Finite element computation of two-phase flow with level set method and explicit interface detection
Finite element computation of two-phase flow
with level set method
and explicit interface detection .
Katsushi OHMORI
.
Faculty of Human Development, Department of Mathematics
University of TOYAMA
A joint work with Atsushi SUZUKI (Osaka University)
December 9, 2016
Université Pierre et Marie Curie (UPMC)
Katsushi OHMORI
Finite element computation of two-phase flow with level set method and explici
Finite element computation of two-phase flow with level set method and explicit interface detection
Toyama est dans la direction de nord-ouest de Tokyo.
”Toyama” veut dire qu’il y a beaucoup de montagnes en japonais.
(mont Tsurugi, 3003m)
Katsushi OHMORI
Finite element computation of two-phase flow with level set method and explici
Finite element computation of two-phase flow with level set method and explicit interface detection
Outline
1
Numerical analysis of two-fluid flows by FreeFem++ and
reinitialization procedure (K. OHMORI)
2
Finite element computation of two-phase flow with level set
method and explicit interface detection (A. SUZUKI)
Katsushi OHMORI
Finite element computation of two-phase flow with level set method and explici
Finite element computation of two-phase flow with level set method and explicit interface detection
Model problem
In this presentation, we present a numerical study of the immscible
incompressible two-fluid flows by FreeFem++.
Ω ⊂ R 2 : a bounded domain such that
Ω = Ω1 (t) ∪ Ω2 (t) and Ω1 (t) ∩ Ω2 (t) = 0.
/
Γ(t) = ∂ Ω1 (t) ∩ ∂ Ω2 (t) : the interface between two fluids
(1)
the density
{ and viscosity
(ρ1 , µ1 ) in Ω1
(ρ , µ ) =
(ρ2 , µ2 ) in Ω2
,where ρ1 > ρ2 and µ1 > µ2
2
u=v=0
fluid1
u=0
0.5
u=0
0.25
fluid2
y
x
O
0.5
1
u=v=0
Katsushi OHMORI
Finite element computation of two-phase flow with level set method and explici
Finite element computation of two-phase flow with level set method and explicit interface detection
Mathematical Model
Immiscible incompressible two-fluid system

∂u


ρ(
+ (u · ∇)u) = −∇p + ∇ · (2µ D(u)) + f + F SF


∂t
∇·u = 0



 u = 0 (no slip condition)
u · n = 0 (free slip condition)
in Ω,
in Ω,
on ΓTB ,
on ΓLR ,
(2)
where D(u) = (∇u + t ∇u)/2 : the viscous stress tensor,
f : the force term (f = t (0 − ρ g))
n : outer normal vector of domain Ω1
κ : the curvature of the interface
σ : the surface tension coefficient
FSF : the surface tension term.
Katsushi OHMORI
Finite element computation of two-phase flow with level set method and explici
Finite element computation of two-phase flow with level set method and explicit interface detection
Transmission conditions on the interface Γ(t):
[u]|Γ = 0,
[2µ D(u)n − pn]|Γ = σ κ n.
Surface tension effects are taken into consideration through the
following force balance at the interface Γ(t).
The surface tension term is expressed as
FSF = σ κ nδΓ
where δΓ is the distribution and it holds
∫
δΓ ϕ dx =
Ω
∫
ϕ dγ
Γ
for any smooth function ϕ .
Katsushi OHMORI
Finite element computation of two-phase flow with level set method and explici
Finite element computation of two-phase flow with level set method and explicit interface detection
Level set method for two-fluid flows
In level set method the interface is evolved by solving the advection
equation.
The interface is define as follows
Γ(t) = {x ∈ Ω ; ϕ (x, t) = 0} for 0 ≤ t < T .
Initial level set function ϕ (x, 0) : a signed distance from the interface
Γ(0)

for x ∈ Ω1 (0),
 dist(x, Γ(t) ∪ ∂ Ω)
0
for x ∈ Γ(0),
ϕ (x, 0) =
(3)

−dist(x, Γ(t) ∪ ∂ Ω) for x ∈ Ω2 (0),
where dist(x, Γ) = min |x − y |.
y ∈Γ
In the level set method, it is common to define the curvature as the
divergence the normal

