lattice mechanics and multiple scale boundary...

1Wing Kam Liu, Eduard G. Karpov, Harold Park, David Farrell, Dept. of Mechanical Engineering, Northwestern University

An Introduction to Lattice Mechanicsand

Multiscale Boundary Conditions In Solids


Introduction to Bridging Scale Concurrent Method

• 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

• Bridging scale used to ensure FEM gives correct coarse scale behavior

• Mathematically consistent treatment for MD boundary condition

• 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


Degree of Freedom Removal

• Short wavelengths are reflected because the fine scale degrees of freedom in the coarse region have been removed

• The correct approach to setting boundary conditions is to retain the effects of the degrees of freedom that have been removed

• The first step is to decompose the displacement into a coarse and fine scale:

• We will treat mathematically the removal of the fine scales u’ from region q2.

uuu ′+=


Generalized Langevin Equation

• The total forcing term consists of four major parts:

– The standard force computed in MD simulation by assuming displacements of all atoms just outside the boundary are given by the coarse scale

– A modified stiffness at the boundary (due to slight difference between total scale and coarse scale)

– Time history-dependent dissipation at the boundary

– Random forcing term at the boundary

• this term can be related to the temperature of the solid:

( ) ( ) ( )( ) ( ) ( ) ( )( ) ( )tRdttttt

A 10 111111

1 0~ +τ−τ−β−−β+= ∫− uququfMq &&&&

( ) ( ) ( ) TkttRtR Bijji βδ−=


References

• L. Brillouin, Wave Propagation in Periodic Structures, Dover Publications, 2003.

• A.A. Maradudin, E.W. Montroll, G.H. Weiss, I.P. Ipatova, Theory of Lattice Dynamics in the Harmonic Approximation, Academic Press, New York, 1971.

• M.T. Dove, Introduction to lattice dynamics, Cambridge University Press, New York, 1993.

• M. Toda, Theory of Nonlinear Lattices, Springer-Verlag, Berlin, 1981.

• Wing Kam Liu, E.G. Karpov, S. Zhang, H.S. Park, An introduction to computational nano mechanics and materials, Computer Methods in Applied Mechanics and Engineering 193, 1529-1578, 2004.

• E.G. Karpov, G.J. Wagner and Wing Kam Liu, A Green's function approach to deriving wave-transmitting boundary conditions in molecular dynamics simulations, accepted by Computational Materials Science, 2004.

• G.J. Wagner, E.G. Karpov, Wing Kam Liu, Molecular dynamics boundary conditions for regular crystal lattices, Computer Methods in Applied Mechanics and Engineering 193, 1579-1601, 2004.

• E.G. Karpov, H. Yu, H.S. Park, Wing Kam Liu, Q.J. Wang, D. Qian, Multiscale Boundary Conditions in Crystalline Solids: Theory and Application to Nanoindentation, submitted to Physical Review B, 2004.


Overview

• Regular lattice structure: general concepts

• Mathematical background (Fourier and Laplace transform)

• Elements of lattice mechanics- potential and kinetic energy of the lattice structure- atomic force elastic constants- lattice Lagrangian and equation of motion- lattice Green’s function - elimination of degrees of freedom

• Multiscale boundary conditions

• Applications


Regular Lattice Structures

The term regular lattice structure refers to any translation symmetric domain

1D lattices(one or several degrees of freedom per lattice site):

2D lattices:

… n-2 n-1 n n+1 n+2 …

… n-2 n-1 n n+1 n+2 …

n-2 n-1 n n+1 n+2 …

(n,m)


Regular Lattice Structures

3D lattices (Bravais crystal lattices)

Bravais lattices represent the existing basic symmetries for one repetitive cell in regular crystalline structures.

The lattice symmetry implies existence of resonant lattice vibration modes.

These vibrations transport energy and are important in the thermal conductivity of non-metals, and in the heat capacity of solids.

The 14 Bravais lattices:

(hyperphysics.phy-astr.gsu.edu)


For the purpose of further discussion we recall the definitions and mathematical properties of

1) general functional operators (transforms)

2) Laplace transform

3) discrete Fourier transform (DFT)

Mathematical Issues


Recall first: A function f assigns to every element x (a number or a vector) from set X a unique element y from set Y.

Function f establishes a rule to map set X to Y

Functional Operators (Transforms)

A functional operator A assigns to every function f fromdomain Xf a unique function F from domain XF .Operator A establishes a transform between domains Xf and XF

X Y

x yy=f(x) Examples:

y=xn

y=sin xy=Bx

Xf XF

f FF=A{f} Examples:

0

( ) ( ( ))

( ) ( )t

F x f f x

F t f dτ τ

=

= ∫


Linear operators are of particular importance:

Functional Operators (Transforms)

{ } { }

{ } { } { }

A C f C A f

A f g A f A g

=+ = +

Examples: { }2 2

2ˆ ˆ, ( )

2

f ga f bg a b

x x x

H E H V xm x

∂ ∂ ∂+ = +

∂ ∂ ∂∂

Ψ = Ψ = − +∂

h

Inverse operator A-1 maps the transform domain XF back to the original domain Xf

Xf XF

f Ff=A-1{F}

1{ { }}A A f f− =


Integral Transforms

Linear convolution with a kernel function K(x):

( ) ( ') ( ') 'K f x K x x f x dx∞

−∞

∗ = −∫

Important properties

{ }{ }

{ }{ }2

0

( ) ( ) ( )'( ) ( ) (0) ( )( )

