Adjoint Flux Research Articles

Evaluation of effective kinetic parameters and adjoint flux distribution using iterated fission probability in the iMC Monte Carlo code

The iterated fission probability (IFP) method enables assessment of adjoint flux-weighted kinetic parameters, i.e., effective kinetic parameters, in Monte Carlo (MC) simulation, an essential capability in modern MC codes. This method can be extended to calculate adjoint flux-weighted quantities within a prescribed phase-space, enabling the estimation of adjoint flux distributions. The iMC Monte Carlo code, developed at the Korea Advanced Institute of Science and Technology (KAIST), is proficient in both calculating effective kinetic parameters and adjoint flux distributions. This paper presents benchmark results verifying the code’s capabilities. Critical device configurations are considered for evaluating kinetic parameters, compared with the Serpent2 code results. Both multi-group and continuous-energy benchmarks are solved to assess IFP-based spatial- and energy-wise adjoint flux distributions, and comparison is made against deterministic transport calculations. Results show that effective kinetic parameters can be accurately estimated, and acceptable adjoint flux distributions can be obtained using the iMC code.

A Stochastic Calculus Approach to Boltzmann Transport

Traditional Monte Carlo methods for particle transport utilize source iteration to express the solution, the flux density, of the transport equation as a Neumann series. Our contribution is to show that the particle paths simulated within source iteration are associated with the adjoint flux density and the adjoint particle paths are associated with the flux density. We make our assertion rigorous through the use of stochastic calculus by representing the particle path used in source iteration as a solution to a stochastic differential equation (SDE). The solution to the adjoint Boltzmann equation is then expressed in terms of the same SDE, and the solution to the Boltzmann equation is expressed in terms of the SDE associated with the adjoint particle process. An important consequence is that the particle paths used within source iteration simultaneously provide Monte Carlo samples of the flux density and adjoint flux density in the detector and source regions, respectively. The significant practical implication is that particle trajectories can be reused to obtain both forward and adjoint quantities of interest. To the best our knowledge, the reuse of entire particles paths has not appeared in the literature. Monte Carlo simulations are presented to support the reuse of the particle paths.

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.

Goal-Oriented Mesh Adaptivity of Three-Dimensional Neutron Transport Calculation Using Weighted Difference Scheme and Dual-Weighted Residual Error Indicators

Maintaining a reasonable balance between computational accuracy and overhead is important for neutron transport simulations of engineering problems. This paper presents a goal-oriented mesh adaptive algorithm applied to the multigroup discrete ordinates equation for fixed source and criticality problems. The dual-weighted residual (DWR) approach estimates numerical solution errors and drives local mesh refinement for specific targets, such as detector response, integral flux, and multiplication factor. We employ a reconstruction method to evaluate the spatial residuals of the fluxes obtained by the weighted difference scheme. To improve the performance of adaptive algorithms, new estimation models are proposed for adjoint fluxes needed by the DWR theory, including a regional goal model for fixed source problems and an inconsistent fission source model for k-eigenvalue problems. Additionally, we analyze the impact of the truncation of flux reconstruction and isotropic approximation of adjoint fluxes on grid error indicators and adaptive calculations. Numerical results demonstrate that for the quantities of interest, our adaptive approach saves more than 70% of computational effort and run time when obtaining a level of high accuracy comparable to that of uniform fine grids.

Full calculation of the long-time-interval limit of noise in reactors

The analysis of the stochastic behavior of detected signals is useful in multiplicative systems such as nuclear reactors. In this work, we derive a simple, exact expression for the long-time limit of the variance-to-mean ratio of neutron detection near a reactor using the probability generating function technique. The obtained result incorporates continuous energy and spatial effects and depends only on the steady-state flux and its adjoint. We then perform MCNP simulations to calculate the adjoint flux in the Israeli Research Reactor no. 1 (IRR-1). This allows us to explicitly calculate the said values for different configurations of varying homogeneity and criticality, and to compare the results of the simulations to the Feynman α point model results. In the comparison we find significant deviations of 10%-80%, with strong dependence on the core's multiplication factor, accentuating the need for spatial corrections.

