A Coupled Sharp-Interface Immersed-Boundary-Finite-Element Method for Flow-Structure Interaction with Application to Human Phonation

Abstract

A new flow-structure interaction method is presented which couples a sharp-interface immersed boundary method (IBM) flow solver with a finite element method (FEM) based solid dynamics solver. The coupled method provides robust and high fidelity solution for complex fluid-structure interaction (FSI) problems, such as those involving three-dimensional flow and viscoelastic solids. The FSI solver is used to simulate flow-induced vibrations of the vocal folds during phonation. Both two- and three-dimensional models have been examined and qualitative as well as quantitative comparisons made with established results in order to validate the solver. The solver is use to study the onset of phonation in a two-dimensional laryngeal model and the dynamics of the glottal jet in a three-dimensional model and results from these studies are also presented.

Introduction

Flow-structure interaction (FSI), which refers to the interaction of a moveable and/or deformable structure with an internal of surrounding fluid, occurs in a number of systems in the human body and is oftentimes critical to the physiology of the system. One obvious example is the cardiovascular system where the pumping of blood from the heart is the result of deformation of the heart wall and the functioning of the heart valves is mostly driven by fluid-structure interaction. Flow-structure interaction is also the key to the physiology of the gastrointestinal system where peristalsis is employed to move food, as well as the respiratory system where contraction/expansion of the lungs leads to respiration.

Another system where FSI is critical is the phonatory system, where airflow induced vibration of the vocal folds in the larynx leads to the production of sound, i.e. to phonation. This system is the primary focus of the current work wherein we are working towards constructing high-fidelity computational tools that model the biomechanics of phonation. The motivation is that such tools will allow us to better understand the fundamental physical mechanisms that underlie phonation. Eventually, such tools could be used for preoperative assessment of laryngeal disorders that affect speech, and even for surgical planning.

Significant advances have been made in the development of methods for simulating fluid-structure interaction in physiological systems. One class of commonly used methods is the so-called Arbitrary-Lagrangian-Eulerian (ALE) methods where the Lagrangian formulation of the Navier equation for solid dynamics is solved in a coupled manner with the Eulerian formulation of the Navier-Stokes equation for flow. The vast majority of commercial FSI software employ the ALE method and these have been used extensively in the biomedical research, especially in the cardiovascular modeling^{14,16,24,32,38,56}. However, the ALE method requires the simulation to be carried out on a body-conformal mesh which has to be modified via a suitable remeshing algorithm at every time-step. Cases involving large deformation and/or topological changes of the boundary pose a severe challenge for the remeshing procedure. Furthermore, the unstructured nature of the mesh eliminates the use of powerful line-iterative and geometric multigrid techniques for the sparse-systems that are required to be solved. Additionally, in order to provide robustness in the presence of distorted grids, the ALE based solvers also employ numerical dissipation through “upwinding”, “stabilizers”, or addition of “artificial-viscosity”, which tends to diminish natural flow-instabilities, spuriously damp vortex structures and hide the effects of under-resolution.

An alternate approach is to employ an immersed-boundary method (IBM) for the flow simulation. This type of method was initially developed by Peskin⁴⁰ to simulate cardiovascular dynamics and this method and its variants (including the so called “fictitious domain method”) have since been used for a variety of flow problems in engineering and biomechanics^{26,36,51,53,54,55}. In this type of method, the governing equations are solved on a fixed Cartesian grid which does not conform to the solid boundaries and this essentially eliminates the grid re-meshing issues encountered with body-conformal grid methods. The structured nature of the mesh also allows for the use of powerful line-iterative schemes as well as geometric multigrid methods, which can provide rapid solution of the discretized equations. However, in this type of method, a discrete delta function forcing term used to model the effect of the boundary is spread across multiple grid cells near the boundary and consequently produces a “diffuse” fluid-solid interface. This diminishes the resolution of the boundary layer and associated vortex dynamics and this can be undesirable at higher Reynolds numbers.

We have recently developed a highly versatile immersed-boundary method for simulating flow with complex moving/deforming boundaries³⁵. This method employs a multi-dimensional ghost-cell methodology in conjunction with a finite-difference scheme to solve the 3D, unsteady, incompressible Navier-Stokes equations on a Cartesian grid. This methodology is categorized as being a “sharp-interface” method in that the imposition of the no-slip, no-penetration boundary condition is localized precisely on the immersed boundary and is not spread (or “diffused”) artificially into the neighborhood of the boundary³⁶. The method also ensures 2^nd-order global and local spatial accuracy and this along with the sharp-interface treatment provides for accurate modeling of the boundary layer. This method is non-dissipative and energy conserving, which is highly sensitive to grid resolution, does not damp out natural flow instabilities and does not spuriously damp out vortex structures.

The flow solution is only one-half of an FSI problem. Also needed is an appropriate technique for solving the solid-dynamics as well as a way of coupling the two solution procedures. There are a number of approaches available for modeling the solid dynamics as well as for coupling the solid and fluid phases in the computation. For instance, Luo et al³⁴ developed a Cartesian grid based immersed boundary method to solve the Navier equations for a viscoelastic solid and coupled it to the immersed boundary solver of Mittal et al³⁵. The use of a Cartesian mesh simplifies the mesh generation problem significantly, and this is the primary advantage of this method. The solid and the fluid use different Cartesian meshes and this provides flexibility in choosing a grid resolution that is appropriate for each phase. However, the method is designed for small solid deformation and is not easily extendable to large-deformation problems which are routinely encountered in biomechanics.

