\section{Gitter}
Alle Gitter bestehen nur aus achsenorientierten Rechtecken.
-Entscheidend für die Tests sind Gitter in einer Ebene und Ganze Körper...
+Entscheidend für die Tests sind Gitter in einer Ebene und Körper...
\chapter{Analytische Berechnung der Integrale}
--- /dev/null
+\documentclass[a4paper,10pt,fleqn]{report}
+\usepackage{fullpage} %Seiten etwas Größer
+\usepackage{amsmath,amssymb} %Mathematische Symbole
+%\usepackage{moreverb}
+\usepackage{graphicx,psfrag,subfig} %Grafiken einbinden/Texte ersetzen/Bilder nebeneinander
+%\usepackage{ifthen}
+%\usepackage{showkeys}
+%\usepackage[final,numbered]{mcode}
+\usepackage{colortbl} %Einfache Färbungen in Tabellen
+\usepackage{color} %Farben benutzen und Definieren
+\usepackage{subfig} %mehrere Figuren in einer
+\usepackage{hyperref} %Links im Inhaltsverzeichnis
+\hypersetup{linkbordercolor={1 1 1},citebordercolor={1 1 1},urlbordercolor={1 1 1}}
+
+
+\usepackage[ngerman]{babel} %Sprachpacket für Überschriften
+\usepackage[utf8]{inputenc} %Eingabekodierung
+\usepackage{fixltx2e} %Deutschsprach Bugs
+
+\definecolor{gray}{gray}{.95}
+
+\def\todo#1{\textcolor{red}{#1}}
+\def\why#1{\textcolor{blue}{#1}}
+\def\Matlab{{\sc Matlab}}
+\def\q{\Q}
+
+\def\Ea{$E$}
+\def\Eb{$\tilde E$}
+\def\v{\boldsymbol{v}}
+\def\D{\mathcal{D}}
+\def\Q{\mathcal{Q}}
+\def\G{\mathcal{G}}
+\def\L{\mathcal{L}}
+\def\T{\mathcal{T}}
+\def\oder{\vee}
+\def\und{\wedge}
+
+\newcommand{\dif}[2]{\frac{\partial#1}{\partial#2}}
+\newcommand{\enorm}[1]{\left| \! \left| \! \left|#1\right| \! \right| \! \right|}
+\newcommand{\norm}[1]{\lVert#1\rVert}
+\newcommand{\abs}[1]{\lvert#1\rvert}
+\newcommand{\showMesh}[2][]{\begin{figure}[ht]
+\caption{#1}
+\label{#2}
+\centering
+\subfloat[Lage]{\includegraphics[width=0.5\textwidth]{fig/#2_ref}}
+\subfloat[Koordinaten]{\input{fig/#2_coo}}\\
+\subfloat[Elemente]{\input{fig/#2_ele}}
+\subfloat[Nachbarn]{\input{fig/#2_nei}}
+\end{figure}}
+
+\def\N{\mathbb{N}}
+\def\R{\mathbb{R}}
+
+\author{P. Schaefer}
+
+\begin{document}
+\tableofcontents
+\clearpage
+\chapter{Implementierung}
+
+
+\section{Datenstruktur}
+Alle Koordinaten werden in einer $ N \times 3,N \in \N$ Matrix abgespeichert.
+\begin{displaymath}
+ COO[i,1:3] = C_i := (x_1,x_2,x_3)^{-1} \text{ wobei } x_1,x_2,x_3 \in \R \und i \in \{1,2 \dots N\}
+\end{displaymath}
+Jedes Element wird dann durch jeweils vier Koordinatenindizes als $M \times 4, M\in\N$ Matrix gespeichert. Wobei die Koordinaten links herum durchgegangen werden.
