Publications

2008

• Analyse asymptotique et numérique de la diffraction d'ondes par des fils minces
  
  • Claeys Xavier
  , 2008. Cette thèse traite de la modélisation de la propagation d'ondes dans des milieux comportant des fils minces i.e. dont l'épaisseur est bien plus petite que la longueur d'onde. En appliquant la méthode des développements raccordés, nous dérivons un développement de la solution de l'équation de Helmholtz en 2D autour d'un petit obstacle avec condition de Dirichlet sur le bord et proposons un modèle approché dans lequel intervient une condition de Dirichlet moyennée. Par ailleurs nous proposons et analysons deux méthodes numériques non standard pour en calculer la solution avec précision : l'une est adaptée de la méthode de la fonction singulière et l'autre est une version scalaire de la méthode de Holland. Nous démontrons la consistance de ces méthodes. Nous effectuons ensuite le même travail en 3D pour le problème de Helmholtz avec condition de Dirichlet sur le bord d'un objet filiforme dont les pointes sont arrondies ellipsoïdalement. Nous dérivons également un modèle approché dont l'étude mène à une justification théorique de l'équation de Pocklington dans sa version scalaire.
• Efficient methods for computing spectral projectors for linearized hydrodynamic equations
  
  • Hechme Grace
  • Nechepurenko Yuri
  • Sadkane Miloud
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2008, 31 (1), pp.667-686. This paper presents efficient methods for computing the spectral projectors for hydrodynamic equations, linearized at a steady state and approximated with respect to space. The focus is on the spectral projectors corresponding to a given part of the finite spectrum. In the case when the size of the problem is not too large, a QR-based method is proposed and compared with the $QZ$ method. In the large scale case, two variants of the Jacobi-Davidson method, with a deflation procedure, are developed. In both cases, the computed spectral projectors can be used to construct low-order models suited for the context of hydrodynamic stability. Numerical results are reported. (10.1137/050648122)
  DOI : 10.1137/050648122
• Construction and analysis of improved Kirchoff conditions for acoustic wave propagation in a junction of thin slots
  
  • Joly Patrick
  • Semin Adrien
  ESAIM: Proceedings, EDP Sciences, 2008, 25, pp.44-67. In this paper, we analyze via the theory of matched asymptotics the propagation of a time harmonic acoustic wave in a junction of two thin slots. This allows us to propose improved Kirchoff conditions for the 1D limit problem, These conditions are analyzed and validated numerically.
• Identification de cavités par la méthode de sensibilité topologique en élastodynamique temporelle
  
  • Bellis Cédric
  • Bonnet Marc
  , 2008.
• Application of Cagniard de Hoop Method to the Analysis of Perfectly Matched Layers
  
  • Diaz Julien
  • Joly Patrick
  , 2008. We show how Cagniard de Hoop method can be used, first to obtain error estimates for the Perfectly Matched Layers in acoustics (PML), then to understand the instabilities of the PML when applied to aeroacousics. The principle of the methods consists in applying to the equations a Laplace transform in time and a Fourier transform in one space variable to obtain an ordinary differential equation which can be explicitely solved.
• Asymptotic expansion of highly conductive thin sheets
  
  • Schmidt Kersten
  • Tordeux Sébastien
  PAMM, Wiley-VCH Verlag, 2008, 7, pp.2040011-2040012. Sensitive measurement and control equipment are protected from disturbing electromagnetic fields by thin shielding sheets. Alternatively to discretisation of the sheets, the electromagnetic fields are modeled only in the surrounding of the layer taking them into account with the so called Generalised Impedance Boundary Conditions. We study the shielding effect by means of the model problem of a diffusion equation with additional dissipation in the curved thin sheet. We use the asymptotic expansion techniques to derive a limit problem, when the thickness of the sheet $\varepsilon$ tends to zero, as well as the models for contribution to the solution of higher order in $\varepsilon$. These problems are posed in limit area of vanishing $\varepsilon$ with condition for the jump of the solution and it's normal derivative, which avoid to mesh the computational domain, even locally, at the scale of $\varepsilon$. We derive the problems for arbitrary order and show their existence and uniqueness. Numerical experiments for the problems up to second order show the asymptotic convergence of the solution of right order in mean of the thickness parameter $\varepsilon$.
• Estimating the eddy-current modelling error
  
