C5G7 Benchmark Research Articles

Linear Sources and Anisotropic Scattering in the Random Ray Method

An enhanced implementation of the random ray method (TRRM), a stochastic adaptation of the method of characteristics, is presented. This implementation generalizes from a traditional flat source (FS) approximation to a linear source (LS) approximation, and from the standard isotropic scattering approximation to an anisotropic approximation, up to P3. On the two-dimensional C5G7 benchmark, LS alone enables a 1.4× and 1.7× enhancement in run time for parallel and serial executions, respectively, without compromising accuracy. Further testing on the three-dimensional C5G7 Rodded B benchmark demonstrates that LS enables significant axial coarsening and reduction in ray populations. This results in a 5.3× speedup in run time while still offering comparable accuracy. On the Babcock and Wilcox 1484 Core II benchmark, incorporating anisotropic scattering in TRRM up to P3 with a FS decreased keff errors (~110 pcm) and maximum and average pin power errors. LS with anisotropic scattering (LSPN), showed similarly improved keff errors while benefiting from a coarser mesh. The linear isotropic flat anisotropic (LIFA) method, relative to LSPN, offers further run time and memory savings of 4.1× and 1.6×, respectively, for P3, while maintaining comparable accuracy to anisotropic FS on more refined meshes.

Coarse mesh finite difference acceleration of the random ray method

This paper presents the development and evaluation of the Coarse Mesh Finite Difference (CMFD) acceleration method to The Random Ray Method (TRRM). We demonstrate its effectiveness in accelerating the convergence of the fission sources and reducing the real statistical error in neutron transport calculations. TRRM treats the angular variable of the neutron flux stochastically and performs batch sampling of characteristic rays to transform the Method of Characteristics (MOC) into a stochastic process, that is capable of making significant strides in memory efficiency and computational performance for some problems. Despite its advantages, TRRM exhibits a potential challenge with a large number of inactive cycles and inherent inter-cycle correlation much like the Monte Carlo method. To address this, the CMFD acceleration method is explored and demonstrated to dramatically reduce the number of required inactive cycles and diminish inter-cycle correlation. Results from the numerical analysis of a 2D C5G7 core problem indicate that the application of CMFD leads to enhanced convergence, with the integration of a CMFD acceleration step every cycle offering the most substantial reduction in statistical noise and error. The study reveals that applying CMFD with every cycle effectively resolves the issue of needing inactive cycles and significantly lowers the inter-cycle correlation, thereby providing a more accurate estimation of standard deviation for pin power distributions. We conclude that using CMFD not only minimizes the number of inactive cycles of TRRM – much like normal Monte Carlo transport – but also lowers real statistical error effectively. For a targeted maximum standard deviation of 0.1% in the pin power, the addition of CMFD can decrease the number of necessary active cycles by 41% compared to standard TRRM, as demonstrated by the 2D C5G7 benchmark analysis.

Mitigating Spatial Error in the Iterative Quasi–Monte Carlo (iQMC) Method for Neutron Transport Simulations with Linear Discontinuous Source Tilting and Effective Scattering and Fission Rate Tallies

The iterative Quasi–Monte Carlo (iQMC) method is a recently proposed method for neutron transport simulations. iQMC can be viewed as a hybrid between deterministic iterative techniques, Monte Carlo simulation, and Quasi–Monte Carlo techniques. iQMC holds several algorithmic characteristics that make it desirable for high-performance computing environments, including an O ( N − 1 ) convergence scheme, a ray-tracing transport sweep, and a highly parallelizable nature similar to analog Monte Carlo. While there are many potential advantages of using iQMC, there are also inherent disadvantages, namely, the spatial discretization error introduced from the use of a mesh across the domain. This work introduces two significant modifications to iQMC to help reduce the spatial discretization error. The first is an effective source transport sweep, whereby the source strength is updated on the fly via an additional tally. This version of the transport sweep is essentially agnostic to the mesh, material, and geometry. The second is the addition of a history-based linear discontinuous source tilting method. Traditionally, iQMC utilizes a piecewise constant source in each cell of the mesh. However, through the proposed source tilting technique, iQMC can utilize a piecewise linear source in each cell and reduce spatial error without refining the mesh. Numerical results are presented from the two-dimensional (2-D) C5G7 and Takeda-1 k-eigenvalue benchmark problems. The results show that the history-based source tilting significantly reduces error in global tallies and the eigenvalue solution in both benchmarks. Through the effective source transport sweep and linear source tilting, iQMC was able to converge the eigenvalue from the 2-D C5G7 problem to less than 0.04% error on a uniform Cartesian mesh with only 204 × 204 cells.

