Fluid–Structure Interaction in Hemodynamics: Modeling, Analysis, and Numerical Simulation - Fluid-Structure Interaction and Biomedical Applications - page 81

in constructing the proof of existence of solutions for this class of problems. In
this chapter we present a stable, convergent, modular scheme with precisely these
properties, called the Kinematically Coupled LJ-Scheme. This scheme was originally
constructed to study FSI problems with a single structural layer, modeled by the
cylindrical Koiter shell equations in [ 85 , 86 ], and then recently improved for higher
accuracy in [ 21 ] (Kinematically Coupled LJ-Scheme). Modifications of this scheme
can be applied to a much larger class of multi-physics problems associated with
FSI, such as FSI involving stent-artery-fluid interaction [ 122 ], FSI involving a
multi-layered elastic porous medium [ 23 ], and FSI involving a non-Newtonian fluid
[ 95 , 96 ].
In this chapter we present a general “recipe” describing the construction of the
main steps of such a scheme that can be used to:

• prove existence of weak solutions, and/or

• construct a numerical solver

to study a class of FSI problems that include:

• problems with viscoelastic and/or purely elastic structural models,

• problems with different coupling conditions (no-slip, slip),

• problems with nonlinear thin structure models,

• 2D and 3D scenarios.

An interesting new feature of the class of problems studied in this chapter is
the fact that the presence of a thin fluid-structure interface with mass regularizes
solutions of this class of FSI problems. More precisely, the energy estimates
presented in this chapter will show that the thin structure inertia regularizes
evolution of the thin structure, which affects the solution of the entire coupled
FSI problem. Namely, if we were considering a problem in which the structure
consisted of only one layer, modeled by the equations of linear elasticity, from the
energy estimates we would not be able to conclude that the fluid-structure interface
is even continuous. With the presence of a thin elastic fluid-structure interface
with mass (modeled, e.g., by the linear wave equation), the energy estimates imply
that the displacement of the thin interface is in H 1 ./, which, due to the Sobolev
embeddings, implies that the interface is Hölder continuous C 0;1=2 ./. The inertia
of the fluid-structure interface with mass serves as a regularizing mechanism for the
entire FSI problem. It will be shown in Sect. 2.7 that numerical simulations confirm
this behavior.
This is reminiscent of the results by Hansen and Zuazua [ 90 ]inwhichthe
presence of a point mass at the interface between two linearly elastic strings with
solutions in asymmetric spaces (different regularity on each side) allowed the proof
of well-posedness due to the regularizing effects by the point mass. More precisely,
they considered two elastic strings modeled by the linear wave equations, connected
by a point mass, with initial data of different regularity on the left or right side
of the point mass. They showed that the rough waves traveling through the point
mass, which served as an interface with mass between the two elastic strings, were
regularized due to the inertia effects of the point mass. See Sect. 2.7.7 for more