The propagation of waves is one of the most common physical phenomena in nature. From the human scale (sounds, vibrations, water waves, telecommunications, radar) to the scales of the universe (electromagnetic waves, gravity waves) and of the atoms (spontaneous or stimulated emission, interferences between particles), the emission and the reception of waves are our privileged way to understand the world that surrounds us. The study and the simulation of wave propagation phenomena constitute a very broad and active field of research in various domains of physics and engineering sciences. The variety and the complexity of the underlying problems, their scientific and industrial interest, the existence of a common mathematical structure to these problems from different areas altogether justify a research project in applied mathematics and scientific computing devoted to this topic.

The activity of the team is oriented towards the design, the analysis and the numerical approximation of mathematical models for all types of problems involving wave propagation phenomena, in mechanics, physics and engineering sciences. Let us briefly describe our core business and current expertise, in order to clarify the new challenges that we want to address in the short and long terms.

Typically, our works are based on boundary value problems established by physicists to model the propagation of waves in various situations. The basic ingredient is a partial differential equation of the
hyperbolic type, whose prototype is the scalar wave equation, or the Helmholtz equation if time-periodic solutions
are considered. More generally, we systematically consider both the transient problem, in the time domain, and the time-harmonic problem, in the frequency domain. Let us mention that, even if different waves share a lot of common properties, the transition from the scalar acoustic equation to the vectorial electromagnetism and elastodynamics systems raises a lot of mathematical and numerical difficulties, and requires a specific expertise.

A notable particularity of the problems that we consider is that they are generally set in unbounded domains: for instance, for radar applications, it is necessary to simulate the interaction of the electromagnetic waves with the airplane only, without any complex environment perturbing the wave phenomena. This raises an intense research activity, both from a theoretical and a numerical point of view. There exist several approaches which all consist in rewriting the problem (or an approximation of it) in a bounded domain, the new formulation being well-suited for classical mathematical and numerical techniques.

One class of methods consists in applying an appropriate condition on some boundary enclosing the zone of interest. In the frequency domain, one can use a non-local transparent condition, which can be expressed by a convolution with a Green function like in integral equation techniques, or by a modal decomposition when a separation of variables is applicable.
But for explicit schemes in the time domain, local radiation conditions at a finite distance are generally preferred (constructed as local approximations at various orders of the exact non-local condition). A second class of methods consists in surrounding the computational domain by so called Perfectly Matched absorbing Layers (PML), which are very popular because they are easy to implement. POEMS members have provided several contributions to these two classes of methods for more than twenty-five years. Among them, on can mention the understanding of the instability of PMLs in anisotropic media and in dispersive media, the derivation of transparent boundary conditions in periodic media or the improvement of Fast Multipole techniques for elastodynamic integral equations.

In addition to more classical domains of applied mathematics that we are led to use (variational analysis and functional analysis, interpolation and approximation theory, linear algebra of large systems, etc...), we have acquired a deep expertise in spectral theory. Indeed, the analysis of wave phenomena is intimately linked to the study of some associated spectral problems. Acoustic resonance frequencies of a cavity correspond to the eigenvalues of a selfadjoint Laplacian operator, modal solutions in a waveguide correspond to a spectral problem set in the cross section. In these two examples, if the cavity or the cross-section is unbounded, a part of the spectrum is a continuum. Again, POEMS has produced several contributions in this field. In particular, a large number of significant results have been obtained for the existence or non-existence of guided modes in open waveguides and of trapped modes in infinite domains.

To end this far from exhaustive presentation of our main expertise domains, let us mention the asymptotic techniques with respect to some small scale appearing in the model: it can be the wavelength compared to the size of the scatterer, or on the contrary, the scale of the scatterer compared to the wavelength, it can be the scale of some microstructure in a composite material or the width of a thin layer or a thin tube. In each case, the objective, in order to avoid the use of costly meshes, is to derive effective simplified models. Our specificity here is that we can combine skills in physics, mathematics and numerics: in particular, we take care of the mathematical properties of the effective model, which are used to ensure the robustness of the numerical method, and also to derive error estimates with respect to the small parameter. There has been a lot of contributions of POEMS to this topic, going from the modeling of electromagnetic coatings to the justification of models for piezoelectric sensors. Let us mention that effective models for small scatterers and thin coatings have been used to improve imaging techniques that we are developing (topological gradient, time reversal or sampling techniques).

In order to consider more and more challenging problems (involving non-deterministic, large-scale and more realistic models), we decided recently to enlarge our domain of expertise in three directions.

Firstly, we want to reinforce our activity on efficient solvers for large-scale wave propagation problems. Since its inception, POEMS has frequently contributed to the development and the analysis of numerical methods that permit the fast solution of large-scale problems, such as high-order finite element methods, boundary elements methods and domain decomposition methods. Nevertheless, implementing these methods in parallel programming environments and dealing with large-scale benchmarks have generally not been done by the team. We want to continue our activities on these methods and, in a more comprehensive approach, we will incorporate modern algebraic strategies and high-performance computing (HPC) aspects in our methodology. In collaboration with academic or industrial partners, we would like to address industrial-scale benchmarks to assess the performance of our approaches. We believe that taking all these aspects into consideration will allow us to design more efficient wave-specific computational tools for large-scale simulations.

Secondly, up to now, probabilistic methods were outside the expertise of POEMS team, restricting us to deterministic approaches for wave propagation problems. We however firmly believe in the importance and usefulness of addressing uncertainty and randomness inherent to many propagation phenomena. Randomness may occur in the description of complex propagation media (for example in the modeling of ultrasound waves in concrete for the simulation of non-destructive testing experiments) or of data uncertainties. To quantify the effect of such uncertainties on the design, behavior, performance or reliability of many systems is then a natural goal in diverse fields of application.

Thirdly and lastly, we wish to develop and strengthen collaborations allowing a
closer interaction between our mathematical, modeling and computing
activities and physical experiments, where the latter may either provide
reality checks on existing models or strongly affect the choice of modeling
assumptions. Within our typical domain of activities, we can mention four areas
for which such considerations are highly relevant. One is musical acoustics,
where POEMS has made several well-recognized contributions dealing with the
simulation of musical instruments. Another area is inverse problems, whose very
purpose is to extract useful information from actual measurements with the help
of (propagation) models. This is a core of our partnership with CEA on
ultrasonic Non Destructive Testing. A third area is the modelling of effective
(acoustic or electromagnetic) metamaterials, where predictions based on
homogenized models have to be confirmed by experiments. Finally, a fourth area
of expertise is the modeling and simulations of waves in reactive media, where
the development of simple mathematical models is of great importance in order to
better understand the complex dynamics of reactive flows.

