Publications

2025

• Limiting absorption principle for a hybrid resonance in a two-dimensional cold plasma
  
  • Kachanovska Maryna
  • Peillon Étienne
  , 2025. An extended version of the manuscript
• Poisson-type problems with transmission conditions at boundaries of infinite metric trees
  
  • Kachanovska Maryna
  • Naderi Kiyan
  • Pankrashkin Konstantin
  , 2026, pp.130261. 46 pages, 4 figures, preliminary version (10.1016/j.jmaa.2025.130261)
  DOI : 10.1016/j.jmaa.2025.130261
• What does it mean for a 3D star-shaped scatterer to be small in the time domain?
  
  • Kachanovska Maryna
  • Savchuk Adrian
  , 2025. In the frequency domain wave scattering problems, obstacles can be effectively replaced by point scatterers as soon as the wavelength of the incident wave exceeds significantly their diameter. The situation is less clear in the time domain, where recent works suggest the presence of an additional temporal scale that quantifies the smallness of the obstacle. In this paper we argue that this is not necessarily the case, and that it is possible to construct asymptotic models with an error that does not deteriorate in time, at least in the case of a sound-soft scattering problem by a star-shaped obstacle in 3D.
• Trapped modes in electromagnetic waveguides
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chesnel Lucas
  • Fliss Sonia
  , 2025. We consider the Maxwell's equations with perfect electric conductor boundary conditions in three-dimensional unbounded domains which are the union of a bounded resonator and one or several semi-infinite waveguides. We are interested in the existence of electromagnetic trapped modes, i.e. $L^2$ solutions of the problem without source term. These trapped modes are associated to eigenvalues of the Maxwell's operator, that can be either below the essential spectrum or embedded in it. First for homogeneous waveguides, we present different families of geometries for which we can prove the existence of eigenvalues. Then we exhibit certain non homogeneous waveguides with local perturbations of the dielectric constants that support trapped modes. Let us mention that some of the mechanisms we propose are very specific to Maxwell's equations and have no equivalent for the scalar Dirichlet or Neumann Laplacians.
• Spurious resonances for substructured FEM-BEM coupling
  
  • Boisneault Antonin
  • Bonazzoli Marcella
  • Claeys Xavier
  • Marchand Pierre
  , 2025. We are interested in time-harmonic acoustic scattering by an impenetrable obstacle in a medium where the wavenumber is constant in an exterior unbounded subdomain and is possibly heterogeneous in a bounded subdomain. The associated Helmholtz boundary value problem can be solved by coupling the Finite Element Method (FEM) in the heterogeneous subdomain with the Boundary Element Method (BEM) in the homogeneous subdomain. Recently, we designed and analyzed a new substructured FEM-BEM formulation, called Generalized Optimized Schwarz Method (GOSM). Unfortunately, it is well known that, even when the initial boundary value problem is well-posed, the variational formulation of classical FEM-BEM couplings can be ill-posed for certain wavenumbers, called spurious resonances. In this paper, we focus on the Johnson-Nédélec and Costabel couplings and show that the GOSM derived from both is not immune to that issue. In particular, we give an explicit expression of the kernel of the local operator associated with the interface between the FEM and BEM subdomains. That kernel and the one of classical FEM-BEM couplings are simultaneously non-trivial.
• Scattering from a random thin coating of nanoparticles: the Dirichlet case
  
  • Boucart Amandine
  • Fliss Sonia
  • Giovangigli Laure
  , 2025. We study the time-harmonic scattering by a heterogeneous object covered with a thin layer of randomly distributed sound-soft nanoparticles. The size of the particles, their distance between each other and the layer's thickness are all of the same order but small compared to the wavelength of the incident wave. Solving the Helmholtz equation in this context can be very costly and the simulation depends on the given distribution of particles. To circumvent this, we propose, via a multi-scale asymptotic expansion of the solution, an effective model where the layer of particles is replaced by an equivalent boundary condition. The coefficients that appear in this equivalent boundary condition depend on the solutions to corrector problems of Laplace type defined on unbounded random domains. Under the assumption that the particles are distributed given a stationary and mixing random point process, we prove that those problems admit a unique solution in the proper space. We then establish quantitative error estimates for the effec tive model and present numerical simulations that illustrate our theoretical results.
• Identification of bottom deformations of the ocean from surface measurements
  
