Coarse Mesh Finite Difference Acceleration Research Articles

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.

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.

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.

The NILO-CMFD Method for Iteratively Solving Coupled Neutron Transport–Thermal Hydraulics Problems

The coarse-mesh finite difference (CMFD) method is commonly used to accelerate the iterative convergence of single-physics neutron transport problems. For multiphysics problems, the neutron cross sections depend on the temperature and density, both of which depend on the fission heat source; the resulting nonlinear feedback can significantly degrade the performance of CMFD and even cause instability. In this paper, we propose, for a class of one-dimensional (1-D) model multiphysics problems, a new nonlinearly implicit low-order (NILO) CMFD (NILO-CMFD) acceleration method to improve the performance of CMFD-based methods for solving loosely coupled multiphysics problems. Our numerical testing and Fourier analysis show that for the 1-D model problems, the new NILO-CMFD method achieves the same rapid convergence rate that CMFD achieves for single-physics problems.

The Legendre Polynomial Axial Expansion Method

This work presents a new formulation of the axial expansion transport method explicitly using Legendre polynomials for arbitrarily high-order expansions. This new formulation also features an alternative method of axial leakage calculation to allow for nonextruded flat source region meshes. This alternative axial leakage is introduced alongside a balance equation requirement to ensure that neutron balance is preserved in the coarse mesh for a given axial leakage formulation, which allows for effective coarse mesh finite difference acceleration. A matrix exponential table method is derived to allow for fast computations of arbitrarily high-order matrix exponentials for this work and precludes the need for further research into matrix exponential calculations for this method. Numerical results are presented that demonstrate the stability of the axial expansion method in systems with voidlike regions, showcase the speedup from matrix exponential tables, and investigate the axial convergence of the method in terms of both expansion order and mesh size.

ARCHER - a new Three-Dimensional method of characteristics neutron transport code for Pebble-bed HTR with coarse mesh finite difference acceleration

A new deterministic code ARCHER using Three-Dimensional (3-D) Method of Characteristics (MOC) has been developed by Institute of Nuclear and new Energy Technology (INET), Tsinghua University, aiming to obtain high-fidelity transport solution of pebble-bed High Temperature gas-cooled Reactor (HTR) with explicit pebble-bed geometry. However, 3-D MOC usually suffers from the slow convergence rate without efficient acceleration methods. In this work, the Coarse Mesh Finite Difference (CMFD) based on regular mesh is implemented to accelerate 3-D MOC calculation, which is available for the problems with cylinder and cuboid geometry. The complex geometry of reactor core with randomly stacked spherical fuel elements and irregular helium flow area, as well as the topological relationships between fine Flat Source Region (FSR) mesh for MOC and coarse CMFD mesh, are the key but challenging tasks for high-fidelity transport solution. Constructive Solid Geometry (CSG) and Boolean operation are constructed to describe problem geometry, which has the ability to deal with complex topological relationships between meshes. In addition, tree grid structure is utilized to reduce the time complexity of ray tracing and obtain the grid mapping relationship between coarse mesh and fine mesh. Spatial domain decomposition parallel based on MPI is utilized to alleviate the memory pressure and pursue high computational performance. Ray parallel is further achieved through OpenMP with shared memory. Several problems have been used to verify the geometric modeling capability and computational performance of this new 3-D MOC code for pebble-bed HTR.

Investigation of Nodal Numerical Adjoints From CMFD-Based Acceleration Methods

In this work, the attribute of mathematical adjoints acquired from various CMFD (coarse-mesh finite difference) accelerated nodal methods based on the nodal expansion method (NEM) is presented. The direct transposition is implemented to the NEM-CMFD system matrix that includes correction factors to acquire the adjoint flux. Three different acceleration schemes are considered in this paper, which are, namely, CMFD, pCMFD, and one-node CMFD methods, and the self-adjoint attribute of migration operator for each acceleration scheme is studied. With regard to the one-node CMFD acceleration, a mathematical description for encountering negative adjoint flux values is given alongside an adjusted one-node CMFD scheme that stifles such an occurrence. The overall features of the aforementioned acceleration methods are recognized through analyses of 2D reactor problems including the KWU PWR 2D benchmark problem. Further systematic assessment is conducted based on the first-order perturbation theory, where the obtained adjoint fluxes are applied as weighting functions. It is clearly shown that the adjusted one-node CMFD scheme results in an improved reactivity estimation by excluding the presence of negative adjoint flux values.