More recently Zhao, et al⁶⁰ have developed a Cartesian grid based method that solves FSI problems in a fully coupled manner. A single set of equations are solved on a single Cartesian grid that cover both the solid and fluid phases wherein body-force terms are used to model the elastic forces inside the fluid. The coupling between fluid and solid effected via transferring of surface stresses from the solid to the fluid. This approach has the advantage that it can address large-deformation problems. However, as mentioned before, the method uses a single Cartesian mesh for solving both the solid and the fluid. While this simplifies the computational methodology, it does not allow much flexibility in separately choosing the grid resolution for the fluid and solid phases. This flexibility is important if one would like to address a large variety of FSI problems. Depending on the rigidity and other material constraints, the resolution requirements for the solid dynamics can be very different from that of the fluid flow. Furthermore, many solids, particularly those encountered in biology can have multi-layered, non-isotropic, non-homogeneous structures which require special attention to the grid resolution. For instance, the vocal folds have a three-layered structure²⁸ which consists of a thin outer layer (lamina propria), intermediate ligament layer and the internal muscle (vocalis) and appropriate resolution of each of these layers is required. Similarly, sometimes the flow may require significantly higher resolution than the solid due to, for instance, the presence of thin boundary layers.

For such general FSI problems, a methodology is needed which allows more flexibility in choosing the grid resolution for the solid and fluids phases, and also has the potential of being extended to large-deformation problems. The finite-element method (FEM) is the de-facto standard for modeling in solid dynamics. This method has been extensively used for a variety of solid dynamics problems and the limitations and capabilities of these methods are well understood⁵. It would therefore seem that an immersed boundary solver (for the fluid) coupled with a finite-element solver (for the solid) would provide all the requisite capabilities for addressing a wide variety of FSI problems in biomechanics.

Motivated by this, we have coupled the sharp-interface immersed boundary method with a finite-element solver that is capable of solving for the deformation in viscoelastic solids. Since most biological materials show some degree of viscoelasticity, including this capability is essential for these types of applications. The FE solver is coupled to the IB fluid solver via an explicit coupling scheme that is shown to be adequate for air-tissue interaction problems encountered in phonation and respiration. Furthermore, the current formulation is limited to small deformation but the underlying finite-element and fluid-solid coupling methodologies could be extended to large deformations.

In the current paper, we present the salient features of the computational methodology. Following this, we describe the use of the method for a two-dimensional model of the human larynx. The model is used to explore a number of issues associated with the biomechanics of phonation including phonation onset, glottal jet dynamics and vocal-fold stresses. Comparisons of computed results to existing data on phonation are made wherever possible. Finally, we show results from a three-dimensional (3D) model of the human larynx which demonstrates the ability of the solver to eventually address more realistic laryngeal models.

Numerical Method

In this section we describe the key elements of the numerical method that we have developed here including the immersed boundary flow solver, the finite-element viscoelastic structural solver and the fluid-structure coupling approach.

Immersed Boundary Flow Solver

The governing equations are the 3D unsteady incompressible Navier-Stokes equations written in the primitive variable form:

$$\frac{\partial u_i}{\partial x_i}=0$$ (1)

$$\frac{\partial u_i}{\partial t}+\frac{\partial u_i{u}_j}{\partial x_j}=-\frac{1}{\rho }\frac{\partial p}{\partial x_i}+\nu \frac{{\partial }^2{u}_i}{\partial x_j\partial x_j}$$ (2)

where u_j are velocity components in three directions, p is pressure, and ρ and υ are flow density and kinematic viscosity.

The Navier-Stokes equations are discretized in space using a cell-centered collocated (non-staggered) arrangement of the primitive variables u_i and p. The fractional step method of Van-Kan⁵¹ is used to integrate the equations in time, which consists of three sub-steps. In the first sub-step, a modified momentum equation is solved to get an intermediate velocity u^*. A second-order, Adams-Bashforth scheme is employed for the convective terms while the implicit Crank-Nicholson scheme is used to discretize the diffusion terms to eliminates the viscous stability constraint. The second sub-step requires the solution of the pressure correction equation which is solved with a highly efficient geometric multigrid method. Once the pressure is obtained, the velocity field is updated to its final value.

The simulation employs a sharp interface immersed boundary method based on a multi-dimensional ghost-cell methodology which has been described in detail in Mittal et al³⁵. In this method, the body whose surface is described by triangular elements is immersed into a Cartesian volume grid. As shown in Figure 1, the Cartesian cells are solid cells or fluid-cells depending on whether they are inside or outside the boundaries of the bodies. This ghost-cell method proceeds by identifying the ghost cells (denoted by “GC”) which are solid cells which have at least one fluid cell neighbor. A “probe” is then extended from one of these ghost cells onto an “image-point” (denoted by “IP”) inside the fluid such that it intersects normal to the immersed boundary and the boundary intercept (denoted by “BI”) is midway between the ghost-node and the image-point. Next, a bi-linear interpolation (tri-linear in 3D) is used to express the value of a generic flow variable at the image-point in terms of the surrounding nodes. Following this, the value of the variable at the ghost-cell is computed by using a central-difference approximation along the normal probe such that the prescribed boundary condition at the boundary intercept is incorporated. Using this procedure, the boundary conditions are prescribed to the second-order accuracy and this along with the second-order accurate discretization of the fluid cells leads to the local and global second-order accuracy in the computations. The formulation for Dirichlet and Neumann boundary conditions are shown as follows

Figure 1.

