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zG~kO6-Ybonl~>G|+<@P(2HzN}ykby%LchzZ2jOrAI#%@{Q@Om_L<3aqs+SMEeDGzu oJ7|zn761QHYT{K@nT_nF@7rhBP0oZ|wyS4mbm?;W$ibQa2a+ZJcmMzZ diff --git a/doc/doc.tex b/doc/doc.tex index 18da785..af8275a 100644 --- a/doc/doc.tex +++ b/doc/doc.tex @@ -26,6 +26,7 @@ \let\mod\relax \DeclareMathOperator{\mod}{mod} +\DeclareMathOperator{\sgn}{sgn} \def\Ta{$T_a$} \def\Tb{$T_b$} @@ -281,7 +282,6 @@ Dann gilt auf isotropen Netzen: \section{Analytische Berechnung der Integrale} -\subsection{Problem} Es seien \Ta, \Tb ~$\subseteq$~ $\R^3$ zwei beschränkte, achsenorientierte rechteckige Seiten in $\R^3$. Berechnet werden soll: \begin{eqnarray*} @@ -306,12 +306,24 @@ Für $p \in \{1/2 , 0 , -1/2 , -1, -3/2\}$ gilt explizit: g(-3/2;y;x;\lambda) &:=& \frac{y-x}{\lambda^2}\{(y-x)^2+\lambda^2\}^{-1/2} \end{eqnarray*} - \subsection{doppel Integral} -\begin{eqnarray*} - 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{eqnarray*} +\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*} Im Zuge der Berechnungen werden auch hier nur die Funktionen für bestimmte Parameter benötigt, die Untersuchung der restlichen werden wir hier nicht durchführen. +\\\noindent +Betrachten wir zunächst den Fall $p = -3/2$ und unterscheiden wir auch zwischen $\lambda = 0 \wedge \lambda \neq 0$, denn alle weiteren für uns relevanten Fälle lassen sich auf diese Beiden zurück führen: +\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|}\\ + &\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 $p = -3/2 +k : k \in \N\backslash0$ lässt sich 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)\\ + &+ (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*} \subsection{Integral über zwei Elemente} Bei der Integration über zwei Seitenelemente \Ta, \Tb $\in \T_{\ell}$ haben wir geometrisch zischen zwei Fällen unterschieden. Entweder die beiden Elemente liegen in parallelen Ebenen oder in orthogonalen Ebenen. @@ -319,7 +331,7 @@ Bei der Integration über zwei Seitenelemente \Ta, \Tb $\in \T_{\ell}$ haben wir \subsubsection{Parallele Elemente} Liegen die beiden Elemente parallel zueinander lassen sie sich Folgendermaßen darstellen: Sei: \Ta = $ \v + [(0,l_1) \times (0,l_2) \times \{0\}]$ und \Tb = $ \tilde \v + [(0,\tilde l_1) \times (0,\tilde l_2) \times \{0\}]$, wobei $\v,\tilde \v \in \R^3$ ist. Weiterhin sei $\boldsymbol{\delta} = \tilde \v - \v \in \R^3$ definiert.\\ -Damit lässt sich zeigen: +Damit können wir zeigen, dass \begin{eqnarray*} &&-\frac{1}{4\pi} \int_T \int_{\tilde T} \frac{1}{|x-y|} ds_y ds_x\\ &=&-\frac{1}{4\pi} \int_T \int_{\tilde T} @@ -331,7 +343,7 @@ Damit lässt sich zeigen: \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*} -Entscheidend ist also die Stammfunktion $F_{par}$ +die Stammfunktion $F_{par}$ das Integral über zwei parallele Elemente löst. \begin{eqnarray*} F_{par}(x_1,x_2,y_1,y_2,\delta_1,\delta_2,\delta_3) &:=& \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*} @@ -585,8 +597,6 @@ $[COO_{fine}, ELE_{fine}, NEI_{fine}, F2S ] = refineQuad(COO, ELE, NEI, marked); \mu_{\ell}(T_j)^2 & = & \frac{ h_{min}(T_j) \abs{T_j}}{4} \sum_{k=1}^4{(x_j^{(k)}-m_j)^2} \end{eqnarray*} -$mu = computeTstSlpMuTilde(x_{fine}, \T, F2S);$\\ -$mu = computeTstSlpMuTilde(x_{fine}, COO, ELE, F2S);$ @@ -650,7 +660,7 @@ $\theta \in (0,1),i =0$ \item $i \mapsto i+1$, gehe zu $(i)$ \end{enumerate} -Zum Plotten (\ref{exmplAA_2DQuad})werden noch folgende Schritte ausgeführt +Zum Plotten (Abb.\ref{exmplAA_2DQuad})werden noch folgende Schritte ausgeführt \begin{itemize} \item Berechne Galerkinlösung $\phi_{l} \in P^0(\T_{\ell}^{(i)})$ % \item $\enorm{\phi_{\ell/2}}$ @@ -663,6 +673,43 @@ Zum Plotten (\ref{exmplAA_2DQuad})werden noch folgende Schritte ausgeführt \item $\kappa3_{i} = \enorm{\phi_{\ell/2}^{(i)}-\phi_{\ell/2}^{(i-1)}}$ \end{itemize} + +\section{Auswertung} +\todo{Viele Bunte Bilder und Testergebnisse} +\begin{itemize} + \item gerechnet wurde $V\phi = 1$ + \item Beispiele für verschiedene Figuren + \begin{itemize} + \item 2D LShape + \item 2D Rechteck + \item 3D Würfel + \item Fischer Würfel + \item L Figur + \end{itemize} + \item Beispiele für verschiedene Berechnungsarten + \begin{itemize} + \item VollAnalytisch + \item SemiAnalytisch über Element + \item SemiAnalytisch über Achse + \item SemiAnalytisch über Seite + \end{itemize} + \item Fehlerschätzer untersuchen und prüfen ob sie sich wie erwartet Verhalten + +\end{itemize} + + +\section{Anhang Code} +\todo{ +Die wichtigsten Funktionen +\begin{enumerate} + \item compute + \item refine + \item mark + \item build\_V + \item plot +\end{enumerate} +} + \begin{figure}[ht] \caption{2D Quad adaptiv anisotrop vollanalytisch $V\phi = 1$} \centering diff --git a/src/A_plots.m b/src/A_plots.m index aa7e0d7..36a9265 100644 --- a/src/A_plots.m +++ b/src/A_plots.m @@ -81,8 +81,8 @@ else % data((end-8):end,[1 [3 4 5]]) % sol = interp1(1./X((round(1)):(end),4)',G_D((round(1)):(end),4)',0,'spline') - sol = 8.28466; % LShape -% sol = 4.609193; % Quad +% sol = 8.28466; % LShape + sol = 4.609193; % Quad % G_D -- 2.47.3