Isogeometric Multi-Level Iterative Solution Algorithms With Applications in Nuclear Reactor Physics

The efficient solution of the neutron diffusion equation for large scale whole core calculations is of paramount importance; especially if the detailed pin-level power distribution and reaction rates are required. For heterogeneous whole core calculations finite element based techniques have been one approach to modelling the detailed pin geometry in whole core calculations. A new approach, pioneered in the last few years, is isogeometric analysis (IGA) methods which enable the exact geometry of fuel pins to be modelled. In order to efficiently solve elliptic partial differential equations (PDEs), such as the neutron diffusion equation, typically multi-grid or multi-level iterative solution techniques are used such as the algebraic multi-grid method. However, using IGA methods it is possible to develop true geometric multi-grid techniques which are potentially much more efficient than the standard algebraic multi-grid methods. In this paper we explore the use of IGA methods to develop a scalable, multi-level, iterative algorithm which is then used to solve the neutron diffusion equation over several geometries. This multilevel solution algorithm utilises a single patch multi-grid framework suggested by Hofreither and Takacs that takes advantage of the tensor product construction to provide scalability with respect to spatial, and polynomial refinement. Furthermore, a two-level balancing Neumann-Neumann solver is used to extend the solver to multiple patches in a scalable way. It is seen that the number of iterations required depends on the mapping between the unit square and the physical geometry, as well as the material coefficients.