Wielandt Method Research Articles

Improved Wielandt method with real-time error tolerance based on NLSP3 code

The Wielandt method reduces the number of outer iterations but increases the number of inner iterations. The number of calculated inner iterations should be reduced to improve the efficiency of the Wielandt method, and then the error tolerance for the Wielandt inner iteration should be larger. We give a new formula to calculate the error tolerance and achieve the real-time adjustment of the error tolerance for the Wielandt inner iteration. The improved Wielandt method is implemented in NLSP3 code, and tested by the 2D-IAEA benchmark and the 3D-IAEA benchmark. The results show that the improved Wielandt method can reduce the total number of calculated inner iterations, thus improving the efficiency of the Wielandt method.

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.

Convergence characteristics and Wielandt acceleration of the time source method for Monte Carlo alpha eigenvalue calculations

This paper discusses the source convergence of Monte Carlo calculations for α-eigenvalues. Compared with the conventional “k–α method”, the “time-source method” has a strong neutronic coupling even in a system where the fission sources are loosely coupled. The dominance ratio of the time source is smaller than that of the k–α method. The time-source method has a drawback in that it requires a much longer computation time. Furthermore, the time-source method cannot be applied to critical or supercritical systems owing to the long fission chain. A new technique is developed to overcome these drawbacks by introducing the Wielandt method for increasing the dominance ratio. This technique, which uses the Wielandt method in an opposite manner, reduces the computation time and expands the availability in a supercritical system. Because the source convergence of the time-source method is inherently stable, the new method causes a limited adverse effect on the source convergence.

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.

Calculation of Fullerene Parameters by the Implemented One-Dimensional Method for Determination of Eigenvalues and Eigenfunctions in One-Dimensional Clusters of Planar, Cylindrical, and Spherical Geometry

The eigenvalues and eigenfunctions for clusters with planar, cylindrical, and spherical geometry with arbitrary potential energy profiles were calculated by means of an implemented algorithm. The results of numerical and analytical solutions for clusters of various geometry were compared. The proposed algorithm for determination of cluster eigenvalues and eigenfunctions shows a power law rate of convergence of the solution towards the target eigenfunction coinciding with the rate of convergence in the modified Wielandt method. This algorithm was used to calculate the geometrical potential of giant fullerene as a function of radius for the state with l = 0. The numerical results are in good agreement with the theoretical results.

Memory-efficient calculations of adjoint-weighted tallies by the Monte Carlo Wielandt method

The current Monte Carlo (MC) adjoint-weighted tally techniques based on the iterated fission probability (IFP) concept require a memory amount which is proportional to the numbers of the adjoint-weighted tallies and histories per cycle to store history-wise tally estimates during the convergence of the adjoint flux. Especially the conventional MC adjoint-weighted perturbation (AWP) calculations for the nuclear data sensitivity and uncertainty (S/U) analysis suffer from the huge memory consumption to realize the IFP concept. In order to reduce the memory requirement drastically, we present a new adjoint estimation method in which the memory usage is irrelevant to the numbers of histories per cycle by applying the IFP concept for the MC Wielandt calculations. The new algorithms for the adjoint-weighted kinetics parameter estimation and the AWP calculations in the MC Wielandt method are implemented in a Seoul National University MC code, McCARD and its validity is demonstrated in critical facility problems. From the comparison of the nuclear data S/U analyses, it is demonstrated that the memory amounts to store the sensitivity estimates in the proposed method become negligibly small.

