Adjoint Flux Research Articles

Monte Carlo Shielding Calculations for a Spent Fuel Transport Cask with Automated Monte Carlo Variance Reduction

For the purpose of performing reasonable shielding calculation of a spent fuel transport cask, the use of Monte Carlo methods has been proposed for solving the radiation transport problem on a detailed structure of the transport cask considering fixed neutron sources. A SMIRE (Simplified MCNP-ANISN_W Variance Reduction) system has been developed in the present study, which is possible to generate automatically the lower weight boundary of the weight window for each mesh based on the Consistent Adjoint Driven Importance Sampling (CADIS). Compared with the case of the importance based on the empirical formula, the figure of merit is increased by a factor of 25. In this system, it is possible to calculate the weight suitable for the distantly-positioned detector point from the fuel effective region to introduce the relaxation factor which relaxes the increase of the particle numbers at the boundary of weight window meshes that are generally caused by the large attenuation of adjoint flux. This system is used for a variety of the radiation transport problems as well as the transport cask.

Prediction of the neutrons subcritical multiplication using the diffusion hybrid equation with external neutron sources

We used the neutron diffusion hybrid equation, in cartesian geometry with external neutron sources to predict the subcritical multiplication of neutrons in a pressurized water reactor, using a 1/ M curve to predict the criticality condition. A Coarse Mesh Finite Difference Method was developed for the adjoint flux calculation and to obtain the reactivity values of the reactor. The results obtained were compared with benchmark values in order to validate the methodology presented in this paper.

Void reactivity decomposition for the Sodium-cooled Fast Reactor in equilibrium fuel cycle

The Sodium-cooled Fast Reactor (SFR) is one of the most promising Generation IV systems with many advantages, but has one dominating neutronic drawback – a positive sodium void reactivity. The aim of this study is to develop and apply a methodology, which should help better understand the causes and consequences of the sodium void effect. It focuses not only on the beginning-of-life (BOL) state of the core, but also on the beginning of open and closed equilibrium (BOC and BEC, respectively) fuel cycle conditions. The deeper understanding of the principal phenomena involved may subsequently lead to appropriate optimization studies. Various voiding scenarios, corresponding to different spatial zones, e.g. node or assembly, have been analyzed, and the most conservative case – the voiding of both inner and outer fuel zones – has been selected as the reference scenario. On the basis of the neutron balance method, the corresponding SFR void reactivity has been decomposed reaction-, isotope-, and energy-group-wise. Complementary results, based on generalized perturbation theory and sensitivity analysis, are also presented. The numerical analysis for both neutron balance and perturbation theory methods has been carried out using appropriate modules of the ERANOS code system. A strong correlation between the flux worth, i.e. the product of flux and adjoint flux, and the void reactivity importance distributions has been found for the node- and assembly-wise voiding scenarios. The neutron balance based decomposition has shown that the void effect is caused mainly by the influence of the sodium removal on the neutron spectrum, and this has been confirmed by the perturbation method. The neutron balance based decomposition has also shown that the prime consequence of the sodium voiding is the reduction of the absorption rate (especially of 238U capture), which provides the strongest positive component of the void reactivity. The changes in neutron production and leakage provide smaller and negative contributions. Sensitivity analysis results for individual isotopic reactions have been found to be, in a relative sense, consistent with the neutron balance based isotope-wise decomposition. Finally, the neutron balance method has been applied to analyze the void reactivity differences between the three different SFR fuel cycle states considered, viz. BOL, BOC and BEC.

Development of analysis method for sodium void reactivity of step type FBR cores by using group-wise Monte Carlo code

A new method is proposed to separate the sodium void reactivity of step type FBR cores to components including non-leakage terms and a leakage term by using a newly developed perturbation code MCPERT where fluxes and adjoint fluxes are derived from a group-wise Monte Carlo code. The step type FBR core is a core where the height of the inner core is smaller than that of the outer core and a large sodium plenum region is located above the core so as to decrease the sodium void reactivity. The conventional diffusion perturbation method cannot treat such a large void region due to the diffusion approximation, while the Monte Carlo code can treat it exactly. In this study, a group-wise Monte Carlo code GMVP with a 70-group constant set JFS-3-J3.3 is employed to evaluate the neutron fluxes and adjoint fluxes which are used as inputs to the MCPERT code to evaluate the non-leakage terms. The leakage term is derived from the difference of the total sodium void reactivity evaluated by the direct calculation of GMVP and the summation of the non-leakage terms. It is found that the proposed method can provide the result approximately consistent to the ratio of the reactivity components derived from the conventional method.

