16/05/2006 cardiff school of mathematics 1 modelling flows of viscoelastic fluids with spectral...

16/05/200616/05/2006 Cardiff School of MathematicsCardiff School of Mathematics 11

Modelling Flows of Viscoelastic Modelling Flows of Viscoelastic Fluids with Spectral Elements:Fluids with Spectral Elements:

a first approach a first approach

Giancarlo Russo, Giancarlo Russo, supervised bysupervised by

Prof. Tim PhillipsProf. Tim Phillips

2216/05/200616/05/2006 Cardiff School of MathematicsCardiff School of Mathematics

The Project - - The development of models to describe and solve the flows of viscoelastic fluids with free surfaces; - a parallel theoretical and numerical (spectral elements) analysis of the problems; - the die-swell problem; - the filament stretching problem.

Work Plan – First Year - - Newtonian fluids; - Analysis of the general formulation of a free surface Stokes problem; - Weak two and three fields formulations: compatible approximation spaces, compatibility conditions and error estimates; - Development of a code to solve 1-D and 2-D steady and unsteady Stokes

flow.


OutlineOutline

The problemsThe problemsTheoretical analysis of the weak formulationTheoretical analysis of the weak formulationResults for the 2-fields problemResults for the 2-fields problemResults for the 3-fields problemResults for the 3-fields problemA stability estimate for the stress tensorA stability estimate for the stress tensorA model to investigate the Extrudate Swell problemA model to investigate the Extrudate Swell problemThe 1-D and 2-D discretization processes : some The 1-D and 2-D discretization processes : some results from the codesresults from the codes


Formulations of the problemFormulations of the problem

2pI % %%

2

0

u p F

u

rr

r

0

2

p T F

u

T

r%

r

% %

•Definition of the stress tensor for a Newtonian fluidDefinition of the stress tensor for a Newtonian fluid

•Two fields formulations Two fields formulations

•Three fields formulations Three fields formulations

•Two free surface boundary conditions Two free surface boundary conditions

2 1

2 11 2

0,

1 1 1.

u n u n

n nR R

r r r r

r r% %

1 2

1 2

,

,

u

p

F

n n

R R

%r

r

r r

%

- Stress tensor

- Velocity

- Pressure

- External Loads

- Normal Vectors

- Curvature Radii

- Rate of Strain

- Kinematic Viscosity

Variables and Parameter

(1)

(2)

(3)

(4)


The Weak FormulationThe Weak Formulation

( , ) ( , ) ( ),

( , ) 0,

( , ) 2 ( , )

b p w d T w l w

b q u

c T t d T u

r r r%r

r% %%

2 1 20

2 4 2 4

2 4 1 2

1 2

: ( ) [ ( )] , ( , ) ( ) ,

:[ ( )] [ ( )] , ( , ) : ,

:[ ( )] [ ( )] , ( , ) : ,

:[ ( )] , ( ) .

b L H b r v v rd

c L L c S s S sd

d L H d S u S ud

l H l u F u

r r

% %% %r r% %rr r

2 1 2 2 40( , , ) ( ) [ ( )] [ ( )]bdryp u T L H L

r %We look for 2 1 2 2 40( , , ) ( ) [ ( )] [ ( )]q w t L H L

r %such that for all

the following equations are satisfied :

where b, c, d, and l are defined as follows :

Remark: 1 2[ ( )]bdryH it simply means the velocity fields has to be chosen according to the

boundary conditions, which in the free surface case are the ones given in (4) .

(5)

(6)


The spurious modes for the pressure: the The spurious modes for the pressure: the compatibility conditioncompatibility condition

1 2 1 20 0:[ ( )] [ ( )] , ( , ) :a H H a u w u wd

r r r r

: ,u wd F w

rr r r 1 20[ ( )]w H

r

20 ( ) : 0mS q L u qd

r

201 2

1 2( )

[ ( )][ ( )]

sup .L

u HH

u qdq

u

r

r

r

The bilinear form in the 2-fields formulation :

The weak problem for the velocity :

Coupling with the pressure : the spurious modes space

The compatibility (LBB) condition (see [1] and [2] ) :

(7)

(8)

(9)

(10)


Some results for the 2-fields formulationSome results for the 2-fields formulation

2( )N NP P 2 2N NP P

1 2 1 2

1 2 20

21 2

2 12

[ ( )] [ ( )]

1

[ ( )] [ ( )]

1

( ) [ ( )]

2

( ) ( )[ ( )]

m

mm

N H H

mN H H

N NL H

mN L HH

u c F

u u cN u

p c F

p p cN u p

rr

r r r

r

r

• A compatibility result (see A compatibility result (see [3]) ) ::

The spaces and as approximating spaces for the

the velocity and pressure fields respectively, are compatible in the sense of

the LBB condition.

