Publications

2011

• Entanglement creation in low-energy scattering
  
  • Weder Ricardo
  Physical Review A : Atomic, molecular, and optical physics [1990-2015], American Physical Society, 2011, 84 (6). We study the entanglement creation in the low-energy scattering of two particles in three dimensions, for a general class of interaction potentials that are not required to be spherically symmetric. The incoming asymptotic state, before the collision, is a product of two normalized Gaussian states. After the scattering, the particles are entangled. We take as a measure of the entanglement the purity of one of them. We provide a rigorous explicit computation, with error bound, of the leading order of the purity at low energy. The entanglement depends strongly on the difference of the masses. It takes its minimum when the masses are equal, and it increases rapidly with the difference of the masses. It is quite remarkable that the anisotropy of the potential gives no contribution to the leading order of the purity, in spite of the fact that entanglement is a second-order effect. © 2011 American Physical Society. (10.1103/physreva.84.062320)
  DOI : 10.1103/physreva.84.062320
• Mathematical analysis of the junction of two acoustic open waveguides
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Goursaud Benjamin
  • Hazard Christophe
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2011, 71 (6), pp.2048-2071. The present paper concerns the scattering of a time-harmonic acoustic wave by the junction of two open uniform waveguides, where the junction is limited to a bounded region. We consider a two-dimensional problem for which wave propagation is described by the scalar Helmholtz equation. The main difficulty in the modeling of the scattering problem lies in the choice of conditions which characterize the outgoing behavior of a scattered wave. We use here modal radiation conditions which extend the classical conditions used for closed waveguides. They are based on the generalized Fourier transforms which diagonalize the transverse contributions of the Helmholtz operator on both sides of the junction. We prove the existence and uniqueness of the solution, which seems to be the first result in this context. The originality lies in the proof of uniqueness, which combines a natural property related to energy fluxes with an argument of analyticity with respect to the generalized Fourier variable. © 2011 Society for Industrial and Applied Mathematics. (10.1137/100811374)
  DOI : 10.1137/100811374
• Analyse mathématique et numérique de quelques problèmes d'ondes en milieu périodique
  
  • Coatléven Julien
  , 2011. De nombreux problèmes physiques sont modélisés par des équations aux dérivées partielles posées dans un domaine pour lesquels la géométrie ainsi que les coefficients sont décrits par des fonctions périodiques, hormis dans certaines régions de taille modeste par rapport à celle du domaine d'intérêt (on parle alors de perturbations pour ces régions). Les caractéristiques du problème sortant très souvent du cadre d'application des méthodes d'homogénéisation, nous avons développé des méthodes alternatives tirant parti de la periodicité afin de restreindre le domaine de calcul à des domaines bornés. Pour cela, nous avons généralisé les approches de type Lippmann-Schwinger, ce qui nous permet de traiter le cas de défauts bornés ou le cas de défauts non bornés structurés, la difficulté tenant au fait que l'on ne dispose pas dans le cas d'un milieu périodique quelconque d'une représentation analytique de la solution en l'absence de perturbation (i.e la fonction de Green est inconnue en général). Notre approche repose sur la connaissance des opérateurs de Dirichlet- to-Neumann (DtN) de bandes périodiques non bornés dans une seule direction. Nous traitons deux grandes familles de problèmes, les problèmes harmoniques, pour lesquels les opérateurs DtN dans les bandes sont connus, et les problèmes d'évolution, pour lesquels nous proposons une méthode de construction de ces opérateurs. Nous traitons dans ces deux situations le cas d'une perturbation bornée ou non, puis nous généralisons les techniques de scattering multiple du milieu homogène au cas périodique, afin de pouvoir traiter le cas de plusieurs perturbations.
• Mathematical and numerical modeling of wave propagation in fractal trees
  
  • Joly Patrick
  • Semin Adrien
  Comptes Rendus. Mathématique, Académie des sciences (Paris), 2011, 349 (19-20), pp.1047-1051. We propose and analyze a mathematical model for wave propagation in infinite trees with self-similar structure at infinity. The emphasis is put on the construction and approximation of transparent boundary conditions. © 2011. (10.1016/j.crma.2011.09.008)
  DOI : 10.1016/j.crma.2011.09.008
• High order transmission conditions for thin conductive sheets in magneto-quasistatics
  
