# Vibrations of Nonlinear Elastic Structure Excited by Compressible Flow
## 5 Realization of the Discrete Nonlinear Elasticity Problem
### Derivatives in the Case of the St. Venant-Kirchhoff Material
The strong coupling algorithm described in Section 4.3 with the prescribed tolerance \(10^{-5}\) is used. The prescribed tolerance was usually reached after two to three coupling subiterations. Figure 5 shows the airflow velocity field in the subglottal and supraglottal regions at five time instants of the VFs self-oscillation. In these time instants, different jet declination behind the channel constriction, i.e., the so-called \"Coanda effect\", can be observed, with the maximum flow velocity ca. 80 m/s. It corresponds to the Mach number \(\text{Ma}=0.23\) \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Layer** & \(\mathbf{E^{b}}\) & \(\mathbf{v^{b}}\) & \(\mathbf{\lambda^{b}}\) & \(\mathbf{\mu^{b}}\) \\ \hline
1. layer (orange) & \(12\cdot 10^{3}\) & 0.4 & 17,143 & 4285 \\ \hline
2. layer (yellow) & \(8\cdot 10^{3}\) & 0.4 & 11,430 & 2857 \\ \hline
3. layer (blue) & \(1\cdot 10^{3}\) & 0.495 & 33,110 & 335 \\ \hline
4. layer (red) & \(100\cdot 10^{3}\) & 0.4 & 142,857 & 35,714 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Prescribed material constants for the VFs model: Young modulus, Poisson ratio and Lamé parameters for different layers, ordered from the lower layer to the upper layer. See Figure 4 for the visualization of the corresponding subdomains. Figure 4: Nonhomogeneous model of VFs formed by four layers with triangulation. Modeled layers: 1. muscle, 2. ligament, 3. superficial lamina propria and 4. epithelium. Similarly as in Figure 5, we see the distribution of the pressure in Figure 6. The maximum air pressure is approximately constant in the subglottic part of the computational model. The minima of the pressure corresponds to centers of the vortices created in the supraglottal region and traveling to the outlet. Figure 7 shows the fluid pressure fluctuations in the middles of the inlet and outlet. The value vibrations are caused by the non-stationary flow behavior. In Figure 8, we can see the time dependence of the average values across the inlet and outlet. The average mean difference between the inlet and outlet pressure is around 1 kPa, which is in the range of subglottic pressure for ordinary phonation. Figure 5: Velocity field in the glottal region at five time instants (\(t=0.0074,0.0084,0.0094,\)\(0.0104,0.0114\,\)s) of the vocal folds self-oscillation. Figure 6: Distribution of the pressure in the glottal region at five time instants (\(t=0.0074,0.0084,0.0094,0.0104,0.0114\) s) of the vocal folds self-oscillation. Figure 9 shows the displacement of the top of the vocal fold in horizontal (\(u1\)) and vertical (\(u2\)) directions. The numerical simulation of the VF vibrations started from zero initial conditions and static fluid forces deformed the VFs statically, predominantly in the horizontal direction. This effect shifted the mean value of VFs self-oscillations downstream. The vibration amplitudes with frequency ca. 100 Hz in the horizontal direction are dominant. Figure 8: Average fluid pressure on the inlet \(\Gamma_{I}\) and on the outlet \(\Gamma_{O}\). Figure 7: The fluid pressure fluctuation at the middle point of inlet \(\Gamma_{I}\) and at the middle point of the outlet of the vocal tract. _Comparison of the FSI Results for Linear and Two Nonlinear Elasticity Models_  
This section will be devoted to the analysis of nonlinear elasticity models in comparison with linear ones. We compare the linear strain tensor \(\mathbf{e}\) and the nonlinear strain tensor \(\mathbf{E}\in\mathbb{R}^{2\times 2}\), defined by (32). For the linear elasticity, the stress tensor depends on the strain tensor \(\mathbf{e}=(e_{ij})_{i,j=1}^{2}\), and in the case of nonlinear elasticity, the stress tensor depends on \(\mathbf{E}=\mathbf{e}+\mathbf{E}^{\star}\), where \(\mathbf{E}^{\star}=(E_{ij}^{\star})_{i,j=1}^{2}\).