FEMxDEM double scale approach with second gradient regularization applied to granular materials modeling

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Stefano Dal Pont Gaël Combe Denis Caillerie Jacques Desrues 16

1 FEMxDEM double scale approach with second gradient regularization applied to granular materials modeling Albert Argilaga Claramunt Stefano Dal Pont Gaël Combe Denis Caillerie Jacques Desrues 16 december 2016 Albert Argilaga Claramunt FEMxDEM Modeling with 1 / 44

2 Outline 1 Introduction 2 3 Random field methods Parametric tests Engineering scale simulation 4 Albert Argilaga Claramunt FEMxDEM Modeling with 2 / 44

Motivation Bridge the gap between continuum and

approaches No need of phenomenological laws Faithful

3 Motivation Bridge the gap between continuum and discrete approaches Continuum approaches Numerically efficient No scale limitations? Figure: Numerical modeling dilemma Discrete approaches No need of phenomenological laws Faithful representation of the microscale Albert Argilaga Claramunt FEMxDEM Modeling with 3 / 44

4 Multiscale coupling FEM: Lagamine (Liège University) A numerically obtained constitutive law is injected into the material points DEM with PBC (in-house 3SR) f n'/m' n m nʼ mʼ f n/m Albert Argilaga Claramunt FEMxDEM Modeling with 4 / 44

5 Introduction Discrete Element Model (DEM) Frictional cohesive contact laws Periodic Boundary Conditions Figure: DEM assembly preparation Albert Argilaga Claramunt FEMxDEM Modeling with 5 / 44

6 Introduction Discrete Element Model (DEM) )/ 11 2 ( Periodic Boundary Conditions Frictional cohesive contact laws (%) Figure: DEM biaxial response Figure: DEM assembly preparation Numerical homogenization 1 X ~m/n m σf = f (~r ~r n ) S (n,m) C Albert Argilaga Claramunt FEMxDEM Modeling with 5 / 44

7 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) Albert Argilaga Claramunt FEMxDEM Modeling with 6 / 44

8 Model performance FEMxDEM issues Bad convergence Computational time Generic issues Mesh dependency Albert Argilaga Claramunt FEMxDEM Modeling with 7 / 44

9 Model performance FEMxDEM issues Bad convergence Computational time Generic issues Mesh dependency FEMxDEM proposed solutions Alternative Operator Generic proposed solutions Regularization Albert Argilaga Claramunt FEMxDEM Modeling with 7 / 44

10 Outline 1 Introduction 2 3 Random field methods Parametric tests Engineering scale simulation 4 Albert Argilaga Claramunt FEMxDEM Modeling with 8 / 44

11 Link with classical geomechanics FEM codes We consider the quasi static evolution of a continuous medium which shares features withelastoplasticity and hypoplasticity. The constitutive equation provides the Cauchy stress σ f as a function of the deformation tensor F f in a loading step: ( ) F f σ f = S F f (1) Albert Argilaga Claramunt FEMxDEM Modeling with 9 / 44

12 Link with classical geomechanics FEM codes We consider the quasi static evolution of a continuous medium which shares features withelastoplasticity and hypoplasticity. The constitutive equation provides the Cauchy stress σ f as a function of the deformation tensor F f in a loading step: ( ) F f σ f = S F f (1) The non linear boundary value problem, is solved using the iterative Newton s method. This needs to differentiate S with respect to F f (1), the gradient of S is often called the consistent operator, it is denoted by C dσ f = S ( F f + df f ) S ( F f ) = C : df f + (2) Albert Argilaga Claramunt FEMxDEM Modeling with 9 / 44

13 Link with classical geomechanics FEM codes We consider the quasi static evolution of a continuous medium which shares features withelastoplasticity and hypoplasticity. The constitutive equation provides the Cauchy stress σ f as a function of the deformation tensor F f in a loading step: ( ) F f σ f = S F f (1) The non linear boundary value problem, is solved using the iterative Newton s method. This needs to differentiate S with respect to F f (1), the gradient of S is often called the consistent operator, it is denoted by C dσ f = S ( F f + df f ) S ( F f ) = C : df f + (2) Albert Argilaga Claramunt FEMxDEM Modeling with 9 / 44

14 Determination of the Consistent Operator Except for some cases for which closed-form expression of the tensor C can be found, the determining of C is performed numerically using a perturbation method: (C ijmn ) = S ( ij F f + ɛλ mn) ( S ij F f ) ɛ Assuming S to be differentiable, that determines C (3) Albert Argilaga Claramunt FEMxDEM Modeling with 10 / 44

