Discrete Ordinates Approximation Research Articles

Multigroup Neutron Transport Using a Collision-Based Hybrid Method

A collision-based hybrid algorithm for the discrete ordinates approximation of the neutron transport equation is extended to the isotropic multigroup setting. The algorithm uses discrete energy and angle grids at two different resolutions and approximates the fission and scattering sources on the coarser grids. The coupling of a collided transport equation, discretized on the coarse grid, with an uncollided transport equation, discretized on the fine grid, yields an algorithm that, in most cases, is more efficient than the traditional multigroup approach. The improvement over existing techniques is demonstrated for time-dependent problems with different materials, geometries, and energy groups.

The Multi-PN angular discretization method of the neutral-particle transport equation for radiation shielding calculations

The well-known Discrete Ordinates (SN) approximation is little effective in multidimensional radiation shielding calculations. In the presence of geometrically large and low-scattering media, discrete-ordinates solutions are generally affected by the ray effect and an adequate refinement of the angular discretization is sometimes prohibitive in terms of calculation cost. The Multi-PN method (MPN) is an angular discretization technique leading to SN-like transport equations but less affected by ray effect. It relies on a solid angle expansion of the interface fluxes, allowing for a local coupling between different angular degrees of freedom within the same solid angle. This coupling enables to solve the equation by means of a standard sweeping algorithm. Furthermore, flux angular moments are directly integrated without any use of quadrature rules. The MPN measured error convergence rate with respect to the number of angular degrees of freedom is significantly larger than the one measured with SN.

On the application of the ADO method to the solution of two-dimensional radiative transfer problems in anisotropic scattering media

In this work, the Analytical Discrete Ordinates (ADO) method is applied to the solution of the discrete ordinates approximation of the two-dimensional radiative transfer equation in Cartesian geometry. The phase function is expanded in terms of Legendre polynomials up to arbitrary order L to model higher-order anisotropy. The discrete ordinates equations are transversally integrated over regions of the domain, yielding one-dimensional equations for average angular intensities in x and y-directions. The ADO method is then applied to the one-dimensional equations, with approximations for the unknown intensities on the contours of the regions, to derive explicit solutions for the spatial variables. This approach, based on the use of the ADO method along with the nodal technique, is usually referred to as ADO-Nodal. The scheme does not use sweeping. Linear systems have to be solved to fully establish the general solutions. Here, using a recent rearrangement of those linear systems, numerical results are provided for more refined domain partitions, into regions, than previously. Higher-order quadrature schemes are also explored. Comparisons with few results available in the literature reinforce the good performance of the method. A detailed analysis is carried out to contribute in the direction of obtaining reference solutions for the class of problems investigated. Furthermore, numerical results for different optical thicknesses and section average intensities are included, that according to our knowledge it has not been presented before in the literature.

Verification of the Parallel Transport Codes Parafish and AZTRAN with the TAKEDA Benchmarks

With the increase in computational resources, parallel computation in neutron transport codes is inherent since it allows simulations with high spatial-angular resolution. Among the different methodologies available for the solution of the neutron transport equation, spherical harmonics (PN) and discrete-ordinates (SN) approximations have been widely used, as they are established classical methods for performing nuclear reactor calculations. This work focuses on describing and verifying two parallel deterministic neutron transport codes under development. The first one is the Parafish code that is based on the finite-element method and PN approximation. The second one is the AZTRAN code, based on the RTN-0 nodal method and SN approximation. The capabilities of these two codes have been tested on the TAKEDA benchmarks and the results obtained show good behavior and accuracy compared to the Monte Carlo reference solutions. Additionally, the speedup obtained by each code in the parallel execution is acceptable. In general, the results encourage further improvement in the codes to be comparable to other well-validated deterministic transport codes.