ˆˆ''( ) ( ) (0) '(0) ( ) ( ) ( )

t K f t K s F sf t s F s f F sf d

f t s F s s f f s K f x K p f pτ τ

∗ == − ⎧ ⎫=⎨ ⎬= − − ∗ =⎩ ⎭

∫LL

LL F

Laplace transform (real t, complex s)

1

0

1: ( ) ( ) : ( ) lim ( )

2

a ibst st

ba ib

F s f t e dt f t F s e dsiπ

∞ +− −

→∞−

= =∫ ∫L L

Fourier transform (real x and p)

1 1ˆ ˆ: ( ) ( ) : ( ) ( )2

ixp ixpf p f x e dx f x f p e dpπ

∞ ∞− −

−∞ −∞

= =∫ ∫F F


Laplace Transform: Illustration

Laplace transform gives a powerful tool for solving ODE

Example: 2( ) ( ) 0

(0) 1

(0) 0 ( ) ?

y t y t

y

y y t

ω+ === =

&&

&

Solution: Apply Laplace transform to both sides of this equation, accounting for linearity of LT and using the property { } 2( ) ( ) (0) (0) :y t s Y s s y y= − −&& &L

{ }

{ }

2 2 2

2 2

1 12 2

( ) ( ) 0 ( ) ( ) 0

( )

( ) ( )

y t y t s Y s s Y s

sY s

ss

y t Y ss

ω ω

ω

ω− −

+ = ⇒ − + =

=+

⎧ ⎫= = ⎨ ⎬+⎩ ⎭

&&L

L L

( ) cosy t tω= 0 2.5 5 7.5 10 12.5 15

-1

-0.5

0

0.5

1

t

y(t)


Numerical Laplace Transform Inversion

( ) ( ) ( )1/ 2

0

/S

c T tf t e b L t Tγ γγ

−

=

≈ ∑

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

( ) ( )0

sin (2 1) arccosS

r tf t b eγγ

γ −

=

≈ +∑

bγ( )L tγ

• Weeks algorithm (J Assoc Comp Machinery 13, 1966, p.419)(Also, see self-study material)

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

Most numerical algorithms for the Laplace transform inversion utilize series decompositions of the original f(t) in terms of functions whose Laplace transform is tabulated. The expansion coefficients are found numerically from F(s).

Examples:


Discrete Fourier Transform (DFT)

Discrete convolution

{ }' ''

ˆ ˆ( ) ( )n n n n nn

K u K u K u K p u p−∗ = ∗ =∑ F

Discrete functional sequences

Infinite: ( / ), 0, 1, 2, ...

Periodic: , is integer, 0, 1, 2, ...n

n kN n

u f nx a n

u u N k+

= = ± ±= = ± ±

1/ 2

1/ 2

1 1ˆ ˆ ˆ( ) ( )

2 2

p

p

v nN ipipn ipn Nn n

n v N

vu p u e u u p e dp u e

N N

ππ

π

ππ

−−

= −−

⎛ ⎞= = = ⎜ ⎟

⎝ ⎠∑ ∑∫

DFT of finite sequences

p and vp are continuous and discrete wavenumbers

1 11ˆ ˆ( ) ( )

2

i pn i pnN NN N

n nn N p N

u p u e u u p eN

π π− −−

=− =−

= =∑ ∑

DFT of periodic sequences

Here, p – integer value between –N and N

Discrete Fourier transform (DFT) reduce solution of a large repetitive structure to the analysis of one representative cell only.

pvp

N

π=


DFT: Illustration

Original n-sequence Transform p-sequence

-20 -10 0 10 20

-1

-0.5

0

0.5

1

-20 -10 0 10 20

-1

-0.5

0

0.5

1

-20 -10 0 10 20

-1

-0.5

0

0.5

1

sin 2 , 4, 40n

nu p p N

Nπ= = =

+ +sin 2 , 11, 40n

nu p p N

Nπ= − = =

⇓

,4 , 4ˆ( ) p pu p δ δ −= −

,11 , 11ˆ( ) p pu p δ δ −= −

⇓

-20 -10 0 10 20

-0.4

-0.2

0

0.2

0.4

-20 -10 0 10 20

-0.4

-0.2

0

0.2

0.4

-20 -10 0 10 20

-0.4

-0.2

0

0.2

0.4


In order to formulate the multiscale boundary conditions, we focus on the following topics from the theory of harmonic lattice structures:

• Potential and kinetic energy of the lattice structure

• Atomic force elastic constants

• Lattice Lagrangian and equation of motion

• Lattice Green’s function

• Elimination of degrees of freedom

Elements of Lattice Mechanics


Periodic Lattice Structure: Potential Energy

Potential energy of a linearized lattice structure:

General form(truncated after the harmonic term):

Repetitive structure(un – displacement vector for one unit cell):

Atomic force elastic coefficients (K-matrices)

fn is the internal atomic force

T0 , ' '

, '

1...

