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Stability of Neutral Delay Differential Equations and Their Discretizations
Dražková, Jana ; Čermák, Libor (oponent) ; Šremr, Jiří (oponent) ; Čermák, Jan (vedoucí práce)
The doctoral thesis discusses the asymptotic stability of delay differential equations and their discretizations. The linear delay differential equations with constant as well as infinite lag are considered. The necessary and sufficient conditions describing the asymptotic stability region of both exact and discretized linear neutral delay differential equation with constant lag are derived. We compare asymptotic stability domains of corresponding exact and discretized equations and discuss properties of derived stability regions with respect to a changing stepsize of the utilized discretization. Further, we investigate the linear delay differential equation with the infinite lag. We present the description of its exact and discrete asymptotic stability regions together with asymptotic estimates of its solutions. The linear delay differential equation with several infinite lags is discussed as well.
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Řešení vývoje nestabilit kapalného filmu s následným odtržením kapek
Knotek, Stanislav ; Kozubková, Milada (oponent) ; Čermák, Libor (oponent) ; Jícha, Miroslav (vedoucí práce)
Dizertační práce se věnuje studiu nestabilit tenkých kapalných filmů až do fáze odtržení kapek. Dle typu struktur a dějů na povrchu filmu byly na základě literární rešerše klasifikovány čtyři typy nestabilit: dvoudimenzionální pomalé vlny, dvoudimenzionální rychlé vlny, třídimenzionální vlny, solitární vlny a fáze odtrhávání kapek z povrchu filmu. Práce analyzuje fyzikální principy vzniku nestabilit a zabývá se matematickou formulací problému. Jako příčina nestabilit jsou identifikovány smykové napětí a tlak působící na povrch filmu. Matematické modely predikce nestabilit jsou demonstrovány pomocí přístupů založených na řešení Orrovy-Sommerfeldovy rovnice a pomocí řešení pohybových rovnic v integrálním tvaru. Dále jsou uvedeny modely fluktuací smykového napětí a tlaku působících na povrch filmu a vybrané modely tloušťky filmu. Těžiště práce je zaměřeno na predikci iniciace dvoudimenzionálních vln pomocí integrálního přístupu. Charakteristiky fluktuací smykového napětí a tlaku na hladině filmu byly modelovány pomocí simulací proudění vzduchu nad pevným povrchem. V závěru práce jsou uvedena kritéria odtržení kapek v závislosti na rychlosti proudění vzduchu a tloušťce filmu.
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Delay Difference Equations and Their Applications
Jánský, Jiří ; Hilscher, Roman Šimon (oponent) ; Čermák, Libor (oponent) ; Čermák, Jan (vedoucí práce)
This thesis discusses the qualitative properties of some delay difference equations. These equations originate from the $\Theta$-method discretizations of the differential equations with a delayed argument. Our purpose is to analyse the asymptotic properties of these numerical solutions and formulate their upper bounds. We also discuss stability properties of the studied discretizations. Several illustrating examples and comparisons with the known results are presented as well.
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Discontinuous Galerkin Methods for Solving Acoustic Problems
Nytra, Jan ; Hlavička, Rudolf (oponent) ; Čermák, Libor (vedoucí práce)
Partial differential equations play an important role in engineering applications. It is often possible to solve these equations only approximately, i.e. numerically. Therefore number of successful discretization techniques has been developed to solve these equations. The presented discontinuous Galerkin method seems to be very general method to solve this type of equations, especially useful for hyperbolic systems. Our aim is to solve aeroacoustic problems, where propagation of acoustic waves is described using linearized Euler equations. This system of equations is indeed hyperbolic and therefore the discontinuous Galerkin method was chosen. The most important aspects of this method is ability to deal with complex geometries, possibility of high-order method and its local character enabling efficient computation parallelization. We first introduce the discontinuous Galerkin method in general for one- and two-dimensional problems. We then test the algorithm to solve advection equation, which was chosen as a model case of hyperbolic equation. The method will be finally tested using number of verification problems, which were formulated to test methods for computational equations, including verification of boundary conditions, which, similarly to computational fluid dynamics, are important part of computational aeroacoustics.
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Numerické modelování šíření zvuku pomocí diferenčních metod
Prochazková, Zdeňka ; Zatočilová, Jitka (oponent) ; Čermák, Libor (vedoucí práce)
Cílem této práce je představit metodu konečných diferencí (FDM) upravenou pro použití v problematice modelování šíření zvuku a další postupy, které se společně s touto metodou používají. To jsou selektivní filtry a časová integrace metodou Runge-Kuttas nízkými nároky na paměť. Důležitou problematikou v modelování šíření zvuku jsou okrajové podmínky. V práci je uvedeno několik typů okrajových podmínek a jejich ověření. Součástí práce je několik vyřešených příkladů, které byly implementovány v Matlabu.
