Keywords
 A6. Modeling, simulation and control
 A6.1. Methods in mathematical modeling
 A6.1.1. Continuous Modeling (PDE, ODE)
 A6.1.2. Stochastic Modeling
 A6.1.4. Multiscale modeling
 A6.1.5. Multiphysics modeling
 A6.1.6. Fractal Modeling
 A6.2. Scientific computing, Numerical Analysis & Optimization
 A6.2.1. Numerical analysis of PDE and ODE
 A6.2.2. Numerical probability
 A6.2.3. Probabilistic methods
 A6.2.7. High performance computing
 A6.3.1. Inverse problems
 A6.3.4. Model reduction
 A6.5.1. Solid mechanics
 A6.5.2. Fluid mechanics
 A6.5.4. Waves
 B2.6. Biological and medical imaging
 B3.3. Geosciences
 B3.3.1. Earth and subsoil
 B3.4. Risks
 B3.4.1. Natural risks
 B3.4.2. Industrial risks and waste
 B5.3. Nanotechnology
 B5.4. Microelectronics
 B5.5. Materials
1 Team members, visitors, external collaborators
Research Scientists
 AnneSophie BonnetBen Dhia [Team leader, CNRS, Senior Researcher, HDR]
 Marc Bonnet [CNRS, Senior Researcher, HDR]
 Eliane Bécache [Inria, Researcher, HDR]
 Stéphanie Chaillat [CNRS, Researcher, HDR]
 Christophe Hazard [CNRS, Researcher, HDR]
 Patrick Joly [Inria, Senior Researcher, HDR]
 Maryna Kachanovska [Inria, Researcher]
 Luiz Maltez Faria [Inria, Researcher]
 Pierre Marchand [Inria, ISFP, from Nov 2021]
 JeanFrançois Mercier [CNRS, Senior Researcher, HDR]
 Axel Modave [CNRS, Researcher]
Faculty Members
 Laurent Bourgeois [École Nationale Supérieure de Techniques Avancées, Professor, HDR]
 Patrick Ciarlet [École Nationale Supérieure de Techniques Avancées, Professor, HDR]
 Sonia Fliss [École Nationale Supérieure de Techniques Avancées, Professor, HDR]
 Laure Giovangigli [École Nationale Supérieure de Techniques Avancées, Associate Professor]
 Eric Lunéville [École Nationale Supérieure de Techniques Avancées, Professor]
PostDoctoral Fellows
 Sara Touhami [École Nationale Supérieure de Techniques Avancées, until Oct 2021]
 Markus Wess [Inria]
PhD Students
 Amond Allouko [CEA, CIFRE]
 Pierre Amenoagbadji [École polytechnique]
 Laura Bagur [École Normale Supérieure de Paris]
 Akram Beni Hamad [Inria, from Oct 2021]
 Amandine Boucart [CEA]
 Damien Chicaud [École Nationale Supérieure de Techniques Avancées, until Dec 2021]
 Jean Francois Fritsch [CEA]
 Quentin Goepfert [École Nationale Supérieure de Techniques Avancées, from Oct 2021]
 Hajer Methenni [CEA, until Mar 2021]
 Alice Nassor [École Nationale Supérieure de Techniques Avancées]
 Etienne Peillon [École Nationale Supérieure de Techniques Avancées]
 Mahran Rihani [École Polytechnique]
 Luis Alejandro Rosas Martinez [Inria]
Technical Staff
 Colin Chambeyron [CNRS, Engineer]
 Nicolas Kielbasiewicz [CNRS, Engineer]
Administrative Assistants
 Corinne Chen [École Nationale Supérieure de Techniques Avancées]
 Marie Enee [Inria]
2 Overall objectives
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.
3 Research program
3.1 Expertises
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 timeperiodic solutions are considered. More generally, we systematically consider both the transient problem, in the time domain, and the timeharmonic 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 wellsuited 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 nonlocal 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 nonlocal 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 twentyfive 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 crosssection 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 nonexistence 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).
3.2 New domains
In order to consider more and more challenging problems (involving nondeterministic, largescale 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 largescale 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 largescale problems, such as highorder finite element methods, boundary elements methods and domain decomposition methods. Nevertheless, implementing these methods in parallel programming environments and dealing with largescale 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 highperformance computing (HPC) aspects in our methodology. In collaboration with academic or industrial partners, we would like to address industrialscale benchmarks to assess the performance of our approaches. We believe that taking all these aspects into consideration will allow us to design more efficient wavespecific computational tools for largescale 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 nondestructive 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 wellrecognized 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.
4 Application domains
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 nondestructive 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 highfrequency problems for radar applications, the multiscale problems that arise for instance in nanotechnologies or the multiphysics 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 ultrasoundbased methods for detection and imaging, which are already intensively used in e.g. medicine, could permit realtime 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 farfromcomprehensive list.
5 Highlights of the year
5.1 Awards
Emile Parolin received the Best Thesis Award Runnerup, awarded by the Institut Polytechnique de Paris.
6 New software and platforms
We are currently developing the softwares COFFEE (a BEM solver for acoustic and elastic waves) and XLiFE++ (a FEMBEM C++ library).
6.1 New software
6.1.1 COFFEE

Keywords:
Numerical simulations, Wave propagation, Boundary element method

Functional Description:
COFFEE is an adapted fast BEM solver to model acoustic and elastic wave propagation (full implementation in Fortran 90). The 3D acoustic or elastodynamic equations are solved with the boundary element method accelerated by the multilevel fast multipole method or a hierarchicalmatrices based representation of the system matrix. The fundamental solutions for the infinite space are used in this implementation. A boundary elementboundary element coupling strategy is also implemented so multiregion 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.

