From: Peter Schaefer Date: Wed, 3 Apr 2013 21:51:35 +0000 (+0200) Subject: [doc] abschlussbemerkung Kap3 Fertig X-Git-Url: https://git.leopard-lacewing.eu/?a=commitdiff_plain;h=4efc1a41afe40839b4b5544eb637725a532e9bb7;p=bacc.git [doc] abschlussbemerkung Kap3 Fertig [res] gutes Technisches Programmieren hinzugefügt [doc] neue Figuren --- diff --git a/doc/code.pdf b/doc/code.pdf index 623dedf..9ddf502 100644 Binary files a/doc/code.pdf and b/doc/code.pdf differ diff --git a/doc/code.tex b/doc/code.tex index ca7de35..60247e6 100644 --- a/doc/code.tex +++ b/doc/code.tex @@ -71,24 +71,24 @@ \tableofcontents \clearpage \section{Anhang Code} - \subsection{C++} - \subsubsection{mex\_build\_V.cpp} - \lstinputlisting[language=C++]{../src/mex_build_V.cpp} - \subsubsection{slpRectangle.hpp} - \lstinputlisting[language=C++]{../src/slpRectangle.hpp} - \subsubsection{slpRectangle.cpp} - \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{C++} +% \subsubsection{mex\_build\_V.cpp} +% \lstinputlisting[language=C++]{../src/mex_build_V.cpp} +% \subsubsection{slpRectangle.hpp} +% \lstinputlisting[language=C++]{../src/slpRectangle.hpp} +% \subsubsection{slpRectangle.cpp} +% \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} % \subsubsection{plotShape.m} % \lstinputlisting[language=M]{../src/plotShape.m} @@ -134,7 +134,7 @@ \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}} - \subfloat[Zeit]{\includegraphics[width=0.5\textwidth]{fig/1tn_2DQuad_time}} +\subfloat[Zeit]{\includegraphics[width=0.5\textwidth]{fig/1tn_2DQuad_time}} \end{figure} \begin{figure}[ht] @@ -151,8 +151,8 @@ \caption{3D FichCube adaptiv anisotrop vollanalytisch $V\phi = 1$} \centering \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[Fehler]{\includegraphics[width=0.7\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} diff --git a/doc/doc.pdf b/doc/doc.pdf index 9730ffb..6ab6bdf 100644 Binary files a/doc/doc.pdf and b/doc/doc.pdf differ diff --git a/doc/doc.tex b/doc/doc.tex index 74979a9..d52b3ca 100644 --- a/doc/doc.tex +++ b/doc/doc.tex @@ -999,16 +999,30 @@ weshalb die Behauptung folgt. Durch ziehen der Wurzel auf beiden Seiten, folgt dann die Behauptung. \end{beweis} - -\noindent -\todo{ -\begin{itemize} - \item approx $A_p$ besser als $A$ - \item aufgrund von Quadratur sehr viele Auswertungsstellen (Aufwändig) - \item Folge : Semianalytisch -\end{itemize} -} - +\begin{bem} + Bei der Randelementmethode mit Galerkin-Verfahren tritt die Stetigkeitsmatrix auf,deren Einträge $A_{jk}$ bis auf einen konstanten Faktor gegeben sind durch + \begin{align}\label{math:sem:approx} + \int_{T_j} \int_{T_k} \frac 1 {\abs{\bs x -\bs y}} ds_{\bs y} ds_{\bs y}. + \end{align} + Wir werden anstelle der exakten Matrix die approximative Matrix $A_p$ berechnen, da hierbei Auslöschungseffekte vermieden werden können, welche insbesondere für kleinen Randelementen auftreten.