2025Activity report​​Project-TeamPOEMS

7.1.1 COFFEE​

• Functional Description:
  COFFEE is​‌ an adapted fast BEM​​ solver to model acoustic​​​‌ and elastic wave propagation​ (full implementation in Fortran​‌ 90). The 3-D acoustic​​ or elastodynamic equations are​​​‌ solved with the boundary​ element method accelerated by​‌ the multi-level fast multipole​​ method or a hierarchical-matrices​​​‌ based representation of the​ system matrix. The fundamental​‌ solutions for the infinite​​ space are used in​​​‌ this implementation. A boundary​ element-boundary element coupling strategy​‌ is also implemented so​​ multi-region problems (strata inside​​​‌ a valley for example)​ can be solved. In​‌ order to accelerate the​​ convergence of the iterative​​​‌ solver, various analytic or​ algebraic preconditioners are available.​‌ Finally, an anisotropic mesh​​ adaptation strategy is used​​​‌ to further reduce the​ computational times.
• URL:
  https://­uma.­ensta-paris.­fr/­soft/­COFFEE/​‌
• Contact:
  Stéphanie Chaillat Loseille​​

7.1.2 Htool-DDM

• Keywords:
  Hierarchical​​​‌ matrices, Domain decomposition, Preconditioner​
• Functional Description:
  Htool-DDM is​‌ a lightweight, header-only C++14​​ library that provides parallel​​​‌ iterative solvers with domain​ decomposition preconditioners, relying on​‌ an in-house hierarchical matrix​​ compression for dense/compressed linear​​​‌ systems.
• URL:
  https://­github.­com/­htool-ddm/­htool
• Contact:​
  Pierre Marchand

7.1.3 HMatrices.jl​‌

• Keywords:
  Boundary element method,​​ Hierarchical matrices
• Functional Description:​​​‌
  This package provides some​ functionality for assembling as​‌ well as for doing​​ linear algebra with hierarchical​​​‌ matrices with a strong​ focus in applications arising​‌ in boundary integral equation​​ methods. It includes shared​​ as well as distributed​​​‌ memory assembly and matrix/vector‌ product, as well as‌​‌ a shared memory LU​​ factorization.
• URL:
  https://­github.­com/­WaveProp/­HMatrices.­jl
• Contact:​​​‌
  Luiz Maltez Faria

7.1.4‌ DDMTool

• Keywords:
  Finite element‌​‌ modelling, C++
• Functional Description:​​
  Finite element library: P0,​​​‌ P1 Lagrange, edge finite‌ elements, in 1D, 2D,‌​‌ 3D volume and surface​​ meshes.
• Contact:
  Xavier Claeys​​​‌

7.1.5 BemTool

• Keyword:
  Boundary‌ element method
• Functional Description:‌​‌
  BemTool is a C++​​ header-only library implementing the​​​‌ boundary element method (BEM)‌ for the discretisation of‌​‌ the Laplace, Helmholtz and​​ Maxwell equations, in 2D​​​‌ and 3D. Its main‌ purpose is the assembly‌​‌ of classical boundary element​​ matrices, which can be​​​‌ compressed and inverted through‌ its interface with the‌​‌ HTool library.
• URL:
  https://­github.­com/­xclaeys/­BemTool​​
• Contact:
  Xavier Claeys

7.1.6​​​‌ XLiFE++

• Name:
  eXtended Library‌ of Finite Elements in‌​‌ C++
• Keywords:
  Finite element​​ modelling, Edge elements, Discontinuous​​​‌ Galerkin
• Functional Description:
  XLiFE++‌ is an FEM-BEM C++‌​‌ library developed by POEMS​​ laboratory, that can solve​​​‌ 1D/2D/3D, scalar/vector, transient/stationary/harmonic problems.‌
• URL:
  https://­xlifepp.­pages.­math.­cnrs.­fr
• Contact:
  Eric‌​‌ Lunéville

Construction of transparent conditions‌​‌ for electromagnetic waveguides

Participants:​​ Anne-Sophie Bonnet-Ben Dhia,​​​‌ Sonia Fliss, Aurélien‌ Parigaux.

This work‌​‌ is done in the​​ framework of the PhD​​​‌ of Aurélien Parigaux, co-advised‌ by Anne-Sophie Bonnet-Ben Dhia‌​‌ and Lucas Chesnel from​​ Inria team IDEFIX, in​​​‌ collaboration with Sonia Fliss.‌

We are particularly interested‌​‌ in computing the electromagnetic​​ field in typically fiber-optic​​​‌ tapers or optical multiplexers,‌ where several semi-infinite waveguides‌​‌ interact in a bounded​​ zone of space. In​​​‌ this context, in order‌ to reduce the computation‌​‌ to the bounded region​​ (using for instance a​​​‌ FE method), one has‌ to truncate the guides‌​‌ and impose adapted transparent​​ conditions on the artificial​​​‌ boundaries to minimize spurious‌ reflections. This question is‌​‌ very well understood for​​ scalar models of acoustic​​​‌ waveguides, but remains a‌ delicate subject for electromagnetic‌​‌ waveguides.

In the case​​ where the truncated waveguide​​​‌ is isotropic and homogeneous,‌ it is known that‌​‌ a transparent boundary condition​​ connecting the tangential electric​​​‌ field to the tangential‌ magnetic field at the‌​‌ artificial boundary, the so-called​​ EtM condition, can be​​​‌ written using a modal‌ expansion on transverse modes‌​‌ (TE, TM and TEM).​​ Another possibility is the​​​‌ use of perfectly matched‌ layers (PML).

The design‌​‌ of suitable transparent boundary​​ conditions is less obvious​​​‌ for guides that are‌ heterogeneous in the cross-section.‌​‌ The difficulties arise from​​ the loss of self-adjointness​​​‌ of the spectral problem,‌ whose modes are the‌​‌ eigensolutions. In particular, the​​ transverse electric fields are​​​‌ no longer orthogonal in‌ $L^2$ of the‌​‌ cross section, modal expansions​​ are no longer available,​​​‌ and inverse modes can‌ occur (with phase and‌​‌ group velocities of opposite​​ signs), which prevents the​​​‌ use of PMLs. We‌ have established a localization‌​‌ result for inverse modes​​ in the $(\mathrm{\omega ‌},\beta )$ plane‌ and an explanation for‌​‌ the mechanism leading to​​ their appearance. Examples of​​​‌ waveguides for which inverse‌ modes appear in some‌​‌ frequency range have been​​​‌ obtained numerically.

Then, based​ on theoretical results of​‌ Kondratiev theory, we have​​ shown that for a​​​‌ heterogeneous isotropic waveguide, it​ is possible to write​‌ an EtM condition with​​ overlap connecting the tangential​​​‌ electric field on an​ inner cross section with​‌ the tangential magnetic field​​ at the artificial boundary.​​​‌ Thanks to bi-orthogonality relations,​ this EtM condition with​‌ overlap can be well​​ approximated by a finite​​​‌ modal sum, where the​ approximation error decreases exponentially​‌ with the size of​​ the overlap. But due​​​‌ to the overlap, the​ equivalence between the problem​‌ in a finite domain​​ and the original problem​​​‌ in the infinite domain​ fails for a sequence​‌ of box eigenfrequencies.

Finally,​​ we derived a second​​​‌ family of transparent conditions,​ which we call CtM​‌ conditions, still with overlap,​​ so that the equivalence​​​‌ holds at all frequencies.​ These conditions link the​‌ currents (jumps of the​​ transverse electric and magnetic​​​‌ fields) on an inner​ cross section to the​‌ tangential magnetic field on​​ the artificial boundary. This​​​‌ approach has several advantages.​ In particular, it can​‌ also be used in​​ the anisotropic case.

All​​​‌ these methods have been​ implemented in the finite​‌ elements library XLiFE++ using​​ Nédélec elements. Numerical results​​​‌ have been obtained for​ homogeneous and heterogeneous isotropic​‌ waveguides, including cases with​​ a non simply connected​​​‌ cross-section.

Trapped modes in​ electromagnetic waveguides

Participants: Anne-Sophie​‌ Bonnet-Ben Dhia, Sonia​​ Fliss.

This is​​​‌ a common work with​ Lucas Chesnel (Inria team​‌ IDEFIX). 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,​​​‌ that is $L^{\mathrm{2}}$ 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​​ continuous 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 show​‌ that certain non homogeneous​​ waveguides with local perturbations​​​‌ of the dielectric constants​ can support trapped modes.​‌ Let us mention that​​ certain mechanisms we propose​​​‌ are very specific to​ Maxwell's equations and have​‌ no equivalent in the​​ classical proofs of existence​​​‌ of trapped modes for​ the scalar Dirichlet or​‌ Neumann Laplacians.

The Half-Space​​ Matching method for the​​​‌ junction of open waveguides​

Participants: Sarah Al Humaikani​‌, Anne-Sophie Bonnet-Ben Dhia​​, Sonia Fliss.​​​‌

This work deals with​ the wave scattering by​‌ a junction of stratified​​ media, also referred to​​​‌ as open waveguides, in​ situations where the stratification​‌ allows propagation without attenuation.​​ More precisely, we consider​​​‌ configurations consisting of three​ or four stratified half-planes,​‌ with a stratification that​​ is orthogonal to the​​​‌ boundary of the half-plane.​ Proposing a characterization of​‌ the outgoing solution for​​ this class of problems​​​‌ is really challenging.

Our​ objective is to characterize​‌ and compute the outgoing​​ solution. We propose to​​ use the HalfSpace Matching​​​‌ (HSM) method. Based on‌ half-plane representations for the‌​‌ solution, the scattering problem​​ can be rewritten as​​​‌ a system coupling (1)‌ a finite element discretisation‌​‌ localized around the junction​​ and (2) integral equations​​​‌ whose unknowns are traces‌ of the solution on‌​‌ the boundaries of the​​ stratified half-planes.

For the​​​‌ scattering problem under consideration,‌ the appropriate functional framework‌​‌ for the traces as​​ well as the well-posedness​​​‌ of the HSM system‌ are not clear. Instead,‌​‌ we derive a similar​​ system, called Complex Scaled-HSM​​​‌ (Cs-HSM) where the unknowns‌ of the integral equations‌​‌ are exponentially decaying analytic​​ extensions of the traces.​​​‌ The key step is‌ to derive half-space representations‌​‌ expressed in terms of​​ the analytical extension of​​​‌ the trace. This is‌ achieved by combining two‌​‌ different formulas: one obtained​​ via a Fourier Transform​​​‌ in the direction of‌ the stratification and another‌​‌ based on a generalized​​ Fourier transform in the​​​‌ direction that is orthogonal‌ to the stratification. Each‌​‌ representation has its own​​ theoretical and numerical advantages​​​‌ and drawbacks.

Finally, we‌ prove that the resulting‌​‌ formulation is of Fredholm-type.​​ Uniqueness remains a conjecture,​​​‌ although it is strongly‌ supported by numerical experiments‌​‌ done with Xlife++ Library.​​

A Rellich type theorem​​​‌ for a class of‌ Helmholtz equations with non-constant‌​‌ coefficients

Participants: Sarah Al​​ Humaikani, Anne-Sophie Bonnet-Ben​​​‌ Dhia, Sonia Fliss‌, Christophe Hazard.‌​‌

Some years ago, the​​ following result was proven:​​​‌ there are no non-trivial‌ square-integrable solutions to the‌​‌ Helmholtz equation in a​​ bidimensional conical domain with​​​‌ opening angle larger than‌ $\pi$. We prove‌​‌ that this result can​​ be generalized to some​​​‌ configurations with non-constant coefficients.‌ More precisely, the conical‌​‌ domain must be replaced​​ by a union of​​​‌ half-planes, such that each‌ half-plane is either homogeneous‌​‌ or stratified with a​​ stratification orthogonal to the​​​‌ boundary of the half-plane.‌ Our proof is based‌​‌ on half-plane representations of​​ the solution derived through​​​‌ a generalization of the‌ Fourier transform adapted to‌​‌ stratified media.

Approximation of​​ reflectionless modes by conjugated​​​‌ PMLs : the zip‌ effect

Participants: Anne-Sophie Bonnet-Ben‌​‌ Dhia, Christophe Hazard​​.

This work is​​​‌ done in collaboration with‌ Lucas Chesnel (Inria team‌​‌ IDEFIX) and Vincent Pagneux​​ from Laboratoire d'Acoustique de​​​‌ l'Université du Maine. We‌ are interested in calculating‌​‌ so-called "reflectionless" frequencies in​​ a locally perturbed infinite​​​‌ waveguide (these can be‌ acoustic or electromagnetic waves).‌​‌ A frequency is said​​ to be reflectionless when​​​‌ there exists at that‌ frequency a wave that‌​‌ is transmitted through the​​ perturbation without producing any​​​‌ reflection. We showed in‌ a previous paper that‌​‌ the search for these​​ frequencies can be mathematically​​​‌ formulated as a non-selfadjoint‌ eigenvalue problem for an‌​‌ unbounded operator. The idea​​ is to introduce, on​​​‌ either side of the‌ perturbation, a complex dilation‌​‌ of the longitudinal variable,​​ with different dilation coefficients​​​‌ on each side of‌ the perturbation, conjugate to‌​‌ each other. This makes​​ the reflectionless wave exponentially​​​‌ decaying at infinity. We‌ can then truncate the‌​‌ waveguide and discretize the​​​‌ problem using finite elements.​ This is the method​‌ of conjugate Perfectly Matched​​ Layers (PMLs).

The spectral​​​‌ problem of reflectionless frequencies​ gives rise to some​‌ rather surprising phenomena. To​​ describe and explain them,​​​‌ it is sufficient to​ consider a 1D Helmholtz​‌ equation with a local​​ variation of the velocity.​​​‌ A first phenomenon concerns​ the problem before the​‌ introduction of PMLs. A​​ trivial but very important​​​‌ observation is the following:​ in the absence of​‌ perturbation (constant velocity), there​​ is never any reflection.​​​‌ In other words, for​ the unperturbed problem, any​‌ real frequency, even a​​ complex one, is reflectionless.​​​‌ Conversely, in the presence​ of a velocity perturbation​‌ constant in an interval,​​ the reflectionless frequencies form​​​‌ a sequence of real​ numbers tending towards infinity.​‌ This is typical of​​ certain non-selfadjoint problems: even​​​‌ the smallest perturbation of​ the unperturbed case transforms​‌ the spectrum of the​​ entire complex plane into​​​‌ a countable sequence of​ real numbers. This surprising​‌ phenomenon gives rise to​​ a second phenomenon concerning​​​‌ the convergence of PMLs​ as the truncation length​‌ L tends towards infinity.​​ We observe that the​​​‌ calculated eigenvalues ​​trace a​ kind of zip in​‌ the complex plane, which​​ opens as $L$ increases.​​​‌ We show how this​ zip effect can be​‌ explained by introducing the​​ notion of a pseudospectrum.​​​‌

Fast​​ boundary element method for​​​‌ lifetime assessment of cracked​ structures

Participants: Marc Bonnet​‌, Luiz Faria.​​

This work, undertaken in​​​‌ the framework of the​ CIFRE Phd of Adrien​‌ Vet supported by Safran​​ Aircraft Engines, is concerned​​​‌ with the development of​ a computational strategy exploiting​‌ boundary elements accelerated with​​ H-matrices towards applications in​​​‌ the fracture mechanics analysis​ of 3D complex engine​‌ structures. The underlying industrial​​ motivation is the availability​​​‌ of a "middle ground"​ computational treatment, where some​‌ modeling simplifications are deemed​​ acceptable (in particular the​​​‌ assumption of linear constitutive​ behavior) in exchange for​‌ reasonably accurate lifetime assessments​​ that avoid performing very​​​‌ demanding full-fledged analyses. Towards​ meeting these objectives, we​‌ have investigated several methodological​​ aspects, in particular the​​​‌ fine-tuning of the quadrature​ method for hypersingular element​‌ integrals originally proposed by​​ Guiggiani, the design of​​​‌ a weight function that​ facilitates the evaluation of​‌ stress intensity factors along​​ the crack front, and​​​‌ streamlining the H-matrix treatment​ of elastostatic integral operators.​‌

Boundary Domain Decomposition method​​ for elastic multiple scattering​​​‌ problems

Participants: Stephanie Chaillat​.