Our research finds applications in many fields where acoustic, elastic, electromagnetic and aquatic waves are involved. Topics that have given rise to industrial partnerships include aircraft noise reduction (aeroacoustics), ultrasonic non-destructive testing of industrial structures, and seismic wave simulations in the subsoil, for the oil exploration.

Nowadays, the numerical techniques for solving the basic academic problems are
well mastered, and significant progress has been made during the last twenty
years for handling problems closer to real applications. But several bottlenecks
remain, among which one can mention the high-frequency problems for radar
applications, the multiscale problems that arise for instance in
nanotechnologies or the multi-physics couplings, like in aeroacoustics.
Moreover, in the recent period, new challenges have emerged, related to new
discoveries in physics (like negative index metamaterials) or to the fantastic
development of information and communication techniques. For example, the
growing development of increasingly connected objects (internet of things) and
the forthcoming availability of autonomous vehicles depend crucially on
electromagnetic waves, raising important issues about radar performance, sensor
reliability, component miniaturization and electromagnetic compatibility.
Generally, there are a lot of application domains which could benefit from
advanced research on waves phenomena. Enhancing ultrasound-based methods for
detection and imaging, which are already intensively used in e.g. medicine,
could permit real-time health monitoring of aircrafts or nuclear plants.
Guarding against seismic risks still requires considerable advances in the
simulation of elastic waves in large and complex media. And many other
applications motivating our research and our prospects could be added to this
far-from-comprehensive list.

Emile Parolin received the Best Thesis Award Runner-up, awarded by the Institut Polytechnique de Paris.

We are currently developing the softwares COFFEE (a BEM solver for acoustic and elastic waves) and XLiFE++ (a FEM-BEM C++ library).

This topic is the subject of our important collaboration with Maxence Cassier (Institut Fresnel).

A huge effort has been devoted to the writing of a long and very technical paper (64 pages) on the limiting absorption and limiting amplitude (understand long time behaviour) principles for the transmission problem between two half-spaces, one made of vacuum and the other filled by a Drude material. This is the second part of a work whose first part, devoted to the spectral theory of the problem, was published in Communications in PDEs. The approach is based on the use of so-called stationary scattering method that must be adapted to the specificities of the constitutive laws of Drude media, specificities that are also the source of new phenomena such as plasmonic interface waves or interface resonances. Our second paper has just been accepted, also in Communications in PDEs.

The second aspect of our research on this topic corresponds the PhD thesis of Alex Rosas Martinez who started in November 2020.

This concerns more precisely the dissipative version of generalized Lorentz
media and more precisely the large time behaviour of the associated Cauchy
problem. It appears that the electromagnetic energy decays in

This work is a continuation of the research done in collaboration with B. Desprès et al. on the degenerate elliptic equations describing plasma heating and is a part of 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.

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. Such solutions can be represented as a product of an

We were able to alter the existing variational formulation in the way that requires a lower regularity of this tangential component of the solution; this allows in particular to be able to use lower order finite elements in the numerical simulation while preserving the general order of convergence of the scheme. Moreover, our numerical experiments indicate that the original regularity assumption for the tangential component is likely to be too restrictive, and does not seem to hold even in the cases when the coefficients of the problem are analytic in both variables. We are currently investigating this phenomenon, both numerically and theoretically.

One considers the equation

in

(plus boundary conditions), where the diffusivity is piecewise constant, with stricly positive values in part of the domain, and stricly negative values elsewhere. When the problem is well-posed in

, meshing rules have been designed in the past, to ensure convergence of the discrete solution towards the exact solution (the so-called T-conform meshes). We have begun to investigate some techniques that allow in principle to compute solutions of problems with sign-changing coefficients without having to comply with those meshing rules. To that aim, ideas borrowed from control theory are used to compute the numerical solution iteratively. The results are very promising, and in particular optimal, monotonic, convergence rates are recovered for families of meshes that are not T-conform.

A collaboration with Juan Pablo Borthagaray (DMEL, Universidad de la República, Montevideo, Uruguay). Consider the equation

in

(plus boundary conditions), where the diffusivity is piecewise constant, and equals

in

(

), with

and

. If

and

have different sign, well-posedness in

may not hold. This occurs when the ratio

belongs to the so-called

. When the interface has a corner, we have observed that this critical interval is shrunk if one replaces the standard

-bilinear forms by corresponding

-forms (

). However, the cost of computing the nonlocal interactions may be prohibitive in applications. Thus, our long term goal is to confine the non-local model to a neighborhood of the interface, while keeping the standard local model in the rest of the domain. A first step in this direction consists in considering the numerical solution of the fractional Laplacian of index

, whose solution a priori belongs to the fractional order Sobolev space

. Under suitable assumptions on the data, its solution is also in

, and we showed that one can derive error estimates in

-norm if one uses the standard Lagrange finite element to discretize the problem.

A smooth metallic particle supports surface plasmonic modes for a discrete sequence of negative permittivity values. For a subwavelenth particle, in the quasi-static approximation, these values are solution of the so-called plasmonic eigenvalue problem. This work investigates the case where the particle has corners. It is well-known that a metallic particle with a non-smooth boundary can exhibit, in some range of permittivity values, strongly-oscillating surface waves whose phase velocities vanish as they reach the corners. This range of permittivity corresponds to the essential spectrum of the former spectral problem.

We are interested by the existence of corner resonances, which are analogous to scattering resonances in the sense that the local behavior at each corner plays the role of the behavior at infinity. Resonant values of the permittivities are sought as eigenvalues of the plasmonic eigenvalue problem with an appropriate complex scaling applied at the corners. The finite element discretization requires a very specific mesh in the vicinity of the surface of the particle, to avoid spurious eigenvalues. Numerical results obtained for an elliptic particle with one corner show that the complex scaling deforms the essential spectrum (associated with the corner) so as to unveil both embedded plasmonic eigenvalues and complex plasmonic resonances.

This is the continuation of many works on transmission problems involving negative materials done in collaboration with Lucas Chesnel from INRIA team IDEFIX.

