A hybrid particle level set method for improved interface capturing

Abstract

In this paper, we propose a new numerical method for improving the mass conservation properties of the level set method when the interface is passively advected in a flow field. Our method uses Lagrangian marker particles to rebuild the level set in regions which are underresolved. This is often the case for flows undergoing stretching and tearing. The overall method maintains a smooth geometrical description of the interface and the implementation simplicity characteristic of the level set method. Our method compares favorably with volume of fluid methods in the conservation of mass and purely Lagrangian schemes for interface resolution. The method is presented in three spatial dimensions.

References

1. 1. D. Adalsteinsson and J. Sethian, A fast level set method for propagating interfaces, J. Comput. Phys.118, 269 (1995).]]
2. 2. A. Amsden, Numerical Calculation of Surface Waves: A Modified ZUNI Code with Surface Particles and Partial Cells, Technical Report LA-5146 (Los Alamos Scientific Laboratory, Los Alamos, NM, 1973).]]
3. 3. J. Bell, P. Colella, and H. Glaz, A second-order projection method for the incompressible Navier--Stokes equations, J. Comput. Phys.85, 257 (1989).]]
4. 4. R. Caiden, R. Fedkiw, and C. Anderson, A numerical method for two-phase flow consisting of separate compressible and incompressible regions, J. Comput. Phys.166, 1 (2001).]]
5. 5. G. Cerne, S. Petelin, and I. Tiselj, Coupling of the interface tracking and the two-fluid models for the simulation of incompressible two-phase flow, J. Comput. Phys.171, 776 (2001).]]
6. 6. S. Chen, D. Johnson, and P. Raad, Velocity boundary conditions for the simulation of free surface fluid flow, J. Comput. Phys.116, 262 (1995).]]
7. 7. S. Chen, D. Johnson, P. Raad, and D. Fadda. The surface marker and micro cell method. Int. J. Numer. Meth. Fluids25, 749 (1997).]]
8. 8. D. Enright, S. Marschner, and R. Fedkiw. Animation and rendering of complex water surfaces, in SIGGRAPH'02 (in press).]]
9. 9. N. Foster and R. Fedkiw, Practical animation of liquids, in SIGGRAPH 01 (2001), p. 15.]]
10. 10. A. Glassner, Principles of Digital Image Synthesis (Morgan Kaufmann San Francisco. CA. 1995).]]
11. 11. F. Harlow, J. Shannon, and J. Welch, THE MAC METHOD: A Computing Technique for Solving Viscous, Incompressible, Transient Fluid-Flow Problems Involving Free Surfaces, Technical Report LA-3425 (Los Alamos Scientific Laboratory, Los Alamos, NM, 1965).]]
12. 12. F. Harlow and J. Welch. Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Phys. Fluids8, 2182 (1965).]]
13. 13. J. Helmsen, A Comparison of Three-Dimensional Photolithography Development Methods, Ph.D. thesis (U.C. Berkeley, Berkeley, CA, 1994).]]
14. 14. C. Hirt and B. Nichols, Volume of fluid (VOF) method for the dynamics of free boundaries. J. Comput. Phys.39, 201 (1981).]]
15. 15. G.-S. Jiang and D. Peng, Weighted ENO schemes for Hamilton-Jacobi equations, SIAM J. Sci. Comput.21, 2126 (2000).]]
16. 16. M. Kang, R. Fedkiw, and X.-D. Liu, A boundary condition capturing method for multiphase incompressible flow, J. Sci. Comput.15, 323 (2000).]]
17. 17. B. Lafaurie. C. Nardone, R. Scardovelli, S. Zaleski, and G. Zanetti, Modelling merging and fragmentation in multiphase flows with SURFER, J. Comput. Phys.113, 134 (1994).]]
18. 18. G. Lapenta and J. Brackbill, Dynamic and selective control of the number of particles in kinetic plama simulations, J. Comput. Phys.115, 213 (1994).]]
19. 19. R. LeVeque, High-resolution conservative algorithms for advection in incompressible flow, SIAM J. Numer. Anal.33, 627 (1996).]]
20. 20. S. Osher and R. Fedkiw, Level set methods: An overview and some recent results, J. Comput. Phys.169, 463 (2001).]]
21. 21. S. Osher and R. Fedkiw, The Level Set Method and Dynamic Implicit Surfaces (Springer-Verlag, New York. 2002).]]
22. 22. S. Osher and J. Sethian, Fronts propagating with curvature dependent speed: Algorithms based on Hamiliton-Jacobi formulations, J. Comput. Phys.79, 12 (1988).]]
23. 23. D. Peng, B. Merriman, S. Osher. H.-K. Zhao. and M. Kang, A PDE-based fast local level set method, J. Comput. Phys.155, 410 (1999).]]
