Discrete Ordinate Code Research Articles

Study on multi-objective optimization for nuclear reactor radiation shield design coupling genetic algorithm with paralleling discrete ordinate code

An optimization method combining a genetic algorithm and the paralleling discrete ordinate code is developed and applied to the shielding design of the Savannah reactor. Several approaches are studied to cope with the multi-objective optimization problem, such as transforming into single-objective optimization approach, non-dominated sorting approach, and the approach of adding constraints on sub-objectives. Comparing with the current design methods, the optimization method developed in this study shows better performance since the discrete ordinate calculations are free of statistical fluctuations. The multi-objective optimization approach with constraints on sub-objectives seems to have the best performance. More optimized individuals satisfying the constraints can be obtained, and the optimized objectives show the best improvements comparing with the initial design.

Numerical results for polarized light scattering in a spherical atmosphere

We report numerical results for polarized light reflection from the top of a Rayleigh scattering spherical atmosphere with height-dependent single scattering albedo over a dark surface. Michael Mishchenko considered this scenario back in the 1990’s, for a plane-parallel atmosphere of unit optical thickness (OT = 1), for which radiance errors arising from neglecting polarization reaches their highest values. To further extend Mishchenko's results, we consider a value of OT = 0.25, for which the effect of atmospheric curvature is pronounced. New results are generated using three state-of-the art radiative transfer (RT) codes. These are: the MYSTIC and MCSSA models, which simulate light scattering in a true-spherical atmosphere using Monte Carlo methods; and the discrete ordinate code VLIDORT, operating with a new multiple-scatter spherical correction designed to deliver reasonable approximations to spherical-medium scattering. In this work, we report results for both single and multiple scattering; this will help to support the validation of existing and future polarized spherical RT codes, especially those using approximative methods to deal with sphericity.

Marvin: A parallel three-dimensional transport code based on the discrete ordinates method for reactor shielding calculations

Marvin is a parallel three-dimensional discrete ordinates (SN) code dedicated for reactor shielding calculations. It employs the multi-group and SN approximations to discretize the energy and angular variables, respectively. Multiple discretization methods, namely weighted diamond difference, theta-weighted diamond difference and step-characteristic, have been implemented to discretize the spatial variables. The within-group transport equation is solved by the source iteration or Krylov subspace solvers, both of which employ the Koch-Baker-Alcouffe (KBA) algorithm to parallelize the transport sweep process. The upscattering in the thermal energy range is treated with thermal upscattering iteration scheme. The code design of Marvin features the mixed programming model and a three-layer architecture design. Two benchmark problems devised from realistic reactor core configurations, namely the VENUS-3 and HBR-2 problems, were calculated to evaluate the correctness, accuracy and parallel computing performance of Marvin. The calculation results were compared against TORT solutions as well as measured data. The validation results indicate that: Marvin is properly implemented and can faithfully produce solutions to the reactor shielding problems; the computing efficiency of Marvin can be significantly enhanced via the KBA parallel transport sweep algorithm.

Steady-state neutronic measurements and comprehensive numerical analysis for the BME training reactor

This paper describes steady-state reactor physics measurements and calculations that were performed for the Training Reactor of Budapest University of Technology and Economics (BME TR) with the purpose of benchmarking. Based on the available geometry specifications and material compositions a model of BME TR was created with the well-validated, general-purpose Serpent 2 Monte Carlo code. Uncertain parameters (such as fuel density and control rod positions) were adjusted to related measurements. The Serpent 2 model was used for the generation of group constants, examining several homogenization schemes. Models were created in the PARCS diffusion code, the SPNDYN diffusion and SP3 code and the PARTISN discrete ordinates code. Various Monte Carlo and deterministic calculations were performed with the adjusted models and the results were then compared with actual measured data. The calculations and measurements show good agreement, this way the Serpent model was successfully validated, while the deterministic models make a good basis for more complex benchmarks in the future, such as transients with thermal–hydraulic feedback.

