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Die x-Achse beschreibt die Entfernung zum Mittelpunkt der Galaxie Die y-Achse beschreibt die Warscheinlichkeit das ein Stern generiert wird \relax }}{6}{figure.caption.4}} +\newlabel{fig:rho}{{3}{6}{Die Rho Funktion im Intervall \( [~0~;~10^7 ~] \) geplottet mithilfe von Logarithmischen Achsen. \\ Die x-Achse beschreibt die Entfernung zum Mittelpunkt der Galaxie \\ Die y-Achse beschreibt die Warscheinlichkeit das ein Stern generiert wird \relax }{figure.caption.4}{}} +\@writefile{toc}{\contentsline {subsubsection}{\numberline {2.1.3}Lookup Tabellen}{7}{subsubsection.2.1.3}} +\newlabel{subsec:lookup}{{2.1.3}{7}{Lookup Tabellen}{subsubsection.2.1.3}{}} +\@writefile{toc}{\contentsline {subsection}{\numberline {2.2}Generierung eines Dunkle-Materie Halos durch Anpassung des NFW-Profils}{7}{subsection.2.2}} +\newlabel{eq:dark_matter}{{3}{7}{Generierung eines Dunkle-Materie Halos durch Anpassung des NFW-Profils}{equation.2.3}{}} +\@writefile{toc}{\contentsline {subsection}{\numberline 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diff --git a/langfassung/docs/4_ergebnisse.tex b/langfassung/docs/4_ergebnisse.tex index f398f55..f398f55 100644..100755 --- a/langfassung/docs/4_ergebnisse.tex +++ b/langfassung/docs/4_ergebnisse.tex diff --git a/langfassung/docs/5_quellen.aux b/langfassung/docs/5_quellen.aux new file mode 100644 index 0000000..f7e47e9 --- /dev/null +++ b/langfassung/docs/5_quellen.aux @@ -0,0 +1,32 @@ +\relax +\providecommand\hyper@newdestlabel[2]{} +\@writefile{toc}{\contentsline {paragraph}{\nonumberline Herrn J\IeC {\"o}rg Thar}{17}{section*.11}} +\@writefile{toc}{\contentsline {paragraph}{\nonumberline Tim Tugendhat}{17}{section*.12}} +\@writefile{toc}{\contentsline {paragraph}{\nonumberline Konstantin Bosbach}{17}{section*.13}} +\@writefile{toc}{\contentsline {paragraph}{\nonumberline Tilman Hoffbauer}{17}{section*.14}} +\@setckpt{docs/5_quellen}{ +\setcounter{page}{18} +\setcounter{equation}{17} +\setcounter{enumi}{0} +\setcounter{enumii}{0} +\setcounter{enumiii}{0} +\setcounter{enumiv}{0} +\setcounter{footnote}{3} +\setcounter{mpfootnote}{0} +\setcounter{part}{0} +\setcounter{section}{4} +\setcounter{subsection}{0} +\setcounter{subsubsection}{0} +\setcounter{paragraph}{0} +\setcounter{subparagraph}{0} +\setcounter{figure}{4} +\setcounter{table}{0} +\setcounter{parentequation}{0} +\setcounter{Item}{0} +\setcounter{Hfootnote}{3} +\setcounter{bookmark@seq@number}{27} +\setcounter{ContinuedFloat}{0} +\setcounter{lstnumber}{15} +\setcounter{section@level}{4} +\setcounter{lstlisting}{0} +} diff --git a/langfassung/docs/5_quellen.tex b/langfassung/docs/5_quellen.tex index 9932028..6219a0a 100644..100755 --- a/langfassung/docs/5_quellen.tex +++ b/langfassung/docs/5_quellen.tex @@ -18,3 +18,5 @@ Die Einführung in die Mathematik bekam ich während meines Praktikums im Zentru \centering \vspace{0.5cm} \textbf{Außerdem gilt mein Dank allen, die mich auf jede nur erdenkliche Weise unterstützt haben.