2D schematic describing ghost-cell methodology

$${\phi }_{IP}=\sum {\beta }_i{\phi }_i$$ (3)

$${\phi }_{GC}+\sum {\beta }_i{\phi }_i=2{\phi }_{BI}$$ (4)

$${\phi }_{GC}-\sum {\beta }_i{\phi }_i=\mathrm{\Delta }l{\left(\frac{\delta \phi }{\delta n}\right)}_{BI}$$ (5)

where φ is the generic flow variable, i is from 1 to 4 (for 2D) or 8 (for 3D) and represents the ith surrouding node for IP, β is the interpolation weigth and Δl is the probe length.

Since the equations are written in the Eulerian form, boundary motion can now be included into this formulation by moving the boundary at a given time-step, recomputing the body-intercepts, image-points and then advancing the flow equations in time. One issue associated with a moving boundary case of these types of methods is the so called “fresh cell” problem^35,50 which refers to the cell that was solid cell at previous time step and became fluid cell at current time step due to the boundary motion. The method for treating such cell is described in detail in Mittal et al³⁵ and will not be repeated here.

Viscoelastic Structural Modeling

The governing equations are the Navier equations written as

$$\rho \frac{{\partial }^2{d}_i}{\partial t^2}=\frac{\partial {\sigma }_{ij}}{\partial x_j}+\rho f_i$$ (6)

where i and j range from 1 to 3, σ is stress tensor, f_i is the body force component in i direction, ρ is the density, and is d_i the displacement component in i direction. Assuming a Kelvin-Voigt model¹⁹ for the viscoelasticity, which has been extensively used in modeling tissue¹⁹, the constitutive law between stress and strain can be written as:

$${\sigma }_{ij}=C_{\mathit{ijkl}}{\epsilon }_{kl}+A_{\mathit{ijkl}}{\stackrel{.}{\epsilon }}_{kl}$$ (7)

where ε is the strain tensor, ε̇ is the strain rate tensor and C and A are fourth-order tensors corresponding to the material constants.

In the current method, a Galerkin formulation⁵ is employed for the finite-element method. This well established⁵ formulation employs virtual displacements to discretize Eq. (6) and represent it in terms of the nodal displacement as follows. The final result of this formulation is the following second-order ordinary differential equation for the nodal displacements:

$$M_{\alpha \beta }{\ddot{D}}^{\beta }+C_{\alpha \beta }{\stackrel{.}{D}}^{\beta }+K_{\alpha \beta }{D}^{\beta }=F_{\alpha }^t+F_{\alpha }^b$$ (8)

where $M_{\alpha \beta }=\underset{v}{\int }\rho N_{j\beta }{N}_{i\alpha }dv,C_{\alpha \beta }=\underset{v}{\int }{A}_{\mathit{ijkl}}{B}_{kl\beta }{B}_{ij\alpha }dv$ and $K_{\alpha \beta }=\underset{v}{\int }{C}_{\mathit{ijkl}}{B}_{kl\beta }{B}_{ij\alpha }dv$ are the mass, damping and stiffness matrices respectively, and $F_{\alpha }^t=\underset{s\sigma }{\int }{\sigma }_s{N}_{i\alpha }ds$ and $F_{\alpha }^b=\underset{v}{\int }\rho f_i{N}_{i\alpha }dv$ are the nodal traction and nodal body force respectively. In the above expressions N_iα is a predefined weight (or “shape”) function, which connects the displacement d_i at any arbitrary point inside an element through a weighted sum of the nodal displacements D^α as follows:

$$d_i=\sum _{\alpha =1}^n{D}^{\alpha }{N}_{i\alpha }$$ (9)

where n is total number of nodes of the element, (3 for triangular and 4 for quadratic and tetrahedral elements). The M, K and C matrices are obtained via an integration over the different elements. The integration is simplified through the use of isoparametric coordinates⁵ Following this, equation (8) is dicretized in time using a Newmark Scheme⁵ leading to the following dicrete equation:

$$\left(K+\frac{1}{\beta \mathrm{\Delta }{t}^2}M+\frac{\gamma }{\beta \mathrm{\Delta }t}C\right)D^{n+1}=F^{n+1}+M\left[\frac{1}{\beta \mathrm{\Delta }{t}^2}{D}^n+\frac{1}{\beta \mathrm{\Delta }t}{\stackrel{.}{D}}^n+\left(\frac{1}{2\beta }-1\right){\ddot{D}}^n\right]+C\left[\frac{\gamma }{\beta \mathrm{\Delta }t}{D}^n+\left(\frac{\gamma }{\beta }-1\right)\stackrel{.}{D}+\left(\frac{\gamma }{2\beta }-1\right)\mathrm{\Delta }t{\ddot{D}}_n\right]$$ (10)

where β, γ are constants. We choose β = 0.25and γ = 0.5, which results in a second-order accurate scheme in time. It should be noted that the Newmark scheme is unconditionally stable⁵.

