3
Tandem Intramolecular Silylformylation and Silicon-Assisted Cross-Coupling Reactions. Synthesis of Geometrically Defined r,-Unsaturated Aldehydes Scott E. Denmark* and Tetsuya Kobayashi Department of Chemistry, Roger Adams Laboratory, University of Illinois, Urbana, Illinois 61801 [email protected] Received January 20, 2003 The palladium- and copper-catalyzed cross-coupling reactions of cyclic silyl ethers with aryl iodides are reported. Silyl ethers 3 were readily prepared by intramolecular silylformylation of homopro- pargyl silyl ethers 2 under a carbon monoxide atmosphere. The reaction of cyclic silyl ethers 3 with various aryl iodides 7 in the presence of [(allyl)PdCl] 2 , CuI, a hydrosilane, and KF2H 2 O in DMF at room temperature provided the R,-unsaturated aldehyde coupling products 8 in high yields. The need for copper in this process suggested that transmetalation from silicon to copper is an important step in the mechanism. Although siloxane 3 and the product 8 are not stable under basic conditions, KF2H 2 O provided the appropriate balance of reactivity toward silicon and reduced basicity. The addition of a hydrosilane to [(allyl)PdCl] 2 was needed to reduce the palladium(II) to the active palladium(0) form. Introduction and Background The formation of carbon-carbon bonds by palladium- or nickel-catalyzed cross-coupling reactions of organo- metallic reagents is one of the most powerful carbon- carbon bond-forming reactions in organic chemistry. 1 Although the Suzuki, Stille, and Negishi couplings are well-known as very useful reactions, they each have distinct drawbacks such as the requirement of forcing conditions, the involvement of toxic reagents and byprod- ucts, as well as oxygen and moisture sensitivity. 1a,2 Recently, the cross-coupling reactions of organosilicon compounds have been extensively developed as an alter- native method which can provide practical solutions to these problems. 3 In the past few years, reports from these laboratories have described new variants of organosilicon cross- coupling reactions for the stereospecific construction of bonds between unsaturated carbon functions. 4 For ex- ample, the synthesis of trisubstituted homoallylic alco- hols by means of either a syn (platinum) 5a and an anti (ruthenium) 5b selective hydrosilylation reaction of alkyn- ylhydrosilanes followed by palladium-catalyzed cross- coupling has recently been reported (Scheme 1). The intramolecular hydrosilylation of alkynylsilyl ethers af- fords cyclic siloxanes having a stereodefined alkenyl moiety. Thus, subsequent stereospecific cross-coupling (1) (a) Metal-catalyzed Cross-coupling Reactions; Diederich, F., Stang, P. J., Eds.; Wiley-VCH: Weinheim, Germany, 1998. (b) Heck, R. F. Palladium Reagents in Organic Syntheses; Academic Press: London, 1985. (c) Tsuji, J. Palladium Reagents and Catalysts. Innova- tions in Organic Synthesis; Wiley: Chichester, UK, 1995. (2) (a) Suzuki, A. In Handbook of Organopalladium Chemistry for Organic Synthesis; Negishi, E., Ed.; Wiley: Hoboken, NJ, 2002; p 249. (b) Miyaura, N.; Suzuki, A. Chem. Rev. 1995, 95, 2457. (c) Suzuki, A. In Metal-catalyzed Cross-coupling Reactions; Diederich, F., Stang, P. J., Eds.; Wiley-VCH: Weinheim, Germany, 1998; Chapter 2. (d) Stille, J. K. Angew Chem., Int. Ed. Engl. 1986, 25, 508. (e) Farina, V.; Krishnamurthy, V.; Scott, W. J. Org. React. 1998, 50, 1. (f) Mitchell, T. N. In Metal-catalyzed Cross-coupling Reactions; Diederich, F., Stang, P. J., Eds.; Wiley-VCH: Weinheim, Germany, 1998; Chapter 4. (g) Negishi, E.; Liu, F. In Metal-catalyzed Cross-coupling Reactions; Diederich, F., Stang, P. J., Eds.; Wiley-VCH: Weinheim, Germany, 1998; Chapter 1. (3) (a) Barcza, S. In Silicon Chemistry; Corey, J. Y., Corey, E. R., Gaspar, P. P., Eds.; Ellis Harwood: Chichester, UK, 1987. (b) Tamao, K.; Kobayashi, K.; Ito, Y. Tetrahedron Lett. 1989, 30, 6051. (c) Hatanaka, Y.; Hiyama, T. Synlett 1991, 845. (d) Hiyama, T. In Metal- Catalyzed Cross-coupling Reactions; Diederich, F., Stang, P. J., Eds.; Wiley-VCH: Weinheim, Germany, 1998; Chapter 10. (e) Mowery, M. E.; DeShong, P. J. Org. Chem. 1999, 64, 1684. (f) Mowery, M. E.; DeShong, P. Org. Lett. 1999, 1, 2137. (g) Mowery, M. E.; DeShong, P. J. Org. Chem. 1999, 64, 3266. (h) Lee, H. M.; Nolan, S. P. Org. Lett. 2000, 2, 2053. (i) Lam, P. Y. S.; Deudon, S.; Averill, K. M.; Li, R.; He, M. Y.; DeShong, P.; Clark, C. G. J. Am. Chem. Soc. 2000, 122, 7600. (j) Denmark, S. E.; Sweis, R. F. Chem. Pharm. Bull. 2002, 50, 1531. (4) (a) Denmark, S. E.; Sweis, R. F. Acc. Chem. Res. 2002, 35, 835. (b) Denmark, S. E.; Yang, S.-M. J. Am. Chem. Soc. 2002, 124, 2102. (c) Denmark, S. E.; Sweis, R. F. Org. Lett. 2002, 4, 3771. (5) (a) Denmark, S. E.; Pan, W. Org. Lett. 2001, 3, 61. (b) Denmark, S. E.; Pan, W. Org. Lett. 2002, 4, 4163. SCHEME 1 10.1021/jo034064e CCC: $25.00 © 2003 American Chemical Society J. Org. Chem. 2003, 68, 5153-5159 5153 Published on Web 06/05/2003