Impact of Angular and Spatial Source Distribution Approximations on Convergence Performance of Nonlinear Acceleration Methods for MOC in Slab Geometry

The convergence performance of nonlinear acceleration methods for the method of characteristics (MOC) with flat source (FS) approximation (FS MOC) or linear source (LS) approximation (LS MOC) is numerically investigated by focusing on the spatial and angular approximations in the acceleration calculations. The convergence of nonlinear acceleration depends on the consistency of the calculation models between the higher-order and lower-order (acceleration) methods. The convergence of four acceleration methods is evaluated to clarify the relationship between model consistency and convergence performance. These methods consist of FS or LS for the spatial source distribution and P1 or discrete angle for the angular distribution, i.e., (1) FS analytic coarse mesh finite difference (ACMFD) acceleration (FS ACMFD), (2) LS ACMFD, (3) FS angular-dependent discontinuity factor MOC (ADMOC) acceleration (FS ADMOC), and (4) LS ADMOC. The ACMFD and ADMOC accelerations are based on P1 and discrete angle approximations, respectively. The FS MOC and LS MOC are considered higher-order methods. The FS MOC and LS MOC with five acceleration methods, i.e., the aforementioned four acceleration methods and the conventional coarse mesh finite difference acceleration method, are used to perform fixed-source calculations in one-group one-dimensional homogeneous slab geometry, and the spectral radii are numerically evaluated. The numerical results indicate that (1) the nonlinear acceleration methods that are unconditionally stable for FS MOC also show similar convergence properties for LS MOC in one-dimensional slab geometry; (2) better convergence is observed when the consistency of higher- and lower-order models is high; and (3) when a coarse mesh is optically thick, the spatial homogenization degrades the convergence performance, even if spatial and angular approximations are consistent between the higher- and lower-order models.

Efficient MC–CMFD multiphysics

Monte Carlo (MC) transport is a promising method for high-fidelity reactor neutronics calculations, but the large computational expense hinders its practicality. To reduce the expense, some researchers have recently applied the CMFD (coarse mesh finite difference) acceleration method to MC solvers. This paper extends that work with two improvements in MC–CMFD methodology. First, instabilities are reduced by a particle weight clipping procedure. Second, multiphysics convergence is accelerated by coupling the CMFD solver directly to the thermal–hydraulics solvers.This methodology is demonstrated on a set of 3D quarter-core PWR simulations with thermal feedback and depletion. Using these methods, an accurate simulation of the BEAVRS benchmark at 1.55 MWd/kg of burnup can be completed with a runtime cost of 2,000 cpu core-hours, suggesting that MC–CMFD solvers are a competitive tool for high-fidelity reactor simulations relative to 3D deterministic methods.

Rasterized coarse mesh finite difference acceleration on method of characteristics for a reactor core with a generalized boundary

The method of characteristics (MOC) is widely employed in neutron transport solvers to obtain accurate solution of a direct transport calculation for a whole reactor core model. Coarse mesh finite difference (CMFD) method is often used to speedup the convergence of MOC solution especially for a large geometry. In this work, a rasterized CMFD method is proposed to improve the convergence rate of the high-order neutron transport calculation for reactor cores with generalized boundaries. The rasterized CMFD method is implemented in a 2D neutron transport MOC code ThorMOC which has been developed to perform the high-fidelity criticality calculation for channel type molten salt reactors (MSRs). The rasterized CMFD method is validated by two benchmarks, a C5G7 hexagonal benchmark and a LFMSR (Liquid-Fueled Molten Salt Reactor) benchmark. The computational result obtained from ThorMOC accelerated by the rasterized CMFD method shows a good agreement with that from the Monte Carlo code OpenMC for the two benchmarks. The rasterized CMFD method can reduce the MOC stand-alone solution time of the LFMSR benchmark by more than 100 times.