} + +\raggedright diff --git a/langfassung/docs/6_abgabe.tex b/langfassung/docs/6_abgabe.tex new file mode 100755 index 0000000..22fe221 --- /dev/null +++ b/langfassung/docs/6_abgabe.tex @@ -0,0 +1,273 @@ +% \subsection{Rotationsmatrizen} +% +% \begin{tabular}{ l | l } +% I & II \\ \hline +% IV & III \\ +% \end{tabular} +% +% \flushleft +% \hfill +% \begin{tabular}{l | l l} +% Clockwise & & \\ \hline +% I & + & + \\ +% II & + & - \\ +% III & - & - \\ +% IV & - & + +% \end{tabular} +% \hfill +% \begin{tabular}{l | l l} +% Anticlockwise & & \\ \hline +% I & - & - \\ +% II & - & + \\ +% III & + & + \\ +% IV & + & - +% \end{tabular} +% \hfill +% +% \subsection{Die Matrizen} +% +% \subsubsection{Rotation um die x-Achse} +% +% \begin{equation} +% R_x (\alpha) = +% \begin{pmatrix} +% 1 & 0 & 0 \\ +% 0 & cos(\alpha) & -sin(\alpha) \\ +% 0 & sin(\alpha) & cos(\alpha) +% \end{pmatrix} +% \end{equation} +% +% \subsubsection{Rotation um die y-Achse} +% +% \begin{equation} +% R_y (\alpha) = +% \begin{pmatrix} +% cos(\alpha) & 0 & sin(\alpha) \\ +% 0 & 1 & 0 \\ +% -sin(\alpha) & 0 & cos(\alpha) +% \end{pmatrix} +% \end{equation} +% +% \subsubsection{Rotation um die z-Achse} +% +% \begin{equation} +% R_z (\alpha) = +% \begin{pmatrix} +% cos(\alpha) & -sin(\alpha) & 0 \\ +% sin(\alpha) & cos(\alpha) & 0 \\ +% 0 & 0 & 1 +% \end{pmatrix} +% \end{equation} + +\subsection{Spiralgalaxien} + +\subsection{Using Object Oriented Programming (OOP) techniques} + +In my case, the objects are galaxies. + +\subsubsection{Initialisation} + +The galaxy is initialised with the following objects: + +\begin{itemize} + \item A list storing the coordinates of each star + \begin{itemize} + \item X Coordinate + \item Y Coordinate + \item Z Coordinate + \end{itemize} + + \item A list storing the individual forces acting on the stars + \begin{itemize} + \item X Force + \item Y Force + \item Z Force + \end{itemize} + + \item A variable storing the number of stars generated in the galaxy + + \item Newtons gravitational constant +\end{itemize} + +\subsection{Generation of new stars} + +The function is given an integer defining the amount of stars that should be +newly generated. + +The newly generated stars are then appended to the list storing the coordinates. + +The counter counting the amount of stars in the galaxy is incremented. + +\subsection{Printing all the coordinates} + +The function cycles through the list storing the star coordinates and prints +them to the command line. + +\subsection{Calculating the Forces acting between the Stars} + +The function recieves two star objects and an axis on which the forces should +be calculated and returns the force acting on the given axis. In case of a +failture (The two given stars have got the same coordiantes), the function +just goes on to the next Star. + +The Forces can be calculated using the equation (\ref{eq:gravitation_law}). + +\subsection{Calculating the forces acting between each star in the galaxy and +each other star} + +To calculate the forces inbetween every star in the galaxy, the function cycles +through every star, looks if the force that should be calculated hat not been +calculated yet and calculates it. This includes testing if the force that should +be calculated is not the force inbetween a star and itself. + +The results of the calculations are stored in a list storinf the forces. + +\subsection{Printing all the individual forces} + +The function is able to print all the forces acting inbetween the stars if no +argument is given. If an argument n is given, the function print out the nth +star in the list. + +\subsection{Spherical cells} + + +\subsubsection{Testing if a point is inside or outside