Coarse Mesh Finite Difference Research Articles

Development of 3-D HCMFD algorithm for efficient pin-by-pin reactor analysis

This paper presents the characteristics and performances of the HCMFD (Hybrid Coarse-Mesh Finite Difference) algorithm for 3-D pin-by-pin reactor core analyses. The HCMFD algorithm has been suggested for an extremely efficient parallel computing of a pin-by-pin neutron diffusion analysis with a pin-level nodal calculation based on a non-linear local-global iterative framework. The fundamental ideas and concepts for the one-node CMFD scheme, in combination with the conventional two-node CMFD scheme using the nodal expansion method, has been described in detail. When the HCMFD algorithm is applied to a 3-D problem, it provides flexibility on the axial mesh refinement so that one can easily optimize the axial mesh size to treat a significant heterogeneity in axial direction. In this paper, some possible variations in axial mesh set up and several numerical strategies for better convergence are discussed. The overall features of the 3-D HCMFD algorithm were identified through analyses of the 3-D EPRI-9R benchmark, and the potential performance of the HCMFD algorithm for a practical pin-by-pin reactor core analysis has been estimated by solving a big PWR problem. Including the sensitivity study regarding the optimization in axial direction, the parallel performances of the 3D HCMFD algorithm were measured and compared with those of a conventional two-node CMFD scheme. It was shown that the HCMFD algorithm has clear advantages over the conventional CMFD approach in 3-D pin-by-pin core analyses.

Stabilization of multi-group neutron transport with transport-corrected cross-sections

Many deterministic neutron transport solvers rely on the source iteration method to solve the multi-group neutron transport equation. Often, these solvers rely on transport corrected cross-sections for accurate prediction of the neutron flux distribution. Transport corrected within-group scattering cross-sections can become negative, and if these negative cross-sections are large in magnitude, source iteration can become unstable and fail to converge for certain cases. In this study, we present evidence of this convergence issue for the method of characteristics (MOC) on full-core PWR problems with common transport correction schemes. A theoretical discussion is presented to illustrate the reason for the convergence issues. Previously established stabilization methods are compared with a newly proposed stabilization method. Results show that the new stabilization method allows for faster convergence than previous techniques. In addition, the effect of Coarse Mesh Finite Difference (CMFD) acceleration on stability is analyzed, showing that CMFD acceleration with full-group structure can overcome the convergence issues, but a stabilization technique is necessary for convergence when a condensed group structure is used.

Performance evaluation of CMFD on inter-cycle correlation reduction of Monte Carlo simulation

This paper presents a performance examination of the Coarse Mesh Finite Difference (CMFD) method for a reduction of inter-cycle correlations in Monte Carlo (MC) active cycle simulations. Sensitivity tests using the UNIST in-house MC code ‘MCS’ reveal the characteristics of CMFD effectiveness depending on problem sizes, dimensions, and cross-section dependency on neutron energy (multi-group vs. continuous). Also, by applying this method to the three-dimensional BEAVRS whole core problem, it is successfully demonstrated that the CMFD can reduce the inter-cycle correlations of practical, large size, and high dominance ratio problems, and that the figure of merit of efficiency for a practical reactor problem increases by a factor of six.

3D whole-core neutron transport simulation using 2D/1D method via multi-level generalized equivalence theory based CMFD acceleration

