femxdem double scale approach with second gradient … · 2018. 1. 17. · albert argilaga...
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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
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IntroductionMethodResults
Conclusions
Outline
1 Introduction
2 MethodNewton OperatorParallelizationSecond Gradient
3 ResultsRandom field methodsParametric testsEngineering scale simulation
4 Conclusions
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IntroductionMethodResults
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
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IntroductionMethodResults
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
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IntroductionMethodResults
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)
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IntroductionMethodResults
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)
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IntroductionMethodResults
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)
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IntroductionMethodResults
Conclusions
Model performance
FEMxDEM issues
Bad convergence
Computational time
Generic issues
Mesh dependency
FEMxDEM proposed solutions
Alternative Operator
Parallelization
Generic proposed solutions
Regularization
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IntroductionMethodResults
Conclusions
Model performance
FEMxDEM issues
Bad convergence
Computational time
Generic issues
Mesh dependency
FEMxDEM proposed solutions
Alternative Operator
Parallelization
Generic proposed solutions
Regularization
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IntroductionMethodResults
Conclusions
Newton OperatorParallelizationSecond Gradient
Outline
1 Introduction
2 MethodNewton OperatorParallelizationSecond Gradient
3 ResultsRandom field methodsParametric testsEngineering scale simulation
4 Conclusions
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IntroductionMethodResults
Conclusions
Newton OperatorParallelizationSecond Gradient
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)
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IntroductionMethodResults
Conclusions
Newton OperatorParallelizationSecond Gradient
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)
Albert Argilaga Claramunt FEMxDEM Modeling with Second Gradient 9 / 44
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IntroductionMethodResults
Conclusions
Newton OperatorParallelizationSecond Gradient
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)
Albert Argilaga Claramunt FEMxDEM Modeling with Second Gradient 9 / 44
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IntroductionMethodResults
Conclusions
Newton OperatorParallelizationSecond Gradient
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
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IntroductionMethodResults
Conclusions
Newton OperatorParallelizationSecond Gradient
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
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IntroductionMethodResults
Conclusions
Newton OperatorParallelizationSecond Gradient
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)
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IntroductionMethodResults
Conclusions
Newton OperatorParallelizationSecond Gradient
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
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IntroductionMethodResults
Conclusions
Newton OperatorParallelizationSecond Gradient
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)
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IntroductionMethodResults
Conclusions
Newton OperatorParallelizationSecond Gradient
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
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IntroductionMethodResults
Conclusions
Newton OperatorParallelizationSecond Gradient
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
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IntroductionMethodResults
Conclusions
Newton OperatorParallelizationSecond Gradient
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
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IntroductionMethodResults
Conclusions
Newton OperatorParallelizationSecond Gradient
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
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IntroductionMethodResults
Conclusions
Newton OperatorParallelizationSecond Gradient
Outline
1 Introduction
2 MethodNewton OperatorParallelizationSecond Gradient
3 ResultsRandom field methodsParametric testsEngineering scale simulation
4 Conclusions
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IntroductionMethodResults
Conclusions
Newton OperatorParallelizationSecond Gradient
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
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IntroductionMethodResults
Conclusions
Newton OperatorParallelizationSecond Gradient
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
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IntroductionMethodResults
Conclusions
Newton OperatorParallelizationSecond Gradient
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
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IntroductionMethodResults
Conclusions
Newton OperatorParallelizationSecond Gradient
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)
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IntroductionMethodResults
Conclusions
Newton OperatorParallelizationSecond Gradient
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
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IntroductionMethodResults
Conclusions
Newton OperatorParallelizationSecond Gradient
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
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IntroductionMethodResults
Conclusions
Newton OperatorParallelizationSecond Gradient
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
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IntroductionMethodResults
Conclusions
Newton OperatorParallelizationSecond Gradient
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
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IntroductionMethodResults
Conclusions
Newton OperatorParallelizationSecond Gradient
Conclusion
The parallelization provides an effective speedup of the model,becoming this competitive with classical FEM codes
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IntroductionMethodResults
Conclusions
Newton OperatorParallelizationSecond Gradient
Outline
1 Introduction
2 MethodNewton OperatorParallelizationSecond Gradient
3 ResultsRandom field methodsParametric testsEngineering scale simulation
4 Conclusions
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IntroductionMethodResults
Conclusions
Newton OperatorParallelizationSecond Gradient
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
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IntroductionMethodResults
Conclusions
Newton OperatorParallelizationSecond Gradient
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
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IntroductionMethodResults
Conclusions
Newton OperatorParallelizationSecond Gradient
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
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IntroductionMethodResults
Conclusions
Newton OperatorParallelizationSecond Gradient
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
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IntroductionMethodResults
Conclusions
Newton OperatorParallelizationSecond Gradient
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
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IntroductionMethodResults
Conclusions
Newton OperatorParallelizationSecond Gradient
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
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IntroductionMethodResults
Conclusions
Newton OperatorParallelizationSecond Gradient
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
!
