TOPICAL PROBLEMS OF FLUID MECHANICS 245 ON NUMERICAL APPROXIMATION OF FLUID-STRUCTURE INTERACTIONS OF AIR FLOW WITH A MODEL OF VOCAL FOLDS

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1 TOPICAL PROBLEMS OF FLUID MECHANICS 245 ON NUMERICAL APPROXIMATION OF FLUID-STRUCTURE INTERACTIONS OF AIR FLOW WITH A MODEL OF VOCAL FOLDS J. Valášek 1, J. Horáček 2, P. Sváček 1 1 Department of Technical Mathematic, Faculty of Mechanical Engineering, Czech Technical Univerity in Prague, Karlovo nám. 13, Praha 2, , Czech Republic 2 Intitute of Thermomechanic, Czech Academy of Science, Dolejškova 5, Praha 8 Abtract Thi paper deal with flow driven vibration of an elatic body. Our goal i to develop and mathematically decribe a implified model of the human vocal fold. The developed numerical cheme for vicou incompreible fluid flow in ALE formulation and the elatic body are implemented by two olver, pecific for each domain. The tudied problem i coupled by Dirichlet-Neumann boundary condition. Both olver are baed on the finite element method. Particularly, for the fluid model the crogrid element are ued. Numerical reult focu on the verification of the developed program. Keyword: finite element method, 2D Navier-Stoke equation, vocal fold, aeroelaticity 1 Introduction The flow driven vibration of elatic bodie i a problem invetigated and olved in many technical application. Thi paper deal biomechanic of human vocal fold, ee e.g. [11]. Human voice i one of baic human being characteritic and it play an important role in the quality of a human life. The air flow from lung excite the vibration of the human fold which etting i influenced by the human mucle and caue the human voice production. Therefore the attention i further devoted for better undertanding of thi complex proce. Thi problem wa tudied in many paper, for example ee [6], [10]. Neverthele, a uitable benchmark for fluid-tructure interaction (FSI) problem i till miing. Here, we focu on the development of numerical method, careful verification of of all it component, and on verification of the olution of the coupled problem. The verification i baed on a benchmark problem introduced here, where only low Reynold number i involved and thu the convergence of the numerical method can be eaily tudied. The numerical method i baed on finite element method (FEM). For the fluid olver the cro-grid P1 element and for the elatic part P1 element are implemented. The ALE method i involved for handling with the time-dependent domain. For the coupled problem the emi-implicit cheme i ued. Figure 1: Schema of vocal fold model with boundarie marked before and after deformation.

2 246 Prague, February 11-13, Mathematical model For the ake of implicity our tudy i retricted to a 2D model problem. A cheme of the ued model i hown in Figure 1, where Ω ref denote the reference repreentation of the tructure, Ωf ref i the domain occupied by the fluid at the time intant t = 0 (alo reference) and Γ Wref = Γ W0 i the common interface. At intant time t the fluid domain Ω f ref turn to Ωf t. The deformated domain Ω = Ω ref at time t i handled in reference coordinate. Elatic tructure. The vocal fold model i conidered a the 2D compliant tructure with the aumption of mall diplacement. Uing the generalized Hook law, the Cauchy tre tenor τij i given by, ee for example [1], τij = C ijkl e kl, (1) where C ijkl denote the fourth order tenor of elatic coefficient and e jk i the train tenor given by e jk = 1 ( uj + u ) k. (2) 2 x k x j Under the aumption of the iotropic body equation (1) can be written in the form τ ij = λ div u + 2µ e ij, (3) where u denote the diplacement, λ, µ are Lame coefficient dependent on the Young modulu of elaticity E and the Poion ratio σ a λ = E σ (1 + σ )(1 2σ ), E µ = 2(1 + σ ). The deformation of the tructure i given by the equation of motion ρ 2 u t 2 τ ij (u) x j = f in Ω (0, T), (4) where the vector f i the volume denity of an acting force and ρ i the tructure denity. Equation (4) i upplied with the following initial and the boundary condition a) u(x, 0) = u 0 (x) for x Ω, u b) t (x, 0) = u 1(x) for x Ω, (5) c) u(x, t) = u Dir (x, t) for x Γ Dir, t (0, T), d) τ ij(x, t) n j(x) = q i (x, t), for x Γ W t, t (0, T), where the boundarie Γ Wt, Γ Dir are hown in Figure 1. ALE method. During the interaction the vocal fold change it hape, thu the fluid domain i alo deformed. Therefore we ue o called Arbitrary-Lagrangian method (ALE) to enable the flow imulation in the time-dependent domain. ALE method i the generalization of the Eulerian and the Lagrangian method. The baic aumption i that there exit difeomorphim A t : Ω f ref Ω f t, e.g. X Ω f ref x = A t (X) Ω f t. In addition mapping A t mut meet the condition A t C(Ω f ref ), A t ( Ω f ref t ) = Ωf t, t (0, T). (6) Uing he ALE mapping the domain velocity of the deformation i defined by w D (x, t) = ŵ D (A 1 t (x), t), t (0, T), x Ω f t, (7) where ŵ D i quantity defined on Ω f ref a ŵ D(X, t) = t A t(x), derivative atifie t (0, T), X Ω f ref. The ALE D A f f(x, t) = Dt t (A t(x), t) = f t (x, t) + w D(x, t) f(x, t), (8) where w D (x, t) i given by (7). Futher information i poible to find for example in the article [7].

