Coarse Mesh Finite Difference Formulation Research Articles

Stability enhancement of planar transport solution based whole-core calculation employing augmented axial method of characteristics

The planar Method of Characteristic (MOC) solution based direct whole core calculation method is stabilized and extended by employing an axial one-dimensional (1D) MOC solver with several augmentations. In order to resolve the inherent instabilities involved in the coarse mesh finite difference formulation employing a nodal simplified P3 (SP3) axial solver, the factors causing negative fluxes are identified and eliminated. The augmentations consist of (1) a subgrid scheme, (2) transverse leakage splitting, and (3) PL scattering and limited transport correction that are selectively applied depending on regional characteristics. Axial domain decomposition is employed for parallel solution of 1D MOC. Various core calculation results demonstrate that the augmented 1D MOC solver considerably enhances stability while retaining comparable accuracy with the SP3 nodal kernel and not causing significant computational overhead.

A Consistent CMFD Formulation for the Acceleration of Neutron Transport Calculations Based on the Finite Element Method

This paper presents the method and performance of a coarse-mesh finite difference (CMFD) scheme for accelerating neutron transport calculations based on the finite element method (FEM). The transport solution based on FEM does not satisfy the neutron balance exactly because FEM yields a nonconservative discretization. A modified CMFD formulation has been developed to correct the limitation of the conventional CMFD that is applicable only to neutronics solvers with a conservative discretization. A consistent CMFD problem for the transport solution based on FEM is constructed by enforcing the neutron balance in each coarse mesh by introducing a pseudo absorption cross section, and the well-established alternating solution process of CMFD and transport calculations is employed to accelerate source convergence. The applicability of the modified CMFD scheme to transport calculation based on FEM was first tested for a one-dimensional, discrete ordinates (SN), discontinuous FEM. The performance of CMFD acceleration was then investigated with a two-dimensional/three-dimensional method of characteristic transport solver for thermal and fast reactor problems with various core sizes. It was observed that the consistent CMFD scheme could improve the computational efficiency of eigenvalue calculation significantly in the framework of a transport solver with fission source iteration.

CMFD and GPU acceleration on method of characteristics for hexagonal cores

Coarse Mesh Finite Difference (CMFD) has been widely adopted as an effective way to accelerate the source iteration of transport calculation. However in a core with hexagonal assemblies there are non-hexagonal meshes around the edges of assemblies, causing a problem for CMFD if the CMFD equations are still to be solved via tri-diagonal matrix inversion by simply scanning the whole core meshes in different directions. To solve this problem, we propose an unequal mesh CMFD formulation that combines the non-hexagonal cells on the boundary of neighboring assemblies into non-regular hexagonal cells. We also investigated the alternative hardware acceleration of using graphics processing units (GPU) with graphics card in a personal computer. The tool CUDA is employed, which is a parallel computing platform and programming model invented by the company NVIDIA for harnessing the power of GPU. To investigate and implement these two acceleration methods, a 2-D hexagonal core transport code using the method of characteristics (MOC) is developed. A hexagonal mini-core benchmark problem is established to confirm the accuracy of the MOC code and to assess the effectiveness of CMFD and GPU parallel acceleration. For this benchmark problem, the CMFD acceleration increases the speed 16 times while the GPU acceleration speeds it up 25 times. When used simultaneously, they provide a speed gain of 292 times.

Improved stiffness confinement method within the coarse mesh finite difference framework for efficient spatial kinetics calculation

An improved Stiffness Confinement Method (SCM) is formulated within the framework of the coarse mesh finite difference (CMFD) formulation for efficient multigroup spatial kinetics calculation. The algorithm for searching for the amplitude frequency that makes the dynamic eigenvalue unity is developed in a systematic way along with the methods for determining the shape and precursor frequencies. A nodal calculation scheme is established within the CMFD framework to incorporate the cross section changes due to thermal feedback and dynamic frequency update. The conditional nodal update scheme is employed such that the transient calculation is performed mostly with the CMFD formulation and the CMFD parameters are conditionally updated by intermittent nodal calculations. A quadratic representation of amplitude frequency is introduced as another improvement. The performance of the improved SCM within the CMFD framework is assessed by comparing the solution accuracy and computing times for the NEACRP control rod ejection benchmark problems with those obtained with the Crank–Nicholson method with exponential transform (CNET). It is demonstrated that the improved SCM is beneficial for large time step size calculations with stability and accuracy enhancement.

