\caption{#1}
\label{#2}
\centering
-\subfloat[Lage]{\includegraphics[width=0.5\textwidth]{fig/#2_ref}}
+\subfloat[Lage]{\includegraphics[width=0.55\textwidth]{fig/#2_ref}}
\subfloat[Koordinaten]{\input{fig/#2_coo}}\\
\subfloat[Elemente]{\input{fig/#2_ele}}
\subfloat[Nachbarn]{\input{fig/#2_nei}}
\subsection{Netze}
Für die Diskretisierung des Problems wollen wir nun einige Begriffe definieren.
-\begin{defi}[Partition]
+\begin{defi}[Rechteck]
Wir nennen $T \subseteq \R^3$ ein achsenorientiertes Rechteck, wenn sich aus einem Eckpunkt $C \in \R^3$, alle weiteren Knoten schreiben lassen als $\{ C + a\cdot A, C + a \cdot A + b \cdot B , C + b \cdot B\}$, wobei Vektoren $A,B \in \{(1,0,0)^T,(0,1,0)^T,(0,0,1)^T\}$. und Skalare $a,b\neq0$ sind und der Flächeninhalt $|T| > 0$. \todo{Kanten/Seiten definieren? Anmerken, dass ein Kante als Vektor nur eine dim $\neq 0$?}\\Weiterhin sei
\begin{align}
C_T &:= \{ c_i:1\leq i \leq 4\}
die Menge der Eckpunkte des Rechtecks.
\end{defi}
-\begin{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 $\abs{T_j \cap T_k}$ ist entweder ein Knoten, eine Seite oder keines von beiden mit $T_j,T_k\in\T_{\ell}$ und $T_j\neq T_k$
+ \item $T_j \cap T_k$ ist entweder ein Knoten, eine Seite oder keines von beiden mit $T_j,T_k\in\T_{\ell}$ und $T_j\neq T_k$
+ \item auf einer Kante liegt maximal ein hängender Knoten
\end{itemize}
\end{defi}
-Den Begriff der Triangulierung wollen wir hier noch stärker definieren, da wir pro Seite eines Rechtecks nur einen hängenden Knoten zulassen wollen.
-\begin{defi}
- Eine Triangulierung $\T$ von $\Gamma$ sei gültig, wenn auf jeder Kante eines Rechtecks maximal ein Knoten liegt. Daraus folgt, dass pro Seite eines Rechtecks
+Daraus folgt, dass pro Seite eines Rechtecks
\begin{itemize}
\item entweder kein Nachbarelement
\item oder genau ein Nachbarelement
\item oder genau zwei Nachbarelemente
\end{itemize}
anliegen können. Weiterhin wollen wir auch verlangen, dass der auf der Kante liegende Knoten die Seite halbiert, die beiden Nachbarlemente also die gleiche Seitenlänge an der anliegenden Kante haben.
-\end{defi}
-\todo{Reguläres Netz auch definieren?}
-\todo{+Bild}
+\begin{figure}[ht]
+\centering
+\label{fig_net}
+\subfloat[gültige Partition]{\includegraphics[width=0.3\textwidth]{fig/net1}}
+\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}
+\begin{figure}[ht]
+\centering
+\label{fig_refType}
+\subfloat[Element]{\includegraphics[width=0.25\textwidth]{fig/refType_full}}
+\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}
+\end{figure}
\begin{defi}
- Sei $\T_{\ell/2}$ die aus der Verfeinerung von $\T_{\ell}$ resultierende gültige Triangulierung.
+ Sei $\T_{\ell/2}$ die aus der Verfeinerung von $\T_{\ell}$ resultierende gültige Partition.
\end{defi}
-\begin{alg}[Verfeinern] \label{alg:refine} Sei $\T_{\ell} = \{T_1,T_2,\dots,T_N\}$ eine gegebenes Netz und $marked$ eine gegebene Markierung wobei $marked_j \hat =$ Markierung von $T_j$. Nun sei $j = 1$ und gehe so vor:
+\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}
-
- markiere zusätzlich den Nachbarn und gehe zu Schritt \ref{alg:refine:last} \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
- \item $j \mapsto j+1$, gehe zu Schritt \ref{alg:refine:checkHN} \label{alg:refine:last}
+ 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}
\end{enumerate}
\end{alg}
\subsection{Fehlerschätzer}
-Um eine Aussage über die Genauigkeit der Galerkinapproximation zu erhalten ist es sinnvoll, dass wir einen A-Posteriori Fehlerschätzer definieren.
