discrete-to-continuum bridging based on multigrid principles

Comput. Methods Appl. Mech. Engrg. 193 (2004) 1693–1711

www.elsevier.com/locate/cma

Discrete-to-continuum bridging based on multigrid principles

Jacob Fish *, Wen Chen

Department of Civil, Mechanical, Aerospace and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180, USA

Received 28 May 2003; received in revised form 29 September 2003; accepted 2 December 2003

Abstract

The paper presents a new concurrent multiscale approach based on multigrid principles with intent of solving large

molecular statics and molecular dynamics systems. We show that the coarse model effective stiffness matrix obtained by

variational restriction of the atomistic model effective stiffness matrix coincides with the effective stiffness matrix of the

equivalent continuum scale model. The equivalent continuum model is defined for both molecular statics and molecular

dynamics cases and theoretical estimates of the rate of convergence for the proposed concurrent multiscale approach

are obtained.

2004 Elsevier B.V. All rights reserved.

Keywords: Molecular dynamics; Implicit time integrator; Multilevel; ARVE; Rate of convergence

1. Introduction

Physical phenomena at different scales can be represented by a hierarchy of mathematical models.

Fortunately, the physical world is dominated by the processes on different scales that are only weakly

coupled. This suggests that hierarchical models representing diverse length and time scales can be bridged ina sequential manner, i.e., calculations at finer scales, and of high-computational complexity, can be used to

evaluate constants for use in a more approximate or phenomenological computational methodology at a

longer length/time scale. This type of scale bridging is often known as sequential, serial, information-

passing, or parameter-passing. For nonlinear problems hierarchical models at different scales are two-way

coupled, i.e., the information continuously flows between the scales. For sequential multiscale methods to

be valid both the temporal and spatial scales have to be separable. For example, if the essential events of

the faster hierarchical model occur on the same time scales as the details of processes computed using the

slower hierarchical model, then the time scales cannot be separated. Likewise, if the wavelength of thetraveling signal is of the order of magnitude of the fine scale features, then the spatial scales cannot be

separated. Sequential scale-bridging methods for linking continuum scales, such as variational multiscale

[1] and the asymptotic multiscale methods [2–4], have their roots in computational mathematics, whereas

* Corresponding author.

E-mail address: [email protected] (J. Fish).

0045-7825/$ - see front matter 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.cma.2003.12.022

mail to: [email protected]

1694 J. Fish, W. Chen / Comput. Methods Appl. Mech. Engrg. 193 (2004) 1693–1711

various engineering homogenization and coarse graining procedures [5–7] typically used to link discretescales are often semi-empirical in nature. One popular example is kinetic Monte Carlo for surface growth,

where barriers to adatom motion are computed with density functional theory. Similarly, for discrete

dislocation dynamics, mobilities are computed using atomistic simulations.

There are many systems, however, which depend inherently on physics at multiple scales. These pose

notoriously difficult theoretical and computational problems. Turbulence, crack propagation, friction, and

problems involving nano-like devices are prime examples. In fracture the crack tip bond breaking can be

described with a quantum mechanical model of bonding, while the rest of the sample is described with

empirical potentials. In friction it might be necessary to describe surface interaction using quantum-chemical approaches while using continuum elasticity to simulate the contact forces. For these types of

problems, multiple scales have to be simultaneously resolved in different portions of the problem domain.

Multiscale methods based on the concurrent resolution of multiple scales are often coined as embedded,

concurrent, integrated or hand-shaking multiscale methods. Various Domain Decomposition (DD)

methods [28–32] are used to communicate the information between the subdomains represented by different

hierarchical models.

This paper presents a new paradigm for a concurrent consideration of multiple spatial scales based on

Multigrid (MG) principles. In principal both MG and DD approaches separate the system response intoglobal (continuum) and local (discrete) effects. While DD based methods may consider continuum and

discrete scales in different subdomains, the proposed multigrid based concurrent method carries out de

facto spectral decomposition where the continuum model is utilized to capture the lower frequency re-

sponse of the discrete scale.

2. Problem statement and objectives

In our investigation, we focus on the class of molecular dynamics [8,9,11,12] (MD) and molecular statics

(MS) problems where the wavelength L of the excitation function is comparable to the characteristic length

of the discrete (atomistic) scale. In this case the sequential multiscale methods are not applicable.

The MD equations can be derived from the Hamiltons principle [19,26]

dZ t2

t1

ðT UÞdt þZ t2

t1

dW dt ¼ 0; ð1Þ

where T and U are the system kinetic and potential energies, respectively; dW is the virtual work done by

external forces. The resulting nonlinear equations of motion can be cast into the following matrix form

Rðd; €dÞ ¼ M€d þNðdÞ F ¼ 0; ð2Þ

where Rðd; €dÞ is the residual vector; M is the mass matrix; NðdÞ is the vector of internal forces and F is the

vector of external forces; and d is the vector representing positions of atoms. The superposed dot denotes

time differentiation.

Eq. (2) can be integrated using the Newmark method, which yields:

_dkþ1 ¼ _dk þ Dt½ð1 cÞ€dk þ c€dkþ1; ð3Þ

dkþ1 ¼ dk þ Dt _dk þ Dt2½ð1=2 bÞ€dk þ b€dkþ1; ð4Þ

where parameters b and c determine stability and accuracy characteristics of the algorithm [10,21,22].

b ¼ 1=4 and c ¼ 1=2 correspond to the trapezoidal rule.