In order to reduce the memory requirement and improve the computational efficiency, a banded LU decomposition is employed to solve Eq. (10). Both Cuthill-Mckee¹⁰ and Gibbs-Poole-Stockmeyer²² methods are used to re-index the nodes and to find the smallest banded matrix.

Contact between bodies is a very common problem in mechanics. This phenomenon is particularly relevant to the problem of vocal fold vibration and phonation since the two vocal folds experience contact during each vibration cycle. In the current study, a penalty coefficient method⁵ is used to model vocal fold contact. According to this method⁵, the contact force is modeled as follows:

$${\overrightarrow{F}}^{\mathit{contact}}=\int \gamma g\overrightarrow{\nabla }\mathit{gdA}$$ (11)

where F^contact is the contact force, A is the contact area, γ is the penalty coefficient (which is always positive), g is the penetration distance. Thus, integrating the contact force over the contact area creates a contact force that opposes penetration of one vocal fold into another, and is proportional to the penetration distance with γ as the constant of proportionality. This integration is performed using shape functions N_a for each surface element which intrudes into the contact plane. By choosing different values for γ, one can enforce different types of contact with higher values of γ leading to a more “hard” contact condition. It should be noted that high values of γ enhance the stiffness of the resulting system and can lead to additional constraints on the size of the time-step. In the normal phonation, the structure and movement of two vocal folds are nearly symmetric, thus, the contact almost occurs at the glottal center-plane., In the current study, to simplify the situation, the contact condition is enforced on the glottal center-plane. It should be noted that in a more general case, which the contact does not necessarily occur on the center plane, more sophisticated searching algorithms are required for the contact detection.

The finite-element solid solver has been validated for 2D and 3D canonical cases and the details for these tests can be found in Zheng⁶³.

Flow-Structure Coupling

A flow-structure coupling scheme has to be implemented through the surface mesh of the solid which is comprised of triangular elements. For the flow solver, no-penetration, no-slip velocity boundary conditions are to be imposed³⁵. According to the ghost-cell methodology described in the earlier section, these boundary conditions are applied at the boundary intercept points (IB). The locations and velocities at these points are directly interpolated from the surface nodal values of finite-element solution. For the solid solver, the traction has to be prescribed at the fluid-solid interface. The normal as well as the shear stress is computed using a tri-linear interpolation (bilinear interpolation for 2D) which is described in Ghias et al²¹. Once pressure and shear stress is computed at the surface nodal points, the total force on a given element is computed using the shape functions.

Usually, there are two coupling strategies for FSI, namely loose coupling and strong coupling. The major difference between loose and strong coupling is that they respectively integrate the governing equations of structure explicitly and implicitly in time. The advantage of the strong coupling scheme is that it is robust and does not usually introduce stability constraints over and above those that exist for the flow and solid solvers. However, strong coupling usually requires an iterative procedure that can significantly increase the computational expense. On the other hand, loose coupling may be subject to additional stability constraints but is computational inexpensive. It should be noted that the flow solver used here employs explicit treatment of the convective terms and is therefore subject to CFL type time-step constraint³⁴. This precludes the use of large-time steps and might eliminate the advantage of using a strong coupling scheme. If it can be shown that loose coupling will not lead to severe stability constraints for the problem at hand, then such a scheme would be idea for our simulations. We now perform a heuristic assessment of the stability of the loose coupling scheme. For this analysis, we choose a flow configuration consisting of an elastically mounted sphere with negligible structural damping in a uniform potential flow. This canonical problem shares many of the features typical of FSI, and the simplicity of the configuration makes it amenable to analysis.

For the above configuration, an explicit coupling leads to the following equation for the motion of the structure in the streamwise direction:

$${M\ddot{X}}^{n+1}+K{X}^{n+1}=F^n$$ (12)

where the fluid traction term Fⁿ is computed using the previous time-step and X is the location of the center of the sphere. If the flow is assumed to be an incompressible potential flow, the fluid dynamic load on the sphere in the streamwise direction is given by³

$$F=-\frac{2}{3}\pi a^3{\rho }_f({\stackrel{.}{U}}_B-\stackrel{.}{U})+\frac{4}{3}\pi a^3{\rho }_f\stackrel{.}{U}$$ (13)

where a is the radius of sphere, ρ_f is the fluid density, U̇_B = Ẍ is the body acceleration and U̇ is free stream flow acceleration. The first term of Eq.(13) is the so called “added mass force” which is due to the sphere acceleration relative to the free stream. The second term is the force due to the imposed freestream pressure gradient. Assuming an explicit coupling between the flow and solid, for a steady free stream, the equation of motion for the sphere becomes

$$\frac{4}{3}\pi a^3{\rho }_s{\ddot{X}}^n+K{X}^n=-\frac{2}{3}\pi a^3{\rho }_f{\ddot{X}}^{n-1}$$ (14)

where ρ_s is the solid density. Using a central difference scheme for the acceleration terms, the resulting finite difference equation is

$$\frac{4}{3}\pi a^3{\rho }_s\frac{X^{n+1}-2X^n+X^{n-1}}{\mathrm{\Delta }{t}^2}+K{X}^n=-\frac{2}{3}\pi a^3{\rho }_f\frac{X^n-2X^{n-1}+X^{n-2}}{\mathrm{\Delta }{t}^2}$$ (15)

