PARALLEL MESHLESS RADIAL BASIS FUNCTION COLLOCATION METHOD FOR NEUTRON DIFFUSION PROBLEMS

Abstract

The meshless global radial basis function (RBF) collocation method is widely used to model physical phenomena in science and engineering. The method produces highly accurate solutions with an exponential convergence rate. However, due to the global approximation structure of the method, dense node distributions lead to long computation times and hinder the applicability of the technique. In order to overcome this issue, this study proposes a parallel meshless global RBF collocation algorithm. The algorithm is applied to 2-D neutron diffusion problems. The multiquadric is used as the RBF. The algorithm is developed with Mathematica and eight virtual processors are used in calculations on a multicore computer with four physical cores. The method provides accurate numerical results in a stable manner. Parallel speedup increases with the number of processors up to five and seven processors for external and fission source problems, respectively. The speedup values are limited by the constrained resource sharing of the multicore computer’s memory. On the other hand, significant time savings are achieved with parallel computation. For the four-group fission source problem, when 4316 interpolation nodes are employed, the utilization of seven processors instead of sequential computation decreases the computation time of the meshless approach by 716 s.

Keywords

• Parallel
• Meshless
• Radial Basis Function
• Collocation
• Neutron diffusion

Nötron difüzyon problemleri için paralel ağsız radyal baz fonksiyonu kollokasyon yöntemi

Abstract

Ağsız global radyal baz fonksiyonu (RBF) kollokasyon yöntemi bilim ve mühendislikte karşılaşılan fiziksel olayların modellenmesinde yaygın bir şekilde kullanılmaktadır. Yöntem, üstel bir yakınsama hızı ile yüksek doğruluğa sahip çözümler üretir. Fakat, yöntemin global yaklaşım yapısı nedeniyle, çok sayıda ayrıklaştırma noktası kullanılması hesaplama sürelerini uzatmakta ve yöntemin uygulanabilirliğini kısıtlamaktadır. Söz konusu sorunun üstesinden gelebilmek için bu çalışmada bir paralel ağsız global RBF kollokasyon algoritması önerilmiştir. Algoritma iki boyutlu nötron difüzyon problemlerine uygulanmıştır. Multikuadrik fonksiyonu RBF olarak kullanılmıştır. Algoritma, Mathematica yazılımı ile geliştirilmiş ve hesaplamalar dört fiziksel çekirdeğe sahip çok çekirdekli bir bilgisayar ile sekiz sanal işlemci ile gerçekleştirilmiştir. Yöntem, doğru sayısal sonuçları kararlı bir şekilde sunmuştur. Paralel hızlanma işlemci sayısıyla, dış ve fisyon kaynağı problemleri için, sırasıyla beş ve yedi işlemciye kadar artmaktadır. Hızlanma değerleri çok çekirdekli bilgisayar belleğinin sınırlı kaynak paylaşımı nedeniyle kısıtlanmıştır. Diğer taraftan, paralel hesaplama ile önemli zaman kazanımları elde edilmiştir. Dört-grup fisyon kaynağı problemi için, 4316 interpolasyon noktası kullanılması durumunda, seri hesaplama yerine yedi işlemci kullanılması ağsız yöntemin hesaplama süresini 716 s kısaltmıştır.

Keywords

• Paralel
• Ağsız
• Radyal baz fonksiyonu
• Kollokasyon
• Nötron difüzyonu

References

1. 1. Alizadeh, A., Abbasi M., Minuchehr, A. and Zolfaghari, A. (2021) A mesh-free treatment for even parity neutron transport equation, Annals of Nuclear Energy, 158, 108292. doi:10.1016/j.anucene.2021.108292
2. 2. Atluri, S.N. and Zhu, T. (1998) A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics, Computational Mechanics, 22, 117-127. doi: 10.1007/s004660050346
3. 3. Barbosa, M., Telles, J.C.F., Santiago, J.A.F., Junior, E.F.F. and Costa, E.G.A. (2021) A parallel implementation strategy for meshless methods based on the functional programming paradigm, Advances in Engineering Software, 151, 102926. doi: 10.1016/j.advengsoft.2020.102926
4. 4. Bassett, B. and Kiedrowski, B. (2019) Meshless local Petrov-Galerkin solution of the neutron transport equation with streamline-upwind Petrov-Galerkin stabilization, Journal of Computational Physics, 377, 1-59. doi:10.1016/j.jcp.2018.10.028
5. 5. Bassett, B. and Owen, J.M. (2022) Meshless discretization of the discrete-ordinates transport equation with integration based on Voronoi cells, Journal of Computational Physics, 449, 110697. doi:10.1016/j.jcp.2021.110697
6. 6. Belytschko, T., Lu, Y.Y. and Gu, L. (1994) Element-free Galerkin methods, International Journal for Numerical Methods in Engineering, 37, 229-256. doi:10.1002/nme.1620370205
7. 7. Cao, C., Chen, H.Q., Zhang, J.L. and Xu, S.G. (2019) A multi-layered point reordering study of GPU-based meshless method for compressible flow simulations, Journal of Computational Science, 33, 45-60. doi:10.1016/j.jocs.2019.04.001
8. 8. Cercos-Pita, J.L. (2015) AQUAgpusph, a new free 3D SPH solver accelerated with OpenCL, Computer Physics Communications, 192, 295-312. doi:10.1016/j.cpc.2015.01.026

