Adjoint Transport Calculations Research Articles

Verification of the Discrete Ordinates Goal-Oriented Multi-Collision Source Algorithm with Neutron Streaming Problems

The shielding calculation of neutron streaming problems with ducts is characterized by the strong anisotropy of angular flux, which poses a challenge for the analysis of nuclear installations. The discrete ordinate method is one of the most commonly deterministic techniques to solve the neutron transport equation, in which the accuracy and efficiency neutron are crucial to ensure the reliability of the streaming shielding simulation. We implemented the goal-oriented multi-collision source algorithm in the 3D transport code ARES. This algorithm can determine the importance factor based on the adjoint transport calculation, obtain the response function to enable problem-dependent, goal-oriented spatial decomposition, and provide the error estimation as a driving force behind the dynamic quadrature to optimize the source iteration. This study focuses on verifying the goal-oriented multi-collision source algorithm under the neutron streaming problems, and the capabilities of the algorithm have been tested on IRI-TUB benchmark of SINBAD database. The numerical results show that the algorithm can effectively control the angular discretization error for the neutron streaming problems, which is more economical than the traditional discrete ordinate calculation.

Goal-oriented multi-collision source algorithm for discrete ordinates transport calculation

Discretization errors are extremely challenging conundrums of discrete ordinates calculations for radiation transport problems with void regions. In previous work, we have presented a multi-collision source method (MCS) to overcome discretization errors, but the efficiency needs to be improved. This paper proposes a goal-oriented algorithm for the MCS method to adaptively determine the partitioning of the geometry and dynamically change the angular quadrature in remaining iterations. The importance factor based on the adjoint transport calculation obtains the response function to get a problem-dependent, goal-oriented spatial decomposition. The difference in the scalar fluxes from one high-order quadrature set to a lower one provides the error estimation as a driving force behind the dynamic quadrature. The goal-oriented algorithm allows optimizing by using ray-tracing technology or high-order quadrature sets in the first few iterations and arranging the integration order of the remaining iterations from high to low. The algorithm has been implemented in the 3D transport code ARES and was tested on the Kobayashi benchmarks. The numerical results show a reduction in computation time on these problems for the same desired level of accuracy as compared to the standard ARES code, and it has clear advantages over the traditional MCS method in solving radiation transport problems with reflective boundary conditions.

Optimization study of pressurized water reactor secondary neutron source location

During the loading and refueling of Pressurized Water Reactor (PWR), Sb-Be secondary neutron source is generally used to induce fission reaction to increase ex-core detector signal, which could eliminate the monitoring blind area. In order to obtain the optimal position of secondary neutron source which would lead a maximum signal, it is necessary to evaluate the contribution of secondary neutron source at different potential positions. Based on CAP1400 PWR model, Discrete ordinate (SN) adjoint transport method has been studied by using 199 groups fine cross section. Fission source contribution is simulated by controlling outer iteration in adjoint transport calculation. The contribution of secondary neutron sources at different potential positions has been analyzed. Monte Carlo (MC) forward calculation has been used as a comparison, and the two results are in good agreement with each other. Through the analysis, secondary neutron source optimal location in reactor core can be obtained. The innovative method can provide a reference for the layout optimization of secondary neutron source.

Improved memory management for solving neutron transport via a novel Modular Ray Tracing (MRT) approach embedded in parallel method of characteristic (MOC) framework

Poor memory management can be considered as the first restrictive barrier to the implementation of the neutron transport equation for large reactors. The current research presents a new MRT with the ability to reduce the required memory and increase cache coherency. Also, given the high impact of the angular quadrature set used in the transport calculation, the effects of the angular set are investigated. As a reliable trial for an angular quadrature set examination, comparisons between the Forward and Adjoint transport calculations are performed.Three types of angular quadrature sets are adopted for our optimized MOC steady-state calculation. Based on two known constraints for the quadrature set evaluation, the accuracy and computational cost of the presented sets are examined. Satisfying these two conditions, our implemented Legendre-Chebyshev quadrature set with S20 angular order results in 7.0000E-06 and the 9.0000E-06 difference with reference Keff of G5G7 benchmark in the forward and adjoint calculation respectively. As well, our optimized parallel algorithm for MOC results in 6.18 speedup with 16 threads.

