Influence of numerical model decisions on the flow-induced vibration of a computational vocal fold model

Abstract

Computational vocal fold models are often used to study the physics of voice production. In this paper the sensitivity of predicted vocal fold flow-induced vibration and resulting airflow patterns to several modeling selections is explored. The location of contact lines used to prevent mesh collapse and assumptions of symmetry were found to influence airflow patterns. However, these variables had relatively little effect on the vibratory response of the vocal fold model itself. Model motion was very sensitive to Poisson’s ratio. The importance of these parameter sensitivities in the context of vocal fold modeling is discussed.

1. Introduction

Typical voiced speech is produced during phonation, the process in which airflow from the lungs drives vocal fold (VF) flow-induced vibration. The VFs are located within the larynx and consist of several layers of tissue of varying degrees of stiffness (see Fig. 1). The glottis is the space between the vocal folds and is bordered inferiorly and superiorly by the subglottic and supraglottic regions, respectively. Phonation involves a complex coupling between VF tissue dynamics, respiratory airway aerodynamics, and acoustics. During phonation the glottis periodically opens and closes on the order of hundreds of times per second, creating pulsatile pressure fluctuations in the supraglottic region. These pressure fluctuations are the primary source of sound for typical voiced speech.

Fig. 1.

Sagittal (left) and coronal (right) views of the larynx (adapted from Gray’s Anatomy of the Human Larynx, www.bartleby.com, used with permission).

The primary goal of voice production research is to enhance our understanding of voice production physics in order to lead to improved prevention, diagnosis, and treatment of voice disorders. In vivo, excised, synthetic, and computational approaches are commonly used. Computational models have advantages of ease of parameterization, accessibility of data, and the ability to study isolated phenomena.

Computational models of phonation have varied in complexity, scope, and accuracy, but have generally included coupling between a model of the VFs and a form of airflow simulation. One of the earliest simulations was that of a system of masses, springs, and dampers to represent the VFs, coupled with a Bernoulli flow solver [1]. Continuum models have been subsequently developed that more accurately represent tissue shape and material properties and include more sophisticated flow models [2–4]. These have usually been based on finite volume or finite element methods, although recently, models using the immersed boundary method have been developed [5].

Many choices must be made when developing, defining, and implementing a computational model. Clearly, for computational models to benefit the voice research community, a necessary level of accuracy, precision, and relevance must be achieved. The following is a brief discussion about modeling choices that have the potential to influence these outcomes. These choices deal with limiting mesh movement, assuming symmetry, and defining material properties.

1.1. Contact line location

During vibration opposing VFs typically collide and remain in contact for a significant duration of each period. This collision temporarily closes the glottis, momentarily halting airflow. In both control volume and immersed boundary method simulations, contact lines or planes have been used to prevent complete glottal closure and associated algorithm failure [4,6–8]. The contact line acts as a barrier through which the solid domain cannot pass. The minimum glottal gap is thus always nonzero and flow leakage during “closure” occurs. In order to reduce the minimum glottal width and minimize flow leakage, yet avoid solver failure, the contact line is typically placed as close to the centerline as possible. While the practice of including contact lines is often used, specific attention has not been given to the effect of its location on predicted VF model response.

1.2. Poisson’s ratio

Another consideration of importance in computational VF modeling is the choice of material properties. Several studies have focused on the influence of tensile or shear modulus, but few have directly considered Poisson’s ratio effects. Human tissue is generally considered to be incompressible at phonation frequencies [9,10]. Depending on the material modeling assumptions, this corresponds to various Poisson’s ratios. In an isotropic material the upper limit of 0.5 corresponds to incompressibility [11]. However, human VF tissue is anisotropic and thus the Poisson’s ratios may exhibit directional dependence. This is often handled by use of a transversely isotropic material model. If the longitudinal Poisson’s ratio (ν′) in this type of material is equal to zero, the transverse Poisson’s ratio (ν) in the can be greater than 0.5, with a limiting value of 1.0 in the incompressible case.

Notwithstanding the incompressibility of VF tissue, computational models have often been defined using Poisson’s ratios that are lower than the corresponding incompressible value. In some instances this has been done to simulate the response of synthetic models, although in many cases the synthetic model material properties have not been measured. In addition, the material properties of a synthetic model are limited by the materials available and do not exactly match those of an actual larynx. One such case is in the use of multi-compound silicone in synthetic models in which the Young’s modulus of the material can be altered over a wide range by selecting a different compound mixing ratio before it cures [4,12,13]. This allows for models with Young’s moduli that approach those measured in human tissue. However, the Poisson’s ratio (ν) of silicone is not necessarily the same as that of human tissue, and cannot generally be controlled.

