Neutral Particle Transport Problems Research Articles

On the Application of the Nyström Method to Two-Dimensional Neutral Particle Transport Problems in Heterogeneous Media

Over time, several methods were developed to deal with neutral particle transport problems. The interest in these problems is related to their wide range of applications, from neutron transport and heat transfer in nuclear reactors to radiative transfer in atmospheric clouds. Unlike the discrete ordinates or discrete ordinates–like methods, integral methods do not require discretization of angular variables. Instead, angular variables are completely eliminated by an integration procedure over the solid angle, which allows elimination of the ray effect. That said, this paper presents a new approach to estimate the scalar flux in two-dimensional fixed-source neutron transport problems in a heterogeneous medium, considering isotropic scattering and vacuum and reflective boundary conditions. Here, the Nyström method is combined with the singularity-subtraction technique to present an integral formulation for the scalar flux in a mesh grid over all regions of the domain. The iterative method of the Neumann series is used as an alternative to direct methods to solve the resulting system of equations generated from the domain discretization. Numerical results are given to verify the offered method.

The Random Ray Method Versus Multigroup Monte Carlo: The Method of Characteristics in OpenMC and SCONE

The Random Ray Method (TRRM) is a recently developed approach to solving neutral particle transport problems based on the Method of Characteristics. While the method previously has been implemented only in closed-source or limited-functionality codes, this work describes its implementation in two open-source Monte Carlo codes: OpenMC and SCONE. The random ray implementations required small modifications to the existing Multigroup Monte Carlo (MGMC) solvers, offering a rare venue for redundant, fine-grained, “apples-to-apples” speed and accuracy comparisons between transport methods. To this end, TRRM and MGMC solvers are evaluated against each other using each code’s native capabilities on reactor eigenvalue problems with different degrees of energy discretization. On the C5G7 benchmark (featuring only seven energy groups), TRRM achieves a maximum pin power error comparable to or lower than that of MGMC for a given run time. On a problem with 69 energy groups, MGMC is found to scale more efficiently, obtaining a lower pin power error for a given run time. However, the defining difference between the two transport methods is found to be their vastly different uncertainty distributions. Specifically, TRRM is found to maintain similar levels of accuracy and uncertainty throughout the simulation domain whereas MGMC can exhibit orders-of-magnitude greater errors in areas of the problem that feature low neutron flux. For instance, TRRM provided an up to 373 times speed advantage compared with MGMC for computing the flux in low-flux regions in the moderator surrounding the C5G7 core.

On a Response Matrix Solver for Slab-Geometry Neutral Particle Transport Problems in the Discrete Ordinates and Energy Multigroup Formulations Considering Non-Uniform Interior Sources

We present in this work an extension of the Response Matrix (RM) method for the numerical solution of slab-geometry neutral particle transport equation in the discrete ordinates (S) and energy multigroup formulations considering non-uniform sources. By using the term non-uniform we mean that the particle source is not spatially uniform inside the regions that compose the domain. The extended RM method, differently from the conventional RM method, is based on the solution of “source-independent” auxiliary problems (Green’s functions). The solution of these auxiliary problems is used in conjunction with the given non-uniform source to generate the sweeping matrices for the extended RM method. Numerical results with respect to both uniform and non-uniform source problems are given to illustrate the efficiency of the offered extended RM method.

The Multi-P N Method: A New Angular Discretization Technique for Neutral-Particle Transport Problems

The Multi-P N Method: A New Angular Discretization Technique for Neutral-Particle Transport Problems

Validation and evaluation of the ADVANTG code on the ICSBEP skyshine benchmark experiment

Validation and evaluation of the recently released ADVANTG code, which combines a well-known Monte Carlo (MC) transport code MCNP with a deterministic neutron transport code Denovo, is presented in the paper. The aim of ADVANTG is to automate the process of generating the variance reduction parameters for the fixed source MCNP calculations, which consequently accelerate the simulations in terms of the required CPU time.Reliability and consistent performance of the ADVANTG code were tested on a computationally demanding benchmark, i.e. the skyshine benchmark experiment from the ICSBEP handbook where neutron and photon scattering in the air above an open operating reactor are simulated. The speed-up factors or the increases in relative efficiency of up to 30,000 and 1400 compared to analog MCNP simulations were achieved using the ADVANTG-generated variance reduction parameters for neutrons and photons respectively. As the mean values obtained by the ADVANTG-accelerated simulation sit within the statistical uncertainties of the analog simulation, it was shown that no additional bias is introduced by the ADVANTG code. This article shows that ADVANTG code is a useful and effective tool for use in solving neutral particle transport problems in large geometries featuring skyshine.

