Journal of KIISE:Computer Systems and Theory (한국정보과학회논문지:시스템및이론)

Korean Institute of Information Scientists and Engineers (한국정보과학회)
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Solver for the Wavier-Stokes Equations by using Initial Guess Velocity

속도의 초기간 추정을 사용한 Navier-Stokes방정식 풀이 기법

  • 김영희 (경북대학교 컴퓨터과학과) ;
  • 이성기 (경북대학교 컴퓨터과학과)
  • Published : 2005.10.01
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Abstract

We propose a fast and accurate fluid solver of the Wavier-Stokes equations for the physics-based fluid simulations. Our method utilizes the solution of the Stokes equation as an initial guess for the velocity of the nonlinear term in the Wavier-Stokes equations. By guessing the initial velocity close to the exact solution of the given nonlinear differential equations, we can develop remarkably accurate and stable fluid solver. Our solver is based on the implicit scheme of finite difference methods, that makes it work well for large time steps. Since we employ the ADI method, our solver is also fast and has a uniform computation time. The experimental results show that our solver is excellent for fluids with high Reynolds numbers such as smoke and clouds.

본 논문은 물리적인 힘을 기반으로 유체의 흐름을 실시간으로 시뮬레이션하기 위하여 유체 의 흐름을 지배하는 Wavier-Stokes 방정식에 대한 빠르고 정확한 풀이 기법을 제안한다 본 논문에서는 Navier-Stokes 방정식에 있는 비선형 항의 속도에 대한 초기값을 Stokes 방정식의 해로써 추정한다. 주어진 비선형 미분방정식의 해에 근사하게 초기값을 추정함으로써 정확하고 안정적인 풀이 기법을 만들 수 있었다. 또한 유한차분법(finite difference method)의 암시적(implicit) 방법 중에서 방대한 계산량을 피할 수 있는 ADI(Alternating Direction Implicit) 방법을 사용함으로써 큰 시간 간격(time-step)에 대해서 시스템이 안정적이며 계산속도 또한 빠르다. 실험 결과들은 특히 연기, 구름과 같이 큰 레이놀드 수(Reynolds number)를 가지는 유체에 대해서 탁월한 성능을 보여주었다.

Keywords

References

  1. Harlow, F.H. and Welch, J.E., 'Numerical Calculation of Time-Dependent Viscous Incompressible Flow of Fluid with a Free Surface,' The Physics of Fluids, 8, pp. 2182-2189, 1965 https://doi.org/10.1063/1.1761178
  2. Kass, M. and Miller, G., 'Rapid, stable fluid dynamics for computer graphics,' ACM Computer Graphics, 24, pp. 49-57, 1990 https://doi.org/10.1145/97879.97884
  3. Chen, J. and Lobo, N., 'Toward Interactive-Rate Simulation of Fluids with Moving Obstacles Using the Navier-Stokes Equations,' Graphical Models and Image Processing, 57, pp. 107-116, 1994 https://doi.org/10.1006/gmip.1995.1012
  4. Foster, N. and Metaxas, D., 'Realistic Animation of Liquids,' Graphical Models and Image Processing, 58, pp. 471-483, 1996 https://doi.org/10.1006/gmip.1996.0039
  5. Stam, J., 'Stable Fluids,' ACM SIGGRAPH 99, pp. 121-128, 1999 https://doi.org/10.1145/311535.311548
  6. Foster, N. and Fedkiw, R., 'Practical Animation of Liquids,' ACM SIGGRAPH 01, pp. 23-30, 200l https://doi.org/10.1145/383259.383261
  7. Treuille, A., McNamara, A., Popovic, Z, and Stam, J., 'Keyframe Control of Smoke Simulations,' ACM SIGGRAPH 03, pp. 716-723, 2003 https://doi.org/10.1145/1201775.882337
  8. Fedkiw, R., Stam, J., and Jensen, H., 'Visual Simulation of Smoke,' ACM SIGGRAPH 01, pp. 15-22, 200l https://doi.org/10.1145/383259.383260
  9. Streeter, V. L., Wylie, E. B., and Bedford, K. W., Fluid Mechanics, McGraw-Hill, 1998
  10. Peaceman, D. W., and H. H. Rachford, J., 'Numerical solution of parabolic and elliptic differential equations,' J. Soc. Indust. Appl. Math., 3, pp. 28-41, 1955 https://doi.org/10.1137/0103003
  11. Girault, V. and Raviart, P. A., Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, Springer-Verlag, New York, 1986
  12. Chorin, A. J., 1968, 'Numerical Solution of the Navier-Stokes Equations,' Math.Comput., Vol. 22, pp. 745-762 https://doi.org/10.2307/2004575
  13. Temam, R., 'Une methode d'approximation de la solution des equations de Navier-Stokes,' Bull. Soc. Math. de France, 98, pp. 115-152, 1968
  14. Shen, J., 'On Error Estimates of the Projection Methods for the Navier-Stokes Equations: Second-Order Schemes,' Math. Comput., 65, pp. 1039-1065, 1996 https://doi.org/10.1090/S0025-5718-96-00750-8
  15. Peyret, R. and Taylor, T. D., Computational Methods for Fluid Flow, Springer-Verlag, New York, 1983
  16. Press, W. H., Teukolsky, S. A., Vetterling, W. T., and Flannery, B. P., Numerical recipes in C. The Art of scientific Computing. Cambridge University Press, 1992
  17. Brian, C., and Leith, L. C., 'Imaging vector fields using line integral convolution,' SIGGRAPH 93, ACM Press, pp. 263-270, 1993 https://doi.org/10.1145/166117.166151
  18. Kilian, S.,and Turek, S. The Virtual Album of Fluid Motion: An Interactive Exploration via Numerical Simulation, DVD, Springer, 2002
  19. Rudiger, W., and Thomas, E. 'Efficiently using graphics hardware in volume rendering applications,' SIGGRAPH 98, ACM Press, pp. 169-177, 1998 https://doi.org/10.1145/280814.280860
  20. Bolz, J., Farmer, I., Grinspun, E., and Schroder, P., 'Sparse matrix solvers on the GPU: conjugate gradients and multigrid,' ACM Trans. Graph. 22, 3, pp. 917-924, 2003 https://doi.org/10.1145/882262.882364
  21. J. Kruger and R. Westermann, 'Linear Algebra Operators for GPU Implementation of Numerical Algorithms,' SIG-GRAPH 2003, 2003 https://doi.org/10.1145/1201775.882363
  22. Harris, M. J., 'Real-Time Cloud Simulation and Rendering,' PhD thesis, University of North Carolina, 2003
  23. Zeller, C., 'Cloth Simulation on the GPU,' SIGGRAPH 2005 Conference Abstracta and Applications, 2005 https://doi.org/10.1145/1187112.1187158