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This paper is concerned with the time-dependent Maxwell's equations for a plane interface between a negative material described by the Drude model and the vacuum, which fill, respectively, two complementary half-spaces. In a first paper, we have constructed a generalized Fourier transform which diagonalizes the Hamiltonian that represents the propagation of transverse electric waves. In this second paper, we use this transform to prove the limiting absorption and limiting amplitude principles, which concern, respectively, the behavior of the resolvent near the continuous spectrum and the long time response of the medium to a time-harmonic source of prescribed frequency. This paper also underlines the existence of an interface resonance which occurs when there exists a particular frequency characterized by a ratio of permittivities and permeabilities equal to $-1$ across the interface. At this frequency, the response of the system to a harmonic forcing term blows up linearly in time. Such a resonance is unusual for wave problem in unbounded domains and corresponds to a non-zero embedded eigenvalue of infinite multiplicity of the underlying operator. This is the time counterpart of the ill-posdness of the corresponding harmonic problem., Comment: 64 pages, 4 figures

This paper is devoted to an acoustic scattering problem in a 2D partially open waveguide, in the sense that the left part of the waveguide is closed, that is with a ‘bounded’ cross-section, while the right part is bounded in the transverse direction by some perfectly matched layers that mimic the situation of an open waveguide, that is with an ‘unbounded’ cross-section. We prove well-posedness of such scattering problem in the Fredholm sense (uniqueness implies existence) and exhibit the asymptotic behaviour of the solution in the longitudinal direction with the help of the Kondratiev approach. Having in mind the numerical computation of the solution, we also propose some transparent boundary conditions in such longitudinal direction, based on Dirichlet-to-Neumann operators. After proving such artificial conditions actually enable us to approximate the exact solution, some numerical experiments illustrate the quality of such approximation.

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A subwavelength metallic particle supports surface plasmons for some negative permittivity values, which are eigenvalues of the self-adjoint quasi-static plasmonic eigenvalue problem (PEP). This work investigates the existence of complex plasmonic resonances for a 2D particle whose boundary is smooth except for one straight corner. These resonances are defined using the multivalued nature of some zeros of the corner dispersion relations and they are shown to be eigenvalues of a PEP that is complex-scaled at the corner, the finite element discretization of which yields a linear generalized eigenvalue problem. Numerical results show that the complex scaling deforms the essential spectrum (associated with the corner) so as to unveil both embedded plasmonic eigenvalues and complex plasmonic resonances. The later are analogous to complex scattering resonances with the local behavior at the corner playing the role of the behavior at infinity. These results corroborate the study of Li and Shipman (J. Integral Equ. Appl. 31(4), 2019), which proved the existence of embedded plasmonic eigenvalues and discussed the construction of particles that exhibit complex plasmonic resonances.

International audience; The purpose of this paper is to investigate the spectral effects of an interface between vacuumand a negative-index material (NIM), that is, a dispersive material whose electric permittivity andmagnetic permeability become negative in some frequency range. We consider here an elementarysituation, namely, 1) the simplest existing model of NIM : the non dissipative Drude model, for whichnegativity occurs at low frequencies; 2) a two-dimensional scalar model derived from the completeMaxwell’s equations; 3) the case of a simple bounded cavity: a polygonal domain partially filled witha portion of Drude material. Because of the frequency dispersion (the permittivity and permeabilitydepend on the frequency), the spectral analysis of such a cavity is unusual since it yields a nonlineareigenvalue problem. Thanks to the use of an additional unknown, we linearize the problem and wepresent a complete description of the spectrum. We show in particular that the interface betweenthe NIM and vacuum is responsible for various resonance phenomena related to various componentsof an essential spectrum.

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International audience; Elastic guided waves are of interest for inspecting structures due to their ability to propagate over long distances. However, waveguides are often embedded in a large solid domain, considered as unbounded. The waves can attenuate strongly along the guide axis due to the energy leakage into the surrounding medium, which reduces the propagating distance. Searching modes with low attenuation becomes necessary. The goal of this work is to propose a numerical approach to compute modes in embedded elastic waveguides (straight and helical structures), combining the so-called semi-analytical finite element method (SAFE) and a perfectly matched layer (PML) method. The application of this work is the non destructive evaluation of multi-wire strands, which constitute cables in civil structures.

International audience; Elastic guided waves are of interest for inspecting structures due to their ability to propagate over long distances. When the guiding structure is embedded into a solid matrix, waveguides are open and waves can be trapped or leaky. With numerical methods, one of the difficulty is that leaky modes attenuate along the axis (complex wavenumber) and exponentially grow along the transverse direction. The goal of this work is to propose a numerical approach for computing modes in open elastic waveguides combining the so-called semi-analytical finite element method (SAFE) and a perfectly matched layer (PML) technique.

International audience; Elastic guided waves are of interest for inspecting structures due to their ability to propagate over long distances. However, guiding structures are often buried in a large domain, considered as unbounded. Waveguides are then open and waves can be trapped or leaky. Analytical tools have been developed to model open solid waveguides but these tools are limited for simple geometries (plates, cylinders). With numerical methods, a difficulty is due to the unbounded geometry. Another issue is due to the presence of leaky modes, which grow exponentially along the transverse directions. The goal of this work is to implement a numerical approach to calculate modes in three dimensional elastic open waveguides, which combines the semi-analytical finite element method and the perfectly matched layers (PML) technique. Both Cartesian and cylindrical PML are implemented.

