A Functional Monte Carlo Method for k-Eigenvalue Problems.
Jinan Yang · Deep Blue (University of Michigan) · 2011
A longstanding problem for Monte Carlo (MC) criticality is the slow convergence of the fission source distribution for systems with a high dominance ratio (DR). In this thesis, we have developed and tested a new hybrid deterministic and Monte Carlo method, called the Functional Monte Carlo (FMC) method, to solve such problems. The FMC method is different from any previous hybrid method. The FMC method does not directly estimate the eigenfunction and eigenvalue via Monte Carlo particle simulation. Instead, the FMC method uses MC techniques to directly estimate certain nonlinear functionals. These estimated functionals are then used in the low-order FMC equations to calculate the k-eigenfunction and eigenvalue. The resulting estimates of the k-eigenfunction and eigenvalue have no spatial or angular truncation errors, and are generally more accurate and have less statistical noise than estimates obtained using conventional Monte Carlo methods. The FMC method is based on two assumptions: 1. The functionals depend weakly on the angular flux and can be evaluated with Monte Carlo more accurately than direct Monte Carlo estimates of the angular or scalar flux. 2. If the low-order FMC equations are solved with small errors in the functionals, the resulting errors in the eigenfunction and eigenvalue will be small. In this work, we have developed the FMC method for monoenergetic, multigroup, and continuous energy k-eigenvalue problems in 1-D planar geometry. We tested the FMC method on various problems, including a 1-D full PWD reactor core, in which standard MC estimates of the eigenfunction tend to ”wobble.” Our numerical simulations indicate that the FMC method offers significant advantages over the traditional Monte Carlo method in solving slow convergence k-eigenvalue problems with high dominance ratios. For future research, it remains to extend the FMC method to include realistic cross sections, and multi-dimensional problems. We see no fundamental impediment to doing this.