\\ + Diese Auslöschungseffekte lassen sich durch ein kleines Beispiel leicht erklären. + Hierzu betrachten wir beispielsweise die Stammfunktion des Integrals über ein Element von \eqref{math:sem:approx} + \begin{align*} + F_{\bs x}(\bs y) = \int_{T_k} \frac 1 {\abs{\bs x -\bs y}} ds_{\bs y} + \end{align*} + für festes $\bs x \in \R^3$, inklusive einer geeigneten Parametrisierung $\gamma_k : [0,1]^2 \to T_k$ + \begin{align*} + \tilde F_{\bs x}(\bs \lambda) = \int_{[0,1]^2} \frac 1 {\abs{\bs x -\gamma_k(\bs \lambda)}} d\bs \lambda. + \end{align*} + Die Stammfunktion ist trotz Parametrisierung stetig. Daher gilt für $T_k$ mit geringem Durchmesser $\diam(T_k)$ aber $\tilde F_{\bs x}(\lambda) \approx \tilde F_{\bs x}(\tilde \lambda)$ für alle $\lambda,\tilde \lambda \in [0,1]^2$, wodurch für den Ausdruck + \begin{align*} + \tilde F_{\bs x}(0,0) - \tilde F_{\bs x}(1,0) -\tilde F_{\bs x}(0,1) + \tilde F_{\bs x}(1,1) = \int_{[0,1]^2} \frac 1 {\abs{\bs x -\gamma_k(\bs \lambda)}} d\bs \lambda = \int_{T_k} \frac 1 {\abs{\bs x -\bs y}} ds_{\bs y} + \end{align*} + starke Auslöschungseffekte auftreten.\\ + Deshalb werden wir für zulässige Randelemente die Integrale geeignet durch Gauss-Quadratur ersetzen. Dabei wird der Integrand mehrfach ausgewehrtet und mit stets positiven Quadraturgewichten multipliziert. Da der Integrand in der Praxis immer das gleiche Vorzeichen hat, werden während der Gauss-Quadratur lediglich Werte gleichen Vorzeichens addiert, wodurch keine Auslöschung auftritt. Deshalb lässt sich beobachten, dass die Auswertung mittels Gauss-Quadratur für zulässige Randstücke genauer ist als die Berechnung mithilfe der Stammfunktion.\\ + Da wir die Integrationsreihenfolge laut Satz \ref{thm:sem:switch} beliebig vertauschen dürfen, ist es sinnvoll über die großen Integrationsbereiche zu erst zu integrieren und die Integration über die kleinen nach außen zu stellen. Denn dadurch wird jene Operation, durch die starke Auslöschungseffekte auftreten, ans Ende der Berechnung gestellt.\\ + Weiterhin kann die Berechnung durch Quadratur abhängig vom gewählten Quadraturgrad sehr aufwändig werden, welches auf die hohe Anzahl der Auswertungsstellen zurückzuführen ist. Für die Quadratur über zwei Integrale mit einem Quadraturgrad von beispielsweise 8 werden dann schon $8^2 = 64$ Auswertungen benötigt. Deshalb werden wir nicht nur die Strategie betrachten in der für alle zulässigen Randelemente alle Integrale durch Quadratur ersetzt werden, sondern auch eine in der nur ein Teil der auftretenden Integrale geeignet durch Gauss-Quadratur ersetzt wird. +\end{bem} % @@ -1684,9 +1698,9 @@ Da wir später den Fehlerschätzer berechnen wollen, ist es wichtig sich zu jede } \subsection{Fehlerschätzer} -Mithilfe des Galerkin-Verfahrens berechnen wir nur eine approximative Lösung, für die wir eine Aussage über die Genauigkeit der Lösung treffen wollen. Da wir für den Fehler $\enorm{\phi - \phi_{\ell}}$, nur die Galerkin-Lösung $\phi_{\ell}$ kennen und die exakte Lösung $\phi$ unbekannt ist, werden wir im Folgenden verschiedene Fehlerschätzer betrachten. +Mithilfe des Galerkin-Verfahrens berechnen wir