Publications

2019

• 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.
• 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.
• Novel Approach Towards Global Optimality of Optimal Power Flow Using Quadratic Convex Optimization
  
  • Godard Hadrien
  • Elloumi Sourour
  • Lambert Amélie
  • Maeght Jean
  • Ruiz Manuel
  , 2019. Optimal Power Flow (OPF) can be modeled as a non-convex Quadratically Constrained Quadratic Program (QCQP). Our purpose is to solve OPF to global optimality. To this end, we specialize the Mixed-Integer Quadratic Convex Reformulation method (MIQCR) to (OPF). This is a method in two steps. First, a Semi-Definite Programming (SDP) relaxation of (OPF) is solved. Then the optimal dual variables of this relaxation are used to reformulate OPF into an equivalent new quadratic program, where all the non-convexity is moved to one additional constraint. In the second step, this reformulation is solved within a branch-and-bound algorithm, where at each node a quadratic and convex relaxation of the reformulated problem, obtained by relaxing the non-convex added constraint, is solved. The key point of our approach is that the lower bound at the root node of the branch-and-bound tree is equal to the SDP relaxation value. We test this method on several OPF cases, from two-bus networks to more-than-a-thousand-buses networks from the MAT-POWER repository. Our first results are very encouraging. (10.1109/CoDIT.2019.8820584)
  DOI : 10.1109/CoDIT.2019.8820584
• Semidefinite programming relaxations through quadratic reformulation for box-constrained polynomial optimization problems
  
  • Elloumi Sourour
  • Lambert Amélie
  • Lazare Arnaud
  , 2019, pp.1498-1503. In this paper we introduce new semidefinite programming relaxations to box-constrained polynomial optimization programs (P). For this, we first reformu-late (P) into a quadratic program. More precisely, we recursively reduce the degree of (P) to two by substituting the product of two variables by a new one. We obtain a quadratically constrained quadratic program. We build a first immediate SDP relaxation in the dimension of the total number of variables. We then strengthen the SDP relaxation by use of valid constraints that follow from the quadratization. We finally show the tightness of our relaxations through several experiments on box polynomial instances. (10.1109/CoDIT.2019.8820690)
  DOI : 10.1109/CoDIT.2019.8820690
• Design of robust networks : application to the design of wind farm cabling networks
  
  • Ridremont Thomas
  , 2019. Nowadays, the design of networks has become a decisive problematic which appears in many fields such as transport or energy. In particular, it has become necessary and important to optimize the way in which networks used to produce, collect or transport energy are designed. We focus in this thesis on electricity produced through wind farms. The production of energy by wind turbines appears more than ever like a good alternative to the electrical production of thermal or nuclear power plants.We focus in this thesis on the design of the cabling network which allows to collect and route the energy from the wind turbines to a sub-station, linking the wind farm to the electrical network. In this problem, we know the location of each wind turbine of the farm and the one of the sub-station. We also know the location of possible inter-connection nodes which allow to connect different cables between them. Each wind turbine produces a known quantity of energy and with each cable are associated a cost and a capacity (the maximum amount of energy that can be routed through this cable). The optimizationproblem that we consider is to select a set of cables of minimum cost such that the energy produced from the wind turbines can be routed to the sub-station in the network induced by this set of cables, without exceeding the capacity of each cable. We focus on cabling networks resilient to breakdowns.
• The status of isochrony in the formation and evolution of self-gravitating systems
  
  • Simon-Petit Alicia
  • Perez Jérôme
  • Plum Guillaume
  Monthly Notices of the Royal Astronomical Society, Oxford University Press (OUP): Policy P - Oxford Open Option A, 2019, 484 (4), pp.Pages 4963–4971. In the potential theory, isochrony was introduced by Michel Hénon in 1959 to characterize astrophysical observations of some globular clusters. Today, Michel Henon's isochrone potential is mainly used for his integrable property in numerical simulations, but is generally not really known. In a recent paper [29], we have presented new fundamental and theoretical results about isochrony that have particular importance in self-gravitating dynamics and which are detailed in this paper. In particular, new characterization of the isochrone state has been proposed which are investigated in order to analyze the product of the fast relaxation of a self-gravitating system. The general paradigm consists in considering that this product is a lowered isothermal sphere (King Model). By a detailed numerical study we show that this paradigm fails when the isochrone model succeeds in reproducing the quasi-equilibrium state obtained just after fast relaxation. (10.1093/mnras/stz351)
  DOI : 10.1093/mnras/stz351
• 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.
• Stochastic Optimization of Braking Energy Storage and Ventilation in a Subway Station
  
