2
:/
binding, TB) (FEM)
(MD) .
,
. MAAD
MD/TB ,
MD
FEM
TB
FEM/MD
Htot = HFE(ua, u˙ a) + HFE/MD(rj , r˙j, ua, u˙ a)+ HMD(rj , r˙j ) + HMD/TB(rj , r˙j)+ HTB(rj , r˙j ) (1)
QC (QC-FNL).
, QC-
FNL
,
,
Cauchy-Born .
,
QC
,
[40].
,
(
)
(
),
“ ”.
QC-
FNL
,
.
131
QC
.
, Phillips
QC
QC
QC
MC
Carlo, QCMC),
.
MD
MD ,
,
.
QC
2.2.3
Cauchy-Born ,
[49,50] 1999
. Shenoy
,
(quasicontinuum Monte
QC
,
, Phillips
QC
[51].
,
.
, MD
[55]
,
. , ,
,
,
.
,
(atomic-scale finite element method, AFEM).
AFEM ,
,
,
[52].
,
lor ,
Tay-
K ·u = P
, K = ∂Utot
,
∂x∂x x=x(0)
, x(0) F
, .
. (27)
E(uk, u˙ k) = 3(N − Nnode)kT +
1 2
(Mjku˙ j · u˙ k + uj · Kjk · uk)
j,k
(20)
,k
,T
,N
, Nnode , Mjk
, u˙ j uj
, Kjk
,
−1
Mjk = m
fjμfμk
μ −1
Kjk =
fj μ Dμ−κ1 fκk
μκ
(21) (22)
,
BSM
.
,
Lagrangian
,
MAq¨ = f (u)
(8)
M d¨ = N Tf (u)
(9)
(8)
(9)
N Tf (u)
.
,
,
,
,
.
,
,
,
,
,
4.
4 BSM
,
[20]
2
:/
BSM
,
(8)
–
.
u¨ l,m,n (t) =
MA−1Kl−l ,m−m ,n−n ×
l ,m ,n
ul ,m ,n (t) +MA−1fle,xmt,n (t)
) , ,
.
, (partitioned-
,
;
.
,
, ,
1
FEAt
[7]
1998 , Abraham Broughton [10,11]
FEAt
.
,
,
/
/
“” ,
, MAAD
(macroscopic, atomistic, ab intio dynamics).
”[12].
,
“
MAAD
(tight-
,
.
BSM
, , Piola-Kirchoff
, BSM
.,
[25].
,
(1)
θ(t − τ )
,
θ(t − τ ) ,
(corner effect), BSM θ(t − τ )
,
,
,
BSM ,
. (2) BSM
,
.
BSM
,
127
,
[26]
,
,
“
”
.
,
scle method, HMM), .
[27]
(heterogeneous multi-
J
λ,
HLBD = HBD + λTg = HBD + λTI gI
I
(17)
HLBD
.
α
λ,
,
,
,
,
[29].
/
. BSM ,
,
,
.
,
,
,
.
BSM ,
,
,
,
,
.
2011
41
2.1.5 CADD
MAAD, BSM, BDM
,
,
(
),
.
, ) .
[31∼33]
( , , Curtin, Shilkrot, Miller
Nloc
, nβ = 1.
, Ortiz
Cauchy-Born ,
, ai ,F
ε
ai = F · Ai
(24)
, Ai
.,
F
,
Nnl
M
Etot,h ≈
nβ Eβ uh +
neαΩ0ε (Fe)
β=1
e=1
, Ω0
.
(25)
2
QC
,
:/
, ,
,
“ ”(ghost force).
,
, Ortiz
Nrep
973
(2007CB814803)
† E-mail: why@
, .
(hierar-
124
2011
41
Clementi ,
[4,5]
. , .
.
, ,, . ,
. ,
,
,
,
/ .
. MD
, .
,
,
.
,
“
”
, .
, , .
. ,
.
2.1
, ,,
.
2.1.1 FEAt
MAAD
cal thermodynamics, CST)
/
(hybrid molecular/cluster stastical thermody-
namics, HMCST)
.
, ,
, .
, ,
. MD , (
,
MST
, ) .
, ,
,
,
,
.
,
,
2011
41
N
A = Φ0 + 3kT ln
|Di|1/6 kT
(coupled atomistic and discrete disloca-
tion plasticity, CADD).
(1)
; (2)
,
, (3)
,
.
CADD
,
,
7
.
Giessen Needleman
[34].
,
(7
I,
σ˜ ε
)
(7
II,
σˆ εˆ
)
.
,
,
,,
,
.
, MAAD
, CADD
,
VFE
=
3
2
(Natom 1
−
Nmesh)kB T +
(5)
VFE + 2 NmeshkBT
, kB , Nmesh
Abraham [14]
,T
, Natom
.
MAAD
, MAAD
(1)
,
,
,
,
. (2) 3
(
)
,
,
2 FEAt
MAAD
[13]
, FE/MD
,
,
,
,
“”
.
,
.,
,
, MAAD
“ ” .“ ”
. ,
,
.
,
, MD
,
MD
,
,
.
,
,
(MD) ,
. MD ,
[2]. ,
/
/
,
/
,
,
109
[3],
.,
chical) [4].
, (concurrent) ,
: 2010-04-29,
: 2011-02-17
∗
(109332011, 10772181, 10732090, 10772012, 10721202),
(bridging scale method, BSM).
—— ,
u(x, t) = u¯(x, t) + u (x, t)
(6)
, u (x, t) u (x, t)
.
,
u = N d, d
,N
. u (x)
, u (x, t)
u (x, t)
.
,
.
q,
u = Nd + q − Pq
(7)
,P
,
.
(10)
, MA
, l, m, n ; l ,m ,n
Kl−l ,m−m ,n−n
fle,xmt,n
.
Fourier ,
,
MD
; ; Laplace
FE
MAq¨ (t) = f (t) + f0im,mp,n(t) + R0,m,n(t)