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Journal of Computational Physics 253 (2013) 64–85

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Journal of Computational Physics

www.elsevier.com/locate/jcp

A multiscale modeling technique for bridgingmolecular dynamics with finite element method

Yongchang Lee ∗, Cemal Basaran

Electronic Packaging Laboratory, Department of Civil, Structural, and Environmental Engineering, State University of New York at Buffalo,United States

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

Article history:Received 1 March 2013Accepted 30 June 2013Available online 12 July 2013

Keywords:Multiscale modelingWeighted averaging momentum methodMolecular dynamicsWave reflection

In computational mechanics, molecular dynamics (MD) and finite element (FE) analysisare well developed and most popular on nanoscale and macroscale analysis, respectively.MD can very well simulate the atomistic behavior, but cannot simulate macroscalelength and time due to computational limits. FE can very well simulate continuummechanics (CM) problems, but has the limitation of the lack of atomistic level degrees offreedom. Multiscale modeling is an expedient methodology with a potential to connectdifferent levels of modeling such as quantum mechanics, molecular dynamics, andcontinuum mechanics. This study proposes a new multiscale modeling technique to coupleMD with FE. The proposed method relies on weighted average momentum principle.A wave propagation example has been used to illustrate the challenges in couplingMD with FE and to verify the proposed technique. Furthermore, 2-Dimensional problemhas also been used to demonstrate how this method would translate into real worldapplications.

© 2013 Elsevier Inc. All rights reserved.

1. Introduction

The insatiate demand for multiscale analysis is not only due to advances in nanotechnology, but also due to experimen-tal results proving that there is a need for connecting nanoscale physics and macroscale continuum analysis. Significantadvancements in computational power make it feasible to link both powerful methods: molecular dynamics (MD) and finiteelement (FE) methods.

MD and FE methods are well suited to a particular level of accuracy on atomistic and continuum simulations, respec-tively. In general, MD cannot be used for macroscale problems due to the restrictions on the number of atoms that can besimulated simultaneously, along with the time scale limit. On the other hand, usage of FE method for atomic scale problemsis not accurate for many reasons mainly because continuum mechanics assumes that the substance of body is distributedcontinuously throughout the space of body and lacks atomic degrees of freedom. These inherent limitations make connect-ing these two methods essential but also challenging. Nevertheless, multiscale modeling will allow us to solve complicatedproblems with a greater accuracy than ever before. It should be pointed out that MD does not have electronic degrees offreedom. However, it is expected that methods like the one proposed here will allow us to connect FE, MD, and quantummechanics, which has electronic degrees of freedom.

* Corresponding author.E-mail address: [email protected] (Y. Lee).

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Y. Lee, C. Basaran / Journal of Computational Physics 253 (2013) 64–85 65

2. Physical background

2.1. Molecular dynamics

Molecular dynamics (MD) is a statistical method about motion and interaction of atoms. The classical mechanics whichgoverns the molecular dynamics of a system can be derived from Hamiltonian formulation. The Hamiltonian H is the totalenergy of a system that is the sum of kinetic energy and potential energy and is given by:

HMD = KMD + V MD (1)

KMD =∑ p2

i

2mi(2)

V MD = Φ(r) (3)

where KMD is kinetic energy of an atom i with mass mi and momentum pi , V MD is potential energy, and Φ(r) is the in-teratomic potential energy of particles with distance, r. The potential energy can be extended depending on the interatomicpotential. The equations of motion in MD are obtained from the Hamiltonian by the following relations:

∂ H

∂ p= x (4)

∂ H

∂x= −p (5)

where x is the coordinate and p is the momentum. In these relations, the total energy of a system is conserved by showingthat the time derivative of Hamiltonian is zero: (dH/dt) = 0. By using Eqs. (4) and (5), the equation of motion is derivedand given by

miqi = f exti − f int

i (6)

An atom of mass, mi , moves as a rigid particle with acceleration, qi . f exti is the external force. When interatomic potential

is known, the internal force f inti acting on a conservative system can be obtained by f int

i = ∇ri V .In classical molecular dynamics, interactions between neighboring atoms are determined by interatomic potential and are

very crucial for acquiring physically meaningful results. For pair potential, Lennard-Jones potential and Morse potential arecommonly used. Simulations of metals require solving the many-body problems, which lead to development of many-bodypotentials. Finnis–Sinclair potential, embedded atom method (EAM), and modified embedded atom method (MEAM) havebeen commonly used for many-body potentials.

2.2. Continuum mechanics

Continuum mechanics (CM) deals with the analysis of the kinetics and the behavior of solid or fluid modeled as contin-uum. The concept of continuum assumes that the substance of body is distributed throughout the space of body and ignoresthe fact that the matter consists of atoms, vacancies and atomic degrees of freedom. Due to the assumption of a continuousand differentiable mass density, a differential equation can be used to solve problems in continuum mechanics. In real life,objects are very large compared to atoms. Thus, at the macroscopic scale, continuum mechanics provides an appropriatestatistical procedure.

The continuum mechanics is based on two types of equations. One is the laws applied to the entire domain such asthe conservation of energy and mass. Another kind of equation describes the behavior of materials such as constitutiveequations. The equation of motion in continuum mechanics can be derived by the Hamiltonian as:

HCM = V CM + KCM (7)

V CM = 1

2

∫Ω

ε · C · ε dΩ (8)

KCM = 1

2

∫Ω

ρu2 dΩ (9)

where ε is the strain tensor, C is the material constitutive tensor, ρ is the material density and u is the nodal velocity. It iswell known that FE can very well simulate the continuum mechanics problems.

66 Y. Lee, C. Basaran / Journal of Computational Physics 253 (2013) 64–85

3. Literature review

Pioneering techniques for multiscale methods are the quasi-continuum (QC) method by Tadmor (1996) [1] and macro-scopic, atomistic, ab initio dynamics (MAAD), or also called as coupling of length scales (CLS), by Abraham et al. (1998) [2].Based on these techniques, various methods for multiscale modeling have been proposed such as coarse-grained molec-ular dynamics (CGMD) by Rudd and Broughton [3], bridging domain method (BDM) by Xiao and Belytschko (2004) [4],and bridging scale method (BSM) by Wagner and Liu (2003) [5] and Park and Liu (2004) [6]. In the following section abrief review of multiscale modeling techniques is presented. We believe this is necessary to put the proposed method in acontext.

