\psfrag{Kondition}{\scriptsize Kondition}
\psfrag{Sekunden}{\scriptsize Zeit in $s$}
+\psfrag{erste Fehler}{\tiny erste Fehler}
+\psfrag{N-12}{\tiny $N^{-1/2}$}
+\psfrag{N-14}{\tiny $N^{-1/4}$}
+\psfrag{N-34}{\tiny $N^{-3/4}$}
+
\psfrag{tmu 1t05n05 A}{\tiny $\tilde \mu$}
\psfrag{eta 1t05n05 A}{\tiny $\eta$}
\psfrag{fehler 1t05n05 A}{\tiny Fehler}
+\psfrag{Zeit 1t05n05 A}{\tiny Zeit}
-\psfrag{erste Fehler}{\tiny erste Fehler}
+\psfrag{tmu 132t05n05 A}{\tiny $\tilde \mu$ analytisch}
+\psfrag{eta 132t05n05 A}{\tiny $\eta$ analytisch}
+\psfrag{fehler 132t05n05 A}{\tiny Fehler analytisch}
+\psfrag{Zeit 132t05n05 A}{\tiny Zeit analytisch}
-\psfrag{Zeit 1t05n05 A}{\tiny Zeit}
+\psfrag{tmu 132t05n05 QEQA}{\tiny $\tilde \mu$ semiquadratur QE}
+\psfrag{eta 132t05n05 QEQA}{\tiny $\eta$ semiquadratur QE}
+\psfrag{fehler 132t05n05 QEQA}{\tiny Fehler semiquadratur QE}
+\psfrag{Zeit 132t05n05 QEQA}{\tiny Zeit semiquadratur QE}
+
+\psfrag{tmu 132t05n05 QA}{\tiny $\tilde \mu$ semiquadratur Q}
+\psfrag{eta 132t05n05 QA}{\tiny $\eta$ semiquadratur Q}
+\psfrag{fehler 132t05n05 QA}{\tiny Fehler semiquadratur Q}
+\psfrag{Zeit 132t05n05 QA}{\tiny Zeit semiquadratur Q}
-\psfrag{N-12}{\tiny $N^{-1/2}$}
-\psfrag{N-14}{\tiny $N^{-1/4}$}
-\psfrag{N-34}{\tiny $N^{-3/4}$}
\usepackage{lstings}
\lstinputlisting[language=C++]{../src/slpRectangle.cpp}
\subsubsection{gauss.hpp}
\lstinputlisting[language=C++]{../src/gauss.hpp}
-% \subsection{Matlab}
-% \subsubsection{compute.m}
-% \lstinputlisting[language=M]{../src/compute.m}
-% \subsubsection{refineQuad.m}
-% \lstinputlisting[language=M]{../src/refineQuad.m}
-% \subsubsection{areaQuad.m}
-% \lstinputlisting[language=M]{../src/areaQuad.m}
-% \subsubsection{mark.m}
-% \lstinputlisting[language=M]{../src/mark.m}
+ \subsection{Matlab}
+ \subsubsection{compute.m}
+ \lstinputlisting[language=M]{../src/compute.m}
+ \subsubsection{refineQuad.m}
+ \lstinputlisting[language=M]{../src/refineQuad.m}
+ \subsubsection{areaQuad.m}
+ \lstinputlisting[language=M]{../src/areaQuad.m}
+ \subsubsection{mark.m}
+ \lstinputlisting[language=M]{../src/mark.m}
% \subsubsection{plotShape.m}
% \lstinputlisting[language=M]{../src/plotShape.m}
\end{figure}
\begin{figure}[ht]
+
\caption{2D Quad adaptiv anisotrop vollanalytisch $V\phi = 1$}
\centering
-\label{fig:exmplAA_2DQuad}
+\label{fig:exmplAA_2DQuad_A}
\subfloat[Fehler]{\includegraphics[width=0.7\textwidth]{fig/1t05n05_2DQuad_error}}\\
\subfloat[Seitenverhältnis]{\includegraphics[width=0.5\textwidth]{fig/1t05n05_2DQuad_hminmax}}
\subfloat[Kondition]{\includegraphics[width=0.5\textwidth]{fig/1t05n05_2DQuad_cond}}
\psfrag{fehler 1t05n0 A}{\tiny Fehler (isotrop)}
\psfrag{Zeit 1t05n0 A}{\tiny Zeit (isotrop)}
-\psfrag*{tmu 1t05n05 A}{\tiny \quad (anisotrop)}
-\psfrag*{eta 1t05n05 A}{\tiny \quad (anisotrop)}
-\psfrag*{fehler 1t05n05 A}{\tiny \qquad\quad (anisotrop)}
-\psfrag*{Zeit 1t05n05 A}{\tiny \qquad (anisotrop)}
+\psfrag{tmu 1t05n05 A}{\tiny $\tilde \mu$ (anisotrop)}
+\psfrag{eta 1t05n05 A}{\tiny $\eta$ (anisotrop)}
+\psfrag{fehler 1t05n05 A}{\tiny Fehler (anisotrop)}
+\psfrag{Zeit 1t05n05 A}{\tiny Zeit (anisotrop)}
\caption{2D Quad im Vergleich vollanalytisch $V\phi = 1$}
\centering
-\label{fig:exmpl_2DQuad_tn}
+\label{fig:exmpl_2DQuad_tn_A}
\subfloat[Fehler]{\includegraphics[width=0.5\textwidth]{fig/1tn_2DQuad_error}}
\subfloat[Seitenverhältnis]{\includegraphics[width=0.5\textwidth]{fig/1tn_2DQuad_hminmax}}\\
\subfloat[Kondition]{\includegraphics[width=0.5\textwidth]{fig/1tn_2DQuad_cond}}
\begin{figure}[ht]
\caption{3D FichCube adaptiv anisotrop vollanalytisch $V\phi = 1$}
\centering
-\label{fig:exmplAA_3DQuad}
+\label{fig:exmplAA_3DQuad_A}
\subfloat[Fehler]{\includegraphics[width=0.5\textwidth]{fig/1t05n05_3DFichCube_error}}
