Florez09

Domain Decomposition Techniques to Domain Decomposition Techniques to Couple Elasticity and Plasticity in Couple Elasticity and Plasticity in

GeomechanicsGeomechanics

By Horacio Florez, M.Sc., M.S.E.By Horacio Florez, M.Sc., M.S.E.

Research work carried out under the supervision of Dr. Mary F. WheelerMary F. Wheeler

The Center for Subsurface ModelingThe University of Texas at Austin

CSM Industrial Affiliates Meeting, Austin, October 14, 2009

Outline Outline

Summary and motivationSummary and motivation

Model problemsModel problems Elasticity and loose coupling with flow (*)Elasticity and loose coupling with flow (*) Computational plasticityComputational plasticity

State-of-the-art State-of-the-art Domain DecompositionDomain Decomposition schemes in schemes in geomechanics: geomechanics: Dirichlet-Neumann (*) Dirichlet-Neumann (*) Mortar FEMMortar FEM

Numerical experiments in geomechanics on multi-Numerical experiments in geomechanics on multi-core processors (*)core processors (*)

Concluding remarks, future work and referencesConcluding remarks, future work and referencesPart of summer internship work at The ConocoPhillips Company (*)Part of summer internship work at The ConocoPhillips Company (*)

SummarySummary

We’re going to discuss

popular Domain DecompositionDomain Decomposition

schemes to couple elasticity

and plasticity in geomechanics.

The algorithm of

computational plasticitycomputational plasticity with

popular failure criteriafailure criteria will be

discussed.

We present the

computer implementation with

FEM FEM and preliminary 2-D

results.

3

1

2

321

Druker-PragerDruker-Prager

Von MisesVon Mises

TopTop: The Druker-Prager and Von

Mises yield surfaces are depicted.

LeftLeft: A plasticity front starts to

propagate due to the fact that the

material is yielding right there.

Motivation: Domain DecompositionMotivation: Domain Decomposition

Reservoir levelReservoir level Near boreholeNear borehole

Elasticity and plasticity in geomechanics: iterative coupling

Outline Outline

Summary and motivation

Model problemsModel problems Elasticity and loose coupling with flow (*)Elasticity and loose coupling with flow (*) Computational plasticityComputational plasticity

State-of-the-art Domain Decomposition schemes in geomechanics: Dirichlet-Neumann (*) Mortar FEM

Numerical experiments in geomechanics on multi-core processors (*)

Concluding remarks, future work and referencesPart of summer internship work at The ConocoPhillips Company (*)

Isotropic Elasticity

dbdsdBBK

K

dbdslda

la

nb

TTT

T

N

N

;

; : :,

,

on ,ˆ ; on ,0 ; in , ND

tfC

fu

vvtvuCvvu

vvu

tu

ext

ext

Fig. Applications include borehole stability and sand production, reservoir compaction and subsidence (loose coupling), among others

FEM formulation:

Loose Coupling: Reservoir CompactionLoose Coupling: Reservoir Compaction

1st problem: at initial equilibrium, conditions are often unknown

2nd problem: the stress changes due to pressure drop and changes in

tractions; no changes in body forces

Superposition principle: solve 2nd problem without knowing the 1st one

dpBdsdBBK

p

ngbb

fTTT

f

N

;

- : ; in , 0

on ,ˆ ; on ,0 ; 0

ND0

tfC

C

tu

ext

Small perturbation assumption :

(*) Part of these ideas come from a geomechanics course under Dr. Yves Leroy

Computational Plasticity AlgorithmComputational Plasticity Algorithm

1111 ,ˆ ; ,ˆ

;

0

nnnnnn

TTT dbdsdBN

tfuf

fuf

extint

extint

Path-dependent materials :

qHqqN

CCp

p

pe

, ; ,

2 ; :

Rate-independent plasticity (small deformation):

Computational Plasticity Cont.Computational Plasticity Cont.

kn

kn

n

epepT

nT

kkn

kn

kn

k

T

DdBDBK

K

1111

11

11

11

ˆ ;

uu

r

uuu

uru

Non-linear solution: The Newton-Raphson scheme

ext

int

int

ffbbtt

tfuuf

fufur

111111

11111

111

;

; ,ˆ

0

next

nnnnn

nTT

next

nnnT

n

ext

nnn

dbdsdBN

The incremental boundary value problem:

Plasticity Return Mapping AlgorithmPlasticity Return Mapping Algorithm

EXIT 4.

system linear-non the Solve mapping.-Return 3.

criterion)(yield ityadmissibil plasticity Check 2.

