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, on can mention the understanding of the instability of PMLs in anisotropic media and in dispersive media, the derivation of transparent boundary conditions in periodic media or the improvement of Fast Multipole techniques for elastodynamic integral equations.

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

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

3.2 New domains

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 Highlights

• During the PhD of Damien Mavaleix-Marchessoux, in collaboration with Naval Group, we have demonstrated the capabilities of fast BEMs coupled with FEM to model underwater explosions, paving the way to the use of the technology at the production level. An important fact is to have proposed developments with various levels of maturity going from TRL1 (new fast BEMs in the time domain) to TRL6 (demonstration of the maturity of the software developed for the application considered).
• As a way to adapt to the sanitary situation, new teaching material was created by many members of the POEMS team for distance learning (tutorials, videos, ...).

6.1 New software

7.1 Modelling complex media

There is a need of a better understanding of wave phenomena in complex media. From a physical point of view, a complex medium is typically a material where the propagation of the waves may be anisotropic and dispersive, such as metamaterials with negative coefficients. These properties are generally the effect of a microstructure, that can be ordered (in e.g. photonic crystals), or disordered (ultrasounds in biological tissues, seismic waves). From a mathematical point of view, one can take into account exactly this microstructure or, at sufficiently low frequency, use effective models justified by the homogenization theory.

Transparent boundary conditions for periodic waveguides: analysis and extensions

Participants: Sonia Fliss, Patrick Joly.

Guided modes in a hexagonal periodic graph-like domain: the zigzag and the armchair cases

Participants: Sonia Fliss.

Wave propagation in quasi periodic media

Participants: Sonia Fliss, Patrick Joly, Pierre Amenoagbadji.

A quasi-periodic medium is an ordered medium without being periodic. A fairly well-known example since the 2011 Nobel Prize in Chemistry is the quasi-crystal. The notion of quasi-periodic and more generally almost periodic function is a very well defined notion in mathematical literature (see in particular the books by Besikovitch and Bohr). To give an idea, a 1D quasi-periodic function of order N is a cut in a non-rational direction of a periodic function of N variables. PDEs with quasi-periodic coefficients have been the subject of a number of theoretical studies (see in particular the work of RM Levitan, VV Zhikov) but it seems that apart in the context of homogenization, there has been much less work on the numerical resolution of these equations.

The objective of this work is to develop original numerical methods for the solution of the time-harmonic wave equation in quasi-periodic media, in the spirit of the methods that we have developed previously for periodic media. The idea is to use, as in the work of Gerard-Varet and Masmoudi, that the study of an elliptic PDE (in the sense that the principal part of the operator is elliptic) with quasi-periodic coefficients comes down to the study of a non-elliptic PDE in higher dimension but whose coefficients are periodic. The periodic nature of the coefficients allows us to use of specific tools adapted for periodic media. However the non-elliptic nature of the equation makes the mathematical and numerical analysis of the PDE difficult.

For now, we have considered the Helmholtz equation, with a not-real frequency, in one dimensional quasi periodic media of order 2. Using the idea explained above, to solve this problem, it suffices to solve a non elliptic PDE with periodic coefficients set in a band with periodic boundary conditions. Our method to solve the Helmholtz equation in periodic waveguides can be extended to the non-elliptic case. The associated numerical method has been already implemented. When the frequency is real, the theoretical justification is less clear even if from a numerical point of view, the method works well.

Wave equation in a weakly randomly perturbed periodic medium

Participants: Sonia Fliss, Laure Giovangigli.

Enriched homogenized model in the presence of boundaries for wave equations

Participants: Clément Bénéteau, Sonia Fliss.

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

Homogenization of resonant or non-resonant thin scatterers arrays

Participants: Jean-François Mercier.

