part 3 - tu/epeters/inj_mould_lectures/im2_tcoupez.pdf · 2007. 4. 27. · – convective...
TRANSCRIPT
![Page 1: PART 3 - TU/epeters/Inj_Mould_Lectures/IM2_TCoupez.pdf · 2007. 4. 27. · – Convective reinitialisation – Sinus level set method ... • SUPG class of method (streamline upwind](https://reader033.vdocuments.mx/reader033/viewer/2022060917/60a9fce802aec852050d6ac6/html5/thumbnails/1.jpg)
1
PART 3 PART 3 DETERMINATION OF THE INTERFACESDETERMINATION OF THE INTERFACES
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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
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Moving free surface and interface flows
• Polymer injection moulding (Rem3D)
• Metal casting• Filling process• Mixing• Foaming
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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.
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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
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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
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Multiphase flow by heterogeneous Navier Stokes Modeling and local extended level set method
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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
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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
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Simulation of the polymer injection moulding process : the mould filling stage (VOF Finite Element method)
Rem3D
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• 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
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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
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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
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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,
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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
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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, …)
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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
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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
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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
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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
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VOF/P0 versus Level Set/P1
VOF with mesh adaptation Level set
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Falling drop
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Exemple
• Fluid ¼ poured in 30 seconds
• 11 bubbles go up into the fluid
• Bubble period of about 2.7 seconds
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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 :
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Distance preserving convective equation in a flow
⎪⎩
⎪⎨⎧
==
=∇++∂∂
)(),0(
).(
0 xxt
sUvt
αα
λαλα
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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
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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
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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
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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
)(
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Falling solid through two fluids
⎩⎨⎧
−+−+=−+−+=
))(1()()))(1()(())(1()()))(1()((
321
321
BeBeAeAe
BeBeAeAe
HHHHHHHH
αηααηαηηαρααραρρ
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RRR benchmark example
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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
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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
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water assisted injection molding simulation
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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
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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
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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
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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
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RemeshingResults on a 2D WAIM example
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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
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Rem3D Recherche 2.0Water assisted injection : Handle example
Remeshing
GLview 3D Plug-in GLview 3D Plug-in
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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
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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∇+
=
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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
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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
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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