Fast BEM techniques​‌ have been developed in​​ the 2000s and have​​​‌ since led to a​ significant reduction in the​‌ computational complexity of each​​ iteration. A complementary strategy​​​‌ to fast BEM approaches​ consists in using domain​‌ decomposition methods (DDM) in​​ order to (i) split​​​‌ the problem into smaller​ subproblems that can be​‌ solved in parallel and​​ (ii) reduce the number​​​‌ of iterations required for​ convergence. While such approaches​‌ are widely used in​​ the finite element community,​​​‌ only a few domain​ decomposition strategies have been​‌ specifically adapted to BEM,​​ leaving substantial room for​​ improvement in this area.​​​‌ In collaboration with Martin‌ Gander (University of Geneva),‌​‌ Laurence Halpern, and Marion​​ Darbas (LAGA), we investigate​​​‌ several ideas aimed at‌ filling this gap.

As‌​‌ a first step, we​​ compare different ways of​​​‌ incorporating domain decomposition concepts‌ into the method of‌​‌ reflections, a domain decomposition​​ approach naturally suited for​​​‌ multiple scattering problems. This‌ method relies on decomposing‌​‌ the total scattered field​​ into contributions associated with​​​‌ each individual obstacle. At‌ each iteration, the diffraction‌​‌ problem is solved independently​​ for each obstacle, with​​​‌ an incident field defined‌ as the superposition of‌​‌ the original incident wave​​ and the fields scattered​​​‌ by the other obstacles‌ at previous iterations. Although‌​‌ this method is independent​​ of the chosen numerical​​​‌ discretization, it is particularly‌ well adapted to BEM‌​‌ due to the homogeneous​​ nature of exterior problems.​​​‌

The main challenge lies‌ in understanding the convergence‌​‌ of this iterative method,​​ which may depend on​​​‌ the frequency and on‌ the geometric configuration, in‌​‌ particular on the distance​​ between obstacles. We have​​​‌ shown that both the‌ sequential and parallel versions‌​‌ of the method of​​ reflections, when based on​​​‌ boundary integral equations, provide‌ an efficient and robust‌​‌ solution for multiple scattering​​ problems. However, we observe​​​‌ that small distances between‌ obstacles may significantly slow‌​‌ down convergence, or even​​ prevent convergence of the​​​‌ parallel version. To address‌ this issue, we have‌​‌ proposed the introduction of​​ overlapping or deflation techniques.​​​‌ A proceedings paper on‌ this topic has been‌​‌ submitted to the 29th​​ Domain Decomposition Methods Conference,​​​‌ and a journal paper‌ is currently being finalized.‌​‌

Fast Preconditioned Boundary Element​​ Methods for piecewise homogeneous​​​‌ elastodynamics problems

Participants: Stephanie‌ Chaillat.

Boundary Element‌​‌ Methods (BEMs) are highly​​ efficient for homogeneous problems;​​​‌ however, significant challenges arise‌ when dealing with layered‌​‌ media, in particular regarding​​ conditioning and the treatment​​​‌ of triple points. While‌ various coupling strategies exist,‌​‌ these issues remain largely​​ open.

The ECOS Chile​​​‌ project, in collaboration with‌ Marion Darbas (LAGA), Carlos‌​‌ Jerez Hanckes and Paul​​ Escapil-Inchauspé (Inria Chile), aims​​​‌ at extending multitrace methods‌ (MTF), originally developed for‌​‌ Helmholtz problems, to two-​​ and three-dimensional elasticity. MTF​​​‌ reformulates the boundary value‌ problem as a well-posed‌​‌ system of first-kind boundary​​ integral equations, naturally suited​​​‌ for parallelization and preconditioning.‌ The approach introduces independent‌​‌ displacement and traction unknowns​​ per subdomain, enforces Calderón​​​‌ identities locally, and weakly‌ imposes transmission conditions across‌​‌ interfaces.

As a first​​ step toward heterogeneous elasticity​​​‌ problems, we focus on‌ the case of a‌​‌ single homogeneous scatterer. All​​ derivations are carried out​​​‌ explicitly in one dimension,‌ with illustrative examples in‌​‌ two dimensions. We analyze​​ the influence of frequency​​​‌ and material contrast on‌ the convergence of the‌​‌ GMRES solver. Preliminary results​​ on an elastic Calderón​​​‌ preconditioner, and a discussion‌ of its potential to‌​‌ further accelerate iterative solvers,​​ are presented in a​​​‌ paper currently under review.‌

Fast, high-order numerical evaluation‌​‌ of volume potentials via​​ polynomial density interpolation

Participants:​​​‌ Luiz Faria, Marc‌ Bonnet.

This research‌​‌ is done in collaboration​​​‌ with Carlos Pérez-Arancibia (University​ of Twente, Netherlands) and​‌ Thomas Anderson (Univ. of​​ Michigan, USA). The proposed​​​‌ method addreses the evaluation​ of (e.g. Newtonian) volume​‌ potentials arising for many​​ classical models of mathematical​​​‌ physics, which include acoustic​ and elastic waves. The​‌ proposed technique relies on​​ using polynomial interpolants of​​​‌ the density function around​ the kernel singularity, polynomial​‌ solutions of the underlying​​ homogeneous PDE with that​​​‌ interpolant in the right-hand​ side and Green's theorem,​‌ and allows to formulate​​ an evaluation scheme that​​​‌ does not entail any​ singular integral. We have​‌ also prpposed a systematic​​ methodology for the construction​​​‌ of polynomial PDE solutions.​ The method is designed​‌ so as to be​​ compatible with the use​​​‌ of fast summation methods​ such as the fast​‌ multipole method. WE obtained​​ error estimates for the​​​‌ regularization and quadrature errors,​ and ran a complete​‌ battery of numerical tests,​​ which include solving Lippmann-Schwinger​​​‌ equations for scattering by​ penetrable objects, for potentials​‌ using the 2D Laplace​​ and Helmholtz Green's functions.​​​‌ Extensions to 3D and​ vector-valued problems are current​‌ underway.

High-order Boundary Integral​​ Equations on implicitly defined​​​‌ surfaces

Participants: Luiz Faria​, Dongchen He.​‌

This research is being​​ done in collaboration with​​​‌ Aline Lefebvre-Lepot (CMAP), and​ in the context of​‌ the PhD thesis of​​ Dongchen He. We are​​​‌ developing a method for​ accurately solving boundary integral​‌ equations on implicitly defined​​ surfaces in ${ℝ}^{\mathrm{d‌}}$. The method relies​ on combining a dimension-indepent​‌ technique for generating a​​ high-order surface quadrature on​​​‌ level-set surfaces, with the​ general-purpose density interpolation method​‌ for handling the singular​​ and nearly-singular integrals ubiquitous​​​‌ in boundary integral formulations.​ The proposed methodology, based​‌ on a Nystrom discretization​​ scheme, bypasses the need​​​‌ for generating a body​ conforming mesh for the​‌ implicit surface, allowing in​​ principle for an efficient​​​‌ coupling between a robust​ dynamic level-set representation of​‌ deforming surfaces, and boundary​​ integral equation solvers. Particular​​​‌ attention is being paid​ to the computation of​‌ singular integrals when only​​ a surface quadrature is​​​‌ available (i.e. in the​ absence of an actual​‌ mesh). We believe such​​ techniques could prove useful​​​‌ in applications involving microscopic​ flows governed by the​‌ Stokes equations; in particular,​​ the simulation of micro-swimmers​​​‌ and droplet microfluidics.

Modelling​ the sound radiated by​‌ a turbulent flow

Participants:​​ Stéphanie Chaillat, Jean-François​​​‌ Mercier, Louise Pacaut​.

The goal of​‌ this PhD study, conducted​​ in collaboration with Gilles​​​‌ Serre (Naval Group), is​ to develop an optimized​‌ numerical method for determining​​ the sound produced by​​​‌ turbulence and scattered by​ a screw propeller. Ultimately,​‌ this research aims to​​ contribute to reducing the​​​‌ noise radiated by ships.​ The study addresses two​‌ challenges: (i) modeling turbulence​​ to derive an acoustic​​​‌ source term, and (ii)​ propagating both direct and​‌ scattered sounds from the​​ source. These challenges are​​​‌ tackled by computing tailored​ Green’s functions, functions satisfying​‌ the natural boundary conditions​​ of obstacles with arbitrary​​​‌ shapes. Building on a​ prior PhD that dealt​‌ with rigid obstacles under​​ Neumann boundary conditions, this​​ study extends the approach​​​‌ to penetrable obstacles. In‌ the fluid-fluid case, such‌​‌ as an air bubble​​ in water, coupled integral​​​‌ equations are derived to‌ express the tailored Green’s‌​‌ function in terms of​​ the free-space Green’s functions​​​‌ of both fluids. A‌ hierarchical matrix-based Boundary Element‌​‌ Method is used to​​ efficiently compute these functions.​​​‌ In the case of‌ a light fluid surrounded‌​‌ by an heavy fluid​​ (air bubble in water),​​​‌ the resolution of the‌ integral equations gave imprecise‌​‌ results and a regularization​​ process had to be​​​‌ introduced. The study also‌ extends to fluid-elastic interactions,‌​‌ where a new challenge​​ arises due to the​​​‌ complexity of the elastic‌ Green’s tensor. Here also‌​‌ a regularization has been​​ required to evaluate correctly​​​‌ the solution close to‌ the elastic medium surface.‌​‌ The validity of these​​ approaches, fluid-fluid and fluid-elastic,​​​‌ is confirmed by testing‌ on a spherical geometry,‌​‌ for which analytical solutions​​ are derived. These results​​​‌ provide a foundation for‌ addressing ship noise reduction‌​‌ in practical applications. The​​ corresponding article is about​​​‌ to be submitted.

Diffraction‌ by fractal screens

Participants:‌​‌ Patrick Joly, Maryna​​ Kachanovska.

This work​​​‌ is done in collaboration‌ with Z. Moitier (currently‌​‌ at IDEFIX, Inria). We​​ develop a new integration​​​‌ technique for computing integrals‌ over self-similar sets, with‌​‌ application to computing discretizations​​ of boundary integral operators​​​‌ over fractal screens. The‌ key idea is inspired‌​‌ by the previous work​​ of Stritchartz, which deals​​​‌ with evaluation of integrals‌ of monomials on fractals‌​‌ based on the self-similarity​​ of the underlying measure,​​​‌ and which we were‌ able to extend to‌​‌ our setting. In particular,​​ the main difficulty in​​​‌ constructing quadratures over the‌ screens lies in evaluation‌​‌ of the integrals of​​ Lagrange polynomials that define​​​‌ quadrature weights. This is‌ now done by a‌​‌ purely algebraic procedure of​​ computing a kernel of​​​‌ an easy-to-compute matrix. The‌ convergence estimates for the‌​‌ new quadrature have been​​ obtained and tested numerically.​​​‌ The results of this‌ research have been accepted‌​‌ for publication in SIAM​​ Journal of Scientific Computing​​​‌ and have been presented‌ at the Conference ICOSAHOM‌​‌ 2025, in July at​​ Montreal, in the framework​​​‌ of a minisymposium "Recent‌ Progress in higher order‌​‌ numerical integration" that we​​ co-organized with Z. Moitier.​​​‌

Integral equation methods for‌ acoustic scattering by fractals‌​‌

Participants: Xavier Claeys.​​

This is a work​​​‌ in collaboration with A.M.‌ Caetano (Universidade de Aveiro,‌​‌ Portugal), S.N. Chandler-Wilde† (University​​ of Reading, United Kingdom),​​​‌ A. Gibbs (University College‌ London, United Kingdom) ,‌​‌ D.P. Hewett (University College​​ London, United Kingdom) and​​​‌ A. Moiola (University of‌ Pavia, Italy).

We study‌​‌ sound-soft time-harmonic acoustic scattering​​ by general scatterers, including​​​‌ fractal scatterers, in 2D‌ and 3D space. For‌​‌ an arbitrary compact scatterer​​ $\Gamma$ we reformulate the​​​‌ Dirichlet boundary value problem‌ for the Helmholtz equation‌​‌ as a first kind​​ integral equation (IE) on​​​‌ $\Gamma$ involving the Newton‌ potential. The IE is‌​‌ well-posed, except possibly at​​ a countable set of​​​‌ frequencies, and reduces to‌ ex- isting single-layer boundary‌​‌ IEs when $\Gamma$ is​​​‌ the boundary of a​ bounded Lipschitz open set,​‌ a screen, or a​​ multi-screen. When $\Gamma$ is​​​‌ uniformly of d-dimensional Hausdorff​ dimension in a sense​‌ we make precise (a​​ d-set), the operator in​​​‌ our equation is an​ integral operator on $\mathrm{\Gamma ‌}$ with respect to d-dimensional​​ Hausdorff measure, with kernel​​​‌ the Helmholtz fundamental solution,​ and we propose a​‌ piecewise-constant Galerkin discretization of​​ the IE, which converges​​​‌ in the limit of​ vanishing mesh width. When​‌ $\Gamma$ is the fractal​​ attractor of an iterated​​​‌ function system of contracting​ similarities we prove convergence​‌ rates under assumptions on​​ $\Gamma$ and the IE​​​‌ solution, and describe a​ fully discrete implementation using​‌ recently proposed quadrature rules​​ for singular integrals on​​​‌ fractals. We present numerical​ results for a range​‌ of examples and make​​ our software available as​​​‌ a Julia code.

Convolution​ Quadrature Methods for Wave​‌ Equation in Non-cylindrical Domains​​

Participants: Maryna Kachanovska.​​​‌

In this work in​ progress, carried out in​‌ collaboration with Lehel Banjai​​ (Heriot-Watt University), we have​​​‌ initiated a new project​ on the design of​‌ convolution quadrature methods for​​ time-domain boundary integral equations​​​‌ for the wave equation​ in non-cylindrical domains. We​‌ have formulated the corresponding​​ boundary integral equations—which, to​​​‌ the best of our​ knowledge, have not appeared​‌ previously in the literature—and​​ proposed a convolution-quadrature-based discretization.​​​‌ We have also performed​ preliminary numerical experiments. At​‌ this stage, the results​​ are not entirely satisfactory:​​​‌ our initial tests indicate​ that stabilization of the​‌ proposed formulation is required.​​

A priori analysis of​​​‌ curved boundary element methods​ for the 3D Laplace​‌ and Helmholtz equations

Participants:​​ Luiz Faria, Pierre​​​‌ Marchand.