Advanced two-level CMFD acceleration method for the 3D whole-core high-fidelity neutron adjoint transport calculation

In the 2D/1D method, a global adjoint CMFD based on the generalized equivalence theory is built to synthesize the 2D radial MOC adjoint and 1D axial NEM adjoint calculation and also to accelerate the iteration convergence of 3D whole-core adjoint transport calculation. Even more important, an advanced yet accurate two-level (TL) CMFD acceleration technique is proposed, in which an equivalent one-group adjoint CMFD is established to accelerate the multi-group adjoint CMFD and then to accelerate the 3D whole-core adjoint transport calculation efficiently. Based on these method, a new code is developed to perform 3D adjoint neutron flux calculation. Then a set of VERA and C5G7 benchmark problems are chosen to verify the capability of the 3D adjoint calculations and the effectiveness of TL CMFD acceleration. The numerical results demonstrate that acceptable accuracy of 2D/1D adjoint calculations and superior acceleration of TL CMFD are achievable.

Inverse estimation methods of unknown radioactive source for fuel debris search

To identify the distribution of fuel debris remaining in the reactor vessel and/or the containment vessel of Fukushima Daiichi NPS, we focused on the inverse estimation of radioactive source distribution using the measured radiation counts. The Maximum Likelihood-Expectation Maximization (ML-EM) and the Moore-Penrose Matrix Inverse (MPMI) methods are examined. The ML-EM method has been used for the image reconstruction of computed tomography, and the MPMI method is one of the solution methods for simultaneous linear equations with the underdetermined condition. A simple calculation model simulating a containment vessel was constructed including detectors and radiation sources. In an actual situation, a sufficient number of radiation measurement positions would not be available owing to the complexity of structures inside the containment vessel. Thus, the number of radiation measurement points (number of constraints) is smaller than that of radiation source positions. It means that an underdetermined inverse problem should be solved. The detection probability of radiation (neutron or photon) is calculated by the adjoint transport calculation since the detection probability is used as the coupling coefficient between radiation counts at a detector and a radioactive source. The result of estimation using the ML-EM or the MPMI method indicates that the accuracy of estimation depends on the distance between a radiation source and a detector, and measurement positions of radiation count. The ML-EM and the MPMI methods show different prediction accuracy depending on the prediction condition. It is found that reasonable prediction accuracy would be obtained when the detectors are placed at the vicinity of radiation sources of interest.

Ex-Core Detector Response Evaluation of the SMART Reactor by Using the DORT Code

The development of a 660 MWt integral reactor SMART (System integrated Modular Advanced ReacTor), called SMART-660, has been initiated in Korea. A preliminary evaluation of the detector response functions was performed for the ex-core detector of the SMART-660. First, a candidate position to install the ex-core detector was selected by considering the reactor configuration, the detector performance, and the thermal neutron flux distribution in the reactor assembly. Second, the detector response functions such as the shape annealing function (SAF) and assembly weighting factor (AWF) for the candidate ex-core detector were calculated by using an adjoint transport calculation with the DORT code and the BUGLE-96 cross-section library. The GEOSHIELD program was used to generate the DORT input file and to process the DORT output file with a graphic visualization. This study has shown that the response functions of the ex-core detector are only sensitive to some parts of the reactor and additional efforts are required to reduce the locality of the ex-core detector response.

Identification of an Unknown Material in a Radiation Shield Using the Schwinger Inverse Method

The Schwinger method for solving inverse gamma-ray transport problems was proposed in a previous paper. The method is iterative and requires a set of uncoupled forward and adjoint transport calculations in each iteration. In this paper, the Schwinger inverse method is applied to the problem of identifying an unknown material in a radiation shield by calculating its total macroscopic photon cross sections. The gamma source (its energy and spatial distribution as well as the composition of the material) is known and the total (angle-independent) gamma leakage is measured. In numerical one-dimensional spherical and slab test problems, the Schwinger inverse method successfully calculated the photon cross sections of an unknown material. Material identification was successfully achieved by comparing the calculated cross sections with those in a precomputed material cross-section library, although there was some ambiguity when realistic measurements were used. The Schwinger inverse method compared very favorably with the standard single-energy transmission technique.

LTS N solution of the adjoint neutron transport equation with arbitrary source for high order of quadrature in a homogeneous slab

In this work we present the LTS N solution of the adjoint transport equation for an arbitrary source, testing the aptness of this analytical solution for high order of quadrature in transport problems and comparing some preliminary results with the ANISN computations in a homogeneneous slab geometry. In order to do that we apply the new formulation for the LTS N method based on the invariance projection property, becoming possible to handle problems with arbitrary sources and demanding high order of quadrature or deep penetration. This new approach for the LTS N method is important both for direct and adjoint transport calculations and its development was inspired by the necessity of using generalized adjoint sources for important calculations. Although the mathematical convergence has been proved for an arbitrary source, when the quadrature order or deep penetration is required the LTS N method presents computational overflow even for simple sources (sin, cos, exp, polynomial). With the new formulation we eliminate this drawback and in this work we report the numerical simulations testing the new approach.