In collaboration with Agnès Maurel (Langevin Institut) and Kim Pham (IMSIA), we developp interface effective models to describe acoustics propagation through thin scatterers arrays. Such effective models involve jump conditions for the fields whose interface parameters are determined thanks to matched asymptotic expansions. We have done non-resonant and resonant homogenization:

• Perfect Brewster transmission through ultrathin perforated films :

  For a plane acoustic wave at oblique incidence on a perforated film, sound penetrable or rigid, we determine the Brewster incidence realizing a perfect transmission. For thick films, the classical volume homogenization provides an accurate prediction of the Brewster angle. However, for thinner films, it deviates from the volumetric prediction. To properly describe this shift, an interface model is built. When varying the contrasts in the material properties of the film and of the surrounding matrix, the model is found to accurately reproduce the spectra of perforated films, from ultrathin to relatively thick.

• Effective transmission conditions across a resonant bubbly metascreen :

  The extension to resonant obstacles has been considered with the study of the acoustic propagation through a thin bubbly screen. The analysis is conducted in the time domain and a difficulty is to preserve the non-linear response of the bubbles in the equivalent interface model. We have been able to provide an effective model involving a jump of the normal velocity coupled to a non-linear equation of the Rayleigh-Plesset's type for the bubble radius.

Mathematical modelling of thin coaxial cables

Participants: Patrick Joly, Akram Beni Hamad.

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

The numerical simulation of the time domain propagation of electromagnetic waves in networks of coaxial electric cables is an important issue in many industrial applications which also represents a scientific challenge given the very different scales present in the problem, in particular the small tranverse dimensions of the cables. Using a direct approach via 3D Maxwell’s equations in 3 dimensions would be quite expensive if not out of reach. This is why it is natural to seek to offer approximate mono-dimensional models which was the subject of the previous PhD thesis of G. Beck. Such models can be derived from an asymptotic analysis of 3D Maxwell’s equatiosn and must handle complex situations, including the highly heterogeneous nature of the internal structure of electric cables as well as geometry imperfections, and junctions of cables.

One of the last theoretical contributions has been concerned with loss phenomena due to skin effects and gave rise to an article to appear in SN Partial Differential Equations and Applications. The effective model involves a fractional time derivatives that ac- counts for the skin effects.

All of these models have been rigorously justified from a mathematical point of view in Geoffrey Beck’s thesis, the truly operational phase of the work, especially everything related to real applications, remained to be done. This is the object of the PhD thesis of Akram Beni Hamad, which started in 2018 that pursues a triple goal :

1. Propose, analyze and implement effective numerical methods for solving the 1D approximate problems.
2. Quantify the errors induced by approximate models, and consequently identify the limits of validity, through the comparison between 1D and 3D simulations.
3. In order to achieve the goal 2, design numerical methods for solving Maxwell 3D equations dedicated to taking into account the specificity of thin electric cables.

This last point led us to propose an original approach consisting in adapting Nédélec's edge elements to elongated prismatic meshes and proposing a hybrid time discretization procedure which is explicit in the longitudinal directions and implicit in the transverse dimensions. On particular, the resulting CFL stability condition is not affected by the smallness of the width of the cable. The first obtained numerical results are quite promising.

Stability and Accuracy of the Time-Domain Foldy-Lax Model

Participants: Maryna Kachanovska.

Modelling a thin layer of randomly distributed nano-particles

Participants: Sonia Fliss, Laure Giovangigli, Amandine Boucart.

Propagation of ultrasounds in complex biological media

Participants: Laure Giovangigli.

This is a joint work with Pierre Millien from Institut Langevin. The project aims at modelling and studying the propagation and diffusion of ultrasounds in complex biological media in order to obtain quantitative images of physical parameters of these media.

The propagation of ultrasounds in biological tissues is a complex multi-scale phenomenon: the scattered wave is produced by small (compared to the wavelength) inhomogeneities randomly distributed throughout the media, but the time of flight is affected by the slow (compared to the wavelength) variations of the medium.

In order to capture both of those effects, we model the biological tissue as a slowly varying medium described by smooth coefficients in which lie small scatterers that are randomly distributed throughout the medium. Under certain assumptions on this distribution, we perform an asymptotic expansion of the scattered wave with respect to the size of the inclusions via stochastic homogenization techniques. Grégoire Cornière, a second year student at ENSTA Paris, studied this model during his research internship in spring and summer 2020 and verified numerically the convergence of the exact solution to its asymptotic expansion in the case where the background medium has constant density and bulk modulus.