Sub-cell balanced nodal expansion methods using S4 eigenfunctions for multi-group SN transport problems in slab geometry

A highly accurate S4 eigenfunction-based nodal method has been developed to solve multi-group discrete ordinate neutral particle transport problems with a linearly anisotropic scattering in slab geometry. The new method solves the even-parity form of discrete ordinates transport equation with an arbitrary SN order angular quadrature using two sub-cell balance equations and the S4 eigenfunctions of within-group transport equation. The four eigenfunctions from S4 approximation have been chosen as basis functions for the spatial expansion of the angular flux in each mesh. The constant and cubic polynomial approximations are adopted for the scattering source terms from other energy groups and fission source. A nodal method using the conventional polynomial expansion and the sub-cell balances was also developed to be used for demonstrating the high accuracy of the new methods. Using the new methods, a multi-group eigenvalue problem has been solved as well as fixed source problems. The numerical test results of one-group problem show that the new method has third-order accuracy as mesh size is finely refined and it has much higher accuracies for large meshes than the diamond differencing method and the nodal method using sub-cell balances and polynomial expansion of angular flux. For multi-group problems including eigenvalue problem, it was demonstrated that the new method using the cubic polynomial approximation of the sources could produce very accurate solutions even with large mesh sizes.

A cell-local finite difference discretization of the low-order quasidiffusion equations for neutral particle transport on unstructured quadrilateral meshes

We present a quasidiffusion (QD) method for solving neutral particle transport problems in Cartesian XY geometry on unstructured quadrilateral meshes, including local refinement capability. Neutral particle transport problems are central to many applications including nuclear reactor design, radiation safety, astrophysics, medical imaging, radiotherapy, nuclear fuel transport/storage, shielding design, and oil well-logging. The primary development is a new discretization of the low-order QD (LOQD) equations based on cell-local finite differences. The accuracy of the LOQD equations depends on proper calculation of special non-linear QD (Eddington) factors from a transport solution. In order to completely define the new QD method, a proper discretization of the transport problem is also presented. The transport equation is discretized by a conservative method of short characteristics with a novel linear approximation of the scattering source term and monotonic, parabolic representation of the angular flux on incoming faces. Analytic and numerical tests are used to test the accuracy and spatial convergence of the non-linear method. All tests exhibit O(h2) convergence of the scalar flux on orthogonal, random, and multi-level meshes.

Generalized discrete ordinates methods for neutral particle transport problems in slab geometry

Generalized discrete ordinates ( S N ) methods for steady-state monoenergetic neutral particle transport problems in slab geometry are developed for arbitrary order L of scattering anisotropy including odd-order N of Gauss–Legendre quadrature sets, provided L ≤ N − 1. The Laplace transform (LT S N ) method is applied to solve analytically the S N problems in homogeneous slabs. Numerical results are given to analyze the accuracy of the odd-order Gauss–Legendre S N formulations.

Comparison via parallel performance models of angular and spatial domain decompositions for solving neutral particle transport problems

A previously reported parallel performance model for angular domain decomposition (ADD) of the discrete ordinates approximation for solving multidimensional neutral particle transport problems is revisited for stronger validation. Three communication schemes, native MPI, the bucket algorithm, and the distributed bucket algorithm, are included in the validation exercise that is successfully conducted on a Beowulf cluster. The parallel component of the parallel performance model is largely independent of the communication scheme, in contrast with the communication component that is strongly dependent on the global reduce algorithm. Correct trends for each component and each communication scheme are measured for the Arbitrarily High Order Transport (AHOT) code, thus validating the performance models. Furthermore, extensive experiments illustrate the superiority of the bucket algorithm, in the sense that it incurs a smaller communication penalty compared to the native MPI and distributed bucket algorithms. The primary question addressed in this work is for a given problem size, which domain decomposition scheme, angular or spatial, is best suited to parallelize discrete ordinates methods on a specific computational platform? We address this question for three-dimensional applications via parallel performance models for the abovementioned ADD, and a previously constructed and validated spatial domain decomposition (SDD) model. The constructed parallel performance models include parameters specifying the problem size and system performance. We conclude that for large problems the parallel component dwarfs the communication component even on moderately large numbers of processors. The main advantages of SDD are (a) scalability to higher numbers of processors of the order of the number of computational cells; (b) smaller memory requirement; (c) better performance than ADD on high-end platforms and large number of processors. On the other hand, the main advantages of ADD are (a) perfect load balance; (b) simple implementation, even on unstructured grids; (c) better performance than SDD on medium- and low-end platforms and large number of discrete ordinates. It follows that programmers and users of discrete ordinates codes must carefully select the appropriate domain decomposition method for the class of problems and multiprocessor platforms they wish to target.

