comparative study of boundary conditions for molecular dynamics simulations of solids at low...

Comparative study of boundary conditions for molecular dynamics simulationsof solids at low temperature

Jerry Z. YangThe Program in Applied and Computational Mathematics, Princeton University, Princeton, New Jersey 08544, USA

Xiantao LiDepartment of Mathematics, Pennsylvania State University, Pennsylvania 16802, USA

�Received 21 February 2006; revised manuscript received 5 May 2006; published 20 June 2006�

We present some comparative studies on various boundary conditions for molecular dynamics simulations ofcrystalline solids. We focus on the effectiveness of these boundary conditions in suppressing boundary reflec-tions. As a quantitative comparison, we compute the reflection coefficients for these methods. Applications tofracture simulations are then demonstrated for problems including brittle cracks in a body-centered-cubiccrystal and ductile cracks in a face-centered-cubic metal.

DOI: 10.1103/PhysRevB.73.224111 PACS number�s�: 31.15.Qg, 62.20.Mk, 46.15.�x, 62.20.Fe

I. INTRODUCTION

In molecular dynamics �MD� simulations, the system isusually truncated to a size that is computationally tractable.As a result, artificial boundaries are introduced, where theboundary condition �BC� has to be provided to take intoaccount the atoms that have been removed from the system.For solids, a particular role of the BC is to prevent artificialreflection of phonons, maintain the external loading, andhence reduce the effect of finite system size. It has beenobserved that the straightforward approach of fixing bound-ary atoms to certain positions results in substantial reflectionof phonons at the boundary. In recent years, there has been agrowing interest in finding effective BCs to prevent wavereflection.

In principle, an exact nonreflecting BC can be derivedanalytically for crystalline solids in the case, where theatomic interaction is linear and the exterior of the system isinitially at rest. This was first done by Adelman and Doll1,2

for the simple one-dimensional chain model. Cai et al.3 dis-cussed how one can, in principle, obtain such exact BCs inthe general case via numerical computations. Karpov et al.4,5

continued with this path and extended this formalism to gen-eral crystal structures.4,5 In principle, for the linearized sys-tems if we neglect corners, these exact BCs produce simula-tion results as if the artificial boundary does not exist.However, obtaining these BCs numerically for general crys-tal structures can be fairly complicated.4,5 More importantly,the exact BCs are nonlocal in both space and time, in thatthey involve all boundary atoms as well as the previous his-tory of the atoms at these points. The decay of the historydependence is rather slow. Hence, implementing such BCs inatomistic simulations may result in additional computationaloverhead.

It is therefore of considerable practical interest to findapproximate BCs which are local. This idea was first pursuedin the work of E and Huang6,7 for simplified models, andlater extended systematically to general crystal structure byLi and E.8 These local BCs, which are obtained from a varia-tional formulation that aims to minimize the boundary reflec-tion, will be referred to as the variational BC �VBC�. Thesuccess of the VBC lies in the fact that it provides approxi-

mate but local BCs, the implementation of which is muchmore efficient than the exact ones. As a result, reasonablecompromise can be reached between computational com-plexity and effectiveness in suppressing phonon reflection.

Another commonly used alternative for preventing pho-non reflection is to append an additional border region to thecomputational domain, which acts as an absorbing layer offinite thickness to annihilate outgoing waves. A commontechnique is to add damping forces in the layer with a damp-ing coefficient d, which is often chosen to be zero at theboundary between the original system and the border region,and positive away from the boundary to damp out the waves,e.g., see Refs. 9–12. Damping method is very convenient touse, but in order to make it effective, a sufficiently largeborder region has to be used.

The problem of BCs for MD simulations is very similar tothe BC problem for solving wave equations. In principle, onecan construct exact BCs using Fourier and Laplace trans-forms for wave equations. But, this kind of BC is nonlocal intime and space which makes it impractical for numericalimplementations. The VBCs are very much similar in spiritto the absorbing boundary conditions13–15 which has beensuccessful in many applications. The idea of appending aborder region in order to damp out the outgoing waves has aclose resemblance to the perfectly matched layers method16

for the wave equations. For continuous wave equations, theborder region and the computational domain can be matchedperfectly, i.e., with no reflection. This, however, cannot beachieved for discrete problems as was observed in Ref. 17.