2K u un n n n

n n

U U= − +∑

… n-2 n-1 n n+1 n+2 …

, ' ' 0

' ''

, 0

1

2T

K K

u K u

n n n n

n

n n n nn n n

U

Uν

ν

−

+

−= −

⇒

≡ =

= − ∑ ∑

' 0 0'

( )| |n

n n nn n

U− = =

∂ ∂= = −∂ ∂u u

f uK f

u u

ν - connectivity number


Periodic Lattice Structure: Lagrange Function

Kinetic energy(M – mass matrix for a unit cell):

Lagrange function:

2D lattices:

3D lattices:

General case of K-matrices:

… n-2 n-1 n n+1 n+2 …

T' '

'

1 1( , )

2 2

n

n n n n n nn n n n

Lν

ν

+

−= −

= +∑ ∑ ∑ Tu u u M u u K u& & &

T1

2k n nn

E = ∑u M u& &

2D T, , , ', ' ', '

, , ', '

3D T, , , , , , ', ', ' ', ', '

, , , , ', ', '

1 1

2 2

1 1

2 2

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

n m l n m l n m l n n m m l l n m ln m l n m l n m l

L

L

− −

− − −

= +

= +

∑ ∑ ∑

∑ ∑ ∑

T

T

u M u u K u

u M u u K u

& &

& &

2

0

, in 1( )

| , ( , ), in 2( , , ), in 3

n DU

n m Dn m l D

=

⎧∂ ⎪= − = ⎨∂ ∂ ⎪⎩n-n' u

n n'

uK n

u u



Lattice Lagrangian, Example 1: Harmonic Potential

General form(nearest neighbors only)

Harmonic potential(R – equilibrium distance)

Potential energyper unit cell

K-matrices

1T

' '' 1

1 1

2 2

n

n n n n n nn n n n

L+

−= −

= +∑ ∑ ∑ Tu M u u K u& &

21( ) ( )

2V r k r R= −

( )

( ) ( )

' '' '

2 2

1 1

(| |) | ' |

1 1

2 2

n n n nn n n n

n n n n

U V V R n n

k k

≠ ≠

− +

= − = − + − =

− + −

∑ ∑r r u u

u u u u

2 2

1 0 0 021

2

1 01

( ) ( )| , | 2 ,

( )| ,

U Uk k

Uk

− = =+

=−

∂ ∂= − = = − = −

∂ ∂ ∂

∂= − =

∂ ∂

u un n n

un n

u uK K

u u u

uK

u u

… n-2 n-1 n n+1 n+2 …

R


Lattice Lagrangian, Example 1: Harmonic PotentialConnection to Finite Element Stiffness Matrix

• K-matrices connected to FEM stiffness (linear elastic)

• K-matrices simply relation of force/displacement at an atom

… n-2 n-1 n n+1 n+2 …

R

1 1

1 1

1 1

1 1

1 1n n

n n

f uk

f u− −

⎧ ⎧⎫ ⎫−⎡ ⎤⎪ ⎪ ⎪ ⎪=⎨ ⎬ ⎨ ⎬⎢ ⎥−⎣ ⎦⎪ ⎪⎪ ⎪⎭ ⎭⎩ ⎩ 1 1

2 2

2 2

1 1

1 1n n

n n

f uk

f u+ +

⎧ ⎧⎫ ⎫−⎡ ⎤⎪ ⎪ ⎪ ⎪=⎨ ⎬ ⎨ ⎬⎢ ⎥−⎣ ⎦⎪ ⎪⎪ ⎪⎭ ⎭⎩ ⎩

FEM Element Equations:

FEM nth equation for assembled system:

1 12assembledn n n nf k u k u k u− += − +

K-1 K0 K+1

Note: have changed from typical FEM sign convention to the lattice mechanics sign convention to show equivalence


Lattice Lagrangian, Example 2: Morse Potential

General form(nearest neighbors only)

Morse potential(R – equilibrium distance)

Potential energyper unit cell

K-matrices

Note: for complicated potentials, the required derivatives of the potential energy can be obtained numerically.

1T

' '' 1

1 1

2 2

n

n n n n n nn n n n

L+

−= −

= +∑ ∑ ∑ Tu M u u K u& &

( )2 ( ) ( )( ) 2R r R rV r e eβ βε − −= −

( )

( ) ( )1 1 1 1

' '' '

2 ( ) ( ) 2 ( ) ( )

(| |) | ' |

2 2n n n n n n n n

n n n nn n n n

U V V R n n

e e e eβ β β βε ε− − + +

≠ ≠

− − − − − − − −

= − = − + − =

− + −

∑ ∑u u u u u u u u

r r u u

2 22 2

1 0 0 021

22

1 01

( ) ( )| 2 , | 4 ,

( )| 2 ,

U U

U

εβ εβ

εβ

− = =+

=−

∂ ∂= − = = − = −

∂ ∂ ∂

∂= − =

∂ ∂

u un n n

un n

u uK K

u u u

uK

u u

… n-2 n-1 n n+1 n+2 …

R


Periodic Lattice Structure: Equation of Motion

The equation of motion has to be identical for all repetitive cells n.