  • Bourgeois Laurent
  • Mercier Jean-François
  • Terrine Raphaël
  Inverse Problems and Imaging, AIMS American Institute of Mathematical Sciences, 2025, 19 (6), pp.1075-1113. In this paper we consider a general scheme to solve two different inverse problems related to oceanography, that is retrieving either a tsunami or the shape of the seabed from the measurement of the free surface perturbation. We consider two dimensional geometries and linear potential models in the frequency regime. Such general scheme consists in firstly recovering the potential in the whole domain and secondly compute the seeked parameter at the bottom of the ocean, which in the two inverse problems is a function involved in a more or less complicated boundary condition. The first step amounts to solve an ill-posed Cauchy problem for the Laplace or Helmholtz equation, which we regularize by using a mixed formulation of the Tikhonov regularization and the Morozov principle to compute the regularization parameter. The computation of such Tikhonov-Morozov solution is based on an iterative method consisting in solving a sequence of weak formulations which are discretized with the help of a simple Lagrange type Finite Element Method. In the particular case of the acoustic model, we need to solve a Laplace-type equation associated with the noisy Neumann boundary data and compute the noise amplitude of its solution. A probabilistic method is proposed to obtain such amplitude of noise. Some numerical experiments show the feasibility of our strategy. (10.3934/ipi.2025008)
  DOI : 10.3934/ipi.2025008
• $T$-coercivity: a practical tool for the study of variational formulations in Hilbert spaces
  
  • Ciarlet Patrick
  , 2025.
• Surface Plasmon Polariton Excitation in Time-modulated Media
  
  • Raziman T. V.
  • Touboul Marie
  • Sapienza Riccardo
  • Craster Richard V.
  • Rodríguez-Fortuño Francisco J.
  , 2025. Surface plasmon polaritons (SPPs) are central to application areas such as sensing, energy harvesting, and nanoscale optics, and are typically excited via spatial structuring -- an approach lacking dynamic control. We demonstrate that step-like time modulation allows the excitation and out-coupling of SPPs through modulating a dispersive Drude-like metal thereby modelling realistic transparent conducting oxides; this establishes a pathway for the active control and extraction of plasmons in experimentally viable time-varying systems. Using finite-difference time-domain simulations we show that time modulation facilitates both the launching and radiation of surface plasmons with frequencies governed by the dispersion of the bounding media. Our results also reveal the generation of time-reflected waves and the emergence of a magnetostatic mode required for matching boundary conditions at the temporal interface.
• Asymptotic models for time-domain scattering by small particles of arbitrary shapes
  
  • Kachanovska Maryna
  • Savchuk Adrian
  , 2025. In this work, we investigate time-dependent wave scattering by multiple small particles of arbitrary shape. To approximate the solution of the associated boundary-value problem, we derive an asymptotic model that is valid in the limit as the particle size tends to zero. Our method relies on a boundary integral formulation, semi-discretized in space using a Galerkin approach with appropriately chosen basis functions, s.t. convergence is achieved as the particle size vanishes rather than by increasing the number of basis functions. Since the computation of the Galerkin matrix involves double integration over particles, the method can become computationally demanding when the number of obstacles is large. To address this, we also derive a simplified model and consider the Born approximation to improve computational efficiency. For each model, we provide an error analysis, supported and validated by numerical experiments.
• Local Multiple Traces Formulation for Transmission Problems in Linear Elasticity
  
