A Coupled Molecular Dynamics/Coarse-Grained-Particle Method

for Dynamic Simulation of Crack Growth at Finite Temperatures

Ryo Kobayashi, Takahide Nakamura and Shuji Ogata

Department of Scientific and Engineering Simulations, Graduate School of Engineering, Nagoya Institute of Technology,Nagoya 466-8555, Japan

A hybrid molecular dynamics/coarse-grained-particle (MD-CGP) method is proposed for dynamic simulation of crack growth at finitetemperatures. In the present method, the MD method is applied for the non-linear elastic region such as the crack tip and dislocation core, whilethe CGP method is applied for the surrounding linear elastic region. For coupling the atomistic and coarse regions, extra atoms and particles areplaced, respectively, beyond the interface of the atomistic and coarse systems. They move according to the Langevin-type equation. Thedissipation term in the Langevin-type equation contains the velocity difference between the atomistic and coarse systems. Hence, the coupling isachieved through the damped oscillation of the difference between the two systems. It is shown in a one-dimensional simulation that the presenthybrid method achieves the non-reflecting boundary condition at the interface for the entire wavenumber range at finite temperatures. Two-dimensional crack growth simulations are performed using both the present hybrid MD-CGP method and full MD method for accuracy check. Inthe hybrid simulation, the location of the atomistic region shifts to follow the moving crack tip. In spite of such drastic operations, the resultsobtained by the present hybrid simulation agree very well with those by the full MD simulation. The hybrid MD-CGP method allows us tosimulate very large systems without losing atomistic information at the singular point and non-linear region, with much lesser degrees offreedom than required for the full MD method. [doi:10.2320/matertrans.M2011116]

(Received April 15, 2011; Accepted May 23, 2011; Published July 6, 2011)

Keywords: multiscale simulation, coarse-graining, molecular dynamics, coupling scales, crack growth

1. Introduction

Materials fail under various loading conditions because ofcrack growth. Stress is highly concentrated at the crack tip,breaking interatomic bonds. Moreover, the stress field isextended, widening considerably around the crack tip, andthus, stress concentration near the crack tip depends mainlyon the boundary conditions.1) Since atomic scale eventsand meso- or macro-scale circumstances are closely related,phenomena such as crack growth are called multiscalephenomena. For crack growth, a finite-temperature effect anddynamics are also important factors; for example, the brittle-ductile transition is inherently temperature dependent, andthe instability of crack propagation depends on the dissipa-tion of energy from the crack-tip area.

Simulating the multiscale phenomena accurately andefficiently has been a topic of research in recent years. Interms of accuracy, the molecular dynamics (MD) method,which affects atoms directly through their interatomicpotentials, simulate bond breakages and non-linearity atand near the crack tip. However, fully atomistic treatment ofmeso- or macro-scale circumstances consumes more compu-tational time and resources, and it is difficult to simulate long-time phenomena which last over nanoseconds. On the otherhand, although the continuum approximation methods suchas the finite element method (FEM) can deal with anextensive stress field around the crack tip, it cannot treat theatomic-scale events and strong non-linearity near the cracktip because it loses atomistic information. Therefore, inrecent years, several hybrid atomistic and continuum simu-lation methods have been proposed, among these, the MDmethod is applied only near the crack tip where the atomisticevents and non-linear effects are significant, while thecontinuum method is used for the surrounding linear elasticregion.2–18)

Although various hybrid methods have been suggested,most methods focus only on zero-temperature or quasi-staticcondition because it is difficult to suppress the reflection oflarge-wavenumber waves at the interface of the atomistic andcoarse (or continuum) regions. Some hybrid methods thateliminate reflections for effective dynamic simulation havebeen proposed; for example, the bridging scale method(BSM),6,19,20) the perfectly matched multiscale simulation(PMMS),21,22) and the bridging domain method with theperfectly matching layer.23) These methods focus on elimi-nating incident large-wavenumber waves from the atomisticto coarse region, and realize transmission of small-wave-number waves and elimination of the reflection of large-wavenumber waves at the interface. However, it is stillunclear whether these methods can effectively eliminate thespurious reflections for the middle-wavenumber waves nearthe Brillouin zone (BZ) edge of the coarse system wheregroup velocities of the atomistic and coarse systems are quitedifferent.

