2024Activity reportProject-TeamPOEMS

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 time-periodic solutions are considered. More generally, we systematically consider both the transient problem, in the time domain, and the time-harmonic problem, in the frequency domain. Let us mention that, even if different waves share a lot of common properties, the transition from the scalar acoustic equation to the vectorial electromagnetism and elastodynamics systems raises a lot of mathematical and numerical difficulties, and requires a specific expertise.

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

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

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

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

3.2 Recent evolutions

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

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

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

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

5.1 Footprint of research activities

Our laboratory is deeply committed to sustainability and social responsibility. We strive to minimize our ecological footprint by carefully considering our travel choices, opting for train travel whenever possible. For instance, this year, several members of the team traveled to Berlin by train to attend the 2024 WAVES conference. We also use our computational resources judiciously to reduce energy consumption.

5.2 Impact of research results

On the societal impact front, while the scope of POEMS spans from theoretical research to numerical experiments, we aim, as much as possible, to collaborate with industry to address meaningful questions that have a tangible impact. We have strengthened our expertise in algorithms and numerical methods, resulting in a greater proportion of our work being linked to practical applications. We have partnered with organizations such as Siemens, Naval Group, and CEA, and contributed to projects under the Interdisciplinary Center for Defense and Security Studies at IP Paris.

Fostering sustainable science is a core value for us. We actively support open science initiatives by promoting the use of open-source software and encouraging the reproducibility of computational codes. We also contribute to Diamond Open Access journals, such as JTCAM, to ensure the free dissemination of scientific knowledge without financial barriers for authors or readers. Additionally, we strive to create a working environment where both permanent staff and PhD/postdoctoral researchers can thrive. To ensure dedicated mentorship, we prioritize co-supervisions and joint PhD programs, allowing us to dedicate meaningful time and resources to nurturing their development.

6.1 Awards

Alice Nassor received the PhD award from the French Computational Structural Mechanics Association (CSMA) and the second prize in the "Maths Entreprises & Société" award from AMIES for her thesis in collaboration with Naval Group and the DGA on the simulation of underwater explosions and their interaction with a submarine. She also represented France for the ECCOMAS PhD award.

6.2 Comments on the COMP

Note : Readers are advised that the Institute does not endorse the text in the “Highlights of the year” section, which is the sole responsibility of the team leader.

At the end of 2024, Inria's top management adopted a new “contrat d'objectifs, de moyens et de performance” (COMP), which defines Inria's objectives for the period 2024–2028. We are very unhappy and concerned about the content of this document and the way it has been enforced.

• Neither the employees nor their representative bodies had the opportunity to to participate in (or influence) the drafting of this document.
• The document defines Inria's main mission as “contributing to the nation's digital sovereignty through research and innovation” and proposes to amend Inria's founding decree to reflect this new definition. We strongly believe that our primary mission is (and should remain) the advancement of human knowledge through research. Research is not a means to achieve “digital sovereignty”, whatever that may mean. Research should not be associated with any particular nation, whatever that nation may be.
• The document announces the creation of a funding agency within Inria. France already has an independent funding agency, the ANR. Creating a new funding agency within a research institute is unnecessary and a waste of resources. It is also likely to lead to confusion, opacity, and conflicts of interest.
• Many aspects of the document reflect the desire to control research from the top down, for example through the selection of “strategic partner institutions” and “strategic topics”. This threatens the fundamental freedom of researchers to choose their research topics and collaborations.
• The document indicates that all of Inria's research should have a “dual character”, that is, both civilian and military applications. Although some of the Institute's research may have military applications, the vast majority of it is, and should remain, independent of the military.
• The document expresses the desire to place all of Inria in a “restricted area” (ZRR), which means that the hiring of researchers and interns will be reviewed and possibly vetoed by the Fonctionnaire Sécurité Défense. This leads to administrative delays, subjects recruitment to opaque criteria and discourages the recruitment of foreign nationals, which is detrimental to research and collaboration.
• The employees' opposition to this policy, which was expressed in several votes and petitions, was largely ignored.

7.1 New software

8.1 Wave propagation in metamaterials and dispersive media

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 of the principal part of the operator changes its sign smoothly along an interface. Such problems are naturally well-posed in a certain weighted Sobolev space; however, the corresponding solutions cannot contribute to the plasma heating, due to their high regularity. It is possible to demonstrate that plasma heating is induced by singular solutions, which are square integrable but do not longer lie in this weighted Sobolev space. From the theoretical viewpoint, the explanation of the plasma heating phenomenon is based on the limiting absorption principle. Previously, we have obtained a proof of the limiting absorption principle which makes use of the refined studies of the regularity of the limiting solution. We have now extended the proof to a more realistic case of 2D vector Maxwell equations. Moreover, we have formulated the limiting problem satisfied by this solution, and have shown its well-posedness by using the machinery of non-local operators. In particular, we have introduced the Neumann-Poincare operator related to this problem, and have proven that it is compact. This allows to put the problem into the Fredholm framework. Finally, the resulting operator appears to be non-self-adjoint, however, has a compact resolvent. The studies of the spectrum of this operator are in progress.

Wave propagation in unbounded hyperbolic media

Participants: Maryna Kachanovska, Dylan Machado.

We study wave propagation in cold plasma, in a regime when it is described by a hyperbolic (Klein-Gordon equation) in the frequency domain, where the role of the time is played by one of the coordinates. The problem possesses an additional difficulty: its coefficients are frequency-dependent. The subject of the internship of D. Machado was to perform a spectral analysis of this problem in a half-space case, depending on the orientation of the interface. This is done by standard Fourier analysis techniques. One of the surprising findings was that at the frequencies when the characteristics of the problem were aligned with an interface, the limiting absorption principle holds true, however, the limiting solution no longer satisfies the boundary condition ! Moreover, for a class of data, the support of the limiting absorption solution is included into the support of the data. We are currently investigating wave propagation in a hyperbolic closed waveguide.

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

Participants: Patrick Ciarlet, Farah Chaaban.

This work is done in collaboration with Mahran Rihani. We started from the scalar equation $-\text{div}\left(\sigma \nabla u\right)-{\omega }^2\eta u=f$ in a domain $\Omega$ (plus boundary conditions), where $\sigma$ and $\eta$ are real-valued, and piecewise constant. Specifically, $\sigma$ is strictly positive in part of the domain, and stricly negative elsewhere. When the problem is well-posed in $H^1\left(\Omega \right)$, meshing rules have been designed in the past to solve the problem with the finite element method (the so-called T-conform meshes), that ensures convergence of the discrete solution towards the exact solution. Following the Master's thesis of David Lassounon (2021) in which the model with $\eta =0$ was addressed, we investigated a method based on control techniques, that allows in principle to compute solutions without having to comply with those meshing rules. The mathematical theory has been completed, and the numerical results confirm theory. Currently, we are investigating similar issues for the 2D Maxwell equation $\mathbf{\text{curl}}\left({\mu }^{-1}\text{curl}𝐄\right)-{\omega }^2\epsilon 𝐄=𝐟$ in $\Omega$ (plus b.c.). The mathematical analysis is under way.