3.1. Macroscopic, atomistic, ab initio dynamics (MAAD)

MAAD by Abraham et al. [2,7] is one of the earlier methods for multiscale simulation. The fundamental idea of thisapproach is to make concurrent links between tight-binding (TB) method, molecular dynamics (MD), and finite elementmethod (FE). In this method, tight-binding method is used for quantum mechanics level degrees of freedom. Molecular dy-namics is used for the representation of atomistic degrees of freedom. Finite element method is used for the deformation ofcontinuum mechanics. Here, all three simulations run at the same time, and dynamically communicate required informationbetween the simulations.

The interactions among three analyses are taken into account by the total Hamiltonian of the system as follows:

Htotal = HFE(d, d) + HFE/MD(d, d, r, r) + HMD(r, r) + HMD/TB(r, r) + HTB(r, r) (10)

In this model, ‘handshake’ region was adopted to couple regions with each other in HFE/MD . A very thin handshake regionis used. FE mesh is graded down to the atomic size for the reduction of wave reflection between MD and FE [8]. However,when connecting molecular dynamics and continuum mechanics by using MD simulation and FE method respectively, thistechnique uses the atomic scale mesh size for FE. However, the latter approach leads to two problems: one numerical andone physical. The numerical issue is that simulation time of FE slows down to picosecond to match the MD time step. AlsoFE time step is governed by the smallest element of FE. The physical issue is that atomic scale FE simulation is physicallyunreasonable because of the fact that constitutive equation of FE is based on continuum mechanics. Because of the fact thatthe time step in FE region depends on the element size, the atomic sized mesh makes the time step too short for realisticengineering problems. Moreover, although Abraham et al. [2,7] mentioned that there is no visible reflection at the FE–MEhandshake region, they did not discuss the error due to reflection of short wavelength in MD region. The reflection of shortwavelength at the interface between the ME and FE region still exists because atoms on the FE mesh side are stationarywhile atoms on the MD side are mobile.

3.2. Coarse-grained molecular dynamics (CGMD)

Rudd and Broughton (1998) developed coarse-grained molecular dynamics (CGMD) approach [3]. CGMD is based on astatistical coarse graining approximation. This approach removes tight-binding method from MAAD, and links FE and MD.A key idea of the method is that degrees of freedom are eliminated by using the coarse-graining approximation that con-verges to the exact atomic energy to reduce the computational cost. The coarse-graining energy for a mono-atomic harmonicsolid of N atoms coarse-grained to Nnode nodes is stated to be

E(uk, uk) = Uint + 1

2

∑j,k

(M jku j uk + u j K jkuk) (11)

Here Uint is the internal energy defined as Uint = 3(N − Nnode)kT where k is Boltzmann constant and T is temperature.The first term of summation M jku j uk is the kinetic energy and the second term of summation u j K jkuk is potential energy.M jk is mass matrix, K jk is stiffness matrix and u j and u j are the displacement and velocity of node j, respectively. In thisapproach, the vital region of simulation is modeled by MD, while peripheral regions are discretized with coarse-graining.The interface of MD and FE regions is modeled such that CGMD imitates the motion of an FE mesh. Rudd (2001) introducedthe generalized Langevin dynamics into the CGMD formulation. The equation of motion is then given by:

Miju j = −G−1ik uk +

t∫−∞

ηik(t − τ )uk(τ )dτ + Fi(t) = Gij (12)

where Gik is elastic Green’s function which is defined as ui = Gij f j in static where ui and f j are the CGMD displacementfield and body force respectively, ηik is memory function or a time history, uk(τ ) is velocity at node k and Fi(t) is a randomforce.

Similar to MAAD, mesh size of CGMD is graded down to atomic scale at the MD region, and coarsened far from the MDregion. Thus, CGMD also experiences the same issues as MAAD such as time step limitation, the wave reflection, and the


Fig. 1. Illustration of Cauchy–Born rule: (left) an element and atoms that are determined by Cauchy–Born rule, and (right) the deformed element andhomogeneously moved atoms by following the deformed shape of an element.

picosecond total simulation time. In order to reduce spurious wave reflection, the additional terms, the second and thirdterms in Eq. (12), are introduced. This technique is very similar to bridging scale method by Liu et al. (2003) which isalso reviewed in the following section. These additional terms lead to additional force calculations in MD simulation whichalready suffers from the limited simulation time due to the computational cost. Considering that the most expensive partof MD simulation is calculations, it is a serious limitation of the method.

3.3. Quasi-continuum (QC) method

Another pioneering approach for multiscale methods is the quasi-continuum (QC) method by Tadmor (1996) [1]. The QCmethod is an approach coupling continuum mechanics with atomistic simulation for the mechanical response of polycrys-talline materials at zero temperature. The QC method is based on an entirely atomistic description of the material domain.To reduce the computational cost, two assumptions are adapted: one is the reduction of degrees of freedom, and anotheris the Cauchy–Born rule: in a crystalline solid subject to a small strain, the positions of the atoms within the crystal latticefollow the overall strain of the medium as depicted in Fig. 1. The Cauchy–Born rule assumes that the continuum energydensity W can be obtained by using an atomistic potential, with the link to the continuum being the deformation gradientF given by:

F = 1 + du

dX(13)

where u is the displacement, dX is an undeformed line segment.By using the Cauchy–Born rule, a continuum stress tensor and tangent stiffness can be acquired from the interatomic

potential W , which allows the usage of nonlinear FE techniques. The continuum stress tensor and tangent stiffness aregiven by:

P = ∂W

∂ F(14)

C = ∂2W

∂ F T ∂ F T(15)

where P is the first Piola–Kirchoff stress tensor and C is the Lagrangian tangent stiffness.The particular representation is determined by the local deformation gradient and dictates a small fraction of the atoms

(called representative atoms or “repatoms”). In this approach, the non-local repatoms are used to represent the atomisticbehaviors, and the local repatoms are used to simulate the continuum domain by using the Cauchy–Born rule in the FEmethod.

