Publications

2007

• Characterization of the kernel of the operator CURL CURL
  
  • Ciarlet Philippe G.
  • Ciarlet Patrick
  • Geymonat Giuseppe
  • Krasucki Françoise
  Comptes Rendus. Mathématique, Académie des sciences (Paris), 2007, 344 (série I), pp.305-308. In a simply-connected domain Ω in R3, the kernel of the operator CURLCURL acting on symmetric matrix fields from L2s (Ω) to H−2 s (Ω) coincides with the space of linearized strain tensor fields. For not simply-connected domains, Volterra has characterized this kernel for smooth fields. Here we extend this result for domains with a Lipschitz-continuous boundary for fields in L2s (Ω). To cite this article: P.G. Ciarlet et al., C. R. Acad. Sci. Paris, Ser. I 344 (2007). © 2007 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved. (10.1016/j.crma.2007.01.001)
  DOI : 10.1016/j.crma.2007.01.001
• Multiscaled asymptotic expansions for the electric potential: Surface charge densities and electric fields at rounded corners
  
  • Ciarlet Patrick
  • Kaddouri Samir
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2007, 17 (6), pp.845-876. We are interested in computing the charge density and the electric field at the rounded tip of an electrode of small curvature. As a model, we focus on solving the electrostatic problem for the electric potential. For this problem, Peek's empirical formulas describe the relation between the electric field at the surface of the electrode and its curvature radius. However, it can be used only for electrodes with either a purely cylindrical, or a purely spherical, geometrical shape. Our aim is to justify rigorously these formulas, and to extend it to more general, two-dimensional, or three-dimensional axisymmetric, geometries. With the help of multiscaled asymptotic expansions, we establish an explicit formula for the electric potential in geometries that coincide with a cone at infinity. We also prove a formula for the surface charge density, which is very simple to compute with the Finite Element Method. In particular, the meshsize can be chosen independently of the curvature radius. We illustrate our mathematical results by numerical experiments. © World Scientific Publishing Company. (10.1142/s0218202507002133)
  DOI : 10.1142/s0218202507002133
• Typer la désérialisation sans sérialiser les types
  
  • Henry Grégoire
  • Mauny Michel
  • Chailloux Emmanuel
  Revue des Sciences et Technologies de l'Information - Série TSI : Technique et Science Informatiques, Lavoisier, 2007, 26 (9), pp.1067-1090. In this paper, we propose a way of assigning static type information to unmarshalling functions and we describe a verification technique for unmarshalled data that preserves the execution safety provided by static type checking. This technique, whose correctness is proven, relies on singleton types whose values are transmitted to unmarshalling routines at runtime, and on an efficient checking algorithm able to deal with sharing and cycles.
• Using a Mixed Integer Quadratic Programming Solver for the Unconstrained Quadratic 0-1 Problem
  
  • Billionnet Alain
  • Elloumi Sourour
  Mathematical Programming Computation, Springer, 2007, 109 (1), pp.55-68. Abstract In this paper, we consider problem (P) of minimizing a quadratic function q(x)=x t Qx+c t x of binary variables. Our main idea is to use the recent Mixed Integer Quadratic Programming (MIQP) solvers. But, for this, we have to first convexify the objective function q(x). A classical trick is to raise up the diagonal entries of Q by a vector u until (Q+diag(u)) is positive semidefinite. Then, using the fact that x i 2=x i, we can obtain an equivalent convex objective function, which can then be handled by an MIQP solver. Hence, computing a suitable vector u constitutes a preprocessing phase in this exact solution method. We devise two different preprocessing methods. The first one is straightforward and consists in computing the smallest eigenvalue of Q. In the second method, vector u is obtained once a classical SDP relaxation of (P) is solved. We carry out computational tests using the generator of (Pardalos and Rodgers, 1990) and we compare our two solution methods to several other exact solution methods. Furthermore, we report computational results for the max-cut problem. (10.1007/s10107-005-0637-9)
  DOI : 10.1007/s10107-005-0637-9
• Non-Spurious Spectral Like Element Methods for Maxwell's equations
  
  • Cohen Gary
  • Duruflé Marc
  Journal of Computational Mathematics -International Edition-, Global Science Press, 2007, pp.282-304. In this paper, we give the state of the art for the so called "mixed spectral elements" for Maxwell's equations. Several families of elements, such as edge elements and discontinuous Galerkin methods (DGM) are presented and discussed. In particular, we show the need of introducing some numerical dissipation terms to avoid spurious modes in these methods. Such terms are classical for DGM but their use for edge element methods is a novel approach described in this paper. Finally, numerical experiments show the fast and low-cost character of these elements.
• The singularity expansion method applied to the transient motions of a floating elastic plate
  
