Publications

2018

• Localization of global norms and robust a posteriori error control for transmission problems with sign-changing coefficients
  
  • Ciarlet Patrick
  • Vohralík Martin
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2018, 52 (5), pp.2037-2064. We present a posteriori error analysis of diffusion problems where the diffusion tensor is not necessarily symmetric and positive definite and can in particular change its sign. We first identify the correct intrinsic error norm for such problems, covering both conforming and nonconforming approximations. It combines a dual (residual) norm together with the distance to the correct functional space. Importantly, we show the equivalence of both these quantities defined globally over the entire computational domain with the Hilbertian sums of their localizations over patches of elements. In this framework, we then design a posteriori estimators which deliver simultaneously guaranteed error upper bound, global and local error lower bounds, and robustness with respect to the (sign-changing) diffusion tensor. Robustness with respect to the approximation polynomial degree is achieved as well. The estimators are given in a unified setting covering at once conforming, nonconforming, mixed, and discontinuous Galerkin finite element discretizations in two or three space dimensions. Numerical results illustrate the theoretical developments. (10.1051/m2an/2018034)
  DOI : 10.1051/m2an/2018034
• An efficient domain decomposition method with cross-point treatment for Helmholtz problems
  
  • Modave Axel
  • Antoine Xavier
  • Geuzaine Christophe
  , 2018. Solving high-frequency time-harmonic scattering problems using finite element techniques is challenging, as such problems lead to very large, complex and indefinite linear systems. Optimized Schwarz domain decomposition methods (DDMs) are currently a very promising approach, where subproblems of smaller sizes are solved in parallel using direct solvers, and are combined in an iterative procedure. It is well-known that the convergence rate of these methods strongly depends on the transmission condition enforced on the interfaces between the subdomains. Local transmission conditions based on high-order absorbing boundary conditions (HABCs) have proved well suited [Boubendir et al, 2012; Gander et al, 2002]. They represent a good compromise between basic impedance conditions (which lead to suboptimal convergence) and the exact Dirichlet-to-Neumann (DtN) map related to the complementary of the subdomain (which is expensive to compute). However, a direct application of this approach for domain decomposition configurations with cross-points, where more than two subdomains meet, does not provide satisfactory results. We present an improved DDM that efficiently addresses configurations with cross points. Noting that these points actually are corners for the subdomains, our strategy consists in incorporating a corner treatment developed for HABCs into the DDM procedure. After a presentation of the key aspects of the methods, the effectiveness of our approach is discussed with two-dimensional finite element results.
• A Sufficient Condition for the Absence of Two-Dimensional Instabilities of an Elastic Plate in a Duct with Compressible Flow
  
  • Mercier Jean-François
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2018, 78 (6), pp.3119-3144. We study the time-harmonic resonance of a finite-length elastic plate in a fluid in 5 uniform flow confined in a duct. Although the resonance frequencies are usually real, the combined 6 effects of plate elasticity and of a flow can create complex frequencies, different from the usual so-7 called scattering frequencies, corresponding to instabilities. We study theoretically the existence of 8 instabilities versus several problem parameters, notably the flow velocity and the ratio of densities 9 and of sound speeds between the plate and the fluid. A 3D-volume in the parameters space is defined, 10 in which no instability can develop. In particular it corresponds to a low enough velocity or a light 11 enough plate. The theoretical estimates are validated numerically. 12 (10.1137/18M1165761)
  DOI : 10.1137/18M1165761
• Metric-based anisotropic mesh adaptation for 3D acoustic boundary element methods
  
  • Chaillat Stéphanie
  • Groth Samuel P
  • Loseille Adrien
  Journal of Computational Physics, Elsevier, 2018, 372, pp.473 - 499. This paper details the extension of a metric-based anisotropic mesh adaptation strategy to the boundary element method for problems of 3D acoustic wave propagation. Traditional mesh adaptation strategies for boundary element methods rely on Galerkin discretizations of the boundary integral equations, and the development of appropriate error indicators. They often require the solution of further integral equations. These methods utilise the error indicators to mark elements where the error is above a specified tolerance and then refine these elements. Such an approach cannot lead to anisotropic adaptation regardless of how these elements are refined, since the orientation and shape of current elements cannot be modified. In contrast, the method proposed here is independent of the discretization technique (e.g., collocation, Galerkin). Furthermore, it completely remeshes at each refinement step, altering the shape, size, and orientation of each element according to an optimal metric based on a numerically recovered Hessian of the boundary solution. The resulting adaptation procedure is truly anisotropic and independent of the complexity of the geometry. We show via a variety of numerical examples that it recovers optimal convergence rates for domains with geometric singularities. In particular, a faster convergence rate is recovered for scattering problems with complex geometries. (10.1016/j.jcp.2018.06.048)
  DOI : 10.1016/j.jcp.2018.06.048
• The interaction of a walking droplet and a submerged pillar: From scattering to the logarithmic spiral
  