Although QC method suggested a new approach for multiscale modeling, this method suffers from the same issues asMAAD, that are spurious wave reflection and the total simulation time limit. In this method, even though Cauchy–Born ruleconnects atoms in MD region with repatoms in FE, in which the mesh size of FE gradually increases from MD region, thespurious wave reflection still exists in MD region [9]. The result leads to spurious energy accumulation in MD region, non-physical heating of the crystal in the MD region, and as a result the solution in MD region becomes unreliable. Moreover,since this method is implemented in MD and FE regions simultaneously, the time step of MD dominates the total simulationtime, which is very short for any practical engineering problem.

3.4. Bridging domain method

Xiao and Belytschko (2004) [4] have developed a coupling method for molecular dynamics and continuum mechanicsbased on a bridging domain method. In this approach, the system consists of three domains: ΩMD (molecular dynamics),ΩCM (continuum mechanics), and ΩHS (handshake region). The main idea of the model is using a linear combination ofHamiltonian on the handshake region, ΩHS . Hamiltonian is defined by:

H = (1 − α)HMD + αHCM

=∑(

1 − α(Xi)) pM

i · pMi

2mi+ (1 − α)V MD +

∑α(Xi)

pCi · pC

i

2Mi+ αV CM (16)

i i


where pMi is momentum of MD region, pC

i is momentum of CM region, parameter α = [0,1] (linear) in ΩHS , α = 0 inΩMD − ΩHS , and α = 1 in ΩCM − ΩHS . Lagrange multipliers method is applied to enforce displacement compatibility in thehandshake region between the molecular and continuum regions by imposing the following equation

gI = {ui(XI ) − diI

} =(∑

J

N J (XI )ui J − diI

)= 0 (17)

i.e. the atomic displacements are required to conform to the continuum displacements at the handshake region. The con-straints are applied to all components of the displacements. In the Lagrange multiplier method, the total Hamiltonian andthe equation of motion are written as:

H L = H + λT g = H +∑

I

λTI gI (18)

miqi = f exti − f int

i − f Li = f ext

i − f inti −

∑J

λTJ∂ g J

∂qi(19)

where λ is a vector of Lagrange multipliers whose components correspond to the components of the displacement qi ofatom i. f L

i is the force due to the constraints enforced by the Lagrange multipliers. ∂ g J /∂qi in Eq. (19) is introduced bysubstituting the Hamiltonian in Eq. (18) into the relations in Eqs. (4) and (5).

As shown above in Eq. (16), the energy within the handshake region goes from entirely atomistic at MD boundaryto entirely continuum at FE boundary. The effect of this energy transition is that short wavelength atomic scale energyis filtered. The idea of spatial filtering is proven by the numerical examples in [4]. The example shows that a minimumhandshake distance is required for the method to eliminate wave reflection effectively. The minimum handshake distance inthis method is relatively long, as a result increasing the computational cost and decreasing the size of MD zone.

3.5. Bridging scale method

The fundamental idea of bridging scale developed by Wagner and Liu (2003) [5], and Park and Liu (2004) [6] is to resolvethe total displacement u(x) in terms of course scale u(x) and fine scale u′(x) at the position x. The coarse scale is governedby the continuum mechanics and simulates the entire field, while the fine scale is used to simulate the region of highinterest.

The total displacement u(x) is resolved into the course scale u(x) and fine scale u′(x) as follows:

u(x) = u(x) + u′(x) (20)

The coarse scale and the fine scale are defined as, respectively

u(Xα) =∑

I

NαI dI (21)

u′ = u − P u (22)

where NαI = NI (Xa) is the shape function on atomic position Xa , dI is the nodal displacement of FE, and P is the projection

matrix which is determined by minimizing the mass-weighted square of the fine scale. This method assumes that the result,q, of any atomistic level simulation could be used to generate an exact solution. Thus, the total displacement is given by:

u = Nd + q − Pq (23)

Hamiltonian formulation generates the coupled multiscale equations of motion as follows:

Md = N T f (u) (24)

maq = f (25)

where M is the mass matrix of FE, and ma is the atomic mass matrix of MD.Bridging scale method starts from an entire molecular system. To save computational time, in this approach, the system

area of MD is reduced from the entire region to a small area of interest. An entire molecular system can be changed intothe reduced MD system along with external forces that act on the boundaries of the reduced lattice. The latter representsthe combined effects of all the atomistic degrees of freedom accounted for by using the generalized Langevin equation(GLE). The effect of using the GLE in conjunction with FE is the dissipation of small wavelengths which FE cannot capturebecause FE can only capture longer wavelengths that are on the order of the FE mesh spacing or larger. The application ofGLE generates the external force on the equation of motion for MD simulation. The additional force f imp is applied at the


Fig. 2. An example of 1-D longitudinal wave; the arrow represents the direction of displacement of particles, and the amplitude of wave represents thedisplacement.

boundary of MD region, defined by:

f impm (t) =

ncrit∑m′=−ncrit

t∫0

θm−m′(t − τ )(q0,m′(τ ) − u0,m′(τ ) − R0,m′(τ )

)dτ (26)

where θm−m′ (t − τ ) is the time history kernel (or memory kernel) function that describes renormalization of the atomicinteraction along the boundary of MD domain, ncrit refers to the maximum number of atomic neighbors with MD bound-ary atoms, and R0,m′ (τ ) is the stochastic displacement that accounts for external force in FE domain. Thus, the coupledmultiscale equations of motion are given by:

maq = f (q) + f imp(t) + R f (t) (27)

Md = N T f (u) (28)

where R f (t) is the random force that accounts for thermal effect in FE region.This approach does not scale down the mesh size of FE to the atomic size, as a result provides different simulation time

scales on both MD and FE sides. It yields excellent wave reflection results because the time history kernel generated bythe generalized Langevin equation (GLE) leads to reduced elastic wave reflection. In comparing bridging scale method to theother multiscale methods (CGMD, MAAD), one clear advantage of this approach is that FEM simulates the entire system, andis not graded down to the atomic scale. The result of this is that long time step used in FE region is not restricted by theatomic sized elements in the mesh. This allows the staggered time integration algorithm. Thus, the coarse scale variablescan evolve on an appropriate time scale, while the fine scale variables can evolve (appropriately) on a much smaller timescale.

However, although the generalized Langevin equation is a mathematically exact representation of the MD degrees offreedom, in multiple dimensions, the time history integral is hard to compute and brings additional computational cost inMD simulation [5], because, in multiple dimensions, the calculation of time history kernel becomes more complex and hasto use numerical techniques increasing the computational cost. The most expensive part of this method is the computationalcost which is computing forces between atoms. When the computational limitation of MD is considered, the calculation ofadditional force can be a significant disadvantage on multiscale modeling.