+\begin{displaymath}
+ ELE[i,1:4] = E_i := (v_1,v_2,v_3,v_4) \text{ wobei } v_1,v_2,v_3,v_4 \in C \und i \in \{1,2 \dots M\}
+\end{displaymath}
+Aufgrund der Netzstabilität wollen wir maximal zwei Nachbarn an einer Kante zulassen, weshalb wir nur $2*4$ Einträge für die Nachbarschaftsrelationen pro Element benötigen. In der $M \times 8$ Matrix sind die Indizes der Nachbarn $n_1,n_2 \in E$ der Kante $(v_k, v_{k^{+1}}) | k\in \{1,2,3,4\}$ des Elements $E_i$ an den Stellen $NEI[i,[k,k+4]]$ gespeichert. Sollte eine Seite keinen Nachbarn haben markieren wir diese mit einer $0$. Außerdem wird an Kanten mit nur einem Nachbarn der erste Index auf den Nachbarn gesetzt und der zweite auf $0$.(Siehe Figur:\ref{exmpl3:nei:part})
+\\\noindent
+Ein ausführliches Beispiel ist in Figure \ref{exmpl3} dargestellt.
+\showMesh[Beispiel 3]{exmpl3}
+\begin{figure}[ht]
+\caption{Nachbarschaftsrelationen Element 4 aus \ref{exmpl3}}
+\label{exmpl3:nei:part}
+\centering
+ \subfloat[Lage]{\includegraphics{fig/Net_Neigh}}
+ \subfloat[Nachbarn]{\input{fig/exmpl3_nei_part}}
+\end{figure}
+
+
+
+
+
+\section{Verfeinern}
+Da wir im weiteren Verlauf sowohl adaptive also auch anisotrope Netzverfeinerung zulassen wollen, ist es sinnvoll eine Verfeinerungsfunktion zu implementieren, die alle Möglichen Teilungsprozesse auf einem Element unterstützt. Dabei sind vier nur wirklich relevant:
+\begin{enumerate}
+ \item keine Teilung
+ \item volle Teilung in vier gleich große Elemente
+ \item halbe Teilung in zwei gleichgroße horizontal liegende Elemente
+ \item halbe Teilung in zwei gleichgroße vertikal liegende Elemente
+\end{enumerate}
+Zusätzlich wurde auch Typ 5. belegt, welcher als Ergebnis eine volle Teilung 2. ausführt, diese aber schrittweise durch eine 3. Teilung und zwei 4. Teilungen.
+
+Damit jedem Element $E_i$ eine Teilungsart zugeordnet werden kann, haben wir einen Markierungsvektor $marked \in {1,2,3,4,5}^M$ eingeführt. Dabei entspricht $marked_i$ der Art der Teilung für das Element $E_i$. Um isotrope und auch uniforme Teilungen zu erleichtern kann statt dem Vektor $marked$ auch nur ein Skalar übergeben werden $marked \in {1,2\dots5}$, wodurch jedes Element mit der gewählten Art verfeinert wird.
+
+Relevant zum Verfeinern eines Netzes sind also die Koordinaten $COO$, Elemente $ELE$ sowie die Nachbarschaftsrelationen $NEI$ und der Markierungsvektor $marked$.
+
+Da wir später einen Fehlerschätzer berechnen wollen, ist es wichtig sich zu jedem Element seine Teilelemente zu merken. Dazu legen wir während der Teilung eine $M \times 4$ Matrix an, in der die maximal vier Elementindizes gespeichert sind. Wenn wir also ein Element in vier gleich große Teile verfeinern, so wird das neue Element links unten das erste sein und alle weiteren folgen im Kreis links herum. Teilen wir ein Element in zwei gleich große Elemente, so werden die doppelt belegten Quadranten auch doppelt eingetragen. Ein gar nicht geteiltes Element wird also vier mal den alten Indizes speichern. Dadurch wird sicher gestellt, dass das arithmetische Mittel über die Elemente immer gültig auszuführen ist.