Assuming numerical error ε_n at time-step ‘n’ propagates as ε_n = λ_n–1 leads to the following characteristic equation:

$${\lambda }^3+\left(\frac{1}{2}\frac{{\rho }_f}{{\rho }_s}+\mathrm{\Delta }{t}^2{\omega }^2-2\right){\lambda }^2+\left(1-\frac{{\rho }_f}{{\rho }_s}\right)\lambda +\frac{1}{2}\frac{{\rho }_f}{{\rho }_s}=0$$ (17)

where $\omega =\sqrt{K/(4/3\pi a^3{\rho }_s)}$ is the natural frequency of the system.

The critia for a stable solution is

$$\mid {\lambda }_i\mid \le 1\text{for}i=1,2,3$$ (21)

which leads to the following constraint for the roots of the cubic characteristic equation:

$$\left|\frac{1}{2}\frac{{\rho }_f}{{\rho }_s}+\mathrm{\Delta }{t}^2{\omega }^2-2\right|=\mid {\lambda }_1+{\lambda }_2+{\lambda }_3\mid \le 3$$ (22)

$$\left|\frac{1}{2}\frac{{\rho }_f}{{\rho }_s}\right|=\mid {\lambda }_1{\lambda }_2{\lambda }_3\mid \le 1$$ (23)

$$\left|1-\frac{{\rho }_f}{{\rho }_s}\right|=\mid {\lambda }_1{\lambda }_2+{\lambda }_2{\lambda }_3+{\lambda }_1{\lambda }_3\mid \le 3$$ (24)

These can be simplified to:

$$\mathrm{\Delta }{t}^2\le \frac{1}{2{\omega }^2}\left(10-\frac{{\rho }_f}{{\rho }_s}\right);\frac{{\rho }_f}{{\rho }_s}\le 2;\frac{{\rho }_f}{{\rho }_s}\le 4;$$ (25)

We are particularly interested in air-tissue interaction, so if we assume that the tissue has a density equivalent to water, then for air-tissue interaction, $\frac{{\rho }_f}{{\rho }_s}\equiv \frac{{\rho }_{\mathit{air}}}{{\rho }_{\mathit{tissue}}}\approx \frac{1kg/m^3}{1000kg/m^3}\sim 0.001$. Thus, the latter two conditions are easily satisfied Furthermore, for air-tissue interaction, the first constraint becomes $\mathrm{\Delta }t\le \sqrt{5}/\omega$, or Δt ≤ 0.36τ, where τ is the time-period of the vibration. Given that we typically have O(1000) time-steps in each vibration cycle, the above condition is also satisfied quite easily. The above implies that the loose-coupling scheme should work quite well for the air-tissue interaction problem associated with phonation. Note that the above analysis neglects viscosity which could potentially modulate the stability behavior. However given that viscosity would likely provide additional damping to the FSI system and that inviscid models have been used to predict basic features of FSI in phonation, we expect that an inviscid analysis will provide a reasonable assessment of the numerical stability.

In the explicit coupling scheme adopted here, the flow is marched by one step with the current deformed shape and velocities of the solid nodes as the boundary conditions. The aerodynamic forces imparted on the VF are then calculated at this current location of the vocal fold surface via an interpolation scheme on the flow grid. Finally, the equation for the solid is marched by one time-step with the updated surface traction, and the deformation and velocities on the solid grid are interpolated onto the vocal fold surface, so that the fluid/solid interface can be updated. This explicit coupling is quite simple, robust and efficient. Implicit coupling can be easily implemented if needed by iterating between the fluid and solid solvers at each time-step.

Computational Modeling of Vocal Fold Dynamics during Phonation

The FSI method developed here has been employed to investigate the phonation problem, which results from a highly coupled biomechanical interaction between glottal airflow and vocal fold tissue. In the past few decades, scientists have achieved various degrees of success in modeling phonation. Approaches have included inviscid, irrotational flow models^30,31, lumped mass vocal fold models^30,31,43,46, and stationary²³ or specified vocal fold motion^59,61. However, these models are inherently low-fidelity and able to capture limited characteristics of phonation. To provide quantitative results for direct clinical diagnosis and treatment as well as the full understanding of the biophysics of phonation, a higher fidelity FSI computational model is required. This is the primary motivation for the current effort.

Model Setup

The dimension and geometry of the laryngeal model used here is based on a laryngeal CT scan of a 30-year old male shown in Figure 4(a). The overall flow domain is a 12 cm long by 2 cm wide channel. The true vocal folds (TVFs) are 1 cm long and extend 0.99 cm towards the glottal midline. The false vocal folds (FVFs) are 2.3 cm long and extend 0.67 cm towards the supraglottal space. The ventricles are about 0.56 cm wide at their widest location and the false vocal fold gap is 0.667cm.

