This paper presents the application of isogeometric analysis (IGA) to the spatial discretisation of the multi-group, self-adjoint angular flux (SAAF) form of the neutron transport equation with a discrete ordinate (S N) angular discretisation. The IGA spatial discretisation is based upon non-uniform rational B-spline (NURBS) basis functions for both the test and trial functions. In addition a source iteration compatible maximum principle is used to derive the IGA spatially discretised SAAF equation. It is demonstrated that this maximum principle is mathematically equivalent to the weak form of the SAAF equation. The rate of convergence of the IGA spatial discretisation of the SAAF equation is analysed using a method of manufactured solutions (MMS) verification test case. The results of several nuclear reactor physics verification benchmark test cases are analysed. This analysis demonstrates that for higher-order basis functions, and for the same number of degrees of freedom, the FE based spatial discretisation methods are numerically less accurate than IGA methods. The difference in numerical accuracy between the IGA and FE methods is shown to be because of the higher-order continuity of NURBS basis functions within a NURBS patch as well as the preservation of both the volume and surface area throughout the solution domain within the IGA spatial discretisation. Finally, the numerical results of applying the IGA SAAF method to the OECD/NEA, seven-group, two-dimensional C5G7 quarter core nuclear reactor physics verification benchmark test case are presented. The results, from this verification benchmark test case, are shown to be in good agreement with solutions of the first-order form as well as the second-order even-parity form of the neutron transport equation for the same order of discrete ordinate (S N) angular approximation.

This paper describes a methodology that enables NURBS (Non-Uniform Rational B-spline) based Isogeometric Analysis (IGA) to be locally refined. The methodology is applied to continuous Bubnov-Galerkin IGA spatial discretisations of second-order forms of the neutron transport equation. In particular this paper focuses on the self-adjoint angular flux (SAAF) and weighted least squares (WLS) equations. Local refinement is achieved by constraining degrees of freedom on interfaces between NURBS patches that have different levels of spatial refinement. In order to effectively utilise constraint based local refinement, adaptive mesh refinement (AMR) algorithms driven by a heuristic error measure or forward error indicator (FEI) and a dual weighted residual (DWR) or goal-based error measure (WEI) are derived. These utilise projection operators between different NURBS meshes to reduce the amount of computational effort required to calculate the error indicators. In order to apply the WEI to the SAAF and WLS second-order forms of the neutron transport equation the adjoint of these equations are required. The physical adjoint formulations are derived and the process of selecting source terms for the adjoint neutron transport equation in order to calculate the error in a given quantity of interest (QoI) is discussed. Several numerical verification benchmark test cases are utilised to investigate how the constraint based local refinement affects the numerical accuracy and the rate of convergence of the NURBS based IGA spatial discretisation. The nuclear reactor physics verification benchmark test cases show that both AMR algorithms are superior to uniform refinement with respect to accuracy per degree of freedom. Furthermore, it is demonstrated that for global QoI the FEI driven AMR and WEI driven AMR produce similar results. However, if local QoI are desired then WEI driven AMR algorithm is more computationally efficient and accurate per degree of freedom.

This paper presents the application of isogeometric analysis (IGA) to the spatial discretisation of the multi-group, source iteration compatible, weighted least squares (WLS) form of the neutron transport equation with a discrete ordinate (S N) angular discretisation. The WLS equation is an elliptic, second-order form of the neutron transport equation that can be applied to neutron transport problems on computational domains where there are void regions present. However, the WLS equation only maintains conservation of neutrons in void regions in the fine mesh limit. The IGA spatial discretisation is based up non-uniform rational B-splines (NURBS) basis functions for both the test and trial functions. In addition a methodology for selecting the magnitude of the weighting function for void and near-void problems is presented. This methodology is based upon solving the first-order neutron transport equation over a coarse spatial mesh. The results of several nuclear reactor physics verification benchmark test cases are analysed. The results from these verification benchmarks demonstrate two key aspects. The first is that the magnitude of the error in the solution due to approximation of the geometry is greater than or equal to the magnitude of the error in the solution due to lack of conservation of neutrons. The second is the effect of the weighting factor on the solution which is investigated for a boiling water reactor (BWR) lattice that contains a burnable poison pincell. It is demonstrated that the smaller the area this weighting factor is active over the closer the WLS solution is to that produced by solving the self adjoint angular flux (SAAF) equation. Finally, the methodology for determining the magnitude of the weighting factor is shown to produce a suitable weighting factor for nuclear reactor physics problems containing void regions. The more refined the coarse solution of the first-order transport equation, the more suitable the weighting factor.