Publications

2017

• Topological derivatives of leading-and second-order homogenized coefficients in bi-periodic media
  
  • Cornaggia Rémi
  • Guzina Bojan B
  • Bonnet Marc
  , 2017. We derive the topological derivatives of the homogenized coefficients associated to a periodic material, with respect of the small size of a penetrable inhomogeneity introduced in the unit cell that defines such material. In the context of an-tiplane elasticity, this work extends existing results to (i) time-harmonic wave equation and (ii) second-order homogenized coefficients, whose contribution reflects the dispersive behavior of the material.
• A nodal discontinuous Galerkin method with high-order absorbing boundary conditions and corner/edge compatibility
  
  • Modave Axel
  • Atle Andreas
  • Chan Jesse
  • Warburton Tim
  , 2017. We present the coupling of a nodal discontinu-ous Galerkin (DG) scheme with high-order absorbing boundary conditions (HABCs) for the simulation of transient wave phenomena. The HABCs are prescribed on the faces of a cuboidal domain in order to simulate infinite space. To preserve accuracy at the corners and the edges of the domain, novel compatibility conditions are derived. The method is validated using 3D computational results.
• Asymptotic analysis for criticality assessment of defects in mechanical structures
  
  • Marenić Eduard
  • Brancherie Delphine
  • Bonnet Marc
  , 2017. The presented work is a step towards designing a numerical strategy capable of assessing the nocivity of a small defect in terms of its size and position in the structure with low computational cost, using only a mesh of the defect-free reference structure. The proposed strategy would allow to assess the criticality of defects by introducing trial micro-defects with varying positions, sizes and me- chanical properties. The main focus of the this work is to present two computational scenarios allowing to efficiently evaluate criticality considering the effect of either a fixed flaw on a region of interest or varying flaws on a fixed evaluation point.
• Calcul des opérateurs d'impédance en Interaction Sol-Structure: méthode éléments de frontière accélérée par méthode multipôle rapide
  
  • Adnani Zouhair
  • Chaillat Stéphanie
  • Bonnet Marc
  • Nieto Ferro Alex
  • Greffet Nicolas
  , 2017. Les effets de site, qu’ils soient d’origine topographique ou lithologique, influencent la propagation des ondes sismiques et peuvent provoquer une amplification ou atténuation du mouvement sismique, ainsi que la modification de son spectre. Ce travail concerne le développement d’une stratégie de calcul numérique pour la prise en compte des effets de sites dans les calculs d’Interaction Sol-Structure. Il repose sur une nouvelle stratégie de couplage des éléments finis aux éléments de frontière accélérés par la méthode multipôle rapide.
• Magnetization moment recovery using Kelvin transformation and Fourier analysis
  
  • Baratchart Laurent
  • Leblond Juliette
  • Lima Eduardo Andrade
  • Ponomarev Dmitry
  , 2017. In the present work, we consider a magnetization moment recovery problem, that is finding integral of the vector function (over its compact support) whose divergence constitutes a source term in the Poisson equation. We outline derivation of explicit asymptotic formulas for estimation of the net magnetization moment vector of the sample in terms of partial data for the vertical component of the magnetic field measured in the plane above it. For this purpose, two methods have been developed: the first one is based on approximate projections onto spherical harmonics in Kelvin domain while the second stems from analysis in Fourier domain following asymptotic continuation of the data. Recovery results obtained by both methods agree and are illustrated numerically by plotting formulas for net moment components with respect to the size of the measurement area.
• Algorithms for job scheduling problems with distinct time windows and general earliness/tardiness penalties
  
  • Alès Zacharie
  • Rosa Bruno Ferreira
  • Souza Marcone Jamilson Freitas
  • de Souza Sérgio Ricardo
  • de França Filho Moacir Felizardo
  • Michelon Philippe Yves Paul
  Computers and Operations Research, Elsevier, 2017, 81, pp.203-215. (10.1016/j.cor.2016.12.024)
  DOI : 10.1016/j.cor.2016.12.024
• Using a Conic Bundle Method to Accelerate Both Phases of a Quadratic Convex Reformulation
  
  • Billionnet Alain
  • Elloumi Sourour
  • Lambert Amélie
  • Wiegele Angelika
  INFORMS Journal on Computing, Institute for Operations Research and the Management Sciences (INFORMS), 2017, 29 (2), pp.318 - 331. (10.1287/ijoc.2016.0731)
  DOI : 10.1287/ijoc.2016.0731
• On the Approximation of Electromagnetic Fields by Edge Finite Elements. Part 2: A Heterogeneous Multiscale Method for Maxwell's equations
  
