Biomedical Engineering Reference
In-Depth Information
5.2.4
Numerical Solution of the Nonlinear Discrete Problem
Oseen Linearization Process
The nonlinear discrete problem (
5.25
) is solved on each time level t
n
C
1
with the aid
of the linearized Oseen iterative process
a.U
.`
h
;U
.`
C
1/
;V
h
/ C
L
h
.U
.`
h
;U
.`
C
1/
;V
h
/ C
P
h
.U
.`
C
1/
;V
h
/ (5.29)
h
h
h
D f.V
h
/ C
F
h
.V
h
/
for all V
h
2 X
h
Q
h
;
where we start from the initial approximation U
.0/
h
D .v
n
; p
n
/ or U
.0/
h
D .2v
n
v
n
1
;2p
n
p
n
1
/. Numerical experiments show that it is usually enough to compute
5-8 Oseen iterations on each time level.
Solution of the Linear Algebraic System
The solution of the linear algebraic system equivalent to (
5.29
) can be realized by
the direct solver UMFPACK [
19
], which works sufficiently fast for systems with
up to 10
5
equations. For larger systems it is necessary to apply more robust and
efficient iterative techniques, such as the domain decomposition approach and/or
the multigrid method.
5.3
Structural Models
5.3.1
Aeroelastic Model of Vocal Folds Vibration with Two
Degrees of Freedom
Original theoretical model for vibration onset of the vocal folds in the airflow
coming from the human subglottal tract allows studying the influence of the physical
properties of the vocal folds (e.g., geometrical shape, mass, and damping) on the nat-
ural frequencies, mode shapes of vibration, and the thresholds of instability [
40
,
41
].
The model of the vocal fold is designed as a simplified dynamic system with
two degrees of freedom (rotation and translation) vibrating on an elastic foundation
in the wall of a channel conveying air. The phonatory airflow is approximated by
unsteady one-dimensional flow theory for inviscid incompressible fluid. A generally
defined shape of the vocal fold surface is considered for expressing the unsteady
aerodynamic forces in the glottis. The parameters of the mechanical part of the
model, i.e., the mass, stiffness, and damping matrices are related to the geometry and
material density of the vocal folds as well as to the fundamental natural frequency
and damping known from the experiments. The coupled numerical solution yields
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