Section: New Results

Domain decomposition methods

Transparent boundary conditions with overlap in unbounded anisotropic media

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

We are interested in acoustic or elastic wave propagation in time harmonic regime in a 2D or 3D medium which is a local perturbation of an infinite anisotropic homogeneous medium. We investigate the question of deriving a formulation which is suitable for numerical computations. This question is difficult due to the anisotropy of the surrounding medium. Our approach consists in coupling several plane-waves representations of the solution in half-spaces surrounding the defect with a FE computation of the solution around the defect. The difficulty is to ensure that all these representations match, in particular in the infinite intersections of the half-spaces. It leads to a formulation which couples, via integral operators, the solution in a bounded domain including the defect and its traces on the edge of the half-planes. We have shown that this formulation has good properties from theoretical and numerical points of view.

Electromagnetic scattering by objects with multi-layered dielectric coatings

Participants : Patrick Joly, Matthieu Lecouvez.

The PhD thesis of Matthieu Lecouvez, undertaken in collaboration with the CEA-CESTA and Francis Collino, has been defended in July. It concerned the diffraction of time harmonic electromagnetic waves by perfectly conducting objects covered by multi-layered (possibly thin) dielectric coatings. This idea was 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). This year, the theoretical aspects of our work have been completed and are the object of an article in preparation.

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 its simplest form, one must solve a neutron diffusion equation with low-regularity solutions, 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 DD method has been addressed for conforming meshes prior to the PhD research of Léandre Giret. The development of non-conforming, hence more flexible, DD methods has recently been finalized.

The optimization of the eigenvalue loop is under way. The current research also focuses on the numerical analysis of the full suite of algorithms to prove convergence for the complete multigroup SPN model (which involves coupled diffusion equations). Realistic computations will be carried out with the APOLLO3 neutronics code.

This topic is developed in partnership with CEA-DEN (Erell Jamelot).