4. Challenges of bridging molecular dynamics with finite element analysis

The main problem on a concurrent multiscale modeling is the spurious wave reflection which is generated in the hand-shake region between MD and FE [3–6,8–25]. Two kinds of wave reflection can occur: one is due to the mesh size in FEregion where representation of short wavelength is not possible [3–6,8–25], and another is due to the different wave dis-persion speed in each domain [24]. Wave reflection causes serious accuracy problems mainly in the MD region becausehigh frequency waves cannot transfer into FE region. They are reflected back into MD region. In order to illustrate the wavereflection problem, an example of 1-Dimensional longitudinal wave is used, Fig. 2. The waves used for the example areillustrated in Fig. 3, and are applied to an atom of which the initial position is zero. Fig. 3(a) is for long wavelength andlow frequency, Fig. 3(b) is for short wavelength and high frequency, and Fig. 3(c) is the sum of these two waves shown inFig. 3(a) and (b).

4.1. Wave reflection due to the different wave dispersion speeds

Wave propagation speed is a crucial quantity to be able to build a suitable handshake region between the continuumand the atomistic description regions. In order to demonstrate the effect of coupling domains with different wave dispersionspeeds, only a low frequency wave shown in Fig. 3(a) is applied in the 1-D example. High frequency wave can cause wavereflection also due to element size. Therefore, it is not used for this example. The details of the wave reflection due to highfrequency wave will be discussed in the following section.

On the one hand, when the wave propagation has the same dispersion speed on both regions, no wave reflection happensas shown in Fig. 4. On the other hand, coupling two domains with different wave dispersion speeds leads to wave reflection


Fig. 3. Applied displacement time-history at x = 0; (a) long wavelength, (b) short wavelength, and (c) the wave that is a summation of (a) and (b).

Fig. 4. Wave with long wavelength travels from left to right with no reflection when the wave propagation speed is same on both sides.

Fig. 5. Wave with long wavelength travels from a faster wave speed region (bold line) to a slower wave speed region.

at the interface as shown in Figs. 5 and 6. Fig. 5 shows the result when wave travels from faster speed region to slowerspeed region. In Fig. 5, in the slower wave speed region, amplitude of displacement decreases and the wavelength is shorter.The faster wave speed region in Fig. 5 exhibits a concave wave reflection. Fig. 6 shows the result for when the wave travelsfrom slower region to faster region. In Fig. 6, in the fast wave speed region, amplitude of displacement increases and thewavelength is bigger. The slower wave speed region in Fig. 6 exhibits a convex wave reflection. These wave reflections cangenerate serious error in MD region. Accordingly, we have to check if the momentum is conserved when discretization scalechanges in a multiscale analysis. Because of these reflections, conservation of momentum will not be satisfied in the MDregion.

4.2. Wave reflection due to the steep change in mesh size between MD and FE

The mesh size for FE is much larger than the interatomic distance used in MD. This different mesh size leads to wavereflection in bridging molecular dynamics with finite element method. For FE scale mesh size, representation of shortwavelength at the atomic scale is not possible, since we need at least two discretization points per wave length.


Fig. 6. Wave with long wavelength travels from a slower wave speed region (bold line) to a faster wave speed region.

Fig. 7. Full MD simulation (entire domain is modeled with MD).

Fig. 8. Equilibrium element size in FE is 4 times larger than equilibrium atomic distance of MD. Straight line is MD region (red on the web version of thisarticle), and circle is FE region (blue on the web version of this article).

Fig. 9. Equilibrium element size in FE is 10 times larger than equilibrium atomic distance of MD. Straight line is MD region (red on the web version of thisarticle), and circle is FE region (blue on the web version of this article).

In order to demonstrate the wave reflection due to relatively large element size in FE region, short wavelength shownin Fig. 3(b) is used. In order to compare the wave transfer between MD and FE, Fig. 7 shows the results when full MDsimulation is done. Figs. 8 and 9 show wave propagation and reflection using a different mesh size in FE domain. In Figs. 8and 9, MD domain is connected with FE domain without using a handshake region. In all cases, the applied wave is movingfrom left to right. When nodal distance in the FE region is 4 times the equilibrium atomic distance, results are shown inFig. 8, where the wave with short wavelength is coarsely transferred into FE domain. In Fig. 8, some part of wave is reflectedat the border between MD and FE domains, and dispersion speed of the transferred wave decreases in FE domain. Fig. 9shows the result in which the nodal distance in FE region is 10 times the equilibrium atomic distance. In Fig. 9, the wavewith short wavelength is perfectly reflected at the border between MD and FE. This reflection happens because the elementsize in FE region is too large to represent the short wavelength of the applied wave.


Fig. 10. Full MD simulation (entire domain is modeled with MD).

Fig. 11. Element size in FE is 4 times the equilibrium atomic distance. Straight line is MD region (red on the web version of this article), and circle is FEregion (blue on the web version of this article).

Fig. 12. Element size in FE is 10 times the equilibrium atomic distance. Straight line is MD region (red on the web version of this article), and circle is FEregion (blue on the web version of this article).

4.3. Wave reflection for long and short wavelengths

In the above example, we can observe two kinds of wave reflection: one is due to large element size in FE region, and theother is due to different wave dispersion speed between the two regions. In this section, a wave with both long and shortwavelengths shown in Fig. 3(c) is applied at an atom, located at x = 0. In this example both domains have the same wavedispersion speed, so we can see the effect of the applied wave with respect to the element size. Fig. 10 shows the results offull MD simulation where both sides of the border are discretized with MD. Figs. 11 and 12 show the results of bridging MDwith FE without using handshake region. The results are similar to the earlier example for short wavelength propagation.The wave with long wavelength is successfully transferred into FE domain from MD region. The short wavelength cannot betransferred into FE region and is reflected at the border of MD and FE regions (Fig. 12). However, Fig. 11 shows that whenelement size in FE region is 4 times the equilibrium atomic distance of MD, additional error is introduced at the borderof MD and FE regions. In Fig. 12, when element size in FE region is 10 times the equilibrium atomic distance, the errordisappears. It becomes more clear when the reader compares the displacement amplitude at the border among Figs. 10, 11,and 12.

