Fission Source Convergence Research Articles

Improving source convergence diagnostics sensitivity using the complex components of Fourier fundamental mode

Previous works have demonstrated that the Fourier fundamental mode coefficient (FFMC) is a potential fission source convergence indicator for Monte Carlo k-eigenvalue calculations. Unlike the conventional Shannon entropy indicator, it requires no spatial meshing, and its computation is computationally inexpensive. However, we found that FFMC is less sensitive to the convergence involving near symmetrical fission distributions. To rectify the problem, we focus on the analysis of the FFMC’s components (the real and imaginary parts) on a simple geometry problem and a full reactor core problem. The usage of the FFMC’s components is more conservative compared to the original FFMC method as supported by the improved results of the fission source convergence detection.

Full Core Criticality Modeling of Gas-Cooled Fast Reactor using the SCALE6.0 and MCNP5 Code Packages

The Gas-Cooled Fast Reactor (GFR) is one of the reactor concepts selected by the Generation IV International Forum (GIF) for the next generation of innovative nuclear energy systems. It was selected among a group of more than 100 prototypes and his commercial availability is expected by 2030. GFR has common goals as the rest GIF advanced reactor types: economy, safety, proliferation resistance, availability and sustainability. Several GFR fuel design concepts such as plates, rod pins and pebbles are currently being investigated in order to meet the high temperature constraints characteristic for a GFR working environment. In the previous study we have compared the fuel depletion results for heterogeneous GFR fuel assembly (FA), obtained with TRITON6 sequence of SCALE6.0 with the results of the MCNPX-CINDER90 and TRIPOLI-4-D codes. Present work is a continuation of neutronic criticality analysis of heterogeneous FA and full core configurations of a GFR concept using 3-D Monte Carlo codes KENO-VI/SCALE6.0 and MCNP5. The FA is based on a hexagonal mesh of fuel rods (uranium and plutonium carbide fuel, silicon carbide clad, helium gas coolant) with axial reflector thickness being varied for the purpose of optimization. Three reflector materials were analyzed: zirconium carbide (ZrC), silicon carbide (SiC) and natural uranium. ZrC has been selected as a reflector material, having the best contribution to the neutron economy and to the reactivity of the core. The core safety parameters were also analysed: a negative temperature coefficient of reactivity was verified for the heavy metal fuel and coolant density loss. Criticality calculations of different FA active heights were performed and the reflector thickness was also adjusted. Finally, GFR full core criticality calculations using different active fuel rod heights and fixed ZrC reflector height were done to find the optimal height of the core. The Shannon entropy of the GFR core fission distribution was proved to be useful technique to monitor both fission source convergence (stationarity) and core eigenvalue convergence (keff) to fundamental eigenmode with MCNP5. All calculations were done with ENDF/B-VII.0 library. The obtained results showed high similarity with reference results.

Single-step Monte Carlo criticality algorithm

A single-step Monte Carlo criticality algorithm is proposed to improve the efficiency of Monte Carlo criticality calculation. This algorithm adopts the strategy of increasing neutron population cycle by cycle and achieves the highest computational efficiency through a weighted technique and mathematical optimization analysis. The inactive cycles are no longer needed, and all neutron histories contribute to the final results, so the fission source's convergence diagnosis is no longer needed. The algorithm's effectiveness is demonstrated with a single-rod model, an infinite medium problem, and the VERA benchmark problem #5 using a modified RMC code.

Dynamic mode decomposition application to dominance ratio assessment in Monte Carlo k-eigenvalue calculation

Dynamic mode decomposition (DMD) is applied to assess the dominance ratio (the ratio of the second-largest to the largest k-eigenvalue) from the fission source convergence during Monte Carlo power iterations. The fission source transition in each discretized tally region toward convergence is viewed as snapshots for DMD analysis. DMD is found to yield satisfactory results for the dominance ratio when various arbitrary parameters are selected, even for systems that have a dominance ratio that is very close to unity (∼0.999). The accuracy of the method depends on the parameters, especially in a system that has a low dominance ratio due to a reduced number of cycles before convergence. The spatial discretization of tally regions where fission sources accumulate can be coarse, in contrast to that in the fission matrix method, which is an advantage of the proposed method.

