Ordinal Problems Research Articles

Analytical solution of the discrete ordinates problem by the decomposition method

In this work is established a closed-form solution for the one-dimensional discrete ordinates problems employing the Decomposition Method. Numerical simulations are reported.

An analytical solution for the two-dimensional discrete ordinates problem in a convex domain

The aim of this work consists in the application of the Laplace transform technique to solve analytically the transverse integrated discrete ordinates equations in curvilinear orthogonal coordinates system for a convex domain.

A spectral nodal method for one-group X,Y,Z-cartesian geometry discrete ordinates problems

The solution of one-group discrete ordinates S N problems with linearly anisotropic scattering in x, y, z - cartesian geometry has been studied by using SGF-CN “spectral Green's function-constant nodal’ method, developed first by De Barros and Larsen (1990–1992) for one dimensional and two dimensional x, y - cartesian geometries. The solutions of S N transverse - integrated nodal equations in which only constant nodal volume leakage approximation is made, are obtained by a simple iterative algorithm. Finally, tabulated numerical results of average cell scalar fluxes are provided.

An analytical solution for the multigroup slab geometry discrete ordinates problems

Abstract In this work an analytical formulation is presented for the solution of the multigroup discrete ordinates problem in planar geometry with isotropic scattering. The approach is based on the application of the Laplace transform to the discrete ordinates equations and analytical inversion. Numerical results are presented for a test problem.

Solution of the three-dimensional one-group discrete ordinates problems by the LTS n method

In this work, the LTS n formulation is extended to the three-dimensional discrete ordinates problems in cartesian geometry. Numerical simulations are presented for problems with cylindrical geometry.

Extension of the LTS N formulation for discrete ordinates problem without azimuthal symmetry

In this work, an extension of the LTS N formulation for the discrete ordinates problem without azimuthal symmetry in a slab is established, considering the one-group and anisotropic scattering model. Numerical simulations are presented.

Automatic Generation of Symbolic Multiattribute Ordinal Knowledge‐Based DSSs: Methodology and Applications*

ABSTRACTA learning‐by‐example algorithm, the ordinal learning model (OLM), that automatically generates symbolic rule‐bases from examples was applied to four real‐world multiattribute ordinal problem domains. The model automatically generates consistent and irredundant symbolic classification rules that mimic, in many aspects, the behavior of human subjects who solved similar problems during empirical studies. The OLM's performance is compared with those of regression analysis and with C4, a well‐known symbolic learning‐by‐example decision tree building algorithm. The OLM uses mainly comparison operations and does not attempt to optimize the rule‐bases it generates. Yet, the results show that the OLM's predictions are very accurate and the resulting rule‐bases are relatively compact. The time required for constructing the rule‐bases via the OLM was very competitive as well.

Iterative Methods for Solvingx-yGeometrySNProblems on Parallel Architecture Computers

In this paper, geometric domain decomposition methods are described for solving x-y geometry discrete ordinates (S[sub N]) problems on parallel architecture computers. First, a parallel source iteration scheme is developed; here, one subdivides the spatial domain of the problem, performs transport sweeps independently in each subdomain, and iterates on the scattering source and the interface fluxes between each subdomain. Second, a parallel diffusion synthetic acceleration (DSA) scheme is developed to speed up the convergence of the parallel source iteration. These schemes have been implemented on the IBM RP3, a shared/distributed memory parallel computer. The numerical results show that the parallel source iteration and DSA methods both exhibit significant speedups over their scalar counterparts, but that a degradation in parallel efficiency occurs due to the geometric domain decomposition (iteration on interface fluxes) and the overhead time required for the communication of data between processors.

A Spectral Nodal Method for One-GroupX, Y-Geometry Discrete Ordinates Problems

This paper reports on a nodal method that is developed for the solution of one-group discrete ordinates (S{sub N}) problems with linearly anisotropic scattering in x, y-geometry. In this method, the spectral Green's function (SGF) scheme, originally developed for solving S{sub N} problems in slab geometry with no spatial truncation error, is generalized to solve the one-dimensional transverse-integrated S{sub N} nodal equations with the constant approximation for the transverse leakage terms. The resulting SGF-constant nodal (SGF-CN) method is more accurate than conventional coarse-mesh methods for deep penetration problems because it treats the scattering source terms implicitly and exactly; the only approximation involves the transverse leakage terms. In conventional S{sub N} nodal methods, the transverse leakage terms and scattering source are both approximated. The authors solve the SGF-CN equations using the one-node block inversion iterative scheme, which uses the best available estimates for the node-entering fluxes to evaluate the node-exiting fluxes in the directions that constitute the incoming fluxes for the adjacent nodes as the equations are swept across the system. Finally, the authors give numerical results that illustrate the accuracy of the SGF-CN method for coarse-mesh calculations.