Comparing different closure relations for cosmic ray hydrodynamics

ABSTRACT Cosmic ray (CR) hydrodynamics is a (re-)emerging field of high interest due to the importance of CRs for the dynamical evolution of the interstellar, the circumgalactic, and the intracluster medium. In these environments, CRs with GeV energies can influence large-scale dynamics by regulating star formation, driving galactic winds, or altering the pressure balance of galactic haloes. Recent efforts have moved the focus of the community from a one-moment description of CR transport towards a two-moment model as this allows for a more accurate description of the microphysics impacting the CR population. Like all hydrodynamical theories, these two-moment methods require a closure relation for a consistent and closed set of evolution equations. The goal of this paper is to quantify the impact of different closure relations on the resulting solutions. To this end, we review the common P1 and M1 closure relations, derive a new four-moment H1 description for CR transport, and describe how to incorporate CR scattering by Alfvén waves into these three hydrodynamical models. While there are significant differences in the transport properties of radiation in the P1 and M1 approximations in comparison to more accurate radiative transfer simulations using the discrete ordinates approximation, we only find small differences between the three hydrodynamical CR transport models in the free-streaming limit when we neglect CR scattering. Most importantly, for realistic applications in the interstellar, circumgalactic, or intracluster medium where CR scattering is frequent, these differences vanish and all presented hydrodynamical models produce the same results.

On the Regularity Order of the Pointwise Uncollided Angular Flux and Asymptotic Convergence of the Discrete Ordinates Approximation of the Scalar Flux

To determine the angular discretization error asymptotic convergence rate of the uncollided scalar flux computed with the discrete ordinates (S) method, a comprehensive theory of the regularity order with respect to the azimuthal angle of the exact pointwise SN uncollided angular flux is derived based on the integral form of the transport equation in two-dimensional Cartesian geometry. With this theory, the regularity order of the pointwise uncollided angular flux can be estimated for a given problem configuration. Our new theory inspired a novel Modified Simpson’s (MS) quadrature that converges the uncollided scalar flux faster than any of the traditional quadratures by avoiding integration across points of irregularity in the azimuthal angle. Numerical results successfully verify our new theory in four variants of a test configuration, and the angular discretization errors in the corresponding scalar flux computed with conventional angular quadrature types and with our new quadrature types are found to converge with different orders. The error convergence rates obtained with traditional quadrature types are limited by the regularity order of the exact angular flux and the quadrature’s integration intervals while our new MS quadrature types converge with order two to four times higher than traditional quadratures. A detailed study of oscillations observed in certain quadrature errors is provided by introducing the effective length of the irregular interval and the associated oscillating function.

Formulações espectrais para solução de problemas de transporte de partículas

In this work, a deterministic approach to the solution of the integro differential linear Boltzmann equation, in one and twodimensional media, is presented. Explicit solutions, in terms of the spatial variables, for the discrete ordinates approximation of the original model are obtained from a spectral formulation. A relevant feature of the methodology is the reduced order of the eigenvalue problems, which is given as half of the number of the discrete directions. Such aspect, along with the use of arbitrary quadrature schemes to represent the integral term of the equation, are fundamental to provide fast, concise and accurate solutions to several problems of interest. In particular, the solutions here are derived for models associated with two different applications: rarefied gas dynamics and neutron transport. Mathematical aspects present in the solution of the two models are emphasized and extensions of the formulation to other applications are commented and referenced.

Recent Studies on Two-Dimensional Radiative Transfer Problems in Anisotropic Scattering Media

In this work, an explicit formulation to solve two-dimensional radiative transfer problems in anisotropic scattering media is developed. A nodal technique along with the Analytical Discrete Ordinates (ADO) method are used to solve the discrete ordinates approximation of the radiative transfer equation, in Cartesian geometry. To make it possible, the discrete ordinates equations are transversally integrated over regions of the domain reducing the complexity of the model, yielding two one-dimensional equations for average angular intensities in x and y directions. The one-dimensional equations, with approximations for the unknown intensities on the contours of the regions, are then explicitly solved by the ADO method, with respect to the spatial variables, whose solutions are written in terms of eigenvalues and eigenfunctions. The phase function is expanded in terms of Legendre polynomials up to arbitrary order L, to model higher order anisotropy. The eigenvalue problem is derived for this general case and it preserves a relevant feature of the ADO method, which is the reduced order equal to half of the number of discrete directions. Numerical results for the average radiation density and radiative heat flux are presented, for test cases in which the degree of anisotropy can be up to twelve and the albedo assumes different values. A comparative analysis with results available in the literature allows the verification of the formulation and indicates a good performance of the proposed method in coarser meshes.