This work​ is a collaboration with​‌ H. Montanelli (Inria Idefix).​​ We established improved convergence​​​‌ rates for curved boundary​ element methods applied to​‌ the three-dimensional Laplace and​​ Helmholtz equations with smooth​​​‌ geometry and data. Our​ analysis relies on a​‌ precise analysis of the​​ errors introduced by the​​​‌ discretization of the solution,​ and the geometry. We​‌ illustrated our results with​​ numerical experiments in 3D​​​‌ based on basis functions​ and curved triangular elements​‌ up to order four​​ using Inti.jl. This led​​​‌ to multiple interesting open​ questions. In particular, using​‌ even order for the​​ curved triangular elements super-converge​​​‌ compared to our analysis,​ which is consistent with​‌ other observation in the​​ literature.

Accelerating​‌ non-local exchange in generalized​​ optimized Schwarz methods

Participants:​​​‌ Xavier Claeys, Roxane​ Delville-Atchekzai.

This is​‌ a joint work with​​ M.Lecouvez (CEA CESTA, Bordeaux).​​​‌ The generalized optimised Schwarz​ method proposed in [Claeys​‌ & Parolin, 2022] is​​ a variant of the​​​‌ Després algorithm for solving​ harmonic wave problems where​‌ transmission conditions are enforced​​ by means of a​​​‌ non-local exchange operator. We​ introduce and analyse an​‌ acceleration technique that significantly​​ reduces the cost of​​​‌ applying this exchange operator​ without deteriorating the precision​‌ and convergence speed of​​ the overall domain decomposition​​​‌ algorithm.

Hierarchical matrix compression​ for inverses of finite​‌ element matrices of convection​​ dominated problems

Participants: Xavier​​ Claeys, Arthur Saunier​​​‌.

This is a‌ joint work with A.Anciaux‌​‌ (IFPEN, Rueil-Malmaison), I.Ben Gharbia​​ (IFPEN, Rueil-Malmaison) and L.Agelas​​​‌ (IFPEN, Rueil-Malmaison). Hierarchical matrices‌ (H-matrices) refer to compression‌​‌ schemes leading to a​​ drastic acceleration of linear​​​‌ algebra operations. They rely‌ on two main ingredients:‌​‌ recursive partitioning of the​​ matrix, and compression of​​​‌ certain so-called admissible blocks‌ of the partition. H-matrices‌​‌ typically target certain class​​ of fully populated matrices​​​‌ stemming from the discretization‌ of PDEs. They perform‌​‌ very well in the​​ case where the underlying​​​‌ PDE is strongly elliptic,‌ which has been well‌​‌ documented and received a​​ solid theoretical justification, but​​​‌ the performance a priori‌ deteriorates when ellipticity is‌​‌ lost. In this work,​​ we shall focus on​​​‌ the case of matrices‌ stemming from the discretization‌​‌ of convection dominated problems.​​ We shall first discuss​​​‌ where the standard proof‌ of approximability fails in‌​‌ the case of dominating​​ convection. Then we shall​​​‌ explain how to modify‌ the partitioning and the‌​‌ adminissibility criterion so as​​ to overcome this issue​​​‌ and restore the performance‌ of H-matrix compression. We‌​‌ also work on obtaining​​ numerical results to illustrate​​​‌ our new approach.

Substructuring‌ based FEM-BEM coupling for‌​‌ Helmholtz problems

Participants: Antonin​​ Boisneault, Xavier Claeys​​​‌, Pierre Marchand.‌

This work is a‌​‌ collaboration with M. Bonazzoli​​ (Inria Idefix) and concerns​​​‌ the solution of the‌ Helmholtz equation in a‌​‌ medium composed of a​​ bounded heterogeneous domain and​​​‌ an unbounded homogeneous one.‌ Such problems can be‌​‌ expressed using classical FEM-BEM​​ coupling techniques. We solve​​​‌ these coupled formulations using‌ iterative solvers based on‌​‌ substructuring Domain Decomposition Methods​​ (DDM), and aim to​​​‌ develop a convergence theory,‌ with fast and guaranteed‌​‌ convergence. A recent article​​ of Xavier Claeys proposed​​​‌ a substructuring Optimized Schwarz‌ Method, with a nonlocal‌​‌ exchange operator, for Helmholtz​​ problems on a bounded​​​‌ domain with classical conditions‌ on its boundary (Dirichlet,‌​‌ Neumann, Robin). The variational​​ formulation of the problem​​​‌ can be written as‌ a bilinear application associated‌​‌ with the volume and​​ another with the surface,​​​‌ for which, under certain‌ sufficient assumptions, convergence of‌​‌ the DDM strategy is​​ guaranteed. We have shown​​​‌ how some specific FEM-BEM‌ coupling methods fit, or‌​‌ not, the previous framework,​​ in which we consider​​​‌ Boundary Integral Equations (BIEs)‌ instead of classical boundary‌​‌ conditions. In particular, we​​ prove that the symmetric​​​‌ Costabel coupling satisfies the‌ assumptions of the proposed‌​‌ framework, which guarantees convergence.​​ We carefully study spurious​​​‌ resonances, which are typical‌ of BEM formulations, and‌​‌ show that the Costabel​​ coupling is robust with​​​‌ respect to them. Numerical‌ experiments are carried out‌​‌ using DDMtool for domain​​ decomposition and Htool-DDM for​​​‌ the hierarchical compression of‌ integral operators. This work‌​‌ is in the framework​​ of OptiGPR3D Exploratory Action​​​‌ and has been presented‌ at DD 29 conference.‌​‌

Convergence study of the​​ iterative finite element solution​​​‌ of Helmholtz problems with‌ near-resonance phenomena

Participants: Pierre‌​‌ Marchand, Axel Modave​​, Timothée Raynaud.​​​‌

This research topic is‌ developed in collaboration with‌​‌ Victorita Dolean (TU/e, The​​​‌ Netherlands) within the framework​ of the ElectroMath CIEDS​‌ project.

We consider the​​ iterative solution of Helmholtz​​​‌ problems discretized using the​ finite element method. For​‌ these problems, the convergence​​ of iterative Krylov methods​​​‌ is usually slow, because​ the matrices of the​‌ resulting linear systems can​​ be indefinite, ill-conditioned, and​​​‌ large. We aim to​ better understand the convergence​‌ of Krylov methods for​​ problems close to resonances,​​​‌ to provide improvements that​ make the iterative solvers​‌ more robust.

Several results​​ characterize the convergence of​​​‌ Krylov methods. These are​ based, for example, on​‌ the distribution of eigenvalues​​ over the spectrum, the​​​‌ notion of pseudospectrum and​ numerical range, or harmonic​‌ Ritz values. Here we​​ have studied convergence results​​​‌ based on harmonic Ritz​ values. We have proved​‌ a new result to​​ better interpret the superlinear​​​‌ convergence of GMRES. We​ have applied this result​‌ for a cavity case​​ (close to resonances) and​​​‌ an open cavity case​ (close to quasi-resonances) implemented​‌ in a MATLAB finite​​ element code. We observed​​​‌ that the superlinear convergence​ behavior is related to​‌ the approximation of the​​ small eigenvalues of the​​​‌ matrix by the small​ harmonic Ritz values computed​‌ during the iterations.

We​​ are studying how deflation,​​​‌ coarse spaces, domain decomposition​ preconditioning techniques, and their​‌ combination influence the convergence​​ of GMRES for configurations​​​‌ with near-resonance phenomena. We​ are carrying out this​‌ study using FreeFEM. Preliminary​​ results have been presented​​​‌ at the 29th International​ Conference on Domain Decomposition​‌ Methods (DD29).

Isometric Arnoldi​​ solvers for domain decomposition​​​‌ methods applied to wave​ propagation problems

Participants: Xavier​‌ Claeys.

Discretization of​​ harmonic wave propagation problems​​​‌ typically leads to non-self-adjoint​ linear systems, which raises​‌ significant challenges from the​​ perspective of linear solvers.​​​‌ In this context, the​ preconditioned conjugate gradient (PCG)​‌ is a priori not​​ applicable, and it is​​​‌ customary to rely on​ GMRes. However, when the​‌ wave propagation problem under​​ consideration does not involve​​​‌ any energy dissipation mechanism,​ the Generalized Optimized Schwarz​‌ Method that we have​​ previously developed leads to​​​‌ linear systems that take​ the form of an​‌ isometric perturbation of the​​ identity operator. Linear systems​​​‌ associated with isometric perturbations​ of the identity can​‌ be solved by means​​ of a very special​​​‌ Krylov linear solver developed​ by Jagels and Reichel​‌ in the 1990s. This​​ special solver, based on​​​‌ an isometric Arnoldi algorithm,​ involves a short-term recurrence,​‌ a desirable property similar​​ to the conjugate gradient.​​​‌ We are currently investigating​ the performance of such​‌ solvers on wave propagation​​ problems.

Hydrid DG-FEM modeling​​​‌ of thin coaxial cables​

Participants: Patrick Joly.​‌

This topic is the​​ subject of a collaboration​​​‌ Sébastien Imperiale (M3disim) and​ constitues the continuation the​‌ PhD thesis of Akram​​ Beni Hamad, defended in​​​‌ September 2023, with which​ we continued to collaborate.​‌ Our morst recent contricution​​ concerns the time domain​​​‌ modeling of deformed thin​ cables, wthere "deformed" refers​‌ to the fact that​​ the cable is not​​​‌ cylindrical. The cylindrical case​ was treated by an​‌ original approach combining Nédélec’s​​ edge elements on elongated​​ prismatic meshes with a​​​‌ hybrid time discretization procedure‌ which is explicit in‌​‌ the longitudinal directions and​​ implicit in the transverse​​​‌ ones. The resulting numerical‌ scheme has the advantage‌​‌ to be stable under​​ a CFL condition that​​​‌ involves only the longitudinal‌ space step, a property‌​‌ which is essential for​​ the efficiency of the​​​‌ method.

The extension of‌ the above method to‌​‌ the non cylindrical case​​ led us to relax​​​‌ the $H(\mathrm{c‌}url)‌‌$ conformity of our finite​​ element spaces and to​​​‌ develop a new hybrid‌ method combining a conforming‌​‌ discretization in the longitudinal​​ variable and a discontinuous​​​‌ Galerkin method in the‌ transverse ones. The resulting‌​‌ method has a complexity​​ which is similar to​​​‌ the one of the‌ cylindrical case. Moreover, and‌​‌ this is the major​​ theoretical result ontained this​​​‌ year, we were able‌ to prove, that the‌​‌ numerical scheme was still​​ stable under a CFL​​​‌ stability condition involving only‌ the longitudinal space step.‌​‌ The corresponding article has​​ been submitted for publication.​​​‌

Hybridizable discontinuous Galerkin (HDG)‌ methods with transmission variables‌​‌ for time-harmonic problems

Participants:​​ Axel Modave, Simone​​​‌ Pescuma, Ari Rappaport‌.

This research topic‌​‌ is developed in collaboration​​ with Théophile Chaumont-Frelet (Inria,​​​‌ Rapsodi) and Gwénaël Gabard‌ (LAUM) within the framework‌​‌ of the WavesDG ANR​​ project and the ElectroMath​​​‌ CIEDS project.

We consider‌ the iterative solution of‌​‌ time-harmonic wave propagation problems​​ discretized with finite element​​​‌ methods. These problems are‌ notoriously difficult to solve‌​‌ iteratively because the matrices​​ of the discrete systems​​​‌ are sparse, complex, and‌ indefinite. We are working‌​‌ on a new hybridizable​​ discontinuous Galerkin method, called​​​‌ the CHDG method, which‌ is based on a‌​‌ standard discontinuous Galerkin scheme​​ with upwind numerical fluxes​​​‌ and high-order polynomial bases.‌ Auxiliary unknowns corresponding to‌​‌ transmission variables are defined​​ at the interface between​​​‌ the elements, and the‌ physical fields are eliminated‌​‌ to obtain a hybridized​​ system. In the case​​​‌ of scalar waves in‌ homogeneous media, it has‌​‌ been observed that the​​ iterative solution of the​​​‌ reduced system (with CGNR‌ and GMRES) is accelerated‌​‌ compared to the standard​​ HDG method where the​​​‌ auxiliary unknowns correspond to‌ a numerical trace.

In‌​‌ the context of Simone​​ Pescuma's PhD thesis, we​​​‌ have extended the CHDG‌ method to scalar heterogeneous‌​‌ problems with piecewise constant​​ coefficients, as well as​​​‌ aeroacoustic problems. Using a‌ series of 2D numerical‌​‌ benchmarks, we systematically studied​​ the standard HDG and​​​‌ CHDG methods. Convergence of‌ standard iterative schemes is‌​‌ always faster with the​​ extended CHDG method than​​​‌ with the standard HDG‌ methods. In the context‌​‌ of Ari Rappaport's postdoctoral​​ research, we have applied​​​‌ the CHDG method to‌ electromagnetic problems in homogeneous‌​‌ media. We validated the​​ approach using an in-house​​​‌ C++ 3D CHDG parallel‌ code.

GPU-accelerated finite element‌​‌ solvers for time-harmonic problems​​

Participants: Ahmed Chabib,​​​‌ Axel Modave.

This‌ research topic is developed‌​‌ in collaboration with Christophe​​ Geuzaine (ULiège, Belgium) and​​​‌ Roland Greffe (ULiège, Belgium).‌

Over the past decade,‌​‌ computing power has shifted​​​‌ from central processing units​ (CPUs) to graphical processing​‌ units (GPUs) in servers​​ and modern supercomputers. Therefore,​​​‌ porting numerical simulation tools​ for GPU computing is​‌ of paramount importance. However,​​ developing numerical tools that​​​‌ efficiently leverage the computing​ power of GPU accelerators​‌ is challenging. This requires​​ not only the fine-tuning​​​‌ of codes but also​ the rethinking of numerical​‌ algorithms.

In this project,​​ we are investigating GPU​​​‌ acceleration for two types​ of finite element solvers​‌ for time-harmonic problems. As​​ part of his postdoctoral​​​‌ research, Ahmed Chabib is​ porting the C++ 3D​‌ CHDG discontinuous finite element​​ code developed in the​​​‌ WavesDG project framework to​ a GPU. In the​‌ context of Roland Greffe's​​ PhD studies, we are​​​‌ investigating GPU implementation strategies​ for a standard continuous​‌ finite element code combined​​ with a domain decomposition​​​‌ substructuring method using GmshFEM.​ Preliminary results were presented​‌ at the ACOMEN 2025​​ conference and the 2025​​​‌ Congrès Français de Mécanique.​

Analysis of a domain​‌ decomposition method for electromagnetic​​ waves in anisotropic media​​​‌

Participants: Patrick Ciarlet,​ Axel Modave, Ari​‌ Rappaport.

This research​​ topic is developed in​​​‌ collaboration with Marcella Bonazzoli​ (IDEFIX, Inria).

