Biomedical Engineering Reference
In-Depth Information
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Wt
~
Wt
ʩ
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ʓ
t
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Wt
x
Wt
2
x
1
Fig. 5.3
Scheme of coupled fluid-structure problem
Adachi et al. [
1
] applied this method to tuning the vocal tract shape. Recently, a
theory of interaction between the source of sound in phonation and the vocal tract
filter was proposed by Titze [
94
]. The 3D FE models of the human vocal tract for
vowels were developed by Švancara et al. [
85
] and Vampola et al. [
96
,
97
] and based
on the MRI and computer tomography measurements during phonation.
Mathematical models for the human phonation process are valuable tools for pro-
viding insight into the basic mechanisms of phonation and in future could help with
surgical planning, diagnostics, and voice rehabilitation. Our goal in this monograph
is to present our original, recently developed numerical methods based on the finite
element simulation of 2D incompressible and compressible laminar viscous flow
described in the glottal region by the Navier-Stokes equations in interaction with a
compliant tissue of the human vocal folds. The vocal folds are either modelled by a
2D elastic layered structure or as a vibrating rigid body. Some methods and results
presented here were obtained in the works [
17
,
33
,
37
,
47
,
48
,
83
,
84
].
5.2
Incompressible Flow in Time-Dependent Domains
The mathematical description of the interaction of incompressible flow and vocal
folds consists of equations of motion for the vocal folds coupled with the incom-
pressible Navier-Stokes equations via interface conditions. First, for simplicity and
clarity of aeroelastic principles, the vocal folds are modelled as rigid bodies with two
degrees of freedom, elastically supported in the glottis, and then as linear 2D elastic
continuum. The solution of the 2D dynamic elasticity equations for the vocal fold
tissue is realized with the aid of conforming finite elements on a reference domain
of
t
. The flow in a time-dependent domain
t
(see Fig.
5.3
) is treated with the
aid of ALE method. The incompressible Navier-Stokes equations are approximated
by the stabilized finite element method. The time discretization based on a semi-
implicit linearized scheme is described. The solution of the coupled problem of
FSI is realized by coupling algorithms enforcing the interface conditions at a fluid-
structure interface.
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