Fully-coupled aeroelastic simulation with fluid compressibility — For application to vocal fold vibration

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Abstract

In this study, a fully-coupled fluid–structure interaction model is developed for studying dynamic interactions between compressible fluid and aeroelastic structures. The technique is built based on the modified Immersed Finite Element Method (mIFEM), a robust numerical technique to simulate fluid–structure interactions that has capabilities to simulate high Reynolds number flows and handles large density disparities between the fluid and the solid. For accurate assessment of this intricate dynamic process between compressible fluid, such as air and aeroelastic structures, we included in the model the fluid compressibility in an isentropic process and a solid contact model. The accuracy of the compressible fluid solver is verified by examining acoustic wave propagations in a closed and an open duct, respectively. The fully-coupled fluid–structure interaction model is then used to simulate and analyze vocal folds vibrations using compressible air interacting with vocal folds that are represented as layered viscoelastic structures. Using physiological geometric and parametric setup, we are able to obtain a self-sustained vocal fold vibration with a constant inflow pressure. Parametric studies are also performed to study the effects of lung pressure and vocal fold tissue stiffness in vocal folds vibrations. All the case studies produce expected airflow behavior and a sustained vibration, which provide verification and confidence in our future studies of realistic acoustical studies of the phonation process.

Introduction

Fluid–structure interactions occur in many engineering applications and systems, as well as innate bio-mechanical functions in a human body. The self-sustained vocal folds vibration is such an application that intrinsically involves fluid (air) and structure (vocal folds) interactions. The vocal folds are a pair of flexible structures at the top of the trachea, where air travels through in the larynx and produces sounds due to its fluctuations.

Over the past two decades, a number of numerical models are built to facilitate the understanding of the self-sustained vocal fold vibration, which span from multi-mass models where the vocal folds are modeled as mass–spring–damper system  [1], [2], [3], [4] to continuum models where the vocal folds are modeled using realistic physical representation. With the dramatic increase in computational capabilities and efficiencies in the recent years, more continuum vocal fold models have been developed using 2-D and 3-D finite elements, some of which are coupled with Bernoulli’s equation for fluid to study the mechanisms behind the self-sustained oscillation process  [5], [6], [7], [8], [9].

These fluid–structure interaction continuum models can typically be categorized into two groups: one-way coupling where one field provides the solution information to another field and repeated over time cycles, and two-way coupling or fully-coupled fluid–structure interaction models where the fluid feeds the information to the solid and the solid feeds information back to the fluid, be it explicit or implicit in time. In general, one-way coupling between the fluid and the solid is where the motion and the deformation of the solid are only determined by the fluid pressure acting on the fluid–structure interface and the solid does not provide any feedback to the fluid domain. The one-way coupling approach includes controlled/manipulated solid motion by some pre-designed spatial–temporal function and superimposed solutions of the fluid and solid  [10], [11]. It is well known that one-way coupling often leads to divergence in the system solution or converging to an unphysical solution.

On the contrary, a fully-coupled fluid–structure system typically yields more stable and more realistic solution comparing to that of one-way coupling. The Immersed Boundary (IB) method is a numerical technique that involves two-way coupling between the fluid and the solid. A first attempt to use the IB method to simulate the airflow interacting with vocal folds using a 2-D model was done by Luo et al.  [10]. In this work, the fluid was incompressible and viscous while the vocal folds were modeled with finite difference method with appropriate visco-elastic material properties. The coupling between the two domains was done via a “sharp interface” methodology, where the solid was driven by the traction boundary condition interpolated from the previous time step and solved as a quasi-steady state. The solid was then updated to a new position and a corresponding velocity (interior and boundary) field was obtained. The velocity at the solid boundary was then interpolated back to the fluid domain and applied as the “boundary” condition for the fluid. This “boundary” is not the boundary of the fluid domain, but rather, the fluid–structure interface inside the fluid domain. Using the IB method, Luo et al.  [12] and Zheng et al.  [13] further studied the effects of false vocal folds on glottal flow and vocal folds vibration. An improved solid model was also presented in  [12], where the viscoelastic solid dynamics was solved using finite element method rather than finite difference as they did in  [10], which provided a better and an easier way of applying solid boundary conditions, especially for complex geometries. A contact model was also included instead of their previous model where the vocal folds displacement was manipulated to avoid overlapping onto each other. More advanced work had also been done by Mittal’s group on representing the acoustic field by performing hydrodynamic/acoustic splitting method to model linearized perturbed compressible fluid equation  [11]. However, this acoustic field was not coupled with a dynamic vocal fold model.

Despite the challenges, the fully-coupled approach to study vocal folds vibration has significantly advanced the capabilities in representing a feasible and realistic glottal system. As of date, there remains a major deficiency in all the sophisticated models to study the full interactions between aeroelasticity and aeroacoustics — they ignored the compressibility of air. The inclusion of compressibility in a fully-coupled fluid–structure glottal system is vitally important since the sound wave generation and propagation during vocal folds vibration are induced and facilitated by the compression of air. In fact, our initial attempt in using incompressible fluid for the coupled system resulted in reasonable magnitude of the vocal folds vibrations, but ultimately failed to reach a sustained vibration. Without a steady-state vibration, it is futile to perform any analysis on the results to understand the fundamental mechanism of sound production.

