Применение графических процессоров для численного моделирования течения вязкой несжимаемой жидкости в областях сложной конфигурации методом погруженной границы
Ключевые слова:
уравнения Навье-Стокса
метод погруженной границы
CUDA
графические процессоры
Аннотация
Рассматривается применение графических процессоров для моделирования вязкой несжимаемой жидкости в областях сложной конфигурации. Метод погруженной границы используется для описания криволинейных границ на прямоугольных сетках. Для оценки эффективности реализации численных методов на архитектуре графических процессоров моделируется течение вокруг кругового цилиндра и группы круговых цилиндров. В качестве примера задачи с подвижными границами численно воспроизводится течение вокруг кругового цилиндра, совершающего вынужденные гармонические колебания.
Раздел
Раздел 1. Вычислительные методы и приложения
Библиографические ссылки
- Mittal R., Iaccarino G. Immersed boundary methods // Annual Review of Fluid Mechanics. 2005. 37. 239-261.
- Peskin C.S. The fluid dynamics of heart valves: experimental, theoretical and computational methods // Annual Review of Fluid Mechanics. 1982. 14. 235-259.
- Tseng Y.-H., Ferziger J.H. A ghost-cell immersed boundary method for flow in complex geometry // J. of Computational Physics. 2003. 192, N 2. 593-623.
- Mohd-Yusof J. Combined immersed boundary/B-spline methods for simulation of flow in complex geometries // CTR Annual Research Briefs, Center for Turbulence Research. Stanford: Stanford University Press, 1997. 317-328.
- Goldstein D., Handler R., Sirovich L. Modeling a no-slip flow boundary with an external force field // J. of Computational Physics. 1993. 105, N 2. 354-366.
- Lai M.-C., Peskin C.S. An immersed boundary method with formal second-order accuracy and reduced numerical viscosity // J. of Computational Physics. 2000. 160, N 2. 705-719.
- Saiki E.M., Biringen S. Numerical simulation of a cylinder in uniform flow: application of a virtual boundary method // J. of Computational Physics. 1996. 123. 450-465.
- Fadlun E.A., Verzicco R. Orlandi P., Mohd-Yusof J. Combined immersed-boundary finite-difference methods for three-dimensional complex flow simulations // J. of Computational Physics. 2000. 161, N 1. 35-60.
- Balaras E. Modeling complex boundaries using an external force field on fixed Cartesian grids in large-eddy simulations // Computers and Fluids. 2004. 33, N 3. 375-404.
- Yang J.M., Balaras E. An embedded-boundary formulation for large-eddy simulation of turbulent flows interacting with moving boundaries // J. of Computational Physics. 2006. 215, N 1. 12-40.
- Su S.-W., Lai M.-C., Lin C.-A. An immersed boundary technique for simulating complex flows with rigid boundary // Computers and Fluids. 2007. 36, N 2. 313-324.
- Yang X., Zhang X., Li Z., He G.-W. A smoothing technique for discrete delta functions with application to immersed boundary method in moving boundary simulations // J. of Computational Physics. 2009. 228, N 20. 7821-7836.
- Van der Vorst H.A. Iterative Krylov methods for large linear systems. Cambridge: Cambridge University Press, 2003.
- Dongarra J.J., Duff I.S., Sorensen D.C., Van der Vorst H.A. Numerical linear algebra for high-performance computers. Philadelphia: SIAM, 1998.
- Ольшанский М.А. Лекции и упражнения по многосеточным методам. М.: Изд-во Моск. ун-та, 2003.
- Шайдуров В.В. Многосеточные методы конечных элементов. М.: Наука, 1989.
- Wesseling P. An introduction to multigrid methods. New York: Wiley, 1992.
- Chronopoulos A.T., Gear C.W. S-step iterative methods for symmetric linear systems // J. of Computational and Applied Mathematics. 1989. 25, N 2. 153-168.
- Demmel J., Heath M., Van der Vorst H. Parallel numerical linear algebra // Acta Numerica. 1993. 2. 111-197.
- Harrar D.L., Ortega J.M. Optimum m-step SSOR preconditioning // J. of Computational and Applied Mathematics. 1988. 24, N 1-2. 195-198.
- Benzi M., Meyer C.D., Tuma M. A sparse approximate inverse preconditioner for the conjugate gradient method // SIAM J. on Scientific Computing. 1996. 17, N 5. 1135-1149.
- Cosgrove J.D. F., Dias J.C., Griewank A. Approximate inverse preconditionings for sparse linear systems // Int. J. of Computer Mathematics. 1992. 44, N 1-4. 91-110.
