Biomedical Engineering Reference
In-Depth Information
A Summary of the Benchmark Problem.
The benchmark problem in fluid-multi-
layered-structure interaction that we are interested in studying is given by the
following:
Find
u
, p, and
d
such that:
in
F
.t/; t 2 .0;T/;
F
.@
t
u
C .
u
r/
u
/ Dr
r
u
D 0
S
@
tt
d
Dr
S
in
S
.0;T/;
9
=
@
t
e
r
D
u
j
R
C
;
e
r
D
d
j
R
;
K
h@
tt
C
L
el
./ C
L
vis
@
@t
DJ.
n
/j
R
C
e
r
C R
S
j
R
e
r
e
r
on .0;T/:
;
L
vis
are defined above. In this formulation, the fluid and thick
structure equations are defined in Cartesian coordinates, while the thin structure
model is given in cylindrical coordinates. Furthermore, the fluid equations are
given in Eulerian framework, while the structure equations are given in Lagrangian
framework. To account for the different coordinates, J in the dynamic coupling
condition denotes the Jacobian of the transformation from the Eulerian to the
Lagrangian framework, and from Cartesian to cylindrical coordinates. Similarly,
the factor R in front of the first Piola-Kirchhoff stress tensor
S
is the Jacobian
of the transformation between the Cartesian and cylindrical coordinates. While the
coupling conditions are calculated at the deformed interface .t/, they are written
in terms of the reference configuration of the fluid-structure interface, namely, they
are written in terms of points on .
Supplemented with initial and boundary conditions, this problem defines a
nonlinear, moving-boundary problem of mixed, parabolic-hyperbolic type. Hyper-
bolicity is associated with the thick structure problem and with the thin structure
problem when no viscoelastic effects are taken into account, i.e., when
L
el
,and
where ,
S
,
L
vis
D 0.
Parabolicity describes the properties of the fluid problem.
We will be studying this class of problems from both numerical, and theoretical
point of view. Numerical method development for this class of problems will be
presented in Sect.
2.7
, while existence of solutions will be studied in Sect.
2.6
.In
those sections, concrete examples of this class of problems will be presented and
studied. A simplified version of these equations in 2D will be presented.
2.4
FSI Literature Review
Fluid-structure interaction problems have been extensively studied for the past 20
years by many authors. The focus has been exclusively on FSI problems with
structures consisting of a single material, except for the numerical simulations
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