In computational models with isotropic materials, Poisson’s ratios of 0.45 [4], 0.3 [8], and 0.49 [14] have been used. Tao and Jiang [15,16] and Zheng et al. [17] used a transversely isotropic material model which utilized a longitudinal Poisson’s ratio of zero and a transverse Poisson’s ratio of 0.3 in Ref. [15,16] and 0.9 in Ref. [17]. Transversely isotropic materials with equal transverse and longitudinal Poisson’s ratios have also been used, in one case with a value of 0.3 [6], and in another case with values for different layers ranging from 0.45 to 0.76 [18]. This range in Poisson’s ratios that has been used in computational VF modeling, as well as differences between the Poisson’s ratios of synthetic model materials and real tissue, provide the motivation for the present study.

1.3. Symmetry assumption

As the flow leaves the glottis, it separates from the VF surface and forms a jet in the supraglottic region. Simulations and experiments have shown that the glottal jet may vary significantly from one cycle to the next [17,19–24] and it has been suggested that such variations could influence speech production [13,19,21,25]. Further, it has been found that this glottal jet often exhibits lateral asymmetry, diverting to one side or the other in the supraglottic region [5,19,20,24]. In computational simulations researchers often employ symmetry assumptions for computational efficiency [4,7,14,26,27]. These provide insight into VF motion and aerodynamics, but do not capture asymmetry in the supraglottic jet or of VF vibration. Researchers have used both half-models with symmetry assumptions, as well as full geometrically-symmetric models where asymmetry of flow and motion is allowed to develop [5,8,20]. However, the degree to which assuming symmetry could influence predicted model response has not been directly studied.

The purpose of this paper is to explore the influence of modeling choices associated with symmetry, contact line location, and Poisson’s ratio on computational vocal fold model response. The influence of these three modeling choices on predicted VF vibration and accompanying flow dynamics is quantified by presenting results of simulations in which these parameters were independently varied. In the following sections the models are described, simulation outputs are presented and discussed, and conclusions regarding modeling choices that can be expected to yield reasonable results are given.

2. Methods

2.1. Computational model

The commercially-available finite element program ADINA was used to simulate the flow-induced vibration of a two-dimensional VF model. This package has been used in previous studies of VF vibration and of other biological flow-structure interaction problems [4,14,28,29]. Fig. 2 illustrates the fluid and solid domain geometries, including dimensions and inlet, exit, and wall boundary condition labels. The geometry was oriented in the yz-plane such that flow was primarily in the y-direction (see Fig. 2). Further details (e.g., meshing, complete geometry definitions) can be found in Ref. [14,30].

Fig. 2.

Airway geometry and boundary conditions.

2.1.1. Fluid domain

The flow was governed by the two-dimensional, laminar, incompressible Navier–Stokes and continuity equations. The fluid density and viscosity values were 1.2 kg/m³ and 1.8 × 10⁻⁵ Pa s, respectively. The interface between the VF and the airway was treated with a fluid-structure interaction (FSI) boundary condition that enforced consistent stress and displacement within the adjoining fluid and solid domains along the wetted interface. The fluid domain was modeled using flow-condition-based interpolation (FCBI) elements [31]. Unless otherwise noted symmetry about the y-axis was assumed and only one vocal fold and one half of the fluid domain was modeled.

2.1.2. Solid domain

The VF solid domain is illustrated in Fig. 3 with reference dimensions (for further details, see [14]). The model included four layers of differing properties: body (muscle), ligament, superficial lamina propria, and epithelium. The model outer geometry was based on the “M5” geometry of Scherer et al. [32] with a convergent included angle of 2° between the medial surface of one fold and the medial plane, for a total intraglottal included angle of 4°. The epithelium layer was 0.05 mm thick.

Fig. 3.

Computational vocal fold solid domain. Radii values (mm) were r₁ = 1.5, r₂ = 0.8, r₃ = 1.0, and r₄ = 0.3. The glottal gap width (G), contact line locations, and minimum allowed gap width (G_min) are exaggerated for visual clarity.

