femxdem double scale approach with second gradient … · 2018. 1. 17. · albert argilaga...

IntroductionMethodResults

Conclusions

FEMxDEM double scale approach with secondgradient regularization applied to granular

materials modeling

Albert Argilaga ClaramuntStefano Dal Pont Gael CombeDenis Caillerie Jacques Desrues

16 december 2016

Albert Argilaga Claramunt FEMxDEM Modeling with Second Gradient 1 / 44


Conclusions

Outline

1 Introduction

2 MethodNewton OperatorParallelizationSecond Gradient

3 ResultsRandom field methodsParametric testsEngineering scale simulation

4 Conclusions



Conclusions

MotivationBridge the gap between continuum and discrete approaches

¿?

Figure: Numerical modeling dilemma

Continuum approaches

Numerically efficient

No scale limitations

Discrete approaches

No need ofphenomenological laws

Faithful representation ofthe microscale



Conclusions

Multiscale coupling

FEM: Lagamine(Liege University)

DEM with PBC(in-house 3SR)

m

n

fn /m

fn '/m '

mʼ

nʼ

A numerically obtained constitutive law isinjected into the material points



Conclusions

Discrete Element Model (DEM)

Frictional cohesive contact laws

Periodic Boundary Conditions

Figure: DEM assembly preparation

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

22(%)

(22

11)/

11

Figure: DEM biaxial response

Numerical homogenization

σf =1

S

∑(n,m)∈C

~f m/n (~rm − ~rn)



Conclusions

State of the Art in FEMxDEM

Early works (Kaneko, 2003; Miehe, 2010; Nitka, 2011)

Study of anisotropy (Guo and Zhao, 2013; Nguyen, 2013)

DEM cohesion (Nguyen, 2014)

Non-local regularization (Liu, 2015)

3D microscale (Liu, 2015; Wang and Sun, 2016)

DEM material heterogeneity (Shahin, 2016)

Macroscale hidro-mechanical coupling(Wang and Sun, 2016; Guo and Zhao, 2016)

Full micro-macro 3D (Guo and Zhao, 2016)



Conclusions

Model performance

FEMxDEM issues

Bad convergence

Computational time

Generic issues

Mesh dependency

FEMxDEM proposed solutions

Alternative Operator

Parallelization

Generic proposed solutions

Regularization



Conclusions

Newton OperatorParallelizationSecond Gradient

Outline

1 Introduction



4 Conclusions



Conclusions


Link with classical geomechanics FEM codes

We consider the quasi static evolution of a continuous mediumwhich shares features withelastoplasticity and hypoplasticity. Theconstitutive equation provides the Cauchy stress σf as a functionof the deformation tensor F f in a loading step:

F f 7→ σf = S(F f)

(1)

The non linear boundary value problem, is solved using theiterative Newton’s method.This needs to differentiate S with respect to F f (1), the gradientof S is often called the ”consistent operator”, it is denoted by C

dσf = S(F f + df f

)− S

(F f)

= C : dF f + · · · (2)



Conclusions


Determination of the Consistent Operator

Except for some cases for which closed-form expression of thetensor C can be found, the determining of C is performednumerically using a perturbation method:

(Cijmn) =Sij(F f + εΛmn

)− Sij

(F f)

ε(3)

Assuming S to be differentiable, that determines C

Issues

Bad convergence

Computational time

Mesh dependency

Proposed solutions

Alternative Operator

Parallelization

Regularization



Conclusions


Alternative Newton Operators for FEMxDEM:

Perturbation based operators

Consistent Tangent (CTO)

Auxiliary Elastic (AEO)

Elastic operators

DEM Quasi Static (DEMQO)

PreStressed Truss-Like (PSTLO)

Kruyt operators

Kruyt Augmented (KAO)

Upper bound Kruyt (UKO)

Upper bound Corrected Kruyt (UCKO - UCKO 2DOF)



Conclusions


Perturbation based operators

Consistent Tangent (CTO)

Classic perturbation approach:

(Cijmn) =Sij(F f + εΛmn

)− Sij

(F f)

ε

Auxiliary Elastic (AEO) (Nguyen, 2013)

Average of the CTO over several macroscopic loading steps



Conclusions


Elastic operators

DEM Quasi Static (DEMQO)

Based on numerical homogenization of an elastic DEM

(Cijmn)DEMQO =SEij (εΛmn)

ε(4)

PreStressed Truss-Like (PSTLO) (Caillerie)