Weakly non-linear waves in reactive media

Participants: Luiz Faria.

Analysis of variational formulations and low-regularity solutions for time-harmonic electromagnetic problems in complex anisotropic media

Participants: Chicaud Damien, Ciarlet Patrick, Modave Axel.

In this work, we have considered the time-harmonic Maxwell's equations with physical parameters, namely the electric permittivity and the magnetic permeability, that are complex, possibly non-hermitian, tensor fields. Both tensor fields verify a general ellipticity condition. We have proven the well-posedness of formulations for the Dirichlet and Neumann problems (i.e. with a boundary condition on the electric field or its curl, respectively) using well-suited functional spaces and Helmholtz decompositions. For both problems, the a priori regularity of the solution and the solution's curl has been analysed. The regularity results have been obtained by splitting the fields and using shift theorems for second-order divergence elliptic operators. Finally, we have considered the discretization of the formulations with a $𝐇\left(\mathrm{𝐜𝐮𝐫𝐥}\right)$-conforming approximation based on edge finite elements. An a priori error estimate has been derived and verified thanks to numerical results with an elementary benchmark.

Mathematical analysis of metamaterials in time domain

Participants: Christophe Hazard, Patrick Joly, Alex Rosas Martinez.

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

An important effort has been devoted to the writing of a long paper (60 pages) on the limiting absorption and limiting amplitude (understand long time behaviour) principles for the transmission problem between two half-spaces, one made of vacuum and the other filled by a Drude material. This is the second part of a work whose first part, devoted to the spectral theory of the problem, was published in Communications in PDEs (where the second part will be submitted too). The approach is based on the use of so-called stationary scattering method that must be adapted to the specificities of the constitutive laws of Drude media, specificities that are also the source of new phenomena such as plasmonic interface waves or interface resonances.

The continuation of the above work is the object of the PhD thesis of Alex Rosas Martinez who started last November. We are working on more general dispersive and passive materials (Drude materials are one example) and try to understand the role of the structure of the measure involved in the Herglotz-Nevanlinna representation of the frequency dependent permittivity and permeability of these media on the apparition of dissipative effects : the conjecture is that these are due to the existence of an absolutely continuous part (with respect to Lebesgue’s measure).

Essential spectrum related to an interface with a negative material

Participants: Christophe Hazard, Sandrine Paolantoni.

Computation of plasmon resonances localized at corners using frequency-dependent complex scaling

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

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

Maxwell's equations in presence of a conical tip with negative electromagnetic constants

Participants: Anne-Sophie Bonnet-Ben Dhia, Mahran Rihani.

Towards non-local interface models

Participants: Patrick Ciarlet.

Wave Propagation in Hyperbolic Metamaterials

Participants: Patrick Ciarlet, Maryna Kachanovska, Etienne Peillon.

7.2 Efficient methods for direct and inverse problems

Numerical methods for wave problems have to deal with two particular difficulties. The first one is the fact that scattering problems are generally posed in unbounded domains. Depending on the complexity of the background, different approaches can be used to formulate them in a bounded domain, suitable for a discretization. Secondly, when the wavelength is very small in comparison to a characteristic length of the scatterer, the linear system resulting from the discretization may be very large and new strategies are needed to reduce the complexity and speed up the resolution. Fast solvers of the direct problem are useful for inverse problems or shape optimization, when iterative procedures are used. An alternative lies in the use of sampling methods, which do not require a priori knowledge of the imaged defect. Also asymptotic approximations for small defects can be exploited.

An automatic PML for acoustic finite element simulations in convex domains of general shape

Participants: Modave Axel.

Stability and Convergence of Perfectly Matched Layers in Waveguides

Participants: Eliane Bécache, Maryna Kachanovska.

