Publications

2024

• Construction of transparent conditions for electromagnetic waveguides
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chesnel Lucas
  • Fliss Sonia
  • Parigaux Aurélien
  , 2024. We are interested in the numerical resolution of diffraction problems in closed electromagnetic waveguides by means of finite elements methods. To proceed, we need to truncate the domain and design adapted transparent conditions on the artificial boundary to avoid spurious reflections. When the guide is homogeneous in the transverse section, this can be done by writing an Electric-to-Magnetic condition based on a modal decomposition of the field. The latter takes a rather simple form thanks to the orthogonality of transverse modes. For guides that are heterogeneous in the transverse section, the transverse modes are no longer orthogonal but satisfy bi-orthogonality relations linked to the Poynting energy flux. Modal decompositions are more delicate to derive and it may happen that certain modes have phase and group velocities of different sign, which prevents the use of Perfectly Matched Layers. Adapting techniques already developed in elasticity, we derive a new transparent condition based on a Poynting-to-Magnetic operator with overlap. To illustrate the method, we present numerical results obtained with Nédélec finite elements using the XLiFE++ library.
• Notes de cours sur les équations de Maxwell et leur approximation
  
  • Ciarlet Patrick
  , 2024, pp.151.
• High-order numerical integration on self-affine sets
  
  • Joly Patrick
  • Kachanovska Maryna
  • Moitier Zoïs
  , 2024. We construct an interpolatory high-order cubature rule to compute integrals of smooth functions over self-affine sets with respect to an invariant measure. The main difficulty is the computation of the cubature weights, which we characterize algebraically, by exploiting a self-similarity property of the integral. We propose an \( h \)-version and a \( p \)-version of the cubature, present an error analysis and conduct numerical experiments.
• Propagation of ultrasounds in random multi-scale media and effecitve speed of sound estimation
  
  • Goepfert Quentin
  , 2024. Ultrasounds are widely used in medical imaging modalities. Originally, the ultrasound devices were built to image the internal structure of the tissues. In recent years, a change of paradigm operated and the goal is now also to assess physical parameters that can be used for medical diagnosis.The speed of acoustic waves inside soft tissues can be used for diagnosis of breast cancers or hepatic steatosis. Moreover, it determines the quality of the tomographic reconstruction of the tissues. Indeed, the images are usually computed by backpropagating the measured echoes at the speed of sound in water. However, the discrepancy between the speed of sound in water and the actual speed of sound inside the tissues results in nonphysical artifacts on the image.In order to establish a quantitative estimator of the propagation speed of sound inside the soft tissues, it is necessary to deeply understand the scattering of the medium. It is commonly admitted that the backscattered echoes are produced by numerous unresolved scatterers inside the medium (cell nuclei, mitochondria...). The scattering is then often modeled by the Born approximation. However, this model does not capture the variation of the effective speed of sound inside the tissue due to the unresolved scatterers. The goal of this thesis is thus to establish a propagation model that takes into account the variations of the effective speed of sound inside the tissues. Then, we will theoretically study the estimators previously introduced by Alexandre Aubry in his work.The tissue is here modeled as a bounded homogeneous mediumin which lie unresolved scatterers. As their distribution is unknown and inaccessible, their number and position is modeled as a random process. To obtain a simple form of the backscattered field, the techniques and tools developed for the quantitative stochastic homogenization theory will be used and a high-order asymptotic expansion will be proven.An asymptotic analysis of the imaging functional is carried out by using the high-order asymptotic expansion. Furthermore, the theoretical study of the estimators introduced by Alexandre Aubry and his team confirms and justifies some of the experimental results. In particular, it is possible to recover the effective speed of sound by a local spatial average of the imaging function.Numerical simulation supports each and every major result proven in this thesis.
• Guided modes in a hexagonal periodic graph like domain
  