- Molemaker J., Cohen J.M., Patel S., Noh J. Low viscosity flow simulations for animation // Proc. of the 2008 ACM SIGGRAPH/Eurographics Symposium on Computer Animation. Aire-la-Ville: Eurographics Associations, 2008. 9-18.
- Bolz J., Farmer I., Grinspun E., Schröder P. Sparse matrix solvers on the GPU: conjugate gradients and multigrid // ACM Transactions on Graphics. 2003. 22, N 3. 917-924.
- Buatois L., Caumon G., Levy B. Concurrent number cruncher: a GPU implementation of a general sparse linear solver // Int. J. of Parallel, Emergent and Distributed Systems. 2009. 24, N 3. 205-223.
- Ament M., Knittel G., Weiskopf D., Strasser W. A parallel preconditioned conjugate gradient solver for the Poisson problem on a multi-GPU platform // Proc. of the 18th Euromicro Conference on Parallel, Distributed and Network-based Processing. Los Amitos: IEEE Computer Society, 2010. 583-592.
- DeLong M.A. SOR as a preconditioner. PhD Thesis. University of Virginia. Charlottesville, 1997.
- Stam J. Stable fluids // Proc. of the 26th Annual Conference on Computer Graphics and Interactive Techniques. New York: ACM Press, 1999. 121-128.
- Harris M.J. Real-time cloud simulation and rendering. PhD Thesis. University of North Carolina. Chapel Hill, 2003.
- Четверушкин Б.Н. Прикладная математика и проблемы использования высокопроизводительных вычислительных систем // Тр. МФТИ. 2011. 3, № 4. 55-67.
- Li W., Wei X., Kaufman A. Implementing lattice Boltzmann computation on graphics hardware // The Visual Computer. 2003. 19, N 7-8. 444-456.
- Rossinelli D., Bergdorf M., Cottet G.-H., Koumoutsakos P. GPU accelerated simulations of bluff body flows using vortex particle methods // J. of Computational Physics. 2010. 229, N 9. 3316-3333.
- Wei X., Zhao Y., Fan Z., Li W., Qui F., Yoakum-Stover S., Kaufman A.E. Lattice-based flow field modeling // IEEE Trans. on Visualization and Computer Graphics. 2004. 10, N 6. 719-729.
- Tölke J., Krafczyk M. TeraFLOP computing on a desktop PC with GPUs for 3D CFD // Int. J. of Computational Fluid Dynamics. 2008. 22, N 7. 443-456.
- Chorin A.J. Numerical solution of the Navier-Stokes equations // Mathematics of Computation. 1968. 22, N 104. 745-762.
- Kim J., Moin P. Application of a fractional-step method to incompressible Navier-Stokes equations // J. of Computational Physics. 1985. 59, N 2. 308-323.
- Brown D.L., Cortez R., Minion M.L. Accurate projection methods for the incompressible Navier-Stokes equations // J. of Computational Physics. 2001. 168, N 2. 464-499.
- Morinishi Y., Lund T.S., Vasilyev O.V., Moin P. Fully conservative higher order finite difference schemes for incompressible flow // J. of Computational Physics. 1998. 143, N 1. 90-124.
- Ol’shanskii M.A., Staroverov V.M. On simulation of outflow boundary conditions in finite difference calculations for incompressible fluid // Int. J. for Numerical Methods in Fluids. 2000. 33, N 4. 499-534.
- Kirkpatrick M.P., Armfield S.W., Kent J.H. A representation of curved boundaries for the solution of the Navier-Stokes equations on a staggered three-dimensional Cartesian grid // J. of Computational Physics. 2003. 184, N 1. 1-36.
- Ingram D.M., Causon D.M., Mingham C.G. Developments in Cartesian cut cell methods // Mathematics and Computers in Simulation. 2003. 61, N 3-6. 561-572.
- Li Z., Lai M.-C. The immersed interface method for the Navier-Stokes equations with singular forces // J. of Computational Physics. 2001. 171, N 2. 822-842.
- Ghias R., Mittal R., Dong H. A sharp interface immersed boundary method for compressible viscous flows // J. of Computational Physics. 2007. 225, N 1. 528-553.
- Lee J., Kim J., Choi H., Yang K.-S. Sources of spurious force oscillations from an immersed boundary method for moving-body problems // J. of Computational Physics. 2011. 230, N 7. 2677-2695.
- Hung Seo J., Mittal R. A sharp-interface immersed boundary method with improved mass conservation and reduced spurious pressure oscillations // J. of Computational Physics. 2011. 230, N 19. 7347-7363.
- Roma A.M., Peskin C.S., Berger M.J. An adaptive version of the immersed boundary method // J. of Computational Physics. 1999. 153, N 2. 509-534.