Generational Variance Reduction in Monte Carlo Criticality Simulations as a Way of Mitigating Unwanted Correlations

Monte Carlo criticality simulations are widely used in nuclear safety demonstrations, as they offer an arbitrarily precise estimation of global and local tallies while making very few assumptions. However, since the inception of such numerical approaches, it is well known that bias might affect both the estimation of errors on these tallies and the tallies themselves. In particular, stochastic modeling approaches developed in the past decade have shed light on the prominent role played by spatial correlations through a phenomenon called neutron clustering. This effect is particularly of great significance when simulating loosely coupled systems (i.e., with a high dominance ratio). In order to tackle this problem, this paper proposes to recast the power iteration technique of Monte Carlo criticality codes into a variance reduction technique called Adaptive Multilevel Splitting. The central idea is that iterating over neutron generations can be seen as pushing a subpopulation of neutrons toward a generational detector (instead of a spatial detector as variance reduction techniques usually do). While both approaches allow for neutron population control, the former blindly removes or splits neutrons. In contrast, the latter optimizes the spatial, generational, and spectral attributes of neutrons when they are removed or split through an adjoint flux estimation, hence tempering both generational and spatial correlations. This is illustrated in the present paper with a simple case of a bare slab reactor in the one-speed theory on which the Adaptive Multilevel Splitting was applied and compared to variations of the Monte Carlo power iteration method used in neutron transport. Besides looking at the resulting efficiency of the methods, this work also aims to highlight the main mechanisms of the Adaptive Multilevel Splitting in criticality calculations.

Comparing the 1st and 2nd moment approaches to variance reduction in Monte Carlo: Two examples

Variance reduction in Monte Carlo particle transport calculations nearly always relies on the generation of information based on the first moment (viz. the adjoint flux or importance). In this paper we revisit the question of the non-optimal nature of this approach. Whilst it has already been shown to be deficient in duct streaming problems, we look at another kind of problem, namely the activation of nuclides whose cross-sections contain resonances. As a benchmark a second moment-based approach that gives near-optimum results is employed. We find important differences in efficiency between the first and second-moment-based approaches. Some discussion is made on the reasons for these differences.

Demonstrating Computational Equivalence Between Continuous and Discrete Adjoint Methods by Calculating Time-Dependent Adjoint Solutions with Neutron Diffusion Models

The continuous adjoint method and the discrete adjoint method are two alternative approaches used to calculate adjoint solutions for adjoint systems. The continuous adjoint method derives adjoint equations analytically from continuous forward equations and then solves the adjoint equations either analytically or numerically in a discretized form whereas the discrete adjoint method calculates the adjoint solutions directly from the discretized forward equations. With regard to the methodology development and calculation procedure, distinct differences are well recognized between the two methods. For certain reasons, both methods are exclusively preferred and commonly used by different computational communities, but limited studies clarify the connections between the two adjoint methods from either of the communities. This paper demonstrates the computational equivalence between the continuous and discrete adjoint methods by investigating time-dependent adjoint solutions to the two-group neutron diffusion model in nuclear reactor analysis problems using both methods. Adjoint solutions can be used to estimate system parameters for reactor safety analysis. Appropriate final state conditions for the adjoint systems are specified in both of the methods, and the conditions are clarified with proper physical explanations. With the help of an event-based case study on neutron diffusion models, the accuracy of the time-dependent adjoint fluxes obtained from both methods is verified, and the pros and cons of both adjoint methods are examined. More importantly, the computational equivalence of both methods is demonstrated when they are applied to multigroup neutron diffusion systems. The advantage of calculating time-dependent adjoint fluxes by directly solving time-dependent adjoint systems rather than taking steady-state approximations as in common practice is also demonstrated.

