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Numerické řešení stlačitelného proudění
Prokopová, Jaroslava ; Feistauer, Miloslav (advisor) ; Dolejší, Vít (referee)
This work deals with the problem of inviscid, compressible flow in a time-dependent 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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Modelování proudění krve v geometrii aneuryzma
Zábojníková, Tereza ; Hron, Jaroslav (advisor) ; Feistauer, Miloslav (referee)
The aim of this work is to find a stable scheme which would solve the Stokes problem of the fluid flow, in which an elastic structure is immersed. Unlike most of the schemes solving fluid-structure interaction problems, in our scheme meshes of fluid and structure do not have to coincide. We have restricted ourselves to two-dimensional domain occupied by fluid with one-dimensional im- mersed structure. To describe a fluid-structure interaction, we have used an Immersed boundary method. At first we consider the strucure to be massless. We have modified an existing scheme and made it unconditionally stable, which was mathematically proven and numerically tested. Then we have proposed a modification where the structure is not massless and also proved the uncondi- tional stability in this case. The proposed schemes were implemented using the Freefem++ software and tested on aneurysm-like geometry. We have tested the behavior of our scheme in case when the qrowing aneurysm touches an obstacle, for example a bone (with no-slip condition on the bone boundary). Powered by TCPDF (www.tcpdf.org)
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Numerical solution of ordinary differential equations
Monhartová, Petra ; Feistauer, Miloslav (advisor) ; Janovský, Vladimír (referee)
In the present work we study numerical methods for the nu- merical solution of initial value problems for ordinary differential equations. With the aid of the Taylor formula we derive several one-step methods. We compare numerical solution computed with explicit and implicit Eu- ler methods. Moreove, we are concerned with second-order and fourth-order Runge-Kutta methods. We find how accurately the numerical methods obta- ined with the aid of these methods approximate the exact solution. Further we estimate the error of these method by the half-step method. 1
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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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Shape Optimization for Navier-Stokes Equations with Viscosity
Stebel, Jan ; Haslinger, Jaroslav (advisor) ; Feistauer, Miloslav (referee) ; Feireisl, Eduard (referee)
We study the shape optimization problem for the paper machine headbox which distributes a mixture of water and wood fibers in the paper making process. The aim is to find a shape which a priori ensures the given velocity profile on the outlet part. The mathematical formulation leads to the optimal control problem in which the control variable is the shape of the domain representing the header, the state problem is represented by the generalised Navier-Stokes system with nontrivial boundary conditions. The objective is to analyze theoretically this problem (proof of the existence of a solution), its discretization and the numerical realization.
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Computational comparison of hp-adaptive approaches
Kubásek, Petr ; Vejchodský, Tomáš (advisor) ; Feistauer, Miloslav (referee)
Cílem této práce je porovnat řízení hp-adaptivního procesu pomocí referenčního řešení a různých aposteriorních odhadů chyby. Tyto přístupy jsou porovnávány z hlediska globální diskretizační chyby a potřebného počtu stupňů volnosti. Konkrétně se zabýváme explicitními residuálními odhady, implicitními residuálními odhady Dirichletova a Neumannova typu a hierarchickými odhady. Všechny odhady jsou v práci podrobně odvozeny včetně jejich nejvýznamnějších vlastností. Jednotlivé přístupy jsou srovnávány pomocí numerických experimentů. Na jejich základě lze ríci, že nejlepších výsledků dosahuje adaptivita řízená pomocí referenčního řešení společně s implicitním Dirichletovým odhadem. Referenční řešení se zdá být nejspolehlivější metodou zatímco implicitní Dirichletův odhad je, s výjimkou některých případů, nejrychlejší.
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Numerical solution of the Navier-Stokes equations with a generalized state eqution
Ullrichová, Martina ; Dolejší, Vít (advisor) ; Feistauer, Miloslav (referee)
Title: Numerical solution of the Navier-Stokes equations with a generalized state eqution Author: Martina Ullrichová Department: Department of numerical Mathematics Supervisor: doc. RNDr. Vít Dolejší, Ph.D., DSc. Supervisor's e-mail address: dolejsi@karlin.mff.cuni.cz Abstract: In the present work we study the simulation of viscous compressible flow. We consider a generalized model of gas described by a generalized state equation. We firstly summarize construction of this generalized equation and its consequences for the existence and uniquenes of the solution of the compressible flow problem. We suggest particular ways how define this generalized relations. Afterwards, we solve the Navier-Stokes problem by the discontinous Galerkin method. Considering the properties of the generalized model we choose explicit time discretization. Finally, we compare new generalized model with the standard model (ideal gas) for different flow regimes around the NACA0012 profil. Keywords: viscous compressible flow, Navier-Stokes equations, discontinous Ga- lerkin method, generalized state equation
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