Generalized quasi-Newtonian approach for modeling and ...€¦ · Page 1 Generalized...

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Generalized quasi-Newtonian approach for modeling and simulating complex flows

A. Ouazzi, A. Afaq, N. Begum, H. Damanik, A. Fatima, S. Turek Institute of Applied Mathematics, LS III,

TU Dortmund University, Dortmund, Germany

A. Ouazzi | Generalized quasi-Newtonian approach for complex flows

4th Indo-German Workshop on Advances in Materials, Reactions and Separation Processes

February 23-26, 2020 Harnack-Haus,

Conference Venue of the Max Planck Society, Berlin, Germany . Max Planck Institute for Dynamics of Complex Technical System Magdeburg

Page 2 Page 2

•  The powder can transit from the quasi-static to the intermediate regime as the shearing rate is increased

Behavior of dense granular material

Shear and pressure dependent viscosity

Axial flow device

Normal Sterss Sensor

Rotating Cylinder

Stationary outer wall

Axial flow experiment in the Couette device: spherical glass beads, 0.1 mm in diameter. (by Gabriel Tardos Group, CCNY)


Page 3 Page 3

Viscoplastic flow

Viscoplastic Lubricate Flow (Yield stress fluids)

Ø  Dependent on the stress field Ø  Constitutive model is dependent on different flow regimes Ø  Non-smooth change in the constitutive relations

Model preserving the sharp changes of the constitutive equations w.r.t. flow regimes


Page 4 Page 4

Thixotropy

Thixotropy concept Ø  Based on viscosity Ø  Flow induced by time-dependent decrease of viscosity Ø  The phenomena is reversible

•  Aging / Build-up

Ø  At rest or under slow flow: fluid ages Increases of the viscosity in time

•  Rejuvenation / Breakdown Ø  “Faster” flow: fluid rejuvenates Decreases of viscosity with acceleration of the flow

Investigation of solid/liquid and liquid/solid transitions with non constant yield stress


Page 5 Page 5

Non-Newtonian phenomena •  Effects due to normal stresses •  Effects due to elongational viscosity •  The drag reduction phenomenon

Differential models


Page 6 Page 6

Realization in FeatFlow

Non-Newtonian flow module: •  generalized Newtonian model

(Power-law, Carreau,...) •  viscoelastic differential model

(Giesekus, FENE, Oldroyd,...)

Multiphase flow module (resolved interfaces): •  l/l – interface capturing (Level Set) •  s/l – interface tracking (FBM) •  s/l/l – combination of l/l and s/l

Numerical features: •  Higher order FEM in space & (semi-) Implicit FD/FEM in time •  Semi-(un)structured meshes with dynamic adaptive grid deformation •  Fictitious Boundary (FBM) methods •  Newton-Multigrid-type solvers

Engineering aspects: •  Geometrical design •  Modulation strategy •  Optimization

Here: FEM-based tools for the accurate simulation of (multiphase) flow problems, particularly with complex rheology

HPC features: •  Moderately parallel •  GPU computing •  Open source Hardware

-oriented Numerics


Page 7 Page 7

Governing equations •  Generalized Navier-Stokes equations

Ø  Viscous stress

Ø  Elastic stress


⇢

✓@

@t+ u ·r

◆u�r · � +rp = ⇢f,

r · u = 0,

� = �s + �p,

�p +We�a�p

�t= 2⌘pD(u).

�s = 2⌘s(DII, p)D(u), DII = tr⇣D(u)2

⌘.

Page 8 Page 8

Quasi-Newtonian models •  Viscous stress

Ø  Power law model Ø  Powder flow in the quasi-static and intermediate regimes


⌘s(z) = ⌘0zr� 1

2 (⌘0 > 0, r > 1)

�s = 2⌘s(DII, p)D(u), DII = tr⇣D(u)2

⌘

8<

:⌘s(z, p) =

p2p

⇣sin�z�

12+ cos�zr�

12

⌘if z 6= 0, r > 1

||�s|| p2p sin� else

(� : the angle of internal friction)

Page 9 Page 9

Quasi-Newtonian models Ø  Yield stress flow (Bingham Model)

Ø  Thixotropic model

Ø Structure parameter equation


@�

@t+ u ·r� = a(1� �)� b�z

12

(a, b are structure parameters)

(⌘s(z,�) = ⌘0 + ⌧0z

� 12 if z 6= 0

||�s|| ⌧0 else

(⌧0 : yield stress)

(⌘s(z,�) = ⌘(�) + ⌧(�)z�

12 if z 6= 0

||�s|| ⌧(�) else

(� : structure parameter)

Page 10 Page 10

Constitutive models •  Elastic stress

Ø  Upper/Lower convective derivative

ga(�,⇥u) =1� a

2��⇥u + (⇥u)T�

⇥

� 1 + a

2�⇥u � + � (⇥u)T

⇥(a = ±1)

�a⇥

�t=

�⇤

⇤t+ u ·�

⇥⇥ + ga(⇥,⇥u)


�p +We�a�p

�t= 2⌘pD(u).

