Journal of Nuclear Science, Engineering and Technology (JONSAT)
In cooperation with the Iranian Nuclear Society

Application of a Mesh Free Method Based on the Wendland Radial Basis Functions to Solve the Neutron Diffusion Equation in Two-Dimensional Geometry

Document Type : Research Paper

Authors

B Rokrok
H Minuchehr
A Zolfaghari
A Movafeghi
Abstract
These days, application of mesh free methods in the areas of numerical analysis and computational sciences has been the subject of many researches. In this paper, the mesh free method based on the point interpolation scheme is used to solve the one-group neutron diffusion equation in a two- dimensional Cartesian coordinate system. The Wendland type radial basis functions were applied to perform the interpolations. The Galerkin method was employed to discretise the weak form of the neutron diffusion equation. In order to calculate the integrations of the weak form of the equations, Gauss-Legendre scheme was applied. The efficiency and accuracy of the method was evaluated through a number of case studies. The results were compared with the analytical solutions. For the cases where the numerical solutions did not exist, the problem was simulated through the Citation code and the results were compared, accordingly. The Reed test problem was solved to show the performance of the developed code. A PWR reactor core was also simulated through the introduced method. The effect of combination of different Wendland functions with polynomial functions on the accuracy of the results was also assessed. There is a good agreement between the numerical and the analytical solutions, and also the result from the Citation code revealed the accuracy of the method, and the good performance of the applied method was also confirmed in this study. At last, the developed method introduced in this work was found to be applicable to implement the desired nuclear computational codes.
 

Highlights

1.    R. Avila, A. Perez, Mesh free methods for partial differential equations IV-A pressure correction approach coupled with the MLPG method for solution of the Navier Stokes Equations, Springer (2008) 19-33.

 2.    H. Ding, C. Shu, K. S. Yeo, D. Xu, Development of least square-based two-dimensional finite-difference and their application to simulate natural convection in a cavity, Computers and Fluids, 33 (2004) 137-154.

 3.    G. R. Liu, M. B. Liu, Smoothed Particle Hydrodynamics, a mesh free practical method, World Scientific Publishing, Singapore (2003).

 4.    T. Belytschko, Y. Y. Lu, L. Gu, Element-free Galerkin methods, Int. Journal of Numerical Methods in Engineering, 37 (1994) 229-256.

 5.    W. Liu, S. Jun, Y. Zhang, Reproducing kernel particle methods, International Journal for Numerical Methods in Fluids, 20 (1995) 1081-1106.

 6.    C. Armando Duarte, J. Tinsley Oden, An h-p adaptive method using clouds, Computer methods in applied mechanics and engineering, 139 (1996) 237-262.

 7.    I. Babuska, B. Uday, E. O. John, Generalized Finite Element Methods: Main Ideas, Results, and Perspective, International Journal of Computational Methods, 1 (1) (2004) 67-103.

 8.    I. Babuska, J. M. Melenk, The Partition of Unity Method, Int. J. Num. Meth. Eng. 40 (1997) 727-758.

 9.    G. R. Liu, Y. T. Gu, A point interpolation method for two dimensional solids, Int. J. Numer. Methods Eng. 50 (2001) 937-951.

 10.  G. R. Liu, Y. T. Gu, An introduction to Mesh free methods and their programming, Springer (2005).

 11.  G. R. Liu, Mesh free methods, Moving Beyond the finite element, CRC Press (2003).

 12.  B. Rokrok, H. Minuchehr, A. Zolfaghari, Appliaction of Radial Point Interpolation Method to Neutron Diffusion field, Trends in applied sciences research, 7(1) (2012) 18-31.

13.T. B. Fowler, D. R. Vondy, G. W. Cunningham, Nuclear Reactor Core Analysis Code: CITATION, ORNL-TM-2496, Rev. 2, with Supplements 1, 2, and 3 (1971).

14.  N. Dyn, D. Levin, S. Rippa, Numerical procedures for surface fitting of scattered data by radial functions, SIAM J. Sci. Stat. Comput. 7 (1986) 639-659.

15. E. J. Kansa, Multiquadrics-A scattered data approximation scheme with applications to computational fluid dynamics I: Solutions to parabolic, hyperbolic, and elliptic partial differential equations, Computers Math. Applic. 19 (8-9) (1990) 147-161.

16. H. Wendland, Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, Advances in Computational Mathematics, 4 (1995) 389-396.

17. O. A. Abuzaid, Discontinuous Finite Elements Solution for Neutron Diffusion and Transport, Ph.D. Thesis, London University (1994).

 

Keywords

Neutron Diffusion Equation
Mesh Free Method
Wendland Radial Basis Functions
Galerkin Method
1.    R. Avila, A. Perez, Mesh free methods for partial differential equations IV-A pressure correction approach coupled with the MLPG method for solution of the Navier Stokes Equations, Springer (2008) 19-33.
 2.    H. Ding, C. Shu, K. S. Yeo, D. Xu, Development of least square-based two-dimensional finite-difference and their application to simulate natural convection in a cavity, Computers and Fluids, 33 (2004) 137-154.
 3.    G. R. Liu, M. B. Liu, Smoothed Particle Hydrodynamics, a mesh free practical method, World Scientific Publishing, Singapore (2003).
 4.    T. Belytschko, Y. Y. Lu, L. Gu, Element-free Galerkin methods, Int. Journal of Numerical Methods in Engineering, 37 (1994) 229-256.
 5.    W. Liu, S. Jun, Y. Zhang, Reproducing kernel particle methods, International Journal for Numerical Methods in Fluids, 20 (1995) 1081-1106.
 6.    C. Armando Duarte, J. Tinsley Oden, An h-p adaptive method using clouds, Computer methods in applied mechanics and engineering, 139 (1996) 237-262.
 7.    I. Babuska, B. Uday, E. O. John, Generalized Finite Element Methods: Main Ideas, Results, and Perspective, International Journal of Computational Methods, 1 (1) (2004) 67-103.
 8.    I. Babuska, J. M. Melenk, The Partition of Unity Method, Int. J. Num. Meth. Eng. 40 (1997) 727-758.
 9.    G. R. Liu, Y. T. Gu, A point interpolation method for two dimensional solids, Int. J. Numer. Methods Eng. 50 (2001) 937-951.
 10.  G. R. Liu, Y. T. Gu, An introduction to Mesh free methods and their programming, Springer (2005).
 11.  G. R. Liu, Mesh free methods, Moving Beyond the finite element, CRC Press (2003).
 12.  B. Rokrok, H. Minuchehr, A. Zolfaghari, Appliaction of Radial Point Interpolation Method to Neutron Diffusion field, Trends in applied sciences research, 7(1) (2012) 18-31.
13.T. B. Fowler, D. R. Vondy, G. W. Cunningham, Nuclear Reactor Core Analysis Code: CITATION, ORNL-TM-2496, Rev. 2, with Supplements 1, 2, and 3 (1971).
14.  N. Dyn, D. Levin, S. Rippa, Numerical procedures for surface fitting of scattered data by radial functions, SIAM J. Sci. Stat. Comput. 7 (1986) 639-659.
15. E. J. Kansa, Multiquadrics-A scattered data approximation scheme with applications to computational fluid dynamics I: Solutions to parabolic, hyperbolic, and elliptic partial differential equations, Computers Math. Applic. 19 (8-9) (1990) 147-161.
16. H. Wendland, Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree, Advances in Computational Mathematics, 4 (1995) 389-396.
17. O. A. Abuzaid, Discontinuous Finite Elements Solution for Neutron Diffusion and Transport, Ph.D. Thesis, London University (1994).
 

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