+In diesem Abschnitt definieren wir die a-posteriori Fehlerschätzer, die wir im Folgenden zur Steuerung des adaptiven Algorithmus einsetzen werden. Wir verwenden dazu die $h-h/2$ Strategie aus Ferraz-Leite (Provet. 108),wo die folgende Aussage bewiesen wird.
+
\begin{defi}[A-Posteriori Fehlerschätzer $\mu$]
sollte
\begin{enumerate}
\subsection{Datenstruktur}
Für die Implementierung in \Matlab~und C++ wollen wollen wir eine einheitliche Datenstruktur einführen.
-Die für die Triangulierung $\mathcal{T}_{\ell} = \{T_1\ldots T_M\}$ benötigen Knoten $\mathcal{K}_{\ell} = \{C_1\ldots C_N\}$ stellen wir in einer $ N \times 3$ Matrix dar. Dabei enthält die $j$-te Zeile die Koordinaten des Knoten $C_j$ im $\R^3$.
+Die für die Partition $\mathcal{T}_{\ell} = \{T_1\ldots T_M\}$ benötigen Knoten $\mathcal{K}_{\ell} = \{C_1\ldots C_N\}$ stellen wir in einer $ N \times 3$ Matrix dar. Dabei enthält die $j$-te Zeile die Koordinaten des Knoten $C_j$ im $\R^3$.
\begin{displaymath}
COO[j,1:3] = C_j := (x_j,y_j,z_j)^{T} \text{ wobei } x,y,z \in \R
\end{displaymath}
Die Knoten wollen wir gegen den Uhrzeigersinn anordnen und der Knoten $C_j$ soll der kleinste bezüglich der Koordinaten sein.
\noindent
-Für die bessere Handhabung der Elemente beim Verfeinern der Triangulierung, 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.
+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}
NEI[i,1:8] = N_i := (n_1,\ldots,n_8)
\end{displaymath}
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{exmpl3:nei:part})
+(Siehe Abb.:\ref{exmpl13:nei:part})
\begin{figure}[ht]
-\caption{Nachbarschaftsrelationen Element 4 aus Abb.\ref{exmpl3}}
-% \label{exmpl3:nei:part}
+\caption{Nachbarschaftsrelationen Element 4 aus Abb.\ref{exmpl13}}
+% \label{exmpl13:nei:part}
\centering
\subfloat[Lage]{\includegraphics{fig/Net_Neigh}}
\subfloat[Nachbarn]{\input{fig/exmpl3_nei_part}}
\end{figure}
\noindent
-Ein ausführliches Beispiel ist in Abb.\ref{exmpl3} dargestellt.
-\showMesh[Beispiel 1.3]{exmpl3}
+Ein ausführliches Beispiel ist in Abb.\ref{exmpl13} dargestellt.
+
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})
+(Siehe Figur:\ref{exmpl13:f2s})
$[\T_{\ell/2}, F2S ] = refineQuad(\T_{\ell}, marked);$\\
$[COO_{fine}, ELE_{fine}, NEI_{fine}, F2S ] = refineQuad(COO, ELE, NEI, marked);$
\begin{figure}[ht]
\caption{VaterSohn}
\centering
-\subfloat[VaterSohn aus \ref{exmpl2}]{\input{fig/exmpl2_f2s}}
-\subfloat[VaterSohn aus \ref{exmpl3}]{\input{fig/exmpl3_f2s}}
+\subfloat[VaterSohn aus \ref{exmpl12}]{\input{fig/exmpl12_f2s}}
+\subfloat[VaterSohn aus \ref{exmpl13}]{\input{fig/exmpl13_f2s}}
\end{figure}
\item SemiAnalytisch über Seite
\end{itemize}
\item Fehlerschätzer untersuchen und prüfen ob sie sich wie erwartet Verhalten
-
\end{itemize}