  • Ciarlet Patrick
  • Fliss Sonia
  • Stohrer Christian
  Computers & Mathematics with Applications, Elsevier, 2017, 73 (9), pp.1900-1919. In the second part of this series of papers we consider highly oscillatory media. In this situation, the need for a triangulation that resolves all microscopic details of the medium makes standard edge finite elements impractical because of the resulting tremendous computational load. On the other hand, undersampling by using a coarse mesh might lead to inaccurate results. To overcome these difficulties and to improve the ratio between accuracy and computational costs, homogenization techniques can be used. In this paper we recall analytical homogenization results and propose a novel numerical homogenization scheme for Maxwell’s equations in frequency domain. This scheme follows the design principles of heterogeneous multiscale methods. We prove convergence to the effective solution of the multiscale Maxwell’s equations in a periodic setting and give numerical experiments in accordance to the stated results. (10.1016/j.camwa.2017.02.043)
  DOI : 10.1016/j.camwa.2017.02.043
• Criticality Computation with Finite Element Method on Non-Conforming Meshes
  
  • Giret L.
  • Ciarlet Patrick
  • Jamelot E.
  , 2017. In this work, we proposed and study a method to use non-conforming meshing for core reactor simulation. This consists in a domain decomposition with Lagrange multipliers of the well known Raviart-Thomas finite element method. Here, we provide an a priori error estimate for criticality computation.
• Optimization of wireless sensor networks deployment with coverage and connectivity constraints
  
  • Elloumi Sourour
  • Hudry Olivier
  • Marie Estel
  • Plateau Agnès
  • Rovedakis Stephane
  , 2017, pp.0336-0341. (10.1109/CoDIT.2017.8102614)
  DOI : 10.1109/CoDIT.2017.8102614
• Stable perfectly matched layers for a class of anisotropic dispersive models. Part II: Energy estimates
  
  • Kachanovska Maryna
  , 2017. This article continues the stability analysis of the generalized perfectly matched layers for 2D anisotropic dispersive models studied in Part I of the work. We obtain explicit energy estimates for the PML system in the time domain, by making use of the ideas stemming from the analysis of the associated sesquilinear form in the Laplace domain. This analysis is based on the introduction of a particular set of auxiliary unknowns related to the PML, which simplifies the derivation of the energy estimates for the resulting system. For 2D dispersive systems, our analysis allows to demonstrate the stability of the PML system for a constant absorption parameter. For 1D dispersive systems, we show the stability of the PMLs with a non-constant absorption parameter.
• Particle system algorithm and chaos propagation related to non-conservative McKean type stochastic differential equations
  
  • Lecavil Anthony
  • Oudjane Nadia
  • Russo Francesco
  Stochastics and Partial Differential Equations: Analysis and Computations, Springer US, 2017, 5 (1), pp.Stochastics and partial differential equations: Analysis and Computation., vol. 5 (1), pp. 1-37, Springer-Verlag, mar, 2017. We discuss numerical aspects related to a new class of nonlinear Stochastic Differential Equations in the sense of McKean, which are supposed to represent non conservative nonlinear Partial Differential equations (PDEs). We propose an original interacting particle system for which we discuss the propagation of chaos. We consider a time-discretized approximation of this particle system to which we associate a random function which is proved to converge to a solution of a regularized version of a nonlinear PDE. (10.1007/s40072-016-0079-9)
  DOI : 10.1007/s40072-016-0079-9
• Perfectly matched layers for convex truncated domains with discontinuous Galerkin time domain simulations
  
  • Modave Axel
  • Lambrechts Jonathan
  • Geuzaine Christophe
  Computers & Mathematics with Applications, Elsevier, 2017, 73 (4), pp.684-700. This paper deals with the design of perfectly matched layers (PMLs) for transient acoustic wave propagation in generally-shaped convex truncated domains. After reviewing key elements to derive PML equations for such domains, we present two time-dependent formulations for the pressure-velocity system. These formulations are obtained by using a complex coordinate stretching of the time-harmonic version of the equations in a specific curvilinear coordinate system. The final PML equations are written in a general tensor form, which can easily be projected in Cartesian coordinates to facilitate implementation with classical discretization methods. Discontinuous Galerkin finite element schemes are proposed for both formulations. They are tested and compared using a three-dimensional benchmark with an ellipsoidal truncated domain. Our approach can be generalized to domains with corners. (10.1016/j.camwa.2016.12.027)
  DOI : 10.1016/j.camwa.2016.12.027
• Reformulation Quadratique Convexe Pour l'Optimisation des Flux de Puissance
  