  • Schmidt Kersten
  • Tordeux Sébastien
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2011, 45 (6), pp.1115-1140. We propose transmission conditions of order 1, 2 and 3 approximating the shielding behaviour of thin conducting curved sheets for the magneto-quasistatic eddy current model in 2D. This model reduction applies to sheets whose thicknesses $\eps$ are essentially smaller or at the order of the skin depth. The sheet has itself not to be resolved, only its midline is represented by an interface. The computation is directly in one step with almost no additional cost. We prove the well-posedness w.r.t.~to the small parameter $\eps$ and obtain optimal bound for the modelling error outside the sheet of order $\eps^{N+1}$ for the condition of order N. Numerical experiments with high order finite elements for sheets with varying curvature verify the theoretical findings. (10.1051/m2an/2011009)
  DOI : 10.1051/m2an/2011009
• Numerical MicroLocal Analysis in Time domain
  
  • Collino Francis
  • Marmorat Simon
  , 2011. Ce rapport présente une étude de la méthode NMLA á partir de données dans le domaine temporel.
• Sur le calcul numérique des modes non linéaires
  
  • Blanc François
  • Ege Kerem
  • Touzé Cyril
  • Mercier Jean-François
  • Bonnet-Ben Dhia Anne-Sophie
  , 2011, pp.1-6. This paper deals with non linear normal modes (NNM) computation. The partial differential equations (PDE) governing the geometry of the NNM in phase space is solved via a finite difference scheme. The PDE is integrated as a transport equation, the initial conditions of which are searched so as to result in periodic solutions. The solving and optimisation algorithms are tested on a 2 dof system with cubic non-linearities. This example let us discuss the algorithms' convergence and implementation. The results are also compared to computations with more standard continuation methods.
• Indirect controllability of locally coupled wave-type systems and applications
  
  • Alabau-Boussouira Fatiha
  • Léautaud Matthieu
  , 2011. We consider symmetric systems of two wave-type equations only one of them being controlled. The two equations are coupled by zero order terms, localized in part of the domain. We prove an internal and a boundary null-controllability result in any space dimension, provided that both the coupling and the control regions satisfy the Geometric Control Condition. We deduce similar null-controllability results in any positive time for parabolic systems and Schrödinger-type systems under the same geometric conditions on the coupling and the control regions. This includes several examples in which these two regions have an empty intersection.
• Modeling and numerical simulation of a grand piano.
  
  • Chabassier Juliette
  • Joly Patrick
  , 2011, pp.00. We consider a complete model of a piano which accounts for the acoustical behavior of the instrument from excitation to soundand, and we propose a numerical discretisation. The model is described as well as the numerical methods used for its discretisation. Nonlinearities and couplings are treated in such a way that energy techniques ensure numerical stability. Numerical results are presented and compared to measurements.
• Identifying cracks in homogeneous and bimaterial bodies using 3D elastodynamic topological sensitivity
  
  • Bellis Cédric
  • Bonnet Marc
  , 2011.
• Wave-based crack imaging in elastic solids using 3D time-domain topological sensitivity
  
  • Bellis Cédric
  • Bonnet Marc
  , 2011.
• Dynamical identification of cracks in 3D elastic bodies using topological sensitivity
  
  • Bellis Cédric
  • Bonnet Marc
  , 2011.
• Coupling between a Particle In Cell method and a H 1-conform mixed spectral finite element approximation of Maxwell's equations
  
  • Cohen Gary
  • Sinding Alexandre
  , 2011. This paper describes the coupling between a Particle In Cell (PIC) method and a H 1-conform mixed spectral finite element approximation of Maxwell's equations for the approximation of low dense plasmas. It uses the H 1-conform mixed spectral method already described in [2]. As particle methods themselves are a classical subject, we mainly focus on the coupling between a finite element method on an unstructured grid and a PIC method. This subject has already been studied for a coupling between a PIC method and a discontin-uous Galerkin scheme [4], but its still a challenging subject and the rehabilitation of continuous methods to this aim hasn't been studied yet. The critical point lies in the coupling between an Eulerian approximation of the fields on an unstructured grid and a Lagrangian description of particles motion. This point can heavily penalize the global cost of the algorithm, if not taken into account carefully. In the next sections, a few techniques are introduced and compared to each other in order to obtain an efficient coupling algorithm between these two methods.
• Fast and accurate point-based method for time-harmonic maxwell problems involving thin layer materials
  
