Virtual element methods for the neutron diffusion and neutron transport equations with applications in nuclear reactor physics
File(s)
Author(s)
Ferguson, John
Type
Thesis
Abstract
In this thesis we propose a new spatial discretisation for the solution of the multigroup neutron diffusion equation (NDE) and the first-order form of the multigroup, discrete-ordinates, neutron transport equation (SN-NTE). The proposed spatial discretisation is called the NURBS- Enhanced virtual element method (NE-VEM). The NE-VEM is a generalisation of the Galerkin finite element method which expands its applicability to meshes comprised of arbitrary polygonal elements with curved edges. Non-uniform rational B-splines (NURBS) are used as the geometric description of curved edges as they are the standard geometry encoding used by the computer-aided geometric design (CAGD) industry.
We developed a continuous-Galerkin VEM (NECG-VEM) to discretise the NDE and a discon- tinuous Galerkin VEM (NEDG-VEM) to discretise the SN-NTE. For both of these discretisatons we prove the well-posedness and derive a priori error estimates. We also present parallelisation strategies for both methods: a distributed memory strategy for the NECG-VEM and shared- memory strategy for the NEDG-VEM. The benefits of maintaining exact geometry are demon- strated through numerical examples. We show that the NE-VEM attains the theoretical order of convergence for problems with curved geometry for both the NDE and the SN-NTE. The convergence rate of conventional, straight-sided VEM however, saturates due to the geometric error induced by approximating curved geometry with line segments.
Additionally, we present a 2D geometric modelling kernel and mesh generator to both model nuclear reactor core geometries and to create polygonal meshes which exactly represent the geometry of the reactor from the lowest level of mesh refinement.
We developed a continuous-Galerkin VEM (NECG-VEM) to discretise the NDE and a discon- tinuous Galerkin VEM (NEDG-VEM) to discretise the SN-NTE. For both of these discretisatons we prove the well-posedness and derive a priori error estimates. We also present parallelisation strategies for both methods: a distributed memory strategy for the NECG-VEM and shared- memory strategy for the NEDG-VEM. The benefits of maintaining exact geometry are demon- strated through numerical examples. We show that the NE-VEM attains the theoretical order of convergence for problems with curved geometry for both the NDE and the SN-NTE. The convergence rate of conventional, straight-sided VEM however, saturates due to the geometric error induced by approximating curved geometry with line segments.
Additionally, we present a 2D geometric modelling kernel and mesh generator to both model nuclear reactor core geometries and to create polygonal meshes which exactly represent the geometry of the reactor from the lowest level of mesh refinement.
Version
Open Access
Date Issued
2022-03
Date Awarded
2022-11
Copyright Statement
Creative Commons Attribution NonCommercial Licence
License URL
Advisor
Eaton, Mathew
Bluck, Michael
Sponsor
Engineering and Physical Sciences Research Council
Rolls-Royce Group plc
Grant Number
EP/R512540/1
Publisher Department
Mechanical Engineering
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)