a waveguide with obstacle. At certain frequencies, for some incident waves, intensity is perfectly transmitted and the reflected field decays exponentially at infinity. In this work, we show that such reflectionless modes can be characterized as eigenfunctions of an original non-selfadjoint spectral problem. In order to select ingoing waves on one side of the obstacle and outgoing waves on the other side, we use complex scalings (or Perfectly Matched Layers) with imaginary parts of different signs. We prove that the real eigenvalues of the obtained spectrum correspond either to trapped modes (or bound states in the continuum) or to reflectionless modes. Interestingly, complex eigenvalues also contain useful information on weak reflection cases. When the geometry has certain symmetries, the new spectral problem enters the class of $\mathcal{PT}$-symmetric problems.

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arXiv2018

Non reflection and perfect reflection via Fano resonance in waveguides

We investigate a time-harmonic wave problem in a waveguide. By means of asymptotic analysis techniques, we justify the so-called Fano resonance phenomenon. More precisely, we show that the scattering matrix considered as a function of a geometrical parameter $\varepsilon$ and of the frequency $λ$ is in general not continuous at a point $(\varepsilon,λ)=(0,λ^0)$ where trapped modes exist. In particular, we prove that for a given $\varepsilon\ne0$ small, the scattering matrix exhibits a rapid change for frequencies varying in a neighbourhood of $λ^0$. We use this property to construct examples of waveguides such that the energy of an incident wave propagating through the structure is perfectly transmitted (non reflection) or perfectly reflected in monomode regime. We provide numerical results to illustrate our theorems.

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arXiv2017

Transmission eigenvalues with artificial background for explicit material index identification

We are interested in the problem of retrieving information on the refractive index $n$ of a penetrable inclusion embedded in a reference medium from farfield data associated with incident plane waves. Our approach relies on the use of transmission eigenvalues (TEs) that carry information on $n$ and that can be determined from the knowledge of the farfield operator $F$. In this note, we explain how to modify $F$ into a farfield operator $F^{art}=F-\tilde{F}$, where $\tilde{F}$ is computed numerically, corresponding to well chosen artificial background and for which the associated TEs provide more accessible information on $n$.

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arXiv2017

Perfect transmission invisibility for waveguides with sound hard walls

We are interested in a time harmonic acoustic problem in a waveguide with locally perturbed sound hard walls. We consider a setting where an observer generates incident plane waves at $-\infty$ and probes the resulting scattered field at $-\infty$ and $+\infty$. Practically, this is equivalent to measure the reflection and transmission coefficients respectively denoted $R$ and $T$. In [9], a technique has been proposed to construct waveguides with smooth walls such that $R=0$ and $|T|=1$ (non reflection). However the approach fails to ensure $T=1$ (perfect transmission without phase shift). In this work, first we establish a result explaining this observation. More precisely, we prove that for wavenumbers smaller than a given bound $k_{\star}$ depending on the geometry, we cannot have $T=1$ so that the observer can detect the presence of the defect if he/she is able to measure the phase at $+\infty$. In particular, if the perturbation is smooth and small (in amplitude and in width), $k_{\star}$ is very close to the threshold wavenumber. Then, in a second step, we change the point of view and, for a given wavenumber, working with singular perturbations of the domain, we show how to obtain $T=1$. In this case, the scattered field is exponentially decaying both at $-\infty$ and $+\infty$. We implement numerically the method to provide examples of such undetectable defects.

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arXiv2017

Oscillating behaviour of the spectrum for a plasmonic problem in a domain with a rounded corner

We investigate the eigenvalue problem $-\text{div}(σ\nabla u) = λu\ (\mathscr{P})$ in a 2D domain $Ω$ divided into two regions $Ω_{\pm}$. We are interested in situations where $σ$ takes positive values on $Ω_{+}$ and negative ones on $Ω_{-}$. Such problems appear in time harmonic electromagnetics in the modeling of plasmonic technologies. In a recent work [15], we highlighted an unusual instability phenomenon for the source term problem associated with $(\mathscr{P})$: for certain configurations, when the interface between the subdomains $Ω_{\pm}$ presents a rounded corner, the solution may depend critically on the value of the rounding parameter. In the present article, we explain this property studying the eigenvalue problem $(\mathscr{P})$. We provide an asymptotic expansion of the eigenvalues and prove error estimates. We establish an oscillatory behaviour of the eigenvalues as the rounding parameter of the corner tends to zero. We end the paper illustrating this phenomenon with numerical experiments.