• Some approximation results (see Some approximation results (see [4])) : :(11)

(12)

(13)

(14)

REMARK : in (14) PN has been used as approximating space for the pressure. It’s non-optimal.


More results for the 3-fields formulationMore results for the 3-fields formulation

2( )N NP P

1 202 4

2 4[ ( )]

[ ( )] [ ( )]

( , )sup .

HL L

d ww

%

r% r

%

4( )N NP P 2 2N NP P • Compatible spaces for the 3-fields formulation (see Compatible spaces for the 3-fields formulation (see [5]) ) :: The space together with the spaces

are compatible spaces for the approximation of the stress, velocity and pressure

fields respectively according to the LBB and the previous condition.

and

• A velocity – stress compatibility condition (see A velocity – stress compatibility condition (see [5]) ) ::

• Some approximation results (see Some approximation results (see [5]) ) ::

2 4 1 4 20

1 41 2 20 0

2 1 4 120

1

[ ( )] [ ( )] [ ( )]

1

[ ( )][ ( )] [ ( )]

3

2( ) [ ( )] ( )[ ( )]

[ ( )

[ ( )

[ ( )

m m

m m

m mm

mN L H H

mN HH H

m

N L H HH

c N u

u u c N u

p p c N u p

r% % %r r r

%

r%

(15)

(16)

(17)

(18)

REMARK : in (18) PN-2 has been used as approximating space for the pressure. It’s non-optimal,But slightly better (a factor ½) than (14).


A stability estimate for the stress tensorA stability estimate for the stress tensor

2 42

1 2

41 [ ( )]

( )[ ( )]

:sup , ( )

N N N

N N

N N N NLv P P N H

vP P

v

r

r%

% %r

The compatibility condition (15) has been derived with an abstract approach, namely, using the Closed Range Theorem . In a similar fashion a stability estimate for the stress tensor is here derived. According to the closed range theorem (see [6] for details) we can swap velocity and stress in (15) without changing our constant β; the same condition still also holds when we approximate the problem in the discrete subspace, and it finally reads as follows :

(19)

We can now replace the integral in (19) using the first equation of (5) , namely, the weak momentum equation,then, exploiting the continuity of the bilinear form (7) and finally the stability estimate for the velocity in (11) we can derive the following estimate:

2 41 2

11[ ( )] [ ( )]

[ ( 2)]uN cont stabL H

C C F

r r

% (20)

Note that in (20) Cconst and Cstab are the continuity constant for the bilinear form in (7) and the stability constant for the velocity in (11) , while β1 =β/C, where C is the continuity constant for the orthogonal projector in

2 ( )L


A model to investigate the Extrudate Swell problem IA model to investigate the Extrudate Swell problem I(see e.g. (see e.g. [7][7] and and [8][8] for an analysis and some data) for an analysis and some data)

Key points: • Tracking the free surface

0( )

zr

diez

ur z D dz

u

• Capturing the large stress gradient at the

exit, analyzing the angle of swelling a;• Estimate the final diameter Dfinal

corresponding to a total stress relaxation.

1 1

1 ( , ) ( , ) (0,0)

1

((1 ) (1 ) ( )) ( ) ( ), ( 1,1), , 1

(1 ) (1 ) ( ) ( ) 0, ( ) ( ).n m N N

x x u x w x u x x

x x P x P x L x P x

• The Sturm-Liouville problem and the Jacobi polynomials


A model to investigate the Extrudate Swell problem IIA model to investigate the Extrudate Swell problem II(see e.g. (see e.g. [7][7] and and [8][8] for an analysis and some data) for an analysis and some data)

Following an approach proposed in [9] for the flow past a sphere, we parametrize dier D by

1dies D and set 0, k such that at each step k the weight function ( ) : (1 ) kw x x

can follow the behaviour of the analytic singularity (while with the parametrization above we approach the geomtrical one).

The main idea is to approximate the free surface problems with a sequence of rigid boundary problems,for which we have the values of the power of the singularity at the die; this values are different from fluid to fluid, but always (approximatively) in the range (-0.3 , -0.7 ).

Pattern of the algorithm : •construct the new boundary from the previous step (the first comes from a fully developed Poiseuille flow, so it’s a stick-slip problem); •find the value of the singular corner as the tangent to the boundary at the die; •find the necessary eigenvalues to compute the power of the singularity (see e.g. [8] ); •Construct the correspondent Jacobi polynomials for the corner element; •Solve the problem with the updated boundary and Jacobi polynomial; •Check the value of the normal velocity.


