Chemistry Reference
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as a heat bath on the MD region. For simplicity, in the earlier applications
of the bridging-scale method this term was set to zero, indicating a zero-
temperature simulation. Recently, though, a finite-temperature form for this
term was derived by Karpov et al.
189
It is worth men
ti
oning that the impe-
dance force also depends on the coarse-scale solution,
u
. This indicates truly
two-way information passing between the two regions; not only does the MD
region affect the FE evolution, but also coarse-scale information originating in
the continuum can be passed onto the MD region. A substantial difference
between this method and many others is that here the FE equation of motion
is solved
everywhere
in the system, i.e., also where MD is applied. This elim-
inates the need to mesh the size of the elements down to the atomic scale, and
therefore allows the use of a staggered time integration scheme.
182
Lastly, it
must be noted that no ad hoc treatment is used to couple the MD and FE
regions, instead the MD lattice behaves as if part of a larger lattice due to
the GLE (the GLE is a mathematically exact representation of the MD degrees
of freedom that are not explicitely solved for), even though ghost atoms are
introduced in the MD simulation to ensure that the reduced atomistic system
does not relax due to spurious free-surface effects. These atoms' displacements
are determined by interpolating the FE displacements of the elements in which
they lie, not by integrating the MD equation of motion. Recent extentions of
the bridging-scale method include the coupling of the continuum region to a
quantum mechanical one (tight binding), instead of a classical atomistic one,
for quasi-static applications.
190
Applications
The bridging method has been successfully used to model both
quasi-static and dynamical phenomena. In its quasi-static formulation it has
been used to model nanoindentation (on a crystalline gold substrate)
191
and
carbon nanostructures: single graphene sheets and multilayered graphite,
192
and buckling of a multiwalled carbon nanotube.
184,193
Dynamically, the
bridging method has been applied to the investigation of crack propagation
183
in fcc materials (3D), and dynamic shear banding
194
(1D and 2D). More
recently the model has been further expanded to describe solids with moving
dislocations in one dimension.
195
Other Adaptive Model Refinement Methodologies
The use of the
adaptive
refinement
idea is absolutely not limited to the two methods presented in
the previous sections. Several other groups have developed methodologies
based upon this principle. Among many, we would like to mention the
coupling scheme proposed by Klein and Zimmerman
196
and the one
suggested by To and co-workers,
197,198
where the bridging-scale method is
combined with the perfectly matched layer method,
199
to eliminate the
spurious wave reflection at the computational boundaries during the fine-
scale part of the simulation. This methodology is then widened to allow the
simulation of nonequilibrium multiscale phenomena.
200
Lastly, a whole
