## 3 Results and Discussion  
### Baseline Case  
In this section, we present the fluid-structure-acoustic interaction (FSAI) simulation results on the original model without introducing subglottic stenosis (SGS). We compared the predicted key phonatory characteristics with the typical physiological range to ensure a reasonable representation of the current model in relation to normal human phonation. The results also serve as baseline values to examine the effects of SGS in the next section. 3.1.1. Baseline Case: Vocal Fold Vibration and Glottal Flow Dynamics  
The vibration pattern of the vocal folds at different instances during a single cycle is shown in Figure 4a, along with two-dimensional cross-sections of the folds in the x-z and y-z planes for additional clarity. During typical human phonation, the vocal folds exhibit alternating convergent-divergent glottal shapes, which promote efficient energy transfer from the airflow to sustain vibration [35]. This alternating pattern is clearly observed in the y-z plane, where the airflow initially forces the vocal folds open into a convergent shape (t = 0.11 T), while also pushing the superior surface of the folds upward. At t = 0.4 T, the glottal opening reaches its maximum, forming a straight channel. Following this, the elastic recoil force causes the folds to close, creating a divergent shape (t = 0.54 T), before they eventually reach full closure. A proper orthogonal decomposition (POD) analysis was further conducted to break down the vibrations into their primary eigenmodes, representing the highest energy contributions [36]. The first mode captures the glottal opening-closing motion, accounting for 66% of the total vibration energy. The second mode corresponds to the convergent-divergent motion of the glottis, contributing 26% of the total energy. Together, these two modes encompass 92% of the overall vibration energy, while the remaining 8% is distributed across higher-order modes. Figure 4: **(a)** The vibration pattern of the vocal fold at different instances during one vibration cycle in the baseline case. The two-dimensional cross-sections of the vocal folds in x-z and y-z planes are also depicted at the same time instances; **(b)** the phase-averaged minimum glottal area waveform and **(c)** the phase-averaged volumetric flow rate waveform over one vibration cycle in the baseline case. The phase-averaged glottal area and volume flow rate are plotted in Figure 4b,c. The glottal area is defined as the minimum cross-sectional area of the glottal channel, while the glottal volume flow rate is measured at the glottal exit. Phase-averaging is performed by dividing each steady cycle into 1000 intervals, and within each interval, values from multiple cycles are averaged. As shown in Figure 4b, the opening phase occurs from 0 to 0.38 T, reaching a maximum glottal area of 0.138 cm\({}^{2}\). The closing phase spans from 0.38 T to 0.69 T, with full closure taking place between 0.69 T and T. During closure, a small gap of approximately 0.02 cm\({}^{2}\) remains, a result of the artificially enforced minimum glottal gap, which is necessary for the successful operation of the ghost cell algorithm in the flow solver. The volume flow rate waveform captures the typical pattern of human phonation, characterized by a slow rise, a rapid decline, and a sustained closure period. The peak and mean values are 388 mL/s and 198 mL/s, respectively. It should be noted that the non-zero flow rate during the glottal closure is due to the enforced artificial minimum gap in the solver. Several key phonatory parameters are further calculated based on the glottal area and volume flow rate. The open quotient \(\tau_{o}\), defined as the ratio of the duration of glottal opening to the full cycle duration, is 0.69, which aligns with typical values for normal phonation [35]. The fundamental vibration frequency of the vocal folds is determined to be 154 Hz from a Fourier transform of the glottal area time history.