Publications

2019

• On well-posedness of time-harmonic problems in an unbounded strip for a thin plate model
  
  • Bourgeois Laurent
  • Chesnel Lucas
  • Fliss Sonia
  Communications in Mathematical Sciences, International Press, 2019, 17 (6), pp.1487-1529. We study the propagation of elastic waves in the time-harmonic regime in a waveguide which is unbounded in one direction and bounded in the two other (transverse) directions. We assume that the waveguide is thin in one of these transverse directions, which leads us to consider a Kirchhoff-Love plate model in a locally perturbed 2D strip. For time harmonic scattering problems in unbounded domains, well-posedness does not hold in a classical setting and it is necessary to prescribe the behaviour of the solution at infinity. This is challenging for the model that we consider and constitutes our main contribution. Two types of boundary conditions are considered: either the strip is simply supported or the strip is clamped. The two boundary conditions are treated with two different methods. For the simply supported problem, the analysis is based on a result of Hilbert basis in the transverse section. For the clamped problem, this property does not hold. Instead we adopt the Kondratiev's approach, based on the use of the Fourier transform in the unbounded direction, together with techniques of weighted Sobolev spaces with detached asymptotics. After introducing radiation conditions, the corresponding scattering problems are shown to be well-posed in the Fredholm sense. We also show that the solutions are the physical (outgoing) solutions in the sense of the limiting absorption principle. (10.4310/CMS.2019.v17.n6.a2)
  DOI : 10.4310/CMS.2019.v17.n6.a2
• Some contributions to the analysis of the Half-Space Matching Method for scattering problems and extension to 3D elastic plates
  
  • Tjandrawidjaja Yohanes
  , 2019. This thesis focuses on the Half-Space Matching Method which was developed to treat some scattering problems in complex infinite domains, when usual numerical methods are not applicable. In 2D, it consists in coupling several plane-wave representations in half-spaces surrounding the obstacle(s) with a Finite Element computation of the solution in a bounded domain. To ensure the matching of all these representations, the traces of the solution are linked by Fourier-integral equations set on the infinite boundaries of the half-spaces. In the case of a dissipative medium, this system of integral equations was proved to be coercive plus compact in an L² framework.In the present thesis, we derive error estimates with respect to the discretization parameters (both in space and Fourier variables). To handle the non-dissipative case, we propose a modified version of the Half-Space Matching Method, which is obtained by applying a complex-scaling to the unknowns, in order to recover the L² framework.We then extend the Half-Space Matching Method to scattering problems in infinite 3D elastic plates for applications to Non-Destructive Testing. The additional complexity compared to the 2D case comes from the decomposition on Lamb modes used in the half-plate representations. Due to the bi-orthogonality relation of Lamb modes, we have to consider as unknowns not only the displacement, but also the normal stress on the infinite bands limiting the half-plates. Some theoretical questions concerning this multi-unknown formulation involving the trace and the normal trace are studied in a 2D scalar case. Connections with integral methods are also addressed in the case where the Green's function is known, at least partially in each subdomain.The different versions of the method have been implemented in the library XLiFE++ and numerical results are presented for both 2D and 3D cases.
• Embedded and high-order meshes : two alternatives to linear body-fitted meshes
  
  • Feuillet Rémi
  , 2019. The numerical simulation of complex physical phenomenons usually requires a mesh. In Computational Fluid Dynamics, it consists in representing an object inside a huge control volume. This object is then the subject of some physical study. In general, this object and its bounding box are represented by linear surface meshes and the intermediary zone is filled by a volume mesh. The aim of this thesis is to have a look on two different approaches for representing the object. The first approach called embedded method consist in integrally meshing the bounding box volume without explicitly meshing the object in it. In this case, the presence of the object is implicitly simulated by the CFD solver. The coupling of this method with linear mesh adaptation is in particular discussed.The second approach called high-order method consist on the contrary by increasing the polynomial order of the surface mesh of the object. The first step is therefore to generate a suitable high-order mesh and then to propagate the high-order information in the neighboring volume if necessary. In this context, it is mandatory to make sure that such modifications are valid and then the extension of classic mesh modification techniques has to be considered.
• Asymptotic stability of the multidimensional wave equation coupled with classes of positive-real impedance boundary conditions
  