Substitute the Lagrangian L into the Lagrange equation of motion

More generally, for the case of non-trivial external forces,

3D generalization:

… n-2 n-1 n n+1 n+2 …

T' '

'

' '( ) ( )'

1 1( , )

2 2

0, 1, 2,..., ( ) ( )

n

n n n n n nn n n n

n

n n n nn nn n

L

d L Ls t t

dt u u

ν

ν

ν

α αν

α

+

−= −

+

−= −

= +

∂ ∂− = = −⇒ =

∂ ∂

→∑ ∑ ∑

∑

Tu u u M u u K u

M u K u 0

& & &

&&&

' ''

( ) ( ) ( )n

extn n n n n

n n

t t tν

ν

+

−= −

− =∑M u K u f&&

, , ', ', ' ', ', ' , .' ' '

( ) ( ) ( )mn l

extn m l n n m m l l n m l n m l

n n m m l l

t t tµν λ

ν µ λ

++ +

− − −= − = − = −

− =∑ ∑ ∑M u K u f&&


Periodic Structure: Dynamic Response Function

Dynamic response function gn(t) is a basic structural characteristic. g describes lattice motion due to a localized external pulse:

,0

, '

( ) ( )

1, ' , 0( ) ( ) 1

0, ' 0, 0

f extn n

n n

t t

n n tt t dt

n n t

δ δ

δ δ δ∞

−∞

=

= ∞ =⎧ ⎧= = =⎨ ⎨≠ >⎩ ⎩ ∫

( ) ( )extn n nt t− ∗ =Mu K u f&&

… n-2 n-1 n n+1 n+2 …

,0( ) ( )M g K gn n nt tδ δ− ∗ =&&

Function g is defined by the following equation:

Solution is compact in terms of Fourier and Laplace transforms


Dynamic Response Function

(“0–” means that before t=0 all atoms are at rest)Apply the Laplace and Fourier transforms:

… n-2 n-1 n n+1 n+2 …

{ }{ } ( )

( ){ }

2,0 ,0

12 2,0

11 1 2

( ) ( ) ( ) ( ) ( )

ˆ( ) ( ) ( , ) ( )

ˆ ˆ( ) ( ) , ( )

Mg K g MG K G

MG K G G M K

g M K K K

n n n n n n

n n n

ipnn n

n

t t t s s s

s s s s p s p

t s p p e

δ δ δ

δ−

−− − −

− ∗ = ⇒ − ∗ =

− ∗ = ⇒ = −

⇓

= − =∑

&&

' '' 0

ˆˆ ˆ( , ) ( , ) ( , ) ( ) ( ) ( )U G F u g ft

ext extn n n n

n

s p s p s p t t dτ τ τ−= ⇒ = −∑ ∫

,0( ) ( ), (0 ) (0 ) 0, ( ) ?n n n n n nt t tδ δ− ∗ = − = − = =M g K g g g g&& &

Using the convolution theorems, solution for a general dynamic load can be given in terms of g:


Dynamic Response Function: 1D Example

Assume first neighbor interaction only:

1

' ' ,0 0 1' 1

( ) ( ) ( ), 2 , , 1Mu K K K Mn

n n n n nn n

t u t t k k kδ δ+

− ±= −

− = = − = = =∑&&

… n-2 n-1 n n+1 n+2 …

( ){ } ( )211 1 2 1 2

2 2( 1)

1( ) ( 2 4

2 4

nip ip

n nM kt s k e e s s

s s

−− − − −

−= =

⎧ ⎫= − − + = + −⎨ ⎬

+⎩ ⎭g ML F L

0 2 4 6 8 10 12 14

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

2

0

( ) (2 )t

n nt J dτ τ= ∫g

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 nt J t=g&

Displacements Velocities

IllustrationTransfer of a unit pulse due to collision (movie):


Time History Kernel (THK)

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

0

0 0 11 1 0 0

1 1

( ) ( ) ( ) , ( ) ( ) ( ), 0, 1,...

( ) ( ) ( )( ) ( ) ( ) (

(

)( ) ( (

)) )

u g v U G V

U G VU G G U

U G VΘ

t

n n n nt t d s s s n

s s s

s

s s s ss s s

τ τ τ

−

= − = = ±

=⎧=⎨ =⎩

⇒ ⇒

∫

1442443

… -2 -1 0 1 2 …

v(t) (force) { }1 0

,0

( ) ( ) , ?

( ) ( )

u θ u θ

f vn n

t t

t tδ= =

=

{ }1 0

01 1

1 0

( ) ( ) ( ) ,

( ) ( ) ( )

u θ u

θ G G

t

t t d

t s s

τ τ τ

− −

= −

=

∫

( )21 2

2

1 2( ) 4 (2 )

4t s s J t

t− ⎧ ⎫= + − =⎨ ⎬⎩ ⎭

θ L

0 2 4 6 8 10 12 14

-0.2

0

0.2

0.4

0.6

2

2( ) (2 )t J t

t=θ

Writing solution for the load fn(t) in terms of g, we can express u1 through u0 via Laplace transform, and let n=0 and 1:

e.g.


Elimination of Dynamic Degrees of Freedom: Example

Equations for the atomic DoF nr1 are no longer required.We have taken them into account implicitly.

1 2 1 0

0 1 0 1 1 2 1 0

0 1 0 01 0 1 20

...

2 0...