The dynamic coupling of two different systems is broadlydefined as the coupling of the phonon modes which existin the two systems. This is true not only for the couplingof different scales but also for the coupling of differentinteratomic potentials in the same scale systems, such as thehybrid quantum-mechanical/molecular-mechanical (QM/MM) method.24) In this paper, we adopt the coarse-grainedparticle (CGP) method to describe the linear elastic regionwhich surrounds the atomistic region, because the CGPmethod accurately reproduces the phonon dispersion relationof the atomistic system.25–27) The fundamental issue in thecoupling of two differently scaled regions is the difference inthe largest possible wavenumber between the two regions,i.e. the difference in the BZ sizes; the coarser system has asmaller BZ. Hence, large-wavenumber waves of the atom-istic system, which are outside the BZ of the coarse system,

Materials Transactions, Vol. 52, No. 8 (2011) pp. 1603 to 1610#2011 The Japan Institute of Metals

do not exist in the coarse system. Moreover, for a wave-number at the BZ edge of the coarse system, the groupvelocity of waves of the coarse system becomes zero becauseof its periodicity, whereas the group velocity in the atomisticsystem at the same wavenumber has non-zero value, andhence, the waves in one system cannot couple with that in theother system. In our former paper,28) we pointed out thisproblem and suggested a new coupling method, called thebridging extra particles (BEP) method, which suppresses thereflection of both short- and middle-wavenumber waves.

Such a non-reflecting boundary condition at the atomistic-to-coarse interface requires to dissipate the energy of thewave-components, which cannot exist in the coarse side,incident from the atomistic side. This acts to lower thetemperature of atomistic system and to prevent finite-temperature simulation. There are some previous studies toachieve finite-temperature simulation in hybrid atomistic/coarse system.29–32) Among them, the methods that use thegeneralized Langevin equation (GLE)31) or Nose-Hooverthermostat32) possibly achieve finite-temperature simulationwith eliminating the reflection of large-wavenumber waves.However these methods conceivably fail to suppress thereflection of middle-wavenumber waves by the reasondiscussed in Sec. 3.1 and Ref. 28). In the present paper, wereformulate the former the BEP method in a Langevin-typeone and show that the present method can achieve finite-temperature simulation with the non-reflecting boundarycondition for both large- and middle-wavenumber waves.

In addition to realize dynamic simulation at finite temper-atures, when we perform the simulation of crack growth, it isuseful to reduce total time by adaptively re-selecting theatomistic region in the total system. An advantage of such ahybrid method is the reduction of computational cost andresources by keeping the size of the atomistic region smalland reducing the degree of freedom (DOF) in the totalsystem. To apply a small atomistic region around the fast-moving crack tip, the atomistic region should shift itsposition frequently to capture the crack tip or non-linearelastic region near the crack tip. In the present paper, weexamine critically the possible artifacts due to the atomistic-region shift, by which creation and annihilation of atoms andcoarse-grained particles occur during the simulation run.

The rest of the paper is organized as follows: In Sec. 2, weprovide a brief review of the CGP method and a descriptionof a new coupling approach of atoms and CG particles. One-dimensional (1D) examples of the boundary conditions at theinterface of the atomistic and coarse regions are shown inSec. 3. In Sec. 4, the two-dimensional (2D) simulation ofcrack growth is described to show the accuracy of the presenthybrid method, with adaptive selection of the atomisticregion in zero- and finite-temperature cases. Section 5presents the conclusions on the present coupling methodand its availability.

2. Methodology

2.1 Coarse-grained particle (CGP) methodAs described above, matching the phonon dispersion

relations between the atomistic and coarse-scale systems isa significant issue in coupling these systems, because phonon

dispersion is a property of dynamics, which depends on notonly the stiffness of the system but also the mass of atoms andparticles, whereas elastic constants depend only on stiffness.The lumped-mass approximation, which is usually used in theFEM and quasi-continuum (QC) method, often fails to repro-duce the phonon dispersion relation of the atomistic system.25)

The CGP method can systematically reproduce the phonondispersion relation by the statistical mechanical average ofall the phonon modes in the atomistic system. Thus, in thispaper, we adopt the CGP method to describe the linear elasticregion which surrounds the non-linear atomistic region.