High-efficiency simulation of VENUS-3 neutron-shielding problem with an automatic and enhanced hybrid Monte-Carlo-Deterministic method

To solve the deep-penetration problem in the Monte-Carlo simulation, the classic consistent adjoint driven importance sampling (CADIS) and forward-weighted CADIS (FW-CADIS) hybrid MC-deterministic method is adopted and enhanced in this paper. An automatic hybrid MC-deterministic code, NECP-MCX, is newly developed. NECP-MCX embeds the discrete-ordinate code NECP-Hydra to automatically perform the forward and adjoint neutron-transport calculations and set up the importance parameters for the MC simulation. In massive parallel calculation for large shielding problems, huge amounts of memory are required to save detailed weight-window information. To mitigate the memory limitation, a mesh-coarsening algorithm is developed based on the contributon theory. It can be applied in the traditional Cartesian-mesh and the nested-mesh structure. NECP-MCX was applied to the VENUS-3 benchmark. The results show that NECP-MCX with the enhanced hybrid method is effective in variance reduction and memory saving for the simulation of large and complicated shielding problems on parallel computer resources.

Computational analysis of the Discrete Ordinates Adjoint Function applicability in VVER Monte Carlo modelling

The possible use of the adjoint function as an automated source for weight windows generation in the case of VVER-1000 benchmark transport calculations is presented in this paper. The numerical solution for obtaining the adjoint function is performed using the Discrete Ordinates code TORT. Interface software is used to translate the adjoint function into weight windows’ input parameters. A deep penetration problem has been modelled for the VVER-1000 Mock Up, built at the NRI/Rez’s LR-0 critical assembly for VVER-1000 benchmark studies. The neutron flux is evaluated in three locations placed on the azimuthal symmetry axis, radially from the reactor core to the shielding. The calculations are performed in three different approaches: by MCNP code using the weight windows obtained through the adjoint function, by MCNP code using weight windows created by the author through the standard procedure and by the TORT code. The results are compared in terms of integral fluxes and the validity of the adjoint function in variance reduction for VVER reactor Monte Carlo calculations is justified.

Application of multiple scattering theory to Doppler velocimetry of ejecta from shock-loaded samples

We address the actual problem of retrieving the physical parameters of ejecta from data of heterodyne Doppler velocimetry. Under the assumption that ejected particles are randomly spaced and their number within the probed volume is great, the noise-free component of the power spectrum of heterodyne beats is shown to be expressed in terms of a solution to the transport equation for the field correlation function which accounts for multiple scattering and absorption of the probing beam in the ejecta cloud. This provides a means for theoretical modeling of experimental Doppler data. The ejecta cloud is considered as a plane layer of particles moving in the air away from the free-surface of the shock-loaded sample. The spatial profile of the scattering coefficient and the cloud thickness are related directly to the distribution of ejected particles over velocities and coordinates. The slowing-down of ejected particles in the air leads to an ambiguous relation between the velocity of particles and their position, resulting in essential complication of the particle distribution. Using a multi-group description of the ejecta cloud, we reduce the transport equation to a system of linked Milne-like equations. The system is solved numerically with the discrete ordinate code. Varying the values of the cloud optical thickness and the parameters of the particle distribution over sizes and initial velocities, we fit the calculated spectrum to the time-resolved data of heterodyne Doppler measurements. Such an approach enables retrieving the primary ejecta characteristics (the total ejected mass, the density-velocity profile and the size distribution of ejected particles) directly from heterodyne experiments. Application of the proposed method to the velocimetry data on ejecta is presented.

High-accuracy neutron diffusion calculations based on integral transport theory

In this paper, the Ronen method is developed, implemented, and applied to resolve the neutron flux and the criticality eigenvalue in simple one-dimensional homogeneous and heterogeneous problems. The Ronen method is based on iterative calculations of correction factors to use in a multigroup diffusion model, where the factors are actually given by the integral transport equation. In particular, spatially dependent diffusion constants are modified locally in order to reproduce new estimates of the surface currents obtained by the integral transport operator. The diffusion solver employed in this study uses finite differences, and the transport-corrected currents are introduced into the numerical scheme as drift terms. The corrected solutions are compared against reference results from a discrete ordinate code. The results match well with the reference solutions, especially in the limit of fine meshes, but slow convergence of the scalar flux is reported.

Development of 3-D parallel first-collision source method for discrete ordinate code JSNT-S

A new three-dimensional parallel first-collision source code named JSNT-FCS is developed to alleviate the ray effect in discrete ordinate method. JSNT-FCS applies the ray trace method to obtain uncollided flux both for the Cartesian and cylindrical geometry. Parallel calculation is implemented based on spatial domain partition algorithm. The ray trace is performed in each domain based on the entire meshes to avoid parallel communication between domains. To decrease the error of FCS method, source region is automatically refined. The reflection boundary condition is handled by adding a symmetric source region at the opposite side of reflection boundary. Several benchmarks are calculated, and all results of calculation prove that the accuracy of JSNT-FCS is high enough. Response calculation of source range detector is performed based on the geometry of HBR-2, and the calculation results indicate that JSNT-FCS code can effectively eliminate the nonphysical oscillations of flux spatial distribution.

