Publications

2022

• Edge states in rationally terminated honeycomb structures
  
  • Fefferman Charles L
  • Fliss Sonia
  • Weinstein Michael I
  Proceedings of the National Academy of Sciences of the United States of America, National Academy of Sciences, 2022, 119 (47), pp.e2212310119. 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. (10.1073/pnas.2212310119)
  DOI : 10.1073/pnas.2212310119
• Windowed Green function method for wave scattering by periodic arrays of 2D obstacles
  
  • Strauszer-Caussade Thomas
  • Faria Luiz
  • Fernandez-Lado Agustín
  • Pérez‐arancibia Carlos
  Studies in Applied Mathematics, Wiley-Blackwell, 2022, 150 (1), pp.277-315. 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. (10.1111/sapm.12540)
  DOI : 10.1111/sapm.12540
• Modélisation et simulation numérique de la propagation d'ondes électromagnétiques dans les câbles coaxiaux.
  
  • Beni Hamad Akram
  , 2022. Dans cette thèse, nous nous intéressons à la propagation des ondes électromagnétiques dans un réseau de câbles coaxiaux minces (constitués d'un matériau diélectrique qui entoure un fil intérieur métallique) avec sections transverses hétérogènes. Le premier objectif, atteint dans la thèse de G. Beck, était de réduire les équations de Maxwell 3D à un graphe quantique dans lequel on se ramène au calcul du potentiel et du courant électriques en résolvant des modèles 1D simplifiés. Ainsi, l'objectif principal de cette thèse est la validation numérique de ces modèles 1D.Dans un premier temps, nous avons proposé, analysé et mis en œuvre des méthodes numériques efficaces pour résoudre les modèles simplifiés 1D. Afin de réaliser la comparaison 1D/3D, un défi majeur est de concevoir des méthodes numériques pour résoudre les équations de Maxwell 3D qui sont adaptées à la spécificité des câbles électriques fins. Une procédure de discrétisation naïve basée sur un schéma explicite saute-mouton peut être vraiment coûteuse en raison de la finesse du câble. Nous avons alors proposé une approche originale consistant à adapter les éléments d'arête "Nedelec" à des mailles prismatiques allongées et à proposer une procédure de discrétisation temporelle hybride, explicite dans les directions longitudinales et implicite dans les directions transversales. En particulier, la condition de stabilité de la CFL qui en résulte n'est pas affectée par l'épaisseur du câble.Cependant, la méthode ci-dessus n'est efficace que pour des câbles parfaitement cylindriques : son extension naïve aux câbles déformés génère un recouplage longitudinal-transversal qui détruit l'efficacité de la méthode. En présence de déformations, la méthode doit donc être modifiée. En conséquence, afin de préserver le découplage longitudinal-transversal, nous proposons une méthode hybride combinant une discrétisation conforme dans les variables longitudinales et une méthode Galerkin discontinue dans les variables transversales. Cette méthode coïncide avec la précédente dans les parties cylindriques du câble.
• Scattering in a partially open waveguide: the inverse problem
  
  • Bourgeois Laurent
  • Fritsch Jean-François
  • Recoquillay Arnaud
  , 2022.
• Long time behaviour for electromagnetic waves in dissipative Lorentz media
  
  • Cassier Maxence
  • Joly Patrick
  • Rosas Martinez Luis Alejandro
  , 2022. 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.
• One-Way methods for wave propagation in complex flows
  
  • Ruello Maëlys
  • Rudel Clément
  • Pernet Sébastien
  • Brazier Jean-Philippe
  , 2022.
• A high-order discontinuous Galerkin Method using a mixture of Gauss-Legendre and Gauss-Lobatto quadratures for improved efficiency
  
  • Chaillat Stéphanie
  • Cottereau Régis
  • Sevilla Ruben
  , 2022. 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.
• An efficient numerical method for time domain electromagnetic wave propagation in co-axial cables
  
  • Beni Hamad Akram
  • Beck Geoffrey
  • Imperiale Sébastien
  • Joly Patrick
  Computational Methods in Applied Mathematics, De Gruyter, 2022. In this work we construct an efficient numerical method to solve 3D Maxwell's equations in coaxial cables. Our strategy is based upon an hybrid explicit-implicit time discretization combined with edge elements on prisms and numerical quadrature. One of the objective 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. (10.1515/cmam-2021-0195)
  DOI : 10.1515/cmam-2021-0195
• Robust treatment of cross points in Optimized Schwarz Methods
  