In the CGP scheme, a CG particle represents the motion ofa group of atoms around the particle; the displacement of theparticle-I is defined as a weighted sum of the displacementsof atoms, UI ¼

Pi wIiui, and for velocity, _UUI ¼

Pi wIi _uui,

where wIi is a weighting function which relates the particlesto atoms. The partition function of the CGP system is definedusing constraints of these relationships as

ZCGP �Z

dfrgdfpge��Hatom

�YI

� UI �Xi

wIiui

!� _UUI �

Xi

wIi

pimi

!; ð1Þ

and we obtain the energy of the CGP system from thederivative of this partition function,

ECGP ¼ �@ZCGP

@�; ð2Þ

where � ¼ 1=kBT , and kB is the Boltzmann factor. Approx-imating the interatomic potential to a harmonic form as,Vatom ¼ 1

2

Pij u

Ti Dijuj, with a force-constant (dynamical)

matrix,Dij, the interparticle potential is analytically obtainedas a harmonic form.25) Then the energy of the CGP systembecomes

ECGP ¼d

�ðNatom � NCGÞ

þ1

2

XIJ

ð _UUT

IMIJ_UUJ þ UT

I KIJUJÞ: ð3Þ

Here d is the dimension of the system and Natom and NCG arethe number of atoms and particles, respectively. The massmatrix and stiffness matrix in eq. (3) are systematicallyobtained using the weighting function as

MIJ ¼ ½ðwm�1wTÞ�1�IJ ; ð4ÞKIJ ¼ ½ðwD�1wTÞ�1�IJ : ð5Þ

Note that the mass matrix has values on non-diagonalelements, indicating that the mass is distributed, unlike thelumped-mass approximation, which is significant for match-ing the phonon dispersion relation with the atomistic system.

The weighting function, w, should satisfy the sum rule,Pi wIi ¼ 1. From the physical meaning of the weighting

function, it also should be localized around the particle I, andhere we adopt w ¼ ½NNT��1N, where N is a linear shapefunction commonly used in the FEM, which is suggested byRudd and Broughton.25) Once finite elements and a shapefunction are defined, corresponding weighting function forparticles positioned at the finite element nodes is constructedfrom the shape function. Therefore the CGP method issystematically applicable to 2D and 3D systems.

1604 R. Kobayashi, T. Nakamura and S. Ogata

2.2 Coupling of atomistic and coarse scalesTo couple the atomistic and coarse scales in the dynamic

and finite-temperature simulation, we consider the followingfeatures important: (i) the small-wavenumber waves aretransmitted at the interface, (ii) there is no artificial reflectionoccurring at the interface, and (iii) the temperature of theatomistic system remains constant even if fine fluctuation iseliminated at the interface. As mentioned in Sec. 1, the BSMis a promising candidate as the coupling method. Since theBSM is based on the GLE that contains the time historykernel function in the dissipation term, it can achieve thenon-reflection boundary condition for wave components thatcannot exist in the coarse system. Adding an appropriatefluctuation to the GLE according to the fluctuation-dissipa-tion theorem, the BSM is applicable to finite-temperaturesimulations. However, there are some drawbacks in the BSM:First, the time history kernel function must be created beforesimulation and it is obtained analytically only for harmonicinteratomic potentials and for simple crystallographic faces,hence it is hard to apply the BSM to the interface that hascomplex shape and changes its location. Second, as pointedout in Sec. 3.1 and Ref. 28), there exists a certain amount ofartificial reflection of waves with wavenumbers close to theBZ edge of the coarse system.

In order to overcome the drawbacks mentioned above, wepropose a novel coupling approach that is based on theLangevin equation but does not use the time history kernelfunction. Avoiding the time history kernel that limitsapplicability of the BSM, the present method can be appli-cable to any interatomic potential and any crystallographicface. In the present method, some atoms and particles areplaced beyond the interface, which we call extra atoms andparticles, as shown in Fig. 1, similar to Ref. 28). Displace-ments with superscripts (r) and (e) indicate respectively thoseof real and extra of atoms or particles, whereas displacementwithout superscript indicates that of any atom or particle.These extra atoms and particles are used as receivers forinformation from the other system and also as dampers forthe wave which is not transmitted to the other system.

Only at the edge of the extra regions, extra atoms (filledcircle in Fig. 1) and particles (filled square in Fig. 1) receivedisplacement information from the other system, using therelationships, ui ¼

PI N

TIiUI and UI ¼

Pi wIiui, where NT

Ii

is a shape function. These operations not only connect thetwo systems by matching displacements but also preventsurface relaxation of atoms and particles at the extra-regionedge.

