Biomedical Engineering Reference
In-Depth Information
V
F
.t/ Df
u
D .
u
z
;
u
r
/ 2 C
1
.
F
.t//
2
Wr
u
D 0;
u
z
D 0 on .t/;
u
r
D 0 on @
F
.t/ n .t/g;
V
F
.t/ D V
F
.t/
H
1
.
F
.t//
:
(2.125)
Using the fact that
F
.t/ is locally a sub-graph of a Hölder continuous function
we can get the following characterization of the velocity solution space
V
F
.t/ (see
[
32
,
79
]):
V
F
.t/ Df
u
D .
u
z
;
u
r
/ 2 H
1
.
.t//
2
Wr
u
D 0;
u
z
D 0 on .t/;
u
r
D 0 on @
.t/ n .t/g:
(2.126)
The function space associated with weak solutions of the 1D linear wave equation
and the thick wall are given, respectively, by
V
W
D H
0
.0;1/;
(2.127)
V
S
Df D .
z
;
r
/ 2 H
1
.
S
/
2
W
z
D 0 on ; D 0 on
in=out
g: (2.128)
Motivated by the energy inequality we also define the corresponding evolution
spaces for the fluid and structure sub-problems, respectively:
W
F
.0;T/ D L
1
.0;T I L
2
.
F
.t/// \ L
2
.0;T I
V
F
.t//;
(2.129)
W
W
.0;T/ D W
1;
1
.0;T I L
2
.0;1// \ L
2
.0;T I
V
W
/;
(2.130)
W
S
.0;T/ D W
1;
1
.0;T I L
2
.
S
// \ L
2
.0;T I
V
S
/:
(2.131)
Finally, we are in a position to define the solution space for the coupled fluid-multi-
layered-structure interaction problem. This space involves the kinematic coupling
condition, which is enforced in strong sense. The dynamic coupling condition will
be enforced in weak sense, through integration by parts in the weak formulation of
the problem. Thus, we define
W
.0;T/ Df.
u
;;d/ 2
W
F
.0;T/
W
W
.0;T/
W
S
.0;T/ W
u
.t;
z
;1C .t;
z
// D @
t
.t;
z
/
e
r
; d.t;
z
;1/D .t;
z
/
e
r
g:
(2.132)
Equality
u
.t;
z
;1C .t;
z
// D @
t
.t;
z
/
e
r
is taken in the sense defined in [
32
,
118
].
The corresponding test space will be denoted by
.0;T/ Df.
q
; ; / 2 C
c
.Œ0;T/I
V
F
V
W
V
S
/ W
q
.t;
z
;1C .t;
z
// D .t;
z
/
e
r
; .t;
z
;1/D .t;
z
/
e
r
g:
Q
(2.133)
Notice the coupling conditions in the test space that are enforced at the fluid-
structure interface.
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