It treats the DEM as a prestressed linearelastic truss withadditional rotational degrees of freedom in the nodes

dσ =1

|Y |∑c∈C

δicd~f c ⊗ ~Y i + σ F−T dFT −

(F−T : dF

)σ (5)



Conclusions


Kruyt operators

Kruyt Augmented (KAO) (Caillerie)

UKO with prestresses and rotational degrees of reedom

Upper bound Kruyt (UKO) (Kruyt & Rothenburg, 1998)

1

|Ya|∑c∈C

(lc)2 (kn (~ec ⊗ ~ec)⊗ (~ec ⊗ ~ec) + kt(~tc ⊗ ~ec

)⊗(~tc ⊗ ~ec

))(6)

Upper bound Corrected Kruyt (UCKO - UCKO 2DOF)

Calibrations of the UKO to fit a CTO



Conclusions


Convergence rates

The convergence rate of the operators is compared in theinitial state and post-peak in a biaxial compression test:

5 10 15 20 25 3010

4

103

102

101

100

iteration

Resid

ual F

NO

RM

/RN

OR

M

OTC

UKO

UCKO

UCKO 2DOF

DEMQO/PTLO

KAO

5 10 15 20 25 3010

4

103

102

101

100

iteration

Resid

ual F

NO

RM

/RN

OR

M

AEO

UKO

UCKO

UCKO 2DOF

DEMQO/PTLO

KAO



Conclusions


Operators performance

Perturbation basedoperators perform well in theinitial stages

Kruyt operators are morerobust in the post-peak part

Elastic operators combinegood performance beforeand after the stress peak

OTC/

AEOUKO UCKO

UCKO

2DOF

DEMQO

/PSTLOKAO

Initial 6 12 9 7 6 9

Post-peak 5 6 4 5 4 6

0

2

4

6

8

10

12

14

# i

te

ra

tio

ns

Figure: Operators comparison:number of iterations



Conclusions


Conclusion

The elastic operatorsovercome the issues givinggood stability andconvergence velocity

OTC/

AEOUKO UCKO

UCKO

2DOF

DEMQO

/PSTLOKAO

Initial 6 12 9 7 6 9

Post-peak 5 6 4 5 4 6

0

2

4

6

8

10

12

14

# i

te

ra

tio

ns

Figure: Operators comparison:number of iterations



Conclusions


Outline

1 Introduction



4 Conclusions



Conclusions


FEMxDEM drawbacks

The evaluation of S is issued from a DEM homogenization:

F f 7→ σf = S(F f)

Issues

Bad convergence

Computational time

Mesh dependency

Proposed solutions

Alternative Newton Operator

Parallelization

Regularization



Conclusions


Element loop parallelization

Amdahl’s law (Amdahl, 1967)

S(n) =1

B + 1n (1− B)

(7)

limB→0

S(n) = n (8)

S: speedupn: number of coresB: non parallelizable part

Other Approaches: MassiveParallelization: MPI (Desrues)

0

20

40

60

80

100

120

140

160

180

200

2x Intel Xeon CPU E5410

2.33GHz (2 x 4 cores)

2x Intel Xeon CPU E5650

2.67GHz (2 x 6 cores)

Wa

llti

me

(m

in)

Sequential

Parallel

Figure: Parallelization speedup:x7.15 (8 cores), x9.06 (12 cores)



Conclusions


Numerical randomness

Theorem

Commutative Law of Addition:a + b = b + a

Figure: Simulated numerical error

Figure: Different possiblesolutions of the same BVP.Cumulative second invariantof strain



Conclusions


Numerical randomness

Does not apply!

Commutative Law of Addition:a + b = b + a

Figure: Simulated numerical error

Figure: Different possiblesolutions of the same BVP.Cumulative second invariantof strain



Conclusions


Conclusion

The parallelization provides an effective speedup of the model,becoming this competitive with classical FEM codes



Conclusions


Outline

1 Introduction



4 Conclusions



Conclusions


First order FEM

0

77

154

232

309

387

464

542

619

697

774

852

* 1. 000E- 03

0

21

43

65

87

109

131

153

174

196

218

240

* 1. 000E- 03

Figure: Non objectivity of the model



Conclusions


First order constitutive relation

Balance equation in virtual power formulation:

∀u∗i ,∫

Ωσijε∗ijdΩ =

∫∂Ω

tiu∗i ds (9)

Where u∗i is the kinematically admissible virtual displacement field,σij the stress field, ε∗ij the virtual strain field, ∂Ω the surface of Ω,ti the surface forces

Issues

Bad convergence

Computational time

Mesh dependency

Proposed solutions

Alternative Newton Operator

Parallelization

Regularization



Conclusions


Second Gradient microstructured materials

Second gradient regularization introduces an internal length thatregularizes the model

∀u∗i ,∫

Ωσijε∗ij + Σijk

∂2u∗i∂xj∂xk

dΩ =

∫∂Ω

tiu∗i ds

First order term

Second order term

where Σijk is the double stress dual of∂2u∗i∂xj∂xk



Conclusions


Second Gradient in a Boundary Value Problem

As it is a local model, it can be injected in the material points inthe same way as the first gradient relation is

!