  • Schmidt Kersten
  • Sterz Oliver
  • Hiptmair Ralf
  IEEE Transactions on Magnetics, Institute of Electrical and Electronics Engineers, 2008, 44 (6), pp.686-689. The eddy-current model is an approximation of the full Maxwell equations. We will give estimates for the modeling error and show how the constants in the estimates are influenced by the geometry of the problem. Additionally, we analyze the asymptotic behavior of the modeling error when the angular frequency tends to zero. The theoretical results are complemented by numerical examples using high order finite elements. These demonstrate that the estimates are sharp. Hence, this work delivers a mathematical basis for assessing the scope of the eddy-current model. (10.1109/TMAG.2008.915834)
  DOI : 10.1109/TMAG.2008.915834
• A non-iterative FEM-based cavity identification method using topological sensitivity for 2D and 3D time domain elastodynamics
  
  • Bellis Cédric
  • Bonnet Marc
  , 2008. This communication addresses the application of topological sensitivity to the numerical solution of cavity identification in elastic media. The topological sensitivity analysis arises in connection with the investigation of the asymptotic behaviour of the featured cost functional (here introduced as a means of formulating the inverse problem in terms of a minimisation) with respect to the creation of a cavity of infinitesimal radius and prescribed location in an otherwise cavity-free solid. Initially developed for the topological optimization of structures, this method provides a non-iterative computational tool for constructing a reliable void indicator function, as previously discussed in e.g. [1,2]. The cost functional used here is classically based on exploiting data about the boundary traces of the mechanical fields arising in wave-imaging processes. It quantifies the gap between quantities (e.g. dis- placements) based on a trial topology domain and on a reference domain. In practice, the reference quantities can be provided by experimental measurements or by numerical simulations. Such problems involve naturally integral formulations. The framework of the topological derivative, ie owning to the infinitesimal size of a cavity, of general functionals is presented in [3] in the linear elasticity case. More details can be found in [2] on the mathematical developements which leads to an analytical first order asymptotic expansion of cost functionals in a frequency domain. The results presented here use the derivation technics based on the use of an adjoint state. This method allows to deal with the topological gradient of general functionals with high simplicity and efficiency. Our aim is to illustrate the efficiency of such non-iterative identification technique implemented in a conventional computational framework (here, the classical displacement-based finite element method together with a Newmark time-stepping algorithm). Results of numerical experiments will be presented for 2-D and 3-D time-domain elastodynamic cases, based on topological sensitivity formulas given in [1], in order to demonstrate the efficiency of the approach. Dynamical simulations will highlight the mechanisms underlying identification methods based on topological sensitivity. As well as other meth- ods such as the linear sampling method [4] (not yet implemented for time-domain problems, to the best of our knowledge), such approach is demonstrated through numerical experiments to provide qualita- tively good identification results while being computationally much more economical than ordinary, iterative, inversion procedures.
• Matching of asymptotic expansions for waves propagation in media with thin slots. II. The error estimates
  
  • Joly Patrick
  • Tordeux Sébastien
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2008, 42 (2), pp.193--221. We are concerned with a 2D time harmonic wave propagation problem in a medium including a thin slot whose thickness $\epsilon$ is small with respect to the wavelength. In Part I [P. Joly and S. Tordeux, Multiscale Model. Simul. 5 (2006), no. 1, 304--336 (electronic); MR2221320 (2007e:35041)], we derived formally an asymptotic expansion of the solution with respect to $\epsilon$ using the method of matched asymptotic expansions. We also proved the existence and uniqueness of the terms of the asymptotics. In this paper, we complete the mathematical justification of our work by deriving optimal error estimates between the exact solutions and truncated expansions at any order. (10.1051/m2an:2008004)
  DOI : 10.1051/m2an:2008004
• Identification of generalized impedance boundary conditions in inverse scattering problems
  
