Nano Mechanics and Materials:Theory, Multiscale Methods and Applications

byWing Kam Liu, Eduard G. Karpov, Harold S. Park

6. Introduction to Bridging Scale

Molecular dynamics to be used near crack/shear band tip, inside shear band, at area of large deformation, etc.

Finite element/meshless “coarse scale” defined everywhere in domain

Two-way coupled MD boundary condition accounts for high frequency wavelengths

G.J. Wagner and W.K. Liu, “Coupling of atomistic and continuum simulations using a bridging scale decomposition”, Journal of Computational Physics 190 (2003), 249-274

Slide courtesy of Dr. Greg Wagner, formerly Research Assistant Professor at Northwestern, currently at Sandia National Laboratories

6.1 Bridging Scale Fundamentals

Based on coarse/fine decomposition of displacement field u(x):

Coarse scale defined to be projection of MD displacements q(x) onto FEM shape functions NI:

P minimizes least square error between MD displacements q(x) and FEM displacements dI

u x u x u x

u x Pq x N I x dII

Bridging Scale Fundamentals

Fine scale defined to be that part of MD displacements q(x) that FEM shape functions cannot capture:

Example of coarse/fine decomposition of displacement field:

u x q x Pq x

= +

xu xu xu

Slide courtesy Dr. Greg Wagner

Multiscale Lagrangian

Total displacement written as sum of coarse and fine scales:

Write multiscale Lagrangian as difference between system kinetic and potential energies:

Multiscale equations of motion obtained via:

,L u u K u V u

0d L Ldt dd

0d L L

dt q q

u Nd q Pq

Coupled Multiscale Equations of Motion

First equation is MD equation of motion Second equation is FE equation of motion with internal force obtained

from MD forces Kinetic energies (and thus mass matrices) of coarse/fine scales

decoupled due to bridging scale term Pq FE equation of motion is redundant if MD and FE exist everywhere

intAM q f

intTMd N f f int r r

LJ r 4r

12

r

6

Bridging Scale Schematic

MD Boundary Condition Approaches

Generalized Langevin Equation (GLE) S.A. Adelman and J.D. Doll, Journal of Chemical Physics 64, 1976. Limited to one-dimensional cases

Minimizing boundary reflections W. Cai, M. de Koning, V.V. Bulatov and S. Yip, Physical Review

Letters 85, 2000. Size of time history kernel related to number of boundary atoms

Matching conditions W.E., B. Engquist and Z. Huang, Physical Review B 67, 2003. Geometry of lattice must be explicitly modeled

Still lacking consistently derived MD boundary condition that is valid for arbitrary lattice structures, interatomic potentials

MD Boundary Condition Assumptions

Utilize inherently periodic/repetitive structure of crystalline lattices Difficult to apply to fluids, amorphous solids (polymers)

Eliminate all MD DOF’s which are assumed to behave harmonically/linear elastically away from nonlinear physics of interest (crack/defects) Work needed to mathematically define where linear/nonlinear

transition actually occurs in practice

Similar to approach by Wagner, Karpov and Liu (2004), Karpov, Wagner and Liu (2004)

Due to reflective boundaries, the wave packages/signals gradually transforms into heat (chaotic motion):

Important information about physics of the process can be lost.

It is required that wave packages propagate to the coarse scale without reflection at the fine/coarse interface. The successive tracking of wave packages is unnecessary.

Transformation of Effective Information into Heat

Spurious wave reflection occurs at the atomistic/continuum interface. For periodic crystal lattices, the response of the coarse can be computed at the atomistic level, without involving the continuum model.

(atomistic solution is not sought on the coarse grain)

The solution for atom 0 can be found without solving the entire domain, if one knows the dependence:

1 0 , aAu u u

is a known coarse scale displacementFor this 1D problem(quasistatic case):

1 01 1

aa

a a

u u u

01 0'( ) ' ,

( ) - equilibrium interatomic distance, '

aU Ua

U rUr

u uf u uThe single equation to solve:

au

fa–1 a…210

MD domain

Coarse grain

……

f10

Multiscale BC

(multiscale boundary condition)

Multiscale Boundary Conditions

1 2 1 0

0 1 0 1

...2 0

2 0u u u u

u u u u

1 00

( ) ( ) ( )t

u t t u d

2. Force boundary conditions(currently used in bridging scale)

1. Displacement boundary conditions

Displacements of the first atom on the coarse scale u1(t) are considered as dynamic boundary conditions for MD simulation:

u1(t) and all other DoF n>1 are eliminated.Their effect is described by an external force term, introduced into the MD equations:

1 2 1 0

0 1 0

...2 0

2 ( )ext

u u u uu u u f t

ext0

0

( ) ( ) ( )t

f t k t u d

… -2 -1 0 1 2 3 4 …

Domain of interest (fine

grain)

Bulk domain (coarse grain)

In both cases, the knowledge of time history kernel Q(t) is important

Dynamic Multiscale Boundary Conditions with a Damping Kernel

1D Illustration: Non-Reflecting MD/FE InterfaceImpedance boundary conditions allows non-reflecting coupling of the fine and coarse grain solutions within the bridging scale method.