+(Siehe Figur:\ref{exmpl3:f2s})
+
+$[COO_{fine}, ELE_{fine}, NEI_{fine}, F2S ] = refineQuad(COO, ELE, NEI, marked);$
+
+\begin{figure}[ht]
+\caption{VaterSohn}
+\centering
+\subfloat[VaterSohn aus \ref{exmpl2}]{\input{fig/exmpl2_f2s}}
+\subfloat[VaterSohn aus \ref{exmpl3}]{\input{fig/exmpl3_f2s}}
+\end{figure}
+
+\section{Fehlerschätzer}
+\begin{eqnarray*}
+\mu_h^2 & = & \norm{\varrho_h^{1/2}(\phi_{\frac h 2}-\Pi_h\phi_{\frac h 2})}_{\L^2(\T)}^2 \\
+& = &\sum_{T\in \T_h}\mu_h(T)^2\\
+\mu_h(T)^2 & = & \norm{\varrho_h^{1/2}(\phi_{\frac h 2}-\Pi_h\phi_{\frac h 2})}_{\L^2(\T)}^2 \\
+& = & h_{min}(T)\norm{\phi_{\frac h 2}-\Pi_h\phi_{\frac h 2}}_{\L^2(\T)}^2 \\
+&& T_j \in \tau_h, t_j^{(1)},\dots,t_j^{(4)} \in \tau_{\frac h 2} \\
+\phi_{\frac h 2}|_{T_j} & = &x_j^{(1)}\chi_{t_j^{(1)}}+\dots+x_j^{(4)}\chi_{t_j^{(4)}}\\
+\Pi_h\phi_{\frac h 2}|_{T_j}&=&\frac 1 {\abs{T_j}}\int_{T_j}\phi_{\frac h 2}d\Gamma\\
+& =&\frac 1 {\abs{T_j}}\sum_{k=1}^4x_j^{(k)}\int_{t^{(k)}}1d\Gamma\\
+& =&\frac 1 {\abs{T_j}}\sum_{k=1}^4x_j^{(k)}\abs{t_j^{(k)}}\\
+& =&\frac 1 {\abs{T_j}}\sum_{k=1}^4x_j^{(k)}\frac {\abs{T_j}} 4 \\
+& =&\frac 1 4 (x_j^{(1)} +\dots+x_j^{(4)} )=: m_j\\
+\norm{\phi_{\frac h 2} - \Pi_h\phi_{\frac h 2}}_{\L^2(T_j)}^2
+&=&\sum_{k=1}^4\norm{\phi_{\frac h 2} - \Pi_h\phi_{\frac h 2}}_{\L^2\left(t_j^{(k)}\right)}^2\\
+&=&\sum_{k=1}^4 \int_{t_j^{(k)}}(\phi_{\frac h 2} - \Pi_h\phi_{\frac h 2})^2d\Gamma\\
+&=&\sum_{k=1}^4 \int_{t_j^{(k)}}(x_j^{(k)}-m_j)^2d\Gamma\\
+&=&\sum_{k=1}^4 \abs{t_j^{(k)}}(x_j^{(k)}-m_j)^2\\
+&=&\frac {\abs{T_j}} 4\sum_{k=1}^4 (x_j^{(k)}-m_j)^2\\
+\mu_e(T_j)^2 & = & \frac{\why{ h_{min}(T_j)} \abs{T_j}}{4} \sum_{k=1}^4{(x_j^{(k)}-m_j)^2}
+\end{eqnarray*}
+
+$mu = computeEstSlpMuTilde(x_{fine}, COO, ELE, F2S);$
+
+\section{Markieren}
+Bestimme $M_e \subseteq T_e$ mit minimaler Kardinalität
+\begin{eqnarray*}
+\theta \sum_{T\in T_e} \mu_e(T)^2 & \leq & \sum_{T\in M_e} \mu_e(T)^2
+\end{eqnarray*}
+Zur anisotropen Verfeinerung wird weiterhin berechnet:
+\begin{eqnarray*}
+\begin{pmatrix}
+ C_j^{(1)}\\ C_j^{(2)}\\C_j^{(3)}\\ C_j^{(4)}
