The three-dimensional neutron diffusion equation has been solved using the boundary element method employing quadratic isoparametric boundary elements. Direct computation of the singular integral which arises in the present formulation can be avoided using standard mapping techniques, i.e., polar coordinates transformation is introduced after the quadrilateral element is subdivided into four triangles. The domain integral in the boundary integral equation corresponding to the neutron diffusion equation is converted into an equivalent boundary one when the integral is related to a uniform source or a slowing-down source. No boundary elements are required on the outer surface of an infinite reflector. The boundary elements for a symmetry plane can be also omitted when the method of images is adopted. Test calculations show that the present techniques provide accurate solutions for problems of irregular geometries.KEYWORDS: three-dimentional neutron diffusion equation, boundary element method, boundary integral equation, irregular geometry, isoparametric elements, discontinuous element, singular integral, Cauchy principal value, Gauss' theorem, Green's second identity, method o f images, parallel processing, accuracy

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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

The multiple reciprocity boundary element method has been applied to three-dimensional two-group neutron diffusion problems. A matrix-type boundary integral equation has been derived to solve the first and the second group neutron diffusion equations simultaneously. The matrix-type fundamental solutions used here satisfy the equation which has a point source term and is adjoint to the neutron diffusion equations. A multiple reciprocity method has been employed to transform the matrix-type domain integral related to the fission source into an equivalent boundary one. The higher order fundamental solutions required for this formulation are composed of a series of two types of analytic functions. The eigenvalue itself is also calculated using only boundary integrals. Three-dimensional test calculations indicate that the present method provides stable and accurate solutions for criticality problems.K E Y WORDS: two-group neutron diffusion equations, matrix-type boundary integral equation, matrix-type fundamental solutions, eigenvalues, source iteration, Wielandt 's algorithm, multiple reciprocity method, criticality

The three-dimensional neutron diffusion equation has been solved using the boundary element method employing quadratic isoparametric boundary elements. Direct computation of the singular integral which arises in the present formulation can be avoided using standard mapping techniques, i.e., polar coordinates transformation is introduced after the quadrilateral element is subdivided into four triangles. The domain integral in the boundary integral equation corresponding to the neutron diffusion equation is converted into an equivalent boundary one when the integral is related to a uniform source or a slowing-down source. No boundary elements are required on the outer surface of an infinite reflector. The boundary elements for a symmetry plane can be also omitted when the method of images is adopted. Test calculations show that the present techniques provide accurate solutions for problems of irregular geometries.KEYWORDS: three-dimentional neutron diffusion equation, boundary element method, boundary integral equation, irregular geometry, isoparametric elements, discontinuous element, singular integral, Cauchy principal value, Gauss' theorem, Green's second identity, method o f images, parallel processing, accuracy