University of Bayreuth master thesis (Scientific Computing)

\chapter{Linear Systems}

Let $A \in \mathbb{R}^{n \times n}$ be a symmetric, positiv definite matrix. The linear system 
\begin{align*}
Ax=b
\end{align*}
is to be solved.

% Starts a new Chapter
\chapter{Some Basic Stuff}\label{Kap1}

In this chapter some basic usage of LaTeX will be shown.

% Use sections uns subsections to structure your thesis
\section{Organize your text}

We start with some basics. 

\subsection{Basics - I}

Let's have a look at the elasticity equations.
Let $ \Omega \subset \mathbb{R}^3 $ with$\Gamma := \partial\Omega$ be given. The solution of the boundary value problem

\begin{align}
	-\sum\limits_{j=1}^{3} \frac{\partial}{\partial x_j} \sigma_{ij}(u,x) &= 0 
	& \text{für } x \in \Omega,\,i=1,2,3, \label{eq_elas}\\
	u(x) &= g_D(x) & \text{für } x \in \Gamma_D, \\
	\sum\limits_{j=1}^{3} \frac{\partial}{\partial x_j} \sigma_{ij}(u,x) n_j(x) &= g_{N,i}(x) & \text{for } x \in \Gamma_N,\,i=1,2,3, 
\end{align}
is to be found.

The homogenous elasticity equations \eqref{eq_elas} describe the deformations of linear elatic and homogenous materials. You can cite literature like this \cite{Beb}.

Let's define something.

\begin{Definition}\label{Sobolev}
Let $\Omega \subset \mathbb{R}^d$ be  $s \in \mathbb{N}_0  $ given.  
The Sobolev Space  $H^s(\Omega)$ is defined as
\begin{equation*}
H^s(\Omega) := \left\{ u \in L_{1}^{loc}(\Omega): D^\alpha u \in L_2(\Omega) \,\text{ for all } 
\alpha \in \mathbb{N}_0^d \text{ mit }  |\alpha| \leq s \right\}.
\end{equation*} 
\end{Definition}

Here we formulate a theorem for Sobolev Spaces defined in Definition \ref{Sobolev}:

\begin{Theorem}[Trace Theorem]\label{Spursatz}
Let $\Omega \subset \mathbb{R}^d$ be a Lipschitz domain with boundary $\Gamma$ and $s\in(\frac{1}{2}, \frac{3}{2})$. The Trace Operator
\begin{equation*}
\gamma_0: H^s(\Omega): \rightarrow H^{s-\frac{1}{2}}(\Gamma)
\end{equation*}
is linear and continous.
\end{Theorem}
\begin{proof}
It's a good excersices to do it for yourself.
\end{proof}

\section{Numerics}

For simulation some numerics are needed.

\subsection{A Very Short Introduction}

Assume there is a force $F$ acting on $\Omega$. For the pressure the condition

\begin{equation}\label{Kräftegleichgewicht}
\int\limits_{A} p (y) dS_y = F
\end{equation}
holds. This is visualized in figure \ref{Figure1}.

\begin{figure}[h]
\begin{subfigure}[c]{5cm}
\centering
\begin{tikzpicture}

\definecolor{darkred}{rgb}{0.6,0.1,0.1}
\definecolor{grey}{rgb}{0.63,0.63,0.63}

\draw[->,line width=1mm,darkred] (0,1.75) -- (0,1);
\node[,text width=1cm, darkred] at (0,1.5) {\textbf{F}};

\fill [ball color = grey](0,0.5) circle (0.5cm);
\fill [ball color = grey](0,-0.5) circle (0.5cm);

\draw[->,line width=1mm,darkred] (0,-1.75) -- (0,-1);
\node[,text width=1cm, darkred] at (0,-1.5) {\textbf{F}};

\end{tikzpicture}
\caption{Before}\label{Spheres_Before}
\end{subfigure} \quad
\begin{subfigure}[c]{5cm}
\centering
\begin{tikzpicture}

\definecolor{darkred}{rgb}{0.6,0.1,0.1}
\definecolor{bruntorange}{rgb}{0.8,0.33,0.0}
\definecolor{grey}{rgb}{0.63,0.63,0.63}

\fill [ball color = grey](0,0.4) ellipse (0.6cm and 0.4cm);
\fill [ball color = grey](0,-0.4) ellipse (0.6cm and 0.4cm);
\draw[line width=1mm,blue] (-0.22,0) -- (0.22,0);
\node[,text width=1cm, blue] at (1,0) {\textbf{$A$}};

\node[,text width=1cm, bruntorange] at (0,1.75) {\textbf{ }};
\node[,text width=1cm, bruntorange] at (0,-1.75) {\textbf{ }};

\draw[->,line width=1mm,darkred] (0,1.55) -- (0,0.8);
\node[,text width=1cm, darkred] at (0,1.3) {\textbf{F}};
\draw[->,line width=1mm,darkred] (0,-1.55) -- (0,-0.8);
\node[,text width=1cm, darkred] at (0,-1.3) {\textbf{F}};

\end{tikzpicture}
\caption{After}\label{Spheres_After}
\end{subfigure}
\caption{Deformation of Spheres}\label{Figure1}
\end{figure}

\FloatBarrier

\subsection{Need for Speed}

The computational methods need to fullfill some criteria.

\begin{itemize}

\item It needs to be fast

\item To be popular it should be easy to use

\end{itemize}

\chapter{Numerical Examples}

In this chapter some numerical examples will be presented.
 
