Whole-core Transport Calculations Research Articles

Stability and Performance of the X-CMFD Method for Multiphysics Reactor Calculations

This paper presents an innovative approach to efficiently perform deterministic direct whole-core transport calculations with multiphysics feedback for steady-state problems. Traditionally, Picard iteration combined with coarse mesh finite difference (CMFD) acceleration has been used, but it can suffer from instability and inefficiency in certain scenarios. In this work, we introduce the X-CMFD method, supported by Fourier analysis, to enhance the stability of the multiphysics iteration scheme. A new and efficient variation of the X-CMFD method for practical simulations is also present. Additionally, we explore the theoretical convergence rates of nonlinear fully coupled diffusion acceleration (NFCDA), a class of diffusion acceleration methods that formalizes similar ideas of previous research. NFCDA uses a low-order diffusion problem that is fully coupled with equivalent nonlinear multiphysics feedback to accelerate the high-order transport problem with feedback. The theoretical analysis shows that NFCDA offers similar convergence rates to nonlinear diffusion acceleration (NDA) in problems without feedback. This provides theoretical support for numerical experiments conducted by other researchers. X-CMFD, which is a discretized form of NFCDA, leverages typical coarse mesh concepts and operators from CMFD while applying feedback to cross sections in the low-order diffusion problem at each power iteration of the low-order problem. To reduce computational costs, we optimize the implementation of X-CMFD in MPACT by introducing an equivalent low-order approximation to the cross-section updates in the nonlinear low-order problem. Numerical results from pressurized water reactor problems demonstrate that X-CMFD, along with its practical implementation, outperforms current relaxed Picard iteration methods in terms of robustness and efficiency, irrespective of the presence of multiphysics feedback.

High-fidelity neutronics simulation of channel-type molten salt reactors

As one of the generation IV reactors, molten salt reactors (MSRs) have attracted much attention in recent years. Generally, molten salt acting as both fuel and coolant circulates through graphite channels and external primary loop in a liquid-fueled MSR with a channel-type core. To accurately simulate the neutronics of channel-type MSRs, a GPU paralleled high-fidelity neutron transport code ThorMOC has been improved by implementing a 3D rasterized coarse mesh finite difference (RCMFD) method and a quasi-2D delayed neutron precursor (DNP) drift model. A Molten Salt Reactor Experiment (MSRE) model and a small Thorium-based MSR (sTMSR) model with stationary and flowing fuel are used for the verification of the improved ThorMOC. The reference results of the two stationary reactor models are obtained from Monte Carlo code OpenMC, which indicates that ThorMOC performs a high accuracy for stationary MSR simulations. The reactivity loss caused by fuel flowing in the steady state MSRE is calculated by ThorMOC and agrees well with the measured value from the MSRE. Moreover, the influences of geometry simplifications of fuel channels and downcomer on the reactivity loss are evaluated by comparing their results with those of the high-fidelity model. The reactivity loss in the sTMSR core is computed, and the effects of fuel velocity in core channels and residence time outside the core on the reactivity loss are also analyzed. The whole-core transport calculations of the 3D MSRE and sTMSR core models with 8 energy groups cost several hundred seconds.

Development of Uncertainty Analysis Techniques for the Fission Matrix–Based Neutron Transport Code RAPID

