Chemistry Reference
In-Depth Information
overlap). Also,
m
i
and
v
i
indicate both the atomic and the nodal masses and velo-
cities, respectively, while
V
ð2Þ
ij
and
V
ð3Þ
ijk
are the two- and three-body terms of the
SW potential. Lastly,
u
l
and
k
l
are the FE displacement vector and stiffness
matrix, respectively, for element
l
. The weights
w
l
,
w
ij
, and
w
ijk
are determined
by matching the elastic constants of the hand-shake region to those of the
''bulk'' MD and FE systems and are given by (see Figure 7 for the definition of
B
1
and B
2
)
8
<
1if
i
or
j
is to the right of B
1
3
4
if
i
and
j
are between B
1
and the FE
=
MD interface
1
2
w
i
¼
if
i
or
j
crosses the FE
=
MD interface
½
28
:
1
4
if
i
and
j
are between B
2
and the FE
=
MD interface
0if
i
or
j
is to the left of B
2
8
<
1if
l
is to the left of B
2
1
2
w
i
¼
MD interface
0if
l
is to the right of the FE
if
l
crosses the FE
=
½
29
:
=
MD interface
The three-body weights,
w
ijk
, are defined similarly to the two-body ones,
w
ij
.
As discussed previously, an important issue that all dynamic multiscale
methodologies must deal with is the phonon spectrum mismatch between the
atomistic and coarse-grained regions. In the CLS method, this effect is handled
by weakly coupling the FE degrees of freedom to a Brownian motion heat bath
that is set to the desired temperature. As a last remark, we need to mention
that this hybrid scheme is computationally very efficient and reasonably simple
to implement, so that several groups other than the original authors have
adopted it.
Applications
The developers of the CLS method applied it to the investigation
of fracture and crack propagation in silicon,
164-167
while Lidorikis et al. used it
to model the strain relaxation and induced-stress distribution in Si/Si
3
N
4
nanopixels.
168,169
Overlapping Domain Decomposition and Edge-to-Edge Decomposition
Methods
In 2003, Belytschko and Xiao introduced the overlapping domain
decomposition (ODD) method and the edge-to-edge decomposition (EED)
method
170
to effectively couple atomistic simulations to continuum ones,
particularly targeting the modeling of large deformations in nanosystems. The
ODD method was later extended to include dynamical effects as well.
171
In the ODD method, the atomistic and continuum regions are coupled
through a standard hand-shake region, and a single energy functional is
considered for the complete domain. As in most methods, the continuum is