To eliminate reflection in the extra region, the gradualdamping of the phonon modes which do not couple to the

other system is important as discussed in Ref. 28). We defineforce formulae acting on the extra atom-i and particle-I basedon the Langevin equation as

f (e)i ¼ �

@Vatom

@r(e)i

� �mi _uu(e)i �

XI

NTIi

_UUI

" #þ fRð�fÞ; ð6Þ

F (e)I ¼ �

@VCGP

@R(e)I

� �XJ

MIJ_UU

(e)

J �Xi

wJi _uui

" #; ð7Þ

where � is the damping constant and fRð�fÞ is the fluctuationrandom force with standard deviation, �f . Note that, in theseformulae, there is no time history kernel that appears in theGLE or the BSM. The first terms on the right-hand side ofeqs. (6) and (7) are forces derived from the interatomic andinterparticle potentials. The brackets in the second termindicate velocity deviations from the other system. Thus, thesecond terms indicate the dissipation of wave componentswhich deviate from the other system. Applying the appro-priate � value, the deviations from the other system areeliminated through damped oscillations. The second termsin eqs. (6) and (7) seem symmetric, and this symmetry isimportant to eliminate the wave components which do notcouple to the other system, and arising from not only theatomistic region but also the coarse region, as discussed inRef. 28). The third term on the right-hand side of eq. (6) isadded to achieve finite-temperature simulations only for thesystem of larger DOF.

The appropriate � value is determined through followingconsideration. Suppose the length of the extra region is le andthe characteristic group velocity is vg. If we assume thevelocity deviation from the other system is reduced to 1%through the damping oscillation by the time the wave reachesthe edge of the extra region and returns to the interface, thedamping constant is determined as

� ¼vg ln 10

le: ð8Þ

The standard deviation �f in eq. (6) is described in Sec. 3.2In the present coupling method, though the characteristic

time scales are different in the atomistic and coarse systems,same time intervals are used for both systems. In many cases,the bottleneck in the hybrid atomistic and coarse simulationis calculation of the atomistic system. Hence the reduction oftotal computation time by applying large time interval onlyfor the coarse system is limited. Moreover the large timeinterval for the coarse system causes large reflection at theinterface.

The total computational cost in the present methoddepends on the total DOF in the system including real andextra atoms and particles. Although there are additionaloperations for the coupling of two system such as

PI N

TIi

_UUI

in eq. (6) andP

i wJi _uui in eq. (7), NTIi and wJi are spatially

localized functions and the computational cost for these arenot significant. To achieve non-reflecting boundary condi-tion, sufficient number of extra atoms/particles are requiredas discussed in Sec. 3.1. This is a disadvantage to thecomputational cost of the present method. However, theseextra atoms/particles enable adaptive selection of the atom-istic region without artificial waves inside the real atoms/particles region as shown in Sec. 4. Hence we can reduce the

atomisticregion

coarseregion

: extra atoms: real atoms

: extra particles : real particles

atomisticsystem

coarsesystem

Fig. 1 Schematic illustration of the interface between the atomistic and

coarse regions in the present coupling method. Open circles (squares) are

real atoms (particles) and filled circles (squares) are extra atoms

(particles).

A Coupled Molecular Dynamics/Coarse-Grained-Particle Method for Dynamic Simulation of Crack Growth 1605

total computational cost by keeping the atomistic regionsmall around strong non-linear region even if it moves duringthe simulation.

3. Numerical Examples in 1D System

3.1 Transmission and reflection waves at the interfaceTo evaluate the effectiveness of the present coupling

method as a non-reflecting boundary condition, transmissionand reflection coefficients for the incident waves of allwavenumbers coming from the atomistic region are eval-uated in a simple 1D beads-spring model. Here the inter-atomic distance a, the atomic mass m and the spring constantk are defined as 1, and the time interval �t as 0.02. Supposethe periodic system of length L ¼ 1600 consists of atomistic(left) and coarse (right) sub-systems as shown in Fig. 1; inthis case the length of the extra region is about 40. In thecoarse sub-system, four atoms are coarse-grained to one CGparticle. Figure 2 shows the dispersion relation curves of theatomistic and coarse systems. The dispersion relation in bothsystems agrees well in a wavenumber range (1), and there areno wave components for a wavenumber range (3) in thecoarse system. There is an intermediate wavenumber range(2), where the group velocities of two systems differsignificantly. The wave components in wavenumber range(3) as well as (2) cannot couple between the atomistic andcoarse systems. Thus we evaluate the transmission andreflection coefficients in the entire range of wavenumbers.

As an initial condition, a wave packet with a characteristicwavenumber q is placed at x0=L ¼ 0:25 in the atomisticregion. The displacements of the atoms are

uiniðxÞ ¼ u0 � gðx� x0Þ � sinðq � ðx� x0ÞÞ ð9Þ

with the envelope function,

gðxÞ ¼1

2½1þ cosð2�x=WÞ�; for �W=2 � x � W=2,

0; otherwise.

8<:

ð10ÞThe width of wave packet W ¼ 400, which is half the sizeof the atomistic region.

