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International audience; In this work we present a method for the computation of numerical solutions of 2D homogeneous isotropic elastodynamics equations by solving scalar wave equations. These equations act on the potentials of a Helmholtz decomposition of the displacement field and are decoupled inside the propagation domain. We detail how these equations are coupled at the boundary depending on the nature of the boundary condition satisfied by the displacement field. After presenting the case of rigid boundary conditions, that presents no specific difficulty, we tackle the challenging case of free surface boundary conditions that presents severe stability issues if a straightforward approach is used. We introduce an adequate functional framework as well as a time domain mixed formulation to circumvent these issues. Numerical results confirm the stability of the proposed approach.

International audience; We consider scalar waves in periodic media through the lens of a second-order effective i.e. macroscopic description, and we aim to compute the sensitivities of the germane effective parameters due to topological perturbations of a microscopic unit cell. Specifically, our analysis focuses on the tensorial coefficients in the governing mean-field equation – including both the leading order (i.e. quasi-static) terms, and their second-order companions bearing the effects of incipient wave dispersion. The results demonstrate that the sought sensitivities are computable in terms of (i) three unit-cell solutions used to formulate the unperturbed macroscopic model; (ii) two adoint-field solutions driven by the mass density variation inside the unperturbed unit cell; and (iii) the usual polarization tensor, appearing in the related studies of non-periodic media, that synthesizes the geometric and constitutive features of a point-like perturbation. The proposed developments may be useful toward (a) the design of periodic media to manipulate macroscopic waves via the microstructure-generated effects of dispersion and anisotropy, and (b) sub-wavelength sensing of periodic defects or perturbations.

This paper presents a computational approach for finding the optimal shapes of peristaltic pumps transporting rigid particles in Stokes flow. In particular, we consider shapes that minimize the rate of energy dissipation while pumping a prescribed volume of fluid, number of particles and/or distance traversed by the particles over a set time period. Our approach relies on a recently developed fast and accurate boundary integral solver for simulating multiphase flows through periodic geometries of arbitrary shapes. In order to fully capitalize on the dimensionality reduction feature of the boundary integral methods, shape sensitivities must ideally involve evaluating the physical variables on the particle or pump boundaries only. We show that this can indeed be accomplished owing to the linearity of Stokes flow. The forward problem solves for the particle motion in a slip-driven pipe flow while the adjoint problems in our construction solve quasi-static Dirichlet boundary value problems backwards in time, retracing the particle evolution. The shape sensitivities simply depend on the solution of one forward and one adjoint (for each shape functional) problems. We validate these analytic shape derivative formulas by comparing against finite-difference based gradients and present several examples showcasing optimal pump shapes under various constraints., Comment: 24 pages, 8 figures

International audience; This paper introduces a novel boundary integral equation (BIE) method for the numerical solution of problems of planewave scattering by periodic line arrays of two-dimensional penetrable obstacles. Our approach is built upon a direct BIE formulation that leverages the simplicity of the free-space Green function but in turn entails evaluation of integrals over the unit-cell boundaries.Such integrals are here treated via the window Green function method. The windowing approximation together with a finite-rank operator correction—used to properly impose the Rayleigh radiation condition—yield a robust second-kind BIE that produces superalgebraically convergent solutions throughout the spectrum,including at the challenging Rayleigh–Wood anomalies. The corrected windowed BIE can be discretized by means of off-the-shelf Nyström and boundary element methods, and it leads to linear systems suitable for iterative linear algebra solvers as well as standard fast matrix–vector product algorithms. A variety of numerical examples demonstrate the accuracy and robustness of the proposed methodology.

In this paper, we study some techniques for solving numerically magnetostatic systems. In particular, we revisit existing classical variational methods and propose new numerical methods. The numerical approximation is either based on the classical edge finite elements, or on continuous Lagrange finite elements. For the first type of discretization, we rely on the design of a new, mixed variational formulation that is obtained with the help of T-coercivity. The numerical method can be related to a perturbed approach for solving mixed problems in electromagnetism. For the second type of discretization, we rely on on an augmented variational formulation obtained with the help of the Weighted Regularization Method.

In this manuscript, we study solutions to resonant Maxwell's equations in heterogeneous plasmas. We concentrate on the phenomenon of upper-hybrid heating, which occurs in a localized region where electromagnetic waves transfer energy to the particles. In the 2D case, it can be modelled mathematically by the partial differential equation − div (α∇u) − ω 2 u = 0, where the coefficient α is a smooth, sign-changing, real-valued function. Since the locus of the sign change is located within the plasma, the equation is non-elliptic, and degenerate. On the other hand, using the limiting absorption principle, one can build a family of elliptic equations that approximate the degenerate equation. Then, a natural question is to relate the solution of the degenerate equation, if it exists, to the family of solutions of the elliptic equations. For that, we assume that the family of solutions converges to a limit, which can be split into a regular part and a singular part, and that this limiting absorption solution is governed by the non-elliptic equation introduced above. One of the difficulties lies in the definition of appropriate norms and function spaces in order to be able to study the non-elliptic equation and its solutions. As a starting point, we revisit a prior work [13] on this topic by A. Nicolopoulos, M. Campos Pinto, B. Després and P. Ciarlet Jr., who proposed a variational formulation for the plasma heating problem. We improve the results they obtained, in particular by establishing existence and uniqueness of the solution, by making a different choice of function spaces. Also, we propose a series a numerical tests, comparing the numerical results of Nicolopoulos et al to those obtained with our numerical method, for which we observe better convergence.

