Differential Operator Sampling Research Articles

Sensitivity and perturbation analysis of alpha-eigenvalue with time-dependent Monte Carlo perturbation techniques

Sensitivity analysis and perturbation calculation methods for prompt neutron decay constants α using the Monte Carlo method, such as iterative fission probability (IFP), differential operator sampling (DOS), and correlated sampling (CS), have been proposed thus far. However, these methods require a significant amount of memory storage. The method developed in this study applies DOS and CS techniques during a time-dependent Monte Carlo simulation for the pulsed neutron method in subcritical systems. The time variation of the neutron flux is fitted to an exponential function during the time range when the flux decays with the fundamental mode α. By applying the least squares method to the results of time-dependent DOS and CS, the sensitivity coefficients with respect to nuclear data and the change in α due to perturbation can be calculated. In contrast to the existing methods that require storing the total fission or time sources per generation over several (∼10) generations, the proposed method only needs to store the weighting coefficient of a single source particle until it dies. The additional memory storage per parameter is comparable to that of conventional time-dependent Monte Carlo simulations and much less than that of existing methods. Numerical tests for the proposed method have demonstrated the accuracy and usefulness of the time-dependent DOS and CS methods.

Development of an Adjoint Flux Calculation Technique for Exact Perturbation Theory in Monte Carlo Code MCS

A calculation technique computing the adjoint flux of perturbed system is developed for the exact perturbation theory in Monte Carlo transport. By using a correlated sampling and iterated fission probability methods together, the adjoint flux of perturbed system is calculated during a forward Mote Carlo simulation of an unperturbed system. In the method, no additional particle simulation is required to compute the adjoint flux of perturbed system. The exact perturbation method is implemented in the Monte Carlo code MCS. The results of perturbation method for k-eigenvalue change are compared against differential operator sampling, adjoint-weighted perturbation, and direct perturbation methods for the Godiva benchmark and PLUS7 fuel assembly.

Monte Carlo Calculation Method for Reactor Period Utilizing the Differential Operator Sampling Technique

The inverse reactor period α is a fundamental mode eigenvalue of the α-mode nonlinear Boltzmann eigenvalue equation that considers delayed neutron contributions. Thus far, several Monte Carlo methods, including the α-k, weight balancing, and transition rate matrix methods, have been developed to calculate α. This study presents a new Monte Carlo method for predicting α by using the derivatives of the k-eigenvalue with respect to α. Formulae are derived to calculate the first and second derivatives using the differential operator sampling method. The key feature of the new proposed method is its ability to estimate the uncertainty of the predicted α by considering the uncertainty of the k-eigenvalue and its derivatives with respect to α.

Research on the calculation method of sensitivity coefficients of reactor power to material density based on Monte Carlo perturbation theory

The ability to calculate the material density sensitivity coefficients of power with respect to the material density has broad application prospects for accelerating Monte Carlo-Thermal Hydraulics iterations. The second-order material density sensitivity coefficients for the general Monte Carlo score have been derived based on the differential operator sampling method in this paper, and the calculation of the sensitivity coefficients of cell power scores with respect to the material density has been realized in continuous-energy Monte Carlo code RMC. Based on the power-density sensitivity coefficients, the sensitivity coefficients of power scores to some other physical quantities, such as power-boron concentration coefficients and power-temperature coefficients considering only the thermal expansion, were subsequently calculated. The effectiveness of the proposed method is demonstrated in the power-density coefficients problems of the pressurized water reactor (PWR) moderator and the heat pipe reactor (HPR) reflectors. The calculations were carried out using RMC and the ENDF/B-VII.1 neutron nuclear data. It is shown that the calculated sensitivity coefficients can be used to predict the power scores accurately over a wide range of boron concentration of the PWR moderator and a wide range of temperature of HPR reflectors.

Monte Carlo sensitivity calculation in fixed source problems with the derivative source method

This paper proposes a new Monte Carlo sensitivity analysis method for fixed source radiation transport problems, which is dubbed the derivative source method (DSM). This method is a variation of the formerly proposed perturbation source method. A transport equation of a derivative particle representing a derivative of a flux with respect to a parameter such as a cross section is derived by differentiating the fixed source radiation transport equation. This paper presents a Monte Carlo algorithm to solve this transport equation. The derivative particle is emitted from a region of interest (ROI) when a radiation particle undergoes a collision in the ROI. The derivative particle is tracked in the same manner as the ordinary radiation particles. The derivative particle that undergoes a collision in the ROI can yield another derivative particle whose derivative order is increased by one from the colliding particle. This higher-order derivative particle can be used for calculating the higher-order derivative of a flux. The differential operator sampling method (DOS), which is commonly used for Monte Carlo sensitivity analyses, suffers from poor performance when an ROI is small. For a small ROI, the efficiency of the DSM can be improved by adding a pseudo scattering cross section in the ROI, thereby increasing the pseudo scatterings and the number of derivative particles emitted from the ROI. The efficiency of the new method is evaluated by comparing the DSM to the DOS. Numerical tests are performed for a two-dimensional geometry containing a small inclusion. The results reveal that the DSM definitely outperforms the DOS for these test problems. The efficiency of the DSM depends strongly on the pseudo scattering cross section added to the inclusion.