+\showMesh[Beispiel 1.1]{exmpl11}
+\showMesh[Beispiel 1.2]{exmpl12}
+\showMesh[Beispiel 1.3]{exmpl13}
+
+
+\begin{figure}[ht]
+\caption{2D Quad adaptiv anisotrop vollanalytisch $V\phi = 1$}
+\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}
+
\section{Anhang Code}
\todo{
Die wichtigsten Funktionen
\end{enumerate}
}
-\begin{figure}[ht]
-\caption{2D Quad adaptiv anisotrop vollanalytisch $V\phi = 1$}
-\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}
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--- /dev/null
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+ 1 & 2 & 5 & 6 & 3\\
+ 2 & 5 & 8 & 9 & 6\\
+ 3 & 4 & 7 & 8 & 5\\
+ 4 & 1 & 4 & 5 & 2
+\end{tabular}
+\label{exmpl11:ele}
\ No newline at end of file
--- /dev/null
+\begin{tabular}{>{\columncolor{gray}}rcccc}
+\rowcolor{gray}
+ Index & e1 & e2 & e3 & e4\\
+\end{tabular}
+\label{exmpl11:f2s}
\ No newline at end of file
--- /dev/null
+\begin{tabular}{>{\columncolor{gray}}rcccccccc}
+\rowcolor{gray}
+ Index & n1 & n2 & n3 & n4 & n5 & n6 & n7 & n8\\
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+ 2 & 3 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\
+ 3 & 0 & 0 & 2 & 4 & 0 & 0 & 0 & 0\\
+ 4 & 0 & 3 & 1 & 0 & 0 & 0 & 0 & 0
+\end{tabular}
+\label{exmpl11:nei}
\ No newline at end of file
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+
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--- /dev/null
+\begin{tabular}{>{\columncolor{gray}}rccc}
+\rowcolor{gray}
+ Index & x1 & x2 & x3\\
+ 1 & 0 & 0 & 0\\
+ 2 & 0 & 0.25 & 0\\
+ 3 & 0 & 0.5 & 0\\
+ 4 & 0 & 0.75 & 0\\
+ 5 & 0 & 1 & 0\\
+ 6 & 0.25 & 0.5 & 0\\
+ 7 & 0.25 & 0.75 & 0\\
+ 8 & 0.25 & 1 & 0\\
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+ 10 & 0.5 & 0.25 & 0\\
+ 11 & 0.5 & 0.5 & 0\\
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+ 13 & 0.5 & 1 & 0\\
+ 14 & 1 & 0 & 0\\
+ 15 & 1 & 0.5 & 0\\
+ 16 & 1 & 0.75 & 0\\
+ 17 & 1 & 1 & 0
+\end{tabular}
--- /dev/null
+\begin{tabular}{>{\columncolor{gray}}rcccc}
+\rowcolor{gray}
+ Index & c1 & c2 & c3 & c4\\
+ 1 & 4 & 7 & 8 & 5\\
+ 2 & 12 & 16 & 17 & 13\\
+ 3 & 9 & 14 & 15 & 11\\
+ 4 & 2 & 10 & 11 & 3\\
+ 5 & 3 & 6 & 7 & 4\\
+ 6 & 11 & 15 & 16 & 12\\
+ 7 & 1 & 9 & 10 & 2\\
+ 8 & 7 & 12 & 13 & 8\\
+ 9 & 6 & 11 & 12 & 7
+\end{tabular}
+\label{exmpl12:ele}
\ No newline at end of file
--- /dev/null
+\begin{tabular}{>{\columncolor{gray}}rcccc}
+\rowcolor{gray}
+ Index & e1 & e2 & e3 & e4\\
+ 1 & 5 & 9 & 8 & 1\\
+ 2 & 6 & 6 & 2 & 2\\
+ 3 & 3 & 3 & 3 & 3\\
+ 4 & 7 & 7 & 4 & 4
+\end{tabular}
+\label{exmpl12:f2s}
\ No newline at end of file
--- /dev/null
+\begin{tabular}{>{\columncolor{gray}}rcccccccc}
+\rowcolor{gray}
+ Index & n1 & n2 & n3 & n4 & n5 & n6 & n7 & n8\\
+ 1 & 5 & 8 & 0 & 0 & 0 & 0 & 0 & 0\\
