Fluid–Structure Interaction in Hemodynamics: Modeling, Analysis, and Numerical Simulation - Fluid-Structure Interaction and Biomedical Applications - page 186

the Lie splitting scheme, also known as the Marchuk-Yanenko scheme. The splitting
discussed in this chapter deals successfully with the added mass effect which is
known to be responsible for instabilities in loosely coupled Dirichlet-Neumann
schemes for FSI problems in which the density of the structure is comparable to
that of the fluid, which is the case in blood flow applications. Particular attention
was payed to a multi-physics FSI problem in which the structure is composed of
multiple structural layers. Problems of this kind arise, for example, in modeling
blood flow through human arteries which are composed of several layers, each
with different mechanical characteristics and thickness. A benchmark problem was
studied in which the structure consists of two layers: a thin layer which is in contact
with the fluid, and a thick layer which sits on top of the thin layer. The thin layer
serves as a fluid-structure interface with mass. Both analytical (existence of a weak
solution) and numerical results were studied for the underlying benchmark problem.
In particular, it was shown that the proposed scheme converges to a weak solution to
the full nonlinear fluid-multi-layered structure interaction problem. Two academic
examples were considered to test the performance of the numerical scheme.
The analytical and numerical methods presented here apply with slight modifi-
cations to a larger class of problems. They include, for example, a study of FSI with
one structural layer (thin [ 20 , 21 , 85 ], or thick [ 22 ]), FSI with poroelastic structures
[ 23 ], FSI between a mechanical device called stent, arterial wall and fluid [ 122 ], and
FSI involving a non-Newtonian fluid [ 94 - 96 ].
This chapter provides the basic mathematical tools for further development of
analytical and computational methods based on the Lie operator splitting approach,
to study various multi-physics problems involving FSI.
Acknowledgements The research of the authors has been supported in part by the National
Science Foundation under the following grants: NIGMS DMS-1263572, DMS-1318763, DMS-
1311709, DMS-1262385, DMS-1109189, and by the Texas Higher Education Board under grant
ARP-003652-0023-2009.
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