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Sváček, P.

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On mathematical models of airflow in a glottal channel model periodically closed by flow induced vocal folds vibration

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2021 November 8 - 10, 2021Srní36   conference with international participationthOn mathematical models of airflow in a glottal channel modelperiodically closed by flow induced vocal folds vibrationP. SváčekCzech Technical University in Prague, Faculty of Mechanical Engineering, Dep. of Technical Mathematics, Czech Republic1. IntroductionIn this paper a two-dimensional mathematical model of human voice creation is addressed.Voice production is a complex process involving airflow induced vibrations of (elastic) vocalfolds, [10]. The vibrations of the vocal folds generate a sound source, which is modified bythe acoustic resonances of vocal tract cavities. The vocal folds oscillations appear at the so-called phonation onset which is basically a flutter type of instability. The phonation onset ischaracterized by certain airflow rate and a certain prephonatory vocal folds position, see [3, 4].During the phonation a higher flow rates causes vibrations with increasing amplitudes, whichleads to collisions of the vocal folds and to the closure of the glottis.Consequently, the mathematical modelling of phonation process is challenging task: it needsto address the flow field, the structural deformation as well as acoustics, see e.g. [6, 11]. Asthe fundamental sound source is significantly influenced by the vocal folds contacts, this phe-nomena needs to be addressed properly in the model. This is rather difficult as it needs to beincluded not only to the structural model as impact forces but also to the fluid model, where itinfluences the geometrical domain deformation with possible topological changes as well as theartificial boundary conditions.As the airflow velocity in the human glottal region is lower than 100 m/s, the air flow ismodelled by the incompressible Navier-Stokes equations. The acoustic part can be modelled byan acoustic model, [9]. The addressed problem is a problem of fluid-structure interaction, wherea simplified model of the elastic structure is used, [3, 5]. The attention is paid to the problemof glottis closure. This is realized by a modification of the computational domain, use of anartificial porous media subdomain and a suitable modification of the inlet boundary condition.The described mathematical model is discretized with the aid of the stabilized finite elementmethod and numerical results are shown.2. Mathematical modelFirst, we consider a simplified two dimensional model of the computational domain during thephonation onset phase, i.e., in the case of vocal folds are vibrating with only small amplitudes.The two-dimensional computational domain Ωt for the fluid flow is considered as shown inFig. 1. In this case the boundary of the computational domain consists of fixed or deformablewalls denoted as ΓWt (it corresponds to the vibrating vocal folds with its upper ΓWt,up and lowerpart ΓWt,down as well as the other fixed walls ΓWf and the inlet ΓI and the outlet ΓO part of theboundary.241The air flow in the computational domain Ωt is modelled as incompressible fluid flow de-scribed by the system of the incompressible Navier-Stokes equations (cf. [2]) written in theALE form (cf. [7])DAuDt+ ((u − wD) · ∇)u = div τ f ,∇ · u = 0, (1)where u = (v1, v2) is the vector of fluid velocity, τ f = (τfij) is the fluid stress tensor given asτ f = −pI+2νD, D is the symmetric gradient tensor D(u) = 12(∇u+∇T u) with componentsdij =12( ∂ui∂xj+∂uj∂xi), p denotes the kinematic pressure (i.e. the pressure divided byt the constantfluid density) and ν > 0 is the constant kinematic fluid viscosity. In order to take into accountthe deformation of the computational domain Ωt the Arbitrary Lagrangian-Eulerian method isused. In this case by At the ALE mapping is denoted, which maps the reference configurationΩref = Ω0 onto the current configuration Ωt at any time t ∈ [0, T ], wD denotes the domainvelocity (i.e. the velocity of the point with a fixed reference), and DAuDtis the ALE derivative,i.e. the derivative with respect to the reference configuration Ωref .OI2g(t)LΓ Ω Γ ΓΓ ΓΓΓ ΓtWtWf WfWtupWfdownWfFig. 1. The computational domain Ωt with the boundary parts representing the (deformable and unde-formable) walls, the inlet and the outlet. The domain is shown in a symmetric configuration with thedashed line showing the axis of symmetryAt the boundary ∂Ωft of the computational domain formed by mutually disjoint parts ∂Ωft =ΓI ∪ ΓO ∪ ΓWt, the following boundary conditions are prescribed:a) u = wD on ΓWt,b)12(u · n)−u − n · τ f = 1ε(u − uI) on ΓI ,v)12(u · n)−u − n · τ f = prefn on ΓO,where n denotes the unit outward normal vector to ∂Ωft , uI is a prescribed inlet velocity, pref isa reference pressure value (pref = 0 in what follows), ε > 0 is a penalization parameter and α−denotes the negative part of a real number α. Let us emphasize that at ΓWf the domain velocityequals zero wD = 0, whereas at the moving wall ΓWt the domain velocity wD is equal to thevelocity of the surface. The boundary condition (2c) weakly