Discontinuous isogeometric analysis methods for the first order form of the neutron transport equation with discrete ordinate angular discretisation

Abstract

This thesis presents the development of a variety of discontinuous isogeometric discretisations of the discrete ordinates equations for neutron transport in two spatial dimensions. Three discretisations are presented of increasing sophistication, and their convergence properties analysed for a wide selection of test cases. The first discretisation uses a conforming mesh approach, which is analogous to many existing discontinuous Galerkin finite element methods for the discrete ordinates equations. This simplifies the analysis of the differences between isogeometric and finite element methods in terms of the numerical upwinding and sweep ordering. The second discretisation extends this approach by introducing hanging-nodes into the mesh. This overcomes the limitation of the tensor-product refinement structure inherent to isogeometric analysis based on non-uniform rational B-splines with a conforming mesh. Adaptive mesh refinement based on element subdivision is also introduced at this point, driven by a selection of heuristic error indicators. In the final discretisation, each energy group has its own associated mesh. The interpolation of functions between meshes is greatly simplified by deriving the mesh in each energy group from a common initial coarsest mesh. To take full advantage of the flexibility of this discretisation, dual weighted residual error metrics are derived for the multigroup discrete ordinates equations for both fixed source and eigenvalue problems. In a representative deep penetration shielding problem, this method is demonstrated to achieve the same level of accuracy in a detector response as uniform, conforming mesh refinement using approximately an order of magnitude less computational effort.