In the PhD of Mahran Rihani, we were interested in the analysis of time-harmonic Maxwell's equations in presence of a conical tip of a material with negative dielectric constants. When these constants belong to some critical range, the electromagnetic field exhibits strongly oscillating singularities at the tip, and the problem is no longer well-posed in the classical framework. Previous results using the T-coercivity approach are not applicable.

Our work is inspired by what is known (thanks to ou rprevious works) for the 2D
scalar case with critical coefficients: it has been proved hat well-posedness
in the classical

For the 3D tip, we established similar results, firstly for the scalar problem: a main difference with the 2D case is the possible existence of several outgoing singularities (the smallest the apex angle, the largest the number of outgoing singularities). These singularities are the eigenfunctions of a Laplace Beltrami spectral problem on the sphere, with sign-changing coefficients. The discreteness of the corresponding spectrum can be proved, for a smooth conical tip, using T-coercivity techniques on the sphere. The so-called critical interval of coefficients, for which propagating singularities exist, has been completely determined for the case of a circular conical tip.

Then we have established a new functional framework for the study of Maxwell's
equations in presence of a conical tip of a negative material, when either the
dielectric permittivity or the magnetic permeability, or both, belong to
critical intervals. The space of electric fields for instance is obtained by
adding to some weighted Sobolev space (included in

In the context of a collaboration with Matias Ruiz (University of Edinburgh) who spent 3 months at POEMS (from october to december, 2021), we have started a work concerning theoretical and numerical aspects of the so-called

. This problem can be formulated as follows, considering time-harmonic electromagnetic scattering (at a fixed frequency) by a metallic particle of given (possibly complex) permittivity

in vacuum. Thanks to integral equation methods or Dirichlet-to-Neumann map techniques, such a problem can be reduced to a problem set on the particle itself. In the absence of incident wave, the reduced problem can be seen as a

eigenvalue problem where

plays the role of an eigenvalue. The eigenfunctions associated to a possible eigenvalue are called

. The questions we are interested in are the following. What can be said about the essential / discrete spectrum? Do the generalized normal modes form a complete family (Riesz basis ?) of the natural energy space? How can we compute these eigenfunctions and associated modes?

In order to deal first with a simple situation for which a dispersion equation can be derived, we have considered the two-dimensional Helmholtz equation in a half uniform waveguide, where the end of the waveguide plays the role of a rectangular metallic particle. The problem has been implemented using the finite element library XLiFE++. Some promising numerical results have already been obtained. They show in particular two categories of modes: bulk modes and plasmonic modes (localized near the interface between the metallic particle and vacuum). A remedy for the occurrence of spurious modes has been tested.

We developed for several years a new method for the solution of 2D scattering problems with complex backgrounds, providing an alternative to Perfectly Matched Layers (PML) or other artificial boundary conditions. This method is called the Half-Space Matching (HSM) method. Based on half-plane representations for the solution, the scattering problem is rewritten as a system coupling (1) a standard finite element discretisation localised around the scatterer and (2) integral equations whose unknowns are traces of the solution on the boundaries of a finite number of overlapping half-planes contained in the domain. While satisfactory numerical results have been obtained for real wavenumbers, well-posedness and equivalence of this HSM formulation to the original scattering problem were established only for complex wavenumbers. Our new results, obtained in collaboration with Simon Chandle-Wilde (Reading University) concern the case of a real wavenumber and a homogeneous background. We proved that the HSM formulation is equivalent to the original scattering problem, and so is well-posed, provided the traces satisfy radiation conditions at infinity analogous to the standard Sommerfeld radiation condition. As a key component of our argument we show that, if the trace on the boundary of a half-plane satisfies our new radiation condition, then the corresponding solution to the half-plane Dirichlet problem satisfies the Sommerfeld radiation condition in a slightly smaller half-plane. We expect that this last result will be of independent interest, in particular in studies of rough surface scattering.

In collaboration with Antoine Tonnoir from INSA of Rouenm, we have extended the Half-Space Matching (HSM) method, first introduced for scalar problems, to elastodynamics, to solve time-harmonic 2D scattering problems, in locally perturbed infinite anisotropic homogeneous media. The HSM formulation couples a variational formulation around the perturbations with Fourier integral representations of the outgoing solution in four overlapping half-spaces. These integral representations involve outgoing plane waves, selected according to their group velocity, and evanescent waves. Numerically, the HSM method consists in a finite element discretization of the HSM formulation, together with an approximation of the Fourier integrals. We have performed numerical results, validating the method for different materials, isotropic and anisotropic, and we have compared them to results obtained with the Perfectly Matched Layers (PML) method. For materials for which PMLs are unstable in the time domain, these results highlight the robustness of the HSM method, contrary to the PML method which is very sensitive to the choice of the parameters.

This work is done in collaboration with M. Halla (MPI for Solar System Research, Göttingen). We study the question of stability of radial PMLs for the simplest case of scalar anisotropic wave equation. Our preliminary numerical experiments on wave scattering in such media indicate time-domain stability of the radial PMLs in the case when geometrical configuration of the PMLs is well-chosen (i.e. the PML is sufficiently far away from the scatterer); however, for some configurations radial PMLs fail. Our analysis in the simplified case (based on the analysis of the fundamental solution for the non-truncated perfectly matched layers, and the construction of the singular sequence for the underlying sesquilinear form) confirm existence of such instability, but for the moment it is unclear why it does not appear in other configurations.

During the post-doc of Markus Wess, we have worked on the extension of the stability and convergence analysis of the generalized perfectly matched layers (GPMLs) for dispersive media, proposed by Bécache, Joly, Vinoles. The idea is to extend the techniques developed for non-dispersive waveguides to the dispersive case. Like in the non-dispersive case, we obtain an explicit representation of the solution in the Laplace domain. The point is to estimate the inverse Laplace integral. One of the difficulties compared to the non-dispersive case is the appearance of various propagation regimes (also in the time domain): there exist backward propagating waves, forward propagating waves, evanescent waves depending on the frequency. We demonstrate that surprisingly, despite the complexity of the physical phenomena, the GPMLs for dispersive media converge as fast as the classical PMLs for non-dispersive media. A paper is in progress.