a sphere} + +In order to test is a point is inside a sphere, one just has to test if the +following conditions are all true: + +\begin{equation} + \begin{split} + S_x - S_r \leq P_x \leq S_x + S_r \\ + S_y - S_r \leq P_y \leq S_y + S_r \\ + S_z - S_r \leq P_z \leq S_z + S_r + \end{split} +\end{equation} + +\( P_x \) , \( P_y \) and \( P_z \) are the coordinates of the point to be tested, +\( S_x \) , \( S_y \) and \( S_z \) are the coordinates of the midpoint of the sphere and +\( S_r \) is the radius of the sphere. + +\subsubsection{Testing if a star is inside or outside of a sphere for a whole galaxy} + +While testting if a star is inside a sphere or not, because of the alignment +of the spheres, a point can be in more than one sphere at the same time. +To get rid of this problem, the software cycles over every star and searches +for matches within the spheres. If a match is found, the next star is tested. +This is pretty much as efficient as it can get. + +\begin{equation} + O(n) = n_{stars} \cdot n_{spheres} +\end{equation} + +\subsubsection{Generate the position of the spheres} + +Generating the position of the spheres is accomplished in the following way: +A 3D-grid is generated and the midpoints of the spheres are positioned on the +gridpoints. + +[Include Graphic] + +The distance the spheres have to each other ist defined using the following +function: + +\begin{equation} \label{sphere_distance} + \texttt{sphere\_distance} = \frac{\texttt{galaxy\_range}}{\texttt{sampling\_rate}} +\end{equation} + +The higher the \texttt{sampling\_rate} is, the more spheres get generated. + +The next goal is to find out the ``sweet spot'' generating the spheres. +When using a very low \texttt{sampling\_rate}, the reault gets inacurate, but +when using a high \texttt{sampling\_rate}, the calculations are not affected +in term of speed and efficiency. The Goal is therefor to find a sampling rate +that enables the generation of accurate but fast results. + +By being able to controll the accuracy anf therefor the time, it is possible to +teach the system to generate a galaxy in like one hour and it will automatically +set the sampling rate so low that the simulation will finish perfectly in time. + +\subsubsection{The radius of the spheres} + +The Radius of the spheres is dynamicly allocated to ensure that the whole galaxy +is covered. + +\begin{equation} \label{sphere_radius} + r = \sqrt{\texttt{sphere\_distance}^2 + \texttt{sphere\_distance}^2 + \texttt{sphere\_distance}^2} +\end{equation} + +\begin{equation*} + 1.7320508075688772 = \sqrt{1^2 + 1^2 + 1^2} +\end{equation*} + +The equation (\ref{sphere_radius}) highly depends on the equation +(\ref{sphere_distance}) and it's parameters. + +\begin{figure} + \includegraphics[width=1\textwidth]{figs/sphere_alignment_cc} + \caption{The Alignment of the spheres\\ + Left: The perfect alignment covering the complete space\\ + Right: A previously generated alignment using small spheres to cover the missing space + } + \label{sphere_alignment} +\end{figure} + +\subsubsection{Calculate the forces acting on the spheres} + +In order to reduce the time that is needed to calculate the forces inbetween +the stars, the stars are subdivided in different cells, in this case spheres. +After all the forces acing inside one sphere are calculated, the forces are +combined and