Space-time discontinuous Galerkin method for the solution of fluid-structure interaction. (English) Zbl 1499.65479

Balázsová, Monika; Feistauer, Miloslav; Horáček, Jaromír; Hadrava, Martin; Kosík, Adam

Appl. Math., Praha 63, No. 6, 739-764 (2018).

Summary: The paper is concerned with the application of the space-time discontinuous Galerkin method (STDGM) to the numerical solution of the interaction of a compressible flow and an elastic structure. The flow is described by the system of compressible Navier-Stokes equations written in the conservative form. They are coupled with the dynamic elasticity system of equations describing the deformation of the elastic body, induced by the aerodynamical force on the interface between the gas and the elastic structure. The domain occupied by the fluid depends on time. It is taken into account in the Navier-Stokes equations rewritten with the aid of the arbitrary Lagrangian-Eulerian (ALE) method. The resulting coupled system is discretized by the STDGM using piecewise polynomial approximations of the sought solution both in space and time. The developed method can be applied to the solution of the compressible flow for a wide range of Mach numbers and Reynolds numbers. For the simulation of elastic deformations two models are used: the linear elasticity model and the nonlinear neo-Hookean model. The main goal is to show the robustness and applicability of the method to the simulation of the air flow in a simplified model of human vocal tract and the flow induced vocal folds vibrations. It will also be shown that in this case the linear elasticity model is not adequate and it is necessary to apply the nonlinear model.

MSC:

65M60	Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M06	Finite difference methods for initial value and initial-boundary value problems involving PDEs
65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
74B05	Classical linear elasticity
74B20	Nonlinear elasticity
74F10	Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
76A15	Liquid crystals

Keywords:

nonstationary compressible Navier-Stokes equations; time-dependent domain; arbitrary Lagrangian-Eulerian method; linear and nonlinear dynamic elasticity; space-time discontinuous Galerkin method; vocal folds vibrations

Software:

UMFPACK