Approximation of The Neutron Diffusion Equation on Hexagonal Geometries Using a h-p finite element method

[EN] The neutron diffusion equation is an approximation of the neutron transport equation that describes the neutron population in a nuclear reactor core. In particular, we will consider here VVER-type reactors which use the neutron diffusion equation discretized on hexagonal meshes. Most of the simulation codes of a nuclear power reactor use the multigroup neutron diffusion equation to describe the neutron distribution inside the reactor core.To study the stationary state of a reactor, the reactor criticality is forced in artificial way leading to a generalized differential eigenvalue problem, known as the Lambda modes equation, which is solved to obtain the dominant eigenvalues of the reactor and their corresponding eigenfunctions. To discretize this model a finite element method with h-p adaptivity is used. This method allows to use heterogeneous meshes, and allows different refinements such as the use of h-adaptive meshes, reducing the size of specific cells, and p-refinement, increasing the polynomial degree of the basic functions used in the expansions of the solution in the different cells. 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 simulate the behaviour of a nuclear power reactor it is necessary to be able to integrate the time-dependent neutron diffusion equation inside the reactor core. The spatial discretization of this equation is done using a finite element method that permits h-p refinements for different geometries. Transients involving the movement of the control rod banks have the problem known as the rod-cusping effect. Previous studies have usually approached the problem using a fixed mesh scheme defining averaged material properties and many techniques exist for the treatment of the rod cusping problem. The present work proposes the use of a moving mesh scheme that uses spatial meshes that change with the movement of the control rods avoiding the necessity of using equivalent material cross sections for the partially inserted cells. The performance of the moving mesh scheme is tested studying different benchmark problems. For reactor calculations, the accuracy of a diffusion theory solution is limited for for complex fuel assemblies or fine mesh calculations. To improve these results a method that incorporates higher-order approximations for the angular dependence, as the simplified spherical harmonics (SPN ) met