Calculation of the Effective Delayed Neutron Fraction by Deterministic and Monte Carlo Methods

The studies on Accelerator‐Driven Systems (ADSs) have renewed the interest in the theoretical and computational evaluation of the main integral parameters characterizing subcritical systems (e.g., reactivity, effective delayed neutron fraction βeff, and mean prompt neutron generation time). In particular, some kinetic parameters, as the effective delayed neutron fraction, are evaluated in Monte Carlo codes by formulations which do not require the calculation of the adjoint flux. This paper is focused on a theoretical and computational analysis about how the different βeff definitions are connected and which are the approximations inherent to the Monte Carlo definition with respect to the standard definition involving weighted integrals. By means of a refined transport computational analysis carried out in a coherent and consistent way, that is, using the same deterministic code and neutron data library for the βeff evaluation in different ways, the theoretical analysis is numerically confirmed. Both theoretical and numerical results confirm the effectiveness of the Monte Carlo βeff evaluation, at least in cases where spectral differences between total and prompt fluxes are negligible with respect to the value of the functionals entering the classical βeff formulation.

Development of Calculation Technique for Iterated Fission Probability and Reactor Kinetic Parameters Using Continuous-Energy Monte Carlo Method

New algorithms and techniques are developed for calculating the iterated fission probability (IFP ) using a Monte Carlo eigenvalue calculation scheme. The proportionality of the calculated IFP to the adjoint flux is confirmed numerically. This IFP is then used with the MCNP code and its point-wise cross section data libraries to calculate kinetic parameters weighted by the adjoint flux. Experimental data for critical cores validate these theoretical estimates.

Simultaneous Optimization of Tallies in Difficult Shielding Problems

Monte Carlo is quite useful for calculating specific quantities in complex transport problems. Many variance reduction strategies have been developed that accelerate Monte Carlo calculations for specific tallies. However, when trying to calculate multiple tallies or a mesh tally, users have had to accept different levels of relative uncertainty among the tallies or run separate calculations optimized for each individual tally. To address this limitation, an extension of the Consistent Adjoint Driven Importance Sampling (CADIS) method, which is used for difficult source/detector problems, has been developed to optimize several tallies or the cells of a mesh tally simultaneously. The basis for this method is the development of an importance function that represents the importance of particles to the objective of uniform Monte Carlo particle density in the desired tally regions. This method utilizes the results of a forward discrete ordinates solution, which may be based on a quick coarse-mesh calculation, to develop a forward-weighted source for the adjoint calculation. The importance map and the biased source computed from the adjoint flux are then used in the forward Monte Carlo calculation to obtain approximately uniform relative uncertainties for the desired tallies. This extension is called forward-weighted CADIS, or FW-CADIS.

On a New Variance Reduction Technique: Neural Network Biasing - A Study of Two Test Cases with the Monte Carlo Code Tripoli4

Various variance-reduction techniques are used in Monte Carlo particle transport. Most of them rely either on a hypothesis made by the user (parameters of the exponential biasing, mesh and weight bounds for weight windows, etc.) or on a previous calculation of the system with, for example, a deterministic solver. This paper deals with a new acceleration technique, namely, autoadaptative neural network biasing. Indeed, instead of using any a priori knowledge of the system, it is possible, at a given point in a simulation, to use the Monte Carlo histories previously simulated to train a neural network, which, in return, should be able to provide an estimation of the adjoint flux, used then for biasing the simulation. We will describe this method, detail its implementation in the Monte Carlo code Tripoli4, and discuss its results on two test cases.

Surface and Volume Integrals of Uncollided Adjoint Fluxes and Forward-Adjoint Flux Products

Ray analysis techniques are standard for computing the uncollided component of a detector response to radiation. In this paper, uncollided adjoint flux integrals and forward-adjoint inner product integrals in volumes and on surfaces are derived for general geometries. In numerical test problems using a one-dimensional sphere and a two-dimensional (r-z) cylinder, deterministic and stochastic evaluations of the integrals yielded the same results. A semianalytic benchmark for the adjoint flux integral on a cylindrical surface is also used.

Calculation of neutron importance function in fissionable assemblies using Monte Carlo method

The purpose of the present work is to develop an efficient solution method for the calculation of neutron importance function in fissionable assemblies for all criticality conditions, based on Monte Carlo calculations. The neutron importance function has an important role in perturbation theory and reactor dynamic calculations. Usually this function can be determined by calculating the adjoint flux while solving the adjoint weighted transport equation based on deterministic methods. However, in complex geometries these calculations are very complicated. In this article, considering the capabilities of MCNP code in solving problems with complex geometries and its closeness to physical concepts, a comprehensive method based on the physical concept of neutron importance has been introduced for calculating the neutron importance function in sub-critical, critical and super-critical conditions. For this propose a computer program has been developed. The results of the method have been benchmarked with ANISN code calculations in 1 and 2 group modes for simple geometries. The correctness of these results has been confirmed for all three criticality conditions. Finally, the efficiency of the method for complex geometries has been shown by the calculation of neutron importance in Miniature Neutron Source Reactor (MNSR) research reactor.

