static and dynamic studies for coupling discrete and continuum media; application to a simple...

International Journal of Solids and Structures 47 (2010) 276–290

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International Journal of Solids and Structures

journal homepage: www.elsevier .com/locate / i jsolst r

Static and dynamic studies for coupling discrete and continuum media;Application to a simple railway track model

Mohammad Hammoud *, Denis Duhamel, Karam SabUniversité Paris-Est, UR Navier, Ecole des Ponts ParisTech, 6 et 8 Avenue Blaise Pascal, Cité Descartes, Champs sur Marne, 77455 Marne La Vallée, Cedex 2, France

a r t i c l e i n f o a b s t r a c t

Article history:Received 4 December 2008Received in revised form 18 September2009Available online 5 October 2009

Keywords:Discrete modelContinuum modelStaticDynamicCouplingMultiscale

0020-7683/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.ijsolstr.2009.09.036

* Corresponding author. Tel.: +33 1 64 15 37 36; faE-mail address: [email protected] (M. Ham

In this paper, we present a formulation for coupling discrete and continuum models for both dynamic andstatic analyses. This kind of formulation offers the possibility of carrying out better simulations of mate-rial properties than the discrete calculations, and with both larger length scales and longer times. Usingonly a discrete approach to simulate a large medium composed of many degrees of freedom seems verydifficult in terms of calculation and implementation. Moreover, using only a continuum approach doesnot give an accurate solution in a zone where particular and localized phenomena can occur. A directapplication of our coupling approach to the case of railway track models subjected to an external load,is proposed for its validation.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

The main objective of the modeling of modern materials is topredict the response and failure of materials which are governedby deformation mechanisms. For many material simulations, it isvery difficult to make a discrete calculation because of large Repre-sentative Volume Element sizes and important computation times.Moreover, using only a continuum approach does not always pro-vide an accurate simulation for the response of the system beingstudied. These two main problems forced researchers to reconsiderthe advantages of each approach and to recognize that a couplingmethodology had to be established that would combine the advan-tages of both discrete and continuum modelings. Although thisfield has been studied in the past, it still remains an active areaof research (see Kohlhoff and Schmauder, 1989; Shenoy et al.,1999; Frangin et al., 2006; Klein and Zimmerman, 2006; Cundalland Stack, 1979; Tadmor et al., 1996; Broughton et al., 1999; Fishand Chen, 2004 and Miller and Tadmor, 2009). In our study, weare interested in using a coupling method applicable to a simplemodel for high-speed train tracks consisting in a beam resting onvery large number of springs (see Figs. 1 and 2).

Existing coupling methodologies can be classified as:Bottom-up methods, top-down methods, and direct methods

ll rights reserved.

x: +33 1 64 15 37 41.moud).

The idea of the bottom-up methods is to solve the non-linearequations at the macroscopic scale by the extraction of the behav-ior laws from an atomic description at the microscopic scale.

Tadmor et al. (1996) developed the quasicontinuum method(QC). The QC uses Finite Elements (FE) representation of the dis-placement field over the entire domain, requiring mesh refinementto the atomic scale in regions of severe deformation. The strainenergy within the element is determined from a single ‘‘represen-tative atom” embedded in a locally constructed crystallite. Consis-tency between refined and coarse areas is achieved by using thefinite deformation elasticity and the Cauchy-Born rule that equatesinteratomic bond energy to continuum potential energy in order todevelop a non-linear continuum constitutive model based on theinteratomic potential used for atomistic simulations. While theQC approach allows a blending between atomistic and continuumregions, it has the disadvantages of relying on an adaptive meshrefinement to the atomic scale, a computationally intensive task,and an inability to eliminate fictitious boundary effects at the lo-cal/non-local boundary.

The bridging scale decomposition (BSD) approach was first usedby Wagner and Liu (2003). The starting point of the method in-volves the use of a bridging subdomain in which the Hamiltonianis chosen as a linear combination of discrete and continuum Ham-iltonians. In the bridging domain, discrete element (DE) degrees offreedom and finite element (FE) ones are linked by Lagrange mul-tipliers. Numerical methods are employed to solve the problem ofspurious wave reflections which appear at the interface due to thesize of the discontinuities of the discretization. The (BSD) is more

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Fig. 1. Proposed model (1D); discrete modelling.

M. Hammoud et al. / International Journal of Solids and Structures 47 (2010) 276–290 277

developed by Xiao and Belytschko (2004) and later Frangin et al.(2006). In the bridging domain, the compatibility is enforced by La-grange multipliers or by an augmented Lagrangian method.

The idea of the top-down method is to treat the atoms per unitcell as ‘‘coarse grains” and to construct the associated energywhich converges to the exact atomic energy in order to derivethe atomic equations. A brief description of some of these methodsfollows.

Rudd and Broughton (1998) developed the coarse-grainedmolecular dynamic method (CGMD) which consists of replacingthe underlying atomic lattice with nodes representing either indi-vidual atoms or a weighted average of a collection of atoms. The to-tal energy of the system is calculated from the potential and kineticenergies of the nodes in addition to a thermal energy term repre-senting the missing degrees of freedom assumed to be at a uniformtemperature.

Broughton et al. (1999) introduced the molecular atomistic ab-initio dynamic (MAAD) approach. The MAAD approach separatesthe physical system into distinct MD and FE regions. The totalHamiltonian of the system consists of contributions from eachindividual region as well as a contribution from the hand shaking(Bridging domain) between regions. The FE mesh in this hand-shaking zone is refined to the atomic scale and the nodes occupythe positions where the atoms would be if the atomic region wereextended into the FE domain. Kinetic energy is attributed to bothnodes and atoms in the hand-shaking zone, while further from thiszone, uniform temperature terms are added to account for themissing degrees of freedom. This approach has successfully per-formed non-reflective transmissions of elastic waves betweenMD and FE regions.

The direct methods consist of the decomposition of a spatial do-main into subdomains; a continuum domain, an atomic domainand finally a hand-shake domain.

(Ben Dhia and Rateau, 2001 and Ben Dhia and Rateau, 2005)introduced the Arlequin method as a flexible engineering designtool. This method is able to introduce local defects (such as cracks,holes or inclusions) with great flexibility in a global existing coarsemodel. Also this method is able to change the local behavior in aglobally simplified model of a given material.

In the Arlequin method the total energy of the system is formu-lated as follows:

0

1

Ω 2Ω sΩ 1

Model 2

Hand shaking domain

Model 1

Fig. 2. Arlequin method: o

Esystem ¼ EðX1nX2Þðu1Þ þ EðX2nX1Þðu2Þ þ a1EðXsÞðu1Þ þ a2EðXsÞðu2Þ

EðX1nX2Þðu1Þ and EðX2nX1Þðu2Þ are the potential energy of the field solu-tions ðu1Þ and ðu2Þ in the domains X1 and X2 without the intersec-tion domain Xs, respectively. EðXsÞðu1Þ and EðXsÞðu2Þ are the potentialenergy of the intersection domain of the field solutions ðu1Þ andðu2Þ, respectively.

In the above paragraphs we presented the coupling methodssuccessfully employed to simulate material deformations such ascrack-grain boundary interactions, dislocation nucleations fromnanoindentation and the dynamic fracture of silicon. However,the weaknesses of these methods shows that more work is neededto develop a coupling atomistic-continuum approach.