  • Godard Hadrien
  • Elloumi Sourour
  • Lambert Amélie
  • Maeght Jean
  • Ruiz Manuel
  , 2017. Mots-clés : Optimisation des flux de puissance, Reformulation quadratique, Programmation quadratique, Programmation semi-définie. Un réseau électrique est constitué d'un ensemble de noeuds reliés par des lignes. L'effet Joule rend non conservatif le transport de l'électricité des noeuds de production aux noeuds de consommation à travers les lignes. L'optimisation des flux de puissance, en anglais Optimal Power Flow (OPF), consiste à déterminer un état du réseau (tension et injections aux noeuds, transits sur les lignes) permettant de satisfaire la demande aux noeuds tout en mi-nimisant un certain critére. Dans ce travail une nouvelle approche permettant de résoudre à l'optimum global l'OPF est introduite. Cette approche consiste à construire une reformulation quadratique [1] de l'OPF en utilisant une relaxation issue de l'état de l'art [4]. 1 Formulation de l'OPF et relaxation semi-définie positive L'OPF s'exprime comme un problème hermitien [5] dont les variables v ∈ C n sont les tensions aux n noeuds du réseau électrique. Il est possible de plonger v dans R 2n en introduisant les variables réelles x tel que x t = [ v t +v t 2 ; v t −v t 2i ] qui représentent les parties réelles et imaginaires des tensions. L'OPF se modélise alors comme un programme quadratique non convexe en variables réelles, avec ∀k ∈ {0,. .. , m}, A k ∈ M 2n (R) : (OP F) min x∈R 2n f 0 (x) = x t A 0 x s.t. f k (x) = x t A k x ≤ a k k = 1,. .. , m En introduisant la matrice variable Y = xx t , (OP F) se reformule ainsi : (RSDP)        min Y ∈M2n(R) A 0 , Y s.t. A k , Y ≤ a k k = 1,. .. , m Y 0, rg(Y) = 1 Une relaxation semi définie positive convexe (RSDP), appelée relaxation du rang, s'obtient en éliminant la contrainte non convexe rg(Y) = 1. 2 Algorithme de résolution de l'OPF Pour résoudre (OP F), nous en construisons dans un premier temps une reformulation qua-dratique à partir de la résolution de (RSDP). Dans un second temps nous résolvons cette reformulation dans un algorithme de branch-and-bound ayant comme borne à la racine la va-leur de (RSDP) qui est de bonne qualité pour le problème de l'OPF [3].
• Optimisation du maillage électrique du parc éoliennes off-shore – projet Stationis
  
  • Gladkikh Egor
  • Lambert Amélie
  • Faye Alain
  • Watel Dimitri
  • Costa Marie-Christine
  , 2017.
• Optimisation de programmes polynomiaux en variables 0-1 et sans contraintes
  
  • d'Ambrosio Claudia
  • Elloumi Sourour
  • Lambert Amélie
  • Lazare Arnaud
  , 2017.
• On the edge capacitated Steiner tree problem
  
  • Bentz Cédric
  • Costa Marie-Christine
  • Hertz Alain
  , 2017.
• Numerical methods for hybrid control and chance-constrained optimization problems
  
  • Sassi Achille
  , 2017. This thesis is devoted to the analysis of numerical methods in the field of optimal control, and it is composed of two parts. The first part is dedicated to new results on the subject of numerical methods for the optimal control of hybrid systems, controlled by measurable functions and discontinuous jumps in the state variable simultaneously. The second part focuses on a particular application of trajectory optimization problems for space launchers. Here we use some nonlinear optimization methods combined with non-parametric statistics techniques. This kind of problems belongs to the family of stochastic optimization problems and it features the minimization of a cost function in the presence of a constraint which needs to be satisfied within a desired probability threshold.
• H-matrix based Solver for 3D Elastodynamics Boundary Integral Equations
  