\subfloat[Seitenverhältnis]{\includegraphics[width=0.5\textwidth]{fig/1t05n05_3DFichCube_hminmax}}\\
\subfloat[Kondition]{\includegraphics[width=0.5\textwidth]{fig/1t05n05_3DFichCube_cond}}
\end{figure}
+
+\begin{figure}[ht]
+\caption{2D Quad adaptiv anisotrop im Vergleich $V\phi = 1$}
+\centering
+\label{fig:exmplAA_2DQuad_QEQA}
+\subfloat[Fehler]{\includegraphics[width=0.5\textwidth]{fig/132t05n05_2DQuad_error}}
+\subfloat[Seitenverhältnis]{\includegraphics[width=0.5\textwidth]{fig/132t05n05_2DQuad_hminmax}}\\
+\subfloat[Kondition]{\includegraphics[width=0.5\textwidth]{fig/132t05n05_2DQuad_cond}}
+ \subfloat[Zeit]{\includegraphics[width=0.5\textwidth]{fig/132t05n05_2DQuad_time}}
+\end{figure}
+
\end{document}
% \numberwithin{bew}{section}
% \numberwithin{sat}{section}
+
+\psfrag{T}{\scriptsize $T$}
+\psfrag{T1}{\scriptsize $T_1$}
+\psfrag{T2}{\scriptsize $T_2$}
+\psfrag{T3}{\scriptsize $T_3$}
+\psfrag{T4}{\scriptsize $T_4$}
+
+\psfrag{x}{\scriptsize $x$}
+\psfrag{y}{\scriptsize $y$}
+\psfrag{z}{\scriptsize $z$}
+
\begin{document}
\todo{
% \input{titelseite}
\begin{align}\label{math:intro:int}
\int_{T_j} \int_{T_k} \kappa(\bs x, \bs y) ds_{\bs y} ds_{\bs x}
\end{align}
-beschäftigen. $\kappa(\bs x, \bs y)$ sei hierbei eine asymptotisch glatte Kernfunktion. Speziell interessieren wir uns hierbei für die asymptotisch glatte Funktion $\abs{\bs x - \bs y}^{-1}$. Ziel wird es sein, das äußere Integral durch eine Gauss-Quadratur zu erstetzen. Unter bestimmten Zulässigkeits\-beding\-ungen werden wir zeigen, dass die Energienorm durch die Gauss-Quadratur bei steigendem Quadraturgrad exponentiell schnell gegen den exakten Wert konvergiert. Weiterhin betrachten wir die in Abschnitt 2 auftretende Matrix $A \in \R^{n\times n}$, deren Einträge $A_{jk}$ durch \eqref{math:intro:int} bestimmt werden. Denn wir werden eine approximative Matrix $A_p$ aufstellen, welche das durch Quadratur approximierte Integral für die zulässigen Einträge und für alle anderen das exakte Integral verwendet. Hiermit können wir dann Zeigen, dass die approximative Matrix unter der Frobeninusnorm exponentiell schnell gegen $A$ konvergiert.\\
-In Abschnitt 4 fassen wir kurz zusammen, wie wir das Doppelintegral in einfache Integrale zerlegen und andschließend voll analytisch berechnen können. Hierzu werden wir uns weitgehend an \cite{mai:3dbem} orientieren.\\
+beschäftigen. $\kappa(\bs x, \bs y)$ sei hierbei eine asymptotisch glatte Kernfunktion. Speziell interessieren wir uns hierbei für die asymptotisch glatte Funktion $\abs{\bs x - \bs y}^{-1}$. Ziel wird es sein, das äußere Integral durch eine Gauss-Quadratur zu ersetzen. Unter bestimmten Zulässigkeits\-beding\-ungen werden wir zeigen, dass die Energienorm durch die Gauss-Quadratur bei steigendem Quadraturgrad exponentiell schnell gegen den exakten Wert konvergiert. Weiterhin betrachten wir die in Abschnitt 2 auftretende Matrix $A \in \R^{n\times n}$, deren Einträge $A_{jk}$ durch \eqref{math:intro:int} bestimmt werden. Denn wir werden eine approximative Matrix $A_p$ aufstellen, welche das durch Quadratur approximierte Integral für die zulässigen Einträge und für alle anderen das exakte Integral verwendet. Hiermit können wir dann Zeigen, dass die approximative Matrix unter der Frobeninusnorm exponentiell schnell gegen $A$ konvergiert.\\
+In Abschnitt 4 fassen wir kurz zusammen, wie wir das Doppelintegral in einfache Integrale zerlegen und anschließend voll analytisch berechnen können. Hierzu werden wir uns weitgehend an \cite{mai:3dbem} orientieren.\\
Abschließend werden wir kurz die numerische Umsetzung der Techniken vorstellen und anhand von numerischen Beispielen vergleichen. Hierbei wird uns die voll analytische und approximative Berechnung, sowie die adaptive und uniforme Netzverfeinerung besonders interessieren.