  • Harris Daniel
  • Brun P.-T.
  • Damiano Adam
  • Faria Luiz
  • Bush John
  Chaos: An Interdisciplinary Journal of Nonlinear Science, American Institute of Physics, 2018, 28 (9), pp.096105. Millimetric droplets may walk across the surface of a vibrating fluid bath, propelled forward by their own guiding or “pilot” wave field. We here consider the interaction of such walking droplets with a submerged circular pillar. While simple scattering events are the norm, as the waves become more pronounced, the drop departs the pillar along a path corresponding to a logarithmic spiral. The system behavior is explored both experimentally and theoretically, using a reduced numerical model in which the pillar is simply treated as a region of decreased wave speed. A trajectory equation valid in the limit of weak droplet acceleration is used to infer an effective force due to the presence of the pillar, which is found to be a lift force proportional to the product of the drop’s walking speed and its instantaneous angular speed around the post. This system presents a macroscopic example of pilot-wave-mediated forces giving rise to apparent action at a distance. (10.1063/1.5031022)
  DOI : 10.1063/1.5031022
• On the spectral theory and limiting amplitude principle for Maxwells equations at the interface of a metamaterial
  
  • Cassier Maxence
  • Hazard Christophe
  • Joly Patrick
  , 2018.
• Une approche nouvelle de la modélisation mathématique et numérique en aéroacoustique par les équations de Goldstein : Applications en aéronautique
  
  • Bensalah Antoine
  , 2018. La problématique du bruit fait par les réacteurs d'avions est un des enjeux majeurs de l’industrie aéronautique.C’est dans ce contexte que l’équipe du centre de recherche d'Airbus travaille au développement du code de calcul Actipole de propagation acoustique en présence d'un écoulement porteur.L'approche consiste en un couplage FEM-BEM entre la zone de propagation loin de l'avion où l'écoulement est supposé uniforme (BEM) et la zone plus proche où l'écoulement est supposé potentiel (FEM).Les équations de l'aéroacoustique en régime harmonique se réduisent alors à la simple équation scalaire d'Helmholtz convectée.Nous étudions une reformulation des équations d'Euler linéarisées, les équations de Goldstein, prenant en compte l'interaction entre l'acoustique et l'hydrodynamique, lorsque l'écoulement n'est plus potentiel, par l'ajout d'une inconnue hydrodynamique localisée aux zones fortement rotationnelles.Les équations de Goldstein peuvent être vues comme une perturbation de l'équation d'Helmholtz convectée, couplée à une équation de transport harmonique.Nos approches théorique et numérique restent dans le cadre de cette vision perturbative en étudiant dans une premier temps la résolution de l'équation de transport.Nous montrons ainsi que sous l'hypothèse d'un écoulement domaine-remplissant, l'équation de transport harmonique peut être inversée et sous contrainte d'un faible rotationnel, le caractère Fredholm de l'équation d'Helmholtz convectée se généralise aux équations de Goldstein.Le cas général est un problème ouvert et difficile, nous montrons que l'équation de transport n'est pas toujours inversible et possède des fréquences de résonance auxquelles les og solutionsfg{} tendent à être singulières le long de lignes de recirculation de l'écoulement.Nous montrons qu'il en est de même des équations couplées qui possèdent en plus des fréquences de résonance du transport d'autres résonances, dites critiques.Nous terminons cette thèse par une étude locale des singularités, par la méthode de Frobenius, des solutions modales obtenues par absorption limite, aux fréquences de résonance du transport et critiques, au voisinage de lignes résonantes, montrant que de telles solutions sortent alors du cadre variationnelle classiques.
• The Halfspace Matching Method : a new method to solve scattering problem in infinite media
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Fliss Sonia
  • Tonnoir Antoine
  Journal of Computational and Applied Mathematics, Elsevier, 2018, 338, pp.44-68. We are interested in acoustic wave propagation in time harmonic regime in a two-dimensional medium which is a local perturbation of an infinite isotropic or anisotropic homogeneous medium. We investigate the question of finding artificial boundary conditions to reduce the numerical computations to a neighborhood of this perturbation. Our objective is to derive a method which can extend to the anisotropic elastic problem for which classical approaches fail. The idea consists in coupling several semi-analytical representations of the solution in halfspaces surrounding the defect with a Finite Element computation of the solution around the defect. As representations of the same function, they have to match in the infinite intersections of the halfspaces. It leads to a formulation which couples, via integral operators, the solution in a bounded domain including the defect and its traces on the edge of the halfspaces. A stability property is shown for this new formulation. (10.1016/j.cam.2018.01.021)
  DOI : 10.1016/j.cam.2018.01.021
• Numerical Analysis of a Non-Conforming Domain Decomposition for the Multigroup SPN Equations
  
