Publications

2023

• Méthode de décomposition de domaine pour les problèmes couplés acoustique-élastique, dans le domaine temporel. Application aux explosions sous-marines.
  
  • Nassor Alice
  , 2023. Ce travail étudie les approches globales en temps de décomposition de domaine pour résoudre des problèmes transitoires d'interaction fluide-structure. Afin de déterminer un algorithme optimal, nous étudions dans un premier temps la solvabilité des problèmes élastodynamiques et acoustiques transitoires avec des conditions aux frontières de type Robin et de Neumann. Nous énonçons des résultats de solvabilité, en soulignant les différentes régularités espace-temps des solutions. Nous étudions également la solvabilité du problème couplé élastodynamique-acoustique transitoire. Puis en nous basant sur ces résultats mathématiques, nous proposons ensuite un algorithme itératif global en temps basé sur les conditions aux limites de type Robin pour le problème couplé et prouvons sa convergence.Ces résultats sont ensuite mis en oeuvre pour coupler deux méthodes numériques efficaces. La réponse du fluide en temps discret est obtenue à l'aide d'une approche Z-BEM qui combine (i) une méthode d'éléments de frontière (BEM) accélérée par la méthode des matrices hiérarchiques dans le domaine de Laplace et (ii) une quadrature de convolution. La réponse de la structure est modélisée à l'aide de la méthode des éléments finis. Nous développons de cette manière une méthode numérique de couplage itérative globale en temps à convergence garantie, permettant en outre d'utiliser deux méthodes numériques distinctes de manière non intrusive.Plusieurs améliorations sont ensuite proposées: une méthode d'accélération de convergence est mise en œuvre et une approximation à haute fréquence est proposée pour améliorer l'efficacité de la Z-BEM. On propose ensuite un deuxième couplage itératif global-en-temps basé sur une interface acoustique-acoustique, dont la convergence est également démontrée. Ce couplage permet ensuite d'introduire des effets non linéaires dus au phénomène de cavitation pour préciser le modèle fluide. La Z-BEM est enfin adaptée en utilisant la méthode des images pour permettre la prise en compte d'une surface libre.Cette méthode est appliquées à des problèmes à dynamique rapide de dispersion d'ondes de choc acoustiques par des structures élastiques immergées et permet de simuler des configurations réalistes rencontrées dans l'industrie navale.
• Wave propagation in quasi-periodic media
  
  • Amenoagbadji Pierre
  , 2023. The goal of this thesis is to develop efficient numerical methods for the solution of the time-harmonic wave equation in quasiperiodic media, in the spirit of methods previously developed for periodic media. The goal is to use as in quasiperiodic homogenization the idea that an elliptic PDE with quasiperiodic coefficients can be interpreted as the cut of a higher-dimensional PDE which is elliptically degenerate, but with periodic coefficients. The periodicity property allows to use adapted tools, but the non-elliptic aspect makes the mathematical and numerical analysis of the PDE delicate. One application concerns transmission problems between periodic half-spaces (typically photonic crystals) when (1) the interface does not cut the periodic half-spaces in a direction of periodicity, or (2) when the periodic media have noncommensurate periods along the interface.
• On the use of a tailored fluid-fluid Green's function to predict scattering from two-phase fluid interfaces
  
  • Pacaut Louise
  • Serre Gilles
  • Mercier Jean-François
  • Chaillat Stéphanie
  , 2023, 268 (8), pp.434-443. In naval applications, the precise knowledge of the acoustical behaviour of two-phase fluids is of interest. Indeed, they appear in many configurations as bubbles curtains, cavitation (along propellers and pipes) or two-phase turbulent boundary layers and wakes (gas exhaust). To model this problem, we focus on a Boundary Element/ Boundary Element coupling for low to moderate frequencies. We propose two approaches: (a) solving the Helmholtz equations in each phase by introducing the free field Green's function and (b) determining a tailored Green's function, taking into account the presence of the two-phase fluid. The determination of a tailored Green's function has two main advantages: (i) it allows a reduction of the numerical model, since it contains all the information on the fluid-fluid coupling, in particular the transmission conditions and resonances and (ii) it is sourceindependent and thus it gives directly the answer to any source. Numerical tests on the evaluation of radiated noise are performed to determine the efficiency of the approach based on a tailored Green's function compared to the free-field based one. (10.3397/IN_2023_0076)
  DOI : 10.3397/IN_2023_0076
• Étude de deux problèmes de propagation d’ondes en milieu électromagnétique dispersif : 1) Stabilité en temps long dans un milieu de Drude-Lorentz; 2) Transmission entre une couche de metamateriau et un diélectrique.
  
