Coarse Mesh Finite Difference Method Research Articles

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).

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.

Implementation and optimization of hybrid parallel strategy in HNET

For the self-developed three-dimensional whole-core High-fidelity NEutron Transport calculation program HNET, although the numerical acceleration algorithms can improve the computation performance of high-fidelity neutron transport in terms of algorithms and models, it still faces some critical issues such as long computing time and enormous memory requirement. The rapid development of high-performance clusters provides a foundation for the application of massively parallel computing. Most current MOC programs are based on a single-type variable to achieve efficient parallelism, and spatial domain decomposition methods are the most common parallel schemes. However, its parallelism is limited and cannot fully utilize the current state-of-the-art computer resources. To solve this problem, hybrid parallel strategies are implemented in HNET to further expand the parallel degree, improve the speed of computation, and reduce the memory requirement. A hybrid MPI/OpenMP method based on domain decomposition and characteristic rays is proposed for the method of characteristics (MOC). For domain decomposition, the simulation domain is divided into spatial subdomains, with each subdomain handled by different processes. On this basis, the characteristic ray parallelism is implemented taking advantage of the inherent parallelism of the characteristic rays. Meanwhile, the optimization of the hybrid parallel strategy further improves the computation speed by eliminating atomic operations and using private pointer arrays and other techniques. In addition, in the framework of generalized equivalence theory (GET) based two-level coarse mesh finite difference method (CMFD), there is also a certain time consumption in solving CMFD linear system. Hence, for CMFD, a hybrid MPI/OpenMP method based on domain decomposition and secondary domain decomposition can be used. Using the secondary domain decomposition method, each subdomain is divided into sub-subdomains, with each sub-subdomain handled by threads, which allows CMFD to utilize the resources of characteristic ray parallelism in MOC, and also increases the speed of CMFD computation. Numerical results show that for both steady-state and transient calculations, the hybrid MPI/OpenMP in HNET can further expand the parallelism and accelerate the computation. It can take full use of parallel resources and achieve large-scale parallelism.

Jacobian-free Newton Krylov coarse mesh finite difference algorithm based on high-order nodal expansion method for three-dimensional nuclear reactor pin-by-pin multiphysics coupled models

The nuclear reactor core is a large-scale, complicated and tight-coupled nonlinear multiphysics coupled system including neutron transport, heat conduction of nuclear fuel pins, coolant flow, heat transfer and so on. To meet the demands of the advanced nuclear reactor analysis and design, there is an urgent need to reduce the computational costs and improve the convergence rate for the three-dimensional (3D) nuclear reactor core multiphysics coupled models, especially for a high-fidelity pin-by-pin (at the level of fuel pin by fuel pin) full core simulation. In this work, we develop CNJ, an efficient Jacobian free Newton Krylov (JFNK) coarse mesh finite difference (CMFD) algorithm based on the nodal expansion method (NEM) to simultaneously solve the complicated pin-by-pin neutronics-thermal hydraulic (N-TH) coupled models. The high-order NEM can improve the numerical accuracy of the CMFD method on a coarse mesh and the JFNK method can ensure the rapid convergence and high efficiency on large-scale and nonlinear coupled discrete systems. By combining the CMFD, NEM and JFNK methods and making full use of their respective advantages, the CNJ algorithm can obtain high accuracy and efficiency even on a coarse mesh to greatly reduce the number of solution variables and the computational cost of the 3D coupled models. Finally, two representative and complicated 3D pin-by-pin N-TH coupled models are analyzed to evaluate the numerical accuracy and efficiency advantage of CNJ in detailed.

A Stable Condition and Adaptive Diffusion Coefficients for the Coarse-Mesh Finite Difference Method

Coarse-mesh finite difference (CMFD) method is a widely used numerical acceleration method. However, the stability of CMFD method is not good for the problems with optically thick regions. In this paper, a stability rule named the “sign preservation rule” in the field of numerical heat transfer is extended to the scheme of CMFD. It is required that the disturbance of neutron current is positively correlated with that of the negative value of flux gradient. A necessary condition for stability of the CMFD method is derived, an adaptive diffusion coefficient equation is proposed to improve the stability of CMFD method, and the corresponding revised CMFD method is called the rCMFD method. With a few modifications of the code, the rCMFD method was implemented in the hexagonal-Z nodal SN (discrete-ordinates) solver in the NECP-SARAX code system. The rCMFD method and other similar acceleration methods were tested by three fast reactor problems which were obtained by modifying the hexagonal pitches of a benchmark problem. The numerical results indicated that the rCMFD method showed better stability than the traditional CMFD method and the artificially diffusive CMFD (adCMFD) method and a better convergence rate than the adCMFD method and the optimally diffusive CMFD (odCMFD) method for these fast reactor problems.