applied to the midpoint of the sphere genrating a new coordinate: +the mean force. The mean force inbetween all the cells can be calculated. + +[Include description binary tree] + +[Include graphic binary tree] + +\subsubsection{Calculate the forces acting on all the spheres together} + +This should be 0. + +\subsubsection{Benchmarks} + +\begin{tabular}{l | l | l | l} + Nr of Stars & Sample rate & Galaxy Range & Time (s) \\\hline + + + 100 & 1 & 100 & 0.0814 \\ + 75 & 1 & 100 & 0.0499 \\ + 50 & 1 & 100 & 0.0295 \\ + 25 & 1 & 100 & 0.0116 \\ \hline + + 100 & 1 & 100 & 0.0828 \\ + 100 & 2 & 100 & 0.0909 \\ + 100 & 4 & 100 & 0.1832 \\ + 100 & 8 & 100 & 1.1114 \\ + 100 & 16 & 100 & 7.6944 \\ + 100 & 32 & 100 & 56.5731 \\ + 100 & 64 & 100 & 217.7768 \\ \hline + + 100 & 1 & 1 & 0.0809 \\ + 100 & 1 & 2 & 0.0844 \\ + 100 & 1 & 4 & 0.0782 \\ + 100 & 1 & 8 & 0.0758 \\ + 100 & 1 & 16 & 0.0847 \\ + 100 & 1 & 32 & 0.0815 \\ + 100 & 1 & 64 & 0.0770 \\ + +\end{tabular} + +The sample rate is the factor that influences the time the most. Knowing this, +it (the sample rate) can be used to controll the time in which a galaxy can +be created. +This is usefull in particular when using some very powerfull mashine with +limited time. + +\subsection{Calculate the Position of a Star after a timestep} + +Not to be considered: +\begin{itemize} + \item drag + \item any kind of resistance + \item acceleration +\end{itemize} diff --git a/langfassung/figs/Jugend_forscht.jpg b/langfassung/figs/Jugend_forscht.jpg index 6d23e96..6d23e96 100644..100755 --- a/langfassung/figs/Jugend_forscht.jpg +++ b/langfassung/figs/Jugend_forscht.jpg Binary files differdiff --git a/langfassung/figs/Untitled Diagram.xml b/langfassung/figs/Untitled Diagram.xml index 7396b9b..7396b9b 100644..100755 --- a/langfassung/figs/Untitled Diagram.xml +++ b/langfassung/figs/Untitled Diagram.xml diff --git a/langfassung/figs/all_forces_5000.png b/langfassung/figs/all_forces_5000.png new file mode 100644 index 0000000..fccab88 --- /dev/null +++ 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If it's 1, each line will be numbered } +% hyperref / hyperlink settings \usepackage{hyperref} \hypersetup{ colorlinks, @@ -35,10 +56,12 @@ linkcolor=black, urlcolor=black } +% renew some commands \newcommand{\bold}{\textbf} \renewcommand{\arraystretch}{1.5} \newcommand{ \rarrow }{\( \rightarrow \)} +% redefine some parameters \setlength\parindent{0pt} \setlength\parskip{10pt} \setlength{\footskip}{30pt} @@ -48,13 +71,16 @@ urlcolor=black \begin{document} +% titlepage settings \title{Galaxy Generation} \subtitle{Jugend Forscht \the\year} \author{ Emile Hansmaennel \texttt{ emile.hansmaennel@gmail.com }} \date{\today} +% generate the title using the informations given above \maketitle +% include a nice image \begin{figure}[h] \centering \includegraphics[width=120mm, trim={0 8.5cm 0 8.5cm}, clip]{figs/galaxy} @@ -62,33 +88,43 @@ urlcolor=black \caption{} \end{figure} +% include the abstract \begin{abstract} -\input{docs/1_kurzfassung} +\include*{docs/1_kurzfassung} \end{abstract} +% don't insert a page number on the first page \thispagestyle{empty} \clearpage \newpage \setcounter{page}{1} +% generate a table of contents \tableofcontents \newpage \section{Einleitung} \label{Einleitung} -\input{docs/2_einleitung} +% \input{docs/2_einleitung} +\include*{docs/2_einleitung} \newpage \section{Hauptteil} \label{Hauptteil} -\input{docs/3_hauptteil} +% \input{docs/3_hauptteil} +\include*{docs/3_hauptteil} \newpage \section{Ergebnisse} \label{ergebnisse} -\input{docs/4_ergebnisse} +% \input{docs/4_ergebnisse} +\include*{docs/4_ergebnisse} \newpage \section{Quellen und Hilfen} \label{quellen} -\input{docs/5_quellen} - +% \input{docs/5_quellen} +\include*{docs/5_quellen} \newpage +\section{Nach der Abgabe...