Параллельная CUDA-реализация алгоритма решения уравнений Навье-Стокса с использованием метода фиктивных областей
DOI:
https://doi.org/10.26577/JMMCS.2021.v109.i1.05Ключевые слова:
уравнения Навье-Стокса, функция тока, вихрь скорости, метод фиктивных областей, граничные условия, CUDA, параллельный алгоритм, высокопроизводительные вычисленияАннотация
Важным направлением развития методов численного моделирования являются исследования приближенных методов решения задач математической физики в сложных многомерных областях. Для решения многих прикладных задач в нерегулярных областях широко применяется метод фиктивных областей, отличающийся высокой степенью автоматизации программирования. Основная идея метода фиктивных областей состоит в том, что задача решается не в исходной сложной области, а в некоторой другой, более простой области. Это позволяет создавать программное обеспечение сразу для достаточно широкого класса задач с произвольными расчетными областями. Возможности применения метода фиктивных областей к задачам гидродинамики в переменных «функция тока, вихрь скорости» рассмотрены во многих работах. В настоящей работе исследуются численный метод решения уравнений Навье-Стокса в двухсвязных областях. Для решения двумерных уравнений Навье-Стокса в нерегулярных областях предложен приближенный метод на основе метода фиктивных областей. Разработан вычислительный конечно-разностный алгоритм решения вспомогательной задачи метода фиктивных областей. Приведены результаты численного моделирования двумерных уравнений Навье-Стокса методом фиктивных областей с продолжением по младшим коэффициентом. Для данной задачи был разработан параллельный алгоритм с использованием архитектуры CUDA, который был протестирован на различных размерностях сетки.
Библиографические ссылки
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28. Temirbekov N. M., Baigereyev D. R. "Modeling of three-phase non-isothermal flow in porous media using the approach of reduced pressure", emphCommunications in Computer and Information Science 549 (2015): 166-176.
2. Cimolin F., Discacciati M., "Navier-Stokes/Forchheimer models for filtration through porous media", emphApplied Numerical Mathematics 72 (2013): 205-224.
3. Temirbekov N. M., Turarov A. K. and Baigereyev D. R. "Numerical modeling of the gas lift process in gas lift wells", emphAIP Conference Proceedings 1739, no. 020067 (2016): 1-9.
4. He Q., Glowinski R. and Wang X., "A least-squares/fictitious domain method for incompressible viscous flow around obstacles with Navier slip boundary condition", emphJournal of Computational Physics 366 (2018): 281-297.
5. Court S., "A fictitious domain approach for a mixed finite element method solving the two-phase Stokes problem with surface tension forces", emphJournal of Computational and Applied Mathematics 359 (2019): 30–54.
6. He Q., Huang J., Shi X., Wang X. and Bi C., "Numerical simulation of 2D unsteady shear-thinning non-Newtonian incompressible fluid in screw extruder with fictitious domain method", emphComputer and Mathematics with Applications 73 (2017): 109-121.
7. Xia Y., Yu Z. and Deng J., "A fictitious domain method for particulate flows of arbitrary density ratio", emphComputers and Fluids 193 (2019): 104293, accessed May 23, 2020, doi:10.1016/j.compfluid.2019.104293.
8. Fournie M., Morrison J., "Fictitious domain for stabilization of fluid-structure interaction", emphIFAC PapersOnLine 50-1 (2017): 12301–12306.
9. Wang Y., Jimack P. and Walkley M., "Energy analysis for the one-field fictitious domain method for fluid-structure interactions", emphApplied Numerical Mathematics 140 (2019): 165–182.
10. Wu M., Peters B., Rosemann T. and Kruggel-Emden H., "A forcing fictitious domain method to simulate fluid-particle interaction of particles with super-quadric shape", emphPowder Technology 360 (2020): 264-277.
11. Zhou G., "The fictitious domain method with H1-penalty for the Stokes problem with Dirichlet boundary condition", emphApplied Numerical Mathematics 123 (2018): 1-21.
12. Mottahedi H., Anbarsooz M. and Passandideh-Fard M., "Application of a fictitious domain method in numerical simulation of an oscillating wave surge converter", emphRenewable Energy 121 (2018): 133-145.
13. Sun P., Wang C., "Distributed Lagrange multiplier/fictitious domain finite
element method for Stokes/parabolic interface problems with jump coefficients", emphApplied Numerical Mathematics 152 (2020): 199-220.