Author:
Stéphanie Chaillat

Contact:
Stéphanie Chaillat
6.1.2 XLiFE++

Name:
eXtended Library of Finite Elements in C++

Keywords:
Numerical simulations, Finite element modelling, Boundary element method

Functional Description:
XLiFE++ is an FEMBEM C++ library developed by POEMS laboratory and IRMAR laboratory, that can solve 1D/2D/3D, scalar/vector, transient/stationary/harmonic problems. Description: https://uma.enstaparis.fr/soft/XLiFE++/

Contact:
Eric Lunéville
7 New results
7.1 Wave propagation in metamaterials and dispersive media
Mathematical analysis of metamaterials in time domain
Participants: Christophe Hazard, Patrick Joly, Alex Rosas Martinez.
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 halfspaces, 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 socalled 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 ${t}^{q}$ with $q>0$ depending on the regularity of the initial data : one speaks of polynomial stability (as opposed to exponential stability for instance). Two methods have been developed. The first one is based on "frequency dependent" Lyapunov estimates. This approach has the advantage to lead to polynomial stability with "little" effort but requires some "strong dissipativity" assumptions. The alternative spectral approach allows us to get rid of these assumptions and leads to sharp estimates, but is technically more involved, in particular because non selfadjoint spectral theory has to be used. Our results generalize and give a new light (in particular from the physical point of view) to previous results of the literature by S. Nicaise et al., for a restricted class of models, or by Figotin et al., for a somewhat larger class of models but without any energy decay estimate.
Wave Propagation in Plasmas
Participants: Patrick Ciarlet, Maryna Kachanovska, Etienne Peillon.
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 wellposed 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 ${L}^{2}$function varying in tangential direction of the interface and a singular function depending on the normal direction. The numerical method suggested in previous works is based on exploiting the plasma heating property, however, leads to the variational formulation which requires high regularity of the tangential component of the singular solution along the interface where the variable coefficient changes its sign.
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.
Optimal controlbased numerical method for problems with signchanging coefficient
Participants: Patrick Ciarlet, David Lassounon, Mahran Rihani.
One considers the equation $\text{div}\left(\sigma \nabla u\right)=f$ in $\Omega $ (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 wellposed in ${H}^{1}\left(\Omega \right)$, meshing rules have been designed in the past, to ensure convergence of the discrete solution towards the exact solution (the socalled Tconform meshes). We have begun to investigate some techniques that allow in principle to compute solutions of problems with signchanging 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 Tconform.Towards nonlocal interface models
Participants: Patrick Ciarlet.
A collaboration with Juan Pablo Borthagaray (DMEL, Universidad de la República, Montevideo, Uruguay). Consider the equation $\text{div}\left(\sigma \nabla u\right)=f$ in $\Omega $ (plus boundary conditions), where the diffusivity is piecewise constant, and equals ${\sigma}_{i}$ in ${\Omega}_{i}$ ($i=\{1,2\}$), with $\overline{{\Omega}_{1}}\cup \overline{{\Omega}_{2}}=\overline{\Omega}$ and ${\Omega}_{1}\cap {\Omega}_{2}=\varnothing $. If ${\sigma}_{1}$ and ${\sigma}_{2}$ have different sign, wellposedness in ${H}^{1}\left(\Omega \right)$ may not hold. This occurs when the ratio ${\sigma}_{2}/{\sigma}_{1}$ belongs to the socalled critical interval. When the interface has a corner, we have observed that this critical interval is shrunk if one replaces the standard ${H}^{1}$bilinear forms by corresponding ${H}^{s}$forms ($s\in (0,1)$). However, the cost of computing the nonlocal interactions may be prohibitive in applications. Thus, our long term goal is to confine the nonlocal 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 $s\in (1/2,1)$, whose solution a priori belongs to the fractional order Sobolev space ${\tilde{H}}^{s}\left(\Omega \right)$. Under suitable assumptions on the data, its solution is also in ${H}^{1}\left(\Omega \right)$, and we showed that one can derive error estimates in ${H}^{1}\left(\Omega \right)$norm if one uses the standard Lagrange finite element to discretize the problem.Computation of plasmon resonances localized at corners using frequencydependent complex scaling
Participants: AnneSophie BonnetBen Dhia, Christophe Hazard, Florian Monteghetti.
A smooth metallic particle supports surface plasmonic modes for a discrete sequence of negative permittivity values. For a subwavelenth particle, in the quasistatic approximation, these values are solution of the socalled plasmonic eigenvalue problem. This work investigates the case where the particle has corners. It is wellknown that a metallic particle with a nonsmooth boundary can exhibit, in some range of permittivity values, stronglyoscillating 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.
Maxwell's equations in presence of a conical tip with negative electromagnetic constants
Participants: AnneSophie BonnetBen Dhia, Mahran Rihani.
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 timeharmonic 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 wellposed in the classical framework. Previous results using the Tcoercivity 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 wellposedness in the classical $H1$ framework is lost. This wellposedness can be recovered by working in an appropriate weighted Sobolev spaces and adding in the space one singular fonction, the socalled outgoing propagating singularity.
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 signchanging coefficients. The discreteness of the corresponding spectrum can be proved, for a smooth conical tip, using Tcoercivity techniques on the sphere. The socalled 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 $L2$) the gradients of outgoing singularities (which do not belong to $L2$). The formulation is proved to be Fredholm, the proof requiring establishing new vector potential decomposition results for non $L2$ vector fields.
Generalized normal modes of a metallic nanoparticle
Participants: AnneSophie BonnetBen Dhia, Christophe Hazard.
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 socalled plasmonic eigenvalue problem. This problem can be formulated as follows, considering timeharmonic electromagnetic scattering (at a fixed frequency) by a metallic particle of given (possibly complex) permittivity $\epsilon $ in vacuum. Thanks to integral equation methods or DirichlettoNeumann 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 non selfadjoint eigenvalue problem where $\epsilon $ plays the role of an eigenvalue. The eigenfunctions associated to a possible eigenvalue are called generalized normal modes. 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 twodimensional 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.
7.2 Methods for unbounded domains, Perfectly Matched Layers and Half Space Matching method
On the Halfspace Matching Method for real wavenumber
Participants: AnneSophie BonnetBen Dhia, Sonia Fliss.
