Coarse Mesh Finite Difference Research Articles

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.

A simplified SP3 NEM solver within a unified formulation for pin-by-pin core multi-group calculations

This study addresses the development and verification of a pin-by-pin core multigroup SP3 solver CTRP-Clouds that employs NEM (Nodal Expansion Method) and three simplified NEM methods within a unified formulation for simultaneously solving the coupling 0th and 2nd SP3 equations. In this work, the solver using this unified formulation does not only include the original NEM and its simplifications but also the EFEN (Exponential Function Expansion Nodal) method and FDM (Finite Difference Method) for the comprehensive evaluation. Also, the solver was accelerated using CMFD (Coarse Mesh Finite Difference) method and parallelized using OpenMP. The computational efficiency of different solution methods was investigated for the 2D KAIST benchmark problems and their modified one for considering 3D extension. The results showed the simplified NEM with flat leakage approximation gives acceptable accuracies of less than 1.2 % in RMS of pin-power discrepancies of all the cases with 1x1 mesh per pin-cell, with a reduction of 20 % computing time compared to the original NEM. Particularly, the calculation time of flat leakage NEM is comparable to EFEN while the pin-wise accuracy is better. Besides, the simplified NEM with 2nd-order flux expansion gives substantially improved accuracy in comparison with FDM within comparable computing time.

Simulating Fuel Assembly Bowing: A Neutron-Diffusion Method Based on Arbitrary Quadrilateral Node and Conformal Mapping Technique

Fuel assembly bowing, widely observed in a pressurized water reactor (PWR), often results in an asymmetrical power distribution. This paper proposes a neutron-diffusion method that integrates the arbitrary quadrilateral node with the conformal mapping technique to characterize the impact of fuel assembly bowing on power distribution. The proposed method involves a nonlinear iteration process to solve the neutron-diffusion equation. The global coarse-mesh finite difference equation is established on the arbitrary quadrilateral nodes, which are redivided in response to fuel assembly bowing. The local two-node nodal expansion method equation is established on the rectangular nodes, which are mapped from the original arbitrary quadrilateral nodes using the conformal mapping technique. The proposed method has improved our self-developed core code, named SPARK, for PWRs. To verify this novel method, two distinct types of fuel assembly bowing are modeled based on the mini core. The reference results for these models were obtained using the Monte Carlo code NECP-MCX. The numerical results suggest a robust agreement between the biases of keff and power distributions and their corresponding reference results.

Solution of OECD/NEA PWR MOX/UO2 benchmark with a high-performance pin-by-pin core calculation code

Expanding upon the framework of the steady-state pin-by-pin 2D/1D decoupling method, a novel and high-performance pin-by-pin transient calculation method has been introduced. This transient method, consistent to the steady-state formulation, is designed for time-dependent calculations utilizing a 3D diffusion-based finite difference method (FDM). The inherent complexity of the large 3D problem is effectively managed by decoupling it into a series of planar (2D) and axial (1D) problems. In addition, tens of thousands of pin-cells are grouped into hundreds of boxes to reduce the computing burden for the 1D calculations without essential loss of the accuracy. Two-level coarse mesh finite difference (CMFD) formulation comprising multigroup nodewise CMFD and two-group assemblywise CMFD is employed as well to accelerate the convergence. Errors originating from the pin-level homogenization, energy group condensation, and the use of lower order calculation methods are simultaneously corrected by the pinwise super homogenization (SPH) equivalence factor.The transient method is evaluated with OECD/NEA PWR MOX/UO2 benchmark. Code-to-code comparison with the nTRACER direct whole core calculation code yielded highly satisfactory results for the transient scenario as well as the steady-state problems. Furthermore, comparative analyses with conventional nodal calculations show superiority of the pin-by-pin calculation.