  • Rosas Martinez Luis Alejandro
  , 2023. Cette thèse traite de deux problèmes indépendants liés aux phénomènes de propagation des ondes dans les milieux dispersifs. Dans la première partie, nous étudions le comportement en temps long des solutions des équations de Maxwell dans des milieux dissipatifs généralisés de Drude-Lorentz. Plus précisément, nous souhaitons quantifier les pertes dans de tels milieux à l'aide du taux de décroissance de l'énergie électromagnétique pour le problème de Cauchy correspondant. Cette première partie est elle-même composée de deux approches. La première, l'approche par fonctions de Lyapunov en fréquence, consiste à obtenir une inégalité différentielle (en temps) pour certaines fonctionnelles de la solution, les fonctions de Lyapunov L(k) où k désigne la fréquence spatiale. Les estimations de stabilité sont ensuite obtenues par l'intégration en temps de l'inégalité différentielle. En développant cette méthode, nous obtenons un résultat de stabilité polynomiale sous des hypothèses de dissipation fortes. La deuxième approche, l'approche modale, exploite les propriétés spectrales de l'opérateur hamiltonien apparaissant dans le problème de Cauchy. Cette dernière approche améliore la première en autorisant des hypothèses de dissipation faibles. Dans la deuxième partie du travail, nous nous intéressons au problème de transmission d'une couche de métamatériau de Drude non dissipatif dans un milieu diélectrique. Dans ce contexte, nous considérons les équations de Maxwell temporelles bidimensionnel en polarisation TM et nous les reformulons en une équation de Schrödinger dont le Hamiltonien, A, est un opérateur autoadjoint non borné. La transformation de Fourier nous permet de travailler avec des Hamiltoniens réduits A(k), k ∈ R. Enfin, nous nous intéressons au spectre ponctuel du Hamiltonien réduit qui est lié aux modes guidés du problème original. Cette étude débouche sur une relation de dispersion dont la difficulté réside dans son caractère hautement non linéaire par rapport au paramètre spectral. Nous prouvons l'existence d'une infinité dénombrable de branches de solutions pour la relation de dispersion : les courbes de dispersion. Nous donnons une analyse précise de ces courbes et mettons en lumière, notamment, l'existence d'ondes guidées correspondant à des palsmons surface.
• An implicit–explicit time discretization for elastic wave propagation problems in plates
  
  • Methenni Hajer
  • Imperiale Alexandre
  • Imperiale Sébastien
  International Journal for Numerical Methods in Engineering, Wiley, 2023, 125, pp.e7393. We propose a new implicit–explicit scheme to address the challenge of modeling wave propagation within thin structures using the time‐domain finite element method. Compared to standard explicit schemes, our approach renders a time marching algorithm with a time step independent of the plate thickness and its associated discretization parameters (mesh step and order of approximation). Relying on the standard three dimensional elastodynamics equations, our strategy can be applied to any type of material, either isotropic or anisotropic, with or without discontinuities in the thickness direction. Upon the assumption of an extruded mesh of the plate‐like geometry, we show that the linear system to be solved at each time step is partially lumped thus efficiently treated. We provide numerical evidence of an adequate convergence behavior, similar to a reference solution obtained using the well‐known leapfrog scheme. Further numerical investigations show significant speed up factors compared to the same reference scheme, proving the efficiency of our approach for the configurations of interest. (10.1002/nme.7393)
  DOI : 10.1002/nme.7393
• A hybridizable discontinuous Galerkin method with characteristic variables for Helmholtz problems
  
  • Modave Axel
  • Chaumont-Frelet Théophile
  Journal of Computational Physics, Elsevier, 2023, 493, pp.112459. A new hybridizable discontinuous Galerkin method, named the CHDG method, is proposed for solving time-harmonic scalar wave propagation problems. This method relies on a standard discontinuous Galerkin scheme with upwind numerical fluxes and high-order polynomial bases. Auxiliary unknowns corresponding to characteristic variables are defined at the interface between the elements, and the physical fields are eliminated to obtain a reduced system. The reduced system can be written as a fixed-point problem that can be solved with stationary iterative schemes. Numerical results with 2D benchmarks are presented to study the performance of the approach. Compared to the standard HDG approach, the properties of the reduced system are improved with CHDG, which is more suited for iterative solution procedures. The condition number of the reduced system is smaller with CHDG than with the standard HDG method. Iterative solution procedures with CGN or GMRES required smaller numbers of iterations with CHDG. (10.1016/j.jcp.2023.112459)
  DOI : 10.1016/j.jcp.2023.112459
• The Half-Space Matching method for elastodynamic scattering problems in unbounded domains
  