The aim of this work is to participate to the popularization of methods for the resolution of wave propagation problems based on integral equations formulations by developping a posteriori error estimates in the context of autoadaptive mesh refinement strategies. The development of such estimates is difficult because of the non-locality of the norms associated to the Sobolev spaces and of the involved integral operators. Estimates from the literature are extended in the case of the propagation of an acoustic wave. The proofs of quasi-optimal convergence of the autoadaptive algorithms are established. We then introduce a new approach with respect to the literature which is based on a new norm-localization technique based on the use of a well-chosen Λ operator and not on inverse inequalities as it was the case previously.We then establish new a posteriori error estimates which are reliable, efficient, local and asymptotically exact with respect to the Galerkin norm of the error. We give a method for the construction of such estimates. Numerical applications on 2D and 3D geometries confirm the asymptotic exactness and the optimality of the autoadaptive algorithm.These estimates are extended in the case of the propagation of an electromagnetic wave. More precisely, we are interested in the EFIE. We suggest generalization of the estimates of the literature. A proof for quasi-optimal convergence is given for an estimate based on a localization of the norm of the residual. The principle of Λ is used to construct the first reliable, efficient, local error estimate for this equation. We give a second forme which is eventually theoretically asymptotically exact., Dans cette thèse, on se propose de participer à la popularisation des méthodes de résolution de problèmes de propagation d'onde basées sur des formulations intégrales en fournissant des indicateurs d'erreur a posteriori utilisable dans le cadre d'algorithmes de raffinement autoadaptatif. Le développement de tels indicateurs est complexe du fait de la non-localité des normes associées aux espaces de Sobolev et des opérateurs entrant en jeu. Des indicateurs de la littérature sont étendus au cas de la propagation d'une onde acoustique. On étend les preuves de convergence quasi-optimale (de la littérature) des algorithmes autoadaptatifs associés dans ce cas. On propose alors une nouvelle approche par rapport à la littérature qui consiste à utiliser une technique de localisation des normes, non pas basée sur des inégalités inverses, mais sur l'utilisation d'un opérateur Λ de localisation bien choisi.On peut alors construire des indicateurs d'erreur a posteriori fiables, efficaces, locaux et asymptotiquement exacts par rapport à la norme de Galerkin de l'erreur. On donne ensuite une méthode pour la construction de tels indicateurs. Les applications numériques sur des géométries 2D et 3D confirment l'exactitude asymptotique ainsi que l'optimalité du guidage de l'algorithme autoadaptatif.On étend ensuite ces indicateurs au cas de la propagation d'une onde électromagnétique. Plus précisément, on s'intéresse au cas de l'EFIE. On propose des généralisations des indicateurs de la littérature. On effectue la preuve de convergence quasi-optimale dans le cas d'un indicateur basé sur une localisation de la norme du résidu. On utilise le principe du Λ pour obtenir le premier indicateur d'erreur fiable, efficace et local pour cette équation. On en propose une seconde forme qui est également, théoriquement asymptotiquement exacte.

This paper presents a numerical framework for designing diffuse fields in rooms of any shape and size, driven at arbitrary frequencies. That is, we aim at overcoming the Schroeder frequency limit for generating diffuse fields in an enclosed space. We formulate the problem as a Tikhonov regularized inverse problem and propose a lowrank approximation of the spatial correlation that results in significant computational gains. Our approximation is applicable to arbitrary sets of target points and allows us to produce an optimal design at a computational cost that grows only linearly with the (potentially large) number of target points. We demonstrate the feasibility of our approach through numerical examples where we approximate diffuse fields at frequencies well below the Schroeder limit.

The well-posedness of the linear evolution problem governing the transient scattering of acoustic waves by an elastic obstacle is investigated. After using linear superposition in the acoustic domain, the analysis focuses on an equivalent causal transmission problem. The proposed analysis provides existence and uniqueness results, as well as continuous data-to-solution maps. Solvability results are established for three cases, which differ by the assumed regularity in space on the transmission data on the acoustic-elastic interface Γ. The first two results consider data with "standard" H −1/2 (Γ) and improved H 1/2 (Γ) regularity in space, respectively, and are established using the Hille-Yosida theorem and energy identities. The third result assumes data with L 2 (Γ) regularity in space and follows by Sobolev interpolation. Obtaining the latter result was motivated by the key role it plays (in a separate study) in the justification of an iterative numerical solution method based on domain decomposition. A numerical example is presented to emphasize the latter point.

In this work we study time-domain sound-soft scattering by small circles. Our goal is to derive an asymptotic model for this problem valid when the size of the particles tends to zero. We present a systematic approach to constructing such models, based on a well-chosen Galerkin discretization of a boundary integral equation. The convergence of the method is achieved by decreasing the asymptotic parameter rather than increasing the number of basis functions. For the case of circular obstacles, we prove the second-order convergence of the field error with respect to the particle size. Our findings are illustrated with numerical experiments.