The combined fission matrix (CFM) method is a newly developed neutron transport theory. This method estimates the fission matrix of the reactor core or spent fuel pool by combining a set of database fission matrices. The RAPID neutron transport code based on the CFM routine was developed originally for the spent fuel storage system and has been applied to the reactor core calculation in recent years. It can perform high-fidelity whole-core transport calculations within minutes. However, since the fission matrix database is obtained from Monte Carlo calculations, the uncertainty in the fission matrix will inevitably pass to its eigenvalue and eigenvector. The RAPID code also uses the fission matrix homogenization and interpolation techniques to further improve the calculation efficiency. Therefore, it is difficult to establish a relationship between the fission matrix elements’ uncertainty and the resulting eigenvalue and eigenvector uncertainties. This paper proposes two uncertainty analysis methods to obtain the eigenvalue and eigenvector uncertainties. The fission matrix resampling method resamples the database fission matrix elements according to each individual uncertainty. This method could generate many fission matrix databases at little additional costs and analyze the eigenvalue and eigenvector uncertainties from these resampled fission matrix coefficients. The analog uncertainty analysis method predicts the eigenvalue uncertainty from the uncertainty of the total fission rate in a fixed-source calculation, which yields a fission matrix column. Both uncertainty analysis methods have been validated against the reference brute-force calculations on a single-pin model and the BEAVRS whole-core model. It shows that the fission matrix resampling method could well estimate the uncertainties in the fission matrix eigenvalue and eigenvector. The analog uncertainty analysis method can accurately predict the eigenvalue uncertainty, which provides a guideline for the number of neutron histories simulated per fixed-source calculation.

The Application of the Combined Fission Matrix Theory in Fast Reactors

The combined fission matrix theory is a recently-developed hybrid neutron transport method. It features high efficiency, fidelity, and resolution whole-core transport calculation. The theory is based on the assumption that the fission matrix element ai,j is dominated by the property of the destination cell i. This assumption can be well explained in thermal reactors, and the combined fission matrix method has been validated in a series of thermal neutron system benchmarks. This work examines the feasibility of the combined fission matrix theory in fast reactors. The European Sodium Fast Reactor is used as the numerical benchmark. Compared to the Monte Carlo method, the combined fission matrix theory reports a 64 pcm keff difference and 8.3% 2D RMS error. The error is much larger than that in thermal reactors, and the correction ratio cannot significantly reduce the material discontinuity error in fast reactors. Overall, the combined fission matrix theory is more suited for thermal reactor transport calculations. Its application in fast reactors needs further developments.

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.

A quasi-3D direct transport method in the direct transport code KuaFu

The 2D/1D transport method is suffered from the stability issues, which affects the application in direct whole-core transport calculation for advanced nuclear reactors. The root of stability issues in the 2D/1D method is researched. To overcome the bottleneck of the stability issues, a quasi-3D method with axial difference relationship is derived, developed and analyzed based on the 2D/1D method. The quasi-3D method is applied in a newly-developed direct transport code KuaFu. Verification of KuaFu is conducted by several benchmarks, such as the C5G7 benchmark. All numerical results show the good accuracy of the quasi-3D method in KuaFu.

Overlapping communication and computation of GPU/CPU heterogeneous parallel spatial domain decomposition MOC method

GPU/CPU heterogeneous parallel calculation has been widely used and great performance has been achieved. Recently it also has been introduced into whole-core transport calculation. Considering the ability to solve the large-scale problem, the spatial domain decomposition (SDD) technique has been applied in the whole-core transport application. However, the communication burden becomes an important bottleneck in heterogeneous parallel SDD calculation and the parallel efficiency decreases dramatically when the number of the calculation domains increasing. GPU brings great acceleration meanwhile the communication time doesn’t change, which makes the parallel efficiency of the heterogeneous parallel calculation worse. In this work, heterogeneous parallel SDD calculation is performed by applying both the spatial domain decomposition and ray parallelization and the parallel performance is analyzed. In addition, a new scheme of the communication is implemented to overlap the MPI communication time and the neutron transport sweep on GPU. In order to accomplish this scheme, both MPI, CUDA protocols are introduced. Numerical results show that the new scheme successfully hide the communication time and the data copy time between GPU and CPU, which brings significant improvement of the parallel efficiency. With the new communication scheme, the large-scale whole-core transport calculation is more practicable.

