Section: New Results

Domain decomposition methods

Transparent boundary conditions with overlap in elastic waveguides

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

This work is a part of the PhD of Antoine Tonnoir and is done in partnership with Vahan Baronian form CEA-LIST. We have conceived new transparent boundary conditions for the time-harmonic diffraction problem in an acoustic or elastic waveguide. These new conditions use the natural modal decomposition in the waveguide and are said “with overlap” by analogy with the domain decomposition methods. Among their main advantages, they can be implemented in general elastic anisotropic waveguides, for which usual Dirichlet to Neumann maps are not available. Moreover, the traditional benefit of the overlap for iterative resolution is obtained, independently of the size of the overlap.

Electromagnetic scattering by objects with multi-layered dielectric coatings

Participants : Patrick Joly, Matthieu Lecouvez.

This is the object of the PhD thesis of Matthieu Lecouvez in collaboration with the CEA-CESTA and Francis Collino. We are interested in the diffraction of time harmonic electromagnetic waves by perfectly conducting objects covered by multi-layered (possibly thin) dielectric coatings. This problem is computationally hard when the size of the object is large (typically 100 times larger) with respect to the incident wavelength. In such a situation, the idea is to use a domain decomposition method in which each layer would constitute a subdomain. The transmission conditions between the subdomains involve some specific impedance operators in order to achieve a geometric convergence of the method (compared to the slow algebraic convergence obtained with standard Robin conditions). We propose a practical solution that uses approximations of nonlocal integral operators with appropriate Riesz potentials.

Domain Decomposition Methods for the neutron diffusion equation

Participants : Patrick Ciarlet, Léandre Giret.

Studying numerically the steady state of a nuclear core reactor is expensive, in terms of memory storage and computational time. In particular, one must solve the neutron diffusion equation discretized by finite element techniques, totaling millions of unknowns or more, within a loop. Iterating in this loop allows to compute the smallest eigenvalue of the system, which determines the critical, or non-critical, state of the 3D core configuration. This problem fits within the framework of high performance computing so, in order both to optimize the memory storage and to reduce the computational time, one can use a domain decomposition method, which is then implemented on a parallel computer. The definition of an efficent DDM has been recently addressed for conforming meshes. The development of non-conforming, hence more flexible, methods is under way. Since one is dealing with highly heterogeneous configurations, the regularity of the exact solution can be very low, which then deteriorates the convergence rate of the discretized solution to the exact one. Next, the optimization of the eigenvalue loop will be studied.

This topic is developed in partnership with CEA-DEN (Erell Jamelot). Realistic computations are carried out with the APOLLO3 neutronics code.