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Extension of smoothed particle hydrodynamics based on Poisson brackets
Thesis title in Czech: Rozšíření metody smoothed particle hydrodynamics s využitím Poissonových závorek
Thesis title in English: Extension of smoothed particle hydrodynamics based on Poisson brackets
Key words: smoothed particle hydrodynamics|SHTC rovnice|supratekuté helium
English key words: smoothed particle hydrodynamics|SHTC equations|superfluid helium
Academic year of topic announcement: 2020/2021
Thesis type: dissertation
Thesis language: angličtina
Department: Mathematical Institute of Charles University (32-MUUK)
Supervisor: doc. RNDr. Michal Pavelka, Ph.D.
Author: hidden - assigned and confirmed by the Study Dept.
Date of registration: 16.07.2020
Date of assignment: 16.07.2020
Confirmed by Study dept. on: 24.09.2020
Date and time of defence: 09.01.2024 14:00
Date of electronic submission:12.10.2023
Date of submission of printed version:17.10.2023
Date of proceeded defence: 09.01.2024
Opponents: prof. Thomas Richter
  prof. Damien Violeau
Advisors: Tomáš Němec
The student should focus on these topics:
1) Further structure of the equations describing continuum thermodynamics like
hyperbolicity, Hamiltonianity, dissipativeness, gauge invariance, etc. Also structure of boundary conditions should be taken into account.

2) The structure should be transmitted to numerical methods solving the equations. Developments of novel numerical methods respecting the structure of the evolution equations are anticipated.

3) The methods should be applied to concrete physical systems. In particular, they should be applied on modelling of electrochemical devices like fuel cells. There are many problems that deserve attention, like proper description of the fundamental processes or complex, e.g. pulsating, flows and their effect on the overall efficiency.
The goal of introducing a pulsating flow is to increase the power output of the fuel cell through enhancement of the electrochemical reactions on the electrodes and removal of the product water from the cathode by fluctuating reactant flow. This part would be studied in collaboration with the Institute of Thermomechanics, ASCR (lab of Dr. Němec) .
[1] F. Maršík, Termodynamika kontinua, Academia 1999.
[2] Hairer, Lubich, Geometric Numerical Integration, Springer 2006.
[3] B. Després, Numerical Methods for Eulerian and Lagrangian Conservation Laws, Springer 2017.
[4] Michal Pavelka, Ilya Peshkov and Václav Klika, On Hamiltonian continuum mechanics, arXiv:1907.03396 [physics.class-ph], 2019, submitted to Physica D.
[5] Ilya Peshkov, Michal Pavelka, Evgeniy Romenski, Miroslav Grmela, Continuum Mechanics and Thermodynamics in the Hamilton and the Godunov-type Formulations, Continuum Mechaanics and Thermodynamics 30(6), 1343-1378, 2018.
[6] Michal Pavelka, Václav Klika and Miroslav Grmela, Ehrenfest regularization of Hamiltonian systems, Physica D: Nonlinear phenomena, 399, 193-210, 2019.
[7] Michal Pavelka, Václav Klika and Miroslav Grmela. Multiscale Thermo-Dynamics, de Gruyter (Berlin), 2018
[8] H. C. Öttinger, GENERIC Integrators: Structure Preserving Time Integration for Thermodynamic Systems, Journal of Non-Equilibrium Thermodynamics 43(2), 2018.
[9] Pavelka, M., Klika, V., Vágner, P., Maršík, F., Generalization of Exergy Analysis, Applied Energy 137 (2015), pp. 158-172.
[10] Václav Tesař: Pressure-Driven Microfluidics, 2007.
[11] Michael Eikerling, Andrei Kulikovski: Polymer Electrolyte Fuel Cells: Physical Principles of Materials and Operation, 2015.
Preliminary scope of work in English
Development of numerical methods solving differential equations is usually based on the properties of the equations. However, when the equations describe physical systems, then they are not just arbitrary equations, but they possess certain structure. One such structure is balance laws [1], which served as motivation for finite volume methods, but one can also see for instance the motion of particles (symplectic and GLACE intergrators) [2,3], gauge freedom of the equations [3,4], the Hamiltonian structure [5] or the structure of General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC) [6,7]. Numerical schemes respecting further structure of the continuum equations have typically advantageous properties, like long-time stability and precision.

The strength of the newly proposed new numerical schemes can be then demonstrated on particular applications in continuum thermodynamics, e.g. fuel cells [8] and evolution of complex fluids and solids.
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