# Vibrations of Nonlinear Elastic Structure Excited by Compressible Flow
## 1 Introduction  
The simulation of an interaction of flow and elastic bodies plays an important role in a number of areas of science, engineering and technology. However, the FSI plays an important role also in bio-medicine. Namely, the flow in veins or the heart or flow-induced vibration of human vocal folds (VFs) producing voice is intensively studied. The problems of FSI have been studied by a number of different methods in several books (e.g., [1, 2, 3, 4, 5, 6, 7]). The mathematical and numerical modeling of hemodynamics uses the fact that blood is an incompressible liquid and its flow can be described by the incompressible Navier-Stokes (N-S) equations. The simulation of the airflow in the vocal tract is usually simulated with the aid of the same system for the description of the airflow in spite of the air being a compressible gas, using the fact that airflow is slow. Maximum glottal jet velocity is ca 45 m/s and computations at low Mach numbers (\(M<0.1\)) are difficult, so the numerical modeling of flow-induced vibrations of the VFs prevails with the usage of incompressible flow; see, e.g., [8, 9, 10]. Nevertheless, in the solution of the compressible N-S equations, weuse a robust numerical method with respect to the Mach number (e.g., [11]). This is the reason that, in our research, we prefer to use the model of compressible flow. As for the motion, vibrations and deformation of VFs induced by the airflow in the vocal tract, several approaches have been used. For example, in [12], the unsteadiness of the flow is modeled by a prescribed periodic motion of a part of the channel wall with large amplitudes, nearly closing the channel. The numerical solution is implemented using the finite volume method and the predictor-corrector MacCormack scheme with artificial viscosity. For computation of the acoustic output, the driven VFs oscillation can be used for modeling the acoustic sources in incompressible, unsteady flow, followed by the usage of an acoustic analogy method; see, e.g., [13]. Various lumped parameter dynamic models of the VFs with several degrees of freedom are used for modeling the phonation onset and the VFs self-sustained vibrations; see, e.g., [14; 15; 16; 17]. The fluid flow in the glottal channel modeled by the incompressible N-S equations in the ALE form, discretized by the finite element (FE) method and in combination with the dynamic lumped model of the VFs with three degrees of freedom, was recently focused on for the introduction of a new inlet boundary condition using a penalization approach. This gives reliable results related to the flutter analysis of the system and physically real values of pressure and flow velocities when the channel is nearly closing during the VFs vibration; see [18]. In [19; 20], the interaction of compressible flow with a linear elastic body is solved. A survey can be found in [21]. In the numerical solution of FSI problems with compressible flow and nonlinear dynamic elasticity, it is necessary to overcome several obstacles: the numerical solution of compressible N-S equations in time-dependent domains, the solution of nonlinear elasticity problems, the realization of the interaction of these problems and the solution of large nonlinear algebraic systems. For the numerical solution of compressible viscous flow, one of the most attractive techniques appears to be the discontinuous Galerkin method (DGM). It is used in a number of works (see, e.g., [22; 23; 24; 25]). The situation is more complicated for the compressible flow in time-dependent domains. One possibility is to combine the DGM with the arbitrary ALE method. In [26], the ALE-DGM is applied to the solution of airfoil vibrations by the compressible flow. In [11; 19], the ALE-DGM is applied to the interaction of compressible flow with the vibrations of a linear dynamic elastic body solved by the combination of the standard FE method for the space discretization and the well-known Newmark time discretization.