Timothy P. Burke

Monte Carlo Perturbation Theory Estimates of Sensitivities to System Dimensions

Abstract Monte Carlo methods are developed using adjoint-based perturbation theory and the differential operator method to compute the sensitivities of the k-eigenvalue, linear functions of the flux (reaction rates), and bilinear functions of the forward and adjoint flux (kinetics parameters) to system dimensions for uniform expansions or contractions. The calculation of sensitivities to system dimensions requires computing scattering and fission sources at material interfaces using collisions occurring at the interface—which is a set of events with infinitesimal probability. Kernel density estimators are used to estimate the source at interfaces using collisions occurring near the interface. The methods for computing sensitivities of linear and bilinear ratios are derived using the differential operator method and adjoint-based perturbation theory and are shown to be equivalent to methods previously developed using a collision history–based approach. The methods for determining sensitivities to system dimensions are tested on a series of fast, intermediate, and thermal critical benchmarks as well as a pressurized water reactor benchmark problem with iterated fission probability used for adjoint-weighting. The estimators are shown to agree within 5% and of reference solutions obtained using direct perturbations with central differences for the majority of test problems.

Kernel Density Estimation of Reaction Rates in Neutron Transport Simulations of Nuclear Reactors

Abstract Kernel density estimators (KDEs) are applied to estimate neutron scalar flux and reaction rate densities in Monte Carlo neutron transport simulations of heterogeneous nuclear reactors in continuous energy. The mean free path (MFP) KDE is introduced in order to handle the issues that arise from estimating the discontinuous reaction rate densities at material interfaces. Results show the MFP KDE is more accurate at estimating reaction rates compared with previous KDE formulations. An approximate MFP (aMFP) KDE is introduced to circumvent several practical issues presented by the MFP KDE. A volume-averaged KDE is derived and used to determine the bias introduced by the aMFP KDE. A KDE is formulated for cylindrical coordinates to better represent the geometry and capture the physics in two-dimensional reactor physics problems. The results indicate that the cylindrical MFP KDE and cylindrical aMFP KDE are accurate tools for capturing reaction rates in heterogeneous reactor physics problems in continuous energy, with local biases of less than 1%.

GPU Acceleration of Mean Free Path Based Kernel Density Estimators for Monte Carlo Neutronics Simulations

Kernel Density Estimators (KDEs) are a non-parametric density estimation technique that has recently been applied to Monte Carlo radiation transport simulations. Kernel density estimators are an alternative to histogram tallies for obtaining global solutions in Monte Carlo tallies. With KDEs, a single event, either a collision or particle track, can contribute to the score at multiple tally points with the uncertainty at those points being independent of the desired resolution of the solution. Thus, KDEs show potential for obtaining estimates of a global solution with reduced variance when compared to a histogram. Previously, KDEs have been applied to neutronics for one-group reactor physics problems and fixed source shielding applications. However, little work was done to obtain reaction rates using KDEs. This paper introduces a new form of the MFP KDE that is capable of handling general geometries. Furthermore, extending the MFP KDE to 2-D problems in continuous energy introduces inaccuracies to the solution. An ad-hoc solution to these inaccuracies is introduced that produces errors smaller than 4% at material interfaces.

GPU Acceleration of Mean Free Path Based Kernel Density Estimators in Monte Carlo Neutronics Simulations with Curvilinear Geometries

KDEs show potential reducing variance for global solutions (flux, reaction rates) when compared to histogram solutions.