Section: New Results

Absorbing boundary conditions and absorbing layers

Evolution problems in perturbed infinite periodic media

Participant : Sonia Fliss.

For parabolic problems set in locally perturbed periodic media, we have developed an approach to determine the time-domain DtN operator. The principle is to apply the Laplace Transform in time to the equation and use the construction of the DtN operator for stationary equations. The main difficulty is the computation of the inverse of the Laplace Transform, more precisely to understand how to deal with the unbounded interval of integration and the choice of the discretization of the laplace variable. To deal with the first difficulty for waveguide problem, we have studied the asymptotic behavior of the DtN operator in the laplace domain when the laplace variable tends to $p_0\pm \infty$. To deal with the second difficulty, we have used the Z-Transformation and its properties. The numerical study is still in progress. This work enters in the framework of the ANR PRoject MicroWave (Sonia Fliss is an external collaborator), in collaboration with Karim Ramdani (Institut Elie Cartan de Nancy, UMR CNRS 7502), Christophe Besse and Ingrid Violet (Laboratoire Paul Painlevé, UMR CNRS 8524).

New transparent boundary conditions for time harmonic acoustic problem in anisotropic media

Participants : Anne-Sophie Bonnet-Ben Dhia, Sonia Fliss, Antoine Tonnoir.

This topis is developed in collaboration with Vahan Baronian (CEA). Many industrial applications require to check the quality of structures such as plates, for instance in aircraft design. A common way to inspect structures is to propagate ultrasonic waves and detect from the experimental results the presence or not of a defect or a crack. However, in aeronautics, structures are often complex media like anisotropic elastic plates for which the interpretation of this results is complicated. Therefore, efficient and accurate numerical methods of simulation are required. In our work, we want to study the diffraction of a time harmonic wave by a bounded defect in an anisotropic elastic media. In order to study the diffraction properties of the defect, we consider it in as infinite. Since the defect has an arbitrary geometry, we want to use a finite element method in a box that surround the defect. On the boundary of this artificial box, we need to find transparent conditions to simulate an infinite domain.

• We first have considered waveguides. The transparent boundary conditions are often written by using the so-called Dirichlet-to-Neumann maps which can be expressed thanks to a modal decomposition. However, classical iterative method does not converge necessarily. In this work, we introduce a new Dirichlet-to-Neumann operator which links the trace of the solution on a section of the waveguide to the normal trace on a different one. This operator can also be expressed analytically via a modal decomposition. Its main advantage is that, because of the overlapping, it becomes compact and this is exactly why we think an iterative resolution has more chance to converge. Other advantages will appear with the elasticity application. Indeed, in the formulation of the transparent boundary condition without overlapping, appears a lagrange multiplier which makes the resolution more costly. This additional unknown will be avoided with an overlap. For now, the theory is done for the scalar acoustic waveguide and the method has been implemented in the Melina code for the acoustic and the elastic case. The redaction of an article is in progress.

  item We then have studied scattering problem in locally perturbed anisotropic plate. The classical methods to derive transparent boundary conditions for acoustic isotropic media are based on the Green function (boundary integral formulation) or Fourier series (to determine DtN operator set on an artificial circle boundary). However, they cannot be extended for anisotropic elastic problems. Using a constructive method to determine transparent boundary conditions for periodic media developped in the laboratory, we were able to propose new exact boundary conditions which are adapted to anisotropic media and for which iterative method could converge rapidly. The numerical study is in progress for acoustic isotropic problem.