Towards non-local interface models

Participants: Patrick Ciarlet.

A collaboration with Juan Pablo Borthagaray (DMEL, Universidad de la República, Montevideo, Uruguay). The long term goal is to better take into account interface transmission conditions between a classical material and a metamaterial. The purely local models have limitations, in the sense that they are not well-posed mathematically in some configurations. On the other hand, nonlocal models allow to take transmission conditions in a more flexible manner but, on the downside, they are much more expensive to solve numerically. So, the thread currently investigated with Juan Pablo Borthagaray is to build a global model that couples local to nonlocal models, the former for their low numerical cost, the latter for their flexibility. As a first step, we focus on the design of a global diffusion model that couples local and nonlocal models, with fixed-sign diffusivity everywhere: there exist several choices to realize the coupling, not all of them being equivalent. Mathematical analysis of the most promising is currently under way.

Generalized normal modes of a metallic nanoparticle

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

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

Asymptotic scattering by metallic particles

Participants: Maryna Kachanovska.

Together with A. Mantile (U. Reims), we consider the resonances for a transmission problem between a small metallic particle filled with lossless Drude media and the vacuum. In the time-harmonic case, for a fixed frequency, the resonances to this problem are known as plasmonic resonances. We consider a more realistic frequency-dependent case. We have obtained asymptotic expansions that allow to determine the resonances of the problem, which are related to a certain non-linear equation involving the Neumann-Poincare operator. Our numerical experiments for a spherical case indicate that plasmonic resonances have imaginary part smaller than zero and are concentrated on a curve approaching the real frequency ${\omega }_p/\sqrt{2}$, with higher spatial harmonics corresponding to resonances closer to the real axis. Our future research is directed towards description of these resonances for the particles of arbitrary shapes.

Mathematical analysis of metamaterials in time domain

Participants: Patrick Joly.

This topic is the subject of our collaboration with Maxence Cassier (Institut Fresnel). This year we have finalized the works initiated during the PhD thesis Alex Rosas Martinez defended in October 2023. First, our second article on the long time behaviour of the solution of the Cauchy problem for dissipative generalized Lorentz media has been accepted for publication. Contrary to the first article where the results were obtained by a "frequency dependent" Lyapunov method, we obtained improved results via a spectral approach. Moreover, we have completed the study of guided waves by a slab of a non dissipative metamaterial (a Drude material) embedded in the vacuum. The corresponding article is in preparation.

Finally, we have been sollicitated to contribute to a volume of the book series "Operator theory" of Springer (edited by Daniel Alpay, Fabrizio Colombo and Irene Sabadini).

We have written two chapers entitled "An Operator Approach to the Analysis of Electromagnetic Wave Propagation in Dispersive Media. Part 1: General Results" and "An Operator Approach to the Analysis of Electromagnetic Wave Propagation in Dispersive Media. Part 2: Transmission Problems". This material will de presented as a plenary talk at the Conference on Mathematics of Wave Phenomena in Karlsruhe (February 2025). The book should be published next year.

Dispersion and space-time modulation

Participants: Marie Touboul.

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

Topology for time-modulated materials

Participants: Marie Touboul.

The occurrence of gaps in wavenumber in time-modulated materials seems to offer a promising avenue for topological protection. The introduction and analysis of well-suited topological invariants for these materials is the subject of an ongoing collaboration with Frank Schindler, Pavez Ignacio and Sébastien Guenneau (Imperial College London). We so far showed that the usual Zak phase is non quantized on the real branches contrary to what was previously stated in the literature. However, it is possible to introduce a quantized quantity provided one works with the imaginary branches.

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

Participants: Marie Touboul.

This is a joint work with Michaël Darch and Bruno Lombard (Laboratoire de Mécanique et d’Acoustique, Marseille), Raphaël Assier (University of Manchester) and Sébastien Guenneau (Imperial College London). The idea is to replace volumetric modulation by imperfect interfaces whose properties depend on time. Experimentally, one could imagine a series of mechanical resonators whose mass and stiffness are modified. The evolution of the mechanical energy in the case of bulk and interface energy shows that the choice of modulation (relative to the considered parameters, properties, and initial state of the media) can attenuate or amplify the scattered waves by modifying the total amount of energy. A numerical method is developed, and numerical studies are performed to understand acoustic wave propagation in 1D media separated by time-varying imperfect interfaces. Moreover, a harmonic balance method is applied in order to understand the generation of harmonics. Asymptotic homogenization is also performed in the case where the microstructure is small compared to the considered wavelength. Based on the homogenized model, the propagation of waves in a time-laminated material (homogeneous in space) is studied to understand the possibilities and limitations that k-gaps offer. Finally, the study of a phase-shifted modulation of the interfaces is conducted.

8.2 Methods for unbounded domains, Perfectly Matched Layers, Dirichlet to Neumann maps and Half Space Matching method

Convergence result of the Perfectly Matched Layers for the anisotropic Helmholtz equation

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

This work is done in collaboration with Maria Kazakova (Univ. Savoie Mont Blanc) and Antoine Tonnoir (INSA Rouen). For general time-dependent hyperbolic problems set in an open geometry, standard cartesian PMLs are known to give rise to instabilities for some anisotropic models, in particular in presence of backward waves in the direction of the absorbing layer (waves for which the projections on the PML direction of the phase and group velocities are of opposite signs). Surprisingly, standard PMLs can be accurate for time-harmonic problems while they are unstable in the time domain.In particular, for the anisotropic scalar wave equation, it is possible to chose the PMLs parameters so that standard PMLs work well numerically. This has been partially supported by a theoretical proof. Indeed, we have extended the convergence results of PMLs obtained in the isotropic case in a paper by Chen et al (2010) to weak anisotropic Helmholtz equation at low frequencies. A paper is in preparation.

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

Participants: Sarah Al Humaikani, Anne-Sophie Bonnet-Ben 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 Half-Space Matching (HSM) method. Based on half-plane representations for the solution, the scattering problem is rewritten as a system coupling (1) a standard finite element discretisation localized around the scatterer and (2) integral equations whose unknowns are traces of the solution on the boundaries of a finite number of overlapping half-planes contained in the domain.

The PhD of Sarah Al Humaikani concerns the application of the HSM method to 2D heterogeneous media that could represent a junction of several open waveguides. More precisely, we consider configurations where each half-plane of the HSM is stratified, with a stratification orthogonal to the boundary of the half-plane. This allows to exploit the so-called generalized Fourier transform to derive half-plane representations. This generalized Fourier transform relies on the diagonalization of the 1D Sturm-Liouville differential operator associated with the stratified medium, and is composed in general of an integral part, like the Fourier transform, and a finite sum associated to the eigenmodes.

As a first step, we consider the case with dissipation. In that case, we can prove that the HSM formulation is of Fredholm type: the arguments are mainly similar to that of the homogeneous case. On the contrary, the proof of uniqueness that we are currently working on is much more difficult than in the homogeneous case, and relies on contour deformations of the generalized Fourier integrals.