We developed for some years a method that we call the Half-Space Matching (HSM) method, to solve scattering problems in unbounded domains, when classical approaches are either not applicable or too expensive. This method is based on an explicit expression of the "outgoing" solution of the problem in half-spaces, by using Fourier, generalized Fourier or Floquet transforms when the background is respectively homogeneous (possibly anisotropic), stratified or periodic. The domain exterior to a bounded region enclosing the scatterers is recovered by a finite number of halfspaces (at least 3). The unknowns of the formulation are the restriction of the solution to the bounded region and the traces of the solution on the infinite boundaries of the halfspaces. The system of equations is derived by writing compatibility conditions between the different representations of the solution. Although the HSM method works in the non-dissipative case, the theoretical and the numerical analysis of the method has been done only in the dissipative case. In the present work done in collaboration with Simon Chandler-Wilde and Karl-Michaël Perfekt form Reading University, we propose a new formulation of the method which is well-suited for the theoretical and numerical analysis of the non dissipative case. In the spirit of PMLs, the idea is to replace the system of equations on the traces by similar equations on exponentially decaying analytical extensions of the traces. In the simple case of the Helmholtz equation, we have proved that this formulation is of Fredholm type and is well-posed. Besides the interest for the theory, this new formulation is also well-suited for numerical purposes. Indeed one can show that the error due to the truncation of the infinite boundaries of the half-spaces decays exponentially with the distance of truncation, which was not the case for the standard method. The analysis requires the study of double-layer potential integral operators on intersecting infinite lines, and their analytic continuations. The effectiveness of the method is validated by preliminary numerical results.

In order to handle infinite interfaces in a boundary integral equation context, a few options are available. For simple geometries, one can construct a problem specific Greens function which incorporates the imposed boundary condition on all but a bounded portion of the interface, thus reducing the problem again to integrals over bounded curves/surfaces. This has the advantage of being conceptually simple provided such problem-specific Greens function can be efficiently computed. Unfortunately, for all but the simplest geometries, the representation of the problem specific Greens function involves challenging integrals which must be approximated numerically.

An alternative approach consists of utilizing the free-space Greens function — readily available for many PDEs of physical relevance — in conjunction with a truncation technique. For non-dissipative problems, the slow (algebraic) decay of the Greens function makes the choice of truncation technique an important aspect which needs to be considered in order to reduce the errors associated with the domain's truncation. An easy-to-implement solution, the so-called windowed Green function approach, has been proposed and validated in several configurations.

We are currently investigating the interest of using instead a complex-scaled Green function, which amounts to combine the method of perfectly-matched-layers (PMLs) and boundary integral equations.

We have applied this idea to the 2D linear time-harmonic water wave problem, in finite or infinite depth, writing a complex-scaled integral equation on the free surface.

The method works well and numerical results show that the error decays exponentially with respect to the distance of truncation. Let us mention that because the water waves are surface waves, the windowed Green function approach does not work for this problem.

This work concerns the Half Space Matching method described just above. A main ingredient of this method is a half-space formula for the outgoing scattered field, as a function of its trace on the boundary of the half-space. For homogeneous isotropic or anisotropic backgrounds, this formula is obtained thanks to a partial Fourier transform and gives raise to oscillatory integrals, hard to compute with standard quadrature formulas. We proposed two ways for improving both the accuracy and the efficiency of the evaluation of this half-space formula. When it is evaluated at a point located far enough from the boundary of the half-space, a simple far-field formula can be used. For other points, a deformation of the Fourier path in the complex plane allows to get rid of the oscillations and of the singularity at the branch point. Numerical experiments for the simple 2D isotropic case confirm that these two ideas drammatically reduce the cost of the method.

This research is done in collaboration with Carlos Pérez-Arancibia (PUC, Chile).

We develop a general high-order kernel regularization technique applicable to all four integral operators of Calderón calculus associated with linear elliptic PDEs in two and three spatial dimensions. The proposed technique relies on interpolating the density function around the kernel singularity in terms of solutions of the underlying homogeneous PDE, so as to recast singular and nearly singular integrals in terms of bounded (or more regular) integrands. We developed a simple interpolation strategy which, unlike previous approaches, does not entail explicit computation of high-order derivatives of the density function along the surface. Furthermore, the proposed approach is kernel- and dimension-independent in the sense that the sought density interpolant is constructed as a linear combination of point-source fields, given by the same Green's function used in the integral equation formulation, thus making the procedure applicable, in principle, to any PDE with known Green's function. In the initial work 23, we have focused on Nyström methods for the (scalar) Laplace and Helmholtz equations and the (vector) elastostatic and time-harmonic elastodynamic equations. The method's accuracy, flexibility, efficiency, and compatibility with fast solvers was demonstrated by means of a variety of large-scale three-dimensional numerical examples.

This study is done in collaboration with Bruno Leblé (Naval Group). It aims at developing computational strategies for modelling the impact of a far-field underwater explosion shock wave on a structure, in deep water. An iterative transient acoustic-elastic coupling is developed to solve the problem. Two complementary methods are used: the Finite Element Method (FEM), that offers a wide range of tools to compute the structure response; and the Boundary Element Method (BEM), more suitable to deal with large surrounding acoustic fluid domains. We concentrate on developing a transient FEM-BEM coupling algorithm with a fast convergence. Since the fast transient BEM is based on a fast multipole-accelerated Laplace-domain BEM (implemented in the in-house code COFFEE), extended to the time domain by the Convolution Quadrature Method (CQM), we have proposed a proof of convergence of global-in-time iterative solution procedures for the coupled transient fluid-structure problem. This proof itself relies on solvability results for the coupled acoustic-elastic problems, some of which we established in the course of this study, and the initial-boundary value problems involved in coupling iterations. We have also implemented a 2D version of this algorithm and assessed its sensitivity to various parameters.