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 HalfSpace Matching (HSM) method. Based on halfplane 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 halfplanes contained in the domain. While satisfactory numerical results have been obtained for real wavenumbers, wellposedness 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 ChandleWilde (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 wellposed, 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 halfplane satisfies our new radiation condition, then the corresponding solution to the halfplane Dirichlet problem satisfies the Sommerfeld radiation condition in a slightly smaller halfplane. We expect that this last result will be of independent interest, in particular in studies of rough surface scattering.
The HalfSpace Matching method for elastodynamic scattering problems in unbounded domains
Participants: Eliane Bécache, AnneSophie BonnetBen Dhia, Sonia Fliss.
In collaboration with Antoine Tonnoir from INSA of Rouenm, we have extended the HalfSpace Matching (HSM) method, first introduced for scalar problems, to elastodynamics, to solve timeharmonic 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 halfspaces. 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.
Radial PML for anisotropic media
Participants: Maryna Kachanovska, Markus Wess.
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 timedomain stability of the radial PMLs in the case when geometrical configuration of the PMLs is wellchosen (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 nontruncated 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.Stability and Convergence of Perfectly Matched Layers in Dispersive Waveguides
Participants: Eliane Bécache, Maryna Kachanovska, Markus Wess.
During the postdoc 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 nondispersive waveguides to the dispersive case. Like in the nondispersive 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 nondispersive 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 nondispersive media. A paper is in progress.The complexscaled HalfSpace Matching Method
Participants: AnneSophie BonnetBen Dhia, Christophe Hazard, Sonia Fliss.
We developed for some years a method that we call the HalfSpace 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 halfspaces, 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 nondissipative 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 ChandlerWilde and KarlMichaël Perfekt form Reading University, we propose a new formulation of the method which is wellsuited 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 wellposed. Besides the interest for the theory, this new formulation is also wellsuited for numerical purposes. Indeed one can show that the error due to the truncation of the infinite boundaries of the halfspaces decays exponentially with the distance of truncation, which was not the case for the standard method. The analysis requires the study of doublelayer potential integral operators on intersecting infinite lines, and their analytic continuations. The effectiveness of the method is validated by preliminary numerical results.PMLBIE methods for unbounded interfaces
Participants: Alexis Anne, AnneSophie BonnetBen Dhia, Luiz Faria.
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 problemspecific 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 freespace Greens function — readily available for many PDEs of physical relevance — in conjunction with a truncation technique. For nondissipative 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 easytoimplement solution, the socalled windowed Green function approach, has been proposed and validated in several configurations.
We are currently investigating the interest of using instead a complexscaled Green function, which amounts to combine the method of perfectlymatchedlayers (PMLs) and boundary integral equations.
We have applied this idea to the 2D linear timeharmonic water wave problem, in finite or infinite depth, writing a complexscaled 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.
Evaluation of oscillatory integrals in the Halfspace Matching Method
Participants: Amond Allouko, AnneSophie BonnetBen Dhia, Sonia Fliss.
This work concerns the Half Space Matching method described just above. A main ingredient of this method is a halfspace formula for the outgoing scattered field, as a function of its trace on the boundary of the halfspace. 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 halfspace formula. When it is evaluated at a point located far enough from the boundary of the halfspace, a simple farfield 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.7.3 Fast solution of boundary integral equations
Generalpurpose kernel regularization of boundary integral equations via density interpolation
Participants: Luiz Faria, Marc Bonnet.
This research is done in collaboration with Carlos PérezArancibia (PUC, Chile).We develop a general highorder 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 highorder derivatives of the density function along the surface. Furthermore, the proposed approach is kernel and dimensionindependent in the sense that the sought density interpolant is constructed as a linear combination of pointsource 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 timeharmonic elastodynamic equations. The method's accuracy, flexibility, efficiency, and compatibility with fast solvers was demonstrated by means of a variety of largescale threedimensional numerical examples.
Boundary Element  Finite Element coupling for transient fluidstructure interaction
Participants: Marc Bonnet, Stéphanie Chaillat, Alice Nassor.
This study is done in collaboration with Bruno Leblé (Naval Group). It aims at developing computational strategies for modelling the impact of a farfield underwater explosion shock wave on a structure, in deep water. An iterative transient acousticelastic 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 FEMBEM coupling algorithm with a fast convergence. Since the fast transient BEM is based on a fast multipoleaccelerated Laplacedomain BEM (implemented in the inhouse code COFFEE), extended to the time domain by the Convolution Quadrature Method (CQM), we have proposed a proof of convergence of globalintime iterative solution procedures for the coupled transient fluidstructure problem. This proof itself relies on solvability results for the coupled acousticelastic problems, some of which we established in the course of this study, and the initialboundary value problems involved in coupling iterations. We have also implemented a 2D version of this algorithm and assessed its sensitivity to various parameters.Improvements of hierarchical matrix based Boundary Element Methods for viscoelastodynamic problems
Participants: Laura Bagur, Stéphanie Chaillat, Patrick Ciarlet, Sara Touhami.
It is well known in the literature that standard $\mathscr{H}$matrix based methods, although very efficient tools for asymptotically smooth kernels, are not optimal for oscillatory kernels. In a previous work, we have shown that the method is already an efficient tool and should be used in the mechanical engineering community due to its straightforward implementation compared to ${\mathscr{H}}^{2}$matrix, or directional, approaches.
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 viscoelastic cases. In this context, we have studied the effect of the introduction of a complex wavenumber on the accuracy and efficiency of hierarchical matrix ($\mathscr{H}$matrix) based fast methods for solving dense linear systems arising from the discretization of the elastodynamic Green's tensors. We have shown that more compression can be achieved when a lot of damping is introduced while for a small amount of damping (of the order of what is typically observed in real soils), no more compression can be obtained. Hopefully, such configurations with a large amount of damping are encountered in the context of the solution of transient purely elastic problems with the convolution quadrature method. We have proposed a new admissibility condition to improve the efficiency of such methods.