The main objective of this paper is to conduct a quantita-tive comparison between the VBC, the damping BC, and thefixed BC. The comparison is first made by computing thereflection coefficients. In addition, a number of numericalexperiments are performed to investigate the problem ofcrack propagation in different systems. We will mainly focuson the computational complexity and the effectiveness ofthese methods in suppressing phonon reflection.

We will organize the paper as follows: the next sectionspresent briefly the formulation and implementation of VBCand damping BC. Then, we evaluate these methods by com-paring the corresponding reflection coefficients in Sec. IV.Finally, we will present the simulations results from somefracture simulations with these BCs applied.

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II. VARIATIONAL BOUNDARY CONDITION

Here, we briefly explain the formulation of the VBC.Consider a boundary with inward normal vector n. We seekBCs in the form of

u0�t� = �rj�J

�0

t0

� j��u j�t − ��d� , �1�

or the discrete version,

u0n+1 = �

j�J�m=1

M

� jmu j

n−m+1�t , �2�

where we have picked the zeroth atom as one of the bound-ary atoms. The set J contains adjacent lattice points inside ofthe system. The matrices �, known as memory kernels, areto be determined. Altogether, the set J and the time interval�0, t0� or m=1,2 , ¯ ,M are called a stencil.

Next, we discuss how phonon reflection takes place, andhow to select the memory kernels to minimize the reflection.Let ri and Ri be the reference and deformed position of theith atom, respectively, and let ui=Ri−ri be the displacementvector. To access the phonon spectrum, the interatomic po-tential is approximated by its second-order Taylor expansion,

V�R1,R2, ¯ ,RN� = V0 +1

2�i�j

uiTDi−ju j . �3�

The Fourier transform of the force constants defines thedynamic matrix:

D�k� = �j

Dje−ik·rj . �4�

The dispersion relation of this system is related to the eigen-values, �s, of the matrix D,

�s2 = �s.

For definiteness, we will take �s=��s. The correspondingeigenvectors �s�k� are the polarization vectors. We will usethe standard normalization,

�s · �s� = �ss�.

The index 1�s�S designates the different phonon branch,and S is the number of branches in the spectrum. A phononmode is designated by a wave number k restricted to the firstBrillouin zone, denoted by B.

When an incident wave arrives at the boundary, reflectionmay occur which produces reflected waves. In order for thedisplacement and strain to be continuous prior to and afterthe phonon reflection, the frequency and the tangential com-ponent of the wave number have to be conserved. To bemore specific, let kI and kR be the wave number correspond-ing to the incident and reflected phonon mode, respectively,and kI�B. Then, kR must satisfy

kI − �kI · n�n = kR − �kR · n�n, �s�k� = �s��kR� , �5�

for some 1�s ,s��S.Solving Eq. �5� can be complicated. An alternative proce-

dure has been found,8 in which the problem of finding the

reflected wave mode was reduced to one of finding the zerosof a polynomial. The degree of the polynomial depends onthe number of phonon branches, and the effective range ofthe interatomic potential, Ne, which indicates along the nor-mal direction the number of layers that have direct interac-tion with the boundary atom. We will denote the reflectedphonon mode by

�kssl�R ,l = 1,2, ¯ ,NR�, NR = SNe.

To estimate the magnitude of the phonon reflection, oneneeds to compute the reflection coefficients. For this pur-pose, we consider the linear superposition of an incidentwave on the branch s and the resulting reflected waves,

u j�t� = csIei�k·rj−�st��s�k� + �

l=1

NR

cssl�R ei�k

ssl�R

·rj−�st��sl��kssl�

R � . �6�

A substitution into Eq. �1� leads to a linear system, fromwhich the reflection coefficients, denoted by Rsl�k� can beobtained. In addition,

cssl�R = �

l

RslcsI . �7�

More details can be found in Ref. 8.Physically, the phonon plays the role of the carrier of

thermal energy, and the propagation of phonon will result inenergy transport.18,19 Phonon reflection is often observedwith an increase in local temperature. It is therefore naturalto formulate the boundary condition from the considerationof energy balance. It was found in Ref. 8 that the energy fluxcan be split and expressed in the wave number space,

Js = JsI + Js

R, �8�

with

JsI · n = �

j� �cs

I�2�s2 � �s · ndk , �9�

and

JsR · n = �

j�

l� �cssl�

R �2�s2 � �s · ndk , �10�

representing, respectively, the energy flux due to the incidentand reflected waves.