Coarse-Mesh Adjoint Biasing of a Monte Carlo Dose Calculation

Abstract Deep penetration radiation transport calculations have been performed to determine the dose rates at a positron emission tomography (PET) facility in China. The radiation source is a cyclotron that is used for the production of the PET isotopes, with a labyrinthine concrete wall arrangement to minimize dose to operation personnel. Since directionally biased Monte Carlo calculations did not reach an acceptably low standard error after a week of CPU time, approximate adjoint fluxes were calculated for each of four detector locations using coarse-mesh adjoint discrete ordinates calculations and were used to bias the Monte Carlo calculations. This biasing resulted in improved computational efficiency of a factor of one to three orders of magnitude. This result shows the value of even approximate adjoint flux biasing to speed up Monte Carlo calculations.

Automated variance reduction for MCNP using deterministic methods

In order to reduce the user's time and the computer time needed to solve deep penetration problems, an automated variance reduction capability has been developed for the MCNP Monte Carlo transport code. This new variance reduction capability developed for MCNP5 employs the PARTISN multigroup discrete ordinates code to generate mesh-based weight windows. The technique of using deterministic methods to generate importance maps has been widely used to increase the efficiency of deep penetration Monte Carlo calculations. The application of this method in MCNP uses the existing mesh-based weight window feature to translate the MCNP geometry into geometry suitable for PARTISN. The adjoint flux, which is calculated with PARTISN, is used to generate mesh-based weight windows for MCNP. Additionally, the MCNP source energy spectrum can be biased based on the adjoint energy spectrum at the source location. This method can also use angle-dependent weight windows.

Shielding for a cyclotron used for medical isotope production in China

Monte Carlo and discrete ordinate calculations have been performed to determine the doses at several locations in a positron emission tomography (PET) facility in China, where the radiation source is a cyclotron that is used for the production of the isotopes necessary for PET scans. The energy-dependent neutron source term is obtained by calculations using the ALICE code, and is interpolated for input to Monte Carlo and discrete ordinate calculations. The building that houses the cyclotron has a labyrinth of walls to minimise dose to operators and to other occupants of the building. Unbiased Monte Carlo calculations did not converge after more than one week of CPU time, whereas direction biasing alone resulted in convergence in several days. A study of several biasing techniques indicated that about a factor of 3 in computational efficiency is obtained using evaluated biasing methods. The use of adjoint fluxes for biasing Monte Carlo calculations can improve computational efficiencies by one or two orders of magnitude for some problems.

The refined shielding design for the cyclotron room of the Buddhist Tzu Chi General Hospital

Full-scale Monte Carlo simulations of the cyclotron room of the Buddhist Tzu Chi General Hospital were carried out to improve the inadequate maze design. The double differential neutron source from the 18O(p,n)18F reaction was adopted for the calculation. The weight window variance reduction technique, where the weight window was set by applying the adjoint flux, has been implemented in the MCNP run to facilitate the calculation of the dose rates outside the cyclotron room. Dose rates including neutron and gamma-ray components were calculated for some maze shielding modifications.

Skyshine analysis using energy and angular dependent dose-contribution fluxes obtained from air-over-ground adjoint calculation

A generalised and convenient skyshine dose analysis method has been developed based on forward-adjoint folding technique. In the method, the air penetration data were prepared by performing an adjoint DOT3.5 calculation with cylindrical air-over-ground geometry having an adjoint point source (importance of unit flux to dose rate at detection point) in the centre. The accuracy of the present method was certified by comparing with DOT3.5 forward calculation. The adjoint flux data can be used as generalised radiation skyshine data for all sorts of nuclear facilities. Moreover, the present method supplies plenty of energy-angular dependent contribution flux data, which will be useful for detailed shielding design of facilities.

Influence of the external neutron sources in the criticality prediction using 1/ M curve

The influence of external neutron sources in the process to obtain the criticality condition is estimated. To reach this objective, the three-dimensional neutron diffusion equation in two groups of energy is solved, for a subcritical PWR reactor core with external neutron sources. The results are compared with the solution of the corresponding problem without external neutron sources, that is an eigenvalue problem. The method developed for this purposes it makes use of both the nodal method (for calculation of the neutron flux) and the finite differences method (for calculation of the adjoint flux). A coarse mesh finite difference method was developed for the adjoint flux calculation, which uses the output of the nodal expansion method. The results regarding the influence of the external neutron source presence for attaining criticality have shown that far from criticality it is necessary to calculate the reactivity values of the system.