As already mentioned, the improper partitioning of the system’spotential energy leads to the appearance of non-existant forcesacting on atoms and nodes within the overlap region. These forcesare often referred to as ‘‘ghost forces” and are often the continueddevelopment of atomistic/continuum coupling methods. For in-stance, the recent review article by (Curtin and Miller (2003) andMiller and Tadmor (2009)) describes the origins and effects ofthe ghost forces that arise from using the QC method. They are alsorevised in an approach by Shenoy et al. (1999) to determine correc-tions that can be introduced to the QC methodology to compensatefor ghost forces. This approach involves the use of dead loads equaland opposite to the ghost forces determined from the undeformedconfiguration of the system. While the introduction of this correc-tion is noteworthy, it inevitably leads to inaccuracies once thecrystal becomes deformed, even for homogeneous loading condi-tions, or if the lattice is subjected to any rotation. Also, Curtin dis-cuss the developpement by Knap and Ortiz (2001) of a fully non-local formulation of the QC method. This approach avoids usingthe Cauchy/Born rule and instead determines nodal forces by con-structing a small cluster of representative atoms surrounding anode and calculating the force using the non-local, atomisticdescription.

In all the approaches presented above, it is evident that the is-sue of how to partition energy within atomistic-continuum overlapregions needs to be addressed properly in order to maintain theintegrity of the two views of material deformation, discrete andcontinuum, and to obtain accurate solutions.

S

α1

Ω 1

α2

Ω 2

α = 1 2 Ω 2 Ω 1\

α = 1 1 Ω 1 Ω 2\

α +1 α = 12 Ω 1 Ω 2

on

on

on

verlapping domains.

278 M. Hammoud et al. / International Journal of Solids and Structures 47 (2010) 276–290

In our proposed coupling approach, due to the reasons men-tioned earlier, the mechanical parameters of the system beingstudied will be calculated in an indirect way that does not requirethe calculation of the energy and avoids the problem of how to par-tition this energy between the discrete and continuum zones.

In this paper, after this introduction of the coupling existingmethodologies, we present a 1D railway track model (see Ricciet al. (2005), Nguyen and Duhamel (2006), Nguyen and Duhamel(2008), Al Shaer et al. (2008) and Bodin-Bourgoin et al. (2006)).This model consists of a beam resting on many elastic springs.The deflection of the beam (as well as the nodal parameters) is cal-culated with two approaches; a discrete approach and a macro-scopic approach deduced from the discrete one. First, acomparison between the responses of the system obtained byusing these two approaches is made in order to determine thecases where the macroscopic approach cannot replace the discreteone. Then, we apply a Discrete/Continuum coupling method tothese cases. Finally, numerical results are presented in order to val-idate and prove the efficiency of the proposed coupling method.

2. Discrete and continuum formulations

2.1. Discrete approach

A beam resting on springs and on which we apply a load F isshown in Fig. 1. It represents a railway under which the track tieand the ballast layer are modeled by elastic springs as the supportsof the beam. Firstly, the applied load is assumed to be fixed, so inthis case a static problem is studied.

2.1.1. Static solutionThe static equilibrium equation of the discrete approach is writ-

ten as follows:

EIuð4ÞðxÞ þXN

i¼1

hkiuðxiÞdðx� xiÞ ¼ Fdðx� DÞ ð1Þ

In Fig. 1, L and xi are the total length of the beam and the posi-tions of the springs along the beam (with x ¼ 0 being the left end-point of the beam) respectively. In Eq. (1) D, h and ki are thedistance of the loading force from the left end of the beam, thespacing between consecutive track ties, and the stiffness of thesprings respectively. N represents the number of track ties.

First, let us consider two adjacent elements of the beam (seeFig. 3). A concentrated force F acts vertically on one of the elementsat a distance Y from the left-hand end point of the element. Thenwe seek to calculate the deflection which minimizes the totalmicroscopic energy of the beam, the springs, and the load.

First a relationship between any two consecutive vectors ofparameters must be established. A vector of parameters consistsof the vector which contains the parameters of each node; thedeflection, rotation, bending moment, and shear force.

2.1.1.1. Formulation of the parameters and of the stiffness matrix. Inthis paragraph, we are interested in solving the 4th order differen-tial equation (1) analytically. The third derivative of the deflectionuðxÞ is not continuous on the segment ½0; h1� on which the load F is

Fig. 3. Two adjacent elements of beam.

applied. It is discontinuous before and after the crossing point be-tween the two segments.

Consequently:

u000ðxÞ ¼ A on ½0; Y��u000ðxÞ ¼ P on ½Yþ; h1�

�ð2Þ

where A and P are two constants. By applying 3 times the integraloperation on u000ðxÞ ¼ A and u000ðxÞ ¼ P, this gives us a system oftwo equations containing eight variables. This system is formulatedas follows:

uðxÞ ¼ Ax3

6þ B

x2

2þ Cxþ D on ½0;Y�� ð3Þ

uðxÞ ¼ Pðx� h1Þ3

6þ Qðx� h1Þ2

2þ Rðx� h1Þ þ S on ½Yþ;h1� ð4Þ

To find the eight unknown coefficients which exist in Eqs. (3)and (4), we suppose that u; u0; u00 and u000 have known values atthe node 0, so u000ð0Þ ¼ A; u00ð0Þ ¼ B; u00ð0Þ ¼ C and uð0Þ ¼ D. Tothese boundary conditions is added the conditions of continuityat the point of load ‘‘Y” on u; u0; u00 and a jump condition on u000.These conditions of continuities are formulated in the system ofEq. (5).

uðYþÞ ¼ uðY�Þu0ðYþÞ ¼ u0ðY�Þu00ðYþÞ ¼ u00ðY�ÞP � A ¼ F

EI

8>>><>>>: ð5Þ

Finally, these conditions added to the relations between differentvectors of parameters (see Appendix A), lead us to the following val-ues of P, Q, R and S.

S ¼ uðhÞ ¼ uð0Þ þ hu0ð0Þ þ h2

2EIu00ð0Þ þ h3

6EIu000ð0Þ � F

ðY � hÞ3

6EI

R ¼ u0ðhÞ ¼ u0ð0Þ þ hu00ð0Þ þ h2

2EIu000ð0Þ þ F

ðY � hÞ2

2EI

Q ¼ u00ðhÞ ¼ u00ð0Þ þ hu000ð0Þ � FðY � hÞ

EIð6Þ

P ¼ u000ðhÞ ¼ u000ð0Þ þ FEI

A jump condition between two consecutive beam elements createdby the spring of stiffness k is given by Eq. (7). It will be used later inthe stiffness matrix.

u000ðh�1 Þ � u000ðhþ1 Þ ¼kEI

uðh�1 Þ ð7Þ

Indeed uð0Þ; u0ð0Þ; ð�EI u00ð0ÞÞ and ð�EIu000ð0ÞÞ represent the deflec-tion u0, the rotation h0, the bending moment M0 and the shear forceT0 respectively.

By using Eqs. (3) and (4) that give the form of the deflection uand by considering the jump condition in Eq. (7), a relationship be-tween the vector of force F01 ¼ T0 M0 T1 M1½ �T and the vectorof displacement U01 ¼ u0 h0 u1 h1½ �T in the element can be for-mulated as follows:

F01 ¼ K01U01 þ R ð8Þ

where the matrix K01 and the vector of R are written as follows:

K01 ¼ EI

12h3

6h2 � 12

h36

h2

6h2

4h � 6

h22h

� 12h3 � 6

h212h3 � k1

EI � 6h2

6h2

2h � 6

h24h

2666664

3777775; R ¼ F

�2Y3þ3Y2h�h3

h3

�Y3þ2Y2h�Yh2

h2

2Y3�3Y2hh3

�Y3þY2hh2

26666664

37777775 ð9Þ

K01 is the stiffness matrix. It depends only on the stiffness of spring‘‘k1” and the spacing between two consecutive springs ‘‘h”. If the


external load is applied within the segment, we assume the exis-tence of the vector R associated with the applied load F.

We can simplify Eq. (8) to a beam resting on N springs by con-sidering that the elements of the connection matrix depend onlyon ki and h.