For the numerical implementation, we generalized to stratified half-planes the HSM software developed previously in the XLiFE++ library for homogeneous half-planes. Up to now, we considered only cases without eigenmodes.

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

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

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

Evaluation of oscillatory integrals in the Half-Space Matching Method

Participants: Amond Allouko, Anne-Sophie Bonnet-Ben Dhia.

This work has been realized during the PhD of Amond Allouko (funded by the European training Network ENHAnCE, in partnership with CEA-List, and defended in March 2024). It focuses on the Half-Space Matching (HSM) method for solving diffraction problems in an unbounded elastic plate to simulate non-destructive testing of composite plates. The HSM method is a hybrid method that couples a finite element calculation in a box containing the defects with semi-analytical representations in four half-plates covering the plate's healthy part. Semi-analytical half-plate representations involve convolution with Green tensors, expressed with Fourier integrals and modal series. However, these expressions can be challenging to evaluate in practice (cost and accuracy).

The difficulties are first analyzed in a two-dimensional scalar (acoustic) case. Two methods are proposed for an efficient evaluation of Fourier integrals: the first uses a far-field type approximation, and the second is based on a deformation of the integration path in the complex plane (complexification method). These two methods are validated in the isotropic and anisotropic scalar case, where we have the exact values of the Fourier integrals expressed using Hankel functions.

They are then generalized to the three-dimensional case of the elastic plate. In this case, the representation formula is obtained by performing a Fourier transform in a direction parallel to the plate, and then, for each value of the Fourier variable $\xi$, a modal decomposition in the thickness. The modes involved called $\xi$-modes, are studied in detail and compared to classical modes (Lamb and SH in the isotropic case). In order to exploit the bi-orthogonality of the $\xi$ modes, the half-plate formula requires the knowledge of both the displacement and the normal stress on the boundary. In the isotropic case, the analytic properties of the $\xi$-modes make it possible to justify and extend the complexification method developed for the 2D scalar case, including in the presence of inverse modes. This reduces the effects of spurious modal coupling induced by the discretization of Fourier integrals.

The complexification method is then used to calculate the operators involved in the HSM method, which derive from the elastic representation formula. Different validations of the HSM method are thus carried out in the isotropic case. The improvements significantly reduce the calculation time and ensure higher accuracy of the HSM method.

Construction of transparent conditions for electromagnetic waveguides

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

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

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

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

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

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

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

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

PML-BIE methods for unbounded interfaces

Participants: Anne-Sophie Bonnet-Ben Dhia, Luiz Faria.

There are several important applications where one must solve a scattering problem in a domain with infinite boundaries; e.g. seismic waves in a stratified medium, or water waves in the ocean. In order to handle such infinite interfaces in a boundary integral equation context, a few options are available. For simple geometries, one can construct a problem specific Green function which incorporates the imposed boundary condition on all but a bounded portion of the interface, thus reducing the problem again to integrals over bounded curves/surfaces. This has the advantage of being conceptually simple provided such problem-specific Green function can be efficiently computed. Unfortunately, that is usually not the case, and the computation of problem-specific Green functions involves challenging integrals which must be approximated numerically.

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

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

We have applied this idea to the 2D linear time-harmonic water wave problem, in finite or infinite depth, writing a complex-scaled integral equation on the free surface. The formulation uses only simple function evaluations (e.g. complex logarithms and square roots). Let us mention that because the water waves are surface waves, the windowed Green function approach does not work for this problem.

Numerical results show that the error decays exponentially with respect to the distance of truncation.

Another advantage of our method is that the formulation has a simple quadratic dependence with respect to the frequency, and is well-suited for computing complex scattering frequencies.

An article is currently under review concerning this work.

Coupling of boundary integral equations and perfectly matched layers

Participants: Maryna Kachanovska, Luiz Faria.

Simulation of time-domain scattering from a rough surface relies on the truncation of the computational domain. One of the ways to do so is to employ the boundary integral equation formulations, which are, in turn, posed on unbounded interfaces, and next apply the PML change of variables to the boundary integal operator. This method was implemented by M. Chaibi in the time-harmonic case during his internship, who has obtained promising results.

8.3 Fast solution of boundary integral equations

Fast Preconditioned Boundary Element Methods for piecewise homogeneous elastodynamics problems

Participants: Stephanie Chaillat.

Boundary Element Methods (BEMs) are highly efficient for homogeneous problems; however, challenges arise when addressing layered homogeneous problems. Various coupling methods exist, but optimizing the conditioning and handling triple points remain significant difficulties. The ECOS Chile project, in collaboration with Marion Darbas (LAGA), Carlos Jerez Hanckes (Universidad Adolfo Ibañez), and Paul Escapil-Inchauspé (INRIA Chile), focuses on extending multitrace methods available for Helmholtz problems for 2D and 3D elasticity problems. We have successfully implemented the 2D elastodynamic case, and an article is currently in preparation.

Windowed Green function method for wave scattering by periodic arrays of 2D obstacles

Participants: Luiz Faria.

This research is done in collaboration with Carlos Pérez-Arancibia (University of Twente, Netherlands), Thomas Strauszer-Caussade (PUC, Chile), and Augustín Fernandez-Lado (Intel Coorporation, Oregon, USA). This work introduces a novel boundary integral equation (BIE) method for the numerical solution of problems of planewave scattering by periodic line arrays of two-dimensional penetrable obstacles. Our approach is built upon a direct BIE formulation that leverages the simplicity of the free-space Green function but in turn entails evaluation of integrals over the unit-cell boundaries. Such integrals are here treated via the window Green function method. The windowing approximation together with a finite-rank operator correction—used to properly impose the Rayleigh radiation condition—yield a robust second-kind BIE that produces superalgebraically convergent solutions throughout the spectrum, including at the challenging Rayleigh–Wood anomalies. The corrected windowed BIE can be discretized by means of off-the-shelf Nyström and boundary element methods, and it leads to linear systems suitable for iterative linear algebra solvers as well as standard fast matrix–vector product algorithms. A variety of numerical examples demonstrate the accuracy and robustness of the proposed methodology.

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

Participants: Luiz Faria, Marc Bonnet.

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

High-order Boundary Integral Equations on implicitly defined surfaces

Participants: Luiz Faria, Dongchen He.

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

Modelling the sound radiated by a turbulent flow

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

The goal of this PhD study, conducted in collaboration with Gilles Serre (Naval Group), is to develop an optimized numerical method for determining the sound produced by turbulence and scattered by a screw propeller. Ultimately, this research aims to contribute to reducing the noise radiated by ships.

The study addresses two challenges: (i) modeling turbulence to derive an acoustic source term, and (ii) propagating both direct and scattered sounds from the source. These challenges are tackled by computing tailored Green's functions, which satisfy the natural boundary conditions of obstacles with arbitrary shapes.