Neutron-gamma flux distributions in cylindrical beam tubes of pool-type research reactor

In nuclear reactor, a variety of ducts are equipped in the radiation shield. Beam tubes, used for experimental facilities for research and development are one of them. It is very important to estimate the neutron and gamma photon flux profile inside as well as at the exit of the duct. This is required from experimental as well as radiation protection point of view. One way to estimate these flux profiles is to use transport method and another way is to use semi-empirical formula which describes energy-space distributions of streamed neutrons and gamma photons in various beam tubes with different shapes. Deterministic transport methodology using discrete ordinates approximation cannot be accurately model the streaming problems. This is due to inability to incorporate sufficient number of angular mesh points in the direction of streaming paths, which leads to ray effect (Miller and Lewis in Computational methods of neutron transport, Wiley, New York, 1985). This ray effect arises in the problem (radiation shielding etc.) having very little scattering and localized sources. In such problems anomalies (unphysical oscillations in the computed scalar flux) likely to arise in the scalar flux distribution. As a result, considerable underestimation occurs. Monte Carlo transport methods seem to be the best method in dealing streaming problems. However, reliable results with small statistical errors are not necessarily obtained in reasonable computation time. So, the use of semi-empirical formula will give the result with suitable accuracy and in less time. In this paper, streaming of neutrons as well as gamma photons through the beam tubes has been calculated using semi-empirical formula and compared with Monte Carlo results. Here, we have considered cylindrical ducts and the results obtained by semi-empirical formula are comparable with those from Monte Carlo code. The results can be verified further when the neutron-gamma flux measurement is done.

Comparisons of radiative transfer schemes for infrared spectra and the region with solar and infrared spectra overlap in RRTMG

Five radiative transfer schemes are compared in infrared spectra using the Rapid Radiative Transfer Model for General Circulation Models Applications (RRTMG). By calculating the root mean squared error of net flux (referred as RMSE(NF)) in various atmosphere, the general accuracy of these schemes ranked from low to high are: non-scattering simplification (NS), the adding method of δ-two-stream discrete ordinate approximation (δ-2DDA), δ-two-stream variational iteration method (δ-2VIM), the adding method of δ-four-stream discrete ordinate approximation (δ-4DDA) and δ-four-stream variational iteration method (δ-4VIM). And their sensitivities to cloud fraction are decreased in the same order. Furthermore, the accuracy and efficiency of these schemes other than NS are studied under the contexts of using two different methods for dealing with the region where the solar and infrared spectra overlap. As one of the two overlap methods, One Band method (OB), which is used by RRTMG, has no advantage in model efficiency and is about 0.34 (0.41) W/m2 higher in mean column RMSE(NF) of the δ-two-stream (δ-four-stream) schemes than Whole Bands method (WB). Moreover, a new scheme, which is simple but adequate to handle the overlap region, is derived to solve the solar energy in longwave spectra.

An analytical discrete ordinates nodal solution to the two-dimensional adjoint transport problem

A solution to the one-group discrete ordinates approximation of the adjoint to the transport equation in two-dimensional Cartesian geometry is developed in this work. Starting from an integration process over a region (node) of the domain, the model is reduced to one-dimensional transverse-integrated equations in the x and y variables which are then solved by the Analytical Discrete Ordinates Method (ADO Method). Particular attention is given to the effect of non-homogeneous boundary conditions in the formulation. Explicit expressions are derived and used for evaluating, in a fast way, absorption rates of internal detectors from the adjoint formulation. Excellent agreement was found between numerical results obtained for the absorption rates evaluated from the forward and adjoint problems.

Light transport with the equation of radiative transfer: The Fourier Continuation – Discrete Ordinates (FC–DOM) Method

We present a method for the treatment of the time dependent radiative transfer equation under the discrete ordinate approximation. The novelty of the proposed approach stems, in part, from the incorporation of a spectral method for the calculation of the spatial differential operators based on the Fourier Continuation procedure introduced recently by Bruno and co–authors. This is a spatially dispersionless and high order method, which can handle arbitrary geometries, including those encountered in the forward model of light transport in optical tomography. We validate our theoretical results by comparison with analytic and experimental outcomes of the fluence measurements on tissue-like phantoms. The method makes it possible to calculate the time of flight of photons in random media efficiently and with high accuracy.

Meshless local Petrov–Galerkin solution of the neutron transport equation with streamline-upwind Petrov–Galerkin stabilization

The meshless local Petrov–Galerkin (MLPG) method is applied to the steady-state and k-eigenvalue neutron transport equations, which are discretized in energy using the multigroup approximation and in angle using the discrete ordinates approximation. To prevent oscillations in the neutron flux, the MLPG transport equation is stabilized by the streamline upwind Petrov–Galerkin (SUPG) method. Global neutron conservation is enforced by using moving least squares basis and weight functions and appropriate SUPG parameters. The cross sections in the transport equation are approximated in accordance with global particle balance and without constraint on their spatial dependence or the location of the basis and weight functions. The equations for the strong-form meshless collocation approach are derived for comparison to the MLPG equations. The method of manufactured solutions is used to verify the resulting MLPG method in one, two and three dimensions. Results for realistic problems, including two-dimensional pincells, a reflected ellipsoid and a three-dimensional problem with voids, are verified by comparison to Monte Carlo simulations.

