On suitable inlet boundary conditions for fluid-structure interaction problems in a channel. (English) Zbl 07088738

Summary: We are interested in the numerical solution of a two-dimensional fluid-structure interaction problem. A special attention is paid to the choice of physically relevant inlet boundary conditions for the case of channel closing. Three types of the inlet boundary conditions are considered. Beside the classical Dirichlet and the do-nothing boundary conditions also a generalized boundary condition motivated by the penalization prescription of the Dirichlet boundary condition is applied. The fluid flow is described by the incompressible Navier-Stokes equations in the arbitrary Lagrangian-Eulerian (ALE) form and the elastic body creating a part of the channel wall is modelled with the aid of linear elasticity. Both models are coupled with the boundary conditions prescribed at the common interface.
The elastic and the fluid flow problems are approximated by the finite element method. The detailed derivation of the weak formulation including the boundary conditions is presented. The pseudo-elastic approach for construction of the ALE mapping is used. Results of numerical simulations for three considered inlet boundary conditions are compared. The flutter velocity is determined for a specific model problem and it is shown that the boundary condition with the penalization approach is suitable for the case of the fluid flow in a channel with vibrating walls.


76D05 Navier-Stokes equations for incompressible viscous fluids
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs


Full Text: DOI


[1] Babuška, I., The finite element method with penalty, Math. Comput. 27 (1973), 221-228 · Zbl 0299.65057
[2] Bodnár, T.; Galdi, G. P.; Nečasová, Š.; (eds.), Fluid-Structure Interaction and Biomedical Applications, Advances in Mathematical Fluid Mechanics, Birkhäuser/Springer, Basel (2014) · Zbl 1300.76003
[3] Braack, M.; Mucha, P. B., Directional do-nothing condition for the Navier-Stokes equations, J. Comput. Math. 32 (2014), 507-521 · Zbl 1324.76015
[4] Curnier, A., Computational Methods in Solid Mechanics, Solid Mechanics and Its Applications 29 Kluwer Academic Publishers Group, Dordrecht (1994) · Zbl 0815.73003
[5] Daily, D. J.; Thomson, S. L., Acoustically-coupled flow-induced vibration of a computational vocal fold model, Comput. Struct. 116 (2013), 50-58
[6] Davis, T. A., Direct Methods for Sparse Linear Systems, Fundamentals of Algorithms 2, Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2006) · Zbl 1119.65021
[7] Diez, N. G.; Belfroid, S.; Golliard, J.; (eds.), Flow-Induced Vibration & Noise. Proceedings of 11th International Conference on Flow Induced Vibration & Noise, TNO, Delft, The Hague, The Netherlands (2016)
[8] Dowell, E. H., A Modern Course in Aeroelasticity, Solid Mechanics and Its Applications 217, Springer, Cham (2004) · Zbl 1297.74001
[9] Feistauer, M.; Hasnedlová-Prokopová, J.; Horáček, J.; Kosík, A.; Kučera, V., DGFEM for dynamical systems describing interaction of compressible fluid and structures, J. Comput. Appl. Math. 254 (2013), 17-30 · Zbl 1290.65089
[10] Feistauer, M.; Sváček, P.; Horáček, J., Numerical simulation of fluid-structure interaction problems with applications to flow in vocal folds, Fluid-Structure Interaction and Biomedical Applications T. Bodnár et al. Advances in Mathematical Fluid Mechanics, Birkhäuser/Springer, Basel (2014), 321-393 · Zbl 1446.76017
[11] Formaggia, L.; Parolini, N.; Pischedda, M.; Riccobene, C., Geometrical multi-scale modeling of liquid packaging system: an example of scientific cross-fertilization, 19th European Conference on Mathematics for Industry 6 pages (2016)
[12] Gelhard, T.; Lube, G.; Olshanskii, M. A.; Starcke, J.-H., Stabilized finite element schemes with LBB-stable elements for incompressible flows, J. Comput. Appl. Math. 177 (2005), 243-267 · Zbl 1063.76054
[13] Girault, V.; Raviart, P.-A., Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms, Springer Series in Computational Mathematics 5, Springer, Cham (1986),\99999DOI99999 10.1007/978-3-642-61623-5 \goodbreak · Zbl 0585.65077
