Publications

2021

• Analysis of time-harmonic electromagnetic problems in elliptic anisotropic media
  
  • Chicaud Damien
  , 2021. 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.
• Outils élémentaires d'analyse pour les Equations aux Dérivées Partielles
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Bourgeois Laurent
  • Hazard Christophe
  , 2021, pp.132. Ce cours présente quelques-uns des principaux outils de l’analyse pour l’étude mathématique des équations aux dérivées partielles issues des sciences physiques ou humaines. Le chapitre 1 rappelle quelques notions essentielles concernant la topologie des espaces vectoriels normés. Le chapitre 2 est une introduction succincte à l’intégrale de Lebesgue, l’idée étant ici de comprendre quels sont les avantages de cette intégrale par rapport à celle de Riemann, et non pas de détailler la théorie sous-jacente. Le chapitre 3 expose les bases de la théorie des distributions, due à Laurent Schwartz, qui généralise la notion de fonction. Le chapitre 4 présente les propriétés fondamentales de la transformation de Fourier pour les fonctions intégrables et les fonctions de carré intégrable. Le chapitre 5 est consacré aux espaces de Hilbert qui, dans le cas de la dimension infinie, fournissent un cadre de travail analogue aux espaces euclidiens. Le chapitre 6 donne des exemples d’espaces de Hilbert qui jouent un rôle important dans l’étude des équations aux dérivées partielles : les espaces de Sobolev. Enfin, le chapitre 7 pose les bases de l’analyse variationnelle des problèmes elliptiques, qui ouvre notamment la porte à la méthode des éléments finis. Mais c’est là une autre histoire..
• Complex-scaling method for the complex plasmonic resonances of planar subwavelength particles with corners
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Hazard Christophe
  • Monteghetti Florian
  Journal of Computational Physics, Elsevier, 2021, 440. 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. (10.1016/j.jcp.2021.110433)
  DOI : 10.1016/j.jcp.2021.110433
• Propagation of elastic waves un buried waveguides: modelling of the forward problem and imaging with sampling methods
  
  • Fritsch Jean-François
  • Bourgeois Laurent
  • Hazard Christophe
  • Baronian Vahan
  • Recoquillay Arnaud
  , 2021.
• Pseudo-compressibility, dispersive model and acoustic waves in shallow water flows
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Bristeau Marie-Odile
  • Godlewski Edwige
  • Imperiale Sébastien
  • Mangeney Anne
  • Sainte-Marie Jacques
  SEMA SIMAI Springer Series, Springer International Publishing, 2021, pp.209--250. In this paper we study a dispersive shallow water type model derived from the free surface compressible Navier-Stokes system. The compressible effects allow to capture the acoustic-like waves propagation and can be seen as a relaxation of an underlying incompressible model. The first interest of such a model is thus to capture both acoustic and water waves. The second interest lies in its numerical approximation. Indeed, at the discrete level, the pseudo-compressibility terms circumvent the resolution of an elliptic equation to capture the non-hydrostatic part of the pressure. This drastically reduces the cost of the numerical resolution of dispersive models especially in 2d and 3d. (10.1007/978-3-030-72850-2_10)
  DOI : 10.1007/978-3-030-72850-2_10
• Non overlapping Domain Decomposition Methods for Time Harmonic Wave Problems
  
  • Claeys Xavier
  • Collino Francis
  • Joly Patrick
  • Parolin Emile
  , 2021. The domain decomposition method (DDM) initially designed, with the celebrated paper of Schwarz in 1870 as a theoretical tool for partial differential equations (PDEs) has become, since the advent of the computer and parallel computing techniques, a major tool for the numerical solution of such PDEs, especially for large scale problems. Time harmonic wave problems offer a large spectrum of applications in various domains (acoustics, electromagnetics, geophysics, ...) and occupy a place of their own, that shines for instance through the existence of a natural (possibly small) length scale for the solutions: the wavelength. Numerical DDMs were first invented for elliptic type equations (e.g. the Laplace equation), and even though the governing equations of wave problems (e.g. the Helmholtz equation) look similar, standard approaches do not work in general.
• Non-local Impedance Operator for Non-overlapping DDM for the Helmholtz Equation
  