  • Delourme Bérangère
  • Fliss Sonia
  Multiscale Modeling and Simulation: A SIAM Interdisciplinary Journal, Society for Industrial and Applied Mathematics, 2024, 22 (3), pp.1196-1245. This paper deals with the existence of guided waves and edge states in particular two-dimensional media obtained by perturbing a reference periodic medium with honeycomb symmetry. This reference medium is a thin periodic domain (the thickness is denoted δ > 0) with an hexagonal structure, which is close to an honeycomb quantum graph. In a first step, we show the existence of Dirac points (conical crossings) at arbitrarily large frequencies if δ is chosen small enough. We then perturbe the domain by cutting the perfectly periodic medium along the so-called zigzag direction, and we consider either Dirichlet or Neumann boundary conditions on the cut edge. In the two cases, we prove the existence of edges modes as well as their robustness with respect to some perturbations, namely the location of the cut and the thickness of the perturbed edge. In particular, we show that different locations of the cut lead to almost-non dispersive edge states, the number of locations increasing with the frequency. All the results are obtained via asymptotic analysis and semi-explicit computations done on the limit quantum graph. Numerical simulations illustrate the theoretical results. (10.1137/23M1600177)
  DOI : 10.1137/23M1600177
• Notes de cours sur les méthodes variationnelles pour l'analyse et la résolution de problèmes non coercifs
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Ciarlet Patrick
  , 2024.
• The Sphericity Paradox and the Role of Hoop Stresses in Free Subduction on a Sphere
  
  • Chaillat Stéphanie
  • Gerardi Gianluca
  • Li Yida
  • Chamolly Alexander
  • Li Zhong‐hai
  • Ribe Neil M.
  Journal of Geophysical Research : Solid Earth, American Geophysical Union, 2024, 129 (9), pp.e2024JB029500. Oceanic plates are doubly curved spherical shells, which influences how they respond to loading during subduction. Here we study a viscous fluid model for gravity‐driven subduction of a shell comprising a spherical plate and an attached slab. The shell is 100–1,000 times more viscous than the upper mantle. We use the boundary‐element method to solve for the flow. Solutions of an axisymmetric model show that the effect of sphericity on the flexure of shells is greater for smaller shells that are more nearly flat (the “sphericity paradox”). Both axisymmetric and three‐dimensional models predict that the deviatoric membrane stress in the slab should be dominated by the longitudinal normal stress (hoop stress), which is typically about twice as large as the downdip stress and of opposite sign. Our models also predict that concave‐landward slabs can exhibit both compressive and tensile hoop stress depending on the depth, whereas the hoop stress in convex slabs is always compressive. We test these two predictions against slab shape and earthquake focal mechanism data from the Mariana subduction zone, assuming that the deviatoric stress in our flow models corresponds to that implied by centroid moment tensors. The magnitude of the hoop stress exceeds that of the downdip stress for about half the earthquakes surveyed, partially verifying our first prediction. Our second prediction is supported by the near‐absence of earthquakes under tensile hoop stress in the portion of the slab having convex geometry. (10.1029/2024JB029500)
  DOI : 10.1029/2024JB029500
• Far-field sound field estimation using robotized measurements and the boundary elements method
  
  • Pascal Caroline
  • Marchand Pierre
  • Chapoutot Alexandre
  • Doaré Olivier
  , 2024, 270 (11), pp.816-827. Sound Field Estimation (SFE) is a numerical technique widely used to identify and reconstruct the acoustic fields radiated by unknown structures. In particular, SFE proves to be useful when data is only available close to the source, but information in the whole space is required. However, the practical implementation of this method is still hindered by two major drawbacks: the lack of efficient implementation of existing numerical methodologies, and the time-consuming and tedious roll-out of acoustic measurements. This paper aims to provide a solution to both issues. First, the measurements step is fully automated by using a robotic arm, able to accurately gather geometric and acoustic data without any human assistance. In this matter, a particular attention has been paid to the impact of the robot on the acoustic pressure measurements. The sound field prediction is then tackled using the Boundary Element Method (BEM), and implemented using the FreeFEM++ BEM library. Numerically simulated measurements have allowed us to assess the method accuracy, and the overall solution has been successfully tested using actual robotized measurements of an unknown loudspeaker (10.3397/IN_2024_2661)
  DOI : 10.3397/IN_2024_2661
• Accelerated iterative DG finite element solvers for large-scale time-harmonic acoustic problems
  
