Geomechanics from Micro to Macro – Soga et al. (Eds)© 2015 Taylor & Francis Group, London, ISBN 978-1-138-02707-7

Bridging the micro and macro for granular media: A computationalmulti-scale paradigm

J.D. Zhao & N. GuoDepartment of Civil and Environmental Engineering, Hong Kong University of Science and Technology,Clearwater Bay, Kowloon, Hong Kong

ABSTRACT: We present a hierarchical multi-scale framework to model geotechnical problems relevant togranular media. The framework employs a rigorous hierarchical coupling between the Finite Element Method(FEM) and the Discrete Element Method (DEM). The FEM is used to discretise the macroscopic geometricdomain of a boundary value problem into a FEM mesh. A DEM assembly with memory of its loading historyis embedded at each Gauss integration point of the mesh to receive the global deformation from the FEM asinput boundary conditions and is solved for the incremental stress-strain relation at the specific material point toadvance the FEM computation. The hierarchical framework helps to avoid the phenomenological nature of con-ventional continuum approaches of constitutive modelling and meanwhile retains the computational efficiencyof the FEM in solving large-scale Boundary Value Problems (BVPs). By virtue of its hierarchical structure,the predictive power of the proposed method can be further unleashed with proper implementation of parallelcomputing techniques. By corroborating the rich information obtained from the particle level with the macro-scopically observed responses, the framework helps to shed light on a cross-scale understanding of granularmedia. We demonstrate the predictive capability of the proposed framework by simulations of shear localisa-tions in monotonic biaxial compression and cavity inflation in a thick-walled cylinder, as well as liquefactionand cyclic mobility in cyclic simple shear tests.

1 INTRODUCTION

Granular materials are multi-scale by nature. Whensubject to shear, a granular material may exhibit com-plicated macroscopic behaviours that are notoriouslydifficult to characterise, such as state dependency,strength anisotropy, strain localisation, non-coaxiality,solid-flow phase transition (e.g., liquefaction) andcritical state (Guo & Zhao 2013b, Zhao & Guo2013). These macroscopic responses reflects highlycomplicated microstructural mechanisms establishedand evolved at the granular particle level during theloading process. While a granular medium has longbeen treated by continuum mechanics, it becomesincreasingly clear now that a better understanding canonly be achieved with the aid of effective bridgingapproaches linking the micro to the macro scales ofthe material. Various homogenisation techniques havebeen developed in material sciences to link differentlength scales of a material for integrated character-isation of the material behaviour. They are targetedat designing engineered or new materials with identi-fiable microstructure to achieve optimal engineeringperformance of various purposes. However, severalintrinsic properties associated with granular mediaexclude the possibility of deriving material proper-ties and predicting the material responses directlyfrom the particle scale through analytical homogeni-sation methods as the material science branch does.

There are two outstanding ones: (1) The lacking ofperiodic microstructure in a granular material pre-vents a general homogenisation method working effec-tively, which is due mainly to the great randomnessand heterogeneity within a granular system; (2) Thebehaviour of a granular material is generally state-dependent and loading-path specific. It is thus difficultto identify a once-for-all microstructure from whichthe macroscopic properties of the material can bederived via homogenisation. To resolve the bridgingissue, a computational multi-scale modelling approachappears to be a viable (if not the only) option. In thisstudy, we propose a hierarchical multi-scale frame-work on micro-macro bridging for granular media.Theframework employs a rigorous hierarchical couplingbetween the finite element method (FEM) and the dis-crete element method (DEM) to solve boundary valueproblems relevant to granular media. The study is inline with a number of previous attempts on this topic(Meier et al. 2008, Meier et al. 2009, Andrade et al.2011, Nitka et al. 2011, Guo and Zhao 2013a).

2 METHODOLOGY AND FORMULATION

2.1 Hierarchical multiscale modelling approach

The multiscale framework employs a rigorous hier-archical coupling between the finite element method

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Figure 1. A hierarchical multiscale modelling frameworkfor granular media based on coupled FEM/DEM.

