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\subsection{Markieren}
Im Adaptiven Algorithmus werden wir die Elemente abhängig vom Fehlerschätzer $\tilde \mu$ verfeinern. Dazu wählen wir mithilfe der Dörfler-Markierung \cite{dor:adapt} eine Teilmenge aus.
\begin{defi}[Dörfler-Markierung]
- Bestimme für feste Konstante $\theta \in (0,1)$ die Menge $M_{\ell} \subseteq T_{\ell}$ mit minimaler Kardinalität
+ Bestimme für feste Konstante $\theta \in (0,1]$ die Menge $M_{\ell} \subseteq T_{\ell}$ mit minimaler Kardinalität
\begin{align*}
\theta \sum_{T\in T_{\ell}} \tilde \mu_{\ell}(T)^2 &\leq \sum_{ T\in M_{\ell}} \tilde \mu_{\ell}(T)^2.
\end{align*}
\begin{align*}
\sum_{T\in M_{\ell}} \tilde \mu_{\ell}(T)^2 - \sum_{T\in \tilde M_{\ell}} \tilde \mu_{\ell}(T)^2 / \sum_{T\in M_{\ell}} \tilde \mu_{\ell}(T)^2 &> 10^{-2}.
\end{align*}
-\end{defi}
+\end{defi}9
Da wir im Adaptiven Algorithmus auch anisotrope Verfeinerungen zulassen werden definieren wir an dieser Stelle eine Auswahlstrategie zum bestimmen der Verfeinerungsart für jedes Element.
\begin{defi}
Seien $\phi_j^{(1)},\ldots, \phi_j^{(4)}$ die Lösungen der isotropen Verfeinerung $T_j^{(1)},\ldots, T_j^{(4)} \in \hat \T_{\ell}$ von $T_j \in T_{\ell}$,
\phi_j^{(1)}\\ \phi_j^{(2)}\\\phi_j^{(3)}\\ \phi_j^{(4)}
\end{pmatrix}.
\end{align*}
-Dann ist mit fester Konstante $\nu \in (0,1)$ die Art der Markierung $marked_j$ zu einem Element $T_j \in \tilde M_{\ell}$ bestimmt durch
+Dann ist mit fester Konstante $\nu \in [0,1)$ die Art der Markierung $marked_j$ zu einem Element $T_j \in \tilde M_{\ell}$ bestimmt durch
\begin{align*}
marked_j :=
\begin{cases} %
\centering
\label{fig: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}}
+% \subfloat[Seitenverhältnis]{\includegraphics[width=0.5\textwidth]{fig/exmplAA_2DQuad_hminmax}}
+\end{figure}
+
+\begin{figure}[ht]
+\caption{3D FichCube adaptiv anisotrop vollanalytisch $V\phi = 1$}
+\centering
+\label{fig:exmplAA_3DQuad}
+\subfloat[Fehler]{\includegraphics[width=0.5\textwidth]{fig/1t05n05_3DFichCube_23_error}}
+\subfloat[Seitenverhältnis]{\includegraphics[width=0.5\textwidth]{fig/1t05n05_3DFichCube_23_hminmax}}
\end{figure}
\begin{figure}[ht]
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%%Title: ../doc/fig/net_wrong.eps
-%%CreationDate: 03/21/2013 23:08:02
+%%CreationDate: 03/29/2013 14:14:48
%%DocumentNeededFonts: Helvetica
%%DocumentProcessColors: Cyan Magenta Yellow Black
%%Extensions: CMYK
portraitMode 0150 5100 csm
916 463 3169 2517 MR c np
-86 dict begin %Colortable dictionary
+83 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
function A_plots(files,printt)
+figure(4)
+figure(5)
+figure(6)
+figure(7)
+figure(8)
+
% type2str = {'Analytisch ' 'Quad Element' 'Quad Achse ' 'Quad Seite '};
type2str = {'A ' 'QA' 'QEQA' 'QA ' 'QS '};
type2sym = {'-o' '-x' '-*' '-+' '-.'};
+type2color = [[0 .3 .3]; [0 .6 .6]; [0 0 .6 ]; [.9 0 0]];
+
+set(0,'DefaultAxesColorOrder',type2color,...
+ 'DefaultAxesLineStyleOrder','-o|-x|-*|-.|--|:')
+
+ set(4,'DefaultAxesColorOrder',type2color(1,:),...
+ 'DefaultAxesLineStyleOrder','-o|-x|-*|-.|--|:')
+ set(6,'DefaultAxesColorOrder',type2color(1,:),...
+ 'DefaultAxesLineStyleOrder','-o|-x|-*|-.|--|:')
G_D=[];
X = [];
sol = 16.604703 % Ferraz-Leite Dipl.
end
+
+
+
% G_D
%% Plotte Fehlerschätzer
figure(4)
i=0;
-
-
% [shift k] = min(G_D(1:end-5,2+1+rows*i)*G_D(1,2)/G_D(1,3)-G_D(1:end-5,2+0+rows*i));
% shift = shift+shift/10;
% shift2 = shift2+shift2/10;
% error*(eta(l)-shift2)/error(l)
+first = find(abs([ 0; (G_D(2:end,2+rows*i)-G_D(1:end-1,2+rows*i))./G_D(2:end,2+rows*i)])>0.5,1);
+Ferr = (X(first,i+1)+X(first-1,i+1))/2;
+
loglog(repmat(X(:,i+1),1,3),[G_D(:,2+rows*i) ...