5. New multiscale modeling approach, WAMM

5.1. Introduction

The atomic motion in MD simulation contains short wavelength that FE region cannot represent. It means the totalmomentum in MD domain cannot be transferred into FE domain in a concurrent multiscale analysis. The undelivered mo-mentum generates noise (spurious additional energy) in the MD region. In this work, to solve this problem, we couple the


Fig. 13. Schematic description for 1-D coupling with handshake region.

momentum of MD and FE in the handshake region ΩHS . The coupled momentum pcoupled is given by:

pcoupled = αpMD + (1 − α)pFE (29)

where pMD and pFE are the momentum of MD and FE region respectively. α is an averaging parameter which is zero inFE domain ΩFE and at the border near FE region in ΩHS , and 1 in MD domain ΩMD and at the border near MD region inΩHS . The schematic description of each domain and parameter α for 1-D case is illustrated in Fig. 13. By using this coupledmomentum, total Hamiltonian can simply be represented as:

Htotal = Kcoupled + V MD + V FE (30)

Kcoupled =N∑i

(pcoupled)2

2mi(31)

V MD =Natoms∑

i

V (ri j) + V extMD (32)

V FE = 1

2

∫ΩFE

σi jεi j dΩFE + V extFE (33)

where N and Natoms are the number of total particles (including atoms and finite element nodes) and atoms, respectively.MD potential V MD includes atoms not only in MD region, but also the atoms in the handshake region. V MD can be extendeddepending on the interatomic potential used for MD simulation. Superscript ext is used for the external effect. In thepotential energy of FE region, V FE , σi j and εi j are stress tensor and strain tensor, respectively.

In the handshake region, atoms of MD and nodes of FE are overlapped and nodal positions of overlapped elementsare determined by the positions of overlapped atoms. The coupling details are shown in Section 5.3. A weighted averagemomentum combination is enforced in the handshake region ΩHS as shown in Eq. (29).

In order to determine the averaging parameter α, a linearly decreasing kinetic energy of MD, KMD , is considered in thehandshake region first.

βKMD = β

Natoms∑i

p2MD,i

2mi=

Natoms∑i

(β0.5i pMD,i)

2

2mi(34)

where the parameter βi varies linearly from zero to one in handshake region ΩHS , pMD,i is the momentum of atom i, andmi is the mass of atom i. In the above form, the decreasing momentum β0.5

i pMD,i can be used for a momentum averagingequation. Introducing (29) into (34) yields the following relationship:

αi = β0.5i (35)

In the following section, we prove that this linear averaging scheme reduces the elastic wave reflection. However, someinsignificant wave reflection still remains. To reduce the wave reflection even more, generalized equation (36) is introducedusing parameter p instead of constant 0.5 that is used in Eq. (35), and is given by:

αi = βpi (36)

The effects of weighted averaging of momentum and parameter p are discussed below.

5.2. Equations of motion

In this work, a new total Hamiltonian is introduced by bridging momenta. Substituting the Hamiltonian in Eq. (30) intothe relations in Eqs. (4) and (5) gives:

∂ H

∂ pcoupled= ∂ Kcoupled

∂ pcoupled= pcoupled

m= x (37)

∂ H = ∂V MD + ∂V FE = −pcoupled (38)
∂x ∂x ∂x


Fig. 14. Application of Cauchy–Born rule for 1-D coupling example; dots are atoms, and circles are finite element nodes.

Fig. 15. The force contributions of atoms in ΩHS and virtual atoms in ΩCM .

When the above equations are considered for each domain, the equations of motion can be given by:

pMD

ma= q,

∂V MD

∂q= −pMD → maq = −∂V MD

∂q= fMD in ΩMD (39)

pcoupled

ma= qhs,

∂V MD

∂qhs= −pcoupled → maqhs = −∂V MD

∂qhs= fhs in ΩHS (40)

pFE

M= d,

∂V FE

∂d= −pFE → Md = −∂V FE

∂d= fFE in ΩFE (41)

where ma and M are corresponding mass of atoms and the nodes. fMD and fFE are corresponding force acting on atomsand nodes. q and d are the displacement of atoms and nodes respectively. qhs and fhs are the displacement and interatomicforce obtained by using coupled momentum pcoupled in handshake region ΩHS .

5.3. Cauchy–Born rule for MD/FE coupling

We now present the details of Cauchy–Born rule to couple MD with FE. In order to exploit the benefit of using anindependent mesh size in MD and FE regions, the nodal space in FE region is not scaled down to interatomic spacing.However, to couple MD with FE using a new multiscale method, FE nodes’ degrees of freedom in handshake region must bethe same as MD’s degrees of freedom. Accordingly, Cauchy–Born rule described in QC method [1] is applied in this study.

The different degrees of freedom are illustrated in Fig. 14. In handshake region ΩHS , MD has 21 atoms, and FE has3 nodes. In order to have the same degrees of freedom in both regions, Cauchy–Born rule is applied to nodes in FE region.This application generates 18 virtual atoms in handshake region ΩHS: 9 virtual atoms per an element. Through Cauchy–Bornrule, the number of total particles of FE in handshake region ΩHS becomes 21 that consists of 3 nodes and 18 virtual atoms.

The virtual atoms outside of handshake region ΩHS , which are also called ghost atoms or pad atoms, are generated forthe force calculation of MD simulation. If there are no virtual atoms outside of handshake region ΩHS , an atom at theboundary of MD region has only the force from the handshake region and as a result cannot be in equilibrium. Fig. 15shows the atomistic 1-D chain with a cutoff radius of 4r0 where r0 is equilibrium interatomic distance. The number ofvirtual atoms outside of handshake region ΩHS is decided by cutoff radius in force calculation of MD. We must have virtualatoms in the FE region to satisfy the equilibrium of forces at zero atom location. When we do not consider virtual atoms,the force acting on atom 0, f0, is given by:


f0 =−1∑

i=−4

f i (42)

where f i is the force contribution of atom i to atom 0. The consequence is imbalance in the forces acting on the atom 0.By introducing virtual atoms in FE region ΩFE , the atoms 1, 2, 3, and 4 can exert force on atom 0. When the distancebetween atoms is the same, the equilibrium of forces at atom 0 is satisfied. The force equilibrium at atom 0 is given by:

f0 =4∑

i=−4, i �=0

f i (43)