  • Collino Francis
  • Joly Patrick
  • Parolin Emile
  , 2021. In the context of time harmonic wave equations, the pioneering work of B. Després [4] has shown that it is mandatory to use impedance type transmission conditions in the coupling of sub-domains in order to obtain convergence of nonoverlapping domain decomposition methods (DDM). In later works [2, 3], it was observed that using non-local impedance operators leads to geometric convergence, a property which is unattainable with local operators. This result was recently extended to arbitrary geometric partitions, including configurations with cross-points, with provably uniform stability with respect to the discretization parameter [1]. We present a novel strategy to construct suitable non-local impedance operators that satisfy the theoretical requirements of [1] or [2, 3]. It is based on the solution of elliptic auxiliary problems posed in the vicinity of the transmission interfaces. The definition of the operators is generic, with simple adaptations to the acoustic or electromagnetic settings, even in the case of heterogeneous media. Besides, no complicated tuning of parameters is required to get efficiency. The implementation in practice is straightforward and applicable to sub-domains of arbitrary geometry, including ones with rough boundaries generated by automatic graph partitioners. We first provide in Section 1 a general definition of this novel transmission operator in a two-domain configuration. In Section 2 we then study more quantitatively the convergence in the geometric configuration of a closed wave-guide. Section 3 illustrates the results using actual finite element computations.
• Analysis of variational formulations and low-regularity solutions for time-harmonic electromagnetic problems in complex anisotropic media
  
  • Chicaud Damien
  • Ciarlet Patrick
  • Modave Axel
  SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2021, 53 (3), pp.2691-2717. We consider the time-harmonic Maxwell's equations with physical parameters, namely the electric permittivity and the magnetic permeability, that are complex, possibly non-Hermitian, tensor fields. Both tensor fields verify a general ellipticity condition. In this work, the well-posedness of formulations for the Dirichlet and Neumann problems (i.e. with a boundary condition on the electric field or its curl, respectively) is proven using well-suited functional spaces and Helmholtz decompositions. For both problems, the a priori regularity of the solution and the solution's curl is analysed. The regularity results are obtained by splitting the fields and using shift theorems for second-order divergence elliptic operators. Finally, the discretization of the formulations with a H(curl)-conforming approximation based on edge finite elements is considered. An a priori error estimate is derived and verified thanks to numerical results with an elementary benchmark. (10.1137/20M1344111)
  DOI : 10.1137/20M1344111
• Modélisation mathématique et méthode numérique pour la simulation du contrôle santé intégré par ultrasons de plaques composites stratifiées
  
  • Methenni Hajer
  , 2021. Ce sujet de thèse s’inscrit dans le contexte du contrôle intégré des structures, ou « Structural Health Monitoring » (SHM). Cette technique de contrôle non-destructif vise à utiliser un ou plusieurs capteurs, installés dans ou sur la structure d’intérêt. Le contrôle se fait in-situ et de façon périodique, afin d’obtenir des informations sur l’éventuelle apparition de défauts, tels que les défauts de corrosion pour les matériaux métalliques ou les défauts de délaminage pour les matériaux composites. Les données recueillies par les capteurs alimentent une analyse statistique dont le but est d’évaluer la santé de la structure au moment du contrôle, d’estimer son temps de vie restant et de faciliter la prise de décision quant à sa maintenance. Le SHM est de plus en plus présent dans de nombreux domaines industriels, en particulier dans le secteur aéronautique. Aussi le développement de modèles numériques pertinents comme performants constitue un atout majeur dans la conception de ces systèmes. Grâce à leur capacité à se propager sur de très grande distance, l’utilisation de capteurs ultrasonores générant des ondes guidées élastiques est une solution attirante. En pratique, des capteurs piézo-électriques fins, disposés à la surface de la structure, ou éventuellement enfouis pendant le procédé de fabrication, sont utilisés. Ils permettent l’émission et la réception des perturbations ultrasonores. Cependant, la nature dispersive des ondes guidées, combinée avec l’anisotropie inhérente aux matériaux composites rend difficile l’analyse des signaux obtenus lors du contrôle. De plus, proposer une modélisation fine de la propagation de ce type d’onde dans des configurations industrielles faisant intervenir des géométries complexes est une tâche difficile. Ces deux points constituent des obstacles non négligeables au développement de la méthodologie SHM, et l’objectif de cette thèse est de constituer l’ensemble des outils numériques qui permettront de proposer des solutions concrètes à ces problèmes.
• Limiting Amplitude Principle for a Hyperbolic Metamaterial in Free Space
  