+ 2 & 6 & 0 & 0 & 8 & 0 & 0 & 0 & 0\\
+ 3 & 0 & 0 & 6 & 7 & 0 & 0 & 0 & 4\\
+ 4 & 7 & 3 & 5 & 0 & 0 & 0 & 9 & 0\\
+ 5 & 4 & 9 & 1 & 0 & 0 & 0 & 0 & 0\\
+ 6 & 3 & 0 & 2 & 9 & 0 & 0 & 0 & 0\\
+ 7 & 0 & 3 & 4 & 0 & 0 & 0 & 0 & 0\\
+ 8 & 9 & 2 & 0 & 1 & 0 & 0 & 0 & 0\\
+ 9 & 4 & 6 & 8 & 5 & 0 & 0 & 0 & 0
+\end{tabular}
+\label{exmpl12:nei}
\ No newline at end of file
--- /dev/null
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+ 4 & 0 & 0.625 & 0\\
+ 5 & 0 & 0.75 & 0\\
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+ 9 & 0.125 & 0.75 & 0\\
+ 10 & 0.25 & 0.25 & 0\\
+ 11 & 0.25 & 0.5 & 0\\
+ 12 & 0.25 & 0.625 & 0\\
+ 13 & 0.25 & 0.75 & 0\\
+ 14 & 0.25 & 1 & 0\\
+ 15 & 0.5 & 0 & 0\\
+ 16 & 0.5 & 0.25 & 0\\
+ 17 & 0.5 & 0.5 & 0\\
+ 18 & 0.5 & 0.75 & 0\\
+ 19 & 0.5 & 1 & 0\\
+ 20 & 1 & 0 & 0\\
+ 21 & 1 & 0.5 & 0\\
+ 22 & 1 & 0.75 & 0\\
+ 23 & 1 & 1 & 0
+\end{tabular}
--- /dev/null
+\begin{tabular}{>{\columncolor{gray}}rcccc}
+\rowcolor{gray}
+ Index & c1 & c2 & c3 & c4\\
+ 1 & 5 & 13 & 14 & 6\\
+ 2 & 18 & 22 & 23 & 19\\
+ 3 & 15 & 20 & 21 & 17\\
+ 4 & 2 & 10 & 11 & 3\\
+ 5 & 4 & 8 & 9 & 5\\
+ 6 & 17 & 21 & 22 & 18\\
+ 7 & 1 & 15 & 16 & 2\\
+ 8 & 13 & 18 & 19 & 14\\
+ 9 & 11 & 17 & 18 & 13\\
+ 10 & 3 & 7 & 8 & 4\\
+ 11 & 8 & 12 & 13 & 9\\
+ 12 & 10 & 16 & 17 & 11\\
+ 13 & 7 & 11 & 12 & 8
+\end{tabular}
+\label{exmpl13:ele}
\ No newline at end of file
--- /dev/null
+\begin{tabular}{>{\columncolor{gray}}rcccc}
+\rowcolor{gray}
+ Index & e1 & e2 & e3 & e4\\
+ 1 & 1 & 1 & 1 & 1\\
+ 2 & 2 & 2 & 2 & 2\\
+ 3 & 3 & 3 & 3 & 3\\
+ 4 & 4 & 12 & 12 & 4\\
+ 5 & 10 & 13 & 11 & 5\\
+ 6 & 6 & 6 & 6 & 6\\
+ 7 & 7 & 7 & 7 & 7\\
+ 8 & 8 & 8 & 8 & 8\\
+ 9 & 9 & 9 & 9 & 9
+\end{tabular}
+\label{exmpl13:f2s}
\ No newline at end of file
--- /dev/null
+\begin{tabular}{>{\columncolor{gray}}rcccccccc}
+\rowcolor{gray}
+ Index & n1 & n2 & n3 & n4 & n5 & n6 & n7 & n8\\
+ 1 & 11 & 8 & 0 & 0 & 5 & 0 & 0 & 0\\
+ 2 & 6 & 0 & 0 & 8 & 0 & 0 & 0 & 0\\
+ 3 & 0 & 0 & 6 & 7 & 0 & 0 & 0 & 12\\
+ 4 & 7 & 12 & 10 & 0 & 0 & 0 & 13 & 0\\
+ 5 & 10 & 11 & 1 & 0 & 0 & 0 & 0 & 0\\
+ 6 & 3 & 0 & 2 & 9 & 0 & 0 & 0 & 0\\
+ 7 & 0 & 3 & 4 & 0 & 0 & 0 & 12 & 0\\
+ 8 & 9 & 2 & 0 & 1 & 0 & 0 & 0 & 0\\
+ 9 & 12 & 6 & 8 & 13 & 0 & 0 & 0 & 11\\
+ 10 & 4 & 13 & 5 & 0 & 0 & 0 & 0 & 0\\
+ 11 & 13 & 9 & 1 & 5 & 0 & 0 & 0 & 0\\
+ 12 & 7 & 3 & 9 & 4 & 0 & 0 & 0 & 0\\
+ 13 & 4 & 9 & 11 & 10 & 0 & 0 & 0 & 0
+\end{tabular}
+\label{exmpl13:nei}
\ No newline at end of file
--- /dev/null
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-/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: 46 201 543 583
-MathWorks begin
-bpage
-%%EndPageSetup
-
-%%BeginObject: obj1
-bplot
-
-/dpi2point 8.33333 def
-portraitMode 0150 5100 csm
-
- 237 236 4142 3188 MR c np