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IntroductionMethodResults
Conclusions
Newton OperatorParallelizationSecond Gradient
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
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428
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600
685
771
857
942
* 1. 000E- 04
0
44
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314
359
404
449
494
* 1. 000E- 03
0
30
60
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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
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IntroductionMethodResults
Conclusions
Newton OperatorParallelizationSecond Gradient
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
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IntroductionMethodResults
Conclusions
Random field methodsParametric testsEngineering scale simulation
Outline
1 Introduction
2 MethodNewton OperatorParallelizationSecond Gradient
3 ResultsRandom field methodsParametric testsEngineering scale simulation
4 Conclusions
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IntroductionMethodResults
Conclusions
Random field methodsParametric testsEngineering scale simulation
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)
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IntroductionMethodResults
Conclusions
Random field methodsParametric testsEngineering scale simulation
Punctual imperfection
The DEM microstructure has a strong influence on the results
Figure: Homogeneous (a) Defect (b). Second invariant of strain
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IntroductionMethodResults
Conclusions
Random field methodsParametric testsEngineering scale simulation
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)
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IntroductionMethodResults
Conclusions
Random field methodsParametric testsEngineering scale simulation
Conclusions
The FEMxDEM approach provides a natural way to embedmaterial heterogeneity into the model
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IntroductionMethodResults
Conclusions
Random field methodsParametric testsEngineering scale simulation
Outline
1 Introduction
2 MethodNewton OperatorParallelizationSecond Gradient
3 ResultsRandom field methodsParametric testsEngineering scale simulation
4 Conclusions
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IntroductionMethodResults
Conclusions
Random field methodsParametric testsEngineering scale simulation
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
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IntroductionMethodResults
Conclusions
Random field methodsParametric testsEngineering scale simulation
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
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IntroductionMethodResults
Conclusions
Random field methodsParametric testsEngineering scale simulation
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
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IntroductionMethodResults
Conclusions
Random field methodsParametric testsEngineering scale simulation
Outline
1 Introduction
2 MethodNewton OperatorParallelizationSecond Gradient
3 ResultsRandom field methodsParametric testsEngineering scale simulation
4 Conclusions
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IntroductionMethodResults
Conclusions
Random field methodsParametric testsEngineering scale simulation
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
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IntroductionMethodResults
Conclusions
Random field methodsParametric testsEngineering scale simulation
Conclusions
Real scale simulations are possible thanks to the secondgradient and parallelization
FEMxDEM allows to calibrate the model directly from themicroscale
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IntroductionMethodResults
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
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IntroductionMethodResults
Conclusions
Perspectives
Calibration of the microscale with experimental data
3D microscale can bring a richer constitutive behabiour
More complex microscale with massive parallelization (MPI)
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IntroductionMethodResults
Conclusions
Thanks!!!
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