Equations �1�, �7�, and �10� constitute the basis for con-structing a BC for MD. In particular, Eq. �10� clearly ex-presses the energy flux induced by the boundary reflection,and Eqs. �1� and �7� indicate how to compute the energy fluxgiven the BC. We now choose the time history kernels byminimizing the energy flux due to phonon reflection. We firstdefine the objective function,

I�� j�;n = �s�

k�B,k·n�0�

l

�Rssl��2��s · n�dk . �11�

Here, we have restricted the integration over half of the Bril-louin zone in which every wave number corresponds to anincident wave. BCs obtained from minimizing this objectivefunction are termed as VBCs.

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In summary, the numerical procedure for finding VBCinvolves the following components:

1. Generate grid points in the first Brillouin zone andcompute the dispersion relation and polarization vector foreach grid point;

2. For each grid point k that represents an incidentwave, find the wave mode for all possible reflected waves�kss�

R �;3. Select the stencil: i.e., the set J for the adjacent

lattice points, and the number of time steps M;4. Initialize the memory kernel � j. As a simple choice,

one may take � j =0;5. Compute the reflection coefficients and the objec-

tive function �11�;6. Use an optimization routine to obtain new values

for the memory kernels; and7. Go to Step 5 unless certain convergence criterion is

met.This procedure can also be carried out for the discretized

MD such as the velocity-Verlet scheme. The only major dif-ference is that the dispersion relation will depend on the sizeof the time step, �t. From our numerical experiments, wehave found that VBC does not affect the stability of thediscrete schemes.

The great advantage of VBC is that it offers the flexibilityof choosing the stencil to achieve the desired accuracy. Forinstance, one can choose to use more lattice points in the setJ but shorter time period t0 for the memory kernels. Moreimportantly, these BCs are local in that they involve a smallnumber of neighboring lattice points and a finite history de-pendence. It has been demonstrated that with VBC, one canachieve almost the same effect as the exact BCs, but at muchless cost, making them attractive for practical use. Finally,calculation of the memory kernels should be seen as an effortof precalculation. Once the BC is obtained, the implementa-tion is rather straightforward, and the application can be ex-tended to many other simulations with the same crystal struc-ture. For more practical issues on VBCs, see Ref. 8.

III. DAMPING REGION BOUNDARY CONDITION

In this section, we discuss the implementation of thedamping BC, which will be later compared to VBC and fixedBC. For simplicity, we assume the periodicity in z directionand consider an essentially two-dimensional �2D� problem.

In damping BC, an additional damping region is attachedto the primary simulation region, see Fig. 1. Hence, theboundary of the original system becomes an interface be-tween the primary and damping regions. The standard MD isemployed in the primary region, and in the damping region,an extra friction term is added to the equation of motion,

mx j = −�V

�x j− x j , �12�

where is the friction coefficient and it increases from zeroat the interface between the primary and damping region toits maximum at the boundary,

= 0�1 −d

w�n

, �13�

TABLE I. Number of atoms in the example of crack in bcccrystals for MD boundary condition methods.

Fixed BC Damping BC VBC

166,464 230,400 166,464

FIG. 1. Illustration of damping region boundary conditionmethod.

FIG. 2. �Color online� 2D triangular lattice, also shown are thebasis vectors.

FIG. 3. �Color online� The first Brillouin zone corresponding tothe 2D triangular lattice. The circles indicate the selected wavenumber at which the reflection coefficients are computed.

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where 0 is the maximum damping coefficient, and the inte-ger n=1,2 ,3 , . . .. In our simulations, we take n=1. w is thewidth of the damped region and d is the minimum distancefrom the atom with equilibrium position �x ,y ,z� to theprimary/damping interface,

d = min�x − xmin�, �x − xmax� + min�y − ymin�, �y − ymax� . �14�

where xmin, xmax, ymin, and ymax are the minimum and maxi-mum positions of the primary region in x and y directions,respectively. This damping method has been used in Ref. 9and 11 and in the former, it was referred to as ramped vis-cous damping method.

The parameter 0 is chosen so that the phonon of allfrequencies is damped out,

0 = 2m�max,

with �max being the highest frequency in the dispersion rela-tion. If 0 is set too small, it will not be able to eliminate allphonon reflection. On the other hand, taking 0 larger than2m�max is not necessary, and it may also cause numericalinstability. In practice, one can choose 0 to be a fraction ofthe Debye frequency �D.