Deterministic and stochastic methods for sensitivity analysis of neutron noise

Neutron noise calculated from the neutron noise equation in the frequency domain is governed by the cross section and kinetic parameters. Deterministic and stochastic methods to obtain the sensitivity coefficient of neutron noise with respect to the abovementioned parameters are proposed. As a deterministic method, a diffusion equation for the first derivative of neutron noise with respect to a cross section or kinetic parameter is derived by differentiating the neutron noise diffusion equation. As a stochastic method, the differential operator sampling method, which is a well-established Monte Carlo technique, is applied to calculate the sensitivity coefficient. Neither method requires adjoint mode calculations and can be expanded to higher-order derivatives. Based on verifications performed in this study, it is discovered that these techniques yield accurate sensitivity coefficients. The methods developed in this study eliminates a large number of calculations that need to be performed in the random sampling method.

Monte Carlo sensitivity analysis method for the effective delayed neutron fraction with the differential operator sampling method

An exact Monte Carlo method for evaluating the sensitivity coefficients of the effective delayed neutron fraction (βeff) with respect to nuclear data is developed using the differential operator sampling (DOS) method. This development includes combining the two characteristic features of the DOS method: the exact evaluation of βeff and the sensitivity analyses of the k- and α-eigenvalues. Numerical tests are performed to calculate the sensitivity coefficients of βeff with the newly developed Monte Carlo method. The new Monte Carlo method’s sensitivities mostly are in accordance with the reference solutions that were obtained by a deterministic discrete ordinates method based on the perturbation theory. In particular, the sensitivities of βeff to the yield and spectrum of delayed neutrons, which are dominant contributors to the uncertainty of βeff, can be predicted precisely by the new method.

Two-step Monte Carlo sensitivity analysis of alpha- and gamma-eigenvalues with the differential operator sampling method

A new Monte Carlo method is developed to calculate the sensitivity coefficients of the α-eigenvalue (the time decay constant) and the γ-eigenvalue (the spatial decay constant in an exponential experiment) with respect to nuclear data. A method that was previously developed for the sensitivity analyses of the α-eigenvalue, which is not based on the normal k-α algorithm, is not applicable to the γ-eigenvalue due to its inability to obtain a converged source distribution. Then, a two-step method in which two sensitivity coefficients are separately calculated using the k-α or k-γ algorithm and the differential operator sampling method is newly developed. The sensitivity coefficient of the α- or γ-eigenvalue is represented by the ratio of the two sensitivity coefficients. Some numerical tests for three-energy group problems are performed using the new method. The sensitivity coefficients that are obtained by the new method are verified by comparing them to the solutions of deterministic transport calculations or to the approximate results that are obtained from the direct perturbations of the cross-sections.

A Monte Carlo technique for sensitivity analysis of alpha-eigenvalue with the differential operator sampling method

A method for Monte Carlo sensitivity analyses of α-eigenvalue (prompt neutron time decay constant) in a subcritical system is developed using the first-order differential operator sampling (DOS) method. The first-order derivative of α-eigenvalue with respect to nuclear data is calculated using the DOS method that includes the capability of calculating perturbed source effect. This paper is an extension of the author’s previous work for development of the sensitivity analysis method for keff-eigenvalue. Unlike the conventional Monte Carlo method for α-eigenvalue calculation that uses the power iteration of fission sources, this paper introduces a recently developed “time source method”. The “time source method” has a weakness for a void-containing subcritical system, which is overcome by assigning a virtual total cross section in the void region. The perturbed source effect, which is caused by the change of nuclear data in a subcritical system, can be calculated by two methods, the source perturbation iteration method and the superhistory method. The source perturbation iteration method is superior in terms of computation efficiency, but a huge computer memory is required. The superhistory method dramatically reduces the memory requirement, although it worsens the variance of the sensitivity coefficients. The method developed in this paper is applied to some numerical tests that use multi-group constants, and it is verified by comparing to the results obtained by a deterministic perturbation theory.