Limited linear source approximation with edge detection for convergence stability of method of characteristics

ABSTRACT A new implementation of the limited linear source approximation (LLSA) is proposed. The LLSA was previously proposed to eliminate local negative source in flux regions to mitigate numerical instability of the method of characteristics (MOC) with linear source approximation (LSA). In the present LLSA implementation, the convex edges of flux regions are used to check the local negative source to decrease the computational load. The present method is implemented in the transport code GENESIS, and its effectiveness is verified through the two dimensional C5G7 benchmark problem and the simplified two-dimensional high-temperature engineering test reactor core. The calculation results indicate that the present LLSA implementation efficiently mitigates the numerical instability of MOC with LSA. Additional computational time is less than 1% of total computation time.

The Random Ray Method Versus Multigroup Monte Carlo: The Method of Characteristics in OpenMC and SCONE

The Random Ray Method (TRRM) is a recently developed approach to solving neutral particle transport problems based on the Method of Characteristics. While the method previously has been implemented only in closed-source or limited-functionality codes, this work describes its implementation in two open-source Monte Carlo codes: OpenMC and SCONE. The random ray implementations required small modifications to the existing Multigroup Monte Carlo (MGMC) solvers, offering a rare venue for redundant, fine-grained, “apples-to-apples” speed and accuracy comparisons between transport methods. To this end, TRRM and MGMC solvers are evaluated against each other using each code’s native capabilities on reactor eigenvalue problems with different degrees of energy discretization. On the C5G7 benchmark (featuring only seven energy groups), TRRM achieves a maximum pin power error comparable to or lower than that of MGMC for a given run time. On a problem with 69 energy groups, MGMC is found to scale more efficiently, obtaining a lower pin power error for a given run time. However, the defining difference between the two transport methods is found to be their vastly different uncertainty distributions. Specifically, TRRM is found to maintain similar levels of accuracy and uncertainty throughout the simulation domain whereas MGMC can exhibit orders-of-magnitude greater errors in areas of the problem that feature low neutron flux. For instance, TRRM provided an up to 373 times speed advantage compared with MGMC for computing the flux in low-flux regions in the moderator surrounding the C5G7 core.

FEMFFUSION and its verification using the C5G7 benchmark

FEMFFUSION is an open source code that solves the multigroup neutron transport equation using the diffusion and the SPN approximations. This code uses the continuous Galerkin finite element method to be able to deal with any type of geometry, structured or unstructured, and problem dimension. To verify the program, a research study of the two-dimensional C5G7 benchmark has been performed, including the static, time-dependent and neutron noise computations. Numerical results include the effective multiplication factor, keff, the spatial neutron fluxes, and neutron power distributions for the static computations. The total neutron power evolution of a given transient and the neutron flux distribution evolution for the different energy groups. In addition, neutron noise frequency domain calculation were completed for perturbations of some cross-section in the reactor core. All computation have been performed using different SPN approximations. FEMFFUSION results are verified against the official OECD/NEA benchmark results.

Multi-objective optimization of method of characteristics parameters based on genetic algorithm

Method of Characteristics (MOC) is a transport-solving method with high numerical accuracy and geometric flexibility, but it is computationally intensive. Since parameters have a significant impact on both computational accuracy and speed, optimization and selection of MOC parameters are necessary. This optimization is challenging due to the large number of parameters involved, the nonlinearities among parameters and objectives, and the contradictory relationship between different objectives. In this study, a genetic algorithm was applied to enable the intelligent and automated optimization process. The results demonstrated its effectiveness in providing the accuracy and speed Pareto front (PF) of MOC when solving C5G7 benchmark problems. And the optimized parameters showed certain applicability in NuScale calculations. Additionally, the PFs can reflect a certain inherent characteristic of MOC. Based on the PFs, the influence of the flat source (FS) approximation, linear source (LS) approximation, and coarse mesh finite difference (CMFD) acceleration on MOC was analyzed.

