Hybrid Monte Carlo - Deterministic Neutron Transport Methods Using Nonlinear Functionals.

Abstract

Several new hybrid Monte Carlo-deterministic methods based on nonlinear functionals are developed in this dissertation. The nonlinear functional approach consists of two fundamental steps: (1) the Monte Carlo estimation of nonlinear functionals, which are ratios of integrals of the particle flux, and (2) the deterministic solution of low-order algebraic equations that contain these functionals as parameters. The nonlinear functionals for each hybrid method are formulated by taking space-angle-energy moments of the transport equation and performing algebraic manipulations to obtain a finite system of “low-order” equations. The stochastic nonlinear functional estimates are used in the low-order equations to solve for the particle flux. If the structure of the low-order equations is favorable, and the functionals are defined appropriately, the solution of the low-order equations will have less variance than the direct Monte Carlo estimate of the solution. Theoretical justification is given that stochastic estimates of the nonlinear functionals should have less variance than direct estimates of standard linear quantities when the same Monte Carlo histories are used to evaluate the numerator and denominator of each functional. The new H-MC-S2 and H-MC-S2X methods incorporate functionals resembling flux-weighted cross sections and quadrature. The low-order equations of these methods resemble the one-group S2 equations, but have no energy, angular, or spatial truncation errors. Simulations show that the variance of the H-MC-S2X final solution is less than the variance of the standard Monte Carlo solution, leading to a reduction in computational cost for several test problems. The new HCMFD-II, HCMFD-III and HCMFD-IV methods improve the previously-published CMFD-Accelerated Monte Carlo method by utilizing angular moments of the transport equation to reduce statistical errors in the CMFD nonlinear functionals. These new methods more efficiently converge the fission source in criticality simulations. Consequently, they require fewer inactive and active cycles, and fewer particles per cycle, leading to a large reduction in computational cost. The techniques in this dissertation are explored for a subset of neutron transport problems including continuous energy fixed source problems and monoenergetic criticality problems. Our numerical results indicate that these nonlinear functional techniques are promising and should be extended to more realistic problems.