From 2dc824e072d684b748690b93a331b5797e28df40 Mon Sep 17 00:00:00 2001 From: treecity Date: Thu, 22 Mar 2012 22:04:18 +0000 Subject: [PATCH] [doc] etwas umstrukturiert [doc] Einleitung vorgeschrieben git-svn-id: https://drops.fb12.tu-berlin.de/svn/bacc/trunk@113 26120e32-c555-405d-b3e1-1f783fb42516 --- doc/doc.pdf | Bin 0 -> 325849 bytes doc/doc.tex | 619 ++++++++++++++++++++++ doc/doc1.dvi | Bin 20496 -> 0 bytes doc/doc1.pdf | Bin 165325 -> 0 bytes doc/doc1.tex | 284 ---------- doc/doc2.pdf | Bin 202690 -> 0 bytes doc/doc2.tex | 220 -------- doc/fig/exmpl1_ref.eps | 127 +++-- doc/fig/exmpl1_ref.pdf | Bin 5485 -> 5548 bytes doc/fig/exmpl2_ref.eps | 175 ++++--- doc/fig/exmpl2_ref.pdf | Bin 5649 -> 5741 bytes doc/fig/exmpl3_ref.eps | 197 +++---- doc/fig/exmpl3_ref.pdf | Bin 5732 -> 5859 bytes doc/fig/exmplAA_2DQuad_error.eps | 867 +++++++++++++++++++++++++++++++ doc/fig/exmplAA_2DQuad_error.pdf | Bin 0 -> 17372 bytes doc/fig/exmplAA_2DQuad_norm.eps | 586 +++++++++++++++++++++ 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+\usepackage{colortbl} %Einfache Färbungen in Tabellen +\usepackage{color} %Farben benutzen und Definieren +\usepackage{subfig} %mehrere Figuren in einer +\usepackage{hyperref} %Links im Inhaltsverzeichnis +\hypersetup{linkbordercolor={1 1 1},citebordercolor={1 1 1},urlbordercolor={1 1 1}} + + +\usepackage[ngerman]{babel} %Sprachpacket für ├Ьberschriften +\usepackage[utf8]{inputenc} %Eingabekodierung +\usepackage{fixltx2e} %Deutschsprach Bugs + +\definecolor{gray}{gray}{.95} + +\def\todo#1{\textcolor{red}{#1}} +\def\why#1{\textcolor{blue}{#1}} +\def\Matlab{{\sc Matlab}} +\def\q{\Q} + +\def\Ta{$T_a$} +\def\Tb{$T_b$} +\def\v{\boldsymbol{v}} +\def\D{\mathcal{D}} +\def\Q{\mathcal{Q}} +\def\G{\mathcal{G}} +\def\L{\mathcal{L}} +\def\T{\mathcal{T}} +\def\oder{\vee} +\def\und{\wedge} + +\newcommand{\dif}[2]{\frac{\partial#1}{\partial#2}} +\newcommand{\ul}[1]{\underline{#1}} +\newcommand{\enorm}[1]{\left| \! \left| \! \left|#1\right| \! \right| \! \right|} +\newcommand{\norm}[1]{\lVert#1\rVert} +\newcommand{\abs}[1]{\lvert#1\rvert} +\newcommand{\showMesh}[2][]{\begin{figure}[ht] +\caption{#1} +\label{#2} +\centering +\subfloat[Lage]{\includegraphics[width=0.5\textwidth]{fig/#2_ref}} +\subfloat[Koordinaten]{\input{fig/#2_coo}}\\ +\subfloat[Elemente]{\input{fig/#2_ele}} +\subfloat[Nachbarn]{\input{fig/#2_nei}} +\end{figure}} + +\def\N{\mathbb{N}} +\def\R{\mathbb{R}} + +\newtheorem{lem}{Lemma} +\newtheorem{defi}{Definition} +\newtheorem{sat}{Satz} +\newtheorem{bew}{Beweis} + +\numberwithin{lem}{section} +\numberwithin{defi}{section} +\numberwithin{bew}{section} +\numberwithin{sat}{section} + +\author{P. Schaefer} + +\begin{document} +\tableofcontents +\clearpage + +\section{Einleitung} + +\subsection{Allgemein} +Gelöst werden soll die Einfachschichtpotential Gleichung. +$\Gamma = \partial \Omega$ +\begin{align} + - \varDelta u &= 0 &\text{ auf } \R^3\backslash \Gamma\\ + u &= f &\text{ in } \Gamma \nonumber \\ +\abs{u(x)} &= O(\abs{x}^{-1}) \nonumber +\end{align} +Daraus folgt: +\begin{align} +V\phi &= f +\end{align} +Sei nun die Fundamentallösung $G$: +\begin{align} +\tilde V \phi (x) &:= \int G (x-y) \phi(y) dy & x\in \R^3\backslash \Gamma +\end{align} +Dabei ist $G(z) := \frac 1 {4 \pi} \frac 1 {\abs{z}}$. +Daraus folgt nun mit $\gamma_0 = \rm{spur}$ : +\begin{align} + V \phi &: \gamma_0 \tilde V \phi_0 +\end{align} +Weiterhin kann man nun Zeigen, dass: +\begin{align} + V : H^{-1/2+s}(\Gamma) &\rightarrow H^{1/2+s}(\Gamma)& \text{mit } s\in [-1/2,1/2] +\end{align} +\begin{lem}[Lax-Milgram] +Sei eine Abbildung $a: X\times X \rightarrow \R$ wobei $X$ ein reflexiver Banachraum. Und gilt: +\begin{itemize} + \item $a$ stetig, d.h. : $\abs{a(x,y)} \leq C \cdot \norm{x} \cdot \norm{y}$ + \item $a$ eliptisch, d.h. : $a(x,x) \geq X \cdot \norm{x}^2$ +\end{itemize} +So folgt daraus +$\forall f \in X'$ $\exists $ eindeutiges $x \in X$ mit $a(x,\cdot) = f$. +\end{lem} +\begin{defi} +Sei $\langle \cdot, \cdot \rangle$ das erweiterte $L_2$ - Skalarprodukt +\end{defi} +Wendet man nun das Lax-Milgram Lemma auf die schwache Formulierung an, +\begin{align} + \langle V \phi, \psi\rangle &= \langle f,\psi\rangle & \psi \in H^{-1/2}\\ +a(\phi,\psi) &:= \langle V\phi,\psi\rangle&:H^{-1/2}(\Gamma)\times H^{-1/2}(\Gamma) \rightarrow \R +\end{align} +zeigen wir noch, dass: +\begin{itemize} + \item $H^{-1/2}(\Gamma)$ ist reflexiver Banachraum, welches aus der Definition von $H^{-1/2}$ folgt + \item $ \abs{\langle V\phi,\psi \rangle} \leq C \cdot \norm{\phi}_{H^{-1/2}} \cdot \norm{\psi}_{H^{-1/2}}$ + \item $ \langle V\phi,\phi \rangle \geq C \cdot \norm{\phi}_{H^{-1/2}}^2$ +\end{itemize} +Daraus folgt nun dass, $\forall f \in H^{-1/2}$ $\exists$ eindeutige Lösung $\phi \in H^{-1/2}(\Gamma)$ von +\begin{align} + \langle V \phi, \psi \rangle &=\langle f, \psi \rangle & \forall \psi \in H^{-1/2}(\Gamma) +\end{align} +Wollen wir nun