A Deterministic Adjoint-Based Semi-Analytical Algorithm for Fast Response Change Computations in Proton Therapy

In this paper we propose a solution to the need for a fast particle transport algorithm in Online Adaptive Proton Therapy capable of cheaply, but accurately computing the changes in patient dose metrics as a result of changes in the system parameters. We obtain the proton phase-space density through the product of the numerical solution to the one-dimensional Fokker-Planck equation and the analytical solution to the Fermi-Eyges equation. Moreover, a corresponding adjoint system was derived and solved for the adjoint flux. The proton phase-space density together with the adjoint flux and the metric (chosen as the energy deposited by the beam in a variable region of interest) allowed assessing the accuracy of our algorithm to different perturbation ranges in the system parameters and regions of interest. The algorithm achieved negligible errors () for small Hounsfield unit (HU) perturbation ranges (–40 HU to 40 HU) and small to moderate errors (3% to 17%) – in line with the well-known limitation of adjoint approaches – for large perturbation ranges (–400 HU to 400 HU) in the case of most clinical interest where the region of interest surrounds the Bragg peak. Given these results coupled with the capability of further improving the timing performance it can be concluded that our algorithm presents a viable solution for the specific purpose of Online Adaptive Proton Therapy.

Preservation of kinetics parameters generated by Monte Carlo calculations in two-step deterministic calculations

The generation of accurate kinetic parameters such as mean generation time Λ and effective delayed neutron fraction βeff via Monte Carlo codes is established. Employing these in downstream deterministic codes warrants another step to ensure no additional error is introduced by the low-order transport operator when computing forward and adjoint fluxes for bilinear weighting of these parameters. Another complexity stems from applying superhomogenization (SPH) equivalence in non-fundamental mode approximations, where reference and low-order calculations rely on a 3D full core model. In these cases, SPH factors can optionally be computed for only part of the geometry while preserving reaction rates and K-effective, but the impact of such approximations on kinetics parameters has not been thoroughly studied. This paper aims at studying the preservation of bilinearly-weighted quantities in the Serpent–Griffin calculation procedure. Diffusion and transport evaluations of IPEN/MB-01, Godiva, and Flattop were carried out with the Griffin reactor physics code, testing available modeling options using Serpent-generated multigroup cross sections and equivalence data. Verifying Griffin against Serpent indicates sensitivities to multigroup energy grid selection and regional application of SPH equivalence, introducing significant errors; these were demonstrated to be reduced through the use of a transport method together with a finer energy grid.

The adjoint transport problem applied to estimate neutral particle leakage in the discrete ordinates formulations

In source–detector problems, neutron leakage is a quantity of interest that could lead to improve shielding structures, thus reducing the dose received by humans. In this work, we apply an adjoint technique with spectral nodal methods to compute neutron leakage in multigroup one– and two–dimensional problems in the discrete ordinates (SN) formulation. The use of the adjoint technique to calculate the leakages due to various source distributions is very convenient as it is possible to run adjoint problems and store the neutron importance maps. Here we solve the homogeneous adjoint SN transport equation by considering unit outgoing adjoint flux at the boundary. In order to numerically solve slab– and X, Y–geometry problems, we use spectral nodal methods that have been widely applied and discussed in the literature. Numerical results are given to illustrate the present adjoint technique to estimate the neutron leakage for each energy group in source–detector problems. For all the test problems, the results obtained by the adjoint technique as described in this paper do agree with the results obtained by solving the analogous forward problem.

Adjoint Flux and Perturbation Theory of γ-Mode Eigenvalue Problem Appearing in Exponential Experiments

The γ-mode eigenvalue problem is investigated to utilize an exponential experiment to validate nuclear data for reactor core analyses. The perturbation of the spatial decay constant γ by the bias of nuclear data is analyzed with the adjoint flux of the γ-mode eigenvalue problem. The adjoint flux at a phase-space position is found to be proportional to the amplitude of the neutron flux on a plane vertically distant from a source placed at the position. The implication of the adjoint flux is numerically demonstrated based on the diffusion theory. The perturbation theory relating the bias of the fission neutron emission to the perturbation of γ is preliminarily justified in the manner of the continuous energy Monte Carlo.