\section{Plots in LaTeX}

It is also possible to create plots directly from data. 

\begin{figure}[h]
\begin{tikzpicture}[scale = 1.1]
\begin{axis}[
 xlabel = x,
 ylabel = y,
 yticklabel style={
        /pgf/number format/fixed,
        /pgf/number format/precision=5},
 scaled y ticks=false,
  xtick distance=1.0,
 grid=major,
 no markers,
 every axis plot/.append style={ultra thick},
 legend style={nodes={scale=0.8, transform shape}},
 legend entries={Data1,Data2}
]
\addplot table [x=xD1, y=yD1, col sep=comma]  {Plots/Data1.csv}; %
\addplot table [x=xD2, y=yD2, col sep=comma] {Plots/Data2.csv};
\end{axis}
\end{tikzpicture}
\caption{Some Plots}
\end{figure}


\FloatBarrier


\section{Test der Randelement-Vorkonditionierer}


\begin{Remark}
Here the usage and style of remark is shown.
\end{Remark}

LaTex also supports the creation of tables. Graphics can be imported to. The package subfigure allows ordering of floating elements inside one figure. 

\begin{figure}[h]
\begin{subfigure}{5cm}
\includegraphics[width=5cm]{Images/Mesh_Cube}
\end{subfigure}
\begin{subfigure}{7cm}
\begin{center}
\begin{tabular}{|c|r|r|}\hline
  Level & Elements & Nodes\\\hline
 $L_1$  & 972      &   488  \\
 $L_2$  & 3 888    & 1 946  \\
 $L_3$  & 15 552   & 7 778  \\
 $L_4$  & 62 208   & 31 106 \\\hline
\end{tabular}
\end{center}
\end{subfigure}
\caption{Mesh of cube $\Omega = [-1,1]^3$}
\label{VorkdWuerfel}
\end{figure} 

% Here the actually thesis begins, so I like the page numbering starting here
\pagenumbering{arabic}

\addcontentsline{toc}{chapter}{Introduction}


\chapter*{Introduction}

\thispagestyle{Introduction}

Write some introduction to your thesis. Normally it should contain a short overview of your work and describtions of the chapters.

\pagenumbering{roman}

 %--- Insert empty page without counting
\newpage 
\thispagestyle{empty}
\quad 
\addtocounter{page}{-1}



\chapter*{Preface}
\thispagestyle{empty}
\addtocounter{page}{-1}

This document is an example of how LaTeX can be used to create a Thesis. Basically it's a 
cut down version of my Master Thesis. It's goal is to provide a template for studens to 
fasciliate their first contact with LaTeX. Therefore, feel free to use this document as a 
starting point for your own work. If you prefere a different layout, adapt it to your liking.

Good luck!


\par \vspace{3cm}

Bayreuth, February 2021 \begin{flushright}
Thomas Rau
\end{flushright}

\addcontentsline{toc}{chapter}{Summary}
\chapter*{Summary}
\thispagestyle{Summary}

Summary of your work. What did you do, what where the central new insights...

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\begin{document}	

% Creates Title page defined below -----------------------------------------
\begin{titlepage}

\begin{flushright}

\includegraphics[width=3cm]{Images/LogoUniBayreuth}
\par\vspace{2cm}

\end{flushright}


\begin{center}
	{\huge\bfseries Thesis Title \par}
	\vspace{1cm}
	
	{\Large \bfseries Master Thesis \par}
	{\large at \par}
	{\Large \bfseries University of Bayreuth \par}
	{\large  Chair of \\ Scientific Computing \\
	         in cooperation with \par}
	{\Large \bfseries Partner's name \par}
    \vspace{4cm}
	
	{\large \begin{tabbing}
	 \quad \qquad \qquad \qquad \qquad\= by Name Name\\
	 \quad \qquad \qquad \qquad \qquad\= Supervised by Prof. Name\\
	 \quad \qquad \qquad \qquad \qquad\= 19 February 2021\\
	 \end{tabbing}}
	 
\end{center}

\end{titlepage}

%----------------------------------------------------------------------------------
% Similiar to programming, it's helpful to split large LaTeX projects into different
% files. 

% Includes the Content of Preface here
\include{Chapter/Preface}

% Creates table of Contents
\tableofcontents

% Here the main part starts.
\include{Chapter/Introduction}

\include{Chapter/Chapter1}
\include{Chapter/Chapter2}

\include{Chapter/Summary}

\appendix
\include{Chapter/Appendix}

% Creates a Bibliography with your References
\nocite{*}
\renewcommand{\bibname}{References}
\printbibliography[heading=bibintoc]

% adds an empty page without counting  
\newpage 
\thispagestyle{empty}
\quad 
\addtocounter{page}{-1}

\end{document}

% Encoding: UTF-8

@Book{Beb,
  author    = {M. Bebendorf},
  date      = {2008},
  title     = {Hierarchical Matrices - Lecture Notes in Computational Science and Engineering},
  publisher = {Springer},
}

@Comment{jabref-meta: databaseType:biblatex;}

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%--- Anpassung der Nummerierungen
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%--- Packages for images
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\usetikzlibrary{snakes}


%--- packages for plotting
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\usepackage{filecontents}
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%--- Packages for creation of plots
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