Deterministic adjoint transport applications for He-3 neutron detector design

This work focuses on the determination of predicted neutron detector response accomplished using neutron importance derived from an adjoint discrete ordinates ( S N) transport calculation. A hypothetical detector apparatus, intended to detect fast neutrons, was modeled using He-3 tubes with graphite moderation using the PENTRAN™ 3-D multi-group discrete ordinates parallel transport code system. The detector geometry was modeled using z-axis symmetry and discretized into 30,280 3-D Cartesian cells. The material spatial mesh was generated using the PENMSH™ code in the PENTRAN system. The 47-group BUGLE-96 neutron cross section library was used for construction of macroscopic neutron cross sections. Results from an S 8 angular quadrature using P 3 anisotropy are presented. An adjoint transport source was established in the model using group dependent He-3 response cross sections. Each He-3 tube contained an adjoint source aliased to group He-3 absorption cross sections to permit assessment of detector performance. The spectrally dependent detector response from neutron capture in He-3 tubes from an arbitrary source can, therefore, be readily determined. This response comes from the complete integral of the actual source strength weighted by the adjoint function at the source location for any source distribution scenario. For selected neutron energies, an equivalent forward MCNP Monte Carlo model was used to demonstrate good agreement with the detector response determined from the adjoint calculation. The graphite used in this design has a large impact on detector performance due to the increasing sensitivity inherent in He-3 gas as neutrons thermalize. Computational adjoint results presented here predict a fast neutron detector design that yields efficiencies between 30 and 50% for neutron energies below 3 keV, and up to 30% efficiencies for neutron energies between 3 keV and 1 MeV. Overall, the methodology applied here highlights the elegant nature of an adjoint transport calculation.

A Hybrid Multigroup/Continuous-Energy Monte Carlo Method for Solving the Boltzmann-Fokker-Planck Equation

A hybrid multigroup/continuous-energy Monte Carlo algorithm is developed for solving the Boltzmann-Fokker-Planck equation. This algorithm differs significantly from previous charged-particle Monte Carlo algorithms. Most importantly, it can be used to perform both forward and adjoint transport calculations, using the same basic multigroup cross-section data. The new algorithm is fully described, computationally tested, and compared with a standard condensed history algorithm for coupled electron-photon transport calculations.

Generation of Spatial Weighting Functions for Ex-Core Detectors by Adjoint Transport Calculation

The response rate of ex-core detectors depends on the power level, power distribution, and reactor configuration. For the analysis of the detector response rate for various core power distributions, it is important to generate spatial weighting functions that are insensitive to small changes in the core power distribution and determined by the reactor configuration. Two-dimensional discrete ordinates adjoint transport calculations are used to calculate the core axial weighting functions. The effects of the reactors operating conditions on the core axial weighting functions are analyzed, and it is found that the soluble boron concentration in the reactor coolant has little effect while the core power level affects the core axial weighting functions significantly. A comparison between the results of the adjoint and forward transport calculations shows an excellent agreement. However, the adjoint transport method provides more detailed data and requires less computing time than forward transport method.

Accounting for Changing Source Distributions in Light Water Reactor Surveillance Dosimetry Analysis

AbstractA technique is described to account for effects of space- and time-dependent core source variations on pressure vessel surveillance dosimetry analysis. The procedure first defines an easily implemented geometry for a single adjoint transport calculation. The results from the adjoint calculation can then be combined with those from a single forward calculation, in conjunction with an adjoint scaling technique, to yield activities and pressure vessel fluxes simultaneously for a wide range of source distributions, dosimeter response functions, and detector locations. This method has been implemented in the LEPRICON code system for vessel fluence determination. An application to an R-θ model of an operating power reactor shows that effects of source perturbations resulting in 20% changes in the core leakage can be predicted within ∼3% at both downcomer and cavity dosimeter locations, for six different dosimeters, by choice of a single suitable adjoint response function.

Development of flat response counters by adjoint transport calculations

Approaches are considered by applying two-dimensional S N methods. Sensitivity vs neutron energy is controlled by changing source distribution. Optimum geometry of drilled holes follows from adjoint angular flux.

Neutron Penetration Factors Obtained by Using Adjoint Transport Calculations

Adjoint transport theory is most widely used in perturbation theory. A most common problem here is the determination of the reactivity change in a self-multiplying system due to the insertion of an...