Investigation of Nodal Numerical Adjoints From CMFD-Based Acceleration Methods

In this work, the attribute of mathematical adjoints acquired from various CMFD (coarse-mesh finite difference) accelerated nodal methods based on the nodal expansion method (NEM) is presented. The direct transposition is implemented to the NEM-CMFD system matrix that includes correction factors to acquire the adjoint flux. Three different acceleration schemes are considered in this paper, which are, namely, CMFD, pCMFD, and one-node CMFD methods, and the self-adjoint attribute of migration operator for each acceleration scheme is studied. With regard to the one-node CMFD acceleration, a mathematical description for encountering negative adjoint flux values is given alongside an adjusted one-node CMFD scheme that stifles such an occurrence. The overall features of the aforementioned acceleration methods are recognized through analyses of 2D reactor problems including the KWU PWR 2D benchmark problem. Further systematic assessment is conducted based on the first-order perturbation theory, where the obtained adjoint fluxes are applied as weighting functions. It is clearly shown that the adjusted one-node CMFD scheme results in an improved reactivity estimation by excluding the presence of negative adjoint flux values.

Adjoint P1 equations for neutron slowing down

Abstract To solve the large-group diffusion adjoint equation the averaged group constants have to be pre-calculated. However, at this point the forward and adjoint fluxes are unknown. Thus, an alternative method, independent of the solution of this equation, has to be provided. This method is divided into two steps. In the first, the forward and adjoint spectra should be obtained for a unit cell assuming infinite medium. The second step consists of the correction for a finite medium by approximating the spatial dependence by a single Fourier fundamental mode in terms of the geometrical buckling B2. This paper is dedicated to the solution of the second step. Therefore, forward and adjoint P1 equations were both obtained and numerically solved in lethargy space in order to provide the neutron fluxes in a finite unit fuel cell. Moreover, the direct, adjoint and bilinear weighting for calculation of macrogroup constants is discussed.

The nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Response-Coupled Forward/Adjoint Linear Systems (nth-CASAM-L): II. Illustrative Application

This work illustrates the application of the nth-order comprehensive adjoint sensitivity analysis methodology for response-coupled forward/adjoint linear systems (abbreviated as “nth-CASAM-L”) to a paradigm model that describes the transmission of particles (neutrons and/or photons) through homogenized materials, as encountered in radiation protection and shielding. The first-, second-, and third-order sensitivities of responses that depend on both the forward and adjoint particle fluxes are obtained exactly, in closed-form, underscoring the principles and methodology underlying the nth-CASAM-L. The results presented in this work underscore the fundamentally important role of the nth-CASAM-L in the quest to overcome the “curse of dimensionality” in sensitivity analysis, uncertainty quantification and predictive modeling.

Comparison of probabilistic and deterministic methods for calculation of kinetic parameters of HWZPR

The accurate calculation of kinetic parameters, including the effective delayed neutron fraction and the neutron mean generation time, is very important in the study of time dependent neutron population behavior. To calculate these parameters, it is necessary to calculate the flux and adjoint flux in the phase space. In this paper, the kinetic parameters of the new core configuration of the heavy water zero power reactor (HWZPR) are calculated using two deterministic and probabilistic methods. The quantitative and qualitative discussion is presented to illustrate the advantages and disadvantages of each method. The maximum difference between the two accounting methods was less than 3%. The effect of different energy structures on the criticality and kinetic parameters has been investigated in the deterministic approach. Additionally, the calculated kinetic parameters of the HWZPR validated experimentally with a reactivity inject response. Good consistency has been observed between different methods of computation and experimental values.