A Robust Relaxation-free Multiphysics Iteration Scheme for CMFD–Accelerated Neutron Transport k–eigenvalue Calculation–II: Numerical Results

In a companion paper, we present the theoretical development of a new robust, relaxation-free iteration scheme for multiphysics -eigenvalue problems. These types of problems are essential to the study of computational reactor physics and in particular whole-core, high-fidelity simulation codes. The deterministic whole-core simulation tools invariably rely on the coarse mesh finite difference (CMFD) acceleration for fast convergence. However, the use of CMFD-accelerated transport in multiphysics problems coupled via Picard iteration is not robust and is frequently treated with relaxation. In this paper, we build on our previous theoretical work that uses Fourier analysis to prove how stability and efficient convergence can be achieved in the multiphysics problem by appropriately loosening the convergence criteria of the low-order diffusion acceleration equations. Specifically, we develop a methodology for estimating a key problem-dependent parameter, the feedback intensity, required by the nearly optimally partially converged coarse mesh finite difference (NOPC-CMFD) method. We then describe the implementation of NOPC-CMFD in the Michigan Parallel Characteristics Transport (MPACT) code and perform several numerical calculations. Problems ranging from a single pressurized water reactor (PWR) fuel rod to a full-core PWR cycle depletion are analyzed to assess the performance and robustness of NOPC-CMFD over a wide range of conditions that consider multiple forms of multiphysics feedback. The results verify the theoretical predictions of our companion paper, illustrating that the NOPC-CMFD is superior to current CMFD or nonlinear diffusion acceleration schemes that use relaxation. Overall, the method is able to recover the performance of traditional CMFD in problems without feedback for a wide range of conditions. This was observed to result in a substantial reduction, up to 40%, of the run time in whole-core cycle depletion problems.

Transient Multilevel Scheme with One-Group CMFD Acceleration

The Transient Multilevel (TML) scheme in the MPACT code has reduced the computational burden for three-dimensional, full-core, time-dependent reactor simulations with pin-resolved detail. However, the total computational cost is still large for practical applications. In this paper, we present a new method that uses a one-group coarse mesh finite difference (1GCMFD) approach to further accelerate the TML scheme. Since multigroup coarse mesh finite difference (MGCMFD) calculations in TML dominate the simulation run time, the 1GCMFD method developed here is shown to reduce the overall computational time by as much as 50% for some large-scale applications. The 1GCMFD method accelerated the calculation primarily by accelerating the convergence of the source for MGCMFD calculations through 1G/MGCMFD iteration. A new 1GCMFD level was implemented in the TML scheme (TML-4) assuming that the energy distribution of the scalar flux shape varies more slowly than the energy-integrated amplitude of the scalar flux. Various numerical cases were used to investigate the practicality of the 1G/MGCMFD iteration and TML-4 scheme. Numerical results show that using the 1G/MGCMFD iteration with a dynamic iteration strategy alone does better capture the evolution of the amplitude function when the scalar flux distribution in energy space varies rapidly and thus provides more accurate results. However, TML-4 is more efficient in capturing the variation of the energy-integrated amplitude when cross-section changes are small and feedback dominates the change of the reactivity. For smaller problems, the 1G/MGCMFD iteration and the new TML-4 scheme can reduce the run time of the coarse mesh finite difference (CMFD) solver by 50% and the total run time by at least 16%. For large-scale, full-core problems, the run time of the CMFD solver can be reduced by 78% and the total run time by as much as 47%.

Numerical stability analysis of lp-CMFD acceleration for the discrete ordinates neutron transport calculation discretized with discontinuous Galerkin Finite Element Method

A projection operator was recently developed to extend the linear prolongation Coarse Mesh Finite Difference acceleration (lp-CMFD) method to accelerate the discrete ordinate (SN) neutron transport calculation discretized with the Discontinuous Galerkin Finite Element Method (DGFEM) based spatial discretization scheme, which is more stable and effective than the conventional scaling based Coarse Mesh Finite Difference (CMFD) method. A thorough investigation of the stability of lp-CMFD was carried out using a series of numerical tests for homogeneous and heterogeneous test problems with vacuum and reflective boundary conditions. It was observed that the lp-CMFD method for the SN calculation discretized with the DGFEM based spatial discretization scheme becomes unstable when the coarse mesh optical thickness becomes large with vacuum boundary conditions and in strongly heterogeneous problems, which was not previously observed. The linear prolongation of the partial current based Coarse Mesh Finite Difference method (lp-pCMFD) is proposed to enhance the stability. Numerical testing showed that lp-pCMFD is stable and effective for the homogeneous and heterogeneous test problems. The linear prolongation update step of lp-pCMFD stabilizes both lp-CMFD and pCMFD for neutron transport problems with the DGFEM spatial discretization scheme.