Computing eigenvalue sensitivity coefficients to nuclear data by adjoint superhistory method and adjoint Wielandt method implemented in RMC code

In the previous work, the continuous-energy Reactor Monte Carlo code RMC has been developed with the capability of calculating eigenvalue sensitivity coefficients with regard to nuclear data based on the iterated fission probability (IFP) method. By applying the physical meaning of adjoint flux, the IFP method computes adjoint-weighted reaction rates directly in the forward transport calculations without any discretization in space, energy and angle. The IFP method is accurate theoretically, however, it may produce poor sensitivity coefficients in practice since its memory requirements are proportional to the number of particle histories in each Monte Carlo cycle and the huge memory requirement further restricts the number of particles due to the limitation of the computer memory capability. To overcome this drawback of the IFP method, the adjoint Wielandt method and the adjoint superhistory method are investigated. Both methods allow computing sensitivity coefficients within a single particle history and thus provide the possibility of reducing memory consumption. Furthermore, they preserve the merits of the IFP method such as no need for any discretization. The accuracy as well as the computational performance of the adjoint Wielandt method and the adjoint superhistory method are compared with the IFP method through several cases such as a multi-group infinite-medium problem, an assembly-level problem and a full-core PWR problem.

ICONE23-1814 AUTOMATIC OPTIMIZED WIELANDT'S ACCELERATION METHOD OF SOURCE CONVERGENCE FOR MONTE CARLO CRITICALITY CALCULATION

Slow source convergence of Monte Carlo criticality calculation is a significant problem. Several acceleration techniques have been proposed, whereas Wielandt's method is an efficient method. However a bad estimated eigenvalue (k_e)which is selected often according experience makes the calculation time consuming. In this study, automatic optimized Wielandt's acceleration method has been developed to get optimized estimated eigenvalue based on dominance ratio (DR) for improving the efficiency of source convergence and implemented in Super Monte Carlo Simulation Program for Nuclear and Radiation Process (SuperMC). Numerical calculated results of 5x5x5 array of metal spheres and Hoogenboom benchmark showed that automatic optimized Wielandt's acceleration method can enhance the convergence efficiency.

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.

Tally efficiency analysis for Monte Carlo Wielandt method

Abstract The Monte Carlo Wielandt method has the potential to eliminate most of a variance bias because it can reduce the dominance ratio by properly controlling the estimated eigenvalue ( k e ). However, it requires increasingly more computation time to simulate additional fission neutrons as the estimated eigenvalue becomes closer to the effective multiplication factor ( k eff ). Therefore, its advantage over the conventional Monte Carlo (MC) power method in the calculation efficiency may not always be ensured. Its efficiency of the tally estimation needs to be assessed in terms of a figure of merit based on a real variance as a function of k e . In this paper, the real variance is estimated by using an inter-cycle correlation of the fission source distribution for the MC Wielandt calculations. Then, the tally efficiency of the MC Wielandt method is analyzed for a 2 × 2 fission matrix system and weakly coupled fissile array problems with different dominance ratios (DRs). It is shown that the tally efficiency of the MC Wielandt method depends strongly on k e , there is a k e value resulting in the best efficiency for a problem with a large DR, and the efficiency curve as a function of L, the average number of fission neutrons per history, follows a long tail after the best efficiency.

Another Look at Sylow's Third Theorem

This result is among the arsenal of tools that every first year algebra student obtains. A group where the order of every element is a power of p is called a p-group; a p-Sylow subgroup of G is a p-subgroup of G of maximal order The idea of Sylow's proof, which was originally stated in terms of permutation groups, is to look at the size of the equivalence classes obtained when all p-Sylow subgroups of G are conjugated by the elements of a fixed p-Sylow subgroup of G. The existence of a p-Sylow subgroup was needed for the proof of the third Sylow theorem, although the conclusion of the theorem certainly implies that there are p-Sylow subgroups. Using the Sylow results, Frobenius, in 1895 [1], proved a generalization: The number of subgroups of G of order ps is congruent to 1 modulo p whenever 1 < s < n. Most current texts show the existence of a p-Sylow subgroup and prove Sylow's third theorem using arguments that involve a group acting on a set. This method of proof for the existence of a p-Sylow subgroup was due to Miller between 1910 and 1915 [8, 9], but, according to Jacobson [9, p. 83], was forgotten until it was rediscovered in 1959 by Wielandt [14]. Krull [7] showed how Wielandt's method could be used to obtain Frobenius' generalization, and Gallagher, in 1967 [2], simplified the argument to one that depends upon the order of G rather than the group itself. Illustrating the combinatorics of finite group actions is part of the motivation to use this method of proof both to demonstrate the existence of p-Sylow subgroups in a finite group and to determine their number. In this note we offer another method to prove these results. Our combinatorial tool will be Mobius inversion on the lattice of subgroups of a finite group. We will see that an application of this method will easily lead to Frobenius' theorem, in fact, a generalization of it. Of course, part of the reason for presenting this proof is to highlight the method.

