Publications

2013

• The topological derivative in anisotropic elasticity
  
  • Bonnet Marc
  • Delgado Gabriel
  Quarterly Journal of Mechanics and Applied Mathematics, Oxford University Press (OUP), 2013, 66, pp.557-586. A comprehensive treatment of the topological derivative for anisotropic elasticity is presented, with both the background material and the trial small inhomogeneity assumed to have arbitrary anisotropic elastic properties. A formula for the topological derivative of any cost functional defined in terms of regular volume or surface densities depending on the displacement is established, by combining small-inhomogeneity asymptotics and the adjoint solution approach. The latter feature makes the proposed result simple to implement and computationally efficient. Both three-dimensional and plane-strain settings are treated; they differ mostly on details in the elastic moment tensor (EMT). Moreover, the main properties of the EMT, a critical component of the topological derivative, are studied for the fully anisotropic case. Then, the topological derivative of strain energy-based quadratic cost functionals is derived, which requires a distinct treatment. Finally, numerical experiments on the numerical evaluation of the EMT and the topological derivative of the compliance cost functional are reported. (10.1093/qjmam/hbt018)
  DOI : 10.1093/qjmam/hbt018
• A Global Steering Method for Nonholonomic Systems
  
  • Chitour Yacine
  • Jean Frédéric
  • Long Ruixing
  Journal of Differential Equations, Elsevier, 2013, 254, pp.1903-1956. In this paper, we present an iterative steering algorithm for nonholonomic systems (also called driftless control-affine systems) and we prove its global convergence under the sole assumption that the Lie Algebraic Rank Condition (LARC) holds true everywhere. That algorithm is an extension of the one introduced in [21] for regular systems. The first novelty here consists in the explicit algebraic construction, starting from the original control system, of a lifted control system which is regular. The second contribution of the paper is an exact motion planning method for nilpotent systems, which makes use of sinusoidal control laws and which is a generalization of the algorithm described in [29] for chained-form systems. (10.1016/j.jde.2012.11.012)
  DOI : 10.1016/j.jde.2012.11.012
• State-constrained Optimal Control Problems of Impulsive Differential Equations
  
  • Forcadel Nicolas
  • Rao Zhiping
  • Zidani Hasnaa
  Applied Mathematics and Optimization, Springer Verlag (Germany), 2013, 68, pp.1--19. The present paper studies an optimal control problem governed by measure driven differential systems and in presence of state constraints. The first result shows that using the graph completion of the measure, the optimal solutions can be obtained by solving a reparametrized control problem of absolutely continuous trajectories but with time-dependent state-constraints. The second result shows that it is possible to characterize the epigraph of the reparametrized value function by a Hamilton-Jacobi equation without assuming any controllability assumption. (10.1007/s00245-013-9193-5)
  DOI : 10.1007/s00245-013-9193-5
• Two-dimensional Maxwell's equations with sign-changing coefficients
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chesnel Lucas
  • Ciarlet Patrick
  Applied Numerical Mathematics: an IMACS journal, Elsevier, 2013. We consider the theoretical study of time harmonic Maxwellʼs equations in presence of sign-changing coefficients, in a two-dimensional configuration. Classically, the problems for both the Transverse Magnetic and the Transverse Electric polarizations reduce to an equivalent scalar Helmholtz type equation. For this scalar equation, we have already studied consequences of the presence of sign-changing coefficients in previous papers, and we summarize here the main results. Then we focus on the alternative approach which relies on the two-dimensional vectorial formulations of the TM or TE problems, and we exhibit some unexpected effects of the sign-change of the coefficients. In the process, we provide new results on the scalar equations. (10.1016/j.apnum.2013.04.006)
  DOI : 10.1016/j.apnum.2013.04.006
• An hp-finite element approximation of guided modes in photonic crystal waveguides using transparent boundary conditions
  
  • Klindworth Dirk
  • Schmidt Kersten
  • Fliss Sonia
  Computers & Mathematics with Applications, Elsevier, 2013. no abstract
• A decomposition technique for pursuit evasion games with many pursuers
  
  • Festa Adriano
  • Vinter Richard
  , 2013. Here we present a decomposition technique for a class of differential games. The technique consists in a decomposition of the target set which produces, for geometrical reasons, a decomposition in the dimensionality of the problem. Using some elements of Hamilton-Jacobi equations theory, we find a relation between the regularity of the solution and the possibility to decompose the problem. We use this technique to solve a pursuit evasion game with multiple agents.
• An adaptive sparse grid semi-lagrangian scheme for first order Hamilton-Jacobi Bellman equations
  
