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Numerical solution of compressible flow
Prokopová, Jaroslava ; Feistauer, Miloslav (advisor)
This work deals with the problem of inviscid, compressible flow in a timedependent domain. We describe mathematical properties of the Euler equations and the system of governing equations is solved with the aid of the discontinuous Galerkin finite element method (DGFEM) in the time-indepentent domain. The main aim of this work is the study of this problem in time-dependent domains. For this reason the Arbitrary Lagrangian-Eulerian (ALE) method is presented. The governing equations are formulated in the ALE formulation and discretized in space and time by the DGFEM. Shortly we mention the shock capturing of the obtained scheme and the solution of the resulting linear system with the aid of Generalized Minimal Residual (GMRES) method. At the end of this work we present and compare results obtained by two different ALE formulations of the governing equations in the rectangular domain with a moving part of lower wall.
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Higher-order approximations in the finite element methods
Kolman, Karel ; Křížek, Michal (advisor) ; Feistauer, Miloslav (referee) ; Mlýnek, Jaroslav (referee)
A survey of existing nite element superconvergence theory is conducted. The Steklov postprocessing operator is presented and its properties are investigated, superconvergence bounds are proved. A two-level method for eigenvalue problems is generalized for application to a nonselfadjoint problem. Applications of superconvergence theory to variationally formulated eigenvalue problems are discussed. Numerical experiments documenting the methods are performed.
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Mathematical Analysis and Numerical Computation of Volume-Constrained Evolutionary Problems Involving Free Boundaries
Švadlenka, Karel ; Feistauer, Miloslav (advisor) ; Křížek, Michal (referee) ; Knobloch, Petr (referee)
The object of study of the present thesis are evolutionary problems satisfying volume preservation condition, i.e., problems whose solution have a constant value of the integral of their graph. In particular, the following types of problems with volume constraint are dealt with: parabolic problem (heat-type), hyperbolic problem (wave-type), parabolic free-boundary problem (heat-type with obstacle) and hyperbolic free-boundary problem (degenerate wave-type with obstacle). The key points are design of equations, proof of existence of weak solutions to them and development of numerical methods and algorithms for such problems. The main tool in both the theoretical analysis and the numerical computation is the discrete Morse flow, a variational method consisting in discretizing time and stating a minimization problem on each time-level. The volume constraint appears in the equation as a nonlocal nonlinear Lagrange multiplier but it can be handled elegantly in discrete Morse flow method by restraining the set of admissible functions for minimization. The theory is illustrated with results of numerical experiments.
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Numerical simulation of interaction of fluids and solid bodies
Dubcová, Lenka ; Knobloch, Petr (referee) ; Feistauer, Miloslav (advisor)
The subject of this thesis is modelling and numerical simulation of the interaction of two-dimensional incompressible viscous flow and a vibrating airfoil. A solid airfoil with two degrees of freedom, which can rotate around the elastic axis and oscillate in the vertical direction, is considered. The numerical simulation consists of the finite element solution of the Navier-Stokes equations coupled with the system of ordinary differential equations describing the airfoil motion. The time dependent computational domain and a moving grid are taken into account with the aid of the Arbitrary Lagrangian-Eulerian (ALE) formulation of the Navier-Stokes equations. High Reynolds numbers up to 106 require the application of a suitable stabilization of the finite element discretization and application of a turbulent model. We apply the algebraic turbulent models, which were designed by Baldwin and Lomax and by Rostand. As a result a sufficiently accurate and robust method is developed, which was tested by the simulation of flow along a flat plate and applied to the computation of pressure distribution along the airfoil with forced vibrations.
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Numerical solution of flow past an airfoil
Prokopová, Jaroslava ; Najzar, Karel (referee) ; Feistauer, Miloslav (advisor)
Nazev prace: Numorioke feseni obtekani leleckeho profilu Autor: .Javoslava Prokopova Katedra (ustav): Katedra numerieke maternal,iky Vedoucf bakalarske prace: Prof- RNDr. Miloslav Feistauer, DrSc. e-mail vedoudho: feiyf'^kaTlin.niff. ouni.cz Abstrakt: Pfedkladaiia prace se vennjo problematic^1 obtekani izolovanelio le- teckeho profilu. .Tso\ zdc popsany rovnice charaktcvizuji nova^kn, Tu^tlacitol- no. novifivc'% stacioiiarni, rovinnr proudorif a nvixUnia koni])lo1,iii cliaraktrriH- lika danclio problonm ]K)inoci ryrhloati i ijroud(jvc tunkc^. Hlavnf uaplni jo ])ak stiulium niol.ody fuukci kuinploxni promonne a melody kouecnych prvku. Pfi aplikaci tfx-hto met.orl so zainonijoino na fescni oljtokaiii Znkov- skcho profilu. Di'ky ostro odtokovo hrano tuhoto profiln jsou zdo studovany odtokovo podmiiiky a jejich vyuziti vo stiulovanych motodacU. Poslodni casti Leto prace jo srovnain' vysledkii dosazeuycli ].)omorf tool)to nu^tod pro zvo- loriy Z\ikovskoho profil. Klicova slova: neva.xke, uoatlaoitolno, lunn'ri^'e, sta.oioiu'i.nii, ruvimio proudoni; mctoda fuukei koinploxni proruonnr; moUnla k(jneciiyoh ])rvkii: Znkovskeho profil; odtokova pochuiuka Title: Numerical .solution of flow past an airfoil Author: Jaroslava Prokopova Department: Department of Numoricn,! Matlieniatica Supervisor: Prof. RNDr. Miloslav...
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Some aspects of the discontinuous Galerkin method for the solution of convection-diffusion problems
Hájek, Jaroslav ; Najzar, Karel (referee) ; Feistauer, Miloslav (advisor)
This work is concerned with the numerical solution of initial-boundary value problems for convection-diffusion partial differential equations. Three methods are studied and compared for this purpose: the combined finite element - finite volume (FE-FV) method, the discontinuous Galerkin finite element (DGFE) method of lines, and the spacetime discontinuous Galerkin method. The combined FE-FV method uses piecewise linear conforming finite elements for the discretization of the diffusion terms and piecewise constant FV approximation of the convective terms. The relation between the FE and FV approximations is determined by the so-called lumping operator. In the DGFE method of lines, the space semidiscretization is carried out by piecewise polynomial functions constructed over a triangular mesh, in general discontinuous on interfaces between neighbouring elements. In the space-time DGFE method, the approximate solution is piecewise polynomial in space as well as in time. We discuss both theoretical and practical aspects of the methods, and present numerical results for each of them. For the DGFE method of lines we derive an a posteriori error estimate.
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