The mathematical​‌ modeling of electromagnetic wave​​ propagation in complex and​​​‌ anisotropic media is an​ active research topic, e.g.,​‌ for designing metamaterials. Fast​​ numerical models accelerated with​​​‌ domain decomposition methods would​ be an asset for​‌ studying complex configurations. However,​​ these methods have mainly​​​‌ been studied for electromagnetic​ problems involving isotropic media,​‌ and few works are​​ available for more complex​​​‌ media. In this project,​ we are extending the​‌ analysis of an overlapping​​ domain decomposition preconditioning method​​​‌ that has already been​ studied for lossy homogeneous​‌ media to anisotropic media.​​

Domain decomposition methods for​​​‌ random multi-scale Helmholtz problems​ arising in ultrasound imaging​‌

Participants: Laure Giovangigli.​​

Together with Emile Parolin​​​‌ (Inria Paris), Laure Giovangigli​ obtained an Exploratory Action​‌ funding from Inria: AEx​​ QUI (Quantitative Ultrasound Imaging)​​​‌ to work on this​ topic. The development of​‌ new quantitative ultrasound imaging​​ algorithms, which aim at​​​‌ reconstructing a map of​ the local speed of​‌ sound in the medium​​ from echoes measurements, requires​​​‌ a validation process that​ can be achieved through​‌ numerical simulation. With this​​ application in mind, we​​​‌ consider the scattering of​ plane waves by a​‌ tissue-mimicking medium where up​​ to a hundred unresolved​​​‌ scatterers per wavelength are​ randomly distributed throughout the​‌ medium. The domains (about​​ a hundred wavelengths in​​​‌ size) require billion degrees​ of freedom in a​‌ simulation, which corresponds to​​ the state of the​​​‌ art in terms of​ direct numerical simulation capacity.​‌ We investigated the efficiency​​ and scalability of one-level​​​‌ and two- levels domain​ decomposition techniques to accurately​‌ solve the full scale​​ model. The primary objective​​​‌ was to validate quantitative​ stochastic homogenization results obtained​‌ in [Garnier et al.,arXiv:2505.07566​​ ] particularly the asymptotic​​​‌ expansions of the scattered​ field with respect to​‌ the size of the​​ scatterers.

Propagation of​ ultrasounds in random multi-scale​‌ media and quantitative medical​​ ultrasound imaging

Participant: Laure​​ Giovangigli, Quentin Goepfert​​​‌.

This work is‌ a joint work with‌​‌ Josselin Garnier (X-CMAP) and​​ Pierre Millien (Institut Langevin)​​​‌ and has been published‌ in IPI. 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 also include at‌​‌ the end of the​​ publication.

We are currently​​​‌ working at extending this‌ study to the case‌​‌ where the medium is​​ characterized by a heteregeneous​​​‌ effective speed of sound.‌ First we study the‌​‌ case of a target​​ embedded in a bilayered​​​‌ medium with a planar‌ interface when the effective‌​‌ speed of sound in​​ each layer and interface​​​‌ depth are not known.‌ We show that we‌​‌ can numerically estimate the​​ two speeds of sound​​​‌ and the interface depth‌ by maximizing the PSF‌​‌ at the imaging point.​​ In the paraxial and​​​‌ broadband regime, we theorically‌ estimate the size (axial‌​‌ and lateral widths of​​ the PSF) and the​​​‌ center of the focal‌ spot. Lastly, we show‌​‌ that it is possible​​ to estimate the three​​​‌ parameters even in the‌ absence of a strong‌​‌ reflector by using the​​ signals reflected by randomly​​​‌ distributed small scatterers. This‌ is a joint work‌​‌ with Sofia Suarez.

Second​​ we quantify the robustness​​​‌ of the effective speed‌ of sound estimator in‌​‌ the presence of medium​​ noise. We suppose that​​​‌ the background speed of‌ sound exhibits random fluctuations‌​‌ around a constant effective​​ speed of sound which​​​‌ we wish to estimate.‌ Assuming that the correlation‌​‌ length of these fluctuations​​ is larger than the​​​‌ wavelength of the incident‌ wave we characterize the‌​‌ law of the estimator​​ in this random medium​​​‌ in the presence of‌ a strong reflector. Using‌​‌ realistic numerically simulated measurements​​ we show that the​​​‌ estimator in a speckle‌ medium without any strong‌​‌ reflector follows the same​​ law. This is a​​​‌ joint work with Zetao‌ Fei (Ecole Polytechnique).

Inverse‌​‌ problems in oceanography

Participant:​​ Laurent Bourgeois, Raphael​​​‌ Terrine.

This work‌ is devoted to an‌​‌ inverse problem arising in​​​‌ oceanography, that is the​ identification of the sea​‌ bottom deformation from measurements​​ of the induced free​​​‌ surface perturbation. An application​ is the early detection​‌ of tsunamis. After studying​​ the problem in the​​​‌ frequency domain, for the​ sake of simplicity, we​‌ have then tackled the​​ problem in the time​​​‌ domain, which is more​ realistic and more difficult.​‌ This is the subject​​ of the PhD Thesis​​​‌ of Raphaël Terrine and​ a collaboration with Philippe​‌ Moireau (Ananke). The underlying​​ model is supposed to​​​‌ be linear, however involving​ both gravity and acoustic​‌ waves. A first issue​​ is proving well-posedness of​​​‌ the forward problem, setting​ it in different frameworks​‌ (for instance the Semi-group​​ theory or Lions' theorem).​​​‌ In order to solve​ the inverse problem, two​‌ strategies were considered. The​​ first method is a​​​‌ space-time mixed formulation of​ the Tikhonov regularization, the​‌ Morozov principle being used​​ to determine the regularization​​​‌ parameter as a function​ of the amplitude of​‌ noise. This determination relies​​ on duality in optimization.​​​‌ The main drawback of​ such one-shot method is​‌ that a large matrix​​ has to be inverted​​​‌ in the presence of​ a small space-time mesh.​‌ To circumvent the problem​​ we have implemented a​​​‌ SOR type method that​ enables us to consider​‌ smaller meshes. The second​​ method is a more​​​‌ classical least square method​ in the spirit of​‌ the optimal control problems​​ à la Lions, an​​​‌ adjoint state being used​ to compute the gradient​‌ of the cost function.​​ In this second method,​​​‌ however, that both the​ control and the observation​‌ be surface functions makes​​ the rigorous justification of​​​‌ the optimal control method​ quite difficult. Besides, while​‌ the discretization of the​​ first method is quite​​​‌ obvious, that of the​ second one is more​‌ challenging. In particular, the​​ convergence analysis of the​​​‌ solution of the discrete​ optimal control problem to​‌ the solution of the​​ continuous one when both​​​‌ the time step and​ the space mesh tend​‌ to 0 is rather​​ technical. It was achieved​​​‌ by carrying out the​ numerical analysis of the​‌ mixed formation formed by​​ the optimality condition satisfied​​​‌ by the solution. Some​ numerical experiments in 2d​‌ have proved the feasibility​​ of both methods, emphasizing​​​‌ their pros and cons.​

Constitutive behavior of linear​‌ viscoelastic solids under the​​ plane stress condition

Participant:​​​‌ Marc Bonnet.

Work​ done in collaboration with​‌ Bojan Guzina, University of​​ Minnesota, USA.

We derive​​​‌ the relationship between the​ plane-stress viscoelastic constitutive parameters,​‌ typically valid for thin​​ solids, and their bulk​​​‌ counterparts. We thus provide​ foundations for reconstructing 3D​‌ constitutive parameters of natural​​ and engineered solids via​​​‌ thin-sheet testing, in particular​ by examining the reduction​‌ of thermodynamic potentials describing​​ linear viscoelasticity under the​​​‌ plane stress condition. The​ analysis is complemented by​‌ a set of analytical​​ and numerical examples, illustrating​​​‌ the effect on the​ plane stress condition on​‌ the behavior of isotropic​​ and anisotropic viscoelastic solids.​​​‌

Early-reverberation imaging functions for​ bounded elastic domains

Participant:​‌ Marc Bonnet.

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 process. Applying​​​‌ classical approaches, derivatives of‌ the relevant data-misfit functional‌​‌ with respect to arbitrary​​ perturbations of the mass​​​‌ density and stiffnesses are‌ expressed using forward and‌​‌ adjoint solutions that incorporate​​ the exponentially decaying weighting.​​​‌ and finite-duration imaging functions‌ are then defined on‌​‌ that basis. 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. We demonstated​​ the potential of the​​​‌ proposed on 2D examples.‌

Shape optimization problems involving‌​‌ slow viscous fluids

Participant:​​ Marc Bonnet.

Work​​​‌ done in collaboration with‌ Shravan Veerapaneni and his‌​‌ group (University of Michigan,​​ USA)

Work done in​​​‌ collaboration with Shravan Veerapaneni‌ and his group (University‌​‌ of Michigan, USA).

This​​ multi-year collaboration addresses the​​​‌ design and implementation of‌ computational methods for solving‌​‌ optimization problems involving slow​​ viscous fluids modelled by​​​‌ the Stokes equations. In‌ particular, we have developed‌​‌ a computational framework that​​ aims at simultaneously optimizing​​​‌ the shape and the‌ slip velocity of an‌​‌ axisymmetric microswimmer suspended in​​ a viscous fluid. We​​​‌ seek swimmer shapes that‌ achieve a given net‌​‌ motion amount while minimizing​​ the incurred power loss.​​​‌ The optimal slip and‌ efficiency (with shape fixed)‌​‌ are here given in​​ terms of two Stokes​​​‌ flow solutions, and we‌ then establish shape sensitivity‌​‌ formulas of adjoint-solution form​​ that provide objective function​​​‌ derivatives with respect to‌ any set of shape‌​‌ parameters on the sole​​ basis of the above​​​‌ two flow solutions. Our‌ computational treatment, which relies‌​‌ on a fast and​​ accurate boundary integral solver​​​‌ for solving all Stokes‌ flow problems, was used‌​‌ on several shape optimization​​ examples. We next extended​​​‌ to 3D swimmers our‌ slip optimization approach by‌​‌ defining a fixed set​​ of (at most) twelve​​​‌ flow problems allowing to‌ determine net motions with‌​‌ minimal power loss, in​​ preparation to addressing the​​​‌ 3D shape optimization in‌ the near future.

Wave diffraction​​​‌ by thin finite periodic‌ layers

Participants: Cédric Baudet‌​‌, Sonia Fliss,​​ Patrick Joly.

This​​​‌ is the subject of‌ the PhD thesis of‌​‌ Cédric Baudet which is​​ part of the HyBox​​​‌ project.

In this work,‌ we consider the diffraction‌​‌ of waves by an​​ object partially covered by​​​‌ a periodic layer whose‌ thickness tends to 0.‌​‌ This situation can model​​ industrial applications where the​​​‌ layer often consists of‌ a metamaterial with unusual‌​‌ wave propagation properties. For​​ layers that cover the​​​‌ entire object, there are‌ already known solutions to‌​‌ this problem. In our​​ case, where the layer​​​‌ is only partial, the‌ difficulty is to treat‌​‌ the tips of the​​ layer, for which no​​​‌ effective model is known‌ yet.

In a previous‌​‌ work, we established a​​​‌ full asymptotic expansion of​ the field when the​‌ thickness of the layer​​ tends to 0, by​​​‌ using matched asymptotic expansions.​ The terms of the​‌ expansion that describe the​​ field far from the​​​‌ corner are built inductively,​ satisfy a boundary condition​‌ replacing the layer and​​ have an intricate singular​​​‌ behavior near the corner.​ This behavior involves algebraic​‌ matching coefficients and coefficients​​ coming from corner profiles.​​​‌ In this work, we​ provide effective models of​‌ order 3, resp. 5.​​ These models replace, far​​​‌ from the corner, the​ layer by an effective​‌ Robin, resp. Ventcel, boundary​​ condition. In addition, we​​​‌ introduce an artificial boundary​ near the corner and​‌ impose a Dirichlet-to-Neumann (DtN)​​ condition. The DtN map​​​‌ relies on a modal​ decomposition of the truncated​‌ asymptotic expansion at order​​ 3, resp. 5. The​​​‌ modes take into account​ the singular behavior, the​‌ algebraic matching coefficients and​​ the corner profiles. We​​​‌ provide error estimates regarding​ the truncation of the​‌ DtN modes. Numerical results​​ illustrates the accuracy of​​​‌ the effective models.

Wave​ propagation in quasi periodic​‌ media

Participants: Sonia Fliss​​, Patrick Joly.​​​‌

This work is done​ in collaboration with Pierre​‌ Amenoagbadji (Columbia University) Our​​ main objective is to​​​‌ develop original numerical methods​ for the solution of​‌ the time-harmonic wave equation​​ where some quasi-periodicity arises​​​‌ in the heterogeneity or​ in the geometry of​‌ the propagation medium. This​​ includes two situations:

• 1D​​​‌ quasi-periodic media: we developed​ an adapted numerical method​‌ based on the so-called​​ lifting approach that was​​​‌ first studied and implemented​ in the case with​‌ absorption. The idea is​​ to interpret the solution​​​‌ of the 1D Helmholtz​ equation as the trace​‌ along the same line​​ of the solution of​​​‌ an augmented degenerate PDE​ in higher dimensions, with​‌ periodic coefficients. The key​​ point is to characterize​​​‌ the transparent boundary condition​ via the DtN operator​‌ associated to the augmented​​ problem through a propagation​​​‌ operator which is the​ solution of a Riccati​‌ equation whose construction is​​ based on the solution​​​‌ of periodicity cell problems..​

  More recent developments concern​‌ the non-absorbing case, for​​ which we have proposed​​​‌ a method based on​ the limiting absorption principle.​‌ For this, one first​​ needs to replace the​​​‌ DtN operator by a​ so-called RtR operator which​‌ associates an incoming Robin​​ trace to an outgoing​​​‌ one. The second difficulty​ consists in selecting the​‌ good physical solution of​​ the Riccati equation for​​​‌ the corresponding propagation operator.​ We have proposed a​‌ method based on an​​ energy-flux criterion to select​​​‌ the physical solution, under​ some technical assumptions.

• Transmission​‌ between two 2D periodic​​ half-spaces: the interesting case​​​‌ is when the two​ structures are not periodic​‌ in the direction of​​ the interface, or when​​​‌ their periods along the​ interface are not commensurate.​‌ However, in this situation,​​ the problem presents a​​​‌ hidden quasi-periodic structure with​ respect to the coordinate​‌ along the interface, in​​ such a way that​​​‌ they fall into the​ scope of the lifting​‌ approach. In a first​​ step, we have considered​​ situations where the structures​​​‌ could be lifted in‌ 3D, that is

  1. the‌​‌ case where the two​​ media are periodic along​​​‌ the interface, but with‌ non-commensurate periods;
  2. the case‌​‌ where one medium is​​ constant, while the other​​​‌ is not periodic with‌ respect to the variable‌​‌ along the interface.

  In​​ each case, the full​​​‌ method couples the DtN‌ (or RtR) approach similar‌​‌ to the 1D case,​​ with the use of​​​‌ the Floquet-Bloch transform with‌ respect to the variable‌​‌ of the lifted interface.​​ An additional difficulty lies​​​‌ in the resolution of‌ 3D cell problems. We‌​‌ have developed a quasi​​ 2D methods which reduces​​​‌ their resolution to a‌ family of independent 2D‌​‌ problems set in rectangles​​ and a non-local problem​​​‌ for an auxiliary unknown‌ set on (a part‌​‌ of) the boundary of​​ the cubic cell. The​​​‌ implementation of this method‌ produces satisfactory results, and‌​‌ an article has benn​​ published.