It is well known in the literature that standard

We are currently investigating two possible improvements of this approach.
Since in practice, not all materials are purely elastic it is important to be able to consider visco-elastic cases. In this context, we have studied the effect of the introduction of a complex wavenumber on the accuracy and efficiency of hierarchical matrix (

Then, since

This was the subject of the PhD study of Nicolas Trafny, done in collaboration with Gilles Serre (Naval Group) and Benjamin Cotté (IMSIA). The aim is to develop an optimized method to determine numerically the 3D Green's function of the Helmholtz equation in presence of an obstacle of arbitrary shape, satisfying the Neumann boundary condition at the boundary surfaces. This so-called rigid Green's function is useful to solve the Lighthill's equation, giving the hydrodynamic noise radiated by a ship. First an integral equation is derived, expressing the rigid Green's function versus the free space Green's function. Then a Boundary Element Method (BEM) implemented in the code COFFEE is used to compute the rigid Green's functions. In order to consider realistic geometries in a reasonable amount of time, fast BEMs are used: fast multipole accelerated BEM and hierarchical matrix based BEM. The efficiency of these two approaches is tested on simple geometries for which analytic solutions can be determined (sphere, cylinder, half plane) and the methods are also compared on realistic geometries of interest for the industrial partner (NACA profiles, boat propeller). A second PhD is scheduled, extending the study to elastic geometries.

This research is developed in collaboration with Marc Lenoir and Daniel Bouche (CEA).

It has recently been realized that the combination of integral and asymptotic methods was a remarkable and necessary tool to solve scattering problems, in the case where the frequency is high and the geometry must be finely taken into account.

In order to implement the high-frequency approximations that we are developing as part of these hybrid HF/BF methods, we have introduced new geometric tools into the XLiFE++ library, in particular splines and B-Splines approximations as well as parameterizations to access quantities such as curvature, curvilinear abscissa, etc. We have also started to interface the OpenCascad library to the XLiFE++ library, which will eventually allow us to manage more complex geometric situations (cylinder and sphere intersection for example). In parallel, we have completed the implementation of 2D HF approximations in the shadow-light transition zone based on the Fock function. Diffraction by a 2D corner is in progress.

In collaboration with Ruiyang Dai (UCLouvain), Christophe Geuzaine (Liège U.) and Anthony Royer (Liège U.), we have worked on non-overlapping domain decomposition methods (DDM) for Helmholtz problems with transmission conditions based on non-reflecting boundary treatments. It is well-known that the convergence rate of non-overlapping DDM applied to the parallel finite-element solution of large-scale time-harmonic wave problems strongly depends on the transmission conditions enforced at the interfaces between the subdomains.

Transmission operators based on perfectly matched layers (PML) have proved to be well-suited for configurations with layered domain partitions. Unfortunately, the extension of the PML-based DDM for more general partitions with cross-points (where more than two subdomains meet) is rather tricky and requires some care. We have proposed a non-overlapping DDM with PML transmission conditions for checkerboard decompositions that takes cross-points into account. In such decompositions, each subdomain is surrounded by PMLs associated to edges and corners. The continuity of Dirichlet traces at the interfaces between a subdomain and PMLs is enforced with Lagrange multipliers. This coupling strategy offers the benefit of naturally computing Neumann traces, which allows to use the PMLs as discrete operators approximating the exact Dirichlet-to-Neumann maps. We considered two possible Lagrange multiplier finite element spaces. We have studied the behavior of the corresponding DDM on several numerical examples.

In parallel, we have explored a family of generalized sweeping preconditionners for Helmholtz problems with non-overlapping checkerboard partition of the computational domain. The domain decomposition procedure relies on high-order transmission conditions and cross-point treatments, which cannot scale without an efficient preconditioning technique when the number of subdomains increases. With the proposed approach, existing sweeping preconditioners, such as the symmetric Gauss-Seidel and parallel double sweep preconditioners, can be applied to checkerboard partitions with different sweeping directions (e.g. horizontal and diagonal). Several directions can be combined thanks to the flexible version of GMRES, allowing for the rapid transfer of information in the different zones of the computational domain, then accelerating the convergence of the final iterative solution procedure. We have proposed several two-dimensional finite element results to study and to compare the sweeping preconditioners, and to illustrate the performance on cases of increasing complexity.

A chapter of this research has been closed with the PhD defense of É. Parolin abd the end of the ANR Project NonlocalDD leaded by X. Claeys (Paris 6 and Alpines). Our research in this domain has recently received several regognitions : Patrick Joly has been invited as a plenary speaker at DDM20 (in the framework of the series of international conferences dedicated to Domain Decomposition Methods) and Émile Parolin has received a Prize from the Institut Polytechnique de Paris for his thesis. In addition, three articles on the subject - included a review article - have been accepted for publication.

Our work will know a new orientation in the framework of the RAPID Project Hybox2. This project, which is headed by T. Abboud and regroups IMACS, Ariane Group and POEMS, started in September. One particular subject of attention will concern the quasi-localization of the communication operator between subdomains in the presence of junction points (see also the previous activity report).

A new activity recently started concerning forward and inverse scattering in junctions of waveguides. It corresponds to the PHD of Jean-François Fritsch and is a collaboration with the CEA-List, in particular Arnaud Recoquillay. Firstly, we have considered the junction between several closed waveguides. It is well-known that defects such as cracks often occur in weld bead of metallic pipes, which can be seen as junctions of waveguides. This explains why it is necessary to adapt Non Destructive Testing procedures to that kind of configuration. Forward scattering problems in junctions of closed waveguides are quite standard and can be solved with classical finite element methods coupled with transparent boundary conditions using Dirichlet to Neumann maps. However, solving inverse scattering problems for such geometries is less standard. In order to cope with those problems, we use a modal version of the classical Colton-Kirsch Linear Sampling Method. The main issue stems from the fact the LSM relies on the fundamental solution, which does not have a closed-form expression in a junction of waveguides, contrary to the case of a homogeneous waveguide. In order to cope with this problem, the main ingredient we introduce is the so-called reference fields, which are the responses of the junction without any defect to the guided modes considered as incident waves. We also use the symmetry property of the fundamental solution. The reference fields enable us to adapt the LSM by paying a reasonable computational cost, in the sense that it is not necessary to actually compute the fundamental solution for each sampling point of the grid. We have shown the feasibility of our method with the help of two-dimensional acoustic numerical experiments, for instance in the case of a junction of three waveguides.

