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Numerical methods for vortex dynamics
Outrata, Ondřej ; Hron, Jaroslav (advisor) ; Šístek, Jakub (referee)
Two aspects of solving the incompressible Navier-Stokes equations are described in the thesis. The preconditioning of the algebraic systems arising from the Finite Element Method discretization of the Navier-Stokes equations is complex due to the saddle point structure of the resulting algebraic problems. The Pressure Convection Diffusion Reaction and the Least Squares Commutator preconditioners constitute two possible choices studied in the thesis. Solving the flow problems in time-dependent domains requires special numerical methods, such as the Fictitious Boundary method and the Arbitrary Lagrangian Eulerian formulation of Navier-Stokes equations which are used in the thesis. The problems examined in the thesis are simulations of experiments conducted in liquid Helium at low temperatures. These simulations can be used to establish a relationship between vorticity and new quantity pseudovorticity in an experiment-like setting.
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Least-squares problems with sparse-dense matrices
Riegerová, Ilona ; Tůma, Miroslav (advisor) ; Tichý, Petr (referee)
Problém nejmenších čtverc· (dále jen LS problém) je aproximační úloha řešení soustav lineárních algebraických rovnic, které jsou z nějakého d·vodu za- tíženy chybami. Existence a jednoznačnost řešení a metody řešení jsou známé pro r·zné typy matic, kterými tyto soustavy reprezentujeme. Typicky jsou ma- tice řídké a obrovských dimenzí, ale velmi často dostáváme z praxe i úlohy s maticemi o proměnlivé hustotě nenulových prvk·. Těmi se myslí řídké matice s jedním nebo více hustými řádky. Zde rozebíráme metody řešení tohoto LS pro- blému. Obvykle jsou založeny na rozdělení úlohy na hustou a řídkou část, které řeší odděleně. Tak pro řídkou část m·že přestat platit předpoklad plné sloupcové hodnosti, který je potřebný pro většinu metod. Proto se zde speciálně zabýváme postupy, které tento problém řeší. 1
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Towards efficient numerical computation of flows of non-Newtonian fluids
Blechta, Jan ; Málek, Josef (advisor) ; Herzog, Roland (referee) ; Süli, Endré (referee)
In the first part of this thesis we are concerned with the constitutive the- ory for incompressible fluids characterized by a continuous monotone rela- tion between the velocity gradient and the Cauchy stress. We, in particular, investigate a class of activated fluids that behave as the Euler fluid prior activation, and as the Navier-Stokes or power-law fluid once the activation takes place. We develop a large-data existence analysis for both steady and unsteady three-dimensional flows of such fluids subject either to the no-slip boundary condition or to a range of slip-type boundary conditions, including free-slip, Navier's slip, and stick-slip. In the second part we show that the W−1,q norm is localizable provided that the functional in question vanishes on locally supported functions which constitute a partition of unity. This represents a key tool for establishing local a posteriori efficiency for partial differential equations in divergence form with residuals in W−1,q . In the third part we provide a novel analysis for the pressure convection- diffusion (PCD) preconditioner. We first develop a theory for the precon- ditioner considered as an operator in infinite-dimensional spaces. We then provide a methodology for constructing discrete PCD operators for a broad class of pressure discretizations. The...
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Diffuse interface models in theory of interacting continua
Řehoř, Martin ; Průša, Vít (advisor) ; Garcke, Harald (referee) ; Rohde, Christian (referee)
We study physical systems composed of at least two immiscible fluids occu- pying different regions of space, the so-called phases. Flows of such multi-phase fluids are frequently met in industrial applications which rises the need for their numerical simulations. In particular, the research conducted herein is motivated by the need to model the float glass forming process. The systems of interest are in the present contribution mathematically described in the framework of the so-called diffuse interface models. The thesis consists of two parts. In the modelling part, we first derive standard diffuse interface models and their generalized variants based on the concept of multi-component continuous medium and its careful thermodynamic analysis. We provide a critical assessment of assumptions that lead to different models for a given system. Our newly formulated class of generalized models of Cahn-Hilliard-Navier-Stokes-Fourier (CHNSF) type is applicable in a non-isothermal setting. Each model belonging to that class describes a mixture of separable, heat conducting Newtonian fluids that are either compressible or incompressible. The models capture capillary and thermal effects in thin interfacial regions where the fluids actually mix. In the computational part, we focus on the development of an efficient and robust...