14. Sun P., "Fictitious domain finite element method for Stokes/elliptic interface problems with jump coefficients", emphJournal of Computational and Applied Mathematics 356 (2019): 81–97.
15. Temirbekov A., Wojcik W., "Numerical Implementation of the Fictitious Domain Method for Elliptic Equations", emphInternational Journal of Electronics and Telecommunications 60, no. 3 (2014): 219-223.
16. Temirbekov A., "Numerical implementation of the method of fictitious domains for elliptic equations", emphAIP Conference Proceedings 1759, no. 020053 (2016): 1-6.
17. Vabischevich P., emphMetod fiktivnyh oblastej v zadachah matematicheskoj fiziki [The fictitious domain method in problems of mathematical physics] (URSS, 2017).
18. Heikkola E., Rossi T. and Toivanen J., "A Parallel Fictitious Domain Method for the Three-Dimensional Helmholtz Equation", emphSIAM Journal on Scientific Computing 24, no. 5 (2000): 1567–1588.
19. Yu Z., Lin Z., Shao X. and Wang L., "A parallel fictitious domain method for the interface-resolved simulation of particle-laden flows and its application to the turbulent channel flow", emphEngineering Applications of Computational Fluid Mechanics 10 (1) (2016): 160-170.
20. Ruess M., Varduhn V., Rank E. and Yosibash Z., "A Parallel High-Order Fictitious Domain Approach for Biomechanical Applications", emph11th International Symposium on Parallel and Distributed Computing, Munich (2012): 279-285.
21. Akhmed-Zaki D., Daribayev B., Imankulov T. and Turar O., "High-performance computing of oil recovery problem on a mobile platform using CUDA technology", emphEurasian Journal of mathematical and computer applications 5, no. 2 (2017): 4-13.
22. Degi D., Starchenko A. and Trunov A., "Realizacija javnoj raznostnoj shemy dlja reshenija dvumernogo uravnenija teploprovodnosti na graficheskom processornom ustrojstve s ispol'zovaniem tehnologii CUDA [Implementation of an explicit difference scheme for solving the two-dimensional heat equation on a graphics processing unit using CUDA technology]", emphScientific service on the Internet: supercomputer centers and tasks (2010): 346-348 [in Russian].
23. Bekibayev T., Asilbekov B., Zhapbasbayev U., Beisembetov I. and Kenzhaliyev B., "Primenenie Cuda dlja rasparallelivanija trehmernoj zadachi fil'tracii nefti [Using Cuda to Parallelize the Three-Dimensional Oil Filtration Problem]", emphVestnik KazNU. Serija mat., mekh., inf. 1 (72) (2012): 65–78 [in Russian].
24. Thibault J., Senocak I., "CUDA Implementation of a Navier-Stokes Solver on Multi-GPU Desktop Platforms for Incompressible Flows" (paper presented at 47th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, 2012).
25. Koldas А., Toleukhanov A., "Aktual'nost' primenenija Cuda tehnologii dlja reshenija zadach podzemnogo hranenija vodoroda [The relevance of applying Cuda technology to solve the problems of underground hydrogen storage]", emphIzvestija NAN RK. Serija fiziko-matematicheskaja 5 (2013): 181–189 [in Russian].
26. Demidov D., Egorov A. and Nuriev A., "Reshenie zadach vychislitel'noj gidrodinamiki s primeneniem tehnologii NVIDIA CUDA [Solving computational fluid dynamics problems using NVIDIA CUDA technology]", emphUchenye zapiski Kazanskogo universiteta. Serija Fiziko-matematicheskie nauki 152 (2010): 142-154 [in Russian].
27. Sirochenko V., "Chislennoe modelirovanie konvektivnyh techenii vjazkoj zhidkosti v mnogosvjaznyh oblastjah [Numerical modeling of convective viscous fluid flow in multiply connected domains]" emphRDAMM-2001 6, no. 2 (2001): 554-562 [in Russian].
28. Temirbekov N. M., Baigereyev D. R. "Modeling of three-phase non-isothermal flow in porous media using the approach of reduced pressure", emphCommunications in Computer and Information Science 549 (2015): 166-176.
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Как цитировать
Temirbekov, A., Malgazhdarov, Y., Kassenov, S., & Urmashev, B. (2021). Параллельная CUDA-реализация алгоритма решения уравнений Навье-Стокса с использованием метода фиктивных областей. Вестник КазНУ. Серия математика, механика, информатика, 109(1). https://doi.org/10.26577/JMMCS.2021.v109.i1.05
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Прикладная математика