:predictor Elastic 1.

and ,,

0

0

0

,

EXIT then and set THEN 0, IF

; ; ;

11

11

1trial

11

1 trial

11

trial11

trial

1

trial1

trial

1

trial

1

trial

1

trial1

trial11

trial1

nen

nn

nnn

nen

en

nnnn

nn

n

ennnen

en

for

qf

H

N

qf

q

Fully implicit elastic predictor/return-mapping:

Outline Outline


Model problems Elasticity and loose coupling with flow (*) Computational plasticity

State-of-the-art State-of-the-art Domain DecompositionDomain Decomposition schemes in schemes in geomechanics: geomechanics: Dirichlet-Neumann (*) Dirichlet-Neumann (*) Mortar FEMMortar FEM



Domain DecompositionDomain Decomposition

kn

Nkn

Nkn

kDkDk

kkn

k

kk

k

uuu

uu

u

ΩΩu

ΩfLu

Ω

u

Ωu

ΩfLu

21

11

2

11

12

1

112

21

2

22

11

11

1

11

1

1

on

on 0

in

on

on 0

in

1

2

Matching gridsMatching grids

Coloring algorithm (3 colors tool)Coloring algorithm (3 colors tool)

Over-relaxation is importantOver-relaxation is important

Iterative coupling by the BC’sIterative coupling by the BC’s

Dirichlet-Neumann (DN)Dirichlet-Neumann (DN)

DN: Coloring and Algorithm DN: Coloring and Algorithm

mn

i j k

1.1. White guys (D-guys) go firstWhite guys (D-guys) go first

2.2. Hybrid grey guys proceed Hybrid grey guys proceed

after D-guysafter D-guys

3.3. Black ones can now goBlack ones can now go

4.4. Feedback displacements to Feedback displacements to

white and grey ones and go to white and grey ones and go to

step 1 if there is a residual in step 1 if there is a residual in

the tractions, stop if not the tractions, stop if not

Fig. General partitioning will require a three-color tool, hybrid sub-domains show up for touching both D- and N- guys

Speedup for Isotropic ElasticitySpeedup for Isotropic Elasticity

Quadrilatheral mesh, nn = 13564, ne =

13280, Intel® Xeon® Processor E5440,

2.83 GHz, Quad Core (Harpertown)

Kirsch’s Benchmark ProblemKirsch’s Benchmark Problem

Exact FEM Q1

FEM P1 FEM P2

xx

(*) Part of internship work at The ConocoPhillips Company

Mortar FEM Method: ElasticityMortar FEM Method: Elasticity

0

0

0

0 ,

,,

; ,

; : :,

2

1

2

1

21

22

11

21

l

l

u

u

u

vvvu

uuuvuvu

vvtvuCvvu

BB

Bk

Bk

b

lba

db

dbdslda

T

T

hh

hhhhh

T

N

1

2

Nonconforming discretizationsNonconforming discretizations:

Fig. Non-matching interfaces and

hanging-nodes are treated properly

Outline Outline




Numerical experiments in geomechanics on multi-Numerical experiments in geomechanics on multi-core processors (*)core processors (*)


Reservoir Cross-Section: Plane StrainReservoir Cross-Section: Plane Strain

0 ,0 xy Fu

0

0

y

x

F

u

0

0

y

x

F

u

0ˆ

n

Fig. The pressure field comes from a 100 x 20 black-oil model, the tensor product mesh propagated in the surroundings is quite inefficient and requires a non-matching treatment

Boundary conditions and conforming meshBoundary conditions and conforming mesh

Fig. The FEM solution shows compaction (in blue) and build-up (in red)

Vertical displacement contourVertical displacement contour

FEM Solution: Conforming Mesh CaseFEM Solution: Conforming Mesh Case

Reservoir Cross-Section: Mortar CaseReservoir Cross-Section: Mortar Case

0 ,0 xy Fu

0

0

y

x

F

u

0

0

y

x

F

u

0ˆ

n

Fig. The same tensor-product mesh is used in the pay-zone while the surroundings are meshed with Delaunay triangulations. The goal is to reduce the computational cost

Boundary conditions and non-conforming meshBoundary conditions and non-conforming mesh

FEM Solution with 4 MortarsFEM Solution with 4 Mortars

Vertical displacement contourVertical displacement contour

Fig. The mortar solution reproduces the same features in the displacement field but the computational cost was reduced by 50% because of the efficient meshing

Strip-Footing: Plasticity ExampleStrip-Footing: Plasticity Example

This problem allows determining the bearing capacity (limit load) of a strip footing before collapsing