Figure 4.

(a) A coronal view of CT scan of human larynx and the current flow domain that attempts to match the key geometrical features in the CT scan. (b) Three layer vocal fold inner structure inside an idealized geometric VF model based on CT scan and triangular elements used in the current solver.

The FVFs are modeled as rigid bodies since they do not move during the normal phonation. The TVFs are viscoelastic bodies with a complex three-layer structure, shown in Figure 4(b). A 17202 triangle mesh is used to represent the entire two dimensional TVF. The material properties of each layer are given in Table 1. Similar properties have been used in the past by Alipour et al¹. It should be noted that the longitudinal Young’s modulus is not given by Alipour et al¹ due to the in-plain motion assumption. It has been shown by Cook et al⁹ that the longitude Young’s modulus must be 10⁴ times of in-plain Young modulus to produce the equivalent vocal fold in-plain motion. Thus in the current study, the longitudinal Young’s modulus is set to be 10⁴ times of in-plain Young modulus. For the 2D simulation, a plain strain assumption has been adopted and the material is assumed to be isotropic with a Poisson’s ratio of 0.3³⁴.

Table 1.

Material properties of the three-layers of the vocal folds

	ρ (g/cm3)	Ep (kPa)	ν p	Epz (kPa)	νpz	Gpz (kPa)	η(poise)
cover	1.043	2.041	0.9	20000	0.0	10	3
ligament	1.043	3.306	0.9	33000	0.0	40	5
body	1.043	3.990	0.9	40000	0.0	20	6

For simulations of normal phonation, constant gauge pressures of 1 kPa and 0kPa are applied at the inlet and exit of the domain respectively, and these approximate typical physiological conditions. A homogeneous normal gradient velocity boundary condition is also applied at the inflow and outflow boundaries. Finally, no-slip and no-penetration boundary conditions are applied on the walls and the flow-tissue interfaces⁴⁴. A 289 × 256 Cartesian grid is chosen for fluid mesh which has a uniform grid in the y-direction and a non-uniform grid in the x-direction with a denser grid in the vicinity of TVFs and FVFs (as shown in Figure 5).

Figure 5.

Grid used in the current simulation. The figure about shows every 3 grid points in each direction.

In the simulation, a time-step corresponding to3.5 × 10⁻³ ms is employed which results in 1000 to 3000 time-steps in every vibration cycle for a typical phonation frequency between 100–200 Hz. Three different studies are reported. The first two studies are based on a 2D laryngeal model and the primary motivation here is to validate computed results both qualitatively and quantitatively against established models and experiments. The grid employed in the 2D simulation is based on our past experience in simulating these flows and has been subjected to grid refinement analysis in previous studies^{34, 62}. The third and final study is based on a 3-D model of the vocal fold where we demonstrate the ability of the current method to compute three-dimensional details of the vocal fold and glottal jet dynamics. The 2D simulations have been carried on a single-node of a 1.96 MHz AMD Opteron^™ workstation whereas the 3D simulations are performed on upto 128 processors of a Cray XT5.

Computational Study of Phonation Onset

We first present results from a phonation onset study where the subglottal pressure is increased systematically from 0.1 KPa, to 2.0 KPa (with intermediate values of 0.3, 0.5, 0.7, 1.0 and 1.5Kpa). It is well known that the onset of sustained vocal fold vibrations occurs via a Hopf’s bifurcation²⁷ once the pressure exceeds a critical value, and we attempt to reproduce this phenomenon with the current computational model. The computations indicate no sustained vibrations for 0.1 KPa and 0.3 KPa cases whereas all other cases exhibit such vibrations. Thus, the phonation onset threshold pressure is predicted to be between 0.3 kPa and 0.5 kPa. A similar range has been reported by Baer⁴ who conducted in-vivo measurement on an excised larynx as well as Titze⁴⁸ who employed a computational with a two-mass vocal fold model coupled with a 1D Bernoulli equation. Thus, the current modeling procedure gives results that are inline with established studies.

For all of cases for which sustained vibrations are achieved, spectral analysis of the time-variation of the glottal gap-width is employed to extract the fundamental phonation frequency F₀. The variation of fundamental frequency F₀ versus sub-glottal pressure P_sub is shown in Figure 6. Just above the threshold pressure, the fundamental frequency is found to increase non-linearly with sub-glottal pressure but becomes nearly constant at higher sub-glottal pressures. It should be noted here that the experimental study of Titze⁴⁸ indicates that the fundamental phonation frequency increases with subglottal pressure when the sub-glottal is pressure significantly larger than normal phonation pressure. However a pressure-frequency relationship similar to what is observed here has been reported by Ishizaka³¹ in one of his two-mass model FSI studies using a linear spring. Thus, it is quite likely that the reason that a behavior similar to Titze⁴⁸ is not observed in the current simulations due to the assumption of material linearity. However, since the focus here is on investigating normal phonatory behavior, we consider the current model adequate for this purpose.

Figure 6.

Variation of fundamental frequency with sub-glottal pressure.