\clearpage
\begin{align*}
\diam (T) &= (a^2+b^2)^{1/2}
\end{align*}
-Durchmesser von T. Weiterhin nennen wir
+Durchmesser von $T$. Weiterhin nennen wir
\begin{align*}
\diam_{\bs a} (T) = a
\end{align*}
\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
+auf achsenorientierten Rechtecken $T_j,T_k \subset \R^3$ beziehungsweise als Spezialfall davon die Berechnung des Integrals
\begin{align}\label{math:gal:kap+}
A_{jk} &= \int_{T_j} \int_{T_k} \frac{1}{|\bs x- \bs y|} ds_{\bs y} ds_{\bs x}.
\end{align}
unter bestimmten Voraussetzungen an die affinen Randstücke $T_j,T_k$ und den asymptotisch glatten Integranden $\kappa : \R^3 \times \R^3 \to \R$.
\subsection{Interpolation}
-An dieser stelle werden wir zunächst den Interpolationsoperator auf dem Intervall $[0,1]$ definieren. Ferner wollen wir mithilfe von Chebyshev-Knoten einen Fehlerschätzer für die Chebyshev'sche Interpolation auf Intervallen $[0,1]^d$ mit $d \in \N$ definieren. Im Folgenden bezeichnet $\P^p$ die Menge aller Polynome vom Grad $\leq p$ auf $[0,1]$.
+An dieser Stelle werden wir zunächst den Interpolationsoperator auf dem Intervall $[0,1]$ definieren. Ferner wollen wir mithilfe von Chebyshev-Knoten einen Fehlerschätzer für die Chebyshev'sche Interpolation auf Intervallen $[0,1]^d$ mit $d \in \N$ definieren. Im Folgenden bezeichnet $\P^p$ die Menge aller Polynome vom Grad $\leq p$ auf $[0,1]$.
\begin{defi}
Für einen festen Grad $p \in \N$ und paarweise verschiedene Knoten $x_j \in [0,1]$ lautet das Lagrange'sche Interpolationsproblem:
\end{sat}
-\begin{beweis} Betrachten wir zunächst die Differenz zwischen $A$ und $A_p$ in einem festen Eintrag $(A-A_p)_{jk}$. Sind $T_j$ und $T_k$ unzulässig, ist die Differenz laut Definition $0$ und die Abschätzung für $\tilde C_{\zeta_Q,j,k} = 0$ erfüllt. Sind $T_j$ und $T_k$ hingegen zulässig, können wir Satz \ref{thm:sem:quad:V} anwenden. Damit erhalten wir
+\begin{beweis} Betrachten wir zunächst die Differenz zwischen $A$ und $A_p$ in einem festen Eintrag $(A-A_p)_{jk}$. Sind $T_j$ und $T_k$ unzulässig, ist die Differenz laut Definition $0$ und die Abschätzung mit $\tilde C_{\zeta_Q,j,k} = 0$ erfüllt. Sind $T_j$ und $T_k$ hingegen zulässig, können wir Satz \ref{thm:sem:quad:V} anwenden. Damit erhalten wir
\begin{align*}
\norm{A-A_p}_F^2 &= \sum_{j,k=1}^n (A_{jk} - (A_p)_{jk})^2\\
&\leq \sum_{j,k=1}^n \left(\tilde C_{\zeta_Q,j,k}\Lambda_{2p+1}^4 2(p+1)\left(1+c_2\zeta_Q\right)^{-2(p+1)}\right)^2\\
\end{sat}
-\begin{beweis} Betrachten wir zunächst die Differenz zwischen $A$ und $A_p$ in einem festen Eintrag $(A-A_p)_{jk}$. Sind $T_j$ und $T_k$ unzulässig, ist die Differenz laut Definition $0$ und die Abschätzung für $\tilde C_{\zeta_E,j,k} = 0$ erfüllt. Sind $T_j$ und $T_k$ hingegen zulässig, unterscheiden wir zwei Fälle. Ist $\diam(T_j) \leq \diam (T_k)$, können wir Satz \ref{thm:sem:quad:E} anwenden. Andernfalls ist durch Lemma \ref{thm:sem:switch} $A_{jk}=A_{kj}$, worauf wir dann Satz \ref{thm:sem:quad:E} anwenden können und dadurch die selbe Abschätzung erhalten. Daraus folgt
