Biomedical Engineering Reference
In-Depth Information
2.6.3
The ALE Formulation and Lie Splitting
First-Order ALE Formulation
As mentioned earlier, since we consider nonlinear coupling between the fluid and
structure, the fluid domain changes in time. To prove the existence of a weak solu-
tionto(
2.91
)-(
2.102
) it is convenient to map the fluid domain onto a fixed domain
F
. The structural problems are already defined on fixed domains since they are
formulated in the Lagrangian framework. We map our fluid domain
F
.t/ onto
F
by using an Arbitrary Lagrangian-Eulerian (ALE) mapping [
21
,
52
,
85
,
132
,
133
]. We
remark here that in our problem it is not convenient to use Lagrangian formulation
for the fluid sub-problem, as is done in, e.g., [
34
,
44
,
102
], since, in our problem, the
fluid domain consists of a fixed, control volume of a cylinder, with prescribed inlet
and outlet pressure data, which does not follow Largangian flow.
We begin by defining a family of ALE mappings A
parameterized by :
;
z
; r/ 2
F
;
z
.1 C .t;
z
//r
A
.t/ W
F
!
F
.t/; A
.t/.
z
; r/ WD
(2.106)
where .
z
; r/ denote the coordinates in the reference domain
F
D .0;1/ .0;1/.
The mapping A
.t/ is a bijection, and its Jacobian is given by
jdetrA
.t/jDj1 C .t;
z
/j:
(2.107)
Composite functions with the ALE mapping will be denoted by
u
.t;:/ D
u
.t;:/ ı A
.t/ and p
.t;:/ D p.t;:/ ı A
.t/:
(2.108)
The derivatives of composite functions satisfy:
@
t
u
D @
t
u
.
w
/
u
;
u
;
r
r
u
Dr
(2.109)
where the ALE domain velocity,
w
, and the transformed gradient, r
,aregiven
by:
0
@
1
A
@
z
r
@
z
1 C
@
r
1
1 C
@
r
w
D @
t
r
e
r
;
r
D
:
(2.110)
v
Dr
v
.rA
/
1
: For the purposes of the existence proof we
also introduce the following notation:
One can see that r
1
2
.r
Dp
I
C 2
D
.
u
/;
D
.
u
/ D
u
/
u
/:
C .r
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