  • Bécache Éliane
  • Bonnet-Ben Dhia Anne-Sophie
  • Fliss Sonia
  • Tonnoir Antoine
  Journal of Computational Physics, Elsevier, 2023, pp.112320. In this paper, the Half-Space Matching (HSM) method, first introduced for scalar problems, is extended to elastodynamics, to solve time-harmonic 2D scattering problems, in locally perturbed infinite anisotropic homogeneous media. The HSM formulation couples a variational formulation around the perturbations with Fourier integral representations of the outgoing solution in four overlapping half-spaces. These integral representations involve outgoing plane waves, selected according to their group velocity, and evanescent waves. Numerically, the HSM method consists in a finite element discretization of the HSM formulation, together with an approximation of the Fourier integrals. Numerical results, validating the method, are presented for different materials, isotropic and anisotropic. Comparisons with the Perfectly Matched Layers (PML) method are performed for several anisotropic materials. These results highlight the robustness of the HSM method compared to the sensitivity of the PML method with respect to its parameters. (10.1016/j.jcp.2023.112320)
  DOI : 10.1016/j.jcp.2023.112320
• Scattered wavefield in the stochastic homogenization regime
  
  • Garnier Josselin
  • Giovangigli Laure
  • Goepfert Quentin
  • Millien Pierre
  , 2023. In the context of providing a mathematical framework for the propagation of ultrasound waves in a random multiscale medium, we consider the scattering of classical waves (modeled by a divergence form scalar Helmholtz equation) by a bounded object with a random composite micro-structure embedded in an unbounded homogeneous background medium. Using quantitative stochastic homogenization techniques, we provide asymptotic expansions of the scattered field in the background medium with respect to a scaling parameter describing the spatial random oscillations of the micro-structure. Introducing a boundary layer corrector to compensate the breakdown of stationarity assumptions at the boundary of the scattering medium, we prove quantitative $L^2$- and $H^1$- error estimates for the asymptotic first-order expansion. The theoretical results are supported by numerical experiments.
• DISCRETE HONEYCOMBS, RATIONAL EDGES AND EDGE STATES
  
  • Fefferman Charles Louis
  • Fliss Sonia
  • Weinstein Michael
  Communications on Pure and Applied Mathematics, Wiley, 2023. Consider the tight binding model of graphene, sharply terminated along an edge l parallel to a direction of translational symmetry of the underlying period lattice. We classify such edges l into those of "zigzag type" and those of "armchair type", generalizing the classical zigzag and armchair edges. We prove that zero energy/flat band edge states arise for edges of zigzag type, but never for those of armchair type. We exhibit explicit formulas for flat band edge states when they exist. We produce strong evidence for the existence of dispersive (non flat) edge state curves of nonzero energy for most l. (10.1002/cpa.22141)
  DOI : 10.1002/cpa.22141
• COMPARISON OF BOUNDARY ELEMENT BASED AND PLANE WAVE APPROXIMATION COMPUTATIONS OF TARGET ECHO STRENGTHS
  
  • Pacaut Louise
  • Mercier Jean-François
  • Serre Gilles
  • Chaillat Stéphanie
  , 2023. In naval defence applications, the knowledge of the Target echo strength (TES) of a submarine is of major interest, in order to optimize the scattered pressure that can be measured by an active sonar. In this contribution, we consider a rigid target and compute the TES using two methods: (i) the solution of the Helmholtz equation by reformulating it into a boundary integral equation with either a full space Green's function or a tailored Green's function, and (ii) the use of a plane wave approximation, well-suited for medium to high frequencies. In the first case, the use of a tailored Green's function adapted to the presence of a target reduces the cost of the numerical model. However, an integral equation still has to be solved. It is not the case with the plane wave approximation where the boundary pressure is not calculated but is considered proportional to the incoming wave. Numerical tests are performed to compare the efficiency and accuracy of each approach with respect to available numerical models developed on the submarine model "BeTSSi" -for Benchmark Target Strength Simulation -, under rigid hypothesis.
• Stability of the P1nc-(P0+P1) element
  