  • Rigaut Tristan
  • Carpentier Pierre
  • Chancelier Jean-Philippe
  • de Lara Michel
  • Waeytens Julien
  IEEE Transactions on Power Systems, Institute of Electrical and Electronics Engineers, 2019, 34 (2), pp.1256-1263. In the Paris subway system, stations represent about one third of the overall energy consumption. Within stations, ventilation is among the top consuming devices; it is operated at maximum airflow all day long, for air quality reasons. In this paper, we present a concept of energy system that displays comparable air quality while consuming much less energy. The system comprises a battery that makes it possible to recover the trains braking energy, arriving under the form of erratic and strong peaks. We propose an energy management system (EMS) that, at short time scale, controls energy flows and ventilation airflow. By using proper optimization algorithms, we manage to match supply with demand, while minimizing energy daily costs. For this purpose, we have designed algorithms that take into account the braking variability. They are based on the so-called Stochastic Dynamic Programming (SDP) mathematical framework. We fairly compare SDP based algorithms with the widespread Model Predictive Control (MPC) ones. First, both SDP and MPC yield energy/money operating savings of the order of one third, compared to the current management without battery. Second, depending on the specific design, we observe that SDP outperforms MPC by a few percent, with an easier online numerical implementation. (10.1109/TPWRS.2018.2873919)
  DOI : 10.1109/TPWRS.2018.2873919
• Réduction des coûts d’adaptation d’un plan de transport ferroviaire à l’aide de solutions adaptative
  
  • Lucas Rémi
  • Alès Zacharie
  • Elloumi Sourour
  • Ramond François
  , 2019.
• Calcul des dates d'atterrissage d'une séquence d'avions pour des fonctions de coût convexes et affines par morceaux
  
  • Diamantini Maurice
  • Faye Alain
  • Khamphousone Julien
  , 2019. Ce papier étudie le problème du séquencement des avions lors de leur arrivée à l'aéroport, problème connu dans la littérature sous le nom de Aircraft Landing Problem [1]. Il s'agit de séquencer les avions arrivant sur la piste d'atterrissage tout en respectant des conditions de sécurité entre les avions. Les avions créent des turbulences et une durée minimum entre deux atterrissages successifs doit être respectée. La durée de séparation dépend du type des avions qui se suivent. Un petit avion qui atterrit derrière un gros avion doit attendre plus longtemps qu'un gros avion qui atterrit à la suite d'un petit. Chaque avion $i$ peut atterrir dans une certaine fenêtre de temps $[E_i , L_i ]$. $E_i$ est la date au plus tôt à laquelle l'avion peut atterrir, $L_i$ est la date au plus tard. Dans cette fenêtre, $T_i$ est la date préférée d'atterrissage qui correspond à la date à laquelle l'avion arriverait sur la piste s'il allait à sa vitesse de croisière. Si l'avion $i$ était seul il atterrirait à la date $T_i$ mais en présence d'autres avions un arbitrage est nécessaire. Les avions doivent soit accélérer pour atterrir plus tôt ou au contraire ralentir voire faire des boucles pour arriver plus tard afin de respecter les contraintes de sécurité. Une déviation par rapport à la date préférée d'atterrissage engendre un coût. Dans la littérature on considère généralement qu'une avance ou un retard engendre un coût linéaire en fonction de l'écartement à la date préférée d'atterrissage. L'objectif est de minimiser le coût total de déviation. Sur chaque piste, le problème se décompose en deux phases : d'abord choisir l'ordre des avions et ensuite calculer les dates d'atterrissage. Ce dernier problème peut se résoudre par un PL (Programme Linéaire) [3, 5, 6, 7]. A. Faye [4] propose un algorithme de complexité quadratique en fonction du nombre d'avions. Cependant, un coût linéaire peut s'avérer assez éloigné des coûts réels encourus. Par exemple, un retard peut avoir des conséquences sur les passagers en correspondance et sur les vols ultérieurs sur lesquels le personnel de bord devra prendre place. On conçoit facilement que plus le retard est grand, plus les complications sont nombreuses et que plus l'impact sur le coût s'accroît. Il est donc légitime de modéliser la fonction coût par une fonction convexe et affine par morceaux centrée en la date préférée d'atterrissage et avec des pentes croissantes de part et d'autre de cette date. Pour des raisons d'équité entre compagnies aériennes, une fonction de ce type a été introduite par Soomer et Franx [6]. Pour un ordre fixé des avions, le coût total était calculé par un PL. Ici, nous proposons un algorithme polynomial dont la complexité dépend à la fois du nombre d'avions et du nombre de pentes de la fonction de coût d'un avion. Ainsi, si n est le nombre d'avions et si b est le nombre maximum de pentes que peut comporter le coût d'un avion, la complexité de l'algorithme est $O(n^2 b^2)$. Cet algorithme est basé sur la programmation dynamique et est une généralisation de l'algorithme proposé dans [4].
• Galaxies et amas globulaires : une diversité régie par l’entropie
  
