Biomedical Engineering Reference
In-Depth Information
11.4 Multiscale analysis
We formulate here different strategies for the interaction of the micro and the
macroscale models. We start from the algorithms that most tightly couple the two
models and we progressively simplify them, in order to balance the accuracy of the
methodology with a reasonable computational cost.
11.4.1 Input/Output description of the models
According to the previous derivation, the macroscale model for polymer mixtures is
not computable, because it needs to determine water and polymer diffisivities in the
mixture to close governing equations and boundary conditions. More precisely, the
input/output structure of the macroscale model can be described as follows:
(
ρ
,
ρ
,...,
ρ
)
(
ρ
,
ρ
,...,
ρ
)
.
Input:
given
D
w
and
D
i
w
1
N
w
1
N
Compute:
for any
(
z
,
t
)
and for
i
=
1
,...,
N
solve the following problem
⎧
⎨
∂
t
ρ
w
=
∇
·
D
w
∇ρ
w
−
k
M
w
N
i
i
−
1
i
ρ
w
ρ
i
,
M
0
x
1
∑
=
1
∂
t
ρ
i
=
∇
D
i
∇ρ
i
−
(
k x
1
ρ
w
ρ
i
+
2
k x
1
ρ
w
j
i
j
ρ
j
,
i
−
1
)
∑
=
i
+
1
(11.4)
∂
z
ρ
w
|
z
=
0
=
0
, −
D
w
∂
z
ρ
w
|
z
=
L
=
π
w
(
ρ
w
|
z
=
L
−
A
)
⎩
∂
z
ρ
i
|
z
=
0
=
0
, −
D
i
∂
z
ρ
i
|
z
=
L
=
π
i
ρ
i
|
z
=
L
w
i
0
˜
ρ
w
(
z
,
0
)=
0
,
ρ
i
(
z
,
0
)=
ρ
.
Output:
determine the mixture partial densities:
that
can be further post-processed to compute other indicators that characterize the
mixture composition.
ρ
(
t
,
z
)
,
ρ
(
t
,
z
)
,...,
ρ
(
t
,
z
)
w
1
N
denotes the state variables of the model, the
macroscale
model can be described by means of the following input/output scheme:
Reminding that
ϒ
=(
ρ
w
,
ρ
1
,...,
ρ
N
)
macro
=
⇒
ϒ
(
D
w
(
ϒ
)
,
D
i
(
ϒ
)
t
,
z
)
.
The solution of problem (11.4) is provided by numerical approximation schemes.
In particular, we have exploited Lagrangian finite elements for the space discretiza-
tion, by resorting to a semi-discrete problem that has been fully discretized with
backward finite difference schemes to advance in time. For further details, we refer
to [15, 39]. The main difficulty in the approximation of the problem at hand consists
in the efficient solution of the nonlinear system of equations arising from the fully
discrete scheme. To this aim, we have applied the damped Newton method proposed
in [8]. For one or two space dimensions, the computational cost of the macroscale
model is almost negligible with respect to the microscale one. Indeed, its characteris-
tic computational cost does not exceeds the scale of minutes of CPU time. A similar
cost can be estimated for three-dimensional applications, for reasonably simple ge-
ometrical configurations.