  • Bourgeois Laurent
  • Haddar Houssem
  , 2008, pp.27. In the context of scattering problems in the harmonic regime, we consider the problem of identification of some Generalized Impedance Boundary Conditions (GIBC) at the boundary of an object (which is supposed to be known) from far field measurements associated with a single incident plane wave at a fixed frequency. The GIBCs can be seen as approximate models for thin coatings, corrugated surfaces or highly absorbing media. After pointing out that uniqueness does not hold in the general case, we propose some additional assumptions for which uniqueness can be restored. We also consider the question of stability when uniqueness holds. We prove in particular Lipschitz stability when the impedance parameters belong to a compact set. We also extend local stability results to the case of back-scattering data.
• Conditional stability for ill-posed elliptic Cauchy problems : the case of Lipschitz domains (part II)
  
  • Bourgeois Laurent
  • Dardé Jérémi
  , 2008. This paper is devoted to a conditional stability estimate related to the ill-posed Cauchy problems for the Laplace's equation in domains with Lipschitz boundary. It completes the results obtained in \cite{bourgeois1} for domains of class $C^{1,1}$. This estimate is established by using an interior Carleman estimate and a technique based on a sequence of balls which approach the boundary. This technique is inspired from \cite{alessandrini}. We obtain a logarithmic stability estimate, the exponent of which is specified as a function of the boundary's singularity. Such stability estimate induces a convergence rate for the method of quasi-reversibility introduced in \cite{lions} to solve the Cauchy problems. The optimality of this convergence rate is tested numerically, precisely a discretized method of quasi-reversibility is performed by using a nonconforming finite element. The obtained results show very good agreement between theoretical and numerical convergence rates.
• Conditional stability for ill-posed elliptic Cauchy problems : the case of $C^{1,1}$ domains (part I)
  
  • Bourgeois Laurent
  , 2008. This paper is devoted to a conditional stability estimate related to the ill-posed Cauchy problems for the Laplace's equation in domains with $C^{1,1}$ boundary. It is an extension of an earlier result for domains of class $C^\infty$. Our estimate is established by using a global Carleman estimate near the boundary in which the exponential weight depends on the distance function to the boundary. Furthermore, we prove that this stability estimate is nearly optimal and induces a nearly optimal convergence rate for the method of quasi-reversibility to solve the ill-posed Cauchy problems.
• A new approach for approximating linear elasticity problems
  
  • Ciarlet Philippe G.
  • Ciarlet Patrick
  Comptes Rendus. Mathématique, Académie des sciences (Paris), 2008, 346 (5-6), pp.351-356. In this Note, we present and analyze a new method for approximating linear elasticity problems in dimension two or three. This approach directly provides approximate strains, i.e., without simultaneously approximating the displacements, in finite element spaces where the Saint Venant compatibility conditions are exactly satisfied in a weak form. To cite this article: P.G. Ciarlet, P. Ciarlet, Jr., C. R. Acad. Sci. Paris, Ser. I 346 (2008). © 2008 Académie des sciences. (10.1016/j.crma.2008.01.014)
  DOI : 10.1016/j.crma.2008.01.014
• The linear sampling method in a waveguide: A modal formulation
  
  • Bourgeois Laurent
  • Lunéville Éric
  Inverse Problems, IOP Publishing, 2008, 24 (1). This paper concerns the linear sampling method used to retrieve obstacles in a 2D or 3D acoustic waveguide. The classical mathematical results concerning the identifiability of the obstacle and the justification of the inverse method are established for this particular geometry. Our main concern is to derive a modal formulation of the linear sampling method that is well adapted to the waveguide configuration. In particular, thanks to such formulation, we highlight the fact that finding some obstacles from remote scattering data is more delicate in a waveguide than in free space. Indeed, the presence of evanescent modes increases the ill posedness of the inverse problem. However, we show that the numerical reconstruction of obstacles by using the far field is feasible, even by using a few incident waves. © 2008 IOP Publishing Ltd. (10.1088/0266-5611/24/1/015018)
  DOI : 10.1088/0266-5611/24/1/015018
• An improved multimodal approach for non-uniform acoustic waveguides
  
