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A meshless local Petrov–Galerkin method for solving the neutron diffusion equation

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Abstract

The goal of this study is to solve the neutron diffusion equation by using a meshless method and evaluate its performance compared to traditional methods. This paper proposes a novel method based on coupling the meshless local Petrov–Galerkin approach and the moving least squares approximation. This computational procedure consists of two main steps. The first involved applying the moving least squares approximation to construct the shape function based on the problem domain. Then, the obtained shape function was used in the meshless local Petrov–Galerkin method to solve the neutron diffusion equation. Because the meshless method is based on eliminating the mesh-based topologies, the problem domain was represented by a set of arbitrarily distributed nodes. There is no need to use meshes or elements for field variable interpolation. The process of node generation is simply and fully automated, which can save time. As this method is a local weak form, it does not require any background integration cells and all integrations are performed locally over small quadrature domains. To evaluate the proposed method, several problems were considered. The results were compared with those obtained from the analytical solution and a Galerkin finite element method. In addition, the proposed method was used to solve neutronic calculations in the small modular reactor. The results were compared with those of the citation code and reference values. The accuracy and precision of the proposed method were acceptable. Additionally, adding the number of nodes and selecting an appropriate weight function improved the performance of the meshless local Petrov–Galerkin method. Therefore, the proposed method represents an accurate and alternative method for calculating core neutronic parameters.

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References

  1. M. Steinhauser, Computational Multi Scale Modeling of Fluids and Solids (Springer, Berlin, 2017)

    Book  Google Scholar 

  2. G.R. Liu, Mesh Free Methods: Moving Beyond the Finite Element Method (Taylor & Francis, Boca Raton, 2009)

    Book  Google Scholar 

  3. T. Belytschko, T. Rabczuk, A. Huerta et al., Mesh free methods, in Encyclopedia of Computational Mechanics, ed. by E. Stein, R. de Borst, J.R. Hughes (Wiley, Chichester, 2004), pp. 1–48

    Google Scholar 

  4. S. Li, W.K. Liu, Mesh free and particle methods and their applications. Appl. Mech. Rev. 55, 1–34 (2002). https://doi.org/10.1115/1.1431547

    Article  Google Scholar 

  5. Y.T. Gu, Mesh free methods and their comparisons. Int. J. Comput. Methods 2, 477–515 (2005). https://doi.org/10.1142/S0219876205000673

    Article  MATH  Google Scholar 

  6. A. Tayebi, Y. Shekari, M.H. Heydari, A meshless method for solving two-dimensional variable-order time fractional advection–diffusion equation. J. Comput. Phys. 340, 655–669 (2017). https://doi.org/10.1016/j.jcp.2017.03.061

    Article  MathSciNet  MATH  Google Scholar 

  7. Q.H. Li, S.S. Chen, G.X. Kou, Transient heat conduction analysis using the MLPG method and modified precise time step integration method. J. Comput. Phys. 230, 2736–2750 (2011). https://doi.org/10.1016/j.jcp.2011.01.019

    Article  MathSciNet  MATH  Google Scholar 

  8. X.H. Wu, W.Q. Tao, S.P. Shen et al., A stabilized MLPG method for steady state incompressible fluid flow simulation. J. Comput. Phys. 229, 8564–8577 (2010). https://doi.org/10.1016/j.jcp.2010.08.001

    Article  MathSciNet  MATH  Google Scholar 

  9. R. Salehi, M. Dehghan, A moving least square reproducing polynomial meshless method. Appl. Numer. Math. 69, 34–58 (2013). https://doi.org/10.1016/j.apnum.2013.03.001

    Article  MathSciNet  MATH  Google Scholar 

  10. R.A. Gingold, J.J. Monaghan, Smoothed particle hydrodynamics: theory and application to non-spherical stars. Mon. Not. R. Astron. Soc. 181, 375–389 (1977). https://doi.org/10.1093/mnras/181.3.375

