Publications

2026

• Poisson-type problems with transmission conditions at boundaries of infinite metric trees
  
  • Kachanovska Maryna
  • Naderi Kiyan
  • Pankrashkin Konstantin
  Journal of Mathematical Analysis and Applications, Elsevier, 2026, 557 (1), pp.130261. The paper introduces a Poisson-type problem on a mixed-dimensional structure combining a Euclidean domain and a lower-dimensional self-similar component touching along a compact surface (interface). The lower-dimensional piece is a so-called infinite metric tree (one-dimensional branching structure), and the key ingredient of the study is a rigorous definition of the gluing conditions between the two components. These constructions are based on the recent concept of embedded trace maps and some abstract machineries derived from a suitable Green-type formula. The problem is then reduced to the study of Fredholm properties of a linear combination of Dirichlet-to-Neumann maps for the tree and the Euclidean domain, which yields desired existence and uniqueness results. One also shows that large finite sections of the tree can be used for an efficient approximation of solutions (10.1016/j.jmaa.2025.130261)
  DOI : 10.1016/j.jmaa.2025.130261
• Discretization in multilayered media with high contrasts: is it all about the boundaries?
  
  • Carvalho Camille
  • Chaillat Stéphanie
  • Tsogka Chrysoula
  • Cortes Elsie A
  , 2026. Wave propagation in multilayered media with high material contrasts poses significant numerical challenges, as large variations in wavenumbers lead to strong reflections and complex transmission of the incoming wave field. To address these difficulties, we employ a boundary integral formulation thereby avoiding volumetric discretization. In this framework, the accuracy of the numerical solution depends strongly on how the material interfaces are discretized. In this work, we demonstrate that standard meshing strategies based on resolving the maximum wavenumber across the domain become computationally inefficient in multilayered configurations, where high wavenumbers are confined to localized subdomains. Through a systematic study of multilayer transmission problems, we show that no simple discretization rule based on the maximum wavenumber or material contrasts emerges. Instead, the wavenumber of the background (exterior) medium plays a dominant role in determining the optimal boundary resolution. Building on these insights, we propose an adaptive approach that achieves uniform accuracy and efficient computation across multiple layers. Numerical experiments for a range of multilayer configurations demonstrate the scalability and robustness of the proposed approach.
• Htool-DDM: A C++ library for parallel solvers and compressed linear systems.
  
  • Marchand Pierre
  • Tournier Pierre-Henri
  • Jolivet Pierre
  Journal of Open Source Software, Open Journals, 2026, 11 (118), pp.9279. (10.21105/joss.09279)
  DOI : 10.21105/joss.09279
• Automated far-field sound field estimation combining robotized acoustic measurements and the boundary elements method
  
  • Pascal Caroline
  • Marchand Pierre
  • Chapoutot Alexandre
  • Doaré Olivier
  Acta Acustica, EDP Sciences, 2026. The identification and reconstruction of acoustic fields radiated by unknown structures is usually performed using either Sound Field Estimation or Near-field Acoustic Holography techniques. The latter turns out to be especially useful when data is only available close to the source, but information throughout the whole space is needed. Yet, the lack of amendable and efficient implementations of state-of-the-art solutions, as well as the laborious and often lengthy deployment of acoustic measurements continue to be significant obstacles to the practical application of such methods. The purpose of this work is to address both problems. First, a completely automated metrology setup is proposed, in which a robotic arm is used to gather extensive and accurately positioned acoustic data without any human intervention. The impact of the robot on acoustic pressure measurements is cautiously evaluated, and proved to remain limited below 1 kHz. The Sound Field Estimation is then tackled using the Boundary Element Method, and implemented using the FreeFEM software. Numerically simulated measurements have allowed us to assess the method accuracy, which matches theoretically expected results and proves to remain robust against positioning inaccuracies, provided that the robot is carefully calibrated. The overall solution has been successfully tested using actual robotized measurements of an unknown loudspeaker, with a reconstruction error of less than 30 %. (10.1051/aacus/2026017)
  DOI : 10.1051/aacus/2026017
• Asymptotic analysis at any order of Helmholtz's problem in a corner with a thin layer: an algebraic approach
  
  • Baudet Cédric
  Asymptotic Analysis, IOS Press, 2026. We consider the Helmholtz equation in an angular sector partially covered by a homogeneous layer of small thickness, denoted ε. We propose in this work an asymptotic expansion of the solution with respect to ε at any order. This is done using matched asymptotic expansion, which consists here in introducing different asymptotic expansions of the solution in three subdomains: the vicinity of the corner, the layer and the rest of the domain. These expansions are linked through matching conditions. The presence of the corner makes these matching conditions delicate to derive because the fields have singular behaviors. Our approach is to reformulate these matching conditions purely algebraically by writing all asymptotic expansions as formal series. By using algebraic calculus we reduce the matching conditions to scalar relations linking the singular behaviors of the fields. These relations have a convolutive structure and involve some coefficients that can be computed analytically. Our asymptotic expansion is justified rigorously with error estimates. (10.1177/09217134251389983)
  DOI : 10.1177/09217134251389983
• Wave propagation in the frequency regime in one-dimensional quasiperiodic media -Limiting absorption principle
  