  • Kachanovska Maryna
  , 2021. Harmonic wave propagation in hyperbolic metamaterials is described by the Maxwell equations with a frequency-dependent tensor of dielectric permittivity. For a range of frequencies, this tensor has eigenvalues of opposite signs, and thus, in two dimensions, the harmonic Maxwell equations can be written as a Klein-Gordon equation. This technical report is mainly dedicated to the proof of the limiting amplitude principle for the simplest case of such a problem, and is a companion to the manuscript.
• Analytical preconditioners for Neumann elastodynamic Boundary Element Methods
  
  • Chaillat Stéphanie
  • Darbas Marion
  • Le Louër Frédérique
  SN Partial Differential Equations and Applications, Springer, 2021, 2 (22). Recent works in the Boundary Element Method (BEM) community have been devoted to the derivation of fast techniques to perform the matrix vector product needed in the iterative solver. Fast BEMs are now very mature. However, it has been shown that the number of iterations can significantly hinder the overall efficiency of fast BEMs. The derivation of robust preconditioners is now inevitable to increase the size of the problems that can be considered. Analytical precon-ditioners offer a very interesting strategy by improving the spectral properties of the boundary integral equations ahead from the discretization. The main contribution of this paper is to propose new analytical preconditioners to treat Neumann exterior scattering problems in 2D and 3D elasticity. These preconditioners are local approximations of the adjoint Neumann-to-Dirichlet map. We propose three approximations with different orders. The resulting boundary integral equations are preconditioned Combined Field Integral Equations (CFIEs). An analytical spectral study confirms the expected behavior of the preconditioners, i.e., a better eigenvalue clustering especially in the elliptic part contrary to the standard CFIE of the first-kind. We provide various 2D numerical illustrations of the efficiency of the method for different smooth and non smooth geometries. In particular, the number of iterations is shown to be independent of the density of discretization points per wavelength which is not the case of the standard CFIE. In addition, it is less sensitive to the frequency. (10.1007/s42985-021-00075-x)
  DOI : 10.1007/s42985-021-00075-x
• A continuation method for building invisible obstacles in waveguides
  
  • Bera Antoine
  • Bonnet-Ben Dhia Anne-Sophie
  • Chesnel Lucas
  Quarterly Journal of Mechanics and Applied Mathematics, Oxford University Press (OUP), 2021, 74 (1), pp.83-116. We consider the propagation of acoustic waves at a given wavenumber in a waveguide which is unbounded in one direction. We explain how to construct penetrable obstacles characterized by a physical coefficient ρ which are invisible in various ways. In particular, we focus our attention on invisibility in reflection (the reflection matrix is zero), invisibility in reflection and transmission (the scattering matrix is the same as if there were no obstacle) and relative invisibility (two different obstacles have the same scattering matrix). To study these problems, we use a continuation method which requires to compute the scattering matrix S(ρ) as well as its differential with respect to the material index dS(ρ). The justification of the method also needs for the proof of abstract results of ontoness of well-chosen functionals constructed from the terms of dS(ρ). We provide a complete proof of the results in monomode regime when the wavenumber is such that only one mode can propagate. And we give all the ingredients to implement the method in multimode regime. We end the article by presenting numerical results to illustrate the analysis. (10.1093/qjmam/hbaa020)
  DOI : 10.1093/qjmam/hbaa020
• Mathematical and numerical analyses for the div-curl and div-curlcurl problems with a sign-changing coefficient
  