Page 11 Page 11

Constitutive models •  Generalized differential constitutive model

Ø  Oldroyd

Ø  Giesekus

Ø  Phan-Thien and Tanner

Ø  White and Metzner

⇤ + We�a⇤

�t+ G(⇤, D) + H(⇤) = 2⇥pD(u)

G = 0, H = 0

G = 0, H = � tr(⇥2)

H = [exp (� tr(⇥))� 1]⇥

G = � (2 D : D)1/2 , H = 0


Page 12 Page 12

Stokes problem

•  Two-field formulation

•  Three-field formulation

(u, p)

(�, u, p)

8>>>>><

>>>>>:

� � 2⌘D(u) = 0 in ⌦

�r ·✓2⌘(1� ↵)D(u) + ↵�

◆+rp = 0 in ⌦

r · u = 0 in ⌦

u = gD on �D

8>>><

>>>:

�r ·✓2⌘D(u)

◆+rp = 0 in ⌦

r · u = 0 in ⌦

u = gD on �D


Page 13 Page 13

Stokes problem •  Two-field formulation

Ø  Set

Ø  Find s.t.

Ø  Compatibily constraints

(u, p)

(u, p) 2 V⇥QDK(u, p), (v, q)

E=

DL, (v, q)

E, 8(v, q) 2 V⇥Q

supv2V

⌦Bv, q

↵

||v||V� � ||q||Q/KerBT , 8q 2 Q

V :=⇥H1

0 (⌦)⇤2

,Q := L20(⌦)


K =

Au BT

B 0

!

Page 14 Page 14

Stokes problem •  Three-field formulation

Ø  Set

Ø  Find s.t.

Ø  Compatibily constraints

(�, u, p)

(�, u, p) 2 T⇥ V⇥QDK(�, u, p), (⌧, v, q)

E=

DL, (⌧, v, q)

E, 8(⌧, v, q) 2 T⇥ V⇥Q

T :=�L2(⌦)

�4sym

,V :=⇥H1

0 (⌦)⇤2

,Q := L20(⌦)


supv2V

⌦Bv, q

↵

||v||V� � ||q||Q/KerBT , 8q 2 Q

supv2V

⌦Cv, ⌧

↵

||v||V� � ||⌧ ||T/KerCT , 8⌧ 2 T

K =

0

B@

A� C 0

CT Au BT

0 B 0

1

CA

Page 15 Page 15

Approximated Stokes problem

•  Conforming approximations

•  Non-conforming approximation

•  Discrete inf-sup condition


supvh2Vh

⌦Bhvh, qh

↵

||vh||V� �h ||qh||Q/KerBT

h

, 8qh 2 Qh

supvh2Vh

⌦Chvh, ⌧h

↵

||vh||V� �h ||⌧h||T/KerCT

h

, 8⌧h 2 Th

Th ⇢ T, Vh 6⇢ V&Vh ⇢ V, Qh ⇢ QA�h = A�, Auh 6= Au, Bh 6= B, Ch 6= C

Th ⇢ T, Vh ⇢ V = V, Qh ⇢ QA�h = A�, Auh = Au, Bh = B, Ch = C

Page 16 Page 16

Robust non-/conforming FEM Qr/P

discr�1 , r � 2

Qr/Pdiscr�1 , r � 2 (u, p)

(�, u, p)Qr/Qr/Pdiscr�1 , r � 2

Ju(uh, vh) = �uX

e2Eh

2⌘↵h

Z

e[ruh] : [rvh] d⌦

The family of non-/conforming FEM and the family of nonconforming FEM for

Ø  Inf-sup stable Ø  Arbitrary order with optimal convergence order Ø  Discontinuous pressure

Ø Good for the solver Ø Element-wise mass conservation

The family of conforming FEM for with stabilization

Ø  Both Inf-sup conditions are satisfied Ø  Highly consistent and symmetric stabilization, penelazing any

spurious current, enhancing the preconditioner, which improve accuracy and efficiency

Ø  None tensorial FEM approximation for the tensorial field Ø Robust solver w.r.t. the monolitic approach Ø Efficient solver w.r.t. multigird solver


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Monolitic-multigrid linear solver

•  Standard geometric multigrid solver

•  Full and restriction and prolongation

•  Local Multilevel Pressure Schur Complement via Vanka-like smoother

Coupled Monolithic Multigrid Solver !