A numerical method for multigroup slab-geometry discrete ordinates problems with no spatial truncation error

Abstract A generalization of the one-group Spectral Green's Function (SGF) method is developed for multigroup, slab-geometry discrete ordinates (SN) problems. The multigroup SGF method is free from spatial truncation errors; it generates numerical values for the cell-edge and cell-average angular fluxes that agree with the analytic solution of the multigroup SN equations. Numerical results are given to illustrate the method's accuracy.

A Numerical Method for One-Group Slab-Geometry Discrete Ordinates Problems with No Spatial Truncation Error

A numerical method that is free from all spatial truncation errors is developed for one-group slab-geometry discrete ordinates problems. The unknowns in the method are the cell-edge and cell-average angular fluxes, and the numerical values obtained for these quantities are those of the analytic solution of the discrete ordinates equations. The method is based on the use of the standard balance equation, which holds in each spatial cell and for each discrete ordinates direction, and a nonstandard auxiliary equation that contains a Green’s function for the cell-average angular flux in terms of the incident angular fluxes on the cell edges and the interior source. Numerical results are given to illustrate the method’s accuracy.

Spatial domain decomposition for Neutron transport problems

Abstract A spatial Domain Decomposition method is proposed for modifying the Source Iteration (SI) and Diffusion Synthetic Acceleration (DSA) algorithms for solving discrete ordinates problems. The method, which consists of subdividing the spatial domain of the problem and performing the transport sweeps independently on each subdomain, has the advantage of being parallelizable because the calculations in each subdomain can be performed on separate processors. In this paper we describe the details of this spatial decomposition and study, by numerical experimentation, the effect of this decomposition on the SI and DSA algorithms. Our results show that the spatial decomposition has little effect on the convergence rates until the subdomains become optically thin (less than about a mean free path in thickness).

On Ordinal Equivalence of Power Indices in Simple Games

The ordinal equivalence of the Shapley-Shubik and Banzhaf-Coleman power indices in a simple game would be important to many applications, where consideration of “how much” more power one member has than another is irrelevant. The reason is that if the ordinal equivalence holds, then another question of which index to use goes out. To the ordinal equivalence problem, this paper gives the following answers. The ordinal equivalence holds true either in the weighted majority games as a subclass of simple games or in the five- or less-member simple games. However, it does not necessarily hold true in the six- or more-member simple games.

Fourier Analysis of the Diffusion Synthetic Acceleration Method for Weighted Diamond-Differencing Schemes in Cartesian Geometries

The diffusion synthetic acceleration (DSA) method is developed for arbitrary weighted diamond-differencing schemes in general rectangular-mesh Cartesian geometry problems and is Fourier analyzed to determine its stability and convergence properties. The spectral radius is computed to be --0.25 for all meshes, angular quadrature sets, and spatial weights, for one-, two-, and three-dimensional problems. The diffusion synthetic acceleration (DSA) method is an iterative procedure for solving discrete ordinates problems.

Modified Diffusion Synthetic Acceleration Algorithms

The diffusion synthetic acceleration (DSA) method is an iterative procedure for obtaining rapidly convergent numerical solutions of discrete ordinates problems. General P/sub L/ acceleration methods for L>1 are first considered to determine if DSA, which is equivalent to P/sub 1/ acceleration, is optimum. After determining that DSA appears to be the optimum P/sub L/ approach, it is then demonstrated that by also using standard techniques of matrix iterative acceleration, modest improvements can be made is DSA performance.

Projected Discrete Ordinates Methods for Numerical Transport Problems

A class of “projected discrete ordinates” (PDO) methods is described for obtaining iterative solutions of discrete ordinates problems with convergence rates comparable to those observed using diffu...

Unconditionally Stable Diffusion-Synthetic Acceleration Methods for the Slab Geometry Discrete Ordinates Equations. Part II: Numerical Results

Diffusion-synthetic acceleration methods that have been proven analytically to be stable for model discrete ordinates problems (for infinite media, with isotropic scattering, constant cross sections, and a uniform spatial mesh) are shown to be experimentally stable for realistic problems (for finite media, with anisotropic scattering, variable cross sections, and a nonuniform spatial mesh). Also, the effect of negative flux fixups on the acceleration methods is discussed.

Ordinal Position and Behavior Problems in Children

There were two major findings. The first was the overrepresentation of the oldest child and the underrepresentation of the youngest in outpatient psychiatric clinics when compared to general population data. The second was the tendency for positional differences to disappear for specific behavioral problems as family size increased. In two-child families, the older was referred more often to a clinic for anxiety, severe psychiatric symptoms, aggression, and difficulties in interpersonal relations; the younger more for mental retardation. In three-child families, only two problems, interpersonal relations and mental retardation, remained significant; in four or more-child families, no problem differentiated position.