Data Assimilation in Cardiovascular Fluid–Structure Interaction Problems: An Introduction - Fluid-Structure Interaction and Biomedical Applications - page 457

ʓ f, 1

ʓ s, 1

ʣ s,in

ʩ f

ʣ f

ʩ s

ʣ s,ext

ʓ f, 2

ʓ s, 2

Fig. 6.10 Representation of the domain of the FSI problem: fluid domain on the left , structure
domain on the right

Problem Formulation

We consider a domain of a vessel (structure domain) perfused by blood (fluid
domain) as depicted in Fig. 6.10 . We make the simplistic assumption that the vessel
is (linearly) elastic, with the stress tensor s depending on the vessel displacement
as

s ./ 1 .r C .r/ T / C 2 .r/I;

where

E
2.1 C / ;

E
.1 C /.1 2/ ;

1 WD 2 WD

are the Lamé constants, I is the identity tensor, E is the Young's modulus, and is
the Poisson's ratio. For the sake of notation, we factor the Young's modulus E out
of the stress tensor, and so that we can write

1
2.1 C / .r C .r/ T / C .1 C /.1 2/ .r/I:

s D E Q s ; Q s WD

The vessel deforms under the stress coming from the blood, and in turn, the
elastic structure of the vessel affects the blood flow. This problem has been largely
investigated in other chapters of this topic (see also, e.g., [ 25 ]). For the sake of
numerical approximation of the problem, the problem is formulated on a frame of
reference moving with the physical wall of the artery and fixed on the artificial
boundaries (inflow/outflow). This approach is known as the Arbitrary Lagrangian
Eulerian (ALE) formulation, see, e.g., [ 20 , 42 ]. We write the problem according to