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Non-equilibrium Thermodynamics of Hyperbolic Systems
Sýkora, Martin ; Pavelka, Michal (advisor) ; Hütter, Markus (referee) ; Lukáčová - Medvidová, Mária (referee)
Continuum mechanics and thermodynamics have been used in many fields from sci- ence to industry to medicine. While simple materials have already been described, for instance by the famous Navier-Stokes equations, more complex materials like superfluids are still not completely understood. Multiple frameworks, often based on balance laws and thermodynamic closures, are used. In this thesis, an alternative General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC) framework is studied. We apply this method, based on Hamiltonian mechanics and gradient dynamics, to a wide range of settings from a simple continuum to mixtures, superfluids and non-Fourier heat conduction. We study the symmetric hyperbolicity of the resulting equations, as it guarantees the well-posedness of the system, and suggest extensions to the already-known symmetric hyperbolic equations. We derive new models, compare them to more classical theories and study what phenomena they can and cannot describe, both analytically and numerically. 1
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Machine learning for recognition of simple physical systems
Benda, Jan ; Pavelka, Michal (advisor) ; Congreve, Scott (referee)
The rise of machine learning, particularly through the use of neural networks, has begun to change how we solve problems, including understanding simple physical systems. This thesis focuses on the Direct Poisson Neural Network (DPNN), a network that uses the structure of Hamilton's equations of motion to learn from data. This method allows us to extract the Hamiltonian and Poisson bivector from the data, helping to identify the type of physical systems. We explore how DPNN works with noisy data and when data is limited, checking its ability to make predictions in challenging conditions. Moreover, we have implemented Energy Ehrenfest regularisation to the model, which helps it recognise and simulate dissipative systems better. 1
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Machine learning through geometric mechanics and thermodynamics
Šípka, Martin ; Pavelka, Michal (advisor) ; Monmarché, Pierre (referee) ; Maršálek, Ondřej (referee)
30. prosinec 2023 This thesis studies novel approaches to learning of physical models, incorporat- ing constraints and optimizing path dependent loss functions. Recent advances in deep learning and artificial intelligence are connected with established knowl- edge about dynamical and chemical systems, offering new synergies and improv- ing upon existing methodologies. We present significant contributions to sim- ulation techniques that utilize automatic differentiation to propagate through the dynamics, showing not only their promising use case but also formulating new theoretical results about the gradient behavior in long evolutions controlled by neural networks. All the tools are carefully tested and evaluated on exam- ples from physics and chemistry, thus proposing and promoting their further applications. 1
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Simulating flows past a mountain range using smoothed particle hydrodynamics
Belán, Kamil ; Pavelka, Michal (advisor) ; Klika, Václav (referee)
The method of Smoothed Particle Hydrodynamics is applied to the phenomenon of mountain waves - atmospheric internal gravity waves generated by flows over topogra- phy. General aspects of the method and the alternative derivations of the theory using Hamiltonian continuum mechanics are discussed. The basic explanation of the physical mechanisms that generate the internal gravity waves and the review of the current state of the numerical simulation of the matter are provided. A code in the Julia programming language is written to simulate the phenomenon of mountain waves using the symplec- ticity of the SPH equations by utilizing a symplectic integrator. The results obtained are compared to those from the literature, and the applicability of the SPH method in meteorology is also discussed. 1
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Extension of smoothed particle hydrodynamics based on Poisson brackets
Kincl, Ondřej ; Pavelka, Michal (advisor) ; Richter, Thomas (referee) ; Violeau, Damien (referee)
The thesis aims to find a generalization of smoothed particle hydrodynamics to fluid models which are compatible with Hamiltonian formulation of physics. We develop an approach based on a particle discretization of Poisson brackets. The main advantage of this approach is easy verification of conservation laws, which are related to the degree of consistency of discrete derivatives. Firstly, we demonstrate our technique on a particle approximation of symmetric hyperbolic thermodynamically compatible equations, which allow for unified description of fluids, viscoelastic materials and solids. Secondly, we develop a novel particle approximation for superfluid helium-4. 1
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Emergence of irreversible dynamics by the lack-of-fit reduction
Mladá, Kateřina ; Pavelka, Michal (advisor) ; Klika, Václav (referee)
