Chemistry Reference
In-Depth Information
atomistic potentials or a continuum finite-element method. However, neither
FEAt nor CLS allow for the existence of continuum dislocations in the finite-
element region, nor can they easily modify the size of the atomistic region to
model a propagating defect. The quasi-continuum method (see above) is able
to follow a moving defect because of an adaptive remeshing algorithm, but it
does not support a continuum description of dislocations. This means that in
the QC model, each dislocation has to be completely embedded in an atomistic
region, therefore substantially limiting the number of dislocations that can be
treated because of their computational cost. In the CADD methodology, not
only the presence and movement of continuum discrete dislocations in the
continuum regime is possible, but also their interactions with each other
and with atoms in the atomistic region.
In its current implementation, the CADD method can only deal with 2D
problems, i.e., problems where the dislocations all have a line direction per-
pendicular to the modeled plane. However, the method does not contain
any limitation on the dislocation character (edge, screw, or mixed) because
periodic boundary conditions along the
z
direction are used in the atomistic
region and three degrees of freedom (displacements u
x
,u
y
, and u
z
) are consid-
ered in the two-dimensional (
x
-
y
) continuum region. The extension of the
methodology to the 3D case is not trivial and is currently being explored.
It must also be noted that this is an iterative method (no unique energy
functional exists) that covers both the atomistic and the continuum regions.
This is because of the methodology adopted to minimize the ghost forces,
which will discussed in more detail below. A modified conjugate-gradient
algorithm is used to search for the point of zero forces on all degrees of free-
dom in order to drive the system to equilibrium. While doing so, atomic coor-
dinates and dislocation positions are updated simultaneously.
There are four main components to the CADD approach: the atomistic
model, the discrete dislocation framework, the coupling between these
regions, and the method for detecting and passing dislocations through the
atomistic/continuum boundary. The atomistic model most commonly used
in CADD simulations is the EAM.
9,10
However, more complex atomistic
approaches could be utilized as well. The adopted discrete dislocation frame-
work is that of van der Giessen and Needleman.
130
It is beyond the scope of
this review to illustrate such a methodology and how it is incorporated into the
CADD model, so we refer the interested reader to the original studies
71,125-128
and references within. That also applies to the description of the algorithm
used to detect and move dislocations through the atomistic/continuum
boundary. In the following we will describe the coupling mechanism between
zones.
The general boundary value problem that the CADD method wishes to
solve is the following: A body, containing
N
continuum dislocations, is divided
into an atomistic region
C
. Such a system is
subject to a known traction
T
¼
T
0
and initial displacements
u
¼
u
0
. The
A
and a continuum region
