A geometry conforming isogeometric method for the self-adjoint angular flux (SAAF) form of the neutron transport equation with a discrete ordinate (SN) angular discretisation

Abstract

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 (SN) 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 (SN) angular approximation.