Results obtained by the present coupling method arecompared with those obtained by the BSM. In the BSM,

an extra atom is placed beyond the interface, which is calledthe ghost atom. The large-wavenumber component (finefluctuation) of the displacement of the ghost atom, �uu(g), isdetermined using the time history kernel, �ðtÞ, and that ofthe atom next to the ghost atom, �uu(r), as

�uu(g)ðtÞ ¼Z t

0

�ðt � �Þ �uu(r)ð�Þd�: ð11Þ

In this case, �ðtÞ is derived analytically assuming �uui ¼ ui � �uuiwhere �uui �

PI; j N

TIiwIjuj ¼

PI N

TIiUI . The large-wavenum-

ber component, �uui, of the wave coming from the atomisticregion is eliminated strictly at the ghost atom position.

Transmission and reflection coefficients at a given wave-number q are calculated in the same way described inRef. 28). Setting a proper time window considering the wavepacket propagation, for the incident wave packet in theatomistic region, first we calculate the maximum intensitiesof the temporal Fourier components of the displacementsof 200 atoms around x=L � 0:375. Furthermore, they areaveraged in 200 atoms by which the numerical uncertaintywith respect to the finite time discretisation in the Fouriertransformation is effectively reduced. Second, the maximumintensities for the reflected wave packet in the atomisticregion (x=L � 0:375) and for the transmitted wave-packetin the coarse region (x=L � 0:625) are obtained similarly.Comparing the squares of the maximum intensities, thetransmission and reflection coefficients for the given q areobtained.

Transmission and reflection coefficients for the entirerange of wavenumbers, 0 � q < �, obtained by the BSM andthe present method are plotted in Fig. 3. The transmissioncoefficients are almost equal; they reach unity in the smallestwavenumber limit, q ¼ 0, and decrease to zero as thewavenumber increases to the BZ edge of the coarse system.At the large wavenumber, q � �, the BSM eliminatesreflection almost completely. The present method leavesreflection at the large wavenumber because the waves withvery slow group velocity receive strong damping at theinterface. The reflection coefficient in the BSM increases toabout 0.3 where the wavenumber is in the range (2), on theother hand, the reflection coefficient in the present methodis less than 0.01 for the wavenumber in the range (2).

The difference in the reflection coefficient between theBSM and present method results from the way how small-wavenumber waves are transfered and large-wavenumberwaves are eliminated. In the BSM, the ghost atom just behindthe interface works to transfer small-wavenumber waves and

0

0.5

1

1.5

2

0 0.5 1 1.5 2 2.5 3Wave number, q

atomisticCGP

(1) (3)(2)

Fre

quen

cy,

0.6

0.7

0.8

0.6 0.7 0.8

Fig. 2 Dispersion relation curves of the atomistic and coarse systems. The

inset shows a magnified view around the BZ edge.

0.5 1 1.5 2 2.5 3Wave number, q

0

0.2

0.4

0.6

0.8

1

0 0 0.5 1 1.5 2 2.5 3

Coe

ffici

ent

Wave number, q

(a) BSM (b) Present method

TransmissionReflection

TransmissionReflection

Fig. 3 Transmission and reflection coefficients with respect to the wave-

number of the incident wave obtained by (a) BSM and (b) the hybrid

MD-CGP method.

1606 R. Kobayashi, T. Nakamura and S. Ogata

to eliminate large-wavenumber waves with using historyinformation. On the other hand, in the present method, small-wavenumber waves are transfered and large-wavenumberwaves are eliminated gradually during their passage throughthe extra region that needs a certain length to eliminatesufficiently. Suppose that the wave with wavenumber close tothe BZ of the coarse system comes from the atomistic region.Its group velocity in the atomistic region is almost the sameas the sound velocity of the atomistic system. On the otherhand, in the coarse region, its group velocity is close to zero.Thus, focusing on the coarse system, the group velocity of thewave coming from the atomistic region suddenly changeswhen the wave goes into the coarse region and the wavebecomes like a standing wave just behind the interface. Thisis an inevitable situation if two regions of different lengthscales are coupled. For such a situation, the ghost atom in theBSM is sensitive to the standing wave existing just behindthe interface, and the artificial reflection waves are created.On the other hand, the extra atoms in the present method areless sensitive to the standing wave and thus the reflectioncoefficients of the present method around the wavenumberregion (2) are less than those of the BSM.

The reflection coefficient in the present method depends onthe length of the extra region. Figure 4 shows the reflectioncoefficients in the present method with respect to the extra-region lengths, 10, 20 and 40. The damping constant, �,is determined by the length, as in eq. (8). The shorter thelength of the extra region (the stronger the damping strength),the larger the reflection coefficient becomes.