+\end{pmatrix}
+& = & \frac 1 4
+\begin{pmatrix}
+1 & 1 & 1 & 1\\
+1 & -1 & 1 & -1\\
+1 & 1 & -1 & -1\\
+1 & -1 & -1 & 1\\
+\end{pmatrix}
+\begin{pmatrix}
+ x_j^{(1)}\\ x_j^{(2)}\\x_j^{(3)}\\ x_j^{(4)}
+\end{pmatrix}
+\end{eqnarray*}
+
+\begin{eqnarray*}
+\eta \abs{ C_j^{(3)}} & \geq & \sqrt{\abs{ C_j^{(2)}}^2 + \abs{ C_j^{(4)}}^2}\\
+\eta \abs{ C_j^{(4)}} & \geq & \sqrt{\abs{ C_j^{(2)}}^2 + \abs{ C_j^{(3)}}^2}
+\end{eqnarray*}
+
+$xF2S := x_{fine}[F2S]$\\
+$marked = computeEstSlpMuTilde(xF2S, mu, theta, eta);$
+\end{document}
--- /dev/null
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+ 8 & 1.0000 & 0.5000 & 0.0000\\
+ 9 & 1.0000 & 1.0000 & 0.0000
+\end{tabular}
--- /dev/null
+\begin{tabular}{>{\columncolor{gray}}rcccc}
+\rowcolor{gray}
+ Index & c1 & c2 & c3 & c4\\
+ 1 & 2 & 5 & 6 & 3\\
+ 2 & 5 & 8 & 9 & 6\\
+ 3 & 4 & 7 & 8 & 5\\
+ 4 & 1 & 4 & 5 & 2
+\end{tabular}
+\label{exmpl1:ele}
\ No newline at end of file
--- /dev/null
+\begin{tabular}{>{\columncolor{gray}}rcccc}
+\rowcolor{gray}
+ Index & e1 & e2 & e3 & e4\\
+ 1 & 1 & 1 & 1 & 1\\
+ 2 & 2 & 2 & 2 & 2\\
+ 3 & 3 & 3 & 3 & 3\\
+ 4 & 4 & 4 & 4 & 4\\
+ 5 & 12 & 14 & 13 & 5\\
+ 6 & 6 & 6 & 6 & 6\\
+ 7 & 7 & 7 & 7 & 7\\
+ 8 & 8 & 8 & 8 & 8\\
+ 9 & 9 & 9 & 9 & 9\\
+ 10 & 10 & 10 & 10 & 10\\
+ 11 & 11 & 11 & 11 & 11
+\end{tabular}
+\label{exmpl1:f2s}
\ No newline at end of file
--- /dev/null
+\begin{tabular}{>{\columncolor{gray}}rcccccccc}
+\rowcolor{gray}
+ Index & n1 & n2 & n3 & n4 & n5 & n6 & n7 & n8\\
+ 1 & 4 & 2 & 0 & 0 & 0 & 0 & 0 & 0\\
+ 2 & 3 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\
+ 3 & 0 & 0 & 2 & 4 & 0 & 0 & 0 & 0\\
+ 4 & 0 & 3 & 1 & 0 & 0 & 0 & 0 & 0
+\end{tabular}
+\label{exmpl1:nei}
\ No newline at end of file
--- /dev/null
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+ 19 & 1.0000 & 1.0000 & 0.0000
+\end{tabular}
--- /dev/null
+\begin{tabular}{>{\columncolor{gray}}rcccc}
+\rowcolor{gray}
+ Index & c1 & c2 & c3 & c4\\
+ 1 & 4 & 9 & 10 & 5\\
+ 2 & 14 & 18 & 19 & 15\\
+ 3 & 11 & 16 & 17 & 13\\
+ 4 & 2 & 7 & 8 & 3\\