Normal Phonation

In all of the current studies, the simulations were continued until the vocal fold vibration reached a stationary state with a limit-cycle type behavior. The characteristics of this stationary state are then examined to further establish the fidelity of the current modeling approach. Figure 7 shows the time history of the glottal airflow volume flux which clearly shows that a stationary vibratory state has been established. The flux shows a phonation frequency of 231 Hz, which is at the upper end of the range associated with normal phonation in humans. The average and peak values of the computed volume flux rate are 0.01619m²/s and 0.04328m²/s respectively. If the vocal fold length is assumed to be 2cm which would be typical for an adult human, the average and peak volume flux are 322 ml/s and 865 ml/s which are inline with in-vivo measurement of excised larynges². The typical value of the Reynolds number for human phonation is around 3000 based on the peak volume flux rate Re_Q = (3/2)Q/υ_a, where υ_a is kinematic viscosity of air. However, as shown in the previous study³⁴, the vibratory features of the vocal folds are relatively insensitive to the Reynolds number. Thus, to alleviate the computational cost, the Reynolds numbers employed in current study is reduced to 454 based on peak value of volume flux rate.

Figure 7.

Time variation of two dimensional glottal volume fluxes in the stationary vibration stage.

Figure 8 shows a sequence of instantaneous spanwise vorticity contours which reveal the details of the flow dynamics during phonation. As the vocal folds open, the air is pushed out by the subglottal pressure into the supraglottal region leading to the formation of the so-called “glottal jet”. The interesting thing is that the jet shows noticeable asymmetry and may be deflected to one side of the channel. This is a consequence of the strong flow recirculation zones in the supraglottal region created in previous cycles which tend to deflect the incipient glottal jet in one direction or the other. The jet deflection is stochastic in nature and does not show any regular cycle-to-cycle behavior. This phenomenon of the so called “biomodal jet” has been reported in several experimental studies^17,39,49 and further details of this phenomenon along with the role of the false-vocal folds in mitigating this bimodality are described in Zheng et al⁶².

Figure 8.

Contours of spanwise vorticity during vocal fold vibration. (a) 0.3384s, (b) 0.3398s, (c) 0.3409s, (d) 0.3454s.

The mechanical stress in the vocal fold tissue is important since vocal folds may experience fatigue and damage due to the excessive localized stress, as well as due to cyclicality of the stress. In extreme cases, excessive and prolonged stress can cause laryngeal pathologies such as vocal fold nodules⁴⁷. Thus, accurate computation of the stress inside the vocal folds can help shed insights into such laryngeal pathologies. In Figure 9 we show contours of mechanical stress in the two vocal folds at two extreme positions during the vibration cycle. When the vocal folds are fully adducted, the contact between the vocal folds produces a large compressive stress (negative σ_yy) on the medial surfaces of the two vocal folds while at the same time, high levels of elongation stress (positive σ_yy) occur at the root of ligament and the maximum shear stress τ_xy occurs on the root of vocal folds. At the maximum abduction position, the shear stress becomes much smaller than during the closing phase. The normal stress σ_yy is mainly compressive during the opening phase and appears at the superior part of the vocal fold body. The normal stress σ_xx is similar at the two positions and is in general smaller than σ_yy.

Figure 9.

Contours of stresses (kPa) in the vocal folds during the open and closed phase of the vibration cycle.

3D Flow-Tissue Interaction

While the majority of computational laryngeal models employed in phonation studies are two-dimensional, the limitations of these models have been well recognized for some time. In particular, the 3-D shape of the glottis and vocal folds are expected to have a significant effect on the vibratory dynamics which cannot be captured in 2D models. Furthermore, 2D models also cannot accurately compute the transition to turbulence which usually occurs in the glottal jet, and which is expected to modify the flow characteristics significantly. In the current study, we have developed a relatively simple 3D model by extruding the 2D model 1.5 cm in anterior and posterior directions. The internal 3 layer structure was kept identical along the longitudinal direction. The anterior, posterior and lateral walls of true vocal folds were fixed to represent the attachment to the cartilage. A 12cm×2.0cm×1.5cm straight rectangular duct was used to mimic the human airway. The locations of vocal folds and false vocal folds inside the airway were kept same as the 2D flow-tissue interaction study. The subglottal and supraglottal pressures were also kept the same as the 2D case and no-slip boundary condition are applied on all of the walls. This simulation employs a non-uniform 256×128×64 Cartesian grid for the flow solver and a 58427 tetrahedral element grid for the solid solver. While this model does not incorporate all the geometrical and structural complexities of the human larynx, it does represent a significant improvement over the 2D model. Similarities between the 2D and 3D geometries employed here also allow us identify the key effects of the inclusion of three-dimensionality. It should be noted that the 3D model is computationally 30 times more expensive than the 2D model and this illustrates the challenge associated with these simulations.

This 3D simulation are carried out on 128 processors on an IBM iDataPlex cluster with Intel(R) Xeon(R) 2.66GHz processors using and continued until the vocal fold vibration reached a stationary state with a limit-cycle type vibration behavior. Each vibration cycle requires 45 hours of CPU time on 128 processors. Figure 12 shows the time history of the glottal airflow volume flux for six vocal fold vibration cycles. The average and peak volume fluxes are 118 ml/s and 300 ml/s which are inline with in-vivo measurement of excised larynges². The Reynolds number based on the peak volume flux rate is 209. The vibration frequency is 242 Hz, which leads to vibration period, about 0.00413s. If we define T as the period of vibration cycle, the maximum flow rate occurs at about 0.4T.