Elastic guided waves are of interest for inspecting structures due to their ability to propagate over long distances. In several applications, the guiding structure is surrounded by a solid matrix that can be considered as unbounded. The physics of waves in open waveguides significantly differs from closed waveguides. Except for trapped modes, part of the energy is radiated in the surrounding medium, yielding attenuated modes along the axis called leaky modes (wavenumbers are then complex). From a numerical modeling point of view, the main difficulty lies in the unbounded nature of the geometry in the transverse direction. This difficulty is particularly severe due to the unusual behavior of leaky modes: while attenuating along the axis, such modes exponentially grow along the transverse direction. This behavior is seldom mentioned in the literature of elastic waveguides. Yet leaky modes have often been considered for NDT applications, which require waves of low attenuation in order to maximize the inspection range. A numerical approach is proposed for computing modes in open elastic waveguides, in the bidimensional case as a first step. The approach combines a semi-analytical finite element method with perfectly matched layers (PML). The technique of absorbing layers (AL) is also implemented, which consists in using large artificial layers of growing viscoelasticity. Numerical results are compared to analytical results. The efficiency of PML is compared to AL and parametric studies are briefly conducted in order to assess the convergence of both techniques. The physical meaning of leaky modes is also highlighted.

International audience; Elastic guided waves are of interest for inspecting structures due to their ability to propagate over long distances. However, when the guiding structure is embedded into a solid matrix, usually considered as unbounded, waveguides are open and waves can be trapped or leaky. In the latter case, the leakage of energy into the surrounding medium yields attenuation along the axis of the waveguide, which can strongly limit the application of guided wave techniques. Analytical tools have been developed for studying open waveguides but they are limited to simple geometry (plates, cylinders). With numerical methods, one of the difficulty is that leaky modes attenuate along the axis (complex wavenumber) and exponentially grow along the transverse direction. A simple procedure used with existing codes consists in using absorbing layers of artificially growing viscoelasticity, but large layers are often required. The goal of this work is to propose a numerical approach for computing modes in open elastic waveguides combining the so-called semi-analytical finite element method and a perfectly matched layer technique. Two-dimensional problems are considered. Numerical solutions are compared to analytical results. The efficiency of both perfectly matched and absorbing layer techniques is evaluated.

International audience; La méthode des éléments finis, apparue dans les années 50 pour traiter des problèmes de mécanique des structures, a connu depuis lors un développement continu et est utilisée, aujourd’hui, dans tous les domaines d’applications : mécanique, physique, chimie, économie, finance et biologie. Elle est maintenant intégrée à la plupart des logiciels de calcul scientifique, et de nombreux ingénieurs y sont confrontés dans le cadre de leur activité de modélisation et de simulation numérique. Cet ouvrage recouvre un cours d’éléments finis avancé dispensé à l’ENSTA Paris depuis plusieurs années et fait suite à un ouvrage introductif à la méthode des éléments finis paru dans la même collection.Le livre aborde les compléments indispensables à connaître dès lors qu’on aborde des problèmes plus réalistes. En particulier, les questions relatives à l’approximation par éléments finis des problèmes spectraux (éléments propres de problèmes elliptiques), des problèmes transitoires (équation de diffusion, équation des ondes) et des problèmes mixtes (équations de Stokes, équations de Maxwell). À l’instar du premier tome, nous présentons à la fois les bases théoriques des méthodes, les aspects de mise en œuvre et de nombreuses illustrations numériques.

The generalized Fourier transform associated with a selfadjoint Sturm--Liouville operator is a unitary transformation which converts the action of this operator into a simple product by a spectral variable. For a particular operator defined on the half-line and which involves a step function, we show how to extend such a transformation to generalized functions, or distributions, with a suitable definition of such distributions. This extension is based essentially on the fact that, as the usual Fourier transform, this transformation has the property to exchange regularity and decay between the physical and spectral variables.

A mathematical study of the focusing properties of acoustic fields obtained by a time-reversal process is presented. The case of time-harmonic waves propagating in a non-dissipative medium containing sound-soft obstacles is considered. In this context,the so-called D.O.R.T. method ("Decomposition of the Time-Reversal Operator" in french) was recently proposed to achieve selective focusing by computing the eigenelements of the time-reversal operator. The present paper describes a justification of this technique in the framework of the far field model, i.e., for an ideal time-reversal mirror able to reverse the far field of a scattered wave. Both cases of closed and open mirrors, that is, surrounding completely or partially the scatterers, are dealt with. Selective focusing properties are established by an asymptotic analysis for small and distant obstacles.

International audience; It is well known that in the case of a regular domain the solution of the time-harmonic Maxwell's equations allows a discretization by means of nodal finite elements: this is achieved by solving a regularized problem similar to the vector Helmholtz equation. The present paper deals with the same problem in the case of a nonconvex polyhedron. It is shown that a nodal finite element method does not approximate in general the solution to Maxwell's equations, but actually the solution to a neighboring variational problem involving a different function space. Indeed, the solution to Maxwell's equations presents singularities near the edges and corners of the domain that cannot be approximated by Lagrange finite elements. A new method is proposed involving the decomposition of the solution field into a regular part that can be treated numerically by nodal finite elements and a singular part that has to be taken into account explicitly. This singular field method is presented in various situations such as electric and magnetic boundary conditions, inhomogeneous media, and regions with screens. Copyright © 1999 Society for Industrial and Applied Mathematics

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