  • Giret Léandre
  , 2018. In this thesis, we investigate the resolution of the SPN neutron transport equations in pressurized water nuclear reactor. These equations are a generalized eigenvalue problem. In our study, we first considerate the associated source problem and after we concentrate on the eigenvalue problem. A nuclear reactor core is composed of different media: the fuel, the coolant, the neutron moderator... Due to these heterogeneities of the geometry, the solution flux can have a low-regularity. We propose the numerical analysis of its approximation with finite element method for the low regular case. For the eigenvalue problem under its mixed form, we can not rely on the theories already developed. We propose here a new method for studying the convergence of the SPN neutron transport eigenvalue problem approximation with mixed finite element. When the solution has low-regularity, increasing the order of the method does not improve the approximation, the triangulation need to be refined near the singularities of the solution. Nuclear reactor cores are well-suited for Cartesian grids, but the refinement of these sort of triangulations increases rapidly their number of degrees of freedom. To avoid this drawback, we propose domain decomposition method which can handle globally non-conforming triangulations.
• Inverse acoustic scattering using high-order small-inclusion expansion of misfit function
  
  • Bonnet Marc
  Inverse Problems and Imaging, AIMS American Institute of Mathematical Sciences, 2018, 12 (4), pp.921-953. This article concerns an extension of the topological derivative concept for 3D inverse acoustic scattering problems involving the identification of penetrable obstacles, whereby the featured data-misfit cost function J is expanded in powers of the characteristic radius a of a single small inhomo-geneity. The O(a 6) approximation J 6 of J is derived and justified for a single obstacle of given location, shape and material properties embedded in a 3D acoustic medium of arbitrary shape. The generalization of J 6 to multiple small obstacles is outlined. Simpler and more explicit expressions of J 6 are obtained when the scatterer is centrally-symmetric or spherical. An approximate and computationally light global search procedure, where the location and size of the unknown object are estimated by minimizing J 6 over a search grid, is proposed and demonstrated on numerical experiments, where the identification from known acoustic pressure on the surface of a penetrable scatterer embedded in a acoustic semi-infinite medium, and whose shape may differ from that of the trial obstacle assumed in the expansion of J, is considered. (10.3934/ipi.2018039)
  DOI : 10.3934/ipi.2018039
• Well-posedness of a generalized time-harmonic transport equation for acoustics in flow
  
  • Bensalah Antoine
  • Joly Patrick
  • Mercier Jean-François
  Mathematical Methods in the Applied Sciences, Wiley, 2018, 41 (8), pp.3117 - 3137. (10.1002/mma.4805)
  DOI : 10.1002/mma.4805
• Imaging defects in an elastic waveguide using time-dependent surface data
  
  • Baronian V
  • Chapuis B
  • Recoquillay A
  • Bourgeois Laurent
  , 2018, 1131, pp.012010 (7 p.). We are interested here in applying the Linear Sampling Method to the context of Non Destructive Testing for waveguides. Specifically the Linear Sampling Method [1] in its modal form [2] is adapted to image defects in an elastic waveguide from realistic scattering data, that is data coming from sources and receivers on the surface of the waveguide in the time domain, as it has already been done in the acoustic case [16]. The obtained method is applied to artificial data and to experimental data in the special case of back-scattering. (10.1088/1742-6596/1131/1/012010)
  DOI : 10.1088/1742-6596/1131/1/012010
• Linear Sampling Method applied to Non Destructive Testing of an elastic waveguide: theory, numerics and experiments
  
