Liam Jongsu Kim
June 04, 2013
350

# Comprehensive Tutorial of Level Set Method

June 04, 2013

## 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

• 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)