5.4. Time integration algorithm

The equations of motion for all domains are derived in the previous section. The fundamental time step algorithmin handshake region is the same as the classical MD simulation. In this work, velocity Verlet algorithm is used for MDsimulations because of its efficiency and accuracy. Through the relation of momentum, p = mv , the coupled momentumequation in Eq. (29) can be represented by

mcoupled vcoupled = αmMD vMD + (1 − α)mFE vFE (44)

where mcoupled , mMD , and mFE are coupled mass of atoms, mass of atoms and mass of virtual atoms in handshake region re-spectively. vcoupled , vMD , and vFE are coupled velocity of atoms, velocity of atoms and velocity of virtual atoms in handshakeregion respectively. If the atomic masses, mcoupled , mMD and mFE , are same, the above equation can be rewritten as:

vcoupled = αvMD + (1 − α)vFE in ΩHS (45)

Utilizing the Cauchy–Born rule at FE mesh in the handshake region, the following relationship can be used to obtain thevirtual atom’s displacement dhs from the nodal displacement d in handshake region:

dhs = Nd (46)

where N is a shape function. Similarly, the virtual atom’s velocity, dhs , and the virtual atom’s acceleration, dhs can becalculated by the equation which is given by:

dhs = Nd (47)

dhs = Nd (48)

where d is the nodal velocity and d is the nodal acceleration in handshake region.In order to utilize the benefit of independent time steps, staggered time integration algorithm is used in the proposed

multiscale modeling approach. The time step is defined by �t = m�tm , �t is used in FE region, and �tm is used in MD andhandshake regions. [ j] will be shorthand for the fractional time step n + ( j/m). Superscripts are used to denote the timestep. In the time integration algorithms, it is assumed that the initial conditions are known.

In MD domain ΩMD , displacement q, velocity q, and acceleration q of atoms are updated via velocity Verlet algorithm asfollows:

q[ j+1] = q[ j] + q[ j]�tm + 1

2q[ j]�t2

m (49)

q[ j+1] = m−1a fMD

(q[ j+1]) (50)

q[ j+1] = q[ j] + 1

2

(q[ j] + q[ j+1])�tm (51)

where ma is mass of atoms, and �tm is time step of MD.Similarly, in handshake region ΩHS , displacement qhs , velocity qhs , and acceleration qhs of atoms are updated as follows:

d[ j+1]hs = d[ j]

hs + dnhs�tm (52)

q[ j+1]hs = q[ j]

hs + q[ j]hs �tm + 1

2q[ j]

hs �t2m (53)

q[ j+1]hs = m−1

a fMD(q[ j+1]

hs

)(54)

q[ j+1]trial = q[ j]

hs + 1

2

(q[ j]

hs + q[ j+1]hs

)�tm (55)

q[ j+1]hs = αq[ j+1]

trial + (1 − α)d[ j+1]hs (56)

where qtrial is a trial velocity.


Fig. 16. Displacement time-history at x = 0.

Once quantities in the MD and handshake regions are obtained using the above algorithm at time n + 1, the nodaldisplacement d, the nodal velocity d, and the nodal acceleration d are updated from time step n to time step n + 1. Theseupdates use a central difference scheme:

dn+1 ={

dn + dn�t + 12 dn�t2 in ΩFE

(N T N)−1N T qn+1hs in ΩHS

(57)

dn+1 = M−1 fFE(dn+1) (58)

dn+1 = dn + 1

2

(dn + dn+1)�t (59)

If we assume that no external force acts upon the system, fFE = f intFE . The internal force in FE region is computed by the

following equation:

f intFE = Kd (60)

where K is the stiffness matrix, and d is the nodal displacement vector. In the handshake region of Eq. (57) we transferatomic positions calculated in MD to nodal positions by assuming that the MD simulation results in the exact displacementsolution to the problem.

5.5. Conservation of energy

So far we described the detailed methodology of a weighted averaging momentum method (WAMM) to couple moleculardynamics with finite element method. We now consider the energy conservation of system in proposed method, WAMM.

Suppose that we have the material at certain temperature T without motion over time. In this material, although kineticenergy at continuum scale is zero, the kinetic energy at atomistic scale has a certain value corresponding to the tempera-ture T . Therefore, we need to use thermal energy in the continuum region to explain the energy conservation that nodaldisplacement cannot represent in finite element analysis. Total energy in FE region will be subdivided by potential energyV FE , kinetic energy KFE , and thermal energy TFE . Note that kinetic energy and potential energy partly overlap with thermalenergy. Thus, total energy Etotal of whole system is given by:

Etotal = EMD + EFE = (KMD + V MD) + (KFE + V FE + TFE − Eoverlap) (61)

Here, Eoverlap is the overlapping energy. The energy term, TFE − Eoverlap , is for the part that nodal displacement in continuummechanics cannot describe but atomistic displacement in molecular dynamics can.

When the wave in Fig. 3(c) is applied to molecular dynamics region in multiscale model, total energy in moleculardynamics region, EMD , can be subdivided by transportable energy Etransportable and reflected energy Ereflected [24]:

EMD = Etransportable + Ereflected (62)

Here, Etransportable and Ereflected are corresponding to KFE + V FE and TFE − Eoverlap in Eq. (61) respectively. Multiscale modelingtechniques, BDM, BSM, and WAMM, remove the reflected energy at the interface during elimination of the wave reflection.In order to conserve the total energy of system in multiscale model, the reflected energy in molecular dynamics regionneeds to be transferred into thermal energy in the handshake region. Otherwise the temperature in the MD region increasescontinuously.

6. 1-D wave propagation example

6.1. Details

In a 1-D wave propagation example, displacement is propagated from MD to FE zone. The prescribed displacement usedin the analysis is plotted in Fig. 16.


Fig. 17. Schematic details of 1-D wave propagation example.

u(t, x = 0) = Ae−((t−7)/xa)

2 − uc

1 − uc

(1 + b cos

(2π

H(t − 7)

))(63)

where uc = e−(Lc/xa) , A = ha = 0.005, xa = 500ha , La = 25xa , b = 0.1, and H = xa/4. Applied units are nanometer (nm) fordisplacement and picosecond (ps) for time. As shown in Fig. 17, the total number of atoms used for MD simulation is 181,the number of (overlapped) atoms in the handshake region is 20, the number of (overlapped) element in the handshakeregion is 2, and the total number of elements for FE is 100. The region designated for MD is 0 � x � 22.23 nm. Thehandshake region is 22.23 nm < x � 24.7 nm. The time step in MD and handshake region is 0.002 ps and the time step inFE is 0.02 ps.