Arbitrary-level coarse mesh finite difference acceleration with hierarchical coarsening in space and energy

In this paper, we present an arbitrary-level coarse mesh finite difference (CMFD) scheme to accelerate discrete ordinates (SN) transport eigenvalue problems. It is a nested iterative scheme where a transport power iteration is accelerated with nested levels of CMFD eigenvalue problems whose space/energy grids can be hierarchically coarsened. The methodology provides a general framework independent of the space/angle discretization used to define the problem of interest. It is also readily extensible to solving diffusion eigenvalue problems. We present 2D and 3D numerical results on a representative quarter-core model from the VERA benchmark suite with different combinations of energy and spatial CMFD grids. We show that the methodology significantly reduces the diffusion and overall computational times required to solve large SN transport eigenvalue problems with CMFD on many parallel processes.

GPU Based Two-Level CMFD Accelerating Two-Dimensional MOC Neutron Transport Calculation

The Graphics Processing Units (GPUs) are increasingly becoming the primary computational platform in the scientific fields, due to its cost-effectiveness and massively parallel processing capability. On the other hand, the coarse mesh finite difference (CMFD) method has been one of the most popular techniques to accelerate the neutron transport calculation. The GPU is employed into the method of characteristics (MOC) accelerated by two-level CMFD to solve the neutron transport equation. In this work, the Jacobi method, the successive over-relaxation (SOR) method with red-black ordering, and the preconditioned generalized minimum residual (PGMRES) method are applied to solve the linear system under the framework of CMFD. The performance of these linear system solvers is characterized on both CPU (Central Processing Unit) and GPU. The two-dimensional (2-D) C5G7 benchmark problem and an extended mock quarter-core problem are tested to verify the accuracy and efficiency of the algorithm with double precision, as well as the feasibility of massive parallelization. Numerical results demonstrate that the desired accuracy is maintained. Moreover, the results show that the few-group CMFD acceleration is effective to accelerate the multi-group CMFD calculation. The PGMRES method shows remarkable convergence characteristics compared to the Jacobi and the SOR methods. However, the SOR method shows better performance on GPU for solving the linear system of CMFD calculation, which reaches about 2400x speedup on GPU with two-level CMFD acceleration compared to the CPU-based MOC calculation.

A Linear Prolongating Coarse Mesh Finite Difference Acceleration of Discrete Ordinate Neutron Transport Calculation Based on Discontinuous Galerkin Finite Element Method

The recently developed linear prolongation Coarse Mesh Finite Difference (lpCMFD) acceleration scheme, which employs a linear additive approach to update the scalar flux, has been shown to be more stable and effective than the conventional scaling-based Coarse Mesh Finite Difference (CMFD) method for accelerating the discrete ordinates (SN) neutron transport calculation using spatial finite difference discretization. In this paper, we study and extend the application of lpCMFD to accelerate the SN neutron transport calculation with spatial discretization using the Discontinuous Galerkin Finite Element Method (DGFEM), which generally involves linear- or higher-order space expansion functions. A function space mapping operator is proposed in this paper to project the lpCMFD linear-order correction flux to an arbitrary-order DGFEM basis function, which is implemented and tested on a one-dimensional (1-D) in-house–developed DGFEM-based SN code. The consistency between the lpCMFD accelerated results and the pure SN results is naturally guaranteed by employing upwind current information from DGFEM-based SN transport calculation to evaluate the drift coefficient. It was found from our numerical testing with the CMFD and the lpCMFD acceleration schemes on single-group fixed-source and k-eigenvalue problems that both acceleration schemes can reproduce the unaccelerated scalar flux and keff, respectively. Further numerical testing on a more realistic case is performed on a 1-D slice multi-energy-group problem based on the three-dimensional C5G7 mixed oxide (MOX) benchmark. It was found that by using the function space projector proposed in this paper, lpCMFD was stable and effective to accelerate the DGFEM-based SN neutron transport calculation for all coarse mesh sizes tested while CMFD diverged for large optical thickness.