Stability and Convergence of Perfectly Matched Layers in Dispersive Waveguides

Participants: Eliane Bécache, Maryna Kachanovska, Markus Wess.

The complex-scaled Halfspace Matching Method

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

The Half-Space Matching method for evolution problems

Participants: Sonia Fliss, Hajer Methenni.

Evaluation of oscillatory integrals in the Halfspace Matching Method

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

PML-BIE methods for unbounded interfaces

Participants: Sonia Fliss, Anne-Sophie Bonnet-Ben Dhia, Luiz Faria, Stéphanie Chaillat.

In order to handle infinite interfaces in a boundary integral equation context, a few options are available. For simple geometries, one can construct a problem specific Greens function which incorporates the imposed boundary condition on all but a bounded portion of the interface, thus reducing the problem again to integrals over bounded curves/surfaces. This has the advantage of being conceptually simple provided such problem-specific Greens function can be efficiently computed. Unfortunately, for all but the simplest geometries, the representation of the problem specific Greens function involves challenging integrals which must be approximated numerically. An alternative approach consists of utilizing the free-space Greens function — readily available for many PDEs of physical relevance — in conjunction with a truncation technique. For non-dissipative problems, the slow (algebraic) decay of the Greens function makes the choice of truncation technique an important aspect which needs to be considered in order to reduce the errors associated with the domain's truncation.

Our objective in this project is to develop an extension of the method of perfectly-matched-layers (PMLs) to boundary integral equations. Building on previous works which consider the application of PMLs to two dimensional Helmholtz equation, the plan is to extend the methodology to consider other PDEs, as well as non-orthogonal PMLs. Such an extension has some important practical consequences, as it allows for the solution to be reconstructed on a much larger domain at the same computational cost, thus enabling in particular for a partial reconstruction of the far field using solely the free-space Greens function.

Non-overlapping Domain Decomposition Method (DDM) using non-local transmission operators for wave propagation problems.

Participants: Patrick Joly, Emile Parolin.

• Integral operator for the electromagnetic setting: the operator is available in closed form and its structure lead naturally to a localizable form via truncation of the kernel to limit the effective computational cost while retaining its good properties. The construction of such an operator turned out to be somewhat difficult due to the particular functional setting of Maxwell's equations.
• DtN based non-local operator: the operator is computed by solving auxiliary coercive problems in the vicinity of the transmission interface. The computational cost remains moderate as the implementation no longer involve dense matrix blocks from the integral operators but rather lead to augmented sparse linear systems. Initially developed for the electromagnetic setting, the approach is appealing as it provided a unified formalism that can be applied both to Helmholtz and Maxwell equations and proved to be efficient in numerical experiments.

Another important research direction is created by the technical and theoretical difficulty posed by junction points, which are points where three or more sub-domains abut. Xavier Claeys recently proposed a method to deal with this specific issue, based on the multi-trace formalism, which led to a joint collaboration on the subject. The main idea is to perform a global exchange operation, on the whole skeleton, rather than a local point-to-point exchange. The preliminary numerical results recently obtained are promising.

General-purpose kernel regularization of boundary integral equations via density interpolation

Participants: Luiz Faria, Marc Bonnet.

We develop a general high-order kernel regularization technique applicable to all four integral operators of Calderón calculus associated with linear elliptic PDEs in two and three spatial dimensions. The proposed technique relies on interpolating the density function around the kernel singularity in terms of solutions of the underlying homogeneous PDE, so as to recast singular and nearly singular integrals in terms of bounded (or more regular) integrands. We developed a simple interpolation strategy which, unlike previous approaches, does not entail explicit computation of high-order derivatives of the density function along the surface. Furthermore, the proposed approach is kernel- and dimension-independent in the sense that the sought density interpolant is constructed as a linear combination of point-source fields, given by the same Green's function used in the integral equation formulation, thus making the procedure applicable, in principle, to any PDE with known Green's function. In the initial work 22, we have focused on Nyström methods for the (scalar) Laplace and Helmholtz equations and the (vector) elastostatic and time-harmonic elastodynamic equations. The method's accuracy, flexibility, efficiency, and compatibility with fast solvers was demonstrated by means of a variety of large-scale three-dimensional numerical examples.

