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 (1/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 of order (1/r)
• stable convergence condition
• Wielandt's algorithm