Goal-based h-adaptivity of the 1-D diamond difference discrete ordinate method.

The quantity of interest (QoI) associated with a solution of a partial differential equation (PDE) is not, in general, the solution itself, but a functional of the solution. Dual weighted residual (DWR) error estimators are one way of providing an estimate of the error in the QoI resulting from the discretisation of the PDE. This paper aims to provide an estimate of the error in the QoI due to the spatial discretisation, where the discretisation scheme being used is the diamond difference (DD) method in space and discrete ordinate (SNSN) method in angle. The QoI are reaction rates in detectors and the value of the eigenvalue (Keff)(Keff) for 1-D fixed source and eigenvalue (KeffKeff criticality) neutron transport problems respectively. Local values of the DWR over individual cells are used as error indicators for goal-based mesh refinement, which aims to give an optimal mesh for a given QoI

NURBS enhanced virtual element methods for the spatial discretization of the multigroup neutron diffusion equation on curvilinear polygonal meshes

The Continuous Galerkin Virtual Element Method (CG-VEM) is a recent innovation in spatial discretization methods that can solve partial differential equations (PDEs) using polygonal (2D) and polyhedral (3D) meshes. Recently, a new formulation of CG-VEM was introduced which can construct VEM spaces on polygons with curvilinear edges. This paper presents the application of the curved VEM to the multigroup neutron diffusion equation and demonstrates its benefits over the conventional straight-sided VEM for a number of benchmark verification test cases with curvilinear domains. These domains were constructed using a topological data-structure developed as part of this paper, based on the doubly-connected edge list, with curves and surfaces both represented using non-uniform rational B-splines (NURBS). This data-structure is used both to specify the geometry of the reactor and to represent the curvilinear polygonal mesh. We also present two separate methods of performing integrations on curvilinear polygons, one for homogeneous functions and one for non-homogeneous functions

Goal-based error estimation for the multi-dimensional diamond difference and box discrete ordinate (SN) methods

Goal-based error estimation due to spatial discretization and adaptive mesh refinement (AMR) has previously been investigated for the one dimensional, diamond difference, discrete ordinate (1-D DD-SN) method for discretizing the Neutron Transport Equation (NTE). This paper investigates the challenges of extending goal-based error estimation to multi-dimensions with supporting evidence provided on 2-D fixed (extraneous) source and Keff eigenvalue (criticality) verification test cases. It was found that extending Hennart’s weighted residual view of the lowest order 1-D DD equations to multi-dimensions gave what has previously been called the box method. This paper shows how the box method can be extended to higher orders. The paper also shows an equivalence between the higher order box methods and the higher order DD methods derived by Hébert et al. Though, less information is retained in the final solution in the latter case. These extensions allow for the definition of dual weighted residual (DWR) error estimators in multi-dimensions for the DD and box methods. However, they are not applied to drive AMR in the multi-dimensional case due to the various challenges explained in this paper

A geometry conforming, isogeometric, weighted least squares (WLS) method for the neutron transport equation with discrete ordinate (SN) angular discretisation

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) 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