Publications

2011

• An alternative to Dirichlet-to-Neumann maps for waveguides
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Legendre Guillaume
  Comptes rendus de l'Académie des sciences. Série I, Mathématique, Elsevier, 2011, 349 (17-18), pp.1005-1009. We are interested by the treatment of the radiation condition at infinity for the numerical solution of a problem set in an unbounded waveguide. We propose an alternative to the classical approach involving a modal expression of Dirichlet-to-Neumann (DtN) operators. This method is particularly simple to implement since it only requires the solution of boundary value problems with local boundary conditions. The corresponding approximate solution is comparable in accuracy to the one obtained by truncating the infinite series in the DtN maps. (10.1016/j.crma.2011.08.006)
  DOI : 10.1016/j.crma.2011.08.006
• 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
• Coupling discontinuous Galerkin methods and retarded potentials for transient wave propagation on unbounded domains
  
  • Abboud Toufic
  • Joly Patrick
  • Rodríguez Jerónimo
  • Terrasse Isabelle
  Journal of Computational Physics, Elsevier, 2011, 230 (15), pp.5877-5907. This work deals with the numerical simulation of wave propagation on unbounded domains with localized heterogeneities. To do so, we propose to combine a discretization based on a discontinuous Galerkin method in space and explicit finite differences in time on the regions containing heterogeneities with the retarded potential method to account the unbounded nature of the computational domain. The coupling formula enforces a discrete energy identity ensuring the stability under the usual CFL condition in the interior. Moreover, the scheme allows to use a smaller time step in the interior domain yielding to quasi-optimal discretization parameters for both methods. The aliasing phenomena introduced by the local time stepping are reduced by a post-processing by averaging in time obtaining a stable and second order consistent (in time) coupling algorithm. We compute the numerical rate of convergence of the method for an academic problem. The numerical results show the feasibility of the whole discretization procedure. © 2011 Elsevier Inc. (10.1016/j.jcp.2011.03.062)
  DOI : 10.1016/j.jcp.2011.03.062
• Time reversal and scattering theory for time-dependent acoustic waves in a homogeneous medium
  
  • Ben-Amar Chokri
  • Hazard Christophe
  IMA Journal of Applied Mathematics, Oxford University Press (OUP), 2011, 76, pp.938-955.
• Numerical solution of Maxwell's equations in axisymmetric domains with the Fourier Singular Complement Method
  
  • Ciarlet Patrick
  • Labrunie Simon
  Journal of Difference Equations and Applications, Taylor & Francis, 2011, 3, pp.113-155. We present an efficient method for computing numerically the solution to the time-dependent Maxwell equations in an axisymmetric domain, with arbitrary (not necessarily axisymmetric) data. The method is an extension of those introduced in [20] for Poisson's equation, and in [4] for Maxwell's equations in the fully axisymmetric setting (i.e., when the data is also axisymmetric). It is based on a Fourier expansion in the azimuthal direction, and on an improved variant of the Singular Complement Method in the meridian section. When solving Maxwell's equations, this method relies on continuous approximations of the fields, and it is both H(curl)- and H(div)-conforming. Also, it can take into account the lack of regularity of the solution when the domain features non-convex edges or vertices. Moreover, it can handle noisy or approximate data which fail to satisfy the continuity equation, by using either an elliptic correction method or a mixed formulation. We give complete convergence analyses for both mixed and non-mixed formulations. Neither refinements near the reentrant edges or vertices of the domain, nor cutoff functions are required to achieve the desired convergence order in terms of the mesh size, the time step and the number of Fourier modes used.
• Non-reflecting boundary conditions for acoustic propagation in ducts with acoustic treatment and mean flow
  