} \label{nach_der_abgabe} +% \input{docs/6_abgabe} +\include*{docs/6_abgabe} + \end{document} diff --git a/langfassung/master.toc b/langfassung/master.toc index 4df4eba..4a85a0f 100644..100755 --- a/langfassung/master.toc +++ b/langfassung/master.toc @@ -5,32 +5,50 @@ \contentsline {section}{\numberline {2}Hauptteil}{3}{section.2} \contentsline {subsection}{\numberline {2.1}Generierung von elliptischen Galaxien}{3}{subsection.2.1} \contentsline {subsubsection}{\numberline {2.1.1}Das Navarro-Frenk-White Profil}{3}{subsubsection.2.1.1} -\contentsline {subsubsection}{\numberline {2.1.2}Random Sampling}{5}{subsubsection.2.1.2} +\contentsline {subsubsection}{\numberline {2.1.2}Random Sampling}{3}{subsubsection.2.1.2} \contentsline {subsubsection}{\numberline {2.1.3}Lookup Tabellen}{5}{subsubsection.2.1.3} \contentsline {subsection}{\numberline {2.2}Generierung eines Dunkle-Materie Halos durch Anpassung des NFW-Profils}{5}{subsection.2.2} -\contentsline {subsection}{\numberline {2.3}Stauchung und Streckung der Galaxie}{6}{subsection.2.3} +\contentsline {subsection}{\numberline {2.3}Stauchung und Streckung der Galaxie}{5}{subsection.2.3} \contentsline {subsection}{\numberline {2.4}Rechenaufwand}{6}{subsection.2.4} -\contentsline {subsection}{\numberline {2.5}Beschleunigung der Generation}{7}{subsection.2.5} -\contentsline {subsubsection}{\numberline {2.5.1}Lookuptable}{7}{subsubsection.2.5.1} +\contentsline {subsection}{\numberline {2.5}Beschleunigung der Generation}{6}{subsection.2.5} +\contentsline {subsubsection}{\numberline {2.5.1}Lookuptable}{6}{subsubsection.2.5.1} \contentsline {subsubsection}{\numberline {2.5.2}Mehr Rechenleistung!}{7}{subsubsection.2.5.2} \contentsline {paragraph}{\nonumberline Amazon Web Services}{7}{section*.5} -\contentsline {subsubsection}{\numberline {2.5.3}Nichts in der Konsole ausgeben}{8}{subsubsection.2.5.3} -\contentsline {subsection}{\numberline {2.6}Nutzung eines neuronalen Netzes zum unbeaufsichtigten generieren von Galaxien}{8}{subsection.2.6} -\contentsline {subsubsection}{\numberline {2.6.1}Aufbau des neuronalen Netzes}{8}{subsubsection.2.6.1} +\contentsline {subsubsection}{\numberline {2.5.3}Nichts in der Konsole ausgeben}{7}{subsubsection.2.5.3} +\contentsline {subsection}{\numberline {2.6}Nutzung eines neuronalen Netzes zum unbeaufsichtigten generieren von Galaxien}{7}{subsection.2.6} +\contentsline {subsubsection}{\numberline {2.6.1}Aufbau des neuronalen Netzes}{7}{subsubsection.2.6.1} \contentsline {paragraph}{\nonumberline Neuronen und Synapsen}{8}{section*.6} \contentsline {subsection}{\numberline {2.7}Spiralgalaxien}{9}{subsection.2.7} \contentsline {subsubsection}{\numberline {2.7.1}Das n-K\IeC {\"o}rper Problem}{9}{subsubsection.2.7.1} \contentsline {subsubsection}{\numberline {2.7.2}Unterteilung des Vektorraumes in verschiedene Zellen}{10}{subsubsection.2.7.2} -\contentsline {subsubsection}{\numberline {2.7.3}Berechnung