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Numerical Simulation of Fluid–Structure Interaction Problems with Applications to Flow in Vocal Folds

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Fluid-Structure Interaction and Biomedical Applications

Part of the book series: Advances in Mathematical Fluid Mechanics ((AMFM))

Abstract

Recently, the numerical solution of FSI problems has become important also in biomechanics, among others in voice modelling. The numerical analysis of this case is very complicated: Human voice is created by passage of air flow between vocal folds, where the constriction formed by the vocal folds induces acceleration of the flow and vocal fold oscillations, which generates the sound. The modelling of such a complex phenomenon encounters many difficulties as it is a result of coupling complex fluid dynamics and structural behavior. We focus on mathematical and numerical modelling of nonlinear coupled problems of fluid–structure interactions (FSI). The main attention is paid to the mathematical description of a relevant problem and to the description of the applied numerical methods. The mathematical description consists of the elasticity equations describing the motion of an elastic structure, and the air flow modelled by the Navier–Stokes equations. Both models are coupled via interface conditions.

The solution of dynamic elasticity equations is realized with the aid of conforming finite elements or the elastic structure motion is modelled by a simplified model of vibrating rigid body. Both compressible and incompressible fluid model is considered. The approximation of flow in moving domains is treated with the aid of the arbitrary Lagrangian–Eulerian method. The incompressible Navier–Stokes equations are approximated by the stabilized finite element method. The compressible Navier–Stokes equations are discretized by the discontinuous Galerkin finite element method. The time discretization based on a semi-implicit linearized scheme is described and the solution of the coupled problem of FSI is realized by a coupling algorithm.

MSC2010: 74F10, 76Z05, 76M10, 74S05, 74H15

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Acknowledgements

This research was supported by grants No. 13-00522S and No. P101/11/0207 of the Czech Science Foundation. M. Feistauer acknowledges his membership in the Nečas Center for Mathematical Modeling (http://ncmm.karlin.mff.cuni.cz).

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Authors and Affiliations

  1. Faculty of Mathematics and Physics, Department of Numerical Mathematics, Charles University in Prague, Prague, Czech Republic

    Miloslav Feistauer

  2. Faculty of Mechanical Engineering, Department of Technical Mathematics, Czech Technical University in Prague, Prague, Czech Republic

    Petr Sváček

  3. Institute of Thermomechanics, Academy of Sciences of the Czech Republic, Prague, Czech Republic

    Jaromír Horáček

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  3. Jaromír Horáček
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Correspondence to Miloslav Feistauer .

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  1. Department of Technical Mathematics, Czech Technical University in Prague, Prague, Czech Republic

    Tomáš Bodnár

  2. Swanson School of Engineering, University of Pittsburgh, Pittsburgh, Pennsylvania, USA

    Giovanni P. Galdi

  3. Institute of Mathematics, Academy of Sciences of the Czech Republic, Prague 1, Czech Republic

    Šárka Nečasová

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Feistauer, M., Sváček, P., Horáček, J. (2014). Numerical Simulation of Fluid–Structure Interaction Problems with Applications to Flow in Vocal Folds. In: Bodnár, T., Galdi, G., Nečasová, Š. (eds) Fluid-Structure Interaction and Biomedical Applications. Advances in Mathematical Fluid Mechanics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0822-4_5

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