3 TOPICAL PROBLEMS OF FLUID MECHANICS 247 Figure 2: The computational domain Ω f t at the time intant t and it boundarie. Flow model. The domain Ω f t at arbitrary time t i hown in Figure 2. The Γ f In i the inlet part of the boundary, Γ f Out i the outlet part of the boundary, Γf Dir repreent olid wall of the glottal channel urface and the only time dependent boundary Γ Wt repreent the interface on the vibrating vocal fold. The motion of the vicou incompreible fluid in the time dependent domain Ω f t i modelled by the Navier-Stoke equation in the ALE form D A v Dt + ((v w D ) )v ν f v + p = g f in Ω f t, div v = 0 in Ω f t, where v denote the fluid velocity and p i the kinematic preure, ν f i the kinematic vicoity of the fluid and g f i vector of acting force. The ytem of equation (9) i equipped with the boundary condition a) v(x, t) = 0 for x Γ f Dir, t (0, T), b) v(x, t) = v Dir (x, t) for x Γ f In, t (0, T), (10) c) (p(x, t) p ref ) n ν f v (x, t) = 0, n for x Γf Out, t (0, T), d) v(x, t) = w D (x, t) for x Γ Wt. The boundary condition on the interface Γ Wt i decribed later. Coupling condition. Deformation and location of interface Γ Wt in time t i dependent on (previou) acting of aerodynamical and elatic force. On the other hand thee force are dependent on poition and velocity of thi interface. Therefore we peak about a coupled problem. The hape of the interface Γ Wt in time t i the reult of force equilibrium between fluid and tructure. It poition i given by the deformation u, o Γ Wt = { x R 2 x = X + u(x, t), X Γ Wref }. (11) On the other hand aerodynamic force work on tructure interface via preure and hear force and the equilibrium i given by the equation 2 τij(x) n j(x) = j=1 2 σ f ij (x) n j(x), i = 1, 2, x Γ Wt, X Γ Wref, (12) j=1 where σ f ij i tre tenor of fluid n j (X) denote the component of the outer normal of the tructure domain to the interface Γ Wt=0 and x i computed from the known deformation u a x = X+u(X, t). The equation (12) can be rewritten a the Neumann boundary condition (5 d) q i (X, t) = 2 σ f ij (x) n j(x), i = 1, 2, x Γ Wt, t (0, T). (13) j=1 (9)

4 248 Prague, February 11-13, Numerical approximation The preented mathematical model repreented by the partial differential equation (4), (9) i dicretized in pace by the finite element method and in time by the Newmark and the backward Euler method, repectively. The time interval (0, T) i divided into the equiditant partition with a contant time tep t. Elatic tructure. The olution u i ought in the pace V = V V, where V = {f W 1,2 (Ω ) f = 0 on Γ Dir }, and W k,p (Ω) denote the Sobolev pace. With the ue of Hook law (1) and the ymmetry of tre tenor equation (4) i reformulated in the weak ene ρ 2u t 2 Φ dx + C ijkl e kl(u)e ij(φ) dx = f Φ dx + q Φ ds. (14) Ω Ω Ω Futhermore, the pace v i approximated by the finite dimenional pace V h V with the dimenion 2N h. Thu the dicrete olution can be written a a linear combination of bai function Φ j V h, i.e. u h (x, t) = 2N h j=1 α j(t)φ j (x). Uing thi and α = (α i ) the equation (3) now ha the form Γ Neu M T α + K T α = b(t), (15) where the element of the matrice M = (b ij ), K = (k ij ) and b = (b i ) are given by m ij = ρ Φ j Φ i dx, k ij = C klmn e mn(φ j )e kl(φ i ) dx, Ω Ω (b(t)) i = b i (t) = f Φ i dx + q Φ i ds. (16) Ω In the practical computation the Lagrange fnite element of the firt order are ued baed on a triangulation, which give the firt order of accuracy in pace. Newmark method. For the time dicretization the Newmark method i ued for the olution of ordinary differential equation of the econd order in a prototype form y (t) = f(t, y(t), y (t)) for t (0, T), y(0) = y 0, y (0) = y 0. (17) Γ Neu It application lead to the numerical cheme y n+1 = y n + ty n + t 2 (βf n+1 + ( 1 2 β)f n ) (18) y n+1 = y n + t (γf n+1 + (1 γ)f n ), (19) where f n = f(t n, y n, y n), f n+1 = f(t n+1, y n+1, y n+1) and the cheme i of econd order accuracy in time for the choice of the parameter β = 1 4, γ = 1 2, ee for example [2]. In the cae of equation (15) the value of y n+1, f n+1 i replaced by y n+1 obtained from the ytem ( M + β( t) 2 K ) y n+1 = gn+1. (20) }{{} =A Sytem of equation (20) i eay to be olved for example by conjugate gradient method, becaue the matrix A i compoed of two poitive defined matrice K, M. ALE method. The ALE mapping i treated uing the elatic analogy, becaue ALE method in fact map Ω f ref to deformated Ωf t, for dicrete contruction of ALE mapping i ued afore mentioned tationary part of the numerical cheme for elatic body, which i implified to K T d = 0, (21)