das Galerkin-Verfahren anwenden benötigen wir die schwache Formulierung: +\begin{align} + \int_{\Gamma} V \phi(x) \cdot \psi(x) dx &= \int_{\Gamma} f(x)\cdot\psi(x) dx +\end{align} +Nun wählen wir einen endlich-dimensionalen Teilraum $P^0(\T_n) \subseteq H^{-1/2}$ und betrachten + +\begin{defi}\label{1} +\begin{align} + \langle V\phi_h,\psi_h \rangle & = \langle f,\psi_h \rangle& \forall \psi_h \in P^0(\T_h) +\end{align} +\end{defi} +Gesucht ist jetzt also $\phi_h \in P^0(\T_h)$ + +\noindent +Aus dem Max-Milgram Lemma und $X = P^0(\T_h)$ folgt wiederum, es $\exists$ eindeutige Lösung $\phi_h \in P^0(\T_h)$, da $\psi_h \in P^0(\T_h),\phi_h \in P^0(\T_h)$. +\begin{defi} + Sei nun die Basis von $P^0(\T_h)$ die charakteristischen Funktionen +\begin{align} + \{\chi_T | T\in\T_h\} &= \{\chi_{T_1},\chi_{T_2},\dots\} +\end{align} +\end{defi} +So können wir mit $N = \dim P^0(\T_h)$ und $\psi_l,\phi_l\in\R$ wobei $l\in \{1\dots N\}$ schreiben +\begin{align} + \psi_h &= \sum_{l=1}^N \psi_l \cdot \chi_{T_l} \\ + \phi_h &= \sum_{l=1}^N \phi_l \cdot \chi_{T_l} +\end{align} +Dadurch können wir Definition \ref{1} nun einfacher Lösen durch: +\begin{align} + \langle V \phi_h,\chi_k\rangle & = \langle f, \chi_k \rangle & k = 1\dots N +\end{align} +Aufgrund der Linearität von $V$ und dem Skalarprodukt schreiben wir: +\begin{align} +\sum_{l=1}^N\langle V\phi_l\chi_l,\chi_k\rangle & = \langle f,\chi_k\rangle +\end{align} +welches sich wiederum so schreiben lässt +\begin{defi}[Galerkinapproximation] +\begin{align} +\ul{\ul{V}} \cdot \ul{\phi} = \ul{f} +\end{align} +wobei $\ul{\ul{V}}\in R^{N \times N},\ul{\phi}\in\R^{N \times 1},\ul{f}\in\R^{N \times 1}$ +\begin{align} + \ul{\ul{V}}_{l,k} &= \langle V \chi_l, \chi_k \rangle\\ + \ul{\phi}_l &= \phi_l \nonumber\\ + \ul{f}_k &= \langle f, \chi_k\rangle \nonumber +\end{align} +Damit ist $\phi_l$ die Galerkinapproximation an $\phi$ +\end{defi} + + +\subsection{Netz} +Alle Elemente bestehen nur aus achsenorientierten Rechtecken. +Entscheidend für die Tests sind Gitter in einer Ebene und Körper... + +\subsubsection{Verfeinern} +Algoritmus zum Verfeinern +\begin{defi} + Sei $\T_{h/2}$ das aus der Verfeinerung von $\T_h$ resultierende Netz +\end{defi} + + +\subsection{Fehlerschätzer} +Um eine Aussage über die Genauigkeit der Galerkinapproximation zu erhalten ist es sinnvoll, dass wir einen A-Posteriori Fehlerschätzer definieren. +\begin{defi}[A-Posteriori Fehlerschätzer $\mu_h$] +sollte + \begin{enumerate} + \item Berechenbar sein, also keine unbekannten enthalten + \item Zuverlässig sein, +\begin{align} + \norm{\phi-\phi_h} &\leq C_{ref} \cdot \mu_h +\end{align} + \item Effizient sein, +\begin{align} + \mu_h &\leq C_{eff} \cdot \norm{\phi - \phi_h} +\end{align} + \end{enumerate} +\end{defi} +\begin{defi}[Saturationsannahme] +\begin{align} + \norm{\phi -\phi_{h/2}} &\leq C_{sat} \cdot \norm{\phi - \phi_h} & 0 < C_{sat} < 1 +\end{align} +\end{defi} + +\begin{defi}[$h-h/2$ - Schätzer] +\begin{align} + \eta_h &:= \norm{\phi_{h/2} - \phi_h}_{H^{-1/2}} +\end{align} +Der Fehlerschätzer ist aber nur unter der Saturationsannahme zuverlässig und ist effizient mit $C_{\tt eff} = 1$ +\end{defi} +Da aber die $\norm{\cdot}_{H^{1/2}}$ schlecht zu berechnen ist, wenden wir die $L_2$-Projektion an. +\begin{defi}[ersetzen von $\norm{\cdot}_{H^{1/2}}$] + \begin{align} + \tilde \eta_h &:= \norm{ \rho^{1/2} (\phi_{h/2} - {\phi_h})}_{L_2} + \end{align} +$\tilde\eta_h$ ist da $\tilde\eta_h \approx \eta_h$ gilt noch immer zuverlässig und effizient. +\end{defi} +Um sich unnötige Berechnungen zu sparen ist es sinnvoll $\phi_h$ zu ersetzen. +\begin{defi}[ersetzen von $\phi_h$] + \begin{align} + \tilde \mu_h &:= \norm{\rho^{1/2}(\phi_{h/2} - \Pi_h\phi_{h/2})}_{L_2} + \end{align} +wobei $\Pi_h$ die $L_2$ Projektion auf $P^0(\T_h)$ ist. +\end{defi} +\begin{sat}[A-posteriori Fehlerschätzer] + Alle vier Schätzer +\end{sat} +\begin{bew} + Siehe S.F. Paper +\end{bew} + + + + +\clearpage + + + + +\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*} +-\frac{1}{4\pi} \int_T \int_{\tilde T} \frac{1}{|x-y|} ds_y ds_x \in \R^3 +\end{eqnarray*} +\subsection{einfach Integral} +Sei: +\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. +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*} +Für $p \in \{1/2 , 0 , -1/2 , -1, -3/2\}$ gilt explizit: +\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|}}{ \lambda}\\ + 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*} +Im Zuge der Berechnungen werden auch hier nur die Funktionen für bestimmte Parameter benötigt, die Untersuchung der restlichen werden wir hier nicht durchführen. + +\subsection{Integral über zwei Elemente} +Bei der Integration über zwei Seitenelemente \Ta, \Tb $\in \T_l$ haben wir geometrisch zischen zwei Fällen unterschieden. Entweder die beiden Elemente liegen in parallelen Ebenen oder in orthogonalen Ebenen. + +\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: +\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} + \left ( (x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2 \right)^{-1/2} ds_y ds_x\\ +&=&-\frac{1}{4\pi} \int_0^{l_1}\int_0^{l_2}\int_0^{\tilde l_1}\int_0^{\tilde l_2} + \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} + dy_2 dy_1 dx_2 dx_1\\ +&=&-\frac{1}{4\pi} \int_0^{l_1}\int_0^{l_2}\int_0^{\tilde l_1}\int_0^{\tilde l_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*} +Entscheidend ist also die Stammfunktion $F_{par}$ +\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*} + +\subsubsection{Orthogonale Elemente} +\Ta = $ \v + [(0,l_1) \times (0,l_2) \times \{0\}]$\\ +\Tb = $ \tilde \v + [ \{0\} \times (0,\tilde l_2) \times (0,\tilde l_3)]$\\ +$\v,\tilde \v \in \R^3$\\ +$\boldsymbol{\delta} = \tilde \v - \v \in \R^3$\\ +\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} + \left ( (x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2 \right)^{-1/2} ds_y ds_x\\ +&=&-\frac{1}{4\pi} \int_0^{l_1}\int_0^{l_2}\int_0^{\tilde l_2}\int_0^{\tilde l_3} + \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} + dy_3 dy_2 dx_2 dx_1\\ +&=&-\frac{1}{4\pi} \int_0^{l_1}\int_0^{l_2}\int_0^{\tilde l_2}\int_0^{\tilde l_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*} +Entscheidend ist also die Stammfunktion $F_{ort}$ +\begin{eqnarray*} +F_{ort}(x_1,x_2,y_1,y_2,\delta_1,\delta_2,\delta_3) &:=& \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*} + +\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} + \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\\ +% +&=& -\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) +- +\dif{}{y_1} \dif{}{x_2} \dif{}{x_1} + 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} + \dif{}{x_2} \dif{}{x_1} + F_{par/ort}(x_1,x_2,\tilde k_1,\tilde k_2) +- + \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) ++ + \dif{}{x_2} \dif{}{x_1} + F_{par/ort}(x_1,x_2,0,0)\big) + dx_2 dx_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) +- +\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) +- +\dif{}{x_1} + F_{par/ort}(x_1,0,0,0)\big) + dx_1\\ +% +&=& -\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) +- + 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) ++ + 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) +- + 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) ++ + F_{par/ort}(0,0,0,\tilde k_2) +- + F_{par/ort}(0,0,0,0)\big) +\end{eqnarray*} + + + +\section{Semi-analytische Berechnung bestimmter Integrale} +\subsection{Gauss-Quadratur} +\begin{displaymath} + \int_0^l f(x) dx \approx \sum_{k=0}^n w_kf(x_k) =: \Q(f) +\end{displaymath} + +\subsection{Quadratur über ein Element} +\begin{displaymath} + \int_{T} \int_{\tilde T} \approx \q_{T} \int_{\tilde T} +\end{displaymath} +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} + \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} + \int_{T} \int_{\tilde T} \approx \q_{I_1} \int_{I_2} \int_{J_1} \int_{J_2} +\end{displaymath} + +$\enorm{a}-\norm{b}$ + + +\section{Zulässigkeitsbedingungen} +\subsection{Lagen der Elemente} +so so oder so... +\subsection{Fehlerschätzer} +\subsection{Möglichkeiten für Kriterien} + +\section{V berechnen} + + +\section{Implementierung} + + +\subsection{Datenstruktur} +Alle Koordinaten werden in einer $ N \times 3,N \in \N$ Matrix abgespeichert. +\begin{displaymath} + COO[i,1:3] = C_i := (x_1,x_2,x_3)^{-1} \text{ wobei } x_1,x_2,x_3 \in \R \und i \in \{1,2 \dots N\} +\end{displaymath} +Jedes Element wird dann durch jeweils vier Koordinatenindizes als $M \times 4, M\in\N$ Matrix gespeichert. Wobei die Koordinaten links herum durchgegangen werden. +\begin{displaymath} + ELE[i,1:4] = T_i := (v_1,v_2,v_3,v_4) \text{ wobei } v_1,v_2,v_3,v_4 \in C \und i \in \{1,2 \dots M\} +\end{displaymath} +Aufgrund der Netzstabilität wollen wir maximal zwei Nachbarn an einer Kante zulassen, weshalb wir nur $2*4$ Tinträe für die Nachbarschaftsrelationen pro Element benötigen. In der $M \times 8$ Matrix sind die Indizes der Nachbarn $n_1,n_2 \in T$ der Kante $(v_k, v_{k^{+1}}) | k\in \{1,2,3,4\}$ des Elements $T_i$ an den Stellen $NTI[i,[k,k+4]]$ gespeichert. Sollte eine Seite keinen Nachbarn haben markieren wir diese mit einer $0$. Außerdem wird an Kanten mit nur einem Nachbarn der erste Index auf den Nachbarn gesetzt und der zweite auf $0$.