Perfectly matched multiscale simulations

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Perfectly matched multiscale simulations

Albert C. To and Shaofan Li*Department of Civil and Environmental Engineering, University of California, Berkeley, California, 94720 USA

�Received 10 November 2004; revised manuscript received 2 May 2005; published 6 July 2005�

A multiscale method is proposed. It combines the so-called bridging scale method and the perfectly matchedlayer method to form a robust and versatile multiscale algorithm. The method can efficiently eliminate thespurious reflections/diffractions from the artificial atomistic/continuum interface by matching the impedance atthe interface of the molecular dynamic region and the perfectly matched layer. Moreover, it is shown in thispaper that the method can capture anharmonic interaction among nonuniformly distributed atoms in a localregion.

DOI: 10.1103/PhysRevB.72.035414 PACS number�s�: 46.15.�x, 02.70.Ns, 05.10.�a, 45.10.�b

I. INTRODUCTION

It has become a general consensus now that within thenear future first-principle-based simulations will still haveinsurmountable difficulties to simulate a nanoscale thermalmechanical system because even a small nanoscale thermalmechanical system may consist of more than billions of de-grees of freedom �109–1012� and have a typical time scale atfemtoseconds �10−15 s�.

An attractive approach is the so-called multiscale simula-tion, which uses first-principle-based methods, such as mo-lecular dynamics �MD� simulations or even quantum me-chanics �QM� based simulations, in a small local area thatcontains localized inhomogeneities �defects�, while it usescontinuum approach to model the rest of the remaining �butvast� area. During the simulation, mechanical and thermody-namic information such as energy, heat, and configurationforce �energy momentum flux� interchange between atomis-tic scale and continuum scale. Coupling these complicatedprocesses in a concurrent multiscale simulation is the chal-lenge of the emerging nanoscale computational mechanics. Ifthe challenge is successfully met, the method can be used tosimulate, for example, DNA supercoiling, crack initiation,and dislocation motion.

The main difficulty in a concurrent multiscale simulationis the spurious reflection of phonons due to the change ofphysical modeling as well as spatial resolution. The spuriousreflection prevents any meaningful evaluation of temperaturefluctuation in the local region, and hence any meaningfulmultiscale simulation. A few techniques have been proposedto solve this problem. We would like to mention the severalfront runners of the current multiscale simulation research:�1� Cai et al.s generalized Langevin approach,1 �2� E’s opti-mal local matching procedure,2–4 and �3� Liu and his co-workers’ bridging scale method.5–8

In this paper, we propose an alternative multiscale ap-proach called the perfectly matched multiscale simulation�PMMS�. The new approach incorporates Berenger’s per-fectly matched layer �PML� into a local MD simulation9 toeliminate the spurious reflections at the atomistic-continuumboundary during the fine scale part of the multiscale simula-tion.

