Biomedical Engineering Reference
In-Depth Information
Therefore, we can proceed as usual, by substituting the test functions in (
2.6.5
)
with structure velocities. More precisely, we replace the test function . ; / by
.v
n
C
2
;V
n
C
2
/ in the first term on the left-hand side, and then replace . ; / by
..
n
C
1
1
2
n
/=t;.d
n
C
2
d
n
/=t/ in the bilinear forms that correspond to the
elastic energy. To deal with the terms .v
n
C
1=2
v
n
/v
n
C
1=2
, .
n
C
1=2
n
/
n
C
1=2
,
.V
n
C
1=2
V
n
/ V
n
C
1=2
,and.d
n
C
1=2
d
n
/ d
n
C
1=2
, we use the algebraic identity
1
2
.jaj
2
/. After multiplying the entire equation by t,
the third equation in (
2.6.5
) can be written as:
2
2
.a b/ a D
Cja bj
jbj
1
2
1
2
1
2
1
2
.kv
n
C
2
L
2
.0;1/
Ckv
n
C
v
n
2
L
2
.0;1/
/ C .kV
n
C
2
L
2
.
S
/
CkV
n
C
V
n
2
k
k
k
k
L
2
.
S
/
/
1
2
1
2
1
2
;d
n
C
1
2
/
k@
z
n
C
2
L
2
.0;1/
Ck@
z
.
n
C
@
z
n
/k
2
L
2
.0;1/
C a
S
.d
n
C
k
C a
S
.d
n
C
2
d
n
;d
n
C
2
d
n
/ Dkv
n
2
L
2
.0;1/
CkV
n
2
L
2
.
S
/
Ck@
z
n
2
L
2
.0;1/
k
k
k
C a
S
.d
n
;d
n
/:
D
u
n
, we can add
f
R
F
.1C
n
/
u
n
C
1=2
on the left-
1
2
Since in this sub-problem
u
n
C
hand side, and
f
R
F
.1C
n
/
u
n
on the right-hand side of the equation. Furthermore,
displacements
d
n
C
1
2
do not change in Problem A2 (see (
2.145
)), and so
we can replace d
n
and
n
on the right-hand side of the equation with d
n
1
2
and
n
C
1
2
and
1
2
, respectively, to obtain exactly the energy equality (
2.144
).
n
t
Semi-discretization of Problem A2
In this step , d and V do not change, and so
1
2
; d
n
C
1
1
2
; V
n
C
1
1
2
:
n
C
1
D
n
C
D d
n
C
D V
n
C
(2.145)
n
Then, define .
u
n
C
1
;v
n
C
1
/ 2
V
F
L
2
.0;1/ to be a weak solution of Problem
A2 (
2.121
) if the following holds for each .
q
; / 2
n
F
V
L
2
.0;1/ such that
q
j
D
e
r
, velocities .
u
n
C
1
;v
n
C
1
/:
.1 C
n
/
u
n
C
1
Z
h
.
u
n
n
i
u
n
C
1
1
2
u
n
C
1
2
1
2
r
e
r
/ r
v
n
C
q
C
q
t
n
i
q
u
n
C
1
Z
h
.
u
n
1
2
1
2
v
n
C
2
r
e
r
/ r
v
n
C
2
u
n
C
1
C
q
C2
Z
.1 C
n
/
D
n
.
u
/ W
D
n
.
q
/
(2.146)
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