... G_D(:,2+8+rows*i)...
G_D(:,2+1+rows*i)...*(G_D(k,2)-shift)/G_D(k,3)...*G_D(1,2)/G_D(1,2+1+rows*i) ...
... [ 0; sqrt(G_D(2:end,2+9+rows*i)-G_D(1:end-1,2+9+rows*i))]...
... [ 0; sqrt(G_D(2:end,2+2+rows*i)-G_D(1:end-1,2+2+rows*i))]...
... G_D(:,2+10+rows*i)...
- ],type2sym{i+1});
+ ],'color', type2color(i+1,:));
hold on
for i = 1:step-1
... [ 0; sqrt(G_D(2:end,2+9+rows*i)-G_D(1:end-1,2+9+rows*i))]...
... [ 0; sqrt(G_D(2:end,2+2+rows*i)-G_D(1:end-1,2+2+rows*i))]...
... G_D(:,2+10+rows*i)...
- ],type2sym{i+1});
+ ],'color', type2color(i+1,:));
end
-loglog(X(:,1),[7*X(:,1).^(-1/2),3*X(:,1).^(-1/4),2*X(:,1).^(-3/4)],'-.')
+
+loglog(X(:,1),7*X(:,1).^(-1/2),'-.',...
+ X(:,1),3*X(:,1).^(-1/4),':',...
+ X(:,1),4*X(:,1).^(-3/4),'--','color', [.9 .6 0])
+
+loglog([Ferr; Ferr],...
+ [min(G_D(:,3+rows*[0:step-1]))*0.6; max(G_D(:,2+rows*[0:step-1]))*1.3],...
+ '--','color',[.9 0 0]);
+
hold off
-title(['Fehler ' t0])
+% title(['Fehler ' t0])
xlabel('Elemente');
ylabel('Schaetzer');
legend({leg0{:} ...
'N^{-1/2}' 'N^{-1/4}' 'N^{-3/4}'...
+ 'erste Fehler' ...
} ,'location','southwest','box', 'off');
print('-r600','-depsc',[printt '_error.eps'])
loglog(repmat(X(:,1),1,1),G_D(:,2+2),type2sym{1});
hold on
for i = 1:step-1
- loglog(repmat(X(:,i+1),1,1),G_D(:,2+2+i*rows),type2sym{i+1});
+ loglog(repmat(X(:,i+1),1,1),G_D(:,2+2+i*rows),'color', type2color(i+1,:));
end
-loglog(X(:,1),repmat(sol,size(X,1),1),'r-.')
+loglog(X(:,1),repmat(sol,size(X,1),1),'-.', 'color', [.9 0 0])
+loglog([Ferr; Ferr],...
+ [min(G_D(:,4+rows*[0:step-1]))*0.6; max(G_D(:,4+rows*[0:step-1]))*1.1],...
+ '--','color',[.9 0 0]);
hold off
ylim([min(min(G_D(:,[4+(0:step-1)*rows]))) 1.005*max(max(G_D(:,[4+(0:step-1)*rows])))])
-title(['Energie Norm ' t0])
+% title(['Energie Norm ' t0])
xlabel('Elemente');
ylabel('eNorm^2');
-legend({leg1{:} , 'extrapoliert' },'location','SouthEast');
+legend({leg1{:} , 'extrapoliert', 'erste Fehler' },'location','SouthEast');
print('-r600','-depsc',[printt '_norm.eps'])
G_D(:,2+4+rows*i)...
G_D(:,2+5+rows*i)...
G_D(:,2+6+rows*i)...
- ],type2sym{i+1});
+ ],'color', type2color(i+1,:));
hold on
for i = 1:step-1
loglog(repmat(X(:,i+1),1,3),[...
G_D(:,2+4+rows*i)...
G_D(:,2+5+rows*i)...
G_D(:,2+6+rows*i)...
- ],type2sym{i+1});
+ ],'color', type2color(i+1,:));
end
% loglog(X(:,1),[7*X(:,1).^(-1/2),3*X(:,1).^(-1/4),2*X(:,1).^(-3/4)],'-.')
+
+loglog([Ferr; Ferr],...
+ [min(G_D(:,6+rows*[0:step-1]))*0.6; max(G_D(:,6+rows*[0:step-1]))*1.3],...
+ '--','color',[.9 0 0]);
+
hold off
-title(['hmin hmax ' t0])
+% title(['hmin hmax ' t0])
xlabel('Elemente');
ylabel('Schaetzer');
legend({leg2{:} ...
+ 'erste Fehler' ...
% 'N^{-1/2}' 'N^{-1/4}' 'N^{-3/4}'...
} ,'location','southwest','box', 'off');
i=0;
loglog(repmat(X(:,i+1),1,1),[...
