Linear Diffusion Acceleration for Neutron Transport Problems

Nuclear engineers are interested in solutions of the Neutron Transport Equation (NTE), with the goal of improving the safety and efficiency of reactors and critical nuclear systems. Complex simulations are used to obtain detailed solutions of the NTE, and can require immense computational resources to execute. A variety of methods have been developed to ease the computational burden of simulating full-scale, whole-core reactor problems. Among these is transport acceleration, which improves the convergence rate of iterative transport calculations. In addition to the use of acceleration methods, certain approximations are often made when solving the NTE. The 2D/1D approximation is used to generate a 3D solution of the NTE by iteratively solving coupled 2D radial and 1D axial equations. This method is one of the foundational techniques used in the neutronics code MPACT. Also, the Transport-Corrected P0 (TCP0) approximation for neutron scattering is often used in reactor analysis codes to simplify higher-order scattering physics. Unfortunately, both of these approximations allow for non-positive flux solutions of the NTE. More importantly, some spatial discretizations of the NTE also permit negative solutions. Under certain conditions, this can cause instability for nonlinear acceleration methods such as Coarse Mesh Finite Difference (CMFD). In this thesis, we propose a novel acceleration scheme called Linear Diffusion Acceleration (LDA) that does not possess the nonlinearities present in CMFD. This thesis work presents LDA as an alternative acceleration scheme to CMFD. As the name suggests, the LDA method is linear with respect to the scalar flux. Therefore, LDA is not susceptible to the same nonlinear modes of numerical failure as CMFD. In addition, LDA is shown to possess similar convergence properties as CMFD for practical problems that have no negative scalar fluxes. Transport acceleration with LDA allows for the use of some of the aforementioned approximations, in which the positivity of the scalar flux is not guaranteed. Fourier analysis of CMFD and LDA is performed to compare the theoretical convergence rates of the two methods for simple, spatially-heterogeneous problems. In addition, simple and practical case studies are presented in which CMFD fails due to nonlinearity. For these cases, LDA is shown to retain stability. Certain other advantages of LDA, which are a consequence of its mathematical structure, are also discussed.PhDNuclear Engineering & Radiological SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/169898/1/zdodson_1.pd