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Comprehensive Tutorial of Level Set Method
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Liam Jongsu Kim
June 04, 2013
Science
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Comprehensive Tutorial of Level Set Method
Liam Jongsu Kim
June 04, 2013
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Transcript
Comprehensive Guide of Level Set Method with Fluid Mechanics Jongsu
Kim Department of Computational Science and Engineering Yonsei University 1
Yonsei University 2/00 Contents • Motivation • Level Set Method
• Level Set Reinitialization • Numerical Schemes • Conclusion
Yonsei University 3/00 Multiphase Phase Flow •
Yonsei University 4/00 • • • • • • •
• Multiphase Phase Flow
Yonsei University 5/00 Contents • Motivation • Level Set Method
• Numerical Schemes • Conclusion • Level Set Reinitialization
Yonsei University 6/00 Modification of Navier-Stokes Equation • = −
+ 2 + = − + 2 + ⋅ = 0 Ω What equation at the interface?
Yonsei University 7/00
Yonsei University 8/00 CSF Model • • 1 − 2
+ = 1 − 2 + 1 − 2 + = 21 1 − 22 2 2 + 2 − 1 + 1 = •
Yonsei University 9/00 CSF Model • = () • ()
() = [] () = 1 ( 1) 2 ( 2) < > = 1 + 2 /2 ( ℎ ) = 2 − 1
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?
Yonsei University 11/00 VOF Method • • • • =
1 ( 2) (0 < < 1) + ⋅ = 0
Yonsei University 12/00 VOF Method • • • • •
Yonsei University 13/00 Level Set Method = =0 = ⋅
=0 , = < 0, ( ) > 0, ( ) = 0 ( ℎ )
Yonsei University 14/00 = , > 0 , ≤ 0
+ ⋅ = 0 , = < 0, ( ) > 0, ( ) = 0 ( ℎ ) Level Set Method
Yonsei University 15/00 + ⋅ = 0 Γ , ,
(, ) , = , , , , , = ( , , , ) ( , , , , ) (, ) Γ , , , , ≡ + + = + + Level Set Method
Yonsei University 16/00 , = < 0, ( ) >
0, ( ) = 0 ( ℎ ) () = , ( ) , ( ) + /2, () () = , ( ) , ( ) + /2, () Level Set Method
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
Yonsei University 18/00 Contents • Motivation • Level Set Method
• Numerical Schemes • Conclusion • Level Set Reinitialization
Yonsei University 19/00 → () Heaviside Function () = 0,
< − 1 2 1 + + 1 sin / , || ≤ 1, > +
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?
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?
Yonsei University 22/00 Level Set Reinitialization = (1 − ∇
) + ⋅ ∇ = = ∇ |∇| •
Yonsei University 23/00 Level Set Reinitialization •
Yonsei University 24/00 Level Set Reinitialization What We Have Done
= − + ⋅ 2 − + + ⋅ = 0 () = 0, < − 1 2 1 + + 1 sin / , || ≤ 1, > + + ⋅ ∇ = Are we done?
Yonsei University 25/00 Contents • Motivation • Level Set Method
• Numerical Schemes • Conclusion • Level Set Reinitialization
Yonsei University 26/00 Back to Equation = − + ⋅
2 − + + ⋅ = 0
Yonsei University 27/00 Back to Equation = Σ | +1
− | +1 ≤ ≤
Yonsei University 28/00 Contents • Motivation • Level Set Method
• Numerical Schemes • Conclusion • Level Set Reinitialization
Yonsei University 29/00 Advantange/Disadvantage • • • Γ , =
0 • •
Yonsei University 30/00 Summary of Algorithm (, 0) ( +
⋅ ∇ = 0) ( ⋅ ∇) = +1 (1 − ∇ )
Yonsei University 31/00 Reference • • • • •
Yonsei University 32/00 Reference (Image) • • • • •
• • • • •
Yonsei University 33/00 Reference (Image) • •