Publications

2020

• Modelling the fluid-structure coupling caused by a far-field underwater explosion
  
  • Mavaleix-Marchessoux Damien
  , 2020. Submarines must withstand the effects of rapid dynamic loads induced by underwater explosions. Due to the very high cost of experimental campaigns, numerical simulations are very important. A remote underwater explosion is a complex event that has two distinct effects: it sends a shock wave, then creates an oscillating gas bubble that sets water in slower motion. The two phenomena have quite different characteristics and time scales. In this work, we consider remote enough underwater explosions so that (i) the presence of the submarine only marginally affects the explosion, and (ii) there is a temporal separation of the two phenomena, as experienced by the ship. Under these conditions, our overall goal is to design, implement (in the context of high performance computing) then validate a computational methodology for the fluid-structure interaction problem, taking into account both phenomena. With this aim, we first study the two perturbations without considering the submarine, to propose appropriate modelling and numerical methods. Then, we design a fast boundary element (BEM) procedure, based on the combination of the convolution quadrature method and an original empirical high frequency approximation. The procedure allows to efficiently simulate 3D rapid transient wave propagation problems set in an unbounded domain, and shows advantageous complexity: O(1) in regards to the time discretisation and O(N log N) for the spatial discretisation. Finally, we implement adequate finite element/boundary element (FEM/BEM) coupling strategies for the shock wave fluid-structure interaction phase (linear acoustics) and that of the gas bubble (incompressible flow). The overall procedure, validated on academic problems, provides very promising results when applied on realistic industrial cases.
• An analytical approach based on tailored Green's functions for flow noise prediction at low Mach number
  
  • Trafny Nicolas
  • Serre Gilles
  • Cotte Benjamin
  • Mercier Jean-François
  , 2020, pp.749-750. The presence of boundary surfaces in a turbulent flow can result in the enhancement of the radiated acoustic field especially for eddies close to any geometrical singularity. At low Mach number, it is well known that the contribution of the diffracted field is dominant. In the present study, we focus on Lighthill?s wave equation solved using a tailored Green?s function and a semi-empirical turbulence model in order to investigate the direct acoustic field produced by a turbulent boundary layer over a flat plate. We consider both the turbulent boundary layer noise and edge noise, and power law results deduced from classical dimensional analysis are recovered. The specular acoustic field, produced by eddies far from any edge increases in proportion to the fourth power of the Mach number and the diffracted field directly in proportion to the Mach number. Finally, we chose a NACA 0012 airfoil in order to validate the noise spectrum prediction for different configurations for which trailing edge noise is the dominant contribution. (10.48465/fa.2020.0912)
  DOI : 10.48465/fa.2020.0912
• Non-overlapping domain decomposition methods with non-local transmission operators for harmonic wave propagation problems
  
  • Parolin Émile
  , 2020. The pioneering work of B. Després then M. Gander, F. Magoulès and F. Nataf have shown that it is mandatory, at least in the context of wave equations, to use impedance type transmission conditions in the coupling of subdomains in order to obtain convergence of non-overlapping domain decomposition methods (DDM). In the standard approach considered in the literature, the impedance operator involved in the transmission conditions is local and leads to algebraic convergence of the DDM in the best cases. In later works, F. Collino, S. Ghanemi and P. Joly then F. Collino, P. Joly and M. Lecouvez have observed that using non local impedance operators such as integral operators with suitable singular kernels could lead to a geometric convergence of the DDM.This thesis extends these works (that mainly concerned the scalar Helmholtz equation) with the extension of the analysis to electromagnetic wave propagation. Besides, the numerical analysis of the method is performed for the first time, proving the stability of the convergence rate with respect to the discretization parameter, hence the robustness of the approach. Several integral operators are then proposed as transmission operators for Maxwell equations in the spirit of those constructed for the acoustic setting. An alternative to integral operators, based on the resolution of elliptic auxiliary problems, is also advocated and analyzed. Extensive numerical results are conducted, illustrating the high potential of the new approach. Based on a recent work by X. Claeys, the last part of this work consists in exploiting the multi-trace formalism to extend the convergence analysis to the case of partitions with junction points, which is a difficult problem that attracted a lot of attention recently. The new approach relies on a new operator that communicates information between sub-domains, which replaces the classical point-to-point exchange operator. A proof of geometrical convergence of the associated iterative algorithm, again uniform with respect to the discretization parameter, is available and we show that one recovers the standard algorithm in the absence of junction points.
• Wind farm cable layout optimization with constraints of load flow and robustness
  