High-frequency effective models​​​‌ for subsonic space-time metamaterials‌

Participants: Marie Touboul.‌​‌

This is a collaboration​​ with Richard Craster (Imperial​​​‌ College London). Laminated materials‌ with space-time modulated properties‌​‌ are known to exhibit​​ unconventional dispersion diagrams with​​​‌ the occurrence of non-symmetric‌ band gaps in the‌​‌ subsonic regime, and of​​ gaps in wavenumber in​​​‌ the supersonic regime. However,‌ these phenomena occur at‌​‌ higher-frequencies for which the​​ low-frequency homogenization is no​​​‌ longer valid. We therefore‌ developed and validated high-frequency‌​‌ homogenization (Craster, 2010) for​​ the subsonic case in​​​‌ order to get an‌ insight on the effective‌​‌ behaviour of the media.​​ We are currently working​​​‌ on exploiting these effective‌ models to better understand‌​‌ the physics of this​​ configuration.

Predicting topologically protected​​​‌ interface state with high-frequency‌ homogenization

Participants: Marie Touboul‌​‌.

This is a​​ collaboration with Bruno Lombard​​​‌ and Antonin Coutant (Laboratoire‌ de Mécanique et d’Acoustique,‌​‌ France).

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 high-frequency 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.

Enriched homogenized model​​ in the presence of​​​‌ boundaries or interfaces

Participants:‌ Laure Giovangigli, Sonia‌​‌ Fliss, Edouard Meddouri-Bernard​​, Marie Touboul.​​​‌

In a first work‌ we study the scalar‌​‌ wave equation (in the​​ time or in the​​​‌ frequency domain) in presence‌ of a periodic medium‌​‌ with a boundary when​​ the period is small​​​‌ compared to the wavelength.‌ High order effective models‌​‌ involving high order differential​​​‌ operators of higher orders​ (at least 4) have​‌ been proposed for infinite​​ periodic media. Proposing boundary​​​‌ conditions for these models​ remain an open question.​‌ Note that one of​​ the difficulty is that​​​‌ one has to derive​ variational conditions for differential​‌ operators of order 4​​ from original variational conditions​​​‌ for operators of order​ 2.

The past few​‌ years, for frequency domain​​ wave equation, we have​​​‌ proposed a new asymptotic​ expansion which takes into​‌ account the microscopic phenomena​​ near the boundary or​​​‌ at the interface with​ a periodic half-space. Our​‌ approach enables to propose​​ appropriate boundary conditions for​​​‌ these models. The objective​ is to apply these​‌ techniques in the context​​ of the long time​​​‌ homogeneization of time-domain wave​ equation. The difficulty is​‌ to propose appropriate boundary​​ conditions that makes the​​​‌ effective problems well-posed, which​ requires new techniques for​‌ time domain problems. We​​ have, for now, studied​​​‌ the presence of a​ Dirichlet boundary, proposed effective​‌ conditions, showed well-posedness and​​ performed the error analysis.​​​‌ We want then to​ tackle similar questions for​‌ Neumann or transmission conditions.​​ This work is done​​​‌ in collaboration with Bruno​ Lombard (LMA, Marseille) and​‌ Remi Cornaggia (Institut d’Alembert,​​ Sorbonne Université).

In a​​​‌ second work we study​ the Helmholtz transmission problem​‌ between two unbounded periodic​​ media when the periods​​​‌ are small compared to​ the wavelength of the​‌ incident wave. We propose​​ effective models whose solutions​​​‌ approximate the solution of​ the Helmholtz equation in​‌ these complex media at​​ any order (with respect​​​‌ to the size $\mathrm{\epsilon }$ of the period). These​‌ approximations remain valid close​​ to the interface. Although​​​‌ the effective problems still​ depend on $\epsilon$,​‌ they do not depend​​ on the microscopic scale​​​‌ and are thus cheap​ to solve numerically. The​‌ constants involved in the​​ transmission conditions across the​​​‌ interface depend on periodic​ Laplace problems posed in​‌ an infinite band of​​ width one. A numerical​​​‌ method to solve these​ corrector problems is proposed​‌ and numerical simulations corroborate​​ the theoretical error estimates.​​​‌ The goal is now​ to extend this work​‌ to quasi-periodic and random​​ media.

Scattering from a​​​‌ random thin coating of​ nanoparticles: the Dirichlet case​‌

Participants: Sonia Fliss,​​ Laure Giovangigli.

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 effective model​​ and present numerical simulations​​​‌ that illustrate our theoretical‌ results. A paper on‌​‌ this subject has been​​ submitted this year.

We​​​‌ are currently working on‌ the Neumann case that‌​‌ is not an easy​​ extension. One of the​​​‌ difficulty is the construction‌ of a stationary corrector‌​‌ term that can be​​ achieved using regularization at​​​‌ the cost of a‌ degradation of the error‌​‌ estimates.

Scattering by electromagnetic​​ waves from an arbitrarily​​​‌ shaped object coated with‌ a random rough thin‌​‌ layer

Participants: Pierre Boulogne​​, Sonia Fliss,​​​‌ Laure Giovangigli.

This‌ work is a joint‌​‌ work with Justine Labat​​ (CEA-CESTA). We are interested​​​‌ in the time-harmonic scattering‌ by a bounded regular‌​‌ object coated with a​​ thin rough layer. Although​​​‌ this problem can be‌ solved numerically, computational costs‌​‌ become prohibitive when the​​ layer's thickness is small​​​‌ compared to the object’s‌ characteristic size and the‌​‌ incident wavelength. However, this​​ scale separation can be​​​‌ exploited to derive effective‌ models that avoid meshing‌​‌ the thin layer.

Effective​​ models for arbitrary objects​​​‌ coated by thin layers‌ of constant thickness on‌​‌ the one hand or​​ for planes coated by​​​‌ rough thin layers on‌ the other hand are‌​‌ well established in the​​ literature. By contrast, objects​​​‌ of arbitrary shapes coated‌ by rough layers require‌​‌ additional tools. Such models​​ can be derived when​​​‌ the roughness (i.e. the‌ surface variations of the‌​‌ layer) is of the​​ same order as the​​​‌ layer's thickness and is‌ either periodic or random‌​‌ and stationary ergodic.

Assuming​​ a two-dimensional setting governed​​​‌ by the Helmholtz equation,‌ we derive homogenized models‌​‌ of any orders for​​ the periodic case and​​​‌ of order 1 and‌ 2 for the random‌​‌ case. The derivation relies​​ on a multi-scale asymptotic​​​‌ method that yields correctors‌ defined on a «‌​‌ cell » and effective​​ solutions, that depend on​​​‌ the roughness properties and‌ on the geometry of‌​‌ the object. The correctors​​ capture the near-field behavior​​​‌ of the solution, while‌ the effective solutions satisfy‌​‌ an equivalent boundary condition​​ around the object, providing​​​‌ accurate approximations of the‌ far field behaviour. We‌​‌ establish quantitative error estimate​​ between the true solution​​​‌ and the effective models‌ at different orders. Numerical‌​‌ simulations for a range​​ of geometries and coatings​​​‌ validate the theoretical convergence‌ rates.

Long time homogeneization‌​‌ in random media

Participants:​​ Laure Giovangigli, Edouard​​​‌ Meddouri-Bernard.

This is‌ a joint work with‌​‌ Mitia Duerinckx (ULB). We​​ study the time-dependent scalar​​​‌ wave equation in presence‌ of a microstrucured random‌​‌ medium. The classical homogenization​​ theory enables to derive​​​‌ an effective model which‌ provides an approximation of‌​‌ the solution. But this​​ effective model does not​​​‌ take into account the‌ long time dispersive effects‌​‌ which appears naturally in​​ microstructured media. This is​​​‌ well known in periodic‌ media since the works‌​‌ of Santosa and Symes​​​‌ in the 90s, but​ in the case of​‌ stationary ergodic media numerous​​ questions remain open: at​​​‌ which time-horizon the homogenization​ remains valid? Can we​‌ quantify the fluctuations of​​ the solution? Using Glauber​​​‌ calculus we prove in​ the case of an​‌ i.i.d. medium that the​​ homogenization theorem remains valid​​​‌ on $[0,T{\epsilon }^{-\mathrm{1‌}}]$. This result​​ is illustrated by numerical​​​‌ simulations in one and​ two dimensions.

Galerkin Foldy-Lax​‌ model for sound-soft scattering​​ by multiple small particles​​​‌

Participants: Maryna Kachanovska,​ Adrian Savchuk.

The​‌ Foldy-Lax model is an​​ asymptotic model used to​​​‌ compute the solution to​ the problem of scattering​‌ by small obstacles. While​​ this subject had been​​​‌ fairly well-studied in the​ frequency-domain, its time- domain​‌ analysis is still in​​ its infancy stage. In​​​‌ our previous work, we​ have suggested a construction​‌ of an asymptotic model​​ as a Galerkin spatial​​​‌ semi-discretization of associated boundary​ integral formulations. The main​‌ idea is to choose​​ the basis functions in​​​‌ a way that the​ convergence of the method​‌ is ensured not by​​ increasing the cardinality of​​​‌ the Galerkin basis, but​ rather by decreasing the​‌ size of the obstacles.​​ We have shown previously​​​‌ that the same choice​ of the Galerkin basis​‌ as for the sphere​​ case cannot yield convergence​​​‌ for particles of arbitrary​ shape (in 3D). This​‌ was confirmed by our​​ numerical experiments. Therefore, we​​​‌ suggested an alternative choice​ of the basis, inspired​‌ by existing works of​​ Sini et al. Namely,​​​‌ now we choose basis​ functions as equilibrium densities.​‌ We have proven the​​ second-order relative convergence, and​​​‌ tested the method numerically.​ Further asymptotic analysis has​‌ allowed us to improve​​ the numerical performance of​​​‌ the method without sacrificing​ the convergence rates. We​‌ have summarized these results​​ in a manuscript. Moreover,​​​‌ we were able to​ extend the asymptotic method​‌ to obtain a higher-order​​ convergence.

Long-time error analysis​​​‌ for scattering by a​ small particle

Participants: Maryna​‌ Kachanovska, Adrian Savchuk​​.

In the frequency​​​‌ domain wave scattering problems,obstacle​ 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 work 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. Our proof uses​ ingredients from the scattering​‌ theory, spectral theory and​​ Laplace countour deformaiton techniques.​​​‌

A non-standard transmission problem​ between an infinite tree​‌ and exterior

Participant: Maryna​​ Kachanovska.

In this​​​‌ theoretical contribution in collaboration​ with K. Naderi and​‌ K. Pankrashkin (University of​​ Oldenburg), we have investigated​​​‌ a non-standard transmission problem​ between a fractal tree​‌ and a standard Eucledian​​ exterior, using the formalism​​ of trace operators we​​​‌ developed previously in our‌ respective works (it seems‌​‌ that the original idea​​ is due to B.​​​‌ Maury and co-workers). The‌ originality here is the‌​‌ coupling between structures of​​ Hausdorff dimension less than​​​‌ the dimension of the‌ ambient space and the‌​‌ exterior domain (of the​​ Hausdorff dimension of the​​​‌ ambient space). The key‌ difficulty was the definition‌​‌ of the conormal trace,​​ which we've done variationally.​​​‌ Once this was done‌ we made use of‌​‌ the boundary integral equations​​ apparatus to conclude about​​​‌ the well-posedness of such‌ non-standard coupled problems.

Wave​​​‌ Propagation in Plasmas

Participants:‌ Manaswinee Bezbaruah, Patrick‌​‌ Ciarlet, Maryna Kachanovska​​.

This work is​​​‌ a continuation of the‌ research done during the‌​‌ PhD thesis of E.​​ Peillon. Plasma heating is​​​‌ modelled by the Maxwell‌ equations with variable coefficients,‌​‌ which, in the simplest​​ 2D setting can be​​​‌ reduced to the 2D‌ Helmholtz equation, where the‌​‌ coefficient the principal part​​ of the operator changes​​​‌ its sign smoothly along‌ an interface. Such problems‌​‌ are naturally well-posed in​​ a certain weighted Sobolev​​​‌ space; however, the corresponding‌ solutions cannot contribute to‌​‌ the plasma heating, due​​ to their high regularity.​​​‌ Actually, it is possible‌ to demonstrate that plasma‌​‌ heating is induced by​​ singular solutions, which are​​​‌ square integrable but do‌ not longer lie in‌​‌ this weighted Sobolev space.​​

Theoretical Aspects: Limiting Absorption​​​‌ Principle and Spectral Analysis‌

From the theoretical viewpoint,‌​‌ the explanation of the​​ plasma heating phenomenon is​​​‌ based on the limiting‌ absorption principle. During the‌​‌ PhD thesis of E.​​ Peillon, we have also​​​‌ obtained a proof of‌ this principle in the‌​‌ case of 2D vector​​ Maxwell-like equations. Moreover, we​​​‌ have formulated the limiting‌ problem satisfied by this‌​‌ solution, and have shown​​ its well-posedness within the​​​‌ Fredholm framework.

Our contributions‌ now are two-fold. First,‌​‌ we have submitted a​​ manuscript summarizing some of​​​‌ our findings about the‌ limiting absorption principle in‌​‌ 2D vector Maxwell equations​​ on general domains (in​​​‌ collaboration with Etienne Peillon,‌ former PhD student). Second,‌​‌ we initiated studies of​​ the spectral properties of​​​‌ the new operator that‌ appears as a result‌​‌ of the LAP, in​​ collaboration with L. Boulton​​​‌ (Heriot-Watt University). This leads‌ us to study the‌​‌ pseudo-spectrum of the operator,​​ which, as we see,​​​‌ fills almost the whole‌ lower half-plane.

Numerical Aspects:‌​‌ An Alternative Numerical Formulation​​

We have formulated an​​​‌ alternative mixed problem for‌ the solution of Maxwell‌​‌ equations in plasmas, and​​ are currently implementing it​​​‌ (this is a post-doc‌ subject of Mansi Bezbaruah).‌​‌ We have finished the​​ first step which was​​​‌ to implement a simplified‌ problem.

Optimal control-based numerical‌​‌ method for problems with​​ sign-changing coefficients

Participants: Patrick​​​‌ Ciarlet, Farah Chaaban‌, Mahran Rihani.‌​‌

During the PhD of​​ F. Chaaban, we have​​​‌ studied the scalar wave‌ equation, with coefficients that‌​‌ are strictly positive in​​ part of the domain,​​​‌ and strictly negative elsewhere.‌ Using an optimal control‌​‌ approach to solve the​​​‌ model, we proved convergence​ of the numerical model​‌ in the supercritical case​​, that is when​​​‌ the contrast across a​ smooth interface between positive​‌ and negative values equals​​ $-1$. The​​​‌ mathematical theory has been​ completed, and the numerical​‌ results confirm theory. We​​ then addressed the same​​​‌ issues for the 2D​ Maxwell equations, using a​‌ formulation via Helmholtz-like decompositions.​​ The analysis of the​​​‌ 3D Maxwell equations is​ under way.