Building on a prior PhD that dealt with rigid obstacles under Neumann boundary conditions, this study extends the approach to penetrable obstacles. In the fluid-fluid case, such as an air bubble in water, coupled integral equations are derived to express the tailored Green's function in terms of the free-space Green's functions of both fluids. A hierarchical matrix-based Boundary Element Method is used to efficiently compute these functions. The study also extends to fluid-elastic interactions, where a new challenge arises due to the complexity of the elastic Green's tensor.

The validity of these approaches—fluid-fluid and fluid-elastic—is confirmed by testing on a spherical geometry, for which analytical solutions are derived. The method is then successfully applied to solve Lighthill's equation, modeling sound generation by turbulent flows, on complex geometries with many degrees of freedom. These results provide a foundation for addressing ship noise reduction in practical applications.

Asymptotic based methods for very high frequency problems.

Participant: Eric Lunéville.

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

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

In order to implement the high-frequency approximations that we are developing as part of these hybrid HF/BF methods, we have introduced new geometric tools into the XLiFE++ library, in particular splines and B-Splines approximations as well as parameterizations to access quantities such as curvature, curvilinear abscissa, etc. We have also achieved the interface between the OpenCascad library and the XLiFE++ library, which allows us to manage complex geometric situations (cylinder and sphere intersection for example). In parallel, we have completed the implementation of 2D HF approximations in the shadow-light transition zone based on the Fock function and the diffraction by a 2D corner using asymptotics approximation. More recently we began to investigate the 3D case. As a first step we developed in XLiFE++ some new tools to compute geodesics on any surfaces (parametrized or only meshed). Now, the work consists in extending the 2D asymptotic expansions along the geodesics.

Quadrature for diffraction by fractal screens

Participants: Patrick Joly, Maryna Kachanovska.

This work is done in collaboration with Z. Moitier (currently at IDEFIX, INRIA). We develop a new integration technique for computing integrals over self-similar sets, with application to computing discretizations of boundary integral operators over fractal screens. The key idea is inspired by the previous work of Stritchartz, which deals with evaluation of integrals of monomials on fractals based on the self-similarity of the underlying measure, and which we were able to extend to our setting. In particular, the main difficulty in constructing quadratures over the screens lies in evaluation of the integrals of Lagrange polynomials that define quadrature weights. This is now done by a purely algebraic procedure of computing a kernel of an easy-to-compute matrix. The convergence estimates for the new quadrature have been obtained and tested numerically. The results of this research are summarized in a submitted manuscript.

Integral equation methods for acoustic scattering by fractals

Participants: Xavier Claeys.

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

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

8.4 Accelerated finite element solvers and domain decomposition methods

Accelerating non-local exchange in generalized optimized Schwarz methods

Participants: Xavier Claeys, Roxane Delville-Atchekzai.

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

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

Participants: Xavier Claeys, Arthur Saunier.

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

Substructuring based FEM-BEM coupling for Helmholtz problems

Participants: Antonin Boisneault, Xavier Claeys, Pierre Marchand.

This work is a collaboration with M. Bonazzoli (Inria Idefix) and concerns the solution of the Helmholtz equation in a medium composed of a bounded heterogeneous domain and an unbounded homogeneous one. Such problems can be expressed using classical FEM-BEM coupling techniques. We solve these coupled formulations using iterative solvers based on substructuring Domain Decomposition Methods (DDM), and aim to develop a convergence theory, with fast and guaranteed convergence. A recent article of Xavier Claeys proposed a substructuring Optimized Schwarz Method, with a nonlocal exchange operator, for Helmholtz problems on a bounded domain with classical conditions on its boundary (Dirichlet, Neumann, Robin). The variational formulation of the problem can be written as a bilinear application associated with the volume and another with the surface, for which, under certain sufficient assumptions, convergence of the DDM strategy is guaranteed. We have shown how some specific FEM-BEM coupling methods fit, or not, the previous framework, in which we consider Boundary Integral Equations (BIEs) instead of classical boundary conditions. In particular, we prove that the symmetric Costabel coupling satisfies the framework assumptions, implying that the convergence is guaranteed. This work is in the framework of OptiGPR3D Exploratory Action and has been presented at WAVES 2024 conference.

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

Participants: Pierre Marchand, Axel Modave, Timothée Raynaud.

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

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

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

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

Participants: Ahmed Chabib, Axel Modave, Simone Pescuma, Ari Rappaport.

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

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

In the context of the PhD thesis of Simone Pescuma, we have extended the CHDG method to scalar problems with piecewise constant physical coefficients. In particular, we have considered formulations with standard upwind and general symmetric fluxes. The CHDG hybridized system can be written as a fixed-point problem, which can be solved with stationary iterative schemes for a class of symmetric fluxes. The standard HDG and CHDG methods have been systematically studied with the different numerical fluxes by considering a series of 2D numerical benchmarks. The convergence of standard iterative schemes is always faster with the extended CHDG method than with the standard HDG methods, with upwind and scalar symmetric fluxes. We are currently extending this work to aeroacoustic and electromagnetic problems as part of Simone Pescuma's PhD thesis and Ari Rappaport's PostDoc, respectively. We are working on an extension of a 3D CHDG parallel code to take advantage of the computing power of GPU clusters (PostDoc by Ahmed Chabib).

Coupling of discontinuous Galerkin and pseudo-spectral methods for time-dependent acoustic problems

Participants: Rose-Cloé Meyer, Axel Modave.

This research topic is developed in collaboration with Hadrien Bériot (SIEMENS) and Gwénaël Gabard (LAUM). It corresponds to the post-doctoral work of Rose-Cloé Meyer, funded in part by the french "Plan de relance" program and SIEMENS. In this project, we have studied the coupling of a pseudo-spectral (PS) method with a nodal discontinuous Galerkin (DG) finite element method to solve time-domain acoustic wave problems. The PS method is very efficient, but is limited to rectangular shaped domains. In contrast, the DG method is easily applied to complex geometries, but can become costly when considering large-scale problems. The idea is to combine the strengths of these two methods: the PS method is used on the part of the domain without geometric constraint, while the DG method is used around the PS region to represent the geometry accurately. An overlap is introduced between the two domains to implement the coupling between the two numerical methods. A windowing in the spectral domain improves the convergence of the overall model. The exchange of information between the two domains is made possible even with non-conforming meshes, through an interpolation of the solution. We have validated the coupling method with one and two-dimensional numerical results, and we have studied the influence of different parameters on the accuracy of the coupling.

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

Participants: Laure Giovangigli.

This work is a joint work with Emile Parolin (Alpines-INRIA). The development of new quantitative ultrasound imaging algorithms, which aim at reconstructing a map of the local ultrasound speed in the medium, requires a validation process that can be achieved through numerical simulation. With this application in mind, we consider the scattering of plane waves by a tissue-mimicking medium where up to a hundred unresolved scatterers per wavelength are randomly distributed throughout the medium. The domains (about a hundred wavelengths in size) require billion degrees of freedom in a simulation, which corresponds to the state of the art in terms of direct numerical simulation capacity.