  • Perez Jérôme
  La Recherche, Sciences et avenir, 2019 (N° 544), pp.P. 45-47. Pour comprendre la structure des grands rassemblements d’étoiles que sont les galaxies et les amas globulaires, la physique statistique est à la manœuvre. Mais il est nécessaire de prendre en compte les spécificités de ces systèmes auto-gravitants, ainsi que leur place dans l’évolution de l’Univers.
• Lagrange et la méthode analytique
  
  • Perez Jérôme
  Bibliothèque Tangente, Editions Pôle Paris, 2019, Hors série 69. A sa mort, Isaac Newton est fier d’avoir trouvé une méthode pour résoudre des problèmes de philosophie naturelle, mais il est conscient de ses limites. La trajectoire de la Lune n’est que très approximativement décrite par le problème des deux corps. Il reviendra à Joseph-Louis Lagrange de réaliser une percée spectaculaire.
• La méthode synthétique de Newton
  
  • Perez Jérôme
  Bibliothèque Tangente, Editions Pôle Paris, 2019 (Hors Série 69). Comment s’effectue le mouvement de la Terre, de masse m, autour du Soleil, de masse M, sous l’effet unique de la gravitation ? Pour répondre à cette question autrement que par des affirmations philosophiques, Newton propose une méthode synthétique inspirée de la description du mouvement.
• Une science en mouvement
  
  • Perez Jérôme
  Bibliothèque Tangente, Editions Pôle Paris, 2019, Hors série 69. Ni les mathématiques, ni la physique ne sont des sciences figées dans le temps ! Jusqu’à récemment, les deux disciplines allaient main dans la main, proposant une philosophie naturelle. Elles essayaient de révéler, de décrire et d’expliquer les lois cachées de la nature.
• Wave propagation in periodic media : mathematical analysis and numerical simulation
  
  • Fliss Sonia
  , 2019.
• Path-dependent Martingale Problems and Additive Functionals
  
  • Barrasso Adrien
  • Russo Francesco
  Stochastics and Dynamics, World Scientific Publishing, 2019, 19 (4), pp.1950027. The paper introduces and investigates the natural extension to the path-dependent setup of the usual concept of canonical Markov class introduced by Dynkin and which is at the basis of the theory of Markov processes. That extension, indexed by starting paths rather than starting points will be called path-dependent canonical class. Associated with this is the generalization of the notions of semi-group and of additive functionals to the path-dependent framework. A typical example of such family is constituted by the laws $({\mathbb P}^{s,η})_{(s,\eta) \in {\mathbb R} \times \Omega}$ , where for fixed time s and fixed path η defined on [0, s], $({\mathbb P}^{s,η})_{(s,\eta) \in {\mathbb R} \times \Omega}$ being the (unique) solution of a path-dependent martingale problem or more specifically a weak solution of a path-dependent SDE with jumps, with initial path η. In a companion paper we apply those results to study path-dependent analysis problems associated with BSDEs. (10.1142/S0219493719500278)
  DOI : 10.1142/S0219493719500278
• Harmonic density interpolation methods for high-order evaluation of Laplace layer potentials in 2D and 3D
  