The material-region-based 2D/1D transport method

The 2D/1D transport method is the dominant method for high-fidelity direct whole-core transport calculations, which attracts a lot of attention in recent decades. In the 2D/1D method, some sources of deviation are introduced, including spatial and angular approximation of leakage term and cross-section homogenization for 1D axial calculation. These approximations are analyzed and a material-region-based 2D/1D transport method with anisotropic leakage term, which avoids cross-section homogenization, as well as reduces spatially flat leakage approximation, are developed to improve accuracy at the expense of acceptable memory and efficiency loss. Finally, a BWR assembly case, the C5G7 benchmark and a rod-cluster assembly case are tested to verify the accuracy and performance of the material-region-based 2D/1D method.

Continuous energy coarse mesh transport (COMET) method

The coarse mesh radiation transport method COMET developed for reactor physics calculations originally treated the phase space variable continuously except for the energy variable. In this work, the method is extended to also treat the energy variable continuously. COMET uses incident flux response expansion coefficients that are generated from local (oarse mesh) fixed source calculations for obtaining the transport solution to whole core problems. In the new method, the energy dependence of the angular flux (spectrum) is factorized into a product of two functions: a strongly varying asymptotic function and a slowly varying (smooth) shape function representing the deviation from the asymptotic function. The method uses spectra from a single assembly or color set calculation as the asymptotic functions and the shape function is expanded by the 0th order B-splines in a set of coarse energy bins. This new factorization allows the use of continuous energy stochastic method for generating the incident flux response expansion coefficients for solving whole-core problems. The new COMET method was tested in a PWR benchmark problem in whole core configurations using UO2 and MOX fuels. It was found that the COMET results agree very well with the corresponding continuous energy Monte Carlo reference solutions. The difference between the COMET and MCNP eigenvalues varied from 29 to 65 pcm, while the mean relative difference in the pin fission density distributions computed by COMET and MCNP was in the range of 0.25%–0.63% for three (namely, controlled, uncontrolled and some-rods-in) core configurations. This difference is less than 2 standard deviations of the MCNP uncertainty. The computational efficiency of the continuous energy COMET was found to be consistent with the multigroup COMET (i.e., more than 4 orders of magnitude faster than MNCP). Based on the benchmark results, it can be concluded that the new continuous energy expansion theory allows COMET to perform high fidelity and highly efficient whole-core transport calculations in heterogeneous reactors.

Evaluation of Pin-Cell Homogenization Techniques for PWR Pin-by-Pin Calculation

In general, spatial homogenization, energy group condensation, and angular approximation are all included in the homogenization process. For the traditional pressurized water reactor (PWR) two-step calculation, the assembly homogenization with assembly discontinuity factors plus two-group (2G) neutron diffusion calculation have been proved to be a very efficient combination. However, this changes and becomes unsettled for the pin-by-pin calculation. Thus, this paper evaluates pin-cell homogenization techniques by comparison with the two-dimensional one-step whole-core transport calculation. For the homogenization, both the generalized equivalence theory (GET) and the superhomogenization (SPH) methods are studied. Considering the spectrum interference effect between different types of fuel pin cells, both 2G and 7-group (7G) structures are condensed from the 69-group WIMS-D4 library structure. For practical reactor core applications, the low-order angular approximations, including the diffusion and the SP3 methods, are compared with each other to determine which one is accurate enough for the PWR pin-by-pin calculation. Numerical results have demonstrated that both the GET and the SPH methods work effectively in pin-cell homogenization. In consideration of the spectrum interference effect, the 7G structure is sufficient for the pin-by-pin calculation. Compared with the diffusion method, the SP3 method can decrease the errors dramatically.

