\inline{General motivation for polyhedral meshes in scientific computing -- advantages w.r.t. tetrahedral or hexahedral meshes: some ideas are in \url{https://www.symscape.com/polyhedral-tetrahedral-hexahedral-mesh-comparison}}
\inline{benefit of MHFEM completely on GPU: coupling with LBM}
Mathematical modeling of fluid dynamics has many ecological, medical and industrial applications and it is one of the central research topics investigated at the Department of Mathematics, FNSPE CTU in Prague in collaboration with prominent domestic as well as foreign institutions.
In order to model complex natural processes governing the behavior of fluids, it is often necessary to run high-resolution simulations on large computational clusters or supercomputers.
However, using the facilities for high performance computing efficiently is non-trivial, as it requires careful management of data in the computer memory and appropriate division of the task between all available processing units.
Especially when designing algorithms for systems with GPU accelerators, which provide significantly more processing units as well as memory levels compared to traditional computational systems, the specifics of the hardware architecture have to be considered.
The purpose of this thesis is the development of efficient data structures and parallel algorithms and using them as the main ingredients in solvers based on, e.g., the lattice Boltzmann method or the mixed-hybrid finite element method, which are used at the Department of Mathematics, FNSPE CTU in Prague.
Development of data structures for organizing structured as well as unstructured data in numerical simulations is necessary in order to match the requirements for efficient utilization of modern hardware architectures such as GPU clusters.
Similarly, high performance parallel algorithms have to be designed specifically for the hardware architectures used by modern GPUs.
\input{content/introduction.tex}
\cleardoublepage
\chapter{Programming Techniques for Modern Parallel Architectures}
\inline{General motivation for polyhedral meshes in scientific computing -- advantages w.r.t. tetrahedral or hexahedral meshes: some ideas are in \url{https://www.symscape.com/polyhedral-tetrahedral-hexahedral-mesh-comparison}}
\inline{benefit of MHFEM completely on GPU: coupling with LBM}
Mathematical modeling of fluid dynamics has many ecological, medical and industrial applications and it is one of the central research topics investigated at the Department of Mathematics, FNSPE CTU in Prague in collaboration with prominent domestic as well as foreign institutions.
In order to model complex natural processes governing the behavior of fluids, it is often necessary to run high-resolution simulations on large computational clusters or supercomputers.
However, using the facilities for high performance computing efficiently is non-trivial, as it requires careful management of data in the computer memory and appropriate division of the task between all available processing units.
Especially when designing algorithms for systems with GPU accelerators, which provide significantly more processing units as well as memory levels compared to traditional computational systems, the specifics of the hardware architecture have to be considered.
The purpose of this thesis is the development of efficient data structures and parallel algorithms and using them as the main ingredients in solvers based on, e.g., the lattice Boltzmann method or the mixed-hybrid finite element method, which are used at the Department of Mathematics, FNSPE CTU in Prague.
Development of data structures for organizing structured as well as unstructured data in numerical simulations is necessary in order to match the requirements for efficient utilization of modern hardware architectures such as GPU clusters.
Similarly, high performance parallel algorithms have to be designed specifically for the hardware architectures used by modern GPUs.
\section*{State of the art}
\addcontentsline{toc}{section}{State of the Art}
TODO
\inline{Data structures for GPUs -- easy to formulate for linear algebra as it provides a well-established framework for representing scientific problems, but for other fields such as mesh computations there is not one obvious way to do things (for example, one problem is that the system of notation is not settled).}
\inline{Linear algebra -- general description, highlight the solution of sparse linear systems (details in \cref{chapter:linear systems})}
\inline{CFD -- FVM, FEM, MHFEM, LBM}
\inline{coupling between different methods -- multiphysics}
\inline{established multiphysics software}
\section*{Research goals}
\addcontentsline{toc}{section}{Research Goals}
Based on the previous section, the following goals were identified for our research.
To the best of our knowledge, they represent unique ideas that push forward the frontiers of the state of the art.
\begin{enumerate}
\item
\inline{develop TNL -- an open-source software library of high performance algorithms and efficient data structures that follows modern software design patterns and simplifies the development of CFD solvers}
\inline{explain what makes it unique}
\inline{note that it is a never ending story (the goal lies in continuous effort rather than reaching a predetermined state)}
\item
\inline{data structure for the representation of unstructured meshes with full support of modern parallel platforms, including GPUs and MPI}
\item
\inline{MHFEM and LBM -- extend the author's previous work}
\item
\inline{LBM -- develop a scalable solver for GPU-based supercomputers}
\item
\inline{develop a high-performance solver for a coupled model for the simulation of vapor transport in air and validate the model using experimentally measured data}
\end{enumerate}
\section*{Achieved results}
\addcontentsline{toc}{section}{Achieved results}
TODO
\section*{Outlook into the future}
\addcontentsline{toc}{section}{Outlook into the future}
TODO
\section*{System of notation}
\addcontentsline{toc}{section}{System of notation}
TODO
\inline{subscripts and superscripts; vectors and matrices/tensors; 3D: $\vec v =[v_x, v_y, v_z]^T$ implies that $v_x \equiv v_1$, $v_y \equiv v_2$, and $v_z \equiv v_3$}
