Masafumi Itagaki

Higher order three-dimensional fundamental solutions to the Helmholtz and the modified Helmholtz equations

Abstract The higher order 3-D fundamental solutions to the Helmholtz and the modified Helmholtz equations have been derived. The Lth order fundamental solution for the 3-D Helmholtz equation has the form of a spherical Bessel function multiplied by a distance to the power L. In the case of the 3-D modified Helmholtz equation a modified spherical Bessel function is required instead of the spherical Bessel function. Each degree of these solutions in the 3-D cases has a singularity of order ( 1 r ). These solutions can be used for applying the multiple reciprocity boundary element method to the 3-D Helmholtz or modified Helmholtz problems.

Space-Dependent Core/Reflector Boundary Conditions Generated by the Boundary Element Method for Pressurized Water Reactors

This paper reports on the boundary element method used to generate energy-dependent matrix-type boundary conditions along core/reflector interfaces and along baffle-plate surfaces of pressurized water reactors. This method enables one to deal with all types of boundary geometries including convex and concave corners. The method is applicable to neutron diffusion problems with more than two energy groups and also can be used to model a reflector with or without a baffle plate. Excellent eigenvalue and flux shape results can be obtained when the boundary conditions generated by this technique are coupled with core-only finite difference calculations.

Generation of higher order fundamental solutions to the two-dimensional modified Helmholtz equation

Abstract The higher order fundamental solutions to the two-dimensional (2-D) modified Helmholtz equation have been derived. The Lth order fundamental solution has the form φ i ∗(L) = A L (kr) L K L (kr) , which satisfies the equation ∂ 2 ∂r 2 + 1 r ∂ ∂r − k 2 φ ∗(L) i +φ ∗(L−1) i =0 and the relationship of the coefficients AL is given as A L = A L−1 (2Lk 2 ) with A 0 = 1 (2π) . These solutions can be used for applying the multiple reciprocity boundary element method to the 2-D modified Helmholtz problems. The derivation procedure shown in the present paper can also be applied to obtaining higher order fundamental solutions to other engineering equations.

Remedy for round-off error accumulation observed in a neutron diffusion calculation using the multiple reciprocity boundary element method

Abstract A round-off error accumulation is observed in a multiple reciprocity computation for a fission neutron source iteration problem when a certain convergence condition is not satisfied. The present paper presents a reformulation of the multiple reciprocity method to remove the numerical problem. The neutron diffusion equation (NDE) is arranged using Wielandts spectral shift technique in such a way that the convergence condition is always satisfied through source iterations. The boundary integral equation in this case requires the fundamental solution to the standard Helmholtz equation, while the fundamental solution corresponding to the correspoding to the original NDE was one to the modified Helmholtz equation. Except for this, the new multiple reciprocity formulation is identical to the original one for the fission source iteration problems. Some test calculation results indicate that a rapid and stable convergence can be realized using the new method and no round-off error accumulation is observed any more.

Boundary element techniques for two-dimensional nuclear reactor calculations

Abstract This paper deals with several techniques for obtaining boundary element solutions of the two-dimensional multi-energy-group neutron diffusion equation which governs the neutronic phenomena in nuclear reactors. In this paper, the modified Bessel function is used as the fundamental solution because the neutron diffusion equation can be classified as a variety of the modified Helmholtz equation. The curvilinear integrals for the matrix element G ii are calculated analytically with the aid of the modified Struve functions. Critical eigenvalue problems are solved using the ‘power method’. The domain integral can be replaced by the equivalent boundary integral for a uniform fixed-source or for a slowing-down source in a non-fuel-bearing region. Furthermore, the method of images is applied to symmetrical geometries to reduce the number of unknowns. In this case the fundamental solution is expressed as the superposition of contributions from the original source point and from its image points.

Multiple reciprocity boundary element formulation for one-group fission neutron source iteration problems

Abstract The multiple reciprocity method has been applied to one-group fission neutron source iteration problems. The domain integral related to the fission neutron source in the m th source iteration stage is transformed into a series of m − 1 boudary integrals. These integrals require the zero order to the m − 1 order fundamental solutions and the boundary neutron fluxes and currents calculated in the previous m − 1 source iterations. Results obtained in a simple test calculation indicate that the source iteration based on the present technique converges rapidly to the desired eigenvalue as long as a certain condition is satisfied.