Fiiþ1 ¼ Kiiþ1Uiiþ1 þ R load is applied within the segmentFiiþ1 ¼ Kiiþ1Uiiþ1 otherwise

�ð10Þ

2.1.2. Dynamic solutionIn this section, a simplified dynamic study is considered. The

applied load is harmonic and we are interested in the harmonic dy-namic response as the real part of uðxÞeixt . So the dynamic equilib-rium equation is formulated as follows:

EIuð4ÞðxÞ þXN

i¼1

hkiuðxiÞdðx� xiÞ � qx2SuðxÞ ¼ Fdðx� DÞ ð11Þ

q; S; x are the density of the steel, the section of the rail and theangular frequency of the wave exciting the beam, respectively.

The characteristic polynomial of the differential equation (11),where the solution takes an exponential form enx, is:

n4 � qx2SEI¼ 0 ð12Þ

Eq. (12) possesses four complex roots; nj ¼ rj þ iqj where rj and qj

represent the attenuation and the propagation part of the waverespectively. The semi analytical solution of the differential Eq.(11) takes the following exponential form:

uðxÞ ¼ aenx þ be�nx þ ceinx þ de�inx ð13Þ

where a; b; c and d are constants that must be calculated at eachelement in order to find the deflection of the discrete approach inthe dynamic case.

Let us consider two adjacent elements of the beam (see Fig. 4).As in the static case we have to consider two cases:

� First case: the load is applied within the beam at the point of load‘‘Y” (see Fig. 4). The three conditions of continuities onuðxÞ; u0ðxÞ and u00ðxÞ and the jump condition on u000ðxÞ are thesame as those in Eq. (5).

A relationship between the vectors g and U0 can be established.U0 consists of the vector of deflection uð0Þ, rotation u0ð0Þ, bendingmoment ð�EIu000Þ and shear force ð�EIu0000 Þ at the first node andg ¼ ½ab c d�.

The relationship between g and U0 is deduced from the solutionof the diffrential Eq. (13) and formulated in matrix form in Eq. (14):

U0 ¼

u0

h0

M0

T0

2666437775 ¼

1 1 1 1n0 �n0 in0 �in0

�EIn20 �EIn2

0 EIn20 EIn2

0

�EIn30 EIn3

0 iEIn30 �iEIn3

0

2666437775

ab

cd

2666437775 ¼ R1g ð14Þ

Similarly, a relationship between the vectors ~g ¼ ½~a ~b ~c ~d� and Uh isestablished. It is written in the following matrix form:

Fig. 4. Conditions of continuity at node 1.

Uh ¼

uh

hh

Mh

Th

2666437775 ¼

a0 b0 c0 d0

n0a0 �n0b0 in0c0 �in0d0

�EIn20a0 �EIn2

0b0 EIn20c0 EIn2

0d0

�EIn30a0 EIn3

0b0 iEIn30c0 �iEIn3

0d0

2666437775

~a~b~c~d

2666437775 ¼ R4~g

ð15Þ

where a0 ¼ en0L1 ; b0 ¼ e�n0L1 ; c0 ¼ ein0L1 and d0 ¼ e�in0L1 .The matrix R4 is calculated using the general solution of the dif-

ferential equation (13) uðxÞ and its derivative u0ðxÞ; u00ðxÞ and u000ðxÞat the node L�1 .

Using the equalities in Eqs. (5), (14) and (15), the following rela-tionship between U0 and Uh is found:

Uh ¼ R4R�13 R2R�1

1 U0 þ R4R�13

FEI

ð16Þ

where R1 and R4 are the matrices calculated in Eqs. (14) and (15), R2

and R3 are the matrices calculated in the Appendix A (see Eq. (44)).In here R2 ¼ R3, thus Eq. (16) is simplified as:

Uh ¼ R4R�11 U0 þ R4R�1

3FEI

� Second case: The load is applied outside the first element

In this case, only Eqs. (14) and (15) are used to link the vector Uh

to U0. This relationship is written as follows:

Uh ¼ R4R�11 U0 ð17Þ

Finally, we generalize the relations in Eqs. (16) and (17) for a beamresting on N springs as follows:

Uiþ1 ¼ R4R�11 Ui þ R4R�1

3FEI

load within the beam

Uiþ1 ¼ R4R�11 Ui otherwise

8<: ð18Þ

The relationship between the vector of force Fiiþ1¼ Ti Mi Tiþ1 Miþ1½ �T

and the vector of displacement Uiiþ1¼ ui hi uiþ1 hiþ1½ �T is calculatedusing numerical methods during the simulation in MATLAB code.

2.2. Macroscopic approach

A macroscopic approach is deduced from the discrete one at themacroscopic scale. In this approach, we proceed by the homogeni-zation of the beam relative to the stiffnesses of the springs (Fig. 5).In the microscopic approach we start with an enormous number ofdegrees of freedom (DoF), whereas homogenisation is used to re-place the zones that have homogeneous DoF with only one DoF,which will then reduce the required computing time.

2.2.1. Static solutionAs with the discrete approach, the static equilibrium equation

of the global system is written as follows:

EIuð4Þh þ kðxÞuh ¼ Fdðx� DÞ ð19Þ

kðxÞ is the microscopic stiffness function of the node positions. To beon the macroscopic scale, it is best to calculate the limit of expres-sions which are functions of hðuð4Þh ;uhÞ in Eq. (19) when h! 0. So, toresolve the 4th order differential Eq. (19) and to calculate these lim-its, let us consider the vector V ¼ ½u u0 u00 u000�T .

By replacing the derivative of V in Eq. (19), the 4th order differ-ential equation is transformed into a 1st order differentialequation:

V 0 ¼ MðxÞV þ �e 2 R4 ð20Þ

MðxÞ and �e are the stiffness matrix and the vector related to the loadF respectively.

Fig. 5. Macroscopic beam with the stiffness of springs.


MðxÞ ¼

0 1 0 00 0 1 00 0 0 1��kðxÞ 0 0 0

2666437775 and �e ¼ F

EI0 0 0 dðx� DÞ½ �T

where �kðxÞ is the microscopic stiffness and is equal to1EI

PNi¼1hkidðx� xiÞ. The general solution VðxÞ of the 1st order differ-

ential equation established in Eq. (20) is formulated as follows:

VðxÞ ¼ expZ x

0MðsÞds

� �aðxÞ ð21Þ

The derivative of VðxÞ in Eq. (21) is written as follows:

dVdx¼ exp

Z x

0MðsÞds

� �a0ðxÞ þMðxÞVðxÞ ð22Þ

By identifying V 0ðxÞ in Eqs. (20) and (22), we obtain the expressionof a0ðxÞ. By applying the integral operation on a0ðxÞ, the functionaðxÞ is written as follows:

aðxÞ ¼ a0 þZ x

0exp �

Z y

0MðsÞds

� �dy �e ð23Þ

The integralR x

0 MðsÞds is approximately equal to x � M �, where� M �¼ 1

x

R x0 MðsÞds is the average of the matrix M on the interval

½0 x�. Finally by introducing the value of aðxÞ in Eq. (21), the generalsolution of the 1st order differential equation VðxÞ becomes:

VðxÞ � expðx � M �Þa0þ � M��1ðexpðx � M �Þ � 1Þ�e ð24Þ

An identification regarding the derivative of VðxÞ calculated in theEq. (24); ðV 0ðxÞ ¼� M � V þ �eÞ and Eq. (20) shows that the generalsolution VðxÞ does not change if we replace MðxÞ by its average� M � on the interval ½x� dx; xþ dx�. So the macroscopic stiffnessis given by KðxÞ ¼� �kðxÞ �¼ 1

2 dx

R xþdxx�dx

�kðxÞdx.Considering the above remarks, if we replace the microscopic

stiffnesses ki by its local average in a defined interval (see Fig. 6),we prove that the solution of the macroscopic approach is closeto the solution of the discrete approach.