  • Jamelot Erell
  • Ciarlet Patrick
  • Sauter Stefan
  , 2024. We solve the Stokes problem numerically. We analyse the P1nc-(P0+P1) mixed finite element method which exhibits interesting numerical features. However, only an incomplete proof of the inf-sup condition is available. We prove here this condition and the stability of the method.
• Fractured meshes
  
  • Averseng Martin
  • Claeys Xavier
  • Hiptmair Ralf
  Finite Elements in Analysis and Design, Elsevier, 2023, 220, pp.103907. This work introduces “generalized meshes”, a type of meshes suited for the discretization of partial differential equations in non-regular geometries. Generalized meshes extend regular simplicial meshes by allowing for overlapping elements and more flexible adjacency relations. They can have several distinct “generalized” vertices (or edges, faces) that occupy the same geometric position. These generalized facets are the natural degrees of freedom for classical conforming spaces of discrete differential forms appearing in finite and boundary element applications. Special attention is devoted to the representation of fractured domains and their boundaries. An algorithm is proposed to construct the so-called virtually inflated mesh, which correspond to a “two-sided” mesh of a fracture. Discrete -differential forms on the virtually inflated mesh are characterized as the trace space of discrete -differential forms in the surrounding volume. (10.1016/j.finel.2022.103907)
  DOI : 10.1016/j.finel.2022.103907
• Eddy-current asymptotics of the Maxwell PMCHWT formulation for multiple bodies and conductivity levels
  
  • Bonnet Marc
  • Demaldent Edouard
  Computers & Mathematics with Applications, Elsevier, 2023, 141, pp.80-101. In eddy current (EC) testing applications, ECs σE (E : electric field, σ: conductivity) are induced in tested metal parts by a low-frequency (LF) source idealized as a closed current loop in air. In the case of highly conducting (HC) parts, a boundary integral equation (BIE) of the first kind under the magneto-quasi-static approximation - which neglects the displacement current - was shown in a previous work to coincide with the leading order of an asymptotic expansion of the Maxwell BIE in a small parameter reflecting both LF and HC assumptions. The main goal of this work is to generalize the latter approach by establishing a unified asymptotic framework that is applicable to configurations that may involve multiple moderately-conducting (σ = O(1)) and non-conducting objects in addition to (possibly multiply-connected) HC objects. Leading-order approximations of the quantities relevant to EC testing, in particular the impedance variation, are then found to be computable from a reduced set of primary unknowns (three on HC objects and two on other objects, instead of four per object for the Maxwell problem). Moreover, when applied to the Maxwell BIE, the scalings suggested by the asymptotic approach stabilize the condition number at low frequencies and remove the low-frequency breakdown effect. The established asymptotic properties are confirmed on 3D numerical examples for simple geometries as well as two EC testing configurations, namely a classical benchmark and a steam generator tube featured in pressurized water reactors of nuclear power plants. (10.1016/j.camwa.2023.03.026)
  DOI : 10.1016/j.camwa.2023.03.026
• Convergence analysis of time-domain PMLs for 2D electromagnetic wave propagation in dispersive waveguides
  
  • Bécache Eliane
  • Kachanovska Maryna
  • Wess Markus
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2023. This work is dedicated to the analysis of generalized perfectly matched layers (PMLs) for 2D electromagnetic wave propagation in dispersive waveguides. Under quite general assumptions on frequency-dependent dielectric permittivity and magnetic permeability we prove convergence estimates in homogeneous waveguides and show that the PML error decreases exponentially with respect to the absorption parameter and the length of the absorbing layer. The optimality of this error estimate is studied both numerically and analytically. Finally, we demonstrate that in the case when the waveguide contains a heterogeneity supported away from the absorbing layer, instabilities may occur, even in the case of the non-dispersive media. Our findings are illustrated by numerical experiments.
• Construction de conditions transparentes pour les guides d’ondes électromagnétiques.
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chesnel Lucas
  • Fliss Sonia
  • Parigaux Aurélien
  , 2023. Nous nous intéressons à la résolution numérique de problèmes de diffraction dans des guides d’ondes électromagnétiques fermés au moyen de méthodes d’Éléments Finis. Pour ce faire, il est nécessaire de tronquer le domaine et de créer une condition transparente adaptée sur la frontière artificielle pour éviter les réflexions parasites. Nous montrons ici comment étendre à ce cas plusieurs techniques développées en acoustique ou en élasticité. Pour écrire la condition transparente, on utilise une décomposition modale. Numériquement, il est nécessaire de tronquer la série correspondante. Pour justifier la convergence de notre approche, on montre que le problème en domaine tronqué est bien posé, et que l’erreur avec la solution exacte décroît exponentiellement avec le rang de troncature. Nous montrerons des résultats numériques obtenus à l’aide de la librairie XliFE++, qui illustrent la résolution des équations de Maxwell 3D en utilisant les éléments finis de Nédélec.
• Propagation des ondes dans les guides partiellement enfouis : résolution du problème direct et imagerie par méthode de type échantillonnage
  
