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

Abstract

A hierarchical domain decomposition boundary element method (HDD-BEM) that was developed to solve a twodimensional neutron diffusion equation has been modified to deal with three-dimensional problems. In the HDD-BEM, the domain is decomposed into homogeneous regions. The boundary conditions on the common inner boundaries between decomposed regions and the neutron multiplication factor are initially assumed. With these assumptions, the neutron diffusion equations defined in decomposed homogeneous regions can be solved respectively by applying the boundary element method. This part corresponds to the "lower level" calculations. At the "higher level" calculations, the assumed values, the inner boundary conditions and the neutron multiplication factor, are modified so as to satisfy the continuity conditions for the neutron flux and the neutron currents on the inner boundaries. These procedures of the lower and higher levels are executed alternately and iteratively until the continuity conditions are satisfied within a convergence tolerance. With the hierarchical domain decomposition, it is possible to deal with problems composing a large number of regions, something that has been difficult with the conventional BEM. In this paper, it is showed that a three-dimensional problem even with 722 regions can be solved with a fine accuracy and an acceptable computation time.KEYWORDS: neutron diffusion equation, boundary element method, domain decomposition, Newton's method, block Jacobi method, high order boundary element, non-conforming element, modern nodal method, three dimensional problems, accuracy