Secondly, we have considered the more challenging case of a closed waveguide which is embedded in a surrounding infinite medium. A typical situation is the case of a steel cable which is partially embedded into concrete or some fluid. It may happen that some defects within the embedded part of the cable or at the interface between the cable and the surrounding medium have to be retrieved from measurements located on the only accessible part of the cable, that is its free part. In a first attempt we have simplified the problem by considering a two-dimensional acoustic problem. Despite such simplification, both the forward and the inverse scattering problems are challenging. We address the forward problem by using Perfectly Matched Layers in the transverse direction, which has the effect of closing the waveguide but of introducing a non-selfadjoint eigenvalue problem in the transverse direction. At the continuous level, we use the Kondratiev approach to establish well-posedness of the forward problem and to specify the asymptotic behaviour of solutions at infinity. In order to compute a numerical solution, we also introduced transparent boundary conditions in the infinite direction of the waveguide, that is a Dirichlet to Neumann map with an overlap. This latter amounts to a transparent thick boundary condition. Using again the Kondratiev approach, we estimated the error introduced by these transparent boundary conditions. We addressed the inverse problem with the help of the modal Linear Sampling Method by using the same ingredient as introduced in the previous case of the junction of closed waveguides. However a new difficulty arises since an open waveguide is characterized by radiation losses, which can be seen as lost information from the point of view of the inverse problem. In the case of a steel cable embedded into a concrete medium or into a fluid, this is all the more difficult as the speed in the core is larger than the speed in the sheath. As a consequence, once we have replaced the infinite medium by PMLs, the modes are either leaky modes or PML modes, both of them being evanescent. In this sense, the inverse problem is more challenging than in the case of closed waveguides, which benefit from a finite number of propagating modes. However, the numerical experiments that we have made show that the LSM is efficient to retrieve defects provided they are sufficiently close to the interface between the closed and the open waveguide. We are currently investigate the more involved case of an elastic waveguide which is embbeded in water, which correspond to a solid-fluid interaction problem.

In the context of time-harmonic scattering by a bounded penetrable scatterer,
interior transmission eigenvalues correspond, when they are real, to discrete
frequencies for which there exists an incident wave which does not scatter. At
such frequencies, inversion algorithms such as the linear sampling method fail.
Real interior transmission eigenvalues are a part of a larger spectrum made of
complex values, which has been largely studied in the case where the difference
between the parameters in the scatterer and outside does not change sign on the
boundary. In collaboration with Lucas Chesnel (INRIA team IDEFIX), we obtained
some years ago some results for a 2D configuration where such sign-change
occurs. The main idea was that, due to very strong singularities that can occur
at the boundary, the problem may lose Fredholmness in the natural

This explains why we decided, in collaboration with Lucas Chesnel and Florian Monteghetti (ISAE), to extend the numerical method we used for plasmonic eigenvalues to interior transmission eigenvalues. It has been already checked that a naive finite element computation does not converge, and that the convergence is restored by using some complex scaling near the singular point.

Work done in collaboration with Jérémi Dardé, from IMT Toulouse.

For a quite long time now, we have developed the notion of mixed formulations of the Tikhonov regularization in collaboration with Jérémi Dardé (IMT Toulouse). This notion has connections with the old concept of quasi-reversibility introduced by Lattès and Lions (67) and is intended to regularize ill-posed PDE problems. Essentially, our mixed formulations enable us to adapt this concept to a friendly Finite Element Method context using simple conforming elements. We adapted the notion of mixed formulations of the Tikhonov regularization to the context of data assimilation problems, in particular by applying the Morozov's discrepancy principle to select the regularization parameter as a function of the amplitude of noise. A specific problem in data assimilation problems is that the underlying operator has not a dense range, which required to extend some well-known results to that situation. A dual problem was introduced in order to compute the regularized solution corresponding to such regularization parameter, possibly by enforcing the solution to satisfy some constraints.

Work done in collaboration with Wilkins Aquino, Duke University, USA.

We investigated the use of adaptive eigenspace bases (AEBs) to reduce design dimension in topology optimization (TO). The concept of AEB, recently shown to be effective for solving medium imaging problems, is here applied to define approximation spaces for density design fields describing optimal topologies. This approach yields low-dimensional approximation spaces and implicitly regularized designs. The known ability of AEBs to approximate piecewise constant fields is especially useful for TO. Performance of the AEBs has been compared against conventional TO implementations in problems for static linear elasticity, showing comparable structural solutions, computational cost benefits, and consistent design dimension reduction.

Work done in collaboration with Marc Deschamps and Eric Ducasse, I2M, Bordeaux.

We establish a leading-order asymptotic model for the scattering of elastodynamic fields by small surface-breaking defects in elastic solids. The asymptotic form of the representation formula of the scattered field is written in terms of the elastodynamic Green's tensor, which is in fact available in semi-analytical form for some geometrical configurations that are of practical interest in ultrasonic NDT configurations. A rigorous proof of the resulting leading asymptotic approximation is obtained. Preliminary numerical examples have been performed on cylindrical elastic pipes with small indentations on the outer surface.

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

This collaboration addresses the design and implementation of computational methods for solving shape optimization problems involving slow viscous fluids modelled by the Stokes equations. We have developed a new boundary integral equation (BIE) approach for finding optimal shapes of peristaltic pumps that transport viscous fluids, as well as dedicated formulas for computing the shape derivatives of the relevant cost functionals and constraints, expressed in boundary-only form. By employing these formulas in conjunction with a BIE approach for solving forward and adjoint problems, we completely avoid the issue of volume remeshing when updating the pump shape as the optimization proceeds. This is especially useful when the fluid carries objects (e.g. particles, deformable vesicles) whose motion is not known beforehand. Significant cost savings are achieved as a result, and we demonstrate the performance on several numerical examples. We also investigate the optimization of the slip velocity (modeling cilia beating) and the shape of self-propelled micro-swimmers achieving self-propelling at least energy expense. Potential applications pertain to the understanding and optimization of various processes involving biological fluids and organisms moving in them.

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

We investigated effective time-harmonic (anti-plane elastic) wave motions in periodic media containing fractures at finite frequencies and wavenumbers, when the latter are close to apexes of the Brillouin zone. Mean motions are defined via inner products with eigenfunctions of the wave equation in Bloch-wave form, and their governing equation is found from asymptotic expansions with respect to frequency-wavenumber perturbations. We considered cases with isolated, repeated or nearby apex eigenvalues. This approach in particular yields approximations of forced wave motions, and of the dispersion surfaces

This is a joint work with Josselin Garnier (X-CMAP) and Pierre Millien (Institut Langevin). This project aim at modelling and studying the propagation and diffusion of ultrasounds in complex multi-scale media in order to obtain quantitive images of physical parameters of these media.

