• Nuclear Engineering and Technology
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• v.56 no.3
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• pp.772-784
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• 2024

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.

• Nuclear Engineering and Technology
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• v.46 no.6
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• pp.783-796
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• 2014

The coarse mesh finite difference (CMFD) method is applied to the discontinuous finite element method based discrete ordinate calculation for source convergence acceleration. The three-dimensional (3-D) DFEM-Sn code FEDONA is developed for general geometry applications as a framework for the CMFD implementation. Detailed methods for applying the CMFD acceleration are established, such as the method to acquire the coarse mesh flux and current by combining unstructured tetrahedron elements to rectangular coarse mesh geometry, and the alternating calculation method to exchange the updated flux information between the CMFD and DFEM-Sn. The partial current based CMFD (p-CMFD) is also implemented for comparison of the acceleration performance. The modified p-CMFD method is proposed to correct the weakness of the original p-CMFD formulation. The performance of CMFD acceleration is examined first for simple two-dimensional multigroup problems to investigate the effect of the problem and coarse mesh sizes. It is shown that smaller coarse meshes are more effective in the CMFD acceleration and the modified p-CMFD has similar effectiveness as the standard CMFD. The effectiveness of CMFD acceleration is then assessed for three-dimensional benchmark problems such as the IAEA (International Atomic Energy Agency) and C5G7MOX problems. It is demonstrated that a sufficiently converged solution is obtained within 7 outer iterations which would require 175 iterations with the normal DFEM-Sn calculations for the IAEA problem. It is claimed that the CMFD accelerated DFEM-Sn method can be effectively used in the practical eigenvalue calculations involving general geometries.

• Nuclear Engineering and Technology
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• v.34 no.3
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• pp.250-258
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• 2002

This Paper is to apply the well-established coarse mesh finite difference(CMFD) method to the method of characteristics(MOC) transport calculation as an acceleration scheme. The CMFD problem is first formulated at the pin-cell level with the multi-group structure To solve the cell- based multi-group CMFD problem efficiently, a two-group CMFD formulation is also derived from the multi-group CMFD formulation. The performance of the CMFD acceleration is examined for three test problems with different sizes including a realistic quarter core PWR problem. The CMFD formulation provides a significant reduction in the number of ray tracings and thus only about 9 ray tracing iterations are enough for the realistic problem. In computing time, the CMFD accelerated case is about two or three times faster than the coarse-mesh rebalancing(CMR) accelerated case.

• Nuclear Engineering and Technology
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• v.54 no.8
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• pp.3059-3072
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• 2022

A Jacobian-Free Newton Krylov Two-Nodal Coarse Mesh Finite Difference algorithm based on Nodal Expansion Method (NEM_TNCMFD_JFNK) is successfully developed and proposed to solve the three-dimensional (3D) and multi-group reactor physics models. In the NEM_TNCMFD_JFNK method, the efficient JFNK method with the Modified Incomplete LU (MILU) preconditioner is integrated and applied into the discrete systems of the NEM-based two-node CMFD method by constructing the residual functions of only the nodal average fluxes and the eigenvalue. All the nonlinear corrective nodal coupling coefficients are updated on the basis of two-nodal NEM formulation including the discontinuity factor in every few newton steps. All the expansion coefficients and interface currents of the two-node NEM need not be chosen as the solution variables to evaluate the residual functions of the NEM_TNCMFD_JFNK method, therefore, the NEM_TNCMFD_JFNK method can greatly reduce the number of solution variables and the computational cost compared with the JFNK based on the conventional NEM. Finally the NEM_TNCMFD_JFNK code is developed and then analyzed by simulating the representative PWR MOX/UO₂ core benchmark, the popular NEACRP 3D core benchmark and the complicated full-core pin-by-pin homogenous core model. Numerical solutions show that the proposed NEM_TNCMFD_JFNK method with the MILU preconditioner has the good numerical accuracy and can obtain higher computational efficiency than the NEM-based two-node CMFD algorithm with the power method in the outer iteration and the Krylov method using the MILU preconditioner in the inner iteration, which indicates the NEM_TNCMFD_JFNK method can serve as a potential and efficient numerical tool for reactor neutron diffusion analysis module in the JFNK-based multiphysics coupling application.