One question which is not answered here is how to deter-mine the size of the damping region. There is no firm answerto this question. In general, the larger the damping region,the better it can reduce the reflection of phonons. But, oneneeds to face the fact that the computational cost is increaseddramatically as the size of damping region increases. For atypical simulation in our latter example of body-centered-cubic �bcc� crystal case �Sec. V�, the damping region is tenunit cells wide. We used four unit cells in the z directionalong which the system is periodic and each unit cell con-tains four atoms. In order to have a primary region with1001004 unit cells, we need to simulate a sample with1201204 unit cells. For fixed BC and VBC methods,only one layer of unit cells need to be placed at the boundary,so the total number of unit cells for these two methods are1021024. Although precalculation of the coefficients forVBC method is required before the simulation starts, that isonly a one-time cost.

In this example, the damping method contains 38.3%more atoms than the fixed BC and VBC methods, see TableI. The situation is even worse for a full three-dimensional�3D� problem.

IV. COMPARISON OF THE REFLECTION COEFFICIENTS

As a first step to compare these methods, we compute thereflection coefficients, which indicate the magnitude of theboundary reflection. For the VBC, this can be done by sub-stituting Eq. �2� into Eq. �6� and solve the resulting linearsystem. For the damping BC, however, there does not seem

FIG. 4. �Color online� Left: A cross section of the initial wavepacket along vertical axis; Right: The reflected waves at the samecross section.

FIG. 5. �Color online� Comparing the total reflection: The totalreflection for the lower branch of the phonon spectrum is plotted forselected wave modes indicated in Fig. 3.

FIG. 6. �Color online� bcc crack: Crack tip position versus time.Only the curve for the fixed BC is plotted here since curves for thefixed BC, damping BC, and VBC are undistinguishable. Cut-offradius of the EAM potential is employed to define broken bonds,and locate the crack tip.


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to be such an analytical means. Therefore, we rely on a nu-merical approach: We conduct a series of MD simulationswith a wave packet as the initial condition. In particular weconsider a 2D Lennard-Jones system with a triangular latticestructure as shown in Fig. 2. In each simulation, the wavepacket is centered at some preselected wave number that isrepresentative of low- and high-frequency phonons as indi-cated in Fig. 3. The system is evolved until the reflectedwaves re-enter the system. This time period can be estimatedfrom the group velocity. A result from such a simulation isshown in Fig. 4, where a fixed BC has been used to make thereflection easily observable. In light of Eq. �6�, we define the

reflection coefficient to be the ratio between the magnitudeof the reflected waves and the incident wave.

In Fig. 5, we plot the reflection coefficients,

�s�

�R1s��2,

for the lower branch for a VBC �2� that involves 11 latticepoints and 20 time steps, and the damping BC with 10 layersof atom in the damping region: w=5a0 with a0 being thelattice parameter. The cost of using these two BCs are com-parable. But in three dimensions, a damping region with such

FIG. 7. �Color online� bcc crack: Snapshots using the fixed BC, VBC, and damping region BC from left to right. The time steps from topto bottom are at 1.65 ps, 3.3 ps, 4.95 ps, and 6.6 ps. 166,464 atoms are used in the fixed BC and VBC methods, and 230,400 atoms are usedin the damping region BC.


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width will cost much more computation. The reflection fromthe fixed BC is close to one, and is not plotted in this figure.For the damping BC, increasing the width of the dampingregion will result in less reflection, especially for the low-frequency regime. Comparing to the damping BC with 10and even 20 layers of atoms, the VBC gives a smaller reflec-tion for most wave numbers. Around Point X, the reflectioncoefficients for the damping BC exhibit a peak. This is be-cause the frequency at this point is the highest.

V. APPLICATIONS TO CRACK PROPAGATION

A. Brittle crack in bcc crystal

In this section, we investigate the Mode I crack in the bcccrystal of �-Fe with different BCs applied. The atomic po-tential used here is the EAM potential developed by Shastryand Farkas.20 The system studied consists of a 3D rectangu-lar sample, with the three orthogonal axis along the 110,110, and 001 directions, respectively. The system is peri-odic in the z direction.

FIG. 8. �Color online� fcc crack Case 1: Snapshots using the fixed BC, VBC, and damping region BC from left to right. The time stepsfrom top to bottom are at 0.85 ps, 1.7 ps, 2.55 ps, and 3.4 ps. 161,376 atoms are used in the fixed BC and VBC methods, and 298,656 atomsare used in the damping region BC.


