• Journal of Nuclear Fuel Cycle and Waste Technology(JNFCWT)
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• v.16 no.2
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• pp.211-221
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• 2018

Nodal transport methods are proposed for solving the simplified even-parity neutron transport (SEP) equation. These new methods are attributed to the success of existing nodal diffusion methods such as the Polynomial Expansion Nodal and the Analytic Function Expansion Nodal Methods, which are known to be very effective for solving the neutron diffusion equation. Numerical results show that the simplified even-parity transport equation is a valid approximation to the transport equation and that the two nodal methods developed in this study also work for the SEP transport equation, without conflict. Since accuracy of methods is easily increased by adding node unknowns, the proposed methods will be effective for coarse mesh calculation and this will also lead to computation efficiency.

• Nuclear Engineering and Technology
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• v.32 no.6
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• pp.605-617
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• 2000

Nodal transport methods are studied for the solution of two dimensional discrete-ordinates, simplified even-parity transport equation(SEP) which is known to be an approximation to the true transport equation. The polynomial expansion nodal method(PEN) and the analytic function expansion nodal method(AFEN）which have been developed for the diffusion theory are used for the solution of the discrete-ordinates form of SEP equation. Our study shows that while the PEN method in diffusion theory can directly be converted without complication, the AFEN method requires a theoretical modification due to the nonhomogeneous property of the transport equation. The numerical results show that the proposed two methods work well with the SEP transport equation with higher accuracies compared with the conventional finite difference method.

• Journal of Advanced Marine Engineering and Technology
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• v.2 no.1
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• pp.3-14
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• 1978

The authors have developed a calculating method of propeller shaft alignment by the finite element method. The propeller shaft is divided into finite elements which can be treated as uniform section bars. For each element, the nodal point equation is derived from the stiffness matrix, the external force vector and the section force vector. Then the overall nodal point equation is derived from the element nodal point equation. The deflection, offset, bending moment and shearing force of each nodal point are calculated from the overall nodal point equation by the digital computer. Reactions and deflections of supporting points of straight shaft are calculated and also the reaction influence number is derived. With the reaction influence number the optimum alignment condition that satisfies all conditions is calculated by the simplex method of linear programming. All results of calculation are compared with those of Det norske Veritas, which has developed a computor program based on the three-moment theorem of the strength of materials. The authors finite element method has shown good results and will be used effectively to design the propeller shaft alignment.

• Nuclear Engineering and Technology
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• v.54 no.9
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• pp.3181-3204
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• 2022

The variational nodal method for solving the neutron transport equation has evolved over 40 years. Based on a functional form of the Boltzmann neutron transport equation, the method now comprises a complete set of variants that can be employed for different problems. This paper presents an extensive review of the development of the variational nodal method. The emphasis is on summarizing the whole theoretical system rather than validating the methodologies. The paper covers the variational nodal formulation of the Boltzmann neutron transport equation, the Ritz procedure for various application purposes, the derivation of boundary conditions, the extension for adjoint and perturbation calculations, and treatments for anisotropic scattering sources. Acceleration approaches for constructing response matrices and solving the resulting system of algebraic equations are also presented.

• Proceedings of the Korea Society for Energy Engineering kosee Conference
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• 1993.11a
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• pp.99-102
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• 1993

A consistent general order nodal method for solving the three-dimensional neutron diffusion equation in (x-y-z) geometry has been derived by using a weighted integral technique and expanding the spatial variable by the Legendre orthogonal series function. The equation set derived can be converted into any order nodal schemes. It forms a compact system for general order of nodal moments. The method utilizes fewer unknown variables in the schemes for iterative-convergence solution than other nodal methods listed in the literatures, and because the method utilizes the analytic solutions of the transverse-integrated one dimensional equations and a consistent approximation for a given spatial variable through all the solution procedures, which renders the use of an approximation for the transverse leakages no longer necessary, we can expect extremely accurate solutions and the solution would converge exactly when the mesh width is decreased or the approximation order is increased.