In the internship of Chris Teku, we investigated the efficiency and scalability of one-level and two-levels domain decomposition techniques to accurately solve the full scale model. The primary objective was to validate quantitative stochastic homogenization results obtained in this paper, particularly the asymptotic expansions of the scattered field with respect to the size of the scatterers. This is still on-going work.

8.5 Inverse Problems, Invisibility and Optimization

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

Participant: Laure Giovangigli, Quentin Goepfert.

This work is a joint work with Josselin Garnier (X-CMAP) and Pierre Millien (Institut Langevin). The technical progress in sensors manufacturing during the last decades and the access to now extensive computational resources constitute a major shift of paradigm for the theory of medical imaging. The fidelity of the images produced from the measurements relies then heavily on the reconstruction algorithm and the underlying mathematical model. In this work, we focus on medical ultrasound imaging where ultrasonic waves are emitted by a transducer array in a region of interest and the echoes generated by the tissues are measured by the same transducers and numerically backpropagated to obtain the image. We aim at constructing an estimator of the effective velocity in a tissue-mimicking medium where echoes come from numerous (up to a few hundred by wavelength) unresolved scatterers randomly distributed throughout the medium.

We use the asymptotics of the scattered field in this regime (derived in a previous work by the same authors to justify the estimators of the effective speed of sound inside biological tissues introduced by A. Aubry and his team. By analyzing the dependence of the imaging functional with respect to the backpropagation speed, we build an estimator of the sound speed in the random multi-scale medium. We then perform a quantitative sensitivity analysis and confront our results with numerical simulations and experimental results.

Inverse problems in oceanography

Participant: Laurent Bourgeois, Jean-François Mercier, Raphael Terrine.

This work is devoted to two different inverse problems which arise in oceanography. The first one is the identification of a tsunami from measurements of the free surface deformation, such tsunami being characterized by a brutal displacement of the bottom surface of the ocean. The second one is the bathymetry problem, which consists in recovering the underwater depth by using the same measurements as in the previous problem. In these two problems, the goal is to retrieve some sea bottom parameters from surface data, but while the loading is passive in the first problem (the tsunami is a natural phenomenon), it is active in the second one (a source generated artificially is required). For simplicity we consider a potential model in two dimensions. Such model, well-designed for tsunamis, contains both the acoustic and the gravity waves. The two inverse problems are severely ill-posed and correspond to the PHD thesis of Raphaël Terrine. As a first attempt, we addressed the resolution of these two problems in the frequency domain. But we now tackle the problem in the time domain in collaboration with Philippe Moireau (Medisim), which is of course a more realistic but more difficult situation. A first issue is proving well-posedness of the forward problem, setting it in different frameworks (for instance the Semi-group theory or Lions'theorem). In order to solve the inverse problems, two strategies are considered. The first method is a space-time mixed formulation of the Tikhonov regularization, the Morozov principle being used to determine the regularization parameter as a function of the amplitude of noise. This determination relies on a new idea based on duality in optimization. The second method is a more classical least square method, an adjoint state being used to compute the gradient of the cost function. In this second method, however, that both the control and the observation be surface functions makes the rigorous justification of the optimal control method quite difficult. Besides, while the discretization of the first method is quite obvious, that of the second one is more challenging. The numerical experiments are still ongoing.

Computation of the interior transmission eigenvalues in presence of strongly oscillating singularities

Participants: Anne-Sophie Bonnet-Ben Dhia, Florian Monteghetti.

This work is done in collaboration with Florian Monteghetti (Université Aix-Marseille). In the context of time-harmonic scattering by a bounded penetrable scatterer, interior transmission eigenvalues correspond, when they are real, to discrete frequencies for which there exists an incident wave which does not scatter. At such frequencies, inversion algorithms such as the linear sampling method fail. Real interior transmission eigenvalues are a part of a larger spectrum made of complex values, which has been largely studied in the case where the difference between the parameters in the scatterer and outside does not change sign on the boundary. In collaboration with Lucas Chesnel (INRIA team IDEFIX), we obtained some years ago some results for a 2D configuration where such sign-change occurs. The main idea was that, due to very strong singularities that can occur at the boundary, the problem may lose Fredholmness in the natural $H^1$ 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 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.

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 multi-year collaboration addresses the design and implementation of computational methods for solving optimization problems involving slow viscous fluids modelled by the Stokes equations. In particular, we have developed a computational framework that aims at simultaneously optimizing the shape and the slip velocity of an axisymmetric microswimmer suspended in a viscous fluid. We seek shapes of a given reduced volume that maximize the swimming efficiency, i.e., the (size-independent) ratio of the power loss arising from towing the rigid body of the same shape and size at the same translation velocity to the actual power loss incurred by swimming via the slip velocity. The optimal slip and efficiency (with shape fixed) are here given in terms of two Stokes flow solutions, and we then establish shape sensitivity formulas of adjoint-solution form that provide objective function derivatives with respect to any set of shape parameters on the sole basis of the above two flow solutions. Our computational treatment relies on a fast and accurate boundary integral solver for solving all Stokes flow problems. We have validated our analytic shape derivative formulas via comparisons against finite-difference gradient evaluations, and obtained several shape optimization examples.

Error-in-constitutive-relation (ECR) framework for the wave-based characterization of linear viscoelastic solids

Participant: Marc Bonnet.

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

In this collaboration, we develop an error-in-constitutive-relation (ECR) approach toward the full-field characterization of linear viscoelastic solids described by free energy and dissipation potentials. Assuming the availability of full-field interior kinematic data, the constitutive mismatch between the kinematic quantities and their “stress” counterparts, commonly referred to as the ECR functional, is established with the aid of Legendre-Fenchel gap functionals linking the thermodynamic potentials to their energetic conjugates. We then proceed by introducing the modified ECR (MECR) functional as a linear combination between its ECR parent and the kinematic data misfit, computed for a trial set of constitutive parameters. The resulting stationarity conditions then yield a coupled forward-adjoint evolution problem, which allows us to establish compact expressions for the MECR functional and its gradient with respect to the viscoelastic constitutive parameters. The formulation is established assuming either time-domain or time-harmonic data. These developments have so far been implemented in a two-dimensional time-harmonic setting, and demonstrated on the multi-frequency MECR reconstruction of a piecewise-homogeneous standard linear solid and a smoothly-varying Jeffreys viscoelastic material.

Active design of diffuse acoustic fields in enclosures

Participant: Marc Bonnet.

Work done in collaboration with Wilkins Aquino (Duke University, USA) and Jerry Rouse (Sandia Natl. Lab., USA).

This collaboration is concerned with the definition and demonstration of a numerical framework for designing diffuse fields in rooms of any shape and size, driven at arbitrary frequencies. In particular, we aim at overcoming the Schroeder frequency lower limit for generating diffuse fields in an enclosed space. We formulate the problem as a Tikhonov regularized inverse problem and propose a low-rank approximation of the spatial correlation that results in significant computational gains. Our approximation is applicable to arbitrary sets of target points and allows us to produce an optimal design at a computational cost that grows only linearly with the (potentially large) number of target points. We demonstrate the feasibility of our approach through numerical examples where we approximate diffuse fields at frequencies well below the Schroeder lower limit.