(FEM) and the discrete element method (DEM) whichis schematically shown in Fig. 1. To solve a boundaryvalue problem, the macroscopic geometric domain isfirst discretized into a FEM mesh. A DEM assem-bly is then embedded at each Gauss integration pointof the mesh serving as a local Representative VolumeElement (RVE). At each load step, the RVE takes itsmemory of the past loading history as initial condi-tions and receives the global deformation from theFEM at the specific Gauss point as input boundaryconditions. It is solved to derive the local incrementalstress-strain relation (e.g., stress and tangential stiff-ness matrix) required for advancing the global FEMcomputation. To be more specific, the FEM is used tosolve the following equation system for the discretiseddomain:

where K , u and f are the stiffness matrix, the unknowndisplacement vector at the FEM nodes, and the nodalforce vector lumped from the applied boundary trac-tion, respectively. Since K is generally dependent onthe state parameters and loading history for a granularmedium, linearisation of the solution and Newton-Raphson iterative method are commonly required. Indoing so, the tangent operator Kt needs to properlyevaluated:

where B is the deformation matrix (i.e. gradient ofthe shape function), and D is the matrix form ofthe rank four tangent operator tensor. During eachNewton-Raphson iteration, both Kt and σ are updatedto minimise the following residual force R to find aconverged solution

Instead of making phenomenological assumptionsfor the instantaneous tangent modulus Dep in Equa-tion 2 to assemble Kt as conventional continuum-based constitutive modelling approaches do (e.g., by

assuming an elasto-plastic stiffness matrix in clas-sic plasticity theory), the coupled FEM/DEM multiscale approach determines the two quantities from theembedded discrete element assembly at each Gausspoint. In doing so each DEM packing receives theboundary condition (deformation) by interpolation ofthe FEM solution (displacement). Upon reaching asolution, each DEM is then homogenised to derive thestress and tangent operator at the material point whichare transferred back to the FEM solver to update thesolution. The Cauchy stress tensor is homogenised bythe following expression

where V is the total volume of the DEM assembly, Ncis the number of contacts within the volume, f c anddc are the contact force vector and the branch vectorconnecting the centres of the two contacted particles,respectively.

In deriving the tangent operator, we use the follow-ing bulk elastic modulus homogenised from the DEMassembly as a first coarse estimation of the the tangentoperator:

where kn and kt are the equivalent normal and tan-gential stiffnesses describing the contact law of theparticles, nc and tc are unit vectors in the outwardnormal and tangential directions of a contact, respec-tively. ‘⊗’ denotes the dyadic product of two vectors.In conduction with an iterative scheme, it works moreefficient in deriving the tangent operator in compar-ison with the alternative perturbation-based method(Meier et al. 2009, Nitka et al. 2011, Guo and Zhao2014).

2.2 Solution procedure

The solution procedure to the hierarchical multi-scalemodeling approach is summarized as follows:

I. Discretise the macro domain by a suitable FEMmesh and embed a DEM assembly with appropriateinitial state at each Gauss point of the mesh as aRVE.

II. Apply one global load step imposed by the bound-ary traction of the FEM domain.

(a) Determine the current tangent operator usingEq. 5 for each RVE.

(b) Assemble the global tangent matrix using Eq. 2and compute a trial solution of displacement uby solving Eq. 1 with FEM.

(c) Interpolate the deformation ∇u at each Gausspoint of the FEM mesh and run the DEM sim-ulation for the corresponding RVE by account-ing for its initial state and using ∇u as the DEMboundary conditions.

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Figure 2. The calibrated RVE comprised of 400 polydis-perse spheres with periodic boundary produces isotropiccontact normal distribution under isotropic compression.

(d) Derive the updated total stress from Eq. 4 foreach RVE and use them to evaluate the residualby Eq. 3 for the FEM domain.

(e) Repeat the above steps from (a) to (d) untilconvergence is reached.

(f) Save the converged solution of each RVE asits new initial state for next step and finish thecurrent load step.