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., 33 pages, 1 figure

We study a transmission problem for the time harmonic Maxwell's equations between a classical positive material and a so-called negative index material in which both the permittivity $\varepsilon$ and the permeability $\mu$ take negative values. Additionally, we assume that the interface between the two domains is smooth everywhere except at a point where it coincides locally with a conical tip. In this context, it is known that for certain critical values of the contrasts in $\varepsilon$ and in $\mu$, the corresponding scalar operators are not of Fredholm type in the usual $H^1$ spaces. In this work, we show that in these situations, the Maxwell's equations are not well-posed in the classical $L^2$ framework due to existence of hypersingular fields which are of infinite energy at the tip. By combining the $\mathrm{T}$-coercivity approach and the Kondratiev theory, we explain how to construct new functional frameworks to recover well-posedness of the Maxwell's problem. We also explain how to select the setting which is consistent with the limiting absorption principle. From a technical point of view, the fields as well as their curls decompose as the sum of an explicit singular part, related to the black hole singularities of the scalar operators, and a smooth part belonging to some weighted spaces. The analysis we propose rely in particular on the proof of new key results of scalar and vector potential representations of singular fields.

International audience; We consider the time-harmonic Maxwell's equations in anisotropic media. The problem to be solved is an approximation of the diffraction problem, or scattering from bounded objects, that is usually set in some exterior domain in $\mathbb{R}^3$. We consider perfectly conducting objects, so the equations are supplemented with a Dirichlet boundary condition on those objects, and we truncate the exterior domain by imposing an impedance condition on an artificial boundary, to model an approximate radiation condition. The resulting problem is then posed in a bounded domain, with Dirichlet and impedance boundary conditions. In this work, we focus on the mathematical meaning of the impedance condition, precisely in which function space it holds. This relies on a careful analysis of the regularity of the traces of electromagnetic fields, which can be derived thanks to the study of the regularity of the solution to second-order surface PDEs. Then, we prove well-posedness of the model, and we determine the a priori regularity of the fields in the domain and on the boundaries, depending on the geometry, the coefficients and the data. Finally, the discretization of the formulations is presented, with an approximation based on edge finite elements. Error estimates are derived, and a benchmark is provided to discuss those estimates.

International audience; 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.

This work is devoted to the resolution of the Helmholtz equation $-(\mu u')' - \rho \omega^2 u = f$ in a one-dimensional unbounded medium. We assume the coefficients of this equation to be local perturbations of quasiperiodic functions, namely the traces along a particular line of higher-dimensional periodic functions. Using the definition of quasiperiodicity, the problem is lifted onto a higher-dimensional problem with periodic coefficients. The periodicity of the augmented problem allows us to extend the ideas of the DtN-based method developed for the elliptic case. However, the associated mathematical and numerical analysis of the method are more delicate because the augmented PDE is degenerate, in the sense that the principal part of its operator is no longer elliptic. We also study the numerical resolution of this PDE, which relies on the resolution of Dirichlet cell problems as well as a constrained Riccati equation., Comment: 45 pages, 15 figures

This paper introduces general methodologies for constructing closed-form solutions to several important partial differential equations (PDEs) with polynomial right-hand sides in two and three spatial dimensions. The covered equations include the isotropic and anisotropic Poisson, Helmholtz, Stokes, and elastostatic equations, as well as the time-harmonic linear elastodynamic and Maxwell equations. Polynomial solutions have recently regained significance in the development of numerical techniques for evaluating volume integral operators and have potential applications in certain kinds of Trefftz finite element methods. Our approach to all of these PDEs relates the particular solution to polynomial solutions of the Poisson and Helmholtz polynomial particular solutions, solutions that can in turn be obtained, respectively, from expansions using homogeneous polynomials and the Neumann series expansion of the operator $(k^2+\Delta)^{-1}$. No matrix inversion is required to compute the solution. The method naturally incorporates divergence constraints on the solution, such as in the case of Maxwell and Stokes flow equations. This work is accompanied by a freely available Julia library, \texttt{PolynomialSolutions.jl}, which implements the proposed methodology in a non-symbolic format and efficiently constructs and provides access to rapid evaluation of the desired solution.

International audience; We analyse $a\ posteriori$ error estimates for the discretization of the neutron diffusion equations with mixed finite elements. We provide guaranteed and locally efficient estimators on a base block equation, the one-group neutron diffusion equation. We pay particular attention to AMR strategies on Cartesian meshes, since such structures are common for nuclear reactor core applications. We exhibit a robust marker strategy for this specific constraint, the $direction\ marker$ strategy. The approach is further extended to a Domain Decomposition Method, the so-called DD+$L^2$ jumps method, as well as to the multigroup neutron diffusion equation.

In this work, we present a new numerical method for solving the scalar transmission problem with sign-changing coefficients. In electromagnetism, such a transmission problem can occur if the domain of interest is made of a classical dielectric material and a metal or a metamaterial, with for instance an electric permittivity that is strictly negative in the metal or metamaterial. The method is based on an optimal control reformulation of the problem. Contrary to other existing approaches, the convergence of this method is proved without any restrictive condition. In particular, no condition is imposed on the a priori regularity of the solution to the problem, and no condition is imposed on the meshes, other than that they fit with the interface between the two media. Our results are illustrated by some (2D) numerical experiments.

