Chemistry Reference
In-Depth Information
Dynamical hybrid methods face even greater challenges than equilibrium
ones. In addition to dealing with the same difficulties in achieving a smooth
matching in the hand-shake region, as discussed earlier, dynamical methods
must also take into consideration the pathological wave reflection that occurs
at the boundary between the discrete and the continuum regions. Such a reflec-
tion occurs because the distribution of wavelengths emitted by the MD region
includes waves significantly shorter than those that can be captured by the
continuum FE region. Because an energy-conserving formulation is usually
used, the wave must go somewhere and, therefore, gets reflected back into
the MD region, leading to fictitious heat generation in the atomistic region.
The ability to minimize (or, even better, eliminate) such a reflection is there-
fore a key point in developing a successful dynamical hybrid methodology.
Lastly, to be effective, dynamical simulations should cover as long a time
as possible. Therefore, the size of the time step is a significant issue in the effec-
tiveness of the methodology. In most hybrid implementations, the size of the
finite-element mesh is graded down to the atomic lattice size at the boundary
between continuum and MD domains to reduce wave reflection (see discus-
sions on domain decomposition and quasi-continuum coarse-grain alternative
below). Because the time step in an FE simulation is governed by the smallest
element in the mesh, this procedure requires the use of the same size time step
in both the discrete and continuum domains, even though the large-scale phy-
sical quantities evolve much more slowly than the short-scale ones. However,
a few methods, such as the bridging scale technique of Wagner and Liu dis-
cussed in the section on adaptive model refinement techniques, have particu-
larly addressed this issue and allow for the possibility of using multiple time
scales in hybrid simulations.
Domain Decomposition Methods
Domain decomposition methods are hybrid methodologies where the var-
ious computational methods are spatially separated, i.e., a higher level of theory
(e.g., atomistic models) is used in the area of interest (near a vacancy, defect,
crack tip, and so on), and a lower level of theory (e.g., continuum models) is
used everywhere else. This is the most intuitive approach to the coupling pro-
blem, and often the easiest to implement computationally. Among these meth-
odologies are some of the most widely used coupling schemes in the solid state
(like the CLS method). However, the existence of spatial boundaries between
computational methods may lead to hand-shake problems larger than those cre-
ated by other approaches. In the following, a few examples of such an approach
will be given.
Coupling of Length Scales Method: Atomistic/Continuum Part
The FE/MD/
TB coupling of length scales (CLS) method,
66,67,160-164
also known as MAAD
(macroscopic, atomistic, ab initio dynamics), is one of the few methodologies
to provide a dynamical coupling of three different regimes: the macroscopic
