This thesis examines uncertainty quantification methods applied to geometric uncertainties in the neutron transport equation. Four methods were studied; perturbation theory, Monte Carlo and intrusive and non-intrusive forms of polynomial chaos.

First, perturbation theory methods were applied to small scale surface roughness on surfaces and interfaces of materials. Several problems were examined in two dimensions, including a homogeneous region with a perturbed boundary, a two region pincell with a perturbed interface and the Feinberg-Galanin model with perturbed rod radius and pitch. Analytic solutions were sought where possible and compared to numerical solutions.

Intrusive and non-intrusive polynomial chaos and Monte Carlo methods were then used to examine geometric uncertainties in a single spatial dimension. These were applied to the mono-energetic neutron diffusion equation with a finite element spatial discretisation. Comparison of these uncertainty quantification methods to analytic solutions showed that they were all capable of converging the result to high accuracy but the intrusive polynomial chaos methods scaled poorly as the number of random variables was increased. Non-intrusive polynomial chaos and Monte Carlo were then used to examine two-dimensional problems using diffusion, discrete ordinates and spherical harmonic forms of the neutron transport equation. It was seen that the non-intrusive polynomial chaos method was capable of producing highly accurate statistical moments and probability density functions using significantly fewer model samples than the Monte Carlo comparisons.

The final cases examined used non-intrusive polynomial chaos and Monte Carlo applied to geometric uncertainties in the neutron transport equation with an isogeometric analysis spatial discretisation. This was used due to the superior representation of conic sections with isogeometric analysis than finite element methods. It was seen that for rectilinear geometries finite elements and isogeometric analysis gave equivalent results. However, for curved surfaces isogeometric analysis was able to reduce the geometric error to a greater extent per degree of freedom. This gives greater confidence that the spatial component of a calculation has converged, enabling greater confidence in the results of uncertainty quantification. This was used to examine a pincell with uncertain spatial dimensions as well as a layer of crud deposition of varying thickness; which is an inherently random process. Finally, a mixed-oxide assembly was examined with a burnable absorber pin and 71 fuel pins with stochastic radii which showed the limits of non-intrusive polynomial chaos to represent geometric uncertainties.