2 0 2 0

2 ( ) ( ) 02 0

...

t

mu u u u

km m

u u u u u u u uk k

mmu u u t u du u u u

kkτ τ τ

− − −

− − − −

−

⎧⎪ ⎧⎪ − + − = ⎪⎪ ⎪⎪⎪ ⎪− + − = ⇒ − + − =⎨ ⎨⎪ ⎪⎪ ⎪ − + − Θ − =− + − =⎪ ⎪

⎩⎪⎪⎩

∫

&&

&& &&

&&&&

1 0

0

( ) ( ) ( )t

u t t u dτ τ τ= Θ −∫

… -2 -1 0 1 2 … … …

Domain of interestEliminated degrees

of freedom

Using the time history kernel:


Several Degrees of Freedom in One Cell

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 n

n n

m k ku f

mu f

κ κκ κ ±

− −⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= = = = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠⎝ ⎠⎝ ⎠ ⎝ ⎠

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 …

10 1 1

2

0 2 0 0 0, , ,

0 2 0 0 0M K K K

m k k k

m k k k−

−⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⇒ = = = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠

… n-2 n-1 n n+1 n+2 …

General definition of K-matrices:2

''

n nn n

U−

∂= −

=∂ ∂K

u 0u u

kκ

k

E.g. 1

E.g. 2


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

2

1

' ' ,0' 1

ˆ ˆ( , ) ( )

( ) ( ) ( ) 0 2 0ˆ ( )0 0 0 0

n

n n n n nip ipn n

s p s p

t t t k k kp e e

δ δ κ κκ κ

−

+

−−= −

= −− = − −⎛ ⎞ ⎛ ⎞ ⎛ ⎞

= + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⇒∑G M K

MG K G IK

&&

Time history kernel:

{ }{ }

1 11 0

11

( ) ( ) ( )

ˆ( ) ( , )

t s s

s s p

− −

−

=

=

Θ G G

G G

L

F

ext1 0 1 0

0 0

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

t d t t dτ τ τ τ τ τ−= − = −∫ ∫Displacement : Force :u Θ u f K Θ u

Multiscale boundary conditions:

(1,1) (1,1)

(2,1) (2,2)

⎛ ⎞Θ Θ→⎜ ⎟

Θ Θ⎝ ⎠Θ

0 5 10 15 20 25

-0.2

0

0.2

0.4

0.6

0.8

(1,1)←Θ

(2,1)←Θ

(1,2) (2,2) 0Θ = Θ =↓

E.g. 2 dof


Quasistatic Formulation

Lagrangian

Variational principle

Equation of the equilibrium state

3D generalization:

Lattice Green’s function

General load solution

… n-2 n-1 n n+1 n+2 …

' ''

1( )

2

n

n n n nn n n

Lν

ν

+

−= −

= ∑ ∑ Tu u K u

', ', ' ', ', ' , .' ' '

mn lext

n n m m l l n m l n m ln n m m l l

µν λ

ν µ λ

++ +

− − −= − = − = −

− =∑ ∑ ∑ K u f

( )

' ''

0, 1, 2,...,n

next

n n n nn n

Ls

u α

ν

ν

α

+

−= −

∂− = =∂

− =⇒ ∑ K u f

{ }1 1ˆ ˆ( ) , ( ) ipnn n

n

p p e− − −= − =∑g K K KF

' ''

extn n n n

n−= ∑u g f



Boundary Condition Operators for Finite Size Domain

The atomic force operators shows dependence of solutions in two adjacent cells. All excitations propagate formally with infinite velocities in the quasistatic case. Therefore, the effect of peripheral boundary conditions, ua, has to be taken into account.

Form of the solution

External force

general load solution:

solving for u1 in terms of u0 and ua givesThe boundary condition operators ( ) ( )

0 0 0

' ' 0 1 1 0 1'

0 0

1

01 1

0

a aext

n n n n n n a a a an

a a a

aa

a

−

− − −

−−

−

= += = + = +

= +

⎛ ⎞

⇒

⇒ = ⎜ ⎟⎝ ⎠

∑u g f g f

u g f g f g f u g f g fu g f g f

g gΘ Ξ g g

g g

{ } { }1 0

,0 0 ,

, ?u Θ u Ξ u Θ Ξ

f f f

a

extn n n a aδ δ

= + =

= +

… -2 -1 0 1 2 … … a-1 a

Domain of interest Eliminated degrees of freedomPeripheral boundary


Finite Size Domain: 1D Example

K-matrices

Fourier image of K-matrices

Lattice Green’s function

Boundary condition operators

Final form of the MSBC

{ } ( )/1/ 2

1 1

1/ 2

1 1ˆ ( ) | |4 2

1 cos

p

p

i v n NN

npv N

ep N n

vNN

π

π

−− −

= −

= − = = −−

∑g KF

1 0 1, 2 ,k k k− = = − =K K K

( ) ( )

( )

1

1 1

1 1

N N aN N a

N a N

a

a a

−−⎛ ⎞= − + − ⎜ ⎟−⎝ ⎠−⎛ ⎞= ⎜ ⎟

⎠⇒

⎝

Θ Ξ

Θ Ξ

ˆ ( ) ( 2) 2 (cos 1)ipn ip ipn

n

p e k e e k p− −= = + − = −∑K K

ua is a known coarse scale displacementa - coarse scale parameter (number of coarse scale cells)

1 0

1 1a

a

a a

−= +u u u

… -2 -1 0 1 2 … … a-1 a


p and vp are continuous and discrete wavenumbers

1,

2pv a

p NN

π += =


Elimination of Quasistatic Degrees of Freedom: Example

Explicit equations for atoms n=1,2,…a-1 are no longer required.We have accounted for these degrees of freedom implicitly.

1 0

2 1 0

2 1 0

2 1

1

1 0 0

...

2 0 ..2 0

.2 0

. 2..