Each VF had approximately 50 × 10³ quadrilateral elements and 10⁵ nodes. Solid domain elements were mixed interpolated elements. The lateral edge of each VF was rigidly fixed. The wetted boundary (epithelial layer) was treated with the same FSI boundary condition as in the fluid domain. To prevent complete collapse of the fluid domain mesh between the vocal folds during collision, a pair of contact lines was utilized in the solid domain. In the symmetry and Poisson’s ratio studies, these lines were located such that the minimum gap width (labeled G_min in Fig. 3) was 50 μm. In the contact line studies, the minimum gap width was varied between 1 and 50 μm. The initial gap width between vocal folds was 100 μm.

All materials were isotropic and had a density of 1070 kg/m³. All layers were modeled as hyperelastic Ogden solids using a total Lagrangian formulation that allowed for large displacement and large strain [33]. The epithelium was modeled using a linear stress-strain relation with a tangent modulus of 50 kPa. For the other layers nonlinear stress-strain relations were given by

$$\sigma (\epsilon )=A(e^{B\epsilon }-1),$$ (1)

where σ is stress, ε is strain, and B = 10.5. A for each nonlinear layer was chosen such that the tangent modulus values at a strain of ε = 0.05 were 0.4 × 10³, 2 × 10³, and 25 × 10³ Pa for the superficial lamina propria, ligament, and body layers, respectively. The high stiffness of the epithelium allowed for an extremely soft cover layer that produced a pronounced mucosal wave. Rayleigh damping with coefficients of α = 56.549 and β = 3.979×10⁻⁵ was used, corresponding to a damping ratio on the order of 0.05 between the frequency range of 100 and 300 Hz.

For each layer the Poisson’s ratio was 0.49 in the symmetry and contact line studies and between 0.4 and 0.49999 in the Poisson’s ratio study. In the Ogden material model these variations in Poisson’s ratios were approximated by varying the bulk modulus according to the small strain relationship

$$\kappa =\frac{E}{3(1-2\nu )},$$ (2)

where κ is bulk modulus, E is Young’s modulus, and ν is Poisson’s ratio. The value of E used to define κ for each layer was the respective layer’s tangent modulus at a strain of 0.05.

2.2. Variable definitions

In the following sections, the following simulation output quantities are used to compare results:

• Fundamental frequency, F₀. The frequency calculated from the inverse of the period of the glottal width data. This was averaged over 10 periods.

• Glottal width, G. The minimum distance between opposing VFs (labeled in Fig. 3).

• Maximum glottal width, G_max. The maximum glottal width, G, over one cycle. This was averaged over 10 periods.

• Average glottal width, G_avg. The average glottal width over one cycle. This was averaged over 10 periods.

• Open quotient, OQ. The fraction of a cycle during which the glottis was not “closed” (i.e., when G > G_min). This was averaged over 10 periods.

• Flow rate, Q. Calculated as Q = 0.015∫vdz where v is the fluid velocity in the superior (y) direction, measured at y = 1.3 cm (just downstream of the glottis). The factor of 0.015 m was used as a rough approximation of the effective three-dimensional flow rate based on the two-dimensional velocity field.

• Maximum flow rate, Q_max. The maximum flow rate, Q, over one cycle. This was averaged over 10 periods.

• Average flow rate, Q_avg. The average of the flow rate, Q, over one cycle. This was averaged over 10 periods.

2.3. Numerical verification

Numerical verification of grid and time step size independence was performed. To ensure time step independence, simulations with time step sizes of 1.25 × 10⁻⁵ and 2.5 × 10⁻⁵ s were performed. The glottal width waveforms over the first 0.05 s were compared and found to be nearly graphically indistinguishable (see Fig. 4). Cases with 68,480 and 259,440 elements in the fluid domain (double the grid density in each dimension, both vocal folds were included without the symmetry assumption) were simulated. The plot of glottal width waveform for the refined mesh was indistinguishable from that of the original grid spacing (Fig. 4). In addition, several other output measures (F₀, G_max, G_avg, OQ, Q_max, and Q_avg) from the three simulations were numerically compared and are listed in Table 1. Variations between cases were about 1% for all variables, with the exception of the Q_max decrease by 8.4% with the smaller time step. Consequently, the simulations reported below were performed using the 68480-element grid and the 2.5 × 10⁻⁵ s time step size.

Fig. 4.

Grid and time step independence study results: — Original grid and time step, - - - finer grid, ··· smaller time step. The three curves are nearly graphically indistinguishable.

Table 1.