Coarse mesh finite difference formulation for accelerated Monte Carlo eigenvalue calculation

An efficient Monte Carlo (MC) eigenvalue calculation method for source convergence acceleration and stabilization is developed by employing the Coarse Mesh Finite Difference (CMFD) formulation. The detailed methods for constructing the CMFD system using proper MC tallies are devised such that the coarse mesh homogenization parameters are dynamically produced. These involve the schemes for tally accumulation and periodic reset of the CMFD system. The method for feedback which is to adjust the MC fission source distribution (FSD) using the CMFD global solution is then introduced through a weight adjustment scheme. The CMFD accelerated MC (CMFD-MC) calculation is examined first for a simple one-dimensional multigroup problem to investigate the effectiveness of the accelerated fission source convergence process and also to analyze the sensitivity of the CMFD-MC solutions on the size of coarse meshes and on the number of CMFD energy groups. The performance of CMFD acceleration is then assessed for a set of two-dimensional and three-dimensional multigroup (3D) pressurized water reactor core problems. It is demonstrated that very rapid convergence of the MC FSD is possible with the CMFD formulation in that a sufficiently converged MC FSD can be obtained within 20 cycles even for large three-dimensional problems which would require more than 600 inactive cycles with the standard MC fission source iteration scheme. It is also shown that the optional application of the CMFD formulation in the active cycles can stabilize FSDs such that the real-to-apparent variance ratio of the local tallies can be reduced. However, due to the reduced importance of the variance bias in fine local tallies of 3D MC eigenvalue problems, the effectiveness of CMFD in tally stabilization turns out to be not so great.

Multi-group unified nodal method with two-group coarse-mesh finite difference formulation

The one-node kernels of the unified nodal method (UNM) which were originally developed for two-group (2 G) problems are extended to solve multi-group (MG) problems within the framework of the 2 G coarse-mesh finite difference (CMFD) formulation. The analytic nodal method (ANM) kernel of UNM is reformulated for the MG application by adopting the Padé approximation to avoid the similarity transform required to diagonalize the G × G buckling matrix. In addition, a one-node semi-analytic nodal method (SANM) kernel which is considered adequate for multi-group calculations is also integrated into the UNM formulation by expressing it in the form consistent with the other UNM kernels. As an efficient global solution framework, the 2 G CMFD formulation with dynamic group condensation and prolongation is established and the performance of the various MG kernels is examined using various static and transient benchmark problems. It turns out that the SANM kernel is the best one for MG problems not only because it retains accuracy comparable to MGANM with a shorter computing time but also because its accuracy or its convergence does not depend on the eigenvalue range of the buckling matrix of the system. The 2 G CMFD formulation with MG one-node UNM kernels turns out to be very effective in that it conveniently accelerates the MG source iteration.

Whole Core Transport Calculation Employing Hexagonal Modular Ray Tracing and CMFD Formulation

A whole core transport module for hexagonal cores is developed and implemented in the DeCART code. The module consists of a whole core ray tracing kernel for solving the two-dimensional (2-D) method of characteristics (MOC) equation and a coarse-mesh finite difference (CMFD) kernel to accelerate the MOC transport iteration. Whole core ray tracing is realized by incorporating a hexagonal-assembly-based modular ray tracing scheme. The complete path linking constraint forthe modular ray is achieved by adjusting the ray angle and the ray spacing in the range of [0, 30°], and the complete reflection constraint at the problem boundary is satisfied by defining the corresponding reflection angles at the reflection surfaces. The hexagonal CMFD kernel employs unstructured nodes that can treat the irregular-shaped gap cells as well as the regular hexagonal cells. Some features such as cell ray approximation and modified cycle ray scheme are employed to reduce the memory requirements for the segment information and the boundary angular fluxes, respectively. The solution accuracy and execution performance of the hexagonal module are examined for the C5G7 hexagonal variation problems that are established by modifying the original C5G7MOX 3-D extension benchmark. The CMFD kernel shows a significant speedup of 60 in the 2-D core problem. The cell ray approximation does not violate the original solution accuracy when using the default ray spacing of the DeCART code. The modified cycle ray scheme shows its superiority over the simple core ray sweeping scheme in terms of the memory requirement and the original cycle ray scheme in terms of the computing time. Compared with the Monte Carlo solutions, the DeCART solution agrees to within 40 pcm for the eigenvalue and 2% for the pin power distribution.