(Siehe Figur:\ref{exmpl3:nei:part}) +\\\noindent +Tin ausführliches Beispiel ist in Figure \ref{exmpl3} dargestellt. +\showMesh[Beispiel 1.3]{exmpl3} +\begin{figure}[ht] +\caption{Nachbarschaftsrelationen Element 4 aus \ref{exmpl3}} +\label{exmpl3:nei:part} +\centering + \subfloat[Lage]{\includegraphics{fig/Net_Neigh}} + \subfloat[Nachbarn]{\input{fig/exmpl3_nei_part}} +\end{figure} + + + +\subsection{Verfeinern} +Da wir im weiteren Verlauf sowohl adaptive also auch anisotrope Netzverfeinerung zulassen wollen, ist es sinnvoll eine Verfeinerungsfunktion zu implementieren, die alle Möglichen Teilungsprozesse auf einem Element unterstützt. Dabei sind vier nur wirklich relevant: +\begin{enumerate} + \item keine Teilung + \item volle Teilung in vier gleich große Elemente + \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. + +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. + +Relevant zum Verfeinern eines Netzes sind also die Koordinaten $COO$, Elemente $ELE$ sowie die Nachbarschaftsrelationen $NTI$ und der Markierungsvektor $marked$. + +Da wir später einen Fehlerschätzer berechnen wollen, ist es wichtig sich zu jedem Element seine Teilelemente zu merken. Dazu legen wir während der Teilung eine $M \times 4$ Matrix an, in der die maximal vier Elementindizes gespeichert sind. Wenn wir also ein Element in vier gleich große Teile verfeinern, so wird das neue Element links unten das erste sein und alle weiteren folgen gegen den Uhrzeigersinn. Teilen wir ein Element in zwei gleich große Elemente, so werden die doppelt belegten Quadranten auch doppelt eingetragen. Ein gar nicht geteiltes Element wird also vier mal den alten Indizes speichern. Dadurch wird sicher gestellt, dass das arithmetische Mittel über die Elemente immer gültig auszuführen ist. +(Siehe Figur:\ref{exmpl3:f2s}) + +$[\T_{1/2}, F2S ] = refineQuad(\T, marked);$\\ +$[COO_{fine}, ELE_{fine}, NTI_{fine}, F2S ] = refineQuad(COO, ELE, NTI, marked);$ + +\begin{figure}[ht] +\caption{VaterSohn} +\centering +\subfloat[VaterSohn aus \ref{exmpl2}]{\input{fig/exmpl2_f2s}} +\subfloat[VaterSohn aus \ref{exmpl3}]{\input{fig/exmpl3_f2s}} +\end{figure} + + +\subsection{Fehlerschätzer} +\begin{eqnarray*} +%\mu_l^2 & = & \norm{\varrho_l^{1/2}(\phi_{\frac l 2}-\Pi_l\phi_{\frac l 2})}_{\L^2(\T)}^2 \\ +%& = &\sum_{T\in \T_l}\mu_l(T)^2\\ +\mu_l(T)^2 & = & \norm{\varrho_l^{1/2}(\phi_{\frac l 2}-\Pi_l\phi_{\frac l 2})}_{\L^2(\T)}^2 \\ +& = & h_{min}(T)\norm{\phi_{\frac l 2}-\Pi_l\phi_{\frac l 2}}_{\L^2(\T)}^2 \\ +&& T_j \in \tau_l, 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_l\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_l\phi_{\frac l 2}}_{\L^2(T_j)}^2 +&=&\sum_{k=1}^4\norm{\phi_{\frac l 2} - \Pi_l\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_l\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_l(T_j)^2 & = & \frac{ h_{min}(T_j) \abs{T_j}}{4} \sum_{k=1}^4{(x_j^{(k)}-m_j)^2} +\end{eqnarray*} + +$mu = computeTstSlpMuTilde(x_{fine}, \T, F2S);$\\ +$mu = computeTstSlpMuTilde(x_{fine}, COO, ELE, F2S);$ + + + + +\subsection{Markieren} +Bestimme $M_l \subseteq T_l$ mit minimaler Kardinalität +\begin{eqnarray*} +\theta \sum_{T\in T_l} \mu_e(T)^2 & \leq & \sum_{T\in M_l} \mu_l(T)^2 +\end{eqnarray*} +Zur anisotropen Verfeinerung wird weiterhin berechnet: +\begin{eqnarray*} +\begin{pmatrix} + C_j^{(1)}\\ C_j^{(2)}\\C_j^{(3)}\\ C_j^{(4)} +\end{pmatrix} +& = & \frac 1 4 +\begin{pmatrix} +1 & 1 & 1 & 1\\ +1 & -1 & 1 & -1\\ +1 & 1 & -1 & -1\\ +1 & -1 & -1 & 1\\ +\end{pmatrix} +\begin{pmatrix} + x_j^{(1)}\\ x_j^{(2)}\\x_j^{(3)}\\ x_j^{(4)} +\end{pmatrix} +\end{eqnarray*} + +\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*} + +$xF2S := x_{fine}[F2S]$\\ +$marked = mark(xF2S, mu, theta, nu);$ + + +\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*} +Wobei $\zeta$ die Zulässigkeitsbedingung und $type$ die Berechnungsart bestimmt. +$V = mex\_build\_AU(\T,zeta,type)$\\ +$V = mex\_build\_AU(COO,ELE,zeta,type)$ + + +\subsection{Vorgehensweise} +Mithilfe der oben Definierten Funktionen ist es uns nun möglich den Ablauf der Verfeinerungen zusammen zu fassen. + +Sei das Netz $\T^{(0)}$ mit $\#{\T^{(0)}}$ Elementen gegeben und iterieren wir über $i \in \N$: + +\begin{enumerate} + \item $(\T^{(i)}_{\frac l 2},f2s^{(i)}_{\frac l 2}) = refineQuad(\T^{(i)},2)$ + \item $V^{(i)}_{\frac l 2} = mex\_build\_AU(\T^{(i)}_{\frac l 2},$ vollAnalytisch $)$ + \item $b^{(i)}_{\frac l 2} = \left\{b_k=size(T_k) | T_k\in\T^{(i)}_{\frac l 2}, k \in \{1,2 \dots\#\T^{(i)}_{\frac l 2}\} \right\}$ + \item Löse: $V^{(i)}_{\frac l 2} \cdot x^{(i)}_{\frac l 2} = b^{(i)}_{\frac l 2}$ + \item $\mu(\T^{(i)})^2 = \left\{\frac{ h_{min}(T_j) \abs{T_j}}{4} \sum_{k=1}^4{(x_{\frac l 2,j}^{(i)(k)}-\frac 1 4\sum_{l=1}^4x_{\frac l 2,j}^{(i)(l)})^2}, j \in \{1,2 \dots\#\T^{(i)}\} \right\}$\\$ = computeTstSlpMuTilde(\phi^{(i)}_{\frac l 2},\T^{(i)},f2s^{(i)}_{\frac l 2})$ + \item $marked^{(i)} = mark(x^{(i)}_{\frac l 2}(f2s^{(i)}_{\frac l 2}),\mu^{(i)}(\T^{(i)})^2$,theta, nu$)$ + \item $\T^{(i+1)} = refineQuad(\T^{(i)},marked)$ +\end{enumerate} +Zum Plotten (\ref{exmplAA_2DQuad}) wird zusätzlich auch folgendes berechnet: +\begin{itemize} + \item $V^{(i)} = mex\_build\_AU(\T^{(i)},$ vollAnalytisch $)$ + \item $b^{(i)} = \left\{b_k=size(T_k) | T_k\in\T^{(i)}, k \in \{1,2 \dots\#\T^{(i)}\} \right\}$ + \item Löse: $V^{(i)} \cdot x^{(i)} = b^{(i)}$ + \item $\enorm{\phi^{(i)}_{\frac l 2}}^2 = x^{(i)\prime}_{\frac l 2}\cdot V^{(i)}_{\frac l 2} \cdot x^{(i)}_{\frac l 2}$ + \item $\enorm{\phi^{(i)}}^2 = x^{(i)\prime}\cdot V^{(i)} \cdot x^{(i)}$ + \item $error = \sqrt{\enorm{\phi}^2 - \enorm{\phi^{(i)}}^2}$ +% \item $error_2 = \sqrt{\enorm{\phi} - \enorm{\phi^{(i)}_{\frac l 2}}}$ + \item $\eta = \sqrt{\enorm{x^{(i)}_{\frac l 2}(f2s^{(i)}_{\frac l 2})}^2 - \enorm{\phi^{(i)}}^2}$ +% \item $error_2 = \enorm{\phi^{(i)}_{\frac l 2}(f2s^{(i)}_{\frac l 2}) - \phi^{(i)}}$ + \item $\mu_2(\T^{(i)})^2 = \left\{\frac{ h_{min}(T_j) \abs{T_j}}{4} \sum_{k=1}^4{(x_{\frac l 2,j}^{(i)(k)}-x_{j}^{(i)})^2}, j \in \{1,2 \dots\#\T^{(i)}\} \right\}$ +\end{itemize} + +\begin{figure}[ht] +\caption{2D Quad adaptiv anisotrop vollanalytisch} +\centering +\label{exmplAA_2DQuad} +\subfloat[Fehler]{\includegraphics[width=0.5\textwidth]{fig/exmplAA_2DQuad_error}} +\subfloat[Energie Norm]{\includegraphics[width=0.5\textwidth]{fig/exmplAA_2DQuad_norm}} +\end{figure} + + +\end{document} \ No newline at end of file diff --git a/doc/doc1.dvi b/doc/doc1.dvi deleted file mode 100644 index 178d6b902372b62040ca5a4a4670ae665cbb35a2..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 20496 zcmeHPdvF`Y89&NCY|YvFokLVaEqs<6o~un-b1gG 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-\usepackage{graphicx,psfrag,subfig} %Grafiken einbinden/Texte ersetzen/Bilder nebeneinander -%\usepackage{ifthen} -%\usepackage{showkeys} -%\usepackage[final,numbered]{mcode} -\usepackage{color} %Farben benutzen und Definieren -\usepackage{hyperref} %Links im Inhaltsverzeichnis -\hypersetup{linkbordercolor={1 1 1},citebordercolor={1 1 1},urlbordercolor={1 1 1}} - - -\usepackage[ngerman]{babel} %Sprachpacket für Überschriften -\usepackage[utf8]{inputenc} %Eingabekodierung -\usepackage{fixltx2e} %Deutschsprach Bugs - - -\def\todo#1{\textcolor{red}{#1}} -\def\Matlab{{\sc Matlab}} -\def\q{\Q} - -\def\Ea{$E$} -\def\Eb{$\tilde E$} -\def\v{\boldsymbol{v}} -\def\D{\mathcal{D}} -\def\Q{\mathcal{Q}} -\def\G{\mathcal{G}} -\def\L{\mathcal{L}} - -\newcommand{\dif}[2]{\frac{\partial#1}{\partial#2}} -\newcommand{\enorm}[1]{\left| \! \left| \! \left|#1\right| \! \right| \! \right|} -\newcommand{\norm}[1]{\lVert#1\rVert} - -\def\N{\mathbb{N}} -\def\R{\mathbb{R}} - -\date{29.08.2011} -\author{P. Schaefer} - -\begin{document} -\tableofcontents -\clearpage -\chapter{Einleitung} - -\section{Allgemein} -Gelöst werden soll eine Einfachschichtpotential Gleichung. -\begin{displaymath} -V\phi = g -\end{displaymath} -Macht man nun den Galerkin-Ansatz mit $P^0(\tau_h)$ -\begin{displaymath} - \phi \approx \phi _h = \sum_{k=1}^N x_k \chi_k \text{ wobei } \tau_h = \{\tau_1,\dots,\tau_N\}\text{ und }\chi_k = \text{ char. Fkt. zu }\tau_k -\end{displaymath} -\begin{eqnarray*} -&& = _{L^2(\Gamma)} \text{~~~} \forall \psi_h \in P^0(\tau_h)\\ -&&{\chi_1,\dots,\chi_N}\subseteq P^0(\tau_h) -\end{eqnarray*} - - - - -\section{Problem} -Es seien \Ea, \Eb ~$\subseteq$~ $\R^3$ zwei beschränkte, achsenorientierte rechteckige Seiten in $\R^3$. -Berechnet werden soll: -\begin{eqnarray*} --\frac{1}{4\pi} \int_E \int_{\tilde E} \frac{1}{|x-y|} ds_y ds_x \in \R^3 -\end{eqnarray*} - - -\section{Gitter} -Alle Gitter bestehen nur aus achsenorientierten Rechtecken. -Entscheidend für die Tests sind Gitter in einer Ebene und Körper... - -\chapter{Analytische Berechnung der Integrale} - -\section{einfach Integral} -Sei: -\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. -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*} -Für $p \in \{1/2 , 0 , -1/2 , -1, -3/2\}$ gilt explizit: -\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|}}{ \lambda}\\ - g(-3/2;y;x;\lambda) &:=& \frac{y-x}{\lambda^2}\{(y-x)^2+\lambda^2\}^{-1/2} -\end{eqnarray*} - - -\section{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*} -Im Zuge der Berechnungen werden auch hier nur die Funktionen für bestimmte Parameter benötigt, die Untersuchung der restlichen werden wir hier nicht durchführen. - -\section{Integral über zwei Elemente} -Bei der Integration über zwei Seitenelemente \Ea, \Eb~ haben wir geometrisch zischen zwei Fällen unterschieden. Entweder die beiden Elemente liegen in parallelen Ebenen oder in orthogonalen Ebenen. - -\subsection{Parallele Elemente} -Liegen die beiden Elemente parallel zueinander lassen sie sich Folgendermaßen darstellen: -Sei: \Ea = $ \v + [(0,l_1) \times (0,l_2) \times \{0\}]$ und \Eb = $ \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: -\begin{eqnarray*} -&&-\frac{1}{4\pi} \int_E \int_{\tilde E} \frac{1}{|x-y|} ds_y ds_x\\ -&=&-\frac{1}{4\pi} \int_E \int_{\tilde E} - \left ( (x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2 \right)^{-1/2} ds_y ds_x\\ -&=&-\frac{1}{4\pi} \int_0^{l_1}\int_0^{l_2}\int_0^{\tilde l_1}\int_0^{\tilde l_2} - \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} - dy_2 dy_1 dx_2 dx_1\\ -&=&-\frac{1}{4\pi} \int_0^{l_1}\int_0^{l_2}\int_0^{\tilde l_1}\int_0^{\tilde l_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*} -Entscheidend ist also