Towards non-local​‌ interface models

Participants: Patrick​​ Ciarlet, Maha Daoud​​​‌.

this work is​ done in collaboration with​‌ Juan Pablo Borthagaray (DMEL,​​ Universidad de la República,​​​‌ Montevideo, Uruguay). The long​ term goal is to​‌ better take into account​​ interface transmission conditions between​​​‌ a classical material and​ a metamaterial. The purely​‌ local models have limitations,​​ see e.g. the previous​​​‌ paragraph. On the other​ hand, nonlocal models allow​‌ to take transmission conditions​​ in a more flexible​​​‌ manner but, on the​ downside, they are much​‌ more expensive to solve​​ numerically. As a first​​​‌ step, we focused on​ the design of a​‌ global diffusion 2D/3D model​​ that couples local and​​​‌ nonlocal models, with fixed-sign​ diffusivity: conclusive results have​‌ been obtained. Then, we​​ studied some simplified 1D​​​‌ models with sign-changing coefficients:​ the results are promising,​‌ and extension to 2D​​ models is under way.​​​‌

Two-level domain decomposition preconditioner​ for the integral fractional​‌ Laplacian

Participants: Pierre Marchand​​.

The fractional Laplacian,​​​‌ and in particular its​ integral representation, shares several​‌ similarities with boundary integral​​ equations. One key feature​​​‌ is its nonlocal nature,​ which leads to dense​‌ matrices after discretization, in​​ contrast to classical partial​​​‌ differential equations that typically​ produce sparse matrices. As​‌ a result, solving finite​​ element discretizations of the​​​‌ integral fractional Laplacian is​ computationally challenging ,and scalable​‌ preconditioners are crucial to​​ enable efficient and parallel​​​‌ solution of large-scale problems.​

We developed a new​‌ preconditioner based on domain​​ decomposition methods for the​​​‌ integral fractional Laplacian, extending​ ideas originally developed for​‌ boundary integral equations using​​ the GenEO coarse space.​​​‌ We provide rigorous bounds​ on the condition number​‌ of the preconditioned system,​​ which guarantees robustness with​​​‌ respect to problem size​ and discretization parameters, and​‌ explains the observed scalability​​ of the method.

Numerical​​​‌ experiments were carried out​ using Htool-DDM for assembling​‌ the preconditioner and PyNucleus​​ for the discretization.

A​​​‌ complex-scaled boundary integral equation​ for the embedded eigenvalues​‌ and complex resonances of​​ the Neumann-Poincaré operator on​​​‌ domains with corners

Participants:​ Luiz Faria.

This​‌ work is done in​​ collaboration with Florian Monteghetti​​​‌ from the I2M at​ Aix Marseille University. The​‌ adjoint of the harmonic​​ double-layer operator, also known​​​‌ as the Neumann-Poincaré (NP)​ operator, is a boundary​‌ integral operator whose real​​ eigenvalues are associated with​​​‌ surface modes that find​ applications in e.g. photonics.​‌ On 2D domains with​​ corners, the NP operator​​​‌ looses its compactness, as​ each corner induces a​‌ bounded interval of essential​​ spectrum, and can exhibit​​​‌ both embedded eigenvalues and​ complex resonances. This work​‌ introduces a non-self-adjoint boundary​​ integral operator whose discrete​​ spectrum contains both embedded​​​‌ eigenvalues and complex resonances‌ of the NP operator.‌​‌ This operator is obtained​​ using a Green’s function​​​‌ that is complex-scaled at‌ each corner of the‌​‌ boundary. Numerical experiments using​​ a Nyström discretization on​​​‌ a graded mesh demonstrates‌ the accuracy of the‌​‌ method and its advantage​​ over a 2D finite​​​‌ element discretization implementing the‌ same complex scaling technique.‌​‌

Generalized normal modes of​​ a metallic nanoparticle

Participants:​​​‌ Anne-Sophie Bonnet-Ben Dhia,‌ Christophe Hazard.

In‌​‌ the context of a​​ collaboration with Matias Ruiz​​​‌ (University of Leicester), we‌ study the question of‌​‌ completeness for a non-standard​​ spectral problem related to​​​‌ the more classical plasmonic‌ eigenvalue problem. Suppose the‌​‌ frequency $\omega$ is given​​ and fixed. Let us​​​‌ consider the time-harmonic electromagnetic‌ scattering by a bounded‌​‌ metallic homogeneous particle of​​ permittivity ε located in​​​‌ vacuum . The spectral‌ problem we are interested‌​‌ in consists in finding​​ the values of ε​​​‌ such that this problem‌ is ill-posed, which means‌​‌ that there is an​​ outgoing solution of the​​​‌ homogeneous equations (in absence‌ of incident wave). The‌​‌ problem can be reformulated​​ as looking for the​​​‌ spectrum of a volume‌ integral operator supported in‌​‌ the particle. This operator​​ is non-compact and non-selfadjoint.​​​‌ When the particle is‌ smooth, it is known‌​‌ that its spectrum is​​ purely discrete with two​​​‌ accumulation points which are‌ -1 and -1/2. Our‌​‌ main result is a​​ condition on the particle,​​​‌ for both the 2D‌ and the 3D cases,‌​‌ such that the system​​ of eigenvectors is complete​​​‌ in $H^1$.‌ Our proof combines variational‌​‌ and layer potentials techniques​​ with the spectral theory​​​‌ of Schatten-class operators and‌ recent results on the‌​‌ spectrum of the Neumann-Poincaré​​ operator.

Wave propagation in​​​‌ unbounded hyperbolic media

Participants:‌ Maryna Kachanovska, Dylan‌​‌ Machado.

We study​​ wave propagation in cold​​​‌ plasma, in a regime‌ when it is described‌​‌ by a hyperbolic (Klein-Gordon​​ equation) in the frequency​​​‌ domain, where the role‌ of the time is‌​‌ played by one of​​ the coordinates. The problem​​​‌ possesses an additional difficulty:‌ its coefficients are frequency-dependent.‌​‌ We have considered the​​ problem in the closed​​​‌ waveguide with geometric perturbations‌ and have proven that‌​‌ the Limiting Absorption Principle​​ holds true for sufficiently​​​‌ large frequencies. The main‌ ingredients of the proof‌​‌ are a certain positivity​​ property of the Dirichlet-to-Neumann​​​‌ operator and the well-posedness‌ of the wave equation‌​‌ in moving, non-cylindrical domains.​​

Mathematical analysis of metamaterials​​​‌ in time domain

Participants:‌ Patrick Joly.

This‌​‌ topic is the subject​​ of our collaboration with​​​‌ Maxence Cassier (Institut Fresnel).‌ of the book series‌​‌ "Operator theory" of Springer​​ (edited by Daniel Alpay,​​​‌ Fabrizio Colombo and Irene‌ Sabadini). We have written‌​‌ two chapers entitled "An​​ Operator Approach to the​​​‌ Analysis of Electromagnetic Wave‌ Propagation in Dispersive Media.‌​‌ Part 1: Gereral Results"​​ and "An Operator Approach​​​‌ to the Analysis of‌ Electromagnetic Wave Propagation in‌​‌ Dispersive Media. Part 2:​​ Transmission Problems" which both​​​‌ have been accepted for‌ publication.

This material has‌​‌ been presented in a​​​‌ plenary talk at the​ Conference on Mathematics of​‌ Wave Phenomena in Karlsruhe​​ (February 2025) and will​​​‌ be the subject on​ an invited conference at​‌ the workshop "The New​​ Frontier of Herglotz-Nevanlinna Functions:​​​‌ Theory, Applications, and Open​ Problems" at BIRS, Banff,​‌ Canada in October 2026.​​

Dispersion and space-time modulation​​​‌

Participants: Marie Touboul.​

This is a collaboration​‌ with T. V. Raziman,​​ Riccardo Sapienza and Richard​​​‌ Craster (Imperial College London).​ In the optical regime,​‌ it becomes crucial to​​ take into account dispersion.​​​‌ The group of Riccardo​ Sapienza has developed some​‌ experiments to modulate the​​ permittivity (described by a​​​‌ Lorentz model). Some work​ has been conducted to​‌ develop adequate models and​​ analyse the occurrence of​​​‌ amplification in these time-modulated​ systems. The creation of​‌ surface plasmons by time​​ modulation has also been​​​‌ investigated.

Topology for time-modulated​ materials

Participants: Marie Touboul​‌.

In this project,​​ we focus on the​​​‌ optical system called photonic​ time crystal (PTC), which​‌ is a temporal analogue​​ of a photonic crystal.​​​‌ A photonic crystal is​ a dielectric material whose​‌ permittivity varies periodically in​​ space. Thus, a dielectric​​​‌ material whose permittivity varies​ periodically (and generally speaking,​‌ also abruptly) in time​​ received the name of​​​‌ photonic time crystal. In​ this system, electromagnetic waves​‌ typically experience time-refraction and​​ time-reflection at each (abrupt)​​​‌ change of the permittivity.​ By the nature of​‌ a PTC, its repeated​​ time-boundaries produce Floquet modes​​​‌ with a band structure​ in wavenumber $k$ (momentum).​‌ Unlike electronic systems, whose​​ band gaps are in​​​‌ energy, the band gaps​ of a PTC are​‌ in wavenumber: there exist​​ some certain $k$-ranges​​​‌ where instead of propagating​ waves, the modes exhibit​‌ decaying or amplification in​​ time.

The introduction and​​​‌ analysis of well-suited topological​ invariants for these materials​‌ is the subject of​​ an ongoing collaboration with​​​‌ Frank Schindler, Pavez Ignacio​ and Sébastien Guenneau (Imperial​‌ College London). We so​​ far showed that one​​​‌ can not only introduce​ a Zak phase for​‌ the real band but​​ also for the imaginary​​​‌ band.

Controlling wave propagation​ by modulating in time​‌ the parameters of imperfect​​ interfaces

Participants: Marie Touboul​​​‌.

This is a​ joint work with Michaël​‌ Darch and Bruno Lombard​​ (Laboratoire de Mécanique et​​​‌ d’Acoustique), Raphaël Assier (University​ of Manchester) and Sébastien​‌ Guenneau (Imperial College London).​​ The idea is to​​​‌ replace volumetric modulation by​ imperfect interfaces whose properties​‌ depend on time. Experimentally,​​ one could imagine a​​​‌ series of mechanical resonators​ whose mass and stiffness​‌ are modified. This interface​​ is modelled by a​​​‌ jump condition for the​ velocity and for the​‌ stress, which involves interfaces​​ parameters (inertia, compliance and​​​‌ dissipation of the interface)​ which depends on time.​‌

Firstly, we studied numerically​​ and theoretically the case​​​‌ of a single modulated​ interface. An energy balance​‌ is conducted, the generation​​ of harmonics is studied​​​‌ through a harmonic balance​ analysis, and the particular​‌ case of reflectionless modulated​​ interface is discussed. A​​​‌ time-domain numerical method is​ also developed and validated​‌ to simulate transient wave​​ phenomena across such a​​ modulated interface. Integration of​​​‌ the momentum equation and‌ of the constitutive law‌​‌ is done by a​​ fourth-order finite-difference ADER scheme.​​​‌ The time-varying jump conditions‌ are discretized by the‌​‌ Explicit Simplified Interface Method​​ (ESIM), requiring new developments​​​‌ of this method. This‌ work has led to‌​‌ a publication in Comptes​​ Rendus. Mécanique.

We then​​​‌ consider a periodic network‌ of modulated interfaces. Each‌​‌ unit cell contains ${\mathrm{N}}_i$ modulated interfaces which​​​‌ properties may differ from‌ one another. This setting‌​‌ is studied through the​​ lens of low-frequency homogenization.​​​‌ The effective model obtained‌ is characterized by effective‌​‌ parameters which are constant​​ in space but depend​​​‌ on time. If the‌ modulations of the interface‌​‌ properties are periodic in​​ time, such are the​​​‌ effective parameters, which therefore‌ leads to the occurrence‌​‌ of gaps in wavenumber.​​ This phenomenon is illustrated​​​‌ numerically together with the‌ validity of the effective‌​‌ leading order model for​​ low values of the​​​‌ source frequency and of‌ the frequency of modulation.‌​‌ A second-order model is​​ then derived to describe​​​‌ the dispersive effects which‌ are missed by the‌​‌ leading-order one. However, even​​ at the second order,​​​‌ the effective models obtained‌ are reciprocal while non-reciprocity‌​‌ is observed numerically for​​ a high-frequency of modulation​​​‌ and a periodic cell‌ containing several interfaces whose‌​‌ modulations are not in​​ phase. To overcome this​​​‌ limitation, we then introduce‌ a fast time scale‌​‌ in the homogenization process.​​ The effective model obtained​​​‌ then presents an effective‌ Willis coupling term.

Theoretical‌​‌ Analysis of Wave Propagation​​ in Time-Dependent Media

Participants:​​​‌ Patrick Joly, Marie‌ Touboul.

Recent advances‌​‌ in the theory of​​ metamaterials have drawn significant​​​‌ attention to wave propagation‌ in media with time-dependent‌​‌ properties, as such materials​​ offer the possibility to​​​‌ overcome certain fundamental limitations‌ of purely spatial metamaterials.‌​‌ However, the fundamental theoretical​​ study of wave equations​​​‌ with space- and time-dependent‌ coefficients has received limited‌​‌ attention from the applied​​ mathematics community.

This work​​​‌ contributes to this area‌ by addressing fundamental questions‌​‌ such as the existence​​ and uniqueness of solutions,​​​‌ the behavior of the‌ associated energy, and the‌​‌ role of non-smooth coefficients,​​ in particular through the​​​‌ analysis of scattering coefficients.‌ Several theoretical results are‌​‌ established in this direction​​ and are illustrated by​​​‌ time-domain numerical simulations.

Special‌ attention is devoted to‌​‌ a one-dimensional transmission problem​​ across a moving interface​​​‌ separating two homogeneous half-spaces.‌ This setting leads to‌​‌ the distinction between subsonic,​​ supersonic, and transsonic regimes,​​​‌ depending on the interface‌ velocity relative to the‌​‌ wave speeds in the​​ two media. In particular,​​​‌ the transsonic regime is‌ associated with ill-posed problems,‌​‌ characterized by non-existence or​​ non-uniqueness of solutions.

Finally,​​​‌ the regularization of the‌ moving interface is addressed,‌​‌ together with an asymptotic​​ analysis of the problem​​​‌ as the regularized velocity‌ profile converges to a‌​‌ discontinuous function.

This work​​ has led to the​​​‌ supervision of two master‌ students.

Hybrid approach​​​‌ to the numerical simulation‌ of ultrasonic NDT experiments‌​‌ on layered structures

Participants:​​​‌ Marc Bonnet.

This​ work is done in​‌ collaboration with Eric Ducasse,​​ Marc Deschamps, Romain Kubecki​​​‌ (I2M, University of Bordeaux).​

We develop a numerical​‌ simulation approach for ultrasonic​​ NDT experiments on layered​​​‌ structures that aims at​ incorporating models for flaws​‌ or other local features​​ (sensors, stiffeners„,) into a​​​‌ semi-analytical computational framework for​ the unperturbed, ideal structure.​‌ The latter takes the​​ form of the existing​​​‌ in-house code TraFiC developed​ at I2M by E.​‌ Ducasse and based on​​ Laplace transform for the​​​‌ time variable and partial​ Fourier transforms along translation-independent​‌ or circumferential spatial coordinates;​​ this code allows to​​​‌ model long-range wave propagation​ in undisturbed structures. The​‌ various flaws or features​​ are then taken into​​​‌ account by using local​ finite element models and​‌ a domain decomposition iterative​​ coupling approach. Regarding the​​​‌ latter, we established the​ convergence of DD iterations​‌ based on Robin boundary​​ conditions on each (TraFiC​​​‌ or FE) subdomain having​ a shared interface. We​‌ formulated projection procedures allowing​​ to convert expansions of​​​‌ fields on the coupling​ interface on the approximation​‌ bases pertaining to either​​ medium, as a crucial​​​‌ ingredient of the overall​ coupling approach. Extensive numerical​‌ tests have been conducted​​ on 2D sample configurations.​​​‌ This work is undertaken​ through the jointly-advised thesis​‌ of Romain Kubecki, whose​​ doctoral grant is co-funded​​​‌ by DGA and CEA​ LIST.

