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Part IV: Monolithic Newton-Multigrid FEM techniques for nonlinear problems with special emphasis on
viscoelastic fluids
S. Turek and the FEAST GroupInstitut für Angewandte Mathematik, LS III, TU Dortmund
http://www.mathematik.tu-dortmund.de/lsiii
S. Turek | Monolithic Newton multigrid FEM
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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
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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,…
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Viscoelastic Fluids• Special effects due to normal stresses• Special effects due to elongational viscosity• The drag reduction phenomenon• …
S. Turek | Monolithic Newton multigrid FEM
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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
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• 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 ,
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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
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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
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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 )(
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• 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
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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
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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
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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
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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 !!
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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
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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
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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
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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 !
S. Turek | Monolithic Newton multigrid FEM
Q2 /Q2 / P1disc
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Variational Formulations
• Standard Navier-Stokes bilinear forms
• Non-symmetric bilinear forms due to LCR
dvDuDdvu
tvua s )(:)(21),(
dvpvpb ),(
duBuc c :),(2),(~
dvDvc LCRLCR )(:)exp(),(
S. Turek | Monolithic Newton multigrid FEM
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Variational Formulations
• Nonlinear tensor variational form due to LCR
• Non-symmetric bilinear forms due to LCR
• Source term
dxuFdx
dxut
d
LCRLCRLCR
LCRLCR
:,:)(
:)(1),(
4
),,,( pul LCR
dxuDdxuDuD
dxkdxut
e
LCRs )exp(:)()(:)(2
1),(
S. Turek | Monolithic Newton multigrid FEM
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Problem Formulation
• Set
• Find such that
Typical saddle point problem !
)(:),()()(: 20
142210 LQHLHX
EEE
DCCA
A
LCRfD
T
0~0
:~
QXpu ),~(
QXqvqvlqvpuK ),~(),~(),~(),,~(
0
~
TBBAK
),,(:~ LCRuu
S. Turek | Monolithic Newton multigrid FEM
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Compatibility Conditions
• Compatibility condition (for classical approach)
What about LCR ?
)(sup 2
0,01,1)(
210
Lqqu
dxqu
Hu
21
0,12,0)(
)(:
sup42
Huudxu
L
S. Turek | Monolithic Newton multigrid FEM
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Compatibility Conditions for LCR
• The non-symmetric bilinear forms due to LCR
such that is positive definite
210,1,02
,1,02
)(,
)exp(
)(:)exp(),(
HvTv
v
dvDvc
PD
210
42,1,02 )(,)(
:),(2),(~
HvLv
duBuc c
42 )( LPDT
S. Turek | Monolithic Newton multigrid FEM
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• Edge-oriented stabilization for
Equal order finite element interpolation for velocity and stress
Convective dominated problem
with , and),,(~ vv),,(~ uu
Efficient Newton-type and multigrid solvers can be easily applied !
[ ˜ u ][ ˜ v ] [ ˜ u i ][ ˜ v i ]i
EO-FEM
S. Turek | Monolithic Newton multigrid FEM
dsvuhhvuJEedge E
EuEpuu ]~][~[,max~,~ 2~~
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Higher Order Nonconforming FEM
• Larger FE space which allows high orderapproximation
• d.o.f.s belong to at most two elements which isgood for parallelisation
• Coupling of different polynomial orders
Mortar condition: test space ≈ order at slave side
No hanging nodes
S. Turek | Monolithic Newton multigrid FEM
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Nonlinear Solver• Damped Newton results in the solution of the form
• Inexact Newton: Jacobian is approximated using finite differences
S. Turek | Monolithic Newton multigrid FEM
),,,(,0)( puxxR LCR
)()(1
1 ll
lll xRxxRxx
2)()()( ijij
ij
l exRexRxxR
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Jacobian Matrix • The Jacobian matrix takes the form
• Generalized non-isothermal non-Newtonian problem
S. Turek | Monolithic Newton multigrid FEM
0
~)(B
BAxxRJ
Tn
A Au
˜ C T ˜ H T
C A 0H 0 A
Typical saddle point problem !
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Linear Solver
• Monolithic multigrid solver
Standard geometric multigrid approach
Full restrictions and prolongations
Local MPSC via Vanka-like smoother
Fully implicit Monolithic FEM-Multigrid Solver !
