Spatially Continuous Depletion Algorithm for Monte Carlo Simulations

To correctly predict reactor behavior during cycle operations, the evolution of nuclide number densities throughout the core must be accurately modeled. The time-varying spatial distribution of nuclide number densities is typically resolved by discretizing the Monte Carlo geometry into smaller cells over which number densities are assumed to be spatially invariant. The nuclide number densities in these smaller cells are integrated through time using reaction rate tallies on the same discretized geometry. However, detailed distributions of nuclide number densities in a full three dimensional simulation can require a prohibitive amount of tallies, and the spatial discretization of the base geometry makes coupling to external multiphysics tools difficult. In this paper a method for solving for spatially continuous number density distributions during depletion calculations will be described. The spatially continuous number densities can be used in the transport method proposed by Brown and Martin which allows for transporting neutrons through a material with continuously varying properties such as temperature and nuclide number densities. Coupled with the ability of Functional Expansion Tallies (FETs) [2] to represent tallied quantities as continuous functions, it is possible to both solve for and make use of spatially continuous nuclide number densities. The need for this capability was alluded to by Brown et. al., but no solution has yet been proposed. With a continuous depletion method, recent work which utilized FETs and continuous material tracking to incorporate multiphysics feedback in Monte Carlo simulations can be extended to simulations that include depletion analysis.United States. Department of Energy (Nuclear Energy University Programs Graduate Fellowship

Predicting Correlation Coefficients for Monte Carlo Eigenvalue Simulations

Monte Carlo methods are most often considered as a reference for neutron transport simulations since very limited approximations are made abount nuclear data and system geometry. To report uncertainty of any tally evaluated as generation averages, the sample variance is divided by the number of active generations, which is based on the assumption that the neutron generations are independent. Correlation effects between neutrons in multiplying systems, particularly when performing power iteration to evaluate eigenvalues have been observed in previous work. Neglecting the correlation effect results in an underestimate of uncertainty reported by Monte Carlo calculations. Previous work has also proposed methods to predict the underestimation ratio. Yamamoto et al expanded the fission source distribution with diffusion equation modes, performed numerical simulation of the AR(autoregressive) process of the expansion coefficients and used the correlation of the AR process to predict that of the Monte Carlo eigenvalue simulation. Sutton applied the discretized phase space (DPS) approach to predict the underestimation ratio but the method cannot predict the ratio when one neutron generates offspring in different phase space regions or generates a random number of offspring. This paper presents a method to predict the correlation effect with the model of multitype branching processes (MBP). The method requires simulations for one generation of neutrons without knowing the source distribution and can predict the underestimation ratio for the cases where the traditional DPS approach does not work. The generation-to-generation correlation determines the convergence rate of active generations, the bias of variance estimator for each generation and the underestimation ratio of variance estimator for tallies averaged over active generations. The generation-to-generation correlation is characterized by the Auto-Correlation Coefficients (ACC) between tallies from different generations.United States. Dept. of Energy (Consortium for Advanced Simulation of Light Water Reactors. Contract DE-AC05-00OR22725

Direct Doppler broadening in Monte Carlo simulations using the multipole representation

A new approach for direct Doppler broadening of nuclear data in Monte Carlo simulations is proposed based on the multipole representation. The multipole representation transforms resonance parameters into a set of poles and residues only some of which exhibit a resonant behavior. A method is introduced to approximate the contribution to the background cross section in an effort to reduce the number of poles needing to be broadened. The multipole representation results in memory savings of 1–2 orders of magnitude over comparable techniques. This approach provides a simple way of computing nuclear data at any temperature which is essential for multi-physics calculations, while having a minimal memory footprint which is essential for scalable high performance computing. The concept is demonstrated on two major isotopes of uranium (U-235 and U-238) and implemented in the OpenMC code. Two LEU critical experiments were solved and showed great accuracy with a small loss of efficiency (10–30%) over a single-temperature pointwise library.United States. Dept. of Energy. Office of Advanced Scientific Computing Research (Contract DE-AC02-06CH11357