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We study the Mode I crack with orientation 001�110�.Similar simulations were conducted in Refs. 21 and 22. Anopening crack with elliptical shape is initiated from the an-isotropic linear elasticity solution,23–26 which will be used asa reference configuration in the VBC. Initially, the crack tipis placed at the center of the sample. This initial configura-tion will allow the system to quickly settle down to an equi-librium or a steady state. In these simulations, we have useda visualization tool, ATOMEYE �Ref. 27�, to demonstrate theatomic configuration of the cracks.

We first test Griffith criterion predicted by the linear elas-ticity solution25 via a number of MD runs with a fixed BCapplied. It works quite well for the stationary crack �within5% of relative error�. In these tests, the surface energy for the110 plane of �-Fe is taken to be �=0.104 eV Å−2, which isclose to the experimentally observed value.21

Next, we set the stress intensity factor to be 1.3KIG inorder to observe the propagation of the crack tip. The systemis started at zero temperature, and the crack quickly developsin a brittle manner. The crack tip moves forward along the110 direction, and the mean propagation speed keeps in-creasing until it reaches a maximum mean value of 973 m/s.Then, it remains steady at this constant mean speed. Theposition of the crack tip versus time is plotted in Fig. 6.

The sound wave speed is calculated as follows:

vs =�B

�, �15�

where � is the density, B= �C11+2C12� /3 is the bulk modu-lus, and C11 and C12 are elastic constants. All of these pa-rameters are directly computed from the EAM potential, andthe sound wave speed of �-Fe is approximately 4709 m/s.Therefore, the crack propagates slower than the sound wavein this example.

FIG. 9. �Color online� fcc crack Case 1: Crack tip configurationsat 5.1 ps using the fixed BC, VBC, and damping BC from top tobottom.

FIG. 10. �Color online� fcc crack Case 2: Crack tip configura-tions at 6.78 ps using the fixed BC, damping BC with large damp-ing region, VBC, and damping BC with small damping region fromtop to bottom.


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Along with the advancing of the crack tip, there are wavesradiating from the crack tip toward the boundary. For a fixedBC method, the waves are reflected back to the bulk of thesimulation sample after they reach the boundary, see Fig. 7.In this figure, the first component of the strain tensor 11 isused as the color map scheme.

For damping BC method, the crack tip advances in thesame manner as in previous simulations, but the waves aregradually damped in the damping region and vanish at theoutermost boundary, see Fig. 8. In these simulations, the sys-

tem is bigger than that of the fixed BC method due to theintroduction of the damping region.

For the VBC method in this example, we used 5 neigh-boring lattice points and 20 time steps as the stencil, and thereflection of waves at the boundary are greatly reduced, seeFig. 7. We have found that increasing the size of the stencilwill further reduce the phonon reflection at the boundary.The computational cost of the VBC is the same as that of thefixed BC once the coefficients are calculated, which do notchange with MD.

FIG. 11. �Color online� fcc crack Case 2: Snapshots using the fixed BC, damping BC with large damping region, VBC, and damping BCwith small damping region from left to right. The time steps from top to bottom are at 0.85 ps, 1.7 ps, 2.55 ps, and 3.4 ps. 161,376 atomsare used in the fixed BC and VBC methods, 298,656 atoms are used in the damping BC with large damping region, and 225,216 atoms areused in the damping BC with small damping region.


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In this example, due to the rapid propagation of the cracktip, none of the BCs changes the behavior of the crack, eventhough the fixed BC generates a considerable amount of re-flection off the boundary.

B. Ductile crack in face-centered-cubic crystal

In this section, we study the crack in a face-centered-cubic �fcc� crystal of aluminum under Mode I loading. Theatomic potential used here is the EAM potential proposedRef. 28. The system studied consists of a 3D rectangularsample, with the three orthogonal axes along the 110,001, and 110 directions, respectively. Again, the systemis periodic in the z direction.

The crack face is set normal to the 001 direction and the

crack front is along the 110 direction. The crack is placedat the 2/3 position of the x dimension. Similarly, the crack isinitialized using the anisotropic linear elasticity solution.25 Inthe following simulation, we set the stress intensity factor as1.6KIG with KIG computed from the linear elasticity solution.

