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The Selected Stochastic Programs in Engineering Design
Čajánek, Michal ; Mrázková, Eva (referee) ; Popela, Pavel (advisor)
Two-stage stochastic programming problem with PDE constraint, specially elliptic equation is formulated. The computational scheme is proposed, whereas the emphasis is put on approximation techniques. We introduce method of approximation of random variables of stochastic problem and utilize suitable numerical methods, finite difference method first, then finite element method. There is also formulated a mathematical programming problem describing a membrane deflection with random load. It is followed by determination of the acceptableness of using stochastic optimization rather than deterministic problem and assess the quality of approximations based on Monte Carlo simulation method and the theory of interval estimates. The resulting mathematical models are implemented and solved in the general algebraic modeling system GAMS. Graphical and numerical results are presented.
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Mixed finite element method for the Poisson equation
Švihlová, Helena ; Knobloch, Petr (advisor) ; Dolejší, Vít (referee)
The aim of this bachelor thesis is the implementation of the mixed element method for the Poisson equation and the comparison with results of the classical finite element method. The thesis is divided into two chapters. In the first chapter there are descriptions of the spaces occurring in the weak formulation of the Poisson equation and descriptions of the spaces which are suitable to approach them. The second chapter studies the existence of the solutions of the approximated tasks and their convergence. The main part of this thesis are schemes of the solutions of both methods and the tables comparing errors of these solutions for three diferent functions. 1
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Study of probe diagnostics of tokamak edge plasma via computer simulation
Podolník, Aleš ; Pánek, Radomír (advisor) ; Cavalier, Jordan (referee) ; Roučka, Štěpán (referee)
The aim of the thesis is to examine plasma-wall interaction using computer modeling. Tokamak- relevant plasma conditions are simulated using the particle-in-cell model family SPICE working in three or two dimensions. SPICE model was upgraded with a parallel Poisson equation solver and a heat equation solver module. Plasma simulation aimed at synthetic Langmuir probe measurements were performed. First set considered a flush-mounted probe and the effect of variable magnetic field angle was studied with aim to compare existing probe data evaluation techniques and assess their operational space, in which the plasma parameters estimation via fit to the current-voltage characteristic is accurate. Second simulation set studied a protruding probe pin. Effective collecting area of such probe was investigated with intentions of density measurement collection. This area was found to be influenced by a combination of two factors. First, the density dampening inside the magnetic pre-sheath of the probe head, and the second, the extension of the area caused by Larmor rotation. A comparison with experimental results obtained at COMPASS tokamak was was performed, confirming these results. Keywords Langmuir probe, simulation, particle-in-cell, tokamak, Poisson equation, COMPASS 1
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Study of probe diagnostics of tokamak edge plasma via computer simulation
Podolník, Aleš ; Pánek, Radomír (advisor) ; Cavalier, Jordan (referee) ; Roučka, Štěpán (referee)
The aim of the thesis is to examine plasma-wall interaction using computer modeling. Tokamak- relevant plasma conditions are simulated using the particle-in-cell model family SPICE working in three or two dimensions. SPICE model was upgraded with a parallel Poisson equation solver and a heat equation solver module. Plasma simulation aimed at synthetic Langmuir probe measurements were performed. First set considered a flush-mounted probe and the effect of variable magnetic field angle was studied with aim to compare existing probe data evaluation techniques and assess their operational space, in which the plasma parameters estimation via fit to the current-voltage characteristic is accurate. Second simulation set studied a protruding probe pin. Effective collecting area of such probe was investigated with intentions of density measurement collection. This area was found to be influenced by a combination of two factors. First, the density dampening inside the magnetic pre-sheath of the probe head, and the second, the extension of the area caused by Larmor rotation. A comparison with experimental results obtained at COMPASS tokamak was was performed, confirming these results. Keywords Langmuir probe, simulation, particle-in-cell, tokamak, Poisson equation, COMPASS 1
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Advanced techniques of computer modelling in low- and high-temperature plasma physics
Pekárek, Zdeněk ; Hrach, Rudolf (advisor) ; Kudrna, Pavel (referee) ; Hron, Jaroslav (referee)
This thesis identifies the obstacles in efficient modelling of interaction of plasma and solid surfaces. It presents an enhanced method of solving the Poisson equation optimized to meet the requirements of the Particle-in-Cell modelling approach. A number of applications are discussed, including models of the plasma-facing wall of the nuclear fusion device, tokamak, and its interaction with particle fluxes driven by the intrinsic magnetic field. Another area of applications covers the modelling of plasma probes deployed to diagnose properties of plasma in various experiments. The thesis also includes the computer library code and instructions enabling a rapid use of the Poisson solver method in a third party computer code which implements the PIC approach in a compatible manner.
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Mixed finite element method for the Poisson equation
Švihlová, Helena ; Knobloch, Petr (advisor) ; Dolejší, Vít (referee)
The aim of this bachelor thesis is the implementation of the mixed element method for the Poisson equation and the comparison with results of the classical finite element method. The thesis is divided into two chapters. In the first chapter there are descriptions of the spaces occurring in the weak formulation of the Poisson equation and descriptions of the spaces which are suitable to approach them. The second chapter studies the existence of the solutions of the approximated tasks and their convergence. The main part of this thesis are schemes of the solutions of both methods and the tables comparing errors of these solutions for three diferent functions. 1
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The Selected Stochastic Programs in Engineering Design
Čajánek, Michal ; Mrázková, Eva (referee) ; Popela, Pavel (advisor)
Two-stage stochastic programming problem with PDE constraint, specially elliptic equation is formulated. The computational scheme is proposed, whereas the emphasis is put on approximation techniques. We introduce method of approximation of random variables of stochastic problem and utilize suitable numerical methods, finite difference method first, then finite element method. There is also formulated a mathematical programming problem describing a membrane deflection with random load. It is followed by determination of the acceptableness of using stochastic optimization rather than deterministic problem and assess the quality of approximations based on Monte Carlo simulation method and the theory of interval estimates. The resulting mathematical models are implemented and solved in the general algebraic modeling system GAMS. Graphical and numerical results are presented.
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Fast direct methods
Práger, Milan
A short survey of standard fast methods for the solution of algebraic systems arising when solving elliptic boundary value problems. It is shown how these methods can be used for the solution of more general problems. AAll polygons and tetrahedra are found where boundary problems can be transformedby reflection and solved with fast methods.
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