Analysis of correlations and their impact on convergence rates in Monte Carlo eigenvalue simulations

This paper provides an analysis of the generation-to-generation correlations as observed when solving full core eigenvalue problems on PWR systems. Many studies have in the past looked at the impact of these correlations on reported variance and this paper extends the analysis to the observed convergence rate on the tallies, the effect of tally size and the effect of generation size. Since performing meaningful analysis on such a large problem is inherently difficult, a simple homogeneous reflective cube problem with analytical solution was developed that exhibits similar behavior to the full core PWR benchmark. The data in this problem was selected to match the dimensionality of the reactor problem and preserve the migration length travelled by neutrons. Results demonstrate that the variance will deviate significantly from the 1/N (N being the number of simulated particles) convergence rate associated with truly independent generations, but will eventually asymptote to 1/N after 1000's of generations regardless of the numbers of neutrons per generation. This indicates that optimal run strategies should emphasize lower number of active generations with greater number of neutrons per generation to produce the most accurate tally results. This paper also describes and compares three techniques to evaluate suitable confidence intervals in the presence of correlations, one based on using history statistics, one using generation statistics and one batching generations to reduce batch-to-batch correlation. Keywords: Monte Carlo, Tally Convergence, Autocorrelation, Confidence IntervalsUnited States. Department of Energy (Consortium for Advanced Simulation of Light Water Reactors. Contract DE-AC05-00OR22725

A Cumulative migration method for computing rigorous transport cross sections and diffusion coefficients for LWR lattices with Monte Carlo

A new method for computing homogenized assembly neutron transport cross sections and diffusion coefficients that is both rigorous and computationally efficient is proposed in this paper. In the limit of a homogeneous hydrogen slab, the new method is equivalent to the long-used, and only-recently-published CASMO transport method. The rigorous method is used to demonstrate the sources of inaccuracy in the commonly applied “out-scatter” transport correction. It is also demonstrated that the newly developed method is directly applicable to lattice calculations performed by Monte Carlo and is capable of computing rigorous homogenized transport cross sections for arbitrarily heterogeneous lattices. Comparisons of several common transport cross section approximations are presented for a simple problem of infinite medium hydrogen. The new method has also been applied in computing 2-group diffusion data for an actual PWR lattice from BEAVRS benchmark.Idaho National Laboratory (Contract DE-AC07-05ID14517

Techniques for Stabilizing Coarse-Mesh Finite Difference (CMFD) in Methods of Characteristics (MOC)

The Coarse-Mesh Finite Difference (CMFD) method has been widely used to effectively accelerate neutron transport calculations. It was however found to be at times unstable in the presence of strong heterogeneities. The common practice to improve stability is to employ a damping factor on the nonlinear diffusion coefficient terms, but there is no method to determine the optimal damping factor for a practical reactor problem prior to the calculation. This paper investigates two problem-agnostic techniques that stabilize reactor calculations that would otherwise diverge with undamped CMFD. The first technique is to perform additional energy sweeps for the upscattering group region during the high-order MOC calculation to generate more accurate information to pass into the CMFD calculation. The second technique extends the traditional scalar flux prolongation to provide spatial variations inside each acceleration cell. This study uses the 2D C5G7 problem and the Babcock & Wilcox 1810 series critical experiment benchmark to evaluate these methods. Numerical simulations showed that both techniques stabilize CMFD, and that the linear prolongation technique did not incur additional computational cost compared to the optimally damped conventional metho