  • Demaldent Edouard
  • Levadoux David P.
  • Cohen Gary
  Journal of Computational Physics, Elsevier, 2011, 230 (14), pp.5774-5786. We present a high-order hybrid boundary-finite elements method well-suited for solving time-harmonic electromagnetic scattering problems. Actually, this method is specially devoted to perfect electric conductors coated with a thin layer material. On such class of problems this method is shown to be fast and accurate. The fast feature is due to the joint use of finite elements of anisotropic order fitting the layer thickness, and of a point-based boundary element method on the skin. The accuracy is ensured, first by a discretization scheme satisfying the Hcurl-Hdiv conformity required by the integro-differential equation and, secondly, by an adaptive technique of integration based on the detection of some local potential trouble on the geometry such as sharp edges or high dilatation of the elements. This algorithm does not need further information from the user and does not deteriorate the computation time. Numerical examples confirm the efficiency of this approach. © 2011 Elsevier Inc. (10.1016/j.jcp.2011.03.060)
  DOI : 10.1016/j.jcp.2011.03.060
• On simultaneous identification of a scatterer and its generalized impedance boundary condition
  
  • Bourgeois Laurent
  • Chaulet Nicolas
  • Haddar Houssem
  , 2011, pp.28. We consider the inverse scattering problem consisting in the identification of both an obstacle and two functional coefficients of a generalized boundary condition prescribed on its boundary, from far--fields due to several plane waves. After proving a uniqueness result for such inverse problem, we define and compute appropriate derivative of the far--field with respect to an obstacle with non constant impedances. A steepest descent method is then applied to retrieve both the obstacle and the functional impedances from the measured far--fields. The feasability of the method is demonstrated with the help of some 2D numerical experiments.
• On the use of T-coercivity to study the interior transmission eigenvalue problem
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chesnel Lucas
  • Haddar Houssem
  Comptes rendus de l'Académie des sciences. Série I, Mathématique, Elsevier, 2011, 349 (11-12), pp.647-651. (10.1016/j.crma.2011.05.008)
  DOI : 10.1016/j.crma.2011.05.008
• Méthode multipôle rapide multi-niveaux en visco-élastodynamique 3D
  
  • Grasso Eva
  • Chaillat Stéphanie
  • Semblat Jean-François
  • Bonnet Marc
  , 2011, pp.8 p. ; Clé USB. See http://hal.archives-ouvertes.fr/docs/00/59/28/83/ANNEX/r_4VPCS0OY.pdf
• Compact imbeddings in electromagnetism with interfaces between classical materials and meta-materials
  
  • Chesnel Lucas
  • Ciarlet Patrick
  , 2011. In a meta-material, the electric permittivity and/or the magnetic permeability can be negative in given frequency ranges. We investigate the solution of the time-harmonic Maxwell equations in a composite material, made up of classical materials, and meta-materials with negative electric permittivity, in a two-dimensional bounded domain Ω. We study the imbedding of the space of electric fields into L²(Ω)². In particular, we extend the famous result of Weber, proving that it is compact. This result is obtained by studying the regularity of the fields. We first isolate their most singular part, using a decomposition à la Birman-Solomyak. With the help of the Mellin transform, we prove that this singular part belongs to Hˢ(Ω)², for some s > 0. Finally, we show that the compact imbedding result holds as soon as no ratio of permittivities between two adjacent materials is equal to −1.
• Remarks on the stability of Cartesian PMLs in corners
  
  • Bécache Eliane
  • Prieto Andres
  , 2011, pp.18. This work is a contribution to the understanding of the question of stability of Perfectly Matched Layers (PMLs) in corners, at continuous and discrete levels. First, stability results are presented for the Cartesian PMLs associated to a general first-order hyperbolic system. Then, in the context of the pressure-velocity formulation of the acoustic wave propagation, an unsplit PML formulation is discretized with spectral mixed finite elements in space and finite differences in time. It is shown, through the stability analysis of two different schemes, how a bad choice of the time discretization can deteriorate the CFL stability condition. Some numerical results are finally presented to illustrate these stability results.
• On the use of Lamb modes in the linear sampling method for elastic waveguides
  
  • Bourgeois Laurent
  • Le Louër Frédérique
  • Lunéville Éric
  Inverse Problems, IOP Publishing, 2011, 27 (5), pp.055001. (10.1088/0266-5611/27/5/055001)
  DOI : 10.1088/0266-5611/27/5/055001
• Lagrange multipliers in intrinsic elasticity.
  
