Biomedical Engineering Reference
In-Depth Information
and the adjoint of problem (
6.70
) reads
13
8
<
f
t
.
n
u
n
1
/
f
..
u
w
/ r/
n
Cr
f
.
n
/ Dt.
meas
n
/
n
in
r
n
in
D 0
n
on †
D 0
s
h
s
t
C E
n
LJt
n
:
on †
n
f
./
n
C
n
D 0
n
D
n
1
C t
n
on †
:
n
The gradient of the cost functional with respect to the parameter E
n
is obtained
using the adjoint variable and relation (
6.38
), which, for the problem at hand
reads
LJ
LJ
LJ
LJ
E
n
D
Z
n
m
DE
LJ
t
u
n
n
C
n
1
.
n
D
J
n
/d:
†
The optimization is performed using the BFGS method. In particular, at each time
step, for a given initial guess of the parameter E
n;.0/
, the BFGS method iteratively
provides parameter guesses E
n;.j/
, based on the values of the cost functional
J
LJ
LJ
LJ
LJ
E
n;.j
1/
. The iterative procedure stops when
m
E
n;.j
1/
and its derivative
D
J
m
DE
n
LJ
LJ
LJ
LJ
E
n;.j
1/
is less than a given tolerance.
Remark 6.9.
BFGS is a method devised for unconstrained optimization, while the
problem at hand features the constraint E>0. Unilateral constraints can be
managed as indicated in [
58
]. Here, we include this, with a simple change of
variable, by using as a control variable D
J
m
DE
the norm of
D
log.E/,sothatE D
exp. / > 0
for every .
Numerical Results on a Simplified Geometry Representing an Abdominal
Aneurysm
In the numerical results presented in this section, we will use the simplified
membrane model (
6.70
). The optimization strategy depicted above, however, can
be equally applied to this simplified problem.
We consider a 2D axisymmetric geometry which represents an abdominal
aneurysm (see Fig.
6.11
, top-left). The radius of the vessel varies from 1 to
2.5 cm and the vessel length is 6 cm. We perform a synthetic simulation in
13
See (
6.37
), and note that here the adjoint variable is denoted with as in this context is used
for the density.
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