Singular solutions of​‌ linear aeroacoustics in recirculating​​ base flows

Participants: Patrick​​​‌ Joly, Jean-François Mercier​.

This is the​‌ continuation of the PhD​​ thesis of A. Bensalah​​​‌ (Airbus) with whom we​ pursue our collaboration. We​‌ recall that aeroacoustics concerns​​ the propagation of sound​​​‌ in a fluid in​ stationnary flow (for classical​‌ acoustics, the flow is​​ at rest). The PhD​​​‌ of A. Bensalah (defended​ in 2018) was devoted​‌ to the Goldstein model​​ in the time harmonic​​​‌ regime, for both mathematical​ and numerical issues. We​‌ were able to prove​​ that the model was​​​‌ well posed in a​ rather standard functional framework​‌ under the essential assumption​​ that the base flow​​​‌ did not contain any​ closed streamline (plus additional​‌ assumption on the size​​ of the vorticity of​​​‌ this flow). The case​ of recirculant flows, i.e.​‌ with closed streamlines, is​​ much more delicate. During​​​‌ his thesis, A. Bensalah​ initiated the study of​‌ a simple model case​​ : a 2D circular​​​‌ flow in an annulus.​ During the past three​‌ years, we have completed​​ this work. We use​​​‌ the method of limiting​ absorption (where $\epsilon >‌0$ is the size​​ of the absorption) and​​​‌ the main technical ingredients​ for the analysis are:​‌

• reduce the problem to​​ a countable family of​​​‌ ODE’s (separation of variables​ in polar coordinates)
• use​‌ Fröbenius method and Fuchs​​ theory for passing to​​​‌ the limit $\epsilon \to 0$. This approach​‌ leads to the apparition​​ of singular solutions that​​​‌ can be fully described.​ These solutions are outside​‌ the functional framework used​​ for the analysis of​​​‌ the non recirculating case.​

The corresponding article has​‌ been already submitted and​​ the referees have raised​​ interesting questions that led​​​‌ us to introduce new‌ developments and to modify‌​‌ the wording of the​​ article. The new article​​​‌ is about to be‌ submitted.

Time stepping methods‌​‌ for linear Friedrichs systems​​

Participants: Patrick Joly.​​​‌

This is a work‌ in collaboration with S.‌​‌ Imperiale (Medisim, Inria) and​​ J. Rodríguez (University of​​​‌ Santiago ce Compostela).

The‌ question we address is‌​‌ a prori very classical​​ and academic : we​​​‌ want to study the‌ stability of explicit numerical‌​‌ schemes for the time​​ discretization of semi-discrete problems​​​‌ issued from the space‌ discretization of first order‌​‌ hyperbolic Friedrichs systems (which​​ include most of relevant​​​‌ linear wave propagation models‌ in physics) with Discontinuous‌​‌ Galerkin Methods, using centered​​ fluxes (which are slightly​​​‌ suboptimal in terms of‌ accuracy but preserve the‌​‌ conservation of energy) or​​ off-centered schemes (which restaure​​​‌ the optimal accuracy but‌ introduce numerical dissipation). This‌​‌ type of method is​​ of particular interest in​​​‌ the context of time‌ domain aeroacoustics.

We have‌​‌ finalized the work initiated​​ in 2023 based on​​​‌ energy techniques. In particular,‌ we have quantified the‌​‌ CFL constants appearing in​​ the stability conditions in​​​‌ terms of the mesh‌ stepsize when ${\mathbb{P}}_{\mathrm{k‌‌}}$ discontinuous are used for​​ the space discretization. The​​​‌ corresponding article is about‌ to be submitted. A‌​‌ companion article devoted to​​ the Von Veumann analysis​​​‌ is in preparation.

10.1.1‌​‌ Visits of international scientists​​

10.1.2 Visits to​​​‌ international teams

10.2.1 Horizon Europe

ANR

DGA​​ / AID

Action‌ Exploratoire Inria

11.1.1​​​‌ Scientific events: organisation

• POEMS‌ organizes, under the responsability‌​‌ of M. Kachanovska, a​​ monthly seminar. One occurrence​​​‌ each semester is co-organized‌ with two other inria‌​‌ teams, IDEFIX and M3DISIM.​​

11.1.2 Journal​‌

11.1.3 Invited talks

11.1.4 Scientific expertise​​​‌

• M. Kachanovska acted as‌ project reviewer for the‌​‌ Swiss National Science Foundation.​​

11.1.5 Research administration

• L.​​​‌ Bourgeois is ENSTA's point‌ of contact for scientific‌​‌ integrity.
• E. Bécache is​​ a deputy chair of​​​‌ the Doctoral School EDMH‌ (École Doctorale Mathématiques Hadamard).‌​‌
• M. Bonnet is since​​ 2019 an appointed member​​​‌ of the COMEVAL, a‌ committee of the Ministry‌​‌ of Ecological and Inclusive​​ Transition (MEIT) similar to​​​‌ a CNRS National Committee‌ section and tasked with‌​‌ the competitive recruitment and​​ career overseeing of the​​​‌ cadre of junior and‌ senior scientists managed by‌​‌ the MEIT. He joined​​ the steering committee of​​​‌ COMEVAL in September 2023.‌
• A.-S. Bonnet-Ben Dhia is‌​‌ a member of the​​ Scientific Council of CNRS​​​‌ since October 2023.
• S.‌ Chaillat is a member‌​‌ of the board of​​ directors of IP Paris​​​‌ (Institut Polytechnique de Paris).‌
• S. Chaillat is the‌​‌ vice president of the​​ (national) Computational Structural Mechanics​​​‌ Association (CSMA)
• P. Ciarlet‌ is a member of‌​‌ the scientific council of​​ the Monalisa federative research​​​‌ project at ONERA (2023-25).‌
• S. Fliss is deputy-chair‌​‌ of the Applied Mathematics​​ Department (UMA) at ENSTA​​​‌ Paris.
• S. Fliss is‌ a member of the‌​‌ scientific committee of the​​ FMJH (Fondation Mathématique Jacques​​​‌ Hadamard).
• M. Kachanovska is‌ a member of the‌​‌ Inria evaluation commission, since​​ September 2025.
• J.-F. Mercier​​​‌ is member of the‌ Academic Council of IP‌​‌ Paris (Institut Polytechnique de​​ Paris).
• J.-F. Mercier is​​​‌ one of the pilots‌ of Axis 1 "Maritime‌​‌ engineering for sustainable ships"​​ of the CIMO (Interdisciplinary​​​‌ Centre for Sea and‌ Ocean)
• A. Modave is‌​‌ assistant director (research support​​ and unit affairs) of​​​‌ the Applied Mathematics Unit‌ of ENSTA, a member‌​‌ of the research council​​ of ENSTA, and a​​​‌ member of the scientific‌ committee of the mesocenter‌​‌ of IP Paris (Institut​​ Polytechnique de Paris).
• P.​​​‌ Marchand is, since September‌ 2023, a member of‌​‌ the Inria Scientific Committee​​ for PhD and Postdoctoral​​​‌ Positions.
• Luiz M. Faria‌ is, since 2025, the‌​‌ Inria Saclay scientific representant​​ for the IES comission.​​​‌
• Luiz M. Faria is,‌ since 2023, a member‌​‌ of the Commission de​​ Développement Technologique (CDT) of​​​‌ Inria Saclay.
• Luiz M.‌ Faria is, since 2025,‌​‌ a member of the​​ selection committe for the​​​‌ FMJH thematic postdoctoral fellowship.‌

11.2.1​​​‌ Administration

Permanent members of‌ POEMS are involved in‌​‌ the management of the​​ engineering program at ENSTA​​​‌ Paris and the master‌ program in applied mathematics‌​‌ at IP Paris and​​​‌ Université Paris-Saclay.

• L. Bourgeois:​ coordinator of the 2nd​‌ year Maths Program at​​ ENSTA; co-head of the​​​‌ M1 Applied Mathematics common​ to IP Paris and​‌ Université Paris-Saclay;
• X. Claeys:​​ coordinator of the 3nd​​​‌ year ENSTA programs on​ modelling and simulation; co-head​‌ of the M2 AMS​​ (Analyse, Modélisation et Simulation)​​​‌ common to IP Paris​ and Université Paris-Saclay;
• S.​‌ Fliss: president of the​​ PhD track of Mathematics​​​‌ of IP Paris;
• L.​ Giovangigli: coordinator of the​‌ 3nd year ENSTA programs​​ on finance and mathematics​​​‌ for life sciences;
• E.​ Lunéville: coordinator of the​‌ apprenticeship training at ENSTA.​​

11.2.2 Courses taught

All​​​‌ permanent members of POEMS,​ as well as most​‌ PhD students and post-docs,​​ are involved in teaching​​​‌ activities. A large fraction​ of these activities is​‌ included in the curriculum​​ of the engineering school​​​‌ ENSTA Paris that hosts​ POEMS team. The 3rd​‌ year of this curriculum​​ is coupled with various​​​‌ research masters, in particular​ the master Analysis, Modelization​‌ and Simulation (denoted below​​ M2 AMS) common to​​​‌ Institut Polytechnique de Paris​ and Université Paris-Saclay.

11.2.3 Supervision

• PhD :‌​‌ Farah Chaaban, "Unconditionally stable​​ numerical methods for solving​​​‌ transmission problems with sign-changing‌ coefficients", defended in December‌​‌ 2025, P. Ciarlet and​​ M. Rihani
• PhD :​​​‌ Dongchen He, "Boundary integral‌ methods for Stokes flows‌​‌ with deformable implicit surfaces",​​ defended in October 2025,​​​‌ L. Faria
• PhD: Roxane‌ Delville-Atchekzai, "Parallelization of the‌​‌ numerical treatment of cross-points​​ in domain decomposition for​​​‌ waves", defended in June‌ 2025, Xavier Claeys and‌​‌ Matthieu Lecouvez
• PhD :​​ Louise Pacaut, "Development of​​​‌ an accelerated numerical BEM/BEM‌ method to determine the‌​‌ Green function of a​​ fluid-structure problem.", defended in​​​‌ January 2025, S. Chaillat‌ and J. F. Mercier‌​‌
• PhD : Aurélien Parigaux,​​ "Construction of transparent boundary​​​‌ conditions for electromagnetic waveguides",‌ defended in December 2025,‌​‌ A.-S. Bonnet-Ben Dhia and​​​‌ L. Chesnel
• PhD :​ Simone Pescuma, "Novel Discontinuous​‌ Finite Elements Methods for​​ Time-Harmonic Wave Propagation", defended​​​‌ in November 2025, G.​ Gabard and A. Modave​‌
• PhD: Arthur Saunier, "Préconditionnement​​ par matrices hiérarchiques pour​​​‌ des problèmes à convection​ dominante", defended in December​‌ 2025, Xavier Claeys, Ani​​ Anciaux, Leo Agelas and​​​‌ Ibtihel Ben Garbia
• PhD​ in progress : Sarah​‌ Al Humaikani « Wave​​ propagation in junction of​​​‌ open waveguides", started October​ 2023, A.-S. Bonnet-Ben Dhia​‌ et S. Fliss
• PhD​​ in progress : Louis​​​‌ AUFFRET, «Advanced Fast BEM​ solver to model long​‌ period seismic waves on​​ realistic configurations», started November​​​‌ 2025, S. Chaillat, J.F.​ Semblat
• PhD in progress​‌ : Cédric Baudet, "Modelisation​​ of partial coatings in​​​‌ electromagnetism", started October 2022,​ S. Fliss and P.​‌ Joly
• PhD in progress​​ : Antonin Boisneault, «​​​‌ Numerical methods and high​ performance simulation for 3D​‌ imaging in complex media​​ », started October 2023,​​​‌ Marcella Bonazzoli, Xavier Claeys​ and Pierre Marchand
• PhD​‌ in progress : Pierre​​ Boulogne « Asymptotic Modelling​​​‌ of a random rough​ thin layer in the​‌ context of electromagnetic wave​​ scattering », started November​​​‌ 2024, S. Fliss and​ L. Giovangigli
• PhD in​‌ progress : Yahya BOYE,​​ «Fast Boundary Element Method​​​‌ for Finite-Geometry Problems in​ Contact Mechanics », started​‌ October 2024, S. Chaillat,​​ V. Yastrebov
• PhD in​​​‌ progress : Mario Gervais,​ "A posteriori estimators of​‌ a nonconforming domain decomposition​​ method", started October 2022,​​​‌ P. Ciarlet and F.​ Madiot
• PhD in progress:​‌ Paul Kaassis, “Modélisation mécanique​​ des tissus biologiques par​​​‌ homogénéisation stochastique pour l'imagerie​ médicale”, started December 2025,​‌ L. Giovangigli, L. Seppecher​​ and G. Vial
• PhD​​​‌ in progress: Romain Kubecki,​ "Development of hybrid numerical​‌ methods for the scattering​​ of ultrasonic waves by​​​‌ obstacles on layered structures,​ and application to nondestructive​‌ testing", started March 2023,​​ M. Bonnet
• PhD in​​​‌ progress : Dylan Machado,​ 'Wave propagation in unbounded​‌ hyperbolic media', started October​​ 2024, M. Kachanovska
• PhD​​​‌ in progress: Edouard Meddouri-Bernard,​ “Modèles homogénéisés enrichis en​‌ présence de bords ou​​ d’interface: cas périodique, quasi-périodique​​​‌ et au delà.”, started​ October 2025, S. Fliss​‌ and L. Giovangigli
• PhD​​ in progress: Yacine Mohammedi,​​​‌ "Discrete adjoint method applied​ to the Ffowcs-Williams Hawkings​‌ integral equation for aeroacoustic​​ shape optimization", started October​​​‌ 2023, M. Bonnet
• PhD​ in progress : Adrian​‌ Savchuk, "Asymptotic modelling of​​ time-domain electromagnetic scattering by​​​‌ small particles", started October​ 2022, M. Kachanovska and​‌ E. Bécache
• PhD in​​ progress: Sofia Suárez, "Problèmes​​​‌ inverses pour des ultrasons​ en milieux multi-échelles, application​‌ à l’imagerie médicale”, started​​ October 2025, J. Garnier,​​​‌ L. Giovangigli and P.​ Millien
• PhD in progress​‌ : Adrien Vet, "Fast​​ boundary element method for​​​‌ simulating 3D cracked structures.​ Implementation and coupling with​‌ the finite element method",​​ started March 2024, M.​​​‌ Bonnet, L. Faria and​ R. de Moura Pinho​‌
• PhD in progress :​​ Timothée Raynaud, « Analysis​​​‌ and acceleration of Krylov​ iterative methods for the​‌ numerical solution of time-harmonic​​ wave problems », started​​​‌ October 2023, Victorita Dolean,​ Pierre Marchand and Axel​‌ Modave
• PhD in progress:​​ Arthur Saunier, "Hierarchical preconditioners​​ applied to advection-diffusion problems",​​​‌ started October 2022, Xavier‌ Claeys, Ani Anciaux, Léo‌​‌ Agelas and Ibtihel Ben​​ Gharbia
• PhD in progress​​​‌ : Raphaël Terrine, "Identification‌ of bottom deformations of‌​‌ the ocean from surface​​ measurements", started October 2023,​​​‌ L. Bourgeois and M.‌ Moireau
• PostDoc : Manaswinee‌​‌ Bezbaruah : "Numerical study​​ of hybrid resonances in​​​‌ cold plasma", started July‌ 2025, P. Ciarlet, M.‌​‌ Kachanovska
• PostDoc : Ahmed​​ Chabib : "GPU-accelerated HDG​​​‌ finite element solver for‌ time-harmonic propagation problems", started‌​‌ September 2024, until September​​ 2025, C. Geuzaine and​​​‌ A. Modave
• PostDoc :‌ Maha Daoud : "Theoretical‌​‌ and numerical study of​​ a nonlocal model with​​​‌ a discontinuous coefficient", started‌ September 2024, until August‌​‌ 2025, P. Ciarlet
• PostDoc​​ : Ari Rappaport :​​​‌ "HDG finite element method‌ and DDM for time-harmonic‌​‌ electromagnetism in complex media",​​ started April 2024, M.​​​‌ Bonazzoli, T. Chaumont-Frelet, P.‌ Ciarlet, A. Modave