Hybrid methods for radiation transport using diagonally implicit Runge–Kutta and space–time discontinuous Galerkin time integration

In this work, we describe extensions of a hybrid method for time-dependent linear, kinetic radiation transport problems to high-order time integration schemes of the diagonally-implicit Runge–Kutta (DIRK) and space–time discontinuous Galerkin (STDG) types. The hybrid methods are constructed by splitting the radiation flux into collided and uncollided components to which low- and high-resolution discrete ordinates approximations are applied. The efficiency of hybrid methods constructed using DIRK, STDG, integral deferred correction (IDC), and implicit Euler schemes is compared using a test problem in one-dimensional slab geometry containing material discontinuities. It is observed that (i) higher-order methods are more efficient than the implicit Euler method, often by an order of magnitude or more; (ii) third-order methods yield solutions of a given error in roughly half the time of the second-order method of the same type; and (iii) for a given order of accuracy it is found that the most efficient class of time integration scheme is STDG, followed by DIRK, with IDC the least efficient, for the test problem considered. Two test problems in two-dimensional xy-geometry are used to compare the computational efficiency of hybrid and standard discrete ordinates methods constructed with DIRK and STDG integrators. We observe that replacing a standard discrete ordinates method using an angular quadrature of order N with a hybrid discrete ordinates method using angular quadratures of order 2N and N/2 for the uncollided and collided fluxes, respectively, usually reduces overall solution time by a factor of 2 or more while simultaneously reducing the resulting solution error by a factor of 2 or more for the test problems considered.

Energy Dependent Source Reconstructions via Explicit Formulations of the Adjoint Particles Flux

An analytical solution to the discrete ordinates approximation of the adjoint to the multigroup transport equation in one-dimensional slab geometry is developed in this work, in order to be used in source estimation problems. The solution is firstly tested in a source-detector problem, where explicit expressions are derived to approximate the absorption rate of particles of internal detectors. Noisy data generated from readings of internal detectors were then used, along with the adjoint formulation, in an iterative process to solve an energy-dependent inverse problem of source reconstruction in the cases of polynomials and piecewise constant sources. The approach is shown to be fast and accurate.

Analysis of radiation heat transfer and temperature distributions of solar thermochemical reactor for syngas production

This paper investigated radiation heat transfer and temperature distributions of solar thermochemical reactor for syngas production using the finite volume discrete ordinate method (fvDOM) and P1 approximation for radiation heat transfer. Different parameters including absorptivity, emissivity, reflection based radiation scattering, and carrier gas flow inlet velocity that would greatly affect the reactor thermal performance were sufficiently investigated. The fvDOM approximation was used to obtain the radiation intensity distribution along the reactor. The drop in the temperature resulted from the radiation scattering was further investigated using the P1 approximation. The results indicated that the reactor temperature difference between the P1 approximation and the fvDOM radiation model was very close under different operating conditions. However, a big temperature difference which increased with an increase in the radiation emissivity due to the thermal non-equilibrium was observed in the radiation inlet region. It was found that the incident radiation flux distribution had a strong impact on the temperature distribution throughout the reactor. This paper revealed that the temperature drop caused by the boundary radiation heat loss should not be neglected for the thermal performance analysis of solar thermochemical reactor.

Double-delta-function adjustment in thermal radiative transfer

For atmospheric scattering, the weak backward scattering peak is always ignored, in contrast with the strong forward scattering peak. In this paper, the backward peak contribution is incorporated in multiple scattering, along with the forward peak contribution. Thus, a new parameterization, the double-δ-function adjustment, is proposed and its application in a two-stream approximation for infrared radiative transfer is shown. The accuracy of the adding method for the double-δ-two-stream discrete-ordinates approximation (2δ-2DDA) is evaluated by the emissivities in a single layer of atmosphere and the fluxes and heating rate in a multi-layer atmosphere with a realistic atmospheric profile. The results show that 2δ-2DDA produces less bias than the δ-two-stream approximation (δ-2DDA) for thick optical depths, such as water cloud conditions. For thin optical depths, such as ice cloud, δ-2DDA and 2δ-2DDA produce similar errors. Generally, 2δ-2DDA is more accurate than δ-2DDA, and it can be easily applied in climate models.