Monte Carlo method for solving a B1 equation with complex-valued buckling in asymmetric geometries and generation of directional diffusion coefficients

Direction-dependent diffusion coefficients generated using a Monte Carlo method that explicitly solves a heterogeneous B1 equation are validated in this paper. The validation is conducted by comparing the direction-dependent neutron leakage using directional diffusion coefficients with the results of a neutron transport calculation. This paper develops a new method for critical buckling search calculation to solve a heterogeneous B1 equation for an asymmetric unit cell in which the critical bucking has a complex value. A complex-valued critical buckling can be searched using the differential operator sampling method for estimating the sensitivity coefficients of keff with respect to the buckling. In an asymmetric unit cell, the neutron flux and the neutron current also have complex values. The real part of the neutron flux obtained by a critical buckling search calculation represents the neutron spectrum in the center portion of the finite geometry composed of the unit cells.

Monte Carlo perturbation methods using “virtual density” theory for calculating reactivity due to geometry change

The “virtual density” theory in the field of neutronics is a method to quantify reactivity of a reactor core due to geometry changes by altering the material density or nuclear cross section instead of explicitly changing the geometry. The Monte Carlo perturbation methods—differential-operator sampling (DOS) and correlated sampling (CS)—can efficiently yield accurate small reactivity for cross section changes. However, Monte Carlo perturbation methods have weakness in their inability to deal with geometry changes. This study incorporates the virtual-density theory into the Monte Carlo perturbation methods. When an entire core expands or swells uniformly, keff of the perturbed core can be calculated by simply changing the material density without changing the geometry. Regarding uniform expansion or swelling, conventional Monte Carlo perturbation methods—where cross sections change proportionally to the material density—can predict the altered keff. According to the virtual-density theory, keff perturbed due to a uniform anisotropic expansion or swelling can be predicted by stretching or shrinking the mean free path of neutrons in only one direction without changing the geometry. A new Monte Carlo algorithm is developed to incorporate the path length-stretching or -shrinking in only one direction into a random-walk process in a Monte Carlo keff-eigenvalue calculation. The new Monte Carlo algorithm provides accurate keff values for uniform anisotropic expansion and swelling regardless of the perturbation degree. New formulations of the Monte Carlo perturbation methods are derived to predict the change in keff due to a uniform anisotropic expansion or swelling. Both Monte Carlo perturbation methods can accurately and efficiently predict changes in keff for a small uniform anisotropic expansion or swelling.

Development of Generalized Perturbation Theory Algorithms for Monte Carlo Eigenvalue Calculations

A generalized perturbation theory (GPT) formulation suited for the Monte Carlo (MC) eigenvalue calculations is newly developed to estimate sensitivities of a general MC tally to input data. In the new GPT formulation, the tally perturbation due to an input parameter change is expressed as a sum of the perturbed operator effect and the perturbed source effect requiring the generalized adjoint function weighting. It is shown that the new GPT formulation is equivalent to the conventional first-order differential operator sampling method augmented by the fission source perturbation method. Because the GPT formulation makes it necessary to compute the generalized adjoint function, MC sensitivity estimation algorithms can consume a huge computer memory space to save historywise estimates of tallies. As a way to alleviate the memory space problem, the MC Wielandt iteration method is incorporated into the MC GPT algorithm. For the purpose of comparison, MC GPT algorithms by both the standard power iteration and the Wielandt iteration methods are implemented in the Seoul National University MC code, McCARD. Their performances are examined in two-group homogeneous problems, Godiva and the TMI-1 pin cell problem. From the nuclear data sensitivity and uncertainty analyses of these problems, it is demonstrated that the new GPT methods can predict the sensitivities of reaction rate tallies to cross-section data very well. It is also demonstrated that the incorporation of the MC Wielandt iteration method into the new GPT calculations consumes a negligibly small amount of memory required for—and thus resolves—the computer memory issue associated with the adjoint function calculations.

Review of Early 21st-Century Monte Carlo Perturbation and Sensitivity Techniques for k-Eigenvalue Radiation Transport Calculations

Since the 1960s, Monte Carlo methods have been used to compute the effect of perturbations on system responses and for computing sensitivity coefficients. This review article focuses on 21st-century developments specific to k-eigenvalue calculations. The theory of correlated sampling, differential operator sampling, and adjoint-based approaches and their historical methods from the 20th century are briefly summarized. Specific focus is given to four recent and significant developments: fission source correction using the correlated sampling and differential operator sampling methods, adjoint-based perturbations for the k eigenvalue using the iterated fission probability method, an extension to reaction rate ratios using generalized perturbation theory, and a recent development using a collision history approach allowing for the calculation of sensitivity coefficients of bilinear ratios and generalized responses. Differences and similarities of the four methods are discussed along with a comparison to the 20th-century approaches. A perspective on future developments is also given.