24. 24. E. Puckett, A. Almgren. J. Bell, D. Marcus, and W. Rider, A high-order projection method for tracking fluid interfaces in variable density incompressible flows, J. Comput. Phys.130. 269 (1997).]]
25. 25. P. Raad, S. Chen, and D. Johnson, The introduction of micro cells to treat pressure in free surface fluid flow problems, J. Fluids Eng.117, 683 (1995).]]
26. 26. W. Rider and D. Kothe, A marker particle method for interface tracking, in Proceedings of the Sixth International Symposium on Computational Fluid Dynamics, edited by H. Dwyer (1995) p. 976.]]
27. 27. W. Rider and D. Kothe, Stretching and tearing interface tracking methods, in 12th AIAA CFD Conference, AIAA 95-1717 (1995).]]
28. 28. W. Rider and D. Kothe, Reconstructing volume tracking, J. Comput. Phys.141, 112 (1998).]]
29. 29. J. Sethian, Curvature and the evolution of fronts, Comm. Math. Phys.101, 487 (1985).]]
30. 30. J. Sethian, Numerical methods for propagating fronts, in Variational Methods for Free Surface Interfaces, edited by P. Concus and R. Finn (Springer-Verlag, Berlin/New York, 1987).]]
31. 31. J. Sethian, A fast marching level set method for monotonically advancing fronts, Proc. Natl. Acad. Sci.93, 1591 (1996).]]
32. 32. J. Sethian, Fast marching methods, SIAM Rev.41, 199 (1999).]]
33. 33. J. Sethian, Level Set Methods and Fast Marching Methods (Cambridge Univ. Press, Cambridge, UK, 1999).]]
34. 34. J. Sethian, Evolution, implementation, and application of level set and fast marching methods for advancing fronts, J. Comput. Phys.169, 503 (2001).]]
35. 35. C. Shu and S. Osher, Efficient implementation of essentially nonoscillatory shock capturing schemes, J. Comput. Phys.77, 439 (1988).]]
36. 36. P. Smolarkiewicz, The multi-dimensional Crowley advection scheme, Month. Weather Rev.110, 1968 (1982).]]
37. 37. M. Sussmam and E. Puckett, A. coupled level set and volume-of-fluid method for computing 3D and axisymmetric incompressible two-phase flows, J. Comput. Phys.162, 301 (2000).]]
38. 38. M. Sussman, A. Almgren, J. Bell, P. Colella, L. Howell, and M. Welcome. An adaptive level set approach for incompressible two-phase flows, J. Comput. Phys.148, 81 (1999).]]
39. 39. M. Sussman and E. Fatemi, An efficient, interface-preserving level set redistancing algorithm and its application to interfacial incompressible fluid flow, SIAM J. Sci. Comput.20, 1165 (1999).]]
40. 40. M. Sussman, E. Fatemi, P. Smereka, and S. Osher. An improved level set method for incompressible two-phase flows, Comput. Fluids27, 663 (1998).]]
41. 41. M. Sussman, P. Smereka, and S. Osher, A level set approach for computing solutions to incompressible two-phase flow, J. Comput. Phys.114, 146 (1994).]]
42. 42. D. Torres and J. Brackbill, The point-set method: Front tracking without connectivity, J. Comput. Phys.165, 620 (2000).]]
43. 43. G. Tryggvason, B. Bunner. A. Esmaeeli, D. Juric, N. Al-Rawahi, W. Tauber, J. Han, S. Nas, and Y.-J. Jan, A front-tracking method for the computations of multiphase flow, J. Comput. Phys.169, 708 (2001).]]
44. 44. S.-O. Unverdi and G. Tryggvason, A front-tracking method for viscous, incompressible, multi-fluid flows, J. Comput. Phys.100, 25 (1992).]]
45. 45. M. Williams, D. Kothe, and E. Puckett, Approximating interfacial topologies with applications for interface tracking algorithms, in 37th AIAA Aerospace Sciences Meeting, AIAA 99-1076 (1999).]]
46. 46. M. Williams, D. Kothe, and E. Puckett, Convergence and accuracy of kernal-based continuum surface tension models, in Fluid Dynamics at Interfaces, edited by W. Shyy (Cambridge Univ. Press, Cambridge, UK, 1999), p. 347.]]
47. 47. S. Zalesak, Fully multidimensional flux-corrected transport algorithms for fluids, J. Comput. Phys.31, 335 (1979).]]
48. 48. Y. Zhang, K. Yeo, B. Khoo, and C. Wang, 3D jet impact and turoidal bubbles, J. Comput. Phys.166, 336 (2001).]]

Index Terms

• Applied computing

  • Physical sciences and engineering

    • Physics

• Mathematics of computing

  • Mathematical analysis

    • Numerical analysis

      • Computations in finite fields

  • Mathematical software