∇ϕ
 n = |∇ϕ | |ϕ =0
 κ = −∇ · n = − ϕy ϕxx −2ϕx ϕy ϕxy +ϕx ϕyy .
(ϕ 2 +ϕ 2 )3/2
2
2
x
Katsushi OHMORI
(4)
y
Finite element computation of two-phase flow with level set method and explici
Finite element computation of two-phase flow with level set method and explicit interface detection
Katsushi OHMORI
Finite element computation of two-phase flow with level set method and explici
Finite element computation of two-phase flow with level set method and explicit interface detection
Variational formulation for two-fluid flows

 V = {v ∈ H 1 (Ω)2 ∫| v = 0 on Γ1 , v · n = 0 on Γ2 } ,
Q = {q ∈ L2 (Ω) | Ω q dx = 0},

Ψ = H 1 (Ω).
(5)
Find (u(t), p(t), ϕ (t)) ∈ V × Q × Ψ for 0 < t < T such that
}
{
 ∫
∫
∂u


ρ
(
ϕ
)
+
(u
·
∇)u
·
v
dx
+
2
µ (ϕ ) D(u) : D(v ) dx


∂t

Ω
Ω∫
∫




 − p ∇ · v dx − q ∇ · u dx
∫
Ω
∫
Ω
(Π)

for ∀(v , q) ∈ V × Q.
= f · v dx + σ κ n · v d γ



Γ
Ω


∫


∂ϕ


{
for ∀ξ ∈ Ψ.
+ u · ∇ϕ }ξ dx = 0
Ω ∂t
(6)
Katsushi OHMORI
Finite element computation of two-phase flow with level set method and explici
Finite element computation of two-phase flow with level set method and explicit interface detection
Reinitialization - Hyperbolic PDE approach
Sussman, Smereka and Osher (1994)
The signed distance function

 dist(x, Γ)
0
d(x) =

−dist(x, Γ)
for x ∈ Ω1 ,
for x ∈ Γ,
for x ∈ Ω2 .
(7)
has the useful property
|∇d(x)| = 1 (Eikonal equation).
However, after the advection step, because of error accumulation the
level set function is no longer a distance function.
d(x)
ψ(x, τ )
ψ0 (x)
x
τ →∞
Katsushi OHMORI
Finite element computation of two-phase flow with level set method and explici
Finite element computation of two-phase flow with level set method and explicit interface detection
Hamilton-Jacobi type equation for pseudo time τ ,

 ∂ ψ + S(ψ )(|∇ψ | − 1) = 0 in Ω,
0
∂τ

ψ (x, 0) = ψ0 = ϕ (x, t),
where S(ψ0 ) is a sign function such that

 −1 for ψ < 0,
0 for ψ = 0,
S(ψ0 ) =

1 for ψ > 0.
(8)
(9)
In practice, S(ψ0 ) is approximate by
ψ
S ∗ (ψ ) = √ 0 .
ψ02 + ε
Katsushi OHMORI
Finite element computation of two-phase flow with level set method and explici
Finite element computation of two-phase flow with level set method and explicit interface detection
If ψ is steady state attained at τ = τ∞ , the redistanced function ψ
would be expected to satisfy
|∇ψ | = 1.
Subsequently, ϕ is recovered as ϕ (x, t) = ψ (x, τ∞ ).
Katsushi OHMORI
Finite element computation of two-phase flow with level set method and explici
Finite element computation of two-phase flow with level set method and explicit interface detection
Semi-discretization
{Th }h>0 : a regular family of triangulation
V h ⊂ V and Qh ⊂ Q : a pair of finite element spaces which satisfies
uniform inf-sup condition
Ψh ⊂ Ψ.
Find (u h , ph , ϕh ) ∈ V h × Qh × Ψh for 0 < t < T such that
{
}
∫
 ∫
∂ uh


ρ (ϕh )
µ (ϕh )D(u h ) : D(v h )dx
+ (u h · ∇)u h · v h dx + 2


∂t
Ω∫
Ω

∫




 − ph ∇ · v h dx − qh ∇ · u h dx
Ω
∫
Ω
∫
′
(Π )

=
f
·
v
dx
+
σ
δε (ϕh )κ n · v h dx for ∀(v h , qh ) ∈ V h × Qh
h


Ω
Ω


}

∫ {


∂ ϕh


for ∀ξh ∈ Ψh .
+ u h · ∇ϕh ξh dx = 0
∂t
Ω
(10)
Katsushi OHMORI
Finite element computation of two-phase flow with level set method and explici
Finite element computation of two-phase flow with level set method and explicit interface detection
The surface integration on Γ for the surface tension term is replaced
by a domain integration :
σ
∫
Γ
.
κn · v dγ = σ
∫
Ω
δε (ϕh )κ n · v h dx
, where