  • Bokanowski Olivier
  • Garcke Jochen
  • Griebel Michael
  • Klompmaker Irene
  Journal of Scientific Computing, Springer Verlag, 2013, 55, pp.pp. 575-605. We propose a semi-Lagrangian scheme using a spatially adaptive sparse grid to deal with non-linear time-dependent Hamilton-Jacobi Bellman equations. We focus in particular on front propagation models in higher dimensions which are related to control problems. We test the numerical efficiency of the method on several benchmark problems up to space dimension d = 8, and give evidence of convergence towards the exact viscosity solution. In addition, we study how the complexity and precision scale with the dimension of the problem. (10.1007/s10915-012-9648-x)
  DOI : 10.1007/s10915-012-9648-x
• Transparent boundary conditions for locally perturbed infinite hexagonal periodic media.
  
  • Besse Christophe
  • Coatléven Julien
  • Fliss Sonia
  • Lacroix-Violet Ingrid
  • Ramdani Karim
  Communications in Mathematical Sciences, International Press, 2013, 11 (4), pp.907-938. In this paper, we propose a strategy to determine the Dirichlet-to-Neumann (DtN) operator for infinite, lossy and locally perturbed hexagonal periodic media. We obtain a factorization of this operator involving two non local operators. The first one is a DtN type operator and corresponds to a half-space problem. The second one is a Dirichlet-to-Dirichlet (DtD) type operator related to the symmetry properties of the problem. The half-space DtN operator is characterized via Floquet-Bloch transform, a family of elementary strip problems and a family of stationary Riccati equations. The DtD operator is the solution of an affine operator valued equation which can be reformulated as a non standard integral equation.
• Recent advances on the fast multipole accelerated boundary element method for 3D time-harmonic elastodynamics
  
  • Chaillat Stéphanie
  • Bonnet Marc
  Wave Motion, Elsevier, 2013, 50, pp.1090-1104. This article is mainly devoted to a review on fast BEMs for elastodynamics, with particular attention on time-harmonic fast multipole methods (FMMs). It also includes original results that complete a very recent study on the FMM for elastodynamic problems in semi-infinite media. The main concepts underlying fast elastodynamic BEMs and the kernel-dependent elastodynamic FM-BEM based on the diagonal-form kernel decomposition are reviewed. An elastodynamic FM-BEM based on the half-space Green's tensor suitable for semi-infinite media, and in particular on the fast evaluation of the corresponding governing double-layer integral operator involved in the BIE formulation of wave scattering by underground cavities, is then presented. Results on numerical tests for the multipole evaluation of the half-space traction Green's tensor and the FMM treatment of a sample 3D problem involving wave scattering by an underground cavity demonstrate the accuracy of the proposed approach. The article concludes with a discussion of several topics open to further investigation, with relevant published work surveyed in the process. (10.1016/j.wavemoti.2013.03.008)
  DOI : 10.1016/j.wavemoti.2013.03.008
• A remark on Lipschitz stability for inverse problems
  
  • Bourgeois Laurent
  Comptes rendus hebdomadaires des séances de l'Académie des sciences, Gauthier-Villars, 2013, 351, pp.187--190. An abstract Lipschitz stability estimate is proved for a class of inverse problems. It is then applied to the inverse medium problem for the Helmholtz equation. (10.1016/j.crma.2013.04.004)
  DOI : 10.1016/j.crma.2013.04.004
• Stability and dispersion analysis of the staggered discontinuous Galerkin method for wave propagation
  
  • Chang Hiu Ning
  • Chung Eric
  • Cohen Gary
  International Journal of Numerical Analysis and Modeling, Institute for Scientific Computing and Information, 2013, 10 (1), pp.233--256. Staggered discontinuous Galerkin methods have been developed recently and are adopted successfully to many problems such as wave propagation, elliptic equation, convection-diffusion equation and the Maxwells equations. For wave propagation, the method is proved to have the desirable properties of energy conservation, optimal order of convergence and blockdiagonal mass matrices. In this paper, we perform an analysis for the dispersion error and the CFL constant. Our results show that the staggered method provides a smaller dispersion error compared with classical finite element method as well as non-staggered discontinuous Galerkin methods.
• Mathematical modeling of electromagnetic wave propagation in heterogeneous lossy coaxial cables with variable cross section
  
