Upgrade to Pro — share decks privately, control downloads, hide ads and more …

Comprehensive Tutorial of Level Set Method

Comprehensive Tutorial of Level Set Method

Liam Jongsu Kim

June 04, 2013
Tweet

Other Decks in Science

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 5/00 Contents • Motivation • Level Set Method

    • Numerical Schemes • Conclusion • Level Set Reinitialization
  4. Yonsei University 6/00 Modification of Navier-Stokes Equation • = −

    + 2 + = − + 2 + ⋅ = 0 Ω What equation at the interface?
  5. Yonsei University 8/00 CSF Model • • 1 − 2

    + = 1 − 2 + 1 − 2 + = 21 1 − 22 2 2 + 2 − 1 + 1 = •
  6. Yonsei University 9/00 CSF Model • = () • ()

    () = [] () = 1 ( 1) 2 ( 2) < > = 1 + 2 /2 ( ℎ ) = 2 − 1
  7. 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?
  8. Yonsei University 13/00 Level Set Method = =0 = ⋅

    =0 , = < 0, ( ) > 0, ( ) = 0 ( ℎ )
  9. Yonsei University 14/00 = , > 0 , ≤ 0

    + ⋅ = 0 , = < 0, ( ) > 0, ( ) = 0 ( ℎ ) Level Set Method
  10. Yonsei University 15/00 + ⋅ = 0 Γ , ,

    (, ) , = , , , , , = ( , , , ) ( , , , , ) (, ) Γ , , , , ≡ + + = + + Level Set Method
  11. Yonsei University 16/00 , = < 0, ( ) >

    0, ( ) = 0 ( ℎ ) () = , ( ) , ( ) + /2, () () = , ( ) , ( ) + /2, () Level Set Method
  12. 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
  13. Yonsei University 18/00 Contents • Motivation • Level Set Method

    • Numerical Schemes • Conclusion • Level Set Reinitialization
  14. Yonsei University 19/00 → () Heaviside Function () = 0,

    < − 1 2 1 + + 1 sin / , || ≤ 1, > +
  15. 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?
  16. 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?
  17. Yonsei University 24/00 Level Set Reinitialization What We Have Done

    = − + ⋅ 2 − + + ⋅ = 0 () = 0, < − 1 2 1 + + 1 sin / , || ≤ 1, > + + ⋅ ∇ = Are we done?
  18. Yonsei University 25/00 Contents • Motivation • Level Set Method

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

    • Numerical Schemes • Conclusion • Level Set Reinitialization
  20. Yonsei University 30/00 Summary of Algorithm (, 0) ( +

    ⋅ ∇ = 0) ( ⋅ ∇) = +1 (1 − ∇ )