    Article  MATH  Google Scholar 

  11. P.W. Cleary, Modeling confined multi-material heat and mass flows using SPH. Appl. Math. Model. 22, 981–993 (1998). https://doi.org/10.1016/S0307-904X(98)10031-8

    Article  Google Scholar 

  12. G.R. Liu, K.Y. Dai, K.M. Lim, A point interpolation mesh free method for static and frequency analysis of two-dimensional piezoelectric structures. Comput. Mech. 29, 510–519 (2002). https://doi.org/10.1007/s00466-002-0360-9

    Article  MATH  Google Scholar 

  13. B. Nayroles, G. Touzot, P. Villon, Generalizing the finite element method: diffuse approximation and diffuse elements. Comput. Mech. 10, 307–318 (1992). https://doi.org/10.1007/BF00364252

    Article  MathSciNet  MATH  Google Scholar 

  14. W.K. Liu, S. Li, T. Belytschko, Moving least-square reproducing kernel methods. Part I: methodology and convergence. Comput. Method Appl. Mech. 143, 113–154 (1977). https://doi.org/10.1016/S0045-7825(96)01132-2

    Article  MATH  Google Scholar 

  15. W.K. Liu, J. Adee, S. Jun et al., Reproducing kernel particle methods for elastic and plastic problems, in Advanced Computational Methods for Material Modeling, ed. by D.A. Siginer, W.E. VanArsdale, C.M. Altan, A.N. Alexandrou (ASME, New Orleans, 1993), pp. 175–189

    Google Scholar 

  16. W.K. Liu, S. Jun, Y.F. Zhang, Reproducing kernel particle methods. Numer. Method Fluids 20, 1081–1106 (1995). https://doi.org/10.1002/fld.1650200824

    Article  MathSciNet  MATH  Google Scholar 

  17. E. Oñate, S. Idelsohn, O.C. Zienkiewicz et al., A finite point method in computational mechanics—applications to convective transport and fluid flow. Int. J. Numer. Methods Eng. 39, 3839–3866 (1996). https://doi.org/10.1002/(SICI)1097-0207(19961130)39:22<3839::AID-NME27>3.0.CO;2-R

    Article  MathSciNet  MATH  Google Scholar 

  18. Y.T. Gu, G.R. Liu, Using radial function basis in a boundrary-type meshless method, boundrary point method, in International Conference on Science & Engineering Computation (Beijing, 2001)

  19. M.D. Buhmann, Radial Basis Functions: Theory and Implementations (Cambridge University, Cambridge, 2003)

    Book  Google Scholar 

  20. H. Mehrabi, B. Voosoghi, On estimating the curvature attributes and strain invariants of deformed surface through radial basis functions. Comp. Appl. Math. (2016). https://doi.org/10.1007/s40314-016-0380-2

    Article  MATH  Google Scholar 

  21. Q. Li, J. Soric, T. Jarak et al., A locking-free meshless local Petrov–Galerkin formulation for thick and thin plates. J. Comput. Phys. 208, 116–133 (2005). https://doi.org/10.1016/j.jcp.2005.02.008

    Article  MATH  Google Scholar 

  22. P. Mycek, G. Pinon, G. Germain, Formulation and analysis of a diffusion-velocity particle model for transport-dispersion equations. Comp. Appl. Math. 35, 447–473 (2014). https://doi.org/10.1007/s40314-014-0200-5

    Article  MathSciNet  MATH  Google Scholar 

  23. S. Abbasbandy, A. Shirzadi, MLPG method for two-dimensional diffusion equation with Neumann’s and non-classical boundary conditions. Appl. Numer. Math. 61, 170–180 (2011). https://doi.org/10.1016/j.apnum.2010.09.002

    Article  MathSciNet  MATH  Google Scholar 

  24. B. Rokrok, H. Minuchehr, A. Zolfaghari, Application of radial point interpolation method to neutron diffusion field. Trend Appl. Sci. Res. 7, 18–31 (2012). https://doi.org/10.3923/tasr.2012.18.31