International audience; We present a computational modeling approach aimed at investigating the effect of fiber grid reinforcement on crack opening displacement and fatigue crack propagation. Grid reinforcements are modeled using elastic membrane finite elements, while the cracked concrete is treated using a symmetric boundary element method (BEM), which in particular allows easy geometrical modelling and meshing of cracks. The BEM is accelerated by the fast multipole method, allowing the handling of potentially large BEM models entailed by three-dimensional configurations hosting multiple cracks. Fatigue crack growth is modelled using the Paris law. The proposed computational approach is first verified on a reinforced cracked beam, and then applied to a three-dimensional configuration featuring a grid-reinforced asphalt pavement.

International audience; We consider a time-harmonic acoustic scattering problem in a 2D infinite waveguide with walls covered with an absorbing material, in the presence of a mean flow assumed uniform far from the source. To make this problem suitable for a finite element analysis, the infinite domain is truncated. This paper concerns the derivation of a non-reflecting boundary condition on the artificial boundary by means of a Dirichlet-to-Neumann (DtN) map based on a modal decomposition. Compared with the hard-walled guide case, several difficulties are raised by the presence of both the liner and the mean flow. In particular, acoustic modes are no longer orthogonal and behave asymptotically like the modes of a soft-walled guide. However, an accurate approximation of the DtN map can be derived using some bi-orthogonality relations, valid asymptotically for high-order modes. Numerical validations show the efficiency of the method. The influence of the liner with or without mean flow is illustrated.

International audience; 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 formulae for flat-band edge states when they exist. We produce strong evidence for the existence of dispersive (nonflat) edge state curves of nonzero energy for most l.

Classically, the well-posedness of variational formulations of mixed linear problems is achieved through the inf-sup condition. In this note, we propose an alternative framework to study such problems by using the T-coercivity approach. This is a constructive approach that leads to the design of suitable approximations in a simple way. In general, the derivation of the uniform discrete inf-sup condition for the approximate problems stems straightforwardly from the study of the original problem. To support our view, we solve a series of classical mixed problems with the T-coercivity approach. Among others, the celebrated Fortin Lemma appears naturally in the numerical analysis of the approximate problems.; Classiquement, le caractère bien posé des formulations variationnelles de problèmes linéaires mixtes est obtenu à l'aide de la condition inf-sup. Dans cette note, nous proposons un cadre alternatif pour étudier de tels problèmes en utilisant la notion de T-coercivité. Il s'agit d'une approche constructive qui permet en outre de concevoir simplement des approximations numériques adaptées car la dérivation de la condition inf-sup discrète uniforme découle en général directement de l'étude du problème continu. Pour appuyer notre propos, nous résolvons une série de problèmes mixtes classiques grâce à la notion de Tcoercivité. Entre autres, le lemme de Fortin apparaît naturellement dans l'analyse numérique des problèmes discrets.

National audience; On s'intéresse au problème de rayonnement acoustique d'une source dans une conduite cylindrique axisymétrique infinie dont la paroi est recouverte d'un matériau absorbant en présence d'un écoulement uniforme. Dans le but d'utiliser la méthode des éléments finis, le domaine infini doit être tronqué par une frontière artificielle sur laquelle une condition limite transparente est introduite. La méthode exposée dans ce travail consiste à écrire la condition transparente sous la forme d'un opérateur Dirichlet to Neumann (DtN) obtenu par la généralisation de travaux réalisés dans le cas d'un guide 2D infini. Cet opérateur est basé sur une décomposition modale qu'il est facile d'expliciter si le guide est rigide: le problème aux valeurs propres associé est alors auto-adjoint. En présence d'un matériau absorbant, modélisé par une impédance locale Z sur la paroi, des difficultés apparaissent car le problème aux valeurs propres n'est plus auto-adjoint. Néanmoins, une relation de bi-orthogonalité existe et permet de définir l'opérateur DtN. Dans le cas plus général d'un guide traité en présence d'un écoulement uniforme, le problème aux valeurs propres est encore non auto-adjoint et il n'existe plus de relation de bi-orthogonalité exacte pour la pression seule. Toutefois, nous montrons qu'il est possible d'écrire une relation d'orthogonalité valide asymptotiquement qui permet d'exprimer une bonne approximation de l'opérateur DtN cherché. La méthode est présentée dans le cas académique d'un guide cylindrique axisymétrique rectiligne, mais elle s'étend à des configurations plus complexes à condition que la partie étudiée soit comprise entre deux tronçons de guide rectiligne dans lesquels l'écoulement est uniforme. En particulier, le cas d'un guide non rectiligne en présence d'un écoulement potentiel sera présenté comme illustration.

International audience; A very general class of models for describing the propagation of waves in dispersive electromagnetic media is provided by generalized Lorentz models. In this work, we study the long time behaviour of the solutions of the dissipative version of these models.

