Fluid–Structure Interaction in Hemodynamics: Modeling, Analysis, and Numerical Simulation - Fluid-Structure Interaction and Biomedical Applications - page 114

The constant C.P in .t/;P out .t/// depends only on the inlet and outlet pressure data,
which are both functions of time.
The expressions for the energy associated with the Koiter shell are given by:

Z Z

h 3
48

h
2

! A ! A

E el ./ D G ./ W G ./R C R ./ W R ./R;

Z Z

h 3
48

h
2

E vis .@ t / D

! B

G .@ t / W G .@ t /R C

! B

R .@ t / W R .@ t /R:

Notice that, due to the presence of an elastic fluid-structure interface with
mass, the kinetic energy term E kin .t/ contains a contribution from the kinetic
energy of the fluid-structure interface k@ t k

2

L 2 ./ incorporating the interface inertia.
Furthermore, the elastic energy E el .t/ of the FSI problem accounts for the elastic
energy k@ z k

2

L 2 ./ of the interface. If an FSI problem between the fluid and a thick
structure was considered without the thin FSI interface with mass, these terms would
not be present. In fact, the traces of the displacement and velocity at the fluid-
structure interface of that FSI problem would not have been even defined for weak
solutions.

Proof. A formal calculation of the energy estimate for this class of problems
typically entails multiplying the fluid and structure equations in differential form
by the fluid and structure velocities, respectively, and performing integration by
parts. Integration by parts of the fluid equations takes into account the boundary
conditions, which are the conditions at the inlet and outlet boundary of the fluid
domain, and the conditions at the lateral boundary of the fluid domain. At the lateral
boundary of the fluid domain, the normal fluid stress is coupled with the structure
equations, and here is where the dynamic and kinematic coupling conditions come
into play. By taking these coupling conditions into account, the energy of the fluid
and the energy of the structure are coupled together into the total energy of the
coupled FSI problem.
More precisely, we first multiply equation ( 2.60 )by u ,integrateover F .t/,and
formally integrate by parts to obtain:
Z

F @ t u u C . u r/ u u C 2 F Z

2

F .t/ j D . u /j

F .t/

Z

.p I C 2 F D . u // n .t/ u D 0: (2.75)

@ F .t/

To deal with the inertia term we first recall that F .t/ is moving in time and that the
velocity of the lateral boundary is given by u j .t/ . The transport theorem applied to
the first term on the left-hand side of the above equation then gives:
Z Z Z

1
2

d
dt

1
2

2 2 u n .t/:

@ t u u D

F .t/ j u j .t/ j u j F .t/