Accelerated Solution of Discrete Ordinates Approximation to the Boltzmann Transport Equation for a Gray Absorbing–Emitting Medium Via Model Reduction

This work applies a projection-based model-reduction approach to make high-order quadrature (HOQ) computationally feasible for the discrete ordinates approximation of the radiative transfer equation (RTE) for purely absorbing applications. In contrast to traditional discrete ordinates variants, the proposed method provides easily evaluated error estimates associated with the angular discretization as well as an efficient approach for reducing this error to an arbitrary level. In particular, the proposed approach constructs a reduced basis from (high-fidelity) solutions of the radiative intensity computed at a relatively small number of ordinate directions. Then, the method computes inexpensive approximations of the radiative intensity at the (remaining) quadrature points of a high-order quadrature using a reduced-order model (ROM) constructed from this reduced basis. This strategy results in a much more accurate solution than might have been achieved using only the ordinate directions used to construct the reduced basis. One- and three-dimensional test problems highlight the efficiency of the proposed method.

An arbitrary-order, fully implicit, hybrid kinetic solver for linear radiative transport using integral deferred correction

In this work, we describe the implementation of an arbitrarily high-order hybrid solver for linear, kinetic, radiative transport equations. The hybrid method is derived from a splitting of the radiative flux into free-streaming and collisional components to which high- and low-resolution discrete ordinates methods are applied, respectively. Arbitrarily high orders of accuracy with respect to time and space are attained by combining an integral deferred correction (IDC) time integration scheme constructed with implicit Euler substepping on Radau II (right biased) nodes with an upwind discontinuous Galerkin (DG) spatial discretization on uniform Cartesian meshes. Numerical experiments are used to demonstrate that the aforementioned IDC methods can be constructed such that they are unconditionally stable (L-stable) to within machine precision. Asymptotic analysis is used to show that such IDC methods also preserve the diffusion limit of the underlying transport equation on the semi-discrete level, in the sense that the semi-discrete transport system under an implicit IDC discretization recovers the same IDC discretization of the limiting diffusion equation. Convergence results in one spatial dimension are provided, and it is found that while the hybrid method exhibits convergence stagnation and order reduction in certain scenarios, the overall accuracy of the hybrid approximation is comparable to a standard discrete ordinates approximation in many cases. Numerical results for two test problems in two spatial dimensions are given to compare the computational efficiency of the hybrid method against a standard discrete ordinates method, and to compare the efficiency of space–time discretizations of different orders of accuracy. The results indicate that a properly chosen hybrid discrete ordinates method can be more efficient than a standard discrete ordinates method by a factor of between 2 and 8 when IDC timestepping methods are used. Further, among the first-, third-, and fifth-order space–time discretizations considered here, the first-order method is the least efficient and the third-order method is the most efficient.

An analytical approach for solving a nodal formulation of two-dimensional fixed-source neutron transport problems with linearly anisotropic scattering

In this work, the formulation of the Analytical Discrete Ordinates (ADO) method for two-dimensional fixed-source neutron transport problems is extended to problems with anisotropic scattering. The methodology is developed from the discrete ordinates approximation of the two-dimensional transport, whose equations are integrated transversally within homogenized regions of the domain, yielding one-dimensional equations for the average angular fluxes. Such one-dimensional equations are solved by the ADO method, after considering approximations for the transverse leakage terms, and their solutions are written in terms of eigenvalues and eigenfunctions. The relevant feature of the ADO method for the isotropic scattering cases, which reduced the order of the eigenvalue problems to half of the number of discrete directions, is also preserved for the present anisotropic cases. Explicit spatial variables expressions are derived for the average angular fluxes in regions of interest defined in the domain. The full domain solution is defined by the coupling of local solutions in different regions, without the use of sweeping procedures. As usual in nodal schemes, auxiliary equations are necessary and, in this case, the unknown region-edge angular fluxes are approximated by constants embedded in the source term. Numerical results for region-average scalar fluxes are obtained and compared with the ones available in the literature to illustrate the feasibility of keeping the computational efficiency already verified in treating problems with isotropic scattering, with the use of the ADO method.