The application of nonclassical orthogonal polynomials in particle transport theory

In this work, the fundamentals of the constructive theory of orthogonal polynomials and their applications are reviewed. The review provides a basis for a subsequent discussion of transport-theory applications of nonclassical orthogonal polynomials that includes solutions to azimuthally-dependent transport problems, chemical kinetics problems, Fokker-Planck equations with nonlinear coefficients, and the problem of neutral particle transport in ducts.

Gamma Ray Albedo Data Generated by the Invariant Embedding Method

The energy-and-angle dependent doubly differential γ-ray albedos for homogeneous semi-infinite medium have been calculated for water, ordinary concrete, soil, heavy concrete, iron, tin and lead. The procedure of calculation employed in the present paper is the invariant embedding method which is being developed to solve neutral particle transport problems in homogeneous one dimensional medium. The calculated γ-ray albedos are stored in the data base. One can easily obtain from this data base the various kinds of albedos (number, dose, energy, etc.) with a simple interface program. The accuracy of the calculated γ-ray albedos is ascertained by comparing with the Monte Carlo calculations (MCNP4A and EGS4).

Gamma Ray Albedo Data Generated by the Invariant Embedding Method.

The energy-and-angle dependent doubly differential γ-ray albedos for homogeneous semi-infinite medium have been calculated for water, ordinary concrete, soil, heavy concrete, iron, tin and lead. The procedure of calculation employed in the present paper is the invariant embedding method which is being developed to solve neutral particle transport problems in homogeneous one dimensional medium. The calculated γ-ray albedos are stored in the data base. One can easily obtain from this data base the various kinds of albedos (number, dose, energy, etc.) with a simple interface program. The accuracy of the calculated γ-ray albedos is ascertained by comparing with the Monte Carlo calculations (MCNP4A and EGS4).

Asymptotic Diffusive Limits for Transport in Markovian Mixtures

We consider the problem of neutral particle transport in a stochastic Markovian mixture consisting of an arbitrary number M of immiscible fluids. The Liouville master equation is used to obtain a model for the ensemble-averaged angular flux. This model consists of M coupled transport equations. If the absorption can be reduced to a diffusive description. Depending upon the scaling of the Markovian transition lengths, the diffusive limit consists of either a single diffusion equation or a set of M coupled diffusion equations. The asymptotic analysis is also used to derive appropriate initial and boundary conditions for each diffusion equation.

Monte Carlo Treatment for Neutral Particle Transport Problems in a Toroidal Plasma

Transport of neutral particles in a toroidal plasma is treated in the same manner as that of neutrons by the Monte Carlo method. A new sampling technique of flight-path length is proposed for the t...

Application of Space-Angle Synthesis to Two-Dimensional Neutral-Particle Transport Problems of Weapon Physics

A space-angle synthesis (SAS) method is developed for the steady-state, two-dimensional transport of neutrons and secondary gamma rays from a point source of simulated nuclear-weapon radiation in air. The method is validated by applying it to the problem of neutron transport from a point source in air over a ground interface, and then comparing the results to those obtained by DOT, a discrete-ordinates code. In the method, the energy dependence of the Boltzmann transport equation is treated in the standard multigroup manner. The angular dependence is treated by expanding the flux in specially tailored trial functions and applying the method of weighted residuals that analytically integrates the transport equation over all angles. The trial functions used in the expansion are composed of combinations of selected trial solutions, the trial solutions being shaped ellipsoids that approximate the angular distribution of the neutron flux in one-dimensional space. Differences between DOT and SAS tissue-dose calculations at distances greater than 60 m from the source were generally under 10 percent and decreased with increasing source or receiver height. Computer computational time was decreased by a factor of approximately 7.