1 A new iterative technique for solving nonlinear coupled equations arising from nuclear waste transport processes H. HOTEIT 1,2, Ph. ACKERER 2, R. MOSE

1

A new iterative technique for solving nonlinear coupled equations arising

from nuclear waste transport processes

H. HOTEIT 1,2, Ph. ACKERER2, R. MOSE2,3

1IRISA-INRIA, Rennes

2Institut de Mécanique des Fluides et des Solides, IMFS, Strasbourg

3Ecole Nationale du Génie de l'Eau et de l'Environnement, ENGEES, Strasbourg

34ème Congrès National d'Analyse Numérique

27 Mai - 31 Mai 2002

2

Outline

Mathematical model of the transport processes.

Numerical methods:

Mixed Hybride Finite Element method (MHFE);

Discontinuous Galerkin method (DG).

Linearization techniques:

Picard (fixed point) method;

Newton-Raphson method.

Some numerical results.

3

Transport Processes

The transport process concerns an isolated nuclide chain :

U Pu 234238

with the following transport mechanisms :

advection, dispersion/diffusion ;

mass production/reduction ;

precipitation/dissolution ;

simplified chemical reactions (sorption).

4

kkkkkkkkkkk

k SCRCRuCCDt

CR

''').(

kkksolkksolk

ksol SFFt

FR

'')1()1()1(

Mathematical model

Transport equation

Sk is a nonlinear precipitation/dissolution term

satkkk

satk

ksatkk

k

CCifCC

FandCCifS

00

)1(

)1(

2,1

)(

kksolkk

ksolkksatke

satk FCR

FCRCC

5

Numerical methods

Operator splitting technique is used by coupling

Diffusion/dispersion by MHFEM

Advection by DGM

Linearization is done by using

Picard (Fixed Point) method

Newton-Raphson method

6

MHFEAdvantages

mass is conserved locally ;

the state head and its gradient are approximated simultaneously ;

velocity is determined everywhere due to Raviart-Thomas space functions; full tensors of permeability are easily approximated ; Fourier BC are easily handled ; it can be simply extended to unstructured 2D and 3D grids ;the linear system to solve is positive definite.

Disadvantages scheme is non monotone ;number of degrees of freedom=number of sides (faces).

7

DGM

Advantages mass is conserved locally ; satisfies a maximum principle (conserves the positively of the solution) ; can capture shocks without producing spurious oscillation ;

ability to handle complicated geometries ;

simple treatment of boundary conditions.

Disadvantages

limited choice of the time-step (explicit time discretization) ;slope (flux) limiting operator stabilize the scheme

but creates small amount of numerical diffusion.

8

''' ,,,,)1( kkkkkkkkk

ksol FFCCSFFAt

FR

''' ,,,, kkkkkkkkk

k FFCCSCCBt

CR

Linearization by the Picard method

The transport system is rewritten in the form

where,

''' )1()1(, kksolkksolkkk FFFFA

'''' ).(),( kkkkkkkkkkkk CRCRuCCDCCB

9


The (m+1)th step of the Picard-iteration process

Stopping criteria

2

1,1

2

,11,1

2

1,1

2

,11,1

mnkba

mnk

mnk

mnkba

mnk

mnk

FFF

CCC

1'

1,11'

,11'

1,11,1

1'

,11'

1,11'

,11,1

,*,,,

,,,,*

)1(

nk

mnk

nk

mnkk

nk

mnkk

nk

mnk

k

nk

mnk

nk

mnkk

nk

mnkk

nk

mnk

ksol

FFCCSCCBt

CCR

FFCCSFFAt

FFR

10


6 7 8 9 10 11 12 13 14 15

0.0

4.0x10-7

8.0x10-7

1.2x10-6

1.6x10-6

2.0x10-6

2.4x10-6

2.8x10-6

Cri

tère

de

co

nve

rge

nce

No d'itération

Erreur sur Fk

Erreur sur Ck

Convergence needs very small time steps, otherwise :

Residual errors for C and F

11

Coupling Picard and Newton-Raphson methods

1'

,11'

,11'

,1,1

,1 ,,,,)1(

n

kmn

knk

mnkk

nk

mnkk

nk

mnk

ksolmn

k FFCCSFFAt

FFRF

Define the residual function

)()( ,1

,1

,1mn

km

kmnk

mnk FF

F

F

By using Taylor’s approximation , we get

By simple differentiating, we obtain

)( ,1,1

,1

,1

,1

,1mn

kmn

kmnk

mnk

mnk

mnkksol FF

F

S

F

A

t

R

12


The iterative process

2

21

1

1

for)1(

f

f)1(

nmt

nmnort

nmort

tn

n

n

n

10

Time steps

1'

1,11'

,11'

1,11,1

,11,1

,1

1

,1

,1

,1

,1,1

,,,,

)(

nk

mnk

nk

mnkk

nk

mnkk

nk

mnk

k

mnk

nk

mnk

mnkmn

k

mnk

mnk

mnkksolmn

k

FFCCSCCBt

CCR

FFF

FF

S

F

A

t

RF

13

3 4 5 6 7 8

0,0

1,0x10-10

2,0x10-10

3,0x10-10

4,0x10-10

5,0x10-10

Erreur sur Fk

Erreur sur Ck

Cri

tère

de

co

nve

rge

nce

No d'itérations


Convergence is attained even with bigger time steps (20 times bigger)

14

Some numerical results

20

40

60

80

100

Z

0

24.8

49.6

X

4.513.5Y

XY

Z

Repository site Network of alveolus Elementary cell

The repository is made up of a big number of alveolus.

Computation is made on an elementary cell .

Periodic boundary conditions are used .

15

106 years105 years104 years

16

0 1x102 2x102 3x102 4x102 5x102 6x102 7x102 8x102 9x102 1x103

0.0

1.0x10-4

2.0x10-4

3.0x10-4

4.0x10-4

5.0x10-4

6.0x10-4

7.0x10-4

8.0x10-4

9.0x10-4

1.0x10-3

Masse dissoute Masse precipité

234 U

( m

ol )

Temps ( années )

0.0 2.0x105 4.0x105 6.0x105 8.0x105 1.0x106

0.00

0.01

0.02

0.03

0.04

0.05

0.06 Masse dissoule + masse présipitée Masse injéctée

234 U

( m

ol )

Temps ( années )

0.0 2.0x105 4.0x105 6.0x105 8.0x105 1.0x106

0.0

3.0x10-5

6.0x10-5

9.0x10-5

1.2x10-4

1.5x10-4

Err

eur r

elat

ive

Temps ( années )

Precipitated and dissolved mass in the domain

Mass balance in the domain Relative error after 106 years

17

Conclusion

Coupling DG and MHEF methods to solve a transport equation with nonlinear precipitation /dissolution function .

By using the Picard method, small time steps should be considered otherwise no convergence is attained.


Newton-Raphson methods is used for solid phase equation.

Picard method methods is used for the transport equation.

Convergence is attained even with bigger time steps

(20 times bigger).

Documents

1 A new iterative technique for solving nonlinear coupled equations arising from nuclear waste transport processes H. HOTEIT 1,2, Ph. ACKERER 2, R. MOSE