International​​​‌ journals

• 1 articleM.‌Michelle Almakari, N.‌​‌N. Kheirdast, C.​​C. Villafuerte, M.​​​‌ Y.Marion Y. Thomas‌, P.P. Dubernet‌​‌, J.J. Cheng​​, A.A. Gupta​​​‌, P.P. Romanet‌, S.S. Chaillat‌​‌ and H. S.Harsha​​ S. Bhat. Spectrum​​​‌ of slip dynamics, scaling‌ and statistical laws emerge‌​‌ from simplified model of​​ fault and damage zone​​​‌ architecture.Journal of‌ Geophysical Research : Solid‌​‌ EarthSeptember 2025HAL​​
• 2 articleP.Pierre​​​‌ Amenoagbadji, S.Sonia‌ Fliss and P.Patrick‌​‌ Joly. Time-harmonic wave​​ propagation in junctions of​​​‌ two periodic half-spaces.‌Pure and Applied Analysis‌​‌72April 2025​​, 299-357HALDOI​​​‌
• 3 articleE.Emmanuel‌ Audusse, V.Virgile‌​‌ Dubos, N.Noémie​​ Gaveau and Y.Yohan​​​‌ Penel. Energy stable‌ and linearly well-balanced numerical‌​‌ schemes for the non-linear​​ Shallow Water equations with​​​‌ Coriolis force.SIAM‌ Journal on Scientific Computing‌​‌4701January 2025​​, A1-A23HALDOI​​​‌
• 4 articleC.Cédric‌ Baudet. Asymptotic analysis‌​‌ at any order of​​ Helmholtz's problem in a​​​‌ corner with a thin‌ layer: an algebraic approach‌​‌.Asymptotic Analysis2025​​. In press. HAL​​​‌
• 5 articleA.-S.Anne-Sophie‌ Bonnet-Ben Dhia, L.‌​‌Lucas Chesnel and M.​​Mahran Rihani. Maxwell's​​​‌ equations with hypersingularities at‌ a negative index material‌​‌ conical tip.Pure​​ and Applied Analysis7​​​‌12025, 127–169‌HAL
• 6 articleL.‌​‌Laurent Bourgeois and J.-F.​​Jean-François Mercier. An​​​‌ inverse problem related to‌ an elasto-plastic beam.‌​‌Inverse Problems4110​​April 2025HALDOI​​​‌
• 7 articleL.Laurent‌ Bourgeois, J.-F.Jean-François‌​‌ Mercier and R.Raphaël​​ Terrine. Identification of​​​‌ bottom deformations of the‌ ocean from surface measurements‌​‌.Inverse Problems and​​ Imaging 196December​​​‌ 2025, 1075-1113HAL‌DOI
• 8 articleM.‌​‌Maxence Cassier, P.​​Patrick Joly and L.​​​‌ A.Luis Alejandro Rosas‌ Martínez. Long time‌​‌ behaviour of the solution​​ of Maxwell's equations in​​​‌ dissipative generalized Lorentz materials‌ (II) A modal approach‌​‌.Journal de Mathématiques​​​‌ Pures et Appliquées2025​. In press. HAL​‌DOI
• 9 articleP.​​Patrick Ciarlet and A.​​​‌Axel Modave. Analysis​ of time-harmonic electromagnetic problems​‌ with elliptic material coefficients​​.Mathematical Methods in​​​‌ the Applied Sciences2025​. In press. HAL​‌DOI
• 10 articleB.​​Bryn Davies, S.​​​‌Stefan Szyniszewski, M.​Marcelo Dias, L.​‌Leo de Waal,​​ A.Anastasia Kisil,​​​‌ V.Valery P Smyshlyaev​, S.Shane Cooper​‌, I.Ilia Kamotski​​, M.Marie Touboul​​​‌, R.Richard Craster​, J.James Capers​‌, S.Simon Horsley​​, R.Robert Hewson​​​‌, M.Matthew Santer​, R.Ryan Murphy​‌, D.Dilaksan Thillaithevan​​, S.Simon Berry​​​‌, G.Gareth Conduit​, J.Jacob Earnshaw​‌, N.Nicholas Syrotiuk​​, O.Oliver Duncan​​​‌, Ł.Łukasz Kaczmarczyk​, F.Fabrizio Scarpa​‌, J.John Pendry​​, M.Marc Martí-Sabaté​​​‌, S.Sébastien Guenneau​, D.Daniel Torrent​‌, E.Elena Cherkaev​​, N.Niklas Wellander​​​‌, A.Andrea Alù​, K.Katie Madine​‌, D.Daniel Colquitt​​, P.Ping Sheng​​​‌, L.Luke Bennetts​, A.Anastasiia Krushynska​‌, Z.Zhaohang Zhang​​, M.Mohammad Mirzaali​​​‌ and A.Amir Zadpoor​. Roadmap on metamaterial​‌ theory, modelling and design​​.Journal of Physics​​​‌ D: Applied Physics58​20April 2025,​‌ 203002HALDOI
• 11​​ articleE.Eric Ducasse​​​‌, S.Samuel Rodriguez​ and M.Marc Bonnet​‌. Early-Reverberation Imaging Functions​​ for Bounded Elastic Domains​​​‌.Acta Acustica10​January 2026, 2​‌HALDOI
• 12 article​​B. B.Bojan B​​​‌ Guzina and M.Marc​ Bonnet. On the​‌ constitutive behavior of linear​​ viscoelastic solids under the​​​‌ plane stress condition.​Journal of Elasticity157​‌2025, 45HAL​​DOI
• 13 articleM.​​​‌Martin Halla, M.​Maryna Kachanovska and M.​‌Markus Wess. Radial​​ Perfectly Matched Layers and​​​‌ Infinite Elements for the​ Anisotropic Wave Equation.​‌SIAM Journal on Mathematical​​ Analysis573June​​​‌ 2025, 3171-3216HAL​DOI
• 14 articleR.​‌Ruowen Liu, H.​​Hai Zhu, H.​​​‌Hanliang Guo, M.​Marc Bonnet and S.​‌Shravan Veerapaneni. Shape​​ optimization of slip-driven axisymmetric​​​‌ microswimmers.SIAM Journal​ on Scientific Computing47​‌22025, A1065-A1090​​HALDOI
• 15 article​​​‌D.Darche Michaël,​ R.Raphaël Assier,​‌ S.S Guenneau,​​ B.Bruno Lombard and​​​‌ M.Marie Touboul.​ Scattering of transient waves​‌ by an interface with​​ time-modulated jump conditions.​​​‌Comptes Rendus. Mécanique335​2025, 923-951HAL​‌
• 16 articleL.Louise​​ Pacaut, S.Stéphanie​​​‌ Chaillat, J.-F.Jean-François​ Mercier and G.Gilles​‌ Serre. Efficient boundary​​ integral method to evaluate​​​‌ the acoustic scattering from​ coupled fluid-fluid problems excited​‌ by multiple sources.​​Journal of Computational Physics​​​‌5242025, 113736​HALDOI
• 17 article​‌S.Simone Pescuma,​​ G.Gwenael Gabard,​​​‌ T.Théophile Chaumont-Frelet and​ A.Axel Modave.​‌ A hybridizable discontinuous Galerkin​​ method with transmission variables​​ for time-harmonic acoustic problems​​​‌ in heterogeneous media.‌Journal of Computational Physics‌​‌5342025, 114009​​HALDOI

International peer-reviewed​​​‌ conferences

• 18 inproceedingsA.-S.‌Anne-Sophie Bonnet-Ben Dhia,‌​‌ L.Lucas Chesnel,​​ S.Sonia Fliss and​​​‌ A.Aurélien Parigaux.‌ Mathematical and numerical analysis‌​‌ of the modes of​​ a heterogeneous electromagnetic waveguide.​​​‌.WAVES 2025 -‌ Conference on Mathematics of‌​‌ Wave PhenomenaKarlsruhe, Germany​​February 2025HAL
• 19​​​‌ inproceedingsA.Ahmed Chabib‌, R.Roland Greffe‌​‌, C.Christophe Geuzaine​​ and A.Axel Modave​​​‌. Portage GPU d'un‌ solveur éléments finis discontinus‌​‌ hybridisé pour les problèmes​​ d'ondes en fréquence.​​​‌CFM 2025 - 26e‌ Congrès Français de Mécanique‌​‌Metz, FranceAugust 2025​​HAL
• 20 inproceedingsY.​​​‌Yacine Mohammedi, M.‌Majd Daroukh, M.‌​‌Martin Buszyk, A.​​Antoine Hajczak, I.​​​‌Itham Salah El Din‌ and M.Marc Bonnet‌​‌. Isolated Rotor Blade​​ Shape Sensitivity for Aeroacoustic​​​‌ Optimization Using a Discrete‌ Adjoint Framework.AIAA‌​‌ AVIATION FORUM AND ASCEND​​ 2025AIAA AVIATION FORUM​​​‌ AND ASCEND 2025Las‌ Vegas, United StatesAmerican‌​‌ Institute of Aeronautics and​​ Astronautics2025HALDOI​​​‌

National peer-reviewed Conferences

• 21‌ inproceedingsR.Romain Kubecki‌​‌, E.Eric Ducasse​​, M.Marc Bonnet​​​‌ and M.Marc Deschamps‌. Méthode hybride de‌​‌ simulation de champs ultrasonores​​ dans une grande structure​​​‌ stratifiée avec des objets‌ au contact.CFA‌​‌ 2025 - 17e Congrès​​ Français d'AcoustiqueParis, France​​​‌2025HAL
• 22 inproceedings‌Y.Yacine Mohammedi,‌​‌ M.Majd Daroukh,​​ M.Martin Buszyk,​​​‌ A.Antoine Hajczak,‌ I.Itham Salah El-Din‌​‌ and M.Marc Bonnet​​. Optimisation par méthode​​​‌ adjointe discrète du bruit‌ tonal d'une hélice estimé‌​‌ par la formulation fréquentielle​​ de Hanson et Parzych​​​‌.CFA 2025 -‌ 17e Congrès Français d'Acoustique‌​‌Paris, France2025HAL​​

Scientific books

• 23 book​​​‌P.Patrick Ciarlet.‌ $T$-coercivity: a practical‌​‌ tool for the study​​ of variational formulations in​​​‌ Hilbert spaces.December‌ 2025HAL

Scientific book‌​‌ chapters

• 24 inbookM.​​Maxence Cassier and P.​​​‌Patrick Joly. An‌ operator approach to the‌​‌ analysis of electromagnetic wave​​ propagation in dispersive media.​​​‌ Part 1: general results.‌.2nd edition of‌​‌ the Springer reference volume​​ Operator Theory, edited by​​​‌ Daniel Alpay, Fabrizio Colombo,‌ and Irene Sabadini.2025‌​‌. In press. HAL​​
• 25 inbookM.Maxence​​​‌ Cassier and P.Patrick‌ Joly. An operator‌​‌ approach to the analysis​​ of electromagnetic wave propagation​​​‌ in dispersive media. Part‌ 2: transmission problems..‌​‌2nd edition of the​​ Springer reference volume Operator​​​‌ Theory, edited by Daniel‌ Alpay, Fabrizio Colombo, and‌​‌ Irene Sabadini.2025.​​ In press. HAL

Reports​​​‌ & preprints

• 26 misc‌S.Sarah Al Humaikani‌​‌, A.-S.Anne-Sophie Bonnet-Ben​​ Dhia, S.Sonia​​​‌ Fliss and C.Christophe‌ Hazard. A Rellich-type‌​‌ theorem for the Helmholtz​​ equation in a junction​​​‌ of stratified media.‌April 2025HAL
• 27‌​‌ miscA.Akram Beni​​ Hamad, S.Sébastien​​​‌ Imperiale and P.Patrick‌ Joly. Hybrid FEM/IPDG‌​‌ semi-implicit schemes for time​​​‌ domain electromagnetic wave propagation​ in non cylindrical coaxial​‌ cables.November 2025​​HAL
• 28 miscA.​​​‌Antonin Boisneault, M.​Marcella Bonazzoli, X.​‌Xavier Claeys and P.​​Pierre Marchand. Discrete​​​‌ FEM-BEM coupling with the​ Generalized Optimized Schwarz Method​‌.January 2026HAL​​
• 29 miscA.Antonin​​​‌ Boisneault, M.Marcella​ Bonazzoli, X.Xavier​‌ Claeys and P.Pierre​​ Marchand. Spurious resonances​​​‌ for substructured FEM-BEM coupling​.November 2025HAL​‌
• 30 miscM.Marc​​ Bonnet, S.Stéphanie​​​‌ Chaillat and A.Alice​ Nassor. A global-in-time​‌ domain decomposition approach for​​ transient acoustic-elastic interaction.​​​‌September 2025HAL
• 31​ miscA.-S.Anne-Sophie Bonnet-Ben​‌ Dhia, L.Lucas​​ Chesnel and S.Sonia​​​‌ Fliss. Trapped modes​ in electromagnetic waveguides.​‌December 2025HAL
• 32​​ miscJ. P.Juan​​​‌ Pablo Borthagaray and P.​Patrick Ciarlet. On​‌ some coupled local and​​ nonlocal diffusion models.​​​‌May 2025HAL
• 33​ miscA.Amandine Boucart​‌, S.Sonia Fliss​​ and L.Laure Giovangigli​​​‌. Scattering from a​ random thin coating of​‌ nanoparticles: the Dirichlet case​​.December 2025HAL​​​‌
• 34 miscF.Farah​ Chaaban, P.Patrick​‌ Ciarlet and M.Mahran​​ Rihani. Solving numerically​​​‌ the two-dimensional time harmonic​ Maxwell problem with sign-changing​‌ coefficients.January 2025​​HAL
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• 42 miscS.​Sébastien Imperiale, P.​‌Patrick Joly and J.​​Jerónimo Rodríguez. Stability​​​‌ of time stepping methods​ for discontinuous Galerkin discretizations​‌ of Friedrichs' systems.​​March 2025HAL
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