J. Fish, W. Chen / Comput. Methods Appl. Mech. Engrg. 193 (2004) 1693–1711 1695

Solving for €dkþ1 from (4) and substituting the result into the residual equation at tkþ1 gives

Rðdkþ1Þ ¼ 0: ð5Þ

Solving the above system of nonlinear equations using Newton method yields (see, e.g., [20])

dsþ1kþ1 ¼ dskþ1 ½ bK ðdskþ1Þ

1Rðdskþ1Þ; ð6Þ

bK ðdskþ1Þ ¼oR

od

dskþ1

: ð7Þ

In this manuscript we focus on the linearized system of Eq. (6) which can be put into the following form:

bKDd ¼ R; Dd 2 Rn; R 2 Rn; bK 2 Rnn: ð8Þ

We define the prolongation operator, Q, as Q : Rm ! Rn, m < n, where m denotes the size of the

auxiliary continuum model. The restriction operator, QT, from the discrete model to continuum model is

conjugated with the prolongation operator, i.e., QT : Rn ! Rm. The coarse model matrix bK 0 is obtained by

variational restriction of bK , i.e.bK 0 ¼ QT bKQ; bK 0 2 Rmm: ð9ÞIn Section 3 we show that the coarse model effective stiffness matrix obtained by variational restriction of

the atomistic model coincides with the effective stiffness matrix of the continuum scale model. In Section 4

we show that while the smoothing preconditioner P 2 Rnn captures the fine (atomistic) scale features, the

equivalent continuum model preconditioner, denoted as C ¼ ðQ bK10 QTÞ1 2 Rnn, can be engineered to

resolve the low frequency response of the atomistic model. Following standard multigrid nomenclature, the

smoothing and coarse model correction iteration matrices are denoted as G ¼ I P1 bK 2 Rnn and,

T ¼ I C1 bK 2 Rnn, respectively, where I is the n n identity matrix. The two-scale iteration matrix isthen given by

L ¼ GtTG t 2 Rnn ð10Þwith t post- and pre-smoothing iterations. For more details we refer to [13–18,23–25].

Convergence studies of the proposed concurrent multiscale method are carried out for a model problem

consisting of a chain of n atoms with alternating masses m1 and m2, as shown in Fig. 1. A time-varying force

f ðtÞ in the negative direction is exerted at atom n, the last atom in the chain. Each atom interacts only with

its nearest neighbors.

m1 m2 m2 m1

x

u1 u2 u3 u4 un 1– un

E1 E2

ui ui i 1+( ), ui 1+

. . . .

ARVE

Fig. 1. The 1D atomistic chain and atomistic representative volume element (ARVE).


The interatomic potentials are modeled by the Lennard-Jones potential:

EkðrÞ ¼ 4ekrkr

12 rk

r

6; ð11Þ

where rk and ek ðk ¼ 1; 2Þ are model parameters and r is the distance between two atoms. Periodicity of

masses and parameters of interatomic potentials is assumed. The distance rk0 for which the interatomic forces

vanish is obtained by setting, dEkðrÞ=dr ¼ 0, ðk ¼ 1; 2Þ, which yields: rk0 ¼ffiffiffi26

prk ¼ 1:12246rk, k ¼ 1, 2.

The total kinetic energy of the atomistic chain is

T ¼ 12m1 _u21 þ 1

2m2 _u22 þ 1

2m1 _u23 þ 1

2m2 _u24 þ þ 1

2m1 _u2n; ð12Þ

where ui (i ¼ 1; 2; . . . ; n) denotes the atom positions. The total potential energy of the atomistic chain is

U ¼Xn1

i¼1

4ekrk

uiþ1 ui

12"(

rkuiþ1 ui

6#)

; k ¼ 1; i is odd;2; i is even:

ð13Þ

Substituting (12) and (13) into (1) yields the MD equations of motion (2), in which

M ¼ diagðm1;m2;m1;m2; . . . ;m2;m1Þ; ð14Þ

d ¼ u1 u2 u3 un1 un½ T; F ¼ 0 0 0 0 f ðtÞ½ T; ð15Þ

NðdÞ ¼ N1 N2 N3 Nn1 Nn½ T: ð16ÞThe elements of NðdÞ and the tangent stiffness matrix, K t, are given in Appendix A.

For the purpose of investigating the rate of convergence of the concurrent multiscale method, we

consider a quadratic approximation of the Lennard-Jones potentials (11)

E1ðrÞ ¼ e1 þ 12aðr r10Þ

2; E2ðrÞ ¼ e2 þ 1

2bðr r20Þ

2; ð17Þ

where rk0 ðk ¼ 1; 2Þ are the interatomic distances at the equilibrium; a and b are constants defined by:

a ¼ 36ffiffi4

3p

e1r21

and b ¼ 36ffiffi4

3p

e2r22

.

Note that periodicity of ek and rk ðk ¼ 1; 2Þ implies periodicity of a and b. The resulting tangent stiffness

matrix is given by

K t ¼ tridiagða; aþ b;bÞ; i is even;tridiagðb; aþ b;aÞ; i is odd;

i ¼ 1; 2; . . . ; n: ð18Þ

3. Equivalent continuum model

The goal of the coarse model is to capture the low frequency response of the fine scale problem. In this

section we define a nearly optimal coarse model for molecular dynamics and statics model problems, and

construct the corresponding prolongation/restriction operators. Convergence studies are then conducted in

Section 4.

Proposition 1. The optimal coarse model for molecular dynamics is a local continuum model which isequivalent to the atomistic model. The definition of equivalence for the model problem is described below.