15 Determination of the Consistent Operator Except for some cases for which closed-form expression of the tensor C can be found, the determining of C is performed numerically using a perturbation method: (C ijmn ) = S ( ij F f + ɛλ mn) ( S ij F f ) ɛ Assuming S to be differentiable, that determines C (3) Issues Bad convergence Computational time Mesh dependency Proposed solutions Alternative Operator Regularization Albert Argilaga Claramunt FEMxDEM Modeling with 10 / 44

16 Alternative s 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) Albert Argilaga Claramunt FEMxDEM Modeling with 11 / 44

17 Perturbation based operators Consistent Tangent (CTO) Classic perturbation approach: (C ijmn ) = S ( ij F f + ɛλ mn) ( S ij F f ) ɛ Auxiliary Elastic (AEO) (Nguyen, 2013) Average of the CTO over several macroscopic loading steps Albert Argilaga Claramunt FEMxDEM Modeling with 12 / 44

18 Elastic operators DEM Quasi Static (DEMQO) Based on numerical homogenization of an elastic DEM (C ijmn ) DEMQO = SE ij (ɛλ mn ) ɛ (4) PreStressed Truss-Like (PSTLO) (Caillerie) It treats the DEM as a prestressed linearelastic truss with additional rotational degrees of freedom in the nodes dσ = 1 Y c C δcd i f c Y ( ) i + σ F T df T F T : df σ (5) Albert Argilaga Claramunt FEMxDEM Modeling with 13 / 44

19 Kruyt operators Kruyt Augmented (KAO) (Caillerie) UKO with prestresses and rotational degrees of reedom Upper bound Kruyt (UKO) (Kruyt & Rothenburg, 1998) 1 Y a (l c ) 2 ( k n ( e c e c ) ( e c e c ( ) + k t t c e c) ( t c e c)) c C Upper bound Corrected Kruyt (UCKO - UCKO 2DOF) Calibrations of the UKO to fit a CTO (6) Albert Argilaga Claramunt FEMxDEM Modeling with 14 / 44

20 Convergence rates The convergence rate of the operators is compared in the initial state and post-peak in a biaxial compression test: OTC UKO UCKO UCKO 2DOF DEMQO/PTLO KAO AEO UKO UCKO UCKO 2DOF DEMQO/PTLO KAO Residual FNORM/RNORM 10 2 Residual FNORM/RNORM iteration iteration Albert Argilaga Claramunt FEMxDEM Modeling with 15 / 44

21 Operators performance Perturbation based operators perform well in the initial stages Kruyt operators are more robust in the post-peak part Elastic operators combine good performance before and after the stress peak # iterations OTC/ AEO UKO UCKO UCKO 2DOF DEMQO /PSTLO Initial Post-peak Figure: Operators comparison: number of iterations KAO Albert Argilaga Claramunt FEMxDEM Modeling with 16 / 44

Conclusion 14 12 10 The elastic operators overcome the issues giving good stability and convergence velocity # iterations 8 6 4 2 0 OTC/ AEO UKO UCKO UCKO 2DOF

22 Conclusion The elastic operators overcome the issues giving good stability and convergence velocity # iterations OTC/ AEO UKO UCKO UCKO 2DOF DEMQO /PSTLO Initial Post-peak KAO Figure: Operators comparison: number of iterations Albert Argilaga Claramunt FEMxDEM Modeling with 17 / 44

23 Outline 1 Introduction 2 3 Random field methods Parametric tests Engineering scale simulation 4 Albert Argilaga Claramunt FEMxDEM Modeling with 18 / 44

24 FEMxDEM drawbacks The evaluation of S is issued from a DEM homogenization: F f σ f = S (F ) f Albert Argilaga Claramunt FEMxDEM Modeling with 19 / 44

25 FEMxDEM drawbacks The evaluation of S is issued from a DEM homogenization: F f σ f = S (F ) f Issues Bad convergence Computational time Mesh dependency Albert Argilaga Claramunt FEMxDEM Modeling with 19 / 44

26 FEMxDEM drawbacks The evaluation of S is issued from a DEM homogenization: F f σ f = S (F ) f Issues Bad convergence Computational time Mesh dependency Proposed solutions Alternative Regularization Albert Argilaga Claramunt FEMxDEM Modeling with 19 / 44

27 Element loop parallelization Amdahl s law (Amdahl, 1967) Sequential Parallel S(n) = 1 B + 1 n (1 B) (7) lim S(n) = n (8) B 0 S: speedup n: number of cores B: non parallelizable part Walltime (min) x Intel Xeon CPU E GHz (2 x 4 cores) 2x Intel Xeon CPU E GHz (2 x 6 cores) Other Approaches: Massive : MPI (Desrues) Figure: speedup: x7.15 (8 cores), x9.06 (12 cores) Albert Argilaga Claramunt FEMxDEM Modeling with 20 / 44