Solution of 3D linearly anisotropic scattering, fixed-source multigroup xyz reactor problems by the [formula omitted] Boundary Element – Response Matrix method

This paper aims at extending the work performed by means of the AN Boundary Element – Response Matrix method in a previous paper, devoted to criticality problems, to non-homogeneous problems (i.e. the problems that involve a fixed external source, as in case of the accelerator driven systems). As in the preceding paper, the interest is given to the 3D, xyz multigroup systems in which linearly anisotropic scattering is allowed. Standard interface conditions have been adopted and no external intervention in order to improve the numerical results, such as some kind of discontinuity factors, has been introduced. The latter choice is motivated by the need of establishing a clean test of the performances of the AN method in itself. Results of 2D and 3D multigroup fixed source problems obtained with the method described in this paper are compared with those obtained by well assessed reference codes such as the MCNP Monte Carlo code and the PARTISN discrete ordinates code. Even in the absence of discontinuity factors and despite of the intrinsically approximate character of the AN method, the accuracy of the numerical results is more than satisfactory. Finally the computational time is in most cases much shorter than that required by the chosen reference code.

Modeling polarized radiative transfer in the ocean-atmosphere system with the GPU-accelerated SMART-G Monte Carlo code

SMART-G (Speed-Up Monte-Carlo Advanced Radiative Transfer code with GPU) is a radiative transfer solver for the coupled ocean-atmosphere system with a wavy interface. It is based on the Monte-Carlo technique, works in either plane-parallel or spherical-shell geometry, and accounts for polarization. The vector code is written in CUDA (Compute Unified Device Architecture) and runs on GPUs (Graphic Processing Units). For typical simulations, the GPU-based code running on the Nvidia GTX 1070 card is shown to be 100 time faster than a state of the art CPU-based code running on an AMD PhenomIIx4 965 at 3.4GHz. This makes SMART-G competitive, in terms of computational efficiency, with codes based on other techniques (e.g., discrete ordinate, doubling-adding, matrix-operator, and successive-orders-of-scattering), while allowing maximum flexibility regarding the scope of the simulations. The monochromatic version of the code without trans-spectral processes (Raman scattering, fluorescence) is described, including the treatment of photon propagation and interactive processes (elastic scattering, absorption, reflection, refraction, but not thermal emission) and variance reduction (local estimate). Benchmark values are accurately reproduced for clear and cloudy atmospheres over a wavy reflecting surface and a black ocean. Results obtained for a diffusely reflecting ocean agree with those from a discrete ordinate code. SMART-G may be used, not only as a reference code, but also to simulate the signal/imagery observed/produced by optical sensors, create accurate look-up tables, and investigate new remote sensing techniques.

Second-Order Sensitivity Analysis of Uncollided Particle Contributions to Radiation Detector Responses

This work presents an application of Cacuci’s Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) to the simplified Boltzmann equation that models the transport of uncollided particles through a medium to compute efficiently and exactly all of the first- and second-order derivatives (sensitivities) of a detector’s response with respect to the system’s isotopic number densities, microscopic cross sections, source emission rates, and detector response function. The off-the-shelf PARTISN multigroup discrete ordinates code is employed to solve the equations underlying the 2nd-ASAM. The accuracy of the results produced using PARTISN is verified by using the results of three test configurations: (1) a homogeneous sphere, for which the response is the exactly known total uncollided leakage, (2) a multiregion two-dimensional (r-z) cylinder, and (3) a two-region sphere for which the response is a reaction rate. For the homogeneous sphere, results for the total leakage as well as for the respective first- and second-order sensitivities are in excellent agreement with the exact benchmark values. For the nonanalytic problems, the results obtained by applying the 2nd-ASAM to compute sensitivities are in excellent agreement with central-difference estimates. The efficiency of the 2nd-ASAM is underscored by the fact that, for the cylinder, only 12 adjoint PARTISN computations were required by the 2nd-ASAM to compute all of the benchmark’s 18 first-order sensitivities and 224 second-order sensitivities, in contrast to the 877 PARTISN calculations needed to compute the respective sensitivities using central finite differences, and this number does not include the additional calculations that were required to find appropriate values of the perturbations to use for the central differences.