Planewave Density Interpolation Methods for the EFIE on Simple and Composite Surfaces

Participants: Luiz Faria.

This research presents an extension of the planewave density interpolation method to the electric-field integral equation (EFIE) for problems of scattering and radiation by perfect electric conducting objects. Relying on the Kirchhoff integral formula and local interpolations of the surface currents that regularize the kernel singularities, the technique enables off- and on-surface EFIE operators to be reexpressed in terms of integrands that are globally bounded over the domain of integration, regardless of the magnitude of the distance between the target and source points. Surface integrals resulting from the application of the method of moments using the Rao–Wilton–Glisson basis functions can then be directly evaluated by means of elementary quadrature rules irrespective of the singularity location. The proposed technique can be applied to simple and composite surfaces comprising two or more overlapping components. The use of composite surfaces can significantly simplify the geometric treatment of complex structures, as the density interpolation method enables the use of separate nonconformal meshes for the discretization of each of the surface components that make up the composite surface. Three-dimensional examples including multiscale and intricate structures were considered in order to validate the method.

Convolution quadrature based boundary integral equations for transient fluid-structure interaction

Participants: Marc Bonnet, Stéphanie Chaillat, Damien Mavaleix-Marchessoux, Alice Nassor.

Modelling the sound radiated by a turbulent flow

Participants: Stéphanie Chaillat, Jean-François Mercier, Nicolas Trafny.

This study is done in collaboration with Gilles Serre (Naval Group) and Benjamin Cotté (IMSIA). The aim is to develop an optimized method to determine numerically the 3D Green's function of the Helmholtz equation in presence of an obstacle of arbitrary shape, satisfying the Neumann boundary condition at the boundary surfaces. This so-called rigid Green's function is useful to solve the Lighthill's equation, giving the hydrodynamic noise radiated by a ship. First an integral equation is derived, expressing the rigid Green’s function versus the free space Green’s function. Then a Boundary Element Method (BEM) is used to compute the rigid Green’s functions (with the code COFFEE). The method is first tested on simple geometries for which analytic solutions can be determined (sphere, cylinder, half plane). Then in order to consider realistic geometries in a reasonable amount of time, fast BEMs are used: fast multipole accelerated BEM and hierarchical matrix based BEM. The efficiencies of these two approaches are compared on realistic geometries of interest for the industrial partner (NACA profiles, boat propeller). Importantly, this first comparative study in an industrial context highlights the robustness of hierarchical matrix based BEM even though there are shown theoretically to be less optimal than fast multipole accelerated BEM. A publication is ongoing.

Improvements of hierarchical matrix based Boundary Element Methods for visco-elastodynamic problems

Participants: Laura Bagur, Stéphanie Chaillat, Patrick Ciarlet, Sara Touhami.

It is well known in the literature that standard $\mathscr{H}$-matrix based methods, although very efficient tools for asymptotically smooth kernels, are not optimal for oscillatory kernels. In a previous work, we have shown that the method is already an efficient tool and should be used in the mechanical engineering community due to its straightforward implementation compared to ${\mathscr{H}}^2$-matrix, or directional, approaches.

We are currently investigating two possible improvements of this approach. Since in practice, not all materials are purely elastic it is important to be able to consider visco-elastic cases. In this context, we study the effect of the introduction of a complex wavenumber on the accuracy and efficiency of hierarchical matrix ($\mathscr{H}$-matrix) based fast methods for solving dense linear systems arising from the discretization of the elastodynamic Green's tensors. Interestingly, such configurations are also encountered in the context of the solution of transient purely elastic problems with the convolution quadrature method. Then, since $\mathscr{H}$-matrices are an automatic tool to remove redundant informations, we are studying its efficiency in the context of realistic sedimentary basins with high velocity contrasts. Fast multipole accelerated Boundary Element Methods have been shown to be inefficient in this context due to the need to use an over refined mesh for one of the two connected domains.