Consider the atomistic and continuum Representative Volume Elements shown in Fig. 2.External forces, F1ðtÞ and F2ðtÞ, act at the two ends of the Atomistic Representative Volume Element

(ARVE). Displacements of the three atoms in ARVE are denoted by u1, u2 and u3. The equivalent continuum

, ,

u1 u2 u3

a b

m1 2/ m2 m1 2/

k ρ Ω

u1 u3

Fig. 2. The atomistic and continuum representative volume elements.


model is characterized by the effective stiffness k and the effective mass density per unit length q. The con-

tinuum model is said to be equivalent to the atomistic model if it produces the same end displacements when

subjected to the same end forces, F1ðtÞ and F2ðtÞ, acting on the atomistic model. The equations of motion for the

atomistic model in ARVE are:

m1=2 0 0

0 m2 0

0 0 m1=2

24 35 €u1€u2€u3

264375þ

a a 0

a aþ b b0 b b

24 35 u1u2u3

24 35 ¼F10

F2

24 35: ð19Þ

The equivalent continuum model is given as

qX6

2 1

1 2

€u1€u3

" #þ k

1 1

1 1

u1u3

¼ F1

F2

: ð20Þ

Proof. Eliminating F1 and F2 from Eqs. (19) and (20) yields

m1

2€u1 þ aðu1 u2Þ ¼

qX6

ð2€u1 þ €u3Þ þ kðu1 u3Þ; ð21Þ

m1

2€u3 þ bðu3 u2Þ ¼

qX6

ð€u1 þ 2€u3Þ þ kðu3 u1Þ; ð22Þ

m2€u2 þ ðaþ bÞu2 ¼ au1 þ bu3: ð23ÞUsing the Newmark integrator the acceleration vector can be expressed as

€ukþ1 ¼1

bDt2ukþ1 þ P ðuk; _uk; €ukÞ; ð24Þ

where Pðuk; _uk; €ukÞ is the term depending on the values at the previous time step. Substituting (24) into (23),

solving for u2, and then inserting the resulting solution into (21) and (22) yields

c2

"þ a k a2

aþ bþ d qX3bDt2

#u1 þ k

" abaþ bþ d

qX6bDt2

#u3

¼ qX3

m1

2

!P1

am2

aþ bþ dP2 þ

qX6P3; ð25Þ


k

" abaþ bþ d

qX6bDt2

#u1 þ

c2

"þ b k b2

aþ bþ d qX3bDt2

#u3

¼ qX6P1

bm2

aþ bþ dP2 þ

qX3

m1

2

!P3; ð26Þ

where Piðuik; _uik; €uikÞ ði ¼ 1; 2; 3Þ are functions computed at the previous step and

c ¼ m1

bDt2; d ¼ m2

bDt2: ð27Þ

As an approximation, we neglect Piðuik; _uik; €uikÞ ði ¼ 1; 2; 3Þ. Since u1 and u3 are non-trivial, the deter-

minant of the coefficient matrix of Eqs. (25) and (26) should vanish, which leads to

c2

"þ a k a2

aþ bþ d qX3bDt2

#c2

"þ b k b2

aþ bþ d qX3bDt2

#

k

" abaþ bþ d

qX6bDt2

#2¼ 0: ð28Þ

It can be easily shown that if

k1 ¼ab

aþ bþ dþ 1

3

adaþ bþ d

þ c2

; q1 ¼

2bDt2

Xad

aþ bþ d

þ c2

ð29Þ

or

k2 ¼ab

aþ bþ dþ 1

3

bdaþ bþ d

þ c2

; q2 ¼

2bDt2

Xbd

aþ bþ d

þ c2

ð30Þ

then Eq. (28) holds. The equivalent continuum model is then defined as an average of the above two

solutions, which yields

k ¼ 1

2ðk1 þ k2Þ ¼ g þ h

6; q ¼ 1

2ðq1 þ q2Þ ¼

bDt2hX

¼ 1

Xm1

þ m2

1þ d=ðaþ bÞ

; ð31Þ

where

g ¼ abaþ bþ d

; h ¼ ðaþ bÞdaþ bþ d

þ c: ð32Þ

Remark 1. It can be observed that parameters defining the interatomic potentials contribute to the

equivalent mass density of the continuum model. Likewise, the mass terms in the atomistic model enter the

expression of the equivalent stiffness of the continuum model.

Discretizing the equivalent continuum model using a consistent mass matrix yields the following stiffnessand mass matrices

K c ¼ k tridiagð1; 2;1Þ; Mc ¼bDt2h6

tridiagð1; 4; 1Þ: ð33Þ

The effective stiffness matrix of the equivalent continuum model becomes

bK c ¼ K c þ1

bDt2Mc ¼ g tridiagð1; 2;1Þ þ hI ; ð34Þ

where I is the identity matrix.


In Section 4, we carry out theoretical convergence studies of the concurrent method which incorporatesthe equivalent continuum model as the auxiliary coarse model.

Proposition 2. The optimal interscale prolongation/restriction operators between the atomistic and equivalentcontinuum models can be constructed by the internal energy minimization of the atomistic model in the ARVEsubjected to the compatibility condition with the equivalent continuum model.

Proof. Consider the representative volume elements of the atomistic and equivalent continuum models

shown in Fig. 2. The continuum model nodal displacements are denoted by u and the atom positions aredenoted by u. In the atomistic model, positions of the atoms on ARVE boundaries are denoted by ui,i ¼ 1; 2; . . . ;m and positions of atoms inside ARVE are denoted by ui;ðiþ1Þ, i ¼ 1; 2; . . . ;m 1.