2 0

0

a a

a

a

u u u

u u u

u u

u u u

u u u u

u

− −

−−

−

−

− −

⎧⎪ ⎧− + =⎪ ⎪⎪ ⇒ − + =⎨ − + =

− + Θ +Ξ⎨

⎪ ⎪⎩⎪

+ =⎪⎩

=

−

{ } { }1 0 a= +u Θ u Ξ u

… -2 -1 0 1 2 … … a-1 a


Using the boundary condition kernels:


The lattice dynamics methods allows introducing the multiscale boundary conditions (e.g. bridging scale analysis, non-reflecting MD boundaries, …).

This is a class of boundary conditions that take into account the effect of degrees of freedom implicitly, without solving for the entire lattice structure.

The dynamic case is often called the impedance boundary condition.

For a wide class of nanoengineering problems, multiscale boundary conditions (MSBC) are ultimately sought to describe the effect “coarse grain” degrees of freedom onto the smaller (“fine grain”) domain of interest.

Below, we discuss some important motivation for the MSBC:

- the bridging scale method and MD domain reduction- handshake-free multiple scale methods- spurious wave-reflection at the boundary of MD domains (the effect

of finite boundaries)- transformation of effective information into heat

Motivation to Multiscale Boundary Conditions


The idea of multiscale boundary conditions can be explained in comparison with traditional atomistic/continuum coupling approach:

Multiscale Boundary Conditions: General Concept

Standard approach Multiscale boundary conditions

The MSBC involve no handshake domain with “ghost” atoms. Positions of the interface atoms are computed based on the boundary condition operators Θ and Ξ. The issue of double counting of the potential energy within the handshake domain does not arise.


Due to reflective boundaries, the wave packages/signals gradually transforms into heat:

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


Impedance (Time Dependent) Boundary Conditions

1 2 1 0

0 1 0 1

...

2 0

2 0

mu u u u

km

u u u uk

− − −

−

⎧⎪⎪⎪ − + − =⎨⎪⎪ − + − =⎪⎩

&&

&&

1 0

0

( ) ( ) ( )t

u t t u dτ τ τ= Θ −∫

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 dynamicboundary conditions for MD simulation:

u1(t) and all other DoF n>1 are eliminated.Their effect is described by an external (impedance) force term:

1 2 1 0

0 1 0

...

2 0

12 ( )ext

mu u u u

km

u u u f tk k

− − −

−

⎧⎪⎪⎪ − + − =⎨⎪⎪ − + =⎪⎩

&&

&&

ext0

0

( ) ( ) ( )t

f t k t u dτ τ τ= Θ −∫

… -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


2-D Lattices

MD DomainReduced MD Domain + Multiscale BC

n,m+1

n,m

n,m-1

n+1,mn-1,m MultiscaleBC

MultiscaleBC

⇒

⇒

The general idea of multiscale boundary conditions for multi-dimensional 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

int,

', '', '

( ) ( ) ( )n m

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

n mn n m m

n m

t t t+ +

− −= − = −

− −

− =

∂=

=∂

∑ ∑Mu K u f

fK

u 0u

&&

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

1

' '' 1

int

''

( ) ( ) ( )n

n n n n nn n

nn n

n

t t t+

−= −

−

− =

∂=

=∂

∑Mu K u f

fK

u 0u

&&


2-D Formulation

1 1

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

( ) ( ) ( )n m


t t t+ +

− −= − = −

− =∑ ∑Mu K u f&&

Equation of motion

Time history kernel - depends on a spatial parameter m:

Response function

( )1 1 1

2, ', ' ', ' ,0 ,0

' 1 ' 1

ˆ ˆ( ) ( ) ( ) ( , , ) ( , )n m


t t t s p q s p qδ δ δ+ + −

− −= − = −

− = ⇒ = −∑ ∑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 ˆ( , ) ( , , )n p ns q s p q−→=G 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

τ τ τ

τ τ τ

+

−= −

+ +

− − −= − = −

= −

= − =

∑ ∫

∑ ∑∫

Displacement : u Θ u

Force : f Θ u Θ K Θ

Multiscale boundary conditions:

n,m+1

n,m

n,m-1

n+1,mn-1,m

n=0 n=1n=-1


Example: Face Centered Cubic Crystal

Numbering of equilibrium atomic positions (n,m,l) in two adjacent planes with l=0 and l=1. (Interplanar distance is exaggerated).

(0,1,1)

(1,0,1) (1,2,1)

(2,1,1)

(0,0,0) (0,2,0)

(1,1,0)

(2,0,0) (2,2,0)

z,l

y,mx,n

Bravais lattice


FCC Lattice: Derivation of K-matrices and Lattice Response Function

int, , int

', ', ' , ,', ', ' , ,

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

1,0, 1 0,1,1

0( )

,

1 1 0 1 1 0 1 0 11 1 0 , 1 1 0 , 0 0 0 ,0 0 0 0 0 0 1 0 1

1 0 1 0 0 00 0 0 , 0 1 11 0 1 0 1 1

|n m ln n m m l l n m l

n m l n m l

U

k k k

k k

− − −

−

−

=∂ ∂

= = −∂ ∂

−⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟ ⎜ ⎟= = − =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

−⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟= =⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠

⇒u

f uK f

u u

K K K

K K 0,1, 1

20,0,0

0 0 0, 0 1 1 ,

0 1 1

1 0 00 1 0 , 20 0 1

k

k k εβ

−

⎛ ⎞⎜ ⎟= −⎜ ⎟−⎝ ⎠

⎛ ⎞⎜ ⎟= − =⎜ ⎟⎝ ⎠

K

K

2 ( ) ( )( ) ( 2 )R r R reU r D e eβ β− −= −

( ), ,

, ,

(cos cos )cos 2 sin sin sin sinˆ ( , , ) 4 sin sin (cos cos )cos 2 sin sin

sin sin sin sin (cos cos )cos 2

i pn qm pln m l

n m l

q r p p q p r

p q r e k p q p r q q r

p r q r p q r

− + +

+ − − −⎛ ⎞⎜ ⎟= = − + − −⎜ ⎟⎜ ⎟− − + −⎝ ⎠

∑K K

{ }1 1ˆ( , ) ( , , )l r lp q p q r− −→= −G K% F

Morse potential

K-matrices

(nearest neighbors only)

Fourier transform

Inverse Fourier transform for r (evaluated numerically for all p,q and l):

z,l

y,mx,n


FCC Lattice: Derivation of Kernel Matrix Q

-10 -8 -6 -4 -2 0 2 4 6 8 10

10 -3

10 -2

10 -1

10 0

m = 0 m = 2 m = 4 m = 6 m = 8

Θn

,m(1

,1)

n

Q n,m , element (1,1)

( ) ( )1

0( ) ( )1 1

0

( , ) ( , )ˆ ˆ( , ) ( , ) ( , ) ( , )( , ) ( , )

G GΘ Ξ G G

G Gaa a

aa

p q p qp q p q p q p q

p q p q

−

−−

⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠

% %% %

% %

( ) ( ){ }( ) ( ) 1 1 ( ) ( ), ,

ˆ ˆ( , ) ( , )Θ Ξ Θ Ξa a a an m n m p n q m p q p q− −

→ →=

redundant block, if , ,n m a =u 0

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

', ' ', '

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

', '

u Θ u Ξ u

u Θ u

a an m n n m m n m n n m m n m a

n m n m

an m n n m m n m

n m

− − − −

− −

+

=⇒

=∑ ∑

∑

This sum can be truncated, because Θ decays quickly with the growth of n and m (see the plot).

Boundary condition operator in the transform domainis assembled from the parametric matrices G (a – coarse scale parameter):

Inverse Fourier transform for p and q

Final form of the boundary conditions


Application: Twisting of Carbon Nanotubes*

The study of twisting performance of carbon nanotubes is important for nanodevices.

The proposed multiscale treatment can predict u1 well at moderate deformation range.

At the same time, it saves the computation for each DOFs that is needed in the range between l = 2 and a (a = 20).

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

' '

u Θ u Ξ ua am m m m m m m a

m m− −= +∑ ∑

Fine grain

Load

(13,0) zigzag

* With Prof Dong Qian, University of Cincinnati.


Multiscale BC

MD Simulation


Nanoindentation of Graphite Sheet.

S.N. Medyanik, E.G. Karpov, Wing Kam Liu, Multiscale mechanics of carbon nanostructures, in preparation.

220 A

80 A

20 A

Fixed b.c.MSBC

Multi-Scale Boundary Conditions (MSBC) are applied at the six sides of the hexagonal reduced domain (blue, small hexagon). The rest of the full domain (red, larger hexagon) is cut-off and calculations are performed only for the reduced domain.

‘a’ is a multi-scale parameter, it defines the size of the cut-off domain and is equal to the number of unit cells between the boundary of the reduced domain and the boundary of the full domain, here a=30.

La = a ä (cell size of crystal lattice)

X

Y

X

ZDeformation of reduced domain, a=30

La =70 A

cut-off domain


Multi-Scale Simulations Results.

Full domain size is 15,000 atoms. Reduced domain – 2,400 atoms.

Step 0.

Step 20,depth=10 A

Step 40,depth=20 A

Step 60,depth=30 A

The plots show the actual scale deformation. Even for such large deformations the maximum error in displacement is less than 10%.


R

C - AuL-J Potential

FCC

12 6

( ) 4U rr r

σ σε⎡ ⎤⎛ ⎞ ⎛ ⎞= −⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦

Au - AuMorse Potential:

2 ( ) ( )( ) ( 2 )R r R reU r D e eβ β− −= −

Diamond Tip

Au

Application: Nanoindentation:

Problem description:


Method Validation

FCC gold

[1] E.G.Karpov, H.Yu, H.S.Park, W.K.Liu, J.Wang, Multiscale boundary conditions in crystalline solids, submitted to PRB 2004.

Kernel matrix Q depends on the interatomic potential only

{ } ( )1, , 0, , , , ', ' 0, ', '

', '

, am n m n a m n m m n n m n

m n

A − −= ≡ ∑u u u Θ u

⇓

0, ,m nu

, ,a m n =u 0

0, ,m nu

1, ,m nu

⇒ 1/4


Compound Interfaces

FCC gold

Fixed boundaries

Multiscale BC at five faces

1/4 of the original volume 1/18 of the original volume

Full domain: 1992 sec/stepMultiscale BC: 31 sec/step

Full domain: 1992 sec/stepMultiscale BC: 468 sec/step


Performance of Multiscale Boundary Conditions

1/18 of the original volume

Back half-domains are shown Radius of Diamond Tip: 1 nmFull MD domain size: 36864 atomsReduced domain size: 3600 atoms

For 1/18 of the original volume:Computation time has been reduced from 73 hours to 1 hour

Lattice deformation pattern is similar for the benchmark and the multiscale simulations:


Load / Indentation Depth Curve

The load/ indentation depth curves are close for the multiscale and benchmark simulations.