  • Fritsch Jean-François
  , 2023. Ce travail de thèse porte sur le contrôle non destructif de structures élancées partiellement enfouies ou immergées, par exemple un câble d'acier partiellement enfoui dans du béton ou une plaque d'acier partiellement immergée dans du sodium liquide. Ces structures peuvent être vues comme la jonction d'un guide fermé et d'un guide ouvert. Pour effectuer des calculs, nous avons tronqué transversalement la partie ouverte de la structure avec des PML finies. Un guide partiellement enfoui peut alors être traité comme la jonction de deux guides fermés, dont la propagation des ondes dans l'un des guides est régie par une équation impliquant des coefficients complexes liés à la présence des PML. Ce constat nous a amené à commencer par traiter dans un premier temps le cas plus simple de la jonction de deux guides acoustiques fermés. Pour ce cas simple, nous avons proposé une démarche de résolution du problème inverse adaptée aux jonctions de guides d'ondes fermés. Elle repose d'une part sur l'introduction des champs de référence, qui sont les réponses de la structure totale sans défaut à un mode provenant d'un des deux demi-guides, et d'autre part sur l'utilisation de la relation de réciprocité de la fonction de Green de la structure sans défaut. Suivant cette démarche, nous avons obtenu une formulation modale efficace de la LSM qui nous a permis d'identifier des défauts. Dans ce cas simple, nous avons tiré parti de la complétude des modes pour analyser les problèmes direct et inverse. Dans un second temps, nous avons traité le cas d'un guide acoustique partiellement enfoui. La perte de complétude des modes dans le demi-guide tronqué transversalement avec des PML nous a amenée à étudier le problème direct à l'aide de la théorie de Kondratiev. Les outils introduits pour la jonction de deux guides fermés ont été ensuite adaptés à la résolution du problème inverse. Dans un troisième temps, nous avons abordé le cas plus réaliste, mais plus complexe, d'un guide élastique partiellement immergé dans un fluide. Pour ce cas difficile, nous avons développé des outils de simulation adaptés et étendus les outils introduits précédemment pour résoudre le problème inverse.
• Homogenization of thin-structured surfaces for acoustics in the presence of a two-dimensional low Mach potential flow
  
  • Mercier Jean-François
  Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Royal Society, The, 2023, 479 (2274). A surface homogenization method for acoustic waves over thin microstructured surfaces in the presence of a fluid in a potential flow is presented. Sound hard surfaces are considered, the flow is considered two-dimensional and slow and a low Mach approximation is introduced. We consider acoustic waves with a typical wavelength 1 / k much larger than the array spacing h and thickness e . Owing to the small parameter ε = k h , with e / h = O ( 1 ) , a matched asymptotic expansion technique is applied to the low Mach potential wave equation in the frequency domain. A boundary condition is obtained on an equivalent flat wall, which links the acoustic velocity to its normal and tangential derivatives (of the Myers type). The accuracy of the effective model is tested numerically for various periodic shapes and the accuracy of the model in O ( ε 2 ) is validated. (10.1098/rspa.2022.0697)
  DOI : 10.1098/rspa.2022.0697
• DataFlowTasks.jl
  
  • Faria Luiz
  • Févotte François
  • Sivadon Vincent
  • Plagne Laurent
  , 2023.
• Long time behaviour of the solution of Maxwell's equations in dissipative generalized Lorentz materials (I) A frequency dependent Lyapunov function approach
  
