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z@&?Cl(z)QURwa@N!%yGHdSC>J@Mpe5ISVT_#F^KyW6mV*yrz#?Xv?*q!sK%8Ag@fCZJN{%A&(i$=iQz^5PA)3iCo(!d(my~NHfX?xr40=1rrAsXA1$VJ AH2?qr diff --git a/doc/doc.tex b/doc/doc.tex index 36a6ad9..dc53b64 100644 --- a/doc/doc.tex +++ b/doc/doc.tex @@ -257,8 +257,7 @@ Abschließend werden wir kurz die numerische Umsetzung der Techniken vorstellen \subsection{Netze} -Für die Diskretisierung des Problems wollen wir nun einige Begriffe definieren. - +Für die Diskretisierung des Problems wollen wir nun einige Begriffe definieren. Zunächst wollen wir ein achsenorientiertes Rechteck beschreiben, dessen Seiten parallel zu den Achsen des kartesischen Koordinatensystems liegen. \begin{defi}\label{math:def:T} Sei $\bs v \in \R^3$, $\bs a,\bs b \in \{(1,0,0)^T,(0,1,0)^T,(0,0,1)^T\}$ mit $\bs a\neq \bs b$ und $a,b \in \R$ mit $a,b > 0$. Dann heißt \begin{align*} T := \{\bs v + \lambda_1 a {\bs a} + \lambda_2 b {\bs b} ~|~ \lambda_1,\lambda_2 \in[0,1]\} @@ -269,17 +268,24 @@ achsenorientiertes Rechteck. Ferner heißt \end{align*} die zu $T$ zugehörige Parametrisierung. \end{defi} -\todo{ -\begin{bem} -Hier Knoten und Seiten definieren mit Text und Skizze +\begin{bem}Weiterhin werden wir die vier Ecken des achsenorientierten Rechtecks, beginnend in $\bs v$, mit $k_1,\ldots,k_4$ bezeichnen, wobei die Menge aller Knoten des Rechtecks $\K_T$ sei. Die Reihenfolge der Knoten sei dabei so gewählt, dass der Normalenvektor $\bs n$ auf das Element $T$ nach außen zeigt. Außerdem benennen wir die Menge der Kanten mit $\E_T$, bestehend aus den vier Kanten $e_1,\ldots,e_4$. In Abb. \ref{fig:net:single} wurde ein Rechteck mit den Bezeichnungen kurz skizziert. \end{bem} \begin{figure}[ht] \centering -\subfloat[gültige Partition]{\includegraphics[width=0.5\textwidth]{fig/net_single}} -\caption{Rechteck} +\psfrag{e1}{\scriptsize $e_1$} +\psfrag{e2}{\scriptsize $e_2$} +\psfrag{e3}{\scriptsize $e_3$} +\psfrag{e4}{\scriptsize $e_4$} +\psfrag{k1}{\scriptsize $k_1$} +\psfrag{k2}{\scriptsize $k_2$} +\psfrag{k3}{\scriptsize $k_3$} +\psfrag{k4}{\scriptsize $k_4$} +\psfrag{n}{\scriptsize $\bs n$} +\includegraphics[width=0.5\textwidth]{fig/net_single} +\caption{achsenorientiertes Rechteck \todo{(Skizze noch doof)}} \label{fig:net:single} -\end{figure}} - +\end{figure} +Des Weiteren werden wir für die Berechnungen noch Aussagen über die Größe eines Elements, sowie über den Abstand zweier Elemente festhalten. \begin{defi}Sei $a,b\in\R$ für $T$ definiert wie in Def. \ref{math:def:T}, dann heißt \begin{align*} \diam (T) &= (a^2+b^2)^{1/2} @@ -296,6 +302,7 @@ die Seitenlänge des Rechtecks $T$ in Richtung $\bs a$. Die Seitenlänge in Rich \end{align*} wobei das Minimum aufgrund der Kompaktheit von $T_j, T_k$ angenommen wird. \end{defi} +Mit diesen Vorüberlegungen definieren wir uns die Diskretisierung des Randes $\Gamma$. \begin{defi} Sei $\T_{\ell} = \{T_1,T_2,\dots,T_N\}$ eine