8.6 Asymptotic analysis, homogenization and effective models

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

Participants: Sonia Fliss, Laure Giovangigli.

We study the time-harmonic scattering by a heterogeneous object covered with a thin layer of randomly distributed sound soft nanoparticles. The size of the particles, their distance between each other and the layer’s thickness are all of the same order but small compared to the wavelength of the incident wave. Solving Maxwell's equations in this context is very costly. To circumvent this, we propose, via a multi-scale asymptotic expansion of the solution, an effective model where the layer of particles is replaced by an equivalent boundary condition. The coefficients that appear in this equivalent boundary condition depend on the solutions to corrector problems of Laplace type defined on unbounded random domains. Under the assumption that the particles are distributed given a stationary and mixing random point process, we prove that those problems admit a unique solution in the proper space. We then establish quantitative error estimates for the effective model and present numerical simulations that illustrate our theoretical results.

Scattering from a random rough thin layer by electromagnetic waves ou asymptotic modeling of a random rough thin layer for the scattering of electromagnetic waves

Participants: Pierre Boulogne, Sonia Fliss, Laure Giovangigli.

This work is a joint work with Justine Labat (CEA-CESTA). We study the time-harmonic scattering by a perfectly conducting object covered with a stack of rough thin layers. The thickness of the different layers is small compared to the wavelength of the incident wave. We aim at deriving an asymptotic model where the heterogeneous object is replaced by an effective boundary condition. In the internship of Pierre Boulogne we consider a stack of periodic thin layers and propose via a multi-scale asymptotic expansion of the solution an effective model at each order in the size of the layer. We establish error estimates between the solution to the original problem and the different effective models. Finally we confront our results with numerical simulations of the true solution and the 5 first orders effective solutions. For the first few months of Pierre’s PhD we have been studying rough layers whose surfaces are realizations of random stationary mixing processes. The objective is to adapt his work in the periodic case to the stationary random setting.

Non-reciprocity for the time-modulated wave equation and diffusion equation through the lens of high-order homogenization

Participants: Marie Touboul.

This is a collaboration with Bruno Lombard (Laboratoire de Mécanique et d’Acoustique, Marseille), Raphaël Assier (University of Manchester), Sébastien Guenneau and Richard Craster (Imperial College London, UK). Laminated media with material properties modulated in space and time in the form of travelling waves have long been known to exhibit non reciprocity. However, when using the method of low-frequency homogenization, it was so far only possible to obtain non-reciprocal effective media when both material properties are modulated in time, in the form of a Willis-coupling in elasticity (or bi anisotropy in electromagnetism) model. If only one of the two properties is modulated in time, while the other is kept constant, it was thought impossible for the method of homogenization to recover the expected non-reciprocity since this Willis coupling coefficient then vanishes. Contrary to this belief, we showed that effective media with a single time-modulated parameter are non-reciprocal, provided homogenization is pushed to the second order. Some further investigation has led us to investigate also the case of the diffusion equation for which the behaviour has turned out to be even more intriguing. Indeed, for the diffusion equation, at leading order, the modulation creates convection in the low-wavelength regime if both parameters are modulated. However, if only one parameter is modulated, which is more realistic, this convective term disappears and one recovers a standard diffusion equation with effective homogeneous parameters; this does not describe the non-reciprocity and the propagation of the field observed from exact dispersion diagrams. This inconsistency was corrected by considering second-order homogenization which resulted in a non reciprocal propagation term that was proved to be non zero for any laminate and verified via numerical simulation. The same methodology was also applied to the case when the density is modulated in the heat equation, leading therefore to a corrective advective term which cancels out non-reciprocity at the leading order but not at the second order.

High-frequency effective models for subsonic space-time metamaterials

Participants: Marie Touboul.

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

High-frequency homogenization for periodic dispersive media

Participants: Marie Touboul.

This is a joint work with Benjamin Vial, Sébastien Guenneau and Richard Craster (Imperial College London) and Raphaël Assier (University of Manchester). The high-frequency homogenization method is used to study dispersive media where the properties of the material depend on the frequency (Lorentz or Drude type). Effective properties are obtained near a given point of the dispersion relation. The method is validated thanks to comparison with FEM simulations and used to predict the hyperbolic or parabolic bahavior of the wave propagation.

Enriched homogenized model in the presence of boundaries for wave equations

Participants: Laure Giovangigli, Sonia Fliss.

We study the time-dependent scalar 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. But this effective model does not take into account the long time dispersive effects which appears naturally in periodic media. This is well known since the works of Santosa and Symes in the 90s and high order effective models involving high order differential operators of higher orders (at least 4) have been proposed for infinite periodic media.

The first question concerns the presence of boundaries or interfaces. Proposing boundary conditions for these models remain open questions. Note that one of the difficulty is that one has to derive variational conditions for differential operators of order 4 from original variational conditions for operators of order 2. The past few years, for frequency domain wave equation, we have proposed a new asymptotic expansion which takes into account the microscopic phenomena near the boundaries. Our approach enables to propose appropriate boundary conditions for these models. The objective is to apply these techniques in the context of the time-domain wave equation. The difficulty is to propose appropriate boundary conditions that makes the effective problems well-posed, which requires new techniques for time domain problems. We have, for now, studied the presence of a Dirichlet boundary, proposed effective conditions, showed well-posedness and performed the error analysis. We want then to tackle similar questions for Neumann or transmission conditions. This work is done in collaboration with Bruno Lombard (LMA, Marseille) and Remi Cornaggia (Institut d’Alembert, Sorbonne Université).

The second question concerns the extension of this long time homogenization to random media. Based on error estimates, one can show that for periodic media, the time scale at which dispersive effects appear is of order (${ϵ}^{-2}$) when $ϵ$ is the period of the medium. In the internship of M. Damak, we quantified this timescale numerically in two different types of one-dimensional random media: constant by parts i.i.d. density and bulk modulus and matched impedance media.

Galerkin Time-Domain Foldy-Lax models

Participants: Maryna Kachanovska, Adrian Savchuk.

The Foldy-Lax model is an asymptotic model used to compute the solution to the problem of scattering by small obstacles. While this subject had been fairly well-studied in the frequency-domain, its time- domain analysis is still in its infancy stage. In our previous work, we have suggested a construction of an asymptotic model as a Galerkin spatial semi-discretization of associated boundary integral formulations. The main idea is to choose the basis functions in a way that the convergence of the method is ensured not by increasing the cardinality of the Galerkin basis, but rather by decreasing the size of the obstacles. We have submitted a manuscript where we presented and analyzed the method for the asymptotic sound-soft scattering by circles. We have shown previously that the same choice of the Galerkin basis as for the sphere case cannot yield convergence for particles of arbitrary shape (in 3D). This was confirmed by our numerical experiments. Therefore, we suggested an alternative choice of the basis, inspired by existing works of Sini et al. Namely, now we choose basis functions as equilibrium densities. We have proven the second-order relative convergence, and tested the method numerically. These results are being summarized in a manuscript. Moreover, we were able to extend the asymptotic method to obtain a higher-order convergence. The implementation in 2D indicates that the idea is promising, and the analysis is on its way. On the other hand, we considered the following question: when the small particles of arbitrary shapes can be replaced by spheres/circles? Our findings indicate that this can be done for the lowest-order method by loosing one order of convergence.