    Article  Google Scholar 

  25. T. Tanbay, B. Ozgener, A comparison of the meshless RBF collocation method with finite element and boundary element methods in neutron diffusion calculations. Eng. Anal. Bound. Elem. 46, 30–40 (2014). https://doi.org/10.1016/j.enganabound.2014.05.005

    Article  MathSciNet  MATH  Google Scholar 

  26. J.R. Xiao, R.C. Batra, D.F. Gilhooley et al., Analysis of thick plates by using a higher-order shear and normal deformable plate theory and MLPG method with radial basis functions. Comput. Methods Appl. Mech. Eng. 196, 979–987 (2007). https://doi.org/10.1016/j.cma.2006.08.002

    Article  MATH  Google Scholar 

  27. S. De, K.J. Bathe, The method of finite spheres. Comput. Mech. 25, 329–345 (2000). https://doi.org/10.1007/s004660050481

    Article  MathSciNet  MATH  Google Scholar 

  28. M. Moradipour, S. Yousefi, Using a meshless kernel-based method to solve the Black-Scholes variational inequality of American options. Comp. Appl. Math. (2016). https://doi.org/10.1007/s40314-016-0351-7

    Article  MATH  Google Scholar 

  29. M. Ebrahimnejad, N. Fallah, A.R. Khoei, Three types of meshless finite volume method for the analysis of two-dimensional elasticity problems. Comp. Appl. Math. 36, 971 (2015). https://doi.org/10.1007/s40314-015-0273-9

    Article  MathSciNet  MATH  Google Scholar 

  30. S. Atluri, T. Zhu, New concepts in meshless methods. Comput. Mech. Int. J. Numer. Methods Eng. 47, 117–127 (2000). https://doi.org/10.1002/(SICI)1097-0207(20000110/30)47:1/3<537::AID-NME783>3.0.CO;2-E

    Article  MathSciNet  MATH  Google Scholar 

  31. M. Dehghan, M. Abbaszadeh, Numerical investigation based on direct meshless local Petrov–Galerkin (direct MLPG) method for solving generalized Zakharov system in one and two dimensions and generalized Gross-Pitaevskii equation. Eng. Comput. 33, 983–996 (2017). https://doi.org/10.1007/s00366-017-0510-5

    Article  Google Scholar 

  32. A. Taleei, M. Dehghan, Direct meshless local Petrov–Galerkin method for elliptic interface problems with applications in electrostatic and elastostatic. Comput. Methods Appl. Mech. Eng. 278, 479–498 (2014). https://doi.org/10.1016/j.cma.2014.05.016

    Article  MathSciNet  MATH  Google Scholar 

  33. M. Dehghan, R. Salehi, A meshless local Petrov–Galerkin method for the time-dependent Maxwell equations. J. Comput. Appl. Math. 268, 93–110 (2014). https://doi.org/10.1016/j.cam.2014.02.013

    Article  MathSciNet  MATH  Google Scholar 

  34. M. Kamranian, M. Dehghan, M. Tatari, An adaptive meshless local Petrov–Galerkin method based on a posteriori error estimation for the boundary layer problems. Appl. Numer. Math. 111, 181–196 (2017). https://doi.org/10.1016/j.apnum.2016.09.007

    Article  MathSciNet  MATH  Google Scholar 

  35. M. Dehghan, M. Abbaszadeh, A. Mohebbi, Meshless local Petrov–Galerkin and RBFs collocation methods for solving 2D fractional Klein-Kramers dynamics equation on irregular domains. Comput. Model. Eng. Sci. 107, 481–516 (2015)

    Google Scholar 

  36. W.H. Chen, X.M. Guo, Element free Galerkin method for three-dimensional structural analysis. Comput. Model. Eng. Sci. 4, 497–508 (2001). https://doi.org/10.3970/cmes.2001.002.497