endliche Menge von achsenorientierten Rechtecken. Es bezeichne $\K_{\ell}:=\bigcup_{T\in\T} k_T$ die Menge der Knoten von $\T_{\ell}$ und $\E_{\ell}:=\bigcup_{T\in\T} e_T$ die Menge der Kanten. Wir nennen $\T_{\ell}$ eine Partition von $\Gamma$, falls \begin{itemize} @@ -321,7 +328,7 @@ Ferner liegen auf jeder Kante von $\T_{\ell}$ maximal 3 Knoten. \subsection{Verfeinern} \begin{defi}[Lokale Verfeinerung] -Ein Element $T \in \T$ wird isotrop in vier Elemente $T_1,\ldots,T_4$ geteilt, wenn $T = \bigcup_{n=1}^4 T_n$ gilt und alle Söhne $T_1,\ldots,T_4$ ähnlich sind zum Vaterelement $T$. Weiterhin wird ein Element $T \in \T$ anisotrop in zwei Elemente $T_1,T_2$ geteilt, wenn ebenfalls $T = T_1 \cup T_2$ gilt und $T_1,T_2$ gleich große Seitenlängen haben. Hierbei kann $T$ entweder horizontal oder vertikal geteilt werden, wie in Abb.~\ref{fig:refType} gezeigt ist. +Ein Element $T \in \T$ wird isotrop in vier Elemente $T_1,\ldots,T_4$ geteilt, wenn $T = \bigcup_{n=1}^4 T_n$ gilt und alle Söhne $T_1,\ldots,T_4$ ähnlich sind zum Vaterelement $T$. Weiterhin wird ein Element $T \in \T$ anisotrop in zwei Elemente $T_1,T_2$ geteilt, wenn ebenfalls $T = T_1 \cup T_2$ gilt und $T_1,T_2$ untereinander gleich sind. Hierbei kann $T$ entweder horizontal oder vertikal geteilt werden, wie in Abb.~\ref{fig:refType} gezeigt ist. \end{defi} \begin{figure}[ht] \centering @@ -336,8 +343,12 @@ Ein Element $T \in \T$ wird isotrop in vier Elemente $T_1,\ldots,T_4$ geteilt, w \begin{defi} Für eine Partition $\T_{\ell}$ bezeichnen wir mit $\widehat \T_{\ell}$ jene Partition die entsteht, wenn alle Elemente isotrop verfeinert werden. \end{defi} -\todo{ -Text der die Idee des Alg beschreibt + Vorüberlegungen} + +Für das Verständnis des folgenden Algorithmus benötigen wir noch einige Beobachtungen: +\begin{itemize} + \item Sind $e,\tilde e \in \E_T$ Kanten von $T \in \T_{\ell}$ mit $e \cap \tilde e = \emptyset$, so liegen diese gegenüber und haben insbesondere dieselbe Länge. Falls $e$ verfeinert werden soll, muss zwingend auch $\tilde e$ verfeinert werden. + \item Es bezeichnet $\S_{\ell} := \{ e_T ~|~ T \in \T_{\ell}, e_T \in \E_T \}$ die Menge aller Kanten von $\T_{\ell}$, wobei es aufgrund der maximal drei Knoten auf einer Kante vorkommen kann, dass eine Kante $e \in \S_{\ell}$, die im inneren von $\Gamma$ liegt, nur zu einem Element $T\in\T_{\ell}$ gehört und nicht zu zwei. In diesem Fall gibt es eine Kante $\tilde e \in \S_{\ell}$ in der $e$ ganz enthalten ist. Sollte $e$ verfeinert werden, so muss zwingend auch $\tilde e$ verfeinert werden, damit nicht mehr als drei Knoten auf einer Kante entstehen. +\end{itemize} \begin{alg}[Verfeinern] \label{alg:refine} Sei $\T_{\ell}$ eine Partition und $\sqcap_{\ell} \subseteq \S_{\ell}$ eine Menge markierter Kanten. Nun sei $\sqcap_{\ell}^{(0)}:=\sqcap_{\ell}$ und $i=0$. Dann gehe so vor: \begin{enumerate} \renewcommand{\theenumi}{(\roman{enumi})} -- 2.47.3