multilevel incomplete factorizations for non-linear fe problems in geomechanics dmmmsa –...

Multilevel Incomplete Factorizations for Multilevel Incomplete Factorizations for Non-Linear FE problems in GeomechanicsNon-Linear FE problems in Geomechanics

DMMMSA – University of PadovaDMMMSA – University of PadovaDepartment of Mathematical Methods and Models for Scientific ApplicationsDepartment of Mathematical Methods and Models for Scientific Applications

Carlo JannaCarlo Janna, Massimiliano Ferronato , Massimiliano Ferronato and Giuseppe Gambolatiand Giuseppe Gambolati

Due Giorni di Algebra Lineare NumericaDue Giorni di Algebra Lineare NumericaBologna, Marzo 6-7, 2008Bologna, Marzo 6-7, 2008

OutlineOutline

IntroductionIntroduction Level structure of the matrixLevel structure of the matrix Multilevel Incomplete Factorization (MIF)Multilevel Incomplete Factorization (MIF) Numerical resultsNumerical results Drawbacks and possible solutionsDrawbacks and possible solutions ConclusionsConclusions

Fluid removal/injection from/to subsurface Fluid removal/injection from/to subsurface

Pore pressure and effective stress variation Pore pressure and effective stress variation

Environmental, geomechanical and geotechnical applicationsEnvironmental, geomechanical and geotechnical applications

Prediction by Prediction by numerical numerical

modelsmodels}

The Geomechanical problemThe Geomechanical problem

Classical Biot’s consolidation theory:Classical Biot’s consolidation theory:

tt

pcpKk

g

zyxii

puG

iG

wbrwrw

w

wi

1

,, 2


The problem can be solved by decoupling:The problem can be solved by decoupling:

The fluid-dynamic problem is solved firstThe fluid-dynamic problem is solved first

The output of the fluid-dynamic part is used as input for the The output of the fluid-dynamic part is used as input for the geomechanical problemgeomechanical problem

The structural part of the problem is solved by FE minimizing The structural part of the problem is solved by FE minimizing

the total potential energythe total potential energy

minu


The arising system of equations is non-linear The arising system of equations is non-linear

because because and and GG are functions of the stress state are functions of the stress state

0

0

0

2

1

u

u

u

nf

f

f

The presence of faults and fractures in the geological media The presence of faults and fractures in the geological media is another source of non-linearityis another source of non-linearity


Constrained Minimization Problem:Constrained Minimization Problem:

0

min

ug

u

n

ugn = opposite of the distance= opposite of the distance


Faults and fractures act as Faults and fractures act as contact contact surfacessurfaces

Interface Elements and Penalty methodInterface Elements and Penalty method

min2 ugu n


Some considerations:Some considerations:

Only FE in the reservoir are subject to a relevant stress changeOnly FE in the reservoir are subject to a relevant stress change

Surrounding FE can be considered to behave elasticallySurrounding FE can be considered to behave elastically

The faults presence involves only few FEThe faults presence involves only few FE

The non-linear quasi static problem is solved by a Newton-like The non-linear quasi static problem is solved by a Newton-like scheme that results in a sequence of linear systems:scheme that results in a sequence of linear systems:

bAx

The Sparse Linear SystemThe Sparse Linear System

The linearized system can be reordered in such a way to The linearized system can be reordered in such a way to show its natural 3-level block structure:show its natural 3-level block structure:

CBB

BKB

BBK

ATT

T

212

2211

1211

Dof linked to linear elementsDof linked to linear elements

Dof linked to non-linear elementsDof linked to non-linear elements

Dof linked to interface elementsDof linked to interface elements

The Sparse Linear SystemThe Sparse Linear System

The system matrix The system matrix AA is Symmetric Positive Definite: use of PCG is Symmetric Positive Definite: use of PCG

The The KK11, , BB1111, , BB1212 blocks do not change during a simulationblocks do not change during a simulation

The The KK22, , BB2222 blocks change whenever a stress perturbation occurs in the reservoirblocks change whenever a stress perturbation occurs in the reservoir

The The CC block change whenever the contact condition varies on the faultsblock change whenever the contact condition varies on the faults

The system is very ill-conditioned due to the penalty approach because ||The system is very ill-conditioned due to the penalty approach because ||CC|| >> || || >> || KK11||, ||||, ||KK22||||

The Multilevel Incomplete FactorizationThe Multilevel Incomplete Factorization

Define a partial incomplete factorization of a matrix Define a partial incomplete factorization of a matrix A:A:

TTT

T

LDLI

HL

S

D

IH

L

MAA

AAA

~~~

00

00 11

1

1

1

1

2212

1211

with:with:

11111 ALDL T 111121 DLAH TT THDHAS 111221


The use of The use of MM-1-1 as a preconditioner requires the solution of: as a preconditioner requires the solution of:

rvLDLMv T ~~~

yvL

zyD

rzL

T

1

1

1

~.3

~.2

~.1

21

12

11

11

2

1

2

1

1

11 0

0~

zSy

zDy

z

z

y

y

S

DzyD

The second step is performed in this way:The second step is performed in this way:

yy22 can be found approximately by using again a partial incomplete factorization of can be found approximately by using again a partial incomplete factorization of SS11


Advantages of the approach:Advantages of the approach:

There is no need to perform the factorization of the There is no need to perform the factorization of the whole matrix whole matrix A A at every non-linear iterationat every non-linear iteration

It is possible to independently tune the fill-in degree of It is possible to independently tune the fill-in degree of each level with 2 parameters each level with 2 parameters ρρi1i1, , ρρi2i2

The unknows linked to the penalty block are kept The unknows linked to the penalty block are kept toghether in a single leveltoghether in a single level

Numerical ResultsNumerical Results

3D Geomechanical problem of faulted rocks 3D Geomechanical problem of faulted rocks discretized with FE and IE discretized with FE and IE


# of unknowns# of unknowns

Level 1Level 1 435,207435,207

Level 2Level 2 163,581163,581

Level 3Level 3 20,01420,014

TotalTotal 618,802618,802

Level 1Level 1Level 2Level 2Level 3Level 3


TH1

1HTL1

1S

A


2STH 2

2HTL2

TL3

1S


Comparison between ILLT and MIF in terms of:Comparison between ILLT and MIF in terms of:

Number of IterationsNumber of Iterations

CPU TimeCPU Time

Memory occupationMemory occupation 21 ,, MIFMIFILLT


145.12145.12T. Tot. [s]T. Tot. [s]

3.923.92

36.4336.43T. CG [s]T. CG [s]

108.69108.69Prec. [s]Prec. [s]

7575# Iterations# Iterations

ILLTILLT

ILLT

2.682.68

72.7672.76T. Tot. [s]T. Tot. [s]

23.1723.17Liv. 1 [s]Liv. 1 [s]

11.2911.29Liv. 2 [s]Liv. 2 [s]

17.0417.04Liv. 3 [s]Liv. 3 [s]

3.293.29

21.2621.26T. CG [s]T. CG [s]

51.5051.50Prec. [s]Prec. [s]

5353# Iterations# Iterations

MIFMIF

1MIF2MIF

Performance in the solution of a single systemPerformance in the solution of a single system


CPU time for Lev. 1 and Lev. 2 (34.46 s) can be made up CPU time for Lev. 1 and Lev. 2 (34.46 s) can be made up for in a few non-linear iterationsfor in a few non-linear iterations

Losing some performance (about 15%) the memory Losing some performance (about 15%) the memory occupation can be further reducedoccupation can be further reduced

Observations:Observations:

3.293.29

1MIF

2MIF

2.682.68

2.352.35

1.891.89

ILLTILLT

CPU PrecCPU Prec 26h 57m 26h 57m 33s33s

T. CGT. CG 3h 45m 42s3h 45m 42s

OverheadOverhead 22m 45s22m 45s

Total CPU Total CPU TimeTime 31h 6m 0s31h 6m 0s

MIFMIF

CPU PrecCPU Prec 4h 44m 10s4h 44m 10s

T. CGT. CG 3h 41m 6s3h 41m 6s

OverheadOverhead 21m 13s21m 13s

Total CPU Total CPU TimeTime 8h 46m 29s8h 46m 29s

28.0ILLT

MIF

T

T54.3 Up-Speed

MIF

ILLT

T

T


Performance in a real whole simulationPerformance in a real whole simulation

DrawbacksDrawbacks

The original matrix is SPD, but an SPD multilevel incomplete The original matrix is SPD, but an SPD multilevel incomplete

factorization is not guaranteed to existfactorization is not guaranteed to exist::

The factorization of the 11 block of the actual level may The factorization of the 11 block of the actual level may be indefinitebe indefinite

The Schur complement of the actual level may be The Schur complement of the actual level may be indefiniteindefinite

Possible solutionsPossible solutions

The use of another solver instead of PCG, i.e. CR or SQMR, The use of another solver instead of PCG, i.e. CR or SQMR, that does not require positive definitenessthat does not require positive definiteness

Allowing for larger fill-in degreesAllowing for larger fill-in degrees

The implementation of special techniques to guarantee the The implementation of special techniques to guarantee the positive definiteness of the factorspositive definiteness of the factors


Procedure of Procedure of Ajiz & JenningsAjiz & Jennings for the 11 block factorization for the 11 block factorization

eee

eee

eee

fff

fff

fff

aaa

aaa

aaa

A TLL E

Diagonal compensation to enforceDiagonal compensation to enforce 0E


Procedure of Procedure of TismenetskyTismenetsky for the Schur complement for the Schur complement computationcomputation

THDHCS

BHLHLBHDHCS TTTT 1

ConclusionsConclusions

The Multilevel Incomplete Factorization has proven to be a The Multilevel Incomplete Factorization has proven to be a robust and reliable tool for the solution of non linear problems robust and reliable tool for the solution of non linear problems in geomechanicsin geomechanics

Part of the preconditioner can be computed at the beginning Part of the preconditioner can be computed at the beginning of the simulation thus reducing the set-up phase during the of the simulation thus reducing the set-up phase during the non-linear iterationsnon-linear iterations

Its level structure allows for a fine tuning of the fill-in degree Its level structure allows for a fine tuning of the fill-in degree and thus of the preconditioner qualityand thus of the preconditioner quality

……and future workand future work

The development of techniques to guarantee the positive The development of techniques to guarantee the positive definiteness of the preconditionerdefiniteness of the preconditioner

Sensitivity analysis of the user-defined parametersSensitivity analysis of the user-defined parameters

Application to other naturally multilevel problems (coupled Application to other naturally multilevel problems (coupled problems such as coupled consolidation, flow & transport..)problems such as coupled consolidation, flow & transport..)

Thank you for your Thank you for your attentionattention

DMMMSA – University of PadovaDMMMSA – University of PadovaDepartment of Mathematical Methods and Models for Scientific ApplicationsDepartment of Mathematical Methods and Models for Scientific Applications

multilevel incomplete factorizations for non-linear fe problems in geomechanics dmmmsa –...

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