The final form of the static differential equation is given asfollows:

uð4ÞðxÞ þ KðxÞuðxÞ � 1EI

Fdðx� DÞ ¼ 0 ð25Þ

For the segments where KðxÞ ¼� �kðxÞ � is constant, the generalsolution of Eq. (25) takes an exponential form:

Fig. 6. Relation between microscopic and macroscopic stiffnesses.

uðxÞ ¼ eðlxÞ½a cosðlxÞ þ b sinðlxÞ� þ eð�lxÞ½c cosðlxÞþ d sinðlxÞ� ð26Þ

a; b; c; d are the numerical parameters that must be calculated ateach element in order to find the deflection of the macroscopic ap-

proach and l ¼ K4

� �14.

We use the same formulation of parameters as with the dy-namic case of the discrete approach, but it has a different solutionfor the differential equation. By considering the same relationestablished in Eq. (14), the matrix R1 becomes:

eR1 ¼

1 0 1 0l0 l0 �l0 l0

0 �2l20EI 0 2l2

0EI

2l30EI �2l3

0EI �2l30EI �2l3

0EI

2666437775 ð27Þ

The relations established between Eqs. (15) and (18) remain validwith the only changes in the form of the matrices R2; R3 and R4.The new form of the matrix R4 is written:

eR4¼

ab ac bd dc

l0ðab�acÞ l0ðacþabÞ �l0ðdbþcdÞ l0ðdb�cdÞ2l2

0EIac �2l20EIab �2l2

0EIcd 2l20EIbd

2l30EIðacþabÞ 2l3

0EIðac�abÞ 2l30EIðcd�dbÞ �2l3

0EIðdbþcdÞ

2666437775

where a¼expðl0L0Þ; b¼cosðl0L0Þ; c¼sinðl0L0Þ and d¼expð�l0L0Þ.The matrices eR2 and eR3 take the same form as the matrix ~R4 by

replacing the term L0 for the variables a, b, c and d, with the term Y.

2.2.2. Dynamic solutionThe resolution of the macroscopic problem in the harmonic dy-

namic case does not change significantly when compared to thedynamic case of the discrete approach. Changes will take place atthe level of the rigidity matrix, especially in the value of the stiff-ness. The dynamic equlibrium equation is written as follows:

EIuð4Þh þ KðxÞuh � qx2Suh ¼ Fdðx� DÞ ð28Þ

The characteristic polynomial of the differential equation is written:

f4 þ K � qx2SEI

¼ 0 ð29Þ

Eq. (29) possesses four complex roots; fj ¼ aj þ ibj where aj and bj

represent the attenuation and the propagation waves respectively.The semi analytical solution of the differential equation in Eq.(28) has the following form:

uðxÞ ¼ Aefx þ Be�fx þ Ceifx þ De�ifx ð30Þ

f is the new form of the wavenumber; it is a function of the averageof the discret stiffness K and the frequency of the wave:

f ¼ �Kþqx2SEI

� �14.

The relations of Eq. (14) through (18) established in the dy-namic case of the discrete approach are still valid. The only change


is for the value of f where the stiffness of a spring ‘‘k” is replaced bythe average of homogeneous stiffness ‘‘K”.

3. Coupling approach

It is expected that there will be a difference between the behav-ior of beam models using the discrete approach and the macro-scopic approach. Those using for both dynamic and static cases.This difference is proved later in the section by the numerical re-sults. For these, a coupling method between these approacheswhich enables us to reproduce similar behavior for both discreteand coupled approach is recommended. This coupling formulationmust take into account several factors: the accurate reproductionof the behavior of the system, the reduction of the number ofDoF and the computation time. Moreover, in the dynamic case,the problem of possible reflection of waves at the interface of thecoupling domains must be considered.

3.1. Numerical tools for the coupling approach

Let us consider an element of the beam composed of two nodesmodelled by the macroscopic approach. Its equivalent in the dis-crete approach depends eventually on the ratio between the sizeof the discrete and macroscopic elements. Initially, we calculateall the nodal parameters using the macroscopic approach. For theelement of the beam under consideration, we take the values ofthe forces on both end nodes, insert these values in the discrete ap-proach, and then run the calculation.

Tjd ¼ Ti

m and Tj�4d ¼ Ti�1

m

Mjd ¼ Mi

m and Mj�4d ¼ Mi�1

m

(ð31Þ

i and j represent the number of a macroscopic node and its discreteequivalentrespectively as shown in Fig. 7. Ti

d and Tjm are the discrete

and macroscopic forces at the ith and jth nodes respectively.Thanks to this easy way, we are able to calculate the values of

the nodal parameters in the discrete approach, for both ends ofthe considered beam element. The absolute error for the deflectionand the rotation is calculated for the beam element ½ðn� ratioÞ; n�by taking the difference between the exact calculation using themacroscopic approach called Uh and the discrete approximatedfrom macroscopic data using the discrete approach called eUd. Thiserror is a function of the DE/ME ratio. The error function is formu-lated as follows:

e ¼P2

i¼1jUih � eUi

djP2i¼1jU

ihj

ð32Þ

Uh is the displacement vector calculated using the macroscopic ap-proach; it is written: Uh ¼ ½uhhh�T and eUd is the displacement vectorcalculated using the discrete approach written: eUd ¼ ½~ud

~hd�T . DE andME are the abbreviations of discrete and macroscopic elementrespectively.

Fig. 7. Beam element ½c � 1; c� for the macroscopic approac

This criterion which is used as a numerical tool is necessary forthe choice of the appropriate approach in each element.

3.2. Algorithm of resolution

Regardless of the problem to be treated, the first stage of thecoupling approach consists of a discrete description of the prob-lem. From the discrete modelling, we deduce the macroscopic ap-proach at the coarse scale. The choice of the ratio between the sizeof the DE and ME is based on the close reproduction of the behaviorby the macroscopic approach. Once this ratio is fixed, the macro-scopic approach becomes the reference for the coupling approach.Firstly, the mechanical parameters on the first element are calcu-lated by applying a criterion of coupling. This criterion can be sum-marized as follows: If the deflection and rotation errors betweenthe discrete and macroscopic approaches are lower than 10%, thescale of computation is not changed, but if this error is higher than10%, the discretization is refined, ie a decrease in the size of themacroscopic element. This procedure of refinement is used as longas is necessary in order to be placed on the scale of the discrete ele-ments. In what follows, the numerical algorithm for the couplingapproach (see Fig. 8) is presented.

The objective of this algorithm is to produce an approach whichcombines between an approach on the macroscopic scale and an-other one on the microscopic scale. The first approach is used tomodel the regular zones, the second one will be used in zoneswhere some irregularities can occur (for example, large variationof the spring stiffnesses).

4. Numerical results

4.1. Comparison between the discrete and macroscopic approaches

We implemented these approaches and elaborated several testbenches in static and dynamic cases in a MATLAB code. We testedmany situations of heterogeneous, homogeneous, or oscillatingstiffnesses. In the case of heterogeneities under the railways, it isclear that the two approaches lead to different behavior, especiallywhen the ratio between the number of discrete and macroscopicelements increases. This difference is illustrated more particularlyin zones with heterogeneities.

In this section, we develop test cases and illustrate the differ-ence in the behavior between these approaches. Table 1 showsthe value of parameters needed in the numerical computation.

4.1.1. Numerical validation of the discrete and macroscopicapproaches

Two numerical tests are carried out to ensure the consistency ofthe proposed model. In the static case, we consider a beam withlength L, fixed at one of its extremities, and with a load F is appliedto its other one. By calculating this structure, the analytical deflec-tion is obtained as follows:

Uanalytical ¼F

6EIx2 3L� xð Þ ð33Þ

h; ½c � 4; c� for the discrete approach. Ratio (ED/ME)=4.