  • Bentz Cédric
  • Costa Marie-Christine
  • Poirion Pierre-Louis
  • Ridremont Thomas
  • Zakour Camille
  , 2020. We consider an offshore wind farm defined by the location of the wind turbines and the amount of energy supplied per turbine, as well as the location of the central station responsible for redistributing the collected energy to users on the power grid. Knowing the power injected at each node, the capacities, susceptance and costs of the cables that can be used, the goal is to determine the least expensive cabling to route the energy supplied by the wind turbines to the central station. This cabling must respect the capacity constraints on the cables as well as the electrical constraints of Load Flow defined at each node of the network. In a second step, we look for a cabling that is also robust in case of failure of a cable, the notion of robustness being seen here as the protection again the worst case of failure. This work was carried out in collaboration with EDF Energies renouvelables. We give a mathematical model of the problem taking into account all the constraints of capacity, connectivity, load flow, cable types and incompatibility between edges, in the form of a mixed integer quadratic program that can be linearized and solved using a MIP solver. We then propose two mathematical models for the robust problem, formulation inspired by the previous one and a bi-level program where the second level is a max min program. Finally we present the results of our tests which provide solutions for real data up to about 50 nodes, before concluding.
• A fast boundary element based solver for localized inelastic deformations
  
  • Ciardo Federico
  • Lecampion Brice
  • Fayard François
  • Chaillat Stéphanie
  International Journal for Numerical Methods in Engineering, Wiley, 2020, 121 (24), pp.5696 - 5718. We present a numerical method for the solution of nonlinear geomechanical problems involving localized deformation along shear bands and fractures. We leverage the boundary element method to solve for the quasi-static elastic deformation of the medium while rigid-plastic constitutive relations govern the behavior of displacement discontinuity (DD) segments capturing localized deformations. A fully implicit scheme is developed using a hierarchical approximation of the boundary element matrix. Combined with an adequate block preconditioner, this allows to tackle large problems via the use of an iterative solver for the solution of the tangent system. Several two-dimensional examples of the initiation and growth of shear-bands and tensile fractures illustrate the capabilities and accuracy of this technique. The method does not exhibit any mesh dependency associated with localization provided that (i) the softening length-scale is resolved and (ii) the plane of localized deformations is discretized a priori using DD segments. (10.1002/nme.6520)
  DOI : 10.1002/nme.6520
• Models and Algorithms for the Product Pricing with Single-Minded Customers Requesting Bundles
  
  • Bucarey Víctor
  • Elloumi Sourour
  • Labbé Martine
  • Plein Fränk
  Computers and Operations Research, Elsevier, 2020. We analyze a product pricing problem with single-minded customers, each interested in buying a bundle of products. The objective is to maximize the total revenue and we assume that supply is unlimited for all products. We contribute to a missing piece of literature by giving some mathematical formulations for this single-minded bundle pricing problem. We first present a mixed-integer nonlinear program with bilinear terms in the objective function and the constraints. By applying classical linearization techniques, we obtain two different mixed-integer linear programs. We then study the polyhedral structure of the linear formulations and obtain valid inequalities based on an RLT-like framework. We develop a Benders decomposition to project strong cuts from the tightest model onto the lighter models. We conclude this work with extensive numerical experiments to assess the quality of the mixed-integer linear formulations, as well as the performance of the cutting plane algorithms and the impact of the preprocessing on computation times. (10.1016/j.cor.2020.105139)
  DOI : 10.1016/j.cor.2020.105139
• Computation of the exact discrete transparent boundary condition for 1D linear equations
  