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Meshfree methods for computational aeroacoustics
Bajko, Jaroslav ; Hlavička, Rudolf (oponent) ; Čermák, Libor (vedoucí práce)
Meshfree methods represent an alternative to the standard mesh-based numerical discretization techniques. Considerable effort has been spent on the verification of the meshless methods capabilities to solve problems from different engineering branches in the past decades. The aim of this master's thesis is an application of a suitable meshfree method in the computational aeroacoustics. Main attention will be focused on the sound propagation problems, which can be modeled using the linearized Euler equations. Necessary theory of the hyperbolic systems will be mentioned with respect to the nature of governing equations. Meshfree Finite point method (FPM) has been chosen due to its achievements in the computational fluid dynamics. The derivation of this meshfree method is presented as well as an accuracy improvements which are necessary for the sound propagation problems. Capabilities of the derived meshfree method will be verified on several benchmark problems using a software which was specially developed for this purpose.
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Bezsíťové metody ve výpočetní dynamice tekutin
Niedoba, Pavel ; Zatočilová, Jitka (oponent) ; Čermák, Libor (vedoucí práce)
Práce se věnuje bezsíťovým metodám, především SPH metodě. Výhradně se práce zabývá problémem konvergence v blízkosti hranice definičního oboru úlohy a jeho následným řešením v podobě použití tzv. fiktivních částic jakožto okrajové podmínky. Dále je zde uvedeno vhodné nastavení parametrů pro shock tube 2D úlohu, které bylo získáno na základě mnoha testů a softwarových úprav.
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Navier-Stokesova rovnice - řešení proudění reálné kapaliny
Krausová, Hana ; Čermák, Libor (oponent) ; Fialová, Simona (vedoucí práce)
Tato práce se zabývá řešením Navier-Stokesových rovnic pro reálnou, stlačitelnou kapalinu s první a druhou viskozitou. Jako metoda řešení je vybrána metoda rozvoje podle vlastních tvarů kmitu. Vztahy pro koeficienty rozvoje a pro vlastní tvary jsou obecně určeny pro 1D, 2D a 3D proudění. Konkrétní vztahy pro vlastní tvary kmitu jsou nalezeny pouze pro 1D proudění. Konečná podoba tlakové funkce je analyzována pomocí softwaru Matlab.
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The nonstationary motion of solid body in a liquid
Stejskal, Jiří ; Hlavička, Rudolf (oponent) ; Čermák, Libor (vedoucí práce)
The subject of this thesis is the numerical simulation of the two-dimensional incompressible viscous flow. We consider a rotating ellipse concentric with a circle. The space between the ellipse and the circle is filled with a fluid. Our goal is to describe the fluid flow caused by the rotating ellipse, i.e., to determine the velocity field and pressure distribution. Further, we want to determine the additional effect of the fluid acting on the ellipse. These results are obtained as a solution of the Navier-Stokes equations by the finite element method. Special emphasis has been put on the derivation of the numerical scheme in a matrix form suitable for algorithmization. The Arbitrary Lagrangian-Eulerian (ALE) method has been used to incorporate the moving domain into the algorithm. A suitable stabilization technique of the finite element method is necessary to obtain relevant outcome. Presented results indicate sufficient robustness and accuracy of the numerical algorithm.
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Numerical study of the fluid motion and mixing processes in the vitreous cavity
Pavlů, Karel ; Čermák, Libor (oponent) ; Repetto, Rodolfo (vedoucí práce)
The vitreous cavity, the largest chamber of the eye, is delimited anteriorly by the lens and posteriorly by the retina and is filled by the vitreous humour. Under normal conditions the vitreous humour has the consistency of a gel, however, typically, with advancing age a disintegration of the gel structure occurs, leading to a vitreous liquefaction. Moreover, after a surgical procedure called vitrectomy the vitreous body may be completely removed and replaced by tamponade fluids. Besides allowing the establishment of an unhindered path of light from the lens to the retina, the vitreous also has important mechanical functions. In particular, it has the role of supporting the retina in contact to the outer layers of the eye, and of acting as a diffusion barrier for molecule transport between the anterior and the posterior segments of the eye. Studying the dynamics of the vitreous induced by eye rotations (saccadic movements) is important in connection of both the above aspects. On the one hand indications exist that the shear stress exerted by the vitreous on the retina may be connected with the occurrence of retinal detachment. On the other hand, if the vitreous motion is intense enough (a situation occurring either when the vitreous is liqueed or when it has been replaced with a uid after vitrectomy), advective transport may be by far more important than diffusion and may have complex characteristics. Advection has indeed been shown to play an important role in the transport phenomena within the vitreous cavity, but, so far, only advection due to the slow overall fluid ux from the anterior to the posterior segments of the eye has been accounted for, while fluid motion due to eye rotations, even if it is generally believed to play an important role, has been invariably disregarded. Some recent contributions have pointed out the importance of accounting for the real vitreous cavity shape in studying uid motion induced by eye rotations. Modelling the vitreous cavity as a deformed sphere, showed that the flow field displays very complex three- dimensional characteristics to which effective fluid mixing is likely to be associated. The purpose of the thesis is to model numerically the motion of the liqueed vitreous within the vitreous cavity induced by different eye movements. Create the model in the Comsol interface, compare the results with theoretical, experimental measurements and do some ow visualizations. Finally show the dependence of the streaming intensity from the amplitude of rotations and the Womersley number .
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