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PART 3 PART 3 DETERMINATION OF THE INTERFACESDETERMINATION OF THE INTERFACES
Plan
• VOF And DG method : – diffusion and mesh adaptation– examples
• Level Set type method : – Transport equation– reinitialisation, – distance property conservation– Convective reinitialisation– Sinus level set method
• Water assisted injection example– Process and computational challenge– Combined sinus level set and anisotropic adaptive
mesh• Conclusion
Moving free surface and interface flows
• Polymer injection moulding (Rem3D)
• Metal casting• Filling process• Mixing• Foaming
Moving free surface and interface calculation with Finite Element
• Tracking : – lagrangian approach, – the free surface or interface is part of the mesh
boundary• Capturing :
– Flow interface are moving through the mesh– Transport equation : – Methods vary with the quantity to be convected :
• VOF : a fill factor , discontinuous, discontinuous Galerkinperforms well. But difusion, element centred, large bandwidth
• LevelSet : a distance, continuous, continuous Galerkintechnique, must stabilsed, must be reinitialised.
Time dependent moving domains
Fixed domain : the entire cavity
)()( tt airfluid ΩΩ=Ω U
fluidΩ airΩ
Ω
Freesurface
Capturing :Free surface = Interface
+∈∀Ω∈∀=∇+∂∂ IRtxv
t,0.αα
The free surface motion by solving a transport equation
)(1),()(
1
αα
α
Hxdxf
f
=Ω∂=
≈
Ω
ΩVolume Of Fluid
Level Set
Unsteady incompressible NavierStokes Multi phase flow
• Navier-Stokes incompressible :
⎩⎨⎧
<=>=
0001
)(αα
αsisi
H))(1()())(())(1()())((
21
21
αηαηαηαραραρ
HHHHHH
−+=−+=
+ mixture low :
( )( )
⎪⎪
⎩
⎪⎪
⎨
⎧
=∇+∂∂
=⋅∇
=∇++⋅∇−
0.
0
))(())((2))((
vt
v
gHpvHdtdvH
αα
αρτεαηαρ
Multiphase flow by heterogeneous Navier Stokes Modeling and local extended level set method
VOF or P0 approximation
KK fluidK
Ω=
Iα
)(1)(1)(
xx KK
Kf
h ∑Ω∈
Ω =τ
α
• Approximation of the domain characteristic function = the fill factor• Weighed weak variationnal formulation• Robust • Discontinuous Galerkin method : conservativity
Drawbacksdiffusionaccuracyneed adaptive meshing
)(,0
)(,1)(1 )(
tx
txx
fluid
fluidtfluid
Ω∉=
Ω∈=Ω
1f=1 0f =1
FE and VOF +DG+ mesh adaptation
Presence function (fill factor). Diffusion limited by mesh adaptation.Solvers :
• flow solver : node centred• transport solver: element centred
breaking dam benchmark
Simulation of the polymer injection moulding process : the mould filling stage (VOF Finite Element method)
Rem3D
• Simple continuous quasi standard finite element simple implementaton P1
• Iterative solver and parallel scalability• Method without diffusion
• Extended Level Set method
Continuous P1 solution for free surface or interface capturing
Level Set method
{ }⎩⎨⎧
=Ω∈=ΓΩ∈Γ=
0)(,,),()(xx
xxdxα
α
⎪⎩
⎪⎨⎧
==
=∇+∂∂
=
)(),0(
0.
0 xxt
vtdt
d
αα
ααα
The distance function
Dynamic : the transport equation
Transport equation
• Transport of the Level Set function• Purely convective scalar equation
– Continuous space interpolation
• Standard Galerkin weak formulation– Stabilization needed
0=dtdα
0. =∇+∂∂ v
tαα
( ) 0,., =∇+⎟⎠⎞
⎜⎝⎛
∂∂
hhhh v
tϕαϕα
α
Stabilization of advection equation for continuous approximation
• SUPG class of method (streamline upwind Petrov-Galerkin )– Brooks and Hughes [1]– Petrov Galerkin approach :
• Test functions differ from the approximation shape function: hϕ
hKhh v ϕτϕϕ ∇+= .~
[1] A. N. Brooks and T. J. R. Hughes,
hϕ~
• RFB-like method (Residual-Free Bubbles)- Inspired from a multiscale approach[2] - fine scale or stabilisation by enrichment and static condensation of bubbles function
K
KK Kk v
hbK 3
ˆ1≈= ∫τ
[2] F. Brezzi and A. Russo,
Level Set Methodadvantages
– Interface or free surface is the 0 of the levelset function– Higher order than VOF : affine recovery– Nice representation of immersed boundary – Smooth gradients ease the convergence and stability of convective schemes
for continuous Finite Element• but
– Distance property not preserved when transported
– Loss of gradient gradient smoothness due to the convective flow : unstable– is initialized as a signed distance function to the interface– Reconstruction of the Level Set needed : the reinitialisation stage