Nonoverlapping local/global iterations with 2-D/1-D fusion transport kernel and p-CMFD wrapper for transient reactor analysis—II: Parallelization and Predictor–Corrector Quasi-Static method application

As modern computing power grows, whole-core transport calculations become more viable with parallel computing architectures. Nonoverlapping local/global (NLG) iterative method has been recently developed for the whole-core transport calculation. The NLG iteration adopts the 2-D/1-D fusion transport kernel as the local solver, and the p-CMFD equation is used as the global wrapper. The NLG iteration is capable of solving 3-D transient heterogeneous problems, and it is naturally parallelizable. In this study, the computationally-expensive local problems in the NLG iteration are parallelized by MPI protocol, and the Predictor–Corrector Quasi-Static (PCQS) method is applied to the transient calculations to reduce the computing time further. The parallelized NLG iteration and the PCQS method have been implemented in an in-house code, CRX-2K. Several numerical problems are computed, and the numerical results reveal that the parallelized NLG iteration has high potential in the realistic whole-core transport calculation, and the computing time in the transient calculations can be reduced by the PCQS method if the reactivity behaves linearly in time for a given macro time-step size.

Physical probability table applied to response matrix-collision probability method for direct whole-core transport calculations

In this study, we applied physical probability table to response matrix-collision probability method to treat and evaluate self-shielded resonance cross-sections for whole-core transport calculations. To attain accurate results, especially in resonance treatment, the conservation laws of probabilities are derived and satisfied using HELIOS normalization scheme. The subgroup weights are efficiently evaluated using the conjugate gradient method based on the normal equations (CGNR). The resonance interference effects are considered. We used Segev method to interpolate the resonance integrals (RIs), in order to accurately obtain the subgroup weights. Finally, a validation is presented where the effective absorption and production cross-sections and the infinite multiplication factors are compared with exact values obtained using NJOY and the equivalence theory for homogeneous media. Other results, such as the Rowland’s benchmark for three assembly production cases, are also presented and compared with those produced by Monte Carlo method. The results indicate the developed method can accurately represent the self-shielding effects.

3D/2D rotational plane slicing method for 3D whole-core transport calculation

The rotational plane slicing (RPS) method is proposed for 3-D whole-core neutron transport calculation. This method takes a view of the 3-D structure with a set of vertical planes that are obtained by slicing along the characteristic lines. In a 3-D structure with a simple axial configuration, the sliced planes consist of rectangular cells. Each sliced plane is then analyzed quite easily with a 2-D transport discrete ordinate solver. Several numerical tests are performed and the results indicate that the RPS method reproduces the reference solution with good accuracy.

A nonoverlapping local/global iterative method with 2-D/1-D fusion transport kernel and p-CMFD wrapper for transient reactor analysis

As modern computing power grows, whole-core transport calculations become more viable. However, the computing time and memory requirement remain major burdens. As a candidate for whole-core transport calculation methods, nonoverlapping local/global (NLG) iteration has recently been developed.In this study, the NLG iteration method is extended to make it capable of transient calculations of 3-D heterogeneous problems. It is then implemented in an in-house code, CRX-2K. Transient NLG iteration uses the 2-D/1-D fusion method as a local transport kernel, with the transient p-CMFD equation adopted as a global wrapper. In addition, a neighboring spectral index (NSI) weighting method is suggested as a tool for correcting the rod cusping phenomenon for rod ejection problems.Four problems, including a 3-D heterogeneous problem, are computed. Numerical results show that the NLG iteration converges to the reference solution obtained by means of the whole-core p-CMFD acceleration, and the NSI weighting method is accurate to correct the rod cusping phenomenon. The NLG iteration in present implementation takes more computing times (still less than 2times) than the whole-core p-CMFD acceleration, but has great potential in parallelization.