 1 [1 + cos( πϕ )] for |ϕ | ≤ ε ,
dHε
2ε
ε
δε (ϕ ) =
(ϕ ) =
,

dϕ
0
for |ϕ | > ε .
and

0





1
ϕ 1
πϕ
Hε (ϕ ) =
)]
[1 + + sin(

ε π
ε
 2



1
Katsushi OHMORI
(11)
for ϕ < −ε ,
for |ϕ | ≤ ε ,
(12)
for ϕ > ε .
Finite element computation of two-phase flow with level set method and explici
Finite element computation of two-phase flow with level set method and explicit interface detection
Full discretization
Finite element triple (u, p, ϕ ) - P2/P1/P2
Stabilization for transport eq. - SUPG method
Find (u nh+1 , phn+1 , ϕhn+1 ) ∈ V h × Qh × Ψh for n ≥ 0 such that
(Πh )
 ∫
∫
)
(

n+1
n 1
n
n

ρ
χ
·
v
dx
+
2
µ n D(u nh+1 ) : D(v h )dx
u
−
u
◦

h
h
h
h

∆t

Ω
Ω
∫
∫




− phn+1 ∇ · v h dx − qh ∇ · u nh+1 dx



∫Ω
Ω ∫




ch n )κhn nnh · v h dx
= f · v h dx + σ δε (ϕ
Ω
Ω
for ∀(v h , qh ) ∈ V h × Qh ,





)
(

∫


(
)
ϕhn+1 − ϕhn

n
n+1


+ u h · ∇ϕn
ψh + α u nh · ∇ψh dx = 0


∆t
Ω



for ∀ψh ∈ Ψh .
(13)
Katsushi OHMORI
Finite element computation of two-phase flow with level set method and explici
Finite element computation of two-phase flow with level set method and explicit interface detection
ch n is removed from ϕhn by the following way
where ϕ