  • Ciarlet Patrick
  , 2021. We study the numerical approximation by edge finite elements of fields whose divergence and curl, or divergence and curl-curl, are prescribed in a bounded set $\Omega$ of $\mathbb{R}^3$, together with a boundary condition. Special attention is paid to solutions with low-regularity, in terms of the Sobolev scale $({\mathbf H}^{s}(\Omega))_{s>0}$. Among others, we consider an electromagnetic-like model including an interface between a classical medium and a metamaterial. In this setting the electric permittivity, and possibly the magnetic permeability, exhibit a sign-change at the interface. With the help of T-coercivity, we address the case of a model with one sign-changing coefficient, both for the model itself, and for its discrete version. Optimal error estimates are derived. Thanks to these results, we are also able to analyze the classical time-harmonic Maxwell equations, with one sign-changing coefficient.
• A DtN approach to the mathematical and numerical analysis in waveguides with periodic outlets at infinity
  
  • Fliss Sonia
  • Joly Patrick
  • Lescarret Vincent
  Pure and Applied Analysis, Mathematical Sciences Publishers, 2021. We consider the time harmonic scalar wave equation in junctions of several different periodic half-waveguides. In general this problem is not well posed. Several papers propose radiation conditions, i.e. the prescription of the behaviour of the solution at the infinities. This ensures uniqueness - except for a countable set of frequencies which correspond to the resonances- and yields existence when one is able to apply Fredholm alternative. This solution is called the outgoing solution. However, such radiation conditions are difficult to handle numerically. In this paper, we propose so-called transparent boundary conditions which enables us to characterize the outgoing solution. Moreover, the problem set in a bounded domain containing the junction with this transparent boundary conditions is of Fredholm type. These transparent boundary conditions are based on Dirichlet-to-Neumann operators whose construction is described in the paper. On contrary to the other approaches, the advantage of this approach is that a numerical method can be naturally derived in order to compute the outgoing solution. Numerical results illustrate and validate the method. (10.2140/paa.2021.3.487)
  DOI : 10.2140/paa.2021.3.487
• Imaging junctions of waveguides
  
  • Bourgeois Laurent
  • Fritsch Jean-François
  • Recoquillay Arnaud
  Inverse Problems and Imaging, AIMS American Institute of Mathematical Sciences, 2021. In this paper we address the identification of defects by the Linear Sampling Method in half-waveguides which are related to each other by junctions. Firstly a waveguide which is characterized by an abrupt change of properties is considered, secondly the more difficult case of several half-waveguides related to each other by a junction of complex geometry. Our approach is illustrated by some two-dimensional numerical experiments. (10.3934/ipi.2020065)
  DOI : 10.3934/ipi.2020065
• Modèles homogénéisés enrichis en présence de bords : Analyse et traitement numérique
  