  • Modave Axel
  , 2024. Finite element methods are widely used to solve time-harmonic wave propagation problems, but solving large cases can be extremely difficult even with the computational power of parallel computers. In this work, the linear system resulting from the finite element discretization is solved with iterative solution methods, which are efficient in parallel but can require a large number of iterations. In standard discontinuous Galerkin (DG) methods, the numerical solution is discontinuous at the interfaces between the elements. In hybridizable DG methods, additional unknowns are introduced at the interfaces between the finite elements, and the physical unknowns are eliminated from the global system, resulting in a hybridized system. We have recently proposed a new strategy, called CHDG, where the additional unknowns correspond to transmission variables, whereas in the standard approach they are numerical fluxes. This strategy improves the properties of the hybridized system for faster iterative solution procedures. In this talk, we present and study a 3D CHDG implementation with nodal finite element basis functions. The resulting scheme has properties amenable to efficient parallel computing. Numerical results are presented to validate the method, and preliminary 3D computational results are proposed. (10.3397/IN_2024_2877)
  DOI : 10.3397/IN_2024_2877
• Computation of Green's functions for the acoustic scattering by an elastic structure excited by a turbulent flow in water
  
  • Pacaut Louise
  • Serre Gilles
  • Mercier Jean-François
  • Chaillat Stéphanie
  • Cotté Benjamin
  , 2024, 270 (5), pp.5995-6006. To model the hydrodynamic noise produced by an elastic ship hull or propeller excited by a turbulent boundary layer, we need an efficient method to compute the acoustic scattering by an elastic body surrounded by a fluid. In 3D, Boundary Element Methods (BEM) are used to reduce the computational costs, for both the fluid and the elastic body. A natural way to compute the boundary integral representation (BIR) of the sound pressure is to use formulations based on the free space acoustic and elastic Green's functions. However, since the turbulent flow along the elastic body is known only statistically, the use of these Green's functions would be too expensive. A remedy is to compute a Green's function adapted to the physical problem, thus satisfying the transmission conditions of the fluid-structure problem. This so-called "tailored Green's function" is determined by solving a coupled acoustic-elastic problem with the BEM, and leads to a simplified BIR of the sound pressure compatible with a stochastic source term. We first validate the computation of the tailored Green's function over a classic spherical geometry. Then we compare the scattering of multiple quadrupoles by elastic or rigid NACA0012 profiles. (10.3397/IN_2024_3671)
  DOI : 10.3397/IN_2024_3671
• Multiscale modeling for a class of high-contrast heterogeneous sign-changing problems
  
  • Ye Changqing
  • Jin Xingguang
  • Ciarlet Patrick
  • Chung Eric T.
  , 2024. The mathematical formulation of sign-changing problems involves a linear second-order partial differential equation in the divergence form, where the coefficient can assume positive and negative values in different subdomains. These problems find their physical background in negative-index metamaterials, either as inclusions embedded into common materials as the matrix or vice versa. In this paper, we propose a numerical method based on the constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) specifically designed for sign-changing problems. The construction of auxiliary spaces in the original CEM-GMsFEM is tailored to accommodate the sign-changing setting. The numerical results demonstrate the effectiveness of the proposed method in handling sophisticated coefficient profiles and the robustness of coefficient contrast ratios. Under several technical assumptions and by applying the T-coercivity theory, we establish the inf-sup stability and provide an a priori error estimate for the proposed method.
• Modeling fluid injection effects in dynamic fault rupture using Fast Boundary Element Methods
  