  • Claeys Xavier
  • Parolin Emile
  Numerische Mathematik, Springer Verlag, 2022, 151 (2), pp.405-442. In the field of Domain Decomposition (DD), Optimized Schwarz Method (OSM) appears to be one of the prominent techniques to solve large scale time-harmonic wave propagation problems. It is based on appropriate transmission conditions using carefully designed impedance operators to exchange information between sub-domains. The efficiency of such methods is however hindered by the presence of cross-points, where more than two sub-domains abut, if no appropriate treatment is provided. In this work, we propose a new treatment of the cross-point issue for the Helmholtz equation that remains valid in any geometrical interface configuration. We exploit the multi-trace formalism to define a new exchange operator with suitable continuity and isometry properties. We then develop a complete theoretical framework that generalizes classical OSM to partitions with cross points and contains a rigorous proof of geometric convergence, uniform with respect to the mesh discretization, for appropriate positive impedance operators. Extensive numerical results in 2D and 3D are provided as an illustration of the performance of the proposed method. (10.1007/s00211-022-01288-x)
  DOI : 10.1007/s00211-022-01288-x
• Convergence d'un couplage élastique-acoustique FEM-BEM itératif, global en temps
  
  • Nassor Alice
  • Chaillat Stéphanie
  • Bonnet Marc
  • Leblé Bruno
  • Barras Guillaume
  , 2022. 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.
• Méthode des éléments de frontière pour la mécanique des failles et le contrôle sismique
  
  • Bagur Laura
  • Chaillat Stéphanie
  • Semblat Jean-François
  • Stefanou Ioannis
  , 2022. 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.
• A non-overlapping domain decomposition method with perfectly matched layer transmission conditions for the Helmholtz equation
  
  • Royer Anthony
  • Geuzaine Christophe
  • Béchet Eric
  • Modave Axel
  Computer Methods in Applied Mechanics and Engineering, Elsevier, 2022, 395, pp.115006. 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. (10.1016/j.cma.2022.115006)
  DOI : 10.1016/j.cma.2022.115006
• Spectral theory for Maxwell's equations at the interface of a metamaterial. Part II: Limiting absorption, limiting amplitude principles and interface resonance
  
  • Cassier Maxence
  • Hazard Christophe
  • Joly Patrick
  Communications in Partial Differential Equations, Taylor & Francis, 2022, 47 (6), pp.1217-1295. 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. (10.1080/03605302.2022.2051188)
  DOI : 10.1080/03605302.2022.2051188
• Modélisation semi-analytique du bruit large bande produit par l’interaction entre un écoulement turbulent et un obstacle rigide de forme complexe
  
  • Trafny Nicolas
  • Serre Gilles
  • Cotté Benjamin
  • Mercier Jean-François
  , 2022. 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.
• Multidirectional sweeping preconditioners with non-overlapping checkerboard domain decomposition for Helmholtz problems
  
  • Dai Ruiyang
  • Modave Axel
  • Remacle Jean-François
  • Geuzaine Christophe
  Journal of Computational Physics, Elsevier, 2022 (453), pp.110887. This paper explores a family of generalized sweeping preconditionners for Helmholtz problems with non-overlapping checkerboard partition of the computational domain. The domain decomposition procedure relies on high-order transmission conditions and cross-point treatments, which cannot scale without an efficient preconditioning technique when the number of subdomains increases. With the proposed approach, existing sweeping preconditioners, such as the symmetric Gauss-Seidel and parallel double sweep preconditioners, can be applied to checkerboard partitions with different sweeping directions (e.g. horizontal and diagonal). Several directions can be combined thanks to the flexible version of GMRES, allowing for the rapid transfer of information in the different zones of the computational domain, then accelerating the convergence of the final iterative solution procedure. Several two-dimensional finite element results are proposed to study and to compare the sweeping preconditioners, and to illustrate the performance on cases of increasing complexity. (10.1016/j.jcp.2021.110887)
  DOI : 10.1016/j.jcp.2021.110887
• Maxwell's equations with hypersingularities at a conical plasmonic tip
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chesnel Lucas
  • Rihani Mahran
  Journal de Mathématiques Pures et Appliquées, Elsevier, 2022, 161, pp.70-110. 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. (10.1016/j.matpur.2022.03.001)
  DOI : 10.1016/j.matpur.2022.03.001
• Mathematical Analysis of Goldstein's Model for time harmonic acoustics in flow
  
