Biomedical Engineering Reference
In-Depth Information
Z
K
h
2
d
dt
k@
t
k
1
2
d
dt
E
el
./ C E
vis
.@
t
/
2
n
.t/
u
D
L
2
.!/
C
.t/
dt
s
k@
t
d
k
L
2
.
S
/
:
1
2
d
2
L
2
.
S
/
C 2k
D
.
d
/k
2
L
2
.
S
/
C kr
d
k
2
(2.83)
By combining (
2.75
) with (
2.76
), (
2.77
), (
2.78
), and (
2.83
), one obtains the
following energy equality:
n
F
k
u
k
1
2
d
dt
2
F
.t/
C
K
hk@
t
k
2
L
2
./
C
S
k@
t
d
k
2
L
2
.
S
/
CE
el
./ C2k
D
.
d
/k
2
L
2
.
S
/
F
.t/
CE
vis
.@
t
/ DǙP
in=out
.t/
Z
L
2
.
S
/
o
C2
F
k
D
.
u
/k
2
2
C kr
d
k
u
z
in=out
Finally, by using the trace inequality and Korn inequality one can estimate:
jP
in=out
.t/
Z
C
2
jP
in=out
j
C
2
k
D
.
u
/k
2
2
u
z
jCjP
in=out
jk
u
k
H
1
.
F
.t//
C
L
2
.
F
.t//
:
in=out
By choosing such that
2
F
we get the energy inequality
n
F
k
u
k
1
2
d
dt
2
F
.t/
C
K
hk@
t
k
2
L
2
./
C
S
k@
t
d
k
2
L
2
.
S
/
CE
el
./ C2k
D
.
d
/k
2
L
2
.
S
/
L
2
.
S
/
o
2
2
C kr
d
k
C
F
k
D
.
u
/k
F
.t/
C E
vis
.@
t
/ C.P
in
.t/;P
out
.t//:
t
2.5.2
ALE Formulation
Since the fluid-structure coupling studied in this chapter is performed along the
moving fluid-structure interface, the fluid domain .t/ is not fixed. This is a
problem from many points of view. In particular, defining the time discretization of
the time derivative @
u
=@t, for example @
u
=@t .
u
.t
n
C
1
;:/
u
.t
n
;://=.t
n
C
1
t
n
/,
is not well defined since
u
.t
n
C
1
;:/and
u
.t
n
;:/are not defined on the same domain
at two different time steps. To resolve this difficulty, often times the fluid domain is
mapped onto a fixed, reference domain via a smooth, invertible ALE mapping [
52
]:
A
W
F
!
F
.t/:
An example of such a mapping is the harmonic extension of the boundary @
F
.t/
onto the fluid domain. See Sect.
2.7
. Another example is a mapping particularly
convenient for the existence proof, presented in Sect.
2.6
. This introduces additional
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