Monte Carlo Solution Research Articles

A meshfree arbitrary Lagrangian-Eulerian method for the BGK model of the Boltzmann equation with moving boundaries

In this paper we present a novel technique for the simulation of moving boundaries and moving rigid bodies immersed in a rarefied gas using an Eulerian-Lagrangian formulation based on least square method. The rarefied gas is simulated by solving the Bhatnagar-Gross-Krook (BGK) model for the Boltzmann equation of rarefied gas dynamics. The BGK model is solved by an Arbitrary Lagrangian-Eulerian (ALE) method, where grid-points/particles are moved with the mean velocity of the gas. The computational domain for the rarefied gas changes with time due to the motion of the boundaries. To allow a simpler handling of the interface motion we have used a meshfree method based on a least-square approximation for the reconstruction procedures required for the scheme. We have considered a one way, as well as a two-way coupling of boundaries/rigid bodies and gas flow. The numerical results are compared with analytical as well as with Direct Simulation Monte Carlo (DSMC) solutions of the Boltzmann equation. Convergence studies are performed for one-dimensional and two-dimensional test-cases. Several further test problems and applications illustrate the versatility of the approach.

Neutronic simulation of the CEFR experiments with the nodal diffusion code system RAST-F

CEFR is a small core-size sodium-cooled fast reactor (SFR) using high enrichment fuel with stainless-steel reflectors, which brings a significant challenge to the deterministic methodologies due to the strong spectral effect. The neutronic simulation of the start-up experiments conducted at the CEFR have been performed with a deterministic code system RAST-F, which is based on the two-step approach that couples a multi-group cross-section generation Monte-Carlo (MC) code and a multi-group nodal diffusion solver. The RAST-F results were compared against the measurement data. Moreover, the characteristic of neutron spectrum in the fuel rings, and adjacent reflectors was evaluated using different models for generation of accurate nuclear libraries. The numerical solution of RAST-F system was verified against the full core MC solution MCS at all control rods fully inserted and withdrawn states. A good agreement between RAST-F and MCS solutions was observed with less than 120 pcm discrepancies and 1.2% root-mean-square error in terms of keff and power distribution, respectively. Meanwhile, the RAST-F result agreed well with the experimental values within two-sigma of experimental uncertainty. The good agreement of these results indicating that RAST-F can be used to neutronic steady-state simulations for small core-size SFR, which was challenged to deterministic code system.

TRANSFER LEARNING ON MULTIFIDELITY DATA

Neural networks (NNs) are often used as surrogates or emulators of partial differential equations (PDEs) that describe the dynamics of complex systems. A virtually negligible computational cost of such surrogates makes them an attractive tool for ensemble-based computation, which requires a large number of repeated PDE solutions. Since the latter are also needed to generate sufficient data for NN training, the usefulness of NN-based surrogates hinges on the balance between the training cost and the computational gain stemming from their deployment. We rely on multifidelity simulations to reduce the cost of data generation for subsequent training of a deep convolutional NN (CNN) using transfer learning. High- and low-fidelity images are generated by solving PDEs on fine and coarse meshes, respectively. We use theoretical results for multilevel Monte Carlo method to guide our choice of the numbers of images of each kind. We demonstrate the performance of this multifidelity training strategy on the problem of estimation of the distribution of a quantity of interest, whose dynamics is governed by a system of nonlinear PDEs (parabolic PDEs of multiphase flow in heterogeneous porous media) with uncertain/random parameters. Our numerical experiments demonstrate that a mixture of a comparatively large number of low-fidelity data and smaller number of high-fidelity data provides an optimal balance of computational speed-up and prediction accuracy. The former is reported relative to both CNN training on high-fidelity images only and Monte Carlo solution of the PDEs. The latter is expressed in terms of both the Wasserstein distance and the Kullback-Leibler divergence.

