Analytical Discrete Ordinates Research Articles

A Double Legendre Polynomial Order N Benchmark Solution for the 1D Monoenergetic Neutron Transport Equation in Plane Geometry

As more and more numerical and analytical solutions to the linear neutron transport equation become available, verification of the numerical results becomes increasingly important. This presentation concerns the development of another benchmark for the linear neutron transport equation in a benchmark series, each employing a different method of solution. In 1D, there are numerous ways of analytically solving the monoenergetic transport equation, such as the Wiener–Hopf method, based on the analyticity of the solution, the method of singular eigenfunctions, inversion of the Laplace and Fourier transform solution, and analytical discrete ordinates in the limit, which is arguably one of the most straightforward, to name a few. Another potential method is the PN (Legendre polynomial order N) method, where one expands the solution in terms of full-range orthogonal Legendre polynomials, and with orthogonality and series truncation, the moments form an open set of first-order ODEs. Because of the half-range boundary conditions for incoming particles, however, full-range Legendre expansions are inaccurate near material discontinuities. For this reason, a double PN (DPN) expansion in half-range Legendre polynomials is more appropriate, where one separately expands incoming and exiting flux distributions to preserve the discontinuity at material interfaces. Here, we propose and demonstrate a new method of solution for the DPN equations for an isotropically scattering medium. In comparison to a well-established fully analytical response matrix/discrete ordinate solution (RM/DOM) benchmark using an entirely different method of solution for a non-absorbing 1 mfp thick slab with both isotropic and beam sources, the DPN algorithm achieves nearly 8- and 7-place precision, respectively.

Technique for reducing time in the calculation of eigenvalues and eigenvectors applied in neutron transport fixed-source problems

In this work, a mathematical technique to accelerate the eigenvalue calculations, originally proposed for the Analytical Discrete Ordinates method (ADO), is applied to the coarse-mesh method Modified Spectral Deterministic (MSD), used in neutron shielding problems (fixed-source) in one-dimensional geometry. This technique consists in an arrangement of the neutron transport equation which leads to a reduced eigenvalue calculation on the intranodal analytical solution, where only the positive half of the eigenvalues and its respective eigenvectors are calculated, hence reducing the methods execution time. Precision and performance tests were performed in 2 model-problems, reducing around 80% of the execution time. All the codes were developed using the programming language C++.

Analytical Discrete-Ordinates Solutions for Improved 1D Models of Particle Transport in Ducts with Wall Migration

Analytical discrete-ordinates (ADO) solutions are developed for two improved one-dimensional (1D) models of particle transport in ducts that include wall migration. One of the studied models is based on an approximation of the transverse and azimuthal dependencies of the angular flux in terms of two basis functions, while the other uses three. Particle migration in the wall is modeled by an exponential displacement kernel. Numerical results for duct reflection and transmission probabilities are reported for various test cases, including some cases of thermal-neutron transport through iron, concrete, and graphite ducts that have been defined and studied by other authors. A comparison is performed with a set of published results comprising realistic results obtained with the Monte Carlo code MCNP and results from a numerical implementation of the discrete-ordinates method for the model based on three basis functions. To resolve some issues raised during the comparison process, a numerical discrete-ordinates solution of the problem has also been implemented in the course of this work.

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.

Three-dimensional analytical discrete-ordinates method for the radiative transport equation

The radiative transport equation in a three-dimensional infinite medium is considered. The coefficients of the radiative transport equation are assumed to be constant. For a pencil beam, we extend the analytical discrete-ordinates method to three dimensions by rotating reference frames. Obtained results are compared to the Monte Carlo simulation.

On the application of the analytical discrete ordinates method to the solution of nonclassical transport problems in slab geometry

In this work we investigate the use of the Analytical Discrete Ordinates (ADO) method when solving the spectral approximation of the nonclassical transport equation. The spectral approximation is a recently developed method based on the representation of the nonclassical angular flux as a series of Laguerre polynomials. This representation generates, as outcome, a system of equations that have the form of classical transport equations and can therefore be solved by current deterministic algorithms. Thus, the investigation of efficient approaches to solve the nonclassical transport equation is of interest and shall be pursued. This is the case of the ADO method which has been successfully used to solve a wide class of problems in the general area of particle transport. Numerical results are presented for two nonclassical test problems in slab geometry. These nonclassical transport problems are chosen in such way that their solution exactly reproduces the solution of the classical diffusion problem. Very accurate results are obtained for both test problems. However, the use of high precision arithmetic is sometimes required as illustrated in the second test problem. Limitations of the spectral approximation are also analyzed and discussed.