die Stammfunktion $F_{par}$ -\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*} - -\subsection{Orthogonale Elemente} -\Ea = $ \v + [(0,l_1) \times (0,l_2) \times \{0\}]$\\ -\Eb = $ \tilde \v + [ \{0\} \times (0,\tilde l_2) \times (0,\tilde l_3)]$\\ -$\v,\tilde \v \in \R^3$\\ -$\boldsymbol{\delta} = \tilde \v - \v \in \R^3$\\ -\begin{eqnarray*} -&&-\frac{1}{4\pi} \int_E \int_{\tilde E} \frac{1}{|x-y|} ds_y ds_x\\ -&=&-\frac{1}{4\pi} \int_E \int_{\tilde E} - \left ( (x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2 \right)^{-1/2} ds_y ds_x\\ -&=&-\frac{1}{4\pi} \int_0^{l_1}\int_0^{l_2}\int_0^{\tilde l_2}\int_0^{\tilde l_3} - \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} - dy_3 dy_2 dx_2 dx_1\\ -&=&-\frac{1}{4\pi} \int_0^{l_1}\int_0^{l_2}\int_0^{\tilde l_2}\int_0^{\tilde l_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*} -Entscheidend ist also die Stammfunktion $F_{ort}$ -\begin{eqnarray*} -F_{ort}(x_1,x_2,y_1,y_2,\delta_1,\delta_2,\delta_3) &:=& \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*} - -\section{Bestimmtes Integral} -\begin{eqnarray*} -&&-\frac{1}{4\pi} \int_E \int_{\tilde E} \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\\ -% -&=& -\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) -- -\dif{}{y_1} \dif{}{x_2} \dif{}{x_1} - 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} - \dif{}{x_2} \dif{}{x_1} - F_{par/ort}(x_1,x_2,\tilde k_1,\tilde k_2) -- - \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) -+ - \dif{}{x_2} \dif{}{x_1} - F_{par/ort}(x_1,x_2,0,0)\big) - dx_2 dx_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) -- -\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) -- -\dif{}{x_1} - F_{par/ort}(x_1,0,0,0)\big) - dx_1\\ -% -&=& -\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) -- - 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) -+ - 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) -- - 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) -+ - F_{par/ort}(0,0,0,\tilde k_2) -- - F_{par/ort}(0,0,0,0)\big) -\end{eqnarray*} - - - -\chapter{Semi-analytische Berechnung bestimmter Integrale} -\section{Gauss-Quadratur} -\begin{displaymath} - \int_0^l f(x) dx \approx \sum_{k=0}^n w_kf(x_k) =: \Q(f) -\end{displaymath} - -\section{Quadratur über ein Element} -\begin{displaymath} - \int_{E} \int_{\tilde E} \approx \q_{E} \int_{\tilde E} -\end{displaymath} -Das Integral über \Eb~ kann durch die analytische Doppelintegral Funktion ersetzt werden. Die Quadratur über \Ea~ wird nun mittels einer doppelten Gauss-Quadratur berechnet. -\subsection{Zulässigkeitsbedingung} -\begin{displaymath} - \D (E, \tilde E) \geq \mu \min\{ \G(E) , \G(\tilde E)\} -\end{displaymath} -Nur wenn die Zulässigkeitsbedingung gültig ist wird die Quadratur verwendet. Sonst wird Analytisch gerechnet. - - -\section{Quadratur über eine Achse} -\begin{displaymath} - \int_{E} \int_{\tilde E} \approx \q_{I_1} \q_{J_1} \int_{I_2}\int_{J_2} -\end{displaymath} -\subsection{Zulässigkeitsbedingung} -\begin{displaymath} - \D (I_1, J_1) \geq \mu \max\{ \L(I_2) , \L(J_2)\} -\end{displaymath} - - -\section{Quadratur über eine Seite} -\begin{displaymath} - \int_{E} \int_{\tilde E} \approx \q_{I_1} \int_{I_2} \int_{J_1} \int_{J_2} -\end{displaymath} - -$\enorm{a}-\norm{b}$ - - -\chapter{Zulässigkeitsbedingungen} -\section{Lagen der Elemente} -so so oder so... -\section{Fehlerschätzer} -\section{Möglichkeiten für Kriterien} - - - -\end{document} diff --git a/doc/doc2.pdf b/doc/doc2.pdf deleted file mode 100644 index 948f3aa6cdbbf7a7ea76bc1a560c9dc91a8c309b..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 202690 zcmc%x1zeO(_dkvc(%l_Phq^4Vv?ATz-Ai|;bc50$tzgmJARr-~(%sT1t;p{}y`S6L 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Version 7.13.0.564 (R2011b). Operating System: Linux 3.0.0-16-generic #29-Ubuntu SMP Tue Feb 14 12:48:51 UTC 2012 x86_64. +%%Title: ./exmplAA_2DQuad_norm.eps +%%CreationDate: 03/18/2012 16:07:35 +%%DocumentNeededFonts: Helvetica +%%DocumentProcessColors: Cyan Magenta Yellow Black +%%Extensions: CMYK +%%Pages: 1 +%%BoundingBox: 44 196 550 602 +%%EndComments + +%%BeginProlog +% MathWorks dictionary +/MathWorks 160 dict begin +% definition operators +/bdef {bind def} bind def +/ldef {load def} bind def +/xdef {exch def} bdef +/xstore {exch store} bdef +% operator abbreviations +/c /clip ldef +/cc /concat ldef +/cp /closepath ldef +/gr /grestore ldef +/gs /gsave ldef +/mt /moveto ldef +/np /newpath ldef +/cm /currentmatrix ldef +/sm /setmatrix ldef +/rm /rmoveto ldef +/rl /rlineto ldef +/s {show newpath} bdef +/sc {setcmykcolor} bdef +/sr /setrgbcolor ldef +/sg /setgray ldef +/w /setlinewidth ldef +/j /setlinejoin ldef +/cap /setlinecap ldef +/rc {rectclip} bdef +/rf {rectfill} bdef +% page state control +/pgsv () def +/bpage {/pgsv save def} bdef +/epage {pgsv restore} bdef +/bplot /gsave ldef +/eplot {stroke grestore} bdef +% orientation switch +/portraitMode 0 def /landscapeMode 1 def /rotateMode 2 def +% coordinate system mappings +/dpi2point 0 def +% font control +/FontSize 0 def +/FMS {/FontSize xstore findfont [FontSize 0 0 FontSize neg 0 0] + makefont setfont} bdef +/reencode {exch dup where {pop load} {pop StandardEncoding} ifelse + exch dup 3 1 roll findfont dup length dict begin + { 1 index /FID ne {def}{pop pop} ifelse } forall + /Encoding exch def currentdict end definefont pop} bdef +/isroman {findfont /CharStrings get /Agrave known} bdef +/FMSR {3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse + exch FMS} bdef +/csm {1 dpi2point div -1 dpi2point div scale neg translate + dup landscapeMode eq {pop -90 rotate} + {rotateMode eq {90 rotate} if} ifelse} bdef +% line types: solid, dotted, dashed, dotdash +/SO { [] 0 setdash } bdef +/DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef +/DA { [6 dpi2point mul] 0 setdash } bdef +/DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 + dpi2point mul] 0 setdash } bdef +% macros for lines and objects +/L {lineto stroke} bdef +/MP {3 1 roll moveto 1 sub {rlineto} repeat} bdef +/AP {{rlineto} repeat} bdef +/PDlw -1 def +/W {/PDlw currentlinewidth def setlinewidth} def +/PP {closepath eofill} bdef +/DP {closepath stroke} bdef +/MR {4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto + neg 0 exch rlineto closepath} bdef +/FR {MR stroke} bdef +/PR {MR fill} bdef +/L1i {{currentfile picstr readhexstring pop} image} bdef +/tMatrix matrix def +/MakeOval {newpath tMatrix currentmatrix pop translate scale +0 0 1 0 360 arc tMatrix setmatrix} bdef +/FO {MakeOval stroke} bdef +/PO {MakeOval fill} bdef +/PD {currentlinewidth 2 div 0 360 arc fill + PDlw -1 eq not {PDlw w /PDlw -1 def} if} def +/FA {newpath tMatrix currentmatrix pop translate scale + 0 0 1 5 -2 roll arc tMatrix setmatrix stroke} bdef +/PA {newpath tMatrix currentmatrix pop translate 0 0 moveto scale + 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill} bdef +/FAn {newpath tMatrix currentmatrix pop translate scale + 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke} bdef +/PAn {newpath tMatrix currentmatrix pop translate 0 0 moveto scale + 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill} bdef +/vradius 0 def /hradius 0 def /lry 0 def +/lrx 0 def /uly 0 def /ulx 0 def /rad 0 def +/MRR {/vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef + /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly + vradius add translate hradius vradius scale 0 0 1 180 270 arc + tMatrix setmatrix lrx hradius sub uly vradius add translate + hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix + lrx hradius sub lry vradius sub translate hradius vradius scale + 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub + translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix + closepath} bdef +/FRR {MRR stroke } bdef +/PRR {MRR fill } bdef +/MlrRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def + newpath tMatrix currentmatrix pop ulx rad add uly rad add translate + rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad + sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix + closepath} bdef +/FlrRR {MlrRR stroke } bdef +/PlrRR {MlrRR fill } bdef +/MtbRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def + newpath tMatrix currentmatrix pop ulx rad add uly rad add translate + rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad + sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix + closepath} bdef +/FtbRR {MtbRR stroke } bdef +/PtbRR {MtbRR fill } bdef +/stri 6 array def /dtri 6 array def +/smat 6 array def /dmat 6 array def +/tmat1 6 array def /tmat2 6 array def /dif 3 array def +/asub {/ind2 exch def /ind1 exch def dup dup + ind1 get exch ind2 get sub exch } bdef +/tri_to_matrix { + 2 0 asub 3 1 asub 4 0 asub 5 1 asub + dup 0 get exch 1 get 7 -1 roll astore } bdef +/compute_transform { + dmat dtri tri_to_matrix tmat1 invertmatrix + smat stri tri_to_matrix tmat2 concatmatrix } bdef +/ds {stri astore pop} bdef +/dt {dtri astore pop} bdef +/db {2 copy /cols xdef /rows xdef mul dup 3 mul string + currentfile exch readhexstring pop + dup 0 3 index getinterval /rbmap xdef + dup 2 index dup getinterval /gbmap xdef + 1 index dup 2 mul exch getinterval /bbmap xdef pop pop}bdef +/it {gs np dtri aload pop moveto lineto lineto cp c + cols rows 8 compute_transform + rbmap gbmap bbmap true 3 colorimage gr}bdef +/il {newpath moveto lineto stroke}bdef +currentdict end def +%%EndProlog + +%%BeginSetup +MathWorks begin + +0 cap + +end +%%EndSetup + +%%Page: 1 1 +%%BeginPageSetup +%%PageBoundingBox: 44 196 550 602 +MathWorks begin +bpage +%%EndPageSetup + +%%BeginObject: obj1 +bplot + +/dpi2point 8.33333 def +portraitMode 0150 5100 csm + + 223 79 4216 3387 MR c np +85 dict begin %Colortable dictionary +/c0 { 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zgmb9>c)@X4ECvgAg|CcZa7d`Q#d*PLKgaMGEL1xB>lg;B02O-vHil99$66#31(hNH zPRA*r{cQ}V2$c-~Hm0Zul_dWJ{dfH@@qsnD*2F5{@tSyynmPul sfzw8#k%~wyv=&-P6R$=5?