Exponential Mixture Density Based Approximation to Posterior Cramér-Rao Lower Bound for Distributed Target Tracking

The posterior Cramr-Rao lower bound (PCRLB) and a number of its extensions including the unconditional PCRLB (U-PCRLB) and the conditional PCRLB (C-PCRLB) have been widely studied in multi-sensor target tracking (MSTT). Previous MSTT works assume that the measurements are conditionally independent, which is often not the case in practice. When the correlations of the measurements are unknown, the standard Bayes update and PCRLB computations cannot be performed. Inspired by geometric average (GA) fusion that implements data fusion with unknown data correlations by employing exponential mixture density (EMD) to compute the global posterior, EMD-based approximations to U-PCRLB and C-PCRLB are derived in this paper for two classic distributed fusion architectures, namely the hierarchical and consensus architectures. The PCRLB can be decomposed into two parts, one coming from the prior information and the other from the information contained in the data. The data information part of PCRLB is approximated via EMD-based posterior, leading to the approximation of the PCRLB. We present a sequential Monte Carlo solution to recursively compute the proposed approximation to the PCRLB for nonlinear non-Gaussian estimation problems. Numerical simulations are provided to show that the proposed approximation to the bound on the estimation mean square error (MSE) is tighter compared to the existing bounds based on likelihood fusion obtained under the measurement independence assumption.

Recovery of thermal load parameters by means of the Monte Carlo method with fixed and meshless random walks

The paper is focused on the numerical analysis of the two-point inverse stationary heat flow problem, namely the identification of the thermal load. Determination of selected parameters is possible on the basis of temperature measurements, done at specified few locations. This problem may be modelled as the optimization problem, standard numerical analysis of which requires multiple solutions of boundary value problems. However, the novel solution approach, proposed here, is based on the well-known concept of the Monte Carlo method with an appropriate random walk technique. It yields explicit stochastic relations combining computed temperatures and all load parameters. Such relations may be directly applied in most standard optimization algorithms, replacing time-consuming solutions of systems of algebraic equations. Moreover, one may construct the semi-analytical approach, in which the unknown load parameters are obtained explicitly, allowing for the elimination of sensitiveness to initial solutions or requirements for admissible load intervals. The paper is illustrated with results of several benchmark problems with simulated measurement data and various numbers of unknown load parameters. The results comparison, between standard element-free methods with selected optimization algorithms as well as the proposed Monte Carlo solution approach is presented and is especially focused on CPU times.

Numerical simulation of pulsed planar evaporation into background gas based on direct Monte Carlo simulation and solution of the BGK model kinetic equation

A numerical study of the planar gas expansion under pulsed evaporation into the background gas is carried out. The chosen conditions are typical for nanosecond laser deposition of thin films and nanostructure synthesis, with the saturated gas pressure at the surface of 5.4 MPa and the background pressure of 50 and 500 Pa. The problem is solved based on the direct simulation Monte Carlo method and direct numerical solution of the BGK model kinetic equation. A generally good agreement was obtained for all computed macroscopic quantities, with the exception of the higher density peak in the compressed layer and a wider shock front in the background gas for the BGK model.

An improved deterministic truncation of Monte Carlo solution for pin-resolved nuclear reactor analysis

An improved deterministic truncation of Monte Carlo (iDTMC) solution method has been developed for acceleration and variance reduction of Monte Carlo (MC) neutronic analysis. A new calculational strategy has been applied by taking advantages of partial-current-based coarse mesh finite difference (p-CMFD) method with two different mesh sizes in a tactical way. Fast convergence of source distribution is attained by a coupled assembly-wise p-CMFD method in inactive cycles, and detailed deterministic solutions are obtained by a decoupled pin-wise p-CMFD method in active cycles. Assistant schemes are also devised to enhance efficiency of the iDTMC method, which include one-node p-CMFD acceleration, new inactive cycle strategy, and real variance estimation method for iDTMC solutions. The iDTMC method is evaluated in a small reactor and a big-size one. The iDTMC solutions are compared with those of the p-CMFD accelerated MC calculation, and they are appraised in terms of source convergence, computing time, and figure-of-merit.