  • Beneteau Clément
  , 2021. Quand on s’intéresse à la propagation des ondes dans un milieu périodique à basse fréquence (i.e. la longueur d’onde est grande devant la période), il est possible de modéliser le milieu périodique par un milieu homogène équivalent ou effectif qui a les mêmes propriétés macroscopiques. C’est la théorie de l’homogénéisation qui justifie d’un point de vue mathématique ce procédé. Ce procédé est très séduisant car les calculs numériques sont beaucoup moins couteux (la petite structure périodique a disparu) et des calculs analytiques sont de nouveau possibles dans certaines configurations. Les ondes dans le milieu périodique et dans le milieu effectif sont très proches d’un point de vue macroscopique sauf en présence de bords ou d’interfaces.En effet, il est bien connu que le modèle homogénéisé est obtenu en négligeant les effets de bords et par conséquent il est beaucoup moins précis aux bords du milieu périodique. Quand les phénomènes intéressants apparaissent aux bords du milieu (comme la propagation des ondes plasmoniques à la surface des métamatériaux par exemple), il semble donc difficile de faire confiance au modèle effectif.En revenant sur le processus d’homogénéisation, nous proposons un modèle homogénéisé qui est plus riche aux niveaux des bords. Le modèle homogénéisé enrichi est aussi simple que le modèle homogénéisé classique loin des interfaces, seule les conditions aux bords changent et prennent mieux en compte les phénomènes. Nous appliquons ce modèle à une équation elliptique dans le cas de la géométrie simple du demi-plan avec des conditions de type Dirichlet ou Neumann. D’un point de vue numérique, en plus des problèmes de cellule classiques qui apparaissent en homogénéisation, des problèmes de bandes périodiques doivent également être résolus. Pour finir, nous appliquons ces résultats à l'homogénéisation de l'équation des ondes en temps long et en présence de bords.
• Propagation dans les guides d'ondes
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Lunéville Éric
  , 2021. On va s'intéresser dans ce cours à la résolution des équations de l'acoustique dans un guide fermé, c'est à dire un milieu cylindrique de section transverse bornée, en régime périodique établi. Dans le premier chapitre, nous nous intéressons à un guide parfait (sans défaut) : nous montrons alors que la propagation peut être décrite à l'aide de solutions particulières, à variables séparées, appelés modes. Dans les chapitres suivants, nous montrons comment étudier ou simuler l'effet d'un défaut ou d'une perturbation du guide sur un tel mode. On présentera en particulier des méthodes permettant de calculer par éléments finis le champ diffracté par le défaut : la difficulté concerne l'écriture de conditions aux limites non réfléchissantes sur les frontières artificielles du domaine de calcul. L'intérêt de la thématique de ce cours est double : D'une part, les guides d'ondes sont présents dans de nombreux domaines d'applications. Ils peuvent être naturels (la mer est un guide acoustique) ou fabriqués par l'homme (ligne co-axiale, plaque élastique etc...). La présence du défaut peut à son tour être accidentelle (fissure dans une plaque élastique) ou voulue (chambre d'expansion jouant le rôle de filtre dans un silencieux d'automobile). Pour un défaut non souhaité, il est intéressant de pouvoir le localiser en mesurant sa réponse à une onde incidente, c'est l'objectif du CND (Contrôle Non Destructif) par ultrasons. Pour une perturbation voulue, l'intérêt de la simulation est d'accéder à une évaluation précise de son effet. D'autre part, nous verrons que les guides d'ondes offrent un cadre assez simple (on utilisera beaucoup la séparation de variables en coordonnées cartésiennes) pour présenter et étudier des méthodes plus générales : en particulier, les techniques de conditions transparentes que nous présenterons (opérateurs DtN et couches PML) sont également utilisées pour des simulations dans des domaines de propagation qui ne sont pas des guides, et peuvent être infinis dans 2 ou 3 directions.
• Scattering of acoustic waves by a nonlinear resonant bubbly screen
  
  • Pham Kim
  • Mercier Jean-François
  • Fuster Daniel
  • Marigo Jean-Jacques
  • Maurel Agnès
  Journal of Fluid Mechanics, Cambridge University Press (CUP), 2021, 906, pp.A19. Some of the authors of this publication are also working on these related projects: PARIS code View project Homogenization of thin and thick microstructured materials View project (10.1017/jfm.2020.799)
  DOI : 10.1017/jfm.2020.799
• Anatomy of Strike Slip Fault Tsunami-genesis
  