0 ,0 xy Fu

0

0

y

x

F

u

0

0

y

x

F

u

0ˆ

nP

xyyx

xyyx

y

pyJf

,,

,,

0.48 ; KPa10E

KPa 7.848

3

criterion yield MisesVon

7

2

Strip-Footing: FEM SolutionStrip-Footing: FEM Solution

Vertical displacement contour and plasticity frontVertical displacement contour and plasticity front

Fig. 1 The elastic trial (top) and the plastic converged (bottom) solutions are shown for a given load increment

Fig. 2 The plasticity front propagates during the incremental loading process

Outline Outline





Concluding remarks, future work and referencesConcluding remarks, future work and referencesPart of summer internship work at The ConocoPhillips Company (*)

Concluding RemarksConcluding Remarks

We have presented:

1. Parallel Finite Element CG-Code was developed and tested

on benchmark problems

2. Domain Decomposition techniques for coupling elasticity

and plasticity with DN and mortars

Scalable speedup obtained for elasticity on 8

processors with DN

Scalable speedup achieved for plasticity up to 4 cores

(multi-threaded ensemble of tangent matrix)

1. Further testing on Linux cluster machines like Bevo, Lonestar,

and Ranger

2. Implement other popular failure criteria such as Druker-Prager

and Cam-Clay

3. Benchmarking with both research and commercials codes

such as HYPLAS, FEAP, Abaqus, etc.

4. Incorporate more physics into the FEM-code: thermal stresses

and coupling with the energy equation

5. We have to try with both Discontinuous Galerkin (DG)

Future WorkFuture Work

References: Domain Decomp. References: Domain Decomp.

1) Toselli, A. and Widlund, O., 2005, “Domain Decomposition Methods – Algorithms and Theory”, Springer Series in computational Mathematics, New York, USA.

2) Quarteroni, A. and Valli A., 1999, “Domain Decomposition Methods for Partial Differential Equations”, Numerical Mathematics and Scientific Computation , Oxford University Press, New York, USA.

3) Girault, V., Pencheva, G., Wheeler, M. and, Wildey, T., 2009, “Domain decomposition for linear elasticity with DG jumps and mortars”, Comput. Methods Appl. Mech. Engrg., 198 (2009) 1751-1765.

4) Girault, V., Pencheva, G., Wheeler, M. and, Wildey, T., 2009, “Domain decomposition for poro-elasticity with DG jumps and mortars”, in preparation.

5) Badia S. et al, 2009, “Robin-Robin preconditioned Krylov methods for fluid-structure interaction problems”, Comput. Methods Appl. Mech. Engrg., 198 (2009) 2768-2784.

6) Discacciati M., et al., 2001, “ROBIN-ROBIN DOMAIN DECOMPOSITION FOR THE STOKES-DARCY COUPLING”, SIAM J. NUMER. ANAL., Vol. 45, No. 3, pp. 1246-1268.

7) Hauret, P. and Le Tallec, P., 2007, “A discontinuous stabilized mortar method for general 3D elastic problems”, Comput. Methods Appl. Mech. Engrg., 196 (2007) 4881-4900.

8) Flemisch B., Wohlmuth, B. I., et al., 2005, “A new dual mortar method for curved interfaces: 2D elasticity”, Int. J. Numer. Meth. Engng. 2005, 68:813-832.

9) Hauret, P. and Ortiz, M., 2005, “BV estimates for mortar methods in linear elasticity”, Comput. Methods Appl. Mech. Engrg., 195 (2006) 4783-4793.

References: Plasticity References: Plasticity

1) Neto, E. A. et al, 2008, “Computational methods for plasticity : theory and applications”, Wiley, UK.2) Simo, J. C. and Hughes T.J.R., 1998, “Computational Inelasticity”, Springer, Interdisciplinary Applied

Mathematics.3) Lubliner, J., 1990, “Plasticity Theory ”, Dover Publications, Inc., New York. 4) Zienkiewicz, O. C. and Cormeau, I.C., 1974, “VISCO-PLASTICITY AND CREEP IN ELASTIC SOLIDS- UNIFIED

NUMERICAL SOLUTION APPROACH”, International Journal of Numerical Methods in Engineering , Vol. 8, pp. 821-845.

5) Cormeau, I.C., 1975, “NUMERICAL STABILITY IN QUASI-STATIC ELASTO/ VISCO-PLASTICITY”, International Journal of Numerical Methods in Engineering , Vol. 9, pp. 109-127.

6) Hughes, T.J.R. and Taylor, R. L., 1978, “UNCONDITIONALLY STABLE ALGORITHMS FOR QUASI-STATIC ELASTO/ VISCO-PLASTIC FINITE ELEMENT ANALYSIS”, Computers & Structures, Vol. 8, pp. 169-173.