S. Turek | Monolithic Newton multigrid FEM
Q2 , P1disc
TTT
llp
llu
Tl
l
l
l
l
hpuRpuR
Jpu
pu
),~(),~(~~
11
1
Page 32Page 32S. Turek | Monolithic Newton multigrid FEM
Flow around Cylinder Benchmark
Local refinement via hanging nodes
Coarse grid and mesh information
Level NEL DOFR3a1 656 15823R3a2 944 22457R3a3 1520 35715R3a4 2672 62221R3a5 4976 115223
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Flow around Cylinder Benchmark• Planar flow around cylinder (Oldroyd-B)
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Flow around Cylinder Benchmark
Efficient continuation for increasing We numbers
We Drag NL We Drag NL We Drag NL0.1 130.366 8 0.8 117.347 4 1.5 125.665 40.3 123.194 4 1.0 118.574 6 1.7 129.494 40.5 118.828 4 1.2 120.919 5 1.9 133.754 40.6 117.779 4 1.3 122.350 4 2.0 136.039 50.7 117.321 4 1.4 123.936 4 2.1 138.438 5
We Drag Peak2 NL We Drag Peak2 NL5 96.943 924.45 14 60 85.859 12010.57 420 89.905 4204.51 12 70 85.365 13773.61 430 88.304 6318.79 5 80 84.937 15502.45 440 87.256 8311.32 5 90 84.585 17207.87 450 86.476 10199.1 4 100 84.287 18897.95 4
• Oldroyd-B
• Giesekus
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Flow around Cylinder Benchmark• Direct steady vs. non-steady approach for Giesekus
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Flow around Cylinder Benchmark• Axial stress w.r.t. X-curved: Oldroyd-B vs. Giesekus
Lack of pointwise mesh convergence due to model
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Level n.o.f. u n.o.f p n.o.f. T n.o.f S VP VPT VSP0 1144 390 572 1716 1534 2106 32501 4368 1560 2184 6552 5928 8112 124892 17056 6240 8528 25584 23296 31824 488803 67392 24960 33696 101088 92352 126048 1934494 267904 99840 133952 401856 367744 501696 7696005 1068288 399360 534144 1602432 1467648 2001792 3070080
• Coarse grid and mesh information
• n.o.f. for different problems
Level NEL NMT NMP1 520 572 10922 2080 2184 42643 8320 8528 168484 33280 33696 669765 133120 133952 267072
Next Problem
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Navier-Stokes Re=20
S. Turek | Monolithic Newton multigrid FEM
Level Drag Lift N/L Drag Lift N/L1 3.112646 2.965870e-2 1/6 5.540999 9.447473e-3 5/22 3.134342 3.005275e-2 1/7 5.566928 1.046885e-2 5/23 3.140327 3.015909e-2 1/7 5.576088 1.056787e-2 5/24 3.141893 3.018665e-2 1/7 5.578652 1.060398e-2 5/25 3.142292 3.019366e-2 1/7 5.579313 1.061503e-2 5/2
Newtonian Problem (VP)
Stokes
Level independent solver !
Page 39Page 39S. Turek | Monolithic Newton multigrid FEM
Power Law Problem (VP)
Level Drag Lift N/L
2 3.26420 -0.01339 4/23 3.27728 -0.01341 3/24 3.27956 -0.01338 2/25 3.28007 -0.01337 2/2
Level Drag Lift N/L
2 13.74280 0.35070 3/23 13.77355 0.34963 3/24 13.78220 0.35062 3/15 13.78445 0.35112 2/2
Level Drag Lift N/L
2 3.26433 -0.01342 4/23 3.27739 -0.01342 3/24 3.27968 -0.01339 2/25 3.28019 -0.01338 2/2
Level Drag Lift N/L
2 13.73800 0.35052 3/23 13.76875 0.34941 3/24 13.77740 0.35040 3/15 13.77970 0.35091 2/2
102 , r 1.5
104 , r 1.5
102 , r 3
104 , r 3
Level and parameter independent solver !
).1,0(,)(),,( 0
)12
(20
rp
r
s
Page 40Page 40S. Turek | Monolithic Newton multigrid FEM
Level Drag Lift N/L
2 6.31313 0.02478 3/13 6.32337 0.02504 3/24 6.32619 0.02509 3/25 6.32691 0.02510 2/2
Level Drag Lift N/L
2 534.29750 6.53247 3/23 535.48500 6.55813 3/34 535.77950 6.56464 3/35 535.84800 6.56621 2/2
Level Drag Lift N/L
2 33.22763 0.81901 2/23 33.29026 0.82140 2/24 33.30657 0.82201 2/25 33.31069 0.82217 2/2
Level Drag Lift N/L
2 15.16395 0.13886 4/33 15.18516 0.13963 4/34 15.19108 0.13978 4/35 15.19262 0.13982 3/3
r 1, r1 1, 0,0 102 r 0, r1 1, 0.1,0 101
r 1, r1 1, 0.1,0 101r 0 r1 1, 0.1,0 102
Cross Model Problem (VP)
103 , a1 a2 0,
Level and model independent solver !
)),(exp())(1()(),,(
3
21
01
aaapp rrs
Page 41Page 41S. Turek | Monolithic Newton multigrid FEM
Level Drag Lift N/L
2 74.29465 1.31636 2/23 74.43290 1.32009 2/24 74.46910 1.32105 2/25 74.47830 1.32129 2/2
Level Drag Lift N/L
2 5579.355 54.79350 3/23 5589.415 54.96815 3/34 5592.050 55.01415 3/35 5592.725 55.02585 3/2
Level Drag Lift N/L
2 53.78930 1.05488 2/23 53.88590 1.05770 3/24 53.91125 1.05844 2/25 53.91770 1.05863 2/2
Level Drag Lift N/L
2 6005.265 59.75125 3/23 6016.220 59.94455 3/24 6019.075 59.99535 3/25 6019.795 60.00820 3/2
r 0, r1 1, 0,0 102 , a2 0.