28 Numerical randomness Theorem Commutative Law of Addition: a + b = b + a Albert Argilaga Claramunt FEMxDEM Modeling with 21 / 44

29 Numerical randomness Does not apply! Commutative Law of Addition: a + b = b + a Albert Argilaga Claramunt FEMxDEM Modeling with 21 / 44

30 Numerical randomness Does not apply! Commutative Law of Addition: a + b = b + a Figure: Simulated numerical error Albert Argilaga Claramunt FEMxDEM Modeling with 21 / 44

numerical error Figure: Different possible solutions of the same

31 Numerical randomness Does not apply! Commutative Law of Addition: a + b = b + a Figure: Simulated numerical error Figure: Different possible solutions of the same BVP. Cumulative second invariant of strain Albert Argilaga Claramunt FEMxDEM Modeling with 21 / 44

32 Conclusion The parallelization provides an effective speedup of the model, becoming this competitive with classical FEM codes Albert Argilaga Claramunt FEMxDEM Modeling with 22 / 44

33 Outline 1 Introduction 2 3 Random field methods Parametric tests Engineering scale simulation 4 Albert Argilaga Claramunt FEMxDEM Modeling with 23 / 44

34 First order FEM * E- 03 * E Figure: Non objectivity of the model Albert Argilaga Claramunt FEMxDEM Modeling with 24 / 44

35 First order constitutive relation Balance equation in virtual power formulation: ui, σ ij ɛ ijdω = t i ui ds (9) Ω Where ui is the kinematically admissible virtual displacement field, σ ij the stress field, ɛ ij the virtual strain field, Ω the surface of Ω, t i the surface forces Ω Albert Argilaga Claramunt FEMxDEM Modeling with 25 / 44

36 First order constitutive relation Balance equation in virtual power formulation: ui, σ ij ɛ ijdω = t i ui ds (9) Ω Where ui is the kinematically admissible virtual displacement field, σ ij the stress field, ɛ ij the virtual strain field, Ω the surface of Ω, t i the surface forces Issues Bad convergence Computational time Mesh dependency Ω Albert Argilaga Claramunt FEMxDEM Modeling with 25 / 44

37 First order constitutive relation Balance equation in virtual power formulation: ui, σ ij ɛ ijdω = t i ui ds (9) Ω Where ui is the kinematically admissible virtual displacement field, σ ij the stress field, ɛ ij the virtual strain field, Ω the surface of Ω, t i the surface forces Ω Issues Bad convergence Computational time Mesh dependency Proposed solutions Alternative Regularization Albert Argilaga Claramunt FEMxDEM Modeling with 25 / 44

38 microstructured materials Second gradient regularization introduces an internal length that regularizes the model First order term ui, Ω σ ij ɛ 2 u i ij + Σ ijk dω = t i ui ds x j x k Ω where Σ ijk is the double stress dual of 2 ui x j x k Albert Argilaga Claramunt FEMxDEM Modeling with 26 / 44

39 microstructured materials Second gradient regularization introduces an internal length that regularizes the model First order term ui, Ω Second order term σ ij ɛ 2 u i ij + Σ ijk dω = t i ui ds x j x k Ω where Σ ijk is the double stress dual of 2 ui x j x k Albert Argilaga Claramunt FEMxDEM Modeling with 26 / 44

40 in a Boundary Value Problem As it is a local model, it can be injected in the material points in the same way as the first gradient relation is! Albert Argilaga Claramunt FEMxDEM Modeling with 27 / 44

41 * E * E * E * E * E * E Introduction Biaxial test enriched with Figure: 128, 512, and 2048 elements biaxial compression tests Albert Argilaga Claramunt FEMxDEM Modeling with 28 / 44

42 With the enrichment the model becomes objective. This allows to use any mesh refinement The code is ready to be used in real scale problems Albert Argilaga Claramunt FEMxDEM Modeling with 29 / 44

43 Random field methods Parametric tests Engineering scale simulation Outline 1 Introduction 2 3 Random field methods Parametric tests Engineering scale simulation 4 Albert Argilaga Claramunt FEMxDEM Modeling with 30 / 44

Random field methods Parametric tests Engineering scale simulation Introduction: Symmetry breaking Material properties in numerical models are spatially homogeneous unless the contrary is defined The

