Numerical Analysis of a Non-Conforming Domain Decomposition for the Multigroup SPN Equations
Thèse de Léandre GIRET
june 21, 2018
ENSTA salle 2.2.34 à 14h30
In this thesis, we investigate the resolution of the SPN neutron transport equations in pressurized water nuclear reactor. These equations are naturally written under a mixed form, the unknowns are the neutron flux and the neutron current, but they can be written under a primal form, where the neutron flux is the only one unknown. The SPN neutron transport equations are a generalized eigenvalue problem. In our study, we first consid- erate the associated source problem and after we concentrate on the eigenvalue problem. A nuclear reactor core is composed of different media: the fuel, the coolant, the neutron moderator... Due to these heterogeneities of the geometry, the solution flux can have a low-regularity. We prove that the problem and its approximation with finite element method are well-posed under its primal and mixed form. Moreover, we find for each form, an a priori error estimate. For the eigenvalue problem under its primal form, we use the theory of eigenvalue problem approximation already developed. But, under its mixed form, we can not rely on the theories already developed. We propose here a new method for studying the convergence of the SPN neutron transport eigenvalue problem approxi- mation with mixed finite element. When the solution has low-regularity, increasing the order of the method does not improve the approximation, the triangulation need to be refined near the singularities of the solution. Nuclear reactor cores are well-suited for Cartesian grids, but the refinement of these sort of triangulations increases rapidly their number of degrees of freedom. To avoid this drawback, we propose domain decomposition method which can handle globally non-conforming triangulations. Finally, we apply this method on a industrial testcase of a pressurized water nuclear reactor.