Reliable Method for Fission Source Convergence of Monte Carlo Criticality Calculation with Wielandt's Method

A new algorithm of Monte Carlo criticality calculations for implementing Wielandt's method, which is one of acceleration techniques for deterministic source iteration methods, is developed, and the algorithm can be successfully implemented into MCNP code. In this algorithm, part of fission neutrons emitted during random walk processes are tracked within the current cycle, and thus a fission source distribution used in the next cycle spread more widely. Applying this method intensifies a neutron interaction effect even in a loosely-coupled array where conventional Monte Carlo criticality calculation methods have difficulties, and a converged fission source distribution can be obtained with fewer cycles. Computing time spent for one cycle, however, increases because of tracking fission neutrons within the current cycle, which eventually results in an increase of total computing time up to convergence. In addition, statistical fluctuations of a fission source distribution in a cycle are worsened by applying Wielandt's method to Monte Carlo criticality calculations. However, since a fission source convergence is attained with fewer source iterations, a reliable determination of convergence can easily be made even in a system with a slow convergence. This acceleration method is expected to contribute to prevention of incorrect Monte Carlo criticality calculations.

Reliable Method for Fission Source Convergence of Monte Carlo Criticality Calculation with Wielandt's Method

Reliable Method for Fission Source Convergence of Monte Carlo Criticality Calculation with Wielandt's Method

Development of BWR regional instability model and verification based on ringhals 1 test

The mathematical model of LAPUR, the well-known frequency domain stability code, was reviewed and modified from the viewpoint of the pure thermohydraulic and neutronic-thermohydraulic interaction mechanisms. Based on the ex-core loop test data, the original combined friction and local pressure loss model was modified taking account of the different dynamic behavior of these two pressure loss mechanisms. That is, the perturbation terms of the two-phase friction multiplier and mass flux correction factor were adjusted semi-empirically. With reference to the neutronic-thermohydraulic interaction, the new regional instability model was introduced considering not only the sub-criticality but also the shape of the higher neutron flux mode. The efficient three-dimensional modal analysis schemes were studied and realized as the independent two energy-groups diffusion code, ACCORD. The power iteration scheme was applied with Wielandt method. In order to resolve tightly clustered harmonic modes, the proper initial guess was given which was derived by the horizontally two-dimensional calculations taking account of the primary important first azimuthal modal shape. The adequacy of the instability evaluation system was verified based on the Ringhals 1 test data which were supplied to participants of the OECD/NEA BWR stability benchmark project. The predominant first azimuthal oscillating power shape was evaluated based on Local Power Range Monitor (LPRM) signals. The prediction by ACCORD shows an excellent agreement both in the LPRM responses and the angle of the neutral line. The measured core-wide and regional instability decay ratios were compared with the prediction by Modified LAPUR. Both the bias and standard deviation of the decay ratios are about 0.1 for both instability modes, which is acceptable in the practical design. An extensive phenomenological study was performed regarding the regional instability predominance. The dependence of the core-wide and regional decay ratios was investigated for several power shape indices. It was indicated that the newly proposed distance-weighted 2nd order axial power momentum will be a good index.

Note on the computation of the maximal eigenvalue of a non-negative irreducible matrix

We prove the convergence of Wielandt's method in the computation of the maximal eigenvalue and eigenvector of a non-negative irreducible matrix.