Practical considerations for the adoption of Anderson acceleration in nonlinear diffusion accelerated transport

The use of the Anderson Acceleration (AA) method within the context of the coarse mesh finite difference (CMFD)-accelerated neutron transport k-eigenvalue calculation is explored for practical problems. Current application of CMFD only has a linear convergence rate. The AA method is similar to a quasi-Newton method and can improve the convergence rate of fixed-point iterations. We therefore implement AA for the 2D calculation in MPACT and study its performance in pressurized water reactor (PWR) problems. The initial results show that the traditional AA method can improve the convergence rate in a trivial problem and improve the stability of the CMFD method for some problems. However, the success of AA in the practical problems, where simulations with symmetry or domain decomposition, relies on one essential factor–the treatment of iterative angular flux boundary conditions in the AA. Additionally, the efficiency of AA is also affected by the computational cost for solving the optimization problem, and the computational cost of the transport solver and the CMFD solver. To address these issues, we propose a new implementation of AA that considers only the mutligroup scalar flux in the optimization problem, but the angular flux and the scalar flux in the mixing. The new approach achieves better stability and efficiency in practical simulations, and can be explained by the theory of Randomized Numerical Linear Algebra (RandNLA).

Convergence Analysis for the CMFD Accelerated Linear Axial Expansion Transport Method Based on Fourier Analysis

The linear axial expansion transport method avoids the negative source problem caused by transverse leakage in the traditional two-dimensional/one-dimensional (2D/1D) transport method and has better stability. However, stability is poor with the coarse-mesh finite difference (CMFD) accelerated linear axial expansion transport method. In this paper, the stability of the partial current–based coarse-mesh finite difference (p-CMFD) method, the optimally diffusive coarse-mesh finite difference (od-CMFD) method, and the linear prolongation coarse-mesh finite difference (lp-CMFD) method is studied based on Fourier analysis. The results of the Fourier analysis indicate that the problem is stable for axial coarse-mesh optical thickness less than 2 or larger than 50; the calculation diverges when the axial coarse-mesh optical thickness is between 2 and 50. The numerical results of the KUCA benchmark problem are the same as the results of the Fourier analysis.

The deterministic neutron transport calculation for the HTR-PM by the three-dimensional method of characteristics code ARCHER

Due to the strong geometric adaptability, the Three-Dimensional (3-D) Method Of Characteristics (MOC) is a promising candidate for 3-D whole-core high-fidelity neutron transport calculations. However, the 3-D MOC can hardly be applied to large-scale full core problems on account of the huge computational costs in memory and time. In the past, the 3-D MOC was mainly used in traditional Light Water Reactor (LWR) calculations, but rarely used for more complex core calculations. For pebble-bed High Temperature gas-cooled Reactors (HTRs), most 3-D MOC codes are out of ability to construct complex pebble beds, let alone efficient acceleration methods. The 3-D MOC code ARCHER can efficiently simulate the large-scale pebble-bed HTRs through using the Linear Source Approximation (LSA), the Coarse Mesh Finite Difference (CMFD) and the hybrid MPI-OpenMP parallel. In this work, the two-level CMFD acceleration and efficient preconditioners on Krylov subspace linear solvers are implemented in the ARCHER code in order to achieve better performance. In addition, the practical HTR-PM criticality problem is calculated by ARCHER, which is the world’s first high-fidelity neutron transport solution of large-scale pebble-bed HTRs by deterministic numerical method. It also shows that the 3-D MOC has great application potential in other complex reactors.

Stability and Performance of the X-CMFD Method for Multiphysics Reactor Calculations

This paper presents an innovative approach to efficiently perform deterministic direct whole-core transport calculations with multiphysics feedback for steady-state problems. Traditionally, Picard iteration combined with coarse mesh finite difference (CMFD) acceleration has been used, but it can suffer from instability and inefficiency in certain scenarios. In this work, we introduce the X-CMFD method, supported by Fourier analysis, to enhance the stability of the multiphysics iteration scheme. A new and efficient variation of the X-CMFD method for practical simulations is also present. Additionally, we explore the theoretical convergence rates of nonlinear fully coupled diffusion acceleration (NFCDA), a class of diffusion acceleration methods that formalizes similar ideas of previous research. NFCDA uses a low-order diffusion problem that is fully coupled with equivalent nonlinear multiphysics feedback to accelerate the high-order transport problem with feedback. The theoretical analysis shows that NFCDA offers similar convergence rates to nonlinear diffusion acceleration (NDA) in problems without feedback. This provides theoretical support for numerical experiments conducted by other researchers. X-CMFD, which is a discretized form of NFCDA, leverages typical coarse mesh concepts and operators from CMFD while applying feedback to cross sections in the low-order diffusion problem at each power iteration of the low-order problem. To reduce computational costs, we optimize the implementation of X-CMFD in MPACT by introducing an equivalent low-order approximation to the cross-section updates in the nonlinear low-order problem. Numerical results from pressurized water reactor problems demonstrate that X-CMFD, along with its practical implementation, outperforms current relaxed Picard iteration methods in terms of robustness and efficiency, irrespective of the presence of multiphysics feedback.