9. 9. Crespo, A.J.C., Domínguez, J.M., Rogers B.D., Gómez-Gesteira, M., Longshaw, S., Canelas, R., Vacondio, R., Barreiro, A. and García-Feal, O. (2015) DualSPHysics: Open-source parallel CFD solver based on Smoothed Particle Hydrodynamics (SPH), Computer Physics Communications, 187, 204-216. doi:10.1016/j.cpc.2014.10.004
10. 10. Danielson, K.T., Hao, S., Liu, W.K., Uras, R.A. and Li, S. (2000) Parallel computation of meshless methods for explicit dynamic analysis, International Journal for Numerical Methods in Engineering, 47, 1323-1341. doi:10.1002/(SICI)1097-0207(20000310)47:7<1323::AID-NME827>3.0.CO;2-0
11. 11. Depolli, M., Slak, J. and Kosec, G. (2022) Parallel domain discretization algorithm for RBF-FD and other meshless numerical methods for solving PDEs, Computers and Structures, 264, 106773. doi:10.1016/j.compstruc.2022.106773
12. 12. Domínguez, J.M., Crespo A.J.C., Valdez-Balderas, D., Rogers, B.D. and Gómez-Gesteira, M. (2013a) New multi-GPU implementation for smoothed particle hydrodynamics on heterogeneous clusters, Computer Physics Communications, 184, 1848-1860. doi:10.1016/j.cpc.2013.03.008
13. 13. Domínguez, J.M., Crespo A.J.C. and Gómez-Gesteira, M. (2013b) Optimization strategies for CPU and GPU implementations of a smoothed particle hydrodynamics method, Computer Physics Communications, 184, 617-627. doi:10.1016/j.cpc.2012.10.015
14. 14. Duan, Y. (2008) A note on the meshless method using radial basis functions, Computers and Mathematics with Applications, 55, 66-75. doi:10.1016/j.camwa.2007.03.011
15. 15. Fedoseyev, A.I., Friedman, M.J. and Kansa, E.J. (2002) Improved multiquadric method for elliptic partial differential equations via PDE collocation on the boundary, Computers and Mathematics with Applications, 43, 439-455. doi:10.1016/S0898-1221(01)00297-8
16. 16. Ferrari, A., Dumbser, M., Toro, E.F. and Armanini, A. (2009) A new 3D parallel SPH scheme for free surface flows, Computers & Fluids, 38, 1203-1217. doi:10.1016/j.compfluid.2008.11.012
17. 17. Griebel, M. and Schweitzer, M.A. (2003). A Particle-Partition of Unity Method-Part IV: Parallelization. In: Griebel, M. and Schweitzer, M.A. (eds) Meshfree Methods for Partial Differential Equations. Lecture Notes in Computational Science and Engineering, vol 26. Springer, Berlin, Heidelberg. doi:10.1007/978-3-642-56103-0_12
18. 18. Günther, F., Liu, W.K., Diachin, D. and Christon, M.A. (2000) Multi-scale meshfree parallel computations for viscous, compressible flows, Computer Methods in Applied Mechanics and Engineering, 190, 279-303. doi:10.1016/S0045-7825(00)00202-4
19. 19. Hardy, R.L. (1971) Multiquadric equations of topography and other irregular surfaces, Journal of Geophysical Research, 76, 1905-1915. doi:10.1029/JB076i008p01905
20. 20. Hu, W., Yao, L.G., Xu, H. and Hua, Z.Z. (2007a) Development of parallel 3D RKPM meshless bulk forming simulation system, Advances in Engineering Software, 38, 87-101. doi:10.1016/j.advengsoft.2006.08.002
21. 21. Hu, W., Yao, L.G. and Hua, Z.Z., (2007b) Parallel point interpolation method for three-dimensional metal forming simulations, Engineering Analysis with Boundary Elements, 31, 326-342. doi:10.1016/j.enganabound.2006.09.012
22. 22. Ihmsen, M., Akinci, N., Becker, M. and Teschner, M. (2011) A parallel SPH implementation on multi-core CPUs, Computer Graphics Forum, 30, 99-112. doi:10.1111/j.1467-8659.2010.01832.x
23. 23. Ingber, M.S., Chen, C.S. and Tanski, J.A. (2004) A mesh free approach using radial basis functions and parallel domain decomposition for solving three-dimensional diffusion equations, International Journal for Numerical Methods in Engineering, 60, 2183-2201. doi:10.1002/nme.1043
24. 24. Kansa, E.J. (1986) Application of Hardy’s multiquadric interpolation to hydrodynamics, Proceedings of the 1986 Summer Computer Simulation Conference, Society for Computer Simulation, San Diego, 4, 111-117.
