Goal-based adaptive methods applied to the spatial and angular dimensions of the transport equation

Abstract

This thesis contains research into adaptive methods for the spatial and angular dimensions of the neutral particle transport equation. Adaptive methods have been developed for two angular discretisations: the spherical harmonics method and an octahedron-based wavelet discretisation. The spatial discretisation used is a sub-grid scale finite element method. The primary focus of the research is goal-based adaptive methods which optimise a particular functional of the solution. The error measures that drive the adaptive methods are presented along with the novel and efficient techniques that are used to approximate them. Adaptive algorithms are first developed and presented for the spatial and angular discretisations separately. The adaptive methods for the angular dimensions produce variable angular resolution across the space and energy dimensions of the equation. The adaptive methods for the spatial dimensions use an anisotropic mesh optimisation algorithm which repositions nodes and elements of the mesh. The adaptive wavelet discretisation allows anisotropic resolution of the angular domain at each point in space and energy which can be very efficient. The ultimate outcome of the research is an algorithm that adapts the angular and spatial resolution simultaneously. This is achieved using the wavelet discretisation by combining the individual adaptive procedures. All adaptive methods developed are shown to produce results with a given accuracy for a smaller number of degrees of freedom. The performance of the methods heavily depends on the physical system that is being modelled. Typically performing best in shielding type calculations. The benefits from the adaptive methods are two-fold: (i) the reduction in degrees of freedom can lead to smaller computational times, and (ii) the automated adaptive process can reduce the overall user time spent performing convergence analysis.