3.2 Temperature control in the atomistic regionIn the present coupling method, the finite-temperature

simulation is achieved by adding the fluctuation term ineq. (6). An appropriate fluctuation under the fluctuationdissipation theorem compensates for the energy dissipatedduring the elimination of large-wavenumber components.The dispersion of the normal distribution random force ineq. (6) is given as �2

f ¼ 2mi�kBT=�t, where T is the target

temperature.During an isothermal simulation, the temperature of

the CGP region is controlled by a simple velocity scalingthermostat such as the Berendsen thermostat.33) Although thisseems inconsistent with the Langevin thermostat applied to

the extra atoms, this inconsistency does not prevent fromachieving the isothermal simulation.

Figure 5 shows the time evolution of the temperature ofreal atoms and particles. The temperature of real atomsbecomes stable near the target temperature T. This impliesthat the hybrid MD-CGP method can achieve an isothermal,finite-temperature simulation. However the temperature ofthe real atoms is slightly higher than the target temperature.The deviation from the target temperature comes from thefact that the fluctuation force in eq. (6) is not perpendicularto the CGP subspace, unlike the dissipation force in eq. (6),it includes components of the CGP subspace. Since thedeviation is less than T= where is coarse-graining ratio, � Natom=NCG, if the ratio is high, i.e. the CGP subspaceis small compared to the full phase space, there are fewcomponents of the CGP subspace included in the fluctuationforce, and thus, the deviation becomes negligibly small.When we consider the hybrid MD-CGP simulation in threedimension, the ratio is about a cube of that in one dimension,3D � 3

1D. Therefore the deviation is small in realistic threedimension systems, and it is not necessary to consider aspecial treatment for correcting the deviation.

4. Adaptive Selection of the Atomistic Region in 2DCrack Growth Problems

4.1 Selection algorithmIn crack growth simulations, since the crack tips move

very fast, and consequently, the atomistic region should shiftand follow the crack tips in order to reduce DOF to be treated.To achieve this, (i) the selection rule of the atomistic regionand (ii) the rule of creation of atoms and particles at the edgeof each system must be defined.

The atomistic region should include not only the crack tipbut also the non-linear field near the crack tip and thedislocation cores emitted from the crack tip. Here the cracktip is defined as the rightmost surface atom except those at theedge of the system. To detect the non-linear field and thedislocation cores, the atomic strain threshold is introduced;using the local strain as the measure for selecting a region ofprecise calculation is appropriate and successful in a QM/MM simulation.34) The atomic strain is calculated accordingto Ref. 35). The atomistic region is fixed if the crack tip,dislocation cores and the non-linear field are sufficientlywithin the atomistic region. When the crack tip, dislocationcore or non-linear field approaches to the interface, theatomistic region is reselected. A new atomistic region isdetermined by extracting the atoms having atomic strainshigher than the threshold, and adding enough space fromthese atoms to avoid shifting the atomistic region sofrequently. The threshold and space to be added depend onthe system (interatomic potential) and they decide the size ofthe atomistic region and thus the computational cost of thepresent hybrid simulation.

When the atomistic region shifts, the creation andannihilation of atoms and particles occur at the edge of theMD and CGP systems. At the front of the crack region, thecoarse system shrinks while the atomistic system expands byannihilating particles and creating atoms. On the other hand,at the back of the crack region, the reverse process with

10-5

10-4

10-3

10-2

10-1

1

0 0.5 1.0 1.5 2.0 2.5 3.0

Coe

ffici

ent

Wave number, q

Reflection (le=40)Reflection (le=20)Reflection (le=10)Transmission

Fig. 4 Reflection coefficients for different lengths of the extra region.

A Coupled Molecular Dynamics/Coarse-Grained-Particle Method for Dynamic Simulation of Crack Growth 1607

creating (annihilating) particles (atoms) occurs. The atomsare created with their displacements according to theinterpolation of displacements of CG particles as ui ¼P

I NTIiUI assuming a perfect crystalline structure. The

displacements of created CG particles are determined usingthe coarse-graining relationship, UI ¼

Pi wIiui. When the

crack grows, there are two surfaces created behind the cracktip, where atoms are annihilated and particles created. Inthat case the simplistic coarse-graining relationship, UI ¼P

i wIiui, sometimes fails at the surface because the relation-ship assumes a crystalline bulk condition. Thus, for the

surface area the weighting function wIi is modified suchthat it does not include atoms on the other surface for theweighted sum as ~wwIi ¼ wIi=

Pi0 wIi0 , where i0 runs over atoms

in the same surface.