  • Fliss Sonia
  • Imperiale Sébastien
  • Tonnoir Antoine
  , 2020. In this work, we are interested in the construction of the exact transparent boundary conditions for a semi-discretized and fully discretized 1D linear PDE. The proposed method is quite general and is based on the computation of a family of canonical functions. Several examples and numerical results to illustrate the method are presented.
• On the edge capacitated Steiner tree problem
  
  • Bentz Cédric
  • Costa Marie-Christine
  • Hertz Alain
  Discrete Optimization, Elsevier, 2020, 38, pp.100607. Given a graph G = (V, E) with a root r ∈ V , positive capacities {c(e)|e ∈ E}, and non-negative lengths { (e)|e ∈ E}, the minimum-length (rooted) edge capacitated Steiner tree problem is to find a tree in G of minimum total length, rooted at r, spanning a given subset T ⊂ V of vertices, and such that, for each e ∈ E, there are at most c(e) paths, linking r to vertices in T , that contain e. We study the complexity and approximability of the problem, considering several relevant parameters such as the number of terminals, the edge lengths and the minimum and maximum edge capacities. For all but one combinations of assumptions regarding these parameters, we settle the question, giving a complete characterization that separates tractable cases from hard ones. The only remaining open case is proved to be equivalent to a long-standing open problem. We also prove close relations between our problem and classic Steiner tree as well as vertex-disjoint paths problems. (10.1016/j.disopt.2020.100607)
  DOI : 10.1016/j.disopt.2020.100607
• A Discrete Domain Decomposition Method for Acoustics with Uniform Exponential Rate of Convergence Using Non-local Impedance Operators
  
  • Claeys Xavier
  • Collino Francis
  • Joly Patrick
  • Parolin Emile
  , 2020, pp.310-317. The relaxed Jacobi algorithm written at the continuous level was proven to converge exponentially. However, it was only a conjecture, hinted at by numerical experiments in [5, Section 8], that the discretized algorithm using finite elements has a rate of convergence uniformly bounded with respect to the discretization parameter (10.1007/978-3-030-56750-7_35)
  DOI : 10.1007/978-3-030-56750-7_35
• Analyse spectrale et simulation numérique de cavités contenant un matériau négatif
  
  • Bernard Paolantoni Sandrine
  , 2020. Cette thèse réalise une étude théorique et numérique du spectre de cavités partiellement composées de matériau négatif, c'est-à-dire de matériau pour lequel la perméabilité magnétique et/ou la permittivité électrique (ou au moins leur partie réelle) deviennent négatives dans certaines plages de fréquences. Cette étude s'inscrit dans la continuité des travaux engagés dans notre laboratoire qui se concentrent sur la propagation des ondes électromagnétiques en présence de matériau négatif, à fréquence fixée. L'objectif de cette thèse est de prendre en compte la dispersion fréquentielle, autrement dit la dépendance en fréquence de la perméabilité et de la permittivité, en considérant la fréquence comme paramètre spectral. Nous mettons en évidence le spectre essentiel résultant de la présence de matériau négatif ainsi que les phénomènes de résonance qui en découlent, pour différents modèles décrivant ce matériau.L'étude théorique se concentre sur le cas de cavités bidimensionnelles polygonales pour les modèles de Drude et de Lorentz (avec et sans dissipation). L'étude théorique du modèle le plus simple (Drude non dissipatif) est étendue au cas d'une interface courbe (mais régulière).Ce modèle fait également l'objet d'une étude numérique, visant à explorer l'effet d'une discrétisation éléments finis du problème théorique, et ainsi mettre en avant les difficultés à observer numériquement certains des phénomènes de résonance.
• A decomposition method by component for the optimization of maintenance scheduling
  
  • Bittar Thomas
  • Chancelier Jean-Philippe
  • Carpentier Pierre
  • Lonchampt Jérôme
  , 2020. We present a decomposition method based on the Auxiliary Problem Principle to design optimal maintenance scheduling policies for systems of physical components (turbines, generators, transformers) sharing a common stock of spare parts. The method outperforms a reference blackbox method on a system with 80 components.
• The K‐partitioning problem: Formulations and branch‐and‐cut
  
