Anisotropic Adaptivity and Subgrid Scale Modelling for the Solution of the Neutron Transport Equation with an Emphasis on Shielding Applications

Abstract

This thesis demonstrates advanced new discretisation and adaptive meshing technologies that improve the accuracy and stability of using finite element discretisations applied to the Boltzmann transport equation (BTE). This equation describes the advective transport of neutral particles such as neutrons and photons within a domain. The BTE is difficult to solve, due to its large phase space (three dimensions of space, two of angle and one each of energy and time) and the presence of non-physical oscillations in many situations. This work explores the use of a finite element method that combines the advantages of the two schemes: the discontinuous and continuous Galerkin methods. The new discretisation uses multiscale (subgrid) finite elements that work locally within each element in the finite element mesh in addition to a global, continuous, formulation. The use of higher order functions that describe the variation of the angular flux over each element is also explored using these subgrid finite element schemes. In addition to the spatial discretisation, methods have also been developed to optimise the finite element mesh in order to reduce resulting errors in the solution over the domain, or locally in situations where there is a goal of specific interest (such as a dose in a detector region). The chapters of this thesis have been structured to be submitted individually for journal publication, and are arranged as follows. Chapter 1 introduces the reader to motivation behind the research contained within this thesis. Chapter 2 introduces the forms of the BTE that are used within this thesis. Chapter 3 provides the methods that are used, together with examples, of the validation and verification of the software that was developed as a result of this work, the transport code RADIANT. Chapter 4 introduces the inner element subgrid scale finite element discretisation of the BTE that forms the basis of the discretisations within RADIANT and explores its convergence and computational times on a set of benchmark problems. Chapter 5 develops the error metrics that are used to optimise the mesh in order to reduce the discretisation error within a finite element mesh using anisotropic adaptivity that can use elongated elements that accurately resolves computational demanding regions, such as in the presence of shocks. The work of this chapter is then extended in Chapter 6 that forms error metrics for goal based adaptivity to minimise the error in a detector response. Finally, conclusions from this thesis and suggestions for future work that may be explored are discussed in Chapter 7.