Biomedical Engineering Reference
In-Depth Information
that will be complemented with the application of the macroscale model in the
lumped state space,
macro
=
⇒
ϒ
=
⇒F
(
ϒ
)
.
D
w
(
F
(
ϒ
))
,
D
i
(
F
(
ϒ
))
We finally notice that the set up of the microscale model is pointwise or local, that
is it applies to any single point
(
t
,
z
)
of the macroscale domain.
11.4.2 Multiscale coupling strategies
First, we address a
fully coupled
algorithm, that best exploits the interaction among
micro and macro-sacles. Let
t
n
the time levels corresponding to time discretization of
the macroscale model, and let
z
j
∈
[
0
,
L
]
the nodes associated to space discretization.
t
n
For any
(
,
z
j
)
, the algorithm consists in the following iterative steps for
k
=
1
,...
:
ϒ
(
0
)
(
t
n
1. Set a guess for the state of the system i.e.
,
z
j
)
.
2. Solve the microscale model:
micro
=
⇒
D
(
k
)
D
(
k
)
i
F
(
ϒ
(
k
−
1
)
(
t
n
t
n
t
n
,
z
j
))
(
,
z
j
)
,
(
,
z
j
)
.
w
3. Solve the macroscale model:
D
(
k
)
i
macro
D
(
k
)
(
t
n
t
n
=
⇒
ϒ
(
k
)
(
t
n
,
z
j
)
,
(
,
z
j
)
,
z
j
)
.
F
(
ϒ
(
k
)
(
t
n
Then, compute
,
z
j
))
.
Reminding that the computational cost of a single solution of the microscale model is
considerable, see Sect. 11.3, this algorithm is not yet affordable with standard com-
putational devices. Furthermore, we notice that the cost of this algorithm exponen-
tially increases with the number of space dimensions accounted by the macroscale
model. Indeed, realistic applications in three space dimensions would be practically
unachievable. For this reason, we consider the following simplification, which can
be classified as a
time staggered and space averaged
coupled algorithm. As it will
be discussed later on, such simplification is acceptable for bulk eroding polymers,
where spatial gradients of densities are almost negligible, i.e.
∇ϒ
(
t
,
x
)
0forany
t
>
0 and for any
x
∈
Ω
. Let us define the following spatial average for
F
:
L
0
F
(
ϒ
(
F
(
ϒ
(
t
,
z
)) =
t
,
z
))
dz
.
t
0
Then, given the initial state of the system i.e.
ϒ
(
,
z
)
,forany
n
>
0 we perform the
following steps:
1. Solve the microscale model at time
t
n
−
1
:
micro
=
⇒
t
n
−
1
t
n
−
1
t
n
−
1
F
(
ϒ
(
,
z
))
D
w
(
)
,
D
i
(
)
.