• Proceedings of the Korean Nuclear Society Conference
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• 1996.05a
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• pp.34-39
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• 1996

A consistent general order nodal method for solving the 3-D neutron diffusion equation in (x-y-z) geometry has ben derived by using a weighted integral technique and expanding the spatial variables by the Legendre orthogonal series function. The equation set derived can be converted into any order nodal schemes. It forms a compact system for general order of nodal moments. The method utilizes the analytic solutions of the transverse-integrated quasi -one dimensional equations and a consistent expansion for the spatial variables so that it renders the use of an approximation for the transverse leakages no necessary. Thus, we can expect extremely accurate solutions and the solution would converge exactly when the mesh width is decreased or the approximation order is increased since the equation set is consistent mathematically.

• Proceedings of the Korea Society for Energy Engineering kosee Conference
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• 2000.11a
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• pp.115-120
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• 2000

The analysis of flux distribution under stead-state in large power reactors with assymetry reactivity insertions requires the use of three-dimensional diffusion calculations. For the purpose, consistently formulated modern nodal methods based on higher order interface techniques have become popular tools for flux distributions in large commercial nuclear reactors. Among the earlier developments, the nodal Green's function method obtains its nodal interface equation from the transverse-integrated integral diffusion equation using a finite-medium Green's function. In this method, the outgoing current from a node surface is formulated as a response of the incoming currents and the spatially integrated neutron source within the same node. The well-known nodal expansion method is also based on an interface partial current formulation. Nodal methods high-level interface variables, i.e., interface net current and flux, may be more computationally efficient than the nodal Green's function method because they have one fewer unknown per interface. The Analytic Nodal Method(ANM), which can be classified as an interface net current technique and, was faster in solving some standard benchmark problems than the other two methods.(omitted)

• Bulletin of the Korean Mathematical Society
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• v.57 no.2
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• pp.331-343
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• 2020

This paper is concerned with the global behavior of components of radial nodal solutions of semilinear elliptic problems -Δv = λh(x, v) in Ω, v = 0 on ∂Ω, where Ω = {x ∈ R^N : r₁ < |x| < r₂} with 0 < r₁ < r₂, N ≥ 2. The nonlinear term is continuous and satisfies h(x, 0) = h(x, s₁(x)) = h(x, s₂(x)) = 0 for suitable positive, concave function s₁ and negative, convex function s₂, as well as sh(x, s) > 0 for s ∈ ℝ \ {0, s₁(x), s₂(x)}. Moreover, we give the intervals for the parameter λ which ensure the existence and multiplicity of radial nodal solutions for the above problem. For this, we use global bifurcation techniques to prove our main results.

• Nuclear Engineering and Technology
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• v.3 no.3
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• pp.121-128
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• 1971

A unified modal-nodal expansion of tile angular distribution of neutron flux in one spatial dimension is considered, following the proposal of Harms. Several standard nodal and/or modal methods of analysis are shown to be specializations of this technique. The modal-nodal moment from of the mono-energetic transport equation with isotropic sources and scattering is derived and the infinite medium eigenvalue problem solved. The technique is shown to yield results which approximate the exact value of the inverse diffusion length in non-multiplying media more accurately than standard methods of equal or somewhat greater computational complexity.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.12 no.4
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• pp.261-269
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• 2008

Let $B_1$ be a unit ball in $R^n(n{\geq}3)$, and $2^*=2n/(n-2)$ be the critical Sobolev exponent for the embedding $H_0^1(B_1){\hookrightarrow}L^{2^*}(B_1)$. By using a variant of Pohoz$\check{a}$aev's identity, we prove the nonexistence of nodal solutions for the Dirichlet problem $-{\Delta}u-{\mu}\frac{u}{{\mid}x{\mid}^2}={\lambda}u+{\mid}u{\mid}^{2^*-2}u$ in $B_1$, u=0 on ${\partial}B_1$ for suitable positive numbers ${\mu}$ and ${\nu}$.