Chord Length Sampling Research Articles

Chord length sampling correction analysis for dispersion fuel in Monte Carlo simulation

Chord length sampling correction analysis for dispersion fuel in Monte Carlo simulation

Chord length sampling with memory effects for spatially heterogeneous Markov media: Application to the rod model.

In this work we propose a modified Chord Length Sampling (CLS) algorithm, endowed with two layers of "memory effects," aimed at solving particle transport problems in one-dimensional spatially nonhomogeneous Markov media. CLS algorithms are a family of Monte Carlo methods which account for the stochastic nature of the media by sampling on-the-fly the random interfaces between material phases during the particle propagation. The possibility for the particles to remember the last crossed interfaces increases the accuracy of these models with respect to reference solutions obtained by solving the Boltzmann equationon a large number of realizations of the Markov media. In previous investigations, CLS models with memory have been tested exclusively for spatially uniform stochastic media: in this paper we extend this class of Monte Carlo methods to the case of spatially nonhomogeneous configurations. The effectiveness and the robustness of the modified CLS are probed considering several benchmark problems with varying material cross sectionsand Markov media densities. The obtained results are a stepping stone towards a generalization to three-dimensional models.

Memory-preserving Chord Length Sampling in three-dimensional spatially heterogeneous Markov media: Preliminary investigations

We report on an investigation of particle transport in spatially heterogeneous Markov media using a memory-preserving Chord Length Sampling (CLS) algorithm. CLS are a family of Monte Carlo methods capable of generating approximate solutions of the transport equations in random geometries by generating material interfaces on-the-fly during particle propagation. Since CLS does not preserve the correlations induced by spatial disorder, the sampled solutions generally present discrepancies with respect to the reference solution obtained by solving the Boltzmann equation in a large ensemble of random media realizations. In order to increase the accuracy of CLS, improved CLS models endowed with spatial memory effects have been proposed. In this work we propose a strategy that allows simultaneously taking into account memory effects and spatial gradients in three-dimensional configurations. Preliminary numerical findings are illustrated and compared to reference solutions.

A Limited-Memory Framework for Conditional Point Sampling for Radiation Transport in 1D Stochastic Media

Conditional Point Sampling (CoPS) is our newly proposed Monte Carlo method for transport in stochastic media that has been demonstrated to achieve highly accurate mean response results and to compute variance of the mean caused by random spatial mixing. The ability of CoPS to efficiently characterize the effects of random spatial mixing beyond the mean is hindered by the algorithm’s potentially unbounded computer memory footprint. Thus, in previous work, we established two limited-memory techniques for CoPS to improve required computer memory, i.e., recent memory (RM) CoPS and amnesia radius (AR) CoPS, the latter of which enables CoPS to tractably compute probability density functions (PDFs) of response. In this work, we create a limited-memory framework that allows CoPS to combine the advantages of limited-memory techniques and populate the framework with the two inaugural techniques of RM and AR. The proposed framework enables the user to control the computational performance of CoPS by making problem-specific trade-offs between accuracy, computer memory footprint, and characterization of response distributions based on input parameters. We present mean leakage results, material-dependent scalar flux, leakage PDFs, and computer memory footprint computed using this new framework. By selecting different input parameters in our proposed limited-memory framework, CoPS is demonstrated to roughly match the accuracy and computer memory footprint of the established approximate method Chord Length Sampling or to provide response distribution information comparable to the brute-force benchmark approach while improving the computer memory footprint compared to the original CoPS algorithm.

Particle transport in Markov media with spatial gradients: Comparison between reference solutions and Chord Length Sampling

Markov media represent a prototype class of stochastic models that are commonly used to describe the statistical distribution of material properties for particle transport applications. Examples are widespread and emerge for instance in radiative transfer or neutron propagation through disordered materials. Particle transport in Markov media can be addressed by two distinct strategies. The first consists in sampling a collection of stochastic geometries, solving the Boltzmann equation for each realisation, and finally taking ensemble averages, which leads to reference solutions for the observables of interest at the expense of high computational cost. The second consists in developing faster but approximate transport models capable of mimicking the effects of disorder within a single transport calculation, such as the celebrated Levermore-Pomraning (LP) equations and their Monte Carlo counterpart, the Chord Length Sampling (CLS) model. In this work we address particle transport in three-dimensional Markov media displaying spatial gradients in their statistical properties, and we compare the CLS model and SN solutions of the LP equations to the reference solutions.