  • Monteghetti Florian
  • Haine Ghislain
  • Matignon Denis
  Mathematical Control and Related Fields, AIMS, 2019, 9 (4), pp.759-791. This paper proves the asymptotic stability of the multidimensional wave equation posed on a bounded open Lipschitz set, coupled with various classes of positive-real impedance boundary conditions, chosen for their physical relevance: time-delayed, standard diffusive (which includes the Riemann-Liouville fractional integral) and extended diffusive (which includes the Caputo fractional derivative). The method of proof consists in formulating an abstract Cauchy problem on an extended state space using a dissipative realization of the impedance operator, be it finite or infinite-dimensional. The asymptotic stability of the corresponding strongly continuous semigroup is then obtained by verifying the sufficient spectral conditions derived by Arendt and Batty (Trans. Amer. Math. Soc., 306 (1988)) as well as Lyubich and Vũ (Studia Math., 88 (1988)). (10.3934/mcrf.2019049)
  DOI : 10.3934/mcrf.2019049
• Outils mathématiques et algorithmiques pour le calcul scientifique
  
  • Ciarlet Patrick
  • Jamelot Erell
  , 2019, pp.1-287. Ce polycopié correspond aux notes du cours "Calcul Scientifique Parallèle", tel qu'enseigné de 2014 à 2019 par les auteurs. Ce cours fait partie du cursus Modélisation et Simulation du M2 Analyse, Modélisation, Simulation de l'Université Paris-Saclay et du cursus de 3ème année ModSim de l'ENSTA Paris. L'objectif principal est de proposer aux étudiants des outils de calcul scientifique permettant d'appréhender les algorithmes adaptés au calcul parallèle, c'est-à-dire pouvant utiliser plusieurs nœuds de calcul simultanément. On abordera essentiellement le calcul parallèle d'un point de vue méthodologique et/ou algorithmique. A partir d'un problème modèle, on présente un certain nombre d'outils et de méthodes permettant de le résoudre numériquement, et on explique comment on peut adapter au calcul parallèle, c'est-à-dire paralléliser, les algorithmes associés. On évoque, sans les occulter, tous les aspects de la résolution, qu'ils soient abstraits (point de vue mathématique) ; discrétisation (point de vue numérique) ; algorithmique (mise en œuvre). A noter : le point de vue informatique du calcul parallèle est développé dans la documentation en ligne disponible à l'adresse https://ams301.pages.math.cnrs.fr/. Le polycopié est composé de trois parties et d'annexes. Dans la première partie, on rappelle quelques problèmes typiques à traiter. Parmi ces problèmes, on se concentrera sur la résolution de l'équation de diffusion des neutrons : résolution mathématique d'une part, et résolution numérique d'autre part. Pour ce second aspect, on introduit deux méthodes de discrétisation : les différences finies et les éléments finis. Les différences finies donnent lieu à des algorithmes de résolution numérique possédant une structure, on parle d'algorithmes structurés, alors que les éléments finis conduisent en général à des algorithmes non-structurés. Après discrétisation, l'opération fondamentale à réaliser est la résolution d'un système linéaire. La seconde partie se concentre donc sur l'algèbre linéaire numérique : éléments d'algorithmique numérique, les méthodes de résolution directes et itératives, les méthodes de Krylov et la méthode de la puissance itérée. La prise en compte de la structure, ou de l'absence de structure, joue un rôle déterminant dans la résolution parallèle. Enfin la troisième partie est une introduction aux méthodes de décomposition de domaine. Le calcul parallèle est naturellement associé à ces méthodes, car on choisit de découper le problème initial en plusieurs sous-problèmes interagissant entre eux, et on discrétise la seconde instance. On reprend comme exemple l'équation de diffusion des neutrons, discrétisée par la méthode des éléments finis. Après une introduction mathématique, on étudiera pour chaque problème deux méthodes de décomposition de domaine : la méthode de Schwarz et la méthode avec contrainte. On s'aidera de l'analyse numérique pour valider nos modèles décomposés. Les annexes comprennent des rappels en algèbre linéaire, les outils de base pour l'étude et l'approximation de formulations variationnelles en dimension infinie, et enfin quelques outils élémentaires sur les distributions et les espaces fonctionnels de type Sobolev.
• Energy Decay and Stability of a Perfectly Matched Layer For the Wave Equation
  
