Chemistry Reference
In-Depth Information
where
p
A
is the atomic momentum,
m
I
is the atomic mass, W
A
is the atomistic
external potential,
p
C
and
M
I
are the nodal momentum and mass for the con-
tinuum model once it is discretized by a finite-element method and
W
C
is the
total potential of the continuum model. The index
I
varies over all of the atoms,
while the index
N
spans all of the nodes covering the continuum region. Explicit
expressions for
W
A
and
W
C
are given in Ref. 171; for the scope of this review it
is sufficient to specify that
W
A
should only be due to a constant external force,
such as electrostatic forces, and a pairwise interatomic potential, while the
constitutive equation for the finite-element method is constructed via the
Cauchy-Born rule as in the quasi-continuum approach, or via the exponential
Cauchy-Born rule
158
if monolayer crystalline membranes such as nanotubes are
simulated. Lastly, the coupling of the two regimes (atomistic and continuum) is
completed by requiring that the atomic displacements should conform to the
continuum displacements at the discrete positions of the atoms.
The constraints are first applied to all components of the displacements
by the Lagrange multiplier method; then the modifications needed for the aug-
mented Lagrangian method are added. Lastly, the equations of motion for the
Lagrange multiplier method are obtained. A detailed derivation of such equa-
tions of motion can be found in Ref. 171, together with the explicit recipe for
the Verlet algorithm used to integrate such equations.
One of the issues in dynamical multiscale coupling is the tailoring of the
time step to the different subdomains. If the same time step is used in both the
atomistic and the continuum regions, computations will be wasted in the con-
tinuum model. However, if in the hand-shake region the size of the FEM
elements is reduced to coincide with the individual atoms, it is difficult to
tailor the time step. Therefore, the authors of the ODD method chose to use
a uniform mesh for the continuum domain, so that a much larger time step
could be used in the continuum model than in the atomistic one. A description
of such a multiple-time-step algorithm is provided in the paper.
169
With regard to the problem of phonon reflections from the interface, the
authors provide test results that show that the ODD method dramatically
reduces spurious wave reflections at the atomistic/continuum interface without
any additional filtering procedure. If the overlapping subdomain is large
enough, they find that the high-frequency wave reflection is almost completely
eliminated.
Lastly, we will briefly describe the EED method introduced in Ref. 170.
While this method is found to be almost as effective as the ODD for static
applications, it is not suited for dynamical ones because it causes significant
reflection of the high-frequency part of the wave at the atomic/continuum
interface. In the EED method, no hand-shake region is defined and the cou-
pling of the two regions occurs through an interface (Figure 9). Three types
of ''particles'' are defined: nodes in the continuum region, atoms in the atomis-
tic domain, and, virtual atoms that are introduced in the continuum domain to
model the bond angle bending for bonds between the continuum and the