1=∇α
Mixing , twin screw extrusionE. Foudrinier et R. Valette :
16
Representation of immersed domain (moving screws)by a Level Set approach
Immersion
To compute complex flows inside extruders (free
surfaces, velocities, strain rates, pressures, filling, …)
Level Set Methodadvantages
– Interface or free surface is the 0 of the levelset function– Higher order than VOF : affine recovery– Nice representation of immersed boundary – Smooth gradients ease the convergence and stability of convective schemes for
continuous Finite Element• but
– Distance property not preserved when transported
– Loss of gradient gradient smoothness due to the convective flow : unstable– Stability of the upwind FE scheme– Reconstruction of the Level Set needed : the reinitialisation stage
1=∇α
Reinitialisation– Idea : propagate the distance value from the 0 level by using
a transient hyperbolic scheme– Solution of Hamilton Jacobi equation
Kh≈Δτ
timepseudo :τ
[3] Sussman, Smereka, Osher, A level set method for computing solutions to incompressible two-phase flow (1994)
( )e
S+
=β
βββ )sgn(
e : thickness of the interface ~ h
⎪⎩
⎪⎨⎧
==
=−∇+∂∂
),(),0(
0)1(
xtx
s
ατβ
βτβ
Reinitialisation
ββββ ∇
∇∇
=∇ .ββ
∇∇
= sU
⎪⎩
⎪⎨⎧
==
=∇+∂∂
),(),0(
.
xtx
sU
ατβ
βτβ
Convective velocity from the gradient of the Level Set
A classical transport equation : U enabling any upwind scheme
Levelset + continuous galerkin + reinitialisationHamilton Jacobi
Flow solver and transport solver are node centred.No diffusion, no mesh adaptation, and accurate surface description.Stabilisation and reinitialisation
VOF/P0 versus Level Set/P1
VOF with mesh adaptation Level set
Falling drop
Exemple
• Fluid ¼ poured in 30 seconds
• 11 bubbles go up into the fluid
• Bubble period of about 2.7 seconds
Something New in the Level Set Method :the Convective reinitialisation
τα
τα
ddt
dtd
dd
=dtdτλ =
0)1( =−∇+ βλβ sdtd
βββ∇+
∂∂
= .vtdt
d
⎪⎩
⎪⎨⎧
==
=−∇+∇+∂∂
)(),0(
0)1(.
0 xx
svt
ατα
αλαα
Physical time and pseudo time link
Convection in time derivative of the Hamilton Jacobi equation
The new equation :
Distance preserving convective equation in a flow
⎪⎩
⎪⎨⎧
==
=∇++∂∂
)(),0(
).(
0 xxt
sUvt
αα
λαλα
Locality and Truncature
• Define the Level set function only in the region of the interface
• Avoid sudden change in gradient : instability • Thickness of the Level set support
• Idea :– Generalisation of Level Set function : not restricted to the
distance function– Self determination of the function by solving an absorbing
hyperbolic equation : stability– Solution : a sinusoidal function
Local sinus level set function
[ ]EExxE
Ex ,;)2
sin(2)( −∈=π
πα
2)2
(1' απαE
−=
2)2
(1 απαE
−=∇E
2E/π
α(x) = xα‘ = 1
α(x) = (2E/π) sin((π /2E) x)α‘ = (1-((π /2E) α)2 )1/2
1≠∇α
Local smooth gradient transition
⎪⎩
⎪⎨
⎧
==
=−−∇+∇+∂∂
)(),0(
0))2
(1(.
0
2
xxtE
svt
αα
απαλαα
E : the thickness of the Level Set function
thΔ
=λ
S= 0 if α lower than h
Multiphase modeling
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
=
===∇
=∇+∇−∇+∂∂
Ω∂0
0)0(0.
))(2.().(
vtvv
gpvvvtv ρηερ
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
⎩⎨⎧
<>
=
−+=−+=
0001
)(
))(1()())(1()(
21
21
αα
α
αηαηηαραρρ
sisi
H
HHHH
⎪⎪
⎩
⎪⎪
⎨
⎧
+<−<
<+
=esiesi
esie
H e
αα
αα
α00
221
)(
Falling solid through two fluids
⎩⎨⎧
−+−+=−+−+=
))(1()()))(1()(())(1()()))(1()((
321
321
BeBeAeAe
BeBeAeAe
HHHHHHHH
αηααηαηηαρααραρρ
RRR benchmark example
High Performance Computing for incompressible flow with
moving free surface
• Continuous galerkin method : simple P1 approximation everywhere
• Advantages: simplicity and efficiency– maillage :2 148 355 nodes et
12 418 472 elements. – 600 time steps– 2 linear systems per
increment of 8.6 millions of unknowns, and of 2.15 millions, respectively : a total of 6 billion
– et 450 million of unknowns. – Cpu time : 5 days and 1 hour
on a cluster of 32 processors• Key points and on going
research : Stability, conservativity, TVD, anisotropic adaptivity
H. DigonnetO. Basset
Forming Process example
Handle bar
Fork
L. Silva W. Zerguine
Water Injection Technology (WIT) or Water Assisted Injection Molding (WAIM).