  • Chaillat Stéphanie
  • Darbas Marion
  • Escapil-Inchauspé Paul
  • Jerez-Hanckes Carlos
  , 2025. We investigate the use of the local Multiple Trace Formulation (MTF) in solving time-harmonic elastic wave transmission problems. Originally devised for heterogeneous acoustic media, MTF recasts the boundary value problem as a well-posed system of first-kind boundary integral equations, naturally amenable to parallelization and preconditioning. The formulation takes independent displacement and traction unknowns per subdomain, enforces Calderón identities locally, and imposes transmission conditions weakly across interfaces. We restrict our analysis to single homogeneous scatterers, representing a foundational step toward the application of MTF techniques in heterogeneous elasticity transmission problems. For the sake of clarity, all derivations in the one-dimensional setting are carried out explicitly, with illustrative examples provided in two dimensions. We analyze the effects of frequency and material contrast on the convergence of the GMRES iterative solver. Finally, we present preliminary results for an elastic Calderón preconditioner and discuss its potential to further accelerate iterative solvers.
• Preconditioning of GMRES for Helmholtz problems with quasimodes
  
  • Dolean Victorita
  • Marchand Pierre
  • Modave Axel
  • Raynaud Timothée
  , 2025. Finite element methods are effective for Helmholtz problems involving complex geometries and heterogeneous media. However, the resulting linear systems are often large, indefinite, and challenging for iterative solvers, particularly at high wave numbers or near resonant conditions. We derive a GMRES convergence bound that incorporates the nonlinear behavior of the relative residual and relates convergence to harmonic Ritz values. This perspective reveals how small eigenvalues associated with quasimodes can hinder convergence, and when they cease to have an effect. These phenomena occur in domain decomposition, and we illustrate them through numerical experiments. We also combine domain decomposition methods with deflation techniques using (approximate) eigenvectors tailored to resonant regimes. Their impact on GMRES performance is evaluated.
• Hybrid FEM/IPDG semi-implicit schemes for time domain electromagnetic wave propagation in non cylindrical coaxial cables
  
  • Beni Hamad Akram
  • Imperiale Sébastien
  • Joly Patrick
  , 2025. In this work, we develop an efficient numerical method for solving 3D Maxwell's equations in non-cylindrical coaxial cables. The main challenge arises from the elongated geometry of the computational domain, which induces strong anisotropy between the longitudinal direction (along the cable) and the transverse directions (within the cross-sections). This leads to the use of highly anisotropic meshes, where the longitudinal mesh size is much larger than the transverse one.<p>Our objective is to design a numerical scheme that is explicit in the longitudinal direction, with a CFL stability condition depending only on the longitudinal mesh size. In a previous work, we achieved this for cylindrical cables by employing prismatic edge elements, 1D quadrature for longitudinal mass lumping, and a hybrid explicit/implicit time discretization. The present paper extends this approach to non-cylindrical cables, addressing several new difficulties with the following key ingredients: (1) representing the cable as a deformation of a reference cylindrical cable and employing mapping techniques between the physical and reference domains; (2) using an anisotropic space discretization that combines an interior penalty discontinuous Galerkin (IPDG) method in the transverse directions with a conforming finite element method in the longitudinal direction; (3) utilizing prismatic edge elements on a prismatic mesh of the reference cable; and (4) adapting the construction of the hybrid explicit-implicit time discretization to the new structure of the semidiscrete problem. From a theoretical perspective, the main difficulty lies in the stability analysis, which requires extending and adapting standard techniques for DG methods in space and energy methods in time.</p>
• On the application of the T-coercivity method for the Helmholtz problem with sign-changing coefficients
  
  • Chaaban Farah
  , 2025. To solve transmission problems with sign-changing coefficients, one can apply the T-coercivity method, which imposes specific mesh conditions near the interface to ensure optimal convergence rates for the finite element approximation. This method was initially proposed and analyzed for the quasi-static case. The aim in this work is to extend it and prove its convergence for the case of non-zero frequency. Additionally, we check its convergence for a general compact perturbation and outline key ideas for a 3D polyhedral interface. Our theoretical results are validated through a numerical test.
• Spectrum of slip dynamics, scaling and statistical laws emerge from simplified model of fault and damage zone architecture
  
