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\def\I{\mathcal{I}} +\def\E{\mathcal{E}} \def\oder{\vee} \def\und{\wedge} @@ -88,6 +89,7 @@ \newtheorem{lem}[defi]{Lemma} \newtheorem{sat}[defi]{Satz} \newtheorem{bew}[defi]{Beweis} +\newtheorem{bem}[defi]{Bemerkung} \newtheorem{alg}[defi]{Algorithmus} \newcommand{\beweis}{{\it Beweis. }} @@ -123,7 +125,7 @@ In dieser Arbeit beschäftigen wir uns mit der Randelementmethode für die homog - \varDelta u &= 0 \quad\text{ in } \Omega \subset \R^3,\\ u &= g \quad \text{ auf }\Gamma, \end{align*} -wobei $\varDelta u := \partial_x^2u+\partial_y^2u$ den Laplace-Operator bezeichnet und $\Omega \subset \R^3$ eine beschränkte Teilmenge von $\R^3$ mit Lipschitz-Rand $\Gamma := \partial \Omega$ ist.\\ +wobei $\varDelta u := \partial_x^2u+\partial_y^2u+\partial_z^2u$ den Laplace-Operator bezeichnet und $\Omega \subset \R^3$ eine beschränkte Teilmenge von $\R^3$ mit Lipschitz-Rand $\Gamma := \partial \Omega$ ist.\\ In Abschnitt 2 stellen wir zunächst die Randelementmethode für die homogene Laplace-Gleichung mit Dirichlet-Randbedingungen vor. Dabei verwenden wir den indirekten Ansatz um anschließend mithilfe des Galerkin-Verfahrens die Gleichung zu lösen. An dieser Stelle werden wir auch kurz die Parametrisierung des Randes vorstellen. Denn wir werden im Folgenden den Rand in affine achsenorientierte Rechtecke $T$ zerlegen, das heißt die Punkte in einem Rechteck liegen in einer zu den Achsen des Koordinatensystem parallelen Ebene.\\ In Abschnitt 3 werden wir uns mit der approximativen Berechnung des Doppelintegrals \begin{align}\label{math:intro:int} @@ -260,16 +262,22 @@ Für die Diskretisierung des Problems wollen wir nun einige Begriffe definieren. \begin{align*} T := \{\bs v + \lambda_1 a {\bs a} + \lambda_2 b {\bs b} ~|~ \lambda_1,\lambda_2 \in[0,1]\} \end{align*} -achsenorientiertes Rechteck. -\end{defi} - -\begin{defi} -Sei $T$ ein Rechteck, Dann heißt +achsenorientiertes Rechteck. Ferner heißt \begin{align*} - \gamma := [0,1]^2 \to T: \lambda_1,\lambda_2 \mapsto \bs v + \lambda_1 a {\bs a} + \lambda_2 b {\bs b} + \gamma_T := [0,1]^2 \to T: \lambda_1,\lambda_2 \mapsto \bs v + \lambda_1 a {\bs a} + \lambda_2 b {\bs b} \end{align*} die zu $T$ zugehörige Parametrisierung. \end{defi} +\begin{bem} +Weiterhin werden wir die vier Ecken des Rechtecks +\begin{align*} + \K_T&:=\{\gamma_T(x,y) ~|~ x,y \in \{0,1\}\} +\end{align*} +als Menge der Knoten des Rechtecks $T$ bezeichnen und als Kanten von $T$ nennen wir die Menge +\begin{align*} + \E_T&:= \{[k_T, \tilde k_T] ~|~ k_T,\tilde k_T\in \K_T, k_T\neq\tilde k_T\}. +\end{align*} +\end{bem} \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} @@ -290,19 +298,19 @@ die Seitenlänge des Rechtecks $T$ in Richtung $\bs a$. Die Seitenlänge in Rich Ist $c \in T_j \cap T_k$ ein Eckpunkt von $T_j$ aber nicht von $T_k$, so nennen wir $c$ einen hängenden Knoten. \end{defi} \begin{defi} -Wir nennen $\T_{\ell} = \{T_1,T_2,\dots,T_N\}$ eine Partition von $\Gamma$ falls: +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} \item $\overline{\Gamma} = \bigcup_{j=1}^NT_j$ -\item alle Elemente aus $T_{\ell}$ sind abgeschlossene achsenorientierte Rechtecke +% \item alle Elemente aus $T_{\ell}$ sind abgeschlossene achsenorientierte Rechtecke \item $\abs{T_j \cap T_k}=0$ für den Schnitt zweier Elemente $T_j,T_k\in\T_{\ell}$ mit $T_j\neq T_k$ \end{itemize} -Hier bezeichne $\abs{\cdot}$ das 2-Dimensionale Oberflächenmaß für Rechtecke. Weiterhin sei der Schnitt $T_j \cap T_k$ zweier Elemente $T_j,T_k\in\T_{\ell}$ mit $T_j\neq T_k$ +Hier bezeichne $\abs{\cdot}$ das 2-dimensionale Oberflächenmaß. Weiterhin sei der Schnitt $T_j \cap T_k$ zweier Elemente $T_j,T_k\in\T_{\ell}$ mit $T_j\neq T_k$ \begin{itemize} \item leer, \item Knoten von $T_j$ und $T_k$, - \item Kante von $T_j$ und $T_k$, - \item Knoten von $T_j$ und $T_k$ und o.B.d.A. Kante von $T_j$, also nur einen hängenden Knoten pro Kante. + \item Kante von $T_j$ oder $T_k$. \end{itemize} +Ferner liegen auf jeder Kante von $\T_{\ell}$ maximal 3 Knoten. \end{defi} \begin{figure}[ht] \centering @@ -311,9 +319,6 @@ Hier bezeichne $\abs{\cdot}$ das 2-Dimensionale Oberflächenmaß für Rechtecke. \caption{Beispiel Partitionen} \label{fig:net} \end{figure} -\begin{defi} -Außerdem wollen wir zu jedem Netz $\T$ die Menge der Seiten $\S$ definieren, wobei zu jeder Seite $e_T \in \S$ eine bezüglich des Elements $T$ gegenüberliegende Seite $\tilde e_T$ existiert. -\end{defi} \subsection{Verfeinern} \begin{defi}[Lokale Verfeinerung] @@ -332,19 +337,22 @@ 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} -\begin{alg}[Verfeinern] \label{alg:refine} Sei $\T_{\ell}$ eine Partition und $\S_{\ell}^{(0)}$ eine Menge markierter Kanten, wobei zu jedem $e_T \in \S_{\ell}^{(0)}$ auch $\tilde e_T \in \S_{\ell}^{(0)}$ ist. Nun sei $i=0$ und gehe so vor: + +\begin{alg}[Verfeinern] \label{alg:refine} Sei $\T_{\ell}$ eine Partition und $\sqcap_{\ell} \subseteq \{e_T ~|~ T \in \T_{\ell},e\in\E_T \}:=\S_{\ell}$ eine Menge markierter Kanten. Nun sei $\sqcap_{\ell}^{(0)}:=\sqcap_{\ell}$ und $i=0$, dann gehe so vor: \begin{enumerate} \renewcommand{\theenumi}{(\roman{enumi})} \item \label{alg:refine:first} - Definiere $\S_{\ell}^{(i+1)} := \{ e_T, \tilde e_T ~|~ T \in \T_{\ell}$ mit $\exists$ hängender-Knoten auf $T$ und $\exists T' \in\T_{\ell}$ sodass $T \cap T' = e_{T'},\in \S_{\ell}^{(i)} \}$ + $\sqcap_{\ell}^{(i+1)} := \sqcap_{\ell}^{(i)} \cup \{ e_{T'}' \in \S_{\ell}\backslash\sqcap_{\ell}^{(i)} ~|~ \exists e_T \in \sqcap_{\ell}^{(i)} e' \supseteq e \} \cup \{\tilde e_T \in \S_{\ell} ~|~ \exists e\in\E_T : e\cap\tilde e=\emptyset\}$ \item - Falls $\S_{\ell}^{(i)} \subsetneq \S_{\ell}^{(i+1)}$, $i = i+1$ und gehe zu Schritt \ref{alg:refine:first} - \item Teile alle Elemente aus $\T_{\ell}$ bezüglich der markierten Kanten $\S_{\ell}^{(0,1,\ldots,i)}$, wie in der lokalen Verfeinerung vorgegeben + Falls $\sqcap_{\ell}^{(i)} \subsetneq \sqcap_{\ell}^{(i+1)}$, $i = i+1$ und gehe zu Schritt \ref{alg:refine:first} + \item Teile alle Elemente aus $\T_{\ell}$ bezüglich der markierten Kanten $\sqcap_{\ell}^{(i)}$, wie in der lokalen Verfeinerung vorgegeben \end{enumerate} + +$\{\tilde e_T ~|~ \exists e \in \E_T : e \cap \tilde e =\emptyset \}$ \end{alg} \clearpage - + \section{Analytische und Semi-analytische Berechnung} \todo{ \scriptsize \begin{itemize} @@ -732,10 +740,10 @@ für orthogonal liegende Elemente schreiben: \end{align*} -% \subsection{Bestimmtes Integral} -% \begin{eqnarray*} -% &&\frac{1}{4\pi} \int_T \int_{\tilde T} \frac{1}{|x-y|} ds_y ds_x\\ -% &\approx& \frac{1}{4\pi} \int_0^{k_1}\int_0^{k_2}\int_0^{\tilde k_1}\int_0^{\tilde k_2} +\subsection{Bestimmtes Integral} +% \begin{align*} +% &\frac{1}{4\pi} \int_T \int_{\tilde T} \frac{1}{|x-y|} ds_y ds_x\\ +% &\approx \frac{1}{4\pi} \int_0^{k_1}\int_0^{k_2}\int_0^{\tilde k_1}\int_0^{\tilde k_2} % \dif{}{y_2} \dif{}{y_1} \dif{}{x_2} \dif{}{x_1} % F_{par/ort}(x_1,x_2,y_1,y_2) % dy_2 dy_1 dx_2 dx_1\\ @@ -754,7 +762,7 @@ für orthogonal liegende Elemente schreiben: % - % \dif{}{x_2} \dif{}{x_1} % F_{par/ort}(x_1,x_2,\tilde k_1,0) \\ -% &&- +% &- % \dif{}{x_2} \dif{}{x_1} % F_{par/ort}(x_1,x_2,0,\tilde k_2) % + @@ -768,19 +776,19 @@ für orthogonal liegende Elemente schreiben: % - % \dif{}{x_1} % F_{par/ort}(x_1,k_2,\tilde k_1,0) \\ -% &&- +% &- % \dif{}{x_1} % F_{par/ort}(x_1,k_2,0,\tilde k_2) % + % \dif{}{x_1} % F_{par/ort}(x_1,k_2,0,0)\\ -% &&- %% +% &- %% % \dif{}{x_1} % F_{par/ort}(x_1,0,\tilde k_1,\tilde k_2) % + % \dif{}{x_1} % F_{par/ort}(x_1,0,\tilde k_1,0) \\ -% &&+ +% &+ % \dif{}{x_1} % F_{par/ort}(x_1,0,0,\tilde k_2) % - @@ -796,7 +804,7 @@ für orthogonal liegende Elemente schreiben: % F_{par/ort}(k_1,k_2,0,\tilde k_2) % + % F_{par/ort}(k_1,k_2,0,0)\\ -% &&- +% &- % F_{par/ort}(k_1,0,\tilde k_1,\tilde k_2) % + % F_{par/ort}(k_1,0,\tilde k_1,0) @@ -804,7 +812,7 @@ für orthogonal liegende Elemente schreiben: % F_{par/ort}(k_1,0,0,\tilde k_2) % - % F_{par/ort}(k_1,0,0,0)\\ -% &&- %% +% &- %% % F_{par/ort}(0,k_2,\tilde k_1,\tilde k_2) % - % F_{par/ort}(0,k_2,\tilde k_1,0) @@ -812,7 +820,7 @@ für orthogonal liegende Elemente schreiben: % F_{par/ort}(0,k_2,0,\tilde k_2) % + % F_{par/ort}(0,k_2,0,0)\\ -% &&- +% &- % F_{par/ort}(0,0,\tilde k_1,\tilde k_2) % + % F_{par/ort}(0,0,\tilde k_1,0) @@ -820,7 +828,7 @@ für orthogonal liegende Elemente schreiben: % F_{par/ort}(0,0,0,\tilde k_2) % - % F_{par/ort}(0,0,0,0)\big) -% \end{eqnarray*} +% \end{align*} \clearpage -- 2.47.3