Chemistry Reference
In-Depth Information
domain with a macrogrid on which a macroscale model is applied. Such a
macroscale model is then locally refined in the area, or areas, of interest.
This kind of approach avoids having to deal with spatial transitions in
the computational
technique and,
therefore, minimizes
the hand-shake
problems.
Heterogeneous Multiscale Method
The heterogeneous multiscale method
(HMM)
175,176
provides a general framework for dealing with multiscale
phenomena and can be easily applied to the coupling of continuum and
atomistic (molecular dynamics) simulations at finite temperature.
The basic goal of this methodology is to enable simulations of macro-
scopic processes in cases where explicit macroscale models are invalid in at
least part of the macroscopic system. In those regions, microscale models
are used to supply the missing data. In the HMM, the computational saving
comes from reducing both the spatial and the temporal domains. The spatial
reduction is, as in all hybrid methodologies, due to the possibility of applying
the higher level of theory only to a limited part of the whole system, while the
reduction in temporal domain originates from the fact that this method natu-
rally decouples the atomistic time scale from the continuum one, therefore
allowing the use of a much larger time step in the macroscopic calculation.
The HMM can also be used to model isolated defects, such as dislocations
or cracks, that require the use of a higher level of theory only in the vicinity
of the defect itself. The different possible applications of the HMM are sche-
matically displayed in Figure 10 and include macroscopic processes with
unknown constitutive relations (a), and isolated defects [(b) and (c)]. How
to treat the problem of an isolated defect depends upon the relationship
between the time scale for the defect dynamics,
T
D
, and the time scale for
the relaxation of the defect structure,
T
r
. For
T
D
T
r
, the simulation time,
t
, for the microscopic model can be less than the macroscopic-scale time
step, (TS) [case (b)]. Conversely, if
T
D
is comparable to
T
r
[case (c)], the whole
time history of the defect should be computed atomistically.
In the following, we discuss in detail the application of the HMM to the
study of macroscopic processes with unknown constitutive relations because
the isolated-defect case can be obtained with easy modifications. Because the
aim of the HMM is to accurately simulate a macroscopic process with state
variable
U
, the main components of the methodology are: (1) a macroscopic
scheme to solve the continuum equations for
U
and (2) a way to estimate the
missing macroscopic data from a microscopic model. Therefore, the key steps
in applying the heterogeneous multiscale procedure to the atomistic/conti-
nuum coupling are as follows. To begin with, because the macroscopic model
is based on the conservation laws of mass, momentum, and energy, the MD
must be expressed in the form of conservation laws as well, i.e., as a set of
partial differential equations (PDEs). Then, the PDEs are solved numerically.