An efficient heterogeneous parallel algorithm of the 3D MOC for multizone heterogeneous systems

Recent innovations in powerful high-performance computing (HPC) architecture have enabled high-fidelity whole-core neutron transport simulations in a reasonable amount of time. Among high-fidelity numerical methods, the one-step 3D Method of Characteristic (MOC) is becoming increasingly popular in the reactor design community due to its numerical accuracy, geometrical flexibility and high degrees of parallelism. In this work, we propose a sophisticated heterogeneous parallel algorithm of the 3D MOC based on a supercomputer prototype consisting of brand-new multizone heterogeneous processors called the MT-3000, which has a peak double precision performance of 11.6 TFLOPS when operating at 1.2 GHz. In the 3D MOC calculation procedure, the total run time is dominated by the so-called MOC transport sweep where large amounts of tracks traverse the mesh grids along different directions. We significantly improve the performance of the MOC transport sweep through a three-level parallel algorithm with multiple hiding latency techniques. We have also implemented heterogeneous level-1 basic linear algebra subprograms (BLAS) that's the second computational hot spot. The 3D C5G7 benchmark problem is employed to verify the correctness and efficiency of the heterogeneous parallel 3D MOC algorithm. Numerical results show that it achieves very high performance on the MT-3000-based supercomputer prototype. Specifically, the three-layer parallel algorithm has proven to be highly scalable for each layer, and the Quarter-MT-3000 performance of the MOC transport sweep has achieved 23.6% of the peak performance, which is close to the theoretical upper limit.

Code Verification and Solution Verification framework in pin-resolved neutron transport code MPACT

Program verification in scientific computing encompasses the application of formal and mathematical techniques to a scientific computing code for its credibility, accuracy, and validity. Code Verification identifies bugs and performance issues in the software development stage. Solution Verification assesses the applicability of the code and the accuracy of the solution to problems of interest. Both activities utilize application cases and quantify the error against prescribed acceptance criteria. However, simply executing more application cases does not guarantee stronger or more comprehensive credibility. In this work, we establish a verification framework that involves Code Verification and Solution Verification, both of which work together such that the overarching goal of “converge to the correct answer for the intended application” can be reasonably inferred. The application of such a verification framework is demonstrated using the pin-resolved neutron transport code MPACT, where standard unit tests and regression tests are covered, and where the Method of Exact Solutions and the Method of Manufactured Solutions are successfully used. Additionally, the applicability of Method of Manufactured Solutions is extended to the OECD/NEA C5G7 benchmark problems of practical material and geometric configurations. Solution Verification activities are demonstrated on a practical hierarchy of application models of increasing complexity ranging from 2D pin cell problems to 3D assembly problems. The convergence behavior and rate of convergence with respect to each individual variable are studied and provided. This framework can be adapted broadly to other fields involving scientific computing codes.

Validation of deterministic code OpenNTP for the analysis of the C5G7-1D benchmark with different configurations

This study was conducted with the aim of verifying that the OpenNTP (Open Neutron Transport Package) code is capable of reproducing high-performance and exact calculation results. Therefore, a thorough analysis of the One-dimensional C5G7 benchmark with different configuration of control rods was performed. This benchmark was chosen because of its strong heterogeneity. For the modelling of the pin cell geometry, different sets of spatial meshes are used to study the sensitivity of the multiplication factors on the spatial and angular discretization. In addition, the influence of the control rods has been studied by analysing the scalar fluxes, the power in each pin cell and the power of the assemblies. It was found that assembly power distribution errors were significant for the Unrodded case of the C5G7 benchmark. The calculation results in the present paper have a good agreement with the reference values, which demonstrates that the OpenNTP code can solve neutronics problems accurately.

A new coupling method based on MOC and the SN nodal method for the neutron transport calculations

The ray spacing must be fine enough to traverse every region of the reactor to get accurate results when using the method of characteristics. However, when calculating some research reactors, the active zone is relatively small compared to the entire problem. It would be time-consuming and memory costing to apply MOC for the entire geometry. A new approach that couples MOC and the SN nodal method is proposed and implemented to solve those issues. The calculation region is divided into the MOC domain, which includes active zone or complex structures, and the SN domain, such as the moderator and reflector region. The MOC and SN nodal methods are coupled by the interface angular flux. The 2D C5G7 benchmark and a 2D JRR-3 plate-type fuels reactor problem are tested to demonstrate the accuracy and efficiency of the coupling method, and the numerical results show that the coupling method achieves superior efficiency.

Development of an improved direct kinetic simulation capability in RMC code

With the continuous development of supercomputing, a direct kinetic simulation based on Monte Carlo method has received more attention owing to its high fidelity. In this paper, an improved direct kinetic simulation capability is developed based on the Reactor Monte Carlo (RMC) code. Several new techniques that are different from other Monte Carlo codes have been proposed, including a new method of obtaining the neutron distribution at the initial moment, and a theoretically more accurate time-dependent tally method. The newly-developed kinetic capability was tested and verified using a point reactor benchmark and a complex three-dimensional C5G7 kinetic benchmark. The numerical results demonstrate that the relative differences of the relative power at all time points between such direct kinetic simulation capability and the reference in the C5G7 benchmark are within 5.15%.