The effective stiffness matrix of the atomistic model in the ARVE can be written as

bK r ¼ K tr þ

1

bDt2M r ¼

aþ c a 0a aþ bþ d b0 b bþ c

24 35; ð35Þ

where K tr and M r are the tangent stiffness and mass matrices, respectively. For each ARVE, the prolon-

gation operator is defined as

1 0h d0 1

24 35 uiuiþ1

¼

uiui;ðiþ1Þuiþ1

24 35; i ¼ 1; 2; . . . ;m 1; ð36Þ

whereby the atom positions on the boundary of the ARVE coincide with the nodal positions in the

equivalent continuum model. Positions of atoms in the interior of the ARVE is determined from the energy

minimization and compatibility with the equivalent continuum model as described below:

Find the positions of interior atoms ui;ðiþ1Þ that minimize the internal energy of the atomistic model in

ARVE

Pr ¼ 12uTrbK rur; ð37Þ

subjected to the compatibility condition with the equivalent continuum model:

ui ¼ ui; uiþ1 ¼ uiþ1; ð38Þwhere ur ¼ ½ui ui;ðiþ1Þ uiþ1T. The solution of the above constrained minimization yields

h ¼ aaþ bþ d

; d ¼ baþ bþ d

: ð39Þ

The global prolongation operator Q is formed by combining the prolongation operators defined over

individual ARVEs

Q ¼

1

h d

1

h d

h d

1

266666666664

377777777775nm

: ð40Þ


The global effective stiffness matrix of the atomistic model is given as

bK ¼

aþ bþ d bb aþ bþ c a

a aþ bþ d b

b aþ bþ c aa aþ bþ d

26666664

37777775ðn2Þðn2Þ

; ð41Þ

where the following Dirichlet boundary conditions have been incorporated:

u1 ¼ 0; un ¼ 0: ð42ÞBased on Eqs. (40) and (41), the coarse model effective stiffness matrix can be obtained by variational

restriction of the atomistic stiffness matrix (41):bK 0 ¼ QT bKQ ¼ g½tridiagð1; 2;1Þ þ hI : ð43ÞComparison of Eqs. (34) and (43) reveals that the effective stiffness matrix of the equivalent continuum

model bK c coincides with the restricted matrix bK 0. Therefore, the prolongation/restriction operators con-

structed on the basis of the constrained minimization problem over ARVE domain coincide with the in-

terscale atomistic-to-continuum model operators. h

Corollary. The optimal coarse model for molecular statics case can be constructed using Cauchy–Bornhypothesis.

Statics is a special case of dynamics with zero inertia yielding c ¼ d ¼ 0. From (31) the equivalent

stiffness constant for the continuum model becomes k ¼ ab=ðaþ bÞ, which is the classical result that can be

constructed using Cauchy–Born hypothesis.

4. Convergence of the concurrent multiscale method

4.1. Eigenpairs of the effective stiffness matrices

We first relate the eigenpairs of the atomistic model effective stiffness matrix bK to those of the equivalent

continuum model matrix bK 0. We note that if / is an eigenvector of bK 01 ¼ g tridiagf1; 2;1g, it is also an

eigenvector of bK 0, i.e.,bK 01/ ¼ k1/; bK 0/ ¼ k/: ð44ÞThe eigenvalues are related by

k ¼ k1 þ h: ð45ÞThe eigenvectors of bK 01 are (cf. Hackbusch [15])

/ki ¼ sin

ði 1Þkpm 1

; 16 i6m; 16 k6m 2; ð46Þ

where the superscript represents the eigenvector count and the subscript denotes the components of a

specific eigenvector. The first equation in (44) can be written as

gð/ki1 þ 2/k

i /kiþ1Þ ¼ kk1/

ki ; 26 i6m 1: ð47Þ


Substituting the eigenvectors in (46) into (47) and using the sine addition formula gives

kk1 ¼ 4g sinkp

2ðm 1Þ

2; kk ¼ 4g sin

kp2ðm 1Þ

2þ h; 16 k6m 2: ð48Þ

The eigenvalue problem of the atomistic model is givenbK/p ¼ kp/p; 16 p6 n 2; ð49Þ

which in view of Eq. (41) can be written as

a/pi þ ðaþ bþ dÞ/p

i;ðiþ1Þ b/piþ1 ¼ kp/p

i;ðiþ1Þ; 16 i6m 1;

b/pði1Þ;i þ ðaþ bþ cÞ/p

i a/pi;ðiþ1Þ ¼ kp/p

i ; 26 i6m 1;

/p1 ¼ 0; /p

m ¼ 0; 16 p6 n 2:

ð50Þ

We denote the prolongation of the eigenvector /k into the interior of ARVE as /ki;ðiþ1Þ and it is pro-

longated in accordance with the prolongation operator defined in (36):

/ki;ðiþ1Þ ¼ h/k

i þ d/kiþ1: ð51Þ

We first solve for the m 2 smallest eigenpairs ðkk;/kÞ, where 16 k6m 2. Assume that the eigen-

vectors of the atomistic model are related to those of the equivalent continuum model by

/ki ¼ /k

i ; 16 i6m;

/ki;ðiþ1Þ ¼ wk/k

i;ðiþ1Þ; 16 i6m 1; 16 k6m 2;ð52Þ

where wk are parameters to be determined. Substituting (52) into (50) yields

ðaþ bþ dÞwk/ki;ðiþ1Þ a/k

i b/kiþ1 ¼ kkwk/k

i;ðiþ1Þ; 16 i6m 1;

ðaþ bþ cÞ/ki awk/k

i;ðiþ1Þ bwk/kði1Þ;i ¼ kk/k

i ; 26 i6m 1;

/k1 ¼ 0; /k

m ¼ 0; 16 k6m 2:

ð53Þ

Inserting (39) and (51) into (53) and using the relation (47) for the eigenpairs of the equivalent con-

tinuum model yields

ðwk 1Þ/ki;ðiþ1Þ ¼

wkkk

aþ bþ d/ki;ðiþ1Þ; 16 i6m 1;

ða(

þ bþ cÞ wka2 þ b2

aþ bþ d

"þ abaþ bþ d

2

kk1g

!#)/ki ¼ kk/k

i ; 26 i6m 1;

/k1 ¼ 0; /k

m ¼ 0; 16 k6m 2:

ð54Þ

The above equations must be satisfied for any eigenvector /k. Since bK is positive definite, all its

eigenvalues should be positive, i.e., kk > 0. It follows that

kk ¼ ðaþ bþ dÞ 2ðaþ bÞ2ð1 qvÞðc dÞ þ R

> 0; for m1 > m2;

wk ¼ ðaþ bþ dÞ½ðc dÞ þ R2ðaþ bÞ2ð1 qvÞ

; 16 k6m 2;

ð55Þ


where

R ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðc dÞ2 þ 4ðaþ bÞ2ð1 qvÞ

q; q ¼ 4ab

ðaþ bÞ2; 0 < q6 1; v ¼ sin

kp2ðm 1Þ

2: ð56Þ

Next, we solve for the m 2 largest eigenpairs ðkrk;/rkÞ, where r ¼ 2ðm 1Þ, 16 k6m 2. Assume

that the eigenvectors of the atomistic model are related to those of the equivalent continuum model by

/rki ¼ /k

i ; 16 i6m;

/rki;ðiþ1Þ ¼ vk/k

i;ðiþ1Þ; 16 i6m 1; 16 k6m 2:ð57Þ

Following a similar procedure as before, we get

krk ¼ ðaþ bþ dÞ þ 2ðaþ bÞ2ð1 qvÞðd cÞ þ R

> 0;

vk ¼ ðaþ bþ dÞ½ðd cÞ þ R2ðaþ bÞ2ð1 qvÞ

; 16 k6m 2:

ð58Þ

The middle eigenpair ðkm1;/m1Þ follows directly from the eigenvalue problem (50) and the symmetry

condition

km1 ¼ aþ bþ d; /m1i ¼ 0; 16 i6m;

/m1i;ðiþ1Þ ¼ b

a/m1

ði1Þ;i; 26 i6m 1:ð59Þ

4.2. Evaluation of the spectral radius of the two-level iteration matrix

Applying the coarse model iteration matrix T to the eigenvectors of the effective stiffness matrix of the

atomistic model yields

T/p ¼ ðI Q bK10 QT bK Þ/p; 16 p6 n 2; ð60Þ

where bK/k ¼ kk/k; bK/rk ¼ krk/rk; r ¼ 2ðm 1Þ;bK/m1 ¼ km1/m1; 16 k6m 2:ð61Þ

Based on the definition of the prolongation operator (40) and the relationship between the eigenvectors

of the atomistic and equivalent continuum models in conjunction with (53) and (57), the restriction of the

eigenvectors takes the following form

½QT/ki ¼2ðaþ bÞ þ cþ d kk

aþ bþ d/ki ; ½QT/rki ¼

krk 2ðaþ bÞ ðcþ dÞaþ bþ d

/ki ;

½QT/m1i ¼ 0; 16 i6m; 16 k6m 2; r ¼ 2ðm 1Þ:ð62Þ

Combining (61) and (62) and using (44) for the eigenpairs of the atomistic model, we can derive

bK10 QT bK/rk ¼ krk

kkkrk 2ðaþ bÞ ðcþ dÞ

aþ bþ d

/k;

bK10 QT bK/k ¼ kk

kk2ðaþ bÞ þ cþ d kk

aþ bþ d

/k; bK1

0 QT bK/m1 ¼ 0:

ð63Þ


The prolongation of the eigenvectors in the equivalent continuum model based on the operator (40) canbe written as

½Q/ki;ðiþ1Þ ¼ h/ki þ d/k

iþ1 ¼ /ki;ðiþ1Þ ¼

1

wk/ki;ðiþ1Þ ¼

1

vk/rki;ðiþ1Þ; 16 i6m 1;

½Q/ki ¼ /ki ¼ /k

i ¼ /rki ; 16 i6m; 16 k6m 2:

ð64Þ

Combining Eqs. (60), (63) and (64) yields

T/k ¼ ð1þ a1Þ/k þ a2/rk; T/rk ¼ a1/k þ ð1 a2Þ/rk; T/m1 ¼ /m1; ð65Þ

where

a1 ¼ð1þ vkÞkk

ðwk þ vkÞkkkk 2ðaþ bÞ ðcþ dÞ

aþ bþ d

;

a2 ¼ð1 wkÞkk

ðwk þ vkÞkkkk 2ðaþ bÞ ðcþ dÞ

aþ bþ d

:

ð66Þ

For simplicity, we consider a two-level cycle with one weighted Jacobi post-smoothing iteration and one

coarse-level correction. The iteration matrix of this two-level cycle becomes

L ¼ GT; ð67Þwhere

G ¼ I x½diagð bK Þ1 bK ; ð68Þwith x being the weighting factor of the Jacobi method.

Let the eigenpairs of this two-level iteration matrix L be ðWk; ckÞ, ðWrk; crkÞ and ðWm1; cm1Þ, wherer ¼ 2ðm 1Þ and 16 k6m 2, giving

LWk ¼ ckWk; LWrk ¼ crkWrk; LWm1 ¼ cm1Wm1: ð69ÞSubstituting the atomistic model effective stiffness matrix bK in (41) into (68) yields

G/p ¼ Ap/p; 16 p6 n 2; ð70Þwhere

Ap ¼ diagðK1;K2;K1;K2; . . . ;K1Þ: ð71Þ

K1 ¼ 1 xkp

aþ bþ d; K2 ¼ 1 xkp

aþ bþ c: ð72Þ

From Eq. (59) and incorporating the Dirichlet boundary conditions (42), we have

/m1 ¼ 1 0 ba

0ba

2

0 ba

3

0ba

4

ba

m2 T

: ð73Þ

Further utilizing Eqs. (65), (67), (70), (71) and (73), we can derive

Am1/m1 ¼ 1

xkm1

aþ bþ d

/m1; ð74Þ

L/m1 ¼ G/m1 ¼ Am1/m1 ¼ 1

xkm1

aþ bþ d

/m1: ð75Þ


Based on Eqs. (59) and (75) it follows:

Wm1 ¼ /m1; cm1 ¼ 1 x: ð76Þ

Let

Ak/k ¼ x1/k þ x2/

rk; Ark/rk ¼ y1/k þ y2/

rk; ð77Þ

where x1, x2, y1 and y2 are coefficients to be subsequently defined. From Eq. (64), we have

/ki;ðiþ1Þ ¼

wk

vk/rki;ðiþ1Þ; /k

i ¼ /rki ; /rk

i;ðiþ1Þ ¼vk

wk/ki;ðiþ1Þ: ð78Þ

From Eqs. (71), (77) and (78) we can derive

x1 ¼ 1 xkk

aþ bþ c1

þ wk

wk þ vkc d

aþ bþ d

; x2 ¼

wk

wk þ vkxkkðd cÞ

ðaþ bþ cÞðaþ bþ dÞ ; ð79Þ

y1 ¼vk

wk þ vkxkrkðd cÞ

ðaþ bþ cÞðaþ bþ dÞ ; y2 ¼ 1 xkrk

aþ bþ c1

þ vk

wk þ vkc d

aþ bþ d

: ð80Þ

Let Wk be expressed in term of the linear combination of /k and /rk, i.e.,

Wk ¼ f1/k þ f2/

rk; ð81Þwhere f1 and f2 are coefficients. Then by exploiting the relations

G/k ¼ Ak/k ¼ x1/k þ x2/

rk; G/rk ¼ Ark/rk ¼ y1/k þ y2/

rk ð82Þand using Eq. (65), we have

LWk ¼ fx1½f1ð1þ a1Þ f2a1 þ y1½f1a2 þ f2ð1 a2Þg/k

þ fx2½f1ð1þ a1Þ f2a1 þ y2½f1a2 þ f2ð1 a2Þg/rk

¼ ckðf1/k þ f2/rkÞ;

from which it follows that

½x1ð1þ a1Þ þ y1a2f1 þ ½y1ð1 a2Þ x1a1f2 ¼ f1ck;

½x2ð1þ a1Þ þ y2a2f1 þ ½y2ð1 a2Þ x2a1f2 ¼ f2ck:ð83Þ

From the first of the above equations, it follows:

ck ¼ ½x1ð1þ a1Þ þ y1a2 þ ½y1ð1 a2Þ x1a1f2f1: ð84Þ

Selecting

f1 ¼ ½y1ð1 a2Þ x1a1; ð85Þwe obtain

ck ¼ x1ð1þ a1Þ þ y1a2 þ f2: ð86ÞTo determine f2 we eliminate ck from the second equation in (83) and (84), and insert (85) into the resulting

equation, which yields

f 22 þ ½x1ð1þ a1Þ þ x2a1 þ y1a2 y2ð1 a2Þf2 þ ½x2ð1þ a1Þ þ y2a2½x1a1 y1ð1 a2Þ ¼ 0: ð87Þ


We get, after some manipulations, the solution of the above quadratic equation:

ðf2Þ1;2 ¼ 1

2Rðaþ bÞ2 4abv

ðaþ bþ cÞðaþ bþ dÞ ½4xða(

þ bÞ 2ðaþ bþ cÞ þ xð3cþ dÞ þ ð1 xÞðc dÞ)

xðaþ bÞ2ð1 qvÞ ð1 xÞðaþ bþ cÞðaþ bþ dÞ2ðaþ bþ cÞðaþ bþ dÞ ; ð88Þ

where R, q and v are given in Eq. (56).

Substituting Eqs. (48), (55), (58), (66), (79), (80) and (88) into (86) yields

ck1 ¼ 0; ck2 ¼ð1 xÞðaþ bþ cÞðaþ bþ dÞ x½ðaþ bÞ2 4abv

ðaþ bþ cÞðaþ bþ dÞ : ð89Þ

Similarly, it can be shown that crk has the same solutions as (89). Therefore, the rate of convergence

expressed in terms of the spectral radius of the two-level iteration matrix L becomes:

qðLÞ ¼ maxfjck2j; 1 xg: ð90ÞSince 06 v6 1, we select the weighting factor x in such a way that all modes excluding the middle mode

have the same rate of convergence for the two extreme values:

jck2jðv¼0Þ ¼ jck2jðv¼1Þ: ð91Þ

The above equation yields

x ¼ ðaþ bþ cÞðaþ bþ dÞðaþ bþ cÞðaþ bþ dÞ þ a2 þ b2

: ð92Þ

It can be seen that 0 < x < 1, since both the numerator and the denominator of (92) are positive and the

denominator is greater than the numerator. Substituting (92) into (89) yields

jck2j ¼2ab

ðaþ bþ cÞðaþ bþ dÞ þ a2 þ b2: ð93Þ

Recall that c ¼ m1=ðbDt2Þ, d ¼ m2=ðbDt2Þ, and select b ¼ 1=4, which yields

jck2j ¼1

1þ k þ 1=k þ 2f2½1þ k þ r þ krð1þ 4f2Þ; ð94Þ

where

k ¼ ab

P 1; r ¼ m1

m2

P 1; f ¼ DtcrDt

ð95Þ

and Dt is the time step employed in the time integration; Dtcr is the critical time step of an explicit scheme,

which is taken approximately as

Dtcr ¼ffiffiffiffiffiffim2

a

r: ð96Þ

The plot of the spectral radius of the two-level iteration matrix (94) for various parameters is given in Fig. 3.

Remark 2. k, r and f in (94) are three non-dimensional parameters which characterize the degree of hetero-

geneity in physical properties and inertia effects, respectively. Note that the value of jck2j decreases with

increasing values of k, r and f. For the case of molecular statics, i.e., f ¼ 0, the rate of convergence reduces

to jck2j ¼ 1=ð1þ k þ 1=kÞ.