  • Cassier Maxence
  • Joly Patrick
  • Martínez Luis Alejandro Rosas
  Zeitschrift für Angewandte Mathematik und Physik = Journal of Applied mathematics and physics = Journal de mathématiques et de physique appliquées, Springer Verlag, 2023, 74 (3), pp.115. It is well-known that electromagnetic dispersive structures such as metamaterials can be modelled by generalized Drude-Lorentz models. The present paper is the first of two articles dedicated to dissipative generalized Drude-Lorentz open structures. We wish to quantify the loss in such media in terms of the long time decay rate of the electromagnetic energy for the corresponding Cauchy problem. By using an approach based on frequency dependent Lyapounov estimates, we show that this decay is polynomial in time. These results extend to an unbounded structure the ones obtained for bounded media in [18] via a quite different method based on the notion of cumulated past history and semi-group theory. A great advantage of the approach developed here is to be less abstract and directly connected to the physics of the system via energy balances. (10.1007/s00033-023-01989-9)
  DOI : 10.1007/s00033-023-01989-9
• A stochastic volume approach based on tailored Green’s functions for airfoil noise prediction at low Mach number
  
  • Trafny Nicolas
  • Serre Gilles
  • Cotté Benjamin
  • Mercier Jean-François
  Journal of Sound and Vibration, Elsevier, 2023, 551, pp.117603. The presence of boundary surfaces in a turbulent flow can result in the enhancement of the radiated acoustic field especially for eddies close to any geometrical singularity. At low Mach number, the best suited prediction methods consist in using an acoustic analogy solved with an integral formulation. In the present study, we focus on the Lighthill's wave equation combined with a tailored Green's function and a new semi-analytical model for the turbulence statistics in the space-frequency domain to extend acoustic analogies to geometries of arbitrary shapes. To validate the model predictions for the leading edge noise and the trailing edge noise, a NACA 0012 airfoil at zero angle of attack is considered and predictions are compared to experimental data. The volume integral approach introduced in this study allows us to study the spatial distribution of the noise sources inside the turbulence volume. In addition, the direct noise radiation associated with the turbulent boundary layer is investigated. (10.1016/j.jsv.2023.117603)
  DOI : 10.1016/j.jsv.2023.117603
• Scaling of Free Subduction on a Sphere
  
  • Ribe Neil M.
  • Chamolly Alexander
  • Gerardi Gianluca
  • Chaillat Stéphanie
  • Li Zhong-Hai
  , 2023. Because Earth's tectonic plates are doubly curved shells, their mechanical behavior during subduction can differ significantly from that of flat plates. We use the boundary-element method (BEM) to study free (gravity-driven) subduction in axisymmetric and 3-D geometry, with a focus on determining the dimensionless parameters that control the dynamics. The axisymmetric model comprises a shell with thickness $h$ and viscosity $\eta_1$ subducting in an isoviscous planet with radius $R_0$ and viscosity $\eta_2$. The angular radius of the trench is $\theta_t$. Scaling analysis based on thin-shell theory reveals two key dimensionless parameters: a `flexural stiffness' $St = (\eta_1/\eta_2)(h/l_b)^3$ that is also relevant for flat plates, and a new `dynamical sphericity number' $\Sigma_D = (l_b/R_0)\cot\theta_t$ that is unique to spherical geometry. Here $l_b$ is the `bending length', or the sum of the lengths of the slab and of the seaward flexural bulge. The definition of $\Sigma_D$ implies that the dynamical effect of sphericity is greater for small plates than for large ones; we call this the `sphericity paradox'. By contrast, the purely geometric effect of sphericity is opposite, i.e. greater for large plates than for small ones. The dynamical and geometrical effects together imply that sphericity significantly influences subduction at all length scales. We confirm the scaling analysis using BEM numerical solutions, which show that the influence of sphericity on the slab sinking speed (up to a few tens of percent) and on the hoop stress (up to a factor of 2-3) are largest for small plates such as the Juan de Fuca, Cocos and Philippine Sea plates. We next study a 3-D model comprising a plate bounded by a ridge and a semicircular trench subducting in a three-layer earth consisting of an upper mantle, a lower mantle and an inviscid core. We examine the linear stability of the shell to longitudinal perturbations corresponding to buckling, and determine a scaling law for the most unstable wavelength that we compare with the observed shapes of northern/western Pacific trenches.
• Diffraction électromagnétique par une couche mince de nanoparticules réparties aléatoirement : développement asymptotique, conditions effectives et simulations.
  