  • Imperiale Sébastien
  • Joly Patrick
  Applied Numerical Mathematics: an IMACS journal, Elsevier, 2013. In this work, we focus on the time-domain simulation of the propagation of electromagnetic waves in non-homogeneous lossy coaxial cables. The full 3D Maxwell equations, that described the propagation of current and elec- tric potential in such cables, are classically not tackled directly, but instead a 1D scalar model known as the telegraphist's model is used. We aim at justifying, by means of asymptotic analysis, a time-domain "homogenized" telegraphist's model. This model, which includes a non-local in time op- erator, is obtained via asymptotic analysis, for a lossy coaxial cable whose cross-section is not homogeneous. (10.1016/j.apnum.2013.03.011)
  DOI : 10.1016/j.apnum.2013.03.011
• Apposition of the topological sensitivity and linear sampling approaches to inverse scattering
  
  • Bellis Cédric
  • Bonnet Marc
  • Guzina B. B.
  Wave Motion, Elsevier, 2013, 50, pp.891-908. The focus of this study is the reconstruction of a penetrable obstacle in acoustic medium from the knowledge of incident time-harmonic waves and corresponding scattered fields. The problem is investigated by way of two competing approaches: the method of topological sensitivity and that of linear sampling, that have been successfully developed for a variety of physical settings (acoustic, electromagnetic, elastodynamic) as non-iterative tools for solving the inverse scattering problem. On adopting a particular scattering configuration -- plane waves impigning on a spherical obstacle -- that permits analytical treatment as the testing platform, a parallel is drawn between the two methods to evaluate their relative performance in reconstructing the obstacle from the scattered field data. For completeness, the comparison is made by considering a range of input parameters in terms of material properties of the scatterer, frequency of illuminating waves, and noise in the data. (10.1016/j.wavemoti.2013.02.013)
  DOI : 10.1016/j.wavemoti.2013.02.013
• Acoustic propagation in non-uniform waveguides: revisiting Webster equation using evanescent boundary modes
  
  • Mercier Jean-François
  • Maurel Agnès
  Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Royal Society, The, 2013, 469. no abstract (10.1098/rspa.2013.0186)
  DOI : 10.1098/rspa.2013.0186
• Strongly oscillating singularities for the interior transmission eigenvalue problem
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chesnel Lucas
  Inverse Problems, IOP Publishing, 2013, 19(10), pp.104004. In this paper, we investigate a two-dimensional interior transmission eigenvalue problem for an inclusion made of a composite material. We consider configurations where the difference between the parameters of the composite material and those of the background changes sign on the boundary of the inclusion. In a first step, under some assumptions on the parameters, we extend the variational approach of the T-coercivity to prove that the transmission eigenvalues form at most a discrete set. In the process, we also provide localization results. Then, we study what happens when these assumptions are not satisfied. The main idea is that, due to very strong singularities that can occur at the boundary, the problem may lose Fredholmness in the natural H1 framework. Using Kondratiev theory, we propose a new functional framework where the Fredholm property is restored. (10.1088/0266-5611/29/10/104004)
  DOI : 10.1088/0266-5611/29/10/104004
• On the use of the Linear Sampling Method to identify cracks in elastic waveguides
  
  • Bourgeois Laurent
  • Lunéville Éric
  Inverse Problems, IOP Publishing, 2013, 29, pp.025017. We consider the identification of cracks in an elastic 2D or 3D waveguide with the help of a modal version of the linear sampling method. The main objective of our paper is to show that since the usual crack in elasticity is traction free, that is, the boundary condition on the lips of the crack is a priori known to be of Neumann type, we shall adapt the formulation of the sampling method to such a boundary condition in order to improve the efficiency of the method. The need for such adaptation is proved theoretically and illustrated numerically with the help of 2D examples. (10.1088/0266-5611/29/2/025017)
  DOI : 10.1088/0266-5611/29/2/025017
• Particle Methods For Stochastic Optimal Control Problems
  
  • Carpentier Pierre
  • Cohen Guy
  • Dallagi Anes
  Computational Optimization and Applications, Springer Verlag, 2013, 56 (3), pp.635-674. To tackle the difficulties faced by both stochastic dynamic programming and scenario tree methods, we present some variational approach for numerical solution of stochastic optimal control problems. We consider two different interpretations of the control problem, an algebraic and a functional one from which we derive optimality conditions. An adaptative mesh discretization method will be used to propose a tractable solution algorithm. An application to a hydro-electric dam production management problem will be presented. (10.1007/s10589-013-9579-y)
  DOI : 10.1007/s10589-013-9579-y
• First and second order optimality conditions for optimal control problems of state constrained integral equations
  