 δε (ϕhn (x)) if dist(x, Γ) > 2ε ,
n
ch (x)) =
δε (ϕ
 0
otherwise.
∂Ω
Γ
Γ
∂Ω
x
φ(0, x)
Figure: Modified contour line of level
set function
Katsushi OHMORI
Finite element computation of two-phase flow with level set method and explici
Finite element computation of two-phase flow with level set method and explicit interface detection
Numerical examples
Skirted bubble
ρ1
1000
ρ2
1
µ1
10
µ2
0.1
g
0.98
σ
1.96
Re
35
Eo
125
Table: Physical parameters and dimensionless numbers
The signed distance function : ϕ0 =
(∆t, ∆τ )
n = 40
(0.025, 1.0e-5)
√
(x − 0.5)2 + (y − 0.5)2 − 0.25,
n = 80
(0.0125, 1.0e-5)
n = 160
(0.00625, 1.0e-5)
Table: Time steps for computation and reinitialization
Katsushi OHMORI
Finite element computation of two-phase flow with level set method and explici
Finite element computation of two-phase flow with level set method and explicit interface detection
Figure: Finite element mesh (n=40)
Katsushi OHMORI
Finite element computation of two-phase flow with level set method and explici
Finite element computation of two-phase flow with level set method and explicit interface detection
Figure: Streamline (t=0.0)
Katsushi OHMORI
Figure: Streamline (n=80, t=3.0)
Finite element computation of two-phase flow with level set method and explici
Finite element computation of two-phase flow with level set method and explicit interface detection
No. of iteration = 10
No. of iteration = 20
No reinitialization
n = 40
1.06535
1.06710
1.53014
n = 80
1.07677
1.06587
1.51965
n = 160
1.07162
1.06584
1.57794
Table: L1 -norm of level set function for reinitialization
No. of iteration = 10
No. of iteration = 20
No reinitialization
n = 40
1.10606
1.10451
1.10299
n = 80
1.08379
1.08257
1.08873
n = 160
1.14104
1.15037
1.14063
Table: Center of mass yc (t = 3), where Ω2 is the region that the bubble
occupies,
∫
The center of mass
Ω
yc = ∫ 2
Ω2
Katsushi OHMORI
y dΩ
1 dΩ
.
Finite element computation of two-phase flow with level set method and explici
Finite element computation of two-phase flow with level set method and explicit interface detection
Time
t=3.0
t=3.0
n = 40
85/120
82/120
n = 80
154/240
154/240
n = 160
314/480
314/480
Table: Number of reinitialization
Time
t=3.0
n = 40
0.200631
n = 80
0.201418
Table: Mass conservation of the bubble
Katsushi OHMORI
Finite element computation of two-phase flow with level set method and explici
Finite element computation of two-phase flow with level set method and explicit interface detection
2
2
"L1norm40-OCT26.txt" using 1:2
"L1norm80-NOV4.txt" using 1:2
1.5
1.5
1
1
0.5
0.5
0
0
0
0.5
1
1.5
2
2.5
3
Figure: L1 −norm (n=40)
Katsushi OHMORI
0
0.5
1
1.5
2
2.5
3
Figure: L1 −norm (n=80)
Finite element computation of two-phase flow with level set method and explici
Finite element computation of two-phase flow with level set method and explicit interface detection
TP2D – TU Dortmund, Inst. Applied Math.
FreeLIFE – EPFL Lausanne, Inst. of Analysis and Sci. Comp.
MooNMD – Uni. Magdeburg, Inst. of Analysis and Num. Math.
Katsushi OHMORI
Finite element computation of two-phase flow with level set method and explici
Finite element computation of two-phase flow with level set method and explicit interface detection
n = 40
n = 80
Katsushi OHMORI
n = 160
Finite element computation of two-phase flow with level set method and explici
Finite element computation of two-phase flow with level set method and explicit interface detection
Conclusion
J. Cahouet and J.-P. Chabard, Some fast 3D finite element solver for
the generalized Stokes, problem, Int. J. Numer. Meth. Fluids, Vol. 8,
pp.869–895(1988)
1
2
3
4
5
Robustness of the algorithm (Good)
Simplicity of the algorithm (Excellent)
No parameter to tune (No good)
Mass balance (Good)
Accuracy (Good)
So far, so good !
Katsushi OHMORI
Finite element computation of two-phase flow with level set method and explici
Finite element computation of two-phase flow with level set method and explicit interface detection
Conclusion
J. Cahouet and J.-P. Chabard, Some fast 3D finite element solver for
the generalized Stokes, problem, Int. J. Numer. Meth. Fluids, Vol. 8,
pp.869–895(1988)
1
2
3
4
5