Then, since $\mathscr{H}$matrices are an automatic tool to remove redundant informations, we are studying its efficiency in the context of realistic sedimentary basins with high velocity contrasts. Fast multipole accelerated Boundary Element Methods have been shown to be inefficient in this context due to the need to use an over refined mesh for one of the two connected domains. We have shown that the use of $\mathscr{H}$matrices even though the method is known to be less efficient than fast multipole methods, enable to consider largescale configurations of sedimentary basins with high velocity contrasts between the different layers of the soil. The capabilities of the methods have been demonstrated on a lot of configurations with semianalytical or reference solutions.
Modelling the sound radiated by a turbulent flow
Participants: Stéphanie Chaillat, JeanFrançois Mercier.
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 socalled 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.
Asymptotic based methods for very high frequency problems.
Participant: Eric Lunéville.
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 highfrequency 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 BSplines 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 shadowlight transition zone based on the Fock function. Diffraction by a 2D corner is in progress.
7.4 Domain decomposition methods
Nonoverlapping DDM with PML transmission conditions and multidirectionnal sweeping preconditioners for Helmholtz problems
Participant: Modave Axel.
In collaboration with Ruiyang Dai (UCLouvain), Christophe Geuzaine (Liège U.) and Anthony Royer (Liège U.), we have worked on nonoverlapping domain decomposition methods (DDM) for Helmholtz problems with transmission conditions based on nonreflecting boundary treatments. It is wellknown that the convergence rate of nonoverlapping DDM applied to the parallel finiteelement solution of largescale timeharmonic 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 wellsuited for configurations with layered domain partitions. Unfortunately, the extension of the PMLbased DDM for more general partitions with crosspoints (where more than two subdomains meet) is rather tricky and requires some care. We have proposed a nonoverlapping DDM with PML transmission conditions for checkerboard decompositions that takes crosspoints 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 DirichlettoNeumann 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 nonoverlapping checkerboard partition of the computational domain. The domain decomposition procedure relies on highorder transmission conditions and crosspoint 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 GaussSeidel 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 twodimensional finite element results to study and to compare the sweeping preconditioners, and to illustrate the performance on cases of increasing complexity.
Nonoverlapping Domain Decomposition Method (DDM) using nonlocal transmission operators for wave propagation problems.
Participants: Patrick Joly, Emile Parolin.
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 quasilocalization of the communication operator between subdomains in the presence of junction points (see also the previous activity report).
7.5 Inverse Problems, Invisibility and Optimization
Imaging junctions of waveguides
Participants: Laurent Bourgeois, Fliss Sonia, Fritsch JeanFrançois, Hazard Christophe.
A new activity recently started concerning forward and inverse scattering in junctions of waveguides. It corresponds to the PHD of JeanFrançois Fritsch and is a collaboration with the CEAList, in particular Arnaud Recoquillay. Firstly, we have considered the junction between several closed waveguides. It is wellknown 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 ColtonKirsch Linear Sampling Method. The main issue stems from the fact the LSM relies on the fundamental solution, which does not have a closedform 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 socalled 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 twodimensional 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 twodimensional 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 nonselfadjoint eigenvalue problem in the transverse direction. At the continuous level, we use the Kondratiev approach to establish wellposedness 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 solidfluid interaction problem.
Computation of the interior transmission eigenvalues in presence of strongly oscillating singularities
Participants: AnneSophie BonnetBen Dhia, Christophe Hazard.
In the context of timeharmonic 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 signchange 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 ${H}^{1}$ framework. Using Kondratiev theory, we proposed a new functional framework where the Fredholm property is restored. This is very similar (while more intricate) to what happens for the plasmonic eigenvalue problem in presence of a corner of negative material.
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.
Mixed formulations of the Tikhonov regularization
Participant: Laurent Bourgeois.
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 quasireversibility introduced by Lattès and Lions (67) and is intended to regularize illposed 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 wellknown 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.
Adaptive eigenspace bases for topology optimization
Participant: Marc Bonnet.
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 lowdimensional 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.
Asymptotic model for elastodynamic scattering by a small surfacebreaking defect
Participant: Marc Bonnet.
Work done in collaboration with Marc Deschamps and Eric Ducasse, I2M, Bordeaux.We establish a leadingorder asymptotic model for the scattering of elastodynamic fields by small surfacebreaking 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 semianalytical 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.
Shape optimization problems involving slow viscous fluids
Participant: Marc Bonnet.
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 boundaryonly 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 selfpropelled microswimmers achieving selfpropelling at least energy expense. Potential applications pertain to the understanding and optimization of various processes involving biological fluids and organisms moving in them.
7.6 Asymptotic analysis, homogenization and effective models
Effective wave motions in periodic media at finite frequencies
Participant: Marc Bonnet.
Work done in collaboration with Bojan Guzina, University of Minnesota, USA.We investigated effective timeharmonic (antiplane 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 Blochwave form, and their governing equation is found from asymptotic expansions with respect to frequencywavenumber 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
Propagation of ultrasounds in complex biological media
Participants: Laure Giovangigli, Quentin Goepfert.
This is a joint work with Josselin Garnier (XCMAP) and Pierre Millien (Institut Langevin). This project aim at modelling and studying the propagation and diffusion of ultrasounds in complex multiscale 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 multiscale 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.
Modelling a thin layer of randomly distributed nanoparticles
Participants: Sonia Fliss, Laure Giovangigli, Amandine Boucart.
This is a joint work with Bruno Stupfel from CEACESTA.We study the timeharmonic 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 multiscale 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 $d\ge 2$) or Neumann boundary conditions (for $d=3$) on the inclusions. We then establish quantitative error estimates for the effective model and present numerical simulations that illustrate our theoretical results.
Stability and Accuracy of the TimeDomain FoldyLax Model
Participant: Maryna Kachanovska.
The FoldyLax model is an asymptotic model used to compute the solution to the problem of scattering by small obstacles. While this subject had been fairly wellstudied in the frequencydomain, its timedomain analysis is still in its infancy stage. Necessity of such analysis is justified by the following observation: the timedomain counterparts of the frequencydomain FoldyLax models can exhibit instabilities. We show that this is in particular the case for the model derived earlier in [Cassier, Hazard, 2014] whose timedomain counterpart admits exponentially growing solutions when the obstacles are located close to each other.A stabilization of the FoldyLax model was obtained by reconsidering it as a Galerkin discretization of the singlelayer 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 timedomain 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 FoldyLax density. This idea can be extended to analyze other problems: transmission problems, soundhard scattering etc.
