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

Simulating deformation of objects with multi‐materials using boundary element method

“…For low latency simulation, capacitance method is a popular method for solving Equation (12). Given a matrix K, K −1 is obtained in a pre-computation.…”

“…Substituting the above two expressions into Equation (25) Equation (26) is in the form of Equation (12), which describes the relationship between the unknown and known displacements of one component. Taking into consideration the effects of common faces and common edges between components, Equation (12) can be extended for simulating the deformation of an object with n components.…”

“…In the area of nuclear science, BEM has been used to solve the multi-region neutron diffusion equation. For objects with a complex geometric shape or if the number of regions is large, a hierarchical domain decomposition BEM (HDD-BEM) is developed for solving the neutron diffusion equation by decomposing the domain into homogeneous regions [12].In using BEM for multi-domain or multi-material problems, an efficient technique to assemble and evaluate the solution of the matrix equations is a critical issue, which has attracted much attention. A popular approach is to use the collocation-based multi-zone boundary element analysis (BEA) [13,14] technique.…”

“…In the area of nuclear science, BEM has been used to solve the multi-region neutron diffusion equation. For objects with a complex geometric shape or if the number of regions is large, a hierarchical domain decomposition BEM (HDD-BEM) is developed for solving the neutron diffusion equation by decomposing the domain into homogeneous regions [12].…”

Solução da Equação de Difusão de Nêutrons Multigrupo Multirregião Estacionária Unidimensional em Geometria Cartesiana pelo Método da Potência via Fronteiras Fictı́cias

“…Dentre os métodos numéricos destacam-se, por exemplo, o método nodal [5], método das diferenças finitas [3], método dos elementos finitos [4] e método dos elementos de contorno [2]. Embora cada um dos métodos mencionados tem seus próprios méritos e deficiências, eles encorrem necessariamente na discretização do operador diferencial da equação de difusão para produzir um conjunto de equações algébricas discretas.…”