III. Advance to next load step and repeat Step II.

2.3 Benchmark and calibration

The proposed hierarchical multi-scale approach hasbeen successfully implemented by two open sourcecodes for FEM and DEM (Guo & Zhao 2014). A sim-ple linear force-displacement contact law in conjunc-tion with Coulomb’s friction is employed in the DEMcode to describe the stick-slip inter particle contact.Polydisperse particles with radii ranging from 3 mmto 7 mm (rmean = 5 mm) are adopted to generate eachDEM assembly. Quasi-static solutions are solved ateach load step for each RVE. To calibrate the sizeof the RVE (e.g., particle number), different RVEswith different particle numbers subject to isotropiccompression have been compared and the resultantcontact normal distribution is examined. A typicalRVE containing 400 particles and with periodic bound-ary is found to provide largely isotropic contactnormal distribution and meanwhile offer reasonablecomputational efficiency (see Fig. 2), which will beadopted for all the simulations in the sequel.

A single-element shear test based on first-orderfour-node quadrilateral element has been used tobenchmark the multi-scale method. All RVEs at thefour Gauss points possess identical initial conditions,which leads to a uniform sample for the FEM solution.The global response of the single element test is com-pared against that obtained from a pure DEM test basedon the RVE in Fig. 3. The global vertical stress σ11 iscalculated from the resultant force exerted on the topboundary and σ00 from the resultant lateral force. Theaxial strain ϵ11 and the volumetric strain ϵv are cal-culated from the overall deformation of the element.Fig. 3 shows that the multi-scale modelling producednearly identical responses with the pure DEM sim-ulation for the single element test, which indicates

Figure 3. Comparison of the macroscopic responses of thesingle-element test by the multiscale method and by a pureDEM test of RVE size.

that the hierarchical multi-scale modelling approachis able to faithfully reproduce the typical behaviour ofgranular media.

Meanwhile, in our hierarchical multi-scale frame-work, the computation of DEM packing attached toeach Gauss point is independent with one another,which makes it ideal for parallelisation of simulations.An effective parallel computing technique has beenimplemented in our multis-cale code and has been usedfor all the subsequent simulations with a cluster atHKUST. The scalability and efficiency of the paral-leled code has been found rather satisfactory (Guo &Zhao 2014).

3 DEMONSTRATIVE EXAMPLES

3.1 Monotonic biaxial compression on sand

We first apply the hierarchical multiscale approach topredicting the response of a sand sample subject tomonotonic biaxial compression. The model setup andboundary conditions are shown in Fig. 4 (left) wherethe FEM mesh consists of four-node linear quadri-lateral elements. Fig. 4 (right) presents the globalstress-strain response measured from the boundary

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Figure 4. Multiscale modelling of monotonic biaxial com-pression on sand: (left) model setup; (right) global stressstrain relationship for the sand sample (calculated from theresultants at the boundary).

Figure 5. Contours of the accumulated deviatoric strain andvoid ratio showing strain localisation in biaxial compressionof sand.

reaction forces and displacements. A comparison caseis also presented for the pure DEM simulation ofthe biaxial shear on a RVE size sample. Notably, theglobal response by the multiscale modelling approachbears great similarity to the RVE response, in partic-ular for the pre-peak stress stage where the materialbehaviour is relatively elastic. While the post-peakresponse of the DEM test shows moderate fluctuations,the multi-scale model results are relatively smooth.

Our multi-scale simulation captures the phe-nomenon of strain localisation that is commonly foundin laboratory biaxial shear tests (see Fig. 5). This isinteresting by the following reason. Our biaxial shearsimulation has been set up with smooth and symmet-ric boundary conditions and loading. The initial statesfor all RVE packings in the FEM mesh are identi-cal and hence the entire sample is homogeneous too.Under such symmetric/homogeneous conditions, con-ventional continuum-model-based FEM approachesare generally unable to capture the strain localisa-tion unless certain artificial imperfection(s) or randomdistributed local properties are added to the sampleto break the symmetry and trigger strain localisation.In our multi-scale modelling of the problem, thoughthe RVEs are identical and homogeneous in the FEMmesh, there is small but observable initial anisotropypresent in the initial RVE packing (see Fig. 2), whichmay serve as the symmetry breaker to trigger the occur-rence in our mutliscale modelling. Similar opinion hasbeen discussed by Gao & Zhao (2013) based on con-tinuum modelling.The localised region observed fromFig. 5 concentrates in the most dilative portion of thesample with large void ratio, which is consistent with

Figure 6. (a) The structure and force chains of the localinitial and deformed RVEs and (b) the corresponding distribu-tions of the contact normals in N◦51 and N◦256. The smoothred curves in (b) are the second-order Fourier approximations.

experimental observations. The localised band of voidratio is found generally wider than that of the deviatoricstrain.