  • Alès Zacharie
  • Knippel Arnaud
  Networks, Wiley, 2020, 76 (3), pp.323-349. (10.1002/net.21944)
  DOI : 10.1002/net.21944
• Asymptotic modelling of Skin-effects in coaxial cables
  
  • Beck Geoffrey
  • Imperiale Sébastien
  • Joly Patrick
  SN Partial Differential Equations and Applications, Springer, 2020. In this work we tackle the modeling of non-perfectly conducting thin coaxial cables. From the non-dimensionnalised 3D Maxwells equations, we derive, by asymptotic analysis with respect to the (small) transverse dimension of the cable, a simplified effective 1D model and an effective reconstruction procedure of the electric and magnetic fields. The derived effective model involves a fractional time derivatives that accounts for the so-called skin effects in highly conducting regions.
• Experimental and theoretical observations on DDT in smooth narrow channels
  
  • Melguizo-Gavilanes J.
  • Ballossier Yves
  • Maltez Faria Luiz
  Proceedings of the Combustion Institute, Elsevier, 2020. A combined experimental and theoretical study of deflagration-to-detonation transition (DDT) in smooth narrow channels is presented. Some of the distinguishing features characterizing the late stages of DDT are shown to be qualitatively captured by a simple one-dimensional scalar equation. Inspection of the structure and stability of the traveling wave solutions found in the model, and comparison with experimental observations, suggest a possible mechanism responsible for front acceleration and transition to detonation. (10.1016/j.proci.2020.07.142)
  DOI : 10.1016/j.proci.2020.07.142
• Time harmonic wave propagation in one dimensional weakly randomly perturbed periodic media
  
  • Fliss Sonia
  • Giovangigli Laure
  SN Partial Differential Equations and Applications, Springer, 2020, 1 (40). In this work we consider the solution of the time harmonic wave equation in a one dimensional periodic medium with weak random perturbations. More precisely, we study two types of weak perturbations: (1) the case of stationary, ergodic and oscillating coefficients, the typical size of the oscillations being small compared to the wavelength and (2) the case of rare random perturbations of the medium, where each period has a small probability to have its coefficients modified, independently of the other periods. Our goal is to derive an asymptotic approximation of the solution with respect to the small parameter. This can be used in order to construct absorbing boundary conditions for such media.
• A Geometrical Approach for the Optimal Control of Sequencing Batch Bio-Reactors
  
  • Abdellatif Nahla
  • Bouhafs Walid
  • Harmand Jérôme
  • Jean Frédéric
  Statistics, Optimization and Information Computing, International Academic Press, 2020, 9 (2), pp.368-382. In this work, we consider an optimal control problem of a biological sequencing batch reactor (SBR) for thetreatment of pollutants in wastewater. This model includes two biological reactions, one being aerobic while the other is anoxic. The objective is to find an optimal oxygen-injecting strategy to reach, in minimal time and in a minimal time/energy compromise, a target where the pollutants concentrations must fulfill normative constraints. Using a geometrical approach, we solve a more general optimal control problem and thanks to Pontryagin’s Maximum Principle, we explicitly give the complete optimal strategy. (10.19139/soic-2310-5070-868)
  DOI : 10.19139/soic-2310-5070-868
• Planification adaptative des ressources ferroviaires
  
  • Lucas Rémi
  , 2020. À la SNCF, la planification de l'exploitation ferroviaire (grilles horaires des trains, roulements des personnels et matériels roulants, maintenances...) suit un processus complexe s'étalant sur plusieurs années. Une fois fixée, cette planification est très souvent réadaptée pour faire face aux aléas. Dans le contexte d'un transport de masse comme c'est le cas en région parisienne, où les marges de manœuvre sont très faibles, le retour au plan de transport nominal est alors d'autant plus difficile. Ainsi, on constate qu'un plan de transport hyperplanifié n'est plus nécessairement adapté : la rigidité de l'exploitation ferroviaire est ici remise en question. De plus, cette planification encore trop fragmentée conduit à des choix non optimaux, au sens où les différentes ressources utilisées pour revenir au plan de transport nominal peuvent être réduites si l'on dispose d'une planification plus agile et d'algorithmes temps réel associés avec une vision plus globale. L'objectif de ma thèse est alors de repenser le paradigme actuel de la planification pour parvenir à une planification adaptative. Quel serait un plan stratégique adaptable par rapport au plan prescripteur très détaillé actuel ?
• Reinforcement Learning for Variable Selection in a Branch and Bound Algorithm
  