Example: Bridging scale simulation of a wave propagation process; ratio of the characteristic lengths at fine and coarse scales is 1:10

Direct coupling with continuum Impedance BC are involved

Over 90% of the kinetic wave energy is reflected back to the fine grain.

Less than 1% of the energy is reflected.

In case of multiple degrees if freedom per unit cell, the equation of motion is still identical for all repetitive cells n, though it takes a matrix form:

(1) (1)1

0 1(2) (2)2

0 2 0, , , ,

0 0 0n n

n nn n

m k ku fmu f

u f M K K

1

' '' 1

( ) ( ) ( )n

n n n n nn n

Mu t K u t f t

… n-2 n-1 n n+1 n+2 …

1

' '' 1

( ) ( ) ( )n

n n n n nn n

t t t

Mu K u f

… n-2 n-1 n n+1 n+2 …

0 1 1

2 0 0 0, ,

2 0 0 0k k k

k k k

K K K

… n-2 n-1 n n+1 n+2 …

General definition of K-matrices:int

int'

'

( )nn n n

n n

U

f uK f

u 0u u

Several Degrees of Freedom in One Cell

1

' '' 1

( ) ( ) ( )n

n n n n nn n

t t t

Mu K u f… n-2 n-1 n n+1 n+2 … Response function

1 12

' ' ,0' 1

?( ) ( ) ( ) ( , ) ( )n

n n n n nn n

t t t s p s p

MG K G I G M K

Time history kernel:

1 11 0( ) ( ) ( )t s s Θ G GL

ext1 0 1 0

0 0

( ) ( ) ( ) ( ) ( )t t

t d t t d Displacement : Force :u Θ u f K Θ u

Multiscale boundary conditions:

(1,1) (1,1)

(2,1) (2,2)

Θ

0 5 10 15 20 25

-0.2

0

0.2

0.4

0.6

0.8

(1,1)

(2,1)

(1,2) (2,2) 0

Several Degrees of Freedom in One Cell

Further Explanation on Assumption of Linearity

Most interatomic potentials function of distance r (LJ 6-12):

Stiffness for a potential can be evaluated as:

Thus, stiffnesses K are function of position r as well But, if K evaluated about equilibrium separation req=2(1/6):

Linearized MD internal force, i.e. fint = Ku Key result from assumption of linearity: constant K Leads to repetitive expression for MD internal force

LJ r 4r

12

r

6

K 2 r r2

624 12

r14 168 6

r8

Theoretical Developments in 1D

1D Lagrangian for linearized lattice:

Equation of motion:

Note equation of motion valid for every atom n (repetitive structure)!

… n-2 n-1 n n+1 n+2 …

' ''

1 1,2 2

nT Tn n n n n n

n n n n

L

u u u Mu u K u

0n n

d L Ldt

u u

' ''

( ) ( ) ( )n

extn n n n n

n n

t t t

M u K u f

Stiffness (K) Matrices (Nearest Neighbors)

Harmonic potential:

Potential energy per unit cell:

K constants:

… n-2 n-1 n n+1 n+2 …

U r k2

r r0 2

U 12

k un un 1 2 12

k un un1 2

K 1 2U (u)unun1

|u0k , K0 2U (u)un

2 |u0 2k,

K1 2U (u)unun 1

|u0k, Kn-n '

2U (u)unun '

|u0

Tie to Finite Elements

Force on atom n becomes:

Equation of motion for three atoms:

The conclusion, if FE nodes = MD atoms

… n-2 n-1 n n+1 n+2 …

fn kun 1 2kun kun1

1 1 1

1 1 1

02

0

extn n n

extn n n

extn n n

u k k u fm u k k k u f

u k k u f

Repetitive, and resultsfrom constant Kassumption

FE extIJ J IJ J I

MD extIJ J IJ J I

M u K u f

M u K u f

K IJ

FE K IJMD

Kn n ' un '(t)

n 'n

n

fn

One Final Comparison

Re-writing the MD equations of motion:

Equations of motion for n>0 atoms no longer necessary; effects implicitly included in time history kernel (t)

… -2 -1 0 1 2 … … …

Domain of interest Eliminated degrees of freedom

u1(t) (t )u0( )d

0

t

1 2 1 0

0 1 0 1

2 0

2 0

m u u u ukm u u u uk

1 2 1 0

0 1 0 00

2 0

2 0t

m u u u ukm u u u t u dk

Final Coupled Equations of Motion

(t-) called “time history kernel”, and acts to dissipate fine scale energy from MD to surrounding continuum; assumptions of linearity only contained within (t-)

Impedance and random forces act only on MD boundary atoms; standard MD equation of motion elsewhere

Stochastic thermal effects captured through random force R(t)

TMd N f

0 00

t

mq t f t q d d R t

Standard MD Impedance Force Random Force

Features of MD Boundary Condition

MD equation of motion is two-way coupled with coarse scale: If information begins in the continuum, can be transferred naturally

to MD as boundary condition has dimensions of minimum number of degrees of freedom in each

unit cell, and is re-used for every boundary atom: Size of remains constant as size of structure grows - leads to

computational scalability for any lattice structure Automated numerical procedure to calculate time history kernel for a

given multi-dimensional lattice structure and potential Standard numerical Laplace and Fourier transform techniques

derived consistently using lattice dynamics principles No ad hoc damping used to eliminate high frequency waves

Ease of implementation: Only additional external force required for MD boundary atoms

MD Domain Reduced MD Domain + Multiscale BC

n, m+1n, mn, m-1

n+1, mn-1, m MultiscaleBC

MultiscaleBC

The general idea of MS boundary conditions for N-D structures is similar to the 1D case. Response of the outer (bulk) material is modeled by additional external forces applied at the MD/continuum interface.

1 1

, ', ' ', ' ,' 1 ' 1

2

', ', ', '

( ) ( ) ( )n m

n m n n m m n m n mn n m m

n n m mn m n m

t t t

U

Mu K u f

Ku 0u u

Update for the equation of motion: 1D lattice: 2D lattice:

1

' '' 1

2

''

( ) ( ) ( )n

n n n n nn n

n nn n

t t t

U

Mu K u f

Ku 0u u

2-D Lattices

1 1

, ', ' ', ' ,' 1 ' 1

( ) ( ) ( )n m

n m n n m m n m n mn n m m

t t t

Mu K u fEquation of motion

Time history kernel - depends on a spatial parameter m:

Response function

1 1 12

, ', ' ', ' ,0 ,0' 1 ' 1

( ) ( ) ( ) ( , , ) ( , )n m

n m n n m m n m n mn n m m

t t t s p q s p q

MG K G I G M K

1 1 11 0( ) ( , ) ( , )m q mt q s q s

Θ G G L F

Mixed real space/Fourier domain function: 1( , ) ( , , )n p ns q s p qG G

F

c

c

c

c

1, ' 0, '' 0

1ext0, ' 0, ' 1, ' '

' ' 10

( ) ( ) ( )

( ) ( ) ( ) ,

tm m

m m m mm m m

tm m m

m m m m m m m mm m m m m

t t d

t t d

Displacement : u Θ u

Force : f Θ u Θ K Θ

Multiscale boundary conditions:

n, m+1n, mn, m-1

n+1, mn-1, m

n=0n=1n=-1

2-D Formulation

Numerical inverse Laplace transform

1/ 2

0

/S

c T tf t e a L t T

– Laguerre polynomials, – coefficients to be computed using F(s)

22 (2 )J tt

0

sin (2 1) arccosS

r tf t a e

a

Papoulis (Quart Appl Math 14, 1956, p.405)

( )L t

Inverse discrete Fourier transform2/ 2 1

/ 2

1 ( )i pnN

Nn

p N

f f p eN

Fast Fourier transform reduce computational cost: 22logN N N

Week (J Assoc Comp Machinery 13, 1966, p.419)

Numerical Transform Inversion

Initial conditions:

2 20 0

, 2

,

( ) ( )(0) exp2

1.25(0) 0

n m

n m

n n m m

u

u

K-matrices and mass matrix

n, m+1

n, m

n, m-1

n+1, m

n-1, m

n+1, m+1

n-1, m+1

n-1, m-1

n+1, m-1

1,0 1,0 0,1 0, 1

1, 1 1,1 1,1 1, 1

0,0

1 0 0 0, ,

0 0 0 1

1 1 1 1, ,

1 1 1 1

1 0 02( ) ;

0 1 0

k

mk

m

K K K K

K K K K

K M

Time history kernel

( )m tΘ

Performance Study: Problem Statement

Reflection coefficient:

N

N

reflected bmT T

incident0

E E ERE E

Performance Study: Size Effect

c

c

1, ' 0, '' 0

( ) ( ) ( )tm m

m m m mm m m

t t d

u Θ u

Temporal andspatial truncation:

Time steps management( and ) :th h

0.05 /th M k

Performance Study: Method Parameters

The impedance boundary conditions were used along the interface between the reduced fine scale domain and the coarse scale domain in dynamic crack propagation problems (H.S. Park, E.G. Karpov, W.K. Liu, 2003).