A major advantage of the multi-scale frameworklies in the rich micro-scale information it can pro-vide when solving an engineering scale problem. Thismay greatly facilitate a better correlation and under-standing of the macroscopic observations. Shown inFig. 6 is a comparison of the contact force networkfor the RVE packings at the three Gauss points indi-cated in Fig. 4 at the initial and final loading states.All RVEs have the same initial isotropic conditionwithout apparent preferably orientated strong forcechains. While the DEM packing at N◦51 does notexperience much deformation, several distinct strongforce chains are observed aligning parallel to the ver-tical shear direction. In contrast, the RVE at N◦256experiences severe shear deformation which resultsin a highly heterogeneous internal structure and moreconcentrated penetrating force chains aligning to thevertical. The packing at N◦422 also deforms notice-ably, but it becomes rather dilute due to remarkablevolumetric expansion. The deformation gradients atN◦256 and N◦422 are consistent with the global shearband inclination. An inspection of the contact normaldistribution (shown in Fig. 6) further confirms that themajor principal directions for both N◦51 and N◦256 areclose to the vertical shear direction.

3.2 Cyclic simple shear test

The multiscale modelling approach has also beenapplied to simulating the hysteresis behaviour of sandsubject to cyclic load. Shown in Fig. 7 is a FEMmesh and boundary conditions for a sand sample sub-ject to maximum shear stress controlled cyclic simpleshear. The initial packing is chosen to be relativelyloose for all RVEs (different than the other exampleswhere dense packing is used). The global stress-strainresponse and loading path are shown in Fig. 8, indi-cating a typical hysteresis behaviour of sand observedin laboratory tests. The contour of the void ratio inFig. 9 indicates that the deformation in the sample

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Figure 7. Discretization (15 × 10 elements) and boundarycondition of the specimen for cyclic simple shear. The N◦390Gauss point will be used for local analyses.

Figure 8. Global responses of the cyclic simple shear testunder maximum shear stress control |σ01| = 30 kPa.

is relatively uniform. The RVE at N◦390 shows sev-eral obvious strong force chains aligning along thediagonal direction of the packing. As the accumu-lated shear strain becomes larger, it is evident thatcyclic mobility occurs for this sample. Not presentinghere due to length limitation, we have also investi-gated a comparison case where the same sample issubject to constant-volume maximum shear strain con-trolled cyclic shear wherein we found liquefaction

Figure 9. (a) Contour of void ratio at the final state after themaximum shear stress controlled cyclic loading and (b) thestructure and the force chains in RVE at N◦390.

occurs. At the liquified material points, the contactnetwork of the RVE become too week to form effec-tive percolating force chains to sustain the externalshear.

3.3 Cavity inflation in thick-walled hollow cylinder

The multi-scale approach has also been applied tomodelling the cavity inflation in thick-walled cylin-der as treated experimentally by Alsiny et al. (1992).Thick-walled hollow cylinder inflation experimentsare commonly performed towards a better understandthe soil behaviour under passive loading conditionswith simultaneous extension in a plane perpendicularto the loading direction. A quarter of a whole thick-walled hollow cylinder is simulated, with an identicalgeometry of that in Alsiny et al. (1992) for the cavityand outer surface of the cylinder: rc = 15 mm and ro =150 mm. Due to symmetry, the displacement of the leftboundary is fixed and the vertical displacement of thebottom boundary is fixed (see Fig. 10a). Eight-nodeelements were used in the study. The outer boundary isprescribed by a constant pressure p0 = 100 kPa. Differ-ent than the inflation pressure boundary inAlsiny et al.(1992), a Neumann boundary with gradually increaseddisplacement uc is applied to the inner surface untiluc = 10 mm. Fig. 10b depicts the inflation-pressureresponse (the differential cavity pressure is defined bythe pressure difference on the inner and outer cylinderwalls), which shows a clear softening portion of thecurve. Our simulation shows that the occurrence ofshear localisation in the cylinder was initiated shortlybeyond the peak inflation pressure, which is consistentwith the experimental finding by Alsiny et al. (1992).Since our simulation uses Neumann boundary at thecavity surface, no diffuse deformation mode has beendetected.