  • Hazard Christophe
  • Loret François
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2007, 41 (5), pp.925-943. In this paper we propose an original approach for the simulation of the time-dependent response of a floating elastic plate using the so-called Singularity Expansion Method. This method consists in computing an asymptotic behaviour for large time obtained by means of the Laplace transform by using the analytic continuation of the resolvent of the problem. This leads to represent the solution as the sum of a discrete superposition of exponentially damped oscillating motions associated to the poles of the analytic continuation called resonances of the system, and a low frequency component associated to a branch point at frequency zero. We present the mathematical analysis of this method for the two-dimensional sea-keeping problem of a thin elastic plate (ice floe, floating runway, ...) and provide some numerical results to illustrate and discuss its efficiency. © EDP Sciences, SMAI 2007. (10.1051/m2an:2007040)
  DOI : 10.1051/m2an:2007040
• A stability estimate for ill-posed elliptic Cauchy problems in a domain with corners
  
  • Bourgeois Laurent
  Comptes Rendus. Mathématique, Académie des sciences (Paris), 2007, 345 (7), pp.385-390. We prove in this Note a stability estimate for ill-posed elliptic Cauchy problems in a domain with corners. This result completes an earlier result obtained for a smooth domain. To cite this article: L. Bourgeois, C. R. Acad. Sci. Paris, Ser. I 345 (2007). © 2007 Académie des sciences. (10.1016/j.crma.2007.09.014)
  DOI : 10.1016/j.crma.2007.09.014
• Convergence analysis of the Jacobi-Davidson method applied to a generalized eigenproblem
  
  • Hechme Grace
  Comptes Rendus. Mathématique, Académie des sciences (Paris), 2007, 345 (5), pp.293-296. In this Note we consider the Jacobi-Davidson method applied to a nonsymmetric generalized eigenproblem. We analyze the convergence behavior of the method when the linear systems involved, known as the correction equations, are solved approximately. Our analysis also exhibits quadratic convergence when the corrections are solved exactly. To cite this article: G. Hechme, C. R. Acad. Sci. Paris, Ser. I 345 (2007). © 2007 Académie des sciences. (10.1016/j.crma.2007.07.003)
  DOI : 10.1016/j.crma.2007.07.003
• Locating an obstacle in a 3D finite depth ocean using the convex scattering support
  
  • Bourgeois Laurent
  • Chambeyron Colin
  • Kusiak Steven
  Journal of Computational and Applied Mathematics, Elsevier, 2007, 204 (2 SPEC. ISS.), pp.387-399. We consider an inverse scattering problem in a 3D homogeneous shallow ocean. Specifically, we describe a simple and efficient inverse method which can compute an approximation of the vertical projection of an immersed obstacle. This reconstruction is obtained from the far-field patterns generated by illuminating the obstacle with a single incident wave at a given fixed frequency. The technique is based on an implementation of the theory of the convex scattering support [S. Kusiak, J. Sylvester, The scattering support, Commun. Pure Appl. Math. (2003) 1525-1548]. A few examples are presented to show the feasibility of the method. © 2006 Elsevier B.V. All rights reserved. (10.1016/j.cam.2006.01.045)
  DOI : 10.1016/j.cam.2006.01.045
• Generalized formulations of Maxwell's equations for numerical Vlasov-Maxwell simulations
  
  • Ciarlet Patrick
  • Barthelmé Régine
  • Sonnendrücker Eric
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2007, 17 (5), pp.657--680. (10.1142/S0218202507002066)
  DOI : 10.1142/S0218202507002066
• Spectral theory for an elastic thin plate floating on water of finite depth
  
  • Hazard Christophe
  • Meylan Michael H.
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2007, 68 (3), pp.629-647. The spectral theory for a two-dimensional elastic plate floating on water of finite depth is developed (this reduces to a floating rigid body or a fixed body under certain limits). Two spectral theories are presented based on the first-order and second-order formulations of the problem. The first-order theory is valid only for a massless plate, while the second-order theory applies for a plate with mass. The spectral theory is based on an inner product (different for the first- and second-order formulations) in which the evolution operator is self-adjoint. This allows the time-dependent solution to be expanded in the eigenfunctions of the self-adjoint operator which are nothing more than the single frequency solutions. We present results which show that the solution is the same as those found previously when the water depth is shallow, and show the effect of increasing the water depth and the plate mass. © 2007 Society for Industrial and Applied Mathematics. (10.1137/060665208)
  DOI : 10.1137/060665208
• Integrability of Bianchi Universes in scalar tensor theory of gravitation
  