Frequency (F₀), maximum glottal width (G_max), average glottal width (G_avg), open quotient (OQ), maximum flow rate (Q_max), and average flow rate (Q_avg) from the grid and time step independence studies. These variables are defined in Section 2.2.

Case	F0 (Hz)	Gmax (mm)	Gavg (mm)	OQ	Qmax (ml/s)	Qavg (ml/s)
Original	234.7	0.829	0.298	0.744	368.1	136.6
Fine grid	234.2	0.827	0.300	0.746	367.0	138.9
Small dt	235.0	0.832	0.298	0.740	337.2	136.8

2.4. Case studies

2.4.1. Contact line location

The influence of contact line location on airflow and VF motion measures was explored. For computational efficiency, only one half of the domain was modeled, with a symmetry condition imposed on the glottis centerline (y-axis; see Fig. 2). The supraglottic width was W= 18.9 mm(9.45 mm half-width). Eight cases focused on the effect of contact line location with values of G_min ranging from 1 to 50 μm.

2.4.2. Poisson’s ratio

The Poisson’s ratio studies were performed using the same fluid and solid domain definitions described in Section 2.4.1. G_min was 50 μm. Twelve cases with Poisson’s ratios ranging from 0.4 to 0.49999 were studied.

2.4.3. Symmetry assumption

One simulation was performed in which the symmetry condition was removed and a full airway with two vocal folds was modeled. For this case, G_min was 50 μm and the Poisson’s ratio was 0.49. The downstream width was W= 18.4 mm, or about 3% narrower than the contact line and Poisson’s ratio studies; previous studies have shown that this small of a change in supraglottic width would not yield significant differences in model response [34]. In the symmetry studies results section (Section 3.4), “half model” is used to denote the model with the symmetry assumption, whereas “full model” denotes the model with both VFs and no symmetry assumption.

3. Results and discussion

3.1. Typical model response

The outlines of the vocal folds at various phases of a single cycle are shown in Fig. 5. Shown are the results of the full model used in the symmetry study. The model response compares favorably with that of the human vocal folds. Qualitatively, an alternating convergent/divergent intraglottal profile during opening/closing and accompanying superiorly-traveling wave along the medial surface can be seen. These features are well-known characteristics of human vocal fold motion. Quantitative agreement was also good. The frequency was 237 Hz. The “closed” portion of the cycle (phases G–A), where the minimum glottal width was at a constant value of G_min (50 μm in this case), lasted for just under one-quarter of the cycle, corresponding to an open quotient near 0.75. These compare favorably with observations of human phonation; for example, Baken and Orlikoff [35] reported a mean OQ of 0.71, with a range of 0.51 to 1.0, for normal speakers with a fundamental frequency of 225 Hz. G_max was 0.837 mm, similar to typical values in humans of approximately 1 mm [36]. From data of y-location of minimum glottal width vs. time, it was also possible to calculate the velocity of the mucosal wave. The wave velocity for this case was 0.85 m/s, which is in the range of 0.5 to 2.0 m/s for excised canine larynges [37].

Fig. 5.

Vocal fold profiles at various phases of one cycle. Light gray denotes undeformed profiles. Phases A through H marked for reference.

3.2. Contact line location

Fig. 6 shows glottal width waveforms for contact lines ranging from 1 to 50 μm for the first 0.05 s of simulation time. All waveforms remain nearly perfectly in phase with each other, with differences primarily only seen during closure. This can be seen in more detail in the phase-aligned glottal width and flow rate waveforms in Fig. 7. The limiting values of G_min were observed for each case. However, the glottal width for the remainder of the cycle was largely unaltered by variations in G_min. Inspection of vocal fold profiles (not shown) revealed vocal fold profile shape during vibration was also only minimally affected by G_min [30].

Fig. 6.

Glottal width vs. time for four contact line cases.

Fig. 7.

Glottal width (left) and flow rate (right) vs. normalized time for four contact line cases.

Values for the quantitative measures introduced in Section 2.2 are listed in Table 2. Less than 1% variation occurred in F₀, G_max, and Q_avg values between all cases. G_avg increased by 20 μm (7.3%) from the 1 μm case to the 50 μm case. The open quotient decreased from 0.795 for the 1 μm case to 0.718 for the 50 μm case, a change in 9.7%. Q_avg increased by 6.8 ml/s (5.3%) from the 1 μm case to the 50 μm case. The slight variations in glottal width for the contact line cases led to similar changes in the flow rate waveforms (see Fig. 7). The phases of the cycle and the degree to which the flow rate was affected was consistent with that seen in the glottal width waveforms.