  • Boucart Amandine
  , 2023. Nous considérons le problème de diffraction, en régime harmonique, d’une onde plane électromagnétique par un objet inhomogène recouvert d’une couche très fine de petites particules parfaitement conductrices distribuées aléatoirement. Nous cherchons à quantifier l’effet de cette couche sur le coefficient de réflexion. La taille des particules, leur espacement et l’épaisseur de la couche sont du même ordre mais petites par rapport à la longueur d’onde incidente et les dimensions de l’objet. Deux difficultés apparaissent : (1) Résoudre numériquement les équations de Maxwell dans ce contexte est extrêmement coûteux en terme de taille mémoire et de temps calcul; (2) la répartition des particules n'est pas connue pour un objet donné. Nous allons supposer que c'est une réalisation d'une répartition supposée aléatoire.Pour contourner ces difficultés, nous proposons alors un modèle effectif, à l’aide d’un développement asymptotique multi-échelle de la solution, où la couche de particules est remplacée par une condition aux bords effective, prescrite sur une surface située au-dessus des particules. Les coefficients qui interviennent dans la condition nécessite la résolution de problèmes, dits de cellule, posés un demi-espace recouvert d'une couche de particules, de taille unitaire, réparties aléatoirement.
• Scattering resonances in unbounded transmission problems with sign-changing coefficient
  
  • Carvalho Camille
  • Moitier Zoïs
  IMA Journal of Applied Mathematics, Oxford University Press (OUP), 2023, 88 (2), pp.215-257. It is well-known that classical optical cavities can exhibit localized phenomena associated to scattering resonances, leading to numerical instabilities in approximating the solution. This result can be established via the ``quasimodes to resonances'' argument from the black-box scattering framework. Those localized phenomena concentrate at the inner boundary of the cavity and are called whispering gallery modes. In this paper we investigate scattering resonances for unbounded transmission problems with sign-changing coefficient (corresponding to optical cavities with negative optical properties, for example made of metamaterials). Due to the change of sign of optical properties, previous results cannot be applied directly, and interface phenomena at the metamaterial-dielectric interface (such as the so-called surface plasmons) emerge. We establish the existence of scattering resonances for arbitrary two-dimensional smooth metamaterial cavities. The proof relies on an asymptotic characterization of the resonances, and showing that problems with sign-changing coefficient naturally fit the black box scattering framework. Our asymptotic analysis reveals that, depending on the metamaterial's properties, scattering resonances situated closed to the real axis are associated to surface plasmons. Examples for several metamaterial cavities are provided. (10.1093/imamat/hxad005)
  DOI : 10.1093/imamat/hxad005
• Combined field-only boundary integral equations for PEC electromagnetic scattering problem in spherical geometries
  
  • Faria Luiz
  • Pérez-Arancibia Carlos
  • Turc Catalin
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2023. We analyze the well posedness of certain field-only boundary integral equations (BIE) for frequency domain electromagnetic scattering from perfectly conducting spheres. Starting from the observations that (1) the three components of the scattered electric field $\mathbf{E}^s(\mathbf{x})$ and (2) scalar quantity $\mathbf{E}^s(\mathbf{x})\cdot\mathbf{x}$ are radiative solutions of the Helmholtz equation, novel boundary integral equation formulations of electromagnetic scattering from perfectly conducting obstacles can be derived using Green's identities applied to the aforementioned quantities and the boundary conditions on the surface of the scatterer. The unknowns of these formulations are the normal derivatives of the three components of the scattered electric field and the normal component of the scattered electric field on the surface of the scatterer, and thus these formulations are referred to as field-only BIE. In this paper we use the Combined Field methodology of Burton and Miller within the field-only BIE approach and we derive new boundary integral formulations that feature only Helmholtz boundary integral operators, which we subsequently show to be well posed for all positive frequencies in the case of spherical scatterers. Relying on the spectral properties of Helmholtz boundary integral operators in spherical geometries, we show that the combined field-only boundary integral operators are diagonalizable in the case of spherical geometries and their eigenvalues are non zero for all frequencies. Furthermore, we show that for spherical geometries one of the field-only integral formulations considered in this paper exhibits eigenvalues clustering at one -- a property similar to second kind integral equations.
• Lecture notes on numerical linear algebra
  
  • Bonnet Marc
  , 2023. Course notes, ENSTA Paris (2nd year and M1 level), 2021.