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.

Triangular Polynomial Expansion Nodal Method for VVER Core Analysis

Due to the low computational cost, nodal diffusion methods are still commonly used to simulate full-core reactor problems. This work represents the developmental effort to build an accurate nodal kernel to treat hexagonal geometry in the core simulator code PARCS. An innovative method called TriPEN-9 has been developed by splitting a hexagonal assembly into six triangular nodes and solved using cubic polynomial expansion for the scalar flux with nine-term expansion coefficients. The nodal diffusion calculation is further accelerated with the multilevel coarse-mesh finite difference method. The verification of the TriPEN-9 method on the VVER full-core problem is provided with the model based on the NURESIM (Nuclear Reactor Simulator)-SP1 V1000-2D-C1-tr benchmark problem. The Serpent Monte Carlo code is used as a reference solution for verification and to generate homogenized group-constants data for PARCS. Exact discontinuity factors were generated in GenPMAXS, a cross-section processing code, using a similar expansion method as the TriPEN-9 core solver method with the utilization of heterogeneous solutions from Serpent. Implementing the TriPEN-9 method in PARCS, this approach can exactly reproduce the solutions from the high-fidelity Serpent calculations.

Convergence analysis for the CMFD accelerated 2D/1D neutron transport calculation method based on Fourier analysis

The two-dimension/one-dimension (2D/1D) transport method and the coarse mesh finite difference (CMFD) method are widely used for high-fidelity transport calculation. The CMFD method could dramatically reduce the calculating time while preserving accuracy. But the CMFD method accelerating the 2D/1D transport calculations may lead to divergence problems in some high scattering rate problems. The results of Fourier analysis indicate that the convergence is improved by increasing transport inner iterations. But more inner iterations lead to time burden in the method of characteristic (MOC) calculation. Fourier analysis is performed on the MOC based 2D/1D method to analyze the convergence feature, including increasing 1D transport inner iterations, iteration sequence of the 2D and 1D calculations, and so on. The results show that the effect of the 1D transport inner iteration on the convergence is related to the coarse-mesh height, and more 1D iteration could improve the convergence rate without introducing much more computation burden. Compared to the 1D2D computation sequence, the 2D1D sequence calculation has better convergence performance.

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 Calculations–I: Theory

This paper presents a new robust scheme for coupled physics nuclear reactor calculations. We focus specifically on high-fidelity whole-core transport calculations with coarse mesh finite difference (CMFD) coupled to thermal hydraulics. These simulations traditionally employ rthe Picard iteration for the coupled solution, where it has been observed that the use of CMFD (or nonlinear diffusion acceleration) is detrimental to the overall convergence of the coupled problem. Moreover, (1) if the acceleration equations are tightly converged every iteration, the overall multiphysics iteration becomes less stable and (2) properly loosening the convergence criteria of the acceleration equations at each iteration can stabilize the overall scheme. In this paper, we develop a Fourier analysis for a simplified CMFD-accelerated neutron transport problem with feedback from flux-dependent cross sections to provide a theoretical explanation for, and gain insight into, the aforementioned observations. Furthermore, we establish the theoretical relationship between relaxation and partial convergence of the low-order problem. Using this result, a relaxation-free iteration scheme is then proposed, with a formula to determine the nearly optimal partial convergence of the low-order diffusion problem. The new CMFD method is called the nearly optimally partially converged coarse mesh finite difference (NOPC-CMFD) method. It is shown theoretically that the NOPC-CMFD method in problems with feedback has stability properties comparable to CMFD in problems without feedback and requires no relaxation factor, i.e., is relaxation free. The results presented in this paper provide a theoretical foundation for the development of a robust multiphysics iteration scheme for nuclear reactor modeling. The implementation of the method and application to various test cases are presented in the companion paper.