International audience; This work is dedicated to the construction and analysis of high-order transparentboundary conditions for the weighted wave equation on a fractal tree, which models sound propaga-tion inside human lungs. This article follows the works [9, 6], aimed at the analysis and numerical treatment of the model, as well as the construction of low-order and exact discrete boundary conditions. The method suggested in the present work is based on the truncation of the meromorphicseries that represents the symbol of the Dirichlet-to-Neumann operator, in the spirit of the absorbingboundary conditions of B. Engquist and A. Majda. We analyze its stability and convergence, as wellas present computational aspects of the method. Numerical results confirm theoretical findings

In discontinuous Galerkin spectral element methods (DGSEM), the two most common approaches to numerically integrate the terms of the weak form are either using Gauss-Legendre or Gauss-Lobatto quadratures. The former yields more accurate results but at a higher computational cost, so that a priori it is not clear whether one approach is more efficient that the other. In this paper, it is shown (theoretically for a particular case and numerically for the general case) that using Gauss-Lobatto quadrature for the convection matrix actually introduces a negligible error. In contrast, using Gauss-Lobatto quadratures for the evaluation of the jump term in the element faces introduces a sizeable error. This leads to the proposal of a new DG approach, where the convection matrix is evaluated using Gauss-Lobatto quadratures, whereas the face mass matrices are integrated using Gauss-Legendre quadratures. For elements with constant Jacobian and constant coefficients, a formal proof shows that no numerical integration error is actually introduced in the evaluation of the residual, even though both the mass and the convection matrices are not computed exactly with Gauss-Lobatto quadratures. For elements with non-constant Jacobian and/or non-constant coefficients, the impact of numerical integration error on the overall error is evaluated through a series of numerical tests, showing that this is also negligible. In addition, the computational cost associated to the matrix-vector products required to evaluate the residual is evaluated precisely for the different cases considered. The proposed approach is particularly attractive in the most general case, since the use of Gauss-Lobatto quadratures significantly speeds-up the evaluation of the residual.

International audience; Une méthode itérative convergente pour le couplage FEM-BEM élastodynamique-acoustique global en temps, permettant de traiter un problème d’interaction fluide-structure est proposée. Les équations structures sont résolues en éléments finis (FEM), tandis que la partie fluide est traitée par éléments de frontière (BEM), formulée en temps discrets par Convolution Quadrature method (CQM). Le couplage présenté se base sur la formulation de conditions de transmission de Robin. La convergence est démontrée et illustrée. Un deuxième couplage itératif en temps à convergence garantie est proposé.

International audience; Ce travail consiste à vérifier numériquement des stratégies de contrôle de séismes par injection de fluide dans le sol. Nous étudions les capacités des méthodes d’éléments de frontière (BEMs) à simuler des séquences de glissements sismiques et asismiques en géomécanique. Un algorithme basé sur les BEMs accélérées par FFT est considéré, validé, et des résultats sont présentés pour un problème simple de mécanique des failles. Les challenges en lien avec l’extension des BEMs accélérées pour incorporer les couplages multi-physiques en jeu sont discutés.

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

International audience; L’interaction entre un écoulement turbulent et un obstacle rigide produit un rayonnement acoustique large bande qui peut avoir un impact significatif dans de nombreuses problématiques industrielles. Les méthodes de prédiction existantes ne sont pour la plupart pas adaptées au contexte des applications navales qui impose trois contraintes : les systèmes considérés sont de formes complexes et les écoulements sont généralement à très bas nombre de Mach et à haut Reynolds. Dans ces conditions, les méthodes de calcul directs du bruit qui reposent sur l’utilisation d’une simulation compressible de l’écoulement sont trop coûteuses. D’autres approches doivent être utilisées. Basées sur les analogies acoustiques, elles reposent sur l’idée de séparer les mécanismes de production et les mécanismes de propagation du bruit. Dans cette étude, l’équation d’onde de Lighthill est résolue grâce à une fonction de Green adaptée qui peut être analytique, pour des géométries canoniques, ou déterminée numériquement grâce à la méthode des éléments de frontière. De plus, un modèle pour l’interspectre des fluctuations turbulentes de vitesse, exprimé en espace-fréquence est introduit. Il peut être construit soit à partir d’une estimation des paramètres de couche limite, soit à partir d’une simulation de l’écoulement moyen. Les prédictions obtenues pour le bruit de bord d’attaque et le bruit de bord de fuite sont validées grâce à des mesures effectuées sur un profil NACA 0012, en air.

This paper is focused on the Morozov’s principle applied to an abstract data assimilation framework, with particular attention to three simple examples: the data assimilation problem for the Laplace equation, the Cauchy problem for the Laplace equation and the data assimilation problem for the heat equation. Those ill-posed problems are regularized with the help of a mixed type formulation which is proved to be equivalent to a Tikhonov regularization applied to a well-chosen operator. The main issue is that such operator may not have a dense range, which makes it necessary to extend well-known results related to the Morozov’s choice of the regularization parameter to that unusual situation. The solution which satisfies the Morozov’s principle is computed with the help of the duality in optimization, possibly by forcing the solution to satisfy given a priori constraints. Some numerical results in two dimensions are proposed in the case of the data assimilation problem for the Laplace equation.

In this work, we construct an efficient numerical method to solve 3D Maxwell’s equations in coaxial cables. Our strategy is based upon a hybrid explicit-implicit time discretization combined with edge elements on prisms and numerical quadrature. One of the objectives is to validate numerically generalized telegrapher’s models that are used to simplify the 3D Maxwell equations into a 1D problem. This is the object of the second part of the article.

International audience; 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.