  • Desiderio Luca
  , 2017. This thesis focuses on the theoretical and numerical study of fast methods to solve the equations of 3D elastodynamics in frequency-domain. We use the Boundary Element Method (BEM) as discretization technique, in association with the hierarchical matrices (H-matrices) technique for the fast solution of the resulting linear system. The BEM is based on a boundary integral formulation which requires the discretization of the only domain boundaries. Thus, this method is well suited to treat seismic wave propagation problems. A major drawback of classical BEM is that it results in dense matrices, which leads to high memory requirement (O (N 2 ), if N is the number of degrees of freedom) and computational costs.Therefore, the simulation of realistic problems is limited by the number of degrees of freedom. Several fast BEMs have been developed to improve the computational efficiency. We propose a fast H-matrix based direct BEM solver.
• A model for Faraday pilot waves over variable topography
  
  • Faria Luiz
  Journal of Fluid Mechanics, Cambridge University Press (CUP), 2017, 811, pp.51-66. (10.1017/jfm.2016.750)
  DOI : 10.1017/jfm.2016.750
• Seismic Wave Amplification in 3D Alluvial Basins: Aggravation factors from Fast Multipole BEM Simulations
  
  • Meza Fajardo Kristel Carolina
  • Semblat Jean-François
  • Chaillat Stéphanie
  • Lenti Luca
  , 2017. In this work, we study seismic wave amplification in alluvial basins having 3D canonical geometries through the Fast Multipole Boundary Element Method in the frequency domain. We investigate how much 3D amplification differs from the 1D (horizontal layering) and the 2D cases. Considering synthetic incident wave-fields, we examine the relationships between the amplification level and the most relevant physical parameters of the problem (impedance contrast, 3D aspect ratio, vertical and oblique incidence of plane waves). The FMBEM results show that the most important parameters for wave amplification are the impedance contrast and equivalent shape ratio. Using these two parameters, we derive simple rules to compute the fundamental frequency for different 3D basin shapes and the corresponding 3D aggravation factor for 5% damping.Effects on amplification due to 3D basin asymmetry are also studied and incorporated in the derived rules.
• Infinite dimensional weak Dirichlet processes and convolution type processes
  
  • Fabbri Giorgio
  • Russo Francesco
  Stochastic Processes and their Applications, Elsevier, 2017, 127 (1), pp.325-357. The present paper continues the study of infinite dimensional calculus via regularization, started by C. Di Girolami and the second named author, introducing the notion of weak Dirichlet process in this context. Such a process X, taking values in a Banach space H, is the sum of a local martingale and a suitable orthogonal process. The concept of weak Dirichlet process fits the notion of convolution type processes, a class including mild solutions for stochastic evolution equations on infinite dimensional Hilbert spaces and in particular of several classes of stochastic partial differential equations (SPDEs). In particular the mentioned decomposition appears to be a substitute of an Itô's type formula applied to f (t, X(t)) where f : [0, T ] × H → R is a C 0,1 function and X a convolution type processes. (10.1016/j.spa.2016.06.010)
  DOI : 10.1016/j.spa.2016.06.010
• Entretien avec...
  
  • Costa Marie-Christine
  , 2017.
• Higher order topological derivatives for three-dimensional anisotropic elasticity
  
  • Bonnet Marc
  • Cornaggia Rémi
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2017 (51), pp.2069-2092. This article concerns an extension of the topological derivative concept for 3D elasticity problems involving elastic inhomogeneities, whereby an objective function J is expanded in powers of the characteristic size a of a single small inhomogeneity. The O(a 6) approximation of J is derived and justified for an inhomogeneity of given location, shape and elastic properties embedded in a 3D solid of arbitrary shape and elastic properties; the background and the inhomogeneity materials may both be anisotropic. The generalization to multiple small inhomogeneities is concisely described. Computational issues, and examples of objective functions commonly used in solid mechanics, are discussed. (10.1051/m2an/2017015)
  DOI : 10.1051/m2an/2017015
• HJB equations in infinite dimension and optimal control of stochastic evolution equations via generalized Fukushima decomposition
  
  • Fabbri Giorgio
  • Russo Francesco
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2017, 55 (6), pp.4072-4091. A stochastic optimal control problem driven by an abstract evolution equation in a separable Hilbert space is considered. Thanks to the identification of the mild solution of the state equation as $\nu$-weak Dirichlet process, the value processes is proved to be a real weak Dirichlet process. The uniqueness of the corresponding decomposition is used to prove a verification theorem. Through that technique several of the required assumptions are milder than those employed in previous contributions about non-regular solutions of Hamilton-Jacobi-Bellman equations. (10.1137/17M1113801)
  DOI : 10.1137/17M1113801