    Article  MATH  Google Scholar 

  37. Q.W. Ma, Meshless local Petrov–Galerkin method for two-dimensional nonlinear water wave problems. J. Comput. Phys. 205, 611–625 (2005). https://doi.org/10.1016/j.jcp.2004.11.010

    Article  MathSciNet  MATH  Google Scholar 

  38. J. Amani Rad, K. Parand, Numerical pricing of American options under two stochastic factor models with jumps using a meshless local Petrov–Galerkin method. Appl. Numer. Math. 115, 252–274 (2017). https://doi.org/10.1016/j.apnum.2017.01.015

    Article  MathSciNet  MATH  Google Scholar 

  39. H. Lin, S.N. Atluri, The Meshless Local Petrov–Galerkin (MLPG) method for solving incompressible Navier–Stokes equations. Comput. Model. Eng. Sci. 2, 117–142 (2001). https://doi.org/10.3970/cmes.2001.002.117

    Article  MathSciNet  Google Scholar 

  40. S.Y. Long, S.N. Atluri, A meshless local Petrov–Galerkin (MLPG) method for solving the bending problems of a thin plate. CMES 3, 53–64 (2002). https://doi.org/10.3970/cmes.2002.003.053

    Article  MATH  Google Scholar 

  41. T. Belytschko, Y.Y. Lu, L. Gu, Element-free Galerkin method. Int. J. Numer. Methods Eng. 37, 229–256 (1994). https://doi.org/10.1002/nme.1620370205

    Article  MathSciNet  MATH  Google Scholar 

  42. G.R. Liu, Y.T. Gu, Meshless local Petrov–Galerkin (MLPG) method in combination with finite element and boundary element approaches. Comput. Mech. 26, 536–546 (2000). https://doi.org/10.1007/s004660000203

    Article  MATH  Google Scholar 

  43. M. Li, F.F. Dou, T. Korakianitis et al., Boundary node Petrov–Galerkin method in solid structures. Comp. Appl. Math. (2016). https://doi.org/10.1007/s40314-016-0335-7

    Article  MATH  Google Scholar 

  44. I. Debbabi, H. BelhadjSalah, Analysis of thermo-elastic problems using the improved element-free Galerkin method. Comp. Appl. Math. (2016). https://doi.org/10.1007/s40314-016-0401-1

    Article  MATH  Google Scholar 

  45. D. Mirzaei, R. Schaback, Direct meshless local Petrov–Galerkin (DMLPG) method: a generalized MLS approximation. Appl. Numer. Math. 68, 73–82 (2013). https://doi.org/10.1016/j.apnum.2013.01.002

    Article  MathSciNet  MATH  Google Scholar 

  46. M. Dehghan, D. Mirzaei, Meshless local Petrov–Galerkin (MLPG) method for the unsteady magnetohydrodynamic (MHD) flow through pipe with arbitrary wall conductivity. Appl. Numer. Math. 59, 1043–1058 (2009). https://doi.org/10.1016/j.apnum.2008.05.001

    Article  MathSciNet  MATH  Google Scholar 

  47. S. Atluri, S. Shen, The basis of meshless domain discretization: the meshless local Petrov–Galerkin (MLPG) method. Adv. Comput. Math. 23, 73–93 (2005). https://doi.org/10.1007/s10444-004-1813-9

    Article  MathSciNet  MATH  Google Scholar 

  48. G.R. Liu, Y.T. Gu, An introduction to mesh free methods and their programming, 1st edn. (Springer, Dordrecht, 2005)

    Google Scholar 

  49. Y.T. Gu, G.R. Liu, A meshless local Petrov–Galerkin (MLPG) method for free and forced vibration analyses for solids. Comput. Mech. 27, 188–198 (2001). https://doi.org/10.1007/s004660100237