  • Baronian Vahan
  • Bourgeois Laurent
  • Chapuis Bastien
  • Recoquillay Arnaud
  Inverse Problems, IOP Publishing, 2018, 34, pp.075006 (34 p.). This paper presents an application of the Linear Sampling Method to ultrasonic Non Destructive Testing of an elastic waveguide. In particular, the NDT context implies that both the solicitations and the measurements are located on the surface of the waveguide and are given in the time domain. Our strategy consists in using a modal formulation of the Linear Sampling Method at multiple frequencies, such modal formulation being justified theoretically in [1] for rigid obstacles and in [2] for cracks. Our strategy requires the inversion of some emission and reception matrices which deserve some special attention due to potential ill-conditioning. The feasibility of our method is proved with the help of artificial data as well as real data. (10.1088/1361-6420/aac21e)
  DOI : 10.1088/1361-6420/aac21e
• Reconstruction of an unknown electrical network from their reflectogram by an iterative algorithm based on local identification of peaks and inverse scattering theory
  
  • Beck Geoffrey
  , 2018, pp.1-6. We aim at recovering the topology of an unknown electrical network made out of a tree of cables with the same characteristics from only the data obtained through reflectometry. The method is based upon an iterative algorithm associating the peaks of a reflectogram with unknown scatterers which can be either junction or terminal end of the network, dispelling the ambiguities caused by the complexity of the reflectogram. To identify the peaks, we propose an new algorithm adapted to our goal. The reconstructed networks are topologically identical to the originals ones in 99 per cent of all cases. Cables lengths and terminal loads are also retrieved with high accuracy, e.g. with typical error respectively less than 5% and less than 15%. (10.1109/I2MTC.2018.8409731)
  DOI : 10.1109/I2MTC.2018.8409731
• Trapped modes and reflectionless modes as eigenfunctions of the same spectral problem
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chesnel Lucas
  • Pagneux Vincent
  Proceedings of the Royal Society of London. Series A, Mathematical and physical sciences, Royal Society, The, 2018. We consider the reflection-transmission problem in a waveguide with obstacle. At certain frequencies, for some incident waves, intensity is perfectly transmitted and the reflected field decays exponentially at infinity. In this work, we show that such reflectionless modes can be characterized as eigenfunctions of an original non-selfadjoint spectral problem. In order to select ingoing waves on one side of the obstacle and outgoing waves on the other side, we use complex scalings (or Perfectly Matched Layers) with imaginary parts of different signs. We prove that the real eigenvalues of the obtained spectrum correspond either to trapped modes (or bound states in the continuum) or to reflectionless modes. Interestingly, complex eigenvalues also contain useful information on weak reflection cases. When the geometry has certain symmetries, the new spectral problem enters the class of PT-symmetric problems.
• Anomalous Chained Turbulence in Actively Driven Flows on Spheres
  
  • Mickelin Oscar
  • Słomka Jonasz
  • Burns Keaton
  • Lecoanet Daniel
  • Vasil Geoffrey
  • Faria Luiz
  • Dunkel Jörn
  Physical Review Letters, American Physical Society, 2018, 120 (16). Recent experiments demonstrate the importance of substrate curvature for actively forced fluid dynamics. Yet, the covariant formulation and analysis of continuum models for non-equilibrium flows on curved surfaces still poses theoretical challenges. Here, we introduce and study a generalized covariant Navier-Stokes model for fluid flows driven by active stresses in non-planar geometries. The analytical tractability of the theory is demonstrated through exact stationary solutions for the case of a spherical bubble geometry. Direct numerical simulations reveal a curvature-induced transition from a burst phase to an anomalous turbulent phase that differs distinctly from externally forced classical 2D Kolmogorov turbulence. This new type of active turbulence is characterized by theself-assembly of finite-size vortices into linked chains of anti-ferromagnetic order, which percolatethrough the entire fluid domain, forming an active dynamic network. The coherent motion of the vortex chain network provides an efficient mechanism for upward energy transfer from smaller to larger scales, presenting an alternative to the conventional energy cascade in classical 2D turbulence. (10.1103/PhysRevLett.120.164503)
  DOI : 10.1103/PhysRevLett.120.164503
• Perfect transmission invisibility for waveguides with sound hard walls
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chesnel Lucas
  • Nazarov Sergei
  Journal de Mathématiques Pures et Appliquées, Elsevier, 2018. We are interested in a time harmonic acoustic problem in a waveguide with locally perturbed sound hard walls. We consider a setting where an observer generates incident plane waves at −∞ and probes the resulting scattered field at −∞ and +∞. Practically, this is equivalent to measure the reflection and transmission coefficients respectively denoted R and T. In [9], a technique has been proposed to construct waveguides with smooth walls such that R = 0 and |T | = 1 (non reflection). However the approach fails to ensure T = 1 (perfect transmission without phase shift). In this work, first we establish a result explaining this observation. More precisely, we prove that for wavenumbers smaller than a given bound k depending on the geometry, we cannot have T = 1 so that the observer can detect the presence of the defect if he/she is able to measure the phase at +∞. In particular, if the perturbation is smooth and small (in amplitude and in width), k is very close to the threshold wavenumber. Then, in a second step, we change the point of view and, for a given wavenumber, working with singular perturbations of the domain, we show how to obtain T = 1. In this case, the scattered field is exponentially decaying both at −∞ and +∞. We implement numerically the method to provide examples of such undetectable defects.
• Modelling resonant arrays of the Helmholtz type in the time domain
  