  • Bagur Laura
  , 2024. Earthquakes due to either natural or anthropogenic sources cause important human and material damage. In both cases, the presence of pore fluids influences the triggering of seismic instabilities.A new and timely question in the community is to show that the earthquake instability could be mitigated by active control of the fluid pressure. In this work, we study the ability 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 mitigation.In a first part, a Fast BEM solver with different temporal integration algorithms is used. We assess the performances of various possible adaptive time-step methods on the basis of 2D seismic cycle benchmarks available for planar faults. We design an analytical aseismic solution to perform convergence studies and provide a rigorous comparison of the capacities of the different solving methods in addition to the seismic cycles benchmarks tested. We show that a hybrid prediction-correction / adaptive time-step Runge-Kutta method allows not only for an accurate solving but also to incorporate both inertial effects and hydro-mechanical couplings in dynamic fault rupture simulations.In a second part, once the numerical tools are developed for standard fault configurations, our objective is to take into account fluid injection effects on the seismic slip. We choose the poroelastodynamic framework to incorporate injection effects on the earthquake instability. A complete poroelastodynamic model would require non-negligible computational costs or approximations. We justify rigorously which predominant fluid effects are at stake during an earthquake or a seismic cycle. To this aim, we perform a dimensional analysis of the equations, and illustrate the results using a simplified 1D poroelastodynamic problem. We formally show that at the timescale of the earthquake instability, inertial effects are predominant whereas 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.
• Construction of transparent conditions for electromagnetic waveguides
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chesnel Lucas
  • Fliss Sonia
  • Parigaux Aurélien
  , 2024. We are interested in the numerical resolution of diffraction problems in closed electromagnetic waveguides by means of finite elements methods. To proceed, we need to truncate the domain and design adapted transparent conditions on the artificial boundary to avoid spurious reflections. When the guide is homogeneous in the transverse section, this can be done by writing an Electric-to-Magnetic condition based on a modal decomposition of the field. The latter takes a rather simple form thanks to the orthogonality of transverse modes. For guides that are heterogeneous in the transverse section, the transverse modes are no longer orthogonal but satisfy bi-orthogonality relations linked to the Poynting energy flux. Modal decompositions are more delicate to derive and it may happen that certain modes have phase and group velocities of different sign, which prevents the use of Perfectly Matched Layers. Adapting techniques already developed in elasticity, we derive a new transparent condition based on a Poynting-to-Magnetic operator with overlap. To illustrate the method, we present numerical results obtained with Nédélec finite elements using the XLiFE++ library.
• Computation of a fluid-structure Green's function using a BEM-BEM coupling
  
  • Pacaut Louise
  • Mercier Jean-François
  • Chaillat Stéphanie
  • Serre Gilles
  , 2024. In order to determine the elasto-acoustic noise produced by a boat hull excited by a turbulent boundary layer, we propose a numerical method to compute the acoustic scattering by an elastic body surrounded by a fluid. To reduce the computational costs a Boundary Element Method (BEM) is used. Since the turbulent flow along the hull is known only statistically, a formulation combining the free field acoustic and elastic Green's functions is not adequate. A better suited choice is to determine a global Green's function satisfying the transmission conditions of the fluid-structure problem. The boundary integral representation of the scattered pressure is then simplified. This so-called tailored Green's function is determined by solving an acoustic/elastic coupled problem with a BEM. Here we focus on a particular difficulty: when the source is close to the surface, the numerical accuracy of the Green's function deteriorates. We describe a method to regularize our BEM scheme in this context. We validate the method for the problem of an elastic sphere in water.
• Evaluation of two boundary integral formulations for the Eddy current nondestructive testing of metal structures
  
  • Demaldent Edouard
  • Bakry Marc
  • Merlini Adrien
  • Andriulli Francesco
  • Bonnet Marc
  , 2024, pp.87-88. We investigate two bounary integral formulations for the resolution of the Maxwell equations in the Eddy Current (EC) regime in a context of nondestructive testing (NdT). The first one, based on an approximation of the Maxwell equations, requires a loop-star decomposition of the surface currents and the global loops are constructed manually for non-simply connected domains. The second formulation is stabilized by using quasi-Helmholtz projectors, thus avoiding the definition of global loops. (10.17617/3.MBE4AA)
  DOI : 10.17617/3.MBE4AA
• Substructuring based FEM-BEM coupling for Helmholtz problems
  
  • Boisneault Antonin
  • Bonazzoli Marcella
  • Claeys Xavier
  • Marchand Pierre
  , 2024. This talk 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. In this presentation we show 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. (10.17617/3.MBE4AA)
  DOI : 10.17617/3.MBE4AA
• Fast and accurate boundary integral equation methods for the multi-layer transmission problem
  