Superphénix Benchmark Part I: Results of Static Neutronics

Abstract In the paper, the specification of a new neutronics benchmark for large sodium cooled fast reactor (SFR) core and results of modeling by different participants are presented. The neutronics benchmark describes the core of the French sodium cooled reactor Superphénix at its startup configuration, which in particular was used for experimental measurement of reactivity characteristics. The benchmark consists of the detailed heterogeneous core specification for neutronic analysis and the results of the reference solution. Different core geometries and thermal conditions from the cold “as fabricated” up to full power were considered. The reference Monte Carlo (MC) solution of serpent 2 includes data on multiplication factor, power distribution, axial and radial reaction rates distribution, reactivity coefficients and safety characteristics, control rods worth, kinetic data. The results of modeling with seven other solutions using deterministic and MC methods are also presented and compared to the reference solution. The comparisons results demonstrate appropriate agreement of evaluated characteristics. The neutronics results will be used in the second phase of the benchmark for the evaluation of transient behavior of the core.

Evaluation of PBMR-400 Core Design Steady State Condition with Serpent and AGREE

The significant recent advances in computer speed and memory have made possible an increasing fidelity and accuracy in reactor core simulation with minimal increase in the computational burden. This has been important for modeling some of the smaller advanced reactor designs for which simplified approximations such as few groups homogenized diffusion theory are not as accurate as they were for large light water reactor cores. For narrow cylindrical cores with large surface to volume ratios such the Ped Bed Modular Reactor (PBMR), neutron leakage from the core can be significant, particularly with the harder neutron spectrum and longer mean free path than a light water reactor. In this paper the core from the OECD PBMR-400 benchmark was analyzed using multigroup Monte Carlo cross sections in the HTR reactor core simulation code AGREE. Homogenized cross sections were generated for each of the discrete regions of the AGREE model using a full core SERPENT Monte Carlo model. The cross sections were generated for a variety of group structures in AGREE to assess the importance of finer group discretization on the accuracy of the core eigenvalue and flux predictions compared to the SERPENT full core Monte Carlo solution. A significant increase in the accuracy was observed by increasing the number of energy groups, with as much as a 530 pcm improvement in the eigenvalue calculation when increasing the number of energy groups from 2 to 14. Significant improvements were also observed in the AGREE neutron flux distributions compared to the SERPENT full core calculation.

Prediction calculations for the first criticality of the HTR-PM using the PANGU code

The high-temperature reactor pebble-bed module (HTR-PM) is a modular high-temperature gas-cooled reactor demonstration power plant. Its first criticality experiment is scheduled for the latter half of 2021. Before performing the first criticality experiment, a prediction calculation was performed using PANGU code. This paper presents the calculation details for predicting the HTR-PM first criticality using PANGU, including the input model and parameters, numerical results, and uncertainty analysis. The accuracy of the PANGU code was demonstrated by comparing it with the high-fidelity Monte Carlo solution, using the same input configurations. It should be noted that keff can be significantly affected by uncertainties in nuclear data and certain input parameters, making the criticality calculation challenge. Finally, the PANGU is used to predict the critical loading height of the HTR-PM first criticality under design conditions, which will be evaluated in the upcoming experiment later this year.

A high-order semi-Lagrangian method for the consistent Monte-Carlo solution of stochastic Lagrangian drift–diffusion models coupled with Eulerian discontinuous spectral element method