25. 25. Kashi, S., Minuchehr, A., Zolfaghari, A. and Rokrok, B. (2017) Mesh-free method for numerical solution of the multi-group discrete ordinate neutron transport equation, Annals of Nuclear Energy, 106, 51-63. doi:10.1016/j.anucene.2017.03.034
26. 26. Karatarakis, A., Metsis, P. and Papadrakakis, M. (2013) GPU-acceleration of stiffness matrix calculation and efficient initialization of EFG meshless methods, Computer Methods in Applied Mechanics and Engineering, 258, 63-80. doi:10.1016/j.cma.2013.02.011
27. 27. Kelly, J.M., Divo, E.A. and Kassab, A.J. (2014) Numerical solution of the two-phase incompressible Navier-Stokes equations using a GPU-accelerated meshless method, Engineering Analysis with Boundary Elements, 40, 36-49. doi:10.1016/j.enganabound.2013.11.015
28. 28. Khuat, Q.H., Hoang, S.M.T., Woo, M.H., Kim, J.H. and Kim, J.K. (2019) Unstructured discrete ordinates method based on radial basis function approximation, Journal of the Korean Physical Society, 75, 5-14. doi: 10.3938/jkps.75.5
29. 29. Khuat, Q.H. and Kim, J.K. (2019) A solution to the singularity problem in the meshless method for neutron diffusion equation, Annals of Nuclear Energy, 126, 178-185. doi:10.1016/j.anucene.2018.10.054
30. 30. Kim, K., Jeong, H.S. and Jo, D. (2017) Numerical analysis for multi-group neutron-diffusion equation using Radial Point Interpolation Method (RPIM), Annals of Nuclear Energy, 99, 193-198. doi:10.1016/j.anucene.2016.08.021
31. 31. Kosec, G., Depolli, M., Rashkovska, A. and Trobec, R. (2014) Super linear speedup in a local parallel meshless solution of thermo-fluid problems, Computers and Structures, 133, 30-38. doi:10.1016/j.compstruc.2013.11.016
32. 32. Li, J., Cheng, A.H.D. and Chen, C.S. (2003) A comparison of efficiency and error convergence of multiquadric collocation method and finite element method, Engineering Analysis with Boundary Elements, 27, 251-257. doi:10.1016/S0955-7997(02)00081-4
33. 33. Liu, G.R. (2010) Meshfree Methods: Moving Beyond The Finite Element Method 2nd Edition, CRC Press, USA.
34. 34. Liu, G.R. and Gu, Y.T. (2005) An Introduction to Meshfree Methods and Their Programming, Springer, Dordrecht.
35. 35. Lucy, L.B. (1977) A numerical approach to the testing of the fission hypothesis, The Astronomical Journal, 82, 1013-1024. doi:10.1086/112164
36. 36. Ma, Z.H., Wang, H. and Pu, S.H. (2014) GPU computing of compressible flow problems by a meshless method with space-filling curves, Journal of Computational Physics, 263, 113-135. doi:10.1016/j.jcp.2014.01.023
37. 37. Ma, Z.H., Wang, H. and Pu, S.H. (2015) A parallel meshless dynamic cloud method on graphic processing units for unsteady compressible flows past moving boundaries, Computer Methods in Applied Mechanics and Engineering, 285, 146-165. doi:10.1016/j.cma.2014.11.010
38. 38. Madych, W.R. (1992) Miscellaneous error bounds for multiquadric and related interpolators, Computers and Mathematics with Applications, 24, 121-138. doi:10.1016/0898-1221(92)90175-H
39. 39. Marrone, S., Bouscasse, B., Colagrossi, A. and Antuono, M. (2012) Study of ship wave breaking patterns using 3D parallel SPH simulations, Computers & Fluids, 69, 54-66. doi:10.1016/j.compfluid.2012.08.008
40. 40. Medina, D.F. and Chen, J.K. (2000) Three-dimensional simulations of impact induced damage in composite structures using the parallelized SPH method, Composites: Part A, 31, 853-860. doi:10.1016/S1359-835X(00)00031-2
41. 41. Ortega, E., Oñate, E., Idelsohn, S. and Flores, R. (2014) Comparative accuracy and performance assessment of the finite point method in compressible flow problems, Computers & Fluids, 89, 53- 65. doi:10.1016/j.compfluid.2013.10.024
42. 42. Rokrok, B., Minuchehr, H. and Zolfaghari, A. (2012) Element-free Galerkin modeling of neutron diffusion equation in X-Y geometry, Annals of Nuclear Energy, 43, 39-48. doi:10.1016/j.anucene.2011.12.032
