Biomedical Engineering Reference
In-Depth Information
2. Advance in time and solve the macroscale model at time
t
n
:
macro
=
⇒
ϒ
(
t
n
−
1
t
n
−
1
t
n
D
w
(
)
,
D
i
(
)
,
z
j
)
.
t
n
Then, compute
F
(
ϒ
(
,
z
))
.
We notice that this algorithm considerably reduces the number of calls to the mi-
croscale model and makes it independent on the number of space dimensions of
the macroscale equations. Since the coupling of the models is sequential in time,
the main limitation of this approach consists in the different characteristic CPU ef-
forts needed for the micro and macroscale models. Considering the aforementioned
promising performance of GPU computing devices for molecular dynamics simu-
lations, future works could be devoted to the development of an hybrid CPU-GPU
coupled computational environment for an efficient implementation of a dynamic
interaction between macroscale and atomistic models.
In alternative to the previous algorithms, which turn out to be computationally
expensive in any case, we propose the following
static coupling strategy
to feed the
macroscale model with data provided by microscale sumulations. For the sake of
simplicity, we only refer to a generic diffusivity
D
(
ϒ
)
that stands for either
D
w
(
ϒ
)
or
P
p
D
i
(
ϒ
)
1
be a collection of different mixture configurations (see Fig. 11.2
for an illustration) and let
D
.Let
{
ϒ
p
}
=
p
=
{
D
(
F
(
ϒ
p
))
}
1
be the vector defined as follows,
=
micro
=
⇒
F
(
ϒ
p
)
D
(
F
(
ϒ
p
))
.
We aim to determine a function
D
(
F
(
ϒ
))
that suitably approximates its discrete
P
. First, a parametrization for
analog
D
∈
R
D
with polynomials is obtained by defin-
M
=
2
0
ing as
q
∈
N
a multi-index and
a
M
q
1
=
0
M
q
2
=
0
a
q
F
2
q
1
1
q
2
D
(
F
)=
D
F
Q
q
2
where
a
is the vector of coefficients
{
a
q
}
1
with
Q
=(
M
+
1
)
=
9. For the well
=
posedness of the forthcoming algorithm we have to make sure that
P
≥
Q
. Then we
introduce a vector function
D
a
:
Q
P
defined as
D
a
D
a
p
R
→
R
=
{
(
F
(
ϒ
p
))
}
1
such
=
that,
a
M
q
1
=
0
M
q
2
=
0
a
q
F
)
.
q
1
1
q
2
2
D
a
(
F
(
ϒ
)) =
D
(
ϒ
)
F
(
ϒ
p
p
p
a
best fits the data
D
are determined as
D
Then, the optimal parameters
a
such that
follows,
a
∗
=
D
a
argmin
a
D
−
.
2
In such way we determine an explicit representation of functions
D
i
(
ϒ
)
representing a closure of the macroscale model (11.4) that is now completely solv-
able.
D
w
(
ϒ
)
and