A new 2D/1D method with multi-level generalized equivalence theory (GET) based coarse mesh finite difference (CMFD) acceleration was proposed for the whole core transport calculation in this paper. Fouressential features of this new 2D/1D methods are as follows: (1) Two new defined factors, nodal discontinuity factor (NDF) and modified diffusion coefficient factor (MDF), were defined in this new GET based CMFD method to achieve equivalence between CMFD and the transport solutions and especially insure that only positive coupling coefficients would occur in the CMFD linear system; (2) For accelerating the convergence of the CMFD, a multi-level acceleration scheme was implemented and an innovative self-developed RSILU preconditioned GMRES solver was developed to solve the CMFD linear system; (3) Within the new CMFD framework, 2D MOC and 1D two-node nodal expansion solvers were developed for 3D whole core transport calculation and the sub-plane technique was applied to minimize the nodal error successfully while maintain a reasonable computing time; (4) For implementation, transverse leakage splitting technique was applied to avoid the total source to become negative, and domain decomposition method based on MPI was implemented to take advantages of high performance clusters. The accuracy of this 2D/1D method via multi-level GET based CMFD and the effectiveness and acceleration performance of the multi-level CMFD method were examined for the well-known C5G7 benchmarks problems. The numerical results demonstrate that superior accuracy is achievable and the multi-level acceleration schemeis efficient and enhances the converge speed of both the GET based CMFD acceleration and the MOC calculation.

Functionalization of the Discontinuity Factor in the Albedo-Corrected Parameterized Equivalence Constants (APEC) Method

The Albedo-corrected Parameterized Equivalence Constants (APEC) method, a new leakage correction method for two-group nodal analysis of light water reactors, has been extended to discontinuity factor (DF) correction. First, the error of nodal calculations induced by an inaccurate assembly discontinuity factor (ADF) is evaluated using the reference two-group cross section (XS) and DF calculated from heterogeneous core transport calculations. Functionalization of DF is performed by finding relationships between surfacewise current-to-flux ratio and change of DF from ADF. The least-squares method is used to fit several candidate functions to various core calculation results. The coefficients of APEC XS and DF correction functions are determined considering several color-set models. In this work, the two-dimensional method of characteristics–based lattice code DeCART2D is used for reference core calculations and lattice calculations. The extended APEC method is implemented in an in-house NEM nodal code using the partial-current coarse mesh finite difference acceleration. A small modular reactor (SMR) initial core benchmark is analyzed to evaluate the performance of the extended APEC method. In addition, the extended APEC method is applied to several variants of the SMR core and large variants to assess its general applicability.

“On the fly” stabilization of the Coarse-Mesh Finite Difference acceleration for multidimensional discrete-ordinates transport calculations

In this paper, the Jacobian matrix of the Coarse-Mesh Finite Difference (CMFD) method is analyzed. Both the homogenization and the current preservation effects are studied in heterogeneous multidimensional configurations. Some bounding values of the spectral radius are also given. An analytical stability analysis is carried on an interface slab problem. This analysis leads to the computation of the stability parameter introduced in the Generalized Coarse Mesh Rebalancing method. A dynamical stabilization technique is proposed for multidimensional neutron lattice calculations. Numerical calculations show that the proposed technique dumps the unstable modes, in particular in optically thick configurations, where the classical CMFD method fails to converge.

Unstructured coarse mesh finite difference method to accelerate k-eigenvalue and fixed source neutron transport calculations

Unstructured coarse mesh finite difference (CMFD) method was implemented to accelerate the k-eigenvalue and fixed source neutron transport calculations in the nodal SN transport code DNTR, which was based on arbitrary triangular-z meshes. The same triangular-z meshes were adopted in the CMFD calculation to accommodate the arbitrary problem boundaries and simplify the coarse mesh generation and mapping processes. Two-level CMFD formulations were derived for the k-eigenvalue problems, and the few-group CMFD calculation was used to speed up the convergence of the multigroup CMFD calculation. For the fixed source problems in the neutronics analyses of subcritical reactors or transient neutron transport calculations, a special ks iteration method by introducing the source multiplication factor ks was employed in the SN transport calculation to remove the dependence of the convergence rate on the subcriticality. Multigroup CMFD formulations were established to accelerate the convergence of the ks iteration procedure in the fixed source neutron transport calculation. The acceleration effects of CMFD on the k-eigenvalue and fixed source calculations were verified using four representative neutron transport problems, i.e. the small fast breeder reactor with Cartesian boundaries, the BN-600 fast reactor with hexagonal boundaries, the space nuclear power reactor with cylindrical boundaries, and the accelerator driven subcritical reactor with hexagonal boundaries. The numerical results showed that for the first three eigenvalue problems, the multigroup CMFD achieved a speedup of total computation time by a factor of 2–5, and the two-level CMFD reduced the computation time by a factor of 3–7. It was also shown that the multigroup CMFD was able to accelerate the ks iteration procedure in the fixed source problems by a factor of about 6. The implementation of the unstructured CMFD method turns the DNTR code into an efficient neutron transport solver with flexible geometry treatment capability for both k-eigenvalue and fixed source problems.