nur eine approximative Lösung, für die wir eine Aussage über die Genauigkeit der Lösung treffen wollen. Da wir für den Fehler $\enorm{\phi - \phi_{\ell}}$ nur die Galerkin-Lösung $\phi_{\ell}$ kennen und die exakte Lösung $\phi$ unbekannt ist, werden wir im Folgenden verschiedene Fehlerschätzer betrachten. % In diesem Abschnitt definieren wir die a-posteriori Fehlerschätzer, die wir zur Steuerung des adaptiven Algorithmus einsetzen werden. -Hierzu verwenden wir die $h-h/2$ Strategie aus \cite{fer:errbem}. +Im Wesentlichen werden wir hierzu die $h-h/2$ Strategie aus \cite{fer:errbem} verwenden. Im Folgenden bezeichnen wir mit $\hat \T_{\ell}$ das Gitter welches entsteht, wenn das Gitter $\T_{\ell}$ uniform, also entlang aller Kanten geteilt wird. Weiterhin bezeichne $\phi$ die exakte Lösung des Galerkin-Verfahrens und $\phi_{\ell}$ die Lösung zum Gitter $\T_{\ell}$, sowie $\hat \phi_{\ell}$ die Lösung zum uniformen Gitter $\hat \T_{\ell}$. \begin{defi}Es bezeichne $\phi$ die exakte Lösung von Formel \todo{ref}, $\phi_{\ell}$ die Galerkin-Lösung auf dem Gitter $\T_{\ell}$ und $\hat \phi_{\ell}$ die Lösung auf dem uniform verfeinerten Gitter $\hat \T_{\ell}$. Dann gilt, der Schätzer \begin{align*} diff --git a/doc/fig/132t05n05_3DFichCube_cond.eps b/doc/fig/132t05n05_3DFichCube_cond.eps new file mode 100644 index 0000000..40c5a34 --- /dev/null +++ b/doc/fig/132t05n05_3DFichCube_cond.eps @@ -0,0 +1,775 @@ +%!PS-Adobe-2.0 EPSF-1.2 +%%Creator: MATLAB, The MathWorks, Inc. Version 7.14.0.739 (R2012a). 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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: 53 195 550 592 +MathWorks begin +bpage +%%EndPageSetup + +%%BeginObject: obj1 +bplot + +/dpi2point 8.33333 def +portraitMode 0150 5100 csm + + 294 165 4145 3302 MR c np +82 dict begin %Colortable 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114 17 152 22 120 18 219 32 142 21 245 37 +235 34 528 78 1095 907 19 MP stroke +DA +113 50 92 41 105 46 82 37 106 47 73 32 99 44 92 41 +97 44 131 57 114 51 152 68 120 53 219 97 142 63 245 109 +235 104 528 235 1095 1542 19 MP stroke +gr + +c11 +DA +0 sg +%%IncludeResource: font Helvetica +/Helvetica /ISOLatin1Encoding 83.3333 FMSR + +2207 3432 mt +(Elementanzahl) s + 388 1926 mt -90 rotate +(Schaetzer) s +90 rotate + 612 3237 mt +( ) s +4333 300 mt +( ) s +SO +1 sg +0 1243 1238 0 0 -1243 665 3166 4 MP +PP +-1238 0 0 1243 1238 0 0 -1243 665 3166 5 MP stroke +2.77778 w +DO +SO +4.16667 w +0 sg + 665 3166 mt 1903 3166 L + 665 3166 mt 665 1923 L +1000 2018 mt +(tmu 132t05n05 A) s +gs 665 1923 1239 1244 MR c np +c8 +257 0 716 1987 2 MP stroke +gs 794 1936 103 103 MR c np + 25 25 845 1987 FO +gr + +gr + +c8 +0 sg +1000 2119 mt +(eta 132t05n05 A) s +gs 665 1923 1239 1244 MR c np +c8 +257 0 716 2089 2 MP stroke +gs 794 2038 103 103 MR c np +0 j +29 50 -58 0 29 -50 845 2122 4 MP +DP +gr + +gr + 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+gr + +gr + +c10 +0 sg +1000 2727 mt +(eta 132t05n05 QA) s +gs 665 