G_D(:,2+7+rows*i)...
- ],type2sym{i+1});
+ ],'color', type2color(i+1,:));
hold on
for i = 1:step-1
loglog(repmat(X(:,i+1),1,1),[...
G_D(:,2+7+rows*i)...
- ],type2sym{i+1});
+ ],'color', type2color(i+1,:));
end
% loglog(X(:,1),[7*X(:,1).^(-1/2),3*X(:,1).^(-1/4),2*X(:,1).^(-3/4)],'-.')
+
+loglog([Ferr; Ferr],...
+ [min(G_D(:,9+rows*[0:step-1]))*0.6; max(G_D(:,9+rows*[0:step-1]))*1.3],...
+ '--','color',[.9 0 0]);
+
hold off
-title(['KonditionierungsZahlen von V ' t0])
+% title(['KonditionierungsZahlen von V ' t0])
xlabel('Elemente');
ylabel('Condition');
legend({leg3{:} ...
+ 'erste Fehler' ...
% 'N^{-1/2}' 'N^{-1/4}' 'N^{-3/4}'...
} ,'location','northwest','box', 'off');
i=0;
loglog(repmat(X(:,i+1),1,1),[...
G_D(:,2+11+rows*i)...
- ],type2sym{i+1});
+ ],'color', type2color(i+1,:));
hold on
for i = 1:step-1
loglog(repmat(X(:,i+1),1,1),[...
G_D(:,2+11+rows*i)...
- ],type2sym{i+1});
+ ],'color', type2color(i+1,:));
end
% loglog(X(:,1),[7*X(:,1).^(-1/2),3*X(:,1).^(-1/4),2*X(:,1).^(-3/4)],'-.')
hold off
-title(['Benoetigte Zeit ' t0])
+% title(['Benoetigte Zeit ' t0])
xlabel('Elemente');
ylabel('Sekunden');
legend({leg4{:} ...
load exmpl_3DFichCube
export_mesh(coordinates,elements,neigh,[],'exmpl_3DFichCube')
+
+%% voll Analytisch
+A_plots({'meshSave/1t05n05_3DFichCube_23'},'../doc/fig/1t05n05_3DFichCube_23')
% save(file,'coordinates', 'elements','neigh','f2s')
%% Übersicht
-plotShape(coordinates,elements,'bdt');
+plotShape(coordinates,elements,'bdtv');
if(length(unique(coordinates(:,3)))==1)
view(2)
% t -> Element Text wird angezeigt (oder Element Index)
% d -> Koordinaten Text wird angezeigt (oder Koordinaten Index)
% e -> dicke linien fuer den Export
+% v -> gestrichelte linien fuer unsichtbare
%
% P.Schaefer
n = 0;
t = [0 0];
f = 0;
+v = 0;
line = 1;
optargin = size(varargin,2);
if(optargin>2)
if(ismember('c',varargin{1}))
c = 1;
end
+
+ if(ismember('v',varargin{1}))
+ v = 1;
+ end
if(ismember('a',varargin{1}))
n = 2;
elseif(ismember('n',varargin{1}))
eles = 1:size(elements,1);
coos = 1:size(coordinates,1);
+anorm = quadNorm(coordinates,elements);
+anorm(:,1:2) = anorm(:,1:2)*-1;
+
if(e==1)
for idx = eles
current = coordinates(elements(idx,[1:4,1])',:);
end
else
for idx = eles
+ if(sum(anorm(idx,:)<0)&&v==1)
+ ltype = ':';
+ else
+ ltype = '-';
+ end
current = coordinates(elements(idx,[1:4,1])',:);
% current(3,:) = current(3,:)-current(1,:)+current(2,:);
- plot3(current(:,1),current(:,2),current(:,3),'LineWidth',line,'color',[0,102,102]/255); % Zeichnet nur Kanten
+ plot3(current(:,1),current(:,2),current(:,3),ltype,'LineWidth',line,'color',[0,102,102]/255); % Zeichnet nur Kanten
hold on
end
end
end
+anorm = quadNorm(coordinates,elements);
+anorm(:,1:2) = anorm(:,1:2)*-1;
+
+
if(t(1))
for idx = eles
current = sum(coordinates(elements(idx,[2,4])',:),1)/2;
- if(e)
- cola = 'bla';
+ if(sum(anorm(idx,:)<0)&&v==1)
+ cola = [.6 .6 .6];
else
- cola = 'bla';
+ cola = [0 0 0];
end
if(isempty(desc))
text(current(1),current(2),current(3),['(' num2str(idx) ')'],'color',cola);
if(t(2))
for idx = coos
current = coordinates(idx,:);
- if(e)
- cola = 'r';
- else
- cola = 'r';
- end
+
+ cola = 'r';
+
if(isempty(desc))
text(current(1)+0.01,current(2)+0.03,current(3)+0.01, num2str(idx) ,'color',cola);
else
% Test ausführen
%Anzahl der Schritte oder wenn groeßer als 40 der Elemente
-steps = 15^4;
+steps = 5;
%Art der Berechnungen
type = [1];