+ 5 & 3 & 8 & 9 & 4\\
+ 6 & 13 & 17 & 18 & 14\\
+ 7 & 7 & 12 & 13 & 8\\
+ 8 & 6 & 11 & 12 & 7\\
+ 9 & 1 & 6 & 7 & 2\\
+ 10 & 9 & 14 & 15 & 10\\
+ 11 & 8 & 13 & 14 & 9
+\end{tabular}
+\label{exmpl2:ele}
\ No newline at end of file
--- /dev/null
+\begin{tabular}{>{\columncolor{gray}}rcccc}
+\rowcolor{gray}
+ Index & e1 & e2 & e3 & e4\\
+ 1 & 5 & 11 & 10 & 1\\
+ 2 & 6 & 6 & 2 & 2\\
+ 3 & 3 & 3 & 3 & 3\\
+ 4 & 9 & 8 & 7 & 4
+\end{tabular}
+\label{exmpl2:f2s}
\ No newline at end of file
--- /dev/null
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+\rowcolor{gray}
+ Index & n1 & n2 & n3 & n4 & n5 & n6 & n7 & n8\\
+ 1 & 5 & 10 & 0 & 0 & 0 & 0 & 0 & 0\\
+ 2 & 6 & 0 & 0 & 10 & 0 & 0 & 0 & 0\\
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+ 9 & 0 & 8 & 4 & 0 & 0 & 0 & 0 & 0\\
+ 10 & 11 & 2 & 0 & 1 & 0 & 0 & 0 & 0\\
+ 11 & 7 & 6 & 10 & 5 & 0 & 0 & 0 & 0
+\end{tabular}
+\label{exmpl2:nei}
\ No newline at end of file
--- /dev/null
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+
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+
+%%Trailer
+%%EOF
--- /dev/null
+\begin{tabular}{>{\columncolor{gray}}rccc}
+\rowcolor{gray}
+ Index & x1 & x2 & x3\\
+ 1 & 0.0000 & 0.0000 & 0.0000\\
+ 2 & 0.0000 & 0.2500 & 0.0000\\
+ 3 & 0.0000 & 0.5000 & 0.0000\\
+ 4 & 0.0000 & 0.6250 & 0.0000\\
+ 5 & 0.0000 & 0.7500 & 0.0000\\
+ 6 & 0.0000 & 1.0000 & 0.0000\\
+ 7 & 0.1250 & 0.5000 & 0.0000\\
+ 8 & 0.1250 & 0.6250 & 0.0000\\
+ 9 & 0.1250 & 0.7500 & 0.0000\\
+ 10 & 0.2500 & 0.0000 & 0.0000\\
+ 11 & 0.2500 & 0.2500 & 0.0000\\
+ 12 & 0.2500 & 0.5000 & 0.0000\\
+ 13 & 0.2500 & 0.6250 & 0.0000\\
+ 14 & 0.2500 & 0.7500 & 0.0000\\
+ 15 & 0.2500 & 1.0000 & 0.0000\\
+ 16 & 0.5000 & 0.0000 & 0.0000\\
+ 17 & 0.5000 & 0.2500 & 0.0000\\
+ 18 & 0.5000 & 0.5000 & 0.0000\\
+ 19 & 0.5000 & 0.7500 & 0.0000\\
+ 20 & 0.5000 & 1.0000 & 0.0000\\
+ 21 & 1.0000 & 0.0000 & 0.0000\\
+ 22 & 1.0000 & 0.5000 & 0.0000\\
+ 23 & 1.0000 & 0.7500 & 0.0000\\
+ 24 & 1.0000 & 1.0000 & 0.0000
+\end{tabular}
--- /dev/null
+\begin{tabular}{>{\columncolor{gray}}rcccc}
