1

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

45

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

46

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

47

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

49

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

53

3) Determination of the interfaces

3.3. Examples of VOF in injection molding

Water assisted