Boundary conditions for molecular dynamics · Boundary conditions for molecular dynamics. Exact...
Transcript of Boundary conditions for molecular dynamics · Boundary conditions for molecular dynamics. Exact...
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
Boundary conditions for molecular dynamics
Exact boundary conditions Variational boundary condition Examples Application to fracture simulations
Motivation
Holian and Ravelo PRB 1994
Xiantao Li and Weinan E
Boundary conditions for molecular dynamics
Exact boundary conditions Variational boundary condition Examples Application to fracture simulations
Outline
I Exact boundary conditions
I Phonon reflection
I Variational formulation
I Examples
I Application to fracture simulation
Xiantao Li and Weinan E
Boundary conditions for molecular dynamics
Exact boundary conditions Variational boundary condition Examples Application to fracture simulations
Problem setup
x
xJ
0
mui = −∑
j Dijuj mui = −∂V∂xi
.
Xiantao Li and Weinan E
Boundary conditions for molecular dynamics
Exact boundary conditions Variational boundary condition Examples Application to fracture simulations
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
Xiantao Li and Weinan E
Boundary conditions for molecular dynamics
Exact boundary conditions Variational boundary condition Examples Application to fracture simulations
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.
Xiantao Li and Weinan E
Boundary conditions for molecular dynamics
Exact boundary conditions Variational boundary condition Examples Application to fracture simulations
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
Xiantao Li and Weinan E
Boundary conditions for molecular dynamics
Exact boundary conditions Variational boundary condition Examples Application to fracture simulations
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.
Xiantao Li and Weinan E
Boundary conditions for molecular dynamics
Exact boundary conditions Variational boundary condition Examples Application to fracture simulations
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.
Xiantao Li and Weinan E
Boundary conditions for molecular dynamics
Exact boundary conditions Variational boundary condition Examples Application to fracture simulations
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.
Xiantao Li and Weinan E
Boundary conditions for molecular dynamics
Exact boundary conditions Variational boundary condition Examples Application to fracture simulations
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.
Xiantao Li and Weinan E
Boundary conditions for molecular dynamics
Exact boundary conditions Variational boundary condition Examples Application to fracture simulations
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.
Xiantao Li and Weinan E
Boundary conditions for molecular dynamics
Exact boundary conditions Variational boundary condition Examples Application to fracture simulations
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.
Xiantao Li and Weinan E
Boundary conditions for molecular dynamics
Exact boundary conditions Variational boundary condition Examples Application to fracture simulations
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.
Xiantao Li and Weinan E
Boundary conditions for molecular dynamics
Exact boundary conditions Variational boundary condition Examples Application to fracture simulations
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).
Xiantao Li and Weinan E
Boundary conditions for molecular dynamics
Exact boundary conditions Variational boundary condition Examples Application to fracture simulations
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
Xiantao Li and Weinan E
Boundary conditions for molecular dynamics
Exact boundary conditions Variational boundary condition Examples Application to fracture simulations
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.
Xiantao Li and Weinan E
Boundary conditions for molecular dynamics
Exact boundary conditions Variational boundary condition Examples Application to fracture simulations
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.
Xiantao Li and Weinan E
Boundary conditions for molecular dynamics
Exact boundary conditions Variational boundary condition Examples Application to fracture simulations
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.
Xiantao Li and Weinan E
Boundary conditions for molecular dynamics
Exact boundary conditions Variational boundary condition Examples Application to fracture simulations
Phonon reflection
Taborek and Goodstein 1979 Incident and reflected waves
1
2
3
ω =ωω =ωω =ω
n
Xiantao Li and Weinan E
Boundary conditions for molecular dynamics
Exact boundary conditions Variational boundary condition Examples Application to fracture simulations
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 .
Xiantao Li and Weinan E
Boundary conditions for molecular dynamics
Exact boundary conditions Variational boundary condition Examples Application to fracture simulations
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).
Xiantao Li and Weinan E
Boundary conditions for molecular dynamics
Exact boundary conditions Variational boundary condition Examples Application to fracture simulations
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).
Xiantao Li and Weinan E
Boundary conditions for molecular dynamics
Exact boundary conditions Variational boundary condition Examples Application to fracture simulations
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).
Xiantao Li and Weinan E
Boundary conditions for molecular dynamics
Exact boundary conditions Variational boundary condition Examples Application to fracture simulations
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).
Xiantao Li and Weinan E
Boundary conditions for molecular dynamics
Exact boundary conditions Variational boundary condition Examples Application to fracture simulations
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.
Xiantao Li and Weinan E
Boundary conditions for molecular dynamics
Exact boundary conditions Variational boundary condition Examples Application to fracture simulations
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.
Xiantao Li and Weinan E
Boundary conditions for molecular dynamics
Exact boundary conditions Variational boundary condition Examples Application to fracture simulations
Symmetry properties
αj ′ = PαjPT ,
Reflection matrices:
R(k; {αj ′}
)= R
(PTk; {αj}
).
αj
αj ′
Xiantao Li and Weinan E
Boundary conditions for molecular dynamics
Exact boundary conditions Variational boundary condition Examples Application to fracture simulations
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
Xiantao Li and Weinan E
Boundary conditions for molecular dynamics
Exact boundary conditions Variational boundary condition Examples Application to fracture simulations
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
Xiantao Li and Weinan E
Boundary conditions for molecular dynamics
Exact boundary conditions Variational boundary condition Examples Application to fracture simulations
Example: 2D triangular lattice
the triangular lattice
t1
t2
the first Brillouin zone
Xiantao Li and Weinan E
Boundary conditions for molecular dynamics
Exact boundary conditions Variational boundary condition Examples Application to fracture simulations
Example: 2D triangular lattice
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
Xiantao Li and Weinan E
Boundary conditions for molecular dynamics
Exact boundary conditions Variational boundary condition Examples Application to fracture simulations
Example: 2D triangular lattice
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
Xiantao Li and Weinan E
Boundary conditions for molecular dynamics
Exact boundary conditions Variational boundary condition Examples Application to fracture simulations
Application to fracture simulations
Fixed BC
Xiantao Li and Weinan E
Boundary conditions for molecular dynamics
Exact boundary conditions Variational boundary condition Examples Application to fracture simulations
Application to fracture simulations
Local BC
Xiantao Li and Weinan E
Boundary conditions for molecular dynamics
Exact boundary conditions Variational boundary condition Examples Application to fracture simulations
More practical issues: external loading
stress free?
The reflection coefficient |R(k)| ≡ 1!
Xiantao Li and Weinan E
Boundary conditions for molecular dynamics
Exact boundary conditions Variational boundary condition Examples Application to fracture simulations
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).
Xiantao Li and Weinan E
Boundary conditions for molecular dynamics
Exact boundary conditions Variational boundary condition Examples Application to fracture simulations
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
Xiantao Li and Weinan E
Boundary conditions for molecular dynamics