Convergence study of CMFD based acceleration schemes for multi-group transport calculations with fission source using fourier analysis

Application of the coarse mesh finite difference (CMFD) method and its variants to accelerate the convergence of criticality calculations has been extremely popular for neutron transport simulations. Despite their widespread use in the multi-group form, the convergence performance of these acceleration schemes is usually studied in the one-group domain and results extended to multi-group problems. Theoretical investigation of the convergence behavior of multi-group CMFD and related methods has not been previously performed and is an important work to complement existing studies. In this paper, we present the Fourier convergence analysis of multi-group CMFD, partial current CMFD (pCMFD), optimally diffusive CMFD (odCMFD) and linear prolongation CMFD (lpCMFD) schemes for acceleration of the fission source neutron transport calculations. The multi-group formulation of the error iteration matrix has been found to be analogous to the results obtained from one-group analysis. It has also been shown that for certain simplified cases, the spectral radius of multi-group CMFD is independent of the group coupling terms and can be determined as the maximum among individual energy groups. The results extend current understanding and provide deeper insight into the application of CMFD based acceleration techniques for more realistic multi-group neutron transport.

MULTILEVEL CMFD ACCELERATION METHODS FOR WHOLE CORE REACTOR SIMULATION IN VERA

Ongoing efforts are being made to improve the performance of MPACT as the deterministic neutron transport solver in the Virtual Environment for Reactor Analysis (VERA). As other parts of the code have been improved, the coarse mesh finite difference method (CMFD) has come to take up a significant portion of the runtime. Multilevel-in-energy CMFD and multilevel-in-space CMFD solvers have been used to improve CMFD solver performance. A new multilevel-in-space-and-energy CMFD solver is being introduced that combines components of these two methods. W-Cycles and partial W-Cycles are being investigated to further improve the efficiency of the multilevel-in-energy CMFD solver. The performance of these methods is demonstrated on full core reactor physics problems of interest to VERA.

DEMONSTRATION OF A LINEAR PROLONGATION CMFD METHOD ON MOC

Coarse Mesh Finite Difference (CMFD) method is a very effective method to accelerate the iterations for neutron transport calculation. But it can degrade and even fail when the optical thickness of the mesh becomes large. Therefore several methods, including partial current-based CMFD (pCMFD) and optimally diffusive CMFD (odCMFD), have been proposed to stabilize the conventional CMFD method. Recently, a category of “higherorder” prolongation CMFD (hpCMFD) methods was proposed to use both the local and neighboring coarse mesh fluxes to update the fine cell flux, which can solve the fine cell scalar flux discontinuity problem between the fine cells at the bounary of the coarse mesh. One of the hpCMFD methods, refered as lpCMFD, was proposed to use a linear prolongation to update the fine cell scalar fluxes. Method of Characteristics (MOC) is a very popular method to solve neutron transport equations. In this paper, lpCMFD is applied on the MOC code MPACT for a variety of fine meshes. A track-based centroids calculation method is introduced to find the centroids coordinates for random shapes of fine cells. And the numerical results of a 2D C5G7 problem are provided to demonstrate the stability and efficiency of lpCMFD method on MOC. It shows that lpCMFD can stabilize the CMFD iterations in MOC method effectively and lpCMFD method performs better than odCMFD on reducing the outer MOC iterations.

Convergence analysis of the non-linear CMFD schemes for acceleration of multi-group fixed source neutron transport calculations

The coarse mesh finite difference (CMFD) and related methods have been some of the most extensively used schemes to accelerate the convergence of neutron transport iterations. Despite its predominant application on multi-group problems, the convergence behavior of this class of acceleration methods has been traditionally characterized for the one-group form and results extended to multi-group domain. Theoretical convergence analysis of the multi-group variants of CMFD scheme has not been previously performed and is an important work related to neutron transport studies. Fourier analysis of the coarse mesh finite difference (CMFD), partial current CMFD (pCMFD) and optimally diffusive CMFD (odCMFD) methods for acceleration of the multi-group fixed source neutron transport calculations has been presented in this paper. The multi-group error transition matrix and intermediate Fourier analysis expressions have been found to be analogous to those for one-group system. Sensitivity studies to characterize the effect of various parameters on the overall convergence behavior have also been presented. The results extend current understanding and provide deeper insight into the application of CMFD class of acceleration schemes for more realistic multi-group calculations.