1 20 40 60 80 1000

0.05

0.1

0.15

0.2

0.25

0.3

m1/m2

Spe

ctra

l rad

ius

of th

e ite

ratio

n m

atrix

∆tcr

/∆t = 1/10← a/b = 1

↓a/b = 10

↓a/b = 100

1 20 40 60 80 1000

0.05

0.1

0.15

0.2

0.25

m1/m2

∆tcr

/∆t = 1/4

← a/b = 1

← a/b = 10

↓a/b = 100

Fig. 3. The rate of convergence of the concurrent multiscale method for the model problem.


4.3. Numerical examples for the anharmonic case

In Section 4.2, we gave a theoretical estimate of the rate of convergence of the proposed concurrent

multiscale method for the harmonic case, i.e. assuming quadratic approximation of the interatomicpotentials, giving rise to constants a and b identical for all ARVEs. This simplification is for the benefit of

theoretical estimate only. For the numerical examples described in this section a general nonlinear case is

considered with the instantaneous prolongation

Q ¼

1

h1 d1

1

h2 d2

hnr dnr

1

2666666664

3777777775nm

ð97Þ

varying in space and time and

hi ¼ai

ai þ bi þ di; di ¼

biai þ bi þ di

; ai ¼ oNpoupþ1

; bi ¼ oNqouqþ1

;

p ¼ 2i 1; q ¼ 2i; i ¼ 1; 2; . . . ; nr; nr ¼ m 1:

ð98Þ

The boundary and loading conditions for the atomistic chain are defined as follows. The first atom is

fixed and the last atom in the chain is free. The interatomic distance for which the interaction forces

vanishes is given as rk0 ¼ffiffiffi26

prk ðk ¼ 1; 2Þ. No external loads are applied, i.e., FðtÞ ¼ 0. At t ¼ 0, the atomic

chain is subjected to the initial pulse disturbance pðxÞ in atom positions, so that the atom positions become,

u0i þ pðxÞ, with u0i being the initial equilibrium positions of the atoms

u0i ¼ði 1Þðr10 þ r20Þ=2; i is odd;

ir10=2þ ði=2 1Þr20; i is even;

(i ¼ 1; 2; . . . ; n

and

pðxÞ ¼ f0a0½x ðx0 dÞ4½x ðx0 þ dÞ4f1 H ½x ðx0 þ dÞg½1 Hðx0 d xÞ;

Table 1

Number of two-level iterations (n) for 1D cases (g ¼ 108)

Iteration numbers CMA (MG-QC) n ¼ 400 n ¼ 1000 n ¼ 2000

m1=m2 ¼ 1, r2=r1 ¼ 1, Dt=Dtcr ¼ 10 7(8) 7(9) 7(14)

m1=m2 ¼ 10, r2=r1 ¼ 1, Dt=Dtcr ¼ 10 4(394) 4(10) 4(5)

m1=m2 ¼ 100, r2=r1 ¼ 1, Dt=Dtcr ¼ 10 3(16) 3(3) 3(3)

m1=m2 ¼ 1000, r2=r1 ¼ 1, Dt=Dtcr ¼ 10 1(2) 1(1) 1(1)

m1=m2 ¼ 1, r2=r1 ¼ 1, Dt=Dtcr ¼ 4 4(4) 4(4) 4(4)

m1=m2 ¼ 1, r2=r1 ¼ 5, Dt=Dtcr ¼ 4 3(4588) 3(6536) 3(6730)

m1=m2 ¼ 1, r2=r1 ¼ 10, Dt=Dtcr ¼ 4 2(>8000) 2(>8000) 2(>8000)

m1=m2 ¼ 10, r2=r1 ¼ 1, Dt=Dtcr ¼ 4 2(2) 2(2) 2(2)

m1=m2 ¼ 100, r2=r1 ¼ 1, Dt=Dtcr ¼ 4 1(1) 1(1) 1(1)


where a0 ¼ 1=d8 and HðxÞ is the Heaviside step function; f0, x0 and d are the magnitude, the location of the

maximum value and the half width of the pulse. This pulse is similar in shape to the Gaussian distribution

function. We set x0 ¼ ðn 1Þðr10 þ r20Þ=4, so that the pulse is centered at the midpoint of the atomic chain.We use three SSOR (Symmetric Successive Overrelaxation) pre- and post-smoothing iterations. The

stopping criterion is taken askrk2kDFk2

¼ g ¼ 108, where krk2 and kDFk2 are the 2-norms of the residual and

right-hand-side vectors, respectively. The energy tolerance for the equilibrium iteration is set as ge ¼ 105.

We study the convergence characteristics of the proposed concurrent multiscale approach (CMA) for

different ratios of masses, interatomic potential parameters and time steps. We keep the interatomic po-

tential parameter e constant, i.e. e1 ¼ e2, and vary the value of r. The results of the concurrent multiscale

approach are compared with the two-grid method where the auxiliary continuum model is constructed

using the Quasi-Continuum method [27,28]. In the Quasi-Continuum approach a subset of representativeatoms is selected to represent the kinematics of the system. The position of the remaining atoms is then

obtained by interpolation. For the model problem considered the representative atoms are selected on the

boundary of ARVE. A linear interpolation is then used to obtain the solution for the interior atoms. Table

1 gives the average number of iterations ðnÞ obtained with the two methods. The iteration numbers in

parentheses given in Table 1 correspond to the two-grid method where the auxiliary coarse model is

constructed using Quasi-Continuum method (MG-QC). For example, 3(6536) denotes that the concurrent

method utilizing the coarse model defined in Section 3 converges in three iterations, while if the Quasi-

