Three-Dimensional Multiple Reciprocity Boundary Element Method for One-Group Neutron Diffusion Eigenvalue Computations.

Abstract

The multiple reciprocity method (MRM) in conjunction with the boundary element method has been employed to solve one-group eigenvalue problems described by the three-dimensional (3-D) neutron diffusion equation. The domain integral related to the fission source is transformed into a series of boundary-only integrals, with the aid of the higher order fundamental solutions based on the spherical and the modified spherical Bessel functions. Since each degree of the higher order fundamental solutions in the 3-D cases has a singularity of order (l/r), the above series of boundary integrals requires additional terms which do not appear in the 2-D MRM formulation. The critical eigenvalue itself can be also described using only boundary integrals. Test calculations show that Wielandt's spectral shift technique guarantees rapid and stable convergence of 3-D MRM computations.KEYWORDS: three dimensional neutron diffusion equation, eigenvalue, source iteration, boundary element method, multiple reciprocity method, higher-order fundamental solution, singularity o f order ( I / r), stable convergence condition, Wielandt's algorithm

Reliable Method for Fission Source Convergence of Monte Carlo Criticality Calculation with Wielandt's Method

“…Many acceleration techniques for deterministic eigenvalue calculations have been developed, and Wielandt's method 7) is one of the acceleration methods, which has been applied to reactor analysis. 8) Using Wielandt's method, rapid and stable convergence is generally guaranteed in deterministic calculation methods.…”

Development of the Hierarchical Domain Decomposition Boundary Element Method for Solving the Three-Dimensional Multiregion Neutron Diffusion Equations.

“…[2][3][4][5][6] In the formulation of BEM, the NDE is transformed into an equivalent boundary integral equation (BIE). This treatment enables a simple and rigorous evaluation of the effect of a boundary, even when the geometric shape is too complex for application of the finite difference method (FDM).…”

“…The application of the BEM to this method usually requires the calculations of domain integrals to evaluate the volume integrals of fission source terms, which considerably impairs the merit of BEM. For overcoming this drawback, Itagaki has proposed two techniques: (1) the dual reciprocity method (DRM) 3) and (2) the multireciprocity method (MRM), 4,6) where the volume integral of the fission source can be converted into equivalent boundary integrals through mathematically sophisticated procedures.…”