In this work, we design a new numerical tool that encompasses the compressibility of the fluid, its duly interactions with aeroelastic materials, as well as other features that are necessary to successfully simulate the laryngeal system. It is built based on a fully-coupled fluid–structure interaction algorithm, the modified immersed finite element method (mIFEM)  [14]. The major distinction between the class of immersed finite element methods and that of the IB methods is that it utilizes the concept of principle of virtual work to remove the artificial effects from the overlapping background fluid, thus providing an independent and accurate solid material description. Both fluid and solid solvers use finite element approach where non-uniform and unstructured meshes can be used to handle complex geometries.

The versatility of the mIFEM provides us the perfect tool in modeling the vocal folds vibration. In this study, the dynamics of the vocal folds motion and deformation are governed by visco-elastic constitutive law and solved on a Lagrangian mesh using the finite element method, while the fluid is governed by Navier–Stokes equations describing compressible isentropic air that are represented using an Eulerian background mesh. We further utilize the numerical tool to perform a series of parametric studies using a range of lung pressure and vocal folds stiffness to evaluate the effects in the fundamental frequency and magnitude of the self-sustained vibration.

In this paper, we will first present the numerical method formulated for compressible fluid interacting with aeroelastic structures in Section  2. The solid model also includes a contact model. The 2-D vocal folds model is presented in Section  3. With a successful mesh convergence study, we first verify the vocal folds vibration in terms of vibration frequency and magnitude. A parametric study is conducted to further analyze the effect of lung pressure and vocal folds stiffness. Finally, the conclusions are drawn in Section  4.

Section snippets

Basic concept of the immersed finite element methods

The immersed finite element method (IFEM) is a volume-based finite element fluid–structure interaction numerical method. Finite elements are not only used as a way of discretization, but the entire IFEM formulation is derived from the weak form, or principle of virtual work, where the total work in the entire fluid–structure coupled system is maintained. Even though the concept of “immersed” is similar to that of the immersed boundary (IB) method  [15] where the solid is immersed in the fluid

Vocal folds physiology

The human vocal folds are a pair of pliable structures located within the larynx at the top of the trachea. In order to produce sound in voice, the lungs blow air against the pair of closed vocal folds causing the immediate increase of the air pressure beneath the vocal folds. This pressure pushes the two vocal folds apart from each other and a jet of air, sometimes called glottal jet, escapes from the gap between the vocal folds, as illustrated in Fig. 5(a). As the air rapidly flows through

Conclusions and outlook

In this work, we successfully developed a numerical tool to simulate a self-sustained vocal fold vibration. This simulation involves a complex fluid–structure interaction process and requires a robust numerical technique. Based on our modified immersed finite element method (mIFEM) that we have been developing over the past decade to study fluid–structure interactions, two additional features are implemented: fluid compressibility and solid contact model. The governing equations for this

Acknowledgments

This work is partially supported by NIH   2R01DC005642-10A1. LTZ would like to acknowledge the supports from NSF-ACI   1126125 and NSFC   11550110185. The computational resources are generously provided by the Center of Computational Innovations (CCI) at Rensselaer Polytechnic Institute. Finally, we would like to thank the reviewers for providing very constructive and valuable suggestions during the revision process.

References (56)

  • D. Torres et al.

    The point-set method: front-tracking without connectivity

    J. Comput. Phys.

    (2000)
  • L. Zhang et al.

    A parallized meshfree method with boundary enrichment for large-scale cfd

    J. Comput. Phys.

    (2002)
  • J. Simo et al.

    An augmented lagrangian treatment of contact problems involving friction

    Comput. Struct.

    (1992)
  • D.K. Chhetri et al.

    Measurement of young’s modulus of vocal folds by indentation

    J. Voice

    (2011)
  • J.J. Jiang et al.

    Measurement of vocal fold intraglottal pressure and impact stress

    J. Voice

    (1994)
  • K. Ishizaka et al.

    Synthesis of voiced sounds from a two-mass model of the vocal cords

    Bell Syst. Tech. J.

    (1972)
  • I. Titze

    Rules for controlling low-dimensional vocal fold models with muscle activation

    J. Acoust. Soc. Am.

    (2002)
  • J. Jiang et al.

    Modeling of chaotic vibrations in symmetric vocal folds

    J. Acoust. Soc. Am.

    (2001)
  • R. Chan et al.

    Dependence of phonation threshold pressure on vocal tract acoustics and vocal fold tissue mechanics

    J. Acoust. Soc. Am.

    (2006)
  • F. Alipour et al.

    A finite-element model of vocal-fold vibration

    J. Acoust. Soc. Am.

    (2000)
  • C. Tao et al.

    Anterior-posterior biphonation in finite element model of vocal fold vibration

    J. Acoust. Soc. Am.

    (2006)
  • C. Tao et al.

    Simulation of vocal fold impact pressure with a self-oscillating finite-element model

    J. Acoust. Soc. Am.

    (2006)
  • S.H.F. Scott L.~Thomson, Luc~Mongeau

    Aerodynamic transfer of energy to the vocal folders

    J. Acoust. Soc. Am.

    (2005)
  • R. Mittal et al.

    Toward a simulation-based tool for the treatment of vocal fold paralysis

    Front. Physiol.

    (2011)
  • H. Luo et al.

    Analysis of flow-structure interaction in the larynx during phonation using an immersed-boundary method

    J. Acoust. Soc. Am.

    (2009)
  • X. Zheng et al.

    A computational study of the effect of false vocal folds on glottal flow and vocal fold vibration during phonation

    Ann. Biomed. Eng.

    (2009)
  • C. Peskin

    The immersed boundary method

    Acta Numer.

    (2002)
  • X.Wang et al.

    Interpolation functions in the immersed boundary and finite element methods

    Comput. Mech.

    (2010)
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    Current affiliation: ANSYS Inc., United States.

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