A modified predictor-corrector quasi-static method in NECP-X for reactor transient analysis based on the 2D/1D transport method

The 2D/1D transport method is a widely applied method for the whole-core heterogeneous transport calculation. The transient simulation based on the pin-resolved 2D/1D method is implemented in the high-fidelity code NECP-X. The 3D transient fixed source problem (TFSP) is transformed into 2D TFSP and 1D TFSP coupled by leakage, and the pin-based coarse mesh finite difference method (CMFD) is used to accelerate convergence of the 2D/1D TFSP. In this paper, a modified predictor-corrector quasi-static (mPCQS) method is developed to provide better reactivity prediction for PK calculations. Furthermore, the simplified source scaling method (SSSM) is used to accelerate the convergence of the CMFD TFSP iterations. The results of three problems are presented including the 2D C5G7-TD benchmark, a MiniCore-3D rod-ejection problem and a 3D heterogeneous rod-ejection problem. Numerical results reveal that the SSSM method can significantly reduce the number of CMFD iterations without increasing the computational complexity, and the mPCQS method can provide better results than the standard PCQS for step perturbation transient event and rod ejection transient event no matter in the homogeneous case or in the heterogeneous case.

Convergence Studies on Nonlinear Coarse-Mesh Finite Difference Accelerations for Neutron Transport Analysis

Application of partial current–based coarse-mesh finite difference (pCMFD) acceleration to a one-node scheme is devised for stability enhancement of the parallel neutron transport calculation algorithm. Conventional one-node coarse-mesh finite difference (CMFD) allows parallel algorithms to be more tractable than two-node CMFD, but it has an inherent stability issue for some problems. In order to overcome this issue, pCMFD is modified to be fitted into the one-node scheme and is tested for both sequential and parallel calculations. The superior stability of the one-node pCMFD is shown by comparing results from analytic and numerical approaches. To investigate the convergence behavior of the acceleration methods in an analytic way, Fourier analysis is applied to an infinite homogeneous slab reactor configuration with the monoenergetic neutron flux assumption, and the spectral radius is calculated as a convergence factor. This paper carefully describes the process of the Fourier analysis on the parallel algorithm for neutron transport and compares it to that of the conventional sequential algorithm.

Adaptive expansion order for diffusion Variational Nodal Method

The Variational Nodal Method (VNM) has been employed as the diffusion module in our PWR core analysis code Bamboo-Core within our PWR fuel management code system NECP-Bamboo. It expands the nodal volumetric flux and surface partial currents into the sums of orthogonal basis functions without using the transverse integration technique. To reduce the extra computing cost by the uniform expansion order setting, an adaptive expansion order technique has been developed in this paper. After estimating the net currents between each pair of neighboring nodes by using the Coarse-Mesh Finite-Difference (CMFD) technique, it estimates the required expansion orders in each node analytically. This technique increases the complexity of the code, but reduces the computational efforts both in computing time and memory storage by a factor of about 5 and 4, respectively. In addition, the CMFD acceleration is also employed to further improve the performance of the code. It is demonstrated by the numerical results that the CMFD acceleration technique can provide a speedup ratio of about 17.

