Discontinuous Galerkin FEMs for Radiation Transport Problems

Widdowson, Richard (2023) Discontinuous Galerkin FEMs for Radiation Transport Problems. PhD thesis, University of Nottingham.

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Abstract

The linear Boltzmann transport equation (LBTE), a high dimensional partial integrodifferential equation, is used to model radiation transport. Radiation transport is an area of physics that is concerned with the propagation and distribution of radiative particles, such as photons and electrons within a material medium.

In this thesis, we present a high-order discontinuous Galerkin finite element method (DGFEM) discretisations of the steady state linear Boltzmann transport equation in the spatial, angular, and energy domains. Comparisons between the tensor discontinuous Galerkin finite element method and discrete ordinates method show that the former is higher order than the latter. A method of block diagonalising the resulting matrix into a sequence of transport equations coupled by the right hand side while retaining high order convergence, is demonstrated for both the angular and energy domains. This new method offers the arbitrary order convergence rates of the discontinuous Galerkin finite element method, but it can be implemented in an almost identical form to standard multigroup discrete ordinates methods.

The assembly of the matrix for the resulting transport equations for a variety of different type of elements is discussed. The generation of meshes formed of general polytopes is discussed, and a comparison between the time to solve the transport equation on meshes formed of the different element types follows.

An efficient implementation of the discontinuous Galerkin finite element method for transport equations is then presented. This algorithm exploits the fixed wind direction of the transport problems resulting from the discontinuous Galerkin finite element method of the LBTE, to solve the transport problem while never forming the matrix. We then compare this algorithm to a direct matrix solver for both convex and non-convex polytopes.

An a posteriori error bound for the discontinuous Galerkin finite element method of the LBTE and the transport problem is then derived. We use this error bound to develop an adaptive framework for the LBTE. Three different adaptive algorithms for the LBTE are then presented and compared. The h-refinement algorithm, which marks the element with the error of each tensor element it is part of, shows a clear advantage over the other methods.

Item Type: Thesis (University of Nottingham only) (PhD) Matthew, HubbardPaul, Houston linear Boltzmann transport equations, Galerkin finite element methods, radiation transport Q Science > QA Mathematics > QA299 Analysis UK Campuses > Faculty of Science > School of Mathematical Sciences 76685 Widdowson, Richard 13 Dec 2023 16:12 13 Dec 2023 16:12 https://eprints.nottingham.ac.uk/id/eprint/76685

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