The rest of the paper is arranged as follows: The generalideas and procedures of PMMS are presented in Sec. II. The

multiscale decomposition of displacement fields is discussedin Sec. III. In Sec. IV, a PML is formulated for an MDsimulation, and, in Sec. V, implementation of the PMMSalgorithm is given and several examples of nonlinear wavepropagation are presented. A few concluding remarks aremade in Sec. VI.

II. THE BASIC IDEAS OF PMMS

A one-dimensional schematic illustration of the PMMS isshown in Fig. 1. A continuum simulation is performed for thewhole domain of interest, while an MD simulation is onlylimited to the pure MD region and the MD-PML region. TheMD simulation is performed to provide information fromatomistic scale to the continuum level, whereas the con-tinuum simulation provides coarse scale information for theMD-PML region. Typically, the continuum simulation is per-formed by using the finite element method �FEM or FE�based on the quasi-continuum modeling.10 The purpose ofthe MD-PML region is to absorb any waves leaving the pureMD region so they will not introduce any artificial reflec-tions.

Since the temporal resolution in the MD simulation ismuch smaller than that in the FE, the MD simulation runsindependently at each fine time step except when the coarsetime step of the FE coincides with the fine step of the MDsimulation. At this coincidence, the fine scale part of the MDdisplacement is added to the FE nodal displacement for eachnode in the MD region for the computation of internal forcein the FE region. At the same time step, the coarse scale partof the displacement and velocity of each atom in the MD-PML region is approximated by the interpolated FE displace-ment and velocity for the corresponding boundary atom. This

FIG. 1. The schematic illustration of PMMS in 1D.

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completes the information exchange between the MD andthe continuum regions. The formulation of the fine-coarsescale decomposition and the PML will be presented in thefollowing two sections, respectively.

The following summarizes the procedure of PMMS:MD �pure MD+MD-PML� region update:1. At each fine step, update displacement, velocity, and

acceleration in the MD region.2. When the coarse step coincides the fine step, the

coarse scale part of the MD displacement and velocity ofeach atom in the MD-PML region is approximated by theinterpolated FE displacement for the corresponding atom.

FE region update:1. At each coarse step, update displacement and velocity.2. Add the fine scale part of the MD displacement to the

displacement for each FE node in the pure MD region.3. Update internal force and acceleration.The numerical implementation of these steps will be

given in Sec. V.

III. FINE-COARSE SCALE DECOMPOSITION

The key issue in multiscale simulations is how to couplethe same physical quantity �i.e., displacement� from the com-putations of different scales by using different modelingschemes. In this paper, we treat the molecular dynamics�MD� simulation as the first principle-based method and seekto replace the MD computation by coarse scale finite elementcomputation in a less crucial region in order to improve com-putational efficiency while maintaining sufficient accuracycompared to an MD only calculation. In the local MD re-gion, information will need to be exchanged between theMD and FE calculations. In the following, a linear relation isderived between the MD displacement and the FE displace-ment to allow exchange between the two regions during aPMMS simulation.

We adopt the Lagrangian mechanics description of mo-lecular dynamics �MD�:

L�q,q� = 12 qTMAq − U�q� + fext

T q , �1�

where q is the MD displacement of the discrete atoms, MA isthe mass matrix, U is the potential energy function, and fext isthe external force.

In principle, the total displacement field of the exact so-lution can be written as the sum of fine scale and coarse scaledisplacements:

u = u� + u , �2�

where u equals q at the atomic positions.The above decomposition is not unique. Assume that the

exact solution of an MD simulation exists. A fine scale solu-tion can be determined if a coarse scale solution has beenassigned.

In this work, the coarse scale solution is taken as an FEMsolution based on the quasi-continuum modeling. Therefore,the coarse scale displacement obtained by the FE at the ini-tial positions of the atoms can be written as

u = Nd , �3�

where d is the nodal displacement and N is a matrix thatconsists of shape functions that interpolate the nodal dis-placement at each initial atomic position. The Lagrangian ofthe FE can then be defined as

L�d,d� = 12 dTMd − U�Nd� + fext

T �Nd� , �4�

where M=NTMAN is the mass matrix.To obtain a relationship between the MD displacement q

and the nodal displacement d in FE in the pure MD region,we assume that a linear mapping P maps the MD displace-ment q to the coarse scale displacement u:

u � Pq , �5�

where the approximation symbol indicates that such projec-tion may not be exact, because first the linear projectionoperator may not be accurate, and second it is possible thatthe FEM solution may not be completely contained in thesolution space of q.