+\begin{beweis} Betrachten wir zunächst die Differenz zwischen $A$ und $A_p$ in einem festen Eintrag $(A-A_p)_{jk}$. Sind $T_j$ und $T_k$ unzulässig, ist die Differenz laut Definition $0$ und die Abschätzung mit $\tilde C_{\zeta_E,j,k} = 0$ erfüllt. Sind $T_j$ und $T_k$ hingegen zulässig, unterscheiden wir zwei Fälle. Ist $\diam(T_j) \leq \diam (T_k)$, können wir Satz \ref{thm:sem:quad:E} anwenden. Andernfalls ist durch Lemma \ref{thm:sem:switch} $A_{jk}=A_{kj}$, worauf wir dann Satz \ref{thm:sem:quad:E} anwenden können und dadurch die selbe Abschätzung erhalten. Daraus folgt
\begin{align*}
\norm{A-A_p}_F^2 &= \sum_{j,k=1}^n (A_{jk} - (A_p)_{jk})^2\\
&\leq \sum_{j,k=1}^n \left(\tilde C_{\zeta_E,j,k}\Lambda_{2p+1}^2 2(p+1)\left(1+\sqrt 2 c_2\zeta_E\right)^{-2(p+1)}\right)^2\\
\begin{align}\label{math:analy:int}
A_{jk} = \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 beschränkten, achsenorientierten Rechtecken $T_j,T_k \subseteq\R^3$ beschäftigen. Die im Folgenden auftretenden Stammfunktionen $\int f(x) dx$ werden wir der Einfachheit halber jeweils mit additiver Verschiebung $0$ schreiben.
+auf zwei beschränkten, achsenorientierten Rechtecken $T_j,T_k \subseteq\R^3$ beschäftigen. Die im Folgenden auftretenden Stammfunktionen $\int f(x) dx$ werden wir der Einfachheit halber jeweils mit additiver Verschiebung $0$ schreiben.
Dazu wollen wir \cite{mai:3dbem} folgend, zwei Stammfunktionen zitieren, welche durch das Aufspalten des Integrals \eqref{math:analy:int} auftreten werden.
\\\noindent
\begin{lem}
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(erste Fehler) s
-gs 665 312 1133 271 MR c np
+gs 665 312 936 233 MR c np
DA
c2
-243 0 713 517 2 MP stroke
+253 0 715 478 2 MP stroke
SO
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4333 300 mt
( ) s
1 sg
-0 168 1086 0 0 -168 665 480 4 MP
+0 131 889 0 0 -131 665 443 4 MP
PP
--1086 0 0 168 1086 0 0 -168 665 480 5 MP stroke
+-889 0 0 131 889 0 0 -131 665 443 5 MP stroke
2.77778 w
DO
SO
4.16667 w
0 sg
- 665 480 mt 1751 480 L
- 665 480 mt 665 312 L
- 979 393 mt
-(cond\(A) s
-%%IncludeResource: font Helvetica
-/Helvetica /ISOLatin1Encoding 66.6667 FMSR
-
-1243 434 mt
-(h/2) s
-%%IncludeResource: font Helvetica
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-
-1335 393 mt
-(\) 1t05n0 A) s
-gs 665 312 1087 169 MR c np
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+ 665 443 mt 665 312 L
+ 992 408 mt
+(cond 1t05n0 A) s
+gs 665 312 890 132 MR c np
c8
-242 0 713 396 2 MP stroke
-gs 783 345 103 103 MR c np
- 25 25 834 396 FO
+251 0 715 377 2 MP stroke
+gs 790 326 103 103 MR c np
+ 25 25 841 377 FO
gr
gr
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4333 300 mt
( ) s
1 sg
-0 168 1040 0 0 -168 665 480 4 MP
+0 131 843 0 0 -131 665 443 4 MP
PP
--1040 0 0 168 1040 0 0 -168 665 480 5 MP stroke
+-843 0 0 131 843 0 0 -131 665 443 5 MP stroke
2.77778 w
DO
SO
4.16667 w
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- 665 480 mt 665 312 L
- 977 393 mt
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-%%IncludeResource: font Helvetica
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-(h/2) s
-%%IncludeResource: font Helvetica
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-
-1333 393 mt
-(\) 1t1n0 A) s
-gs 665 312 1041 169 MR c np
+ 665 443 mt 1508 443 L
+ 665 443 mt 665 312 L
+ 990 408 mt
+(cond 1t1n0 A) s
+gs 665 312 844 132 MR c np
c8
-240 0 713 396 2 MP stroke
-gs 782 345 103 103 MR c np