  • Pérez-Arancibia Carlos
  • Faria Luiz
  • Turc Catalin
  Journal of Computational Physics, Elsevier, 2019, 376, pp.411-434. We present an effective harmonic density interpolation method for the numerical evaluation of singular and nearly singular Laplace boundary integral operators and layer potentials in two and three spatial dimensions. The method relies on the use of Green’s third identity and local Taylor-like interpolations of density functions in terms of harmonic polynomials. The proposed technique effectively regularizes the singularities present in boundary integral operators and layer potentials, and recasts the latter in terms of integrands that are bounded or even more regular, depending on the order of the density interpolation. The resulting boundary integral scan then be easily, accurately, and inexpensively evaluated by means of standard quadrature rules. A variety of numerical examples demonstrate the effectiveness of the technique when used in conjunction with the classical trapezoidal rule (to integrate over smooth curves) in two-dimensions, and with a Chebyshev-type quadrature rule (to integrate over surfaces given as unions of non-overlapping quadrilateral patches) in three-dimensions. (10.1016/j.jcp.2018.10.002)
  DOI : 10.1016/j.jcp.2018.10.002
• Analysis of topological derivative as a tool for qualitative identification
  
  • Bonnet Marc
  • Cakoni Fioralba
  Inverse Problems, IOP Publishing, 2019, 35 (104007). The concept of topological derivative has proved effective as a qualitative inversion tool for a wave-based identification of finite-sized objects. Although for the most part, this approach remains based on a heuristic interpretation of the topological derivative, a first attempt toward its mathematical justification was done in Bellis et al. (Inverse Problems 29:075012, 2013) for the case of isotropic media with far field data and inhomogeneous refraction index. Our paper extends the analysis there to the case of anisotropic scatterers and background with near field data. Topological derivative-based imaging functional is analyzed using a suitable factorization of the near fields, which became achievable thanks to a new volume integral formulation recently obtained in Bonnet (J. Integral Equ. Appl. 29:271-295, 2017). Our results include justification of sign heuristics for the topological derivative in the isotropic case with jump in the main operator and for some cases of anisotropic media, as well as verifying its decaying property in the isotropic case with near field spherical measurements configuration situated far enough from the probing region. (10.1088/1361-6420/ab0b67)
  DOI : 10.1088/1361-6420/ab0b67
• A Fourier-accelerated volume integral method for elastoplastic contact
  
  • Frérot Lucas
  • Bonnet Marc
  • Molinari Jean-François
  • Anciaux Guillaume
  Computer Methods in Applied Mechanics and Engineering, Elsevier, 2019, 351, pp.951-976. The contact of solids with rough surfaces plays a fundamental role in physical phenomena such as friction, wear, sealing, and thermal transfer. However, its simulation is a challenging problem due to surface asperities covering a wide range of length-scales. In addition, non-linear local processes, such as plasticity, are expected to occur even at the lightest loads. In this context, robust and efficient computational approaches are required. We therefore present a novel numerical method, based on integral equations, capable of handling the large discretization requirements of real rough surfaces as well as the non-linear plastic flow occurring below and at the contacting asperities. This method is based on a new derivation of the Mindlin fundamental solution in Fourier space, which leverages the computational efficiency of the fast Fourier transform. The use of this Mindlin solution allows a dramatic reduction of the memory in-print (as the Fourier coefficients are computed on-the-fly), a reduction of the discretization error, and the exploitation of the structure of the functions to speed up computation of the integral operators. We validate our method against an elastic-plastic FEM Hertz normal contact simulation and showcase its ability to simulate contact of rough surfaces with plastic flow. (10.1016/j.cma.2019.04.006)
  DOI : 10.1016/j.cma.2019.04.006
• An efficient preconditioner for adaptive Fast Multipole accelerated Boundary Element Methods to model time-harmonic 3D wave propagation
  
  • Amlani Faisal
  • Chaillat Stéphanie
  • Loseille Adrien
  Computer Methods in Applied Mechanics and Engineering, Elsevier, 2019, 352 (1), pp.189-210. This paper presents an efficient algebraic preconditioner to speed up the convergence of Fast Multipole accelerated Boundary Element Methods (FM-BEMs) in the context of time-harmonic 3D wave propagation problems and in particular the case of highly non-uniform discretizations. Such configurations are produced by a recently-developed anisotropic mesh adaptation procedure that is independent of partial differential equation and integral equation. The new preconditioning methodology exploits a complement between fast BEMs by using two nested GMRES algorithms and rapid matrix-vector calculations. The fast inner iterations are evaluated by a coarse hierarchical matrix (H-matrix) representation of the BEM system. These inner iterations produce a preconditioner for FM-BEM solvers. It drastically reduces the number of outer GMRES iterations. Numerical experiments demonstrate significant speedups over non-preconditioned solvers for complex geometries and meshes specifically adapted to capture anisotropic features of a solution, including discontinuities arising from corners and edges. (10.1016/j.cma.2019.04.026)
  DOI : 10.1016/j.cma.2019.04.026