Chemistry Reference
In-Depth Information
cell,
E
, by applying the deformation gradient
A
to the basis vectors of the
undeformed cell,
E
:
E
¼
AE
½
37
Several other requirements have to be taken into account to keep the MD
simulations consistent with the macroscopic fields, and they are discussed in
Ref. 176. The final step is to average the microscopic fluxes to obtain the fluxes
needed by the macroscopic scheme.
Two final remarks need to be made. First, even when the MD simulation
time is short compared to the macroscopic time scale, it can still be very long
when compared with the microscopic time step if a substantial energy barrier
must be overcome during the microscopic relaxation. Second, the choice of the
MD cell size is a delicate balance between accuracy and computational effi-
ciency; as the cell size increases, the error decreases (as
L
1:5
, for a cell of
volume
V
L
3
), but the computational time significantly increases.
¼
Applications
The HMM has been applied to the study of friction between
two-dimensional atomically flat crystal surfaces, dislocation dynamics in the
Frenkel-Kontorova model (i.e., considering a one-dimensional chain of
atoms in a periodic potential, coupled by linear springs
181
), and crack
propagation in an inhomogeneous medium.
175,176
Bridging Scale
The bridging scale technique by Wagner and Liu
73,182-184
is a
multiple-scale method explicitly developed to eliminate the elastic wave
reflection at the continuum-discrete interface. At the same time, because this
methodology couples finite elements, or other continuous interpolation
functions, to MD simulations without grading the continuum nodal spacing
down to the atomic lattice size, it permits the use of a larger time step in
the FE simulation than in the MD one.
The key idea of this technique is to decompose the total displacement
field
u
into a coarse and a fine scale:
u
¼
u
þ
u
0
½
38
where
u
is the coarse-scale component and
u
0
is the fine-scale one. The coarse
scale is that part of the solution that can be represented by a set of basis func-
tions (finite element or mesh-free shape functions):
X
N
I
d
I
u
ð
X
a
Þ¼
ð
or
u
¼
Nd
in matrix representation
Þ
½
39
I
where
N
I
¼
) is the shape function associated with node
I
evaluated at
the initial atomic position
X
a
N
I
ð
X
a
, and
d
I
is the FE nodal displacement associated