4.2 Configuration of the systemTo show that the present coupling method and the adaptive

shifting of the atomistic region are effective for the crackgrowth simulation, we compared the full MD and hybridMD-CGP simulations of the 2D crack growth. Atoms in a 2Dtriangle lattice interact with each other by the Lennard-Jones(LJ) potential. The energy, length and mass parameters in theLJ potential are denoted as "LJ, �LJ and mLJ, respectively, andthe time interval is chosen as �t ¼ 2:3� 10�3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimLJ�2

LJ="LJ

p.

The system size ðLx;LyÞ ¼ ð450�LJ; 210ffiffiffi3p�LJÞ in which

there are about 180,000 atoms in case of fully atomisticsystem. As an initial condition, the crack of one atomic layerwidth and length a0 ¼ 150�LJ along x direction at the left sideof the system is introduced at y ¼ Ly=2, and the entire systemis homogeneously strained 0.5% in the y direction. In case ofhybrid MD-CGP simulations, 36 atoms are coarse grained to1 CG particle. For simplicity, the atomistic region is chosento be a rectangular shape. In this case, we set the thresholdof the atomic strain as 0.05 considering only the diagonalelements, xx and yy, of the strain tensor. Then therectangular-shape atomistic region is re-selected to includeextra space of 90�LJ to each x and y direction from theoutermost highly-strained region of atoms. The presentvalues of the threshold and extra space are determinedphenomenologically. The numbers of atoms and particles in

(a) (b) 0ΔΔt (relaxed)0Δt (relaxed)

Fig. 6 Relaxed configurations of a 2D crack problem by (a) the full MD method and (b) the hybrid MD-CGP method.

0

1

0 100 50

Real atomsReal particles

Elapsed time, t /1000

Tem

pera

ture

, T/T

*

Δt

Fig. 5 Time evolution of temperature of real atoms and particles.

(a) (b)90,000ΔΔt 90,000Δt

Fig. 7 Snapshots of the crack growth simulations at 90;000�t after applying tensile loading by (a) the full MD method and (b) the hybrid

MD-CGP method. The insets show magnified views around the crack tips of both simulations.

1608 R. Kobayashi, T. Nakamura and S. Ogata

the hybrid MD-CGP system are about 20,000 and 5,000,respectively, in the same system size.

First, the system is relaxed by velocity damping dynamicsin which the positions of atoms (particles) on the top andbottom layers are fixed and those at the rightmost layer arerestricted to move only in the y direction. After relaxation,tensile loading is applied by moving the top and bottomlayers toward opposite direction, upward and downward. Toavoid shockwaves created by the sudden movement of thetop and bottom layers, loading speed is gradually increasedand then decreased to stop the loading. This tensile loading iscontrolled to make the final strain 1.5%.

In case of finite-temperature simulations, the Berendsen(velocity scaling) thermostat is applied for the region awayfrom the crack tip, which is a coarse region in the hybridsimulation. For the region near the crack tip, atoms movewith the interatomic potential without any control.

4.3 Simulation resultsFigures 6(a) and 6(b) show snapshots of strain distribution

of the full MD and hybrid MD-CGP simulations in fixedloading after relaxation. The colours indicate strain distribu-tions; red for tensile and blue for compressive. The strainfield in the hybrid system seems smooth at the interface of theatomistic and coarse regions, and the overall strain fields inboth simulations agree very well.

Figures 7(a) and 7(b) show snapshots of the full MDand hybrid MD-CGP simulations in zero-temperature at90;000�t after applying the dynamic tensile loading. Theatomistic region in the hybrid simulation shifts 10 times by90;000�t to follow the moving crack tip. The overall strainfields and near the crack tip shown in the insets are almostthe same in both simulations. This implies that large-wavenumber waves occurring at the crack tip are eliminatedat the interface of the atomistic and coarse regions and arenot reflected in the atomistic region even after the atomisticregion shifts. Time evolution of the crack tip x-positions inthe full MD and hybrid simulations are plotted in Fig. 8.Timing and speed of the crack growth are in good agreementbetween the full MD and hybrid simulations.

When the atomistic region shifts, some extra atoms areremoved and edge positions of the extra region are shifted.Since atoms newly treated as the edge of the extra region

are suddenly controlled by fixing their displacements asui ¼

PI N

TIiUI , their positions may jump, causing artificial

waves at the edge. However, the artificial waves created atthe edge of the extra region are eliminated as the waves passthrough the extra region. Thus, the hybrid simulation with theadaptive change of the atomistic region provides almost thesame result as that of the full MD simulation.