Sub-space iteration based Monte Carlo scheme for simultaneous computation of multiple dominant k-eigenmodes and acceleration of fission source iterations

Monte Carlo neutron transport codes are widely used for criticality simulation of full reactor core owing to their ability to model complex geometry in exact details, and of using the best available nuclear data. These codes are generally based on the Power Iteration (PI) method which is used to estimate only the fundamental eigenmode, and suffers from poor convergence rate specially when the dominance ratio is close to unity. Estimation of higher modes are required for various applications such as modal analysis, perturbation theory, etc. Computation of the dominance ratio is essential for determining the convergence rate of the power iterations. To overcome these issues, we have utilized a more robust Sub-Space Iteration (SSI) based Monte Carlo algorithm for simultaneous and efficient computation of multiple dominant eigenmodes for the general geometry full-core reactor problems. We have done an extensive investigation of the scheme on variety of problems with multi-group cross-section data, including the realistic full-core reactor benchmark (OECD/NEA 3D C5G7) to exemplify its validity, accuracy, and effectiveness. It was observed that the scheme works well for various kinds of systems, with good accuracy and has distinctly superior convergence properties than the PI method. The SSI based Monte Carlo scheme is built on top of the PI based neutron transport algorithm, and hence can be easily incorporated in the existing PI-based Monte Carlo codes for accelerating the fission source convergence and also for simultaneous computation of multiple eigenmodes.

Source extrapolation scheme for Monte Carlo fission source convergence based on RMC code

The source extrapolation method had been used to accelerate the convergence of fission sources in the deterministic methods. However, it has not been effectively used for the Monte Carlo criticality calculation. We apply the source extrapolation method to the Monte Carlo criticality calculation to accelerate the fission source convergence. An explicit and straightforward formula for determining the extrapolation factor is given based on the mathematical derivation and numerical calculations, and a new source extrapolation scheme for the Monte Carlo criticality calculation is proposed. This source extrapolation scheme is easily implemented and makes the fission sources converge twice as fast as the standard Monte Carlo criticality calculation. The validity and optimality of the source extrapolation scheme are tested by a single fuel rod model and the OECD source convergence benchmark. The results show that the new scheme can accelerate the fission source convergence.

Optimal Batch Size Growth for Wielandt Method and Superhistory Method

In a high dominance ratio system, the problem of slow fission source convergence is faced during the Monte Carlo criticality calculation. The Wielandt method and the Superhistory method have been proven to reduce the dominance ratio. Still, the Wielandt method and the Superhistory method have also proven to be unable to accelerate the convergence of the fission source. With the estimation of errors in the cumulative fission source and the batch size optimization methodology, the optimal batch size growth for the Wielandt method and the Superhistory method is proposed. Compared with the direct simulation with the optimal batch size growth, the Wielandt simulation and the Superhistory simulation better use the optimal batch size growth. A single fuel rod model was tested, and the results show that the new method is helpful for the acceleration of fission source convergence.

Numerical evidence on the formulation of the Fourier fundamental mode coefficient as a source convergence indicator

In our previous work, it is theoretically proven that Fourier Fundamental Mode Coefficient (FFMC) is a potential fission source convergence indicator for the application of the Monte Carlo k-eigenvalue calculation. The advantage of using this novel approach is that its computation does not require spatial discretization. Thus, it eliminates the burden of sorting fission sites into their respective bin and also avoids the subjectivity of choosing the number of spatial bins as imposed by the calculation of the Shannon entropy indicator. Unfortunately, the theoretical formulation of FFMC as a source convergence indicator is purely based on mathematical deduction. The mathematical theorems used in the formulation of FFMC is still yet to be verified numerically via Monte Carlo k-eigenvalue calculation. In this paper, we provide the numerical evidence that justifies the formulation of FFMC. The outcome achieved from this work confirms that the theoretical formulation of FFMC is valid.

Forced propagation method for Monte Carlo fission source convergence acceleration in the RMC

In loosely coupled or large-scale problems with high dominance ratios, slow fission source convergence can take extremely long time, reducing Monte Carlo (MC) criticality calculation efficiency. Although various acceleration methods have been developed, some methods cannot reduce convergence times, whereas others have been limited to specific problem geometries. In this study, a new fission source convergence acceleration (FSCA) method, the forced propagation (FP) method, has been proposed, which forces the fission source to propagate and accelerate fission source convergence. Additionally, some stabilization techniques have been designed to render the method more practical. The resulting stabilized method was then successfully implemented in the MC transport code, and its feasibility and effectiveness were tested using the modified OECD/NEA, one-dimensional slab benchmark, and the Hoogenboom full-core problem. The comparison results showed that the FP method was able to achieve efficient FSCA.

ENHANCING THE PERFORMANCE OF FISSION SOURCE CONVERGENCE WITH FUNCTIONAL EXPANSION TALLIES IN SERPENT 2 MONTE CARLO CODE

This paper presents an upgrade to the built-in response matrix based solver implemented in Serpent 2 Monte Carlo code aiming to improve the fission source convergence when obtaining the forward solution to the k-eigenvalue criticality source problems. The functional expansion tallies are introduced in an attempt to improve the accuracy of the cell-wise form factors that feed the response matrix solver, replacing the current mesh-based approach. The functional expansion tallies reconstruct the binning surface and collision tallies, by using high-order series expansion to represent the original and continuous spatial distributions. This new feature is implemented to Serpent 2 and tested by single-assembly and full-core PWR calculations (BEAVRS benchmark). The results show enhanced performance of the convergence acceleration methodology based on an improved initial guess of the fission source.

