Chemistry Reference
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with node
I
. The sum is over all of the coarse-scale nodes. The fine-scale
u
0
is
defined as that part of the solution whose projection onto the coarse scale is
zero; in other words, it is the part of the total solution that the coarse scale
cannot represent. Because the coarse and fine scales are orthogonal to each
other, it is possible to obtain a set of multiscale equations of motion for the
MD and FE systems that are coupled only through the interatomic forces
f
(the negative derivative of the interatomic potential energy
U
):
M
A
q
¼
f
½
40
Md
¼
N
T
f
½
41
where
q
is the MD displacement,
M
A
is a diagonal matrix with the atomic
masses on the diagonal,
d
is the coarse scale displacement, and
M
is the
coarse-scale mass matrix defined as
M
¼
N
T
M
A
N
½
42
Equations [40] and [41] were obtained considering that the coarse and
the fine scales coexist everywhere in the system. However, the goal is to expli-
citly simulate the fine scale only in a small region of the domain, while solving
the FE equation of motion
everywhere
in the system, and including the effects
of the fine scale that lies outside the MD region, at least in an average way.
This reduction of the fine-scale degrees of freedom can be achieved in the
bridging-scale technique by applying a generalized Langevin equation
(GLE)
185-188
boundary condition. A detailed derivation of the final equations
of motion is beyond the scope of this review and can be found in Refs. 73 and
182; here, it will suffice to say that the process of eliminating the unnecessary
fine-scale degrees of freedom results in a modified MD equation of motion that
includes an external force, named the impedance force,
f
imp
, which contains
the time history kernel
and acts to dissipate fine-scale energy from
the MD simulation into the surrounding continuum. The numerical result is
a highly desiderable nonreflective boundary between the MD and FE regions.
The final form for the coupled equations of motion for the two regions is
y
ð
t
t
Þ
M
A
q
¼
f
þ
f
imp
þ
R
½
43
Md
¼
N
T
f
½
44
Clearly, Eq. [43] is the modified MD equation of motion; the first term is the
standard nonlinear interatomic force, the second term is the impedance force
discussed above, and the third term is a stochastic force representing the
exchange of thermal energy between the MD region and the surrounding
eliminated fine-scale degrees of freedom. In other words, this last term acts