A Linear Prolongation Approach to Stabilizing CMFD

In this paper, we propose a new prolongation method to replace the conventional flat flux ratio–based scaling approach of coarse-mesh finite difference (CMFD) for updating the flux. The new prolongation method employs a linear interpolation of the scalar flux differences at the coarse-mesh cell edges between the neutron transport and CMFD calculations. This linear prolongation scheme, called lpCMFD, can greatly improve the stability of CMFD, particularly for problems with large optical thickness. A detailed convergence study of lpCMFD based on Fourier analysis and numerical testing shows that lpCMFD is unconditionally stable and effective for a wide range of optical thicknesses.

Multi-level coarse mesh finite difference acceleration with local two-node nodal expansion method

A computationally efficient and effective multi-level coarse mesh finite difference (CMFD) acceleration is proposed using the local two-node nodal expansion method for solving the three-dimensional multi-group neutron diffusion equation. The multi-level CMFD acceleration method consists of two essential features: (1) a new one-group (1G) CMFD linear system is established by using cross sections, flux and current information from the multi-group (MG) CMFD to accelerate the MG CMFD calculation, (2) an adaptive Wielandt shift method is proposed to accelerate the inverse power iteration of 1G CMFD in order to provide an accurate estimate of the eigenvalue at the beginning of the iteration for both 1G and MG CMFD linear system. Additionally, a nodal discontinuity factor and a diffusion coefficient correction factor are defined to achieve equivalence of the 1G and MG CMFD system. The accuracy and acceleration performance of multi-level CMFD are examined for a variety of well-known multi-group benchmarks problems. The numerical results demonstrate that superior accuracy is achievable and the multi-level CMFD acceleration method is efficient, particularly for the larger, multi-group systems.

One-node and two-node hybrid coarse-mesh finite difference algorithm for efficient pin-by-pin core calculation

Abstract This article presents a new global–local hybrid coarse-mesh finite difference (HCMFD) method for efficient parallel calculation of pin-by-pin heterogeneous core analysis. In the HCMFD method, the one-node coarse-mesh finite difference (CMFD) scheme is combined with a nodal expansion method (NEM)–based two-node CMFD method in a nonlinear way. In the global-local HCMFD algorithm, the global problem is a coarse-mesh eigenvalue problem, whereas the local problems are fixed source problems with boundary conditions of incoming partial current, and they can be solved in parallel. The global problem is formulated by one-node CMFD, in which two correction factors on an interface are introduced to preserve both the surface-average flux and the net current. Meanwhile, for accurate and efficient pin-wise core analysis, the local problem is solved by the conventional NEM-based two-node CMFD method. We investigated the numerical characteristics of the HCMFD method for a few benchmark problems and compared them with the conventional two-node NEM-based CMFD algorithm. In this study, the HCMFD algorithm was also parallelized with the OpenMP parallel interface, and its numerical performances were evaluated for several benchmarks.

A Local Adaptive Coarse-Mesh Nonlinear Diffusion Acceleration Scheme for Neutron Transport Calculations

In order to improve the effectiveness and stability of the coarse-mesh finite difference method (CMFD), we developed a new nonlinear diffusion acceleration scheme for solving neutron transport equations. This scheme, called LR-NDA, employs a local refinement approach on the framework of CMFD by solving a local boundary value problem of the scalar flux on the coarse-mesh structure to replace the piecewise constant scalar flux obtained by CMFD. The refined flux is then used to update the scalar flux in the neutron transport source iteration. In this paper, a detailed convergence study of LR-NDA is carried out based on a two-dimensional fixed-source problem, and it shows that LR-NDA is much more effective and stable than CMFD for a wide range of optical thicknesses. In addition, we demonstrate that LR-NDA is a local adaptive method. LR-NDA does not necessarily require local refinement for all the coarse-mesh cells on the problem domain, i.e., it can be used only for relatively optically thick regions where the standard CMFD scheme would encounter the convergence problem.