  • Bonnans J. Frédéric
  • de La Vega Constanza
  • Dupuis Xavier
  Journal of Optimization Theory and Applications, Springer Verlag, 2013, 159 (1), pp.1-40. This paper deals with optimal control problems of integral equations, with initial-final and running state constraints. The order of a running state constraint is defined in the setting of integral dynamics, and we work here with constraints of arbitrary high orders. First and second-order necessary conditions of optimality are obtained, as well as second-order sufficient conditions. (10.1007/s10957-013-0299-3)
  DOI : 10.1007/s10957-013-0299-3
• Negative materials and corners in electromagnetism
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chesnel Lucas
  • Ciarlet Patrick
  • Claeys Xavier
  • Nazarov Serguei
  , 2013. no abstract
• A Hamilton-Jacobi approach to junction problems and application to traffic flows
  
  • Imbert Cyril
  • Monneau Régis
  • Zidani Hasnaa
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2013, 19 (01), pp.pp 129-166. This paper is concerned with the study of a model case of first order Hamilton-Jacobi equations posed on a ''junction'', that is to say the union of a finite number of half-lines with a unique common point. The main result is a comparison principle. We also prove existence and stability of solutions. The two challenging difficulties are the singular geometry of the domain and the discontinuity of the Hamiltonian. As far as discontinuous Hamiltonians are concerned, these results seem to be new. They are applied to the study of some models arising in traffic flows. The techniques developed in the present article provide new powerful tools for the analysis of such problems. (10.1051/cocv/2012002)
  DOI : 10.1051/cocv/2012002
• Conjugate-cut loci and injectivity domains on two-spheres of revolution
  
  • Bonnard Bernard
  • Caillau Jean-Baptiste
  • Janin Gabriel
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2013, 19 (2), pp.533-554. In a recent article \cite{BCST2009}, we relate the computation of the conjugate and cut loci of a family of metrics on two-spheres of revolution whose polar form is $g=\d\vp^{2}+m(\vp)\d\th^{2}$ to the period mapping of the $\vp$-variable. One purpose of this article is to use this relation to evaluate the cut and conjugate loci for a family of metrics arising as deformation of the round sphere and to determine the convexity properties of the injectivity domains of such metrics related to applications to optimal control in space mechanics, quantum control and optimal transport. (10.1051/cocv/2012020)
  DOI : 10.1051/cocv/2012020
• Domain decomposition for the neutron SPN equations
  
  • Jamelot Erell
  • Ciarlet Patrick
  • Baudron Anne-Marie
  • Lautard Jean-Jacques
  , 2013. Studying numerically the steady state of a nuclear core reactor is expensive, in terms of memory storage and computational time. In order to address both requirements, one can use a domain decomposition method, implemented on a parallel computer. We present here such a method applied to the neutron SPN equations, which are an approximation of the transport neutron equation. This method is based on the Schwarz iterative algorithm with optimized Robin interface conditions to handle communications. From a computational point of view, this method is rather easy to implement. We give some numerical results in highly heterogeneous 3D configurations. Computations are carried out with the MINOS solver, which is a multigroup SPN solver of the APOLLO3® neutronics code. Numerical experiments show that the method is robust and efficient, and that our choice of the Robin parameters is satisfactory.no abstract
• Plasmonic cavity modes: black-hole phenomena captured by Perfectly Matched Layers
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Carvalho Camille
  • Chesnel Lucas
  • Ciarlet Patrick
  , 2013. no abstract
• Scalar transmission problems between dielectrics and metamaterials: T-coercivity for the Discontinuous Galerkin approach.
  
  • Chung Eric T.
  • Ciarlet Patrick
  Journal of Computational and Applied Mathematics, Elsevier, 2013, 239, pp.189--207. no abstract
• A Dirichlet-to-Neumann approach for the exact computation of guided modes in photonic crystal waveguides
  
  • Fliss Sonia
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2013, 35 (2), pp.B438 - B461. This work deals with one-dimensional infinite perturbation---namely, line defects---in periodic media. In optics, such defects are created to construct an (open) waveguide that concentrates light. The existence and computation of the eigenmodes is a crucial issue. This is related to a self-adjoint eigenvalue problem associated to a PDE in an unbounded domain (in the directions orthogonal to the line defect), which makes both the analysis and the computations more complex. Using a Dirichlet-to-Neumann approach, we show that this problem is equivalent to one set on a small neighborhood of the defect. Contrary to existing methods, this one is exact, but there is a price to be paid: the reduction of the problem leads to a nonlinear eigenvalue problem of a fixed point nature. © 2013, Society for Industrial and Applied Mathematics (10.1137/12086697X)
  DOI : 10.1137/12086697X