Publications

2008

• 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
• Singular trajectories of control-affine systems
  
  • Chitour Yacine
  • Jean Frédéric
  • Trélat Emmanuel
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2008, 47 (2), pp.1078--1095. When applying methods of optimal control to motion planning or stabilization problems, some theoretical or numerical difficulties may arise, due to the presence of specific trajectories, namely, singular minimizing trajectories of the underlying optimal control problem. In this article, we provide characterizations for singular trajectories of control-affine systems. We prove that, under generic assumptions, such trajectories share nice properties, related to computational aspects; more precisely, we show that, for a generic system -- with respect to the Whitney topology --, all nontrivial singular trajectories are of minimal order and of corank one. These results, established both for driftless and for control-affine systems, extend previous results. As a consequence, for generic systems having more than two vector fields, and for a fixed cost, there do not exist minimizing singular trajectories. We also prove that, given a control system satisfying the LARC, singular trajectories are strictly abnormal, generically with respect to the cost. We then show how these results can be used to derive regularity results for the value function and in the theory of Hamilton-Jacobi equations, which in turn have applications for stabilization and motion planning, both from the theoretical and implementation issues. (10.1137/060663003)
  DOI : 10.1137/060663003
• Complexity results for the horizontal bar packing problem
  
  • Costa Marie-Christine
  • Jarray Fethi
  • Picouleau Christophe
  Information Processing Letters, Elsevier, 2008, 108 (6), pp.356-359. The paper deals with the problem of packing a set of horizontal bars by taking into account some constraints on the distance between two consecutive bars on the same row. This problem is deeply connected with Discrete Tomography and it finds application in workforce scheduling. We study the complexity of the problem and show that, depending on the kind of constraints considered, the problem can be polynomial or NP-Complete. We analyze in details the case where a minimal distance between consecutive bars is required and propose a greedy algorithm which solves this problem in polynomial time. (10.1016/j.ipl.2008.07.007)
  DOI : 10.1016/j.ipl.2008.07.007
• 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.
• Numerical solution of large-scale Lyapunov equations, Riccati equations, and linear-quadratic optimal control problems
  
  • Benner Peter
  • Li Jing-Rebecca
  • Penzl Thilo
  Numerical Linear Algebra with Applications, Wiley, 2008, 15 (9), pp.755-777. We study large-scale, continuous-time linear time-invariant control systems with a sparse or structured state matrix and a relatively small number of inputs and outputs. The main contributions of this paper are numerical algorithms for the solution of large algebraic Lyapunov and Riccati equations and linearquadratic optimal control problems, which arise from such systems. First, we review an alternating direction implicit iteration-based method to compute approximate low-rank Cholesky factors of the solution matrix of large-scale Lyapunov equations, and we propose a refined version of this algorithm. Second, a combination of this method with a variant of Newton's method (in this context also called Kleinman iteration) results in an algorithm for the solution of large-scale Riccati equations. Third, we describe an implicit version of this algorithm for the solution of linear-quadratic optimal control problems, which computes the feedback directly without solving the underlying algebraic Riccati equation explicitly. Our algorithms are efficient with respect to both memory and computation. In particular, they can be applied to problems of very large scale, where square, dense matrices of the system order cannot be stored in the computer memory. We study the performance of our algorithms in numerical experiments. (10.1002/nla.622)
  DOI : 10.1002/nla.622
• Generalized impedance boundary conditions for scattering problems from strongly absorbing obstacles: the case of Maxwell's equations
  
  • Haddar Houssem
  • Joly Patrick
  • Nguyen Hoai-Minh
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2008, 18 (10), pp.1787-1827. (10.1142/S0218202508003194)
  DOI : 10.1142/S0218202508003194
• A characterization of singular electromagnetic fields by an inductive approach
  
  • Assous F.
  • Ciarlet Patrick
  • Garcia E.
  International Journal of Numerical Analysis and Modeling, Institute for Scientific Computing and Information, 2008, 5 (3), pp.491-515. In this article, we are interested in the mathematical modeling of singular electromagnetic fields, in a non-convex polyhedral domain. We first describe the local trace (i. e. defined on a face) of the normal derivative of an L2 function, with L2 Laplacian. Among other things, this allows us to describe dual singularities of the Laplace problem with homogeneous Neumann boundary condition. We then provide generalized integration by parts formulae for the Laplace, divergence and curl operators. With the help of these results, one can split electromagnetic fields into regular and singular parts, which are then characterized. We also study the particular case of divergence-free and curl-free fields, and provide non-orthogonal decompositions that are numerically computable. © 2008 Institute for Scientific Computing and Information.