Resumen de Development of a finite element method for neutron transport equation approximations

A. Vidal Ferràndiz

• The neutron transport equation describes the neutron population and the nuclear reactions inside a nuclear reactor core. First, this equation is introduced and its assumptions are stated. Then, the stationary neutron diffusion equation which is the most useful approximation of this equation, is studied. This approximation leads to a differential eigenvalue problem. To solve the neutron diffusion equation, a h-p finite element method is investigated. To improve the efficiency of the method a Restricted Additive Schwarz preconditioner is implemented.

  Once the solution for the steady state neutron distribution is obtained, it is used as initial condition for the time integration of the neutron diffusion equation. To test the behaviour of the method, rod ejection accidents are numerically simulated. However, a non-physical behaviour appears when a cell is partially rodded: this is, the rod cusping effect, which is solved by using a moving mesh scheme. In other words, the mesh follows the movement of the control rod. Numerical results show that the rod cusping effect is corrected with this scheme.

  After that, the simplified spherical harmonics approximation, SPN, is developed to solve the steady state problem. This approximation extends the spherical harmonics approximation, PN, in one dimensional geometries to multidimensional geometries with strong assumptions. It improves the diffusion theory results but does not converge as N tends to infinity. The advantages and limitations of this approximation are tested on several one-, two- and three-dimensional reactors.

  Finally, the spatial homogenization in the context of the finite element method is studied. Homogenization consists in replacing heterogeneous subdomains by homogeneous ones, in such a way that the homogenized problem provides fast and accurate average results. Discontinuous solutions were proposed in the Generalized Equivalence Theory. Here, a discontinuous Galerkin finite element method where the jump condition for the neutron flux is imposed in a weak sense using interior penalty terms is introduced. Also, the use of discontinuity factors for the correction of the homogenization error when using the SPN equations is investigated.