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.

Nodal method for handling irregularly deformed geometries in hexagonal lattice cores

The hexagonal nodal code RENUS has been enhanced to handle irregularly deformed hexagonal assemblies. The underlying RENUS methods involving triangle-based polynomial expansion nodal (T-PEN) and corner point balance (CPB) were extended in a way to use line and surface integrals of polynomials in a deformed hexagonal geometry. The nodal calculation is accelerated by the coarse mesh finite difference (CMFD) formulation extended to unstructured geometry. The accuracy of the unstructured nodal solution was evaluated for a group of 2D SFR core problems in which the assembly corner points are arbitrarily displaced. The RENUS results for the change in nuclear characteristics resulting from fuel deformation were compared with those of the reference McCARD Monte Carlo code. It turned out that the two solutions agree within 18 pcm in reactivity change and 0.46% in assembly power distribution change. These results demonstrate that the proposed unstructured nodal method can accurately model heterogeneous thermal expansion in hexagonal fueled cores.

High-Fidelity Transient Analysis with STREAM Based on the Predictor-Corrector Quasi-Static Method

The predictor-corrector quasi-static method (PCQM) is used to solve the transient problem in the STREAM code, a steady-state and transient reactor analysis code with the method of characteristics. In PCQM, the angular neutron flux undergoes a factorized split to form the product of shape and amplitude functions. The time-dependent neutron transport equation is solved to obtain the shape function whereas the amplitude function is obtained by resolving the exact point kinetics equations (EPKEs). A two-level coarse mesh finite difference technique is implemented to reduce the transient running time of the transport solution. Moreover, high-order polynomial interpolation is applied to the kinetics parameters utilized in EPKEs to reduce the error when the reactivity insertion is nonlinear. Several numerical benchmarks are solved to justify the application of the procedure, proving that the method maintains solution accuracy.

High fidelity transient solver in STREAM based on multigroup coarse-mesh finite difference method

This study incorporates a high-fidelity transient analysis solver based on multigroup CMFD in the MOC code STREAM. Transport modeling with heterogeneous geometries of the reactor core increases computational cost in terms of memory and time, whereas the multigroup CMFD reduces the computational cost. The reactor condition does not change at every time step, which is a vital point for the utilization of CMFD. CMFD correction factors are updated from the transport solution whenever the reactor core condition changes, and the simulation continues until the end. The transport solution is adjusted once CMFD achieves the solution. The flux-weighted method is used for rod decusping to update the partially inserted control rod cell material, which maintains the solution's stability. A smaller time-step size is needed to obtain an accurate solution, which increases the computational cost. The adaptive step-size control algorithm is robust for controlling the time step size. This algorithm is based on local errors and has the potential capability to accept or reject the solution. Several numerical problems are selected to analyze the performance and numerical accuracy of parallel computing, rod decusping, and adaptive time step control. Lastly, a typical pressurized LWR was chosen to study the rod-ejection accident.

The improved odCMFD method based on the 2D/1D Fourier analysis

The coarse mesh finite difference (CMFD) method is widely used to accelerate neutron transport calculation. The Fourier analysis shows that approximately three times of 1D inner iteration can improve the convergence and efficiency in CMFD accelerated two-dimension/one-dimension (2D/1D) neutronic transport simulation. To further enhance the convergence and efficiency of the 2D/1D neutronic transport simulation, the improved odCMFD method is studied based on the 2D/1D Fourier analysis, and the scattering ratio correction term is implemented in the improved odCMFD method. The results of the VERA benchmark and the SPEPT whole-core problem indicate that the improved odCMFD method has better convergence than the traditional odCMFD method, which is of great significance to further improve the computational efficiency of the three-dimension (3D) full core transport calculation.