- 25 25 833 396 FO
+250 0 714 377 2 MP stroke
+gs 788 326 103 103 MR c np
+ 25 25 839 377 FO
gr
gr
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%%Title: ../doc/fig/1tn_2DQuad_cond.eps
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4333 300 mt
( ) s
1 sg
-0 447 1132 0 0 -447 665 759 4 MP
+0 333 935 0 0 -333 665 645 4 MP
PP
--1132 0 0 447 1132 0 0 -447 665 759 5 MP stroke
+-935 0 0 333 935 0 0 -333 665 645 5 MP stroke
2.77778 w
DO
SO
4.16667 w
0 sg
- 665 759 mt 1797 759 L
- 665 759 mt 665 312 L
- 982 392 mt
-(cond\(A) s
-%%IncludeResource: font Helvetica
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-1246 433 mt
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-%%IncludeResource: font Helvetica
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-
-1338 392 mt
-(\) 1t1n0 A) s
-gs 665 312 1133 448 MR c np
+ 665 645 mt 1600 645 L
+ 665 645 mt 665 312 L
+ 994 407 mt
+(cond 1t1n0 A) s
+gs 665 312 936 334 MR c np
c8
-243 0 713 395 2 MP stroke
-gs 784 344 103 103 MR c np
- 25 25 835 395 FO
+253 0 715 376 2 MP stroke
+gs 791 325 103 103 MR c np
+ 25 25 842 376 FO
gr
gr
c8
0 sg
- 982 532 mt
-(cond\(A) s
-%%IncludeResource: font Helvetica
-/Helvetica /ISOLatin1Encoding 66.6667 FMSR
-
-1246 573 mt
-(h/2) s
-%%IncludeResource: font Helvetica
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-
-1338 532 mt
-(\) 1t05n0 A) s
-gs 665 312 1133 448 MR c np
+ 994 509 mt
+(cond 1t05n0 A) s
+gs 665 312 936 334 MR c np
c9
-243 0 713 535 2 MP stroke
-gs 784 484 103 103 MR c np
+253 0 715 478 2 MP stroke
+gs 791 427 103 103 MR c np
0 j
-29 50 -58 0 29 -50 835 568 4 MP
+29 50 -58 0 29 -50 842 511 4 MP
DP
gr
c9
0 sg
- 982 672 mt
-(cond\(A) s
-%%IncludeResource: font Helvetica
-/Helvetica /ISOLatin1Encoding 66.6667 FMSR
-
-1246 713 mt
-(h/2) s
-%%IncludeResource: font Helvetica
-/Helvetica /ISOLatin1Encoding 83.3333 FMSR
-
-1338 672 mt
-(\) 1t05n05 A) s
-gs 665 312 1133 448 MR c np
+ 994 611 mt
+(cond 1t05n05 A) s
+gs 665 312 936 334 MR c np
c10
-243 0 713 675 2 MP stroke
-gs 784 624 103 103 MR c np
- 818 658 mt 852 692 L
- 852 658 mt 818 692 L
+253 0 715 580 2 MP stroke
+gs 791 529 103 103 MR c np
+ 825 563 mt 859 597 L
+ 859 563 mt 825 597 L
gr
gr
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portraitMode 0150 5100 csm
334 236 4045 3206 MR c np
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916 463 3169 2517 MR c np
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623 840 mt 574 817 L
506 837 mt
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(x) s
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-
2467 909 mt 2501 943 L
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-
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2279 3096 mt 4034 2621 L
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0 sg
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+(e4) s
1363 3346 mt
(y) s
gs 624 269 3721 2937 MR c np
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380 1673 mt -90 rotate
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90 rotate
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}';
leg3 = {leg3{:}...
- ['cond(A_{h/2}) ' l0 l1{i}]...
+ ['cond ' l0 l1{i}]...
}';
leg4 = {leg4{:}...
['Zeit ' l0 l1{i}]...