  • Almakari Michelle
  • Kheirdast N.
  • Villafuerte C.
  • Thomas Marion Y.
  • Dubernet P.
  • Cheng J.
  • Gupta A.
  • Romanet P.
  • Chaillat S.
  • Bhat Harsha S.
  Journal of Geophysical Research : Solid Earth, American Geophysical Union, 2025. <div><p>Seismological and geodetic observations of a fault zone reveal a wide range of slip dynamics, scaling, and statistical laws. However, the underlying physical mechanisms remain unclear. In this study, we show that incorporating an off-fault damage zone-characterized by distributed fractures surrounding a main fault-can reproduce many key features observed in seismic and geodetic data. We model a 2D shear fault zone in which off-fault cracks follow power-law size and density distributions, and are oriented either optimally or parallel to the main fault. All fractures follow the rate-and-state friction law with parameters chosen such that each can host slip instabilities. We do not introduce spatial heterogeneities in the frictional properties of the fault. Using quasi-dynamic boundary integral simulations accelerated by hierarchical matrices, we simulate slip dynamics of this system and analyze the events produced both on and off the main fault. Despite the spatially uniform frictional properties, we observe a natural continuum from slow to fast ruptures, as observed in nature. Our simulations reproduce the Omori law, the inverse Omori law, the Gutenberg-Richter scaling, and the moment-duration scaling. We also observe seismicity localizing toward the main fault when an event is about to nucleate on the main fault. During slow slip events, off-fault seismicity migrates in a pattern resembling a fluid diffusion front, despite the absence of fluids in the model. We also show that tremors, Very Low Frequency Earthquakes (VLFEs), Low Frequency Earthquakes (LFEs), Slow Slip Events (SSEs), and earthquakes (EQs) can all emerge naturally in the ‘digital twin’ framework.</p></div>
• A global-in-time domain decomposition approach for transient acoustic-elastic interaction
  
  • Bonnet Marc
  • Chaillat Stéphanie
  • Nassor Alice
  , 2025. This work develops a global-in-time iterative domain decomposition approach for transient fluid-structure interaction problems involving acoustic scattering by elastic obstacles. The proposed method, inspired by optimized Schwarz waveform relaxation algorithms, proceeds by iteratively exchanging Robin boundary conditions, enabling the non-intrusive coupling of distinct fluid and structure solvers, and works for arbitrary transient incident acoustic fields. We prove the convergence of the proposed coupling iterations in continuous form. A BEM-FEM coupling implementation of the method is then validated against a reference analytical solution, and its efficiency, accuracy and robustness demonstrated through numerical experiments on configurations representative of potential industrial applications.
• Portage GPU d'un solveur éléments finis discontinus hybridisé pour les problèmes d'ondes en fréquence
  
  • Chabib Ahmed
  • Greffe Roland
  • Geuzaine Christophe
  • Modave Axel
  , 2025. Dans ce travail, nous nous intéressons à la résolution par éléments finis de problèmes de propagation d'ondes en régime harmonique de très grande taille. L'utilisation de cartes graphiques (GPU) permet d'accélérer les calculs, mais il est difficile d'en exploiter pleinement la puissance. Nous considérons une méthode d'éléments finis discontinus de type Galerkin (DG) avec des flux amonts, hybridisée utilisant des variables de transmission définies aux faces des éléments. L'élimination des variables physiques conduit à un système linéaire adapté pour une résolution itérative et une implémentation parallèle efficace sur GPU. Après une description de la méthode, appelée CHDG, nous présentons quelques stratégies de mise en oeuvre sur GPU et nous comparons et discutons leurs performances.
• Isolated Rotor Blade Shape Sensitivity for Aeroacoustic Optimization Using a Discrete Adjoint Framework
  