  • Cortes Elsie A
  • Carvalho Camille
  • Chaillat Stéphanie
  • Tsogka Chrysoula
  , 2024. We consider a multi-layer transmission problem, which can be used for example to describe the light scattering in meta-materials (assemblings of various concentric penetrable materials). Our goal is to solve the multi-layer problem accurately with optimal discretization. Generally, the costs to solve this problem grow as more layers are introduced - solving this problem is thus particularly challenging for 3D models. For this reason, we use boundary integral equation (BIE) methods: they reduce the dimensionality of the problem and can provide high order accuracy. However, BIE methods suffer from the so-called close evaluation problem. We address it using modified representations. We further examine how to improve the speed of our method by optimizing the accuracy over number of discretization points ratio. In particular, we investigate whether the usual rule of thumb to mesh interfaces, based on the most constraining material, is necessary for the multi-layer transmission problem. (10.17617/3.MBE4AA)
  DOI : 10.17617/3.MBE4AA
• Efficient methods for the solution of boundary integral equations on fractal antennas
  
  • Joly Patrick
  • Kachanovska Maryna
  • Moitier Zoïs
  , 2024. This work focuses on construction of efficient numerical methods for wave scattering by fractal antennas, see [3]. It builds on the theoretical basis proposed in the recent work [1], which establishes boundary integral (BIE) formulations for solving sound-soft Helmholtz scattering problems on fractal screens. An important feature of such formulations is the use of the Hausdorff measure on fractals instead of the standard Lebesgue’s measure. This adds an extra dimension to the two classical difficulties encountered with numerical BEM simulations, namely the evaluation of boundary integrals and the fact that the underlying matrices are dense. Our idea is to exploit the Hausdorff measure’s self-similar structure in order to deal with these difficulties. (10.17617/3.MBE4AA)
  DOI : 10.17617/3.MBE4AA
• Contributions to Efficient Finite Element Solvers for Time-Harmonic Wave Propagation Problems
  
  • Modave Axel
  , 2024. The numerical simulation of wave propagation phenomena is of paramount importance in many scientific and engineering disciplines. Many time-harmonic problems can be solved with finite elements in theory, but the computational cost is a strong constraint that limits the size of the problems and the accuracy of the solutions in practice. Ideally, solution techniques should provide the best accuracy at minimal computational cost for real-world problems. They should take advantage of the power of modern parallel computers, and they should be as easy as possible to use for the end user. In this HDR thesis, contributions are presented on three topics: the improvement of domain truncation techniques (i.e. high-order absorbing boundary conditions and perfectly matched layers), the acceleration of substructuring and preconditioning techniques based on domain decomposition methods (i.e. non-overlapping domain decomposition methods with interface conditions based on domain truncation techniques), and the design of a new hybridization approach for efficient discontinuous finite element solvers.
• Heat and momentum losses in H 2 –O2 –N 2/Ar detonations: on the existence of set-valued solutions with detailed thermochemistry
  
  • Veiga-López F.
  • Faria Luiz
  • Melguizo-Gavilanes J.
  Shock Waves, Springer Verlag, 2024, 34 (3), pp.273-283. The effect of heat and momentum losses on the steady solutions admitted by the reactive Euler equations with sink/source terms is examined for stoichiometric hydrogen–oxygen mixtures. Varying degrees of nitrogen and argon dilution are considered in order to access a wide range of effective activation energies, $$E_{\textrm{a,eff}}/R_{\textrm{u}}T_{0}$$ E a,eff / R u T 0 , when using detailed thermochemistry. The main results of the study are discussed via detonation velocity-friction coefficient ( D – $$c_{\textrm{f}}$$ c f ) curves. The influence of the mixture composition is assessed, and classical scaling for the prediction of the velocity deficits, $$D(c_{\textrm{f,crit}})/D_{\textrm{CJ}}$$ D ( c f,crit ) / D CJ , as a function of the effective activation energy, $${E}_{\textrm{a,eff}}/R_{\textrm{u}} T_{0}$$ E a,eff / R u T 0 , is revisited. Notably, a map outlining the regions where set-valued solutions exist in the $$E_{\textrm{a,eff}}/R_{\textrm{u}}T_{0}\text {--}{\alpha }$$ E a,eff / R u T 0 -- α space is provided, with $$\alpha $$ α denoting the momentum–heat loss similarity factor, a free parameter in the current study. (10.1007/s00193-024-01182-5)
  DOI : 10.1007/s00193-024-01182-5
• Generalized impedance boundary conditions with vanishing or sign-changing impedance
  