The propagation of ultrasounds in biological tissues or composite materials is a complex multi-scale phenomenon : the scattered wave is produced by small (compared to the wavelength) inhomogeneities randomly distributed throughout the media, but the time of flight is affected by the slow (compared to the wavelength) variations of the medium. To capture both of those effects, the medium is described as a slowly varying or homogeneous by parts medium in which lie randomly distributed small scatterers. In order to characterize the response of this medium to an incident plane wave, we perform an asymptotic expansion of the scattered wave using stochastic homogenisation techniques. The difficulties lie in the slow varying background and the transmission conditions at the boundary of the medium or between the different components. We then derive quantitative error estimates given that the random distribution of inhomogeneities in the different components verify mixing conditions. Finally we present numerical simulations to illustrate and validate our results. This constitutes the first part of this project.

This is a joint work with Bruno Stupfel from CEA-CESTA.We study the
time-harmonic scattering by a heterogeneous object covered with a thin layer of
randomly distributed 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 numerically Maxwell's
equation in this context is very costly. 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 spaces with either homogeneous Dirichlet (for

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. Necessity of such analysis is justified by the following observation: the time-domain counterparts of the frequency-domain Foldy-Lax models can exhibit instabilities. We show that this is in particular the case for the model derived earlier in [Cassier, Hazard, 2014] whose time-domain counterpart admits exponentially growing solutions when the obstacles are located close to each other.

A stabilization of the Foldy-Lax model was obtained by reconsidering it as a Galerkin discretization of the single-layer boundary integral equation with a specula basis consisting of functions which are constant on each of the obstacles. Let us remark that the model of [Cassier, Hazard, 2014] fits into this framework, however, the respective Galerkin matrix is perturbed; this perturbation is responsible for occurrence of time-domain instabilities. We have proven the convergence of this new asymptotic method when the size of the obstacles tends to zero. An interesting side effect of our analysis is the proof of the superconvergence of the approximation of the scattered field, compared to the approximation of its Foldy-Lax density. This idea can be extended to analyze other problems: transmission problems, sound-hard scattering etc.

We are currently summarizing our results in a manuscript.

We study the wave equation in presence of a periodic medium when the period is small compared to the wavelength. The classical homogenization theory enables to derive an effective model which provides an approximation of the solution. However, it is well known that these models are not accurate near the boundaries. In this work, we propose an enriched asymptotic expansion which enables to derive high order effective models at any order, when the geometry of the periodic medium is simple — absence of corners and its boundary (or the interface with other media) must lie in a direction of periodicity. For the model at order 1, the volume equation is the same than the classical one, but the boundary/transmission conditions is modified. Let us mention that the model of order 2 is particularly relevant when one is interested in the long time behaviour of the solution of the time-dependent wave equation. Indeed, it is well-known that the classical homogenized model does not capture the long time dispersion of the exact solution. In several works, homogenized models involving differential operators of high order (at least 4), are proposed for the wave equation in infinite domains. Dealing with boundaries and proposing boundary conditions for these models were open questions. Our approach enables to propose appropriate and accurate boundary conditions for these models. The analysis of such model and its implementation is under investigation. This work is the fruit of a long time collaboration with Xavier Claeys (LJLL, Sorbonne University ) and a recent one with Timothée Pouchon (EPFL). Clement Beneteau has defended his PhD thesis in January 2021.

In collaboration with Agnès Maurel (Langevin Institut), Kim Pham (IMSIA) and B. Delourme (LAGA, Paris 13), we study effective transmission conditions capable of reproducing the effect of a periodic array of Dirichlet wires on wave propagation, in particular when the array delimits a Faraday cage capable of resonating. In the literature, different transmission conditions are obtained by homogenization, the validity of which depends on the frequency. In practice, it is difficult to deal with such piecewise conditions, especially if the problem is in the time domain. By using an homogenization process at higher orders, we have developed a new interface effective model, involving jump conditions for the fields whose interface parameters are determined thanks to matched asymptotic expansions. We have demonstrated the validity of this new model, called unified because it is valid at all frequencies and therefore easy to use in time. To illustrate the accuracy of the model, a formulation guaranteeing the stability of a numerical scheme based on finite elements has been implemented.

This is work in collaboration with Josue Melguizo-Gavilanes and Ferando Veiga-Lopez (institute PPrime) in the context of the ANR-JCJC FASTD.

The steady propagation of hydrogen-fueled detonation waves with momentum and
heat losses is analyzed including detailed kinetics by means of the detonation
velocity - friction coefficient (choking regime. For
choking
detonations with a realistic chemistry model and a popular mixture. We also
assessed the effect of the mixture composition (nitrogen and argon were added)
on the choking regime for very low

This topic is the subject of a long term collaboration with Sébastien Imperiale (M3disim).

We are interested in the electromagnetic wave propagation in a network of thin coaxial cables (made of a dielectric material which surrounds a metallic inner wire) with heterogeneous cross section. The first goal, achieved in the PhD thesis of G. Beck few years ago, was to reduce 3D Maxwell’s equations to a quantum graph in which, along each edge, one is reduced to compute the electrical potential and current by solving 1D wave equations coupled by vertex conditions. We obtained various effective 1D like models.

Since two years, in the framework of the PhD thesis of Akram Beni Hamad, we work on the numerical validation and quantitative evaluation of these models.

In a first step we have proposed, analysed and implemented efficient numerical methods for solving the 1D approximate problems. However, In order to achieve the 1D/3D comparison, a major challenge is to design numerical methods for solving 3D Maxwell’s equations dedicated to taking into account the specificity of thin electric cables. A naive discretization procedure based on a leap-frog explicit scheme can be really costly because of the thinness of the cable. In the case have then proposed an original approach consisting in adapting Nédélec’s edge elements to elongated prismatic meshes and proposing a hybrid time discretization procedure which is explicit in the longitudinal directions and implicit in the transverse ones. In particular, the resulting CFL stability condition is not affected by the small thickness of the cable.

However the above is only effective for perfectly cylindrical cables : its naive extension do deformed cables generates longitudinal / transverse recoupling that destroys the efficiency of the method. Thus, In the presence of deformations, the method needs to be modified. In order to preserve the longitudinal / transverse decoupling, we propose a hybrid method combining a conforming discretization in the longitudinal variable and a discontinuous Galerkin method in the transverse ones. This method is designed in order to coincide with the previous one in the cylindrical parts of the cable. The analysis ans implementation of this method are currently under way.