Conditional Point Sampling: A stochastic media transport algorithm with full geometric sampling memory

Current methods for stochastic media transport are either computationally expensive or, by nature, approximate. Moreover, none of the well-developed, benchmarked approximate methods can compute the variance caused by the stochastic mixing, a quantity especially important to safety calculations. Therefore, we derive and apply a new conditional probability function (CPF) for use in the recently developed stochastic media transport algorithm Conditional Point Sampling (CoPS), which 1) leverages the full intra-particle memory of CoPS to yield errorless computation of stochastic media outputs in 1D, binary, Markovian-mixed media, and 2) leverages the full inter-particle memory of CoPS and the recently developed Embedded Variance Deconvolution method to yield computation of the variance in transport outputs caused by stochastic material mixing. Numerical results demonstrate errorless stochastic media transport as compared to reference benchmark solutions with the new CPF for this class of stochastic mixing as well as the ability to compute the variance caused by the stochastic mixing via CoPS. Using previously derived, non-errorless CPFs, CoPS is further found to be more accurate than the atomic mix approximation, Chord Length Sampling (CLS), and most of memory-enhanced versions of CLS surveyed. In addition, we study the compounding behavior of CPF error as a function of cohort size (where a cohort is a group of histories that share intra-particle memory) and recommend that small cohorts be used when computing the variance in transport outputs caused by stochastic mixing.

Burnup analysis of the pebble-bed fluoride-salt-cooled high-temperature reactor based on the Chord Length Sampling method

The neutronics calculation of the PB-FHR with the fraction ranging from 5% to 30% are carried out by the CLS method and ERM method. It can be observed that the infinite multiplication factor obtained with the CLS method are consistently smaller than those obtained with the ERM method when the fraction is less than 15%. Moreover, the differences of the two methods basically increases with the increasing burnup level when the fraction is less than 15%. For the fraction up to 30%, the CLS method shows significant bias of 520 pcm with the explicit method. After the CLS method with a modified packing fraction correction is adopted, the largest difference drops to 220 pcm when the fraction is 30%.

Poisson-Box Sampling algorithms for three-dimensional Markov binary mixtures

Particle transport in Markov mixtures can be addressed by the so-called Chord Length Sampling (CLS) methods, a family of Monte Carlo algorithms taking into account the effects of stochastic media on particle propagation by generating on-the-fly the material interfaces crossed by the random walkers during their trajectories. Such methods enable a significant reduction of computational resources as opposed to reference solutions obtained by solving the Boltzmann equation for a large number of realizations of random media. CLS solutions, which neglect correlations induced by the spatial disorder, are faster albeit approximate, and might thus show discrepancies with respect to reference solutions. In this work we propose a new family of algorithms (called ’Poisson Box Sampling’, PBS) aimed at improving the accuracy of the CLS approach for transport in d-dimensional binary Markov mixtures. In order to probe the features of PBS methods, we will focus on three-dimensional Markov media and revisit the benchmark problem originally proposed by Adams, Larsen and Pomraning [1] and extended by Brantley [2]: for these configurations we will compare reference solutions, standard CLS solutions and the new PBS solutions for scalar particle flux, transmission and reflection coefficients. PBS will be shown to perform better than CLS at the expense of a reasonable increase in computational time.

