Numerical Solution of the Multigroup Neutron Diffusion Equation by the Meshless RBF Collocation Method

Tanbay, Tayfun; Ozgener, B.

doi:10.3390/mca18030399

Cited by 6 publications

(4 citation statements)

References 11 publications

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“…In case of the constant source problem the nuclear data and source term are D = 1.77764 cm, Σ r = 0.0143676 cm −1 , and S = 1 n /(cm 3 s), respectively, and the analytical solution of this problem is given in [3]. For the fission source case the nuclear data are given in [2] which results with a k value of 0.75024. The following error criteria are used to test the accuracy of the numerical method: for external source…”

Section: Resultsmentioning

confidence: 99%

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A Meshless Method Based on Symmetric RBF Collocation for Neutron Diffusion Problems

Tanbay

Ozgener

2019

Acta Phys. Pol. A

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In this study we have worked on the numerical solution of the multigroup neutron diffusion equation with the symmetric radial basis function collocation method. For the spatial approximation of the neutron flux, multiquadric, inverse multiquadric, and Gaussian basis functions are used as the interpolation functions. To test the performance of the method, both external and fission source problems are considered in two-dimensional Cartesian geometry. The effect of the shape parameter on the convergence and stability of the numerical algorithm is also investigated. The results have shown that, when the multiquadric is chosen, the symmetric RBF collocation method converges exponentially, and it is possible to obtain highly accurate multiplication factors and neutron flux distributions with this algorithm.

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Section: Resultsmentioning

confidence: 99%

“…Kansa's method is also known as the asymmetric collocation method due to its asymmetric collocation matrix. The characteristics of the asymmetric RBF collocation method for the numerical solution of neutron diffusion equation is investigated in [2,3].…”

Section: Introductionmentioning

confidence: 99%

A Meshless Method Based on Symmetric RBF Collocation for Neutron Diffusion Problems

Tanbay

Ozgener

2019

Acta Phys. Pol. A

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“…The radiation diffusion and transport equations have also been solved using meshless methods other than SPH, including for coupled radiative transport and conductive heat transfer [21,22,23], neutron transport [24], and neutron diffusion [25,26]. Many of these discretizations involve either relatively flat meshless functions (which increases accuracy but makes the system ill-conditioned) or integration of the meshless functions.…”

Section: Introductionmentioning

confidence: 99%

Efficient smoothed particle radiation hydrodynamics I: Thermal radiative transfer

Bassett,

Owen,

Brunner

2020

Preprint

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This work presents efficient solution techniques for radiative transfer in the smoothed particle hydrodynamics discretization. Two choices that impact efficiency are how the material and radiation energy are coupled, which determines the number of iterations needed to converge the emission source, and how the radiation diffusion equation is solved, which must be done in each iteration. The coupled material and radiation energy equations are solved using an inexact Newton iteration scheme based on nonlinear elimination, which reduces the number of Newton iterations needed to converge within each time step. During each Newton iteration, the radiation diffusion equation is solved using Krylov iterative methods with a multigrid preconditioner, which abstracts and optimizes much of the communication when running in parallel. The code is verified for an infinite medium problem, a one-dimensional Marshak wave, and a two and three-dimensional manufactured problem, and exhibits first-order convergence in time and second-order convergence in space. For these problems, the number of iterations needed to converge the inexact Newton scheme and the diffusion equation are independent of the number of spatial points and the number of processors.

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“…The neutron diffusion equation governs the behavior of neutrons in a multiplying or non-multiplying nuclear system. This equation was solved by the RBF collocation method [10][11][12]. Although an exponential convergence rate and better accuracy than linear finite and boundary element solutions were found [11], the use of constant shape parameters led to an ill-conditioning and accuracy degradation when a certain shape parameter value was exceeded.…”

Section: Introductionmentioning

confidence: 99%

Meshless solution of the neutron diffusion equation by the RBF collocation method using optimum shape parameters

Tanbay

2019

Journal of Innovative Science and Engineering (JISE)

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The meshless radial basis function collocation method is an efficient numerical technique for solving partial differential equations. The multiquadric is the most widely utilized radial function for this purpose; but it contains a shape parameter, which has a significant effect on the performance of the method. In this study, the meshless collocation method employing multiquadric as the radial function with optimum shape parameters is applied to the numerical solution of the multigroup neutron diffusion equation. The optimization of the shape parameter is performed by minimizing the Madych-Nelson function. One external and two fission source problems are solved to investigate the performance of the method. The results show that the meshless collocation method with optimized shape parameters yield a high level of accuracy with an exponential convergence rate.

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Numerical Solution of the Multigroup Neutron Diffusion Equation by the Meshless RBF Collocation Method

Cited by 6 publications

References 11 publications

A Meshless Method Based on Symmetric RBF Collocation for Neutron Diffusion Problems

A Meshless Method Based on Symmetric RBF Collocation for Neutron Diffusion Problems

Efficient smoothed particle radiation hydrodynamics I: Thermal radiative transfer

Meshless solution of the neutron diffusion equation by the RBF collocation method using optimum shape parameters

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