We are currently summarizing our results in a manuscript.
Enriched homogenized model in the presence of boundaries for wave equations
Participant: Clément Bénéteau, Sonia Fliss.
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 timedependent wave equation. Indeed, it is wellknown 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.
A stable and unified model for Faraday cages
Participants: Eric Lunéville, JeanFrançois Mercier.
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.
7.7 Waves in quasi 1D or 2D domains
Heat and momentum losses effect on hydrogen detonations
Participants: Luiz Faria.
This is work in collaboration with Josue MelguizoGavilanes and Ferando VeigaLopez (institute PPrime) in the context of the ANRJCJC FASTD.
The steady propagation of hydrogenfueled detonation waves with momentum and heat losses is analyzed including detailed kinetics by means of the detonation velocity  friction coefficient ($D{c}_{f}$) curves. We show that for undiluted H${}_{2}$O${}_{2}$ mixtures the heat losses (${c}_{q}=\alpha {c}_{f}$ with $\alpha $ a similarity factor) yield strong changes on the $D{c}_{f}$ steady solutions, moving the critical point, ${c}_{f}^{\u2606}$, towards faster detonations propagating in smoother tubes (i.e., lower ${c}_{f}$), which also limits their propagation at low velocities. Thus we found that for a high enough sim. factor $\alpha >{\alpha}^{\u2606}$ no detonations may propagate at the choking regime. For $\alpha <{\alpha}^{\u2606}$, we found several solutions given a fixed $D$, confirming the existence of setvalued solutions only 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 $D{c}_{f}$ curves choosing different similarity factors: (i) nitrogen strongly reduces ${c}_{f}^{\u2606}$ and moves it to faster $D$; no detonations were found at the choking regime for very low $\alpha $, limiting the appearance of the setvalues; (ii) argon presents a peculiar nonlinear effect on the ${c}_{f}^{\u2606}$ but always moves it to slower $D$; the setvalued regions widen and higher $\alpha $ are required to avoid them. We finally relate our results to realistic configurations, comparing them to previous works: good agreements were found. We asses the uncertainties that appear due to the models applied; the detailed chemical schemes used strongly modify the predictions.
Mathematical modelling of thin coaxial cables
Participants: Patrick Joly, Akram Beni Hamad.
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 leapfrog 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.
7.8 From periodic to random media
Guided modes in a hexagonal periodic graphlike domain: the zigzag and the armchair cases
Participant: Sonia Fliss.
In collaboration with Bérangère Delourme (LAGA, Paris 13), we have studied the spectrum of periodic operators in thin graphlike 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.
Discrete honeycombs, rational edges and edge states
Participant: Sonia Fliss.
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 zigzagtype edges support zero energy / flat bands and armchairtype edges support no zero energy / flat bands. We also perform a careful numerical investigations showing very strong evidence for the existence of nonzero energy (dispersive / nonflat) 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 wavepackets (superpositions of edge states) of rationally terminated structures.
Wave propagation in quasi periodic media
Participants: Sonia Fliss, Patrick Joly, Pierre Amenoagbadji.
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 quasiperiodic 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 quasiperiodic 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 quasiperiodic 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 wellposed 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.
 The 2D periodic halfspace. We are studying the propagation of waves in presence of a 2D periodic halfspace. When the interface lies in a direction of periodicity of the halfspace, it suffices (and this is what we have done few years ago) to apply the Floquet Bloch Transform in the direction of the interface, and then solve a family of waveguide problems. Dealing with an interface which does not lie in any direction of periodicity was an open question until few months ago. This problem does not concern, properly speaking a quasiperiodic medium bur the problem can be seen as quasiperiodic in the coordinate along the boundary. That is why we can use a similar but more subtle approach than above, to interpret the solution of our problem as the trace on a halfplane of the solution of a 3D PDE with periodic coefficients. The theoretical justification and the implementation of the method are in progress.
8 Bilateral contracts and grants with industry
8.1 Bilateral Contracts with Industry

Contract with DGA and Naval Group on transient fluidstructure 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: JF Mercier, B. Cotté, N. Trafny
Start: 04/2018. End: 03/2021. Administrator: ENSTA
9 Partnerships and cooperations
9.1 National initiatives
ANR

ANR project MODULATE (Modeling lOngperioD groUnd motions, and assessment of their effects on Largescale 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 (Wavespecific Discontinuous Galerkin Finite Element Methods for TimeHarmonic 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)
DGA

Contract on boundary element methods and highfrequency 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
 DGA provides partial funding for several PhD students:
 D. Chicaud on domain decomposition methods for timeharmonic electromagnetic wave problems with complex media (Start: 10/2018)
 A. Nassor on transient fluidstructure coupling caused by remote underwater explosions, including cavitation effects (Start: 10/2020)
10 Dissemination
10.1 Promoting scientific activities
10.1.1 Scientific events: organisation
 Most of the permanent members of POEMS are involved in the organization of the 15th International Conference on Mathematical and Numerical Aspects of Wave Propagation (organizing committee chaired by Sonia Fliss and Christophe Hazard). The conference was initially scheduled for July 2630, 2021. This particular edition will be an opportunity to celebrate a double anniversary: the 15th occurrence of the conference, which coincides with its 30 years birthday! Unfortunately, due to the Covid situation, we have decided to postpone it to July 2529, 2022. The conference will be held at ENSTAParis. Our wish is that this scientific meeting will also be a human meeting, taking advantage of the structure of ENSTAParis to receive on site in “full board” all participants who wish it. Website: www.waves2022.fr.
 S. Chaillat is a coanimator of the topic “Modeling and simulation” of the GDR Ondes (gdrondes.cnrs.fr). Within this framework, she coorganizes a webinar series.
 J.F. Mercier is a coanimator of the topic “Effective dynamics of microstructured media” of the GDR MecaWave (mecawave.cnrs.fr).
