Chemistry Reference
In-Depth Information
To connect the two regions while making sure that the real atoms near
and at the interface are correctly coordinated, pad atoms are added to the sys-
tem in the continuum region [see Figure 5(b)]. This treatment of the hand-
shake region is inspired by the one used in the FEAt method. The pad thickness
is crucial for obtaining correct results, and not only should it be larger than the
range of the atomistic interactions
r
cut
, but the real atoms must continue to be
correctly coordinated even after dislocations have crossed the boundary
between the atomistic and continuum zones, a phenomenon that generates
slip steps at the interface. The positions,
r
P
, of the pad atoms are obtained
by interpolating the FE nodal displacements to their reference positions, i.e.,
they are considered a ''continuum region'' quantity and are instantaneously
perceived as fixed by the real atoms. However, the expression for the total
energy of the atomistic domain includes interactions with the pad atoms:
X
E
a
¼
E
i
ð
r
A
;
r
I
;
r
P
Þ
f
A
u
A
½
11
2ð
;
;
Þ
i
A
I
P
where
u
A
are the atomic displacements,
f
A
are the lumped forces, and the sum
includes the pad atoms even though they are not degrees of freedom. The full
atomistic forces on the atoms are then obtained by taking the partial deriva-
tives of
E
a
with respect to positions
r
A
and
r
I
. It is important to notice that
only the partial derivatives of
E
a
with respect to the pad atom positions would
contain unphysical forces, but they are not needed because the pad atom
displacements are completely determined by those of the FE nodes. The total
energy functional for the continuum domain is given by a sum over the
energies,
E
m
, of the finite elements,
m
, plus boundary work terms:
ð
X
;
d
i
E
c
¼
E
m
ð
U
I
;
U
C
Þ
½
12
T
0
u dA
q
T
m
It is important to mention that the displacement of the interfacial nodes,
U
I
,
are not degrees of freedom in the continuum calculation: Because the interfa-
cial nodes coincide with interface atoms,
r
I
, they are moved as atoms and
appear as fixed nodes with prescribed displacements to the other elements of
the continuum region. Lastly, it must be remarked that this formulation of the
continuum/atomistic coupling does not allow for the use of a unique energy
functional because
E
a
includes the elastic energy of the pad atoms, whose
energy is already implicitly contained in the continuum energy,
E
c
.
Before concluding the analysis of the hybrid coupling utilized in the
CADD method, it is important to briefly discuss the issue of ghost forces.
As in all of the other coupled methodologies, in CADD the main cause for
such spurious forces is that the atomistic region is inherently nonlocal since
the range of realistic interatomic potentials is larger than the nearest-neighbor