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.

A linearized coarse mesh finite difference preconditioner for the within-group Krylov subspace iteration based on two-dimensional method of characteristics

In this work, a preconditioner is developed based on the linear coarse mesh finite difference (CMFD) formulation for the flexible generalized minimal residual (FGMRES) algorithm, and applied to accelerate within-group Krylov iterations based on the two-dimensional (2-D) method of characteristics (MOC). The conventional CMFD method is linearized by replacing the multiplicative updating operator with an additive correction operator in the prolongation step. The effectiveness of the linearized CMFD preconditioner for problems featuring steep flux gradients and high scattering ratios can be demonstrated by the numerical results for the IAEA LWR pool reactor problem. The total FGMRES iterations and computing time were decreased by 52.7% and 41.8%, respectively. However, only modest efficiency improvements were achieved for the 2-D C5G7 and the KAIST-2A benchmark problems, revealing the degraded performance of the linearized CMFD preconditioner for problems with strong local heterogeneities.

Enhancing lpCMFD Acceleration with Successive Overrelaxation for Neutron Transport Source Iteration

We present the new iterative method lpCMFD-SOR, which combines the linear prolongation coarse-mesh finite difference (lpCMFD) scheme with the method of successive overrelaxation (SOR) for neutron transport source iteration (SI). The lpCMFD method is the latest coarse-mesh finite difference (CMFD)–type acceleration scheme and is unconditionally stable and more effective than the standard CMFD method. The SOR method is a variant of the Gauss-Seidel method for solving a linear system of equations, resulting in faster convergence. The idea is to update the scattering source with overrelaxation to speed up the coupled transport-diffusion SI. Fourier analysis shows that the lpCMFD-SOR method converges for a relaxation parameter in the range of . It becomes less effective when underrelaxed (i.e., ) and increasingly more effective as increases above 1 until reaching the optimal overrelaxation value, which is, however, problem dependent. The optimal overrelaxation parameter increases with both the scattering ratio and the optical thickness of the problem. Numerical experiments have confirmed the Fourier analysis results. In general, the SOR method can further enhance the convergence rate of the lpCMFD method by more than 40% for neutron transport problems.

Toward Asymptotic Diffusion Limit Preserving High-Order, Low-Order Method

Recent development of the high-order, low-order (HOLO) method has shown promising results for solving thermal radiative transfer problems. The HOLO algorithm is a moment-based acceleration, similar to the well-known nonlinear diffusion acceleration and coarse-mesh finite difference methods. In this work, we introduce a new spatial-differencing scheme for the low-order (LO) system based on the corner-balance method and analyze an asymptotic diffusion property for a one-dimensional gray equation. An asymptotic analysis indicates that the new spatial-differencing scheme possesses the equilibrium diffusion limit. Numerical examples demonstrate significant improvements in the solution accuracy compared to the LO finite-volume discretization with a discontinuous source reconstruction.

The response matrix acceleration: A new non-linear method for the 3D discrete-ordinate transport equation

In this paper, we propose a new non-linear technique for accelerating the solution of the discrete ordinates transport equation. The new method, called Response Matrix Acceleration (RMA), has been designed to complement the Coarse-Mesh Finite Difference method (CMFD) by offering better stability and improved performance in cases where CMFD fails. To accomplish this, RMA uses knowledge of the transport operator along with nonlinear coefficients and solves for the interface partial currents to maintain consistency with the transport operator. Two distinct variants of RMA are derived. The convergence properties of both variants of RMA applied the source iteration schemes are investigated for the one-group transport operator. Analysis of the results indicates that both variants of RMA have improved effectiveness and stability relative to CMFD, for optically diffusive materials. To achieve optimal numerical performance, a combination of RMA and CMFD is suggested. Improvements in the performance of RMA are expected with ongoing development and optimization. Further investigation into the use of RMA for accelerating outer iterations, parallel problems, and different transport operators is proposed. The results of a spectral radius analysis are presented, along with a strong scaling benchmark using the 3D C5G7 MOX problems. Furthermore, two real-scale problems, the whole-core EOLE reactor simulation and a PWR assembly simulation, are studied to assess the performances of the new method in a parallel computing framework using the constant and linear short characteristics of the IDT solver in APOLLO3®.

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.