0

B@

�l+1

ul+1

pl+1

1

CA =

0

B@

�l

ul

pl

1

CA+ !lX

T2Th

0

BB@⇣Kh + Ju

⌘

|T

1

CCA

�1 0

B@R�l

Rul

Rpl

1

CA

|T

Qr P discr�1


Page 18 Page 18

Flow around cylinder Benchmark tests (by Aaqib Afaq)

Ø  Two field formulation versus three field formulation

Ø  Consistency of the stabilization for three field formulation

Monolithic Multigrid Solver

Three fields Stokes vs Stokes solver in primitive variables

Level Lift Drag NL/LL Lift1 Drag NL/LL

1 0.009498 5.5550 7/2 0.009498 5.5550 9/22 0.010601 5.5722 7/2 0.010601 5.5722 9/23 0.010616 5.5776 7/2 0.010616 5.5776 9/14 0.010618 5.5791 7/1 0.010618 5.5791 8/15 0.010619 5.5794 6/2

Three field solver performance e�cient as primitive Stokes solver

Check robustness and consistency!

1Damanik. H ”FEM Simulation of Non-isothermal Viscoelastic fluids”, PhD ThesisMotivation Governing Equations Variational Formulation Finite Element Approximation Numerical Results Summary

EO-FEM : Consistency

Consistency for case – = 0

No stabilization With stabilizationLevel – Lift Drag NL/LL Lift Drag NL/LL

2 0 0.010601 5.5722 7/2 0.010702 5.5674 7/23 0 0.010616 5.5776 7/2 0.010619 5.5757 7/24 0 0.010618 5.5791 7/1 0.010617 5.5782 7/25 0 0.010619 5.5794 6/2 0.010618 5.5790 6/3

Edge Oriented FEM is consistent

Side e�ect neither on solution nor on the solver

Motivation Summary


Monolithic Multigrid Solver

Three fields Stokes vs Stokes solver in primitive variables

Three-field Two-field1

Level Lift Drag NL/LL Lift Drag NL/LL

1 0.009498 5.5550 7/2 0.009498 5.5550 9/22 0.010601 5.5722 7/2 0.010601 5.5722 9/23 0.010616 5.5776 7/2 0.010616 5.5776 9/14 0.010618 5.5791 7/1 0.010618 5.5791 8/1

Three field solver performance e�cient as primitive Stokes solver

Check robustness and consistency!

1Damanik. H ”FEM Simulation of Non-isothermal Viscoelastic fluids”, PhD ThesisMotivation Summary


Page 19 Page 19

Flow around cylinder Benchmark tests (by Aaqib Afaq)

Ø  Robustness and efficiency of the stabilization for three field formulation

without any viscous contribution

Accurate, robust and efficient monolitic-multigrid Stokes solver in two-field and three-field formulations !

EO-FEM :Robustness

Two extreme cases–=0 æ viscous contribution

–=1 æ no viscous contribution

No stabilization With stabilizationLevel – Lift Drag NL/LL Lift Drag NL/LL

2 0 0.010601 5.5722 7/2 0.010702 5.5674 7/23 0 0.010616 5.5776 7/2 0.010619 5.5757 7/24 0 0.010618 5.5791 7/1 0.010617 5.5782 7/25 0 0.010619 5.5794 6/2 0.010618 5.5790 6/32 1 ———– —— — 0.010588 5.5520 7/23 1 ———– —— — 0.010600 5.5698 7/24 1 ———– —— — 0.010612 5.5756 7/25 1 ———– —— — 0.010617 5.5778 7/3

Robustness w.r.t. problem!