- Pourquie M., Breugem W.P., Boersma B.J. Some issues related to the use of immersed boundary methods to represent square obstacles // Int. J. for Multiscale Computational Engineering. 2009. 7, N 6. 509-522.
- Domenichini F. On the consistency of the direct forcing method in the fractional step solution of the Navier-Stokes equations // J. of Computational Physics. 2008. 227, N 12. 6372-6384.
- Guy R.D., Hartenstine D.A. On the accuracy of direct forcing immersed boundary methods with projection methods // J. of Computational Physics. 2010. 229, N 7. 2479-2496.
- Taira K., Colonius T. The immersed boundary method: a projection approach // J. of Computational Physics. 2007. 225, N 2. 2118-2137.
- Mori Y., Peskin C.S. Implicit second-order immersed boundary methods with boundary-mass // Computer Methods in Applied Mechanics and Engineering. 2008. 197, N 25-28. 2049-2067.
- Pinelli A., Naqavi I.Z., Piomelli U., Favier J. Immersed-boundary methods for general finite-difference and finite-volume Navier-Stokes solvers // J. of Computational Physics. 2010. 229, N 24. 9073-9091.
- Винников В.В., Ревизников Д.Л. Применение декартовых сеток для решения уравнений Навье-Стокса в областях с криволинейными границами // Математическое моделирование. 2005. 17, № 8. 15-30.
- Мортиков Е.В. Применение метода погруженной границы для решения системы уравнений Навье-Стокса в областях сложной конфигурации // Вычислительные методы и программирование. 2010. 11, № 1. 36-46.
- Gao T., Tseng Y.-H., Lu X.-Y. An improved hybrid Cartesian/immersed boundary method for fluid-solid flows // Int. J. for Numerical Methods in Fluids. 2007. 55, N 12. 1189-1211.
- Udaykumar H.S., Mittal R., Rampunggoon P., Khanna A. A sharp interface Cartesian grid method for simulating flows with complex moving boundaries // J. of Computational Physics. 2001. 174, N 1. 345-380.
- Braza M., Chassaing P., Minh H.H. Numerical study and physical analysis of the pressure and velocity fields in the near wake of a circular cylinder // J. of Fluid Mechanics. 1986. 165. 79-130.
- Liu C., Zheng X., Sung C.H. Preconditioned multigrid methods for unsteady incompressible flows // J. of Computational Physics. 1998. 139, N 1. 35-57.
- Calhoun D. A Cartesian grid method for solving the two-dimensional streamfunction-vorticity equations in irregular regions // J. of Computational Physics. 2002. 176, N 2. 231-275.
- Herfjord K. A study of two-dimensional separated flow by a combination of the finite element method and Navier-Stokes equations. Dr. Ing. Thesis. University of Trondheim. Trondheim, 1996.
- Berthelsen P.A., Faltinsen O.M. A local directional ghost cell approach for incompressible viscous flow problems with irregular boundaries // J. of Computational Physics. 2008. 227, N 9. 4354-4397.
- Wu Y.L., Shu C. Application of local DFD method to simulate unsteady flows around an oscillating circular cylinder // Int. J. for Numerical Methods in Fluids. 2008. 58, N 11. 1223-1236.
- Williamson C.H. K. Oblique and parallel modes of vortex shedding in the wake of a circular cylinder at low Reynolds numbers // J. of Fluid Mechanics. 1989. 206. 579-627.
- Schäfer M., Turek S. The benchmark problem flow around a cylinder // Flow Simulation with High-Performance Computers II. Notes on Numerical Fluid Mechanics. Hirschel (ed.). 52. Vieweg: Wiesbaden, 1996. 547-566.
- Ongoren A., Rockwell D. Flow structure from an oscillating cylinder. Part 1. Mechanisms of phase shift and recovery in the near wake // J. of Fluid Mechanics. 1988. 191. 197-223.
- Bishop R.E. D., Hassan A.Y. The lift and drag forces on a circular cylinder oscillating in a flowing fluid // Proc. of the Royal Society of London. Series A. Mathematical and Physical Sciences. 1964. 277, N 1368. 51-75.
- Guilmineau E., Queutey P. A numerical simulation of vortex shedding from an oscillating circular cylinder // J. of Fluids and Structures. 2002. 16, N 6. 773-794.
- Глазунов А.В. Вихреразрешающее моделирование турбулентности с использованием смешанного динамического локализованного замыкания. Часть I. Формулировка задачи, описание модели и диагностические численные тесты // Изв. РАН. Физика атмосферы и океана. 2009. 45, № 1. 7-28.
- Sagaut P. Large eddy simulation for incompressible flows. An introduction. Berlin: Springer, 2006.