Lennard-Jones (LJ) potential, ΦLJ , is used for the interatomic potential which is given by:

ΦLJ(ri j) = 4ε

[(σ

ri j

)12

−(

σ

ri j

)6](64)

Here, ri j is interatomic distance between atoms i and j, ε = 0.2 J and σ = 0.11 nm and the equilibrium bond length isr0 = 21/6σ . The cutoff radius rc is 2.5r0. The interatomic force in MD simulation can be obtained by following equation:

f (ri j) = −∂Φ(ri j)

∂ri j= 48ε

ri j

[(σ

ri j

)12

− 1

2

(σ

ri j

)6](65)

In continuum mechanics, the energy density W is obtained by using LJ 6-12 potential. As following the stiffness defini-tion in Cauchy–Born rule in Eq. (15), the stiffness k0 between two nodes in 1-D is defined as:

k0 = ∂2W

∂ F T ∂ F T

∣∣∣∣r=r0

= r0

l0

(624ε

214/6σ 2− 168ε

28/6σ 2

)(66)

where l0 is the initial nodal distance.

6.2. Results

Weighted averaging momentum in the handshake region yields the different levels of reduction in the high frequencywave reflection depending on the value of parameter p. To optimize the value of parameter p, a parametric study fordifferent p values was conducted. Fig. 18 shows the displacement at 17 ps with two overlapped elements in the handshakeregion. Normalized kinetic energy in MD region at 17 ps is plotted in Fig. 19. MD region kinetic energy is normalized withrespect to the maximum value. As shown in Figs. 18, 19, and 20, the wave reflection and normalized kinetic energy havethe minimum value for p = 0.01. Parameter p below and above 0.01 has a larger wave reflection and larger kinetic energy.

To optimize the length of handshake region, various lengths of handshake region are studied and presented in Figs. 21,22, and 23. Figs. 21 and 22 show that using only one overlapped element in handshake region shows the worst kineticenergy transfer. Increasing the number of overlapped elements in handshake region shows excellent kinetic energy transfer.Among them, the case of two overlapped elements in the handshake region has the minimum wave reflection with thebenefit of shortest handshake region, Fig. 23.

The amplitude of the input low frequency wave is A = 0.005 nm, and amplitude of input high frequency is 2Ab =0.001 nm. In Fig. 24, influence of a weighted averaging momentum in handshake region is compared with and withouthandshake region. When handshake region is not used, the maximum displacement in FE is 5.024 × 10−3 nm, and themaximum displacement in MD is 0.9479 × 10−3 nm. It is obvious that the deformation and associated energy of shortwavelength is not transferred into the FE region. When handshake region is introduced, for the case of p = 0.01 which hasthe minimum wave reflection, the maximum displacement in FE is 5.13 × 10−3 nm, and the maximum displacement of MDis 7.266 × 10−5 nm. When handshake region is used, the maximum displacement in FE with handshake region increases


Fig. 18. Parametric study for different p values at 17 ps with two overlapped elements in handshake region.

Fig. 19. Normalized kinetic energy in MD region versus parameter p values at 17 ps with two overlapped elements in handshake region.


Fig. 20. Time history of normalized kinetic energy in MD region for different p values.

by 2.11%, while the high frequency wave’s amplitude in MD region is reduced by 92.73%. It implies that when WAMMtechnique is used, the most of the high frequency wave kinetic energy is dissipated.

After 18 ps, kinetic energy in MD region with p = 0.01 is almost zero, while that of MD without handshake region isstill large. This result indicates that kinetic energy of high frequency motion is almost dissipated in the handshake region,while kinetic energy of high frequency wave in MD without handshake region is reflected at the boundary of MD regionand remains in MD region until the end of simulation. The spurious kinetic energy in MD region, without handshakeregion, leads to noise in the computation of MD region. In contrast, the values of kinetic energy in FE region with andwithout handshake region are almost same because high frequency energy cannot be transferred to the FE region becauseof element size. In Fig. 25, the curve of normalized kinetic energy of FE using a weighted averaging momentum shows aparallel shift because the wave arrives later in FE region due to the handshake region. In Fig. 25 after 18 ps, kinetic energy inMD region when there is no handshake region periodically fluctuates near a certain large energy value due to the reflectedhigh frequency energy.

6.3. Comparison with other multiscale modeling techniques

MAAD [2] and QC method [1] have two main issues: one is the short wavelength reflection at the interface and anotheris the time step limitation that the time step of FE must be the same scale as that in MD region. BDM [4], BSM [5,6],and the technique proposed in this paper, weighted averaging momentum method (WAMM), solve the short wavelengthreflection problem and show similarly excellent results in eliminating the wave reflection. The direct comparison of 1-Dwave propagation example is shown in Figs. 26 and 27 for BDM and BSM respectively. In Fig. 26, the results of BSM andthe proposed technique WAMM are almost same. The results in Fig. 27 are very similar but have a small difference incontinuum region. It may be because BSM uses meshfree method for the analysis in continuum region while we use finiteelement methods for WAMM. The details of 1-D example are in Refs. [4,5].

Among all the multiscale techniques discussed above only, BDM, BSM, and the proposed technique WAMM use separatetime steps in FE and MD regions. In spite of these independent time steps, computational cost of the multiscale method isstill the most important criterion in choosing a multiscale approach. To estimate the increasing computing load due to themultiscale modeling, we consider a typical case with nhs atoms in handshake region and nB atoms at MD boundary. In oneMD time step �tm , BDM increases computational cost on the order of O(nhs), BSM increases computational cost on the orderof O((2ncrit + 1)nHnB) where the period of time history kernel is nH�tm [22] and ncrit is referred in Eq. (26), and WAMMproposed here increases computational cost on the order of O(nhs). When these values are normalized by nB , the normalizedvalue, nhs/nB , of BDM is about 64 [4], the normalized value of BSM is (2ncrit + 1)nH [22], and the normalized value ofWAMM is 20. In this comparison BSM could theoretically have a smaller number of normalized value of computations byusing smaller ncrit . However, it has to be emphasized that the calculations for each atom in WAMM are much simpler thanBDM and BSM. While BDM and BSM have to execute complex force calculations, WAMM has a simple weighted averagingvelocity calculation. For this reason, we believe that the proposed technique WAMM has a big advantage, over BSM andBDM, in reducing the computational cost of multiscale analysis.