Different from the example of bcc crack in last section,the behavior of the crack in aluminum at the early stage withthe current setup depends on the details of the initialization:The crack is more brittle if we initially put the crack face atthe middle of two neighboring 001 planes, and it is ductileif the initial crack face coincides with one such plane.

Figure 9 shows the crack tip configuration of the first caseat the early stage. In this figure, 12 is used as the colorscheme. The crack tip opens along the 111 plane to theupper right-hand side direction. There is a dislocation nucle-ated along the upper left 111 direction. They select the111 plane and 111 direction because �111� planes are theclosed-packed planes along which the �-surface energy andsurface energy are the smallest. Different methods of BCsprovide the same structure of the crack tip, see Fig. 9. Thereflected waves have reached the crack tip at this time. Thecrack becomes ductile at a later time.

In the process of the crack extension, waves are generatedfrom the crack tip as a result of energy release. Phonon re-flection is clearly observed at the boundary for the fixed BCmethod, see Fig. 8, in which v1 is used as color map scheme.

For the damping BC method, we put 20 unit cells in thedamping region with 20 layers of atoms. The waves aregradually damped in this region and vanish at the boundaryas shown in Fig. 8. Meanwhile, Fig. 8 shows the snapshots ofthe simulation using VBC with 8 neighboring lattice pointsand 20 time steps in the stencil. The outgoing waves areabsorbed at the boundary with no reflection observed.

Figure 10 shows the crack tip configurations of the secondcase, where the initial crack face coincides with one of the001 atomic planes. The crack is ductile in this case fromthe very beginning. Dislocations are nucleated along thelower left and lower right 111 directions. This is due to thesame reason for the low-�-surface energy of the �111�planes. However, there is a third dislocation nucleated alongthe lower left 111 direction for the VBC method and damp-ing region BC method, but not for the fixed BC method. This

is observed after the waves reflected at the right boundaryarrive at the crack tip. So, in this case, the phonon reflectiondoes change the configuration of the crack tip. This is notobserved in the previous examples because the cracks thereare brittle and open up rapidly. The effects of reflected wavesare not big enough to change the crack tip behavior. But, inthis case of ductile crack, a small change around the crack tipcan dramatically change its configuration. We also performsimulation with a damping BC, but for the smaller dampingregion; the wave reflection result is almost the same as thatwith the large damping region, see Fig. 11, but there is noadditional dislocation nucleated along the lower left 111direction. When a dislocation is about to be nucleated, theatoms around the nucleation point vibrate until they over-come the energy barrier for the dislocation nucleations. Aclose investigation shows that the reflection waves changethe vibration manner of those atoms, which prevents thenucleation of the third dislocation.

Figure 10 also shows that the dislocations in the VBCmethod go further than those in damping region BC methodat the same time step. This is due to the fact that the dampingregion cools down the system by introducing the frictionterm. An investigation shows that the local temperature nearthe boundary for the VBC is higher than that of the corre-sponding position for the damping BC.

Similar to previous examples, there are waves coming outfrom the crack tip and propagating to the boundary with acylindrical wave front. The waves are reflected at the bound-ary for fixed BC method, see Fig. 11. VBC method reducedthe reflection at the boundary, see Fig. 11, and with dampingregion BC, the waves are damped in the damping region, seeFig. 11.

VI. CONCLUSION

We have considered three boundary conditions for MDsimulations in crystalline solids: A straightforward BC thatfixes the boundary atom at certain position, a VBC whichexpresses the displacement of an atom at the boundary interms of the previous position of its neighboring atoms withweights chosen to minimize the phonon reflection, and thedamping BC which introduces a damping region to absorb-ing phonons that come to the boundary. Both the VBC andthe damping BC are very effective in suppressing phononreflections. In the damping BC, no precalculation is needed;but introducing a damping region may considerably increasethe overall computational complexity. The VBC requiressome preprocessing steps in order to compute the weightsused in the BCs, but once it is obtained, the implementationis much more efficient computationally.

For crack dynamics in simple crystals, brittle cracks aresimple and ductile cracks are more complicated. The behav-ior of the ductile crack is difficult to predict because thecrack has no strong preference about how to propagate as inbrittle crack case. A small perturbation of the environmentaround the ductile crack tip may change its behavior dramati-cally.


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comparative study of boundary conditions for molecular dynamics simulations of solids at low...

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