Biomedical Engineering Reference
In-Depth Information
improved this result in [
79
] to hold for a 2D elastic plate. These results were
extended to a more general geometry in [
106
], and then to the case of generalized
Newtonian fluids in [
105
], and to a non-Newtonian shear-dependent fluid in [
94
,
95
].
In these works existence of a weak solution was proved for as long as the elastic
boundary does not touch “the bottom” (rigid) portion of the fluid domain boundary.
Muha and Canic recently proved existence of weak solutions to a class of FSI
problems modeling the flow of an incompressible, viscous, Newtonian fluid flowing
through a cylinder whose lateral wall was modeled either by the linearly viscoelastic
or by the linearly elastic Koiter shell equations [
119
], assuming nonlinear coupling
at the deformed fluid-structure interface. The fluid flow boundary conditions were
not periodic, but rather, the flow was driven by the dynamic pressure drop data.
The methodology of proof in [
119
] was based on a semi-discrete, operator splitting
Lie scheme which we discuss later in this chapter, and which was also used in
[
85
] to design a stable, loosely coupled partitioned numerical scheme, called the
kinematically coupled scheme (see also [
21
]). Ideas based on the Lie operator
splitting scheme were also used by Temam in [
140
] to prove the existence of a
solution to the nonlinear Carleman equation.
Finally, we also mention the results in [
73
] where a free-boundary problem for
a steady flow of the incompressible, viscous fluid past a three-dimensional elastic
body was studied, and the results in [
18
] where the authors consider a rigid body
floating on the free surface of the fluid.
2.4.2
Literature on Numerical Simulation of FSI Problems
The development of numerical solvers for FSI problems has become particularly
active since the 1980s. Among the most popular techniques are the Immersed
Boundary Method [
57
,
66
,
80
-
84
,
111
,
117
,
127
,
128
] and the Arbitrary Lagrangian-
Eulerian (ALE) method [
52
,
92
,
93
,
104
,
110
,
132
,
133
]. We further mention the
Fictitious Domain Method in combination with the mortar element method or ALE
approach [
7
,
142
], and the methods recently proposed for the use in the blood flow
application such as the Lattice Boltzmann method [
56
,
59
,
99
,
100
], the Level Set
Method [
42
], and the Coupled Momentum Method [
65
].
Until recently, only monolithic algorithms seemed applicable to blood flow
simulations [
15
,
16
,
47
,
65
,
77
,
126
,
144
]. These algorithms are based on solving
the entire nonlinear coupled problem as one monolithic system. They are, however,
generally quite expensive in terms of the computational time, programming time,
and memory requirements, since they require solving a sequence of strongly coupled
problems using, e.g., the fixed point and Newton's methods [
31
,
47
,
62
,
92
,
115
,
126
],
or the Steklov-Poincaré-based domain decomposition methods [
48
].
The multi-physics features of the blood flow problem strongly suggest to employ
partitioned (or staggered) numerical algorithms, where the coupled fluid-structure
problem is separated into a pure fluid sub-problem and a pure structure sub-problem.
The fluid and structure sub-problems are integrated in time in an alternating way,
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