Wave diffraction by thin finite periodic layers

Participants: Cédric Baudet, Sonia Fliss, Patrick Joly.

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

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

In a paper submitted to Asymptotic Analysis in May 2024, we proposed an asymptotic expansion of the solution at any order when the thickness of the layer tends to 0, in the case of a homogeneous layer. We use matched asymptotic expansions, which consist in introducing two different asymptotic expansions of the solution when the thickness of the layer tends to 0: one near the corner and another far away. Moreover, the asymptotic behavior of the near fields at infinity must coincide with that of the far fields in the corner. In our case, these behaviors are particularly complicated, which led us to introduce a new algebraic formalism that represents them by formal series. In this way, we can avoid cumbersome computations and replace them with abstract, generalizable computations that allow a better understanding of the structure of the asymptotic expansion at any order. Moreover, the expressions of the algebraic formal series are entirely explicit and can be computed exactly and very quickly. This formalism has helped us to develop an inductive algorithm to form the first terms of the expansion for any given order. We have justified it with error estimates.

Recently, we have been working on numerically validating the above asymptotic expansion. We first concentrated on solving each individual near-field problem. The near-fields diverge at infinity, but their diverging part can be computed explicitly, so that it is sufficient to compute the remainder, which tends to 0 at infinity. Two main difficulties remain. First, the solution tends slowly to 0 at infinity, so truncating the domain with a Dirichlet or Neumann condition is not very accurate. Moreover, the solution has no modal decomposition with variable separation that would help us to develop a DtN operator. Second, in the inductive construction of the terms, the far fields depend on the asymptotic behavior at infinity of the previous near fields at infinity, so we need a way to obtain these behaviors. Actually, we know that these are linear combinations of explicit corner singularities, and we only need the coefficients of the linear combinations.

We solved both issues by using an enriched Galerkin space made of functions defined on the unbounded domain : the finite elements on a truncation of the domain, extended by 0 on the rest of the domain, and the span of the corner singularities that appear in the solution’s behavior at infinity. We proved the convergence and provide error estimates for both the solution and the corner singularities coefficients when the size of the truncated domain tends to infinity. The more singularities there are in Galerkin space, the better the convergence rates.

8.7 Waves in quasi 1D or 2D domains

Mathematical modelling of thin coaxial cables

Participants: Patrick Joly.

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

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

8.8 From periodic to random media

Guided modes in a hexagonal periodic graph-like domain

Participant: Sonia Fliss, Nicolas Kielbasiewicz.

In collaboration with Bérangère Delourme (LAGA, Paris 13), we have studied the spectrum of periodic operators in thin graph-like domains: more precisely Neumann-Laplacian defined in periodic media which are close to quantum graphs. Moreover, we exhibit situations where the introduction of line defects in the geometry of the domain leads to the appearance of guided modes. A few years ago we have dealt with rectangular lattices and more recently we have studied hexagonal lattices. In this last case, we have shown that the dispersion curves exhibit 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 also proved the stability of the guided modes when the position of the edge varies (while keeping the same direction). As part of the Mattheo Giovaninni internship, we have simulated the behavior of the solution of the wave equation in a hexagonal graph like domain when the excitation is a time-harmonic source term for a given frequency. The behavior of such a non-stationary solution is well known when the frequency lies in a «regular" »part of the continuous spectrum: it behaves over a long time like a time-harmonic solution with the same frequency as the excitation. This is the so-called limiting absorption principle. However, if the frequency is exactly a Dirac point of the spectrum, the behavior of the solution is not clear. Numerical simulations could help us to understand this behavior. The first numerical simulations were not really conclusive, this is still under investigation.

Wave propagation in quasi periodic media

Participants: Sonia Fliss, Patrick Joly.

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

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

  More recent developments concern the non-absorbing case, for which we have proposed a heuristic method based on the ideas of the limiting absorption principle. For this, one first needs to replace the DtN operator by a so-called RtR operator which associates an incoming Robin trace to an outgoing one. The second difficulty consists in selecting the good physical solution of the Riccati equation for the corresponding propagation operator. This has led us to look at the spectral theory of weighted shift operators, a class of operators to which the propagation operator belongs. This helped us in improving the initially designed method, by computing the so-called principal “eigenpair” of the propagation operator. This also allowed us to make significative advances in the theory of limiting absorption, still incomplete. The corresponding article should be submitted soon.

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

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

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

  The most recent and last aspect of the work concerns the generalization of the previous study to a transmission problem between two arbitrary periodic media. A priori this is treatable with the lifting method but with an augmented problem in dimension 5 whose solution is a priori too costly to compute. To overcome this, we have proposed a domain decomposition appraoch in which two different lifting approaches in 3D are applied in each subspace, involving two distinct Floquet-Bloch transforms. This leads to a non-local 1D problem whose unknown is the trace of the solution on the interface.

Wave scattering by a quasi periodic surface or a quasi periodic layer

Participants: Sonia Fliss.

This this work is done in collaboration with Pierre Amenoagbadji (Columbia University), Tilo Arena (KIT), Fioralba Cakoni (Rutgers University) and have been initiated by the stay of Sonia Fliss at Rutgers University in Summer 2024. The aim of this work is to propose a new formulation of wave scattering by quasi-periodic layers that is in some sense and suitable for numerical simulations. The well-posedness requires some classical «non-trapping" » conditions for the layer. This issue is well understood for periodic layers: In general, one can restrict the problem to a periodicity cell and use transparent boundary conditions to reduce it to a bounded domain. Using Fredholm theory, one can show the well-posedness under certain conditions and use the integral formulation or the finite element method to simulate the problem. There are only a few formulations for general rough layers in 2D or in 3D and certain source terms. Moreover, these formulations were not suitable for simulation. We wanted to investigate whether we could take advantage of the quasiperiodicity assumption to complement previous works.

Inspired by our previous work on quasi-periodicity, we are able to lift the problem to a higher dimension to recover the periodicity properties, however, the principal part of the operator involved in the equation is no longer elliptic. This means that we can still restrict the new problem to a periodicity cell, we can introduce appropriate but non-standard transparent boundary conditions to reduce it to a bounded domain. However, the lack of ellipticity prevents us from using Fredholm theory. By using an appropriate functional framework, we were able to derive an inf-sup condition under certain conditions for the layer. Moreover, using the theory of the family of collective compact operators, we were able to propose an integral formulation at low frequency under other conditions for the layers. Many questions remain open: What happens at higher frequencies? Is it only technical? Can we study the existence of guided modes in quasi-periodic layers? Can we describe the scattering by a quasi-periodic layer in presence go guided modes?