We can then write the MD displacement as

q = Pq + �I − P�q . �6�

Define QªI−P. This suggests that inside the MD zone, theexact displacement field may be approximated as

u � Nd + Qq , �7�

where Qq is used to approximate the fine scale solution of aFEM node inside the MD zone, i.e., u��Qq, and hence totalsolution for a FEM node inside the MD zone. This cannot beachieved without the coarse scale FE calculation inside theMD zone. Note that inside the MD zone, a FEM node doesnot necessarily coincide with an atom’s position, thereby onecannot directly use the MD solution q to find u withoutinterpolation.

The coarse scale calculation based on the FE is an ap-proximation to the MD calculation, but it should be close toit. Thus we find the projection map Q by minimizing thedifference of the MD and the FE Lagrangians,

C�d,d� = mind,d

�L„q�d�,q�d�… − L�d,d�� . �8�

Substituting in the respective Lagrangians, we obtain

C�d,d� = mind,d

� 12 �Nd + Qq�TMA�Nd + Qq� − U�Nd + Qq�

+ fextT �Nd + Qq� −

1

2dTMd + U�Nd� − fext

T �Nd�� .

�9�

Let partial derivatives of the objective function C�d , d� tozero,

�C

�d= − NT�U�q�

�q+ fext + NT�U�u�

�u− fext = 0, �10�

which implies that we shall choose U�q�= U�u�. In addition,we also have

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�C

�d= NTMANd + NTMAQq − NTMANd = 0, �11�

which yields

d = M−1NTMAq �12�

where we have made use of Eq. �7�:

Nd = �I − Q�q . �13�

Comparing �13� with �12�, one sees that

Q = I − NM−1NTMA. �14�

It is simple to check that QQ=Q, which means that Q is aprojector. Also, the projector P is

P = NM−1NTMA. �15�

Although the linear projectors Q and P obtained here areidentical to those obtained for the bridging scale method,6

the derivation here is physically better justified by minimiz-ing the difference between the Lagrangian of the MD and theFE. Also, a linear mapping is assumed in this work; however,a nonlinear operator can also be assumed, but analytical re-sults will be difficult to obtain, and the resulting operatormay only be obtained numerically.

More precise physical meaning can be attached to theterms “fine scale displacement” and “coarse scale displace-ment.” Say, the nodal spacing in the FE simulation is h andthe atomic spacing is ha, and h�ha is assumed as usual. Theminimum wavelength that the FE can support is �min�4hwhile the MD simulation can support down to a wavelengthof �min

a �4ha. According to the decomposition in Eq. �2�, thecoarse scale displacement u calculated by the FE includes allwavelengths that are greater than �min while the fine scaledisplacement u� calculated by the MD simulation representswavelengths between �min

a and �min. These more precise defi-nitions will help to explain the ideas of PMMS.

IV. PERFECTLY MATCHED LAYER

The perfectly matched layer �PML� was developed byBerenger9 to damp electromagnetic waves without introduc-ing artificial reflections at the boundary of the model domain.The technique has since been applied to many different tran-sient problems and is highly successful since it damps bodywaves and evanescent waves at all frequencies except at zerofrequency with minimal reflections.11–14 It can also be ap-plied to anisotropic and inhomogeneous media, and itsimplementation is quite simple. The PML equation for themolecular dynamics �MD� simulation is derived below and itshows that anharmonic waves in MD can also be treated.

We begin the derivation with the classical Newton’s sec-ond law:

F� = mq� = −�U�rN�

�r�

, �16�

where the subscript � denotes the �th atom, and U is thepotential function that depends on the positions rN of the Nth

atom that interact with the �th atom. In Cartesian coordi-nates, the forces in each coordinate are independent of thosein another coordinate; it is sufficient to look at, say, the xcoordinate, and Newton’s law becomes

m�q� = −�U�rN�

�x�

. �17�

Denote

S ª −� U dx�. �18�

Equation �17� can be rewritten as

m�q� =�2S�x�

2 . �19�

This equation can be split into two first-order equations bydefining an auxiliary variable:

�p�

�x�

ª q�. �20�

Its dual equation is

p� =1

m�

�S�x�

. �21�

One constructs a perfectly matched layer by convertingthe real coordinates into complex coordinates as a functionof frequency �. In other words, the following operation isperformed to the dual equations �20� and �21� to convert theequations

�x� → 1 +d�xp�− i�

� x�, �22�

where d�xp� is the damping function that controls the damp-ing with propagating distance and xp is the distance awayfrom the MD and PML boundary. Applying the coordinatestretching to the Fourier transformed dual equations, one canobtain

− i�1 +d�xp�− i�

q� =� p�

�x�

, �23�

− i�1 +d�xp�− i�

p� =1

m�

�S�x�

, �24�

where f denotes the Fourier transform of f . Performing in-verse transform, the perfectly matched layer equations areobtained:

q� =�p�

�x�

− d�xp�q�, �25�

p� = −U

m�

− d�xp�p�. �26�

Combining �25� and �26� yields the modified equation ofmotion for PML:

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m�q� = −�U�rN�

�x�

− m�d�xp�2q� − 2m�d�xp�q�. �27�

Note that the effect of coordinate stretch is not equivalent tosimply adding a damping term; it also changes the stiffnessof the discrete system in the PML zone in order to create anoptimal match for impedance forces.

Since the derivation of the MD-PML equation does notinvolve or require any linearization of the atomistic potential,the PML technique can handle spurious reflections of bothharmonic waves and anharmonic waves. Note that when thedamping function d�xp�=0, the MD-PML equation �27� re-covers the original Newton’s equation.

In this paper, we adopt the semi-empirical damping func-tion proposed by Collino and Tsogka15 :

d�xp� = d0xp

2

�, �28�

d0 = log 1

R3V

2�, �29�

where � is the PML thickness, R is a free parameter less thanunity, and V is the wave velocity.

The success of the absorbing boundary conditions hasvery important implications in PMMS since artificial reflec-tions are of primary concern. PML formulation for other par-tial differential equations can be shown to be nonreflecting atthe boundary in its continuum form, but once the equationsare discretized as in finite difference or finite element meth-ods, artificial reflected waves may exist due to discretization.Nevertheless, the PML is still much better than absorbingboundary conditions that use linear damping mechanismsthat is only effective in a certain range of frequencies whilethe PML formulation damps waves of all frequencies exceptzero frequency �static�. For elastodynamic problems that usethe PML in finite difference, the PML only requires five toten layers of nodes for 2D and 3D problems to prevent anyartificial reflections. The current PML formulation for mo-

lecular dynamics is inherently discretized since the potentialfunction U depends on the neighboring atoms, but from thediscussion above, the PML for molecular dynamics shouldalso perform well as will be seen.

V. IMPLEMENTATION

The implementation of the PMMS is much simpler thanother existing multiscale methods. As stated in Sec. II, thewhole simulation consists of running the FE at coarse timesteps �t and running the MD simulation at fine time steps�tm=�t /m, where m is an integer. Therefore, at the nth stepof the FE, information from the j= �n�m�th step of the MDcalculation is projected onto the FE nodes for FE update.Immediately after the FE update, information is interpolatedat the MD boundary atoms. We use the mixed time integra-tion scheme pioneered by Liu and Belytschko for solvingdynamic fluid-structure problems.16

The Verlet algorithm is chosen for the MD �including thePML� simulation update at the jth step:

q j+1 = q j + �tmq j + 12�tm

2 q j , �30�

q j+1/2 = q j + 12�tmq j , �31�

q j+1 = MA−1−

�U

�q− MAD2q j+1 − 2MADq j+1/2 + fext

j ,

�32�

q j+1 = q j+1/2 + 12�tmq j+1, �33�

where the matrix D is a diagonal matrix consisting of thedamping function d as defined in �28�, which is set to zero inthe pure MD region. Note that Eq. �32� is the discretizedversion of �27� for all the atoms. At the nth step of the FEupdate, we have

dn+1 = dn + �tvn + 12�t2an, �34�

FIG. 2. Initial displacement for the 1D lattice problem withLennard-Jones potential function obtained by �a� PMMS and �b�full MD simulation.

FIG. 3. Displacement at t=10 for the 1D lattice problem withthe Lennard-Jones potential function obtained by �a� PMMS and �b�full MD simulation.

