Publications

2019

• 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.
• 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.
• 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.
• 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
• 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
• 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.
• Forward Feynman-Kac type representation for semilinear nonconservative Partial Differential Equations
  
  • Lecavil Anthony
  • Oudjane Nadia
  • Russo Francesco
  Stochastics: An International Journal of Probability and Stochastic Processes, Taylor & Francis: STM, Behavioural Science and Public Health Titles, 2019, 91 (8). We propose a nonlinear forward Feynman-Kac type equation, which represents the solution of a non-conservative semilinear parabolic Partial Differential Equations (PDE). We show in particular existence and uniqueness. The solution of that type of equation can be approached via a weighted particle system. (10.1080/17442508.2019.1594809)
  DOI : 10.1080/17442508.2019.1594809
• Pareto Front Characterization for Multiobjective Optimal Control Problems Using Hamilton--Jacobi Approach
  
  • Desilles Anna
  • Zidani Hasnaa
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2019, 57 (6), pp.3884-3910. (10.1137/18M1176993)
  DOI : 10.1137/18M1176993
• On the convergence in $H^1$-norm for the fractional Laplacian
  
  • Borthagaray Juan Pablo
  • Ciarlet Patrick
  SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2019, 57, pp.1723-1743. We consider the numerical solution of the fractional Laplacian of index $s \in (1/2, 1)$ in a bounded domain $\Omega$ with homogeneous boundary conditions. Its solution a priori belongs to the fractional order Sobolev space $\widetilde{H}^s(\Omega)$. For the Dirichlet problem and under suitable assumptions on the data, it can be shown that its solution is also in $H^1(\Omega)$. In this case, if one uses the standard Lagrange finite element to discretize the problem, then both the exact and the computed solution belong to $H^1(\Omega)$. A natural question is then whether one can obtain error estimates in $H^1(\Omega)$-norm, in addition to the classical ones that can be derived in the $\widetilde{H}^s(\Omega)$ energy norm. We address this issue, and in particular we derive error estimates for the Lagrange finite element solutions on both quasi-uniform and graded meshes. (10.1137/18M1221436)
  DOI : 10.1137/18M1221436
• Optimality and modularity in human movement: from optimal control to muscle synergies
  
  • Berret B.
  • Delis Ioannis
  • Gaveau Jeremie
  • Jean Frédéric
  , 2019, 124. In this chapter, we review recent work related to the optimal and modular control hypotheses for human movement. Optimal control theory is often thought to imply that the brain continuously computes global optima for each motor task it faces. Modular control theory typically assumes that the brain explicitly stores genuine synergies in specific neural circuits whose combined recruitment yields task-effective motor inputs to muscles. Put this way, these two influential motor control theories are pushed to extreme positions. A more nuanced view, framed within Marr’s tri-level taxonomy of a computational theory of movement neuroscience, is discussed here. We argue that optimal control is best viewed as helping to understand “why” certain movements are preferred over others but does not say much about how the brain would practically trigger optimal strategies. We also argue that dimensionality reduction found in muscle activities may be a by-product of optimality and cannot be attributed to neurally hardwired synergies stricto sensu, in particular when the synergies are extracted from simple factorization algorithms applied to electromyographic data; their putative nature is indeed strongly dictated by the methodology itself. Hence, more modeling work is required to critically test the modularity hypothesis and assess its potential neural origins. We propose that an adequate mathematical formulation of hierarchical motor control could help to bridge the gap between optimality and modularity, thereby accounting for the most appealing aspects of the human motor controller that robotic controllers would like to mimic: rapidity, efficiency, and robustness. (10.1007/978-3-319-93870-7_6)
  DOI : 10.1007/978-3-319-93870-7_6
• On projective and affine equivalence of sub-Riemannian metrics
  
  • Jean Frédéric
  • Maslovskaya Sofya
  • Zelenko I.
  Geometriae Dedicata, Springer Verlag, 2019, 203 (1), pp.279-319. Consider a smooth manifold $M$ equipped with a bracket generating distribution $D$. Two sub-Riemannian metrics on $(M,D)$ are said to be projectively (resp. affinely) equivalent if they have the same geodesics up to reparameterization (resp. up to affine reparameterization). A sub-Riemannian metric $g$ is called rigid (resp. conformally rigid) with respect to projective/affine equivalence, if any sub-Riemannian metric which is projectively/affinely equivalent to $g$ is constantly proportional to $g$ (resp. conformal to $g$). In the Riemannian case the local classification of projectively and affinely equivalent metrics is classical (Levi-Civita, Eisenhart). In particular, a Riemannian metric which is not rigid satisfies the following two special properties: its geodesic flow possesses nontrivial integrals and the metric induces certain canonical product structure on the ambient manifold. These classification results were extended to contact and quasi-contact distributions by Zelenko. Our general goal is to extend these results to arbitrary sub-Riemannian manifolds, and we establish two types of results toward this goal: if a sub-Riemannian metric is not projectively conformally rigid, then, first, its flow of normal extremals has at least one nontrivial integral quadratic on the fibers of the cotangent bundle and, second, the nilpotent approximation of the underlying distribution at any point admits a product structure. As a consequence we obtain two types of genericity results: first, we show that a generic sub-Riemannian metric on a fixed pair $(M,D)$ is projectively conformally rigid. Second, we prove that, except for special pairs $(m,n)$, every sub-Riemannian metric on a rank $m$ generic distribution in an $n$-dimensional manifold is projectively conformally rigid. For the affine equivalence in both genericity results conformal rigidity can be replaced by usual rigidity. (10.1007/s10711-019-00437-1)
  DOI : 10.1007/s10711-019-00437-1