The explicit semi-Lagrangian method for solution of Lagrangian transport equations as developed in Natarajan and Jacobs (2020) is adopted for the solution of stochastic differential equations that is consistent with Discontinuous Spectral Element Method (DSEM) approximations of Eulerian conservation laws. The method extends the favorable properties of DSEM that include its high-order accuracy, its local and boundary fitted properties and its high performance on parallel platforms for the concurrent Monte-Carlo, semi-Lagrangian and Eulerian solution of a class of time-dependent problems that can be described by coupled Eulerian–Lagrangian formulations. Such formulations include the probabilistic models used for the simulation of chemically reacting turbulent flows or particle-laden flows. Consistent with an explicit, DSEM discretization, the semi-Lagrangian method seeds particles at Gauss quadrature collocation nodes within a spectral element. The particles are integrated explicitly in time according to a drift velocity and a Wiener increment forcing and form the nodal basis for an advected interpolant. This interpolant is mapped back in a semi-Lagrangian fashion to the Gauss quadrature points through a least squares fit using constraints for element boundary values. Stochastic Monte-Carlo samples are averaged element-wise on the quadrature nodes. The stable explicit time step Wiener increment is sufficiently small to prevent particles from leaving the element’s bounds. The semi-Lagrangian method is hence local and parallel and does not have the grid complexity, and parallelization challenges of the commonly used Lagrangian particle solvers in particle-mesh methods for solution of Eulerian–Lagrangian formulations. Formal proof is presented that the semi-Lagrangian algorithm evolves the solution according to the Eulerian Fokker–Planck equation. Numerical tests in one and two dimensions for drift–diffusion problems show that the method converges exponentially for constant and non-constant advection and diffusion velocities.

Upscaling of transport through discrete fracture networks via random walk: A comparison of models.

Many models have been created for upscaled transport modeling in discrete fracture networks (DFNs). Random walk examples of these are the Markov directed random walk (MDRW), Monte Carlo solution of the Boltzmann transport equation (BTE), and the spatial Markov model (SMM). Each model handles the correlation between the random walk steps using different techniques and has successfully reproduced the results of full-resolution transport simulations in DFNs. However, their predictive capabilities under different modeling scenarios have not been compared. We construct a set of random 2D DFNs for three different fracture transmissivity distributions to comparatively evaluate model performance. We focus specifically on random walk models to determine what aspects of the space and time step distributions (e.g., correlation and coupling) must be accounted for to get the most accurate predictions. For DFNs with low heterogeneity in fracture transmissivity, accounting for correlation generally leads to less accurate predictions of transport behavior, but as the fracture transmissivity distribution widens, preferential pathways form and correlation between modeling steps becomes important, particularly for early breakthrough predictions.

Monte Carlo and Milne-like solutions for temporal correlation function in a wide temporal range

The Milne problem for the temporal correlation function of radiation backscattered from a half space is solved analytically. The angular and temporal dependencies calculated with the Monte Carlo simulations coincide with high precision with the Milne-like solution for a near isotropic scattering; for a highly anisotropic scattering the Monte Carlo results turn to be in a fair agreement with the known measurement, while the Milne-like solution greatly disagree with the experiment. Effects of finite thickness and non-zero absorption are also considered.

MCBTE: A variance-reduced Monte Carlo solution of the linearized Boltzmann transport equation for phonons

MCBTE solves the linearized Boltzmann transport equation for phonons in three-dimensions using a variance-reduced Monte Carlo solution approach. The algorithm is suited for both transient and steady-state analysis of thermal transport in structured materials with size features in the nanometer to hundreds of microns range. The code is portable and integrated with both first-principles density functional theory calculations and empirical relations for the input of phonon frequency, group velocity, and mean free path required for calculating the thermal properties. The program outputs space- and time-resolved temperature and heat flux for the transient study. For the steady-state simulations, the frequency-resolved contribution of phonons to temperature and heat flux is written to the output files, thus allowing the study of cumulative thermal conductivity as a function of phonon frequency or mean free path. We provide several illustrative examples, including ballistic and quasi-ballistic thermal transport, the thermal conductivity of thin films and periodic nanostructures, to demonstrate the functionality and to benchmark our code against available theoretical/analytical/computational results from the literature. Moreover, we parallelize the code using the Matlab Distributed Computing Server, providing near-linear scaling with the number of processors. Program summaryProgram Title:MCBTECPC Library link to program files:https://doi.org/10.17632/rzn86w7t3p.1Developer's repository link:https://github.com/abhipath90/MCBTECode Ocean capsule:https://codeocean.com/capsule/7196257Licensing provisions: GPLv3Programming language: MATLABNature of problem: Calculation of time- and space-dependent temperature and heat flux profiles, and frequency-resolved effective thermal conductivity in structured systems where heat is carried by phonons.Solution method: Solution of linearized Boltzmann transport equation for phonons, variance-reduced Monte Carlo approach.Additional comments including restrictions and unusual features: Runtime about 1 to 10 hours on a personal computer.

