# Examining the Quasi-Steady Airflow Assumption in Irregular Vocal Fold Vibration
## 2 Methods
### Simulation Setup and Case Summary  
The computational model setup is shown in Figure 3. Figure 3a is the setup with a supraglottic tract. The vocal fold surfaces were immersed into a 1.5 cm \(\times\) 21.8 cm \(\times\) 1.5 cm rectangular computational domain, which consists of a 17 cm-long supraglottic vocal tract, a 3.6 cm-long trachea, and a pair of 0.8 cm-thick vocal folds. The computational domain was discretized by a 128 \(\times\) 128 \(\times\) 96 (x \(\times\) y \(\times\) z) Cartesian grid, and grid independence was achieved with this grid. Dirichlet boundary conditions for pressure of 1.0 kPa and 0 kPa were applied at the inlet and exit of the computational domain, respectively. No-slip and no-penetration boundary conditions were imposed on the trachea and vocal tract walls. Figure 3b is the setup without the supraglottic tract. The vocal folds were located at the bottom center of a cubic computational domain, 30 cm \(\times\) 30 cm \(\times\) 30 cm, mimicking a large open environment. A 256 \(\times\) 128 \(\times\) 128 (x \(\times\) y \(\times\) z) Cartesian grid was used to discretize the computational domain. The grid around the vocal folds was 164 \(\times\) 72 \(\times\) 92 (x \(\times\) y \(\times\) z), which provided a similar grid resolution as the setup with the supraglottic tract. Similar to the former setup, constant pressure boundary conditions were applied at the entrance of the trachea and top surface of the computational domain,0 kPa and 0 kPa, respectively. The left, right, front, and back surfaces, as well as the bottom surface outside the trachea area, were treated as no-slip and no-penetration boundaries. For both setups, the surface of each vocal fold was discretized into 71,400 triangular elements with 0.01 cm resolution. In the dynamic simulations, a 0.006 cm minimum space between the left and right medial surface in contact area was enforced, which was necessary for the success of airflow simulation. In the dynamic simulations, the time steps of 1.184 \(\times\) 10\({}^{-6}\) s and 1.016 \(\times\) 10\({}^{-6}\) s were used for the case of 100 Hz and 500 Hz vibration frequency, respectively. Such time steps were chosen for (1) outputting the results at the specified 16 phases and (2) complying with the Courant-Friedrichs-Lewy condition. The same time steps were used for pseudo-static and quasi-steady simulations. For a few cases that experienced numerical convergence problems, the time step was appropriately reduced. Time independence was achieved for all three types of simulations. For each dynamic simulation, glottal airflow was simulated for two cycles, reaching steady cycles based on the observation that cycle-to-cycle flow variations were very small. For each pseudo-static or quasi-steady simulation, glottal airflow was calculated until a sustained steady state was reached. The simulations were run on the XSEDE COMET cluster, which used Intel's Xeon Processor E5-2600 v3 family. One hundred and twenty-eight processors were used for each dynamic or pseudo-static simulation, and sixty-four processors were used for each quasi-steady simulation. Thecomputational time of the dynamic simulations ranged from 40 h to 240 h, depending on the vibration frequency and number of grids. The average computational time for a pseudo-static or quasi-steady simulation was around 20 h.  
The total number of simulations included in this study was 200, with 8 dynamic, 128 pseudo-static, and 64 quasi-steady cases. For each of the two sets of glottal wall motions, setups were implemented with two different supraglottic configurations (with and without the supraglottic tract) under two different vibration frequencies (100 and 500 Hz), which adds up to the 8 dynamic cases. Based on each dynamic case, 16 pseudo-static simulations, each with a glottal wall velocity extracted from the dynamic simulation, were built, resulting in 128 pseudo-static cases.