=l!NkS2jJ09^4`==Fo&W#< literal 0 HcmV?d00001 diff --git a/src/A_plots.m b/src/A_plots.m index 2eb5c15..f86082c 100644 --- a/src/A_plots.m +++ b/src/A_plots.m @@ -41,7 +41,7 @@ for i = 1:length(files) l1 = {type2str{data(1,[2+(0:step-1)*rows])}}; for i = 1:step leg0 = {leg0{:} ['\mu ' l0 l1{i}] ['\eta ' l0 l1{i}]... - ['error ' l0 l1{i}] ['\mu2 ' l0 l1{i}] ['error2 ' l0 l1{i}]}'; + ['error ' l0 l1{i}] ['\mu2 ' l0 l1{i}]}'; leg1 = {leg1{:} [ l0 l1{i}]}'; sym = {sym{:} type2sym{data(1,[2+(i-1)*rows])}}' end @@ -64,11 +64,10 @@ else % G_D figure(4) i=0; -loglog(X(:,[2+(0:rows-2)]+rows*i),[G_D(:,2+rows*i) ... +loglog(X(:,[2+(0:rows-3)]+rows*i),[G_D(:,2+rows*i) ... G_D(:,2+1+rows*i)...*G_D(1,2)/G_D(1,2+1+rows*i) ... sqrt(abs(sol - G_D(:,2+2+rows*i)))...*G_D(1,2)/sqrt(abs(sol - G_D(1,2+2+rows*i)))... G_D(:,2+3+rows*i)...*G_D(1,2)/G_D(1,2+3+rows*i) ... - G_D(:,2+4+rows*i)...*G_D(1,2)/G_D(1,2+3+rows*i) ... ],sym{i+1}); hold on for i = 1:step-1 diff --git a/src/A_step.m b/src/A_step.m index cf7e677..32e5924 100644 --- a/src/A_step.m +++ b/src/A_step.m @@ -97,7 +97,7 @@ time = zeros(1,3); - data = [data type(i) sqrt(sum(ind)) sqrt(ind2) xe_fine sqrt(sum(ind3)) sqrt(abs(xe-xe_fine))]; + data = [data type(i) sqrt(sum(ind)) sqrt(ind2) xe_fine sqrt(sum(ind3))]; end time(2) = toc; diff --git a/src/export_mesh.m b/src/export_mesh.m index ff4f5b2..67227f7 100644 --- a/src/export_mesh.m +++ b/src/export_mesh.m @@ -1,6 +1,6 @@ function export_mesh(coo, ele, nei, f2s, file) -plotShape(coo,ele,'tb');view(2); +plotShape(coo,ele,'db');view(2); print('-r600','-depsc',['../doc/fig/' file '_ref.eps']) %% Koordinaten diff --git a/src/mark.m b/src/mark.m index 458ee49..17ba062 100644 --- a/src/mark.m +++ b/src/mark.m @@ -1,9 +1,9 @@ -function REF = mark(xF2S,ind,theta,eta) -% function REF = mark(xF2S,ind,theta,eta) +function REF = mark(xF2S,ind,theta,nu) +% function REF = mark(xF2S,ind,theta,nu) % xF2S - Father son relation % ind - error estimator % theta - refine element? (0..1, 1 = All) -% eta - refine how? (0...1, 0 = Isotrop) +% nu - refine how? (0...1, 0 = Isotrop) % REF - vector with entries [1 : 4] if(size(xF2S,1)==1) @@ -34,10 +34,10 @@ end %% Wie muss verfeinert werden -if(eta > 0) % Horizontal oder Vertikal - t3 = (eta*abs(Ct(3,:)) >= sqrt(Ct(2,:).^2+Ct(4,:).^2)); +if(nu > 0) % Horizontal oder Vertikal + t3 = (nu*abs(Ct(3,:)) >= sqrt(Ct(2,:).^2+Ct(4,:).^2)); REF(t3-t1==1) = 3; - t4 = (eta*abs(Ct(4,:)) >= sqrt(Ct(2,:).^2+Ct(3,:).^2)); + t4 = (nu*abs(Ct(4,:)) >= sqrt(Ct(2,:).^2+Ct(3,:).^2)); REF(t4-t1==1) = 4; end REF(~(t4+t3+t1)) = 2; % Rest wird Horizontal UND Vertikal geteilt diff --git a/src/plotShape.m b/src/plotShape.m index 8273433..d2f0d56 100644 --- a/src/plotShape.m +++ b/src/plotShape.m @@ -10,6 +10,8 @@ function plotShape(coordinates, elements, varargin) % n -> Normen auf ein Element einzeichnen (mit Laenge 1) % a -> Normen auf ein Element einzeichnen (Laenge durch Flaecheninhalt) % s -> Flaechen werden gefaerbt, wobei VEC die Farben der Elemente angibt +% t -> Element Text wird angezeigt (oder Element Index) +% d -> Koordinaten Text wird angezeigt (oder Koordinaten Index) % % P.Schaefer @@ -56,12 +58,16 @@ elseif(optargin>=1) n = 2; elseif(ismember('n',varargin{1})) n = 1; - elseif(ismember('t',varargin{1})) + end + if(ismember('t',varargin{1})) desc = {}; t = 1; if(optargin==2 && length(varargin{2})==size(elements,1)) desc = varargin{2}; end + elseif(ismember('d',varargin{1})) + desc = {}; + t = 2; end end @@ -74,6 +80,7 @@ end % [b eles] = unique(elements,'rows'); %% Flächen eles = 1:size(elements,1); +coos = 1:size(coordinates,1); if(e==1) for idx = eles @@ -127,9 +134,26 @@ if(t) cola = 'bla'; end if(isempty(desc)) - text(current(1),current(2),current(3),num2str(idx),'color',cola); + text(current(1),current(2),current(3),['(' num2str(idx) ')'],'color',cola); + else + text(current(1),current(2),current(3),['(' desc{idx} ')'],'color',cola); + end + hold on + end + end + +if(t>=2) + for idx = coos + current = coordinates(idx,:); + if(e) + cola = 'w'; + else + cola = 'r'; + end + if(isempty(desc)) + text(current(1)+0.01,current(2)+0.03,current(3)+0.01, num2str(idx) ,'color',cola); else - text(current(1),current(2),current(3),desc{idx},'color',cola); + text(current(1)+0.01,current(2)+0.03,current(3)+0.01,desc{idx},'color',cola); end hold on end @@ -145,6 +169,19 @@ end % hold on % end +xdif = max(coordinates(:,1)) - min(coordinates(:,1)); +ydif = max(coordinates(:,2)) - min(coordinates(:,2)); +zdif = max(coordinates(:,3)) - min(coordinates(:,3)); + +if(ydif) + ylim([min(coordinates(:,1))-ydif/10 max(coordinates(:,1))+ydif/10]) +end +if(xdif) + xlim([min(coordinates(:,2))-xdif/10 max(coordinates(:,2))+xdif/10]) +end +if(zdif) + zlim([min(coordinates(:,3))-zdif/10 max(coordinates(:,3))+zdif/10]) +end xlabel 'x' ylabel 'y' zlabel 'z' -- 2.47.3