  • Hazard Christophe
  • Lunéville Éric
  IMA Journal of Applied Mathematics, Oxford University Press (OUP), 2008, 73 (4), pp.668-690. This paper explores from the point of view of numerical analysis a method for solving the acoustic time-harmonic wave equation in a locally non-uniform 2D waveguide. The multimodal method is based on a spectral description of the acoustic field in each transverse section of the guide, using Fourier-like series. In the case of sound-hard boundaries, the weak point of the method lies in the poor convergence of such series due to a poor approximation of the field near the boundaries. A remedy was proposed by Athanassoulis & Belibassakis (1999, J. Fluid Mech., 389, 275-301), where the convergence is improved thanks to the use of enhanced series. The present paper proposes a theoretical analysis of their idea and studies further improvements. For a general model which includes different kinds of non-uniformity (varying cross section, bend and inhomogeneous medium), a hybrid spectral/variational formulation is introduced. Error estimates are provided for a semi-discretized problem which concerns the effect of spectral truncation. These estimates are confirmed by numerical results. © The Author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. (10.1093/imamat/hxn006)
  DOI : 10.1093/imamat/hxn006
• Computing electromagnetic eigenmodes with continuous Galerkin approximations
  
  • Ciarlet Patrick
  • Hechme Grace
  Computer Methods in Applied Mechanics and Engineering, Elsevier, 2008, 198 (2), pp.358-365. Costabel and Dauge proposed a variational setting to solve numerically the time-harmonic Maxwell equations in 3D polyhedral geometries, with a continuous approximation of the electromagnetic field. In order to remove spurious eigenmodes, three computational strategies are then possible. The original method, which requires a parameterization of the variational formulation. The second method, which is based on an a posteriori filtering of the computed eigenmodes. And the third method, which uses a mixed variational setting so that all spurious modes are removed a priori. In this paper, we discuss the relative merits of the approaches, which are illustrated by a series of 3D numerical examples. © 2008 Elsevier B.V. All rights reserved. (10.1016/j.cma.2008.08.005)
  DOI : 10.1016/j.cma.2008.08.005
• Higher order time stepping for second order hyperbolic problems and optimal CFL conditions
  
  • Gilbert Jean Charles
  • Joly Patrick
  , 2008, 16, pp.67-93. We investigate explicit higher order time discretizations of linear second order hyperbolic problems. We study the even order (2m) schemes obtained by the modified equation method. We show that the corresponding CFL upper bound for the time step remains bounded when the order of the scheme increases. We propose variants of these schemes constructed to optimize the CFL condition. The corresponding optimization problem is analyzed in detail and the analysis results in a specific numerical algorithm. The corresponding results are quite promising and suggest various conjectures. (10.1007/978-1-4020-8758-5_4)
  DOI : 10.1007/978-1-4020-8758-5_4
• Local time stepping and discontinuous Galerkin methods for symmetric first order hyperbolic systems
  
  • Ezziani Abdelaâziz
  • Joly Patrick
  Journal of Computational and Applied Mathematics, Elsevier, 2008. We present a new non conforming space-time mesh refinement method for symmetric first order hyperbolic system. This method is based on the one hand on the use of a conservative higher order discontinuous Galerkin approximation for space discretization and a finite difference scheme in time, on the other hand on appropriate discrete transmission conditions between the grids. We use a discrete energy technique to drive the construction of the matching procedure between the grids and guarantee the stability of the method.
• A spurious-free space-time mesh refinement for elastodynamics
  
  • Rodríguez Jerónimo
  International Journal for Multiscale Computational Engineering, Begell House, 2008, 6 (3), pp.263-279. We propose a generalization of the space-time mesh refinement technique for elastodynamics presented by 14 to the case where the discretization step (in space and time) on the fine grid is q N times finer than the one on the coarse grid. This method uses the conservation of a discrete energy to ensure the stability under the usual CFL condition. Some numerical examples show that the method is only first order accurate (and thus suboptimai with respect to the second-order interior scheme we have used) when the ratio of refinement is higher than 2. A Fourier analysis of the computed signals exhibits the presence of high-frequency waves (aliasing phenomena) polluting the fields on the fine grid. Those results provide valuable information with which to build a postprocessing by averaging that removes the spurious phenomena. Finally, we introduce a new numerical scheme, computing the postprocessed solution directly. This method is stable and second-order consistent, regardless of the ratio of refinement. Its performance is shown through a numerical simulation of the diffraction of elastic waves by small cracks. © 2008 by Begell House, Inc. (10.1615/intjmultcompeng.v6.i3.60)
  DOI : 10.1615/intjmultcompeng.v6.i3.60
• Spectral elements for the integral equations of time-harmonic Maxwell problems
  