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an+1 = M−1NTf�Ndn+1 + Qq j+1� , �35�

vn+1 = vn + 12�t�an + an+1� . �36�

At the same j= �m�n�th fine step, the coarse scale part ofthe MD displacements and velocities are approximated bythe FE displacements and velocities through interpolation foronly the atoms in the PML region:

q j+1 = Ndn+1 + Qq j+1, �37�

q j+1 = Nvn+1 + Qq j+1, �38�

where Qq j+1 and Qq j+1 are the fine scale part of the dis-placements and velocities in the PML region having beencalculated in Eqs. �30� and �33�. Equations �37� and �38�, ineffect, allow the damping of all the fine scale waves in thePML region while maintaining sufficient accuracy at theboundary of the pure MD region.

To demonstrate the proposed perfectly matched multiscalemethod, numerical simulations are carried out for the follow-ing one-dimensional problem. Consider a one-dimensionallattice. The Lennard-Jones �LJ� potential function is used forthe nearest neighboring atom interaction:

U�rij� = 4� 1

rij12

− 1

rij6� , �39�

where rij the position difference between atom i and j. Aninitial displacement is given,

u�x,t = 0� =�Ae−�x/�2

−uc

1−uc�1 + b cos� 2x

H �� , �x� � Lc,

0, �x� � Lc,

�40�

which is a Gaussian pulse of amplitude A and width . Thepulse is truncated at x= ±Lc. The initial displacement is plot-ted in Fig. 2 in the 1D lattice. The lattice is modeled by 60elements in the FE simulation, 181 atoms spaced at 21/6 in

the full MD simulation, of which 60 atoms are in the MD-PML region. In addition, the fine and coarse time steps are�tm=0.002 and �t=0.1, respectively. The values used for theparameters in Eq. �40� are =20.0, H= /4, A=0.01, b=0.2, and Lc=4.

Figures 3 and 4 show the displacement at t=10 and t=20 coarse time steps, respectively. There are very fewwaves reflected back into the pure MD region when thewaves propagate into the uncoupled FE region. The wavesthat pass into the MD-PML region from the pure MD regionare being damped quickly with very few reflections. Thetime-displacement histories for PMMS and full MD simula-tion are plotted in Fig. 5, and their general shapes agree well.A quantitative analysis of the current method is performed byplotting the Fourier spectrum of the displacement-time his-tory of the incident wave and reflected wave measured in theMD region at x=56.12 and that of the transmitted wave mea-sured in the FE region at x=101.02, as shown in Fig. 6. Theenergy of the incident wave is concentrated at three peaks inthe figure, and the transmitted wave spectrum captures the

FIG. 4. Displacement at t=20 for the 1D lattice problem withLennard-Jones potential function obtained by �a� PMMS and �b�full MD simulation.

FIG. 5. Time-displacement history for the 1D lattice problemwith Lennard-Jones potential function obtained by �a� PMMS and�b� MD.

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lowest frequency peak almost exactly, but completely missesthe second and third peaks, which represent the higher fre-quency waves at the back end of the main wave front seenclearly in Fig. 4 at around x= ±120. The minimum wave-length that the FE region can support is �min=40 and thewave travels at velocity v=9, and thus the highest frequencythat the FE region can support is approximately fmax=v /�min=0.23, which is right after the first peak. In the fullMD simulation, there is no reflected wave, however, inPMMS the reflected wave is approximately two orders ofmagnitude less than the incident wave in amplitude, but re-sembles the incident wave in frequency content for the firsttwo peaks.

In the second example, the collision of a pair of sine-Gordon solitons is considered. The potential function used inthis case is

U�rij� = 12rij

2 + 1 − cos�rij − r0� , �41�

where r0 is the equilibrium atomic spacing. We only considerthe nearest neighboring interaction among atoms.

The initial displacement and velocity used for the colli-sion of two solitons are

u�x,0� = 4 tan−1�expb�x + x0��1 − �2 �

+ 4 tan−1�expb�− x + x0��1 − �2 � , �42�

v�x,0� = − 4 b��1−�2 exp� b�x+x0�

�1−�2 �1 + exp� b�x+x0�

�1−�2 �2 − 4 b��1−�2 exp� b�−x+x0�

�1−�2 �1 + exp� b�−x+x0�

�1−�2 �2 .

�43�

where the initial displacement field in a 1D lattice is plottedin Fig. 7. The configuration of the 1D lattice is identical to

FIG. 6. Fourier spectrum of the incident wave in the pure MDregion, the reflected wave in the pure MD region after reaching thehandshake region, and the transmitted wave in the FE region cor-responding to the Lennard-Jones example.