Table 2.

Measured values of frequency (F₀), maximum glottal width (G_max), average glottal width (G_avg), open quotient (OQ), maximum flow rate (Q_max), and average flow rate (Q_avg) for the contact line study.

Gmin (μm)	F0 (Hz)	Gmax (mm)	Gavg (mm)	OQ	Qmax (ml/s)	Qavg (ml/s)
1	235.6	0.839	0.273	0.795	363.6	129.0
3	236.0	0.839	0.275	0.789	363.5	129.7
5	236.0	0.839	0.276	0.789	363.6	130.1
7	236.1	0.839	0.277	0.788	363.6	129.2
10	236.1	0.839	0.278	0.776	363.6	129.7
20	236.4	0.838	0.283	0.765	363.8	131.3
30	236.7	0.836	0.286	0.753	363.8	132.8
50	236.5	0.836	0.293	0.718	363.9	135.8

The change in the intraglottal flow as contact line location is shown in Fig. 8. A steady decrease in the velocity was seen with decreasing G_min during the closed phase of the cycle as G_min decreased from 50 μm to 7 μm. With G_min = 50 μm the leakage velocity during the closed phase was 23.1 m/s, while with G_min = 7 μm it was only 1.63 m/s. These were 43.7% and 3.1% of the respective maximum velocities. With G_min < 7 μm, the leakage velocity was essentially unchanged, although the duration over which the minimum leakage velocity prevailed during the closed phase was slightly greater for G_min = 1 μm than G_min = 7 μm.

Fig. 8.

(Top) Velocity plots at four phases of several contact line cases. The top row is when the glottis has first closed and the bottom row is just after the glottis has opened. (Bottom left) Maximum glottal velocity over time and (Bottom right) leakage velocity vs. minimum glottal width.

In theory, with a completely closed glottis, there will be a step change from subglottic to supraglottic pressure. However, because the glottis could not fully close in these simulations, the pressure drop was gradual (see Fig. 9). The more completely the glottis closed (i.e., smaller G_min), the sharper the transition became. However, additional features in the pressure profiles were seen. With G_min = 50 μm the pressure profile was smooth and exhibited a pronounced minimum pressure, as expected. As G_min was reduced, the general path of the curve became closer to the expected step discontinuity in upstream and downstream pressures (see Fig. 9(c) and (d)). However, with G_min < 20 μm, the curve began to exhibit spurious pressure fluctuations, with the most severe case being G_min = 1 μm.

Fig. 9.

Pressure profiles for four contact line cases at different phases: (a) just after the glottis has opened, (b) glottis at its maximum width, (c) when the glottis has first closed, and (d) in the middle of the closed portion of the cycle. Note in plots (a) and (b) the pressure profiles for the 50, 30, and 10 μm cases are directly over one another.

These spurious pressure fluctuations were very ordered, with adjacent nodes alternating in pressure increase or decrease and the maximum fluctuations occurring at the location of contact. These are attributed to numerical instabilities associated with excessive fluid element compression and resulting high aspect ratios. Recalling that the initial gap width was 100 μm, the minimum gap width in the G_min = 1 μm case was only 1% of the initial gap width. The pressure profiles more than 1 mm upstream or down-stream of the constriction do not appear to contain significant spurious pressure fluctuations. This is consistent with the assumption that the fluctuations arose from the highly compressed control volumes, since the control volumes further than about 1 mm from the point of contact were not significantly compressed beyond their initial width.

By analyzing these trends, an estimate of the suitable value of G_min can be made. For most measures for this model, G_min = 50 μm was sufficient: G_avg, OQ, and Q_avg were within 10% of the G_min = 1 μm values, and the F₀, G_max, and Q_max were within 1%. The glottal profile, glottal width, and flow rate responses were all very similar. In addition, the G_min = 50 μm case elements were able to collapse within the glottal gap without introducing the spurious pressure profiles seen for smaller G_min. If higher accuracy is desired in closed glottis pressure profile predictions, as well as in other reported values, a G_min of 10 μm was feasible without introducing significant fluctuations in the pressure data. Using this G_min value increased the consistency these measured values to within approximately 1% of the 1 μm results. Larger G_min values will often be preferred from a computational standpoint because they are less prone to solver failure due to mesh movement problems. It should be noted, however, that these simulations did not include acoustic predictions, and it would be important to perform a similar study to determine the degree to which contact line location could be expected to influence radiated sound production.