Derivation of optimal process MOC parameters and analysis of the 2D C5G7 MOX benchmark using DRAGON5 code

This work uses the Characteristics Methods (MOC) implemented in the lattice DRAGON5 code to analyze the 2D C5G7 MOX Benchmark. This method was performed using the MCCGT: module of DRAGON5, this module includes parameters that affect the results. So, a sensitivity study was necessary to find the optimum parameters. The eigenvalues calculated by DRAGON5 code agree well with the reference Monte Carlo code MCNP, and also with the OpenMC code with a percent error of just 0.001% and 0.004% respectively. Compared to the Monte Carlo code OpenMC, the maximum percentage error of assembly power, the maximum and the average percentage error of pin power are 0.07%, 0.85% and 0.13%, respectively.Based on the accuracy of the results and on the computational cost, the sensitivity study of various MOC parameters reveals that the 2D C5G7 benchmark problem requires: between 16 and 32 for the azimuthal angles, more than 10 cm-1 for the ray density and 8 × 8 for the special mesh.

Methodology for Discontinuity Factors Generation for Simplified P3 Solver Based on Nodal Expansion Formulation

The Simplified Spherical Harmonic (SPN) approximation was first introduced as a three-dimensional (3D) extension of the plane-geometry Spherical Harmonic (PN) equations. A third order SPN (SP3) solver, recently implemented in the Nodal Expansion Method (NEM), has shown promising performance in the reactor core neutronics simulations. This work is focused on the development and implementation of the transport-corrected interface and boundary conditions in an NEM SP3 solver, following recent published work on the rigorous SPN theory for piecewise homogeneous regions. A streamlined procedure has been developed to generate the flux zero and second order/moment discontinuity factors (DFs) of the generalized equivalence theory to minimize the error introduced by pin-wise homogenization. Moreover, several colorset models with varying sizes and configurations are later explored for their capability of generating DFs that can produce results equivalent to that using the whole-core homogenization model for more practical implementations. The new developments are tested and demonstrated on the C5G7 benchmark. The results show that the transport-corrected SP3 solver shows general improvements to power distribution prediction compared to the basic SP3 solver with no DFs or with only the zeroth moment DF. The complete equivalent calculations using the DFs can almost reproduce transport solutions with high accuracy. The use of equivalent parameters from larger size colorset models show a slightly reduced prediction error than that using smaller colorset models in the whole-core calculations.

Analysis and comparison of the 2D/1D and quasi-3D methods with the direct transport code SHARK

The 2D/1D method has become the mainstream of the direct transport calculation considering the balance of accuracy and efficiency. However, the 2D/1D method still suffers from stability issues. Recently, a quasi-3D method has been proposed with axial Legendre expansion. Analysis and comparison of the 2D/1D and quasi-3D method is conducted in theory from the equation derivation. Besides, the C5G7 benchmark, the KUCA benchmark and the macro BEAVRS benchmark are calculated to verify the theory comparisons of these two methods with the direct transport code SHARK. All results show that the quasi-3D method has better stability and accuracy than the 2D/1D method with worse efficiency and memory cost. It provides a new option for direct transport calculation with the quasi-3D method.

Uniform variance method for accelerated Monte Carlo criticality calculation

The problem of the non-uniform distribution of relative errors is encountered during the Monte Carlo criticality calculation, which not only reduces the efficiency and the accuracy of some local tallies, but also affects the stability of the Monte Carlo multi-physics coupling calculation. To solve this problem, the uniform variance method is proposed. In this method, the global information is recorded and updated during the inactive cycles, and the source bias technique and the weight windows technique are combined to achieve the uniform distribution of relative errors. The modified C5G7 benchmark is tested by the uniform variance method, and it is found that the variance is reduced in almost all tallies compared with that of the direct simulation. So the uniform variance method is helpful for the global variance reduction, and can be used for the acceleration of the Monte Carlo criticality calculation.

A linear source scheme for the 2D/1D transport method in SHARK

The 2D/1D transport method has already become the most popular direct transport method these several years considering the balance of efficiency and accuracy. Linear source scheme has been researched and applied in method of characteristics (MOC), which shows great advantages in efficiency and accuracy compared to the traditional flat source approximation. Therefore, to improve the calculation performance of the 2D/1D method further, the linear source scheme is applied in the 2D/1D method with a newly-developed direct transport code SHARK. The equation derivation is conducted and the framework of the linear source scheme in SHARK is introduced. Two macro cases, including C5G7 benchmark and macro BEAVRS benchmark, are adopted to show the performance improvement of the linear source scheme in the 2D/1D method.