  • Baffet Daniel Henri
  • Grote Marcus J.
  • Imperiale Sébastien
  • Kachanovska Maryna
  Journal of Scientific Computing, Springer Verlag, 2019. In [25, 26], a PML formulation was proposed for the wave equation in its standard second-order form. Here, energy decay and L 2 stability bounds in two and three space dimensions are rigorously proved both for continuous and discrete formulations. Numerical results validate the theory. (10.1007/s10915-019-01089-9)
  DOI : 10.1007/s10915-019-01089-9
• Mathematical models for dispersive electromagnetic waves
  
  • Cassier Maxence
  • Joly Patrick
  • Kachanovska Maryna
  , 2019.
• Modeling multicrack propagation by the fast multipole symmetric Galerkin BEM
  
  • Dansou Anicet
  • Mouhoubi Saida
  • Chazallon Cyrille
  • Bonnet Marc
  Engineering Analysis with Boundary Elements, Elsevier, 2019, 106, pp.309-319. The Fast Multipole Method coupled with the Symmetric Galerkin BEM is employed in this work to simulate fatigue crack growth. The resulted crack propagation code is accelerated with a fast matrix update, a parallel implementation and a sparse matrix format. By using multiple nodes, this code accommodates also multiple surface-breaking cracks. The numerical tests presented herein allow the propagation of multiple cracks in single or multilayer domains. (10.1016/j.enganabound.2019.05.019)
  DOI : 10.1016/j.enganabound.2019.05.019
• An efficient domain decomposition method with cross-point treatment for Helmholtz problems
  
  • Modave Axel
  • Antoine Xavier
  • Royer Anthony
  • Geuzaine Christophe
  , 2019. The parallel finite-element solution of large-scale time-harmonic scattering problems is addressed with a non-overlapping domain decomposition method (DDM). It is well known that the efficiency of this method strongly depends on the transmission condition enforced on the interfaces between the subdomains. Local conditions based on high-order absorbing boundary conditions (HABCs) are well suited for configurations without cross points (where more than two subdo-mains meet). In this work, we extend this approach to efficiently deal with cross points. Two-dimensional finite-element results are presented.
• Effective models for non-perfectly conducting thin coaxial cables
  
  • Beck Geoffrey
  • Imperiale Sebastien
  • Joly Patrick
  , 2019. Continuing past work on the modelling of coax-ial cables, we investigate the question of the modeling of non-perfectly conducting thin coax-ial cables. Starting from 3D Maxwell's equations , we derive, by asymptotic analysis with respect to the (small) transverse dimension of the cable, a simplified effective 1D model. This model involves a fractional time derivatives that accounts for the so-called skin effects in highly conducting regions.
• Invisible floating objects
  
  • Chesnel Lucas
  • Rihani Mahran
  , 2019. We consider a time-harmonic water waves problem in a 2D waveguide. The geometry is symmetric with respect to an axis orthogonal to the direction of propagation of waves. Moreover, the waveguide contains two floating obstacles separated by a distance L. We study the behaviours of R (the reflection coefficient) and T (the transmission coefficient) as L tends to +∞. From this analysis, we exhibit situations of non reflectivity (R = 0, |T | = 1) or perfect invisibility (R = 0, T = 1). (10.34726/waves2019)
  DOI : 10.34726/waves2019
• Recovering underlying graph for networks of 1D waveguides by reflectometry and transferometry
  
  • Beck Geoffrey
  • Bonnaud Maxime
  • Benoit Jaume
  , 2019. We present a method for blind recovery of network made out of a tree of 1D homogeneous waveguides with the same physical characteristics using reflectogram and transferogram(s).
• Convolution quadrature methods for time-domain scattering from unbounded penetrable interfaces
  
