Boundary conditions for molecular dynamics · Boundary conditions for molecular dynamics. Exact...

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Exact boundary conditions Variational boundary condition Examples Application to fracture simulations

Boundary conditions for molecular dynamics

Xiantao LiInstitute for Math and its Applications

joint work withWeinan E

Princeton University

Xiantao Li and Weinan E



Motivation

Holian and Ravelo PRB 1994




Outline

I Exact boundary conditions

I Phonon reflection

I Variational formulation

I Examples

I Application to fracture simulation




Problem setup

x

xJ

0

mui = −∑

j Dijuj mui = −∂V∂xi

.




MD boundary condition: a 1D example

Exact boundary condition can beobtained by solving the half spaceproblem:

uj = uj+1 − 2uj + uj−1, j ≤ 0uj(0) = 0, vj(0) = 0, j ≤ 0ub(t) = u1(t).

Boundary condition

j=1

Computational domain

j=0




1D example: exact boundary condition

Exact boundary condition (Adelman et. al. 1974, 1976) :

u0(t) =

∫ t

0β(t − s)u1(s)ds, j ≤ 0

β(t) =J2(2t)

t.




Faster decay by using more atoms

u0 =J∑

j=1

∫ t

0αj(t − s)uj(s)ds

α(t) ∼ C

tJ, t → +∞.

J=2

alpha0alpha1beta0

J=3




General latticeLinearized equation of motion,

Mui ,j ,k =∑l ,m,n

Di−l ,j−m,k−nul ,m,n,

Fourier transform in the j and k direction,

MUi (η, ζ) =∑

l

Di−l(η, ζ)Ul(η, ζ).

U = Fj→η,k→ζ [u],D = Fj→η,k→ζ [D].

Exact boundary condition:

u0,j ,k(t) =∑m

∑n

∫ t

0θl ,j−m,k−n(t − τ)ul ,m,n(τ)dτ.




Existing work

I W. Cai, M. de Koning, V. V. Bulatov and S. Yip 2000,

I G.J. Wagner, G.K. Eduard and W.K. Liu 2004,

I H.S. Park, E.G.Karpov, W.K. Liu and P.A. Klein 2004.




Practical issue

Exact boundary conditions:

I nonlocal in both space and time

I premature truncation leads to large reflection

I not feasible in a multiscale method

Objectives:

I local boundary conditions

I given stencil, find the BC with minimal phonon reflection

I take into account of external loading

An analogy: boundary condition for the wave equation: ABC(Engquist and Majda 1979), and many other methods.




Practical issue





Objectives:








Practical issue





Objectives:








Practical issue





Objectives:








Practical issue





Objectives:








Phonon spectrum

Harmonic approximation:

mi ui = −∑

j

Di−juj .

Dynamic matrix:

MD(k) =∑

j

Dje−irj ·k,

Phonon spectrum

D(k)εs(k) = λsεs(k).




Brillouin zone: triangular lattice

the first Brillouin zone

t1

t2

Phonon spectrum

−4

−2

0

2

4

−4

−2

0

2

40

5

10

15

20

25




Brillouin zone

BZ for BCC lattice

Symmetry of the spectrum:

ω(k) = ω(Pk), ε(Pk) = Pε(k).

Integration over the BZ: K pointmethod (Monkhorst and Pack 1976).

ui = (2i − nq − 1)/(2nq)kijl = uibi + ujbj + ulbl .

The symmetry of the grid pointssubstantially reduces thecomputation.




Brillouin zone

BZ for BCC lattice


ω(k) = ω(Pk), ε(Pk) = Pε(k).







Brillouin zone

BZ for BCC lattice


ω(k) = ω(Pk), ε(Pk) = Pε(k).







Phonon reflection

Taborek and Goodstein 1979 Incident and reflected waves

1

2

3

ω =ωω =ωω =ω

n




Phonon reflection:

kI − (kI · n) n = kR − (kR · n) nωs(kI ) = ωs′(k

R).