Monte Carlo chord length sampling for d-dimensional Markov binary mixtures

The Chord Length Sampling (CLS) algorithm is a powerful Monte Carlo method that models the effects of stochastic media on particle transport by generating on-the-fly the material interfaces seen by the random walkers during their trajectories. This annealed disorder approach, which formally consists of solving the approximate Levermore–Pomraning equations for linear particle transport, enables a considerable speed-up with respect to transport in quenched disorder, where ensemble-averaging of the Boltzmann equation with respect to all possible realizations is needed. However, CLS intrinsically neglects the correlations induced by the spatial disorder, so that the accuracy of the solutions obtained by using this algorithm must be carefully verified with respect to reference solutions based on quenched disorder realizations. When the disorder is described by Markov mixing statistics, such comparisons have been attempted so far only for one-dimensional geometries, of the rod or slab type. In this work we extend these results to Markov media in two-dimensional (extruded) and three-dimensional geometries, by revisiting the classical set of benchmark configurations originally proposed by Adams, Larsen and Pomraning [1] and extended by Brantley [2]. In particular, we examine the discrepancies between CLS and reference solutions for scalar particle flux and transmission/reflection coefficients as a function of the material properties of the benchmark specifications and of the system dimensionality.

Benchmark solutions for transport in d-dimensional Markov binary mixtures

Linear particle transport in stochastic media is key to such relevant applications as neutron diffusion in randomly mixed immiscible materials, light propagation through engineered optical materials, and inertial confinement fusion, only to name a few. We extend the pioneering work by Adams, Larsen and Pomraning [1] (recently revisited by Brantley [2]) by considering a series of benchmark configurations for mono-energetic and isotropic transport through Markov binary mixtures in dimension d. The stochastic media are generated by resorting to Poisson random tessellations in 1d slab, 2d extruded, and full 3d geometry. For each realization, particle transport is performed by resorting to the Monte Carlo simulation. The distributions of the transmission and reflection coefficients on the free surfaces of the geometry are subsequently estimated, and the average values over the ensemble of realizations are computed. Reference solutions for the benchmark have never been provided before for two- and three-dimensional Poisson tessellations, and the results presented in this paper might thus be useful in order to validate fast but approximated models for particle transport in Markov stochastic media, such as the celebrated Chord Length Sampling algorithm.

On the improvements in neutronics analysis of the unit cell for the pebble-bed fluoride-salt-cooled high-temperature reactor

The stochastic characteristics of the randomly distributed coated-fuel particles, the thermal-neutron scattering effect of the fluoride-salt and the resonance elastic scattering effect of heavy nuclides were neglected in early neutronics studies on the Pebble-Bed Fluoride-salt-cooled High-temperature Reactor (PB-FHR). In order to assess the impact of these effects on the neutronics calculation, the stochastic effect is analyzed by applying the explicit random modeling approach and Chord Length Sampling method, the thermal-neutron scattering effect of the fluoride-salt is quantified by evaluating and processing a new thermal-neutron scattering data library, and the resonance elastic scattering effect is covered by using the Doppler Broadening Rejection Correction (DBRC) method in this work. According to the critical calculations of the PB-FHR pebble unit cells with different TRISO Packing Factor (TPF), the different stochastic modeling methods lead to a difference of kinf by tens of pcm, the thermal scattering effect of 2LiF-BeF2 results in a decreasement of kinf by 104–290 pcm and the resonance elastic scattering effect leads to a decreasement of kinf by 107–437 pcm, respectively. In addition, the thermal-neutron scattering effect and resonance elastic scattering effect can be addible, the total impact of the thermal scattering effect of 2LiF-BeF2 and the resonance elastic scattering effect of heavy nuclide leads to a decreasement of kinf by 204–747 pcm.

Depletion benchmarks calculation of random media using explicit modeling approach of RMC

Monte Carlo method plays an important role in accurate simulation of random media, owing to its advantages of the flexible geometry modeling and the use of continuous-energy nuclear cross sections. Three stochastic geometry modeling methods including Random Lattice Method, Chord Length Sampling and explicit modeling approach with mesh acceleration technique, have been implemented in RMC to simulate the particle transport in the dispersed fuels, in which the explicit modeling method is regarded as the best choice. In this paper, the explicit modeling method is applied to the depletion benchmark for HTGR fuel element, and the method of combination of adjacent burnup regions has been proposed and investigated. The results show that the explicit modeling can provide detailed burnup distribution of individual TRISO particles, and this work would serve as a supplement for the HTGR fuel depletion benchmark calculations. The combination of adjacent burnup regions can effectively reduce the memory footprint while keeping the computational accuracy.