Two-Level Coarse Mesh Finite Difference Formulation with Multigroup Source Expansion Nodal Kernels

As an effort to establish a fast, yet accurate multigroup nodal solution method that is crucial in repeated static and transient calculations for advanced reactors, the source expansion form of the semi-analytic nodal method (SANM) is introduced within the framework of the coarse mesh finite difference (CMFD) formulation. The source expansion is to expand the analytic form of the source appearing in the groupwise neutron diffusion equation with a set of orthogonal polynomials in order to obtain a group decoupled analytic solution. Both one- and two-node formulations are examined to determine the best nodal kernel. For the acceleration, a two-level CMFD scheme is established employing a multigroup and two-group CMFD. In addition, an alternative two-node direct SANM formulation with a quartic polynomial is examined to assess the direct vs. iterative resolution of the group coupling. The performance of the CMFD formulation with three different multigroup SANM nodal kernels is examined for a wide variety of multigroup benchmark problems including several MOX-loaded LWR cores and large FBR cores. It is demonstrated that superior accuracy is achievable with all the SANM kernels while the iterative two-node SANM kernel outperforms the others in the multigroup calculations employing more than two groups, and the two-level CMFD formulation is quite efficient in the acceleration of the outer iteration.

Implementation of Two-Level Coarse-Mesh Finite Difference Acceleration in an Arbitrary Geometry, Two-Dimensional Discrete Ordinates Transport Method

The coarse-mesh finite difference (CMFD) formulation is applied as an efficient means of acceleration of the heterogeneous whole-core transport calculation. The CMFD formulation enables dynamic homogenization of the cells during the iterative solution process such that the heterogeneous transport solution can be preserved. Dynamic group condensation is also possible with a two-level CMFD formulation involving alternate multigroup and two-group calculations. The two-dimensional discrete ordinates (SN) method is used as the kernel to generate the heterogeneous solution; the CMFD solution provides the SN kernel with much faster convergence of fission and scattering source distributions. In this paper, the two-level CMFD acceleration has been tested using the VENUS-2 two-dimensional whole-core model; it is shown that the number of SN transport sweeps can be reduced by a factor of about 10 while exactly reproducing the original transport solution. The second level of CMFD acceleration is also significant in reducing the computation time. The application of the CMFD formulation in arbitrary geometry demonstrates that CMFD also works well for irregular geometries.

Two-Level Coarse Mesh Finite Difference Formulation with Multigroup Source Expansion Nodal Kernels

As an effort to establish a fast, yet accurate multigroup nodal solution method that is crucial in repeated static and transient calculations for advanced reactors, the source expansion form of the semi-analytic nodal method (SANM) is introduced within the framework of the coarse mesh finite difference (CMFD) formulation. The source expansion is to expand the analytic form of the source appearing in the groupwise neutron diffusion equation with a set of orthogonal polynomials in order to obtain a group decoupled analytic solution. Both one- and two-node formulations are examined to determine the best nodal kernel. For the acceleration, a two-level CMFD scheme is established employing a multigroup and two-group CMFD. In addition, an alternative two-node direct SANM formulation with a quartic polynomial is examined to assess the direct vs. iterative resolution of the group coupling. The performance of the CMFD formulation with three different multigroup SANM nodal kernels is examined for a wide variety of multigroup benchmark problems including several MOX-loaded LWR cores and large FBR cores. It is demonstrated that superior accuracy is achievable with all the SANM kernels while the iterative two-node SANM kernel outperforms the others in the multigroup calculations employing more than two groups, and the two-level CMFD formulation is quite efficient in the acceleration of the outer iteration.

A Hybrid Nodal Diffusion/SP3 Method Using One-Node Coarse-Mesh Finite Difference Formulation

A hybrid nodal diffusion/simplified P3 (SP3) method was implemented within the framework of a one-node coarse-mesh finite difference formulation. The one-node formulation enables the use of various combinations of space, energy, and angular approximations within the framework of the one-node global/local solution approach. Spatial approximations include advanced nodal methods and fine-mesh finite difference methods. Energy approximations involve conventional two-group and multiple energy groups. Angular approximations contain both the diffusion and SP3 methods. Partial-moment boundary conditions are used to solve the one-node problems since they simplify the formulation of consistent interface conditions for the various methods. All directional moments are determined simultaneously to stabilize convergence of the one-node global/local solution approach. Results for a light water reactor mixed-oxide benchmark problem indicate that the hybrid application of the one-node-based nodal SP3 method developed here can provide substantial reductions in the computational time without compromising the accuracy of the solution.