A new three-level CMFD method based on the loosely coupled parallel strategy

In the whole-core heterogeneous transport calculations with massive parallel, the computational cost of the Coarse Mesh Finite Difference (CMFD) method might become the most time-consuming part if directly using the traditional CMFD method. To accelerate the CMFD calculations, a new three-level CMFD method based on the loosely coupled parallel strategy is developed in the high-fidelity neutronics code NECP-X. Two aspects are considered to accelerate the CMFD calculations, one is reducing the CMFD matrix size and another is minimizing the number of outer iteration numbers. The three-level CMFD acceleration method includes: (1) the multi-group pin-level CMFD (MG-Pin-CMFD) for accelerating transport calculations, in which the cross sections and currents are obtained from the 2D-1D transport calculation; (2) the one-group pin-level CMFD (1G-Pin-CMFD) for accelerating the MG-Pin-CMFD calculations, and the 1G-Pin-CMFD is established by the group-condensed cross sections, flux and currents from the MG-Pin-CMFD; (3) the one-group assembly-level CMFD (1G-ASY-CMFD) for accelerating 1G-Pin-CMFD, and the assembly-wise parameters are from the spatial homogenization of the 1G-Pin-CMFD calculations. The three-level CMFD method can reduce the outer iterations of CMFDs, such as MG-Pin-CMFD and 1G-Pin-CMFD. In every level of CMFD, the group-wised CMFD linear system is constructed based on the loosely coupled spatial domain decomposition, and the serial Generalized Minimum RESsidual (GMRES) method is implemented to solve the linear system instead of the parallel GMRES method to minimize the communications. A set of benchmark problems are tested and the results show good performance and good accuracy in transport calculations.

Improvements of the CMFD acceleration capability of OpenMOC

Due to its computational efficiency and geometrical flexibility, the Method of Characteristics (MOC) has been widely used for light water reactor lattice physics analysis. Usually acceleration methods are necessary for MOC to achieve acceptable convergence on practical reactor physics problems. Among them, Coarse Mesh Finite Difference (CMFD) is very popular and can drastically reduce the number of transport iterations. In OpenMOC, CMFD acceleration was implemented but had the limitation of supporting only a uniform CMFD mesh, which would often lead to splitting MOC source regions, thus creating an unnecessary increase in computation and memory use. In this study, CMFD acceleration with a non-uniform Cartesian mesh is implemented into OpenMOC. We also propose a quadratic fit based CMFD prolongation method in the axial direction to further improve the acceleration when multiple MOC source regions are contained in one CMFD coarse mesh. Numerical results are presented to demonstrate the improvement of the CMFD acceleration capability in OpenMOC in terms of both efficiency and stability.

Impact of Various Parameters on Convergence Performance of CMFD Acceleration for MOC in Multigroup Heterogeneous Geometry

The impact of various parameters in the coarse mesh finite difference (CMFD) acceleration method on overall convergence behavior is investigated through numerical calculations using the method of characteristics (MOC). Four parameters appearing in the CMFD acceleration with MOC, i.e., scalar flux distribution in flat flux regions (FFRFlux), the scalar flux distribution in CMFD meshes (CMFDFlux), homogenized cross sections (HXSs) in CMFD meshes, and current correction factors (CCFs), are considered. Parts of these four parameters are fixed to the converged values throughout iterations in order to estimate their impact on convergence. Numerical calculations are carried out for Korea Advanced Institute of Science and Technology’s (KAIST’s) benchmark problem KAIST-2A, which is a heterogeneous and multigroup problem, and the number of outer iterations to reach convergence is evaluated. The impact of geometric heterogeneity and cross-section homogenization in the CMFD acceleration has not been considered in linearized Fourier analysis so far. The calculation results indicate that (1) convergence of HXS has little impact on the overall convergence, (2) convergence of FFRFlux is dominant followed by CCF when a CMFD mesh is optically thin, and (3) convergence of FFRFlux is dominant when a CMFD mesh is optically thick and contains many flat flux regions.

Transport consistent diffusion coefficient for CMFD acceleration and comparison of convergence properties

ABSTRACT A diffusion coefficient for the coarse mesh finite difference (CMFD) acceleration is derived from the semi-analytic solution of one-group, one-dimensional, even-parity transport equation. The derived diffusion coefficient, i.e., the transport consistent diffusion coefficient (TCD), depends on the optical length of a mesh and shows similar behavior with the artificial grid diffusion (AGD) and the effective diffusion (EffD) coefficients for an optically thick mesh. Convergence properties of typical diffusion coefficients are evaluated using the linearized Fourier analysis. Analyses of the C5G7 3D benchmark problems with and without voided region are carried out to compare the convergence properties. The number of transport sweeps to reach convergence using TCD is smaller than that using EffD or AGD.