Conclusions


Biaxial test enriched with Second Gradient

0

77

154

232

309

387

464

542

619

697

774

852

* 1. 000E- 03

0

21

43

65

87

109

131

153

174

196

218

240

* 1. 000E- 03

0

85

171

257

342

428

514

600

685

771

857

942

* 1. 000E- 04

0

44

89

134

179

224

269

314

359

404

449

494

* 1. 000E- 03

0

30

60

91

121

152

182

213

243

273

304

* 1. 000E- 03

0

14

28

42

56

70

84

98

112

126

140

154

* 1. 000E- 03

Figure: 128, 512, and 2048 elements biaxial compression tests



Conclusions


Conclusions

With the Second Gradient enrichment the model becomesobjective. This allows to use any mesh refinement

The code is ready to be used in real scale problems



Conclusions

Random field methodsParametric testsEngineering scale simulation

Outline

1 Introduction



4 Conclusions



Conclusions


Introduction: Symmetry breaking

Material properties in numericalmodels are spatially homogeneousunless the contrary is defined

The use of a catalyzer to break thesymmetry can improve the modelperformance, both physically andnumerically

Figure: A CT scan of a sandsample (Ando, 2013)



Conclusions


Punctual imperfection

The DEM microstructure has a strong influence on the results

Figure: Homogeneous (a) Defect (b). Second invariant of strain



Conclusions


Full field DEM random variability

Figure: (a) FEMxDEM and pure DEM responses under a biaxial loading.(b) and (c), cumulative deviatoric strain field (Shahin, 2016)



Conclusions


Conclusions

The FEMxDEM approach provides a natural way to embedmaterial heterogeneity into the model



Conclusions


Outline

1 Introduction



4 Conclusions



Conclusions


Outer radius

The outer radius has an influenceon the results, a compromise sizeis chosen

isa outer radious 100m aargilagaclara

50maargilagac

Figure: Radius=50, 100 and 200 m

Figure: Outer radius=50, 100 and200 m. Cumulative second invariantof strain

M.S. research project

Nurisa, 2015



Conclusions


Micro/macro

DEM allows to tune thecoordination independently fromthe density

Figure: Coordination number =4.10, 3.67 and 3.14. Cumulativesecond invariant of strain

The model can cope with anymacroscopic boundary conditions

Figure: Stress ratio σ0H : σ0V = 1:1,1:1.3 and 1:2. Cumulative secondinvariant of strain



Conclusions


Intrinsic anisotropy

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010

0.5

1

1.5

2

2.5

3

3.5

X: 0.003001Y: 2.33

εyy

q

X: 0.006305Y: 2.892

Figure: DEM preparation

Figure: Precharge 0%, 0.3% and 0.6%.Cumulative second invariant of strain



Conclusions


Outline

1 Introduction



4 Conclusions



Conclusions


Hollow cylinder with Second Gradient

X

Y

0

44

88

132

176

220

264

308

352

396

440

484

* 1.000E-04

Figure: Localization around a gallery. FEM:4950 elements, 18900 nodes, 66297 DOF

Figure: 400 particles DEMmicroscale at the end of theloading



Conclusions


Conclusions

Real scale simulations are possible thanks to the secondgradient and parallelization

FEMxDEM allows to calibrate the model directly from themicroscale



Conclusions

General Conclusions

A FEMxDEM approach allows to use a constitutive law builtvia numerical homogenization in the DEM microscale

An alternative Newton operator provides the model with goodconvergence rate and robustness

Parallelization allows the model to compete with classicalFEM approaches in terms of walltime

A Second gradient regularization assures the meshindependence

Real scale problems with fine meshes can be considered



Conclusions

Perspectives

Calibration of the microscale with experimental data

3D microscale can bring a richer constitutive behabiour

More complex microscale with massive parallelization (MPI)



Conclusions

Thanks!!!


femxdem double scale approach with second gradient … · 2018. 1. 17. · albert argilaga...

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