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Negative Weights and Weight Cancellation to Treat Anisotropic Scattering in Multigroup Monte Carlo Simulations

Parth Singh, Hunter Belanger

TL;DR

This work addresses the challenge of sampling anisotropic scattering in multigroup Monte Carlo simulations, where truncated Legendre expansions can yield negative angular distributions. By combining importance sampling with a positive surrogate distribution and a weight-cancellation mechanism, the authors enable unbiased sampling from signed scattering distributions while maintaining convergence. The method is validated on analytic benchmarks, where it reproduces the exact eigenvalues and fluxes more accurately than positive-only approximations, and is demonstrated on a realistic 2D reactor model (RCF) where it compares favorably with deterministic anisotropic-solvers. The results suggest that importance sampling with weight cancellation provides a robust, GPU-friendly path for accurate multigroup MC with anisotropic scattering and offers a valuable reference for deterministic solvers.

Abstract

The Monte Carlo method is typically considered the gold standard for simulating reactor physics problems, as it does not require discretization of the phase space. This is not necessarily true though when simulating multigroup problems, as it has traditionally been a challenge to model anisotropic scattering in such simulations. Multigroup data used in reactor simulations generally uses low order Legendre expansion for scattering distributions, often stopping at the third Legendre moment. With so few terms, the angular distribution can easily have negative regions for highly anisotropic energy transfers, which makes it impossible to use standard Monte Carlo methods to sample a scattering angle. Multigroup Monte Carlo codes therefore often resort to only using isotropic scattering with the transport correction (which is not always possible), or approximating the angular distribution with discrete angles. Neither case is ideal, and makes it impossible to accurately model such problems or verify deterministic codes that can and do make use of the low order Legendre expansions without issue. This is addressed in the present study by using importance sampling in conjunction with negative particle weights to sample scattering angles from negative scattering distributions. It is demonstrated that such an approach necessitates the use of weight cancellation methods in order to be stable and converge to a solution. The technique is tested on two simple analytic benchmark problems, and then further demonstrated by modeling a small zero power research reactor, comparing results against a deterministic solver which can treat anisotropic scattering. Comparison of the simulation results indicates that importance sampling for anisotropic scattering with weight cancellation can be used to obtain reference Monte Carlo results for multigroup problems despite negative scattering distributions.

Negative Weights and Weight Cancellation to Treat Anisotropic Scattering in Multigroup Monte Carlo Simulations

TL;DR

This work addresses the challenge of sampling anisotropic scattering in multigroup Monte Carlo simulations, where truncated Legendre expansions can yield negative angular distributions. By combining importance sampling with a positive surrogate distribution and a weight-cancellation mechanism, the authors enable unbiased sampling from signed scattering distributions while maintaining convergence. The method is validated on analytic benchmarks, where it reproduces the exact eigenvalues and fluxes more accurately than positive-only approximations, and is demonstrated on a realistic 2D reactor model (RCF) where it compares favorably with deterministic anisotropic-solvers. The results suggest that importance sampling with weight cancellation provides a robust, GPU-friendly path for accurate multigroup MC with anisotropic scattering and offers a valuable reference for deterministic solvers.

Abstract

The Monte Carlo method is typically considered the gold standard for simulating reactor physics problems, as it does not require discretization of the phase space. This is not necessarily true though when simulating multigroup problems, as it has traditionally been a challenge to model anisotropic scattering in such simulations. Multigroup data used in reactor simulations generally uses low order Legendre expansion for scattering distributions, often stopping at the third Legendre moment. With so few terms, the angular distribution can easily have negative regions for highly anisotropic energy transfers, which makes it impossible to use standard Monte Carlo methods to sample a scattering angle. Multigroup Monte Carlo codes therefore often resort to only using isotropic scattering with the transport correction (which is not always possible), or approximating the angular distribution with discrete angles. Neither case is ideal, and makes it impossible to accurately model such problems or verify deterministic codes that can and do make use of the low order Legendre expansions without issue. This is addressed in the present study by using importance sampling in conjunction with negative particle weights to sample scattering angles from negative scattering distributions. It is demonstrated that such an approach necessitates the use of weight cancellation methods in order to be stable and converge to a solution. The technique is tested on two simple analytic benchmark problems, and then further demonstrated by modeling a small zero power research reactor, comparing results against a deterministic solver which can treat anisotropic scattering. Comparison of the simulation results indicates that importance sampling for anisotropic scattering with weight cancellation can be used to obtain reference Monte Carlo results for multigroup problems despite negative scattering distributions.

Paper Structure

This paper contains 13 sections, 32 equations, 16 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Methods and Theory
  3. Importance Sampling
  4. Power Iteration Stability In The Presence of Negative Weights
  5. Weight Cancellation
  6. Simple Analytic Test Problems
  7. Problem-37: Infinite Cylinder
  8. Problem-71: Infinite Slab
  9. Research Reactor Model
  10. Geometry Description
  11. Multigroup Cross Section Generation
  12. Results
  13. Conclusions

Figures (16)

  • Figure 1: Scattering Distribution and its CDF for in-group scattering within the energy interval $9.999987M\eV - 19.6403M\eV$ in water with $L=3$ for RPI's Research Critical Facility (RCF).
  • Figure 2: Shapes of the approximated angular distributions proposed by Brown and Barnett to avoid negative weights brown2008.
  • Figure 3: Cancellation mesh convergence study for problem-37
  • Figure 4: Estimated flux and % relative error in the flux for each angular distribution representation in problem-37
  • Figure 5: Cancellation mesh convergence study for problem-71.
  • ...and 11 more figures