“Virtual density” and traditional boundary perturbation theories: Analytic equivalence and numeric comparison

We compare and contrast “virtual density” perturbation theory with the traditional boundary perturbation theory developed by Pomraning, Larsen, and Rahnema in the context of diffusion theory. First, after reviewing that literature, we mathematically prove that virtual density perturbations and traditional boundary perturbations are precisely equivalent for arbitrary 1-D problems, which constitute non-uniform isotropic expansions. We also mathematically prove that these two perturbation theories are equivalent for 2-D boundary shift problems, which constitute non-uniform anisotropic expansions. Extension of this proof to swellings or 3-D problems is straightforward. We compare the two theories numerically for a series of alternating uranium and sodium 1-D slabs in finite difference diffusion, and we show that virtual density theory predicts reactivities much more accurately and efficiently than traditional boundary perturbation theory. Boundary perturbation theory is often very inaccurate on a coarse mesh but converges to the virtual density solution as the mesh becomes finer. We also compare the two theories for axial assembly swelling in an abbreviated FFTF benchmark with a coarse mesh. Here we find that reactivity coefficients obtained via virtual density perturbation theory agree with reference solutions to within 0.1%, while those obtained via boundary perturbation theory exhibit sporadic accuracy – sometimes in the range of 1–5% error, more frequently in the range 5–20% error, and occasionally well over 100% error in control rod assemblies. We conclude that although virtual density perturbation theory and boundary perturbation theory are analytically equivalent, boundary perturbations in diffusion theory are often thwarted in coarse mesh finite difference solutions due to inaccurate flux gradients along mesh cell surfaces in heterogeneous cores.

Acceleration and Real Variance Reduction in Continuous-Energy Monte Carlo Whole-Core Calculation via p-CMFD Feedback

In the three-dimensional (3-D) continuous-energy whole-core reactor analysis, the partial current–based coarse mesh finite difference (p-CMFD) feedback was applied to the Monte Carlo (MC) k-eigenvalue problem simulation for both inactive and active iterations (cycles). To reduce the stochastic errors in the p-CMFD parameters and their biases due to the ratio-type estimators, the first-in-first-out (FIFO) accumulation scheme was introduced in the MC/p-CMFD procedure. The MC/p-CMFD procedure was tested on a typical pressurized water reactor 3-D continuous-energy whole-core problem while varying the FIFO queue lengths and the results were compared with the conventional power iteration. The Shannon entropy analysis showed that MC/p-CMFD accelerates the convergence of the fission source distributions and mitigates the spatial clustering phenomenon. The real variance analysis also showed that MC/p-CMFD reduces the interiteration correlation, leading to the most real variance reduction in the local MC tallies at the optimum queue length (L = 5). It was also shown that a nontrivial bias was introduced by the p-CMFD feedback, especially for the global tally (keff) with L = 1. However, the bias decreased as the tally bin size became smaller and it was effectively reduced by increasing the queue length (L ≥ 5). In conclusion, the MC/p-CMFD procedure showed promising capability for 3-D continuous-energy whole-core reactor analysis by MC simulation.

Practical environment-corrected discontinuity factors and homogenized parameters for improved PT-SCWR neutron diffusion solutions

A novel application of practical discontinuity factors in coarse-mesh finite-difference solutions of heterogeneous multi cells with Pressure Tube Supercritical Water-cooled Reactor (PT-SCWR) type fuel and moderator cells has been investigated. In addition to discontinuity factors, homogenization schemes have also been studied and applied to reflector-adjacent-fuel cells and reflector cells. A 49 node inner-core multi cell model containing only fuel and a 40 node outer-core multi cell model containing reflector and fuel cells were simulated. In addition to nominal conditions, various discontinuity factors and homogenization techniques, along with conventional techniques, were applied to static beyond-nominal condition scenarios to evaluate the error associated with neutron power changes from nominal conditions, relative to reference transport solutions. The use of novel mean reference discontinuity factors and environment homogenized reflector-adjacent-fuel cells, over conventional methods, reduces core-wide reactivity errors, RMS neutron power errors, and maximum channel specific power errors by up to 2.6mk, 2.9%, and 6.7%, respectively.