43. 43. Sefidgar, S.M.H., Firoozjaee, A.R. and Dehestani, M. (2022) Sparse discrete least squares meshless method on multicore computers, Journal of Computational Science, 62, 101686. doi:10.1016/j.jocs.2022.101686
44. 44. Shirazaki, M. and Yagawa, G. (1999) Large-scale parallel flow analysis based on free mesh method: a virtually meshless method, Computer Methods in Applied Mechanics and Engineering, 174, 419-431. doi:10.1016/S0045-7825(98)00307-7
45. 45. Singh, I.V. and Jain, P.K. (2005a) Parallel EFG algorithm for heat transfer problems, Advances in Engineering Software, 36, 554-560. doi:10.1016/j.advengsoft.2005.01.009
46. 46. Singh, I.V. and Jain, P.K. (2005b) Parallel meshless EFG solution for fluid flow problems, Numerical Heat Transfer, Part B: Fundamentals, 48, 45-66. doi:10.1080/10407790590935993
47. 47. Tanbay, T. (2018) On the accuracy and stability of the meshless RBF collocation method for neutron diffusion calculations, Journal of Innovative Science and Engineering, 2, 8-18.
48. 48. 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, 23-31. doi:10.38088/jise.570328
49. 49. Tanbay, T. and Ozgener, B. (2013) Numerical solution of the multigroup neutron diffusion equation by the meshless RBF collocation method, Mathematical and Computational Applications, 18, 399-407. doi:10.3390/mca18030399
50. 50. Tanbay, T. and Ozgener, B. (2014) A comparison of the meshless RBF collocation method with finite element and boundary element methods in neutron diffusion calculations, Engineering Analysis with Boundary Elements, 46, 30-40. doi:10.1016/j.enganabound.2014.05.005
51. 51. Tanbay, T. and Ozgener, B. (2019) A meshless method based on symmetric RBF collocation for neutron diffusion problems, Acta Physica Polonica A, 135, 661-663. doi:10.12693/APhysPolA.135.661
52. 52. Tanbay, T. and Ozgener, B. (2020) Fully meshless solution of the one-dimensional multigroup neutron transport equation with the radial basis function collocation method, Computers and Mathematics with Applications, 79, 1266-1286. doi:10.1016/j.camwa.2019.08.037
53. 53. Tayefi, S., Pazirandeh, A. and Saadi, M.K. (2018) A meshless local Petrov-Galerkin method for solving the neutron diffusion equation, Nuclear Science and Techniques, 29, 169. doi:10.1007/s41365- 018-0506-x
54. 54. Trobec, R., Šterk, M. and Robic, B. (2009) Computational complexity and parallelization of the meshless local Petrov-Galerkin method, Computers and Structures, 87, 81-90. doi:10.1016/j.compstruc.2008.08.003
55. 55. Ullah, Z., Coombs, W. and Augarde, C. (2016) Parallel computations in nonlinear solid mechanics using adaptive finite element and meshless methods, Engineering Computations, 33, 1161-1191. doi:10.1108/EC-06-2015-0166
56. 56. Yokota, R., Barba, L.A. and Knepley, M.G. (2010) PetRBF – A parallel O(N) algorithm for radial basis function interpolation with Gaussians, Computer Methods in Applied Mechanics and Engineering, 199, 1793-1804. doi:10.1016/j.cma.2010.02.008
57. 57. Zhang, L.T., Wagner, G.J. and Liu, W.K. (2002) A parallelized meshfree method with boundary enrichment for large-scale CFD, Journal of Computational Physics, 176, 483-506. doi:10.1006/jcph.2002.6999
58. 58. Zhang, J.L., Ma, Z.H., Chen, H.Q. and Cao, C. (2018a) A GPU-accelerated implicit meshless method for compressible flows, Journal of Computational Physics, 360, 39-56. doi:10.1016/j.jcp.2018.01.037
59. 59. Zhang, Y.N. Zhang, H.C. Zhang, X. Yu, H.X. and Zhao, G.B. (2018b) Block Radial Basis Function Collocation Meshless method applied to steady and transient neutronics problem solutions in multi- material reactor cores, Progress in Nuclear Energy, 109, 83-96. doi:10.1016/j.pnucene.2018.08.010
60. 60. Zhang, J.L., Chen, H.Q., Xu, S.G. and Gao, H.Q. (2020) A novel GPU-parallelized meshless method for solving compressible turbulent flows, Computers and Mathematics with Applications, 80, 2738- 2763. doi:10.1016/j.camwa.2020.08.030