  • Elbanna Ahmed
  • Abdelmeguid Mohamed
  • Ma Xiao
  • Amlani Faisal
  • Bhat Harsha S.
  • Synolakis Costas
  • Rosakis Ares
  , 2021. Tsunami generation from earthquake induced seafloor deformations has long been recognized as a major hazard to coastal areas. Strike—slip faulting has generally been believed as insufficient for triggering large tsunamis, except through the generation of submarine landslides. Herein, we demonstrate that ground motions due to strike–slip earthquakes can contribute to the emergence of large tsunamis (>1m) underrather generic conditions. To this end, we have developed a computational framework that integrates models for earthquake rupture dynamics with models of tsunami generation and propagation. The three-dimensional time-dependent vertical and horizontal ground motions from spontaneous dynamic rupture models are used to drive boundary motions in the tsunami model. Our results suggest that super shearruptures propagating along strike–slip faults, traversing narrow and shallow bays are prime candidates for tsunami generation. We show that dynamic focusing and the large horizontal displacements, characteristic of strike-slip earthquakes on long faults, are critical drivers for the tsunami hazard. These findings point to intrinsic mechanisms for sizeable tsunami generation by strike–slip faulting, which do not require complex seismic sources, landslides, or complicated bathymetry. Furthermore, our model identifies three distinct phases in the tsunamic motion; an instantaneous dynamic phase, a lagging coseismic and a classical postseismic phase, each of which may affect coastal areas differently. We conclude that near-source tsunami hazards and risk from strike-slip faulting need to be re–evaluated. (10.31223/X57G72)
  DOI : 10.31223/X57G72
• General-purpose kernel regularization of boundary integral equations via density interpolation
  
  • Maltez Faria Luiz
  • Pérez-Arancibia Carlos
  • Bonnet Marc
  Computer Methods in Applied Mechanics and Engineering, Elsevier, 2021, 378, pp.113703. This paper presents a general high-order kernel regularization technique applicable to all four integral operators of Calder\'on calculus associated with linear elliptic PDEs in two and three spatial dimensions. Like previous density interpolation methods, the proposed technique relies on interpolating the density function around the kernel singularity in terms of solutions of the underlying homogeneous PDE, so as to recast singular and nearly singular integrals in terms of bounded (or more regular) integrands. We present here a simple interpolation strategy which, unlike previous approaches, does not entail explicit computation of high-order derivatives of the density function along the surface. Furthermore, the proposed approach is kernel- and dimension-independent in the sense that the sought density interpolant is constructed as a linear combination of point-source fields, given by the same {Green's function} used in the integral equation formulation, thus making the procedure applicable, in principle, to any PDE with known {Green's function}. For the sake of definiteness, we focus here on Nystr\"om methods for the (scalar) Laplace and Helmholtz equations and the (vector) elastostatic and time-harmonic elastodynamic equations. The method's accuracy, flexibility, efficiency, and compatibility with fast solvers are demonstrated by means of a variety of large-scale three-dimensional numerical examples. (10.1016/j.cma.2021.113703)
  DOI : 10.1016/j.cma.2021.113703
• Effective wave motion in periodic discontinua near spectral singularities at finite frequencies and wavenumbers
  