Fig. 21. Time history of normalized kinetic energy in MD region along the number of elements in handshake region using p = 0.01.

Fig. 22. Normalized kinetic energy in MD region versus the number of overlapped elements in handshake region with p = 0.01 at 17 ps.

Fig. 23. Amplitude of the first reflected waves depending on the number of overlapped elements in handshake region.


Fig. 24. Effect of using a weighted averaging momentum method.

Fig. 25. Energy transfer from MD to FE for two overlapped elements and p = 0.01.

Fig. 26. Comparison of the proposed weighted averaging momentum method (WAMM) with BDM [4] for 1-D wave propagation example [4].

7. 2-D wave propagation example

In a 2-D wave propagation example, displacement is propagated from MD to FE region. The displacement used in theanalysis is applied to one atom at the center of model in x direction, and is given by:

u(t, x = 0, y = 0) = Ae−((t−7)/xa)

2 − uc(

1 + b cos

(2π

(t − 7)

))(67)

1 − uc H


Fig. 27. Comparison of the proposed weighted averaging momentum method (WAMM) with BSM [5,6] for 1-D wave propagation example [6].

Fig. 28. Schematic details of the model used for 2-D wave propagation example.

where uc = e−(Lc/xa) , A = ha = 0.005, xa = 500ha , La = 25xa , b = 0.3, and H = xa/4. Applied units are nanometer (nm) fordisplacement and picosecond (ps) for time.

As shown in Fig. 28, the total number of atoms used for 2-D simulation is 154 241, the number of (overlapped) atoms inthe handshake region is 23 512, the number of (overlapped) element in the handshake region is 112, and the total numberof elements for FE is 32 000. The region designated for MD is |x| � 64

√3r0 and |y| � 216r0. The equilibrium bond length

r0 is same as in 1-D example. The region designated for handshake is 64√

3r0 < |x| � 80√

3r0 and 216r0 < |y| � 240r0. Thetime step in MD and handshake region is 0.002 ps and the time step in FE is 0.02 ps. Parameter p is 0.01.

In molecular dynamics and handshake region, Lennard-Jones (LJ) potential ΦLJ is used for the interatomic potential givenby Eq. (64) and same parameters and cutoff radius as in 1-D example are used. The interatomic force in MD simulation canbe obtained by Eq. (65). In continuum mechanics, the constitutive relationship and constitutive matrix in 2-D are definedas:


Fig. 29. A four-noded element and 6 atoms of distance r0 with the atom 0 of which interatomic stiffness is k.

Fig. 30. The cases of force generation for an atom in x and y directions: (a) force generation at the surface in x direction by missing atoms 4 and 5, (b) forcegeneration in y direction at the surface by missing atoms 2, 3, and 4, and (c) force generation in y direction at the surface by missing atom 3.

σi j = [C]εi j (68)

[C] =⎡⎣ c11 c12 c13

c21 c22 c23c31 c32 c33

⎤⎦ (69)

To obtain the components of constitutive matrix, following Cauchy–Born rule, atomic positions in an element are il-lustrated in Fig. 29. Because the interatomic distance r0 dominates the force contribution, the three cases in Fig. 30 areconsidered. We assume that the set of atoms are connected with a spring of stiffness k. The force acting on an atom fromthe surface by strain εi j can be calculated and be translated to the stress σi j . The constitutive components ci j are obtainedby using the constitutive relationship given by Eq. (68).

k = ∂2Φ(ri j)

∂r2i j

∣∣∣∣ri j=r0=21/6σ

= 624εσ 12

r140

− 168εσ 6

r80

= 72ε

21/3σ 2(70)

c11 = c22 = 3√

3

4k, c21 = c12 = c33 =

√3

4k, c13 = c31 = c23 = c32 = 0 (71)

The calculated constitutive matrix shows isotropy and plane stress conditions (elastic modulus E = (2/√

3)k, Poisson’s ratioν = 1/3). For a 2-D linear elastic system, the stiffness matrix can be defined as:

K =∫

[B]T [C][B]dV (72)

In this 2-D example, a four-noded rectangular finite element is used with the size, lx = 8√

3r0, ly = 12r0.Fig. 31 shows the contour of the result from 2-D wave propagation at 16 ps. As Fig. 31 shows, when a handshake region

is not used, while long wavelength is transferred from MD to FE, short wavelength is reflected at |x| = 80√

3r0 ≈ 17.11 nm


Fig. 31. Contour of 2-D wave propagation example at 16 ps: (left) direct coupling without the handshake region, and (right) the proposed weightedaveraging momentum method (WAMM).

that is the border of MD and FE regions. The reflected wave travels in MD region resulting in spurious energy accumulationin MD region. However, when WAMM is utilized, the short-wavelength wave reflection is successfully eliminated duringwave propagation. This result indicates that the parameter p and handshake length, which are determined in 1-D example,work successfully at 2-D example as well.

Comparison with other methods at 2-D is not possible, because there is no published data of 2-D wave reflection ex-amples in the literature. Programming other multiscale methods for a 2-D example is almost impossible due to the lack ofdetails in the published papers.

8. Conclusions

In this study, a weighted averaging momentum method, WAMM is introduced for multiscale modeling and shows excel-lent coupling results in 1-D and 2-D examples through the transfer of the energy and displacement. This method does notrequire scaling down FE mesh size to MD atomic resolution. As a result MD region and FE region have independent timesteps. We use staggered time integration algorithms. Moreover, because this approach has a short handshake region, it hasthe benefit of reducing computational cost significantly. The biggest advantage of WAMM is the fact that it is a simple wayto link MD with FE. On handshake region, FE nodes coarsely represent the MD energy, and MD velocity is averaged with FEvelocity. By using this approach, the spurious wave reflection at the interface is dramatically reduced.

Acknowledgements

The research project has been sponsored by US Navy Office of Naval Research (ONR) Advanced Electrical Power Systemsprogram under the direction of program director Dr. Peter Cho.

Appendix A. Supplementary material

Supplementary material related to this article can be found online at http://dx.doi.org/10.1016/j.jcp.2013.06.039.

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