Biomedical Engineering Reference
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the structure displacement
n
C
1
onto the whole domain
F
n
C
1
A
D 0 in
F
;
n
C
1
j
D
n
C
1
;
A
n
C
1
A
j
@
F
n
D 0:
n
C
1
@t
, based on the updated
location of the structure, and use it in the advection problem, Problem A2(b) below.
A
@
From here we calculate the domain velocity
w
n
C
1
D
Remark 1.
Note that in Problem A1, we can rewrite the membrane equations by
using the kinematic coupling conditions in the following way:
@
z
C
1
@
2
d
z
K
h
@V
z
@t
C
2
@d
r
on .t
n
;t
n
C
1
/;
C S
e
r
e
z
D 0
@
z
2
K
h
@V
r
@t
C C
0
d
r
C C
2
@d
z
on .t
n
;t
n
C
1
/:
@
z
C Se
r
e
r
D 0
In this way the membrane equations serve as Robin boundary conditions for the
thick structure problem.
Problem A2(a): The Stokes Problem
This step involves solving a time-dependent Stokes problem on .t
n
;t
n
C
1
/, with a
Robin-type boundary condition involving the thin structure inertia and Part I of the
fluid stress. This problem is solved on the fixed fluid domain
F
.t
n
/, determined
by the structure position in the previous time step. Using the updated fluid domain
calculated in Problem A1 is also an option. In the proof of stability of this scheme,
using
F
.t
n
/ is more convenient for the proof. In this step the structure position
and the velocity of the thick structure do not change, and so
n
C
2=3
D
n
C
1=3
;
d
n
C
2=3
D
d
n
C
1=3
;V
n
C
2=3
D V
n
C
1=3
:
The problem reads as follows:
Find
u
;p,and
v
such that for t 2 .t
n
;t
n
C
1
/, with p
n
denoting the pressure
obtained at the previous time step, the following holds:
LJ
LJ
LJ
LJ
F
Dr;
F
@
u
@t
in
F
.t
n
/ .t
n
;t
n
C
1
/;
r
u
D 0
@t
C J
nj
.t/
C LJp
n
nj
.t/
D 0
K
h
@.
u
j
.t/
/
on .t
n
;t
n
C
1
/;
on .t
n
;t
n
C
1
/;
v
D
u
j
.t/
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