8.9 Coupled phenomena for waves in fluids and solids

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

Participants: Marc Bonnet.

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

We develop a numerical simulation approach for ultrasonic NDT experiments on layered structures that aims at incorporating models for flaws or other local features (sensors, stiffeners,,,) into a semi-analytical computational framework for the unperturbed, ideal structure. The latter takes the form of the existing in-house code TraFiC developed at I2M by E. Ducasse and based on Laplace transform for the time variable and partial Fourier transforms along translation-independent or circumferential spatial coordinates; this code allows to model long-range wave propagation in undisturbed structures. The various flaws or features are then taken into account by using either small-size asymptotic models (which exploit the Green's tensors already implemented in TraFiC) or local finite element models and a domain decomposition iterative coupling approach. Regarding the latter, we established the convergence of DD iterations based on Robin boundary conditions on each (TraFiC or FE) subdomain having a shared interface. This work is undertaken through the jointly-advised thesis of Romain Kubecki, whose doctoral grant is co-funded by DGA and CEA LIST.

Singular solutions of linear aeroacoustics in recirculating base flows

Participants: Patrick Joly, Jean-François Mercier.

This is the continuation of the PhD thesis of A. Bensalah (Airbus) with whom we pursue our collaboration. We recall that aeroacoustics concerns the propagation of sound in a fluid in stationnary flow (for classical acoustics, the flow is at rest). The PhD of A. Bensalah (defended in 2018) was devoted to the Goldstein model in the time harmonic regime, for both mathematical and numerical issues.

We were able to prove that the model was well posed in a rather standard functional framework under the essential assumption that the base flow did not contain any closed streamline (plus additional assumption on the size of the vorticity of this flow). The case of recirculant flows, i.e. with closed streamlines, is much more delicate. During his thesis, A. Bensalah initiated the study of a simple model case : a 2D circular flow in an annulus.

During the past two years, we have completed this work. We use the method of limiting absorption (where $\epsilon >0$ is the size of the absorption) and the main technical ingredients for the analysis are

• reduce the problem to a countable family of ODE's (separation of variables in polar coordinates)
• use Fröbenius method and Fuchs theory for passing to the limit $\epsilon \to 0$.

This approach leads to the apparition of singular solutions that can be fully described. These solutions are outside the functional framework used for the analysis of the non recirculating case. The corresponding article has been finalized and is about to be submitted.

Time stepping methods for linear Friedrichs systems

Participants: Patrick Joly.

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

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

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

Modelling fluid injection in seismic cycles

Participants: Laura Bagur, Stéphanie Chaillat.

This work is done in collaboration with J.F. Semblat (ENSTA Paris) and I. Stefanou (Ecole Centrale Nantes). Earthquakes due to either natural or anthropogenic sources cause important human and material damage. In both cases, the presence of pore fluid influences the triggering of seismic instabilities. A timely question in the scientific community is to show that the earthquake instability could be avoided by an active control of the fluid pressure.

We use the capabilities of Fast Boundary Element Methods (Fast BEMs) to provide a multi-physic large-scale robust solver required for modeling earthquake processes, human induced seismicity and their control. Fast BEMs are combined with a rate-and-state friction law and different adaptive time stepping algorithms available in the literature. In a first step, we have checked the capabilities of all these algorithms in terms of accuracy, stability and computational times. These methods have been compared on different benchmarks. We have derived an analytic aseismic solution to perform the convergence study.

Once the optimal solver determined, we have considered poro-elastodynamic effects. Since the use of the complete poro-elastic model would lead to unacceptable computational costs to consider realistic 3D configurations, we have performed a dimensional analysis. It allows to determine which of the poroelastodynamic effects are predominant depending on the observation time of the fault. We have rigorously justified the predominant fluid effects at stake during an earthquake or a seismic cycle. We have shown that at the timescale of the earthquake instability, inertial effects are predominant. On the other hand a combination of diffusion and elastic deformation due to pore pressure change should be privileged at the timescale of the seismic cycle instead of the diffusion model mainly used in the literature. We have illustrated these effects on a simplified crustal faulting problem with fluid-injection at the timescale of the earthquake instability.

Designing Efficient Neural Networks for PDEs Using Fast Boundary Element Methods

Participants: Stéphanie Chaillat.

This research is conducted in collaboration with Ian McBrearty (Stanford University). Airgun-type seismic sources release compressed air to generate underwater acoustic waves, which are used in seismic imaging to analyze the composition of the seabed based on acoustic reflections.

Our work focuses on modeling the gas bubbles generated by these sources and determining their acoustic signatures. Unlike in many other contexts, these bubbles cannot be assumed to remain spherical, which significantly complicates the physical modeling. Instead of solving a simple ordinary differential equation, we must address a coupled vector problem. To tackle this, we assume that the liquid near the source is incompressible, inviscid, and subject to an irrotational flow, allowing us to describe the velocity field as the gradient of a velocity potential. Given that the flow is incompressible, this potential satisfies the Laplace equation, effectively decoupling spatial and temporal variables.

The evolution of the bubble’s geometry is determined by solving the Laplace equation, while the Bernoulli equation updates the boundary conditions at each time step. Since the bubble exists in an infinite medium (the ocean), the Boundary Element Method (BEM) is a natural choice for solving the Laplace problem. However, a major challenge arises: the computational cost. Solving a Laplace problem on an evolving geometry at each time step quickly becomes prohibitive. Even with fast BEMs, simulations become impractical if the computation time exceeds one second per step, limiting precision.

To address this challenge, our work explores an innovative approach leveraging graph neural networks (GNNs) tailored for dynamic geometries. Unlike conventional GNNs, which suffer from oversmoothing, our method integrates BEM-inspired features to maintain sharp representations while ensuring computational efficiency. This novel framework paves the way for accelerating simulations while preserving both accuracy and stability.

9.1 Bilateral Contracts with Industry

• Contract and CIFRE PHD with CEA CESTA on Asymptotic Modelling of a random rough thin layer in the context of electromagnetic wave scattering

  Participants: Pierre Boulogne, Sonia Fliss, Laure Giovangigli.

  Start:: 11/2024. End: 10/2027. Administrator: ENSTA.

• Contract with SIEMENS on GPU accelerated discontinuous Galerkin finite element solver for aeroacoustics

  Participants: Rose-Cloe Meyer, Axel Modave.

  Start: 01/2022. End: 01/2024. Administrator: ENSTA Paris

• Contract and CIFRE PhD with Naval Group on flow noise prediction

  Participants: Stéphanie Chaillat, Jean-Francois Mercier, Laure Pacaut.

  Start: 02/2022. End: 01/2025. Administrator: CNRS

• Contract with Sercel on the modelling of non-spherical bubbles of gaz generated by airguns

  Participants: Stéphanie Chaillat.

  Start: 01/2023. End: 12/2024. Administrator: CNRS

10.1 International initiatives

10.2 International research visitors

10.3 National initiatives

11.1 Promoting scientific activities

11.2 Teaching - Supervision - Juries

12.1 Publications of the year