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Anmerken, dass ein Kante als Vektor nur eine dim $\neq 0$?} +\begin{defi}[Rechteck] Sei $\bs x \in \R^3$, $\bs a,\bs b \in \{(1,0,0)^T,(0,1,0)^T,(0,0,1)^T\}$ mit $\bs a\neq \bs b$ und $a,b \in \R$ mit $a,b > 0$. Dann heißt +\begin{align} + T := \{\bs x + \lambda a {\bs a} + \gamma b {\bs b} ~|~ \lambda,\gamma \in[0,1]\} +\end{align} +achsenorientiertes Rechteck. +\end{defi} + +\begin{defi}[hängender Knoten] + 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}[Partition] Wir nennen $\T_{\ell} = \{T_1,T_2,\dots,T_N\}$ eine Partition von $\Gamma$ falls: \begin{itemize} \item $\overline{\Gamma} = \bigcup_{j=1}^NT_j$ - \item alle Elemente aus $T_{\ell}$ 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. -\end{defi} -Weiterhin wollen wir in dieser Arbeit Aufgrund der Netzstabilität noch fordern, dass 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 für Rechtecke. Weiterhin sei der Schnitt $T_j \cap T_k$ zweier Elemente $T_j,T_k\in\T_{\ell}$ mit $T_j\neq T_k$ \begin{itemize} - \item leer ist, - \item oder Knoten von $T_j$ und $T_k$, - \item oder Kante von $T_j$ und $T_k$, - \item oder o.B.d.A. Kante von $T_j$ und Kantenhälfte von $T_k$. + \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. \end{itemize} - - +\end{defi} \begin{figure}[ht] \centering \subfloat[gültige Partition]{\includegraphics[width=0.3\textwidth]{fig/net1}} @@ -258,14 +268,29 @@ Weiterhin wollen wir in dieser Arbeit Aufgrund der Netzstabilität noch fordern, \caption{Beispiel Partitionen} \label{fig:net} \end{figure} -\subsection{Verfeinern}\todo{ +\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} + + +\todo{ +\subsection{Abstand und Größe der Elemente} +\begin{defi}[dist] + \end{defi} +\begin{defi}[diam] + \end{defi} +\begin{defi}[hmin] + \end{defi} +} + +\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 die Seitenlängen der Elemente $T_1,\ldots,T_4$ gleich große sind. 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. -\end{defi}} +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 die Seitenlängen der Elemente $T_1,\ldots,T_4$ gleich groß sind. 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. +\end{defi} \begin{figure}[ht] \centering \subfloat[Element]{\includegraphics[width=0.25\textwidth]{fig/refType_full}} -\subfloat[Isotrop]{ \includegraphics[width=0.25\textwidth]{fig/refType_2}} +\subfloat[Isotrop]{\includegraphics[width=0.25\textwidth]{fig/refType_2}} \subfloat[Vertikal]{\includegraphics[width=0.25\textwidth]{fig/refType_3}} \subfloat[Horizontal]{\includegraphics[width=0.25\textwidth]{fig/refType_4}} \caption{Teilungsarten} @@ -275,17 +300,14 @@ 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 $\T_{\ell} = \{T_1,T_2,\dots,T_N\}$ eine Partition und $marked$ eine gegebene Markierung wobei $marked_j \hat =$ Markierung von $T_j$. Gehe so vor: +\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{enumerate} \renewcommand{\theenumi}{(\roman{enumi})} - \item ermittle ein markiertes Element $T_j$, welches noch nicht geteilt wurde \label{alg:refine:first} - \item - falls Eckpunkt von $T_j$ Hängender-Knoten