  • Maurel Agnès
  • Marigo Jean-Jacques
  • Mercier Jean-François
  • Pham Kim
  Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Royal Society, The, 2018, 474 (2210), pp.20170894. We present a model based on a two-scale asymptotic analysis for resonant arrays of the Helmholtz type, with resonators open at a single extremity (standard resonators) or open at both extremities (double-sided resonators). The effective behaviour of such arrays is that of a homogeneous anisotropic slab replacing the cavity region, associated with transmission, or jump, conditions for the acoustic pressure and for the normal velocity across the region of the necks. The coefficients entering in the effective wave equation are simply related to the fraction of air in the periodic cell of the array. Those entering in the jump conditions are related to near field effects in the vicinity of the necks and they encapsulate the effects of their geometry. The effective problem, which accounts for the coupling of the resonators with the surrounding air, is written in the time domain which allows us to question the equation of energy conservation. This is of practical importance if the numerical implementations of the effective problem in the time domain is sought. (10.1098/rspa.2017.0894)
  DOI : 10.1098/rspa.2017.0894
• Méthodes d'échantillonnage appliquées à l'imagerie de défauts dans un guide d'ondes élastiques
  
  • Recoquillay Arnaud
  , 2018. De nombreuses structures utilisées industriellement peuvent être considérées comme des guides d'ondes, comme les plaques, les tuyaux ou encore le rails. La maintenance de ces structures nécessite de pouvoir détecter efficacement des défauts internes par le Contrôle Non Destructif. Nous nous intéressons dans ce manuscrit à l'application d'une méthode d'échantillonnage, la Linear Sampling Method, au CND des guides d'ondes élastiques, qui en particulier impose des sollicitations et des mesures à la surface du guide en régime temporel. La stratégie choisie repose sur une formulation modale et multi-fréquentielle de la LSM, spécifique aux guides d'ondes, qui permet une régularisation efficace et de nature physique du problème inverse, qui est par nature mal posé. Cette stratégie permet par ailleurs une optimisation du nombre et de la position des émetteurs et des récepteurs. Nous nous limitons dans un premier temps au cas scalaire du guide d'ondes acoustiques, pour ensuite s'attaquer au cas vectoriel, et par conséquent plus complexe, du guide d'ondes élastiques.L'efficacité de la méthode inverse est dans un premier temps démontrée sur des données artificielles (obtenues numériquement), puis sur des données réelles obtenues à l'aide d'expériences réalisées sur des plaques métalliques. Ces expériences confirment la faisabilité du CND par méthode d'échantillonnage dans un cadre industriel. Dans le cas où une seule sollicitation est réalisée, l'utilisation de la LSM est exclu. Nous utilisons une approche tout à fait différente et dite "extérieure", couplant une formulation mixte de quasi-réversibilité et une méthode de lignes de niveau, pour reconstruire le défaut.
• Sub-wavelength sensing of bi-periodic materials using topological sensitivity of second-order homogenized model
  