Development and validation of multiphysics PWR core simulator KANT

KANT (KAIST Advanced Nuclear Tachygraphy) is a PWR core simulator recently developed at Korea Advance Institute of Science and Technology, which solves three-dimensional steady-state and transient multigroup neutron diffusion equations under Cartesian geometries alongside the incorporation of thermal-hydraulics feedback effect for multi-physics calculation. It utilizes the standard Nodal Expansion Method (NEM) accelerated with various Coarse Mesh Finite Difference (CMFD) methods for neutronics calculation. For thermal-hydraulics (TH) calculation, a single-phase flow model and a one-dimensional cylindrical fuel rod heat conduction model are employed. The time-dependent neutronics and TH calculations are numerically solved through an implicit Euler scheme, where a detailed coupling strategy is presented in this paper alongside a description of nodal equivalence, macroscopic depletion, and pin power reconstruction. For validation of the steady, transient, and depletion calculation with pin power reconstruction capacity of KANT, solutions for various benchmark problems are presented. The IAEA 3-D PWR and 4-group KOEBERG problems were considered for the steady-state reactor benchmark problem. For transient calculations, LMW (Lagenbuch, Maurer and Werner) LWR and NEACRP 3-D PWR benchmarks were solved, where the latter problem includes thermal-hydraulics feedback. For macroscopic depletion with pin power reconstruction, a small PWR problem modified with KAIST benchmark model was solved. For validation of the multi-physics analysis capability of KANT concerning large-sized PWRs, the BEAVRS Cycle1 benchmark has been considered. It was found that KANT solutions are accurate and consistent compared to other published works.

Geometry Extension and Assemblywise Domain Decomposition of nTRACER for Direct Whole-Core Calculation of VVERs

The capability and performance of the hexagonal version of the nTRACER direct whole-core calculation code are enhanced for VVER applications by extending the geometry-handling features and also by implementing assemblywise parallelization of the planar method of characteristics (MOC) calculation with higher-order scattering. The geometry-handling methods for the VVER hexagonal geometry having various special constituents are presented with detailed illustrations. The assemblywise domain decomposition (ADD) scheme is established under the hexagonal coarse-mesh finite difference formulation, which is exploited to update the incoming angular flux needed for the ADD parallelization. The solution accuracy and parallel performance are assessed for various hexagonal core problems, including the VVER benchmarks. It is shown that the hexagonal geometry solutions of nTRACER match with the reference Monte Carlo solutions within about 50 pcm in reactivity and 1% in pin power distribution and that the hexagonal ADD can reduce the computing time of the planar MOC calculation by up to 53% when compared to the anglewise parallelization.

Hybrid parallel reduction algorithms for the multi-level CMFD acceleration in the neutron transport code PANDAS-MOC

The coarse mesh finite difference (CMFD) technique is considered efficiently in accelerating the convergence of the iterative solutions in the computational intensive 3D whole-core pin-resolved neutron transport simulations. However, its parallel performance in the hybrid MPI/OpenMP parallelism is inadequate, especially when running with larger number of threads. In the original Whole-code OpenMP threading hybrid model (WCP) model of the PANDAS-MOC neutron transport code, the hybrid MPI/OpenMP reduction has been determined as the principal issue that restraining the parallel speedup of the multi-level coarse mesh finite difference solver. In this paper, two advanced reduction algorithms are proposed: Count-Update-Wait reduction and Flag-Save-Update reduction, and their parallel performances are examined by the C5G7 3D core. Regarding the parallel speedup, the Flag-Save-Update reduction has attained better results than the conventional hybrid reduction and Count-Update-Wait reduction.

Operator split, Picard iteration and JFNK methods based on nonlinear CMFD for transient full core models in the coupling multiphysics environment

Operator Split (OS), Picard iteration and Jacobian-free Newton Krylov (JFNK) methods based on nonlinear coarse mesh finite difference (CMFD) with nodal expansion methods (NEM) are respectively introduced to solve the transient full core coupled models in the COupling Multiphysics Environment (COME). In the COME, in addition to the traditional OS and Picard coupled methods, we develop two different JFNK-based coupled methods with physics and algebraic-based hybrid preconditioners at each time step: OS_JFNK and JFNK coupled methods. Specifically, the efficient JFNK method is applied to solve CMFD discrete models of only neutronics in the OS framework for OS_JFNK coupled strategies or whole Neutronics-Thermal Hydraulic (N-TH) coupled models for JFNK coupled methods. Then nonlinear corrective coupling coefficients are updated by the two-node NEM every few Newton steps at each time step. Six cases of NEACRP 3D core transient benchmarks are analyzed and numerical solutions of OS, OS_JFNK, Picard and JFNK methods in the COME generally agree well with other published solutions. JFNK and Picard coupled methods can track the reference solutions better than OS and OS_JFNK coupled methods due to multiple iterations or simultaneous solutions between neutronics and TH models in the tight coupled form. Furthermore, OS_JFNK or JFNK methods can obtain higher computational efficiency than OS or Picard methods with or without the fission source extrapolation acceleration, respectively, which indicates the potential of the developed COME using different coupled methods for the transient full core multiphysics coupled problems.