Calculation of reactor kinetics parameters with Monte Carlo differential operator sampling

The methods to calculate the reactor kinetics parameters of βeff and Λ with the differential operator sampling have been reviewed. The comparison of the results obtained with the differential operator sampling and iterated fission probability approaches has been performed. It is shown that the differential operator sampling approach gives the same results as the iterated fission probability approach within the statistical uncertainty. In addition, the prediction accuracy of the evaluated nuclear data library JENDL-4.0 for the measured βeff/Λ and βeff values is also examined. It is shown that JENDL-4.0 gives a good prediction except for the uranium-233 systems. The present results imply the need for revisiting the uranium-233 nuclear data evaluation and performing the detailed sensitivity analysis.

Uncertainty Propagation Analysis for PWR Burnup Pin-Cell Benchmark by Monte Carlo Code McCARD

In the Monte Carlo (MC) burnup analyses, the uncertainty of a tally estimate at a burnup step may be induced from four sources: the statistical uncertainty caused by a finite number of simulations, the nuclear covariance data, uncertainties of number densities, and cross-correlations between the nuclear data and the number densities. In this paper, the uncertainties ofkinf, reaction rates, and number densities for a PWR pin-cell benchmark problem are quantified by an uncertainty propagation formulation in the MC burnup calculations. The required sensitivities of tallied parameters to the microscopic cross-sections and the number densities are estimated by the MC differential operator sampling method accompanied by the fission source perturbation. The uncertainty propagation analyses are conducted with two nuclear covariance data—ENDF/B-VII.1 and SCALE6.1/COVA libraries—and the numerical results are compared with each other.

Adjoint Sensitivity and Uncertainty Analyses in Monte Carlo Forward Calculations

The adjoint-weighted perturbation (AWP) method, in which the required adjoint flux is estimated in the course of Monte Carlo (MC) forward calculations, has recently been proposed as an alternative to the conventional MC perturbation techniques, such as the correlated sampling and differential operator sampling (DOS) methods. The equivalence of the first-order AWP method and first-order DOS method with the fission source perturbation taken into account is proven. An algorithm for the AWP calculations is implemented in the Seoul National University MC code McCARD and applied to the sensitivity and uncertainty analyses of the Godiva and Bigten criticalities.

Estimation of Sample Reactivity Worth with Differential Operator Sampling Method

Estimation of Sample Reactivity Worth with Differential Operator Sampling Method

Calculation of effective delayed neutron fraction with Monte Carlo perturbation techniques

New methods are proposed to estimate the effective delayed neutron fraction β eff in Monte Carlo calculations: the eigenvalue methods jointly used with the differential operator sampling and correlated sampling techniques. In particular, the eigenvalue method with the differential operator sampling technique has a distinct feature that it theoretically gives an exact β eff value. To verify the proposed methods, Monte Carlo calculations are performed for several systems with simple geometry. It is found that the results obtained with the proposed methods agree with the reference deterministic results within sufficiently small statistical uncertainties. The indirect perturbed source effect must be taken into account to estimate an exact β eff value.

Impact of Perturbed Fission Source on the Effective Multiplication Factor in Monte Carlo Perturbation Calculations

A new method to estimate a change in the effective multiplication factor due to the perturbed fission source distribution has been proposed for Monte Carlo perturbation calculations with the correlated sampling and differential operator sampling techniques. The method has been implemented into the MVP code for verification. Simple benchmark problems have been set up for fast and thermal systems and the applicability of the method has been verified with the problems. Consequently, it has been confirmed that the method is very effective to estimate the change. It has been also shown that there are some cases where the perturbed source effect is significant and the change in reactivity cannot be estimated accurately without taking the effect into account. Even in such cases, the new method can estimate the perturbed source effect and the estimation of the change in reactivity has been remarkably improved.

Impact of Perturbed Fission Source on the Effective Multiplication Factor in Monte Carlo Perturbation Calculations

A new method to estimate a change in the effective multiplication factor due to the perturbed fission source distribution has been proposed for Monte Carlo perturbation calculations with the correlated sampling and differential operator sampling techniques. The method has been implemented into the MVP code for verification. Simple benchmark problems have been set up for fast and thermal systems and the applicability of the method has been verified with the problems. Consequently, it has been confirmed that the method is very effective to estimate the change. It has been also shown that there are some cases where the perturbed source effect is significant and the change in reactivity cannot be estimated accurately without taking the effect into account. Even in such cases, the new method can estimate the perturbed source effect and the estimation of the change in reactivity has been remarkably improved.