Robustness of the algorithm (Good)
Simplicity of the algorithm (Excellent)
No parameter to tune (No good)
Mass balance (Good)
Accuracy (Good)
So far, so good !
Katsushi OHMORI
Finite element computation of two-phase flow with level set method and explici
Finite element computation of two-phase flow with level set method and explicit interface detection
Conclusion
J. Cahouet and J.-P. Chabard, Some fast 3D finite element solver for
the generalized Stokes, problem, Int. J. Numer. Meth. Fluids, Vol. 8,
pp.869–895(1988)
1
2
3
4
5
Robustness of the algorithm (Good)
Simplicity of the algorithm (Excellent)
No parameter to tune (No good)
Mass balance (Good)
Accuracy (Good)
So far, so good !
Katsushi OHMORI
Finite element computation of two-phase flow with level set method and explici
Finite element computation of two-phase flow with level set method and explicit interface detection
Conclusion
J. Cahouet and J.-P. Chabard, Some fast 3D finite element solver for
the generalized Stokes, problem, Int. J. Numer. Meth. Fluids, Vol. 8,
pp.869–895(1988)
1
2
3
4
5
Robustness of the algorithm (Good)
Simplicity of the algorithm (Excellent)
No parameter to tune (No good)
Mass balance (Good)
Accuracy (Good)
So far, so good !
Katsushi OHMORI
Finite element computation of two-phase flow with level set method and explici
Finite element computation of two-phase flow with level set method and explicit interface detection
Conclusion
J. Cahouet and J.-P. Chabard, Some fast 3D finite element solver for
the generalized Stokes, problem, Int. J. Numer. Meth. Fluids, Vol. 8,
pp.869–895(1988)
1
2
3
4
5
Robustness of the algorithm (Good)
Simplicity of the algorithm (Excellent)
No parameter to tune (No good)
Mass balance (Good)
Accuracy (Good)
So far, so good !
Katsushi OHMORI
Finite element computation of two-phase flow with level set method and explici
Finite element computation of two-phase flow with level set method and explicit interface detection
Conclusion
J. Cahouet and J.-P. Chabard, Some fast 3D finite element solver for
the generalized Stokes, problem, Int. J. Numer. Meth. Fluids, Vol. 8,
pp.869–895(1988)
1
2
3
4
5
Robustness of the algorithm (Good)
Simplicity of the algorithm (Excellent)
No parameter to tune (No good)
Mass balance (Good)
Accuracy (Good)
So far, so good !
Katsushi OHMORI
Finite element computation of two-phase flow with level set method and explici
Finite element computation of two-phase flow with level set method and explicit interface detection
Conclusion
J. Cahouet and J.-P. Chabard, Some fast 3D finite element solver for
the generalized Stokes, problem, Int. J. Numer. Meth. Fluids, Vol. 8,
pp.869–895(1988)
1
2
3
4
5
Robustness of the algorithm (Good)
Simplicity of the algorithm (Excellent)
No parameter to tune (No good)
Mass balance (Good)
Accuracy (Good)
So far, so good !
Katsushi OHMORI
Finite element computation of two-phase flow with level set method and explici
Finite element computation of two-phase flow with level set method and explicit interface detection
Conclusion
J. Cahouet and J.-P. Chabard, Some fast 3D finite element solver for
the generalized Stokes, problem, Int. J. Numer. Meth. Fluids, Vol. 8,
pp.869–895(1988)
1
2
3
4
5
Robustness of the algorithm (Good)
Simplicity of the algorithm (Excellent)
No parameter to tune (No good)
Mass balance (Good)
Accuracy (Good)
So far, so good !
Katsushi OHMORI
Finite element computation of two-phase flow with level set method and explici
Finite element computation of two-phase flow with level set method and explicit interface detection
Je vous remercie de votre attention.
Katsushi OHMORI
Finite element computation of two-phase flow with level set method and explici
weak formulation
V = {v ∈ H 1 (Ω)2 ; v · n = 0 on ΓRL , v = 0 on ΓTB },
Q = {q ∈ L2 (Ω)2 ; (q, 1) = 0},
Ψ = H 1 (Ω) .
initial condition : u(0, x), ϕ(0, x)
find {u(t), p(t), ϕ(t)} ∈ V × Q × Ψ
Z
Z
Z
Z
ρ(ϕ)(∂t u + u · ∇u) · v + 2
µ(ϕ)D(u) : D(v) −
p∇ · v −
q∇ · u =
Ω
Ω
Ω
Ω
Z
Z
−
ρ(ϕ)g e2 · v + σ κ n · v
∀(v, q) ∈ V × Q ,
Ω
Γ
Z
∂t ϕψ + u · ∇ϕψ = 0
∀ψ ∈ Ψ .
Ω
ρ1 ϕ(t, x) > 0
ρ(ϕ)(t, x) =
, µ(ϕ) as same as ρ(ϕ) .
ρ2 ϕ(t, x) < 0.
0
e2 =