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Approximations by low-rank matrices and their applications
Outrata, Michal ; Tůma, Miroslav (advisor) ; Rozložník, Miroslav (referee)
Consider the problem of solving a large system of linear algebraic equations, using the Krylov subspace methods. In order to find the solution efficiently, the system often needs to be preconditioned, i.e., transformed prior to the iterative scheme. A feature of the system that often enables fast solution with efficient preconditioners is the structural sparsity of the corresponding matrix. A recent development brought another and a slightly different phe- nomenon called the data sparsity. In contrast to the classical (structural) sparsity, the data sparsity refers to an uneven distribution of extractable information inside the matrix. In practice, the data sparsity of a matrix ty- pically means that its blocks can be successfully approximated by matrices of low rank. Naturally, this may significantly change the character of the numerical computations involving the matrix. The thesis focuses on finding ways to construct Cholesky-based preconditioners for the conjugate gradi- ent method to solve systems with symmetric and positive definite matrices, exploiting a combination of the data and structural sparsity. Methods to exploit the data sparsity are evolving very fast, influencing not only iterative solvers but direct solvers as well. Hierarchical schemes based on the data sparsity concepts can be derived...
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Incomplete Cholesky factorization
Hoang, Phuong Thao ; Tůma, Miroslav (advisor) ; Tichý, Petr (referee)
The thesis is about the incomplete Cholesky factorization and its va- riants, which are important for preconditioning a system with symmetric and positive definite matrix. Our main focus is on solving these systems, which arise in many technical applications and natural sciences, using preconditioned Con- jugate Gradients. Besides many other ways we can apply Cholesky factorization approximately, incompletely. In this thesis we study existence of the incomplete Cholesky factorization and we evaluate behaviour and potential of different vari- ants of the generic algorithm. 1
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Optimation of inoculation process of ductile iron
Ulm, Daniel ; Pecina, Vladimír (referee) ; Roučka, Jaromír (advisor)
The master‘s thesis deals in theoretical part with the casting classification, ductile iron casting, its inoculation and modification and quality evaluation using thermal analysis, image analysis and testing of mechanical properties. The aim of the practical part was to test the effects of preconditioning on the properties of ductile iron and to find out whether it is able to replace the current method of inoculation or to increase the mechanical properties of ductile iron castings. The state of the ductile iron was under control by spectral and thermal analysis. The mechanical properties and image analysis were checked on finished casting.
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Algebraic preconditioning for Biot-Barenblatt poroelastic systems
Blaheta, Radim ; Luber, Tomáš
Poroelastic systems describe fluid flow through porous medium coupled with deformation of the porous matrix. In this paper, the deformation is described by linear elasticity, the fluid flow is modelled as Darcy flow. The main focus is on the Biot-Barenblatt model with double porosity/double permeability flow, which distinguishes flow in two regions considered as continua. The main goal is in proposing block diagonal preconditionings to systems arising from the discretization of the Biot-Barenblatt model by a mixed finite element method in space and implicit Euler method in time and estimating the condition number for such preconditioning. The investigation of preconditioning includes its dependence on material coefficients and parameters of discretization.
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Cardioprotection induced by changes in abiotic environmental factors
Kubátová, Hana ; Vybíral, Stanislav (advisor) ; Žurmanová, Jitka (referee)
Cardiovascular diseases are one of the most frequent causes of death worldwide. Methods of limiting the ischemia-reperfusion injury, which occurs as a result of acute myocardial infarction (MI), have therefore been receiving significant attention. Preconditioning through ischemia application, cold exposure or physical exercise reduces the size of an infarction and the occurrence of ventricular arrhythmias. The exact mechanism behind this process has not yet been fully comprehended; nevertheless important roles are played by factors such as the maintenance of calcium homeostasis, the opening of ATP-dependent potassium channels (KATP) and the prevention of the Mitochondrial Permeability Transition pore (MPTP) opening. Chronic hypoxia and exercise have also been proven to have a cardioprotective effect, which is to be at least partially attributed to the intensified activity of antioxidant enzymes and the opening of KATP. During winter the risk of MI is increased for a wide range of patients. In winter swimmers, the stress caused by a cold stimulus is reduced. However, a mild stress is necessary to initiating cardioprotection. At present, there is unfortunately insufficient information available regarding the effects of long term cold adaptation on the size reduction of MI. Some studies suggest,...
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Cardiac ischemic tolerance of hypertensive rats
Jelínek, Jan ; Neckář, Jan (advisor) ; Sotáková, Dita (referee)
The aim of this thesis is to summarize current knowledge about the influence of the ischemic- reperfusion injury at the myocard of hypertensive subjects. First part of this thesis is focused on the description of ischemia, reperfusion and changes in the myocardial metabolism during these processes. These changes in the myocardial metabolism are for example necrosis or apoptosis of the myocardial cells. The second part describes the currently known cardioprotective phenomena. This part also compares their effects. The signalization of preconditioning, the second window of preconditioning and the postconditioning are described here in more details. Third part is focused on the description of the risk factors connected to the ICHS and hypertension. It describes also classes of hypertension, clinical and experimental methods of hypertension treatment, description of the laboratory breeds of hypertensive rats. In the last part of this thesis I describe the influence of hypertension on the I-R injury in current laboratory studies. In the most studies spontaneously hypertensive rats (SHR) were used. As a normotensive controls Wistar-Kyoto rats were mostly used. For some other experiments transgenic genetic rats (TGR) were used. Powered by TCPDF (www.tcpdf.org)
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