Associatedmacroscopic approach

Application of numerical tools

Localisation

discrete elementsloop over all new

NextElement

Integration of a differentialsystem (Internal variables)

loop over alldiscrete elements

of the macroscopic approachDiscrete computation by refinement

if e>10%if e<10%

Fig. 8. Numerical algorithm for the coupling approach.

Table 1Mechanical and numerical parameters used in the numerical simulations.

Parameters Values Units

Young’s modulus of steel Al Shaer et al. (2005) Esteel ¼ 210 GPaQuadratic moment of a section I ¼ 1:65� 10�5 m4

Applied load Nguyen and Duhamel (2008),Al Shaer et al. (2008)

F = 80 KN

Beam length L = 120 mSpace between tie track h = 0.6 mDiscrete stiffness Nguyen and Duhamel (2006) k ¼ 104—5� 105 N/m

Macroscopic stiffness K ¼� k � N/mRatio between DE and ME numbers R ¼ 1; 2; 3; 4; 7; 9Mass per unit of surface qS ¼ 60:3 kg/mFrequence of the propagating wave f ¼ 0:5—1 HzDamping of the steel Young’s modulus m ¼ 0:05—0:1Damping of the spring’s stiffness n ¼ 0:1—0:3


In the dynamic case, we consider the same beam, but we applythe load at the mid point of the beam. This beam is fixed at bothends. The analytical solution has the same form as Eq. (13). So toevaluate the solution we have to calculate the vector of constants½A B C D�. It is calculated by using the following equality:

A

B

C

D

2666437775 ¼

a b �ac �bc

�fd fa fdc �fac

f2EIc f2EId �f2EIcc �f2EIdc

�f3EIb f3EIc f3EIbc �f3EIcc

2666437775�1 0

00F

2666437775 ð34Þ

where c is the point of the load,

a ¼ cosðfcÞ � coshðfcÞ; ac ¼ cosðfðc � LÞÞ � coshðfðc � LÞÞ;b ¼ sinðfcÞ � sinhðfcÞ; bc ¼ sinðfðc � LÞÞ � sinhðfðc � LÞÞ;c ¼ cosðfcÞ þ coshðfcÞ; cc ¼ cosðfðc � LÞÞ þ coshðfðc � LÞÞ;d ¼ sinðfcÞ þ sinhðfcÞ and dc ¼ sinðfðc � LÞÞ þ sinhðfðc � LÞÞ:

In this example, the discrete and macroscopic stiffnesses areequal 0 and the boundary conditions are represented by the block-ing of the rotation and the deflection at the first node. In Fig. 9, aperfect match is observed. In Fig. 9(b), we can see the influenceof the damping of Young’s modulus and the stiffness of the springson the amplitude of the wave.

4.1.2. Cases of good matching between the two approaches4.1.2.1. Homogeneous stiffness; low and high values. In this first typeof test, we examined two subcases:

� Low value of stiffness: Its value is considered 100 times smallerthan its real value (Boussinesq solution Ricci and Sab, 2006). Thedeflection and the rotation at the two ends of the beam areblocked.

� High value of stiffness: Its value is the same as the value of theBoussinesq solution (Ricci and Sab, 2006). Similar to the preced-ing test, the deflection and the rotation at the two ends of thebeam are blocked.

0 10 20 30 40 50 60 70 80 90 100−18

−16

−14

−12

−10

−8

−6

−4

−2

0D

efle

ctio

n va

lue

(mm

)

Node position (m)

Comparison between deflection in different approaches

Discrete deflectionMacroscopic deflectionAnalytical deflection

0 20 40 60 80 100 120−8

−6

−4

−2

0

2

4

6

8 x 10−5

Node position (m)

Def

lect

ion

valu

e (m

)

No Damped DiscreteDamped DiscreteDamped Analytical

a b

Fig. 9. Numerical validation of the proposed model with the analytical solution: (a) static case and (b) dynamic case.


In these subcases, there is always a perfect match between themacroscopic and discrete approaches for all node parameters(deflection, rotation, bending moment and constraints in spring)as we can see in the tables of Fig. 10 for the static and dynamiccases. This good match is not valid for the shear forces due to thediscontinuity at the third derivative of the deflection uðxÞ.

In the case of high values for stiffness, the deflection takes asharp form as shown in Fig. 10(a) and the affected zone is nearthe applied load F. In the case of lower values for stiffness,Fig. 10(b) shows that the affected zone is very large compared tothat obtained in the case of high values stiffness.

4.1.3. Cases of significant difference between approaches4.1.3.1. Stiffness with an area of weakness around the applied load. Inthis case, an area of stiffness with low values around the appliedload F is considered. Furthermore the stiffness is assumed homog-enous with high values. In situ, the zone of weakness can exist dueto an absence or bad distribution of the ballast under the tie track,or due to a worn tie track. Several positions of the load F are alsotested.

Fig. 11(a) shows the difference from the deflection of the beamin the static case. Fig. 11(b) shows the variation of the error be-tween different parameters calculated using macroscopic and dis-crete approaches. For example, the error on the deflection isevaluated as follows:

e ¼PN

i¼1judi � uh

i jPNi¼1jud

i jð35Þ

uh and ud are the macroscopic and discrete deflection respectively.We also tested the influence of the size of the zone of heteroge-

neity on the behavior of the ballast studied via discrete and macro-scopic approaches. When the ratio between the number of DE andME was increasing, the weakness zone was fixed just at the pointof load, the difference between the approaches becomes signifi-cant. However, when the size of the weakness zone around the ap-plied load was increasing, the difference decreased substantially asshown in Fig. 11(b). We can obtain a good agreement when the sizeof the weakness zone becomes significant compared to the lengthof the beam studied.

4.1.3.2. Oscillating stiffness. In this case, the stiffness is consideredan oscillating function, as written in Eq. (36):

Ki ¼ cos2pih

3

� �þ 1

� �105 þ 103 ðN m�1Þ ð36Þ

This oscillating function has a period of 3. Firstly, tests are doneby considering the same number of DoF in the two approaches andby supposing that the macroscopic stiffness oscillates identically tothe microscopic stiffness.

A good match between all node parameters is shown inFig. 12(a). However by increasing the value of the DE/ME ratio,we can conclude a difference as shown in Fig. 12(b) where this ra-tio increases progressively.

4.1.4. Error evolution for the criterion of the coupling methodFig. 13 shows the evolution of the error of the deflection and

rotation of the beam element considered in Section 3.1 which isnecessary for the numerical criterion of the coupling apporach.

4.2. Numerical validation of the coupling approach

4.2.1. Test benchesBased on the algorithm of resolution, we implement the static

and the harmonic dynamic studies of the coupling approach in aMATLAB code. In this section we present some cases studied. Acomparison between discrete and coupling solution is always donein order to prove the efficiency of the coupling approach. We studythe cases where the macroscopic approach cannot reproduce thesame behavior as the discrete one.

4.2.2. Validation exampleFirstly a sample test is done to validate the numerical imple-

mentation of the coupling approach. We consider the exampledealt with Section 4.1.3.1, the case where we have a homogeneousstiffness with a zone of heterogeneity at the point of load.

In this example we implement the coupling approach where weuse the discrete approach in the zone of heterogeneity ð45—55 mÞand further we use the macroscopic approach. The size of the dis-crete zone is between ð40 and 60 mÞ. A good match between thecoupling and the discrete behaviors is shown in Fig. 14.