Continuum method is used instead the two-level method would converge in 6536 iterations. In all the casesit can be seen that the proposed concurrent multiscale approach converges to the prescribed tolerance in

only few iterations and the number of iterations reduces with decreasing time step size and increasing ratios

of masses m1=m2 and interatomic potential parameters r2=r1, while the behavior of the MG-QC deterio-

rates with increase in heterogeneity and inertia effects. These observations are consistent with our theo-

retical estimates for the harmonic case given in Eq. (94). Recall that the ratio r2=r1 and e1 ¼ e2 is equivalentto

ffiffiffiffiffiffiffiffia=b

pfor the harmonic case. From Table 1, it can be seen that the rate of convergence is independent of

the size of the problem for the concurrent multiscale approach. In the numerical simulations, we observed

that it takes in average two equilibrium iterations for the Newmark predictor–corrector algorithm toconverge to the prescribed energy tolerance of ge ¼ 105.

5. Convergence studies for the 2D harmonic case

We consider a 2D complex lattice composed of atoms with masses m1 and m2 (see Fig. 4). The atoms are

assumed to interact only with their nearest neighbors in both horizontal and vertical directions. The atoms

Fig. 4. The 2D complex lattice and the corresponding ARVE.


on the external boundary are constrained. The interatomic potentials are assumed to be identical for all

atomic pairs and take the form of the quadratic approximation of the Lennard-Jones potential with

r ¼ 3:405 1010, e ¼ 1:6572 1021. The ARVE consists of 16 atoms as illustrated in Fig. 4.

The prolongation operator eQa is constructed based on the constrained minimization problem over

ARVE defined as:

Pa ¼ 12/T bK a/; subjected to k/k2 ¼ 1; ð99Þ

where / and bK a ¼ K ta þ ½1=ðbDt2ÞMa are the displacement vector and the effective stiffness matrix of the

ARVE, respectively. This leads to the eigenproblem:

bK a/ ¼ ka/; k/k2 ¼ 1: ð100Þ

The prolongation operator eQa is constructed from the eigenvectors corresponding to the m smallesteigenvalues of bK a. In the numerical examples considered we selected m ¼ 12.

The global prolongation operator eQ is formed by combining the local prolongation operators computed

over individual ARVEs. We consider the two-level iteration process consisting of three SSOR pre- and

post-smoothing iterations and a coarse-level correction. The spectral radius of this two-level iteration

matrix qðG3TG3Þ is evaluated numerically and the results are summarized in Table 2.

Table 2

Spectral radius of the 2D two-level iteration matrix qðG3TG3ÞqðG3TG3Þ n ¼ 14 14 n ¼ 30 30

Dt=Dtcr ¼ 4 Dt=Dtcr ¼ 10 Dt=Dtcr ¼ 4 Dt=Dtcr ¼ 10

m1=m2 ¼ 1 0.0130 0.1760 0.0146 0.2152

m1=m2 ¼ 10 6.002· 104 0.0614 6.626· 104 0.0727

m1=m2 ¼ 100 7.979· 109 2.531· 104 8.536· 109 3.010· 104

m1=m2 ¼ 1000 4.574· 1015 6.422· 1010 4.628· 1015 6.774· 1010


6. Conclusions

A novel concurrent multiscale approach based on multigrid principles is developed. The two-level method

consists of smoothing, which captures the high frequency response of the atomistic scale and the equivalent

continuum model engineered to resolve the low frequency response of the atomistic model. Both the the-

oretical studies on a model problem as well as the numerical experiments conducted for the anharmonic case

in 1D and 2D revealed that the method converges in just one or two iterations, and that the iteration count

decreases with increase in material heterogeneity and inertia effects. The rate of convergence has been foundto be insensitive to the problem size. Numerical studies show that the convergence of the two-level method is

highly sensitive to the choice of the coarse model. For example, if the coarse model is constructed using the

Quasi Continuum method the number of iterations may increase by a factor of 4000 in some cases.

Acknowledgements

The financial support of Sandia National Laboratory under contract Sandia 84211 and Office of NavalResearch under contract ONR N00014-97-1-0687 are gratefully acknowledged.

Appendix A

A.1. The internal force

N1 ¼24e1r1

2u2 u1

r1

13"

u2 u1r1

7#;

Nn ¼24e2r2

un un1

r2

7"

2un un1

r1

13#;

Ni ¼

24e1r1

ui ui1

r1

7

2ui ui1

r1

13" #

24e2r2

uiþ1 uir2

7

2uiþ1 ui

r2

13" #

; i is even;

24e2r2

ui ui1

r2

7

2ui ui1

r2

13" #

24e1r1

uiþ1 uir1

7

2uiþ1 ui

r1

13" #

; i is odd;

2

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

6 i6 n 1:

A.2. The tangent stiffness matrix

Kt11 ¼

24e1r21

26u2 u1

r1

14"

7u2 u1

r1

8#;

Ktnn ¼

24e2r22

26un un1

r2

14"

7un un1

r2

8#;


Ktii ¼

24e1r21

26ui ui1

r1

14

7ui ui1

r1

8" #

þ 24e2r22

26uiþ1 ui

r2

14

7uiþ1 ui

r2

8" #

; i is even;

24e2r22

26ui ui1

r2

14

7ui ui1

r2

8" #

þ 24e1r21

26uiþ1 ui

r1

14

7uiþ1 ui

r1

8" #

; i is odd;

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

26 i6 n 1;

Kti;iþ1 ¼

24e2r22

7uiþ1 ui

r2

8

26uiþ1 ui

r2

14" #

; i is even;

24e1r21

7uiþ1 ui

r1

8

26uiþ1 ui

r1

14" #

; i is odd;

1

8>>>><>>>>: 6 i6 n 1:

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