• Nuclear Engineering and Technology
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• v.53 no.5
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• pp.1403-1415
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• 2021

The transient capability of a SP₃ based pin-wise core analysis code SPHINCS is developed and verified through the analyses of the C5G7-TD benchmark. Spatial discretization is done by the fine mesh finite difference method (FDM) within the framework of the coarse mesh finite difference (CMFD) formulation. Pin size fine meshes are used in the radial fine mesh kernels. The time derivatives of the odd moments in the time-dependent SP₃ equations are neglected. The pin homogenized group constants and Super Homogenization (SPH) factors generated from the 2D single assembly calculations at the unrodded and rodded conditions are used in the transient calculations via proper interpolation involving the approximate flux weighting method for the cases that involve control rod movement. The simplifications and approximations introduced in SPHINCS are assessed and verified by solving all the problems of C5G7-TD and then by comparing with the results of the direct whole core calculation code nTRACER. It is demonstrated that SPHINCS yields accurate solutions in the transient behaviors of core power and reactivity.

• Nuclear Engineering and Technology
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• v.53 no.11
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• pp.3543-3562
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• 2021

The methods and performance of a 3D pin-by-pin neutronics code based on the 2D/1D decoupling method are presented. The code was newly developed as an effort to achieve enhanced accuracy and high calculation performance that are sufficient for the use in practical nuclear design analyses. From the 3D diffusion-based finite difference method (FDM) formulation, decoupled planar formulations are established by treating pre-determined axial leakage as a source term. The decoupled axial problems are formulated with the radial leakage source term. To accelerate the pin-by-pin calculation, the two-level coarse mesh finite difference (CMFD) formulation, which consists of the multigroup node-wise CMFD and the two-group assembly-wise CMFD is implemented. To enhance the accuracy, both the discontinuity factor method and the super-homogenization (SPH) factor method are examined for pin-wise cross-section homogenization. The parallelization is achieved with the OpenMP package. The accuracy and performance of the pin-by-pin calculations are assessed with the VERA and APR1400 benchmark problems. It is demonstrated that pin-by-pin 2D/1D alternating calculations within the two-level 3D CMFD framework yield accurate solutions in about 30 s for the typical commercial core problems, on a parallel platform employing 32 threads.

• Nuclear Engineering and Technology
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• v.56 no.9
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• pp.3654-3667
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• 2024

Expanding upon the framework of the steady-state pin-by-pin 2D/1D decoupling method, a novel and highperformance 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 twogroup assemblywise CMFD is employed as well to accelerate the convergence. Errors originating from the pinlevel 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.

• Nuclear Engineering and Technology
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• v.43 no.6
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• pp.531-546
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• 2011

The backward differentiation formula (BDF) method is applied to a three-dimensional reactor kinetics calculation for efficient yet accurate transient analysis with adaptive time step control. The coarse mesh finite difference (CMFD) formulation is used for an efficient implementation of the BDF method that does not require excessive memory to store old information from previous time steps. An iterative scheme to update the nodal coupling coefficients through higher order local nodal solutions is established in order to make it possible to store only node average fluxes of the previous five time points. An adaptive time step control method is derived using two order solutions, the fifth and the fourth order BDF solutions, which provide an estimate of the solution error at the current time point. The performance of the BDF- and CMFD-based spatial kinetics calculation and the adaptive time step control scheme is examined with the NEACRP control rod ejection and rod withdrawal benchmark problems. The accuracy is first assessed by comparing the BDF-based results with those of the Crank-Nicholson method with an exponential transform. The effectiveness of the adaptive time step control is then assessed in terms of the possible computing time reduction in producing sufficiently accurate solutions that meet the desired solution fidelity.