  • Etheve Marc
  • Alès Zacharie
  • Bissuel Côme
  • Kedad-Sidhoum Safia
  • Juan Olivier
  , 2020, pp.176-185. (10.1007/978-3-030-58942-4_12)
  DOI : 10.1007/978-3-030-58942-4_12
• Mixed Spatial and Temporal Decompositions for Large Scale Multistage Stochastic Optimization Problems
  
  • Carpentier Pierre
  • Chancelier Jean-Philippe
  • de Lara Michel
  • Pacaud François
  Journal of Optimization Theory and Applications, Springer Verlag, 2020, 186 (3), pp.985-1005. We consider multistage stochastic optimization problems involving multiple units. Each unit is a (small) control system. Static constraints couple units at each stage. We present a mix of spatial and temporal decompositions to tackle such large scale problems. More precisely, we obtain theoretical bounds and policies by means of two methods, depending whether the coupling constraints are handled by prices or by resources. We study both centralized and decentralized information structures. We report the results of numerical experiments on the management of urban microgrids. It appears that decomposition methods are much faster and give better results than the standard Stochastic Dual Dynamic Programming method, both in terms of bounds and of policy performance. (10.1007/s10957-020-01733-7)
  DOI : 10.1007/s10957-020-01733-7
• Transparent boundary conditions for wave propagation in fractal trees: convolution quadrature approach
  
  • Joly Patrick
  • Kachanovska Maryna
  Numerische Mathematik, Springer Verlag, 2020, 146(2), pp.281-334. In this work we propose high-order transparent boundary conditions for the weighted wave equation on a fractal tree, with an application to the modeling of sound propagation in a human lung. This article follows the recent work [29], dedicated to the mathematical analysis of the corresponding problem and the construction of low-order absorbing boundary conditions. The method proposed in this article consists in constructing the exact (trans-parent) boundary conditions for the semi-discretized problem, in the spirit of the convolution quadrature method developed by Ch. Lubich. We analyze the stability and convergence of the method, and propose an efficient algorithm for its implementation. The exposition is concluded with numerical experiments.
• Mixed Spatial and Temporal Decompositions for Large-Scale Multistage Stochastic Optimization Problems
  
  • Carpentier Pierre
  • Chancelier Jean-Philippe
  • de Lara Michel
  • Pacaud François
  Journal of Optimization Theory and Applications, Springer Verlag, 2020, 186 (3), pp.985-1005. (10.1007/s10957-020-01733-7)
  DOI : 10.1007/s10957-020-01733-7
• Discrete-type Approximations for Non-Markovian Optimal Stopping Problems: Part II
  
  • Bezerra Sérgio
  • Ohashi Alberto
  • Russo Francesco
  • de Souza Francys
  Methodology and Computing in Applied Probability, Springer Verlag, 2020, 22 (3), pp.1221-1255. (10.1007/s11009-019-09764-y)
  DOI : 10.1007/s11009-019-09764-y
• A non-overlapping domain decomposition method with high-order transmission conditions and cross-point treatment for Helmholtz problems
  
