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Solution of inverse problem for a flow around an airfoil
Šimák, Jan ; Feistauer, Miloslav (advisor) ; Felcman, Jiří (referee) ; Sváček, Petr (referee)
Title: Solution of inverse problem for a flow around an airfoil Author: Mgr. Jan Šimák Department: Department of Numerical Mathematics Supervisor: prof. RNDr. Miloslav Feistauer, DrSc., dr. h. c., Department of Numerical Mathematics Abstract: The method described in this thesis deals with a solution of an inverse problem for a flow around an airfoil. It can be used to design an airfoil shape according to a specified velocity or pressure distribution along the chord line. The method is based on searching for a fixed point of an operator, which combines an approximate inverse and direct operator. The approximate inverse operator, derived on the basis of the thin airfoil theory, assigns a corresponding shape to the specified distribution. The resulting shape is then constructed using the mean camber line and thickness function. The direct operator determines the pressure or velocity distribution on the airfoil surface. We can apply a fast, simplified model of potential flow solved using the Fredholm integral equation, or a slower but more accurate model of RANS equations with a k-omega turbulence model. The method is intended for a subsonic flow.
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A posteriori error estimates for numerical solution of convection-difusion problems
Šebestová, Ivana ; Dolejší, Vít (advisor) ; Sváček, Petr (referee) ; Brandts, Jan (referee)
This thesis is concerned with several issues of a posteriori error estimates for linear problems. In its first part error estimates for the heat conduction equation discretized by the backward Euler method in time and discontinuous Galerkin method in space are derived. In the second part guaranteed and locally efficient error estimates involving algebraic error for Poisson equation discretized by the discontinuous Galerkin method are derived. The technique is based on the flux reconstruction where meshes with hanging nodes and variable polynomial degree are allowed. An adaptive strategy combining both adaptive mesh refinement and stopping criteria for iterative algebraic solvers is proposed. In the last part a numerical method for computing guaranteed lower and upper bounds of principal eigenvalues of symmetric linear elliptic differential operators is presented. 1
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Discontinuous Galerkin method for the solution of compressible viscous flow
Česenek, Jan ; Feistauer, Miloslav (advisor) ; Najzar, Karel (referee) ; Sváček, Petr (referee)
Title: Discontinuous Galerkin method for solving compressible viscous flow Author: Jan Česenek Department: Department of Numerical Mathematics Supervisor: prof. RNDr. Miloslav Feistauer, DrSc., dr.h.c., Department of Numerical Mathematics Abstract: The subject of this PhD thesis is the numerical simulation of the interaction of two-dimensional compressible viscous flow and a vibrating airfoil. We consider a solid airfoil with two degrees of freedom which can rotate around the elastic axis and oscillate in the vertical direction. The numerical simulation of this problem consist of the dis- continuous Galerkin finite element method solving Navier-Stokes equations coupled with a system of nonlinear ordinary differential equations describing the airfoil motion. The time-dependent domain is taken into account with the aid of the Arbitrary Lagrangian- Eulerian(ALE) formulation. Theoretical part of this paper is concerned with error esti- mates of the space-time discontinuous Galerkin method for scalar nonstationary equations with nonlinear convection and nonlinear diffusion. Keywords: convection-diffusion problems, discontinuous Galerkin method, interaction of a fluid with a vibrating airfoil, ALE method
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Numerical Solution of a Fredholm Integral Equation of the Second Kind Related to Induction Heating
Rak, Josef ; Kofroň, Josef (advisor) ; Feistauer, Miloslav (referee) ; Sváček, Petr (referee)
This thesis deals with numerical solution of an integral equation of the second kind with special singular kernel function related to induction heating. The numerical solution is based on collocation and Nyström methods. The idea of collocation methods is to choose a finite-dimensional space of candidate solutions (usually polynomials up to a certain degree). The Nyström methods are based on approximation of the integral in equation by numerical integration rule. This thesis describes and gives error estimates of both methods. Error estimates are compared to the exact solutions in simple cases.
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On numerical approximation of fluid-structure interactions of air flow with a model of vocal folds
Valášek, J. ; Horáček, Jaromír ; Sváček, P.
This paper deals with flow driven vibration of an elastic body. Our goal is to develop and mathematically describe a simplified model of the human vocal fold. The developed numerical schemes for viscous incompressible fluid flow in ALE formulation and the elastic body are implemented by two solvers, specific for each domain. The studied problem is coupled by Dirichlet-Neumann boundary conditions. Both solvers are based on the finite element method. Particularly, for the fluid model the crossgrid elements are used. Numerical results focus on the verification of the developed program.
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On application of finite element method for approximation of 3D flow problems
Sváček, P. ; Horáček, Jaromír
This paper is interested to the interactions of the incompressible flow with a flexibly supported airfoil. The bending and the torsion modes are considered. The problem is mathematically described. The numerical method is based on the finite element method. A combination of the streamline-upwind/Petrov-Galerkin and pressure stabilizing/Petrov-Galerkin method is used for the stabilization of the finite element method. The numerical results for a three-dimensional problem of flow over an airfoil are shown.
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Finite element simulation of aeroelasticity problems
Horáček, Jaromír ; Sváček, P. ; Feistauer, M.
The paper presents results achieved by the authors in development of in-house codes based the finite element (FE) method and applied to solution of fluid-structure problems in aeroelasticity of airfoils. We consider flexibly supported airfoil with two- or three degrees of freedom (2- or3-DOF) in two-dimensional (2D) incompressible viscous flow. The airfoil vibration is described by nonlinear ordinary differential equations of motion for large vibration amplitudes. The flow is medeled by the Navier-Stokes equations for laminar flow or by the Reynolds averaged Navier-Stokes (RANS) equations for turbulent model.
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On mathematical modelling of gust response using the finite element method
Sváček, P. ; Horáček, Jaromír
In this paper the numerical approximation of aeroelastic response to sudden gust is presented. The fully coupled formulation of two dimensional incompressible viscous fluid flow over a flexibly supported structure is used. The flow is modelled with the system of Navier-Stokes equations written in Arbitrary Lagrangian-Eulerian form and coupled with system of ordinary differential equations describing the airfoil vibrations with two degrees of freedom. The Navier-Stokes equations are spatially discretized by the fully stabilized finite element method. The numerical results are shown.
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On higher-order space-time discretization of an nonlinear aeroelastic problem with the consideration of large displacements
Sváček, P. ; Horáček, Jaromír
This paper focuses on the mathematical and numerical modelling of interaction of the two-dimensional incompressible fluid flow and a flexibly supported airfoil section wit control section. A simplified problem is considered: The flow is modelled by the system of Navier-Stokes equations and the structure motion is described with the aid of nonlinear ordinary differential equations. The time-dependent computational domain is taken into account by the Arbitrary Lagrangian-Eulerian method. Higher order time discretization is considered within the stabilized finite element method. The application of the described method is shown.
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