7) Simo, J. C. and Taylor, R. L., 1985, “CONSISTENT TANGENT OPERATORS FOR RATE INDEPENDENT ELASTOPLASTICITY”, Computer Methods in Applied Mechanics and Engineering, Vol. 48, pp. 101-118.

8) Simo, J. C. and Taylor, R. L., 1986, “A RETURN MAPPING ALGORITHM FOR PLANE STRESS ELASTOPLASTICITY”, International Journal of Numerical Methods in Engineering, Vol. 22, pp. 649-670.

9) Wilkins, M.L., 1964, “Calculation of Elasto-Plastic Flow”, In Methods of Computational Physics 3, eds. , B. Alder et. al., Academic Press, New York.

10) Clausen, J., et al., 2007, “An efficient return mapping algorithm for non-associated plasticity with linear yield criteria in principal stress plane”, Computers & Structures, Vol. 85, pp. 1975-1807.

References: Poroelasticity References: Poroelasticity

1) Kim, J. et al., 2009, “Stability, Accuracy and Efficiency of Sequential Methods for Coupled Flow and Geomechanics”, SPE Paper 119084.

2) Liu R., 2004, “Discontinuous Galerkin Finite Element Solution for Poromechanics”, PhD thesis, The University of Texas at Austin .

3) Gai X., 2004, “A Coupled Geomechanics and Reservoir Flow Model on Parallel Computers”, PhD thesis, The University of Texas at Austin .

4) Han G. et al., 2002, “Semi-Analytical Solutions for the Effect of Well Shut Down on Rock Stability”, Canadian International Petroleum Conference, Calgary, Alberta .

5) Chen Z, et al, 2006, “Computational Methods for Multiphase Flows in Porous Media, SIAM, pp. 57; 247-258 .

6) Du J, and Olson J., 2001, “A poroelastic reservoir model for predicting subsidence and mapping subsurface pressure fronts”, Journal of Petroleum Technology & Science, Vol. 30, pp. 181-197.

7) Grandi, S. and Nafi M., 2001, “Geomechanical Modeling of In-situ Stresses around a Borehole”, MIT, Cambridge, MA.

8) Charlez A., 1999, “The concept of Mud Window Applied to Complex Drilling”, SPE Paper 56758 .

End of presentationThanks for your attention

Contact Us:Contact Us:

Visit us: http://www.ices.utexas.edu/subsurface/

e-Mail: [email protected]

Any Questions?

Rate Independent PlasticityRate Independent Plasticity

We just follow the approach by Simo and Hughes (1998) and Lubliner (1990):

211E ; 1E

2 ; :

CC p

pe

Elastic domain and yield criterion:

0, | ,: qfxSq me

Flow rule and hardening law:

qHq

qNp

,

,

Kuhn-Tucker complementary conditions:

0, and ,0, ,0 qfqf


Interpretation of the Kuhn-Tucker complementary conditions:

0

loading) (Plastic 0 ,0

unloading) (Neutral 0 ,0

unloading) (Elastic 00

E 0

(Elastic) 0Eint ,0

f

f

f

f

f

qf

Consistency condition and elastoplastic tangent moduli:

0::::

:

:

Hq

fNC

fC

ff

qq

fC

ff

qq

fff

p


Assumption for the flow rule, hardening law, and yield condition satisfy:

2 ;

::

::0

0::

xxx

HfNCf

Cff

HfNCf

q

q

Finally the so called tensor of tangent elastoplastic moduli becomes:

0 if ::

:: 0 if

:::

HfNCf

CfNCC

C

C

CNCC

q

ep

epp

For the special case of associative flow rule we have:

qfqN ,,

Failure CriteriaFailure Criteria

We just follow the approach by Zienkiewicz and Cormeau (1974) and Hughes (1978):

n

p

xx

fQ

fQ

Q

f

f

kyff

plasticity eassociativnon

plasticity eassociativ

0,,

0

The visco-plastic strain rate law:

III

m

m

m

p

m

MJ

QM

J

QM

QQ

J

J

QJ

J

QQQ

QQ

f

f

JJff

32

0

3

3

2

2

0

32

,,

Druker-Prager Yield SurfaceDruker-Prager Yield Surface

Common expressions are given by:

stress yield Uniaxial

; c23

criterion MisesVon 0

sin3

cosc63

sin3

6sin

22

2

y

yJJf

Jf m

3

1

2

321

Druker-PragerDruker-Prager

Von MisesVon Mises

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Florez09