Publications

2016

• Iterative methods for scattering problems in isotropic or anisotropic elastic waveguides
  
  • Baronian Vahan
  • Bonnet-Ben Dhia Anne-Sophie
  • Fliss Sonia
  • Tonnoir Antoine
  Wave Motion, Elsevier, 2016. We consider the time-harmonic problem of the diffraction of an incident propagative mode by a localized defect, in an infinite straight isotropic elastic waveguide. We propose several iterative algorithms to compute an approximate solution of the problem, using a classical finite element discretization in a small area around the perturbation, and a modal expansion in unbounded straight parts of the guide. Each algorithm can be related to a so-called domain decomposition method, with or without an overlap between the domains. Specific transmission conditions are used, so that only the sparse finite element matrix has to be inverted, the modal expansion being obtained by a simple projection, using the Fraser bi-orthogonality relation. The benefit of using an overlap between the finite element domain and the modal domain is emphasized, in particular for the extension to the anisotropic case. The transparency of these new boundary conditions is checked for two- and three-dimensional anisotropic waveguides. Finally, in the isotropic case, numerical validation for two- and three-dimensional waveguides illustrates the efficiency of the new approach, compared to other existing methods, in terms of number of iterations and CPU time.
• Uniqueness results for inverse Robin problems with bounded coefficient
  
  • Baratchart Laurent
  • Bourgeois Laurent
  • Leblond Juliette
  Journal of Functional Analysis, Elsevier, 2016. In this paper we address the uniqueness issue in the classical Robin inverse problem on a Lipschitz domain $\Omega\subset\RR^n$, with $L^\infty$ Robin coefficient, $L^2$ Neumann data and isotropic conductivity of class $W^{1,r}(\Omega)$, $r>n$. We show that uniqueness of the Robin coefficient on a subpart of the boundary given Cauchy data on the complementary part, does hold in dimension $n=2$ but needs not hold in higher dimension. We also raise on open issue on harmonic gradients which is of interest in this context. (10.1016/j.jfa.2016.01.011)
  DOI : 10.1016/j.jfa.2016.01.011
• A numerical study of the solution of x-mode equations around the hybrid resonance
  
  • Caldini-Queiros Céline
  • Després Bruno
  • Imbert-Gérard Lise-Marie
  • Kachanovska Maryna
  ESAIM: Proceedings and Surveys, EDP Sciences, 2016, 53, pp.1-21. Hybrid resonance is a physical phenomenon that appears for example in the heating of plasma, and as such is of scientific interest in the development of the ITER project. In this paper we focus some solutions with low regularity of Maxwell equations in plasmas under strong background magnetic field. Our purpose is two-fold. On one hand we investigate the finite element approximation of the one dimensional problem written in the frequency domain, and on the other hand we investigate two different finite difference approximations of the one dimensional time dependent problem. We will also compare the results of these different methods.
• Solvability of a Volume Integral Equation Formulation for Anisotropic Elastodynamic Scattering
  
  • Bonnet Marc
  Journal of Integral Equations and Applications, Rocky Mountain Mathematics Consortium, 2016, 28, pp.169-203. This article investigates the solvability of volume integral equations arising in elastodynamic scattering by penetrable obstacles. The elasticity tensor and mass density are allowed to be smoothly heterogeneous inside the obstacle and may be discontinuous across the background-obstacle interface, the background elastic material being homogeneous. Both materials may be anisotropic, within certainme limitations for the background medium. The volume integral equation associated with this problem is first derived, relying on known properties of the background fundamental tensor. To avoid difficulties associated with existing radiation conditions for anisotropic elastic media, we also propose a definition of the radiating character of transmission solutions. The unique solvability of the volume integral equation (and of the scattering problem) is established. For the important special case of isotropic background properties, our definition of a radiating solution is found to be equivalent to the Sommerfeld-Kupradze radiation conditions. Moreover, solvability for anisotropic elastostatics, directly related to known results on the equivalent inclusion method, is recovered as a by-product.
• Existence of guided waves due to a lineic perturbation of a 3D periodic medium
  
  • Delourme Bérangère
  • Joly Patrick
  • Vasilevskaya Elizaveta
  Applied Mathematics Letters, Elsevier, 2016. In this note, we exhibit a three dimensional structure that permits to guide waves. This structure is obtained by a geometrical perturbation of a 3D periodic domain that consists of a three dimensional grating of equi-spaced thin pipes oriented along three orthogonal directions. Homogeneous Neumann boundary conditions are imposed on the boundary of the domain. The diameter of the section of the pipes, of order ε > 0, is supposed to be small. We prove that, for ε small enough, shrinking the section of one line of the grating by a factor of √ µ (0 < µ < 1) creates guided modes that propagate along the perturbed line. Our result relies on the asymptotic analysis (with respect to ε) of the spectrum of the Laplace-Neumann operator in this structure. Indeed, as ε tends to 0, the domain tends to a periodic graph, and the spectrum of the associated limit operator can be computed explicitly. (10.1016/j.aml.2016.11.017)
  DOI : 10.1016/j.aml.2016.11.017