-85 dict begin %Colortable dictionary
-/c0 { 0.000000 0.000000 0.000000 sr} bdef
-/c1 { 1.000000 1.000000 1.000000 sr} bdef
-/c2 { 0.900000 0.000000 0.000000 sr} bdef
-/c3 { 0.000000 0.820000 0.000000 sr} bdef
-/c4 { 0.000000 0.000000 0.800000 sr} bdef
-/c5 { 0.910000 0.820000 0.320000 sr} bdef
-/c6 { 1.000000 0.260000 0.820000 sr} bdef
-/c7 { 0.000000 0.820000 0.820000 sr} bdef
-c0
-1 j
-1 sg
- 0 0 4801 3603 PR
-4.16667 w
-0 1623 -1616 -744 0 -1622 4344 2635 4 MP
-PP
-1616 743 0 1623 -1616 -744 0 -1622 4344 2635 5 MP stroke
-1616 743 2105 -570 -1616 -743 2239 3205 4 MP
-PP
--2105 570 1616 743 2105 -570 -1616 -743 2239 3205 5 MP stroke
-0 1623 2105 -571 0 -1622 623 2462 4 MP
-PP
--2105 570 0 1623 2105 -571 0 -1622 623 2462 5 MP stroke
-2.77778 w
-DO
-SO
-4.16667 w
-0 sg
-2239 3205 mt 4344 2635 L
-2239 3205 mt 623 2462 L
- 623 2462 mt 623 840 L
-2414 3158 mt 2464 3181 L
-%%IncludeResource: font Helvetica
-/Helvetica /ISOLatin1Encoding 83.3333 FMSR
-
-2485 3268 mt
-(0) s
-2765 3063 mt 2814 3086 L
-2836 3173 mt
-(0.2) s
-3116 2968 mt 3165 2991 L
-3187 3078 mt
-(0.4) s
-3466 2873 mt 3516 2896 L
-3538 2983 mt
-(0.6) s
-3817 2778 mt 3867 2801 L
-3889 2888 mt
-(0.8) s
-4168 2683 mt 4218 2706 L
-4239 2793 mt
-(1) s
-2104 3144 mt 2051 3158 L
-1982 3241 mt
-(0) s
-1835 3020 mt 1782 3034 L
-1644 3117 mt
-(0.2) s
-1566 2896 mt 1513 2910 L
-1375 2993 mt
-(0.4) s
-1296 2772 mt 1244 2786 L
-1106 2870 mt
-(0.6) s
-1027 2648 mt 975 2662 L
- 836 2746 mt
-(0.8) s
- 758 2524 mt 705 2538 L
- 636 2622 mt
-(1) s
- 623 2462 mt 574 2440 L
- 457 2459 mt
-(-1) s
- 623 2057 mt 574 2034 L
- 388 2054 mt
-(-0.5) s
- 623 1651 mt 574 1628 L
- 506 1648 mt
-(0) s
- 623 1245 mt 574 1223 L
- 437 1243 mt
-(0.5) s
- 623 840 mt 574 817 L
- 506 837 mt
-(1) s
-gs 624 269 3721 2937 MR c np
-/c8 { 0.000000 0.000000 1.000000 sr} bdef
-c8
-673 310 -878 237 -673 -309 878 -238 1606 1975 5 MP stroke
-673 309 -877 238 -673 -310 877 -237 2484 1737 5 MP stroke
-672 310 -877 237 -673 -310 878 -237 3156 2047 5 MP stroke
-673 310 -878 238 -672 -310 877 -238 2279 2285 5 MP stroke
-gr
-
-c8
-0 sg
-1709 1731 mt
-(\(1\)) s
-2586 1494 mt
-(\(2\)) s
-3259 1804 mt
-(\(3\)) s
-2381 2041 mt
-(\(4\)) s
-/c9 { 1.000000 0.000000 0.000000 sr} bdef
-c9
-2257 2283 mt
-(1) s
-1584 1974 mt
-(2) s
- 911 1664 mt
-(3) s
-3134 2046 mt
-(4) s
-2461 1736 mt
-(5) s
-1788 1426 mt
-(6) s
-4011 1808 mt
-(7) s
-3338 1498 mt
-(8) s
-2665 1189 mt
-(9) s
-0 sg
- 331 1673 mt -90 rotate
-(z) s
-90 rotate
-3194 3389 mt
-(x) s
-1363 3346 mt
-(y) s
-
-end %%Color Dict
-
-eplot
-%%EndObject
-
-epage
-end
-
-showpage
-
-%%Trailer
-%%EOF
--- /dev/null
+%% Example 1
+
+load exmpl_2DQuad
+
+export_mesh(coordinates,elements,neigh,[],'exmpl11')
+
+
+marked = [2 3 1 3];
+[coordinates elements neigh f2s sites] =...