 A.S. BonnetBen Dhia is a member of the Organization Committee of the Scientific Committee of the hybrid conference on "HerglotzNevanlinna Functions and their Applications to Dispersive Systems and Composite Materials" which will be held in May 2022 in CIRM.
 POEMS organizes, under the responsability of M. Kachanovska, a monthly seminar. One occurrence each semester is coorganized with two other inria teams, IDEFIX and M3DISIM.
10.1.2 Journal
Member of the editorial boards
 A. S. BonnetBen Dhia is a member of the Editorial Board of the SIAM journal of applied mathematics.
 M. Bonnet is a member in the Editorial Board of Computational Mechanics (Comput. Mech.), Engineering Analysis with Boundary Elements (EABE), J. of Integral Equations and Applications (JIE), J. Optimization Theory and Applications (JOTA), and Inverse Problems.
 P. Ciarlet is a member in the Editorial Board of ESAIM:M2AN (Mathematical Modeling and Numerical Analysis).
 P. Joly is a member of the Editorial Board of the Book series "Scientific Computing" of Springer,
Reviewer  reviewing activities
 The team members regularly review papers for many international journals.
10.1.3 Research administration
 E. Bécache is a deputy chair of the Doctoral School EDMH.
 M. Bonnet is a 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 recruiting and career overseeing of the cadre of junior and senior scientists managed by MEIT.
 A.S. BonnetBen Dhia is deputychair of the Applied Mathematics Department (UMA) at ENSTA Paris. She is a member of the Scientific Council of the Doctoral School EDMH, and of the BCEP (Bureau du Comité des Equipes Projets) at INRIA Saclay from 2018. She is a member of the exterior scientific councils of Institut Fresnel and Laboratoire de Mécanique et d'Acoustique, both in Marseille.
 A.S. BonnetBen Dhia and S. Chaillat are members of the Academic Council of IP Paris (Institut Polytechnique de Paris).
 S. Chaillat has been a member of the section 09 (Solid mechanics, materials and structures, biomechanics, acoustics) of the CoCNRS from January 2020 to September 2021. She is a member of the Scientific Council of CNRS from October 2021.
 P. Ciarlet is coordinator of the Mathematics in Computational Science and Engineering Program of the Mathematics Hadamard Labex (LMH).
 M. Kachanovska is a member of the INRIA Scientific Committee for PhD and Postdoctoral Positions, from 2017.
 A. Modave is a member of the scientific committee of the mesocenter of IP Paris (Institut Polytechnique de Paris).
10.2 Teaching  Supervision  Juries
10.2.1 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 ParisSaclay.
 L. Bourgeois: coordinator of the 2nd year Maths Program at ENSTA; cohead of the M1 Applied Mathematics common to IP Paris and Université ParisSaclay;
 P. Ciarlet: coordinator at Institut Polytechnique de Paris of the Mathematics and Applications Master's Program;
 S. Fliss: coordinator of the 3nd year ENSTA programs on modelling and simulation; cohead of the M2 AMS (Analyse, Modélisation et Simulation) common to IP Paris and Université ParisSaclay;
 L. Giovangigli: coordinator of the 3nd year ENSTA programs on finance and mathematics for life sciences.
10.2.2 Courses taught
All permanent members of POEMS, as well as most PhD students and postdocs, 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 ParisSaclay.
Teaching activities of the permanent members of POEMS
 Eliane Bécache
 Introduction à la discrétisation des équations aux dérivées partielles, ENSTA (1st year)
 Analyse et approximation par éléments finis d'EDP, ENSTA (2nd year) and Master Applied Math (M1)
 Equations intégrales de frontière, ENSTA (3rd year) and Master AMS (M2)
 Marc Bonnet
 Calcul scientifique à haute perforance, ENSTA (2nd year) and Master Applied Math (M1)
 AnneSophie BonnetBen Dhia
 Fonctions de variable complexe, ENSTA (1st year)
 Théorie spectrale des opérateurs autoadjoints, ENSTA (2nd year) and Master Applied Math (M1)
 Méthodes variationnelles pour l'analyse et la résolution de problèmes non coercifs, ENSTA (3rd year) and Master AMS (M2)
 Problèmes de diffraction en domaines non bornés, ENSTA (3rd year) and Master AMS (M2)
 Laurent Bourgeois
 Outils élémentaires pour l'analyse des équations aux dérivées partielles, ENSTA (1st year)
 Fonctions de variable complexe, ENSTA (1st year)
 Problémes inverses pour des systémes gouvernés par des EDPs, ENSTA (3rd year) and Master AMS (M2)
 Stéphanie Chaillat
 Techniques numériques et algorithmiques pour les équations intégrales, ENSTA (3rd year) and Master AMS (M2)
 Éléments finis et éléments de frontière : parallélisation, couplage et performance, ENSTA (3rd year) and Master AMS (M2)
 Colin Chambeyron
 Remise à niveau en maths, Licence (1st year), ParisDauphine University
 Outils mathématiques, Licence (L1), ParisDauphine University
 Analyse  Optimisation, Licence (L1), ParisDauphine University
 Algèbre linéaire, Licence (L2), ParisDauphine University
 Patrick Ciarlet
 Méthodes variationnelles pour l'analyse et la résolution de problèmes non coercifs, ENSTA (3rd year) and Master AMS (M2)
 Modèles mathématiques et leur discrétisation en électromagnétisme, ENSTA (3rd year) and Master AMS (M2)
 Luiz Faria
 Programmation scientifique en C++, ENSTA (2nd year) and Master Applied Math (M1)
 Projet de simulation numérique, ENSTA (2nd year) and Master Applied Math (M1)
 Calcul scientifique à haute perforance, ENSTA (2nd year) and Master Applied Math (M1)