The main subject of this thesis is the study of time-harmonic electromagnetic waves in a heterogeneous medium composed of a dielectric and a negative material (i.e. with a negative dielectric permittivity ε and/or a negative magnetic permeability μ) which are separated by an interface with a conical tip. Because of the sign-change in ε and/or μ, the Maxwell’s equations can be ill-posed in the classical L2 −frameworks. On the other hand, we know that when the two associated scalar problems, involving respectively ε and μ, are well-posed in H1, the Maxwell’s equations are well-posed. By combining the T-coercivity approach with the Mellin analysis in weighted Sobolev spaces, we present, in the first part of this work, a detailed study of these scalar problems. We prove that for each of them, the well-posedeness in H1 is lost iff the associated contrast belong to some critical set called the critical interval. These intervals correspond to the sets of negative contrasts for which propagating singularities, also known as black hole waves, appear at the tip. Contrary to the case of a 2D corner, for a 3D tip, several black hole waves can exist. Explicit expressions of these critical intervals are obtained for the particular case of circular conical tips. For critical contrasts, using the Mandelstam radiation principle, we construct functional frameworks in which well-posedness of the scalar problems is restored. The physically relevant framework is selected by a limiting absorption principle. In the process, we present a new numerical strategy for 2D/3D scalar problems in the non-critical case. This approach, presented in the second part of this work, contrary to existing ones, does not require additional assumptions on the mesh near the interface. The third part of the thesis concerns Maxwell’s equations with one or two critical coefficients. By using new results of vector potentials in weighted Sobolev spaces, we explain how to construct new functional frameworks for the electric and magnetic problems, directly related to the ones obtained for the two associated scalar problems. If one uses the setting that respects the limiting absorption principle for the scalar problems, then the settings provided for the electric and magnetic problems are also coherent with the limiting absorption principle. Finally, the last part is devoted to the homogenization process for time-harmonic Maxwell’s equations and associated scalar problems in a 3D domain that contains a periodic distribution of inclusions made of negative material. Using the T-coercivity approach, we obtain conditions on the contrasts such that the homogenization results is possible for both the scalar and the vector problems. Interestingly, we show that the homogenized matrices associated with the limit problems are either positive definite or negative definite., Le sujet principal de cette thèse est l’étude de la propagation des ondes électromagnétiques, en régime harmonique, dans un milieu hétérogène composé d’un diélectrique et d’un matériau négatif (c’est-à-dire avec une permittivité diélectrique négative ε et/ou une perméabilité magnétique négative μ) qui sont séparés par une interface avec une pointe conique. En raison du changement de signe de ε et/ou μ, les équations de Maxwell peuvent être mal posées dans les cadres classiques (basés sur l’espace L2). D’autre part, nous savons que lorsque les deux problèmes scalaires associés, impliquant respectivement ε et μ, sont bien posés dans H1, les équations de Maxwell sont bien posées. En combinant la méthode de la T-coercivité avec l’analyse de Mellin dans les espaces de Sobolev à poids, nous présentons, dans la première partie de ce travail, une étude détaillée de ces problèmes scalaires. Nous prouvons que pour chacun d’entre eux, le caractère bien posé dans H1 est perdu si et seulement si le contraste associé appartient à un ensemble critique appelé intervalle critique. Ces intervalles correspondent aux ensembles de contrastes négatifs pour lesquels des singularités propagatives, aussi appelées ondes de trou noir, apparaissent à l’extrémité de la pointe. Contrairement au cas d’un coin 2D, pour une pointe 3D, plusieurs ondes de trou noir peuvent exister. Des expressions explicites de ces intervalles critiques sont obtenues pour le cas particulier des pointes coniques circulaires. Pour les contrastes critiques, en utilisant le principe de radiation de Mandelstam, nous construisons des cadres fonctionnels dans lesquels le caractère bien posé des problèmes scalaires est restauré. Le cadre physiquement pertinent est sélectionné par un principe d’absorption limite. En outre, nous présentons, dans la deuxième partie de ce travail, une nouvelle méthode numérique pour les problèmes scalaires dans le cas des contrastes non-critiques. Cette approche, contrairement aux techniques existantes, ne nécessite pas d’hypothèses supplémentaires sur le maillage au voisinage de l’interface. La troisième partie de la thèse concerne l’étude des équations de Maxwell avec un ou deux coefficients critiques. En utilisant de nouveaux résultats de potentiels vecteurs dans des espaces de Sobolev à poids, nous expliquons comment construire de nouveaux cadres fonctionnels pour les problèmes électrique et magnétique, qui sont directement liés à ceux obtenus pour les deux problèmes scalaires associés. Si l’on utilise le cadre qui respecte le principe d’absorption limite pour les problèmes scalaires, alors les cadres fournis pour les problèmes électrique et magnétique sont également cohérents avec le principe d’absorption limite. Enfin, la dernière partie porte sur des résultats d’homogénéisation des équations de Maxwell harmoniques et des problèmes scalaires associés dans un domaine 3D qui contient une distribution périodique d’inclusions faites de matériau négatif. En utilisant l’approche de la T-coercivité, nous obtenons des conditions sur les contrastes telles que le processus d’homogénéisation est possible pour les problèmes scalaires et vectoriels. De façon peu intuitive, nous montrons que les matrices homogénéisées associées auxproblèmes limites sont soit définies positives, soit définies négatives.