Analytical Discrete Ordinates Solution for a 1D Model of Particle Transport in Ducts that Includes Wall Migration

The analytical discrete ordinates (ADO) method is used to develop a solution to a one-dimensional model of particle transport in ducts that includes wall migration. Particle reemission from the wall is described by a nonlocal, exponential displacement kernel. Since the governing transport equation of the model is not directly amenable to a solution by the ADO method, an alternative transport equation is derived first. For an approximation based on a half-range quadrature of order , the ADO solution of the alternative equation becomes available once techniques of linear algebra are used to solve a quadratic eigenproblem of order for the eigenvalues and eigenvectors. The solution is expressed as a superposition of 4N modes, which are constructed from 2N positive/negative pairs of separation constants (the reciprocals of the square roots of the eigenvalues) and associated eigenvectors. Compatibility conditions that the solution must satisfy in order to also solve the governing equation of the model result in a reduction of the number of relevant modes to 2N + 2, just two in excess of the number of modes in the solution of the problem without wall migration. Highly accurate numerical results for the reflection and transmission probabilities are reported for isotropic and monodirectional incidence.

On computing the effective multiplication factor using the ADO method

In this work, the Analytical Discrete Ordinates method, ADO method, is applied to evaluate the effective multiplication factor in criticality problems of a nuclear reactor. To address this class of problems, the multigroup ADO formulation in multislab geometry is here derived taking into account real and complex spectrum. Two approaches are presented to find the desired dominant k-eigenvalue.In the first one, a criticality (characteristic) equation is derived from the complete original equation defined in multiplying media and a combined scheme, using regula-falsi and secant root-finding methods, is used to extract a root of that equation. Such scheme is shown to be very efficient and provided high-quality benchmark results, from lower quadrature order approximations than other schemes available in the literature, as found by the application of the ADO formulation also in several other transport problems.As a second proposed approach, the ADO solution is used along with the usual power iteration method in a simpler procedure. This simpler procedure avoids complex eigenvalues, but it is shown to be less accurate due to required approximations in the source approximation procedure.

Linear Transport in Porous Media

The linear transport theory is developed to describe the time dependence of the number density of tracer particles in porous media. The advection is taken into account. The transport equation is numerically solved by the analytical discrete ordinates method. For the inverse Laplace transform, the double-exponential formula is employed. In this paper, we consider the travel distance of tracer particles whereas the half-space geometry was assumed in our previous paper [Amagai et al. (2020). Trans. Porous Media 132:311–331].

Calculation of radiative heat flux on irregular boundaries in participating media* *Project supported by the Startup Foundation for Introducing Talent of Nanjing University of Information Science and Technology, the Anhui Provincial Natural Science Foundation, China (Grant No. 2008085ME151), and the National Natural Science Foundation of China (Grant Nos. 51976057 and 51827808).

Radiative heat flux at wall boundaries is important for its thermal design. Numerical methods based on structured grids are becoming trendy due to their simplicity and efficiency. Existing radiative transfer equation solvers produce oscillating radiative heat flux at the irregular boundary if they are based on structured grids. Reverse Monte Carlo method and analytical discrete ordinates method are adopted to calculate the radiative heat flux at complex boundaries. The results show that the reverse Monte Carlo method can generate a smooth radiative heat flux profile and it is smoother with larger energy bundles. The results from the analytical discrete ordinates method show that the fluctuations are due to the ray effect. For the total or the mean radiative heat flux, the results from the analytical discrete ordinates method are very close to those from the reverse Monte Carlo method.

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.

A Bayesian approach for neutral particles source estimation

In this work, the adjoint to the transport operator is used to estimate the spatial distribution of an energy dependent neutral particles source in a homogeneous one-dimensional medium, from measurements of internal detectors. An analytical discrete ordinates formulation, the ADO method, is applied to derive a spatially explicit solution for the adjoint flux. A Bayesian approach is considered to relate the measurements with prior information and then the Metropolis-Hastings algorithm is used do draw samples of the posterior distributions. Explicit expressions for the measurements are derived, allowing relevant reduction in computational time. Simulations are performed for successfully estimating polynomial and piecewise constant localized energy dependent sources. Two and six energy group problems are considered.

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.

