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Comprehensive Tutorial of Level Set Method

Comprehensive Tutorial of Level Set Method

Liam Jongsu Kim

June 04, 2013
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  1. Comprehensive Guide
    of Level Set Method
    with Fluid Mechanics
    Jongsu Kim
    Department of Computational Science and Engineering
    Yonsei University
    1

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  2. Yonsei
    University 2/00
    Contents
    • Motivation
    • Level Set Method
    • Level Set Reinitialization
    • Numerical Schemes
    • Conclusion

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  3. Yonsei
    University 3/00
    Multiphase Phase Flow

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  4. Yonsei
    University 4/00








    Multiphase Phase Flow

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  5. Yonsei
    University 5/00
    Contents
    • Motivation
    • Level Set Method
    • Numerical Schemes
    • Conclusion
    • Level Set Reinitialization

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




    = −
    + 2
    +




    = −
    + 2
    +

    ⋅ = 0 Ω
    What equation at the interface?

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  7. Yonsei
    University 7/00

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  8. Yonsei
    University 8/00
    CSF Model


    1
    − 2
    +
    = 1
    − 2

    +


    1
    − 2
    + = 21



    1
    − 22



    2
    2



    +


    2
    − 1



    +


    1
    =



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  9. Yonsei
    University 9/00
    CSF Model

    =
    ()

    ()
    ()

    =

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

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  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?

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  11. Yonsei
    University 11/00
    VOF Method




    = 1
    ( 2)
    (0 < < 1)


    + ⋅ = 0

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  12. Yonsei
    University 12/00
    VOF Method





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  13. Yonsei
    University 13/00
    Level Set Method
    =
    =0
    = ⋅
    =0
    , =
    < 0, ( )
    > 0, ( )
    = 0 ( ℎ )

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  14. Yonsei
    University 14/00
    =

    , > 0

    , ≤ 0


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

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  15. Yonsei
    University 15/00


    + ⋅ = 0
    Γ , , (, )
    ,

    = , , , ,
    ,

    = ( , , , )
    ( , , , , ) (, ) Γ
    , , , ,






    +




    +


    =
    +
    +
    Level Set Method

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

    , ( )

    , ( )

    +
    /2, ()
    () =

    , ( )

    , ( )

    +
    /2, ()
    Level Set Method

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  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

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  18. Yonsei
    University 18/00
    Contents
    • Motivation
    • Level Set Method
    • Numerical Schemes
    • Conclusion
    • Level Set Reinitialization

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  19. Yonsei
    University 19/00
    → ()
    Heaviside Function
    () =
    0, < −
    1
    2
    1 +


    +
    1

    sin / , || ≤
    1, > +

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  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?

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  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?

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  22. Yonsei
    University 22/00
    Level Set Reinitialization


    = (1 − ∇ )

    + ⋅ ∇ = =

    |∇|

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  23. Yonsei
    University 23/00
    Level Set Reinitialization

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



    = − + ⋅ 2 − +


    + ⋅ = 0
    () =
    0, < −
    1
    2
    1 +


    +
    1

    sin / , || ≤
    1, > +

    + ⋅ ∇ =
    Are we done?

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  25. Yonsei
    University 25/00
    Contents
    • Motivation
    • Level Set Method
    • Numerical Schemes
    • Conclusion
    • Level Set Reinitialization

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  26. Yonsei
    University 26/00
    Back to Equation



    = − + ⋅ 2 − +


    + ⋅ = 0

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

    |
    +1 ≤

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  28. Yonsei
    University 28/00
    Contents
    • Motivation
    • Level Set Method
    • Numerical Schemes
    • Conclusion
    • Level Set Reinitialization

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  29. Yonsei
    University 29/00
    Advantange/Disadvantage


    • Γ
    , = 0


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

    + ⋅ ∇ = 0)
    ( ⋅ ∇)

    = +1
    (1 − ∇ )

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  31. Yonsei
    University 31/00
    Reference





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  32. Yonsei
    University 32/00
    Reference (Image)










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  33. Yonsei
    University 33/00
    Reference (Image)


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