%
% [area, vector, sites] = area(coordinates, elements)
%
-% Diese Funktion Berechnet den Flächeninhalt zu jedem Element (area) und
+% Diese Funktion Berechnet den Flaecheninhalt zu jedem Element (area) und
% zu jedem Element die Orthogonalen mit Laenge 1, des Weiteren werden auch
% die Seitenlaengen zu jedem Element gespeichert
%
% Datei laden
load(file)
+% Kontainer fuer Messungen initialisieren
if(~exist('data','var'))
data=[];
end
+
kap3 = 0;
-%times mal verfeinern
+
+%times mal verfeinern oder bis Elementanzahl fuer times > 40
for j = 1:times
%Beende Schleife wenn Element Anzahl erreicht
if(times>40 && size(elements,1) > times)
break;
end
+
+ %altes Netz mit Loesung merken
if(exist('coo_fine','var'))
old_C_fine = coo_fine;
old_F_fine = f2s;
old_E_fine = ele_fine;
old_x_fine = x_fine;
end
-
-% figure(10)
-% plotShape(coordinates,elements,'');
-
+
%uniformIsotrop Verfeinern
[coo_fine,ele_fine,neigh_fine,f2s,sit_fine]...
=refineQuad(coordinates,elements,neigh,sites,2);
-
-% figure(11)
-% plotShape(coordinates,elements,'');
%Flaecheninhalte Berechnen (rhs)
b_fine = areaQuad(sit_fine);
clear V_fine
- % \tilde \mu ( \Pi h -h + L_2 )
+ % \tilde \mu ( \Rho h -h + L_2 )
tmu = hmin.* b .* ...
sum((x_fine(f2s)'-repmat(sum(x_fine(f2s)',1)/4,4,1)).^2)' /4;
- % Export für Valgrind
-% z = zeta{i};
-% t = typ(i);
-% save('input.mat','coordinates','elements','z','t');
-
-
%Fehlerschaetzer 2 aufbauen
V = mex_build_V(coordinates,elements,zeta{i},typ(i));
-
if(~vcon)
x = V\b;
else
xo_fine(f2s) = repmat(x,1,4);
xd_fine = abs(xo_fine'-x_fine);
-
-
- % \tilde \mu ( h/2 -h + L_2 )
+ % \mu ( h/2 -h + L_2 )
mu = hmin.*b.*sum((x_fine(f2s)'-repmat(x',4,1)).^2)'/4;
%Enorm^2 Elementweise Vergleich
-
if(exist('old_F_fine','var'))
-% xf_fine = zeros(size(x_fine));
for k = 1:size(ele_fine,1)
e_f = find(sum(k==f2s,2));
e_f_p = find(f2s(e_f,:)==k);
inh = [1 2 4 3];
e_p = inh(e_ff_p(mod(floor((e_f_p)/2),2)+1));
end
- end
-% figure(11)
-% plotShape(old_C,old_E,'b');view(2);
-% plotMark(e_ff,coo_fine,ele_fine);
-%
-% figure(12)
-% plotShape(old_C_fine,old_E_fine,'b'); view(2);
-% plotMark(old_F_fine(e_ff,e_p),coo_fine,ele_fine);
-%
-%
-% figure(13)
-% plotShape(coordinates,elements,'b');view(2);
-% plotMark(e_f,coo_fine,ele_fine);
-%
-%
-% figure(14)
-% plotShape(coo_fine,ele_fine,'b');view(2);
-% plotMark(k,coo_fine,ele_fine);
-
+ end
xf_fine(k,1) = abs(x_fine(k) - old_x_fine(old_F_fine(e_ff,e_p)));
end
kap3 = b_fine'*(xf_fine);
end
-
-% xe_fine = b_fine'*x_fine;
-
%\tilde \mu 2 = ( ||\Pi h|| - ||h||)
% |||h/2 -h|||
% eta = xd_fine'*A_fine*xd_fine;
eta = abs(xe_fine-xe);
-
-% save_A_fine{i} = V_fine;
-% save_x_fine{i} = x_fine;
-%
-% save_A{i} = V;
-% save_x{i} = x;
-
-
-end_time = toc - start_time;
+
+ end_time = toc - start_time;
dataS = [dataS ...
typ(i) ... berechnet mit Typ
sqrt(sum(tmu2))... tilde mu 2
xe_fine... (kappa)
kap3 ... kappa 3
- end_time ... benötigte Zeit (Aufbauen, Berechnen)
+ end_time ... benoetigte Zeit (Aufbauen, Berechnen)
];
end
-
- % nur RandElemente Verfeinern
-% marked = ones(1,size(G_E,1));
-% marked(find(sum((G_N(:,1:4)==0),2))) = 2;
-
% Markieren mit gewaehlten Parametern
marked = mark(x_fine(f2s)',tmu,theta,nu);
old_E = elements;
old_S = sites;
-% figure(1)
-% plotShape(coordinates,elements(:,:),'db');view(2);
-
%Netz Verfeinern, wie durch marked bestimmt
[coordinates, elements, neigh, f, sites, er]...
= refineQuad(coordinates,elements,neigh,sites,marked);
-% figure(2)
-% plotShape(coordinates,elements(:,:),'db');view(2);
-
-% f
-
%Vater Sohn test
assert(sum(areaQuad(old_S))==sum(areaQuad(sites)),...
