Research Article

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

Volume: 3 Number: 1 June 24, 2019
Journal of Innovative Science and Engineering Cover
Journal of Innovative Science and Engineering
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Meshless solution of the neutron diffusion equation by the RBF collocation method using optimum shape parameters

Abstract

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.

Keywords

References

  1. [1] Kansa, E.J. (1986). Application of Hardy’s multiquadric interpolation to hydrodynamics, In: R. Crosbie, P. Luker (Eds.), Proceedings of the 1986 Summer Computer Simulation Conference, San Diego, Society for Computer Simulation, 4:111-117.
  2. [2] Madych, W.R. (1992). Miscellaneous error bounds for multiquadric and related interpolators. Computers and Mathematics with Applications, 24:121-138.
  3. [3] Hardy, R.L. (1971). Multiquadratic equations for topography and other irregular surfaces. Journal of Geophysical Research, 76:1905-1915.
  4. [4] Franke, R. (1982). Scattered data interpolation tests of some methods. Mathematics of Computation, 38:181-200.
  5. [5] Fasshauer, G.E. (2002). Newton iteration with multiquadrics for the solution of nonlinear PDEs. Computers and Mathematics with Applications, 43:423-438.
  6. [6] Rippa, S. (1999). An algorithm for selecting a good value for the parameter c in radial basis function interpolation. Advances in Computational Mathematics, 11:193-210.
  7. [7] Esmaeilbeigi, M. and Hosseini, M.M. (2014). A new approach based on the genetic algorithm for finding a good shape parameter in solving partial differential equations by Kansa’s method. Applied Mathematics and Computation, 249:419-428.
  8. [8] Koupaei, J.A., Firouznia, M. and Hosseini, M.M. (2018). Finding a good shape parameter of RBF to solve PDEs based on the particle swarm optimization algorithm. Alexandria Engineering Journal, 57:3641-3652.

Details

Primary Language

English

Subjects

Engineering

Journal Section

Research Article

Authors

Publication Date

June 24, 2019

Submission Date

May 26, 2019

Acceptance Date

June 17, 2019

Published in Issue

Year 2019 Volume: 3 Number: 1

DOI

https://doi.org/10.38088/jise.570328

IZ

https://izlik.org/JA58WC27BL

Cite

RIS / Bibtex
APA
Tanbay, T. (2019). Meshless solution of the neutron diffusion equation by the RBF collocation method using optimum shape parameters. Journal of Innovative Science and Engineering, 3(1), 23-31. https://doi.org/10.38088/jise.570328
AMA
1.Tanbay T. Meshless solution of the neutron diffusion equation by the RBF collocation method using optimum shape parameters. JISE. 2019;3(1):23-31. doi:10.38088/jise.570328
Chicago
Tanbay, Tayfun. 2019. “Meshless Solution of the Neutron Diffusion Equation by the RBF Collocation Method Using Optimum Shape Parameters”. Journal of Innovative Science and Engineering 3 (1): 23-31. https://doi.org/10.38088/jise.570328.
EndNote
Tanbay T (June 1, 2019) Meshless solution of the neutron diffusion equation by the RBF collocation method using optimum shape parameters. Journal of Innovative Science and Engineering 3 1 23–31.
IEEE
[1]T. Tanbay, “Meshless solution of the neutron diffusion equation by the RBF collocation method using optimum shape parameters”, JISE, vol. 3, no. 1, pp. 23–31, June 2019, doi: 10.38088/jise.570328.
ISNAD
Tanbay, Tayfun. “Meshless Solution of the Neutron Diffusion Equation by the RBF Collocation Method Using Optimum Shape Parameters”. Journal of Innovative Science and Engineering 3/1 (June 1, 2019): 23-31. https://doi.org/10.38088/jise.570328.
JAMA
1.Tanbay T. Meshless solution of the neutron diffusion equation by the RBF collocation method using optimum shape parameters. JISE. 2019;3:23–31.
MLA
Tanbay, Tayfun. “Meshless Solution of the Neutron Diffusion Equation by the RBF Collocation Method Using Optimum Shape Parameters”. Journal of Innovative Science and Engineering, vol. 3, no. 1, June 2019, pp. 23-31, doi:10.38088/jise.570328.
Vancouver
1.Tayfun Tanbay. Meshless solution of the neutron diffusion equation by the RBF collocation method using optimum shape parameters. JISE. 2019 Jun. 1;3(1):23-31. doi:10.38088/jise.570328

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