Adam Kosík

DGFEM for dynamical systems describing interaction of compressible fluid and structures

Abstract The paper is concerned with the numerical solution of flow-induced vibrations of elastic structures. The dependence on time of the domain occupied by the fluid is taken into account with the aid of the ALE (Arbitrary Lagrangian–Eulerian) formulation of the compressible Navier–Stokes equations. The deformation of the elastic body, caused by aeroelastic forces, is described by the linear dynamical elasticity equations. These two systems are coupled by transmission conditions. The flow problem is discretized by the discontinuous Galerkin finite element method (DGFEM) in space and by the backward difference formula (BDF) in time. The structural problem is discretized by conforming finite elements and the Newmark method. The fluid–structure interaction is realized via weak or strong coupling algorithms. The developed technique is tested by numerical experiments and applied to the simulation of vibrations of vocal folds during phonation onset.

On the stability of the space–time discontinuous Galerkin method for the numerical solution of nonstationary nonlinear convection–diffusion problems

Abstract The subject of this paper is the analysis of the space-time discontinuous Galerkin method for the solution of nonstationary, nonlinear, convection-diffusion problems. In the formulation of the numerical scheme, the nonsymmetric, symmetric and incomplete versions of the discretization of diffusion terms and interior and boundary penalty are used. Then error estimates are briefly characterized. The main attention is paid to the investigation of unconditional stability of the method. An important tool is the concept of the discrete characteristic function. Theoretical results are accompanied by numerical experiments.

Numerical Simulation of Interaction of Human Vocal Folds and Fluid Flow

Our goal is to simulate airflow in human vocal folds and their flow-induced vibrations. We consider two-dimensional viscous incompressible flow in a time-dependent domain. The fluid flow is described by the Navier-Stokes equations in the arbitrary Lagrangian-Eulerian formulation. The flow problem is coupled with the elastic behaviour of the solid bodies. The developed solution of the coupled problem based on the finite element method is demonstrated by numerical experiments.

Space-Time Discontinuous Galerkin Method for the Problem of Linear Elasticity

The subject of this paper is the numerical solution of the problem of dynamic linear elasticity by several time-discretization techniques based on the application of the discontinuous Galerkin (DG) method in space. In the formulation of the numerical scheme, the nonsymmetric, symmetric and incomplete versions of the discretization of the elasticity term and the interior and boundary penalty are used. The DG space discretization is combined with the backward-Euler, second-order backward-difference formula and DG time discretization. Finally, we present some test problems.

Two Dimensional Compressible Fluid-Structure Interaction Model Using DGFEM

The subject of this paper is the numerical solution of the interaction of compressible flow and an elastic body with a special emphasis on the simulation of vibrations of vocal folds during phonation onset. The time-dependence of the domain occupied by the fluid is treated by the ALE (Arbitrary Lagrangian-Eulerian) method and the compressible Navier-Stokes equations are written in the ALE form. The deformation of the elastic body, caused by the aeroelastic forces, is described by the linear dynamical elasticity equations. Both these systems are coupled by transmission conditions. For the space-discretization of the flow problem the discontinuous Galerkin finite element method (DGFEM) is used. The time-discretization is realized by the backward difference formula (BDF). The structural problem is discretized by the conforming finite element method and the Newmark method. The results of the use of two different couplings and their comparison are presented.

Discontinuous Galerkin Method for the Solution of Elasto-Dynamic and Fluid-Structure Interaction Problems

This paper is concerned with the numerical solution of dynamic elasticity by the discontinuous Galerkin (dG) method. We consider the linear and nonlinear St. Venant-Kirchhoff model. The dynamic elasticity problem is split into two systems of first order in time. They are discretized by the discontinuous Galerkin method in space and backward difference formula in time. The developed method is tested by numerical experiments. Then the method is combined with the space-time dG method for the solution of compressible flow in a time dependent domain and used for the numerical simulation of fluid-structure interaction.

Analysis of Space-Time DGFEM for the Solution of Nonstationary Nonlinear Convection-Diffusion Problems

The subject of this paper is the analysis of the space-time discontinuous Galerkin method for the solution of nonstationary, nonlinear, convection-diffusion problems. In the formulation of the numerical scheme, the nonsymmetric, symmetric and incomplete versions of the discretization of diffusion terms and interior and boundary penalty are used. Then error estimates derived under a sufficient regularity of the exact solution are briefly characterized. The main attention is paid to the investigation of unconditional stability of the method. An important tool is the concept of the discrete characteristic function. The dominating convection case is not considered. Theoretical results are accompanied by numerical experiments.

The Interaction of Compressible Flow and an Elastic Structure Using Discontinuous Galerkin Method

In this paper we are concerned with the numerical simulation of the interaction of fluid flow and an elastic structure in a 2D domain. For each individual problem we employ the discretization by the discontinuous Galerkin finite element method (DGM). We describe the application of the DGM to the problem of compressible fluid flow in a time-dependent domain and also to the dynamic problem of the deformation of an elastic body. Finally, we present our approach to the coupling of these two independent problems: both are solved separately at a given time instant, but we require the approximate solutions to satisfy certain transient conditions. These transient conditions are met through several inner iterations. In each iteration a calculation of both the elastic body deformation problem and the problem of the compressible fluid flow is performed. The presented method can be applied to solve a selection of problems of biomechanics and aviation. Our numerical experiments are inspired by the simulation of airflow in human vocal folds, which implies the choice of the properties of the flowing fluid and the material properties of the elastic body. The results are post-processed in order to get a visualization of the approximate solution. We are especially interested in the visualization of the elastic body deformation and the visualization of some chosen physical quantities of the flow.

Numerical simulation of the interaction between a nonlinear elastic structure and compressible flow by the discontinuous Galerkin method

This paper is concerned with the numerical simulation of the interaction of compressible viscous flow with a nonlinear elastic structure. The flow is described by the compressible Navier-Stokes equations written in the arbitrary Lagrangian-Eulerian (ALE) form. For the elastic deformation the St. Venant-Kirchhoff model is used. In the space discretization the discontinuous Galerkin finite element method (DGM) is applied both for the flow problem in a time-dependent domain and for the dynamic nonlinear elasticity system. We show that the DGM is applicable to the discretization of both problems. As a new result we particularly present the application of the DGM to the discretization of the dynamic nonlinear elasticity problem and the DGM solution of the fluid-structure interaction (FSI). The applicability of the developed technique is demonstrated by several numerical experiments. The main novelty of the paper is the application of the DGM to the FSI problem using the model of compressible flow coupled with nonlinear elasto-dynamic system.

Numerical Solution of Fluid-Structure Interaction by the Space-Time Discontinuous Galerkin Method

This paper is devoted to the numerical solution of the interaction of compressible viscous flow with elastic structures. The flow in a time-dependent domain is described by the compressible Navier-Stokes equations written in the ALE formulation and the deformation of elastic structures is described by the dynamic linear elasticity system. For each individual problem we employ the discretization by the space-time discontinuous Galerkin finite element method (ST-DGM). The flow and elasticity problems are coupled via transmission conditions. The developed method is tested by numerical experiments.