, g : gravity acceleration coefficient
1
1 / 17
discretization scheme
∆t : time step, Th mesh decomposition with triangle K.
2
Vh = {u|K ∈ (P1 (K))2 ; v · n = 0 on ΓRL , v = 0 on ΓTB } ⊂ H 1 (Ω) ,
R
Qh = {p|K ∈ P1 (K); Ω p = 0} ⊂ L20 (Ω),
Ψh = {φ|K ∈ P1 (K)} ⊂ H 1 (Ω).
characteristic finite element method + penalty stabilization / SUPG
(um , ϕm ) : known, find (um+1 , pm+1 , ϕm+1 ) ∈ Vh × Qh × Ψh ,
Z
Z
1 m+1
m+1
m
m
+2
µm
(u
−
u
) : D(vh )
ρm
◦
X
)
·
v
h
h D(uh
h
h
h
h
∆t
Ω
Ω
Z
Z
Z
X
m+1
2
−
pm+1
∇
·
v
−
q
∇
·
u
−δ
h
∇pm+1
· ∇qh =
h
h
K
h
h
h
Ω
Ω
Z
−
ρm
h g e2 · vh +σ
Ω
Z
Ω
ϕ
fh
m+1
− ϕm
m
h
+ um
h · ∇ϕh
∆t
ZK
K
κm nm · vh
∀(vh , qh ) ∈ Vh × Qh ,
Γm
h
!
(ψh + αum
h · ∇ψh ) = 0 ∀ψh ∈ Ψh .
δ > 0 : stability parameter, hK = diam(K), α > 0
2 / 17
computation of the surface tension effect
Z
κn · v
how to compute surface integration? σ
Γ
∇ϕ
|ϕ=0 , ∇S : tangential gradient
|∇ϕ|
Question ? ϕ : approximated by P1 ⇒ κ = 0?
surface divergence formula with integration by part
Z
Z
Z
κv · n = − ∇S · n v · n = − ∇S · v .
κ = −∇S · n = −∇ · n, n =
Γ
Γ
Γ
cf. J.-F. Gerbeau, T. Lelièvre, C. Le Bris, JCP, 184 (2003)
m
Γm
:
P1-interpolation
h
[
[of zero-level set {ϕh (x) = 0},
Γh ∩ K l =
γl . time step m : fixed.
Γh =
l
l
in 2-D case, surface gradient is easily obtained as
∇ = ∇N + ∇S = n(n · ∇) + t(t · ∇). n = (n1 , n2 )T , t = (n2 , −n1 )T
Z
XZ
−
∇S · vh = −
tl (tl · ∇) · vh
Γh
l
γl
X Z n2 v
=−
(n2 ∂1 − n1 ∂2 ) · 1
−n
v2
1
γl
l
3 / 17
reinitialization of the level set?
Eikonal eq. |∇ϕ| = 1 from signed distance function
|∇ϕm+1 | = 1 is not satisfied in vicinity of Γ(t)
⇒ reinitialization procedure by solving PDE
cf. M. Sussman, P. Smereka, S. Osher, JCP, 114 (1994)
mathematical consideration by Dr. N. Hamamuki
Reinitialization process with hyperbolic PDE by Sussman et al. does
not change zero-level set of the signed distance function itself.
m+1
reconstruction of signed distance function
fh
⇒ ϕh m+1
S m+1ϕ
from zero-level set in finite elements, l Γh
∩ Kl
cf. M. K. Touré, A. Soulaïmani, doi:10.1016/j.camwa.2016.02.028
ϕm+1
(P ) = dist(P, Γm+1
) = min dist(P, γlm+1 )
h
h
l
with reduction of computational cost by selecting FE nodes P in
neighborhood of Γm
h
W m (ζ) = {x ∈ Ω; |ϕm (x)| < ζ},
ζ>0
4 / 17
reconstruction of signed distance function
will be repalced by distance()
315.62 sec/140.67 sec @ t=0.25
e
P
Q
ϕm+1
(P ) = dist(P, Γm+1
)
h
h
= min dist(P, γlm+1 )
l
Ω1
ψ (x,y)>0
γl
ψ(x,y)=0
ψ (x,y )<0
Ω2
5 / 17
isotropic mesh refinement to save DOF for linear solver
mesh around the bubble by FreeFem++
layers are defined by W m (ζk ), ζ1 < ζ2 < ζ3 .
The signed distance function is totally reconstructed and re-meshing
when center of mass of the bubble moves a lot, e.g. 0.01
6 / 17
numerical setting of a test problem
density ρk
viscosity µk
σ
# of K
# of vertex
hmin
1,000 1.0 10
0.1
1.96 32,656
16,479
0.00356
mesh around the bubble at time t = 2.5. Ω = (0, 1) × (0, 2).
hmax
∆t
0.0279
0.005
7 / 17
time evolution of a test problem
8 / 17
preliminary results : 1/3
center of mass
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5
0.4
0
0.5
1
1.5
time
2
2.5
3
9 / 17
preliminary results : 2/3
satisfuction of Eikonal eq.
1.025
1.02
|\nabla\varphi|
1.015
1.01
1.005
1
0.995
0.99
0.985
0.98
0.975
0
0.5
1
1.5
2
2.5
3
time
10 / 17
preliminary results : 3/3
red
green
blue
TP2D
FreeLIFE
MooNMD
S. Turek, D. Kuzmin, S. Hysing
E. Burman, N. Parolini
L. Tobiska, S. Ganesan
level set
level set
ALE
S. Hysing, S. Turek, D. Kuzmin, N. Parolini, E. Burman,
S. Ganesan, L. Tobiska, Int. J. Numer. Meth. Fluids. 60 (2009)
11 / 17
conclusion
I
P1/P1/P1 finite element for velocity/pressure/levelset unknowns
I
surface tension is treated by a surface divergence formula
reconstruction of level set function from geometrical information
I
I
I
isotropic mesh refinement around the zero-level set boundary
Dissection direct solver is used to solve generalized Stokes eqs.,
which indefinite and singular
ongoing
I
qualitative comparison with other schemes in the benchmark
paper
future work
I
P2 approximation to level set
iso-parametric approximation to zero-level set curve in each finite
element
12 / 17