[14] Horáček, J.; Radolf, V. V.; Bula, V.; Košina, J., Experimental modelling of phonation using artificial models of human vocal folds and vocal tracts, V. Fuis Engineering Mechanics 2017 Brno University of Technology, Faculty of Mechanical Engineering (2017), 382-385
[15] Horáček, J.; Šidlof, P.; Švec, J. G., Numerical simulation of self-oscillations of human vocal folds with Hertz model of impact forces, J. Fluids Struct. 20 (2005), 853-869
[16] Horáček, J.; Švec, J. G., Aeroelastic model of vocal-fold-shaped vibrating element for studying the phonation threshold, J. Fluids Struct. 16 (2002), 931-955
[17] Horáček, J.; Švec, J. G., Instability boundaries of a vocal fold modelled as a flexibly supported rigid body vibrating in a channel conveying fluid, ASME 2002 International Mechanical Engineering Congress and Exposition American Society of Mechanical Engineers (2002), 1043-1054
[18] Johnson, C., Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press, Cambridge (1987) · Zbl 0628.65098
[19] Kaltenbacher, M.; Zörner, S.; Hüppe, A., On the importance of strong fluid-solid coupling with application to human phonation, Prog. Comput. Fluid Dyn. 14 (2014), 2-13 · Zbl 1400.76041
[20] Link, G.; Kaltenbacher, M.; Breuer, M.; Döllinger, M., A 2D finite-element scheme for fluid-solid-acoustic interactions and its application to human phonation, Comput. Methods Appl. Mech. Eng. 198 (2009), 3321-3334 · Zbl 1230.74188
[21] Sadeghi, H.; Kniesburges, S.; Kaltenbacher, M.; Schützenberger, A.; Döllinger, M., Computational models of laryngeal aerodynamics: Potentials and numerical costs, Journal of Voice (2018)
[22] Seo, J. H.; Mittal, R., A high-order immersed boundary method for acoustic wave scattering and low-Mach number flow-induced sound in complex geometries, J. Comput. Phys. 230 (2011), 1000-1019 · Zbl 1391.76698
[23] Šidlof, P.; Kolář, J.; Peukert, P., Flow-induced vibration of a long flexible sheet in tangential flow, D. Šimurda, T. Bodnár Topical Problems of Fluid Mechanics 2018 Institute of Thermomechanics, The Czech Academy of Sciences, Praha (2018), 251-256
[24] Slaughter, W. S., The Linearized Theory of Elasticity, Birkhäuser, Boston (2002) · Zbl 0999.74002
[25] Sváček, P.; Horáček, J., Numerical simulation of glottal flow in interaction with self oscillating vocal folds: comparison of finite element approximation with a simplified model, Commun. Comput. Phys. 12 (2012), 789-806
[26] Sváček, P.; Horáček, J., Finite element approximation of flow induced vibrations of human vocal folds model: effects of inflow boundary conditions and the length of subglottal and supraglottal channel on phonation onset, Appl. Math. Comput. 319 (2018), 178-194 · Zbl 1426.76311
[27] Takashi, N.; Hughes, T. J. R., An arbitrary Lagrangian-Eulerian finite element method for interaction of fluid and a rigid body, Comput. Methods Appl. Mech. Eng. 95 (1992), 115-138 · Zbl 0756.76047
[28] Valášek, J.; Kaltenbacher, M.; Sváček, P., On the application of acoustic analogies in the numerical simulation of human phonation process, Flow, Turbul. Combust. (2018), 1-15
[29] Valášek, J.; Sváček, P.; Horáček, J., Numerical solution of fluid-structure interaction represented by human vocal folds in airflow, EPJ Web of Conferences 114 (2016), Article No. 02130, 6 pages
[30] Valášek, J.; Sváček, P.; Horáček, J., On finite element approximation of flow induced vibration of elastic structure, Programs and Algorithms of Numerical Mathematics 18. Proceedings of the 18th Seminar (PANM), 2016 Institute of Mathematics, Czech Academy of Sciences, Praha (2017), 144-153 · Zbl 1413.65374
[31] Venkatramani, J.; Nair, V.; Sujith, R. I.; Gupta, S.; Sarkar, S., Multi-fractality in aeroelastic response as a precursor to flutter, J. Sound Vib. 386 (2017), 390-406
[32] Zorner, S., Numerical Simulation Method for a Precise Calculation of the Human Phonation Under Realistic Conditions, Ph.D. Thesis, Technische Uuniversität Wien (2013)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.