New dynamic method to measure rod worths in zero power physics test at PWR startup

To measure and validate the worth of control (or shutdown) bank in zero power physics test at PWRs, a dynamic control rod reactivity measurement (DCRM) technique has been developed and applied to six startups of Westinghouse plants as well as Korea Standard Nuclear power Plants based on the Combustion Engineering System 80 NSSS. With this technique, just one test bank is inserted into the bottom of the core at maximum stepping rate and withdrawn immediately to the all rod-out position. Specially designed inverse point kinetics equations are used to determine the test bank worth from the measured ex-core detector signals, which are controlled by the neutron-to-response conversion factor and the dynamic-to-static conversion factor. These two parameters are predetermined by the three-dimensional neutron adjoint flux distribution for both the top and bottom ex-core detector and the three-dimensional steady and transient core power distribution for test bank movement. To eliminate the gamma-ray effect on ex-core detector signals, a simple method, using reactivity curve characteristics, was also developed. To verify the DCRM method, a total of 28 bank worths of six different PWRs was measured by the DCRM and compared with those of conventional method. Results show that the DCRM method has a similar accuracy as the conventional technique. However, with the DCRM method, it only takes ∼15 min per bank from the beginning of rod insertion to the determination of measured static worth. From its performance, one can conclude that the DCRM method is an effective replacement for the conventional rod worth measurement method.

Kinetics models for safety studies of accelerator driven systems

Safety analyses of Accelerator Driven Systems (ADSs) are mainly performed by codes developed in the past for critical reactors, a point-kinetics model for computing the transient power being often employed. It is shown that this model – that assumes time-independent neutron direct and adjoint flux shapes – may be inaccurate in the ADS case even if it is acceptable for a similar “critical” case. Although the material distribution remains almost unchanged, flux (power) shape variations may be significant in the first case due to external source related effects. An option for a more refined modelling of the neutron adjoint flux in ADS analyses is discussed. This option is shown to be rather complicated in the general case: it may give rise to involvement of several adjoint flux shapes and several sets of related point-kinetics parameters (reactivity, etc.). To improve the accuracy of the point-kinetics treatment, an extended point-kinetics model, which employs several flux (or power) shapes precomputed at steady-state conditions, is proposed in the paper. This approach may help to avoid the spatial kinetics treatment if no strong material movement occurs. The reactivity and other point-kinetics parameters are defined similarly to a critical reactor.

Proposal of Direct Calculation of Kinetic Parameters βeff and Based on Continuous Energy Monte Carlo Method

Direct calculation methods of kinetic parameters are proposed based on the continuous energy Monte Carlo method. In the proposed methods, the effective delayed neutron fraction βeff and the neutron generation time ∧ are estimated using eigenvalue calculations. The expected number of fission neutrons in the next generation is newly applied to the proposed methods instead of the adjoint flux that has been conventionally used. The algorithms to estimate the kinetic parameters are established and incorporated into the continuous energy Monte Carlo transport calculation code MCNP-4C, which is versatile for eigenvalue calculations of nuclear reactor cores with various types of neutron energy spectrum and geometry. The proposed methods were validated since the calculated values agreed with the experimental data of βeff and βeff/∧ for the critical cores within accuracies of 4.5% and 10%, respectively.

Adjoint Monte Carlo method for prostate external photon beam treatment planning: an application to 3D patient anatomy

Recently, the theoretical framework of the adjoint Monte Carlo (AMC) method has been developed using a simplified patient geometry. In this study, we extended our previous work by applying the AMC framework to a 3D anatomical model called VIP-Man constructed from the Visible Human images. First, the adjoint fluxes for the prostate (PTV) and rectum and bladder (organs at risk (OARs)) were calculated on a spherical surface of 1 m radius, centred at the centre of gravity of PTV. An importance ratio, defined as the PTV dose divided by the weighted OAR doses, was calculated for each of the available beamlets to select the beam angles. Finally, the detailed doses in PTV and OAR were calculated using a forward Monte Carlo simulation to include the electron transport. The dose information was then used to generate dose volume histograms (DVHs). The Pinnacle treatment planning system was also used to generate DVHs for the 3D plans with beam angles obtained from the AMC (3D-AMC) and a standard six-field conformal radiation therapy plan (3D-CRT). Results show that the DVHs for prostate from 3D-AMC and the standard 3D-CRT are very similar, showing that both methods can deliver prescribed dose to the PTV. A substantial improvement in the DVHs for bladder and rectum was found for the 3D-AMC method in comparison to those obtained from 3D-CRT. However, the 3D-AMC plan is less conformal than the 3D-CRT plan because only bladder, rectum and PTV are considered for calculating the importance ratios. Nevertheless, this study clearly demonstrated the feasibility of the AMC in selecting the beam directions as a part of a treatment planning based on the anatomical information in a 3D and realistic patient anatomy.