  • Redon Emmanuel
  • Bonnet-Ben Dhia Anne-Sophie
  • Mercier Jean-François
  • Sari Poernomo
  International Journal for Numerical Methods in Engineering, Wiley, 2011, 86 (11), pp.1360-1378. We consider a time-harmonic acoustic scattering problem in a 2D infinite waveguide with walls covered with an absorbing material, in the presence of a mean flow assumed uniform far from the source. To make this problem suitable for a finite element analysis, the infinite domain is truncated. This paper concerns the derivation of a non-reflecting boundary condition on the artificial boundary by means of a Dirichlet-to-Neumann (DtN) map based on a modal decomposition. Compared with the hard-walled guide case, several difficulties are raised by the presence of both the liner and the mean flow. In particular, acoustic modes are no longer orthogonal and behave asymptotically like the modes of a soft-walled guide. However, an accurate approximation of the DtN map can be derived using some bi-orthogonality relations, valid asymptotically for high-order modes. Numerical validations show the efficiency of the method. The influence of the liner with or without mean flow is illustrated. (10.1002/nme.3108)
  DOI : 10.1002/nme.3108
• 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. This paper is devoted to a modal formulation of the linear sampling method in elastic 2D or 3D waveguide, that is we use the guided modes (called Lamb modes in 2D) as incident waves and the corresponding far fields in order to retrieve some obstacles. We provide the mathematical background to tackle the problem of identifiability and the justification of the linear sampling method for such a case. The elastic waveguide raises a specific issue: it concerns the projection of the scattered field on a transverse basis, which requires the introduction of new variables that mix displacement and stress components and satisfy the so-called Fraser's biorthogonality condition. Some numerical experiments in 2D show the feasibility of the reconstruction in the case of a finite number of incident waves formed by Lamb modes. © 2011 IOP Publishing Ltd. (10.1088/0266-5611/27/5/055001)
  DOI : 10.1088/0266-5611/27/5/055001
• About identification of defects in an elastic-plastic medium from boundary measurements in the antiplane case
  
  • Bourgeois Laurent
  • Dardé Jérémi
  Applicable Analysis, Taylor & Francis, 2011, 90 (10), pp.1481-1497. This article is devoted to a new method to find defects in some elasticplastic structure in the antiplane case from overdetermined data on a subpart of its boundary. This iterative method merges the method of quasireversibility and a simple level set technique. The foundations of the method are justified from a theoretical point of view and its efficiency is shown with the help of numerical experiments. © 2011 Taylor & Francis. (10.1080/00036811.2010.549481)
  DOI : 10.1080/00036811.2010.549481
• Stable reconstruction of generalized impedance boundary conditions
  
  • Bourgeois Laurent
  • Chaulet Nicolas
  • Haddar Houssem
  Inverse Problems, IOP Publishing, 2011, 27 (9), pp.095002. (10.1088/0266-5611/27/9/095002)
  DOI : 10.1088/0266-5611/27/9/095002
• 10e colloque national en calcul des structures (CSMA 2011)
  
  • Bonnet Marc
  • Cornuault Christian
  • Pagano Stéphane
  , 2011, pp.273 articles. Ce dixième colloque a vocation d'aborder tous les thèmes en rapport avec le calcul des structures : Modélisation et comportement : Structures minces, élasto-visco-plasticité, rupture, fatigue, endommagement, matériaux composites, mise en forme, procédés de fabrication et d'assemblage. Méthodes et techniques de résolution : Parallélisme, sous-structuration, contact, frottement, approches multiéchelles, approches stochastiques, prise en compte des incertitudes, méthodes alternatives aux éléments finis. Contrôle et optimisation : Estimation d'erreur et applications, maillage adaptatif, problèmes inverses, identification, recalage, optimisation de forme et de processus, structures adaptatives, contrôle actif, structures intelligentes. Dynamique, interactions et couplages : Dynamique rapide et vibrations non linéaires, crash, vibroacoustique, interactions multiphysiques, sol-structure, fluide-structure. Applications : Industries mécaniques, biens d'équipement, aéronautique et spatial, véhicules et transport, construction navale, génie civil, énergie, nanotechnologies, biomécanique.
• Stability of acoustic propagation in 2D duct flows: A low frequency approach
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Duruflé Marc
  • Joly Patrick
  • Joubert Lauris
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2011, 21 (5), pp.1121-1151. In this paper, we derive a simplified "quasi-1D" limit model for the propagation of low frequency acoustic waves in a laminar flow filling a 2D duct. We analyze the well-posedness of this model in function of the Mach profile of the flow. This can be reduced to the study of the spectrum of a bounded non-normal operator. As a by-product of this analysis, we establish new results for hydrodynamic instabilities of KelvinHelmholtz type in compressible fluids. © 2011 World Scientific Publishing Company. (10.1142/s0218202511005271)
  DOI : 10.1142/s0218202511005271