The 1-D discretization processThe 1-D discretization process(note: all the results are obtained for N=5 and an error tolerance of 10°-05 in the CG routine)(note: all the results are obtained for N=5 and an error tolerance of 10°-05 in the CG routine)

• The spectral (Lagrange) basis :The spectral (Lagrange) basis :

2(1 ) ( )( ) , 0,

( 1) ( )( )N

iN i i

Lh i N

N N L

0

1

1

0

( ) ( ), 1, 2

( ) ( ), 1

( ) ( ), , 1, 2

i

i

i

Nk kN i

i

Nk kN i

i

Nkl klN i

i

u u h k

p p h k

h k l

• Approximating the solution: replacing velocity, pressure and stress by the following expansions and the integral by Approximating the solution: replacing velocity, pressure and stress by the following expansions and the integral by a Gaussian quadrature on the Gauss-Lobatto-Legendre nodes, namely the roots of L’(x), a Gaussian quadrature on the Gauss-Lobatto-Legendre nodes, namely the roots of L’(x), (5)(5) becomes a linear becomes a linear system: system:

( ) , ( 3) (7) 12xu x e u u


The 2-D discretization process IThe 2-D discretization process I

, 0

1

, 1

, 0

( , ) ( ) ( ), 1, 2

( , ) ( ) ( ), 1

( , ) ( ) ( ), , 1, 2

i

i

i

Nk kN ij j

i j

Nk kN ij j

i j

Nkl klN ij j

i j

u u h h k

p p h h k

h h k l

• The 2-D spectral (tensorial) expansion :The 2-D spectral (tensorial) expansion :


The 2-D discretization process IIThe 2-D discretization process II

2

1

3 32

, ( 1,1)

,x y

u f

u e eu f

u x y

1 1 2 2

0 0

( , ) ( ) ( ) ( , ) ( ) ( )N N

N ij i j N ij i ji i

u u h h u u h h

1 1 1

2 2 2

H u l

H u l

, 0

( ) ( ) ( )

( ) ( , )

Nk kij lj l m i l j m

l m

k kij i j ij k i j

H w h h

l w f

• The problem :The problem :

• The 2-D expansion :The 2-D expansion :

• The discrete problem:The discrete problem:

Approxim

ation of u

1A

pproximation of

u 2


Coming soon (hopefully…)Coming soon (hopefully…)

Complete the codes including pressure gradient and Complete the codes including pressure gradient and stress fieldstress field

Adapt them to the Extrudate Swell problem (in the Adapt them to the Extrudate Swell problem (in the newtonian case)newtonian case)

Generalize to the unsteady cases Generalize to the unsteady cases

Try some preconditioners on the CG routineTry some preconditioners on the CG routine

Start with non-newtonian fluidsStart with non-newtonian fluids


ReferencesReferences[1] BABUSKA I., The finite element method with Lagrangian multipliers, Numerical Mathematics, 1973, 20: 179-192.

[2] BREZZI F., On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers, R.A.I.R.O., Anal. Numer., 1974, R2, 8, 129-151.

[3] MADAY Y, PATERA A.T., RØNQUIST E.M., The PN ×PN−2 method for the approximation of the Stokes problem. Technical Report 92025, Laboratoire d‘Analyse Numrique, Universitet Pierre et Marie Curie, 1992.

[4] BERNARDI C., MADAY Y. Approximations spectrales de problemes aux limites elliptiques, Springer Verlag France, Paris, 1992.

[5] GERRITSMA M.I., PHILLIPS T.N., Compatible spectral approximation for the velocity-pressure-stress formulation of the Stokes problem, SIAM Journal of Scientific Computing, 1999, 20 (4) : 1530- 1550.

[6] SHWAB C., p- and hp- Finite Element Methods, Theory and Applications in Solid and Fluid Mechanics, Oxford University Press, New York, 1998.

[7] OWENS R.G., PHILLIPS T. N. Computational Rheology, Imperial College Press, 2002.

[8] TANNER R.I., Engineering Rheology - 2nd Edition, Oxford University Press, 2000.

[9] GERRITSMA M.I., PHILLIPS T.N., Spectral elements for axisymmetric Stokes problem, Journal of Computational Physics 2000, 164 : 81-103.

[10] KARNIADAKIS G., SHERWIN S.J., Spectral / hp elements for computational fluid dynamics - 2nd Edition, Oxford University Press, 2005.

16/05/2006 cardiff school of mathematics 1 modelling flows of viscoelastic fluids with spectral...

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pressure fields

velocity fields

fields problemresults

fields problema stability

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compatibility conditions

approximation results

approximating spaces