  • Demaldent Édouard
  • Levadoux David
  • Cohen Gary
  IEEE Transactions on Antennas and Propagation, Institute of Electrical and Electronics Engineers, 2008, 56 (9), pp.3001-3010. We present a novel high-order method of moments (MoM) with interpolatory vector functions, on quadrilateral patches. The main advantage of this method is that the Hdiv conforming property is enforced, and at the same time it can be interpreted as a point-based scheme. We apply this method to field integral equations (FIEs) to solve time-harmonic electromagnetic scattering problems. Our approach is applied to the first and second Nédélec families of Hdiv conforming elements. It consists in a specific choice of the degrees of freedom (DOF), made in order to allow a fast integral evaluation. In this paper we describe these two sets of DOF and their corresponding quadrature rules. Sample numerical results on FIE confirm the good properties of our schemes: faster convergence rate and cheap matrix calculation. We also present observations on the choice of the discretization method, depending on the FIE selected. © 2008 IEEE. (10.1109/tap.2008.927551)
  DOI : 10.1109/tap.2008.927551
• Vector and scalar potentials, Poincaré's theorem and Korn's inequality
  
  • Amrouche Chérif
  • Ciarlet Philippe G.
  • Ciarlet Patrick
  , 2008. In this Note, we present several results concerning the vector potentials and the scalar potentials in a bounded, not necessarily simply-connected, three-dimensional domain. We consider also singular potentials corresponding to data in negative order Sobolev spaces. We also give some applications to Poincaré's theorem and to Korn's inequality.
• A new compactness result for electromagnetic waves. Application to the transmission problem between dielectrics and metamaterials
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Ciarlet Patrick
  • Zwölf Carlo Maria
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2008, 18 (9), pp.1605-1631. We consider the time-harmonic Maxwell equations, involving wave transmission between media with opposite sign dielectric and/or magnetic coefficients. We prove that, in the case of sign-shifting dielectric coefficients, the space of electric fields is compactly embedded in L 2. We build a three-field variational formulation equivalent to Maxwell system for sign-shifting magnetic coefficients and show that, under some suitable conditions, the formulation fits into the coercive plus compact framework. © 2008 World Scientific Publishing Company. (10.1142/s0218202508003145)
  DOI : 10.1142/s0218202508003145
• The linear sampling method in a waveguide: A formulation based on modes
  
  • Bourgeois Laurent
  • Lunéville Éric
  Journal of Physics: Conference Series, IOP Science, 2008, 135 (-), pp.012023. This paper concerns the Linear Sampling Method to retrieve obstacles in a 2D or 3D acoustic waveguide. We derive a modal formulation of the LSM which is suitable for the waveguide configuration. Despite the ill-posedness of the inverse problem is increased owing to the evanescent modes, numerical experiments show good reconstruction of obstacles by using the far field. © 2008 IOP Publishing Ltd. (10.1088/1742-6596/135/1/012023)
  DOI : 10.1088/1742-6596/135/1/012023
• A monotonic evaluation of lower bounds for inf-sup stability constants in the frame of reduced basis approximations
  
  • Chen Yanlai
  • Hesthaven Jan S.
  • Maday Yvon
  • Rodríguez Jerónimo
  Comptes Rendus. Mathématique, Académie des sciences (Paris), 2008, 346 (23-24), pp.1295-1300. For accurate a posteriori error analysis of the reduced basis method for coercive and non-coercive problems, a critical ingredient lies in the evaluation of a lower bound for the coercivity or inf-sup constant. In this short Note, we generalize and improve the successive constraint method first presented by Huynh (2007) by providing a monotonic version of this algorithm that leads to both more stable evaluations and fewer offline computations. © 2008 Académie des sciences. (10.1016/j.crma.2008.10.012)
  DOI : 10.1016/j.crma.2008.10.012
• Propagation of an acoustic wave in a junction of two thin slots
  
  • Joly Patrick
  • Semin Adrien
  , 2008, pp.61. In this research report, we analyze via the theory of matched asymptotics the propagation of a time harmonic acoustic wave in a junction of two thin slots. This allows us to propose improved Kirchoff conditions for the 1D limit model. These conditions are analyzed and validated numerically.