+ refineQuad(coordinates,elements,neigh,sites,marked);
+
+export_mesh(coordinates,elements,neigh,f2s,'exmpl12')
+
+marked = ones(1,9);
+marked(5) = 2;
+% marked([4,7]) = 3;
+[coordinates elements neigh f2s sites] =...
+ refineQuad(coordinates,elements,neigh,sites,marked);
+
+export_mesh(coordinates,elements,neigh,f2s,'exmpl13')
+
+%% Example Right&Wrong
+
+plotShape(coordinates,elements(:,:),'be');
+view(2);
+axis off;
+print('-r600','-depsc',['../doc/fig/net1.eps'])
+system(['epstopdf ../doc/fig/net1.eps']);
+
+load exmpl_2DQuad_wrong
+
+plotShape(coordinates,elements(:,:),'be');
+view(2);
+axis off;
+print('-r600','-depsc',['../doc/fig/net_wrong.eps'])
+system(['epstopdf ../doc/fig/net_wrong.eps']);
\ No newline at end of file
-function export_mesh(coo, ele, nei, f2s, file)
+function export_mesh(coordinates, elements, neigh, f2s, file)
-plotShape(coo,ele,'db');
+%% Daten
+save(file,'coordinates', 'elements','neigh','f2s')
+
+%% Übersicht
+plotShape(coordinates,elements,'db');
+
+if(length(unique(coordinates(:,3)))==1)
+ view(2)
+end
print('-r600','-depsc',['../doc/fig/' file '_ref.eps'])
system(['epstopdf ../doc/fig/' file '_ref.eps']);
fprintf(fid,'\\begin{tabular}{>{\\columncolor{gray}}rccc}\n');
fprintf(fid,'\\rowcolor{gray}\n Index & x1 & x2 & x3\\\\');
-[m n] =size(coo);
+[m n] =size(coordinates);
for i = 1:m
if(i~=1)
end
fprintf(fid,['\n ' int2str(i) ...
- ' & ' num2str(coo(i,1),4)...
- ' & ' num2str(coo(i,2),4)...
- ' & ' num2str(coo(i,3),4)]);
+ ' & ' num2str(coordinates(i,1),4)...
+ ' & ' num2str(coordinates(i,2),4)...