imposes the Dirichlet boundarycondition u = uI using the penalization parameter ε > 0.2.1 Vocal fold vibrationsThe elastic deformation of the vibrating vocal fold is described using a simplified model withtwo degrees of freedom, i.e. the displacements w1(t) and w2(t) of the masses m1 and m2,242respectively (see Fig. 2). These displacements are governed by the equation of motion (see [3]for details)Mẅ + Bẇ + Kw = −F , (2)where M is the mass matrix of the system, K = diag(k1, k2, k4, k5) is the diagonal stiffnessmatrix of the system characterized by spring constants k1, k2, and B = ε1M+ ε2K is the matrixof the proportional structural damping, ε1, ε2 are the constants of the proportional damping.The mass matrix is given byM =m1 +m34m340 0m34m2 +m340 00 0 m4 +m64m640 0 m64m5 +m64 , (3)where m1,m2,m3 are the three masses shown in Fig. 2 (left) for the lower part of the vocalfold model and m4,m5,m6 are the three masses shown in Fig. 2 (right) for the upper part ofthe vocal fold model. The vector F = Fimp + Faero consists of the impact forces Fimp and theaerodynamical forces Faero = (F1, F2, F4, F5)T .The aerodynamical forces are evaluated as surface integrals from the aerodynamical quan-tites computed in the fluid part of the model, i.e. by the (kinematic) pressure p and by thegradient of the flow velocity u = (u1, u2). The displacement of the surface ΓdownWt is determinedin terms of w1, w2, and it is used as boundary condition for the displacement of any point of thecomputational domain Ωref0 onto the domain Ωt. Similarly the displacement of the surface ΓupWtis determined in terms of the displacements w4, w5.21121221FFmkkmwCoGm3wFFmkkmwCoGmw545445465Fig. 2. The aeroelastic model of the down part (left) and the upper part (right) of the vocal fold2.2 Contact problemIn order to model the contact of the vocal folds the flow of the air is approximated on a modi-fication of the computational domain still denoted by Ωt, which is formed of the original fluiddomain Ωft and of the part of the domain ΩPt , which is the part of the domain Ωt assumed to beoccupied by vocal folds. In practical problem it is realized by determination of the gap 2g(t)between the two vocal folds. In case when g(t) ≥ gmin > 0 no modification of the computa-tional domain is needed (here gmin a suitable minimal gap threshold). For the case g(t) ≤ gmin,the displacements of the boundary parts ΓupWt and ΓdownWt is modified, which leads to artificiallycreated fictitious porous media domain Ωpt . The flow in ΩPt is then governed by the modifiedequationsDAuDt+ ((u − wD) · ∇)u + σP u = div τ f ,where the coefficient σP corresponds to the artificial permeability P of the fictitious porousmedia. This approach can be also understand as penalization, see [1].243Glottal area detail Glottal area detailFig. 3. The flow velocity distribution in the computational domain during two time instants for the casecorresponding to post-flutter simulation from [8]AcknowledgmentThis work was supported by the Czech Science Foundation under the Grant No. 19 - 04477S.References[1] Angot, P., Bruneau, C. H., Fabrie, P., A penalization method to take into account obstacles inincompressible viscous flows, Numerische Mathematik 81 (1999) 497-520.[2] Feistauer, M., Mathematical methods in fluid dynamics, Longman Scientific & Technical, Harlow,1993.[3] Horáček, J., Šidlof, P., Švec, J. G., Numerical simulation of self-oscillations of human vocal foldswith Hertz model of impact forces, Journal of Fluids and Structures 20 (6) (2005) 853-869.[4] Horáček, J., Švec, J. G., Aeroelastic model of vocal-fold-shaped vibrating element for studyingthe phonation threshold, Journal of Fluids and Structures 16 (7) (2002) 931-955.[5] Ishizaka, K., Flanagan, J. L., Synthesis of voiced sounds from a two-mass model of the vocalcoords, The Bell System Technical Journal 51 (1972) 1233-1268.[6] Link, G., Kaltenbacher, M., Breuer, M., Döllinger, M., A 2D finite element scheme for fluid-solid-acoustic interactions and its application to human phonation, Computation Methods in AppliedMechanical Engineering 198 (2009) 3321-3334.[7] Nomura, T., Hughes, T. J. R., An arbitrary Lagrangian-Eulerian finite element method for inter-action of fluid and a rigid body, Computer Methods in Applied Mechanics and Engineering 95(1992) 115-138.[8] Sváček, P., Horáček, J., Numerical simulation of glottal flow in interaction with self oscillating vo-cal folds: Comparison of finite element approximation with a simplified model, Communicationsin Computational Physics 12 (3) (2012) 789-806.[9] Šidlof, P., Zörner, S., Hüppe, A., A hybrid approach to computational aeroacoustics of humanvoice production, Biomechanics and Modeling in Mechanobiology 14 (2015) 473-488.[10] Titze, I. R., The myoelastic aerodynamic theory of phonation, National Center for Voice andSpeech, U.S.A., 2006.[11] Valášek, J., Kaltenbacher, M., Sv’aček, P., On the application of acoustic analogies in the numer-ical simulation of human phonation process, Flow Turbulence Combust 102 (2019) 129-143.244

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On mathematical models of airflow in a glottal channel model periodically closed by flow induced vocal folds vibration

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