Figure 12.

Iso-surface of swirl strength at six different time instants over one vocal fold vibration cycle. (a) 0.0245s, (b) 0.02625s, (c) 0.02695s, (d) 0.02800s, (i)-(j) 0.02870s

Particular focus with this simulation is to determine the extent to which the inclusion of three-dimensionality affects the glottal jet. Figure 12(a) shows an iso-surface of the swirl strength⁸ which is used to identify vortex structures in flows. The plots indicate that the glottal jet becomes highly three-dimensional in a very early stage in its formation. The three-dimensionality in the glottal jet amplifies further as the jet develops into the supraglottal region and is characterized by a complex conglomeration of vortex structures. The development of these three-dimensional vortex structures is the first stage in the transition to turbulence of the glottal jet. As shown in the 2D simulation and other PIV experiments^17,39,49, the glottal jet exhibits a “bimodal behavior” which has a prominent effect on flow impedance, turbulent characteristics and vocal fold vibrations. This bimodal phenomenon also can be clearly observed in this 3D simulation. In the particular cycles shown in Figure 13(c), once the jet flow exits the glottis, it starts to deflect towards the lower false vocal fold (shown in Figure 13(d)). In subsequent cycles, the jet stochastically deflects towards one or the other false vocal folds.

Figure 13.

Iso-surface of the turbulent kinetic energy corresponding to a value of K_T =0.15 and contour of turbulent kinetic energy at the center plane (z = 0.75cm).

To investigate the turbulence characteristics of the glottal jet we plot the distribution of the turbulence kinetic energy K_T in Figure 13. The turbulence kinetic energy is computed as:

$$K_T=\frac{1}{U_{\mathit{jet}}^2}\frac{1}{2m\tau }\underset{t}{\overset{t+m\tau }{\int }}{V}_i^{\prime }{V}_i^{\prime }dt$$ (26)

where τ is the vibration peroid, m is number of cycles over which averaging is done, $V_i^{\prime }=V_i-{\stackrel{\sim }{U}}_i$ denotes the turbulent fluctutation of velocity, V_i denotes the instantenous velocity, Ũ_i is the phase-averaged velocity^29,37(which is 47 m/s for the current simulation) and U_jet is the peak jet velocity during the cycle. It should be noted that K_T contains the effects of both the turbulent fluctuation as well as the cycle-to-cycle deflection of jet. The distribution of the turbulent kinetic energy shows that the turbulence intensity is high in the region just downstream of the glottis and this is due to the the rapid breakdown of the jet into small scale vortex structures. In addition, K_T also shows two lobes downstrem of the false vocal folds which are likely due to the jet deflection. The peak value of the tubulent kinetic energy is found to be 0.545.

In order to further understand the transition to turbulence in the glottal jet we examine the frequency spectra of the velocity in the jet core. Past experiments³⁹ indicate that the glottal jet has a laminar core which rapidly transitions to turbulence further downstream. We have computed the frequency spectra at the five locations along the center line (shown in figure 14(a). The spectra are also averaged along the spanwise direction in the core of the glottal jet. Also plotted in the spectra is a line with a slope of −5/3 which represents the inertial subrange. The plots in Figure 14(b)-(f) show the most clear presence of a substantial inertial subrange at location 4 indicating that the transition to turbulence occurs between locations 3 and 4. Thus, the current simulations allow us to capture the phenomenon of transition to turbulence in the glottal jet.

Figure 14.

(a) Five locations along the center line in X-Y plane chosen to perform the flow spectrum analysis. (b)-(f) span-averaged stream-wise velocity spectrum, (b): point 1 (x=3.025cm), (c) point 2 (x=3.175cm), (d) point 3 (x=3.325cm), (e) point 4 (x=3.575cm), (f) point5 (x=3.975cm). Dash-dot line corresponds to k^−5/3

Conclusions

A flow-structure interaction method has been developed which couples a sharp-interface immersed boundary method (IBM) with a finite-element method based solid dynamics solver. The coupled method provides robust and high-fidelity solution for a highly complex FSI system, such involving three-dimensional flows and viscoelastic solids. The FSI solver is used to simulate flow-induced vibrations of the vocal folds during phonation. Both two- and three-dimensional models have been examined qualitative and quantitative comparisons with established results indicate that the solver is able to reproduce the salient features of phonatory dynamics. To our knowledge, this is the first time a computational model has been used to capture the phenomena of transition in the glottal jet. The solver is currently being used for detailed investigation of the dynamics in a realistic 3D laryngeal model and these results will be presented in the near future.

Figure 2.

2D illustration of contact force. The upper vocal fold intrudes into the center contact plane.

Figure 3.

Schematic flow past an elastically mounted sphere.

Figure 10.

Flow domain in the 3D flow structure simulation and finite element mesh for the true vocal folds.

Figure 11.

Time variation of three dimensional glottal volume flux in the stationary vibration stage