Biomedical Engineering Reference
In-Depth Information
terest
Ω
(see Fig. 12.1, right) features an inflow boundary
Γ
in
, an outflow boundary
Γ
out
and the physical wall of the vessel
Γ
out
can possibly consist of
more parts (like in a vascular bifurcation). Variables of interest are the velocity
u
and the pressure
P
Γ
wall
.
Γ
in
and
=
ρ
f
p
which are assumed to obey the incompressible Navier-
Stokes equations in
ρ
f
the fluid density. At this stage,
we consider the steady problem. We assume to have velocity measures (
Data
)
u
m
available at
N
s
sites
1
x
i
∈
Ω
Ω
. Here we have denoted by
. Following the general description given in the Intro-
duction, we assume that the CV is represented by the inflow normal stress
h
. This is
an arbitrary choice, an extensive comparison with other choices is still to be done.
Post-processing
f
(
·
)
in this case is given by the Dirac delta distributions, such that
is the vector of the values of the computed velocity at the measurement sites.
Then, the distance
dist
f
(
u
)
N
s
i
u
m
u
m
2
. The control
(
f
(
u
)
,
)
is defined as
1
(
u
(
x
i
)
−
(
x
i
))
∑
=
problem reads: Find
u
m
mi
h
J
(
u
,
h
)=
dist
(
f
(
u
)
,
)+
Regularization
(
h
)
⎧
⎨
T
u
−
μ∇
·
(
∇
u
+
∇
)+(
u
·
∇
)
u
+
∇
p
=
s
in
Ω
,
∇
·
u
=
0
in
Ω
,
(12.2)
s.t.
u
=
0
on
Γ
wall
,
⎩
T
u
−
μ
(
∇
u
+
∇
)
·
n
+
p
n
=
h
on
Γ
in
,
T
u
−
μ
(
∇
u
+
∇
)
·
n
+
p
n
=
g
on
Γ
out
,
where
n
denotes the outward unit vector normal to the boundary. A Newtonian rhe-
ology is supposed to hold, since it is a common assumption in large and medium
vessels [16] and
is the kinematic viscosity. Since we are considering fixed geome-
tries, we assume homogeneous Dirichlet boundary conditions on
μ
Γ
wall
. When solving
problems in the form (12.2) there are in general two possibilities. In the first one, we
first write the necessary conditions associated with the continuous constrained opti-
mization problem, the so called
Karush Kuhn Tucker
(KKT) system [6, 12]. These
are obtained by augmenting the original functional with the (variational formula-
tion of) the constraint given by FW (in this case the steady Navier-Stokes problem),
weighted by unknown multipliers and then by setting to zero the derivatives of the
augmented functional with respect to the multipliers (so to obtain the
state problem
),
to the variables (
adjoint problem
)andtoCV(
optimality conditions
). Successively,
the resulting problem is discretized (
Optimize then Discretize
- OD - approach). In
the second approach, we first discretize the different components of the problem (the
functional to be minimized and the constraints) and then perform the optimization
of the discrete system (
Discretize then Optimize
- DO - approach). In [55] we com-
pared the two strategies, and found that the DO is more efficient for the problem at
hand. For this reason we proceed with the latter approach.
1
Notice that we use the word “sites” for the location of measurements, as opposed to the word
“nodes” for points where velocities are computed. We do not assume at this level particular posi-
tions for the sites, even though in the applications it is reasonable to assume that they are located
on planes transverse to the blood stream.