Asymptotic based methods for very high frequency problems.

Participants: Eric Lunéville.

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

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

Solving elastodynamics equations with potentials and finite elements.

Participants: Joly Patrick.

Work done in collaboration with Sebastien Imperiale (Inria EPI M3DISIM) and Alexandre Imperiale (CEA-LIST).

This work is the major theme of our collaboration with the University of Saint-Jacques de Compostelle (Jeronimo Rodríguez, Jorge Albella – whose PhD thesis was defended in March 2019). When solving 2D linear elastodynamic equations in a homogeneous isotropic media, a Helmholtz decomposition of the displacement field decouples the equations into two scalar wave equations that only interact at the boundary. It is then natural to look for numerical schemes that independently solve the scalar equations and couple the solutions at the boundary. This is expected to be very efficient to handle soft tissues for instance.

The case of rigid boundary condition was treated first and, in a second step, the case of free surface boundary condition was proven to be unstable if a straightforward approach is used. Then an original and adequate functional framework as well as a time domain mixed formulation to circumvent these issues were proposed.

Subsequently, we proposed a discrete formulation based on finite elements. We provided the complete stability analysis of the corresponding numerical scheme. The work has been accepted in Mathematics of Computation, in which numerical results that illustrate the theory are also shown.

The most recent developments concern the transmission problems between homogeneous media, which was our initial and ultimate goal, and for which the above mentioned approach has been successfully adapted to the treatment of interface conditions. The elaboration of this work and the design and implementation of efficient algorithms will be the subject of forthcoming developments.

Implicit-explicit scheme for elastodynamic equations in plates

Participants: Sonia Fliss, Hajer Methenni.

Imaging junctions of waveguides

Participants: Laurent Bourgeois, Fliss Sonia, Fritsch Jean-François, Hazard Christophe.

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

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

Modified forward and inverse Born series for the Calderon and diffuse-wave problems

Participants: Marc Bonnet.

This investigation proposes a new direct reconstruction method based on series inversion for Electrical Impedance Tomography (EIT) and the inverse scattering problem for diffuse waves. The standard Born series for the forward problem has the limitation that the contrast lies within a certain radius for convergence. Here, we instead propose a modified Born series, based on our previous work on alternative formulations of volume integral equations, which converges for the forward problem unconditionally on the contrast. We then invert this modified Born series and compare reconstructions with the usual inverse Born series, showing that the proposed modified inverse Born series has a larger radius of convergence.

Asymptotic model for elastodynamic scattering by a small surface-breaking defect

Participants: Marc Bonnet.

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

Shape optimization problems involving slow viscous fluids

Participants: Marc Bonnet.

This collaboration addresses various shape optimization problems involving slow viscous fluid flows modelled by the Stokes equations. We have developed a new boundary integral approach for finding optimal shapes of peristaltic pumps that transport a viscous fluid. Formulas for computing the shape derivatives of the standard cost functionals and constraints, expressed in boundary-only form, are derived. They involve evaluating physical variables (traction, pressure, etc.) on the boundary only. By employing these formulas in conjunction with a boundary integral approach for solving forward and adjoint problems, we completely avoid the issue of volume remeshing when updating the pump shape as the optimization proceeds. This leads to significant cost savings and we demonstrate the performance on several numerical examples. We also investigate the optimization of the slip velocity (modeling cilia beating) and the shape of self-propelled micro-swimmers, so as to achieve self-propelling at least energy expense.

8.1 Bilateral Contracts with Industry

• Contract and CIFRE PhD with Naval Group on modelling the fluid-structure coupling caused by a far-field underwater explosion

  Participants: M. Bonnet, S. Chaillat, D. Mavaleix-Marchessoux

  Start: 11/2017. End: 10/2020. Administrator: ENSTA

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

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

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

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

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

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

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

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

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

9.1 National Initiatives

10.1 Promoting Scientific Activities

10.2 Teaching - Supervision - Juries

11.1 Publications of the year