Finally, in the framework of a recent collaboration with G. Beck, A. Beni Hamad is deveploping adapted numerical methods for higher order effective models.

In collaboration with Bérangère Delourme (LAGA, Paris 13), we have studied the spectrum of periodic operators in thin graph-like domains: more precisely Neumann Laplacian defined in periodic media which are close to quantum graphs. Moreover, we exhibit situations where the introduction of lineic defects into the geometry of the domain leads to the appearance of guided modes. We dealt with rectangular lattices few years ago and more recently we are studying hexagonal lattices. In this last case, we have shown that the dispersion curves have conical singularities called Dirac points. Their presence is linked to the invariance by rotation, symmetry and conjugation of the model. We have also observed that the direction of the line defect leads to very different properties of the guided modes. Finally, we have also proven the stability of the guided modes when the position of the edge varies (keeping the same direction). We want now to (1) open gaps (around the Dirac points) in the spectrum of the periodic operator by breaking one of the invariance of the problem (2) study the effect on guided waves.

This work is done in collaboration with C.L. Fefferman (Princeton University) and M.I. Weinstein (Columbia University). We Consider the tight binding model of graphene terminated along a rational edge, i.e. an arbitrary line in a direction of periodicity of the structure.
We present a comprehensive rigorous study of zero energy / flat band edge states; all zigzag-type edges support zero energy / flat bands
and armchair-type edges support no zero energy / flat bands. We also perform a careful numerical investigations showing very strong evidence for the existence of non-zero energy (dispersive / non-flat) edge states. We are investigating now the existence of states which are bounded and oscillatory parallel to an irrational edge
and which decay into the bulk. The idea is to construct an « edge » state for irrational termination as the limit of a sequence of edge state wave-packets
(superpositions of edge states) of rationally terminated structures.

This is the subject of the PhD thesis of P. Amenoagbadji. We refer the reader to the previous activity report for a general presentation of the thematic and the related scientific context.

Our main objective is to develop original numerical methods for the solution of the time- harmonic wave equation in quasi-periodic media, in the spirit of the methods that we have developed previously for periodic media.

We began with the 1D case, in the presence of absorption. More precisely, we consider the 1D Helmholtz equation whose coefficients are quasi-periodic functions, namely the traces along a particular line of a periodic functions in higher dimensions (such functions are no longer periodic in general). Accordingly, 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 PDE in higher dimensions, with periodic coefficients. This allows us to extend the ideas of the DtN based methods previously developed at POEMS for periodic media. However the associated mathematical and numerical analysis of the method is more difficult because the augmented PDE is anisotropically degenerate in the sense that the principal part of the equation is no longer elliptic. The associated numerical method, based on the solution of Dirichlet cell problems and a Ricatti equation, has been implemented, an article is being written.

The continuation of this work separates in two directions :

The 1D non absorbing case. The natural idea is to pass to the limit is the previous method when the absorption goes to 0. Understanding the limit process is related to the spectral analysis of a 1D differential operator along the real line with quasi-periodic coefficients. Most often, this spectrum could have an absolutely continuous part, a singular continuous part or even a point part. For now, we can conclude easily on the limiting absorption principle when the spectral variable (i.e. the square of the frequency) is not in the spectrum. When it is in the point part of the spectrum, the limiting absorption principle cannot hold in a classical setting. In all the other cases, the question is still open. Even if from a theoretical point of view, the answer of the limiting absorption principle is not clear, we can use a numerical procedure thanks to the method described above. The first difficulty is that we have shown that the Dirichlet cell problems are not well-posed for intervals of frequencies. The solution is to solve Robin cell problems instead and extend our method to construct DtN transparent boundary conditions or RtR boundary conditions. The second difficulty concerns the solution of the Ricatti equations. This work is in progress.

Contract with DGA and Naval Group on transient fluid-structure coupling caused by remote underwater explosions, including cavitation effects

Participants: M. Bonnet, S. Chaillat, A. Nassor

Start: 10/2020. End: 09/2023. Administrator: CNRS

Contract and CIFRE PhD with CEA on Modelling of thin layers of randomly distributed nanoparticles for electromagnetic waves

Participants: A. Boucart, S. Fliss, L. Giovangigli

Start: 10/2019. End: 09/2022. Administrator: ENSTA

Contract and CIFRE PhD with Naval Group on flow noise prediction

Participants: J-F Mercier, B. Cotté, N. Trafny

Start: 04/2018. End: 03/2021. Administrator: ENSTA

ANR project MODULATE (Modeling lOng-perioD groUnd motions, and assessment of their effects on Large-scale infrAsTructurEs)

Partners: ENSTA (UME), Inria POEMS, CentraleSupelec, BRGM, GDS

Start: 11/2018. End: 10/2021. Administrator: ENSTA

Participant of POEMS: S. Chaillat

Coordinator: K. Meza Fajardo (BRGM)

ANR JCJC project FAsTD (Flame Acceleration and Transition to Detonation in Narrow Channels)

Partners: INRIA (POEMS), CNRS (Institut Pprime)

Start: 12/2020. End: 12/2024. Administrator: CNRS

Participant of POEMS: L. Faria

Coordinator: J. Melguizo Gavilanes (Institut Pprime)

ANR JCJC project WavesDG (Wave-specific Discontinuous Galerkin Finite Element Methods for Time-Harmonic Problems)

Partners: POEMS (CNRS, INRIA, ENSTA), Atlantis (INRIA), LAUM (U. Le Mans), U. Liège

Start: 10/2021. End: 12/2025. Administrator: CNRS

Coordinator: A. Modave (POEMS, CNRS)

Other participant of POEMS: P. Ciarlet (POEMS, ENSTA)

Contract on boundary element methods and high-frequency problems

Participants: E. Lunéville, M. Lenoir, N. Kielbasiewicz

Start: 10/2018. End: 09/2021. Administrator: ENSTA

In partnership with F. Alouges and M. Aussal (CMAP, Ecole Polytechnique)

Projet RAPID HyBOX (Hybridization toolbox for complex materials and metamaterials)

Partners: IMACS, ARIANEGROUP, ENSTA

Start: 10/2020. End: 09/2023. Administrator: ENSTA

Participants of POEMS: Patrick Joly, Sonia Fliss, Maryna Kachanovska, Axel Modave, Pierre Marchand

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.

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.