  • Bourgeois Laurent
  • Chesnel Lucas
  SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2024, 56 (3), pp.4223-4251. We consider a Laplace type problem with a generalized impedance boundary condition of the form ∂_νu = −∂_x(g∂_xu) on a flat part Γ of the boundary. Here ν is the outward unit normal vector to ∂Ω, g is the impedance function and x is the coordinate along Γ. Such problems appear for example in the modelling of small perturbations of the boundary. In the literature, the cases g=1 or g=−1 have been investigated. In this work, we address situations where Γ contains the origin and g(x)=1_{x>0}(x)x^\alpha or g(x)=−sign(x)|x|^\alpha with \alpha≥ 0. In other words, we study cases where g vanishes at the origin and changes its sign. The main message is that the well-posedness in the Fredholm sense of the corresponding problems depends on the value of \alpha. For \alpha∈ [0,1), we show that the associated operators are Fredholm of index zero while it is not the case when \alpha=1. The proof of the first results is based on the reformulation as 1D problems combined with the derivation of compact embedding results for the functional spaces involved in the analysis. The proof of the second results relies on the computation of singularities and the construction of Weyl's sequences. We also discuss the equivalence between the strong and weak formulations, which is not straightforward. Finally, we provide simple numerical experiments which seem to corroborate the theorems. (10.1137/23M1604217)
  DOI : 10.1137/23M1604217
• Optimal computation of integrals in the Half-Space Matching method for modal simulation of SHM/NDE in 3D elastic plate
  
  • Allouko Amond
  • Bonnet-Ben Dhia Anne-Sophie
  • Lhémery Alain
  • Baronian Vahan
  Journal of Physics: Conference Series, IOP Science, 2024, 2768, pp.012004. Simulating structural health monitoring (SHM) or nondestructive evaluation (NDE) methods based on elastic guided waves (GW) is very helpful to handle their complexity (co-existence of several GW modes, frequency dependence of wavespeed) and to further design optimal methods of inspection offering high sensitivity to the sought flaws. The half-space matching (HSM) method has been established for the development of a model that hybridizes local finite element (FE) computations for GW scattering by a flaw, with a modal semi-analytical model for GW radiation and propagation in flawless plate-like structures. Highly oscillatory Integral formulae appear in the HSM method that radiate the scattered field away from the FE zone as the superimposition of modal contributions, which computation can be time-consuming. The present work is concerned with their optimal computation. Integral of this form can be efficiently computed under the far-field approximation but this classical technique fails at predicting accurately wavefields at relatively short distances (small number of wavelengths). The method developed herein relies on the complexification of the integrals to be computed and on specific deformation of integration paths in the complex plane, as detailed in the paper. It allows the evaluation of the integrals without approximation other than that of numerical quadratures, ensuring high accuracy while offering high computing performances. It indifferently applies in the far-field and in the near-field. The method of computation is validated by comparing its predictions with a reference solution of GW scattering. Its computational performances are also demonstrated, compared to those of the standard computation of the HSM integral formulae to be computed and on specific deformation of integration paths in the complex plane, as detailed in the paper. It allows the evaluation of the integrals without approximation other than that of numerical quadratures, ensuring high accuracy while offering high computing performances. It indifferently applies in the far-field and in the near-field. The method of computation is validated by comparing its predictions with a reference solution of GW scattering. Its computational performances are also demonstrated, compared to those of the standard computation of the HSM integral formulae. (10.1088/1742-6596/2768/1/012004)
  DOI : 10.1088/1742-6596/2768/1/012004
• Simulation and analysis of sign-changing Maxwell’s equations in cold plasma
  