+ ' & ' num2str(coordinates(i,3),4)]);
end
fprintf(fid,'\n\\end{tabular}');
% fprintf(fid,['\n\\caption{' file ' - Koordinaten}']);
fprintf(fid,'\\begin{tabular}{>{\\columncolor{gray}}rcccc}\n');
fprintf(fid,'\\rowcolor{gray}\n Index & c1 & c2 & c3 & c4\\\\');
-[m n] =size(ele);
+[m n] =size(elements);
for i = 1:m
if(i~=1)
fprintf(fid,'\\\\');
end
- fprintf(fid,'\n %d & %d & %d & %d & %d',i,ele(i,1),ele(i,2),ele(i,3),ele(i,4));
+ fprintf(fid,'\n %d & %d & %d & %d & %d',i,elements(i,1),elements(i,2),elements(i,3),elements(i,4));
end
fprintf(fid,'\n\\end{tabular}');
% fprintf(fid,['\n\\caption{' file ' - Elemente}']);
fprintf(fid,'\\begin{tabular}{>{\\columncolor{gray}}rcccccccc}\n');
fprintf(fid,'\\rowcolor{gray}\n Index & n1 & n2 & n3 & n4 & n5 & n6 & n7 & n8\\\\');
-[m n] =size(nei);
+[m n] =size(neigh);
for i = 1:m
if(i~=1)
fprintf(fid,'\\\\');
end
- fprintf(fid,'\n %d & %d & %d & %d & %d & %d & %d & %d & %d',i,nei(i,1),nei(i,2),nei(i,3),nei(i,4),nei(i,5),nei(i,6),nei(i,7),nei(i,8));
+ fprintf(fid,'\n %d & %d & %d & %d & %d & %d & %d & %d & %d',i,neigh(i,1),neigh(i,2),neigh(i,3),neigh(i,4),neigh(i,5),neigh(i,6),neigh(i,7),neigh(i,8));
end
fprintf(fid,'\n\\end{tabular}');
% fprintf(fid,['\n\\caption{' file ' - Nachbarn}']);
--- /dev/null
+
+file = 'refType';
+input = 'exmpl_2DQuad2';
+load(input);
+
+plotShape(coordinates,elements(:,:),'bet',{'T'});
+
+view(2);
+axis off;
+% set(gcf, 'paperpositionmode', 'auto');
+print('-r600','-depsc',['../doc/fig/' file '_full.eps'])
+system(['epstopdf ../doc/fig/' file '_full.eps']);
+
+
+for i= 1:4
+
+load(input)
+
+[coordinates elements neigh f2s sites] = ...
+ refineQuad(coordinates,elements,neigh,sites,i);
+
+plotShape(coordinates,elements(:,:),'bet',{'T1','T2','T3','T4'});
+view(2);
+axis off;
+% set(gcf, 'paperpositionmode', 'auto');
+print('-r600','-depsc',['../doc/fig/' file '_' num2str(i) '.eps'])
+system(['epstopdf ../doc/fig/' file '_' num2str(i) '.eps']);
+
+end
\ No newline at end of file
% 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)
+% e -> thick lines for export
%
% P.Schaefer
e = 1;
n = 0;
t = 0;
+line = 1;
optargin = size(varargin,2);
if(optargin>2)
error('Zu viele Argumente');
% % error('plotShape:COLOR','Optionale Parameter fuer Faerbung sind Falsch')
% end
end
+
+ if(ismember('e',varargin{1}))
+ line = 2;
+ end
+
if(ismember('c',varargin{1}))
c = 1;
end
if(ismember('t',varargin{1}))
desc = {};
t = 1;
- if(optargin==2 && length(varargin{2})==size(elements,1))
+ if(optargin==2 && length(varargin{2})>=size(elements,1))
desc = varargin{2};
end
elseif(ismember('d',varargin{1}))
for idx = eles
current = coordinates(elements(idx,[1:4,1])',:);
% current(3,:) = current(3,:)-current(1,:)+current(2,:);
- plot3(current(:,1),current(:,2),current(:,3),'color',[0,100,100]/255); % Zeichnet nur Kanten
+ plot3(current(:,1),current(:,2),current(:,3),'LineWidth',line,'color',[0,100,100]/255); % Zeichnet nur Kanten
hold on
end
end
if(isempty(desc))
text(current(1),current(2),current(3),['(' num2str(idx) ')'],'color',cola);
else
- text(current(1),current(2),current(3),['(' desc{idx} ')'],'color',cola);
+ text(current(1),current(2),current(3),[desc{idx}],'color',cola);
end
hold on
end
while(1==1)
%jeden 4ten 2er durch schrittweise Verfeinerung (5) ersetzen
t_ref=find(G_ref_t==2);
- G_ref_t(t_ref(1:4:end)) = 5;
+ G_ref_t(t_ref(2:4:end)) = 5;
%Welche Elemente müssen bearbeitet werden
ref = find(G_ref_t>1);
sit = G_ref_S;
%Doppelte Koordinaten loeschen
- [coo l pos] = unique(coo,'rows');
- ele = pos(ele);
+[coo l pos] = unique(coo,'rows');
+pos = pos';
+ele = pos(ele);
%INTERNE Globale Variablen freigeben
clear G_ref_E G_ref_C G_ref_N G_ref_f2s G_ref_t G_ref_tD G_ref_s
projects / bacc.git / commitdiff