3.3. Poisson’s ratio

Of the three modeling choices discussed in this paper, the Poisson’s ratio, ν, had the most pronounced effect on model response. Fig. 10 shows the model profiles for cases with ν = 0.4, 0.46, 0.49, and 0.4995 at eight different phases throughout the cycle. Larger amplitude motion in both the superior and medial directions for lower Poisson’s ratios is evident. In the cases with ν > 0.495, the model did not contact the contact line, and therefore had no “closed” portion in the cycle. In addition to never fully closing, the ν = 0.499 and 0.4995 cases exhibited a form of period doubling in which each complete cycle had two distinct peaks (see Fig. 11).

Fig. 10.

Vocal fold profiles at eight phases throughout one cycle for several Poisson’s ratios. Dotted lines represent original profile.

Fig. 11.

Glottal width over each model’s respective period for four Poisson’s ratio cases.

The quantitative measures are summarized in Table 3. The vibratory amplitudes were somewhat similar between 0.4 and 0.48. There was a rapid decrease in G_max as ν approached 0.5. In the limiting case of 0.49999, self-sustained vibration was no longer achieved, hence G_avg = G_max. The vibration frequency was also greatly affected by Poisson’s ratio. With ν < 0.499, F₀ gradually decreased with ν from F₀ = 275.9 Hz with ν = 0.4 to F₀ = 225.5 Hz with ν = 0.495. The cases with ν = 0.499 and 0.4995 were quite different, with F₀ values around one third of the other cases. As ν increased from 0.4 to 0.48, the open quotient decreased from 0.828 to 0.66, but then increased with further increases with ν, eventually reaching OQ = 1.0 for ν ≥ 0.499.

Table 3.

Measured values of frequency (F₀), maximum glottal width (G_max), average glottal width (G_avg), open quotient (OQ), maximum flow rate (Q_max), and average flow rate (Q_avg) for the Poisson’s ratio study.

Poisson’s ratio	F0 (Hz)	Gmax (mm)	Gavg (mm)	OQ	Qmax (ml/s)	Qavg (ml/s)
0.4	275.9	1.151	0.433	0.828	512.0	209.3
0.43	256.9	1.011	0.353	0.783	451.3	164.8
0.46	243.0	1.001	0.341	0.771	428.7	158.8
0.48	248.6	0.970	0.310	0.660	419.6	140.7
0.485	243.6	0.883	0.297	0.691	379.1	137.1
0.4875	240.7	0.858	0.295	0.689	373.1	136.8
0.49	236.5	0.836	0.293	0.718	363.9	135.8
0.4925	231.7	0.788	0.286	0.747	338.9	133.7
0.495	225.5	0.734	0.277	0.787	316.6	129.4
0.499	83.0	0.617	0.290	1.000	240.8	132.5
0.4995	83.7	0.460	0.298	1.000	158.0	137.0
0.49999	0	0.279	0.279	1.000	126.2	126.2

That the model ceased to vibrate at higher Poisson’s ratios is interesting considering that vocal fold tissue is generally considered to be incompressible, which would seem to suggest that the ideal Poisson’s ratio would be 0.5. Insight into the lack of motion for ν close to 0.5 can be gained by investigating the relevant equations. The compliance matrix [C] is used in the three-dimensional form of Hooke’s law to relate stresses and strains, in which the strain tensor {ε} is equal to the product of [C] and the stress tensor {σ}. For an isotropic, plane-strain material with the longitudinal axis in the 3-direction, this takes the form [38]

$$\left\{\begin{array}{c}{\epsilon }_{11}\\ {\epsilon }_{22}\\ 0\\ {\epsilon }_{12}\end{array}\right\}=\left[\begin{array}{llll}\hfill \frac{1}{E}\hfill & \hfill -\frac{\nu }{E}\hfill & \hfill -\frac{\nu }{E}\hfill & \hfill 0\hfill \\ \hfill -\frac{\nu }{E}\hfill & \hfill \frac{1}{E}\hfill & \hfill -\frac{\nu }{E}\hfill & \hfill 0\hfill \\ \hfill -\frac{\nu }{E}\hfill & \hfill -\frac{\nu }{E}\hfill & \hfill \frac{1}{E}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{1+\nu }{E}\hfill \end{array}\right]\left\{\begin{array}{c}{\sigma }_{11}\\ {\sigma }_{22}\\ {\sigma }_{33}\\ {\sigma }_{12}\end{array}\right\},$$ (3)

where E and ν denote the Young’s modulus and Poisson’s ratio, respectively. The axis orientation is such that the 1-direction is along the inferior-superior anatomical plane (the y-direction in Fig. 2), the 2-direction is along the medial-lateral direction (z-direction in Fig. 2), and the 3-direction is along the anterior-posterior direction (out-of-plane direction in Fig. 2).