  • Perez Jérôme
  • Larena Julien
  Classical and Quantum Gravity, IOP Publishing, 2007, 24 (11), pp.2901. In this paper, we develop a method based on the analysis of the Kovalewski exponents to study the integrability of anisotropic and homogeneous Universes. The formalism is developed in scalar-tensor gravity, the general relativistic case appearing as a special case of this larger framework. Then, depending on the rationality of the Kovalewski exponents, the different models, both in the vacuum and in the presence of a barotropic matter fluid, are classified, and their integrability is discussed. (10.1088/0264-9381/24/11/008)
  DOI : 10.1088/0264-9381/24/11/008
• Augmented Galerkin Schemes for the Numerical Solution of Scattering by Small Obstacles.
  
  • Claeys Xavier
  • Collino Francis
  , 2007. Dans le contexte de la propagation des ondes electromagnétiques, nous nous intéressons au problème de diffraction par des fils minces parfaitement conducteurs. Si l'on suppose que leur épaisseur est bien plus petite que la longueur d'onde caractéritique de l'onde incidente, il n'est pas posible de prendre en compte des fils minces sans faire face à un problème de verrouillage numérique. Le modèle de Holland, largement utilisé dans les codes différences finis, fournit une solution pragmatique à ce problème, en modifiant le schéma numérique sur quelques noeuds du maillage avoisinant les fils. Jusqu'à présent ce modèle n'a pas re\c cu de justification théorique solide, et il implique un paramètre appelé l'inductance linéique, qu'il doit être choisi suivant des considértions heuristiques. Nous nous intéressons ici au problème modèle de la diffraction acourtique par un petit obstacle, avec condition de Dirichlet au bord, en deux dimensions dans un milieu homogène. Nous présentons et analysons un schéma numérique qui est compatible avec les méthodes éléments finis standards (sans raffinement de maillage) et ne souffre de verrouillage numérique. Ce schéma mélange des techniques d'analyse asymptotique avec une formulation de type domaine fictif. Suivant les résultats que nous démontrons sur ce schéma, nous aboutissons à une généralisation du modèle de Holland et à un calcul automatique de l'inductance linéique. Notre analyse amène, à notre connaissance, à la première justification théorique de ce type de modèle.
• Asymptotic analysis for the solution to the Helmholtz problem in the exterior of a finite thin straight wire
  
  • Claeys Xavier
  , 2007. In this document we are interested in the solution of the Helmholtz equation with Dirichlet boundary condition in the exterior of a thin elongated body. We suppose that the geometry is well described in ellipsoidal coordinates. We propose an asymptotic analysis of this problem, using matched expansions. This leads to the construction of an approximate field with more explicit expression. The approximate field is composed of the first terms of the asymptotic expansion of the exact solution. Our study also leads to a validation of an acoustic version of the Pocklington's equation.
• Anti-dissipative schemes for advection and application to Hamilton-Jacobi-Bellmann equations
  
  • Bokanowski Olivier
  • Zidani Hasnaa
  Journal of Scientific Computing, Springer Verlag, 2007, 30 (1), pp.1-33. We propose two new antidiffusive schemes for advection (or linear transport), one of them being a mixture of Roe's Super-Bee scheme and of the "Ultra-Bee" scheme. We show how to apply these schemes to treat time-dependent first order Hamilton-Jacobi-Bellman equations with discontinuous initial data, possibly infinitely-valued. Numerical tests are proposed, in one and two space dimensions, in order to validate the methods. (10.1007/s10915-005-9017-0)
  DOI : 10.1007/s10915-005-9017-0
• Asymptotic expansion of highly conductive thin sheets
  
  • Schmidt Kersten
  • Tordeux Sébastien
  , 2007, 7 issue 1, pp.2040011-2040012. (10.1002/pamm.200700278)
  DOI : 10.1002/pamm.200700278
• Continuous Galerkin methods for solving the time-dependent Maxwell equations in 3D geometries
  
  • Ciarlet Patrick
  • Jamelot Erell
  Journal of Computational Physics, Elsevier, 2007, 226 (1), pp.1122-1135. A few years ago, Costabel and Dauge proposed a variational setting, which allows one to solve numerically the time-harmonic Maxwell equations in 3D geometries with the help of a continuous approximation of the electromagnetic field. In this paper, we investigate how their framework can be adapted to compute the solution to the time-dependent Maxwell equations. In addition, we propose some extensions, such as the introduction of a mixed variational setting and its discretization, to handle the constraint on the divergence of the field. © 2007 Elsevier Inc. All rights reserved. (10.1016/j.jcp.2007.05.029)
  DOI : 10.1016/j.jcp.2007.05.029