'Gesamtinhalt Fehlerhaft')
else
p2(2) = p2(2) -2;
end
-
-
%ErgebnisWerte Speichern
data(size(data,1)+1,1:length(dataS)) = dataS;
end
-% kappa:
-% kappa = ||| phi_{h/2}^{\ell} - \phi_{h/2}^{\ell-1} |||
-% ist die differenz der eNormen von den h/2 Netzen?
-%
-% tildeMu:
-% tilde-mu(T)^2 = hmin(T) || (1-Pi_h) phi_{h/2} ||^2_{L2(T)}
-% = hmin(T) * ( ||phi_{h/2}||^2_{L2(T)} - ||Pi_h
-% phi_{h/2}||^2_{L2(T)}
-
-
function str = t2str(time)
type = 's';
load exmpl_2DQuad2
plotShape(coordinates,elements(:,:),'ben');
+hold on
+text(-.05,-.05,0,'k1');
+text(1.05,0,0,'k2');
+text(1.05,1.05,0,'k3');
+text(-.05,1.05,0,'k4');
+
+text(.5,-.05,0,'e1');
+text(1.05,.5,0,'e2');
+text(.55,1.1,0,'e3');
+text(-.05,.5,0,'e4');
+
+text(.55,.5,.7,'n');
+hold off
+
+
% view(2);
% axis off;
print('-r600','-depsc',['../doc/fig/net_single.eps'])
'meshSave/1t05n05_2DQuad_29'},'../doc/fig/1tn_2DQuad')
%% Semianalytisch
-A_plots({'meshSave/132t05n05_2DQuad_29'},'../doc/fig/132t05n05_2DQuad')
+A_plots({'meshSave/132t05n05_2DQuad_30'},'../doc/fig/132t05n05_2DQuad')
% [coo,ele,nei,f2s,sit,err] = refineQuad(coordinates,elements,type)
%
% Verfeinert die markierten Elemente mit dem entsprechenden TYP und gibt
-% auch die F2S Beziehungen zurück. type muss von der Länge der Anzahl der
-% Elemente entsprechen oder genau 1 und die Einträge können 1,2,3,4,5 sein.
+% auch die F2S Beziehungen zurueck. type muss von der Laenge der Anzahl der
+% Elemente entsprechen oder genau 1 und die Eintraege koennen 1,2,3,4,5 sein.
%
% der Typ zu jedem Element entspricht dabei:
% 1 - keine Verfeinerung
% 2 - 4 neue Elemente
-% 3 - 2 neue Elemente, übereinander
+% 3 - 2 neue Elemente, uebereinander
% 4 - 2 neue Elemente, nebeneinander
% 5 - 4 neue Elemente, wird erst (3) und dann beide Elemente (4) geteilt
%
global G_ref_f2s; %Finale Beziehung (VaterSohn)
global G_ref_t; %wie soll verfeinert werden
global G_ref_tD; %wie wurde bereits verfeinert (in diesem Durchlauf)
-global G_ref_f2sT; %Temporäre Beziehung
+global G_ref_f2sT; %Temporaere Beziehung
%INTERNE Globale Variablen zuweisen
G_ref_E = elements;
G_ref_N = neigh;
G_ref_S = sites;
G_ref_t = typ;
-G_ref_f2s = repmat([1:size(elements,1)]',1,4);
+G_ref_f2s = repmat((1:size(elements,1))',1,4);
G_ref_tD = ones(size(elements,1),1);
%Parameter Freigeben (Speicher...)
t_ref=find(G_ref_t==2);
G_ref_t(t_ref(2:4:end)) = 5;
- %Welche Elemente müssen bearbeitet werden
+ %Welche Elemente muessen bearbeitet werden
ref = find(G_ref_t>1);
ref = reshape(ref,1,length(ref));
break;
end
-
-% figure(6)
-% plotShape(G_ref_C,G_ref_E)
-% plotMark(ref,G_ref_C,G_ref_E,'xg')
-% title('Zum Verfeinern Markierte Elemente')
-
% Elementeweise Bearbeiten
for ele = ref % ref(randperm(length(ref)))
-
-% figure(5)
-% plotShape(G_ref_C,G_ref_E)
-% plotMark(ele,G_ref_C,G_ref_E,'xg')
-% view(2)
-% title('Entscheidene Nachbarelemente')
-
- % # HangingNode Check
+
+ % HangingNode Check
Nt = find(G_ref_N(ele,5:8)==0);
N = G_ref_N(ele,Nt);
N2t = find(N~=0);
elseif(G_ref_t(ele)==4)
N2 = N2(mod(N2tt,2)==1);
end
-
-% plotMark(N2,G_ref_C,G_ref_E,'or')
-
+
%Hat noch zu teilende Nachbarn?
