Biomedical Engineering Reference
In-Depth Information
momentum equation has been solved, the pressure can be recovered by solving the
least square problem in the full Finite Element space, that is,
p
h
D
q
h
2
Q
h
jjf
h
C
u
h
B
T
qjj
2
:
min
(6.76)
The solution to this problem exists and is unique, provided that the velocity and
pressure FE spaces satisfy the
inf-sup
condition, which guarantees that B
T
has
full column rank. In order to have a representation of the pressure in the reduced
space, one has to make sure that the reduced saddle point problem is non-singular.
In literature this issue has been tackled by enriching the velocity reduced space [
64
].
We therefore construct the reduced basis only for the fluid velocity and mem-
brane displacement fields. To this end, we solve the forward problem for a given
set of Young moduli E
1
;:::;E
M
and store the corresponding solutions (snapshots)
u
h;i
;
h;i
. In order to deal with nonhomogeneous boundary conditions at the
inflow/outflow sections, we modify the velocity snapshots in the following way
u
h;i
D
u
h;i
u
`
(6.77)
where
u
`
is the solution of a
steady
rigid-wall Stokes problem used as a
lifting
function
for the nonhomogeneous boundary conditions. This choice allows us
to preserve the divergence-free nature of the snapshots which are then collected
(amended by the lifting) in the snapshots matrices X
u
and X
. We compute the SVD
of these matrices and let W
Ǜ
be the matrices containing the first k
Ǜ
left singular
vectors of X
Ǜ
(Ǜ D
u
;), with k
Ǜ
such that
i
N
Ǜ
X
k
Ǜ
X
i
(6.78)
i
D
1
i
D
1
where
i
are the singular values of X
Ǜ
, is the fraction of data variability that we
want to capture (typically we take D 0:9;0:95 or 0:99)and
N
Ǜ
is the dimension
of the FE space. The columns of W
u
and W
form the reduced basis for the fluid
velocity and membrane displacement spaces.
If we project the IFMI problem (
6.74
) onto the reduced space, we then obtain
1
2
jj
r;h
d
r
jj
E
h
D arg min
E
h
2R
r
.E
h
/ D
2
k
J
†
C
R
.E
h
/
"
f
r;h
n
1
r;h
#
s.t.
C
r
O
tP
r
I
u
r
r
(6.79)
D
where C
r
D W
u
CW
u
,M
r
D W
T
M
†
W
,P
r
D W
T
PW
u
, f
r;h
D W
u
.f
h
B
T
p
h
/,
d
r;h
D W
T
h
meas
, and the dependence of C and f
h
on E is understood for brevity.
The minimization problem is then solved as in the previous section, using the
BFGS method. In particular, in order to evaluate the functional and its gradient, we
Search WWH ::
Custom Search