ist, - markiere zusätzlich den Nachbarn, gehe zu Schritt \ref{alg:refine:first} \label{alg:refine:checkHN} - \item teile $T_j$ wie in $marked_j$ vorgegeben, -% \item aktualisiere Nachbarinformationen -% \item halte fest in welche Elemente $T_j$ zerlegt wurde - gehe zu Schritt \ref{alg:refine:first} \label{alg:refine:last} + \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 $T \cap T' = e_{T'},\in \S_{\ell}^{(i)} \}$ + \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 \end{enumerate} \end{alg} @@ -308,48 +330,38 @@ analytisch/semidiskret/volldiskret \noindent Ziel dieses Abschnitts ist die approximative Berechnung des Integrals -\begin{eqnarray}\label{math:gal:kap} -A_{jk} &=& \int_{T_j} \int_{T_k} \kappa(\bs x,\bs y) ds_{\bs y} ds_{\bs x}. -\end{eqnarray} +\begin{align}\label{math:gal:kap} +A_{jk} &= \int_{T_j} \int_{T_k} \kappa(\bs x,\bs y) ds_{\bs y} ds_{\bs x}. +\end{align} beziehungsweise als Spezialfall davon die Berechnung des Integrals -\begin{eqnarray}\label{math:gal:frac} -A_{jk} &=&\frac{1}{4\pi} \int_{T_j} \int_{T_k} \frac{1}{|\bs x- \bs y|} ds_{\bs y} ds_{\bs x}. -\end{eqnarray} +\begin{align}\label{math:gal:frac} +A_{jk} &=\frac{1}{4\pi} \int_{T_j} \int_{T_k} \frac{1}{|\bs x- \bs y|} ds_{\bs y} ds_{\bs x}. +\end{align} unter bestimmten Vorraussetzungen an die affinen Randstücke $T_j,T_k$ und den asymptotisch glatten Integranden $\kappa : \R^3 \times \R^3 \to \R$. + \subsection{Gauss-Quadratur} -\begin{displaymath} - \int_0^s f(x) dx \approx \sum_{k=0}^n w_kf(x_k) =: \Q(f) -\end{displaymath} +\begin{align} + \Q(f) := \sum_{k=0}^n w_kf(x_k) \approx \int_0^s f(x) dx +\end{align} + +\subsection{Bedingung} +\begin{align} + dist (T, \tilde T)&\geq \mu \min\{ diam(T) , diam(\tilde T)\}\\ + dist (I_1, J_1) &\geq \mu \max\{ len(I_2) , len(J_2)\} +\end{align} -\subsection{Quadratur über ein Element} -\begin{displaymath} +\subsection{Semianalytisch} +\begin{align} \int_{T} \int_{\tilde T} \approx \q_{T} \int_{\tilde T} -\end{displaymath} +\end{align} Das Integral über \Tb~ kann durch die analytische Doppelintegral Funktion ersetzt werden. Die Quadratur über \Ta~ wird nun mittels einer doppelten Gauss-Quadratur berechnet. -\subsubsection{Zulässigkeitsbedingung} -\begin{displaymath} - \D (T, \tilde T) \geq \mu \min\{ \G(T) , \G(\tilde T)\} -\end{displaymath} -Nur wenn die Zulässigkeitsbedingung gültig ist wird die Quadratur verwendet. Sonst wird Analytisch gerechnet. - - -\subsection{Quadratur über eine Achse} -\begin{displaymath} +\begin{align} \int_{T} \int_{\tilde T} \approx \q_{I_1} \q_{J_1} \int_{I_2}\int_{J_2} -\end{displaymath} -\subsubsection{Zulässigkeitsbedingung} -\begin{displaymath} - \D (I_1, J_1) \geq \mu \max\{ \L(I_2) , \L(J_2)\} -\end{displaymath} - - -\subsection{Quadratur über eine Seite} -\begin{displaymath} +\end{align} +\begin{align} \int_{T} \int_{\tilde T} \approx \q_{I_1} \int_{I_2} \int_{J_1} \int_{J_2} -\end{displaymath} - -$\enorm{a}-\norm{b}$ +\end{align} \section{Analytische Berechnung vom Integral im Fall des @@ -363,52 +375,64 @@ Einfachschichtpotentials} } \noindent In diesem Abschnitt wollen wir uns mit der analytischen