Pressurized Water Reactor Mixed Oxide/UO2 Transient Benchmark Calculations Using Monte Carlo Serpent 2 Code and Open Nodal Core Simulator ADPRES

Abstract This paper presents static and transient solutions for the pressurized water reactor (PWR) MOX/UO2 transient benchmark by Serpent 2 Monte Carlo (MC) code and open nodal core simulator called Abu Dhabi Polytechnic Reactor Simulator (ADPRES). The presence of mixed oxide (MOX) fuels and burn-up variation in the benchmark pose challenges for reactor simulators due to severe flux gradient across fuel assemblies. In this work, the two-step method was used, in which the assembly level two-group constants were generated from single assembly calculations with zero net current boundary conditions using Serpent 2 Monte Carlo code, and later the core calculation was performed using ADPRES open nodal core simulator. Two types of diffusion coefficients were generated: the conventional B1 leakage corrected and the cumulative migration method (CMM). Finally, the solutions of Serpent 2/ADPRES, including multiplication factor, power distribution, control rod (CR) worth, and critical boron concentration (CBC) using both diffusion coefficients, were compared against solutions from heterogeneous Serpent 2 calculations where the fuel and cladding are explicitly modeled. The reactor power during transients was also compared qualitatively against other nodal core simulators. The results showed that Serpent 2/ADPRES were able to predict the heterogeneous Monte Carlo solutions very well with reasonable differences. The transient solutions were also quite accurate compared to other nodal core simulators. For the diffusion coefficients comparison, it was found that the CMM diffusion coefficient provides more accurate solutions for the benchmark compared to the B1 leakage corrected diffusion coefficients.

A nodal transport method for coupled fast-thermal reactor analysis

For efficient core analyses of coupled fast-thermal reactors, a new deterministic method has been developed based on the variational nodal transport method of VARIANT. A new multigroup cross section generation procedure was devised by combining Monte Carlo lattice calculations for thermal assemblies and the two-step procedure of the MC2-3 code for fast assemblies. To reduce assembly homogenization errors in nodal transport calculations, a new nodal equivalence method was developed based on the partial current discontinuity factor (PCDF) and incorporated in VARIANT. A practical procedure to calculate PCDFs with fixed source supercell calculations was also devised. The performance of the proposed method was investigated using a test problem derived from the versatile coupled test reactor (VCTR) design. The new procedure produced multigroup cross sections accurately for all assemblies. VARIANT transport calculations with PCDFs produced accurate multiplication factor and power distribution compared to reference Serpent-2 Monte Carlo solutions.

Influence of gas transmission network failure on security of supply

This paper investigates the influence of failure of transmission network key elements on security of natural gas delivery. The proposed framework combines the probabilistic structure integrity study and the hydraulic analysis. A factor called a volumetric loss was introduced to quantify the effect of network failure. Based on the magnetic flux leakage diagnostics, the rupture probability of steel tube plastic collapse was estimated using a random variable method and Monte Carlo solution. In-line inspections should monitor only the rate of tube wall corrosion, whereas the proposed methodology enables the pipeline operator to evaluate the system ability to deliver gas at the customer nodes. For the hydraulic analysis, an unsteady gas flow model was applied using conservation principles of mass, momentum, and energy. The flow model was tuned to real pressure and temperature data using a multi-objective optimization approach. To demonstrate the practical value of the methodology, the time and amount of gas shortage, line pack, and a volumetric loss were computed for a real tree-topological pipeline network. Simulations demonstrated that the highest volumetric loss is experienced when both compressor stations are shut down. The aim of the proposed algorithm is identification of critical nodes as well as evaluation of measures for improving the gas supply reliability to the customers.