An Analytic Benchmark for Neutron Boltzmann Transport with Downscattering—Part II: Flux and Eigenvalue Sensitivities to Nuclear Cross Sections and Resonance Parameters

A benchmark to verify the accuracy of neutron transport criticality solvers along the energy dimension was established. For the first time, the analytic solution of the flux amplitude was derived in the particular case of an infinite-homogeneous medium with isotropic scattering in the center of mass and an arbitrary number of no-threshold, neutral particle reaction resonances (e.g., radiative capture, fission, and resonance scattering). In this paper, the benchmark is extended to the adjoint transport problem, and a solution to the adjoint flux is derived. The adjoint flux solution is then combined with the forward flux to obtain expressions for an arbitrary-order cross section and resonance parameter sensitivity coefficients. Finally, numerical solutions are provided for a benchmark problem constituted of the first resonance of 239Pu, the 6.67-eV resonance of 238U, and a scattering isotope with a flat cross section, allowing for computational verification of the sensitivity coefficients and nuclear data uncertainty of current neutron transport criticality codes. Through these novel results, this analytic benchmark can serve as a reference to verify the sensitivity analysis of neutron transport criticality calculations.

A HYBRID MONTE-CARLO-DETERMINISTIC METHOD FOR AP1000 EX-CORE DETECTOR RESPONSE SIMULATION

The ex-core detector-response calculation is a typical deep-penetration problem, which is challenging for the Monte Carlo method. The response of the ex-core detector is an important parameter for the safe operation of the nuclear power plants. Meanwhile, evaluation of the ex-core detector response during each step of fuel-loading is used to guide the fuel-loading sequence. The response can also be used to reconstruct core-power distribution for online monitoring of long-term power. The detector used for the ex-core response is the source-range detector which is sensitive to thermal neutrons. For a Monte Carlo shielding calculation of the above detector response, the thermal flux under 0.625eV is needed, which is too small to be tallied by traditional Monte Carlo simulations. In practice, the tally results are close to zero in the detector region under direct Monte Carlo calculation. Even if the number of particles is increased to a significant amount, the statistical variance is still very large. The high variance along with a significant calculation time leads to a small Figure Of Merit (FOM). In order to solve this problem and to improve the tally efficiency of the ex-core detector response, a hybrid Monte-Carlo-deterministic method is employed in this study, and an in-house hybrid Monte-Carlo-deterministic particle transport code, NECP-MCX, is developed in this paper. The method takes the space-energy-dependent adjoint fluxes to generate importance parameters for the mesh-based weight window in the Monte Carlo calculation. Simultaneously, the mesh-based source biasing is performed with the consistent importance parameters to make the starting weight of neutrons matching with the survival weight of the weight windows. As the mesh used in the hybrid Monte-Carlo-deterministic method is superimposed, the mesh of the weight window will not be affected by the complex geometry model. The adjoint flux is obtained by the efficient SN method with the multi-group cross-section data. The whole toolset is convenient to use with single set of the modelling data for both Monte Carlo and deterministic simulations. Compared with the direct Monte Carlo simulation, the hybrid Monte-Carlo-deterministic method has a higher efficiency for a typical deep-penetration problem such as the AP1000 ex-core detector-response simulation.

The heuristically-based generalized perturbation theory

The generalized perturbation method is described relevant to ratios of bi-linear functionals of the real and adjoint neutron fluxes of critical multiplying systems. Simple linear analysis for optimization and sensitivity studies are then feasible relative to spectrum and space-dependent quantities, such as Doppler and coolant void reactivity effects in fast reactors.

Sensitivities of delayed neutron fractions in the framework of SCALE 6.2

The paper presents different methods for evaluating sensitivities and uncertainties of the effective delayed neutron fraction (βeff) with respect to microscopic nuclear data in the framework of the SCALE 6.2 code system. The prompt k ratio method is used to calculate βeff approximately. Sensitivities of βeff to microscopic reaction data are calculated with deterministic linear perturbation theory from sensitivities of the multiplication factor, and by performing linear regression analysis on results obtained from random sampling. The latter approach is also applied for estimating sensitivities of kinetic parameters in six delayed neutron groups, as obtained by adjoint flux weighting in lattice calculations, to nuclear reactions. Finally, it is demonstrated that the uncertainties of βeff strongly depend on the choice of the covariance data set; in particular, the use of SCALE 6.2 covariance data for the average number of delayed neutrons per fission in some cases leads to very high βeff uncertainties.