Chemistry Reference
In-Depth Information
To do so, a Taylor expansion of the energy is considered, up to the quadratic
term:
x
¼
x
0
ð
x
x
0
Þ
E
tot
ð
x
0
Þþ
q
E
tot
q
x
E
tot
ð
x
Þ
x
¼
x
0
ð
x
x
0
Þ
2
E
tot
q
x
q
x
1
2
ð
x
x
0
Þ
q
T
þ
½
14
where
x
0
¼ð
x
ð0Þ
1
T
is an initial guess of the equilibrium state.
Substituting Eq. [14] into Eq. [13], and defining
u
¼
x
x
0
as the atomic
displacement, produces an equation typical of the continuum FEM:
;
x
ð0Þ
2
; ...;
x
ð0Þ
N
Þ
Ku
¼
P
½
15
2
E
tot
=
q
x
q
x
j
x
¼
x
0
where, following the FEM nomenclature,
K
¼
q
is the stiffness
matrix, and
P
¼
q
E
tot
=
q
x
j
x
¼
x
0
is the nonequilibrium force vector. The main
difference between the AFEM and the standard FEM is in the definition and
properties of the elements. In AFEM an element
i
is defined as the ensemble of
atoms that contribute to the computation of the atomic energy
E
i
[similar to
the definition of crystallite in the fully nonlocal quasi-continuum method (dis-
cussed above)]. Clearly, the size and shape of the elements depend on the
atomic structure and chosen interatomic potential. However, in AFEM the
''elements'' overlap in space, to account for multibody atomistic interactions,
while in the FNL-QC method the crystallites were suitably truncated to avoid
overlapping. This means that, in AFEM, the energy is not partitioned into
elements, as it is in standard FEM, and that all of the atoms inside an element
contribute to the energy calculations, not only the nodes. Equation [15] is then
solved iteratively until
P
reaches zero. When both atomistic and continuum
region are considered (i.e., the AFEM is used in an hybrid methodology),
the total energy of the system is minimized simultaneously with respect to
both atomic positions (in AFEM) and FEM nodes (in continuum FEM).
Applications
The AFEM methodology has been applied to the investigation
of properties of single carbon nanotubes (e.g., deformation under
compression), and of woven nanostructures of carbon nanotubes.
134
The
same methodology has also been applied to simulate postbuckling
135
and
critical strain
136
of carbon nanotubes.
Green's Function Boundary Condition Methods
In the following, we explore the possibility of imposing boundary condi-
tions as a way to relieve incompatibility stresses at the interfaces between
computational domains, while, at the same time, allowing the use of a minimal
atomistic region. More specifically, boundary condition (BC) methods can be