• Further Study Materials


Wave Number and Dispersion Law

Wave number p is defined through the inverse wave length λ(d – interatomic distance):

The waves are physical only in the Brillouin zone (range),

The dispersion law shows dependence of frequency on the wave number:

0Here, /k mω =-1 -0.5 0.5 1

0.5

1

1.5

2

2.5

3

/p π

0/ω ωcontinuum

λ = 10d, p = ≤π/5

λ = 4d, p = ≤π/2

λ = 2d, p = ≤π

λ = 10/11d, p = ≤11π/5 (NOT PHYSICAL)

2 /p dπ λ= ±

pπ π− ≤ ≤


Phase Velocity of Waves

The phase velocity, with which the waves propagate, is given by

Dependence on the wave number:

Value v0 is the phase velocity of the longest waves (at p Ø 0).

-1 -0.5 0.5 1

0.5

1

1.5

2

2.5

3

/p π

0/ω ωcontinuum

vp

ω=

-6 -4 -2 2 4 6

0.2

0.4

0.6

0.8

1

/p π

continuum

0/v v

0 0

12sin sin

2 2

p v p

v p

ωω

=⇒=

⇓


Numerical Laplace Transform Inversion: Weeks Algorithm

Analysis of regular lattice structures require numerical inversion of the Laplace transform for non-tabulated functions.

Weeks algorithm(Lγ – Laguerre polynomials,F is the Laplace image of f):

Example: compute the inverse LT of

( )2

0

,0

0

max

max

( ) /

2Re ( ) cot Re ( ) cos( )

2 ( 1) 2

(2 1) 1, cot , ,

2( 1) 2 2

t SctT

S

f t e b L t T

xb F s F s x

T S

x tix s c T c

S T S t

γ γγ

γ αγ α α α

α

αα α

δγ

π α

−

=

=

≈

− ⎛ ⎞= −⎜ ⎟+ ⎝ ⎠+

= = + = =+

∑

∑

( )21( ) 4

2F s s s= − + +

exact 1(2 )( )

J tf t

t=

Abs[ ]s

( )Abs[ ]F s

⇒maxt


Performance Study: Problem Statement

-10

-5

0

5

10-10

-5

0

5

10

0

0.25

0.5

0.75

1

-10

-5

0

5

10

Initial conditions:

2 20 0

, 2

,

( ) ( )(0) exp

2

1.25

(0) 0

n m

n m

n n m m

σσ

⎛ ⎞− + −= −⎜ ⎟

⎝ ⎠=

=

u

u&

K-matrices and mass matrix

n, m+1

n, m

n, m-1

n+1, mn-1, m

n+1, m+1n-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

κ

κ κ

κ

− −

− − − −

⎛ ⎞ ⎛ ⎞= = = =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠−⎛ ⎞ ⎛ ⎞

= = = =⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠−⎛ ⎞ ⎛ ⎞

= + =⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠

K K K K

K K K K

K M

Time history kernel

( )m tΘ


Performance Study: Size Effect

Reflection coefficient:

N

N

reflected bmT T

incident0

E E ER

E E

−=


Performance Study: Method Parameters

c

c

1, ' 0, '' 0

( ) ( ) ( )tm m

m m m mm m m

t t dτ τ τ+

−= −

= −∑ ∫u Θ u

Temporal andspatial truncation:

Time steps management( and ) :th hτ

0.05 /th M k=


Overview of the Bridging Scale Method

( ) ( ) ( )x x x′= +u u u

( )( ) ( )( ) ( ) ( ) ( )x xx x x x=

′ = − = −u Puu u Pu I P u

= +

( )xu ( )xu ( )x′u+=

• Does not involve separation of variable for the fine and coarse grains.

• Instead assumes a single solution u(x) for the entire domain. This solution is decomposed into the fine and coarse scale.


Within the bridging scale method, the MD and FE formulation exist simultaneously over the entire computational domain:

+ =

The total displacement is a combination of the FE and MD solutions:

Lagrangian formulation gives a set of coupled coarse and fine scale equations of motion:

= +u Nd Qq

T

T

( )

( )A

U= ∂=∂=

Md N f uf

uM q Q f u

&&

&&

Bridging-Scale Equations of Motion

MD, q FE, d MD + FE, (q, d)

The bridging scale method is discussed in more detail in forthcoming lectures.


MD Domain Reduction

MD degrees of freedom outside the localized domain are solved implicitly

+ =

Due to the atomistic character of simulations, the structural impedance has to be computed at the molecular level.

( )

T

0

( )

( ) ( ) ( ) ( )t

A t dτ τ τ τ

=

= + − −∫

Md N f u

M q f u Θ q u

&&

&&

The MD domain is too large to solve, so that we eliminate the MD degrees of freedom outside the localized domain of interest.

Collective atomic behavior of in the bulk material is represented by an impedance forceapplied at the formal MD/continuum interface:

FE + Reduced MD + Impedance BC

MD FE

⇒


1D Application: Non-Reflecting Bridging Scale Interface

The multiscale 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 (fine grain only).

Direct coupling with the 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.


Effect of Boundary Conditions in MD simulations

Dissipation of the high frequency waves outside the MD domain should be simulated correctly.

Spurious waves reflection from the domain interface lead to unphysical behavior in ion beam deposition, nanoindentation and failure.


Application: Bridging Scale Simulation of Crack Growth

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

lattice mechanics and multiple scale boundary...

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