The thesis studies theories of dimensional reduction on the example of the Kac- Zwanzig (heat bath) model. The studied methods are the Mori-Zwanzig projection for- malism and the lack-of-fit reduction, both applied for two sets of resolved variables. The methods give integro-differential and ordinary differential evolution equations re- spectively. For the Mori-Zwanzig formalism, a limit of the number of particles going to infinity is made, which leads to an exponential memory kernel and consequently to a set of stochastic differential equations. The evolution equations of the two methods are compared using numerical simulations. 1
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The Connection between Continuum Mechanics and Riemannian Geometry
Burýšek, Miroslav ; Pavelka, Michal (advisor) ; Klika, Václav (referee)
We investigate the systems of quasi-linear partial differential equations of hydrody- namic type. These equations occur mainly in hydrodynamics and continuum mechanics, but they arise in other various applications. In the study of such systems, one finds an intersection of Poisson and pseudo-Riemannian geometry. The Poisson bracket is deter- mined by functions that turn out to be metrics and Christoffel symbols. If the metric is non-degenerate, the existence of Poisson structure is equivalent to the existence of flat metric and Levi-Civita covariant derivative with zero curvature. Moreover, one can find special flat coordinates where the bracket is trivial. This result was found in the eighties by Dubrovin and Novikov for the one-dimensional case and later on extended to more dimensions. In this thesis we provide the proof of the Dubrovin-Novikov theorem, which was only sketched in the original paper. We also conducted an overview of current knowledge in the multi-dimensional case, where the theory gets much more complicated. In particular, the link between the compatible brackets and the possibility of finding flat coordinates is discussed. The Riemannian character of the Hamiltonian equations of hydrodynamic type can be used to prove their symmetric hyperbolicity, even when the equations are not in the...
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Consistent non-equilibrium thermodynamic modeling of hydrogen fuel cells
Jambrich, Jakub ; Pavelka, Michal (advisor) ; Souček, Ondřej (referee)
At first, a classical irreversible thermodynamic model of the hydrogen pump is derived in this thesis. The model is then numerically implemented by using the Finite volume method in the VoronoiFVM library in Julia. The numerical implementation is further used to explain the measured experimental data from [1]. The plateau observed in the Voltage-current figure could not be explained in the original work, as the membrane was approximated with a single point. Such a zero-dimensional model did not predict the plateau, and it was believed to originate from some Interfacial effects. This work will focus on the correct implementation of the equations inside the membrane and try to explain the observed effects using no additional assumptions.
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Thermodynamic modelling of hydrogen fuel cells
Nováček, Marek ; Pavelka, Michal (advisor) ; Němec, Tomáš (referee)
In this thesis, proton exchange membrane fuel cells are studied. At the beginning, the ideas underlying their function are exposed and some possibilities of usage are pre- sented. Thereafter, we aim to describe the processes inside the fuel cells with the aid of thermodynamics and in agreement with constitutive relations that have been obtained experimentally. Namely, we are interested in the fluxes of water and protons inside the membrane, where they are acted upon by thermodynamic forces, and the electrochemical reactions at the electrodes, which can be described by the Butler-Volmer equations. Also do we study the efficiency of the fuel cell by evaluating the production of entropy due to the diverse processes that take place in the fuel cell. It is the goal of the computational part of this thesis to propose a zero-dimensional model and compare it with the results provided in the supervisor's doctoral thesis. 1
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Thermodynamic analysis of processes in Hydrogen fuel cells.
Pavelka, Michal ; Maršík, František (advisor) ; Grmela, Miroslav (referee) ; Sciacovelli, Adriano (referee)
Non-equilibrium thermodynamics, which serves as a framework for formulating evolution equations of macroscopic and mesoscopic systems, is briefly reviewed and further developed in this work. For example, the relation between the General Equation for the Nonequilibrium Reversible- Irreversible Coupling (GENERIC) and (ir)reversibility is elucidated, and Onsager-Casimir reciprocal relations are shown to be an implication of GENERIC. Non-equilibrium thermodynamics is then applied to describe fuel cells and related devices, and theoretical conclusions are compared to experimental data. Moreover, a generalization of standard exergy analysis is developed bringing a new method for revealing a map of useful work losses in electricity producing devices. This method requires a non-equilibrium thermodynamic model, and so the general theory of non- equilibrium thermodynamics and optimization of real power generating devices stand side by side.
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