  • Labarca Ignacio
  • Faria Luiz
  • Pérez-Arancibia Carlos
  Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Royal Society, The, 2019, 475 (2227), pp.20190029. This paper presents a class of boundary integral equation methods for the numerical solution of acoustic and electromagnetic time-domain scattering problems in the presence of unbounded penetrable interfaces in two spatial dimensions.The proposed methodology relies on convolution quadrature (CQ) schemes and the recently introduced windowed Green function (WGF) method. As in standard time-domain scattering from bounded obstacles, a CQ method of the user’s choice is used to transform the problem into a finite number of (complex) frequency-domain problems posed, in our case, on the domains containing unbounded penetrable interfaces. Each one of the frequency-domain transmission problems is then formulated as a second-kind integral equation that is effectively reduced to a bounded interface by means of the WGF metho—which introduces errors that decrease super-algebraically fast as the window size increases.The resulting windowed integral equations can then be solved by means of any (accelerated or unaccelerated) off-the-shelf Nyström or boundary element Helmholtz integral equation solvers capable of handling complex wavenumbers with large imaginary part. A high-order Nyström method based on Alpert’s quadraturerules is used here. A variety of CQ schemes and numerical examples, including wave propagation inopen waveguides as well as scattering from multiplelayered media, demonstrate the capabilities of the proposed approach. (10.1098/rspa.2019.0029)
  DOI : 10.1098/rspa.2019.0029
• Planewave Density Interpolation Methods for 3D Helmholtz Boundary Integral Equations
  
  • Pérez-Arancibia Carlos
  • Turc Catalin
  • Faria Luiz
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2019, 41 (4), pp.A2088-A2116. This paper introduce planewave density interpolation methods for the regularization of weakly singular, strongly singular, hypersingular and nearly singular integral kernels present in 3D Helmholtz surface layer potentials and associated integral operators. Relying on Green’s third identity and pointwise interpolation of density functions in the form of planewaves, these methods allow layer potentials and integral operators to be expressed in terms of integrand functions that remain smooth (at least bounded) regardless the location of the target point relative to the surface sources. Common challenging integrals that arise in both Nyström and boundary element discretization of boundary integral equation, can then be numerically evalu-ated by standard quadrature rules that are irrespective of the kernel singularity. Closed-formand purely numerical planewave density interpolation procedures are presented in this paper, which are used in conjunction with Chebyshev-based Nyström and Galerkin boundary element methods. A variety of numerical examples—including problems of acoustic scattering involving multiple touching and even intersecting obstacles, demonstrate the capabilities of the proposed technique. (10.1137/19M1239866)
  DOI : 10.1137/19M1239866
• Impact of the Green function in acoustic analogies for flow noise predictions at low Mach number
  
  • Trafny Nicolas
  • Serre Gilles
  • Cotté Benjamin
  • Mercier Jean-François
  , 2019. It is known that hydrodynamic noise can be a major contribution to the total sound radiated by a ship. It is in part attributed to the interaction between turbulent eddies with appendages and marine propeller blades. Because hydrodynamics is associated with very low Mach numbers, direct noise computation methods are too expensive. Other approaches must be chosen, based on acoustic analogies which consist first in modeling the incompressible turbulent flow and then in computing the noise radiated by this flow. We focus on Lighthill's wave equation, solved using the free space Green function or a tailored Green's function in presence of an arbitrary geometry. Unlike many studies from the literature where the impact of the chosen turbulent model is evaluated over a semi-infinite plate, the objective of this study is to evaluate the impact of the chosen Green function on the predicted broadband flow noise for a fixed semi-empirical turbulence model. The impact of the chosen tailored Green function on the radiated noise spectra and directivity diagrams is evaluated considering various analytical and numerical tailored Green's functions.
• Wave propagation in fractal trees. Mathematical and Numerical Issues
  