  • Bonnet Marc
  • Cornaggia Rémi
  • Guzina Bojan B
  Journal of Physics: Conference Series, IOP Science, 2018, 1131, pp.012008. We aim to detect defects or perturbations of periodic media, e.g. due to a defective manufacturing process. To this end, we consider scalar waves in such media through the lens of a second-order macroscopic description, and we 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. Then, we present a method that permits sub-wavelength sensing of periodic media, given the (anisotropic) phase velocity of plane waves illuminating the considered medium for several angles and wavenumbers. (10.1088/1742-6596/1131/1/012008)
  DOI : 10.1088/1742-6596/1131/1/012008
• On the analysis of perfectly matched layers for a class of dispersive media and application to negative index metamaterials
  
  • Bécache Eliane
  • Joly Patrick
  • Vinoles Valentin
  Mathematics of Computation, American Mathematical Society, 2018, 87, pp.2775-2810. This work deals with Perfectly Matched Layers (PMLs) in the context of dispersive media, and in particular for Negative Index Metamaterials (NIMs). We first present some properties of dispersive isotropic Maxwell equations that include NIMs. We then demonstrate numerically the inherent instabilities of the classical PMLs applied to NIMs. We propose and analyse the stability of very general PMLs for a large class of dispersive systems using a new change of variable. We give necessary criteria for the stability of such models. For dispersive isotropic Maxwell equations, this analysis is completed by giving necessary and sufficient conditions of stability. Finally, we propose new PMLs that satisfy these criteria and demonstrate numerically their efficiency. (10.1090/mcom/3307)
  DOI : 10.1090/mcom/3307
• Solving 2D linear isotropic elastodynamics by means of scalar potentials: a new challenge for finite elements
  
  • Albella Martínez Jorge
  • Imperiale Sébastien
  • Joly Patrick
  • Rodríguez Jerónimo
  Journal of Scientific Computing, Springer Verlag, 2018. 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. (10.1007/s10915-018-0768-9)
  DOI : 10.1007/s10915-018-0768-9
• Mathematical foundations of computational electromagnetism
  
  • Assous Franck
  • Ciarlet Patrick
  • Labrunie Simon
  , 2018. This book presents an in-depth treatment of various mathematical aspects of electromagnetism and Maxwell's equations: from modeling issues to well-posedness results and the coupled models of plasma physics (Vlasov-Maxwell and Vlasov-Poisson systems) and magnetohydrodynamics (MHD). These equations and boundary conditions are discussed, including a brief review of absorbing boundary conditions. The focus then moves to well‐posedness results. The relevant function spaces are introduced, with an emphasis on boundary and topological conditions. General variational frameworks are defined for static and quasi-static problems, time-harmonic problems (including fixed frequency or Helmholtz-like problems and unknown frequency or eigenvalue problems), and time-dependent problems, with or without constraints. They are then applied to prove the well-posedness of Maxwell’s equations and their simplified models, in the various settings described above. The book is completed with a discussion of dimensionally reduced models in prismatic and axisymmetric geometries, and a survey of existence and uniqueness results for the Vlasov-Poisson, Vlasov-Maxwell and MHD equations. The book addresses mainly researchers in applied mathematics who work on Maxwell’s equations. However, it can be used for master or doctorate-level courses on mathematical electromagnetism as it requires only a bachelor-level knowledge of analysis. (10.1007/978-3-319-70842-3)
  DOI : 10.1007/978-3-319-70842-3
• Mesh requirements for the finite element approximation of problems with sign-changing coefficients
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Carvalho Camille
  • Ciarlet Patrick
  Numerische Mathematik, Springer Verlag, 2018, 138, pp.801-838. Transmission problems with sign-changing coefficients occur in electromagnetic theory in the presence of negative materials surrounded by classical materials. For general geometries, establishing Fredholmness of these transmission problems is well-understood thanks to the T-coercivity approach. Moreover, for a plane interface, there exist meshing rules that guarantee an optimal convergence rate for the finite element approximation. We propose here a new treatment at the corners of the interface which allows to design meshing rules for an arbitrary polygonal interface and then recover standard error estimates. This treatment relies on the use of simple geometrical transforms to define the meshes. Numerical results illustrate the importance of this new design. (10.1007/s00211-017-0923-5)
  DOI : 10.1007/s00211-017-0923-5
• Microstructural topological sensitivities of the second-order macroscopic model for waves in periodic media
  
  • Bonnet Marc
  • Cornaggia Rémi
  • Guzina Bojan B
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2018, 78 (4), pp.2057-2082. 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. (10.1137/17M1149018)
  DOI : 10.1137/17M1149018