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arXiv2016

On the use of Perfectly Matched Layers at corners for scattering problems with sign-changing coefficients

We investigate in a $2$D setting the scattering of time-harmonic electromagnetic waves by a plasmonic device, represented as a non dissipative bounded and penetrable obstacle with a negative permittivity. Using the $\textrm{T}$-coercivity approach, we first prove that the problem is well-posed in the classical framework $H^1_{\text{loc}} $ if the negative permittivity does not lie in some critical interval whose definition depends on the shape of the device. When the latter has corners, for values inside the critical interval, unusual strong singularities for the electromagnetic field can appear. In that case, well-posedness is obtained by imposing a radiation condition at the corners to select the outgoing black-hole plasmonic wave, that is the one which carries energy towards the corners. A simple and systematic criterion is given to define what is the outgoing solution. Finally, we propose an original numerical method based on the use of Perfectly Matched Layers at the corners. We emphasize that it is necessary to design an $\textit{ad hoc}$ technique because the field is too singular to be captured with standard finite element methods.

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arXiv2016

Team organization may help swarms of flies to become invisible in closed waveguides

We are interested in a time harmonic acoustic problem in a waveguide containing flies. The flies are modelled by small sound soft obstacles. We explain how they should arrange to become invisible to an observer sending waves from $-\infty$ and measuring the resulting scattered field at the same position. We assume that the flies can control their position and/or their size. Both monomodal and multimodal regimes are considered. On the other hand, we show that any sound soft obstacle (non necessarily small) embedded in the waveguide always produces some non exponentially decaying scattered field at $+\infty$ for wavenumbers smaller than a constant that we explicit. As a consequence, for such wavenumbers, the flies cannot be made completely invisible to an observer equipped with a measurement device located at $+\infty$.

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arXiv2015

Non-scattering wavenumbers and far field invisibility for a finite set of incident/scattering directions

We investigate a time harmonic acoustic scattering problem by a penetrable inclusion with compact support embedded in the free space. We consider cases where an observer can produce incident plane waves and measure the far field pattern of the resulting scattered field only in a finite set of directions. In this context, we say that a wavenumber is a non-scattering wavenumber if the associated relative scattering matrix has a non trivial kernel. Under certain assumptions on the physical coefficients of the inclusion, we show that the non-scattering wavenumbers form a (possibly empty) discrete set. Then, in a second step, for a given real wavenumber and a given domain D, we present a constructive technique to prove that there exist inclusions supported in D for which the corresponding relative scattering matrix is null. These inclusions have the important property to be impossible to detect from far field measurements. The approach leads to a numerical algorithm which is described at the end of the paper and which allows to provide examples of (approximated) invisible inclusions.

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arXiv2015

Spectrum of a diffusion operator with coefficient changing sign over a small inclusion

We study a spectral problem $(\mathscr{P}^δ)$ for a diffusion like equation in a 3D domain $Ω$. The main originality lies in the presence of a parameter $σ^δ$, whose sign changes on $Ω$, in the principal part of the operator we consider. More precisely, $σ^δ$ is positive on $Ω$ except in a small inclusion of size $δ>0$. Because of the sign-change of $σ^δ$, for all $δ>0$ the spectrum of $(\mathscr{P}^δ)$ consists of two sequences converging to $\pm\infty$. However, at the limit $δ=0$, the small inclusion vanishes so that there should only remain positive spectrum for $(\mathscr{P}^δ)$. What happens to the negative spectrum? In this paper, we prove that the positive spectrum of $(\mathscr{P}^δ)$ tends to the spectrum of the problem without the small inclusion. On the other hand, we establish that each negative eigenvalue of $(\mathscr{P}^δ)$ behaves like $δ^{-2}μ$ for some constant $μ<0$. We also show that the eigenfunctions associated with the negative eigenvalues are localized around the small inclusion. We end the article providing 2D numerical experiments illustrating these results.

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arXiv2014

Construction of invisible conductivity perturbations for the point electrode model in electrical impedance tomography

We explain how to build invisible isotropic conductivity perturbations of the unit conductivity in the framework of the point electrode model for two-dimensional electrical impedance tomography. The theoretical approach, based on solving a fixed point problem, is constructive and allows the implementation of an algorithm for approximating the invisible perturbations. The functionality of the method is demonstrated via numerical examples.

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arXiv2014

A numerical approach for the Poisson equation in a planar domain with a small inclusion

We consider the Poisson equation in a domain with a small hole of size $δ$. We present a simple numerical method, based on an asymptotic analysis, which allows to approximate robustly the far field of the solution as $δ$ goes to zero without meshing the small hole. We prove the stability of the scheme and provide error estimates. We end the paper with numerical experiments illustrating the efficiency of the technique.

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