Let λ = e i(kR ·n)an

det(D(kR)− ω(kI )I

)= 0,

leads to a polynomial for λ. The roots of the polynomial come inpairs (λ, 1/λ∗). The degree of the polynomial: Nd × Ne × Na.The wavenumber:

kRs,s′ · n = kr + iki .




Phonon reflection: reflection coefficientsBoundary condition:

u0(t) =∑

j

∫ t0

0αj(τ)uj(t − τ)dτ.

Incident and reflected waves:

uj(t) = c Is e

i(rj ·k−ωs t)εs(k)

+ cRss′e

i(rj ·kRss′−ωs′ t)εs′(k

Rss′)

cR = RcI ,

Linear system:

(I −A(k))εs(k) +∑s′

Rss′(I −A(kRss′))εs′(k).





u0(t) =∑

j

∫ t0



uj(t) = c Is e


+ cRss′e


Rss′)

cR = RcI ,

Linear system:

(I −A(k))εs(k) +∑s′






u0(t) =∑

j

∫ t0



uj(t) = c Is e


+ cRss′e


Rss′)

cR = RcI ,

Linear system:

(I −A(k))εs(k) +∑s′






u0(t) =∑

j

∫ t0



uj(t) = c Is e


+ cRss′e


Rss′)

cR = RcI ,

Linear system:

(I −A(k))εs(k) +∑s′





Thermal fluxEnergy flux at the atomic scale:

J =1

2

∑(ui + uj)Di−j(ui − uj)rij .

Convert to Fourier space:

J = J I + JR .

JR =

∫|cR

ss′ |2ωs′∇λs′(k)dk,

JRn = 2

∑s

∫k∈BZ , k·n≤0

|∑s′

c Is′Rss′ |2ω2

s (∇ωs · n)dk.

This is the thermal flux due to the applied boundary condition.




Variational boundary conditions

Discrete boundary condition:

un+10 =

J∑j=1

M∑m=1

αmj un−m

j ∆t,

Variational formulation:

min{αm

j }

∑s

∫k∈BZ , k·n≤0

∑s′

|Rss′ |2 |(∇ωs · n)|dk

subject to certain constraints.




Symmetry properties

αj ′ = PαjPT ,

Reflection matrices:

R(k; {αj ′}

)= R

(PTk; {αj}

).

αj

αj ′




Example: 1D chain

Reflection (E and Huang 2001, 2002)

R(k) =1−

∑j e ijk

∫ t00 αj(τ)e iωτdτ

1−∑

j e−ijk∫ t00 αj(τ)e iωτdτ

.

exact kernel

0 2 4 6 8 10 12 14 16 18 20−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

time

exact kernelapproximate




Example: 1D chain

Premature truncation vs variational BC

−3 −2.5 −2 −1.5 −1 −0.5 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

k

|R|2




Example: 2D triangular lattice

the triangular lattice

t1

t2

the first Brillouin zone





neighbor atoms

MD boundary

nearest neighbors2nd nearest neighbors3rd nearest neighbors

phonon reflection for J = 26,M = 2

−4

−3

−2

−1

0

−4

−2

0

2

40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

k1

k2

|R|2





size of the stencil

1 2 5 8 10 12 16 20 25 300

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

M

J R

J=8J=17J=26

phonon reflection along θ = 2π/3

−2 −1.8 −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 00

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

k1

|R|2

J=26, M=2J=8, M=30




Application to fracture simulations

Fixed BC




Application to fracture simulations

Local BC




More practical issues: external loading

stress free?

The reflection coefficient |R(k)| ≡ 1!




More practical issues: external loading

External loading

Jxx0

na

Applied deformation:

W = Fn− n =∂u

∂n,

Boundary condition:

w0 =∑

j

∫ t0

0αj(s)wj(t − s)ds.

In Fourier space:

w(k) = u(k)(1− e iak·n).




Summary

1. analysis of phonon reflection

2. variational formulation to minimize phonon reflection

3. local boundary condition

4. implementation issues

5. application to fracture simulations



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