Consistent and grid independent solverMotivation Summary



Page 20 Page 20

Multiphase flow problem

Incompressible N-S Equation)

Ø  Viscous stress

Ø  Interface boundary conditions


⇢(�)

✓@

@u+ u ·r

◆u�r · �s +rp = 0

�s = 2⌘s(�)D(u)

[u]|� = 0

�[pI + �s]|� · n = �n

Page 21 Page 21

Multiphase flow problem •  New extra stress for multiphase flow

•  Full set of equations for multiphase flow


8>>>>>>>>>>>>>><

>>>>>>>>>>>>>>:

⇢( )

✓@u

@t+ u ·ru

◆�r · ⌧ +rp = 0

r · u = 0

⌧ � 2⌘s( )D(u) + �

✓r ⌦r

|r |

◆= 0

@

@t+ u ·r +r ·

✓�nc (1� )

r |r |

◆

�r ·✓�nd

✓r · r

|r |

◆r |r |

◆= 0

�m = ��✓r ⌦r

|r |

◆

Page 22 Page 22

Monolitic approach

Oscillating bubble (by Aaqib Afaq)

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

100 101 102 103 104

Time

rxrySurface Area


Page 23 Page 23

Generalized Newton’s method Robust nonlinear solver based on Newton’s method following a specefic path of convergence using the residual’s convergence

Ø  Robust w.r.t. starting guesses

Ø  Dealing with Jacobian’s singularities using generalized deriviatives or approximated one

Ø  Full benefit from the quadratic convergence’s region of classical Newton’s method


Page 24 Page 24

Newton’s method Let , , or and be the continuous or the discrete corresponding system’s residum.

Ø  Update of the nonlinear iteration with the correction i.e.

Ø  The linearization of the residual provides

Ø  The Newton’s method assuming invertible Jacobian

(�, u, p,')U = (u, p) (�, u, p) RU (U)

�UUN = U + �U

UN = U � J�1⇣U⌘· RU

⇣U⌘


RU

⇣UN

⌘=RU

⇣U + �U

⌘

=RU

⇣U⌘+ J

⇣U⌘· �U

Page 25 Page 25

Generalized Newton’s method Jacobian calculations

Ø  Exact G-Newton based on a priori study of Jacobian’s properties and decompositions

Ø  Inexact G-Newton based on the residum‘s convergence

J⇣U⌘=

@RU

�U�

@U

!


J⇣U⌘=

@RU

�U�

@U

!+ �

@RU

�U�

@U

!

@ ¯RU

�U�

@U

!

ij

⇡

¯RU

�U + ✏+ej

�� ¯RU

�U � ✏�ei

�

( ✏+ + ✏�)

!,

¯R = R, ˆR, or

˜R .

Page 26 Page 26

Three-field viscoplastic application •  Viscoplatic constitutive law

Ø  Bingham constitutive law

Ø  New extra stress for viscoplastic flow s.t. •  Three-field viscoplastic set of equations


8>>><

>>>:

||D(u)||�Y0�D(u) = 0 in ⌦

�r ·✓2⌘D(u) + ⌧0�Y0

◆+rp = 0 in ⌦

r · u = 0 in ⌦

||D(u)||�Y0= D(u)

8<

:� = 2⌘D(u) + ⌧0

D(u)

||D(u)|| if ||D(u)|| 6= 0

||�|| ⌧0 if ||D(u)|| = 0

�Y0

Page 27 Page 27

Generalized Newton’s method

Dynamic path versus static one w.r.t. number of iterations, and the corresponding convergence of the residium (by Arooj Fatima)

10-5

10-4

10-3

10-2

10-1

100

101

0 2 4 6 8 10

spdp

10-8

10-6

10-4

10-2

100

102

104

106

108

1 10 100

sp1sp2sp3sp4sp5dp


Page 28 Page 28

Lid driven cavity benchmark

Unyielded zone for two different yield stresses, , and (by Arooj Fatima)

⌧0 = 2 ⌧0 = 5


Page 29 Page 29

Constitutive models •  Generalized differential constitutive model

Ø  Oldroyd

Ø  Giesekus

Ø  Phan-Thien and Tanner

Ø  White and Metzner

⇤ + We�a⇤

�t+ G(⇤, D) + H(⇤) = 2⇥pD(u)

G = 0, H = 0

G = 0, H = � tr(⇥2)

H = [exp (� tr(⇥))� 1]⇥

G = � (2 D : D)1/2 , H = 0


Page 30 Page 30

Viscoelastic benchmark •  Planar flow around cylinder Oldroyd-B (by Hogenrich Damanik)


Page 31 Page 31

The numerical method do not introduce errors !