Solving the third row of Eq. (3) for σ₃₃ results in

$${\sigma }_{33}=\nu ({\sigma }_{11}+{\sigma }_{22}).$$ (4)

Using this result, the first and second rows can be written as

$${\epsilon }_{11}=\frac{(1-{\nu }^2){\sigma }_{11}-\nu (1+\nu ){\sigma }_{22}}{E},$$ (5)

and

$${\epsilon }_{22}=\frac{-\nu (1+\nu ){\sigma }_{11}+(1-{\nu }^2){\sigma }_{22}}{E}.$$ (6)

If ν = 0.5 Eqs. (5) and (6) become

$${\epsilon }_{11}=\frac{0.75{\sigma }_{11}-0.75{\sigma }_{22}}{E},$$ (7)

and

$${\epsilon }_{22}=\frac{-0.75{\sigma }_{11}+0.75{\sigma }_{22}}{E}.$$ (8)

These results show that for equal magnitude normal stresses in the 1 and 2 directions, the normal components of strain will be zero. In the vocal fold case, the model is loaded primarily by pressure in the airway, with only a small contribution from viscous forces [4]. Considering this to lead to very similar normal stresses in all directions on any given point the vocal fold surface, it can be seen that the resulting strains should indeed be very small for ν close to 0.5.

If a lower value such as 0.49 is used for Poisson’s ratio, Eqs. (5) and (6) become

$${\epsilon }_{11}=\frac{0.7599{\sigma }_{11}-0.7301{\sigma }_{22}}{E},$$ (9)

and

$${\epsilon }_{22}=\frac{-0.7301{\sigma }_{11}+0.7599{\sigma }_{22}}{E}.$$ (10)

Under these conditions even equal normal stresses in the 1 and 2 directions will lead to normal strain in both the 1 and 2 directions. In the dynamic model of the vocal folds with ν = 0.49999, there was very little deformation, and the model did not vibrate. With slightly lower ν, and therefore larger deformation, the model self-oscillated.

For the present material formulation (two-dimensional plane strain), Poisson’s ratios above 0.495 did not yield not model motion that was similar to that of the human vocal folds. Clearly the ideal solution is to model the vocal fold in three dimensions using a transversely isotropic material. However, in cases in which two-dimensional models are suitable, a Poisson’s ratio in the range of 0.43 to 0.495 resulted in motion characteristics that were similar to those of human vocal folds.

3.4. Symmetry assumption

The assumption of symmetry led to significant differences in the supraglottic flow field, but the changes in solid model vibration pattern were very small. This can be seen in the glottal width waveforms in Fig. 12 in which the glottal width waveforms are very similar between the full and half model responses. The difference in model position is further evident in the snapshot of the model positions shown in Fig. 13. The slight variations in vocal fold motion were reflected by similarly small variations in flow rate (see Fig. 14). The flow rate was slightly higher for the full model from the time that the glottis opened until the time of maximum glottal width. It was then lower by about the same amount during the closed portion of the cycle.

Fig. 12.

Glottal width waveforms for a full and half models.

Fig. 13.

Vocal fold profiles at the same phase for full and half models, showing only slightly different model responses.

Fig. 14.

Volumetric flow rate vs. normalized time for full and half models.

As can be seen in Table 4, the quantitative output measures were negligibly affected. The fundamental frequency was higher by 0.9 Hz (0.38%) for the full model. G_max and G_avg were higher by 0.001 mm (0.12%) and 0.004 mm (1.3%), respectively. The changes in Q_max and Q_avg were 3.9 ml/s (1.1%) and 1.4 ml/s (1.0%), respectively. The largest effect was seen in OQ, where using a full model increased OQ by 0.023 (3.2%), which is still relatively small.

Table 4.

Measured values of frequency (F₀), maximum glottal width (G_max), average glottal width (G_avg), open quotient (OQ), maximum flow rate (Q_max), and average flow rate (Q_avg) for the symmetry assumption study.