  • Joly Patrick
  • Kachanovska Maryna
  • Semin Adrien
  Networks and Heterogeneous Media, American Institute of Mathematical Sciences, 2019, 14 (2). We propose and analyze a mathematical model for wave propagation in infinite trees with self-similar structure at infinity. This emphasis is put on the construction and approximation of transparent boundary conditions. The performance of the constructed boundary conditions is then illustrated by numerical experiments. (10.3934/nhm.2019010)
  DOI : 10.3934/nhm.2019010
• Modélisation de l'interaction fluide-structure lors d'une explosion sous-marine lointaine par méthode des éléments de frontière accélérée
  
  • Mavaleix-Marchessoux Damien
  • Chaillat Stéphanie
  • Leblé Bruno
  • Bonnet Marc
  , 2019. Cette contribution concerne la modélisation de l’impact de l’onde de choc d’une explosion sous-marine sur une structure située loin de la source, en eau profonde. Pour rendre compte du phénomène, un couplage est mis en place : les équations structures sont résolues en éléments finis, tandis que la partie fluide est traitée en éléments de frontière. La présente contribution met en avant la résolution côté fluide, avec l’extension de la méthode des éléments de frontière, accélérée par la méthode multipôle rapide, au domaine temporel par Convolution Quadrature Method.
• Recent developments on adaptive fast Boundary Element Methods to model elastic wave propagation in sedimentary basins
  
  • Amlani Faisal
  • Chaillat Stéphanie
  • Loseille Adrien
  , 2019. The main advantage of the Boundary Element Method (BEM) is that only the domain boundaries are discretized. It is thus well-suited to study site effects. This advantage is offset by the full BEM matrix. In the last couple of years, fast BEMs have been proposed to overcome this drawback. If the efficiency of fast BEMs has been demonstrated, the iteration count becomes now the main limitation to consider realistic problems. Mesh adaptation is an additional technique to reduce the computational cost and number of iterations of the BEM. In this contribution, we give an overview of recent works to speed-up fast BEMs, i.e. an anisotropic metric-based mesh adaptation technic.
• An efficient domain decomposition method with cross-point treatment for Helmholtz problems
  
  • Modave Axel
  • Antoine Xavier
  • Geuzaine Christophe
  , 2019. The parallel finite-element solution of large-scale time-harmonic scattering problems is addressed with a non-overlapping domain decomposition method. The efficiency of this method strongly depends on the transmission condition enforced on the interfaces between the subdomains. Conditions based on high-order local absorbing boundary conditions have proved well suited for configurations without cross points (i.e. points where more than two subdomains meet). In this work, we extend this approach to efficiently deal with cross points. Two-dimensional numerical results are presented.
• A parallel boundary element method code to simulate multicracked structures
  
  • Dansou Anicet
  • Mouhoubi Saïda
  • Chazallon Cyrille
  • Bonnet Marc
  , 2019. This paper presents the parallel version of a boundary element method code to simulatecrack problems. The code is based on the symmetric Galerkin boundary element method and takes alsoadvantage of the fast multipole method. The time-consuming phases of the code are accelerated by ashared memory parallelization using OpenMP. The performance of the new code is shown through manysimulations including crack problems involving thousands of cracks.
• Contact élastoplastique : équations intégrales accélérées par une approche Fourier
  
  • Frérot Lucas
  • Bonnet Marc
  • Molinari Jean-François
  • Anciaux Guillaume
  , 2019. Une approche par équations intégrales volumiques du problème de contact élastoplastique périodique est présentée. Elle repose sur la formulation des fonctions de Green nécessaires au calcul des opérateurs intégraux directement dans l’espace de Fourier. cela permet d’utiliser l’algorithme de la transformée de Fourier rapide pour l’application des opérateurs intégraux, d’éviter le stockage coûteux des fonctions de Green qui peuvent être évaluées à la volée et d’optimiser l’application des opérateurs intégraux dans la direction non transformée via l’exploitation de la structure des fonctions de Green dans l’espace de Fourier. Ces avancées permettent une exploitation plus efficace des ressources de calcul et la simulation du contact élastoplastique de surfaces rugueuses, dont les caractéristiques influencent de nombreux phénomènes, tels que le frottement ou l’usure.
• Numerical analysis of the Half-Space Matching method with Robin traces on a convex polygonal scatterer
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Fliss Sonia
  • Tjandrawidjaja Yohanes
  , 2019, 24. We consider the 2D Helmholtz equation with a complex wavenumber in the exterior of a convex polygonal obstacle, with a Robin type boundary condition. Using the principle of the Half-Space Matching method, the problem is formulated as a system of coupled Fourier-integral equations, the unknowns being the Robin traces on the infinite straight lines supported by the edges of the polygon. We prove that this system is a Fredholm equation of the second kind, in an $L^2$ functional framework. The truncation of the Fourier integrals and the finite element approximation of the corresponding numerical method are also analyzed. The theoretical results are supported by various numerical experiments.
• An inverse obstacle problem for the wave equation in a finite time domain
  