5 TOPICAL PROBLEMS OF FLUID MECHANICS 249 where d tay for the diplacement of the vertice of the triangulation τ h at the time intant t n+1. A boundary condition it i precribed Dirichlet boundary condition, which equal deformation of common interface on Γ Wtn+1 and otherwie equal zero d i = u(a 1 t n+1 (X i ), t n+1 ), X i Γ Wtn+1. (22) The Lame coefficient, which are ued during aembling of the matrix K T artificial and baed on article [5], according to thi formula in (21), i choen λ + µ = 0, λ = C avgdiam τ h diam K, (23) where C > 0 i a uitable contant, diam K i the diameter of the triangle K τ h and avgdiam τ h i the average element diameter of the triangulation τ h. So Lame contant are recaled on every triangle with repect to it area. Thi choice enure better behaviour at the interface, where a refined meh i ued. Flow model. For the flow model we ue the invere procedure - firtly the equation (9) i dicretized in time and then in pace. For time dicretization we ued backward Euler method uitable for ordinary differential equation of firt order. So the ALE derivative i approximated by D A v Dt (x n+1, t n+1 ) vn+1 (x n+1 ) v n (x n ). (24) t Denoting v n (x n+1 ) = v n (x n ), where x n = A tn (A 1 t n+1 (x n+1 )). the cheme read v n+1 v n t + ((v n+1 w n+1 D ) )vn+1 ν f v n+1 + p n+1 = g f,n+1, (25) v n+1 = 0. Then we continue with the weak formulation, where the velocity olution v in time t n+1 i ought in the functional pace X = W 1,2 (Ω f t n+1 ) and q M = L 2 (Ω f t n+1 ). Further, the pace of the tet function i defined by X = {f W 1,2 (Ω f t n+1 ) f = 0 on Γ f Dir Γf In Γf W tn+1 } W 1,2 (Ω f t n+1 ). For better arrangement it i written v n+1 a v and Ω f = Ω f t n+1. The weak formulation in pace i acquired by multiplication of the firt equation (25) by Φ X, integration over the whole domain Ω f t n+1 and by uing the Green theorem, that reult in the following equation ( v v n ), ϕ + ((v w D ) )v, ϕ) t Ω f + ν f ( v, ϕ) Ω f (p, div ϕ) Ω f = Ω f (q, div v) Ω f = 0, = (g f, ϕ) Ω f (p ref, ϕ n) L 2 (Γ f Out ), (26) where (, ) Ω tay for the calar product of L 2 (Ω f ) or [L 2 (Ω f )] 2 pace. The FEM then approximate pace X and M by the finite dimenion pace X h and M h, o the olution v v h can be expreed a v h (x) = 2Nh vel j=1 β j ϕ j (x), p h (x) = 2N vel h +N p h j=2n vel h +1 γ j q j (x). (27) Now by uing relation (27) in equation (26) we get the ytem ( ( ) ( A(v h ) B β g B 0) T =, (28) γ 0)