 Éléments finis et éléments de frontière : parallélisation, couplage et performance, ENSTA (3rd year) and Master AMS (M2)
 Sonia Fliss
 La méthode des éléments finis, ENSTA (2nd year) and Master Applied Math (M1)
 Introduction à la discrétisation des équations aux dérivées partielles, ENSTA (1st year)
 Homogénéisation périodique, ENSTA (3rd year), ENSTA (3rd year) and Master AMS (M2)
 Laure Giovangigli
 Introduction aux probabilités et aux statistiques, ENSTA (1st year)
 Martingales et algorithmes stochastiques, ENSTA (2nd year)
 Calcul stochastique, ENSTA (3rd year) and Master MMMEF (M2)
 Introduction à l’imagerie médicale, ENSTA (3rd year) and Master AMS and MSV (M2)
 Homogénéisation stochastique, ENSTA (3rd year) and Master AMS and MSV (M2)
 Christophe Hazard
 Outils élémentaires d'analyse pour les équations aux dérivées partielles, ENSTA (1st year)
 Théorie spectrale des opérateurs autoadjoints, ENSTA (2nd year) and Master Applied Math (M1)
 Patrick Joly
 Introduction à la discrétisation des équations aux dérivées partielles, ENSTA (1st year)
 Analyse fonctionnelle, ENSTA (2nd year) and Master AMS (M2)
 Techniques de discrétisation avancées pour les problèmes d'évolution, ENSTA (3rd year) and Master AMS (M2)
 Maryna Kachanovska
 Analyse fonctionnelle, ENSTA (2nd year) and Master Applied Math (M1)
 Modèles mathématiques et leur discrétisation en électromagnétisme, ENSTA (3rd year) and Master AMS (M2)
 Equations intégrales de frontière, ENSTA (3rd year) and Master AMS (M2)
 Nicolas Kielbasiewicz
 Programmation scientifique en C++, ENSTA (2nd year) and Master Applied Math (M1)
 Projet de simulation numérique, ENSTA (2nd year) and Master Applied Math (M1)
 Calcul scientifique parallèle, ENSTA (3rd year) and Master AMS (M2)
 Eric Lunéville
 Introduction au calcul scientifique, ENSTA (2nd year).
 Programmation scientifique en C++, ENSTA (2nd year) and Master Applied Math (M1)
 Projet de simulation numérique, ENSTA (2nd year) and Master Applied Math (M1)
 Problèmes de diffraction en domaines non bornés, ENSTA (3rd year) and Master AMS (M2)
 Pierre Marchand
 Introduction à MATLAB, ENSTA (1st year)
 JeanFrançois Mercier
 Outils élémentaires d'analyse pour les équations aux dérivées partielles, ENSTA (1st year)
 Fonctions de variable complexe, ENSTA (1st year)
 Théorie spectrale des opérateurs autoadjoints, ENSTA (2nd year) and Master Applied Math (M1)
 Axel Modave
 Calcul scientifique à haute performance, ENSTA (2rd year) and Master Applied Math (M1)
 Calcul scientifique parallèle, ENSTA (3rd year) and Master AMS (M2)
 Éléments finis et éléments de frontière : parallélisation, couplage et performance, ENSTA (3rd year) and Master AMS (M2)
10.2.3 Supervision
 PhD: Clément Bénéteau, "Asymptotic analysis of time harmonic Maxwell equations in presence of metamaterials", January 2021, S. Fliss and X. Claeys
 PhD: Hajer Methenni, "Mathematical modelling and numerical method for the simulation of ultrasound structural health monitoring of composite plates", March 2021, S. Fliss and S. Impériale
 PhD: Nicolas Trafny, "Development of semianalytical models to predict the noise produced by turbulenceedges interactions", November 2021, J.F. Mercier and B. Cotté
 PhD: Damien Chicaud, "Méthodes de décomposition de domaine pour la résolution de problèmes harmoniques d'ondes électromagnétiques en milieux complexes", December 2021, P. Ciarlet and A. Modave
 PhD in progress: Mahran Rihani, "Équations de Maxwell en présence de métamatériaux", November 2018, A.S. BonnetBen Dhia and L. Chesnel
 PhD in progress: Akram Beni Hamad, "Propagation d'ondes électromagnétiques dans les cables coaxiaux", Septembre 2019, S. Imperiale, P. Joly and M. Khenissi
 PhD in progress: JeanFrançois Fritsch, "Imagerie dans les guides d'ondes enfouis", Octobre 2019, L. Bourgeois and C. Hazard
 PhD in progress: Amandine Boucart "Modélisation d'une couche mince de nanoparticules réparties aléatoirement pour les ondes électromagnétiques, Octobre 2019, S. Fliss and L. Giovangigli
 PhD in progress: Amond Allouko, "Approche semianalytique hybride utilisant les guides d’ondes et la méthode des éléments finis pour le contrôle de santé intégrée de plaques composites", September 2020, A.S. Bonnet and A. Lhemery
 PhD in progress: Laura Bagur, "Three dimensional modeling of seismic and aseismic slip using Fast Boundary Element Methods", September 2020, S. Chaillat, J.F. Samblat and I. Stéfanou
 PhD in progress: Pierre Amenoagbadji, "Propagation des ondes dans des milieux quasipériodiques", Octobre 2020, S. Fliss and P. Joly
 PhD in progress: Etienne Peillon, "Justification et analyse mathématique de modèles de métamatériaux hyperboliques", Octobre 2020, P. Ciarlet and M. Kachanovska
 PhD in progress: Alice Nassor, "Transient fluidstructure coupling caused by remote underwater explosions, including cavitation effects", Octobre 2020, S. Chaillat and M. Bonnet
 PhD in progress: Alejandro Rosas Martinez Luis, "Dispersive electromagnetic media: mathematical and numerical analysis", November 2020, M. Cassier and P. Joly
 PhD in progress: Quentin Goepfert, "Inverse problems in ultrasonic imaging", October 2021, J. Garnier, L. Giovangigli and P. Millien
10.3 Popularization
10.3.1 Interventions
 Luiz Faria has given a conference "La simulation numérique au service des phénomènes physiques” in ”Unithé ou Café” at INRIA Saclay in June 2021.