The Lennard-Jones potential is utilized.

The 2D time history kernel represents the effect of eliminated fine scale degrees of freedom.

Problem statement

v

FE + MD

FE

FE

Pre-crack

Model description

Application: Bridging Scale Simulation of Crack Growth

Results of the simulations, compared with benchmark (full atomistic solution):

Full atomistic domainFine grain

(coupled MD/FE region)

Crack propagation speeds are virtually identical in the benchmark and multiscale simulations:

Crack tip position vs. time

Application: Bridging Scale Simulation of Crack Growth

Removing Fine Scale Degrees of Freedom in Coarse Scale Region

Equation of motion is identical for all repetitive cells n

Introduce the stiffness operator K

int

int1 1

2 2

1 1 2 2

( ) ( ) ( )

( )

...

2 2 ...

n n n

n n n n n

n n n n

n n n n n n

Mu t f t f t

f t k u u k u u

u u u u

k u u u u u u

int' ' 0 1 2

'

( ) ( ) ( ), 2( ), , , ...

n

n n n n nn n

f t K u t K u t K k K k K

( ) ( ) ( )n n nMu t K u t f t

… n-2 n-1 n n+1 n+2 …

Dynamic response function Gn(t) is a basic structural characteristic. G describes lattice motion due to an external, unit momentum, pulse:

,0

, '

( ) ( )

1, ' , 0( ) ( ) 10, ' 0, 0

n n

n n

f t t

n n tt t dtn n t

( ) ( )n n nMu t K u f t … n-2 n-1 n n+1 n+2 …

2,0 ,0

12 2,0

( ) ( ) ( ) ( ) ( )

ˆ( ) ( ) ( , ) ( )

n n n n n n

n n n

MG t K G t t s MG s K G s

s MG s K G s G s p s M K p

LF

11 1 2

'' 0

ˆ( ) ( )

ˆˆ ˆ( , ) ( , ) ( , ) ( ) ( ) ( )

n

text

n n n nn

G t s M K p

U s p G s p F s p u t G t f d

L F

Periodic Structure: Response Function

Assume first neighbor interaction only:1

' ' ,0 0 1' 1

( ) ( ) ( ), 2 ,n

n n n n nn n

Mu t K u t t K k K k

… n-2 n-1 n n+1 n+2 …

211 1 2 1 2

2 2( 1)

1( ) ( 2 42 4

nip ip

n nM kG t s M k e e s s

s s

L F L

0 2 4 6 8 10 12 140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

20

( ) (2 )t

n nG t J d 0 2 4 6 8 10 12 14

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

2( ) (2 )n nG t J t

Displacements Velocities

Illustration(transfer of a unit pulse due to collision):

Response Function: Example

The time history kernel shows the dependence of dynamics in two distinct cells.Any time history kernel is related to the response function.

1' 1 1 0 0

0

( ) ( ) ( ) , ( ) ( ) ( ), ( ) ( ) ( ) ( )t

n n n n nu t G t f d U s G s F s U s G s G s U s

… -2 -1 0 1 2 …

f(t) 1 0

,0

( ) ( ) , ?( ) ( )n n

u t A u t Af t f t

1 11 0 1 0

0

( ) ( ) ( ) , ( ) ( ) ( )t

u t t u d t G s G s L

21 2

21 2( ) 4 (2 )4

t s s J tt

L

0 2 4 6 8 10 12 14

-0.2

0

0.2

0.4

0.6

22( ) (2 )t J tt

Time History Kernel (THK)

Equations for atoms n > 0 are no longer required

1 2 1 0

0 1 0 1 1 2 1 0

1 0 1 20 1 0 0

0

...2 0 ...

2 0 2 02 0 2 ( ) ( ) 0

...

t

u u u uu u u u u u u u

u u u u u u u t u d

1 00

( ) ( ) ( )t

u t t u d

… -2 -1 0 1 2 …

Domain of interest

Elimination of Degrees of Freedom