  • Bensalah Antoine
  • Joly Patrick
  • Mercier Jean-François
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2022, 56 (2), pp.451-483. Goldstein’s equations have been introduced in 1978 as an alternative model to linearized Euler equations to model acoustic waves in moving fluids. This new model is particularly attractive since it appears as a perturbation of a simple scalar model: the potential model. In this work we propose a mathematical analysis of boundary value problems associated with Goldstein’s equations in the time-harmonic regime. (10.1051/m2an/2022007)
  DOI : 10.1051/m2an/2022007
• Maxwell's equations in presence of metamaterials
  
  • Rihani Mahran
  , 2022. 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.
• Improvement of hierarchical matrices for 3D elastodynamic problems with a complex wavenumber
  
  • Bagur Laura
  • Chaillat Stéphanie
  • Ciarlet Patrick
  Advances in Computational Mathematics, Springer Verlag, 2022, 48 (9). It is well known in the literature that standard hierarchical matrix (H-matrix) based methods, although very efficient for asymptotically smooth kernels, are not optimal for oscillatory kernels. In a previous paper, we have shown that the method should nevertheless be used in the mechanical engineering community due to its still important data-compression rate and its straightforward implementation compared to H 2-matrix, or directional, approaches. Since in practice, not all materials are purely elastic it is important to be able to consider visco-elastic cases. In this context, we study the effect of the introduction of a complex wavenumber on the accuracy and efficiency of H-matrix based fast methods for solving dense linear systems arising from the discretization of the elastodynamic (and Helmholtz) Green's tensors. Interestingly, such configurations are also encountered in the context of the solution of transient purely elastic problems with the convolution quadrature method. Relying on the theory proposed in [12] for H 2-matrices for Helmholtz problems, we study the influence of the introduction of damping on the data compression rate of standard H-matrices. We propose an improvement of H-matrix based fast methods for this kind of configuration. This work is complementary to the recent work [12]. Here, in addition to addressing another physical problem, we consider standard H-matrices, derive a simple criterion to introduce additional compression and we perform extensive numerical experiments. (10.1007/s10444-021-09921-3)
  DOI : 10.1007/s10444-021-09921-3
• Efficient evaluation of three-dimensional Helmholtz Green's functions tailored to arbitrary rigid geometries for flow noise simulations
  
  • Chaillat Stéphanie
  • Cotté Benjamin
  • Mercier Jean-François
  • Serre Gilles
  • Trafny Nicolas
  Journal of Computational Physics, Elsevier, 2022, 452. The Lighthill's wave equation provides an accurate characterization of the hydrodynamic noise due to the interaction between a turbulent flow and an obstacle, that is needed to get in many industrial applications. In the present study, to solve the Lighthill's equation expressed as a boundary integral equation, we develop an efficient numerical method to determine the three-dimensional Green's function of the Helmholtz equation in presence of an obstacle of arbitrary shape, satisfying a Neumann boundary condition. This so-called tailored Green's function is useful to reduce the computational costs to solve the Lighthill's equation. The first step consists in deriving an integral equation to express the tailored Green's function thanks to the free space Green's function. Then a Boundary Element Method (BEM) is used to compute tailored Green's functions. Furthermore, an efficient method is performed to compute the second derivatives needed for accurate flow noise determinations. The proposed approach is first tested on simple geometries for which analytical solutions can be determined (sphere, cylinder, half plane). In order to consider realistic geometries in a reasonable amount of time, fast BEMs are used: fast multipole accelerated BEM and hierarchical matrix based BEM. A discussion on the numerical efficiency and accuracy of these approaches in an industrial context is finally proposed. (10.1016/j.jcp.2021.110915)
  DOI : 10.1016/j.jcp.2021.110915
• The Morozov's principle applied to data assimilation problems
  