Acceleration method of fission source convergence based on RMC code

To improve the efficiency of MC criticality calculation, an acceleration method of fission source convergence which gives an improved initial fission source is proposed. In this method, the MC global homogenization is carried out to obtain the macroscopic cross section of each material mesh, and then the nonlinear iterative solution of the SP3 equations is used to determine the fission source distribution. The calculated fission source is very close to the real fission source, which describes its space and energy distribution. This method is an automatic computation process and is tested by the C5G7 benchmark, the results show that this acceleration method is helpful to reduce the inactive cycles and overall running time.

An efficient Monte Carlo fission source convergence acceleration strategy adapted from the survival biasing technique

The current effective methods for accelerating fission source convergence in criticality calculation such as the super-history powering and Wielandt method are proven to reduce the dominance ratio. As a result, source convergence can be achieved with a smaller number of iteration cycles. However, none of these methods could improve the convergence time in most of the cases, despite the reduced number of iteration cycles. In particular, the conventional source acceleration mechanism adapted in the super-history powering and Wielandt method involve banking fission neutrons for later tracking. Here, reactive fission sites are only stored during tracking higher-generation fission neutrons. We postulated that the conventional acceleration strategy is computationally inefficient and will consequently result in a remarkable increase in the convergence time. To rectify the problem, we present an acceleration strategy which is adapted from the survival biasing technique. In this work, the new strategy is targeted to increase the chance of searching for reactive high-generation fission sites whilst eliminating most of the computational cost of simulating neutrons that are not able to survive until higher generation fission events. Numerical verification results show that the new method is feasible and effective, which can save up to 87% of the convergence time according to the selected acceleration parameter.

Acceleration of fission source convergence in the Serpent 2 Monte Carlo code using a response matrix based solution for the initial source distribution

This paper presents a new response matrix based solver implemented in the Serpent 2 Monte Carlo code for the purpose of producing an improved initial guess to accelerate source convergence in criticality source simulations. The solver obtains coupling coefficients required for the response matrix solution from Monte Carlo simulations, and provides a spatial distribution that approximates the converged fission source. The implemented methodology is demonstrated by single-assembly and full-core PWR calculations. The results show that the improved initial guess leads to faster source convergence in terms of both inactive cycles and overall running time.

Stability of Monte Carlo k-eigenvalue simulations with CMFD feedback

In this paper we perform a Fourier stability analysis of MC-CMFD, a hybrid Monte Carlo k-eigenvalue method that utilizes coarse mesh finite difference (CMFD) feedback. The MC-CMFD method is nonlinear and contains random statistical errors; both of these features are inconsistent with the direct application of a Fourier stability analysis. To accomplish this analysis, we first formulate a non-random iteration method that approximates MC-CMFD, by assuming an infinite number of Monte Carlo particles per cycle. Then we (i) linearize this method, and (ii) Fourier-analyze the linearized method to theoretically predict its convergence properties. Finally, we demonstrate by direct numerical simulations that the Fourier analysis of the linearized non-random method accurately predicts the stability and fission source convergence rate (during the inactive cycles) of the original nonlinear MC-CMFD method. We do this by comparing the predictions of the Fourier analysis to simulations that utilize (i) a high-fidelity SN-CMFD code (which has no random statistical errors), and (ii) a MC-CMFD code (which has random statistical errors). The Fourier analysis and our two test codes confirm that the MC-CMFD method is stable if the optical thickness of the coarse grid (in the low-order CMFD calculation) is sufficiently small. However, the spectral radius increases monotonically with the coarse grid size, and if the latter exceeds a critical value, the MC-CMFD method becomes unstable. We discuss some implications of our results for practical MC-CMFD simulations.

Monte Carlo criticality calculations accelerated by a growing neutron population

We propose a fission source convergence acceleration method for Monte Carlo criticality simulation. As the efficiency of Monte Carlo criticality simulations is sensitive to the selected neutron population size, the method attempts to achieve the acceleration via on-the-fly control of the neutron population size. The neutron population size is gradually increased over successive criticality cycles so that the fission source bias amounts to a specific fraction of the total error in the cumulative fission source. An optimal setting then gives a reasonably small neutron population size, allowing for an efficient source iteration; at the same time the neutron population size is chosen large enough to ensure a sufficiently small source bias, such that does not limit accuracy of the simulation.