  • Guzina Bojan B
  • Bonnet Marc
  Wave Motion, Elsevier, 2021, 103, pp.102729. We consider the effective wave motion, at spectral singularities such as corners of the Brillouin zone and Dirac points, in periodic continua intercepted by compliant interfaces that pertain to e.g. masonry and fractured materials. We assume the Bloch-wave form of the scalar wave equation (describing anti-plane shear waves) as a point of departure, and we seek an asymptotic expansion about a reference point in the wavenumber-frequency space-deploying wavenumber separation as the perturbation parameter. Using the concept of broken Sobolev spaces to cater for the presence of kinematic discontinuities, we next define the "mean" wave motion via inner product between the Bloch wave and an eigenfunction (at specified wavenumber and frequency) for the unit cell of periodicity. With such projection-expansion approach, we obtain an effective field equation, for an arbitrary dispersion branch, near apexes of "wavenumber quadrants" featured by the first Brillouin zone. For completeness, we investigate asymptotic configurations featuring both (a) isolated, (b) repeated, and (c) nearby eigenvalues. In the case of repeated eigenvalues, we find that the "mean" wave motion is governed by a system of wave equations and Dirac equations, whose size is given by the eigenvalue multiplicity, and whose structure is determined by the participating eigenfunctions, the affiliated cell functions, and the direction of wavenumber perturbation. One of these structures is shown to describe the so-called Dirac points-apexes of locally conical dispersion surfaces-that are relevant to the generation of topologically protected waves. In situations featuring clusters of tightly spaced eigenvalues, the effective model is found to entail a Diraclike system of equations that generates "blunted" conical dispersion surfaces. We illustrate the analysis by numerical simulations for two periodic configurations in R 2 that showcase the asymptotic developments in terms of (i) wave dispersion, (ii) forced wave motion, and (iii) frequency-and wavenumber-dependent phonon behavior. (10.1016/j.wavemoti.2021.102729)
  DOI : 10.1016/j.wavemoti.2021.102729
• On a surprising instability result of Perfectly Matched Layers for Maxwell's equations in 3D media with diagonal anisotropy
  
  • Bécache Eliane
  • Fliss Sonia
  • Kachanovska Maryna
  • Kazakova Maria
  Comptes Rendus. Mathématique, Académie des sciences (Paris), 2021. The analysis of Cartesian Perfectly Matched Layers (PMLs) in the context of time-domain electromagnetic wave propagation in a 3D unbounded anisotropic homogeneous medium modelled by a diagonal dielectric tensor is presented. Contrary to the 3D scalar wave equation or 2D Maxwell's equations some diagonal anisotropies lead to the existence of backward waves giving rise to instabilities of the PMLs. Numerical experiments confirm the presented result. (10.5802/crmath.165)
  DOI : 10.5802/crmath.165
• Stability and Convergence Analysis of Time-domain Perfectly Matched Layers for The Wave Equation in Waveguides
  
  • Bécache Eliane
  • Kachanovska Maryna
  SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2021. This work is dedicated to the proof of stability and convergence of the Bérenger's perfectly matched layers in the waveguides for an arbitrary L ∞ damping function. The proof relies on the Laplace domain techniques and an explicit representation of the solution to the PML problem in the waveguide. A bound for the PML error that depends on the absorption parameter and the length of the PML is presented. Numerical experiments confirm the theoretical findings. (10.1137/20M1330543)
  DOI : 10.1137/20M1330543
• An automatic PML for acoustic finite element simulations in convex domains of general shape
  
  • Bériot Hadrien
  • Modave Axel
  International Journal for Numerical Methods in Engineering, Wiley, 2021, 122 (5), pp.1239-1261. This article addresses the efficient finite element solution of exterior acoustic problems with truncated computational domains surrounded by perfectly matched layers (PMLs). The PML is a popular nonreflecting technique that combines accuracy, computational efficiency, and geometric flexibility. Unfortunately, the effective implementation of the PML for convex domains of general shape is tricky because of the geometric parameters that are required to define the PML medium. In this work, a comprehensive implementation strategy is proposed. This approach, which we call the automatically matched layer (AML) implementation, is versatile and fully automatic for the end‐user. With the AML approach, the mesh of the layer is extruded, the required geometric parameters are automatically obtained during the extrusion step, and the practical implementation relies on a simple modification of the Jacobian matrix in the elementwise integrals. The AML implementation is validated and compared with other implementation strategies using numerical benchmarks in two and three dimensions, considering computational domains with regular and nonregular boundaries. A three‐dimensional application with a generally shaped domain generated using a convex hull is proposed to illustrate the interest of the AML approach for realistic industrial cases. (10.1002/nme.6560)
  DOI : 10.1002/nme.6560
• Local transparent boundary conditions for wave propagation in fractal trees (I). Method and numerical implementation
  
  • Joly Patrick
  • Kachanovska Maryna
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2021. 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 (10.1137/20M1362334)
  DOI : 10.1137/20M1362334