FIG. 7. Initial displacement for the 1D lattice problem withsine-Gordon potential function obtained by �a� PMMS and �b� fullMD simulation.

FIG. 8. Displacement at t=40 for the 1D lattice problem withsine-Gordon potential function obtained by �a� PMMS and �b� fullMD simulation.

FIG. 9. Displacement at t=80 for the 1D lattice problem withsine-Gordon potential function obtained by �a� PMMS and �b� fullMD simulation.

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the previous example except that the atomic spacing is takenas unity in this example. The fine and coarse time steps arechosen as follows: �tm=0.04 and �t=0.4. The values usedfor the parameters in Eqs. �42� and �43� are x0=0.02, b=0.2, and �=0.2.

Figures 8–10 show the displacement distribution at vari-ous coarse time steps: t=40, t=80, and t=120, respectively.The time-displacement histories obtained from both PMMSand full MD simulation are displayed in Fig. 11. Similar tothe previous example, the solution of PMMS smooths out thesharp features that exist in full MD simulation, and there isstill a small number of waves being reflected back into theMD region. A Fourier spectral analysis of the incident wave,reflected wave, and transmitted wave is performed, and itsresults are shown in Fig. 12. The displacement-time historiesof the incident wave and the reflected wave are measured atx=30 in the MD region and that of the transmitted wave ismeasured at x=100 in the FE region. The energy of the in-cident wave decays almost monotonically with an increase infrequency. The incident wave and the transmitted wave Fou-rier spectrum is almost identical. Similar to the Lennard-Jones potential example, the reflected wave is approximatelytwo orders of magnitude less than the incident wave in am-plitude, but their frequency content is similar.

VI. CONCLUDING REMARKS

Both the idea and implementation of the perfectlymatched multiscale simulations �PMMS� are simple. Onlythe local heterogeneous region is modeled by using the MDsimulation, while the rest �but vast area� of the medium ismodeled by quasi-continuum approach. To match the MDsimulation and quasi-continuum modelings, the perfectlymatched layer �PML� is employed to absorb outgoing tran-sient waves from the MD zone, which leaves very few arti-ficial reflections back into the MD region. With the PML,impedance force or matching boundary conditions at the

FIG. 10. Displacement at t=120 for the 1D lattice problem withsine-Gordon potential function obtained by �a� PMMS and �b� fullMD simulation.

FIG. 11. Time-displacement history for the 1D lattice problemwith sine-Gordon potential function obtained by �a� PMMS and �b�full MD simulation.

FIG. 12. Fourier spectrum of the incident wave in the pure MDregion, the reflected wave in the pure MD region after reaching thehandshake region, and the transmitted wave in the FE region cor-responding to the sine-Gordon potential example.

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outer boundary of the MD region are not needed for PMMSas required in some other multiscale methods.1,2,6 As a matterof fact, the PML can also be thought of as another way toimpose matching atomistic/continuum boundary conditions,but, clearly, the advantages in utilizing the PML are its gen-erality and flexibility.

We would like to point out that the PML/MD interfacereflection coefficient used in this paper has not been opti-mized yet. To derive optimal reflection matching coefficientsat the atomistic/continuum interface, which balance numeri-cal accuracy, efficiency, and stability, is still a task on hand.We do not anticipate any major problems in doing so, be-cause this and other issues, e.g., extending the PMMS tomultiple dimensions, have been well understood and havebeen resolved in solving various other types of differentialequations.11–14

The proposed method has not been thermalized in thiswork, but one may design a thermodynamics-based PMLreflection coefficient, for instance, one can let,

d�xp,T� = T0

T− 1d�xp� , �44�

where T0 is the temperature of the heat bath and T is thetemperature of the MD region. By doing so, the PMMS al-gorithm may preserve thermodynamic equilibrium.

On the other hand, the projector obtained from the multi-scale decomposition for mapping the MD displacement tocoarse scale displacement may also play a significant role inthe accuracy of the method. These topics will be addressed inseparate works.

ACKNOWLEDGMENTS

This work is supported by a grant from NSF �Grant No.CMS-0239130�, which is greatly appreciated.

*Electronic address: [email protected] W. Cai, M. deKoning, V. V. Bulatov, and S. Yip, Phys. Rev. Lett.

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�2003�.5 W. K. Liu, E. G. Karpov, S. Zhang, and H. S. Park, Comput.

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