Isogeometric analysis for second-order forms of the neutron transport equation with applications to reactor physics

Abstract

Development of computational methods to solve reactor physics and shielding problems has been a constant area of research since the 1940s and the Manhattan project. As technology and improved methods have been developed more detailed simulations have been possible. However, solving the neutron transport equation for full-core heterogeneous problems is still a challenging proposition, even for modern numerical methods and petaflop high performance computers. The high dimensionality of the physical problem combined with the complex geometries of reactor cores and their shields has meant that approximations in all dimensions have to be made in order to make finding a solution tractable. Such approximations include using a simple diffusion approximation and coarse geometric approximations typically discretised with polygonal finite elements. This thesis details the investigation into the application of Isogeometric Analysis (IGA) to second-order forms of the neutron transport equation with application to reactor physics and the development of a new parallel reactor physics code PIRANA. Isogeometric Analysis is a generalisation of the finite element method which uses the mathematical basis of computer aided design (CAD) for numerical analysis. By using the CAD model directly in analysis the geometry is modelled exactly with no approximation. The results of this study have found that the exact geometric representation of the IGA method can significantly improve the accuracy of solutions to the neutron transport equation compared to the finite element method. Additionally, the exact geometric representation has allowed for local refinement to be performed for non-Cartesian geometries within the analysis program which can result in a significant reduction of the computational cost. Finally, IGA allows for optimisation of the multigroup form of the neutron transport equation by solving each group on a uniquely refined spatial mesh significantly reducing the total number of spatial degrees-of- freedom for the same overall solution accuracy.