    Article  MATH  Google Scholar 

  50. P. Lancaster, K. Salkauskas, Surfaces generation by moving least squares methods. Math. Comput. 37, 141–158 (1981). https://doi.org/10.1090/S0025-5718-1981-0616367-1

    Article  MATH  Google Scholar 

  51. V.R. Hosseini, E. Shivanian, W. Chen, Local radial point interpolation (MLRPI) method for solving time fractional diffusion-wave equation with damping. J. Comput. Phys. 312, 307–332 (2016). https://doi.org/10.1016/j.jcp.2016.02.030

    Article  MathSciNet  MATH  Google Scholar 

  52. Y.T. Gu, G.R. Liu, A local point interpolation method for static and dynamic analysis of thin beams. Comput. Methods Appl. Mech. Eng. 190, 5515–5528 (2001). https://doi.org/10.1016/S0045-7825(01)00180-3

    Article  MATH  Google Scholar 

  53. G. Liu, Mesh Free Methods: Moving Beyond the Finite Element Method (CRC Press, Boca Raton, 2003)

    MATH  Google Scholar 

  54. M. Dehghan, D. Mirzaei, Numerical solution to the unsteady two-dimensional Schrödinger equation using meshless local boundary integral equation method. Int. J. Numer. Methods Eng. 76, 501–520 (2008). https://doi.org/10.1002/nme.2338

    Article  MATH  Google Scholar 

  55. J. Zheng, S. Long, G. Li, Topology optimization of free vibrating continuum structures based on the element free Galerkin method. Struct. Multidiscip. Optim. 45, 119–127 (2012). https://doi.org/10.1007/s00158-011-0667-2

    Article  MathSciNet  MATH  Google Scholar 

  56. M. Sterk, B. Robic, R. Trobec, Mesh free method applied to the diffusion equation. Parallel Numer. 5, 57–66 (2005). https://doi.org/10.1007/s00158-011-0667-2

    Article  Google Scholar 

  57. T. Tanbay, B. Ozgener, Numerical solution of the multigroup neutron diffusion equation by the meshless RBF collocation method. Math. Comput. Appl. 18, 399–407 (2013). https://doi.org/10.3390/mca18030399

    Article  MathSciNet  MATH  Google Scholar 

  58. D. Mirzaei, M. Dehghan, Meshless local Petrov–Galerkin (MLPG) approximation to the two dimensional sine-Gordon equation. J. Comput. Appl. Math. 233, 2737–2754 (2010). https://doi.org/10.1016/j.cam.2009.11.022

    Article  MathSciNet  MATH  Google Scholar 

  59. D. Jeong, J. Kim, A Crank–Nicolson scheme for the Landau–Lifshitz equation without damping. J. Comput. Appl. Math. 234, 613–623 (2010). https://doi.org/10.1016/j.cam.2010.01.002

    Article  MathSciNet  MATH  Google Scholar 

  60. O. Abbasi, A. Rostami, G. Karimian, Identification of exonic regions in DNA sequences using cross-correlation and noise suppression by discrete wavelet transform. BMC Bioinformatics 12, 430 (2011). https://doi.org/10.1186/1471-2105-12-430

    Article  Google Scholar 

  61. O.A. Abuzaid, A.H.M. Gashut, Discontinuous finite element methods for reactor calculations, in Third Arab Conf. Peac. Uses At. Energy (Damascus, 1996), pp. 39–46

  62. IAEA, Status of small and medium sized reactors (2011)

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  1. Department of Nuclear Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran

    Shima Tayefi, Ali Pazirandeh & Mohsen Kheradmand Saadi

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  1. Shima Tayefi
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  2. Ali Pazirandeh
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Tayefi, S., Pazirandeh, A. & Kheradmand Saadi, M. A meshless local Petrov–Galerkin method for solving the neutron diffusion equation. NUCL SCI TECH 29, 169 (2018). https://doi.org/10.1007/s41365-018-0506-x

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