A hybrid neutronics method with novel fission diffusion synthetic acceleration for criticality calculations

A novel Fission Diffusion Synthetic Acceleration (FDSA) method is developed and implemented as a part of a hybrid neutronics method for source convergence acceleration and variance reduction in Monte Carlo (MC) criticality calculations. The acceleration of the MC calculation stems from constructing a synthetic operator and solving a low-order problem using information obtained from previous MC calculations. By applying the P1 approximation, two correction terms, one for the scalar flux and the other for the current, can be solved in the low-order problem and applied to the transport solution. A variety of one-dimensional (1-D) and two-dimensional (2-D) numerical tests are constructed to demonstrate the performance of FDSA in comparison with the standalone MC method and the coupled MC and Coarse Mesh Finite Difference (MC-CMFD) method on both intended purposes. The comparison results show that the acceleration by a factor of 3–10 can be expected for source convergence and the reduction in MC variance is comparable to CMFD in both slab and full core geometries, although the effectiveness of such hybrid methods is limited to systems with small dominance ratios.

A leakage correction with burnup-dependent GPS method in two-step core depletion analysis

The aim of this study is to introduce a burnup-dependent GPS† method in the pin-resolved two-step reactor depletion analysis and to verify how effectively leakage correction works in terms of reactor eigenvalue (keff) and power distribution. The burnup-dependent GPS function sets were generated based on the different color-set models with various conditions, considering the adjacent non-target fuels with fresh or burned fuels at each burnup step. In the functionalization stage, the cross-section-dependent SPH factors were parameterized considering the current-to-flux ratio (CFR) indicating the integrated leakage information and the spectral index (SI) representing the fluctuations in the neutron spectrum. The computational performance for the burnup-dependent GPS method was evaluated with a modified PWR benchmark problem for a UOX-loaded small reactor. The lattice calculations for the group constants generation and the reference core calculations were performed using the DeCART2D transport code. The nodal depletion calculations were performed with an in-house nodal expansion method (NEM) code implemented with a hybrid coarse mesh finite difference (HCMFD) solver. The two-step macroscopic depletion calculations were performed based on 4 types of conditions named; (1) w/o GPS, (2) w/BOC-GPS, (3) w/BU-GPS(F), and (4) w/BU-GPS(F&B), where letters ‘F’ and ‘B’ within the parenthesis indicate ‘Fresh’ and ‘Burned’ respectively. As a result, by individually applying two different GPS functions in the boundaries and internal area of the assembly (i.e., w/BU-GPS(F&B)), it shows a good performance with low errors at a sufficiently acceptable level in terms of reactivity and the pin-wise power distribution.†GPS: Generalized equivalence theory (GET) Plus Super-homogenization (SPH).

High-Fidelity Neutron Transport Solution of High Temperature Gas-Cooled Reactor by Three-Dimensional Linear Source Method of Characteristics

Recently, a three-dimensional method of characteristics (MOC) code called Advanced Reactor CHaracteristics tracER (ARCHER) has been developed by the Institute of Nuclear and New Energy Technology, Tsinghua University, to solve the neutron transport problem in high-temperature gas-cooled reactors (HTRs) with explicit pebble-bed geometry. Although the spatial domain decomposition using the message passing interface (MPI) and the ray parallel using OpenMP have been implemented in the previous version of ARCHER, in order to simulate practical HTR problems it is still necessary to reduce the great computational burden through efficient algorithms. Therefore, the linear source approximation (LSA) scheme, which allows coarser transport calculation grids while maintaining high accuracy, has been added in the latest version of ARCHER to relieve memory pressure together with the MPI-based spatial domain decomposition. Moreover, on-the-fly calculation of the relative position coordinates of the ray segment center can further reduce the memory for storing segment information under LSA. In addition, time-consuming MOC transport sweeps can be reduced greatly with coarse-mesh finite difference (CMFD) acceleration. Numerical results show that both LSA and CMFD acceleration contribute to simulate the practical HTR-10 problem successfully.