Biomedical Engineering Reference
In-Depth Information
Now, we multiply the first and second equation in (
5.12
) by any function ' 2 X
and q 2 Q, respectively, sum them, integrate over , transform the viscous term
containing second-order derivatives of v and the term rp with the aid of Green's
theorem and use the boundary conditions (
5.4
), (c)(ii), (d). We define the weak
solution of the problem (
5.12
), (
5.4
) as a couple U D .v;p/ 2 W Q satisfying
the conditions (
5.4
), (a), (b), (c)(i) and the identity
a.U;U;V/ D f.V/;
for all V D .';q/2 X Q:
(5.15)
We use the notation
Z
.Ǜ;LJ/
!
D
Ǜ LJdx;
(5.16)
!
for the scalar product in L
2
.!/ for a set !. It generates the norm kk
L
2
./
. The form
a.U
;U;V/is defined by
2
rv Cr
T
'
C c.v
;v;'/
3
2t
.v;'/
C
a.U
;U;V/D
T
v; r' Cr
z
n
C
1
rv;'
.p; r'/
C .rv;q/
; (5.17)
Z
4v
n
v
n
1
;'
2t
f.V/D
p
ref
' ndS;
O
U D .v;p/; U
D .v
;p/2 W; V D .';q/2 X;
where the convective form c.v
;v;'/ reads
Z
2
.v
r/v;'
2
.v
r/';v
C
1
1
1
2
.v
n/
C
v ' dS:
(5.18)
c.v
;v;'/ D
O
The form c.v
;v;'/ is obtained from the convective term .v
rv;'/
by
integration by parts, using the boundary conditions e (
5.4
) (d) at the outlet and the
definition of the space X (i.e., ' D 0 on
I
[
Wt
and ' n D 0 on
S
). For Ǜ 2 R
we set Ǜ
C
D max.0;Ǜ/.
Space Discretization and Stabilization
In order to apply the Galerkin finite element method (FEM) to the discretization of
the problem (
5.15
), we approximate the spaces W , X, Q from the weak formulation
by finite dimensional subspaces W
h
, X
h
, Q
h
, h 2 .0;h
0
/, h
0
>0,
X
h
Dfv
h
2 W
h
Iv
h
j
I
\
Wt
D 0;v
2
j
S
D 0g:
(5.19)
Search WWH ::
Custom Search