An efficient hybrid multi-level CMFD in space and energy for accelerating the high-fidelity neutron transport calculation

The high-fidelity neutron transport calculations accelerated by the cell based homogenized multi-group CMFD method still incurs a significant computational burden because its iterations will reduce to a certain number and cannot continue to decrease. To further reduce the iterations of transport calculations, a more efficient hybrid multi-level gCMFD in space and energy method is proposed and implemented in this study. This hybrid multi-level gCMFD acceleration method mainly includes two parts: (1) a new 1/4 cell based homogenized multi-group gCMFD, which is built for directly accelerating the neutron transport calculation; (2) a cell based homogenized one-group gCMFD is used for accelerating 1/4Cell-MG-gCMFD. Then a set of C5G7 benchmark problems are chosen to verify the acceleration performance and the numerical results indicate that the excellent acceleration of the hybrid multi-level gCMFD has achieved.

Solution of the Neutron Transport Equation on Unstructured Grids Using the Parallel Block Jacobi-Integral Transport Matrix Method via the Novel Green’s Function ITMM Construction Algorithm on Massively Parallel Computer Systems

The Parallel Block Jacobi (PBJ) spatial domain decomposition is well suited for implementation on massively parallel computers to solve the neutron transport equation on unstructured grids due to the simple scheduling policy that arises from the PBJ’s iterative asynchronicity. The Parallel Block Jacobi-Integral Transport Matrix Method (PBJ-ITMM) is an iterative method that utilizes the PBJ decomposition and resolves local within-group scattering in a single iteration, but requires a matrix-vector iterative solution. This work details the development, implementation, and testing of the novel Green’s Function ITMM Construction (GFIC) algorithm. The GFIC constructs the matrices required for the PBJ-ITMM’s iterative solution on unstructured grids, utilizing the physical interpretation of these matrices as discretized response functions to create a local problem with a Green’s Function–like source. Conducting a set of mesh sweeps over all angles on this local problem yields the ITMM matrix elements. On unstructured grids, this approach utilizes the kernel calculation and fundamental solution algorithm present in an existing transport code, thus avoiding reimplementation of code functionality. Using the GFIC, the PBJ-ITMM is implemented in THOR, a tetrahedral mesh transport code, along with the Inexact Parallel Block Jacobi (IPBJ) method for performance comparison. This comparison involves strong and weak scaling studies of the Godiva and C5G7 benchmark problems using up to 32 768 processors. These studies establish that the PBJ-ITMM executes faster than the IPBJ when the number of cells per subdomain falls below a problem-dependent threshold, ~128 cells for Godiva, >256 cells for C5G7. The largest problem tested, comprising more than 6.8 billion unknowns, solves in <30 min with the IPBJ and <20 min with the PBJ-ITMM, using 32 768 processors. These results demonstrate the PBJ-ITMM as a viable approach for solving neutron transport problems on unstructured grids using massively parallel computers. Additionally, this study illustrates the range of number of cells per subdomain over which this method is favorable.

Rattlesnake: A MOOSE-Based Multiphysics Multischeme Radiation Transport Application

Advanced reactor concepts span the spectrum from heat pipe–cooled microreactors, through thermal and fast molten-salt reactors, to gas- and salt-cooled pebble bed reactors. The modeling and simulation of each of these reactor types comes with their own geometrical complexities and multiphysics challenges. However, the common theme for all nuclear reactors is the necessity to be able to accurately predict neutron distribution in the presence of multiphysics feedback. We argue that the current standards of modeling and simulation, which couple single-physics, single-reactor-focused codes via ad hoc methods, are not sufficiently flexible to address the challenges of modeling and simulation for advanced reactors. In this work, we present the Multiphysics Object Oriented Simulation Environment (MOOSE)–based radiation transport application Rattlesnake. The use of Rattlesnake for the modeling and simulation of nuclear reactors represents a paradigm shift away from makeshift data exchange methods, as it is developed based on the MOOSE platform with its very natural form of shared data distribution. Rattlesnake is well equipped for addressing the geometric and multiphysics challenges of advanced reactor concepts because it is a flexible finite element tool that leverages the multiphysics capabilities inherent in MOOSE. This paper focuses on the concept and design of Rattlesnake. We also demonstrate the capabilities and performance of Rattlesnake with a set of problems including a microreactor, a molten-salt reactor, a pebble bed reactor, the Advanced Test Reactor at the Idaho National Laboratory, and two benchmarks: a multiphysics version of the C5G7 benchmark and the LRA benchmark.