6 250 Prague, February 11-13, 2015 Figure 3: The crogrid finite element for incompreible problem (P1-crogrid/P1). vertice repreent preure variable and arrow are component of velocity. Dot at where A(v h ) = 1 t M + C(v h ) + D. The element of the matrice M = (m ij), C = (c ij ), D = (d ij ) are given by m ij = (ϕ j, ϕ i ) Ω f, c ij = ((v h w D ) )ϕ j, ϕ i ) Ω f, d ij = ν f ( ϕ j, ϕ i ) Ω f, (29) b ij = ( q j, div ϕ i ) Ω f, g i = (g f, ϕ i ) Ω f (p ref, ϕ i n) L2 (Γ f Out ) + ( un t, ϕ i) Ω f. The ytem of equation (28) i non-linear. For it olution the linearization v h = vn i ued. For the olution of the ytem (28) the mathematical library UMFPACK i employed, ee [3]. One of the apect of the FEM i that pace X h, M h cannot be choen randomly, but they mut atify the well-known Babuka-Brezzi condition, ee [4]. In thi article P1-crogrid/P1 element (ee Figure 3) are ued, which according to [8] atify it. Coupled problem. Lat part of our algoritm i the evaluation of the aerodynamical force. For that the velocity and preure value at adjacent triangle are ued. It mean, that we have according to (13) ( qi (X, t) = σ f ij (x) n j(x) = p + 2ν f 1 ( vi + v )) j n 2 x j x j, (30) i where n = (n 1, n 2) i the outer normal to boundary of tructure Γ Wt. Then it i evaluated in the middle of the common ide and with weight of one half added to the value in the vertice on thi ide. The value of aerodynamical force computed in vertice on the interface are then ued a dicrete verion of Neumann boundary condition of elatic olver. The algoritm olving coupled problem i implemented in emi-implicit form: We tart with the initial value v 0, p 0, u 0, q 0, A t0 and Ω f t 0. Then for n = 0, 1,... we proceed in computation in the following tep: 1. Baed on the preented cheme (20) it i acquired new olution u n+1 on n+1-th time layer, where q n+1 i extrapolated from q n. 2. From the known deformation u n+1 the ALE mapping A tn+1 i contructed by (21) and Ω f t n+1 i determined. Afterward we et w n+1 D (x) At (X) Atn (X) n+1 t, where x = A tn+1 (X). 3. We olve (29) and get v n+1, p n+1 defined on Ω f t n The value of the aerodynamical force on the interface are determined by (30) at time t n+1 from the known value v n+1, p n We et n := n + 1 and continue with the firt tep.

7 TOPICAL PROBLEMS OF FLUID MECHANICS Numerical reult The numerical reult for fluid flow interacting with the vocal fold model M5 uggeted by paper [9] are preented. The model M5 together with the triangulation i hown in Figure 4. Here, only one half of the channel wa ued a the computational domain with ymmetric boundary condition precribed at the top of the fluid domain (y = m). Figure 4: The triangulation of the computational domain Ωf0 and of the vocal fold model M5 (dimenion in [m]). Modal analyi. The eigenfrequencie of the vocal fold model wa determined by modal analyi. The olution of the ytem of ordinary differential equation Mu + Ku = 0, where the matrice M and K are given by (16), it i ought in the form u = ei ωj t uj, which lead to a generalized eigenvalue problem (K ωj2 M)uj = 0 = det(k ωj2 M) = 0. (31) The reult are hown in Table 1, where the Young modulu of elaticity wa E = 12 kpa, σ = 0, 4 inide Ωref and E = 100 kpa, σ = 0, 4 in a thin layer along the interface ΓWref. Frequency f1 f2 f3 f4 f5 [Hz] Frequency f1 f2 f3 x-direction y-direction Table 1: Left: The lowet eigenfrequencie obtained from the modal analyi of model M5. Right: The ignificant eigenfrequencie obtained from the pectral analyi of the time ignal at the point A. Dynamic of elatic tructure, energy conervation tet. Futhermore, the olution of the ytem (15) with zero body force and non-zero initial condition wa approximated by FEM, and the time ignal of a chooen point A [x = m, y = m] wa analyzed by Fourier tran 0 formation, ee Table 2. The initial condition were u1 = 0, u0 (x, y) = (y ) The frequencie obtained by the Fourier tranformation are hown in Table 2, where the motion of the point A wa analyzed. The frequencie agree well with the reult ummarized in Table 1. Another poibility, how to verify olver of elatic body, i to control energy conervation of vibrating tructure. The total energy E conit of um of kinetic and potential energy E = Ekin + Epot. Thee energie Ekin, Epot are given and then approximated in dicrete form a Z Z 1 T Ekin = ρ u dx u Mu, Epot = Cijkl eij (u)ekl (u) dx ut Ku. (32) Ω Ω

