Spectral Nodal Method Research Articles

A new spectral nodal method based on Larsen’s extended diamond approach

In this article, we describe a new spectral nodal method for solving discrete ordinates ( S N ) neutron transport problems with anisotropic scattering for arbitrary order N of angular quadrature. The key to our new spectral nodal method is a consistent derivation of nonstandard auxiliary equations that relate angular neutron fluxes only in the upwind directions. These nonstandard equations are angularly coupled extensions of very basic auxiliary equations proposed by Edward W. Larsen in his extended diamond scheme of solving S 2 problems in the presence of scattering and free from spatial truncation error. The resulting method here is also free from spatial truncation error and, in contrast to previously developed spectral nodal methods, it is compatible with an efficient use of iteration on the scattering source and is free from the storage of cell-edge angular fluxes.

Recent Advances in Spectral Nodal Methods for Numerically Solving Neutron‐Diffusion Eigenvalue Problems

In this paper we present the recent advances in spectral nodal methods for numerically solving neutron-diffusion eigenvalue problems in Cartesian geometry. For one‐dimensional two‐energy group diffusion criticality problems, we describe the use of nonconventional albedo boundary conditions that substitute approximately the reflective properties of the nonmultiplying media around the core. We also present the Chebyshev acceleration scheme for the power method used in the outer iterations. For three‐dimensional one‐speed diffusion eigenvalue problems, we present a spectral nodal method that is based on transverse‐integrating the diffusion equation separately in X, Y, and Z directions, and then considering flat approximations for the transverse leakage terms. Numerical results for typical model problems are given to illustrate the accuracy.

An exponential spectral nodal method for one-speed X,Y-geometry deep penetration discrete ordinates problems

Presented here is an exponential spectral nodal method applied to deep penetration X,Y-geometry heterogeneous neutron transport problems in the discrete ordinates ( S N) formulation. This numerical method uses the spectral Green's function (SGF) scheme for solving the one-dimensional transverse-integrated S N exponential nodal equations with no spatial truncation error. Based on the physics of deep penetration problems, we approximate the transverse leakage terms by exponential functions. We show in two numerical experiments that the SGF-exponential nodal method (SGF-ExpN) generates very accurate results when compared to the conventional transport nodal methods for coarse-mesh deep penetration S N problems, specially in highly absorbing media.

Analytical Coarse-Mesh Method for Monoenergetic Slab-Geometry Reactor Kinetics using Neutron Diffusion Model

Presented here is an analytical spectral nodal method for slab-geometry one-speed kinetics diffusion model with a given number N of groups of delayed neutron precursors. Our method is based on a spectral analysis of the nodal kinetics diffusion equat

A coarse-mesh numerical method for one-speed neutron transport eigenvalue problems in two-dimensional Cartesian geometry

We describe a hybrid spectral nodal method applied to one-speed S N eigenvalue problems in x, y-geometry for nuclear reactor global calculations. To solve the transverse-integrated S N nodal equations, we generalize the spectral diamond (SD) method, that we developed for numerically solving slab-geometry S N eigenvalue problems with no spatial truncation error. In the present generalization, we approximate the transverse leakage through the edges of each spatial node by constants, so we call our method the SD-constant nodal (SD-CN) method that we use in the fuel regions of the nuclear reactor core. In the non-multiplying regions, e.g., reflector and baffle, we use the spectral Green's function-constant nodal (SGF-CN) method; hence the hybrid characteristic of our method. In order to converge the numerical solution for each S N “fixed source” problem (inner iterations) in each outer iteration (power method), we use the one-node block inversion (NBI) scheme. We show some numerical results to a typical model problem to illustrate the method's accuracy in coarse-mesh calculations.

Recent advances in spectral nodal methods for X,Y - geometry discrete ordinates deep penetration and eigenvalue problems

We describe the recent advances in a class of nodal methods applied to multidimensional discrete ordinates (S N) transport problems in Cartesian geometry. This class of coarse-mesh methods is referred to as spectral nodal methods. The basic numerical schemes that we present are the spectral Green's function (SGF) nodal method and the spectral diamond (SD) nodal method. First we describe a spectral nodal method applied to monoenergetic X,Y-geometry deep penetration S N problems with flat approximations for the transverse leakage terms of the transverse integrated S N nodal equations. This method is referred to as the SGF constant nodal (SGF-CN) method. Furthermore, we describe the SGF exponential nodal (SGF-ExpN) method, wherein the transverse leakage terms are approximated by exponential functions. Next, we describe a hybrid spectral nodal method applied to monoenergetic X,Y-geometry S N eigenvalue problems with flat approximations for the transverse leakage terms of the transverse integrated S N nodal equations. For the multiplying regions of the nuclear reactor core, e.g. the fuel regions, we use the SD constant nodal (SD-CN) method, and for the non-multiplying regions, e.g. the reflector regions, we use the SGF-CN method. Numerical results are given to illustrate the accuracy of each method presented.

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.