Berechnung der Galerkin-Matrix -\begin{eqnarray*}\label{math:V} -A_{jk} &=& \frac{1}{4\pi} \int_{T_j} \int_{T_k} \frac{1}{|\bs x- \bs y|} ds_{\bs y} ds_{\bs x} \in \R^3. -\end{eqnarray*} +\begin{align*}\label{math:V} +A_{jk} &= \frac{1}{4\pi} \int_{T_j} \int_{T_k} \frac{1}{|\bs x- \bs y|} ds_{\bs y} ds_{\bs x} \in \R^3. +\end{align*} mit zwei beschränkte, achsenorientierte Rechtecke $T_j,T_k \subseteq\R^3$ beschäftigen. Dazu wollen wir angelehnt an \cite{mai:3dbem} zwei Integrale anschauen, welche durch das Aufspalten des Integrals $A_{jk}$ auftreten. \\\noindent \begin{lem} Für das Integral -\begin{eqnarray*} - g(p;y;x;\lambda) &:=& \int \{(x-y)^2 + \lambda^2 \}^p dy -\end{eqnarray*} +\begin{align*} + g(p;y;x;\lambda) &:= \int \{(x-y)^2 + \lambda^2 \}^p dy +\end{align*} mit $\lambda = 0$ gilt die Formel mit beliebigen Skalaren $x,y,p \in \R$ -\begin{eqnarray*} +\begin{align*} (2p+1)g(p;y;x;0) = \begin{cases} \sgn(y-x) \log \abs{y-x}^{2p} & p=-1/2 \\(y-x) \abs{y-x}^{2p} & sonst \end{cases} -\end{eqnarray*} +\end{align*} falls für $p\leq-/2$ $x$ außerhalb des betrachteten Integrationsbereichs liegt. Für beliebige $x,p,y, \lambda \in \R$ mit $\lambda \neq 0$ gilt die Rekursionsformel: -\begin{eqnarray*} - (2p+1)g(p;y;x;\lambda) = (y-x) \{(y-x)^2+\lambda^2\}^p + 2p\lambda g(p-1;y;x;\lambda) -\end{eqnarray*} +\begin{align*} + (2p+1)g(p;y;x;\lambda) &= (y-x) \{(y-x)^2+\lambda^2\}^p + 2p\lambda g(p-1;y;x;\lambda) +\end{align*} Im weiteren werden wir noch die Formeln für $p \in \{1/2 , 0 , -1/2 , -1, -3/2\}$ -\begin{eqnarray*} - g(1/2;y;x;\lambda) &=& \frac{y-x}{2}\{(y-x)^2+\lambda^2\}^{1/2} + \frac{\lambda^2}{2} \text{arsinh} \frac{y-x}{|\lambda|}\\ - g(0;y;x;\lambda) &=& y-x\\ - g(-1/2;y;x;\lambda) &=& \text{arsinh} \frac{y-x}{|\lambda|}\\ - g(-1;y;x;\lambda) &=& \frac{\text{arctan} \frac{y-x}{|\lambda|}}{ \todo{\lambda}}\\ - g(-3/2;y;x;\lambda) &=& \frac{y-x}{\lambda^2}\{(y-x)^2+\lambda^2\}^{-1/2} -\end{eqnarray*} +\begin{align*} + g(1/2;y;x;\lambda) &= \frac{y-x}{2}\{(y-x)^2+\lambda^2\}^{1/2} + \frac{\lambda^2}{2} \arsinh \frac{y-x}{|\lambda|}\\ + g(0;y;x;\lambda) &= y-x\\ + g(-1/2;y;x;\lambda) &= \arsinh \frac{y-x}{|\lambda|}\\ + g(-1;y;x;\lambda) &= \frac 1 {\abs{\lambda}} \arctan \frac{y-x}{|\lambda|}\\ + g(-3/2;y;x;\lambda) &= \frac{y-x}{\lambda^2}\{(y-x)^2+\lambda^2\}^{-1/2} +\end{align*} benötigen, welche alle in \cite[Seite 2-3]{mai:3dbem} bewiesen wurden. \end{lem} -\todo{Maischak Fehler in $g(-1;y;x;\lambda)$} +Hierbei wollen wir darauf hinweisen, dass wir für $p=-1$ nicht die in \cite[Seite 3]{mai:3dbem} vorgeschlagene Formel verwenden, sondern die selbst hergeleitete: +\begin{align*} + \int \frac 1 {(x-y)^2 +\lambda^2} dy &= \int \frac 1 {\lambda^2((\frac{y-x}{\abs{\lambda}})^2 +1)} dy + = \left| + \begin{array}{c} + z = \frac{y-x}{\abs{\lambda}}\\ + dz = \frac 1 {\abs{\lambda}} dy + \end{array} + \right| + = \frac 1 {\abs{\lambda}} \int \frac 1 {z^2+1} dz\\ + &= \frac 1 {\abs{\lambda}} \arctan (z) +c = \frac 1 {\abs{\lambda}} \arctan (\frac{y-x}{\abs{\lambda}}) +c +\end{align*} + \begin{lem} Des Weiteren werden wir das Integral \begin{align*} G(p;y_1;y_2;x_1;x_2;\lambda) &:= \int \int \{(x_1-y_1)^2 + (x_2-y_2)^2 + \lambda^2\}^p dy_2 dy_1 \end{align*} -betrachten, welches sich für den den Fall $p = -3/2$ und $\lambda = 0 \wedge \lambda \neq 0$ schreiben lässt als: +benötigen, welches sich für den den Fall $p = -3/2$ und $\lambda = 0 \wedge \lambda \neq 0$ schreiben lässt als: \begin{align*} - G(-3/2;y_1,y_2;x_1,x_2,0) =& - \frac{\sqrt{(y_1-x_1)^2+(y_2-x_2)^2}}{(y_1-x_1)(y_2-x_2)}\\ - G(-3/2;y_1,y_2;x_1,x_2,\lambda) = &- \frac{\sgn\{(y_1-x_1)(y_2-x_2)\}}{2|\lambda|}\\ + G(-3/2;y_1,y_2;x_1,x_2,0) &= - \frac{\sqrt{(y_1-x_1)^2+(y_2-x_2)^2}}{(y_1-x_1)(y_2-x_2)}\\ + G(-3/2;y_1,y_2;x_1,x_2,\lambda) &=- \frac{\sgn\{(y_1-x_1)(y_2-x_2)\}}{2|\lambda|}\\ &\cdot \arccos\left( \frac{-2(y_1-x_1)^2(y_2-x_2)^2}{\{(y_1-x_1)^2+\lambda^2\}\{(y_2-x_2)^2+\lambda^2\}} +1 \right). \end{align*} Für alle weiteren relevante Fälle $p = -3/2 +k : k \in \N\backslash0$ können wir folgende Rekursionsformel aufstellen: \begin{align*} - (2p+2)G(p;y_1,y_2;x_1,x_2,\lambda) =& 2p\lambda^2 G(p-1;y_1,y_2;x_1,x_2,\lambda)\\ + (2p+2)G(p;y_1,y_2;x_1,x_2,\lambda) &= 2p\lambda^2 G(p-1;y_1,y_2;x_1,x_2,\lambda)\\ &+ (y_1-x_1)g(p;y_2;x_2;\sqrt{(y_1-x_1)^2+\lambda^2})\\ &+ (y_2-x_2)g(p;y_1;x_1;\sqrt{(y_2-x_2)^2+\lambda^2}), \end{align*} welche ebenfalls in \cite[Seite 5-7]{mai:3dbem} bewiesen wurden. @@ -417,60 +441,79 @@ Für alle weiteren relevante Fälle $p = -3/2 +k : k \in \N\backslash0$ können \subsection{Integral über zwei Elemente} Bei der Berechnung von \eqref{math:V} werden wir geometrisch zwischen zwei Fällen zu unterscheiden. Entweder die beiden Elemente liegen in parallelen Ebenen oder in orthogonalen Ebenen. + \subsubsection{Parallele Elemente} Liegen die beiden Elemente parallel zueinander, können wir sie wie in \cite[Seite 13]{mai:3dbem} gezeigt, darstellen als: -\begin{eqnarray*} - T_j &=& \v + [(0,s_1) \times (0,s_2) \times \{0\}]\\ - T_k &=& \tilde \v + [(0,\tilde s_1) \times (0,\tilde s_2) \times \{0\}], \text{ mit } \v,\tilde \v \in \R^3. -\end{eqnarray*} -Wir setzen $\bs{\delta} = \tilde \v - \v \in \R^3$, dann können wir -\begin{eqnarray*} +\begin{align*} + T_j &= \v + [(0,s_1) \times (0,s_2) \times \{0\}]\\ + T_k &= \tilde \v + [(0,\tilde s_1) \times (0,\tilde s_2) \times \{0\}], \text{ mit } \v,\tilde \v \in \R^3. +\end{align*} +Wir setzen $\bs{\delta} = \tilde \v - \v \in \R^3$, dann können wir das gesuchte Integral +\begin{align*} \int_{T_j} \int_{T_k} \frac{1}{|\bs x- \bs y|} ds_{\bs y} ds_{\bs x}\\ -\end{eqnarray*} umformen zu: -\begin{eqnarray*} -&=& \int_{T_j} \int_{T_k} +\end{align*} umformen zu: +\begin{align*} +&= \int_{T_j} \int_{T_k} \left ( (x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2 \right)^{-1/2} ds_{\bs y} ds_{\bs x}\\ -&=& \int_{T_j} \int_{T_k} +&= \int_{T_j} \int_{T_k} \left ( ((x_1 + v_1)-(y_1 +\tilde v_1))^2+((x_2 + v_2)-(y_2 +\tilde v_2))^2+( v_3- \tilde v_3)^2 \right)^{-1/2} ds_{\bs y} ds_{\bs x}\\ -&=& \int_0^{s_1}\int_0^{s_2}\int_0^{\tilde s_1}\int_0^{\tilde s_2} +&= \int_0^{s_1}\int_0^{s_2}\int_0^{\tilde s_1}\int_0^{\tilde s_2} \left ( (x_1-y_1-\delta_1)^2+(x_2-y_2-\delta_2)^2+\delta_3^2 \right)^{-1/2} dy_2 