Development of random geometry capability in RMC code for stochastic media analysis

Simulation of particle transport in random media poses a challenge for traditional deterministic transport methods, due to the significant effects of spatial and energy self-shielding. Monte Carlo method plays an important role in accurate simulation of random media, owing to its flexible geometry modeling and the use of continuous-energy nuclear cross sections. Three stochastic geometry modeling methods including Random Lattice Method, Chord Length Sampling and explicit modeling approach with mesh acceleration technique, have been developed in RMC to simulate the particle transport in the dispersed fuels. The verifications of the accuracy and the investigations of the calculation efficiency have been carried out. The stochastic effects of the randomly dispersed fuel particles are also analyzed. The results show that all three stochastic geometry modeling methods can account for the effects of the random dispersion of fuel particles, and the explicit modeling method can be regarded as the best choice.

A proposal on alternative sampling-based modeling method of spherical particles in stochastic media for Monte Carlo simulation

Chord length sampling method in Monte Carlo simulations is a method used to model spherical particles with random sampling technique in a stochastic media. It has received attention due to the high calculation efficiency as well as user convenience; however, a technical issue regarding boundary effect has been noted. In this study, after analyzing the distribution characteristics of spherical particles using an explicit method, an alternative chord length sampling method is proposed. In addition, for modeling in finite media, a correction method of the boundary effect is proposed. Using the proposed method, sample probability distributions and relative errors were estimated and compared with those calculated by the explicit method. The results show that the reconstruction ability and modeling accuracy of the particle probability distribution with the proposed method were considerably high. Also, from the local packing fraction results, the proposed method can successfully solve the boundary effect problem. It is expected that the proposed method can contribute to the increasing of the modeling accuracy in stochastic media.

A novel extension of chord length sampling method for TRISO-type fueled reactor applications

A new chord length distribution model is proposed to characterize the stochastic distribution of TRISO fuel particles in nuclear reactor systems and is used in the chord length sampling (CLS) method to analyze the neutronic behavior of TRISO fuel systems. In this model, the coating layers of fuel particles are homogenized with the background matrix region. The probability density function (PDF) of the chord length between fuel kernels, instead of fuel particles, is developed and is used in the CLS method. We first apply the new CLS model to solving one-group eigenvalue problems in a simplified 3-D stochastic medium system. Good accuracy is obtained in predicting the multiplication factor and fission density distribution. The relative differences are within 1.0% for both the multiplication factor and the total fission density in all the studied scenarios. We then apply the new CLS model to analyzing three realistic reactor designs: two Very High Temperature Gas-cooled Reactor unit cells and one fuel pin unit cell of an innovative light water reactor design with accident tolerant fuel. Infinite multiplication factor and intra-Dancoff factor are evaluated for the three unit cells respectively. Compared with the reference results, predictions from CLS simulations show a relative difference of less than 0.26% for infinite multiplication factors and less than 1.0% for intra-Dancoff factors in all the studied cases. Meanwhile, a significant improvement in the computational efficiency has been observed for the CLS new model compared with the old model, with a speedup of at least 1.70. The CLS method with the new chord length distribution model is verified to be an efficient Monte Carlo method, superior over the old model and without sacrificing the accuracy.