INVESTIGATION ON DETERMINISTIC TRUNCATION TO CONTINUOUS ENERGY MONTE CARLO NEUTRON TRANSPORT CALCULATION

This paper presents the application and evaluation of a deterministic truncation of Monte Carlo (DTMC) solution method in a whole core reactor problem based on a continuous energy transport calculation. The DTMC method has been studied and developed as a systematic way to truncate the high-fidelity Monte Carlo (MC) solution to reduce the computational cost without compromising the essential reliability of the solution. Its fea-sibility and capability were preliminarily validated in several benchmark problems using a multi-group energy MC code. In this paper, further study has been conducted in the more practical application with the continuous-energy based MC calculation. The con-cept of the DTMC method is briefly described. Improvements to enhance the numerical stability and efficiency are specified in details. The DTMC method is applied to an SMR problem, in which reactor parameters are estimated to characterize the numerical per-formance and are compared to the standard MC method. Last, the computing time and corresponding figure-of-merit are evaluated.

VERIFICATION OF THE OpenMC HOMOGENIZED MYRRHA-1.6 CORE MODEL

The OpenMC code is being employed both as a multi-group nodal macroscopic cross-section generator and a reference multi-group Monte Carlo (MGMC) solution. The aim is to do a neutronic benchmark verification study versus a deterministic model (based on the MYRRHA-1.6 core) performed by the PHISICS simulator. MYRRHA, a novel research accelerator driven system concept that is also foreseen to work as a critical configuration, offers a rich opportunity of testing state-of-the art methods for reactor physics analysis due to its strong heterogeneous configuration utilized for both thermal and fast spectra irradiation purposes. The original core configuration representing MYRRHA-1.6 and formed by 169 assemblies, was launched in OpenMC for producing a homogenous nodal model that, when executed in its multi-group Monte Carlo mode, it produced a keff that differs in almost 500 pcm from the original case. This means that in the future, such approximation should correct the nodal cross-sections to preserve the reaction rates in order to match those ones from the heterogeneous model. Nevertheless, such MGMC mode of operation offered by OpenMC could be exploited in order to verify deterministic core simulators. By inputting the same nodal multi-group cross-section model into the transport solver of the PHISICS toolkit, the neutronic benchmark showed a difference of 171 pcm in eigenvalue while comparing it to its OpenMC MGMC counterpart. Also, local multi-group and energy-integrated nodal profiles of the neutron flux showed a maximum relative difference between methodologies of 15% and 1%, respectively. This means that the MGMC capabilities offered by OpenMC can be employed to verify other deterministic methodologies.

MODELLING ASTRID-LIKE SODIUM-COOLED FAST REACTOR WITH SERPENT-DYN3D CODE SEQUENCE

This study explores the feasibility of applying the Serpent-DYN3D sequence to the analysis of Sodium-cooled Fast Reactors (SFRs) with complex core geometries, such as the ASTRIDlike design. The core is characterised by a highly heterogeneous configuration and was likely to challenge the accuracy of the Serpent-DYN3D sequence. It includes axially heterogeneous fuel assemblies, non-uniform fuel assembly heights and large sodium plena. Consequently, the influence of generation and correction methods of various homogenised, few-group crosssections (XS) on the accuracy of the full-core nodal diffusion DYN3D calculations is presented. An attempt to compare the approximate time effort spent on models preparation against the accuracy of the result is made. Results are compared to reference full-core Serpent MC (Monte Carlo) solutions. Initially, XS data was generated in Serpent using traditional methods (2D single assemblies and 2D super-cells). Full core calculations and MC simulations offered a moderate agreement. Therefore, XS generation with 2D fuel-reflector models and 3D single assembly models was verified. Super-homogenisation (SPH) factors for XS correction were applied. In conclusion, the performed work suggests that Serpent-DYN3D sequence could be used for the analysis of highly heterogeneous SFR designs similar to the studied ASTRID-like, with an only small penalty on the accuracy of the core reactivity and radial power distribution prediction. However, the XS generation route would need to include the correction with SPH factors and generation of XS with various MC models, for different core regions. At a certain point, there are diminishing returns to using more complex XS generation methods, as the accuracy of full-core deterministic calculations improves only slightly, while the time effort required increases significantly.