der wirkenden Kr\IeC {\"a}fte}{11}{subsubsection.2.7.3} -\contentsline {paragraph}{\nonumberline Masse der Sterne}{11}{section*.7} -\contentsline {paragraph}{\nonumberline Abstand der Sterne}{11}{section*.8} -\contentsline {subsection}{\numberline {2.8}Weiteres}{11}{subsection.2.8} -\contentsline {section}{\numberline {3}Ergebnisse}{12}{section.3} -\contentsline {subsection}{\numberline {3.1}Simulations Geschwindigkeit}{12}{subsection.3.1} -\contentsline {subsection}{\numberline {3.2}Lookuptabellen Geschwindigkeit}{12}{subsection.3.2} -\contentsline {subsection}{\numberline {3.3}Fazit}{13}{subsection.3.3} -\contentsline {section}{\numberline {4}Quellen und Hilfen}{14}{section.4} -\contentsline {paragraph}{\nonumberline Herrn J\IeC {\"o}rg Thar}{14}{section*.11} -\contentsline {paragraph}{\nonumberline Tim Tugendhat}{14}{section*.12} -\contentsline {paragraph}{\nonumberline Konstantin Bosbach}{14}{section*.13} -\contentsline {paragraph}{\nonumberline Tilman Hoffbauer}{14}{section*.14} +\contentsline {subsubsection}{\numberline {2.7.3}Berechnung der wirkenden Kr\IeC {\"a}fte}{10}{subsubsection.2.7.3} +\contentsline {paragraph}{\nonumberline Masse der Sterne}{10}{section*.7} +\contentsline {paragraph}{\nonumberline Abstand der Sterne}{10}{section*.8} +\contentsline {subsection}{\numberline {2.8}Weiteres}{10}{subsection.2.8} +\contentsline {section}{\numberline {3}Ergebnisse}{11}{section.3} +\contentsline {subsection}{\numberline {3.1}Simulations Geschwindigkeit}{11}{subsection.3.1} +\contentsline {subsection}{\numberline {3.2}Lookuptabellen Geschwindigkeit}{11}{subsection.3.2} +\contentsline {subsection}{\numberline {3.3}Fazit}{11}{subsection.3.3} +\contentsline {section}{\numberline {4}Quellen und Hilfen}{13}{section.4} +\contentsline {paragraph}{\nonumberline Herrn J\IeC {\"o}rg Thar}{13}{section*.11} +\contentsline {paragraph}{\nonumberline Tim Tugendhat}{13}{section*.12} +\contentsline {paragraph}{\nonumberline Konstantin Bosbach}{13}{section*.13} +\contentsline {paragraph}{\nonumberline Tilman Hoffbauer}{13}{section*.14} +\contentsline {section}{\numberline {5}Nach der Abgabe...}{14}{section.5} +\contentsline {subsection}{\numberline {5.1}Spiralgalaxien}{14}{subsection.5.1} +\contentsline {subsection}{\numberline {5.2}Using Object Oriented Programming (OOP) techniques}{14}{subsection.5.2} +\contentsline {subsubsection}{\numberline {5.2.1}Initialisation}{14}{subsubsection.5.2.1} +\contentsline {subsection}{\numberline {5.3}Generation of new stars}{14}{subsection.5.3} +\contentsline {subsection}{\numberline {5.4}Printing all the coordinates}{14}{subsection.5.4} +\contentsline {subsection}{\numberline {5.5}Calculating the Forces acting between the Stars}{14}{subsection.5.5} +\contentsline {subsection}{\numberline {5.6}Calculating the forces acting between each star in the galaxy and each other star}{15}{subsection.5.6} +\contentsline {subsection}{\numberline {5.7}Printing all the individual forces}{15}{subsection.5.7} +\contentsline {subsection}{\numberline {5.8}Spherical cells}{15}{subsection.5.8} +\contentsline {subsubsection}{\numberline {5.8.1}Testing if a point is inside or outside a sphere}{15}{subsubsection.5.8.1} +\contentsline {subsubsection}{\numberline {5.8.2}Testing if a star is inside or outside of a sphere for a whole 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