Fluid–Structure Interaction in Hemodynamics: Modeling, Analysis, and Numerical Simulation - Fluid-Structure Interaction and Biomedical Applications - page 103

In the rest of this chapter we will be working with the linearized Saint Venant-
Kirchhoff model.

2.3 A Benchmark Problem

In this section we focus on a benchmark problem in fluid-multi-layered structure
interaction. The problem consists of studying FSI between an incompressible,
viscous fluid, and a structure consisting of two layers: a thin layer modeled by the
Koiter shell equations, and a thick layer modeled by the equations of linear elasticity.
The methods presented in this chapter work for an entire class of problems in which
the thin structural layer can be described by either the full cylindrical linearly elastic
Koiter shell model, described in Examples 1 and 2 of Sect. 2.2.1 , the linearly elastic
membrane equations, presented in Example 2 of Sect. 2.2.1 ,thenonlinearly elastic
Koiter membrane/shell model, described in Example 3 of Sect. 2.2.1 , the cylindrical
linearly viscoelastic Koiter shell model, presented in Example 4 of Sect. 2.2.1 ,or
the elastic string model described in Example 5 of Sect. 2.2.1 .

2.3.1 The Model Equations

The thin structural layer is modeled by the reduced equations of linear

(visco)elasticity, discussed in Sect. 2.2.1 , which take the general form:

K h @ 2

@t 2 C L el ./ C L vis @

THIN STRUCTURE W

@t D f; on .0;T/;
(2.58)

These equations are defined on the reference domain which is a cylinder of radius R:

3

Df.R cos;Rsin ; z / 2 R W z 2 .0;L/; 2 .0;2/g:

L el may be a linear or a nonlinear operator modeling
the elastic properties of shells or membranes, and

As discussed in Sect. 2.2.1 ,

L visc denotes a linear operator

L vis may be equal to the zero
operator. The methodology presented in this chapter is robust in the sense that it can
be applied to solving both the viscoelastic and purely elastic thin structure models.
The thick structural layer is modeled by the equations of linear elasticity,
discussed in Sect. 2.2.2 :

modeling their viscoelastic properties. Operator

s @ tt d Dr S ; on S .0;T/; where
S D .r d C .r d / T / C .r d / I :

THICK STRUCTURE W (2.59)