Biomedical Engineering Reference
In-Depth Information
is implemented to discretize the time-derivatives. For the space discretization, we
use the finite element method approach. Thus, we define the finite element spaces
V
h
F
.t
n
/
V
F
.t
n
/;Q
h
.t
n
/ Q.t
n
/;
h
h
V
K
V
K
and
V
S
V
S
, and introduce the
following bilinear forms
a
F
.
u
;'/ WD 2
F
Z
D.
u
/ W D.'/;
F
.t
n
/
Z
b
F
.p;'/ WD
pr';
F
.t
n
/
a
K
.
r
;
r
/ WD C
0
Z
L
0
r
r
;
a
S
.
d
; / WD 2
Z
D.
d
/ W D. / C
Z
.r
d
/.r /:
S
S
A weak formulation of the fully discrete loosely coupled algorithm applied to the
simplified problem is given as follows:
Problem A1 (The Structure Problem).
To discretize the structure problem in
time we use the second-order Newmark scheme. The problem reads as follows: Find
.
d
n
C
1=3
h
;V
n
C
1=3
h
/ 2
V
S
V
S
such that for all .
h
;
h
/ 2
V
S
V
S
S
Z
h
C
Z
V
n
C
1=3
h
d
h
C
d
n
C
1=3
V
h
h
h
t
2
S
S
C
K
h
Z
V
n
C
1=3
r;h
r;h
C a
K
.
d
r;h
C
d
n
C
1=3
V
r;h
t
;
r;h
/ C a
S
.
d
h
C
d
n
C
1=3
r;h
h
;
h
/
2
2
C
S
Z
.
V
h
C
V
n
C
1=3
d
n
C
1=3
h
d
h
h
/
h
2
t
S
C
K
h
Z
.
V
r;h
C
V
n
C
1=3
d
n
C
1=3
r;h
d
r;h
t
r;h
/
r;h
D 0:
(2.201)
2
Note that in this step we take all the kinematic coupling conditions into account.
More precisely:
1.
Initially we set V
r;h
j
D v
r;h
D
u
r;h
j
.t
n
/
;
2.
Once
d
n
C
1=3
h
and V
n
C
1=3
h
are computed,
n
C
1=3
r;h
;v
n
C
1=3
r;h
and
u
n
C
1=3
r;h
j
.t
n
/
are
recovered via
n
C
1=3
r;h
D d
n
C
1=3
r;h
j
;v
n
C
1=3
r;h
D
u
n
C
1=3
r;h
j
.t
n
/
D V
n
C
1=3
r;h
j
:
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