حل معادله‌ی پخش نوترون با استفاده از روش بدون مش بر پایه‌ی تابع‌های شعاعی وِندلند در مختصات دو بعدی کارتزین

نوع مقاله: مقاله پژوهشی

نویسندگان

1 دانشکده مهندسی هسته‌ای، دانشگاه شهید بهشتی، صندوق پستی: 1983963113، تهران ـ ایران

2 پژوهشگاه علوم و فنون هسته‌ای، سازمان انرژی اتمی ایران، صندوق پستی: 836-14395، تهران ـ ایران

چکیده

امروزه در حوزه­ی تحلیل عددی و علوم محاسباتی پژوهش­های گسترده­ای در زمینه­ی استفاده از روش­های بدون مش در حال انجام است. در این مقاله از یک روش بدون مش بر پایه­ی درون­یابی نقطه‌ای برای حل معادله­ی پخش یک- گروهی نوترون در مختصات دو بعدی کارتزین استفاده شده است. برای درون­یابی، از تابع­های شعاعی وندلند استفاده شد. از روش گالرکین برای گسسته­سازی شکل ضعیف معادله­ی پخش نوترون استفاده شد و انتگرال­های موجود در معادله با روش عددی گوس- لژاندر محاسبه شدند. برای ارزیابی روش معرفی شده، نمونه­های مختلفی از معادله­ی پخش نوترون در دو بعد حل و نتایج با پاسخ تحلیلی مقایسه شد. در مواردی که پاسخ تحلیلی وجود نداشت، مسئله با کد محاسباتی سایتیشن شبیه­سازی و نتایج مقایسه شد. برای ارزیابی کارآیی روش، مسئله­ی آزمون رید حل شد. به عنوان یک مسئله­ی کاربردی، معادله­ی پخش نوترون برای ربع قلب یک رآکتور آب تحت فشار (PWR) در دو بعد حل شد. هم­چنین ترکیب تابع­های شعاعی مختلف وندلند با تابع­های چندجمله‌ای از نظر دقت نتایج با یک­دیگر مقایسه شدند. مقایسه­ی نتایج به دست آمده با پاسخ­های تحلیلی و نتایج حاصل از شبیه­سازی با کد سایتیشن نشان داد که روش استفاده شده از دقت و کارآیی مطلوبی برخوردار است و می‌تواند برای تولید کدهای هسته‌ای مورد استفاده قرار گیرد.
 

تازه های تحقیق

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).

 

کلیدواژه‌ها


عنوان مقاله [English]

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

نویسندگان [English]

  • B Rokrok 1
  • H Minuchehr 1
  • A Zolfaghari 1
  • A Movafeghi 2
چکیده [English]

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.
 

کلیدواژه‌ها [English]

  • 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).