Chemistry Reference
In-Depth Information
Figure 10
Schematic representation of HMM applications. In (a) simulation of a
macroscopic process for which the constitutive relations have to be obtained from
modeling at the microscale. The macroscopic system is solved using a grid (
x
k
), and only
a small region around each macroscopic-solver grid point is used for the atomistic
calculation (the shaded area represents the atomic cell at grid point
x
k
). The time step
(TS) used for the macroscopic calculations is much larger than the microscopic one (ts),
and times,
TS are necessary to equilibrate the atomistic calculations. In (b) and (c)
isolated defect calculations, i.e., problems where the coupling with the microscale model
is needed only in a limited part of the system (near the defect itself). If the time scale for
the defect dynamics is much larger than the time scale for the relaxation of the defect
structure (case b), then only a short time
t
TS is simulated using the atomistic model
for each macroscopic time step, otherwise (case c) the whole time history of the defect
should be computed atomistically.
t
At each macroscopic time step, MD calculations, constrained by the local
macrostate of the system, are used to compute the missing data needed in
the numerical solution of the PDEs. These data are obtained as time averages
of microscopic variables after the MD simulation has equilibrated. The
physical requirement to minimize/eliminate wave reflection at the boundary
between the atomistic and the continuum treatment is achieved via well-
chosen boundary conditions for the MD simulational cell.
177,178
Choosing to work in Lagrangian coordinates, the conservation laws at
the basis of the continuum model are
q
t
A
r
x
0
v
¼
0
time evolution of the deformation
q
t
q
r
x
0
s
¼
0
conservation of momentum
½
33
r
0
q
t
e
r
x
0
j
¼
0
conservation of energy
where
A
,
v
,
q
,
e
are the deformation gradient, velocity, momentum, and total
energy per atom, respectively,
is the first Piola-
Kirchhoff stress tensor,
j
is the energy flux, and
x
0
is the reference coordinate
of the solid. The position after deformation is then
r
0
is the initial density,
s
x
¼
x
0
þ
u
ð
x
0
;
t
Þ
.To
