Biomedical Engineering Reference
In-Depth Information
Computational and algorithmic aspects of the numerical solution of the minimiza-
tion problem are challenging. Here we resort to the workflow
Time Discretize, then
Optimize, then Space Discretize
. This means that we first discretize in time the prob-
lem by collocating the minimization process at selected time instants
t
k
. Then we
perform the minimization, by computing the KKT system for the space-continuous
problem. Finally, we discretize the KKT system. In this way, the variational proce-
dure for the minimization does not involve adjoint backward-in-time problems (see
[12]) and the differentiation of the Lagrangian functional does not require to per-
form differentiation of the domain
Ω
(shape derivatives), since at each instant the
domain
is frozen. The anticipated drawback of this approach is that the effect of
noise over the time interval is not damped by a least square minimization, being the
problem collocated pointwise at
t
k
. Post-processing for the estimates of
E
obtained
at each step is required for filtering out the error. An extensive analysis of different
solution methods for this problem is however an important follow up in this context.
After time discretization, at each instant
t
k
the problem reads (hereafter we omit
to specify the time index
k
for the sake of readability): Find the piecewise constant
function
E
defined on
Ω
Γ
w
that minimizes the functional
log
E
E
re f
2
2
d
x
m
J
=
(
η
−
η
)
+
α
max
x
∈
Γ
w
Γ
w
γ
1
u
+(
u
−
w
)
∇
·
u
−
μ∇
·
∇
u
+
∇
u
T
+
∇
p
=
g
1
⎧
⎨
⎩
in Ω
,
∇
·
u
=
0
in Ω
,
(
γ
2
+
γ
3
E
θ
)
u
·
n
−
p
+
μ
(
∇
u
+
∇
u
T
)
·
n
·
n
=
g
2
,
s
.
t
.
Γ
w
,
on
n
×
u
×
n
=
0
+
Boundary Conditions
o
n
Γ
in
and
Γ
out
.
(12.14)
The functions
g
1
,
g
2
and
γ
i
depend on the time discretization. Notice that for the fluid
at the interface with the wall, here we prescribe null tangential velocity conditions.
Moreover, in the previous system, the FSI problem has been simplified by eliminat-
ing the displacement, leading to a fluid problem with Robin boundary conditions, as
proposed in [82].
The explicit computation of the KKT system for this problem and its generaliza-
tion to the case of a 3D thick elastic structure are reported in [83]. We have analyzed
different choices for the space of the admissible CV (
E
). In particular, for the piece-
wise constant and piecewise linear cases, we can prove the following Proposition.
Proposition 2.
For
0
the KKT system associated with the minimization problem
has at least one solution.
α
>
After the space discretization, the KKT system yields a non-linear algebraic min-
imization problem. In particular we can use again the gradient-based BFGS method
(see e.g. [6]). For more details, see [83].