•  Experimental and numerical results for dry, frictional powder flows in the quasi-static and intermediate regimes

Quasi-Newtonian model for powder flow


Page 32 Page 32

Quasi-Newtonian thixotropic model Ø  Viscosity model for thixotropic flow i.e. extended viscosity defined

on all domaine s.t.

Ø  Structure equation

Ø  Full set of equations


@�

@t+ u ·r� = a(1� �)� b�z

12

(a, b are structure parameters)

8>>>><

>>>>:

✓@

@t+ u ·r

◆u�r ·

⇣2⌘s(||D(u)||,�)D(u)

⌘+rp = 0 in ⌦

r · u = 0 in ⌦

@�

@t+ u ·r�� a(1� �) + b�||D(u)|| = 0 in ⌦

(⌘s(||D(u)||,�) = ⌘(�) + ⌧(�)||D(u)||�

12 if ||D(u)|| 6= 0

||�s|| ⌧(�) else

(� : structure parameter)

Page 33 Page 33

Thixotropic flow Shear rate in a couette w.r.t. breakdown parameter (by Naheed Begum)


Page 34 Page 34

Thixotropic flow


Structure parameter in a couette w.r.t. breakdown (by Naheed Begum)

Page 35 Page 35

ü  shear history effect

ü  time history effect

ü Hysteresis

ü  stress overhoots

Thixotropy flow

A quasi-Newtonian model for thixotropic phenomena via a time and shear dependent viscosity


Page 36 Page 36

Generalized quasi-Newtonian approach

Ø  Include the non-Newtonian stress or any extra stress in diffusion operator

Ø  Get rid of a tensorial field

Ø  Less constraints for the choices of FE approximation

Ø Robust and efficient numerical algorithms

Ø Simple evolution equations !

Ø  Proof of the concept and validation


Page 37 Page 37

Laplacian operator •  Divergence form

•  Weak form

Benefit of the weak form representation !

Lu =nX

i,j=1

@

@xi

✓aij u

@

@xi

◆

LW u =NX

i,j=1

Aij :⇣r · ej ⌦r · ei

⌘u


Page 38 Page 38

Stokes problem Weak form representation 2D

•  Gradient formulation

•  Deformation formulation

Different deriviatives combinations accessibilities !

A11 =

"1 0

0 0

#, A12 =

"0 0

0 0

#

A21 =

"0 0

0 0

#, A22 =

"0 0

0 1

#

A11 =

2

41 0

01

2

3

5 , A12 =

2

4 01

20 0

3

5

A21 =

2

40 0

1

20

3

5 , A22 =

2

41

20

0 1

3

5

LW u =2X

k,l=1

NX

i,j=1

[Akl]ij :⇣r · ej ⌦r · ei

⌘ulj , k = 1, 2


Page 39 Page 39

Generalized quasi-Newtonian approach Weak form representation 2D

•  Generalized formulation I

More deriviatives combinations accessibilities are allowed !

A11 =

2

64a11

1

2a21

1

2a12

1

4(a11 + a22)

3

75 , A12 =

2

64

1

2a12

1

4(a11 + a22)

01

2a12

3

75

A21 =

2

64

1

2a21 0

1

4(a11 + a22)

1

2a21

3

75 , A22 =

2

64

1

4(a11 + a22)

1

2a21

1

2a12 a22

3

75

LW u =2X

k,l=1

NX

i,j=1

[Akl]ij :⇣r · ej ⌦r · ei

⌘ulj , k = 1, 2


Page 40 Page 40

Viscoelastic benchmark •  Planar flow around cylinder Oldroyd-B (by Hogenrich Damanik)

115

120

125

130

135

140

0 0.5 1 1.5 2

Dra

g co

effic

ient

We number

Drag coefficient planar flow around cylinder

H. Damanik, PhD thesis TU Dortmund Generalized quasi-Newtonian approach

Genalized quasi-Newtonian approach for non-Newtonian problem i.e. Oldroyd-B !


Page 41 Page 41

Summary

New generalized quasi-Newtonian approach for modeling and simulating complex flows is introduced and validated. Based on new numerical and algorithmic tools using

ü  Monolithic FEM two-field and three-field Stokes solver

ü  Generalized Newton’s method w.r.t. singularities with global convergent property

ü  Edge Oriented stabilization (EO-FEM)

ü  Fast Multigrid Solver with local MPSC smoother Extensively tested from numerical and physical perspectives via the simulations of different flow problems in different formulations to motivate the newly introduced generalized quasi- Newtonian approach.


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