Fig. 11 presents the localised distributions of shearstrain, void ratio and average rotation (obtained fromthe RVE) within the cylinder at the end of the inflation.The three contours show apparently good correlations.Within the shear bands the soil is severely sheared,highly dilated (initial void ratio is 0.177) and expe-riences considerable rotation of soil particles. Theshear bands originated from the cavity surface formcross-shaped patterns which have been observed inlaboratory tests. The soil particles within the bands ofdifferent orientation have been rotated in totally differ-ent direction. Other than those close to the cavity, thereare two less developed, notable shear bands touching

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Figure 10. Multiscale simulation of cavity inflationof a thick-walled hollow cylinder in sand: (a) mesh;(b) pressure-inflation curve.

Figure 11. Localised distributions for (a) shear strain,(b) void ratio and (c) average rotation in the cylinder.

the outer surface of the cylinder. They are however notoriginated directly from the cavity surface, but fromthe two fixed boundaries.They appear to be reflectionsof the two major shear bands initiated from the cavitysurface. Further investigation based on a full cylinder

rather than its quarter will be carried out to see if this isthe case. The shear band orientation will also be anal-ysed based on more refined model of the problem inthe future.

4 CONCLUSIONS

A hierarchical multi-scale framework has been devel-oped to bridge the micro and macro behaviours ofgranular media for practical modelling of engineeringscale problems. Based on a rigorous coupling betweenFEM and DEM, it circumvents the phenomenolog-ical assumptions commonly required in continuumconstitutive modelling and meanwhile retains a goodpredictive capability on solving practical problemswhich purely micromechanics-based approaches areinherently restrained to solve.The framework has beenbenchmarked, calibrated and enhanced with parallelcomputing techniques. Its predictive capability hasbeen demonstrated with three illustrative examples,including monotonic biaxial compression test, cyclicsimple shear tests and cavity inflation in thick-walledhollow cylinder.

REFERENCES

Alsiny,A., I.Vardoulakis, &A. Drescher (1992). Deformationlocalisation in cavity inflation experiments on dry sand.Géotechnique 42(3): 395–410.

Andrade, J. E., C. F. Avila, S. A. Hall, N. Lenoir, &G. Viggiani (2011). Multiscale modeling and charac-terization of granular matter: from grain kinematics tocontinuum mechanics. Journal of the Mechanics andPhysics of Solids 59: 237–250.

Gao, Z. & J. Zhao (2013). Strain localization and fabricevolution in sand. International Journal of Solids andStructures 50: 3634–3648.

Guo, N. & J. Zhao (2013a). A hierarchical model for cross-scale simulation of granular media. In AIP ConferenceProceedings, Volume 1542, pp. 1222–1225.

Guo, N. & J. Zhao (2013b). The signature of shear-inducedanisotropy in granular media. Computers and Geotech-nics 47: 1–15.

Guo, N. & J. Zhao (2014). A coupled FEM/DEM approachfor hierarchical multiscale modelling of granular media.International Journal for Numerical Methods in Engi-neering In press. DOI: 10.1002/nme.4702.

Meier, H., P. Steinmann, & E. Kuhl (2009). On the multiscalecomputation of confined granular media. In J. Eberhard-steiner, C. Hellmich, H. Mang, and J. Périaux (Eds.),ECCOMAS Multidisciplinary Jubilee Symposium, Vol-ume 14 of Computational Methods in Applied Sciences,pp. 121–133. Springer Netherlands.

Meier, H. A., P. Steinmann, & E. Kuhl (2008). Towardsmultiscale computation of confined granular media–contact forces, stresses and tangent operators. TechnischeMechanik 28(1): 32–42.

Nitka, M., G. Combe, C. Dascalu, & J. Desrues (2011).Two-scale modeling of granular materials: a DEM-FEMapproach. Granular Matter 13: 277–281.

Zhao, J. & N. Guo (2013). Unique critical state character-istics in granular media considering fabric anisotropy.Géotechnique 63(8): 695–704.

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