Page 1 Page 1 Page 1 Monolithic Newton-Multigrid FEM techniques for nonlinear problems with special emphasis on viscoelastic fluids H. Damanik, J. Hron, A. Ouazzi, S. Turek Institut für Angewandte Mathematik, LS III, TU Dortmund http://www.mathematik.tu-dortmund.de/lsiii Modelling, Optimization and Simulation of Complex Fluid Flow June 20 – 22, 2012 TU Darmstadt Darmstadt, Germany S. Turek | Monolithic Newton multigrid FEM
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Page 1Page 1Page 1
Monolithic Newton-Multigrid FEM techniques for nonlinear problems with special
emphasis on viscoelastic fluids
H. Damanik, J. Hron, A. Ouazzi, S. Turek Institut für Angewandte Mathematik, LS III, TU Dortmund
http://www.mathematik.tu-dortmund.de/lsiii
Modelling, Optimization and Simulation of Complex Fluid Flow June 20 – 22, 2012
TU DarmstadtDarmstadt, Germany
S. Turek | Monolithic Newton multigrid FEM
Page 2Page 2
Multiscale CFD Problems
Turbulence flow inside a pipe. From ProPipe
S. Turek | Monolithic Newton multigrid FEM
• Characteristics: Complex temporal behaviour and spatially disordered Broad range of spatial/temporal scales
• Inertia turbulence Re>>1 Numerical instabilities and problems
Special turbulence models required Special stabilization techniques required
Page 3Page 3
Multiscale CFD Problems
From physics.ucsd.edu
S. Turek | Monolithic Newton multigrid FEM
• Elastic turbulence Re<<1, We>>1 (less inertia, more elasticity) Numerical instabilities and problems (HWNP)
Special flow models: Oldroyd-B, Giesekus, Maxwell,… Special stabilization: EEME, EEVS, DEVSS/DG, SD, SUPG,…
Page 4Page 4
Viscoelastic Fluids• Special effects due to normal stresses• Special effects due to elongational viscosity• The drag reduction phenomenon• …
S. Turek | Monolithic Newton multigrid FEM
Page 5Page 5S. Turek | Monolithic Newton multigrid FEM
Application: Twinscrew Extruder• Viscoelastic rheological models (additionally shear & temperature
dependent) with non-isothermal conditions (cooling from outside, heat production, melting, solidification)
• Multiphase flow behaviour due to partially filled and transported granular material
• Complex time dependent geometry and meshes
Page 6Page 6S. Turek | Monolithic Newton multigrid FEM
FeatFlow Simulations with FBM
Page 7Page 7
• Generalized Navier-Stokes equations
• Viscous stress
• Elastic stress
).)((tr ,),,( 2 2uDDpss
( t u )u p , u 0,
c p ( t u ) k1
2 k2D :D,
s p .
Governing Equations
S. Turek | Monolithic Newton multigrid FEM
).(2 )(),(),( p321 uDFDFLf ppppp
D(u) 12
u (u)T ,
Page 8Page 8
Constitutive Models (I)• Viscous stress
Power Law model
Generalized Cross model
S. Turek | Monolithic Newton multigrid FEM
).)((tr ,),,( 2 2uDDpss
).1,0(,)(),,( 0
)12
(20
rp
r
s
).0,1 ,0(
)),(exp())(1()(),,(
0
3
21
01
ra
aapp rrs
Page 9Page 9
Constitutive Models (II)
• Generalized upper convective constitutive model
S. Turek | Monolithic Newton multigrid FEM
),(2 )(),(),),(,( p321 uDFDFtrLf ppppppk
.: Tppp
pp uuu
t
Oldroyd-B/UCM
Giesekus
FENE-P/-CR
White & Metzner
PTT
Pom-Pom ),,( 23 pGF
)( DD pp )),(,(1 pp trf
1 0 02 p1 0
))(,(1 pk trLf 0 01 0 0
0)),((1 ptrf ),,(2 pGF
1f 2F 3F
)(),( pp
Page 10Page 10
Constitutive Models (III)
• Exemplary model: White-Metzner
Larson:
Cross:
Carreau-Yasuda:
S. Turek | Monolithic Newton multigrid FEM
),()(2 )( p uDpp )(:)(2 uDuD
a1
)(
a1 )( p
p
n
1)L(1 )(
mp
p 1)k(1 )(
bn
b1
)L(1 )(
am
app
1
)k(1 )(
Page 11Page 11
• Discretizations have to handle the following challenges points Stable FEM spaces for velocity/pressure and velocity/stress
interpolation or or the new Special treatment of the convective terms: edge-oriented/interior
penalty (EO-FEM), TVD/FCT High Weissenberg number problem (HWNP): LCR
• Solvers have to deal with different sources of nonlinearity Nonlinearity: Newton method Strong coupling of equations: monolithic multigrid approach
• Complex geometries (and meshes) FBM + distance based Level Set FEM for free interfaces
S. Turek | Monolithic Newton multigrid FEM
Numerical Challenges
Q2 /Q2 / P1disc ˜ Q 1 / ˜ Q 1 / P0
˜ Q 2 / ˜ Q 2 / P1disc
Page 12Page 12
Problem Reformulation (I)Elastic stress
(1)
Replace in (1) with )( Icp
p
(2)
Conformation stress is positive definite by design !!