  • Iosifescu Oana
  • Ciarlet Philippe G.
  • Ciarlet Patrick
  • Sauter Stefan
  • Jun Zou
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2011, 21, pp.651-666. In an intrinsic approach to three-dimensional linearized elasticity, the unknown is the linearized strain tensor field (or equivalently the stress tensor field by means of the constitutive equation), instead of the displacement vector field in the classical approach. We consider here the pure traction problem and the pure displacement problem and we show that, in each case, the intrinsic approach leads to a quadratic minimization problem constrained by Donati-like relations (the form of which depends on the type of boundary conditions considered). Using the Babuška-Brezzi inf-sup condition, we then show that, in each case, the minimizer of the constrained minimization problem found in an intrinsic approach is the first argument of the saddle-point of an ad hoc Lagrangian, so that the second argument of this saddle-point is the Lagrange multiplier associated with the corresponding constraints. Such results have potential applications to the numerical analysis and simulation of the intrinsic approach to three-dimensional linearized elasticity. (10.1142/S0218202511005167)
  DOI : 10.1142/S0218202511005167
• Numerical MicroLocal Analysis Revisited
  
  • Benamou Jean-David
  • Collino Francis
  • Marmorat Simon
  , 2011, pp.62. The report bundles a theoretical and a numerical papier presenting a stable version of the {\tt NMLA} algorithm as well a a new curvature estimation method and a linearization error correction method.
• A multiscale hp-FEM for 2D photonic crystal bands
  
  • Brandsmeier Holger
  • Schmidt Kersten
  • Schwab Christoph
  Journal of Computational Physics, Elsevier, 2011, 230 (2), pp.349-374. A multiscale generalised hp-finite element method (MSFEM) for time harmonic wave propagation in bands of locally periodic media of large, but finite extent, e.g., photonic crystal (PhC) bands, is presented. The method distinguishes itself by its size robustness, i.e., to achieve a prescribed error its computational effort does not depend on the number of periods. The proposed method shows this property for general incident fields, including plane waves incident at a certain angle to the infinite crystal surface, and at frequencies in and outside of the bandgap of the PhC. The proposed MSFEM is based on a precomputed problem adapted multiscale basis. This basis incorporates a set of complex Bloch modes, the eigenfunctions of the infinite PhC, which are modulated by macroscopic piecewise polynomials on a macroscopic FE mesh. The multiscale basis is shown to be efficient for finite PhC bands of any size, provided that boundary effects are resolved with a simple macroscopic boundary layer mesh. The MSFEM, constructed by combing the multiscale basis inside the crystal with some exterior discretisation, is a special case of the generalised finite element method (g-FEM). For the rapid evaluation of the matrix entries we introduce a size robust algorithm for integrals of quasi-periodic micro functions and polynomial macro functions. Size robustness of the present MSFEM in both, the number of basis functions and the computation time, is verified in extensive numerical experiments. © 2010 Elsevier Inc. (10.1016/j.jcp.2010.09.018)
  DOI : 10.1016/j.jcp.2010.09.018
• Time-domain study of the Drude-Born-Fedorov model for a class of heterogeneous chiral materials
  
  • Ciarlet Patrick
  • Legendre Guillaume
  • Nicaise Serge
  , 2011. We deal with the well-posedness of the transient Maxwell equations in a particular class of heterogeneous chiral material modeled by the Drude-Born-Fedorov constitutive relations. A new formulation of the underlying evolution problem allows us to correct a previous result establishing the existence and uniqueness of the electromagnetic fields in a homogeneous medium.
• Accelerated Boundary Element Method for Diffuse Optical Imaging
  
  • Elisee Josias
  • Bonnet Marc
  • Arridge Simon
  Optics Letters, Optical Society of America - OSA Publishing, 2011, 36, pp.4101-4103. The boundary element method (BEM) is a useful tool in Diffuse Optical Imaging (DOI) when modelling large optical regions whose parameters are piecewise constant, but is computationally expensive. We present here an acceleration technique, the single-level Fast Multipole Method, for a highly lossy medium. The enhanced practicability of the BEM in DOI is demonstrated through test examples on single-layer problems, where order of magnitude reduction factors on solution time are achieved, and on a realistic three-layer model of the neonatal head. Our experimental results agree very closely with theoretical predictions of computational complexity. (10.1364/OL.36.004101)
  DOI : 10.1364/OL.36.004101