Hybrid space–angle adaptivity for whole-core particle transport calculations

Adaptive refinement is a powerful method for efficiently solving physical problems. In this paper we present a new coupled space–angle adaptive algorithm for neutron transport calculations. The scheme is specifically employed for the solution of the integral form of transport equation based on the collision probability–response matrix method. The adaptive algorithm is started by first applying angular adaptivity and then projecting the solution to the spatial mesh refinement. A posteriori error estimate is derived by utilizing the flux gradient approach based on the net current leakage of nodes. A new approach is used to apply continuity of flux in the interface between nodes by escalating the order of spherical harmonics expansions of entrance response matrix to the same order of spherical harmonics expansions of outgoing angular flux at the neighboring node. Using an integral transport method within the node and refined space and angle variables, a new method for whole-core transport calculations is introduced. The validity of the developed adaptive strategy is assessed by a series of numerical experiments. Comparisons indicate that the space–angle adaptivity framework is capable of resulting acceptable solution with less number of the degrees of freedom (DOFs).

High-Order Perturbation Theory for Incident Flux Response Expansion Methods

In this work, a high-order perturbation method is developed to generate/compute response functions for the coarse mesh transport (COMET) method, which provides whole-core neutron transport solutions to various reactor core types. In this approach, the response functions are first generated at a reference eigenvalue point, and the response functions at an arbitrary eigenvalue are computed as a high-order perturbation from the reference point. The method has been tested at both the lattice level (response function generation) and the core level (whole-core transport calculations). At the lattice level, it is found that the response functions predicted by the perturbation method agree very well with those directly computed by the Monte Carlo method. The average relative difference in the surface-to-surface response functions is 0.29% to 0.46% when the eigenvalue k ranges from 0.6667 to 1.5. In whole-core transport calculations, the COMET calculations using the high-order perturbation method are almost identical to those using the interpolation method. The eigenvalue, assembly, and pin fission densities predicted by COMET agree very well with the MCNP reference solution. This indicates that the high-order perturbation response function generation method can achieve the same accuracy as the interpolation method while significantly improving the computational efficiency of the precomputation phase.

PROSPECTS IN DETERMINISTIC THREE-DIMENSIONAL WHOLE-CORE TRANSPORT CALCULATIONS

The point we made in this paper is that, although detailed and precise three-dimensional (3D) whole-core transport calculations may be obtained in the future with massively parallel computers, they would have an application to only some of the problems of the nuclear industry, more precisely those regarding multiphysics or for methodology validation or nuclear safety calculations. On the other hand, typical design reactor cycle calculations comprising many one-point core calculations can have very strict constraints in computing time and will not directly benefit from the advances in computations in large scale computers. Consequently, in this paper we review some of the deterministic 3D transport methods which in the very near future may have potential for industrial applications and, even with low-order approximations such as a low resolution in energy, might represent an advantage as compared with present industrial methodology, for which one of the main approximations is due to power reconstruction. These methods comprise the response-matrix method and methods based on the two-dimensional (2D) method of characteristics, such as the fusion method.

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.

06/01977 Two-dimensional whole-core transport calculations of the OECD benchmark problem C5G7 MOX by the CRX code: Lee, G. S. and Cho, N. Z. Progress in Nuclear Energy, 2004, 45, (2–4), 169–177

06/01977 Two-dimensional whole-core transport calculations of the OECD benchmark problem C5G7 MOX by the CRX code: Lee, G. S. and Cho, N. Z. Progress in Nuclear Energy, 2004, 45, (2–4), 169–177

Multigroup multi-layer models of neutron reflection and transmission for reactor transport calculations with anisotropic scattering

In this article, we extend the one-speed multi-layer models to neutron reflection and transmission developed in our earlier work (de Abreu, M.P., 2005. Multi-layer models to neutron reflection and transmission for whole-core transport calculations, Annals of Nuclear Energy 32, 215) to multigroup transport theory. We begin by considering a two-layer boundary region, and we develop for such a region discrete ordinates models to the diffuse reflection and transmission of neutrons for multigroup nuclear reactor core problems with anisotropic scattering. We perform numerical experiments to show that our models to neutron reflection and transmission can be used to replace efficiently and accurately two nonactive boundary layers in whole-core transport calculations. We conclude this article with an inductive extension of our two-layer results to a boundary region with an arbitrary number of layers.