special discretization: TVD p
(u , p, c )
(u , p, p )
S. Turek | Monolithic Newton multigrid FEM
)(2 )(),(),(
0 ,2
p321 uDFDFLf
uDpuut
ppppp
ps
0,
0,u ,12u
4
uF
Dput
cc
cps
Page 13Page 13
Conformation Tensor Properties
2 Observations:- positive definite special discretizations like FCT/TVD- exponential behaviour approximation by polynomials???
c(t) 1We2
t exp (t s)We
F(s,t) F(s,t)T ds
Positive by design, so we can take its logarithm
S. Turek | Monolithic Newton multigrid FEM
Page 14Page 14
Exponential Behaviour
Old Formulation Vs Lcr
-5495995
1495199524952995
0,00 0,20 0,40 0,60 0,80 1,00
x
stre
ss_1
1
We= 0.5 We= 1.5
Driven Cavity:as We number changes fromWe=0.5 to We=1.5, the stress value jumps significantly
Cutline of Stress_11 component at y = 1.0
S. Turek | Monolithic Newton multigrid FEM
Page 15Page 15
Problem Reformulation (II)
S. Turek | Monolithic Newton multigrid FEM
1 cNBu
cLCR log
TLCR RR
c)log(
• Experience: Stresses grow exponentially Conformation tensor is positive by design
• Fattal and Kupferman: Take the logarithm as a new variable using the
eigenvalue decomposition
Decompose the velocity gradient inside the stretching part
Remark for PTT only DuLNBL c ,1
LCR can be applied to all upper convective models !!
Page 16Page 16
LCR Reformulation
TLCR RR
c)log(
1 cNBu
ccccc But
I1 2
LCRc exp
S. Turek | Monolithic Newton multigrid FEM
)(2 )(),(),( p321 uDFDFLf ppppp
0,4
uF cc )( Icp
p
., 2 4 uFBut LCRLCRLCRLCR
Page 17Page 17
Full Set of Equations (LCR)• Generalized Newtonian (VP)
• Non-isothermal effect (T)
• LCR equation (S)
., 2 4 uFBut LCRLCRLCRLCR
S. Turek | Monolithic Newton multigrid FEM
c p ( t u ) k1
2 k2D :D,
0, ,1)(),,(2
ueuDppuut
LCRps
uF LCR ,4 Refers to all upper convectiveconstitutive models
Page 18Page 18
Examplary Models (LCR)
S. Turek | Monolithic Newton multigrid FEM
)(14 IeF LCR
))((1 24 IeeIeF LCRLCRLCR
))((1))((144 IRfeFIeRfF LCRLCR
)(14 IeF LCR
))))(3((exp(1
))))(3((1(1
4
4
IeetrF
IeetrF
LCRLCR
LCRLCR
))1(]2)(([1 24 IeefF LCRLCR
LCR
Oldroyd-B/UCM
Giesekus
FENE-P/-CR
White-Metzner
Linear PTT
Exponential PTT
Pom-Pom
Page 19Page 19
FEM Discretization• High order for velocity-stress-pressure
Advantages: Inf-sup stable for velocity and pressure
High order: good for accuracy
Discontinuous pressure: good for solver & physics
Disadvantages: Stabilization for same spaces for stress-velocity
a single d.o.f. belongs to four elements (in 2D)
Compatibility condition between the stress and velocity spaces via EO-FEM !