  • Peillon Etienne
  , 2024. Nowadays, plasmas are mainly used for industrial purpose. One of the most frequently cited examples of industrial use is electric energy production via fusion nuclear reactors. Then, in order to contain plasma properly inside the reactor, a background magnetic field is imposed, and the density and temperature of the plasma must be precisely controlled. This is done by sending electromagnetic waves at specific frequencies and directions depending on the characteristics of the plasma.The first part of this PhD thesis consists in the study of the model of plasma in a strong background magnetic field, which corresponds to a hyperbolic metamaterial. The objective is to extend the existing results in 2D to the 3D-case and to derive a radiation condition. We introduce a splitting of the electric and magnetic fields resembling the usual TE and TM decomposition, then, it gives some results on the two resulting problems. The results are in a very partial state, and constitute a rough draft on the subject.The second part consists in the study of the degenerate PDE associated to the lower-hybrid resonant waves in plasma. The associated boundary-value problem is well-posed within a ``natural'' variational framework. However, this framework does not include the singular behavior presented by the physical solutions obtained via the limiting absorption principle. Notice that this singular behavior is important from the physical point of view since it induces the plasma heating mentioned before. One of the key results of this second part is the definition of a notion of weak jump through the interface inside the domain, which allows to characterize the decomposition of the limiting absorption solution into a regular and a singular parts.
• Modélisation hybride modale-éléments finis pour le contrôle ultrasonore d'une plaque élastique. Traitement des intégrales oscillantes de la méthode HSM
  
  • Allouko Amond
  , 2024. Cette thèse porte sur la méthode Half-Space Matching (HSM) pour la résolution de problèmes de diffraction dans une plaque élastique non-bornée, en vue de la simulation du contrôle non-destructif de plaques composites. La méthode HSM est une approche hybride qui couple un calcul éléments finis dans une boite contenant les défauts, avec des représentations semi-analytiques dans quatre demi-plaques qui recouvrent la partie saine de la plaque. Les représentations semi-analytiques de demi-plaques font intervenir des tenseurs de Green, exprimés à l'aide d'intégrales de Fourier et de séries modales. Or ces expressions peuvent être délicates à évaluer en pratique (coût et précision), rendant la méthode HSM inexploitable industriellement. Les difficultés sont d'abord analysées dans un cas scalaire bidimensionnel (acoustique). Deux méthodes sont proposées pour une évaluation efficace des intégrales de Fourier : la première exploite une approximation de type champ lointain et la seconde repose sur une déformation du chemin d'intégration dans le plan complexe (méthode de la complexification). Ces deux méthodes sont validées dans les cas scalaires isotrope et anisotrope où l'on dispose des valeurs exactes des intégrales de Fourier exprimées à l'aide de fonctions de Hankel. Elles sont ensuite généralisées au cas tridimensionnel de la plaque élastique. Dans ce cas, la formule de représentation est obtenue en faisant une transformée de Fourier suivant une direction parallèle à la plaque, puis, pour chaque valeur de la variable de Fourier ξ, une décomposition modale dans l'épaisseur. Les modes mis en jeu, appelés ξ-modes, sont étudiés en détail et comparés aux modes classiques (Lamb et SH dans le cas isotrope). Afin d'exploiter la bi-orthogonalité des ξ-modes, la formule de demi-plaque requiert la connaissance à la fois du déplacement et de la contrainte normale sur la frontière. Dans le cas isotrope, les propriétés d'analyticité des ξ-modes permettent de justifier et d'étendre la méthode de la complexification, y compris en présence de modes inverses. Ceci réduit les effets de couplage modal parasite induits par la discrétisation des intégrales de Fourier. La méthode de la complexification est ensuite utilisée pour le calcul des opérateurs intervenant dans la méthode HSM, qui dérivent tous de la formule de demiplaque. Différentes validations de la méthode HSM sont ainsi effectuées dans le cas isotrope. Des résultats préliminaires encourageants sont également obtenus pour une plaque orthotrope. Les améliorations réalisées ont permis à la fois de réduire significativement le temps de calcul et d'assurer une plus grande précision de la méthode HSM, permettant d'envisager son exploitation systématique dans un cadre de simulation industrielle.
• Introduction aux équations aux dérivées partielles hyperboliques et à leur approximation numérique
  
  • Fliss Sonia
  • Bonnet-Ben Dhia Anne-Sophie
  • Joly Patrick
  • Moireau Philippe
  , 2024.