Symmetry condition	F0 (Hz)	Gmax (mm)	Gavg (mm)	OQ	Qmax (ml/s)	Qavg (ml/s)
Full	237.4	0.837	0.297	0.741	367.8	137.2
Half	236.5	0.836	0.293	0.718	363.9	135.8

Fig. 15 shows vorticity plots of the two cases for selected phases of one cycle. The jet in the half model was, by definition, symmetric. It therefore exited the glottis and remained straight until it exited the airway. On the other hand, the jet in the full model exhibited strong asymmetries. As the glottis first opened, the jet exited and remained attached to one vocal fold surface during the first two open phases (B and C). It then detached from the vocal fold surface and exited the glottis in a slightly straighter path (phase D), but a large recirculation region from the previous cycle caused it to deflect towards one wall of the supraglottic airway. It maintained this configuration until the glottis again closed at phase G. More rapid mixing and dissipation was observed in the full model.

Fig. 15.

Vorticity contours for the half model (left) and the full model (right).

It is interesting that despite the obvious differences in the downstream region of the glottal jet, there was little effect on the rest of the model. The changes in F₀, G_max, G_avg, OQ, Q_max, and Q_avg values were less than 3.2% when a full model was used. Only very small differences in model motion were observed. Therefore, for research focusing primarily on vocal fold motion and not flow patterns, for cases with symmetric geometry and material properties, a half model would likely be adequate. A full model would clearly be necessary if details of the asymmetries present in the glottal jet or asymmetries of pressures within the glottis are of interest, or if the nature of the airflow patterns themselves are to be studied. It is important to note that since acoustic sound production is closely tied to the jet dynamics, a half model would likely be inadequate for predicting radiated sound.

4. Conclusions

Computational models of the flow-induced vibrations of the vocal folds are powerful tools that can be used in conjunction with physical experiments to better understand voice production. An improved understanding of the physics of voice production can be obtained by using models that are designed with proper parameter assumptions. In this paper, the influence of three important computational modeling parameters – contact line location, Poisson’s ratio, and symmetry assumptions – were systematically varied to determine how they influenced a vocal fold model’s response.

For the model used here, a minimum glottal gap width of 50 μm was sufficient for predicting most features of the model response. Smaller values down to 10 μm yielded better results. Models with gaps smaller than this were found to produce spurious pressure fluctuations.

While vocal fold tissue is typically assumed to be incompressible, most computational models have been defined using a Poisson’s ratio (ν) lower than the incompressible limit. There have been a wide variety of Poisson’s ratios used, yet no systematic evaluation of its effect on a vocal fold model has previously been performed. In this research the Poisson’s ratio in the solid model was varied from 0.4 to 0.49999, where 0.5 corresponds to incompressibility. The model motion was strongly influenced by variations in Poisson’s ratio, with larger amplitudes resulting with lower ν values. For ν ≥ 0.499, the glottis did not close completely, and for ν = 0.49999 the model did not vibrate. This pattern was discussed in the context of the plane strain model formulation. The model motion reasonably approximated that of human vocal folds for Poisson’s ratios between 0.43 and 0.495.

For computational efficiency, often only one half of the larynx is modeled and a symmetry boundary condition is used. However, asymmetry in the glottal jet is usually present. The results of a full model, consisting of two vocal folds, and a half model, consisting of one vocal fold and a symmetry boundary condition, were compared. Only slight differences in the profiles of the vocal folds were observed. In the full model, the frequency, glottal width, and flow rate measures were all very similar, in spite of significant variations in the glottal jets. For studies not dependent on glottal jet characteristics (such as acoustic sound production), or for studies that do not include inherent asymmetry in geometry or material properties, these results suggest that a half model would likely be sufficient.

Several areas of future work related to this research are suggested. Because of the inherent three-dimensional nature of the glottis and the glottal jet, a study that incorporates three-dimensional vocal folds and airway is necessary. Because a two-dimensional flow model does not include energy dissipation in the third dimension, the effects of the glottal jet may be artificially amplified in a two-dimensional simulation. It is not anticipated that the primary conclusions of this paper regarding symmetry and contact line location would change if a three-dimensional solid model were to be incorporated. However, the conclusions regarding Poisson’s ratio values are possibly only applicable to the two-dimensional model, since in the three-dimensional case, additional displacement in the third dimension could be possible and full anisotropy of tissue properties (including Poisson’s ratios) could be modeled. Additional related topics of potential interest include the effect of different lung pressures, the use of turbulent models, and the effect of these modeling choices on radiated acoustic sound production.