  • Modave Axel
  • Royer Anthony
  • Antoine Xavier
  • Geuzaine Christophe
  Computer Methods in Applied Mechanics and Engineering, Elsevier, 2020, 368, pp.1131622020. A non-overlapping domain decomposition method (DDM) is proposed for the parallel finite-element solution of large-scale time-harmonic wave problems. It is well-known that the convergence rate of this kind of method strongly depends on the transmission condition enforced on the interfaces between the subdomains. Local conditions based on high-order absorbing boundary conditions (HABCs) have proved to be well-suited, as a good compromise between basic impedance conditions, which lead to suboptimal convergence, and conditions based on the exact Dirichlet-to-Neumann (DtN) map related to the complementary of the subdomain — which are too expensive to compute. However, a direct application of this approach for configurations with interior cross-points (where more than two subdomains meet) and boundary cross-points (points that belong to both the exterior boundary and at least two subdomains) is suboptimal and, in some cases, can lead to incorrect results. In this work, we extend a non-overlapping DDM with HABC-based transmission conditions approach to efficiently deal with cross-points for lattice-type partitioning. We address the question of the cross-point treatment when the HABC operator is used in the transmission condition, or when it is used in the exterior boundary condition, or both. The proposed cross-point treatment relies on corner conditions developed for Padé-type HABCs. Two-dimensional numerical results with a nodal finite-element discretization are proposed to validate the approach, including convergence studies with respect to the frequency, the mesh size and the number of subdomains. These results demonstrate the efficiency of the cross-point treatment for settings with regular partitions and homogeneous media. Numerical experiments with distorted partitions and smoothly varying heterogeneous media show the robustness of this treatment. (10.1016/j.cma.2020.113162)
  DOI : 10.1016/j.cma.2020.113162
• On the efficiency of nested GMRES preconditioners for 3D acoustic and elastodynamic H-matrix accelerated Boundary Element Methods
  
  • Kpadonou Félix D.
  • Chaillat Stéphanie
  • Ciarlet Patrick
  Computers & Mathematics with Applications, Elsevier, 2020, 80 (3). This article is concerned with the derivation of fast Boundary Element Methods for 3D acoustic and elastodynamic problems. In particular, we are interested in the acceleration of Hierarchical matrix (-matrix) based iterative solvers. While H-matrix representations allow to reduce the storage requirements and the cost of a matrix–vector product, the number of iterations for an iterative solver, as the frequency or the problem size increases, remains an issue. We consider an inner–outer preconditioning strategy, i.e., the preconditioner is applied through an iterative solver at the inner level. The preconditioner is defined as a H-matrix representation of the system matrix with a given accuracy. We investigate the influence of various parameters of the preconditioner, i.e., the H-matrix accuracy, the GMRES threshold and the maximum number of iterations of the inner solver. Different numerical results are presented to compare the efficiency of the preconditioner with respect to the unpreconditioned reference system. Finally, we propose a way to define the optimal setting for this preconditioner. (10.1016/j.camwa.2020.03.021)
  DOI : 10.1016/j.camwa.2020.03.021
• Planewave Density Interpolation Methods for the EFIE on Simple and Composite Surfaces
  
  • Pérez-Arancibia Carlos
  • Turc Catalin
  • Faria Luiz
  • Sideris Constantine
  IEEE Transactions on Antennas and Propagation, Institute of Electrical and Electronics Engineers, 2020. This paper presents an extension of the recently introduced planewave density interpolation (PWDI) method to the electric field integral equation (EFIE) formulation of problems of scattering and radiation by perfect electric conducting (PEC) objects. Relying on Kirchhoff integral formula and local interpolation of surface current densities that regularize the kernel singularities, the PWDI method enables off-and on-surface EFIE operators to be re-expressed in terms of integrands that are globally bounded (or even more regular) over the whole domain of integration, regardless of the magnitude of the distance between target and source points. Surface integrals resulting from the application of the method-of-moments (MoM) using Rao-Wilton-Glisson (RWG) basis functions, can then be directly and easily evaluated by means of elementary quadrature rules irrespective of the singularity location. The proposed technique can be applied to simple and composite surfaces comprising two or more simply-connected overlapping components. The use of composite surfaces can significantly simplify the geometric treatment of complex structures, as the PWDI method enables the use of separate non-conformal meshes for the discretization of each of the surface components that make up the composite surface. A variety of examples, including multi-scale and intricate structures, demonstrate the effectiveness of the proposed methodology.