Discrete ordinates characteristics solution for phonon transport in an anisotropic plant canopy

Radiation-vegetation canopy interaction is analyzed using a semi-analytic discrete ordinates characteristics solution. The plant canopy is considered as a single layer containing a set of leave ensembles of bi-Lambertian surfaces and the radiation-plant canopy interaction is described by a radiative transfer equation in which the plant canopy radiative properties depend on the incident radiation direction. An analytical expression for anisotropic plant canopy radiance is derived. Radiance and hemispherical reflectance/transmittance for different boundary conditions and the plant canopy in the visible and near infrared are predicted. Results show that the discrete ordinates characteristics solution using the moderate number of double-Gauss ordinates direction compete well with the FN or “facile” and analytical discrete ordinates methods, which are two other literature quasi-analytical methods for phonon transport in the anisotropic plant canopy. Results also reveal that the proposed method matches exact Chandrasekhar benchmark, except for view cosine direction µ≥0.9, where the accuracy on transmitted radiance is less than 0.3% for low angular discretization. Comparisons between numerical predictions and soybean reflectance experiments indicate that the leaf area index and soil reflectance effects are significant in the near infrared plateau.

THE ADO METHOD FOR SOLVING ONE-DIMENSIONAL DEEP-PENETRATION TRANSPORT PROBLEMS

In this work, the Analytical Discrete Ordinate method (ADO method) is used to solve deep-penetration transport problems in one-dimensional Cartesian geometry, subject to isotropic and linear anisotropic scattering effects. The regime is considered permanent, the media are homogeneous, and the fluxes are caused by sources located on the boundaries of the domain. In addition, the energy fluctuations will be considered as not significant, characterizing the phenomena as monoenergetic problems. In order to validate the code, method and provide benchmark results, some test problems will be treated and results will be discussed. In particular, the ADO method generated fairly accurate results when compared to other methods based on SN approaches, at a relatively low computational cost.

The influence of half-range quadrature scheme on ADO method convergence

In this work, a discrete ordinates solution for a neutron transport problem in one-dimensional Cartesian geometry is presented. In order to evaluate the efficiency of the half-range quadrature scheme, the Analytical Discrete Ordinates method (ADO) is used to solve two classes of problems in finite and homogeneous media (with isotropic and linear anisotropic scattering), for steady-state regime, without inner source and prescribed boundary conditions. Numerical results for the scalar fluxes were obtained and comparisons with other works in the literature were made. The versatility of the use of quadratures has always been seen as an advantage of the ADO method which, besides providing accurate results at a low computational cost, has a simpler approach, allowing the use of free software distribution for the simulations. In the results analysis, it was verified that the use of the half-range quadrature was able to accelerate the convergence, mainly in linearly anisotropic problems.

On the use of the adjoint operator for source reconstruction in particle transport problems

ABSTRACTIn this work, the adjoint to the transport operator is used to estimate the spatial distribution of an isotropic neutral particles source in a homogeneous one-dimensional medium, from readings of internal detectors. An analytical discrete ordinates formulation, the ADO method, is applied to derive a spatially explicit solution for the adjoint flux. Simulations are performed for reconstructing Gaussian and piecewise localized sources. Numerical results indicate that the source estimates may be considered satisfactory although scalar fluxes are reconstructed more precisely.

Application of the ADO method on disadvantage-factor calculation for heterogeneous slab cells considering linear anisotropic scattering effects

In this work, the disadvantage-factor calculation is performed by a version of the Analytical Discrete Ordinates method (ADO method). For this, a class of one-dimenisional neutron transport problems in Cartesian geometry, with linearly anisotropic scattering effects in layered heterogeneous media, will be considered. The process consists in applying the angular discretization SN, converting the integro-differential transport equation into an ordinary differential equations system, which homogeneous solution is obtained by a quadratic eigenvalues problem with reduced order. To establish a closed form solution for this class of problems, particular solutions in terms of constants are used, as well as reflective boundary conditions and interface conditions. In order to validate the code, the method and to provide benchmark results, a class of two-layered problems is treated. A brief discussion on the influence of anisotropy is made and numerical aspects are discussed.

Particle Transport in a 3D Duct by Adding and Doubling

ABSTRACTHow particles are transported through a duct by Lambertian reflection from duct walls is again considered. This popular transport example has been solved with most numerical transport methods except notably one—the method of adding and doubling. We shall show that the method of doubling provides every bit as, or more, accurate reflectances and transmittances as the numerical discrete ordinates and analytical discrete ordinates methods with generally less mathematical and numerical effort.

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.