11 Scientific production
11.1 Publications of the year
International journals
 1 articleNumerical Analysis of a Method for Solving 2D Linear Isotropic Elastodynamics with Free Boundary Condition using Potentials and Finite Elements.Mathematics of Computation2021
 2 articleOn a surprising instability result of Perfectly Matched Layers for Maxwell's equations in 3D media with diagonal anisotropy.Comptes Rendus. Mathématique2021
 3 articleStability and Convergence Analysis of Timedomain Perfectly Matched Layers for The Wave Equation in Waveguides.SIAM Journal on Numerical Analysis2021
 4 articleA continuation method for building invisible obstacles in waveguides.Quarterly Journal of Mechanics and Applied MathematicsFebruary 2021
 5 articleAn automatic PML for acoustic finite element simulations in convex domains of general shape.International Journal for Numerical Methods in Engineering12252021, 12391261
 6 articleOn the justification of topological derivative for wavebased qualitative imaging of finitesized defects in bounded media.Engineering Computations2021
 7 articlePseudocompressibility, dispersive model and acoustic waves in shallow water flows.SEMA SIMAI Springer SeriesMay 2021, 209250
 8 articleThe ComplexScaled HalfSpace Matching Method.SIAM Journal on Mathematical Analysis2021
 9 articleComplexscaling method for the complex plasmonic resonances of planar subwavelength particles with corners.Journal of Computational Physics440September 2021
 10 articleImaging junctions of waveguides.Inverse Problems and Imaging February 2021
 11 articleLimiting amplitude principle and resonances in plasmonic structures with corners: numerical investigation.Computer Methods in Applied Mechanics and EngineeringOctober 2021
 12 articleAnalytical preconditioners for Neumann elastodynamic Boundary Element Methods.SN Partial Differential Equations and Applications222March 2021
 13 articleAnalysis of variational formulations and lowregularity solutions for timeharmonic electromagnetic problems in complex anisotropic media.SIAM Journal on Mathematical Analysis533May 2021, 26912717
 14 articleA mathematical study of a hyperbolic metamaterial in free space.SIAM Journal on Mathematical Analysis2021
 15 articleModelling of the fatigue cracking resistance of grid reinforced asphalt concrete by coupling fast BEM and FEM.Road Materials and Pavement Design2022
 16 articleA stable, unified model for resonant Faraday cages.Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences4772245January 2021, 20200668
 17 articleA DtN approach to the mathematical and numerical analysis in waveguides with periodic outlets at infinity.Pure and Applied AnalysisFebruary 2021
 18 articleOptimal Ciliary Locomotion of Axisymmetric Microswimmers.Journal of Fluid Mechanics9272021, A22
 19 articleOptimal slip velocities of microswimmers with arbitrary axisymmetric shapes.Journal of Fluid Mechanics9102021, A26
 20 articleEffective wave motion in periodic discontinua near spectral singularities at finite frequencies and wavenumbers.Wave Motion1032021, 102729
 21 articleLocal transparent boundary conditions for wave propagation in fractal trees (I). Method and numerical implementation.SIAM Journal on Scientific Computing2021
 22 articleLocal transparent boundary conditions for wave propagation in fractal trees (ii): error and complexity analysis.SIAM Journal on Numerical Analysis2021
 23 articleGeneralpurpose kernel regularization of boundary integral equations via density interpolation.Computer Methods in Applied Mechanics and Engineering3782021, 113703
 24 articleScattering of acoustic waves by a nonlinear resonant bubbly screen.Journal of Fluid Mechanics906January 2021, A19
 25 articleAn Adaptive Eigenfunction Basis Strategy to Reduce Design Dimension in Topology Optimization.International Journal for Numerical Methods in Engineering1222021, 74527481
Doctoral dissertations and habilitation theses
 26 thesisEnriched homogenized models in presence of boundaries : analysis and numerical treatment.Institut Polytechnique de ParisJanuary 2021
 27 thesisMathematical modelling and numerical method for the simulation of ultrasound structural health monitoring of laminated composite plates.Institut Polytechnique de ParisMarch 2021
Reports & preprints
 28 miscImprovement of hierarchical matrices for 3D elastodynamic problems with a complex wavenumber.June 2021
 29 miscShape optimization of peristaltic pumps transporting rigid particles in Stokes flow.October 2021
 30 miscON THE HALFSPACE MATCHING METHOD FOR REAL WAVENUMBER *.November 2021
 31 miscMaxwell's equations with hypersingularities at a conical plasmonic tip.December 2021
 32 miscThe Morozov's principle applied to data assimilation problems.September 2021
 33 miscScattering in a partially open waveguide: the forward problem.October 2021
 34 miscSpectral theory for Maxwell's equations at the interface of a metamaterial. Part II: Limiting absorption, limiting amplitude principles and interface resonance.October 2021
 35 miscEfficient evaluation of threedimensional Helmholtz Green’s functions tailored toarbitrary rigid geometries for flow noise simulations.October 2021
 36 miscMathematical and numerical analyses for the divcurl and divcurlcurl problems with a signchanging coefficient.February 2021
 37 miscOn the approximation of electromagnetic fields by edge finite elements. Part 4: analysis of the model with one signchanging coefficient.June 2021
 38 miscNon overlapping Domain Decomposition Methods for Time Harmonic Wave Problems.May 2021
 39 miscRobust treatment of cross points in Optimized Schwarz Methods.January 2021
 40 miscNonlocal Impedance Operator for Nonoverlapping DDM for the Helmholtz Equation.May 2021
 41 miscMultidirectionnal sweeping preconditioners with nonoverlapping checkerboard domain decomposition for Helmholtz problems.May 2021
 42 miscAnatomy of Strike Slip Fault Tsunamigenesis.January 2021
 43 reportLimiting Amplitude Principle for a Hyperbolic Metamaterial in Free Space.InriaMarch 2021
 44 miscA nonoverlapping domain decomposition method with perfectly matched layer transmission conditions for the Helmholtz equation.November 2021
11.2 Other
Educational activities
 45 unpublishedLecture notes on numerical linear algebra.March 2021, Engineering schoolFrance
 46 unpublishedNotes de cours sur les équations de Maxwell et leur approximation.November 2021, MasterFrance