Cross Model problem (VTP)
103 , a1 0, a3 1, k1 k2 102 ,
r 0.1, r1 1, 0,0 101 , a2 0.
r 0.1, r1 1, 0,0 102 , a2 1. r 0.1, r1 1, 103 ,0 101 , a2 1.
Non heated cylinder
Level and model independent solver !
)),(exp())(1()(),,(
3
21
01
aaapp rrs
Page 42Page 42S. Turek | Monolithic Newton multigrid FEM
Cross Model Problem (VTP)
103 , r 0.1, r1 1, 103 , a1 0, a2 1, a3 1,
Level Drag Lift N/L
2 45.26969 0.90303 3/23 45.35251 0.90563 3/24 45.37431 0.90632 2/25 45.37988 0.90649 2/2
Level Drag Lift N/L
2 512.7765 5.37640 3/23 513.7120 5.39301 3/34 513.9585 5.39743 3/35 514.0215 5.39856 3/2
Level Drag Lift N/L
2 464.07865 4.90752 2/23 464.93045 4.92313 3/24 465.15470 4.92724 2/25 465.21195 4.92828 2/2
Level Drag Lift N/L
2 5528.860 53.11685 3/23 5539.025 53.28790 3/24 5541.690 53.33335 3/25 5542.365 53.34490 3/2
0 102 , k1 k2 102 . 0 101 , k1 k2 103.
0 101 , k1 k2 102 . 0 1.0, k1 k2 103.
Heated cylinder
Level and model independent solver !
)),(exp())(1()(),,(
3
21
01
aaapp rrs
Page 43Page 43S. Turek | Monolithic Newton multigrid FEM
Viscoelastic Fluids (VSP)
Level Drag Lift N/L
2 5.57150 0.01031 2/23 5.58032 0.01047 3/24 5.58285 0.01051 2/25 5.58351 0.01052 2/2
Level Drag Lift N/L
2 5.56474 0.01053 2/23 5.57511 0.01064 2/24 5.57936 0.01062 2/25 5.58131 0.01059 2/2
Level Drag Lift N/L
2 20.8412 0.32761 7/73 17.7123 0.22910 6/84 15.0096 0.14311 6/95 12.58895 0.07002 6/9
Level Drag Lift N/L
2 5.03961 -0.00172 4/23 4.93834 -0.00210 4/34 4.84483 -0.00252 3/35 4.77541 -0.00276 3/4
We 0.002 We 0.002
We 1.0 We 1.0
Level independent solver !
Oldroyd-B Giesekus
Oldroyd-B Giesekus
Page 44Page 44S. Turek | Monolithic Newton multigrid FEM
Viscoelastic Fluids (VSP)
Lower We vs. higher We !
We=0.1 We=1
Page 45Page 45S. Turek | Monolithic Newton multigrid FEM
Viscoelastic Fluids (VSP)
Lower We vs higher We !
We 0.1 0.2 0.3 0.4 0.5 0.6 … 1.0Oldroyd-B 5 [130.06] 4 [130.06] 4 [130.06] 5 [120.40] 5 [118.67] 5 [117.66] … 7 [119.33]
Giesekus 4 [129.37] 4 [124.41] 4 [119.86] 3 [116.31] 3 [113.67] 3 [111.71] … 3 [107.29]
Fene-P 4 [128.91] 4 [124.62] 4 [120.70] 3 [117.67] 3 [115.51] 4 [114.02] … 3 [111.84]
Fene-CR 4 [130.15] 4 [126.72] 4 [123.68] 3 [121.51] 3 [120.14] 3 [119.42] … 3 [120.13]
Pom-pom 3 [137.17] 4 [127.23] 4 [119.78] 4 [114.51] 4 [110.66] 5 [107.76] … 3 [101.00]
Page 46Page 46S. Turek | Monolithic Newton multigrid FEM
Outlook: Multiphase Flow• Distanced based FEM Level Set formulation
• Test cases
surfaceCSF nFut
,10,
Test case1. Viscoelastic 10 0.1 10 1 9.8 0.2452. Newtonian 10 0.1 10 1 9.8 0.2453. Viscoelastic 10 0.1 2 1 9.8 0.2454. Newtonian 10 0.1 2 1 9.8 0.245
1 2 1 2 g 10
100
0
Page 47Page 47S. Turek | Monolithic Newton multigrid FEM
Preliminary Results
Typical cusp formation
• Rising bubble surrounded by viscoelastic fluids
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Summary
Robust numerical and algorithmic tools are available using Monolithic Finite Element Method (M-FEM)
Classical and Log Conformation Reformulation (LCR)
Edge Oriented stabilization (EO-FEM)
Fast Newton-Multigrid Solver with local MPSC smoother
for the simulation of nonlinear flow with viscoelastic behaviour
Advantages No CFL-condition restriction due to the fully implicit coupling
Positivity preserving
Higher order and local adaptivity
S. Turek | Monolithic Newton multigrid FEM