+\rowcolor{gray}
+ Index & c1 & c2 & c3 & c4\\
+ 1 & 5 & 14 & 15 & 6\\
+ 2 & 19 & 23 & 24 & 20\\
+ 3 & 16 & 21 & 22 & 18\\
+ 4 & 2 & 11 & 12 & 3\\
+ 5 & 4 & 8 & 9 & 5\\
+ 6 & 18 & 22 & 23 & 19\\
+ 7 & 11 & 17 & 18 & 12\\
+ 8 & 10 & 16 & 17 & 11\\
+ 9 & 1 & 10 & 11 & 2\\
+ 10 & 14 & 19 & 20 & 15\\
+ 11 & 12 & 18 & 19 & 14\\
+ 12 & 3 & 7 & 8 & 4\\
+ 13 & 8 & 13 & 14 & 9\\
+ 14 & 7 & 12 & 13 & 8
+\end{tabular}
+\label{exmpl3:ele}
\ No newline at end of file
--- /dev/null
+\begin{tabular}{>{\columncolor{gray}}rcccc}
+\rowcolor{gray}
+ Index & e1 & e2 & e3 & e4\\
+ 1 & 1 & 1 & 1 & 1\\
+ 2 & 2 & 2 & 2 & 2\\
+ 3 & 3 & 3 & 3 & 3\\
+ 4 & 4 & 4 & 4 & 4\\
+ 5 & 12 & 14 & 13 & 5\\
+ 6 & 6 & 6 & 6 & 6\\
+ 7 & 7 & 7 & 7 & 7\\
+ 8 & 8 & 8 & 8 & 8\\
+ 9 & 9 & 9 & 9 & 9\\
+ 10 & 10 & 10 & 10 & 10\\
+ 11 & 11 & 11 & 11 & 11
+\end{tabular}
+\label{exmpl3:f2s}
\ No newline at end of file
--- /dev/null
+\begin{tabular}{>{\columncolor{gray}}rcccccccc}
+\rowcolor{gray}
+ Index & n1 & n2 & n3 & n4 & n5 & n6 & n7 & n8\\
+ 1 & 13 & 10 & 0 & 0 & 5 & 0 & 0 & 0\\
+ 2 & 6 & 0 & 0 & 10 & 0 & 0 & 0 & 0\\
+ 3 & 0 & 0 & 6 & 8 & 0 & 0 & 0 & 7\\
+ 4 & 9 & 7 & 12 & 0 & 0 & 0 & 14 & 0\\
+ 5 & 12 & 13 & 1 & 0 & 0 & 0 & 0 & 0\\
+ 6 & 3 & 0 & 2 & 11 & 0 & 0 & 0 & 0\\
+ 7 & 8 & 3 & 11 & 4 & 0 & 0 & 0 & 0\\
+ 8 & 0 & 3 & 7 & 9 & 0 & 0 & 0 & 0\\
+ 9 & 0 & 8 & 4 & 0 & 0 & 0 & 0 & 0\\
+ 10 & 11 & 2 & 0 & 1 & 0 & 0 & 0 & 0\\
+ 11 & 7 & 6 & 10 & 14 & 0 & 0 & 0 & 13\\
+ 12 & 4 & 14 & 5 & 0 & 0 & 0 & 0 & 0\\
+ 13 & 14 & 11 & 1 & 5 & 0 & 0 & 0 & 0\\
+ 14 & 4 & 11 & 13 & 12 & 0 & 0 & 0 & 0
+\end{tabular}
+\label{exmpl3:nei}
\ No newline at end of file
--- /dev/null
+\begin{tabular}{>{\columncolor{gray}}rcccccccc}
+\rowcolor{gray}
+ Index & n1 & n2 & n3 & n4 & n5 & n6 & n7 & n8\\
+ 4 & 9 & 7 & 12 & 0 & 0 & 0 & 14 & 0
+\end{tabular}
+\label{exmpl3:nei}
--- /dev/null
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+
+eplot
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+
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+%%Trailer
+%%EOF
projects / bacc.git / commitdiff