International audience; In this paper we consider an inverse scattering problem which consists in retrieving obstacles in a partially embedded waveguide in the acoustic case, the measurements being located on the accessible part of the structure. Such accessible part can be considered as a closed waveguide (with a finite cross section), while the embedded part can be considered as an open waveguide (with an infinite cross section). We propose an approximate model of the open waveguide by using Perfectly Matched Layers in order to simplify the resolution of the inverse problem, which is based on a modal formulation of the Linear Sampling Method. Some numerical results show the efficiency of our approach. This paper can be viewed as a continuation of the article [11], which was focused on the forward problem.

PurposeThis work contributes to the general problem of justifying the validity of the heuristic that underpins medium imaging using topological derivatives (TDs), which involves the sign and the spatial decay away from the true anomaly of the TD functional. The author considers here the identification of finite-sized (i.e. not necessarily small) anomalies embedded in bounded media and affecting the leading-order term of the acoustic field equation.Design/methodology/approachTD-based imaging functionals are reformulated for analysis using a suitable factorization of the acoustic fields, which is facilitated by a volume integral formulation. The three kinds of TDs (single-measurement, full-measurement and eigenfunction-based) studied in this work are given expressions whose structure allows to establish results on their sign and decay properties. The latter are obtained using analytical methods involving classical identities on Bessel functions and Legendre polynomials, as well as asymptotic approximations predicated on spatial scaling assumptions.FindingsThe sign component of the TD imaging heuristic is found to be valid for multistatic experiments and if the sought anomaly satisfies a bound (on a certain operator norm) involving its geometry, its contrast and the operating frequency. Moreover, upon processing the excitation and data by applying suitably-defined bounded linear operatirs to them, the magnitude component of the TD imaging heuristic is proved under scaling assumptions where the anomaly is small relative to the probing region, the latter being itself small relative to the propagation domain. The author additionally validates both components of the TD imaging heuristic when the probing excitation is taken as an eigenfunction of the source-to-measurement operator, with a focusing effect analogous to that achieved in time-reversal based methods taking place. These findings extend those of earlier studies to the case of finite-sized anomalies embedded in bounded media.Originality/valueThe originality of the paper lies in the theoretical justifications of the TD-based imaging heuristic for finite-sized anomalies embedded in bounded media.

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.

International audience; The limiting amplitude principle states that the response of a scatterer to a harmonic light excitation is asymptotically harmonic with the same pulsation. Depending on the geometry and nature of the scatterer, there might or might not be an established theoretical proof validating this principle. In this paper, we investigate a case where the theory is missing: we consider a two-dimensional dispersive Drude structure with corners. In the non lossy case, it is well known that looking for harmonic solutions leads to an ill-posed problem for a specific range of critical pulsations, characterized by the metal’s properties and the aperture of the corners. Ill-posedness is then due to highly oscillatory resonances at the corners called black-hole waves. However, a time-domain formulation with a harmonic excitation is always mathematically valid. Based on this observation, we conjecture that the limiting amplitude principle might not hold for all pulsations. Using a time-domain setting, we propose a systematic numerical approach that allows to give numerical evidences of the latter conjecture, and find clear signature of the critical pulsa-tions. Furthermore, we connect our results to the underlying physical plasmonic resonances thatoccur in the lossy physical metallic case.

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.

International audience; 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.

International audience; The problem of the steady propagation of detonation waves with friction losses is revisited including detailed kinetics. The derived formulation is used to study the influence of chemical modeling on the steady solutions and reaction zone structures obtained for stoichiometric hydrogen-oxygen. Detonation velocity-friction coefficient (D − c f) curves, pressure, temperature, Mach number, thermicity and species profiles are used for that purpose. Results show that both simplified kinetic schemes considered (i.e., one-step and three-step chainbranching), fitted using standard methodologies, failed to quantitatively capture the critical c f values obtained with detailed kinetics; moreover one-step Arrhenius chemistry also exhibits qualitative differences for D/D CJ ≤ 0.55 due to an overestimation of the chemical time in this regime. An alternative fitting methodology for simplified kinetics is proposed using detailed chemistry D − c f curves as a target rather than constant volume delay times and ideal Zel'dovich-von Neumann-Döring profiles; this method is in principle more representative to study non-ideal detonation propagation. The sensitivity of the predicted critical c f value, c f,crit , to the detailed mechanisms routinely used to model hydrogen oxidation was also assessed; significant differences were found, mainly driven by the consumption/creation rate of the HO 2 radical pool at low postshock temperature.

International audience; In this work, we are interested in the analysis of time-harmonic Maxwell's equations in presence of a conical tip of a material with negative dielectric constants. When these constants belong to some critical range, the electromagnetic field exhibits strongly oscillating singularities at the tip which have infinite energy. Consequently Maxwell's equations are not well-posed in the classical $L^2$ framework. The goal of the present work is to provide an appropriate functional setting for 3D Maxwell's equations when the dielectric permittivity (but not the magnetic permeability) takes critical values. Following what has been done for the 2D scalar case, the idea is to work in weighted Sobolev spaces, adding to the space the so-called outgoing propagating singularities. The analysis requires new results of scalar and vector potential representations of singular fields. The outgoing behaviour is selected via the limiting absorption principle.