dy_1 dx_2 dx_1. -\end{eqnarray*} +\end{align*} Dies können wir als Stammfunktion -\begin{eqnarray*} -F_{par}(x_1,x_2,y_1,y_2,\bs \delta) &:=& \int \int \int \int \left\{(x_1-y_1-\delta_1)^2+(x_2-y_2-\delta_2)^2+\delta_3^2\right\}^{-1/2}dy_2 dy_1 dx_2 dx_1 -\end{eqnarray*} +\begin{align*} +F_{par}(x_1,x_2,y_1,y_2,\bs \delta) &:= \int \int \int \int \left\{(x_1-y_1-\delta_1)^2+(x_2-y_2-\delta_2)^2+\delta_3^2\right\}^{-1/2}dy_2 dy_1 dx_2 dx_1 +\end{align*} schreiben, welche das Integral über zwei parallele Elemente löst. +\begin{align*} +F_{par}(x_1&,x_2,y_1,y_2,\bs \delta)\\ + =& (x_1-y_1-\delta_1)(x_2-y_2-\delta_2)G(-1/2;x_1,x_2;y_1+\delta_1,y_2+\delta_2,\delta_3)\\ + &-(x_1-y_1-\delta_1) g(1/2;x_1;y_1+\delta_1;\{(x_2-y_2-\delta_2)^2 + \delta_3^2\}^{1/2})\\ + &-(x_2-y_2-\delta_2) g(1/2;x_2;y_2+\delta_2;\{(x_1-y_1-\delta_1)^2 + \delta_3^2\}^{1/2})\\ + &+\frac 1 3 \{(x_1-y_1-\delta_1)^2+(x_2-y_2-\delta_2)^2+\delta_3^2\}^{3/2} +\end{align*} + \subsubsection{Orthogonale Elemente} Analog können wir orthogonal liegende Elemente wie in \cite[Seite 14]{mai:3dbem} gezeigt, schreiben als: -\begin{eqnarray*} - T_j &=& \v + [(0,s_1) \times (0,s_2) \times \{0\}]\\ - T_k &=& \tilde \v + [ \{0\} \times (0,\tilde s_2) \times (0,\tilde s_3)], \text{ mit } \v,\tilde \v \in \R^3 -\end{eqnarray*} und setzen wir $\bs{\delta} = \tilde \v - \v \in \R^3$. +\begin{align*} + T_j &= \v + [(0,s_1) \times (0,s_2) \times \{0\}]\\ + T_k &= \tilde \v + [ \{0\} \times (0,\tilde s_2) \times (0,\tilde s_3)], \text{ mit } \v,\tilde \v \in \R^3 +\end{align*} und setzen wir $\bs{\delta} = \tilde \v - \v \in \R^3$. Dann können wir -\begin{eqnarray*} +\begin{align*} \int_{T_j} \int_{T_k} \frac{1}{|\bs x- \bs y|} ds_{\bs y} ds_{\bs x}\\ -\end{eqnarray*} umformen zu: -\begin{eqnarray*} -&=& \int_{T_j} \int_{T_k} +\end{align*} umformen zu: +\begin{align*} +&= \int_{T_j} \int_{T_k} \left ( (x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2 \right)^{-1/2} ds_{\bs y} ds_{\bs x}\\ -&=& \int_{T_j} \int_{T_k} +&= \int_{T_j} \int_{T_k} \left ( ((x_1 + v_1)-\tilde v_1)^2+((x_2 + v_2)-(y_2 +\tilde v_2))^2+( v_3- (y_3 +\tilde v_3))^2 \right)^{-1/2} ds_{\bs y} ds_{\bs x}\\ -&=& \int_0^{s_1}\int_0^{s_2}\int_0^{\tilde s_2}\int_0^{\tilde s_3} +&= \int_0^{s_1}\int_0^{s_2}\int_0^{\tilde s_2}\int_0^{\tilde s_3} \left ( (x_1-\delta_1)^2+(x_2-y_2-\delta_2)^2+(y_3-\delta_3)^2 \right)^{-1/2} dy_3 dy_2 dx_2 dx_1 -\end{eqnarray*} +\end{align*} Dies können wir ebenfalls als Stammfunktion -\begin{eqnarray*} -F_{ort}(x_1,x_2,y_1,y_2,\bs \delta) &:=& \int \int \int \int \left\{(x_1-\delta_1)^2+(x_2-y_2-\delta_2)^2+(-y_3-\delta_3)^2\right\}^{-1/2} +\begin{align*} +F_{ort}(x_1,x_2,y_2,y_3,\bs \delta) &:= \int \int \int \int \left\{(x_1-\delta_1)^2+(x_2-y_2-\delta_2)^2+(-y_3-\delta_3)^2\right\}^{-1/2} dy_3 dy_2 dx_2 dx_1 -\end{eqnarray*} +\end{align*} für orthogonal liegende Elemente schreiben. +\begin{align*} +2F_{ort}(x_1&,x_2,y_2,y_3,\bs \delta)\\ +=&-G(1/2;y_3,x_1;-\delta_3,\delta_1,x_2-y_2-\delta_2)\\ +&-(x_1-\delta_1)(x_2-y_2-\delta_2)G(-1/2;x_2,y_3;y_2+\delta_2,-\delta_3,x_1-\delta_1)\\ +&+(x_1-\delta_1)g(1/2);y_3;-\delta_3,\{(x_1-\delta_1)^2+(x_2-y_2-\delta_2)^2\}^{1/2})\\ +&-(x_3-\delta_3)(x_2-y_2-\delta_2)G(-1/2;x_1,x_2;\delta_1,y_2+\delta_2,-y_3-\delta_3)\\ +&+(x_3-\delta_3)g(1/2);x_1;\delta_1,\{(x_2-y_2-\delta_2)^2+(y_3+\delta_3)^2\}^{1/2}) +\end{align*} + + % \subsection{Bestimmtes Integral} % \begin{eqnarray*} % &&\frac{1}{4\pi} \int_T \int_{\tilde T} \frac{1}{|x-y|} ds_y ds_x\\ @@ -479,7 +522,7 @@ für orthogonal liegende Elemente schreiben. % F_{par/ort}(x_1,x_2,y_1,y_2) % dy_2 dy_1 dx_2 dx_1\\ % % -% &=& \frac{1}{4\pi}\big( \int_0^{k_1}\int_0^{k_2}\int_0^{\tilde k_1} +% &= \frac{1}{4\pi}\big( \int_0^{k_1}\int_0^{k_2}\int_0^{\tilde k_1} % \dif{}{y_1} \dif{}{x_2} \dif{}{x_1} % F_{par/ort}(x_1,x_2,y_1,\tilde k_2) % - @@ -487,7 +530,7 @@ für orthogonal liegende Elemente schreiben. % F_{par/ort}(x_1,x_2,y_1,0) \big) % dy_1 dx_2 dx_1\\ % % -% &=& \frac{1}{4\pi}\big( \int_0^{k_1}\int_0^{k_2} +% &= \frac{1}{4\pi}\big( \int_0^{k_1}\int_0^{k_2} % \dif{}{x_2} \dif{}{x_1} % F_{par/ort}(x_1,x_2,\tilde k_1,\tilde k_2) % - @@ -501,7 +544,7 @@ für orthogonal liegende Elemente schreiben. % F_{par/ort}(x_1,x_2,0,0)\big) % dx_2 dx_1\\ % % -% &=& \frac{1}{4\pi}\big( \int_0^{k_1} +% &= \frac{1}{4\pi}\big( \int_0^{k_1} % \dif{}{x_1} % F_{par/ort}(x_1,k_2,\tilde k_1,\tilde k_2) % - @@ -527,7 +570,7 @@ für orthogonal liegende Elemente schreiben. % F_{par/ort}(x_1,0,0,0)\big) % dx_1\\ % % -% &=& \frac{1}{4\pi}\big( +% &= \frac{1}{4\pi}\big( % F_{par/ort}(k_1,k_2,\tilde k_1,\tilde k_2) % - % F_{par/ort}(k_1,k_2,\tilde k_1,0) @@ -562,6 +605,7 @@ für orthogonal liegende Elemente schreiben. % \end{eqnarray*} +\clearpage \section{fasträguläre Partionierung in \Matlab} \todo{ \begin{itemize} @@ -573,22 +617,22 @@ für orthogonal liegende Elemente schreiben. \subsection{Datenstruktur} Für die Implementierung in \Matlab~und C++ wollen wollen wir eine einheitliche Datenstruktur einführen. -Die für die Partition $\mathcal{T}_{\ell} = \{T_1\ldots T_N\}$ benötigten Knoten $\mathcal{K}_{\ell} = \{C_1\ldots C_M\}$ stellen wir in einer $M \times 3$ Matrix $COO$ dar. Dabei enthält die $j$-te Zeile die Koordinaten des Knoten $C_j$, d.h. : -\begin{displaymath} +Die für die Partition $\T_{\ell} = \{T_1\ldots T_N\}$ benötigten Knoten $\K_{\ell} = \{C_1\ldots C_M\}$ stellen wir in einer $M \times 3$ Matrix $COO$ dar. Dabei enthält die $j$-te Zeile die Koordinaten des Knoten $C_j$, d.h. : +\begin{align} COO[j,1:3] = C_j := (x_j,y_j,z_j)^{T} \text{ wobei } x_j,y_j,z_j \in \R. -\end{displaymath} -Die Elemente $\mathcal{T}_{\ell}$ werden wir ebenfalls Zeilenweise in einer $N \times 4$ Matrix $ELE$ abspeichern. Dabei soll die $i$-te Zeile den Indizes der Knoten $\{C_j,C_k,C_{\ell},C_m\}$ des Elements $T_i$ entsprechen, also: -\begin{displaymath} +\end{align} +Die Elemente $\T_{\ell}$ werden wir ebenfalls Zeilenweise in einer $N \times 4$ Matrix $ELE$ abspeichern. Dabei soll die $i$-te Zeile den Indizes der Knoten $\{C_j,C_k,C_{\ell},C_m\}$ des Elements $T_i$ entsprechen, also: +\begin{align} ELE[i,1:4] = T_i := (j,k,l,m). -\end{displaymath} +\end{align} Die Knoten wollen wir gegen den Uhrzeigersinn anordnen \todo{und der Knoten $C_j$ soll der kleinste bezüglich der Koordinaten sein}. \noindent Für die bessere Handhabung der Elemente beim Verfeinern der Partition, wollen wir auch die Nachbarschaftsrelationen geeignet abspeichern. Dazu überlegen wir uns, dass wir Aufgrund der Netzstabilität maximal zwei Nachbarn pro Kante eines Elements zulassen wollen. Wir legen also eine $M \times 8$ Matrix für die Indizes der Nachbarelemente an, wobei die $i$-te Zeile die Nachbarelemente $\{T_{n_1},\ldots,T_{n_8}\}$ zum Element $T_i$ enthält. -\begin{displaymath} +\begin{align} NEI[i,1:8] = N_i := (n_1,\ldots,n_8) -\end{displaymath} +\end{align} Offensichtlich ist $i \notin N_i$. Wir wollen uns aber noch genauer eine geeignete Anordnung für die Nachbarelemente überlegen. Hierbei bezeichnen wir die Seite $[j,k]$ eines Elements als Seite 1 und gegen den Uhrzeigersinn alle weiteren mit $\{2,3,4\}$. Für den einfacheren Zugriff auf die Elemente einer Seite $s$, wollen wir die Nachbarelemente zur Seite $s$ unter den Indizes $n_s$ und $n_{s+4}$ abspeichern. Die Nachbarelemente $\{T_{n_s}, T_{n_{s+4}}\}$ liegen also an der Seite $s$ des Elements. Für Seiten die nur einen Nachbarn $T_{n_s}$ besitzen setzen wir $n_{s+4}=0$ und für Seiten mit keinem Nachbarn setzen wir $n_s = n_{s+4} = 0$. Daraus folgt unmittelbar, dass für $n_s =0$ auch $n_{s+4} = 0$ gilt, die Seite $s$ also keine Nachbarelemente besitzt und umgekehrt folgt aus $n_{s+4} \neq 0$ $n_s \neq 0$, womit die Seite $s$ genau zwei Nachbarelemente hat. (Siehe Abb.:\ref{exmpl13:nei:part}) \begin{figure}[ht] @@ -639,15 +683,15 @@ $[COO_{fine}, ELE_{fine}, NEI_{fine}, F2S ] = refineQuad(COO, ELE, NEI, marked); \subsection{Markieren} \cite{dor:adapt} Bestimme $M_{\ell} \subseteq T_{\ell}$ mit minimaler Kardinalität -\begin{eqnarray*} -\theta \sum_{T\in T_{\ell}} \mu_{\ell}(T)^2 & \leq & \sum_{T\in M_{\ell}} \mu_{\ell}(T)^2 -\end{eqnarray*} +\begin{align*} +\theta \sum_{T\in T_{\ell}} \mu_{\ell}(T)^2 &\leq \sum_{T\in M_{\ell}} \mu_{\ell}(T)^2 +\end{align*} Zur anisotropen Verfeinerung wird weiterhin berechnet: -\begin{eqnarray*} +\begin{align*} \begin{pmatrix} C_j^{(1)}\\ C_j^{(2)}\\C_j^{(3)}\\ C_j^{(4)} \end{pmatrix} -& = & \frac 1 4 +&= \frac 1 4 \begin{pmatrix} 1 & 1 & 1 & 1\\ 1 & -1 & 1 & -1\\ @@ -657,12 +701,12 @@ Zur anisotropen Verfeinerung wird weiterhin berechnet: \begin{pmatrix} x_j^{(1)}\\ x_j^{(2)}\\x_j^{(3)}\\ x_j^{(4)} \end{pmatrix} -\end{eqnarray*} +\end{align*} -\begin{eqnarray*} -\nu \abs{ C_j^{(3)}} & \geq & \sqrt{\abs{ C_j^{(2)}}^2 + \abs{ C_j^{(4)}}^2}\\ -\nu \abs{ C_j^{(4)}} & \geq & \sqrt{\abs{ C_j^{(2)}}^2 + \abs{ C_j^{(3)}}^2} -\end{eqnarray*} +\begin{align*} +\nu \abs{ C_j^{(3)}} &\geq \sqrt{\abs{ C_j^{(2)}}^2 + \abs{ C_j^{(4)}}^2}\\ +\nu \abs{ C_j^{(4)}} &\geq \sqrt{\abs{ C_j^{(2)}}^2 + \abs{ C_j^{(3)}}^2} +\end{align*} $xF2S := x_{fine}[F2S]$\\ $marked = mark(xF2S, mu, theta, nu);$ @@ -671,9 +715,9 @@ $marked = mark(xF2S, mu, theta, nu);$ \subsection{Assemblieren} %Es seien \Ta, \Tb ~$\subseteq$~ $\R^3$ zwei beschränkte, achsenorientierte rechteckige Seiten in $\R^3$. %Berechnet werden soll: -\begin{eqnarray*} -V(j,k) &=& \frac{1}{4\pi} \int_{T_j} \int_{T_k} \frac{1}{|x-y|} ds_y ds_x \in \R^3,T_j,T_k\in\T -\end{eqnarray*} +\begin{align*} +V(j,k) &= \frac{1}{4\pi} \int_{T_j} \int_{T_k} \frac{1}{|x-y|} ds_y ds_x \in \R^3,T_j,T_k\in\T +\end{align*} Wobei $\zeta$ die Zulässigkeitsbedingung und $type$ die Berechnungsart bestimmt. $V = mex\_build\_AU(\T,zeta,type)$\\ $V = mex\_build\_AU(COO,ELE,zeta,type)$ @@ -709,7 +753,7 @@ Zum Plotten (Abb.