Parallel CUDA implementation of the algorithm for solving the Navier-Stokes equations using the fictitious domain method
DOI:
https://doi.org/10.26577/JMMCS.2021.v109.i1.05Keywords:
Navier-Stokes equations, stream function, velocity vortex, fictitious domains method, boundary conditions, CUDA, parallel algorithm, high performance computingAbstract
An important direction in the development of numerical simulation methods is the study of approximate methods for solving problems of mathematical physics in complex multidimensional fields. To solve many applied problems in irregular areas, the method of fictitious areas is widely used, which is characterized by a high degree of automation of programming. The main idea of the method of fictitious domains is that the problem is solved not in the original complex domain, but in some other, simpler domain. This allows you to create software immediately for a fairly wide class of problems with arbitrary computational domains. The possibilities of applying the method of fictitious domains to the problems of hydrodynamics in the variables “stream function, velocity vortex” have been considered in many works. In this paper, we study a numerical method for solving the Navier-Stokes equations in doubly connected domains. To solve the two-dimensional Navier-Stokes equations in irregular domains, an approximate method based on the method of fictitious domains is proposed. A computational finite difference algorithm for solving an auxiliary problem of the method of fictitious domains has been developed. The results of numerical simulation of two-dimensional Navier-Stokes equations by the method of fictitious domains with continuation by the lowest coefficient are presented. For this problem, a parallel algorithm was developed using CUDA technologies, which was tested on various mesh dimensions
References
1. Zhumagulov B., Monakhov V., emphFluid dynamics of oil production (Elsevier, 2014), 177-178.
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.
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.
Downloads
How to Cite
Temirbekov, A., Malgazhdarov, Y., Kassenov, S., & Urmashev, B. (2021). Parallel CUDA implementation of the algorithm for solving the Navier-Stokes equations using the fictitious domain method. Journal of Mathematics, Mechanics and Computer Science, 109(1). https://doi.org/10.26577/JMMCS.2021.v109.i1.05
Issue
Section
Applied Mathematics