  • Bourgeois Laurent
  • Ponomarev Dmitry
  • Dardé Jérémi
  Inverse Problems and Imaging, AIMS American Institute of Mathematical Sciences, 2019, 19 (2), pp.377-400. We consider an inverse obstacle problem for the acoustic transient wave equation. More precisely, we wish to reconstruct an obstacle characterized by a Dirichlet boundary condition from lateral Cauchy data given on a subpart of the boundary of the domain and over a finite interval of time. We first give a proof of uniqueness for that problem and then propose an " exterior approach " based on a mixed formulation of quasi-reversibility and a level set method in order to actually solve the problem. Some 2D numerical experiments are provided to show that our approach is effective. (10.3934/ipi.2019019)
  DOI : 10.3934/ipi.2019019
• Enhanced resonance of sparse arrays of Helmholtz resonators—Application to perfect absorption
  
  • Maurel Agnès
  • Mercier Jean-François
  • Pham Trung Kien
  • Marigo J.-J
  • Ourir Abdelwaheb
  Journal of the Acoustical Society of America, Acoustical Society of America, 2019, 145 (4), pp.2552-2560. We inspect the influence of the spacing on the resonance of a periodic arrangement of Helmholtz resonators. An effective problem is used which captures accurately the properties of the resonant array within a large range of frequency, and whose simplified version leaves us with an impedance condition. It is shown that the strength of the resonance is enhanced when the array becomes sparser. This degree of freedom on the radiative damping is of particular interest since it does not affect the resonance frequency nor the damping due to losses within each resonator; besides, it does not affect the total thickness of the array. We show that it can be used for the design of a perfect absorbing walls. (10.1121/1.5098948)
  DOI : 10.1121/1.5098948
• Contributions to the modelling of acoustic and elastic wave propagation in large-scale domains with boundary element methods
  
  • Chaillat Stéphanie
  , 2019. The main advantage of the BEM is that only the domain boundaries (and possibly interfaces) are discretized leading to a drastic reduction of the total number of degrees of freedom. In traditional BE implementation the dimensional advantage with respect to domain discretization methods is offset by the fully-populated nature of the BEM matrix, with setup and solution times rapidly increasing with the problem size. In the last couple of years, fast BEMs have been proposed to overcome the drawback of the fully populated matrix. The Fast Multipole Method (FMM) is a fast, reliable and approximate method to compute the linear integral operator and is defined together with an iterative solver. The efficiency of the method has been demonstrated for 3D wave problems. However, the iteration count becomes the main limitation to consider realistic problems. Other accelerated BEMs are based on hierarchical matrices. When used in conjunction with an efficient rank revealing algorithm, it leads to a data-sparse and memory efficient approximation of the original matrix. Contrary to the FM-BEM it is a purely algebraic tool which does not require a priori knowledge of the closed-form expression of the fundamental solutions and it is possible to define iterative or direct solvers. Mesh adaptation is an additional technique to reduce the computational cost of the BEM. The principle is to optimize (or at least improve) the positioning of a given number of degrees of freedom on the geometry of the obstacle, in order to yield simulations with superior accuracy compared to those obtained via the use of uniform meshes. If an extensive literature is available for volume methods, much less attention has been devoted to BEMs. In this document, I give an overview of recent works to speed-up the solution of 3D acoustic and elastodynamic BEMs.