Simple benchmark for evaluating self-shielding models

Accounting for self-shielding effects is paramount to accurate generation of multigroup cross sections for use in deterministic reactor physics neutronics calculations. Historically, equivalence in dilution and subgroup techniques have been the preeminent means of accounting for these effects, but recent work has proposed new solutions, including the Embedded Self-Shielding Method (ESSM). This paper presents a very simple benchmark problem to compare these and future self-shielding methods. The benchmark is perhaps the simplest problem in which both energy and spatial self-shielding effects are important, a two-region problem with a lumped resonant material. A single resonance in a single energy group is considered. Scattering is approximated using the narrow resonance approximation, decoupling each energy value and allowing an easily-computed reference solution to be obtained. Equivalence in dilution using two-term rational expansions and the subgroup method were both found to give very accurate solutions on this benchmark, with errors less than 1% in nearly all cases. One-term rational expansions and ESSM showed much larger errors

Accelerated sampling of the free gas resonance elastic scattering kernel

In this work, we present the derivation and investigation of a new Doppler broadening rejection sampling approach for the exact treatment of resonance elastic scattering in Monte Carlo neutron transport codes. Implemented in OpenMC, this method correctly accounts for the energy dependence of cross sections when treating the thermal motion of target nuclei in elastic scattering events. The method is verified against both stochastic and deterministic reference results in the literature for ²³⁸U resonance scattering. Upscatter percentages and mean scattered energies calculated with the method are shown to agree well with the reference scattering kernel results. Additionally, pin cell and full core k[subscript eff] results calculated with this implementation of the exact resonance scattering kernel are shown to be in close agreement with those in the literature. The attractiveness of the method stems from its improvement upon a computationally expensive rejection sampling procedure employed by an earlier stochastic resonance scattering treatment. With no loss in accuracy, the accelerated sampling algorithm is shown to reduce overall runtime by 3–5% relative to the Doppler broadening rejection correction method for both pin cell and full core benchmark problems. This translates to a 30–40% reduction in runtime overhead.United States. Department of Energy (DE-AC05-00OR22725

Progress toward Monte Carlo–thermal hydraulic coupling using low-order nonlinear diffusion acceleration methods

A new approach for coupled Monte Carlo (MC) and thermal hydraulics (TH) simulations is proposed using low-order nonlinear diffusion acceleration methods. This approach uses new features such as coarse mesh finite difference diffusion (CMFD), multipole representation for fuel temperature feedback on microscopic cross sections, and support vector machine learning algorithms (SVM) for iterations between CMFD and TH equations. The multipole representation method showed small differences of about 0.3% root mean square (RMS) error in converged assembly source distribution compared to a conventional MC simulation with ACE data at the same temperature. This is within two standard deviations of the real uncertainty. Eigenvalue differences were on the order of 10 pcm. Support vector machine regression was performed on-the-fly during MC simulations. Regression results of macroscopic cross sections parametrized by coolant density and fuel temperature were successful and eliminated the need of partial derivative tables generated from lattice codes. All of these new tools were integrated together to perform MC-CMFD-TH-SVM iterations. Results showed that inner iterations between CMFD-TH-SVM are needed to obtain a stable solution

Quantifying Transient Uncertainty in the BEAVRS Benchmark Using Time Series Analysis Methods

Introduction - Advances in computation have brought about significant improvements in creating fast-running high-fidelity simulations of nuclear cores. The BEAVRS benchmark [1] is a highly-detailed PWR specification with two cycles of measured operational data used to validate high-fidelity core analysis methods. This PWR depletion benchmark captures the fine details of the LWR fuel assemblies, burnable absorbers, in-core fission detectors, core loading and shuffling patterns. Specifically, 58 of the 193 assemblies contain in-core detectors with measurements taken over 61 axial positions every month. These detectors are U-235 fission chambers with slightly varying mass of U-235. The collected signals are normalized on a given assembly permitting full core comparisons. The fuel layout for cycle 1 and instrument tube locations for the reactor are given in figures 1 and 2 respectively. Through a series of data processing and comparisons, it was shown [2] that axially integrated radial maps of reaction rates were in close agreement between provided detector data and calculated dataUnited States. Department of Energy (Nuclear Energy University Program Grant