  • Mohammedi Yacine
  • Daroukh Majd
  • Buszyk Martin
  • Hajczak Antoine
  • Salah El Din Itham
  • Bonnet Marc
  , 2025. A discrete adjoint framework is developed to optimize rotor self-noise from steady fluid simulations in the rotating frame. To this end, a simplified expression of the off-body frequency-domain Ffowcs-Williams and Hawkings (FW-H) equation is derived for far-field observers, following the model of Hanson and Parzych (1993) originally written for on-body surfaces. The latter is implemented and compared against the results given by an established time-domain FW-H solver. Far-field acoustic pressure sensitivities are derived analytically and validated by comparison with second-order accurate finite differences. The sensitivities of any objective function expressed in terms of the acoustic pressure can therefore be reconstructed. Then the discrete adjoint of a Reynolds-averaged Navier-Stokes solver provides the objective function gradients with respect to the blade shape parameters. The complete workflow is validated against finite difference evaluations on an isolated open rotor in cruise conditions. (10.2514/6.2025-3367)
  DOI : 10.2514/6.2025-3367
• Convergence rates of curved boundary element methods for the 3D Laplace and Helmholtz equations
  
  • Faria Luiz
  • Marchand Pierre
  • Montanelli Hadrien
  , 2025. We establish improved convergence rates for curved boundary element methods applied to the three-dimensional (3D) Laplace and Helmholtz equations with smooth geometry and data. Our analysis relies on a precise analysis of the consistency errors introduced by the perturbed bilinear and sesquilinear forms. We illustrate our results with numerical experiments in 3D based on basis functions and curved triangular elements up to order four.
• Convergence analysis of GMRES applied to Helmholtz problems near resonances
  
  • Dolean Victorita
  • Marchand Pierre
  • Modave Axel
  • Raynaud Timothée
  , 2025. In this work we study how the convergence rate of GMRES is influenced by the properties of linear systems arising from Helmholtz problems near resonances or quasi-resonances. We extend an existing convergence bound to demonstrate that the approximation of small eigenvalues by harmonic Ritz values plays a key role in convergence behavior. Next, we analyze the impact of deflation using carefully selected vectors and combine this with a Complex Shifted Laplacian preconditioner. Finally, we apply these tools to two numerical examples near (quasi-)resonant frequencies, using them to explain how the convergence rate evolves.
• On some coupled local and nonlocal diffusion models
  
  • Borthagaray Juan Pablo
  • Ciarlet Patrick
  , 2025. We study problems in which a local model is coupled with a nonlocal one. We propose two energies: both of them are based on the same classical weighted $H^1$-semi norm to model the local part, while two different weighted $H^s$-semi norms, with $s \in (0, 1)$, are used to model the nonlocal part. The corresponding strong formulations are derived. In doing so, one needs to develop some technical tools, such as suitable integration by parts formulas for operators with variable diffusivity, and one also needs to study the mapping properties of the Neumann operators that arise. In contrast to problems coupling purely local models, in which one requires transmission conditions on the interface between the subdomains, the presence of a nonlocal operator may give rise to nonlocal fluxes. These nonlocal fluxes may enter the problem as a source term, thereby changing its structure. Finally, we focus on a specific problem, that we consider most relevant, and study regularity of solutions and finite element discretizations. We provide numerical experiments to illustrate the most salient features of the models.
• Probing the speckle to estimate the effective speed of sound, a first step towards quantitative ultrasound imaging
  
  • Garnier Josselin
  • Giovangigli Laure
  • Goepfert Quentin
  • Millien Pierre
  , 2025. <div><p>In this paper, we present a mathematical model and analysis for a new experimental method [Bureau and al., arXiv:2409.13901, 2024] for effective sound velocity estimation in medical ultrasound imaging. We perform a detailed analysis of the point spread function of a medical ultrasound imaging system when there is a mismatch between the effective sound speed in the medium and the one used in the backpropagation imaging functional. Based on this analysis, an estimator for the speed of sound error is introduced. Using recent results on stochastic homogenization of the Helmholtz equation, we provide a representation formula for the field scattered by a random multi-scale medium (whose acoustic behavior is similar to a biological tissue) in the time-harmonic regime. We then prove that statistical moments of the imaging function can be accessed from data collected with only one realization of the medium. We show that it is possible to locally extract the point spread function from an image constituted only of speckle and build an estimator for the effective sound velocity in the micro-structured medium. Some numerical illustrations are presented at the end of the paper.</p></div>
• A hybridizable discontinuous Galerkin method with transmission variables for time-harmonic electromagnetic problems
  