The numerical simulation of electromagnetic problems in complex physical settings is a trending topic which conveys many scientific and industrial applications, such as the design of optical metamaterials, or the study of cold plasmas. The mathematical and numerical analysis of Maxwell problems is wellknown in simple physical contexts, when the material parameters are isotropic. Some results in anisotropic media exist, but they generally tend to focus on the case where the material tensors are real symmetric (or complex) Hermitian) definite positive. However, problems in more complex media are not covered by the standard theory. Therefore, new mathematical tools need to be developped to analyse thses problems. This thesis aims at analysing time-harmonic electromagnetic problems for a general class of complex anisotropic material tensors. These are called ellopptic materials. We derive an extended functional framework well-suited for these anisotropic problems, generalizing well-known results. We study the well-posedness of Maxwell boundary value problems for Dirichlet, Neumann, and Robin boundary conditions. For the Robin case, the characterization of appropriate function spaces for Robin traces is addressed. The regularity of the solution and its curl is studied, and elements of numerical analysis for edge finite elements are provided. In the perspective of the use of Domain Decomposition Methods (DDM) for accelerated numerical computing, various decomposed formulations are proposed and studied, focusing on their right meaning in terms of function spaces and equivalence with the global problem. These results are complemented with some numerical DDM experimentations in anisotropic media., La simulation numérique de problèmes électromagnétiques dans des configurations physiques complexes est largement utilisée pour de nombreuses applications scientifiques et industrielles, telles que la conception de métamatériaux optiques ou l'étude des plasmas froids. L'analyse mathématique et numérique des problèmes de Maxwell est bien connue dans des contextes physiques simples, où les paramètres du milieu sont isotropes. Des résultats en milieux anisotropes existent, mais se limitent généralement au cas des tenseurs réels symétriques (ou complexes hermitiens) définis positifs. Cependant, pour certains milieux plus complexes, les problèmes ne sont pas couverts par la théorie standard. De nouveaux outils mathématiques doivent donc être développés pour analyser ces problèmes.Dans cette thèse, nous analysons des problèmes électromagnétiques harmoniques en temps pour une classe générale de tenseurs matériels anisotropes, appelés elliptiques. Nous développons un cadre fonctionnel étendu adapté à ces problèmes avec conditions limites de Dirichlet, Neumann ou Robin. Dans le cas de Robin, un intérêt particulier est porté à la caractérisation des espaces pour les traces de Robin. Nous étudions la régularité de la solution et de son rotationnel, et donnons des éléments d'analyse numérique. Dans la perspective de l'utilisation de méthodes de décomposition de domaine (DDM) pour une résolution accélérée, nous proposons et étudions différentes formulations décomposées, en nous focalisant sur leurs espaces fonctionnels et leur équivalence avec le problème global. Quelques expérimentations numériques sur la DDM complètent ce travail.

International audience; It is well-known that the convergence rate of non-overlapping domain decomposition methods (DDMs) applied to the parallel finite-element solution of large-scale time-harmonic wave problems strongly depends on the transmission condition enforced at the interfaces between the subdomains. Transmission operators based on perfectly matched layers (PMLs) have proved to be well-suited for configurations with layered domain partitions. They are shown to be a good compromise between basic impedance conditions, which lead to suboptimal convergence, and computational expensive conditions based on the exact Dirichlet-to-Neumann (DtN) map related to the complementary of the subdomain. Unfortunately, the extension of the PML-based DDM for more general partitions with cross-points (where more than two subdomains meet) is rather tricky and requires some care.In this work, we present a non-overlapping substructured DDM with PML transmission conditions for checkerboard (Cartesian) decompositions that takes cross-points into account. In such decompositions, each subdomain is surrounded by PMLs associated to edges and corners. The continuity of Dirichlet traces at the interfaces between a subdomain and PMLs is enforced with Lagrange multipliers. This coupling strategy offers the benefit of naturally computing Neumann traces, which allows to use the PMLs as discrete operators approximating the exact Dirichlet-to-Neumann maps. Two possible Lagrange multiplier finite element spaces are presented, and the behavior of the corresponding DDM is analyzed on several numerical examples.

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.

The modelling of resonant waves in 2D plasma leads to the coupling of two degenerate elliptic equations with a smooth coefficient $\alpha $ and compact terms. The coefficient $\alpha $ changes sign. The region where $\{\alpha>0\}$ is propagative, and the region where $\{\alpha

International audience; In bounded domains, the regularity of the solutions to boundary value problems depends on the gometry, and on the coefficients that enter into the definition of the model. This is in particular the case for the time-harmonic Maxwell equations, whose solutions are the electromagnetic fields. In this paper, emphasis is put on the electric field. We study the regularity in terms of the fractional order Sobolev spaces $H^s$, $s\in[0,1]$. Precisely, our first goal is to determine the regularity of the electric field and of its curl, that is to find some regularity exponent $\tau\in(0,1)$, such that they both belong to $H^s$, for all $s\in[0,\tau)$. After that, one can derive error estimates. Here, the error is defined as the difference between the exact field and its approximation, where the latter is built with N\'ed\'elec's first family of finite elements. In addition to the regularity exponent, one needs to derive a stability constant that relates the norm of the error to the norm of the data: this is our second goal. We provide explicit expressions for both the regularity exponent and the stability constant with respect to the coefficients. We also discuss the accuracy of these expressions, and we provide some numerical illustrations.

International audience; A subwavelength metallic particle supports localized 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 solutions 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