Study on variance reduction technique based on adjoint Discrete Ordinate method

For deep-penetration shielding calculation, Monte Carlo (MC) method requires simulating a great number of particles to obtain reliable results, thus huge amount of computation time is the main problem of the MC method. Source biasing and weight window technique effectively decrease the tally error of deep-penetration problem. This paper studies variance reduction techniques (VRTs) based on adjoint fluence rate from a Discrete Ordinate (SN) code JSNT, generates source biasing factors and weight window parameters for a MC code JMCT, develops source sampling subroutine for JMCT, and verifies the VRT on measurements, finally applies the VRT to reactor pressure vessel fast neutron fluence rate and cavity neutron and photon dose rate calculation. Numerical results show that the VRT increases calculation efficiency by 1–2 orders for deep-penetration shielding calculation with high precision compared with unbiased MC method.

A Detailed Comparison of Multidimensional Boltzmann Neutrino Transport Methods in Core-collapse Supernovae

Abstract The mechanism driving core-collapse supernovae is sensitive to the interplay between matter and neutrino radiation. However, neutrino radiation transport is very difficult to simulate, and several radiation transport methods of varying levels of approximation are available. We carefully compare for the first time in multiple spatial dimensions the discrete ordinates (DO) code of Nagakura, Yamada, and Sumiyoshi and the Monte Carlo (MC) code Sedonu, under the assumptions of a static fluid background, flat spacetime, elastic scattering, and full special relativity. We find remarkably good agreement in all spectral, angular, and fluid interaction quantities, lending confidence to both methods. The DO method excels in determining the heating and cooling rates in the optically thick region. The MC method predicts sharper angular features due to the effectively infinite angular resolution, but struggles to drive down noise in quantities where subtractive cancellation is prevalent, such as the net gain in the protoneutron star and off-diagonal components of the Eddington tensor. We also find that errors in the angular moments of the distribution functions induced by neglecting velocity dependence are subdominant to those from limited momentum-space resolution. We briefly compare directly computed second angular moments to those predicted by popular algebraic two-moment closures, and we find that the errors from the approximate closures are comparable to the difference between the DO and MC methods. Included in this work is an improved Sedonu code, which now implements a fully special relativistic, time-independent version of the grid-agnostic MC random walk approximation.

A Deep Penetration Problem Calculation Using AETIUS:An Easy Modeling Discrete Ordinates Ṯransport Code UsIng Unstructured Tetrahedral Mesh, Shared Memory Parallel

As computing power gets better and better, computer codes that use a deterministic method seem to be less useful than those using the Monte Carlo method. In addition, users do not like to think about space, angles, and energy discretization for deterministic codes.However, a deterministic method is still powerful in that we can obtain a solution of the flux throughout the problem, particularly as when particles can barely penetrate, such as in a deep penetration problem with small detection volumes.Recently, a new state-of-the-art discrete-ordinates code, ATTILA, was developed and has been widely used in several applications. ATTILA provides the capabilities to solve geometrically complex 3-D transport problems by using an unstructured tetrahedral mesh.Since 2009, we have been developing our own code by benchmarking ATTILA. AETIUS is a discrete ordinates code that uses an unstructured tetrahedral mesh such as ATTILA. For pre- and post- processing, Gmsh is used to generate an unstructured tetrahedral mesh by importing a CAD file (*.step) and visualizing the calculation results of AETIUS. Using a CAD tool, the geometry can be modeled very easily.In this paper, we describe a brief overview of AETIUS and provide numerical results from both AETIUS and a Monte Carlo code, MCNP5, in a deep penetration problem with small detection volumes. The results demonstrate the effectiveness and efficiency of AETIUS for such calculations.

Comparison of Standard Light Water Reactor Cross-Section Libraries using the United States Nuclear Regulatory Commission Boiling Water Reactor Benchmark Problem

This paper describes a comparison of contemporary and historical light water reactor shielding and pressure vessel dosimetry cross-section libraries for a boiling water reactor calculational benchmark problem. The calculational benchmark problem was developed at Brookhaven National Laboratory by the request of the U. S. Nuclear Regulatory Commission. The benchmark problem was originally evaluated by Brookhaven National Laboratory using the Oak Ridge National Laboratory discrete ordinates code DORT and the BUGLE-93 cross-section library. In this paper, the Westinghouse RAPTOR-M3G three-dimensional discrete ordinates code was used. A variety of cross-section libraries were used with RAPTOR-M3G including the BUGLE93, BUGLE-96, and BUGLE-B7 cross-section libraries developed at Oak Ridge National Laboratory and ALPAN-VII.0 developed at Westinghouse. In comparing the calculated fast reaction rates using the four aforementioned cross-section libraries in the pressure vessel capsule, for six dosimetry reaction rates, a maximum relative difference of 8% was observed. As such, it is concluded that the results calculated by RAPTOR-M3G are consistent with the benchmark and further that the different vintage BUGLE cross-section libraries investigated are largely self-consistent.