4.2.3. Stiffness with a zone of heterogeneity around the applied loadThis proposed test is very close to reality. This situation often

occurs when a tie is either broken or moved under the railway,as well as when a vacuum is created under a tie. Numerically, thesereal problems are replaced by heterogeneities in the stiffnesses of

0 20 40 60 80 100 120−0.2

−0.15

−0.1

−0.05

0

0.05

Def

lect

ion

valu

e (m

m)

Node position (m)

Discrete approachMacroscopic approach

0 20 40 60 80 100 120−7

−6

−5

−4

−3

−2

−1

0

1

Def

lect

ion

valu

e (m

m)

Node position (m)


0 20 40 60 80 100 120−10

−8

−6

−4

−2

0

2

4

6x 10 −3

Node position (m)

Def

lect

ion

valu

e (m

)


a

c

b

Fig. 10. (a) and (b) Deflection in case of high and low stiffness values in the static case and (c) deflection in case of low stiffness in the dynamic case.

0 20 40 60 80 100 120−4.0

−3.5

−3.0

−2.5

−2.0

−1.5

−1.0

−0.5

0

0.5

Def

lect

ion

valu

e (m

m)

Node position (m)


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Size of the unhomogeneous area side of the applied load

erro

r co

mm

ited

betw

een

para

met

ers

Ratio between Discrete and Homogeneous Elements; ED/EF = 4

Deflection errorRotation errorBending moment errorForce error

a b

Fig. 11. (a) Deflection in two approaches; ratio DE/ME = 3 and (b) error evolution versus the ratio.


0 20 40 60 80 100 120−20000

−15000

−10000

−5000

0

5000Fo

rce

in th

e sp

ring

s (N

)

Node position (m)


0 20 40 60 80 100 120−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5 x 104

Forc

e in

the

spri

ngs

(N)

Node position (m)


a b

Fig. 12. (a) and (b) Forces in the springs; case of oscillating stiffness; ratio = 1 and ratio = 4 respectively.

1 2 3 4 5 6 7 8 90

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Err

or o

n th

e be

am e

lem

ent o

n w

hich

the

load

is a

pplie

d

Ratio between the number of DE and ME

Deflection errorRotation error

1 2 3 4 5 6 7 8 90

0.1

0.2

0.3

0.4

0.5

0.6

0.7E

rror

on

the

beam

ele

men

t on

whi

ch th

e lo

ad is

app

lied

Ratio between the number of DE and ME

Deflection error Rotation error

a b

Fig. 13. Evolution of the deflection and rotation errors, function of the ratio between the DE and ME: (a) static case and (b) dynamic case.


the springs which replace the ties and the grains of ballast. In theprevious comparison, we concluded on a perceptible difference be-tween the discrete behavior and the macroscopic one. It is pro-posed to study this case via the coupling approach.

In the numerical computation, we conclude that the couplingapproach is able to detect the place of these heterogeneities. In thiscase, a refinement of the scale of computation is also observed atthese heterogeneities as we can see in Fig. 15(b). Indeed, in theexample simulated in Fig. 15(b), we have employed some local het-erogeneties where the stiffness is considered to be between 10 and30 times less than that of the rest of the beam. The coupling ap-proach has succesfully detected their locations. In Fig. 15(a), a goodmatch between the discrete deflection and the coupling one in thedynamic case is observed. This result is similar to that of the staticcase.

4.2.4. Stiffness with arbitrary valuesIn this case, we consider the stiffness of the discrete approach to

be an arbitrary function which varies between two values (kmin andkmax).

ki ¼ ðkmax � kminÞ � randðn;1Þ þ kmin ð37Þ

The macroscopic stiffness is the average of the local microscopicstiffnesses as proved in Section 2.2.1. By comparing the discrete

and the macroscopic solution in the static and dynamic cases, wehave observed a difference between the different node parameters.Then, the coupling approach is employed to make a better simula-tion than the macrosopic one. Fig. 16 shows the good matching be-tween the discrete and coupling solutions.

It should be noted that the coupling approach is applied in thecase of homogeneous stiffnesses. In the comparison of the discreteand macroscopic solutions, it is concluded that the macroscopicapproach adequatly replaces the discrete one. Once this couplingapproach is applied, we find that no refinement of the macroscopicscale computation is needed.

4.2.5. Error evolutionHere, we consider the example dealt within Section 4.2.3. We

calculate the error at each node between the discrete and the cou-pled deflection and rotation. At the first iteration, this error is max-imized, because the coupling approach still the same as themacroscopic with coarse element. While increasing the iterationnumber, we can observe that this error decreases substantially.Fig. 17 shows the evolution of the deflection and rotation errorsin the static case. In the dynamic case, the evolution of the deflec-tion and rotation errors is similar to that of the static case.

0 20 40 60 80 100 120−0.4

−0.35

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05D

efle

ctio

n va

lue

(mm

)

Node position (m)

Discrete approachMacroscopic approachCoupling approach 0 20 40 60 80 100 120

−12

−10

−8

−6

−4

−2

0

2

4

6x 10−3

Node position (m)

Def

lect

ion

valu

e (m

)

Discrete approach

Coupling approach

Macroscopic approach

a b

Fig. 14. Validation of the numerical implementation of the coupled approach: (a) static case and (b) dynamic case.

0 20 40 60 80 100 120−12

−10

−8

−6

−4

−2

0

2

4

6x 10−3

Node position (m)

Def

lect

ion

valu

e (m

)

Discrete approach

Coupling approach

0 20 40 60 80 100 1200

1

Type

of u

sed

appr

oach

Node positions (m)

Discrete = 1Macroscopic = 0

Discrete dof=36Macroscopic dof=41

a b

Fig. 15. Example of heterogeneties in the stiffnesses of the springs: (a) deflection calculated via discrete and coupling approaches (dynamic case) and (b) type of scale used ateach node in the coupled approach (static case).


4.2.6. Spurious wave reflectionAn obvious challenge that arises from the problem of coupling

between fine and coarse scales, is that of the reflexion of the waveat the coupling interface. For the correct propagation and represen-tation of a wave in an element, it is necessary that its wavelengthbe at least 5 times larger than the size of this element. Then, if awave carries out a brutal passage from a fine scale to a coarseone, the problem of the reflexion wave must be taken into account.In our case, the coupled approach is at its base a macroscopic ap-proach with coarse elements. Thus, the propagation of a wavethrough the elements of this mesh means that its wavelength isadapted to this type of mesh. When its obliged to become refinedin mesh, the wave will not have a problem to continuing to prop-agate in the new mesh, because the wavelength in this case is rep-resented by a number of elements higher than that of the startingdiscretization.

In conclusion, the reflexion of the wave of scale variations be-tween coarse-fine is not a problem in this study. If a problem ofreflexion arises, it means that from the begining the wavelength

is not adapted to the mesh and the problem does not come fromthe coupling scale. The absence of spurious waves in the couplingapproach is viewed as an advantage.

4.2.7. Gain in the number of DoFsIn this section, we elaborate the same test benches elaborated

in Section 4.1.1. In all these tests, the starting approach is macro-scopic with coarse elements. A comparison between discrete andcoupling approaches is always done. The objective of this approachis to reproduce accurately the behavior of the structure whiledecreasing the degrees of freedom (DoFs) and the computing time.A good match between the mechanical parameters is obtained. Ineach test, the ratio between the number of CE (Coupling Element)and DE (Discrete Element) is calculated. Table 2 shows the evolu-tion of this important factor. It also shows that the value of thisgain ratio is oscillating between 2 and 3.

Heterogeneity and r mean the test case where we have hetreo-geneity in the stiffness and the ratio of the DE to the FErespectively.

0 20 40 60 80 100 120−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08N

ode

rota

tion

(Â°)

Node position (m)

Discrete approachCoupling approach

0 20 40 60 80 100 120−8

−6

−4

−2

0

2

4x 10−3

Node position (m)

Def

lect

ion

valu

e (m

)

Discrete approach

Coupling approach

a b

Fig. 16. Comparison between deflection and rotation calculated in a discrete and coupling approaches: (a) rotation in static case and (b) deflection in dynamic case.