  • Bourgeois Laurent
  • Dardé Jérémi
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2022. 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. (10.1051/m2an/2022061)
  DOI : 10.1051/m2an/2022061
• On the approximation of electromagnetic fields by edge finite elements. Part 4: analysis of the model with one sign-changing coefficient
  
  • Ciarlet Patrick
  Numerische Mathematik, Springer Verlag, 2022, 152, pp.223-257. In electromagnetism, in the presence of a negative material surrounded by a classical material, the electric permittivity, and possibly the magnetic permeability, can exhibit a sign-change at the interface. In this setting, the study of electromagnetic phenomena is a challenging topic. We focus on the time-harmonic Maxwell equations in a bounded set $\Omega$ of ${\mathbb R}^3$, and more precisely on the numerical approximation of the electromagnetic fields by edge finite elements. Special attention is paid to low-regularity solutions, in terms of the Sobolev scale $({\boldsymbol{H}}^{\mathtt{s}}(\Omega))_{\mathtt{s}>0}$. With the help of T-coercivity, we address the case of one sign-changing coefficient, both for the model itself, and for its discrete version. Optimal a priori error estimates are derived. (10.1007/s00211-022-01315-x)
  DOI : 10.1007/s00211-022-01315-x
• On the justification of topological derivative for wave-based qualitative imaging of finite-sized defects in bounded media
  
  • Bonnet Marc
  Engineering Computations, Emerald, 2022, 39 (1), pp.313-336. The concept of topological derivative (TD) is known to provide, through its heuristic interpretation involving its sign and its spatial decay away from the true anomaly, a basis for the qualitative imaging of finite-sized anomalies. The TD imaging heuristic is currently partially backed by conditional mathematical justifications. Continuing earlier efforts towards the justification of TD-based identification, this work investigates the acoustic wave-based imaging of finite-sized (i.e. not necessarily small) medium anomalies embedded in bounded domains and affecting the leading-order term of the acoustic field equation. Both the probing excitation and the measurement are assumed to take place on the domain boundary. We extend to this setting the analysis approach previously used for unbounded media with either refraction-index anomalies and far-field measurements (Bellis et al., \emph{Inverse Problems} \textbf{29}:075012, 2013) or mass-density anomalies and meaurements at finite distance (Bonnet, Cakoni, \emph{Inverse Problems} \textbf{35}:104007, 2019). Like in the latter work, TD-based imaging functionals are reformulated for analysis using a suitable factorization of the acoustic fields, facilitated by a volume integral formulation. Our results, which echo corresponding results of our earlier investigations, conditionally validate the TD imaging heuristic. Moreover, we show on a geometrically simple configuration that the spatial behavior of the TD associated with standard $L^2$ cost functionals is degraded by ``echoes'' of the true anomaly, an aspect specific to the present bounded-domain framework. This undesirable effect is removed by a combination of (i) post-processing the measurements by application of a suitable integral operator (a treatment introduced by Ammari et al., 2011, for the analysis of TD-based imaging involving true flaws modelled using small-anomaly asymptotics), and (ii) expressing the background field as an incoming single-layer potential defined in the full space (after an idea used in Bonnet, Cakoni, 2019). Finally, we also show that selecting eigenfunctions of the source-to-measurement operator as excitations enhances the spatial decay properties of the TD functionals (10.1108/EC-08-2021-0471)
  DOI : 10.1108/EC-08-2021-0471
• Influence of chemistry on the steady solutions of hydrogen gaseous detonations with friction losses
  
  • Veiga-Lopez Fernando
  • Maltez Faria Luiz
  • Melguizo-Gavilanes J.
  Combustion and Flame, Elsevier, 2022, 240, pp.112050. 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. (10.1016/j.combustflame.2022.112050)
  DOI : 10.1016/j.combustflame.2022.112050
• Limiting amplitude principle and resonances in plasmonic structures with corners: numerical investigation
  
  • Carvalho Camille
  • Ciarlet Patrick
  • Scheid Claire
  Computer Methods in Applied Mechanics and Engineering, Elsevier, 2022, 388, pp.114207. 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 that occur in the lossy physical metallic case. (10.1016/j.cma.2021.114207)
  DOI : 10.1016/j.cma.2021.114207