44 Random field methods Parametric tests Engineering scale simulation Introduction: Symmetry breaking Material properties in numerical models are spatially homogeneous unless the contrary is defined The use of a catalyzer to break the symmetry can improve the model performance, both physically and numerically Figure: A CT scan of a sand sample (Andò, 2013) Albert Argilaga Claramunt FEMxDEM Modeling with 31 / 44

45 Random field methods Parametric tests Engineering scale simulation Punctual imperfection The DEM microstructure has a strong influence on the results Figure: Homogeneous (a) Defect (b). Second invariant of strain Albert Argilaga Claramunt FEMxDEM Modeling with 32 / 44

Random field methods Parametric tests Engineering scale simulation Full field DEM random variability Figure: (a) FEMxDEM and pure DEM responses

46 Random field methods Parametric tests Engineering 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) Albert Argilaga Claramunt FEMxDEM Modeling with 33 / 44

47 Random field methods Parametric tests Engineering scale simulation The FEMxDEM approach provides a natural way to embed material heterogeneity into the model Albert Argilaga Claramunt FEMxDEM Modeling with 34 / 44

48 Random field methods Parametric tests Engineering scale simulation Outline 1 Introduction 2 3 Random field methods Parametric tests Engineering scale simulation 4 Albert Argilaga Claramunt FEMxDEM Modeling with 35 / 44

49 Random field methods Parametric tests Engineering scale simulation Outer radius The outer radius has an influence on the results, a compromise size is chosen Figure: Outer radius=50, 100 and 200 m. Cumulative second invariant of strain Figure: Radius=50, 100 and 200 m M.S. research project Nurisa, 2015 Albert Argilaga Claramunt FEMxDEM Modeling with 36 / 44

Figure: Coordination number = 4.10, 3.67 and 3.14.

50 Random field methods Parametric tests Engineering scale simulation Micro/macro DEM allows to tune the coordination independently from the density The model can cope with any macroscopic boundary conditions Figure: Coordination number = 4.10, 3.67 and Cumulative second invariant of strain Figure: Stress ratio σ 0H : σ 0V = 1:1, 1:1.3 and 1:2. Cumulative second invariant of strain Albert Argilaga Claramunt FEMxDEM Modeling with 37 / 44

q Introduction Random field methods Parametric tests Engineering

892 2.5 X: 0.003001 Y: 2.33 2 1.5 1 0.5 0 0 0.001 0.002 0.003 0.

51 q Introduction Random field methods Parametric tests Engineering scale simulation Intrinsic anisotropy X: Y: X: Y: Figure: Precharge 0%, 0.3% and 0.6%. Cumulative second invariant of strain ε yy Figure: DEM preparation Albert Argilaga Claramunt FEMxDEM Modeling with 38 / 44

52 Random field methods Parametric tests Engineering scale simulation Outline 1 Introduction 2 3 Random field methods Parametric tests Engineering scale simulation 4 Albert Argilaga Claramunt FEMxDEM Modeling with 39 / 44

53 Random field methods Parametric tests Engineering scale simulation Hollow cylinder with * 1.000E-04 Y X Figure: Localization around a gallery. FEM: 4950 elements, nodes, DOF Figure: 400 particles DEM microscale at the end of the loading Albert Argilaga Claramunt FEMxDEM Modeling with 40 / 44

54 Random field methods Parametric tests Engineering scale simulation Real scale simulations are possible thanks to the second gradient and parallelization FEMxDEM allows to calibrate the model directly from the microscale Albert Argilaga Claramunt FEMxDEM Modeling with 41 / 44

55 General A FEMxDEM approach allows to use a constitutive law built via numerical homogenization in the DEM microscale An alternative Newton operator provides the model with good convergence rate and robustness allows the model to compete with classical FEM approaches in terms of walltime A Second gradient regularization assures the mesh independence Real scale problems with fine meshes can be considered Albert Argilaga Claramunt FEMxDEM Modeling with 42 / 44

56 Perspectives Calibration of the microscale with experimental data 3D microscale can bring a richer constitutive behabiour More complex microscale with massive parallelization (MPI) Albert Argilaga Claramunt FEMxDEM Modeling with 43 / 44

57 Thanks!!! Albert Argilaga Claramunt FEMxDEM Modeling with 44 / 44

Hierarchical Multiscale Modeling of Strain Localization in Granular Materials: ACondensedOverviewandPerspectives

Hierarchical Multiscale Modeling of Strain Localization in Granular Materials: ACondensedOverviewandPerspectives Jidong Zhao Abstract This paper presents a brief overview on hierarchical multiscale modeling