0 20 40 60 80 100 1200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Abs

olut

e er

ror

on th

e de

flec

tion

Node positions (m)

1st iteration2nd iteration4th iteration

0 20 40 60 80 100 1200

0.1

0.2

0.3

0.4

0.5

0.6

0.7A

bsol

ute

erro

r on

the

rota

tion

Node positions (m)

1st iteration2nd iteration4th iteration

Fig. 17. Evolution of the deflection and rotation errors, function of the iteration numbers; Static case.


5. Extension to 2D problems

After validation of our proposed coupling method on the 1Drailway model, an extension to a 2D problem is in progress. Westudy a 2D model which consists of a regular lattice of square rigidgrains interacting by their elastic interfaces (see Fig. 18).

Two models have been developed, a discrete one and a contin-uous one. In the discrete model, the grains which form the latticeare modeled as rigid bodies connected by elastic interfaces (elasticthin joints). In other words, the lattice is seen as a ‘‘skeleton” inwhich the interactions between the rigid grains are representedby forces and moments which depend on their relative displace-

Table 2Influence of the coupling approach on the number of DoFs.

Test cases Dis approach Mac approach

DE ME

Heterogeneity (r = 4) 200 50Heterogeneity (r = 7) 211 31Heterogeneity (r = 9) 217 25Oscillating (r = 4) 200 50Oscillating (r = 7) 211 31

ments and rotations. The continuous model is based on the homog-enization of the discrete model (Cecchi and Sab (2009), Cecchi andSab (2006)). Considering the case of singularities within the lattice(a crack for example), we will develop a coupling model which usesthe discrete model in singular zones (zones where the discretemodel cannot be homogenized), and the continuous model else-where. Here a brief description of the coupling model is presented.

5.1. Principle of the coupling method

The medium is decomposed in two regions. The first one is thecontinuum region modeled by finite elements (rectangular with 2

Coupling approach

Total DE ME Gain

77 36 41 2.673 50 23 2.973 54 19 385 50 35 2.479 57 22 2.7


DoFs by node), the second is the discrete region where the DE arethe centre of grains (3 DoFs at the center of grains). At the interfacebetween these zones, interpolated DE are used to link the FE of thecontinuum zone to the DE of the discrete zone (see Figs. 19 and 20).

Noting Ediscrete the elastic energy of the discrete zone, Econtinuum

the elastic energy of the continuum zone and Einteraction the energyof the interaction between the DE and the FE, the total energy ofthe coupled medium is written:

Etotal ¼ Ediscrete þ Econtinuum þ Einteraction ð38Þ

The interaction energy between two DEs (- and +) is written asfollows:

Einteraction ¼ 12

U�

Uþ

� �T

½Kinterface�U�

Uþ

� �ð39Þ

Kinterface is the stiffness matrix of the interface between two adja-cents grains. U� and Uþ are the vectors of displacements and rota-tion of the grains (�) and (+), respectively.

If we consider a FE modeled by DEs, a relationship between thedisplacement of the FE node’s (h) and the displacement of the DE( created inside the FE) can be established by interpolation, usingthe shape functions. By noting ½U;V ;W�T the vector of displace-ment and rotation of a DE and ½u1;v1;u2;v2;u3;v3;u4;v4�T the vec-tor of nodal displacement of a FE, the relationship writes:

U;V ;W½ �T ¼ D½u1;v1;u2;v2;u3; v3;u4;v4�T ð40Þ

D is an interpolation matrix.At the same time, each DE located at the edge of the discrete

zone ðBDÞ is connected to an interpolated DE located at the edgeof the continuum zone ðBCÞ by adding half of the interaction energy(39) to the total elastic energy.

Thus, from these two relationships, a DE located in the discretezone is linked to a FE in the continuum zone. If we use Eq. (40) forthe interpolated DE (U� or Uþ), then the interaction energy (39) be-tween the DE and the FE will be a quadratic function of UD and UC.

UD and UC are the global displacements vector of the discreteand continuum zones respectively. By designing KD and KC the dis-crete and continuum stiffness matrices, the total energy of themedium will be:

y2

Fig. 18. Square grains form

ð41Þ

KC-D is the global matrix of interaction which is calculated by thesummation of all elementary interaction energies between DE andFE.

As like as for the 1D methodolgy, a criterion of coupling isdeveloped to limit the size of the discrete zone used in the singularzone. The idea is to apply discrete external forces and moments onthe DE located at the edege of a FE near the interface zone and tocompare the discrete responses of the grains inside the FE to theirinterpolated FE responses.

The external loading is computed as follows: Using Eq. (40), thedisplacements at the center of the interpolated DE created in the FEcan be calculated. From the interaction energy formulated in (39),we calculate the interaction forces and moment between these 2DE using the relation ðF ¼ ½Kinterface�:½Uþ;U��TÞ. All the interactionforces between a DE (�) and an external interpolated DE () atthe edge of FE are computed and assembled to form the externalglobal load applied on the discrete zone included in the FE.

Using the discrete model, we calculate the discrete displace-ments of the DE noted as Udiscrete

approximated. After that, we calculate thedifference between the interpolated continuum displacements inEq. (40) ðU continuum

interepolatedÞ at the center of grains and Udiscreteapproximated. This

difference will be the criterion for coupling. It is formulated asfollows:

error ¼Udiscrete

approximated � Ucontinuuminterpolated

Udiscreteapproximated

ð42Þ

If this error is more than 10%, the scale of computation will be thatof the discrete model. In the other case, the continuum scale of com-putation is adapted. Due to this criterion, the size of the discrete re-gion is controled and the number of DoF is reduced.

More details of discrete and continuum models will be the ob-ject of a future publication.

y1

ing the regular lattice.

Fig. 19. Regular lattice of square grains modeled by a coupling discrete/continuum model; (d) are the DE of the region ðBDÞ, (s) are the interpolated DE of the ðBIÞ and (h) arethe finite element nodes of the region ðBCÞ.

Fig. 20. ðf ; mÞ are the forces and the moments of interaction between DEs inside the considered FE (d) and interpolated DEs (s) inside adjacent FEs.


6. Conclusion

In this paper, we presented a formulation used for coupling dis-crete and continuum mediums. This coupling approach can beused to study models which represent some irregularities (hetero-geneity, crack and so on). For a direct application of this approach,we considered a beam resting on tie tracks and on grains of ballastmodelled by springs with elastic behavior. This model is studied inboth dynamic and static cases. First, this model is calculated viadiscrete and macroscopic approaches. We proved the existence ofseveral cases where the macroscopic approach cannot replace thediscrete one. Due to this difference, a coupling approach betweendiscrete and continuum ones was proposed. This approach consistsof a continuum approach derived from the discrete one at macro-

scopic scale. The macroscopic elements have a coarse size com-pared to the discrete ones. Using a criterion of coupling, the sizeof these elements should be refined when necessary. This proce-dure should be repeated until the size of the ME is the same as thatof a DE.

After applying the coupling approach in the cases where themacroscopic and discrete approaches do not give an identicalbehavior of the ballast, we could show the efficiency of this ap-proach and summarize it in some points. Firstly, we observe a goodmatch between the discrete and the coupling behaviors. Secondly,we observe a gain in the number of elements, which implies areduction in the computation time compared to that needed inthe discrete approach. This approach can also detect the locationsof heterogeneities. In addition to these conclusions, the absence of


the spurious wave at the interface of coupling is also proved. In fu-ture works, this coupling approach will be extended to 2 and 3dimensional problems.

