Finite Element Method with Spectral Green's Function in Slab Geometry for Neutron Diffusion in Multiplying Media and One Energy Group

Abstract

ABSTRACT. The physical phenomenon of neutrons transport associated with eigenvalue problems appears in the criticality calculations of nuclear reactors and can be treated as a diffusion process. This paper presents a method to solve eigenvalue problems of neutron diffusion in slab geometry and one energy group. This formulation combines the Finite Element Method, considered an intermediate mesh method, with the Spectral Green's Function Method, which is free of truncation errors, and it is considered a coarse mesh method. The novelty of this formulation is to approach the spatial moments of the neutron flux distribution by the first-order polynomials obtained from the spectral analysis of diffusion equation. The approximations provided by the proposed formulation allow obtaining accurate results in coarse mesh calculations. To validate the proposed method, we compared its results with methods described in the literature. The accuracy and computational performance of our formulation were characterized by solving a benchmark problem with a high degree of heterogeneity.

An Extended Linear Discontinuous Method for One-group Fixed Source Discrete Ordinates Problems with Isotropic Scattering in Slab Geometry

“…In this paper, we develop the hybrid Extended Linear-Discontinuous method, for discrete ordinates formulation (ELD-SN), to solve fixed source problems in slab geometry, considering one energy group approximation and an isotropic scattering. This method extends the prime work in hybrid methods, proposed by [24] to solve eigenvalue diffusion problems in one-dimensional Cartesian geometry. The ELD-SN combines a linear discontinuous approximation inside the spatial cell, common in finite element method, and a quasi-analytic approach, characteristic in SGF methods for preserving the analytical spectral solutions inside the cell.…”

“…al. (2016) [24]. Specifically, that work described the Finite Element Spectral Green's Function method (FEM-SGF) to solve the neutron diffusion equation in non-multiplying media for slab geometry.…”

“…Recently, some authors developed numerical approaches that exploit the advantages of SGF methods, minimizing its deficiencies. Among these, stand out the composite spatial grid methods [14] and the finite element spectral nodal hybrid methods [24].…”

A multi-group extended linear discontinuous method for fixed-source discrete ordinates problems in slab geometry

“…Therefore, it is necessary to specify some auxiliary equations to transform the system into a fully determined system with a unique solution. The ELD method auxiliary equations for several energy groups use the linear discontinuous approximation of the finite element methods (Hill, 1975;Rocha et al, 2016) introducing two additional spectral parameters. These parameters are calculated using the general quasi-analytical solution of the S N one-dimensional multi-group spectral nodal method (Barros and Larsen, 1991).…”

“…Recently, several publications focus on the development of new numerical approaches that offer an acceptable compromise between accuracy and computational cost (Dominguez et al, 2010;Rocha et al, 2016;Dominguez et al, 2018). Rivas-Ortiz et al (2019) proposed the Extended Linear Discontinuous (ELD) method in discrete ordinates formulation for the resolution of one-energy group fixed-source problems.…”