Chord length sampling method for analyzing stochastic distribution of fuel particles in continuous energy simulations

The Chord Length Sampling method (CLS) is studied in continuous energy simulations by analyzing two types of Very High Temperature Gas-cooled Reactor (VHTR) unit cells: the fuel compact cell in the prismatic type VHTR and the fuel pebble cell in the pebble-bed type VHTR. Infinite multiplication factors of the unit cells are calculated by the CLS and compared to benchmark simulations at different volume packing fractions from 5% to 30%. It is shown that the accuracy of the CLS is affected by the boundary effect, which is induced by the CLS procedure itself and results in a reduction in the total volume packing fraction of the fuel particles. To mitigate the boundary effect, three correction schemes based on the research of (1) Murata et al., (2) Ji and Martin and (3) Griesheimer et al. are used to improve the accuracy by applying a corrected value of the volume packing fraction to the CLS. These corrected values are calculated based on (1) a simple linear relationship, (2) an iterative self-consistent simulation correction method and (3) a theoretically derived non-linear relationship, respectively. The CLS simulation using the corrected volume packing fraction shows excellent improvements in the infinite multiplication factors for the VHTR unit cells. Ji and Martin’s self-consistent correction method shows the best improvement.

On the Chord Length Sampling in 1-D Binary Stochastic Media

When the chord length sampling method is applied to solve radiation transport problems in random media where inclusions are randomly distributed in a background region, inaccuracy occurs due to two major factors: memory effect and boundary effect. In this article, by applying chord length sampling to solve fixed source and eigenvalue problems in 1-D binary stochastic media, an investigation on how and why these two effects give rise to the inaccuracy in the final solutions is performed. The investigation is based on a series of radiation transport simulations for the calculation of reflection rate, flux distribution, and effective multiplication factor.

Investigation of Stochastic Radiation Transport Methods in Binary Random Heterogeneous Mixtures

This paper reviews existing Monte Carlo techniques for performing neutron transport simulations in binary random heterogeneous fissile fuels and presents a new approach offering superior efficiency at little cost in fidelity for problems involving densely packed, optically thick absorbers. The accuracy of the chord-length sampling technique is demonstrated to be a function of the total optical thicknesses and optical scattering thickness of the constituent materials as well as the packing density of the fissile kernels. The results of this parameter assessment provide a foundation for an original hybrid algorithm that combines homogeneous and explicit geometry models within a single Monte Carlo simulation. The geometry model utilized is selected according to the energy-dependent optical thickness. By partitioning the geometry representation within a single Monte Carlo simulation into homogenous and heterogeneous energy-dependent models, acceptable ensemble average results are obtained in a fraction of the run time of the detailed explicit geometry benchmark method.

Hemisphere Sizing from Linear Intercept Measurement

The stereological problem of unfolding the hemisphere radius distribution from the length distribution is analyzed. Let a stationary isotropic process of hemispheres be given. The hemispheres have random diameters and are isotropically uniformly randomly orientated in space. A straight line probe yields a process of intercepts. The inverse problem of re-obtaining the size distribution of the hemispheres in terms of an experimental intercept length distribution is solved. The length distribution of a single hemisphere, known analytically, is approximated by piecewise polynomials in two intervals. The solution of the inverse problem is traced back to a simple recurrence equation. Numerical checks with exact and simulated data are performed to demonstrate the applicability. Data of chord length sampling, resulting from image analysis procedures, from scattering methods or from other appropriate physical apparatuses, are applicable.

Application of Monte Carlo Chord-Length Sampling Algorithms to Transport through a Two-Dimensional Binary Stochastic Mixture

Monte Carlo algorithms are developed to calculate the ensemble-average particle leakage through the boundaries of a two-dimensional binary stochastic material. The mixture is specified within a rectangular area and consists of a fixed number of disks of constant radius randomly embedded in a matrix material. The algorithms are extensions of the proposal of Zimmerman et al., using chord-length sampling (CLS) to eliminate the need to explicitly model the geometry of the mixture. Two variations are considered. The first algorithm uses CLS for both material regions. The second algorithm employs limited CLS (LCLS), using only CLS in the matrix material. Ensemble-average leakage results are computed for a range of material interaction coefficients and compared against benchmark results for both accuracy and efficiency. Both algorithms are exact for purely absorbing materials and provide decreasing accuracy as scattering is increased in the matrix material. The LCLS algorithm shows a better accuracy than the CLS algorithm for all cases while maintaining an equivalent or better efficiency. Accuracy and efficiency problems with the CLS algorithm are due principally to assumptions made in determining the chord-length distribution within the disks.