Appendix A

In the Section 2.1.1, we have concluded on a relationship thatgives the value of parameters P; Q ; R and S. Here, we developthe method which is used to prove this relationship. Using Eqs.(3) and (4) we conclude on the following equalities:

U0þ ¼ T1½A B C D�T

UY� ¼ T2½A B C D�T

UYþ ¼ T3½P Q R S�T

Uh� ¼ T4½P Q R S�T

8>>>><>>>>:where

T1 ¼ T4 ¼

0 0 0 10 0 1 00 1 0 01 0 0 0

2666437775; T2 ¼

Y3

6Y2

2 Y 1Y2

2 Y 1 0Y 1 0 01 0 0 0

266664377775

T3 ¼

ðY�hÞ36

ðY�hÞ22 ðY � hÞ 1

ðY�hÞ22 ðY � hÞ 1 0

ðY � hÞ 1 0 01 0 0 0

266664377775

U0þ ¼ ½uð0þÞ u0ð0þÞ u00ð0þÞ u000ð0þÞ�T

UY� ¼ ½uðY�Þ u0ðY�Þ u00ðY�Þ u000ðY�Þ�T

UYþ ¼ ½uðYþÞ u0ðYþÞ u00ðYþÞ u000ðYþÞ�T

Uh� ¼ ½uðh�Þ u0ðh�Þ u00ðh�Þ u000ðh�Þ�T

8>>>><>>>>:Reminding the conditions of continuity at the point of load ‘‘Y”

on u; u0 and u00 and the jump condition on u000 (Eq. (5)) and usingthe above relations, the relationship between the vectors Uh� andU0þ is written in the following matrix form:

Uh� ¼ T4T�13 T2T�1

1 U0þ þ T4T�13

FEI

ð43Þ

Replacing the value of the matrix elements in Eq. (43), we deducethe value of parameters P; Q ; R; S as written in Eq. (6).

In the Section 2.1.2, we have concluded on two relationshipsthat link UY� to the vector g and UYþ to the vector ~g. They are rep-resented in the following system:

UY� ¼ R2½a b c d�T

UYþ ¼ R3½~a ~b ~c ~d�T

(where

R2 ¼ R3 ¼

a1 b1 c1 d1

n1a1 �n1b1 in1c1 �in1d1

�EIn21a1 �EIn2

1b1 EIn21c1 EIn2

1d1

�EIn31a1 EIn3

1b1 iEIn31c1 �iEIn3

1d1

2666437775 ð44Þ

where a1 ¼ en1Y ; b1 ¼ e�n1Y ; c1 ¼ ein1Y and d1 ¼ e�in1Y .These relationships are used to prove the relation between Uh

and U0 established in Eq. (16).

References

Al Shaer, A., Duhamel, D., Sab, K., Foret, G., Schmitt, L., 2008. Experimentalsettlement and dynamic behavior of a portion of ballasted railway track underhigh speed trains. Journal of Sound and Vibration 316, 211–233.

Al Shaer, A., Duhamel, D., Sab, K., Cottineau, L.M., Hornych, P., Schmitt, L., 2005.Dynamical experiment and modeling of a ballasted railway track bank.EURODYN 2005: Sixth European Conference on Structural Dynamics 3, 2065–2070.

Ben Dhia, H., Rateau, G., 2001. Analyse mathématique de la méthode Arlequinmixte. Comptes Rendus de I’Académie des Sciences – Series 1 – Mathematics332 (7), 649–654.

Ben Dhia, H., Rateau, G., 2005. The Arlequin method as a flexible engineering designtool. International Journal for Numerical Methods in Engineering 62, 1442–1462.

Bodin-Bourgoin, V., Tamagny, P., Sab, K., Gautier, P.E., 2006. Experimentaldetermination of a settlement of the ballast portion of railway trackssubjected to lateral charging. Canadian Geotechnical Journal 43 (10), 1028–1041.

Broughton, J.Q., Abraham, F.F., Bernstein, N., Kaxiras, E., 1999. Concurrent couplingof length scales: methodology and application. Physical Review B 60, 2391–2403.

Cecchi, A., Sab, K., 2009. Discrete and continuous models for in plane loaded randomelastic brickwork. European Journal of Mechanics – A/Solids 28, 610–625.

Cecchi, A., Sab, K., 2006. Corrigendum to A comparison between a 3D discrete modeland two homogenised plate models for periodic elastic brickwork. InternationalJournal of Solids and Structures 43, 390–392 [International Journal of Solids andStructures 41 2004 2259–2276].

Cundall, P.A., Stack, O.D.L., 1979. A discrete numerical model for granularassemblies. Geotechnique 29, 47–65.

Curtin, W.A., Miller, R.A., 2003. Atomistic/continuum coupling in computationalmaterials science. Modelling and Simulation in Materials Science andEngineering 11, R33–R68.

Fish, J., Chen, W., 2004. Discrete-to-continuum bridging based on multigridprinciples. Computer Methods in Applied Mechanics and Engineering 193(17-20), 1693–1711.

Frangin, E., Marin, P., Daudeville, L., 2006 Coupled finite/discrete elements methodto analyze localized impact on reinforced concrete structure. In: ProceedingsEURO-C.

Klein, P.A., Zimmerman, J.A., 2006. Coupled atomistic-continuum simulationsusing arbitrary overlapping domains. Journal of Computational Physics 213,86–116.

Knap, J., Ortiz, M., 2001. An analysis of the quasicontinuum method. Journal of theMechanics and Physics of Solids 49, 1899–1923.

Kohlhoff, S., Schmauder, S., 1989. A new method for coupled elastic-atomisticmodelling, atomistic simulation of materials. In: Vitek, V., Srolovitz, D.J. (Eds.),Beyond Pair Potentials. Plenum Press, New York, pp. 411–418.

Miller, R.E., Tadmor, E.B., 2009. Topical review: a unified framework andperformance benchmark of fourteen multiscale atomistic/continuum couplingmethods. Modelling and Simulation in Material Science and Engineering 17(053001), 51.

Nguyen, Vu-Hieu, Duhamel, Denis, 2006. Finite element procedures for nonlinearstructures in moving coordinates. Part 1: infinite bar under moving axial loads.Computers and Structures 84, 1368–1380.

Nguyen, Vu-Hieu, Duhamel, Denis, 2008. Finite element procedures for nonlinearstructures in moving coordinates. Part II: infinite beam under moving harmonicloads. Computers and Structures 86, 2056–2063.

Ricci, L., Nguyen, V.H., Sab, K., Duhamel, D., Schmitt, L., 2005. Dynamic behaviour ofballasted railway tracks: a discrete/continuous approach. Computers andStructures 83 (28-30), 2282–2292.

Ricci, L., Sab, K., 2006. Influence des plates-formes sur le tassement des voies ferréesballastées. PhD thesis, Ecole Nationale des Ponts et Chaussées.

Rudd, R.E., Broughton, J.Q., 1998. Coarse-grained molecular dynamics and theatomic limit of finite elements. Physical Review B 58, R5893–R5896.

Shenoy, V.B., Miller, R., Tadmor, E.B., Rodney, D., Philips, R., Ortiz, M., 1999. Anadaptative finite element approach to atomic-scale mechanics – thequasicontinuum method. Journal of the Mechanics and Physics of Solids 47,611–642.

Tadmor, E.B., Ortiz, M., Philips, R., 1996. Quasicontinuum analysis of defects insolids. Philosophical Magazine A 73 (6), 1529–1563.

Wagner, G.J., Liu, W.K., 2003. Coupling of atomistic and continuum simulationsusing a bridging scale decomposition. Journal of Computational Physics 190,249–274.

Xiao, S.P., Belytschko, T., 2004. A bridging domain method for coupling continuawith molecular dynamics. Computer Methods in Applied Mechanics andEngineering 193, 1645–1669.

static and dynamic studies for coupling discrete and continuum media; application to a simple...

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