# Voice Simulation: The Next Generation
## 2 Materials and Methods--Criteria for Selecting Simulation Models  
It is probably wise to distinguish between a _simulation model_ and a _computational prototype_. A model usually simplifies the real world. It does not need to be a virtual representation that can replace the original system in all its dimensionality and all its complexity. It is often a caricature of a phenomenon or feature of interest, specific to a question. Finally, and perhaps most importantly, a model has the potential of being predictive of what cannot be observed or measured. In that sense, a model is highly educational. A computational prototype can have all these features, but it attempts to include most of the structural detail with equal weighting so that a physical implementation is possible. It does not reduce complexity non-uniformly to feature a specific phenomenon. Some preliminary considerations are appropriate for selecting a simulation model. How accurate do the results need to be to answer the question? Sometimes ratios of variables are good enough, such as ratios of impedances, ratios of physical dimensions, or ratios of energies. Much of current simulation is based on imaging, from which ratios of physical dimensions are much easier to obtain than absolute dimensions. Further, if the simulation contains a parameter with questionable accuracy, a sensitivity analysis should be done on this parameter to avoid contamination of the results. For intelligent interpretation of simulation results, a simplified analytical analysis can be done to determine how variables relate to each based on dimensional analysis. For this reason, low-dimensional models are insightful for guiding high-dimensional computational prototypes. All digital models require decisions about temporal and spatial resolution. The Courant Criteria for time and space discretization are useful when wave propagation in elastic media (air or tissue) is involved. In bounded media, modes of vibration are defined by standing waves, which in turn are described by wavelengths. The Courant Criteria are:  
\[\Delta\mathrm{x}<\lambda/20\qquad\text{for spatial accuracy} \tag{1}\]  
\[\Delta\mathrm{t}<\Delta\mathrm{x}/2\mathrm{c}\qquad\text{for stability} \tag{2}\]  
The \(\lambda/20\) spatial discretization is considered an intermediate mesh for quantifying a full period of a sinusoid, while \(\lambda/10\) is considered a course mesh and \(\lambda/30\) a fine mesh, e.g., [1]. The \(\lambda/20\) rule provides on the order of 1% accuracy in sinusoidal wave displacement at all points. For example, if only one half-wavelength is of interest in vocal fold vibration in both the anterior-posterior and inferior-superior directions [known as the (1,1) normal mode on the surface], then 10 sections, or 10 finite elements, are needed for sufficient accuracy in the anterior-posterior direction. If the elastic wave propagates at 100 cm/s in the vertical direction and the vocal fold thickness is 0.5 cm, the sampling interval \(\Delta t\) for stability is 0.00025 s. Mesh and time-sampling independence, often mandated in computational prototypes, can sometimes be relaxed if approximate solutions are sufficient. This is the case with all low-dimensional vocal fold models. Another example is spatial discretization in vocal tract wave propagation modeling. The number of sections needed depends on the number of modes (formants) of interest. Figure 1 shows that 15 sections are sufficient for the first formant, while at least 40-50 sections are needed for an equivalent accuracy for the fourth formant. Airflow through the glottis is characterized by some fundamental dimensionless numbers [3,4]. The _Strouhal number_ (St) pertains to oscillatory flows.