This 2D system with the LJ interatomic potential becomesductile at very low temperatures by emitting dislocationsfrom the crack tip. The target temperature is 0.01 "LJ=kB

which is enough for this system to be ductile. Figure 9(a)shows the magnified snapshot at 120;000�t in the full MDsimulation at the target temperature; the upward red line andshort downward red line from the crack tip are the trails ofdislocations. Here, not only the crack tip and dislocationscores, but also the trails of dislocations are considered to behighly strained atoms. They are treated within one atomisticregion, which extends as the dislocations move away fromthe crack tip. The first dislocation is emitted at about80;000�t and the second one at about 110;000�t, as seenin Fig. 8. In addition, in the hybrid simulation, the firstdislocation is emitted upward and the second one downward(Fig. 9(b)); their starting positions are similar to those in thefull MD simulation. As seen in Fig. 8, timing of the crack

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 2 4 6 8 10 12

Cra

ck-t

ip p

ositi

on, x

/Lx

Elapsed time, t /10000Δ

MD (ZT)MD-CGP (ZT)

MD (FT)MD-CGP (FT)

t

Fig. 8 Time evolution of crack tips obtained by the simulations of zero

temperature (ZT) and finite temperature (FT) by the full MD and the

hybrid MD-CGP methods.

(a) (b)120,000ΔΔt 120,000Δt

Fig. 9 Magnified views around the crack tips of the finite-temperature crack growth simulations at 120;000�t after applying tensile

loading by (a) the full MD method and (b) the hybrid MD-CGP method.

A Coupled Molecular Dynamics/Coarse-Grained-Particle Method for Dynamic Simulation of Crack Growth 1609

propagation and dislocation emissions are in good agreementbetween the full MD and hybrid simulations. In both simu-lations, the crack growth stops at almost the same time in thesame length because of the relief of strain by dislocationemissions. This indicates that the hybrid MD-CGP methodachieves non-reflecting boundary condition at the interfaceand finite-temperature simulation simultaneously.

5. Conclusion

A hybrid molecular dynamics/coarse-grained-particle(MD-CGP) method is proposed. To couple the atomisticand coarse scales smoothly, extra atoms and particles areplaced beyond the interface of the atomistic and coarseregions, and they are defined as moving according to theLangevin eqs. (6) and (7). The dissipation terms in theLangevin equations contain velocity differences between theatomistic and coarse systems. The differences are eliminatedthrough damped oscillation, and a non-reflecting boundarycondition is achieved for waves including any wavenumbercomponents. By adding the fluctuation term to the atomisticsystem, the present hybrid method achieves finite-temper-ature simulations.

Both 1D and 2D simulations are performed using thehybrid MD-CGP method. Calculated reflection coefficientsfor the incident waves of various wavenumbers in the 1Dsimulation are less than those obtained by preceding studies.Crack growth simulations in 2D LJ system by the hybridmethod and full MD method are performed to compare straindistributions around the crack tip and time evolution of thecrack tip position. In the hybrid MD-CGP simulation, theatomistic region shifts to follow the moving crack tip; thisshifting involves creation and annihilation of atoms andparticles. Despite such drastic operations, the results obtainedby the present hybrid simulation agree very well with thoseby the full MD simulation.

There are various problems in the field of materialsstrength and toughness in which differences in scale,temperature effects, and dynamics (or energy dissipation)play important roles not only in the crack growth but also inphenomena such as nano-indentation, friction and irradiationdamage. The present hybrid method is an accurate andefficient tool for simulating such complex phenomena.

We should take into account the difference of thermalexpansion rate between the atomistic and coarse systems inthe simulation at high temperatures. While the atomisticsystem expands, the coarse system does not do even at hightemperatures in the present formulation. At high temperature,substantial thermal stress should therefore be stored at theatomistic-to-coarse interface. To avoid the problem, we areworking to include the anharmonic effects including thethermal expansion in the CGP method. Its detail will beexplained in a separate paper.

Acknowledgements

This study was supported in part by Grant-in-Aid forScientific Research on Priority Areas ‘‘Nano MaterialsScience for Atomic Scale Modification 474’’ from theMinistry of Education, Culture, Sports, Science and Tech-

nology (MEXT) of Japan. The authors thank Dr. Seiji Kajitaand Dr. Shi-aki Hyodo at Toyota Central R&D Laboratory,and Prof. Kenji Tsuruta at Okayama University for fruitfuldiscussions.

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