Advantages:- cooling and reduction of the cycle times (when compared with GAI) –- water incompressibility gives higher pressures, longueur channels with more important and uniform thicknesses and smoother surfaces - lower costsInconv:- technological: water leakage, removal of water from the part, design of thewater
water assisted injection molding simulation
Computing difficulties of the process
• High difference in viscosity between polymer and water : 106
• High Reynolds• Thermal shock
• Need enrichment in the vicinity of the interface• One way : Adaptive anisotropic mesh refinement
Need of anisotropic remeshing :The problem of the gap in viscosity
Fluid/fluidinterface
Entry 2nd fluid
Outlet 1st fluid
1st fluid• t=0 : filled cavity with fluid 1• varying η1/η2
Influence on the interface evolution
η1/η2 decreases (0.1, 0.01, 0.001)
Need of refinement near the interface
RemeshingComputation of the metrics field
IAmM 221 ε+= TA αα ∇⊗∇=
ελ /1/1 01 ==⊥h
⊥∇= α0v
α∇=1v
20 ελ =
2221 εαλ +∇= m
⊥∇α
α∇
221 /1 ε+= mh
22211 /1/1 εαλ +∇== mh
the mesh size is
Computation of the multidomain metrics using a classical Level Set function
with
Let us define
In the direction
If we have our LLS function , and we can define
with
1≠∇α IBmM 222 ε+=
2ααα
∇
∇⊗∇=
T
B
RemeshingComputation of the metrics field
e
N is the number of elements in the thickness
⎪⎩
⎪⎨⎧
+−
>=
onIBeN
esiIM
sin)/(
2/22
2
3εε
αε
e is the thickness
We can choose to control the number of elements in a certain remeshing thickness
RemeshingResults on a 2D WAIM example
Rem3D_RWater assisted injection : Handle example
Two Level Set functions and anisotropic mesh adaptation
Injection poignée
Polymer injection (until~50%)
Water injection
Volume, %
2 50
100 Calcul isotherme
Visco pol ~1000 Pas
Visco eau ~1 Pas
Maillage initiale :
~70 000 nœuds
Cas avec remaillage :
~200 000 noeuds
Rem3D Recherche 2.0Water assisted injection : Handle example
Remeshing
GLview 3D Plug-in GLview 3D Plug-in
Rem3D Recherche 2.0Water assisted injection : Handle example
Isosurfaces
GLview 3D Plug-in GLview 3D Plug-in
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3) Determination of the interfaces
3.2. LevelSet I
Interface capturing techniqueis a signed distance function to the interface
is continuous
Advantages:• Node centred (P1) : better interface definition• Faster• The coupled system is entirely continuous (P1)• No diffusion no need of mesh adaptation
Inconvenients:• Continuous Galerkin is not well-suited for this type of problem (pure convection) must be stabilized• The distance function deteriorates quickly must be reinitialised
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3) Determination of the interfaces
3.2. LevelSet II
This boundary value is generally converted into a non-steady problem
With the initial condition and where
Remarks:– The normal to the interface can be computed thanks to
– We can also deduce
Resolution of the transport equation:– Standard Galerkin– SUPG– RFB or Macroscale
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3) Determination of the interfaces
3.2. LevelSet III
Standard Galerkin:
We consider the general convection diffusion problem
And its variational form: find such that
SUPG: find such that
We introduce the local Peclet number
To define
Error
48
3) Determination of the interfaces
3.2. LevelSet III
RFB: in this case, we redefine . Let us consider the bubble space such that
The variational form is
And we solve
where
find and
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3) Determination of the interfaces
3.2. LevelSet IV
Reinitialization : Hamilton-Jacobi equation ( )( )αατα
∇−=∂∂ 1S [ ]ετ ,0∈
Kh≈Δτ
reset being is whichinside zone
αε =
timeartificial :τ
( )222
KhS
αα
αα∇+
=
50
3) Determination of the interfaces
3.3. VOF versus LevelSet I
Presence function (fill factor). Diffusion limited by mesh adaptation.Solvers :
• flow solver : node centred• transport solver: element centred
Results are a courtesy of O. Basset
51
3) Determination of the interfaces
3.3. VOF versus LevelSet II
Results are a courtesy of O. Basset
Flow solver and transport solver are node centred.No diffusion, no mesh adaptation, and accurate surface description.But stabilization, and reinitialization
52
3) Determination of the interfaces
3.3. Examples of VOF in injection molding
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3) Determination of the interfaces
3.3. Examples of VOF in injection molding
Water assisted