Space-Dependent Wielandt Shifts for Multigroup Diffusion Eigenvalue Problems

We present a new concept—the space-dependent Wielandt shift (SDWS)—for accelerating the convergence of the power iteration (PI) scheme for multigroup diffusion k-eigenvalue problems. The SDWS improves on standard Wielandt shift (WS) techniques, which are empirical in nature and are typically effective only when the current estimate of the solution is reasonably converged. By accounting for the physics of the problem through SDWS, we are able to improve the acceleration for the initial iterates when the current estimate of the solution is not close to convergence. Numerical results from one-dimensional problems suggest that, compared to standard WS techniques, the new SDWS techniques can provide upward of a 46% reduction in the number of PIs required for convergence and a 40% reduction in the computational time required. This improvement is sensitive to several problem-dependent factors, such as the geometry and energy-dependence of the problem, the spatial discretization, and the initial guess. The reduction in computational time is also dependent on the linear solver in the PI scheme, as it is well known that WSs can significantly worsen the conditioning of the diffusion linear system. In this paper, we provide a detailed study of the impact of WSs on the performance of several iterative linear solvers. Results from our implementation of SDWS in the three-dimensional (3D) code MPACT show that SDWS can provide similar speedups for 3D multigroup diffusion eigenvalue problems. These results also show that moderate speedups can be obtained by applying SDWS to the coarse mesh finite difference (CMFD) solver in a CMFD-accelerated transport scheme. However, the benefit of doing this may be limited because all but the first few CMFD solves are relatively easy to converge, regardless of the WS used.

A multilevel in space and energy solver for multigroup diffusion eigenvalue problems

Abstract In this paper, we present a new multilevel in space and energy diffusion (MSED) method for solving multigroup diffusion eigenvalue problems. The MSED method can be described as a PI scheme with three additional features: (1) a grey (one-group) diffusion equation used to efficiently converge the fission source and eigenvalue, (2) a space-dependent Wielandt shift technique used to reduce the number of PIs required, and (3) a multigrid-in-space linear solver for the linear solves required by each PI step. In MSED, the convergence of the solution of the multigroup diffusion eigenvalue problem is accelerated by performing work on lower-order equations with only one group and/or coarser spatial grids. Results from several Fourier analyses and a one-dimensional test code are provided to verify the efficiency of the MSED method and to justify the incorporation of the grey diffusion equation and the multigrid linear solver. These results highlight the potential efficiency of the MSED method as a solver for multidimensional multigroup diffusion eigenvalue problems, and they serve as a proof of principle for future work. Our ultimate goal is to implement the MSED method as an efficient solver for the two-dimensional/three-dimensional coarse mesh finite difference diffusion system in the Michigan parallel characteristics transport code. The work in this paper represents a necessary step towards that goal.

Recent developments in the GENESIS code based on the Legendre polynomial expansion of angular flux method

Abstract This paper describes recent development activities of the GENESIS code, which is a transport code for heterogeneous three-dimensional geometry, focusing on applications to reactor core analysis. For the treatment of anisotropic scattering, the concept of the simplified Pn method is introduced in order to reduce storage of flux moments. The accuracy of the present method is verified through a benchmark problem. Next, the iteration stability of the GENESIS code for the highly voided condition, which would appear in a severe accident (e.g., design extension) conditions, is discussed. The efficiencies of the coarse mesh finite difference and generalized coarse mesh rebalance acceleration methods are verified with various stabilization techniques. Use of the effective diffusion coefficient and the artificial grid diffusion coefficients are found to be effective to stabilize the acceleration calculation in highly voided conditions.