  • Amenoagbadji Pierre
  • Fliss Sonia
  • Joly Patrick
  , 2026. <div><p>We study the one-dimensional Helmholtz equation with (possibly perturbed) quasiperiodic coefficients. Quasiperiodic functions are the restriction of higher dimensional periodic functions along a certain (irrational) direction. In classical settings, for real-valued frequencies, this equation is generally not well-posed: existence of solutions in L 2 is not guaranteed and uniqueness in L ∞ may fail. This is a well-known difficulty of Helmholtz equations, but it has never been addressed in the quasiperiodic case. We tackle this issue by using the limiting absorption principle, which consists in adding some imaginary part (also called absorption) to the frequency in order to make the equation well-posed in L 2 , and then defining the physically relevant solution by making the absorption tend to zero. In previous work, we introduced a definition of the solution of the equation with absorption based on Dirichlet-to-Neumann (DtN) boundary conditions. This approach offers two key advantages: it facilitates the limiting process and has a direct numerical counterpart. In this work, we first explain why the DtN boundary conditions have to be replaced by Robin-to-Robin boundary conditions to make the absorption go to zero. We then prove, under technical assumptions on the frequency, that the limiting absorption principle holds and we propose a numerical method to compute the physical solution.</p></div>
• Predicting topologically protected interface state with high-frequency homogenization
  
  • Lombard Bruno
  • Coutant Antonin
  • Touboul Marie
  , 2026. When two semi-infinite periodic media are joined together, a localized interface mode may exist, whose frequency belongs to their common band gap. Moreover, if certain spatial symmetries are satisfied, this mode is topologically protected and thus is robust to defects. A method has recently been proposed to identify the existence and the frequency of this mode, based on the computation of surface impedances at all the frequencies in the gap. In this work, we approximate the surface impedances thanks to highfrequency effective models, and therefore get a prediction of topologically protected interface states while only computing the solution of an eigenvalue problem at the edges of the bandgaps. We also show that the nearby eigenvalues high-frequency effective models give rise to a better approximation of the surface impedance.
• Discrete FEM-BEM coupling with the Generalized Optimized Schwarz Method
  
  • Boisneault Antonin
  • Bonazzoli Marcella
  • Claeys Xavier
  • Marchand Pierre
  , 2026. The present contribution aims at developing a non-overlapping Domain Decomposition (DD) approach to the solution of acoustic wave propagation boundary value problems based on the Helmholtz equation, on both bounded and unbounded domains. This DD solver, called Generalized Optimized Schwarz Method (GOSM), is a substructuring method, that is, the unknowns of an iteration are associated with the subdomains interfaces. We extend the analysis presented in a previous paper of one of the author to a fully discrete setting. We do not consider only a specific set of boundary conditions, but a whole class including, e.g., Dirichlet, Neumann, and Robin conditions. Our analysis will also cover interface conditions corresponding to a Finite Element Method - Boundary Element Method (FEM-BEM) coupling. In particular, we shall focus on three classical FEM-BEM couplings, namely the Costabel, Johnson-Nédélec and Bielak-MacCamy couplings. As a remarkable outcome, the present contribution yields well-posed substructured formulations of these classical FEM-BEM couplings for wavenumbers different from classical spurious resonances. We also establish an explicit relation between the dimensions of the kernels of the initial variational formulation, the local problems and the substructured formulation. That relation especially holds for any wavenumber for the substructured formulation of Costabel FEM-BEM coupling, which allows us to prove that the latter formulation is well-posed even at spurious resonances. Besides, we introduce a systematically geometrically convergent iterative method for the Costabel FEM-BEM coupling, with estimates on the convergence speed.
• Early-Reverberation Imaging Functions for Bounded Elastic Domains
  
  • Ducasse Eric
  • Rodriguez Samuel
  • Bonnet Marc
  Acta Acustica, EDP Sciences, 2026, 10, pp.2. For the ultrasonic inspection of bounded elastic structures, finite-duration imaging functions are derived in the Fourier-Laplace domain.The signals involved are exponentially windowed, so that early reflections are taken into account more strongly than later ones in the imaging methodology.Applying classical approaches to the general case of anisotropic elasticity, we express the Fréchet derivatives of the relevant data-misfit functional with respect to arbitrary perturbations of the mass density and stiffnesses in terms of forward and adjoint solutions.Their definitions incorporate the exponentially decaying weighting. The proposed finite-duration imaging functions are then defined on that basis.As some areas of the structure are less insonified than others, it is necessary to define normalized imaging functions to compensate for these variations.Our approach in particular aims to overcome the difficulty of dealing with bounded domains containing defects not located in direct line of sight from the transducers and measured signals of long duration.For this initiation work, we demonstate the potential of the proposed method on a two-dimensional test case featuring the imaging of mass and elastic stiffness variations in a region of a bounded isotropic medium that is not directly visible from the transducers. (10.1051/aacus/2025069)
  DOI : 10.1051/aacus/2025069
• Crouzeix-Raviart elements on simplicial meshes in $d$ dimensions
  
  • Bohne Nis-Erik
  • Ciarlet Patrick
  • Sauter Stefan
  Foundations of Computational Mathematics, Springer Verlag, 2026. In this paper we introduce Crouzeix-Raviart elements of general polynomial order $k$ and spatial dimension $d\geq2$ for simplicial finite element meshes. We give explicit representations of the non-conforming basis functions and prove that the conforming companion space, i.e., the conforming finite element space of polynomial order $k$ is contained in the Crouzeix-Raviart space. We prove a direct sum decomposition of the Crouzeix-Raviart space into (a subspace of) the conforming companion space and the span of the non-conforming basis functions. Degrees of freedom are introduced which are bidual to the basis functions and give rise to the definition of a local approximation/interpolation operator. In two dimensions or for $k=1$, these freedoms can be split into simplex and $(d-1)$ dimensional facet integrals in such a way that, in a basis representation of Crouzeix-Raviart functions, all coefficients which belong to basis functions related to lower-dimensional faces in the mesh are determined by these facet integrals. It will also be shown that such a set of degrees of freedom does not exist in higher space dimension and $k&gt;1$.