Chemistry Reference
In-Depth Information
where
u
is the atomic displacement field for the entire sample and
N
is the total
number of atoms in the system. The first approximation of the QC method is
to replace Eq. [3] with
X
N
E
tot;
h
ð
u
h
¼
E
i
Þ
½
4
¼
1
i
where
u
h
are exactly calculated displacements for the repatoms (if coinciding
with the FEM nodes) and interpolated displacements for all other atoms. An
important point is that the displacement of nonrepresentative atoms inside an
element (atom P in Figure 3, e.g.) is completely determined by the displace-
ments of the nodes defining the element itself (atoms A, B, and C in the figure).
Thus,
X
u
h
ð
X
P
Þ¼
N
j
ð
X
P
Þ
u
j
j
¼
A
;
B
;
C
½
5
j
where
N
j
are the finite-element shape functions and
X
P
is the position of such a
non-representative atom. This makes computing the total energy of the system
(Eq. [4]) much faster than in the explicit case (Eq. [3]). However, because the
summation in Eq. [4] still includes all of the atoms in the system, further
approximations are needed to make the computations feasible for large
systems. Depending upon which approximation is chosen, different formula-
tions of the method (local, nonlocal, or mixed) are obtained.
Local QC
The local formulation of the QC method
65,87,88
is the most
straightforward and computationally efficient way to relate the atomic
positions to the continuum displacement field. This formulation is based on
two main approximations. First, the total energy of the system is obtained by
adding the energies of each element, instead of each atom. Second, the
element energy is computed using the Cauchy-Born rule,
100,101
i.e., assuming
uniform deformation inside the element. More specifically, the Cauchy-Born
rule states that the deformed structure of a crystal subject to a uniform
deformation gradient
F
can be found by simply applying
F
to the undeformed
crystal lattice basis
A
i
and then reconstructing the crystal from the altered
basis vectors
a
i
(i.e.,
a
i
¼
FA
i
). Therefore, one atom (repatom) is chosen
inside each element (near the quadrature point), such that its crystallite is
completely enclosed in the element itself [see Figure 4(a)], so that the same
deformation gradient can reasonably be assumed for the entire crystallite.
Then, its energy is found by calculating the energy of the corresponding
perfect, periodic crystal that is generated using the deformed basis vectors
a
i
¼
FA
i
, where
F
is the deformation gradient for the chosen repatom. Finally,