1923 1239 1244 MR c np +c10 +257 0 716 2696 2 MP stroke +gs 794 2645 103 103 MR c np +0 j +0 -40 -40 0 0 40 40 0 825 2676 5 MP +DP +gr + +gr + +c10 +0 sg +1000 2828 mt +(fehler 132t05n05 QA) s +gs 665 1923 1239 1244 MR c np +c10 +257 0 716 2797 2 MP stroke +gs 794 2746 103 103 MR c np +0 j +25 33 -25 33 -25 -33 25 -33 845 2830 5 MP +DP +gr + +gr + +c10 +0 sg +1000 2930 mt +(N-12) s +gs 665 1923 1239 1244 MR c np +DD +c11 +257 0 716 2898 2 MP stroke +SO +gr + +c11 +0 sg +1000 3031 mt +(N-14) s +gs 665 1923 1239 1244 MR c np +DO +c11 +257 0 716 2999 2 MP stroke +SO +gr + +c11 +0 sg +1000 3132 mt +(N-34) s +gs 665 1923 1239 1244 MR c np +DA +c11 +257 0 716 3101 2 MP stroke +SO +gr + +c11 + +end %%Color Dict + +eplot +%%EndObject + +epage +end + +showpage + +%%Trailer +%%EOF diff --git a/doc/fig/132t05n05_3DFichCube_hminmax.eps b/doc/fig/132t05n05_3DFichCube_hminmax.eps new file mode 100644 index 0000000..65608fe --- /dev/null +++ b/doc/fig/132t05n05_3DFichCube_hminmax.eps @@ -0,0 +1,1049 @@ +%!PS-Adobe-2.0 EPSF-1.2 +%%Creator: MATLAB, The MathWorks, Inc. Version 7.14.0.739 (R2012a). Operating System: Linux 3.2.0-40-generic #64-Ubuntu SMP Mon Mar 25 21:22:10 UTC 2013 x86_64. +%%Title: ../doc/fig/132t05n05_3DFichCube_hminmax.eps +%%CreationDate: 04/03/2013 12:40:08 +%%DocumentNeededFonts: Helvetica +%%DocumentProcessColors: Cyan Magenta Yellow Black +%%Extensions: CMYK +%%Pages: 1 +%%BoundingBox: 53 195 550 592 +%%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: 53 195 550 592 +MathWorks begin +bpage +%%EndPageSetup + +%%BeginObject: obj1 +bplot + +/dpi2point 8.33333 def +portraitMode 0150 5100 csm + + 294 165 4145 3302 MR c np +81 dict begin %Colortable 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Operating System: Linux 3.2.0-40-generic #64-Ubuntu SMP Mon Mar 25 21:22:10 UTC 2013 x86_64. +%%Title: ../doc/fig/132t05n05_3DFichCube_time.eps +%%CreationDate: 04/03/2013 12:40:08 +%%DocumentNeededFonts: Helvetica +%%DocumentProcessColors: Cyan Magenta Yellow Black +%%Extensions: CMYK +%%Pages: 1 +%%BoundingBox: 53 195 550 592 +%%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: 53 195 550 592 +MathWorks begin +bpage +%%EndPageSetup + +%%BeginObject: obj1 +bplot + +/dpi2point 8.33333 def +portraitMode 0150 5100 csm + + 294 165 4145 3302 MR c np +82 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 2936 3720 0 0 -2936 624 3206 4 MP +PP +-3720 0 0 2936 3720 0 0 -2936 624 3206 5 MP stroke +2.77778 w +DO +SO +4.16667 w +0 sg + 624 3206 mt 4344 3206 L + 624 270 mt 4344 270 L + 624 3206 mt 624 270 L +4344 3206 mt 4344 270 L + 624 3206 mt 4344 3206 L + 624 3206 mt 624 270 L + 624 3206 mt 624 3187 L + 624 270 mt 624 288 L + 624 3206 mt 624 3168 L + 624 270 mt 624 307 L +%%IncludeResource: font Helvetica +/Helvetica /ISOLatin1Encoding 83.3333 FMSR + + 563 3332 mt +(10) s +%%IncludeResource: font Helvetica +/Helvetica /ISOLatin1Encoding 55.5556 FMSR + + 655 3281 mt +(1) s + 997 3206 mt 997 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