Hybrid Delta Tracking Schemes Using a Track-Length Estimator

TL;DR

The paper addresses efficiency in Monte Carlo neutron transport by integrating Woodcock delta tracking with a track-length flux estimator on a structured mesh and by developing two hybrid delta-tracking schemes (hybrid-in-material and hybrid-in-energy). A voxelized tally framework enables track-length tallies with delta tracking, even in moving-surfaces scenarios, and is implemented in MC/DC for CPU and GPU performance testing. Across four time-dependent benchmarks, voxelized delta tracking with track-length tallies shows FoM improvements in several regimes, while hybrid-in-energy yields substantial gains (7x–11x) in continuous-energy reactor problems; delta tracking with collision estimators can perform poorly in void-dominated regions. The results demonstrate that mixing tracking strategies according to problem physics can yield meaningful performance gains and that delta tracking can be extended to moving-surface and energy-dependent applications with appropriate tallying infrastructure.

Abstract

In Monte Carlo radiation transport calculations, Woodcock-delta tracking is a common alternative to the more popular surface tracking technique. In this work we introduce a delta-tracking algorithm that tallies fluxes to a structured rectilinear mesh using the track-length estimator. This development also enables hybrid surface-delta tracking algorithms, because the track-length tally can be used everywhere for scalar flux estimation regardless of which tracking algorithm is employed. We use this tallying technique to develop a novel hybrid-in-energy method. We also implement a hybrid-in-material method, like what is implemented in Serpent2. We demonstrate that these delta tracking algorithms can be used in conjunction with continuously moving surfaces. We compare these methods showing figures of merit on four time-dependent problems (multi-group and continuous energy) solved with CPU- and GPU-based computers. Our implementation of delta tracking with a track length tally modestly improves figures of merit compared to standard delta tracking with a collision estimator and surface tracking with a track length estimator (1.5X 2.5X) for a problem with significant void regions. For both multi-group and continuous energy pressurized water reactor benchmarks, standard delta tracking with a collision estimator performs best. Hybrid-in-energy methods show significant improvements (7X-11X) for a continuous energy reactor benchmark problem.

Figures (12)

• Figure 1: Continuous energy macroscopic majorant cross section $\Sigma_{\text{maj}}(E)$ and other material cross sections for the continuous energy version of C5G7 pressurized water reactor described in Appendix \ref{['app:c5ce_mat']}.
• Figure 2: A particle tallying to multiple tally mesh bins (shown in red). Implemented as a single operation for both surface and delta tracking in MC/DC.
• Figure 3: Convergence rate verification of AZURV1 ganapol_2001_homogeneous, showing the expect Monte Carlo convergence rate ($N^{-1/2}$).
• Figure 4: Convergence rate of an infinite pin using the SHEM 361 group cross section library hfaiedh_2005_shem, showing the expected Monte Carlo convergence rate ($N^{-1/2}$).
• Figure 5: Convergence rate of flux from Reed's problem reed_difference_1971 showing the expect Monte Carlo convergence rate ($N^{-1/2}$).