A general solution strategy of modified power method for higher mode solutions

A general solution strategy of the modified power iteration method for calculating higher eigenmodes has been developed and applied in continuous energy Monte Carlo simulation. The new approach adopts four features: 1) the eigen decomposition of transfer matrix, 2) weight cancellation for higher modes, 3) population control with higher mode weights, and 4) stabilization technique of statistical fluctuations using multi-cycle accumulations. The numerical tests of neutron transport eigenvalue problems successfully demonstrate that the new strategy can significantly accelerate the fission source convergence with stable convergence behavior while obtaining multiple higher eigenmodes at the same time. The advantages of the new strategy can be summarized as 1) the replacement of the cumbersome solution step of high order polynomial equations required by Booth's original method with the simple matrix eigen decomposition, 2) faster fission source convergence in inactive cycles, 3) more stable behaviors in both inactive and active cycles, and 4) smaller variances in active cycles. Advantages 3 and 4 can be attributed to the lower sensitivity of the new strategy to statistical fluctuations due to the multi-cycle accumulations. The application of the modified power method to continuous energy Monte Carlo simulation and the higher eigenmodes up to 4th order are reported for the first time in this paper. Modified power method is applied to continuous energy Monte Carlo simulation.Transfer matrix is introduced to generalize the modified power method.All mode based population control is applied to get the higher eigenmodes.Statistic fluctuation can be greatly reduced using accumulated tally results.Fission source convergence is accelerated with higher mode solutions.

Large-scale Monte Carlo neutron transport calculations with thermal hydraulic feedback

The Monte Carlo method provides the most accurate description of the particle transport problem. The criticality problem is simulated by following the histories of individual particles without approximating the energy, angle or the coordinate dependence. These calculations are usually done using homogeneous thermal hydraulic conditions. This is a very crude approximation in the general case. In this paper, the method of internal coupling between neutron transport and thermal hydraulics is presented. The method is based on dynamic material distribution, where coordinate dependent temperature and density information is supplied on the fly during the transport calculation. This method does not suffer from the deficiencies characteristic of the external coupling via the input files. In latter case, the geometry is split into multiple cells having distinct temperatures and densities to supply the feedback. The possibility to efficiently simulate large scale geometries at pin-by-pin and subchannel level resolution was investigated. The Wielandt shift method for reducing the dominance ratio of the system and accelerating the fission source convergence was implemented. During the coupled iteration a detailed distribution of the fission heat deposition is required by the thermal hydraulics calculation. Providing reasonable statistical uncertainties for tallies having large numbers of bins, is a complicated task. This problem was resolved by applying the Uniform Fission Site method. Previous investigations showed that the convergence of the coupled neutron transport/thermal hydraulics calculation is limited by the statistical uncertainty and exhibits strong nonuniform behavior. The stochastic approximation scheme was used to stabilize the convergence. In combination with the Uniform Fission Site method, uniform convergence was achieved.

A New Approach to Check and Diagnose the Fission Source Convergence in Monte Carlo Criticality Calculations

In order to avoid calculation bias and increase calculation efficiency, convergence of the fission source distribution (FSD) in Monte Carlo simulations is important. Numerous analysis methods have been developed and used for checking the fission source convergence. However, such schemes have low applicability to Monte Carlo codes or give low diagnostic accuracy. To address these limitations, a method to verify the fission source convergence using the average and standard deviation of fission source positions is proposed. To collect the fission site information, the MCNP5 code was modified, and position information was extracted. The accuracy and advantages of the proposed method were verified by solving Organisation for Economic Co-operation and Development/Nuclear Energy Agency benchmark problems with the modified MCNP5 code and comparing the results to those obtained with the Shannon entropy and the nine center distance sum approaches. The analysis shows that the proposed method has good applicability and exhibits high accuracy for verifying the convergence of the FSD. It is expected that the proposed scheme will be a valuable contribution to the field of the fission source convergence.

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.

Asymptotic Wielandt Method and Superhistory Method for Source Convergence in Monte Carlo Criticality Calculation

In loosely coupled systems and large-scale systems, Monte Carlo criticality calculation suffers from slow fission source convergence because of the high dominance ratio (DR). In previous work, the Wielandt method and the superhistory method have been separately proposed to accelerate source convergence. However, although both methods decrease the DR, they are found not able to sufficiently accelerate fission source convergence. In this paper, the effective DR is defined and used to analyze the effectiveness of the Wielandt method and the superhistory method and to theoretically prove that they cannot reduce the computational time to converge the fission source. Accordingly, both methods are modified by adjusting the source population of inactive cycles, and their efficiency after adjustment is also compared. Moreover, the asymptotic Wielandt method (AWM) and the asymptotic superhistory method (ASM) are proposed, and the rules of deciding asymptotic parameters are also discussed. The new methods are implemented into the RMC code and validated by calculating loosely coupled problems and large-scale problems. Numerical calculation results show that AWM and ASM are practical and efficient for source convergence acceleration, which can save 75% to 90% of the computational time to reach a converged fission source.