Comprehensive Tutorial of Level Set Method

Transcript

1. Comprehensive Guide of Level Set Method with Fluid Mechanics Jongsu

   Kim Department of Computational Science and Engineering Yonsei University 1

2. Yonsei University 2/00 Contents • Motivation • Level Set Method

   • Level Set Reinitialization • Numerical Schemes • Conclusion

3. Yonsei University 3/00 Multiphase Phase Flow •

4. Yonsei University 4/00 • • • • • • •

   • Multiphase Phase Flow

5. Yonsei University 5/00 Contents • Motivation • Level Set Method

   • Numerical Schemes • Conclusion • Level Set Reinitialization

6. Yonsei University 6/00 Modification of Navier-Stokes Equation • = −

   + 2 + = − + 2 + ⋅ = 0 Ω What equation at the interface?

7. Yonsei University 7/00

8. Yonsei University 8/00 CSF Model • • 1 − 2

   + = 1 − 2 + 1 − 2 + = 21 1 − 22 2 2 + 2 − 1 + 1 = •

9. Yonsei University 9/00 CSF Model • = () • ()

   () = [] () = 1 ( 1) 2 ( 2) < > = 1 + 2 /2 ( ℎ ) = 2 − 1

10. Yonsei University 10/00 What We Have Done • = −

    + 2 + , = 0 = − + 2 + , = 0 ⋅ = 0 Ω 2 − = − + ul = ug , x ∈ Γ Three equations.. Should we solve these equation separately? And how we know the interface?

11. Yonsei University 11/00 VOF Method • • • • =

    1 ( 2) (0 < < 1) + ⋅ = 0

12. Yonsei University 12/00 VOF Method • • • • •

13. Yonsei University 13/00 Level Set Method = =0 = ⋅

    =0 , = < 0, ( ) > 0, ( ) = 0 ( ℎ )

14. Yonsei University 14/00 = , > 0 , ≤ 0

    + ⋅ = 0 , = < 0, ( ) > 0, ( ) = 0 ( ℎ ) Level Set Method

15. Yonsei University 15/00 + ⋅ = 0 Γ , ,

    (, ) , = , , , , , = ( , , , ) ( , , , , ) (, ) Γ , , , , ≡ + + = + + Level Set Method

16. Yonsei University 16/00 , = < 0, ( ) >

    0, ( ) = 0 ( ℎ ) () = , ( ) , ( ) + /2, () () = , ( ) , ( ) + /2, () Level Set Method

17. Yonsei University 17/00 = − + ⋅ 2 − +

    Level Set Method () () = + − , = + − () What We Have Done () = 0, < 0 1 2 , = 0 1, > 0 However, we still have a discontinuity … + ⋅ = 0

18. Yonsei University 18/00 Contents • Motivation • Level Set Method

    • Numerical Schemes • Conclusion • Level Set Reinitialization

19. Yonsei University 19/00 → () Heaviside Function () = 0,

    < − 1 2 1 + + 1 sin / , || ≤ 1, > +

20. Yonsei University 20/00 Distance Function • • 2 |∇| ()

    = 0, < − 1 2 1 + + 1 sin / , || ≤ 1, > + • ∇ = 1 ≤ • ∇ = 1 ∈ Ω = 0 ∈ Γ The distance function = The level set function?

21. Yonsei University 21/00 Level Set Reinitialization = (1 − ∇

    ) , 0 = () sign = −1, < 0 0, = 0 1, > 0 ∇ = 1 ∈ Ω = 0 ∈ Γ |∇| = 1 |∇| = 1 ≤ Do we solve this equation each iteration?

22. Yonsei University 22/00 Level Set Reinitialization = (1 − ∇

    ) + ⋅ ∇ = = ∇ |∇| •

23. Yonsei University 23/00 Level Set Reinitialization •

24. Yonsei University 24/00 Level Set Reinitialization What We Have Done

    = − + ⋅ 2 − + + ⋅ = 0 () = 0, < − 1 2 1 + + 1 sin / , || ≤ 1, > + + ⋅ ∇ = Are we done?

25. Yonsei University 25/00 Contents • Motivation • Level Set Method

    • Numerical Schemes • Conclusion • Level Set Reinitialization

26. Yonsei University 26/00 Back to Equation = − + ⋅

    2 − + + ⋅ = 0

27. Yonsei University 27/00 Back to Equation = Σ | +1

    − | +1 ≤ ≤

28. Yonsei University 28/00 Contents • Motivation • Level Set Method

    • Numerical Schemes • Conclusion • Level Set Reinitialization

29. Yonsei University 29/00 Advantange/Disadvantage • • • Γ , =

    0 • •

30. Yonsei University 30/00 Summary of Algorithm (, 0) ( +

    ⋅ ∇ = 0) ( ⋅ ∇) = +1 (1 − ∇ )

31. Yonsei University 31/00 Reference • • • • •

32. Yonsei University 32/00 Reference (Image) • • • • •

    • • • • •

33. Yonsei University 33/00 Reference (Image) • •