8 252 Prague, February 11-13, 2015 Figure 5: Graph of energy conervation, potential energy Epot i plotted with red colour and croe, kinetic energy Ekin i pictured by blue colour and behaviour of total energy E i hown with black colour. The reult hown in Figure 5 confirm energy conervation of implemented procedure (no dumping wa conidered). Flow in the channel with the fixed model of vocal fold. The flow olver wa teted for flow in the domain Ωfref without any interaction. Figure 6 how the approximation of the development of x-component velocity between time 0.05 and 0.3. The kinematic vicoity wa et ν f = 1, m/2 and the inlet boundary condition wa a parabolic profile: (Yi )(0.017 Yi ) vdir = 0.025, [Xi, Yi ] ΓfIn. (33) 0 Figure 6: The flow x-velocity pattern around the fixed model at ix different time intant t = to + j 0.05, (j = 0,..., 5). Figure are ordered from the left to the right.

9 TOPICAL PROBLEMS OF FLUID MECHANICS 253 The flow acceleration can be een in Figure 6 in the narrowet part of the channel. Reynold number for thi etting wa approximately 10. FSI tet. For teting interaction between fluid and elatic body, we adopted previou tet and to the elatic body we precribed all initial and boundary condition equal zero. The interaction wa enabled after time t = 0.1, when the velocity profile behind the vocal fold wa developed. The flow excited a periodic vibration of vocal fold with mall amplitude. Figure 7: Time evolution of x-diplacement of point A for the FSI tet (left) and it Fourier tranformation with dominant frequency f = 51 Hz (right). Figure 8: Time evolution of y-diplacement of point A for the FSI tet (left) and it Fourier tranformation with dominant frequency f = 51 Hz (right). Figure 9: Comparion of the x-diplacement of the point A computed with different time tep, t = (blue), t = (red with croe), for the FSI tet.

10 254 Prague, February 11-13, 2015 Figure 7 and 8 how time ignal of x-diplacement and y-diplacement of the point A together with their Fourier tranformation, repectively. The comparion of the time ignal computed by uing two different time tep in numerical procedure are in hown in Figure 9 demotrating a very good agreetment between thee olution. 5 Concluion The paper decribed the formulation of the FSI problem demontrated by the example of human fold vibration in airflow. The formulation ue FEM and the ALE method to imulate behaviour of the coupled ytem. The developed numerical cheme were implemented in an own program, which i able to handle complex geometrie. In the end the each part of the olver wa teted by imply tet, which verifie it baic functionality in the interaction problem. The acquired reult proved convergence of the computational cheme for low Reynold number and mall diplacement. Acknowledgment The financial upport for the preent project wa partly provided by the Czech Science Foundation under the Grant No. 101/11/0207 and project S GS 13/174/OHK2/3T/12. Reference [1] Brdička, M., Samek, L., Sopko, B.:Mechanika kontinua. Academia, Praha, [2] Curnier, A., Computational Method in Solid Mechanic, Kluwer Academic Publihing Group, Dodrecht, [3] Davi, T.: Umfpack. [mathematical library] Verion Univerity of Florida, USA. Available at: [4] Girault, V., Raviart, P., A.:Finite Element Method for Navier-Stoke Equation. Springer, Berlin, [5] Hadrava, M. & Feitauer M. & Horáček & Koík, A.: Dicontinuou Galerkin Method for the Problem of Linear Elaticity with Application to the Fluid-Structure Interaction. AIP Conference Proceeding, vol. 1558, pp ; [6] Koík, A. & Feitauer, M. & Horáček, J. & Sváček, P.: Numerical Simulation of Interaction of Human Vocal Fold and Fluid Flow Springer Proceeding in Phyic, vol. 139, pp , [7] Nomura, T. & Hughe, T. J. R.: An arbitrary Lagrangian-Eulerian finite element method for interaction of fluid and a rigid body. Computer Method in Applied Mechanic and Engineering, vol. 95, pp , [8] Quarteroni, A. & Valli, A.: Numerical Approximation of Partial Differential Equation. Springer, Berlin, [9] Scherer et al.: Intraglottal preure profile for a ymmetric and oblique glotti with a divergence angle of 10 degree Journal Acoutic Society of America, 4, 109, April [10] Sváček, P. & Horáček, J.: Numerical Simulation of Glottal Flow in Interaction with Self Ocillating Vocal Fold: Comparion of Finite Element Approximation with a Simplified Model, Communication in computational phyic, vol. 12, Iue: 3, pp , [11] Titze, I. R.: The Myoelatic Aerodynamic Theory of Phonation. National Center for Voice and Speech, USA, 2006.