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Weiterhin wollen wir auch verlangen, dass der auf der Kante li \subfloat[ungültige Partition]{\includegraphics[width=0.3\textwidth]{fig/net_wrong}} \caption{Beispiel Partitionen} \end{figure} -\psfrag{T}{\scriptsize $T$} -\psfrag{T1}{\scriptsize $T_1$} -\psfrag{T2}{\scriptsize $T_2$} -\psfrag{T3}{\scriptsize $T_3$} -\psfrag{T4}{\scriptsize $T_4$} \subsubsection{Verfeinern} \begin{defi}[Lokale Verfeinerung] \end{defi} @@ -252,17 +254,14 @@ anliegen können. Weiterhin wollen wir auch verlangen, dass der auf der Kante li \end{figure} \begin{defi} - Sei $\T_{\ell/2}$ die aus der Verfeinerung von $\T_{\ell}$ resultierende gültige Partition. + Für eine Partition $\T_{\ell}$ bezeichnen wir mit $\hat \T_{\ell}$ 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{enumerate} \renewcommand{\theenumi}{(\roman{enumi})} \item ermittle ein markiertes Element $T_j$, welches noch nicht geteilt wurde \label{alg:refine:first} \item - \begin{itemize} - \item falls ein Eckpunkt von $T_j$ in der Kante eines Nachbarelements liegt - \item falls Eckpunkt von $T_j$ Hängender-Knoten ist, - \end{itemize} + 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 @@ -343,17 +342,18 @@ Dann gilt auf isotropen Netzen: \section{Analytische Berechnung der Integrale} -Es seien \Ta, \Tb ~$\subseteq$~ $\R^3$ zwei beschränkte, achsenorientierte rechteckige Seiten in $\R^3$. -Berechnet werden soll: +Es seien $T_j,T_k \subseteq\R^3$ zwei beschränkte, achsenorientierte Rechtecke in $\R^3$. +Die Berechnung der Matrix für das Galerkin-Verfahren benötigt die Auswertung von \begin{eqnarray*} --\frac{1}{4\pi} \int_{T_i} \int_{T_j} \frac{1}{|\bs x- \bs y|} ds_{\bs y} ds_{\bs x} \in \R^3 +\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*} +Wir betrachten zunächst die Berechnung von zwei Integralen, die im Zuge der Berechnungen auftreten werden. \subsection{einfach Integral} -Sei: +Wir bezeichnen \begin{eqnarray*} g(p;y;x;\lambda) &:=& \int \{(x-y)^2 + \lambda^2 \}^p dy \end{eqnarray*} -Da $g$ nur für bestimmte Parameter $p$ und $\lambda$ benötigt wird, werden hier nur die entscheidenden Fälle betrachtet. +Da $g(p;y;x;\lambda)$ nur für bestimmte Parameter $p$ und $\lambda$ benötigt wird, werden hier nur die entscheidenden Fälle betrachtet. 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) @@ -614,7 +614,7 @@ Da wir im weiteren Verlauf sowohl adaptive also auch anisotrope Netzverfeinerung \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 Trgebnis eine volle Teilung 2. ausführt, diese aber schrittweise durch eine 3. Teilung und zwei 4. Teilungen. Aus Sicherheitsgründen wird auch jede vierte volle 2. Teilung durch eine 5. Teilung ausgeführt, da sonst kurzzeitig Seiten mit mehr als zwei Elementen auftreten könnten. +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. Aus Sicherheitsgründen wird auch jede vierte volle 2. Teilung durch eine 5. Teilung ausgeführt, da sonst kurzzeitig Seiten mit mehr als zwei Elementen auftreten könnten. Damit jedem Element $T_i$ 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 $T_i$. 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. -- 2.47.3