if(~isempty(N2))
N3t = mod(find((G_ref_N(N2',:)==ele)')-1,4)+1; %Nachbarseiten
end
- % Da Nachbarn noch Verfeinert werden müssen erst mal weiter
+ % Da Nachbarn noch Verfeinert werden muessen erst mal weiter
continue;
end
end
- % Wenn Alle Überprüfungen durchgelaufen sind
+ % Wenn Alle Ueberpruefungen durchgelaufen sind
assert(G_ref_tD(ele)~=2,'Element ist schon verfeinert')
assert(G_ref_t(ele)>1,'Element ist nicht Markiert')
G_ref_f2sT = ones(1,4)*ele;
refineE(ele); %Element Teilen
updateN(ele); %Nachbarn des Elements aktualisieren
updateF2S(ele); %VaterSohn Relation setzen
-% plotShape(G_ref_C,G_ref_E)
end
end
sit = G_ref_S;
%Doppelte Koordinaten loeschen
-[coo l pos] = unique(coo,'rows');
+[coo , ~, pos] = unique(coo,'rows');
pos = pos';
ele = pos(ele);
clear G_ref_E G_ref_C G_ref_N G_ref_f2s G_ref_t G_ref_tD G_ref_s
end
-%% Element Verfeinern ! sollte nur ausgeführt werden wenn wirklich möglich
+%% Element Verfeinern ! sollte nur ausgefuehrt werden wenn wirklich moeglich
function refineE(ele)
% Element wird gnadenlos Verfeinert
end
-%% Aktualisieren der Nachbarn ! sollte nur ausgeführt werden wenn wirklich möglich
+%% Aktualisieren der Nachbarn ! sollte nur ausgefuehrt werden wenn wirklich moeglich
function updateN(ele)
% Nachbarschaften werden neu gesetzt (nach N und f2s)
%An welchen Kanten habe ich Nachbarn
S = find(mod((this(1:4)~=0).*(this(5:8)==0),2))'; %Einen Nachbar (Single)
D = find(this(5:8)~=0)'; %Zwei Nachbarn (Double)
-% G_ref_N([this(S) this(D)],:);
%An welchen Kanten bin ich Nachbar
MSt = mod(find((G_ref_N(this(S),:)==ele)')-1,8)+1;
% Beziehungen fuer Kanten mit einem Nachbar
for i = 1:length(S)
- if(mod(S(i),2)==0) %TODO was ist wenn ich selbst doppelNachbar an der Seite bin???
+ if(mod(S(i),2)==0)
G_ref_N(this(S(i)),[MS(i) MS(i)+4]) = [G_ref_f2sT(S(i)) G_ref_f2sT(mod(S(i),4)+1)];
G_ref_N([G_ref_f2sT(S(i)) G_ref_f2sT(mod(S(i),4)+1)]',S(i))=this(S(i));
else
end
-%% Aktualisieren der VaterSohn Beziehung ! sollte nur ausgeführt werden wenn wirklich möglich
+%% Aktualisieren der VaterSohn Beziehung ! sollte nur ausgefuehrt werden wenn wirklich moeglich
function updateF2S(ele)
%Vater Sohn Beziehungen richtig Setzen
global G_ref_f2s;
global G_ref_f2sT;
global G_ref_t;
global G_ref_tD;
-% global G_ref_C;
-% global G_ref_E;
-
if(G_ref_tD(ele)==1) %Wenn Element zum ersten Mal verfeinert wird
G_ref_f2s(ele,:) = G_ref_f2sT;
pos=find(G_ref_f2s(org,:)==ele,1);
if(G_ref_t(ele)==3)
-% G_ref_f2s(org,:)
-% pos=find(G_ref_f2s(org,:)==ele,1)
-% G_ref_f2sT
-
if(pos==1)
G_ref_f2s(org,[1 4]) = G_ref_f2sT([1 3]);
else
G_ref_f2s(org,[2 3]) = G_ref_f2sT([3 1]);
end
-% G_ref_f2s(org,:)
-% plotShape(G_ref_C,G_ref_E(G_ref_f2s(org,:),:),'t')
else
if(pos==1)
G_ref_f2s(org,[1 2]) = G_ref_f2sT([1 2]);
else
G_ref_f2s(org,[3 4]) = G_ref_f2sT([2 1]);
end
-% G_ref_f2s(org,:)
-% plotShape(G_ref_C,G_ref_E(G_ref_f2s(org,:),:),'t')
end
G_ref_t(G_ref_f2sT) = 0;
end
end
-%% Berechnet die Flächeninhalte der Elemente
+%% Berechnet die Flaecheninhalte der Elemente
% function updateS(ele)
% %Globale Variabelen aufbauen
% global G_ref_E;
% Test ausführen
%Anzahl der Schritte oder wenn groeßer als 40 der Elemente
-steps = 10^4;
+steps = 4;
%Art der Berechnungen
type = [1 3 2];
zeta = { [2 2 2] [2 2 2] [2 2 2]};
projects / bacc.git / commitdiff