\ref{exmplAA_2DQuad})werden noch folgende Schritte ausgeführt \item $\kappa3_{i} = \enorm{\hat \phi_{\ell}^{(i)}-\hat \phi_{\ell}^{(i-1)}}$ \end{itemize} - +\clearpage \section{Numerische Experimente} \todo{ \begin{itemize} @@ -771,26 +815,26 @@ Dann gilt auf isotropen Netzen: \end{bew} -\begin{eqnarray*} -%\mu_{\ell}^2 & = & \norm{\varrho_{\ell}^{1/2}(\phi_{\frac l 2}-\Pi_{\ell}\phi_{\frac l 2})}_{\L^2(\T)}^2 \\ -%& = &\sum_{T\in \T_{\ell}}\mu_{\ell}(T)^2\\ -\mu_{\ell}(T)^2 & = & \norm{\varrho_{\ell}^{1/2}(\phi_{\frac l 2}-\Pi_{\ell}\phi_{\frac l 2})}_{\L^2(\T)}^2 \\ -& = & h_{min}(T)\norm{\phi_{\frac l 2}-\Pi_{\ell}\phi_{\frac l 2}}_{\L^2(\T)}^2 \\ +\begin{align*} +%\mu_{\ell}^2 &= \norm{\varrho_{\ell}^{1/2}(\phi_{\frac l 2}-\Pi_{\ell}\phi_{\frac l 2})}_{\L^2(\T)}^2 \\ +%&=\sum_{T\in \T_{\ell}}\mu_{\ell}(T)^2\\ +\mu_{\ell}(T)^2 &= \norm{\varrho_{\ell}^{1/2}(\phi_{\frac l 2}-\Pi_{\ell}\phi_{\frac l 2})}_{\L^2(\T)}^2 \\ +&= h_{min}(T)\norm{\phi_{\frac l 2}-\Pi_{\ell}\phi_{\frac l 2}}_{\L^2(\T)}^2 \\ && T_j \in \tau_{\ell}, t_j^{(1)},\dots,t_j^{(4)} \in \tau_{\frac l 2} \\ -\phi_{\frac l 2}|_{T_j} & = &x_j^{(1)}\chi_{t_j^{(1)}}+\dots+x_j^{(4)}\chi_{t_j^{(4)}}\\ -\Pi_{\ell}\phi_{\frac l 2}|_{T_j}&=&\frac 1 {\abs{T_j}}\int_{T_j}\phi_{\frac l 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\\ +\phi_{\frac l 2}|_{T_j} &=x_j^{(1)}\chi_{t_j^{(1)}}+\dots+x_j^{(4)}\chi_{t_j^{(4)}}\\ +\Pi_{\ell}\phi_{\frac l 2}|_{T_j}&=\frac 1 {\abs{T_j}}\int_{T_j}\phi_{\frac l 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 l 2} - \Pi_{\ell}\phi_{\frac l 2}}_{\L^2(T_j)}^2 -&=&\sum_{k=1}^4\norm{\phi_{\frac l 2} - \Pi_{\ell}\phi_{\frac l 2}}_{\L^2\left(t_j^{(k)}\right)}^2\\ -&=&\sum_{k=1}^4 \int_{t_j^{(k)}}(\phi_{\frac l 2} - \Pi_{\ell}\phi_{\frac l 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_{\ell}(T_j)^2 & = & \frac{ h_{min}(T_j) \abs{T_j}}{4} \sum_{k=1}^4{(x_j^{(k)}-m_j)^2} -\end{eqnarray*} +&=\sum_{k=1}^4\norm{\phi_{\frac l 2} - \Pi_{\ell}\phi_{\frac l 2}}_{\L^2\left(t_j^{(k)}\right)}^2\\ +&=\sum_{k=1}^4 \int_{t_j^{(k)}}(\phi_{\frac l 2} - \Pi_{\ell}\phi_{\frac l 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_{\ell}(T_j)^2 &= \frac{ h_{min}(T_j) \abs{T_j}}{4} \sum_{k=1}^4{(x_j^{(k)}-m_j)^2} +\end{align*} diff --git a/src/slpRectangle.cpp b/src/slpRectangle.cpp index 37db16d..5283e4c 100644 --- a/src/slpRectangle.cpp +++ b/src/slpRectangle.cpp @@ -164,7 +164,7 @@ double inline g_AY(double p, double y, double x, double l) { else if (p == -0.5) sol = asinh((y - x) / fabs(l)); else if (p == -1) - sol = atan((y - x) / fabs(l)) / l; + sol = atan((y - x) / fabs(l)) / fabs(l); else if (p == -1.5) sol = (y - x) / ((l * l) * sqrt((y - x) * (y - x) + l * l)); else -- 2.47.3