Changes to Irradiation Conditions of VVER-1000 Surveillance Specimens Resulting from Fuel Assemblies with Greater Fuel Height

The goal of the work was to obtain experimental data on the influence of newtype fuel assemblies with higher fuel rods on the irradiation conditions of surveillance specimens installed on the baffe of VVER-1000. For this purpose, two surveillance sets with container assemblies of the same design irradiated in reactors with different fuel assemblies in the core were investigated. Measurements of neutron dosimeters from these sets and retrospective measurements of 54 Mn activity accumulated in each irradiated specimen allow a detailed distribution of the fast neutron flux in the containers to be obtained. Neutron calculations have been done using 3D discrete ordinate code KATRIN. On the basis of the obtained results, a change of the lead factor due to newtype fuel assemblies was evaluated for all types of VVER-1000 container assemblies.

ACS Algorithm in Discrete Ordinates for Pressure Vessel Dosimetry

The Adaptive Collision Source (ACS) method can solve the Linear Boltzmann Equation (LBE) more efficiently by adaptation of the angular quadrature order. This is similar to, and essentially an extension of, the first collision source method. Previously, the ACS methodology has been implemented into the TITAN discrete ordinates code, and has shown speedups of 2-4 on a simple test problem, with very little loss of accuracy (within a provided adaptive tolerance). This work examines the use of the ACS method for a more realistic problem: pressure vessel dosimetry with the VENUS-2 MOX-fuelled reactor dosimetry benchmark. The ACS method proved to be able to obtain accurate results while being approximately twice as efficient as using a constant quadrature in a standard source iteration scheme.

Preliminary Study on Applying Discrete Ordinates Code Supporting Unstructured Tetrahedral Mesh to the 40-Degree Toroidal Segment ITER Model

The discrete ordinates code under development by KAERI uses an unstructured tetrahedral mesh, and can thus be applied to solve the radiation transport in a complicated geometry. In addition, the geometry modeling process has become much easier because computational tetrahedral meshes are generated based on the CAD file by Gmsh. This program has been enhancing its performance and adding functions for each application.In previous research, it was applied in a neutronics analysis for the Korea Helium Cooled Ceramic Reflector (HCCR) TBM. The total neutron fluxes were compared with the results from MCNPX and showed good agreement.In this paper, we applied our program to a simplified ITER model which is a 40-degree toroidal segment. The zone averaged total fluxes were compared with those of MCNPX, and total neutron flux distribution was visualized in a three-dimensional system domain.

Improvement of a calculation procedure of neutron-flux distribution for radioactivity inventory estimation for decommissioning of nuclear power plants

We improved a calculation procedure of neutron-flux distribution in the Primary Containment Vessel (PCV) to get reliable estimation of radioactivity inventory, which is crucial for specification of a decommissioning plan of nuclear power plants. We calculated two-dimensional (2D) distribution of neutron-flux in cylindrical coordinate (r–z) with a 2D discrete ordinate (Sn) code DORT by utilizing knowledge of the characteristics of neutron-flux distribution in the PCV of Tsuruga Nuclear Power Plant unit 1 (TS-1). The knowledge was obtained from measurements and three-dimensional (3D) calculation of neutron-flux distribution in cylindrical coordinate (r–θ–z) with a 3D Sn code TORT. The measurements were performed by using activation foils at 30 locations where the distribution characteristics could be observed in the PCV. The 3D calculation provided us better qualitative understandings about the characteristics of neutron-flux distribution in the PCV. An improved model of 2D calculation of neutron-flux distribution was generated by utilizing the knowledge from the measurements and the 3D calculation in the PCV. The reliability of 2D calculation was confirmed by comparing calculated fluxes with measured ones at locations where measurements were performed. The comparison between them showed that neutron-flux distribution calculated by the improved 2D model has met our criteria of the reliability. Therefore, the improved procedure provided a reliable 2D distribution of neutron-flux.