  • Rappaport Ari
  • Chaumont-Frelet Théophile
  • Modave Axel
  , 2025. The CHDG method is a hybridizable discontinuous Galerkin (HDG) finite element method suitable for the iterative solution of time-harmonic wave propagation problems. Hybrid unknowns corresponding to transmission variables are introduced at the element interfaces and the physical unknowns inside the elements are eliminated, resulting in a hybridized system with favorable properties for fast iterative solution. In this paper, we extend the CHDG method, initially studied for the Helmholtz equation, to the time-harmonic Maxwell equations. We prove that the local problems stemming from hybridization are well-posed and that the fixed-point iteration naturally associated to the hybridized system is contractive. We propose a 3D implementation with a discrete scheme based on nodal basis functions. The resulting solver and different iterative strategies are studied with several numerical examples using a high-performance parallel C++ code.
• Open Review of "Normal form analysis of nonlinear oscillator equations with automated arbitrary order expansions
  
  • de Figueiredo Stabile André
  • Touzé Cyril
  • Vizzaccaro Alessandra
  • Römer Ulrich
  • Raze Ghislain
  • Chaillat Stéphanie
  , 2025.
• Méthode hybride de simulation de champs ultrasonores dans une grande structure stratifiée avec des objets au contact
  
  • Kubecki Romain
  • Ducasse Eric
  • Bonnet Marc
  • Deschamps Marc
  , 2025. Ce travail a pour objectif de simuler la propagation d'ultrasons dans une structure stratifiée de grande taille comportant des objets au contact (de type traducteur, raidisseur, ou autre), dans un contexte de contrôle non destructif. La taille modérée des objets permet leur simulation par éléments finis, qui est par contre prohibitive pour la structure stratifiée de base. Si cette dernière est de géométrie canonique (plane ou tubulaire à symétrie de révolution), les champs peuvent en revanche être calculés par une méthode semi-analytique rapide utilisant des transformées de Laplace en temps et de Fourier par rapport aux coordonnées « longitudinales » (plan de la plaque ou positions axiale et azimutale dans le tube). En effet, dans le domaine <latex>(k,r,s)</latex> (<latex>k~vecteur</latex> d'onde, <latex>r~position</latex> dans l'épaisseur, <latex>s~variable</latex> de Laplace), le problème de propagation peut être résolu de manière exacte, et massivement parallélisable. Pour exploiter les atouts des deux méthodes, nous proposons une approche itérative de couplage par <i>décomposition de domaine</i> (DDM), reposant sur une suite de problèmes de propagation dans chaque sous-domaine comprenant sur leur interface commune des conditions aux limites dépendant des solutions de l'itération précédente. La littérature montre que le choix de conditions de Robin (de type impédance) entre deux domaines couplés garantit dans beaucoup de situations la convergence des itérations de couplage. Nous prouvons que cette convergence a bien lieu pour notre contexte particulier et présentons une validation numérique préliminaire en configuration 2D. Le caractère spatialement non-local du traitement semi-analytique de la structure stratifiée nous conduit ensuite à construire des fonctions de base négligeables en-dehors d'un voisinage de l'interface et à développer un protocole spécifique pour leur couplage avec les éléments finis. Ces deux aspects constituent les principaux ingrédients de la méthode hybride proposée ici. <latex>\medskip\hspace20mm</latex><i>Ce travail est financé par la DGA-AID et le CEA-List.</i>