kolloquium.tex

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\title[Masterthesis ]{\small{Implementierung und Untersuchung \\einer hoch effizienten Methode \\zur Druck-Geschwindigkeits-Kopplung}}
\subtitle{\small{Masterthesis}}
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\author[Fabian Gabel ]{Fabian Gabel}

\institute[FNB, TU Darmstadt]
%{Fachgebiet für Numerische Berechnungsverfahren im Maschinenbau\\
%Technische Universität Darmstadt}

\date[\today]
{Kolloquium, 02.04.2015}

\subject{seminar presentation}

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\section{Motivation}
\begin{frame}

\frametitle{Motivation}
    \vspace{0.5cm}
    Herausforderungen für CFD-Applikationen:
    \begin{itemize}
    \item \small{schnelle Verfügbarkeit von Simulationsergebnissen}
    \item \small{Ergebnisse mit hoher Genauigkeit}
    \item \small{Umgang mit komplexen Geometrien}
    \item \small{multiphysikalische Problemstellungen}
    \end{itemize}

    \begin{columns}
      \vspace{-0.5cm}
        \begin{column}{6cm}
            \begin{figure}
            \centering
            \includegraphics[width= 0.8\linewidth]{./img/turbine.jpg}
            \setlength{\abovecaptionskip}{-5pt plus 3pt minus 2pt}
            \caption{Gasturbine (VDI)}
            \end{figure}
        \end{column}
        \begin{column}{8cm}
            \begin{itemize}
                \item \small{Adaptivität}
                \item \small{Parallelrechner (Skalierbarkeit)}
                  \item \small{robuste Algorithmen}
                \item \small{Variablenkopplung berücksichtigen}
            \end{itemize}
        \end{column}
    \end{columns}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}

\frametitle{Motivation}
    Vollständig gekoppelter Lösungsansatz \\für Navier-Stokes Gleichungen (Darwish et al. 2009):
    \begin{itemize}
    \item simultane Lösung des Geschwindigkeits- und Druckfelds
    \item semi-implizite Druck-Geschwindigkeits-Kopplung 
    \item robuster Algorithmus ohne Unterrelaxation
    \end{itemize}

    \vspace{1cm}
    Neue Herausforderung:
    \begin{itemize}
      \item Umgang mit Speicheranforderungen
      \item Auswahl geeigneter skalierbarer Gleichungslöser
    \end{itemize}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Aufgabenstellung und Bearbeitung}
\begin{frame}

\frametitle{Aufgabenstellung und Bearbeitung}
  \begin{itemize}
    \item Implementierung eines vollständig gekoppelten Lösungsansatzes 
      \begin{itemize}\scriptsize
        \item Finite-Volumen Methode für 3d Navier-Stokes Gleichungen 
        \item blockstrukturierte, lokal verfeinerte Gitter mit hängenden Knoten
        \item MPI-Parallelisierung des Lösers
        \item Kopplungsansätze für Temperaturgleichung 
      \end{itemize}
    \item Skalierbarkeitsuntersuchung auf HHLR
      \begin{itemize}\scriptsize
        \item Manufactured Solution
      \end{itemize}
    \item Performancevergleich mit herkömmlichem SIMPLE-Verfahren
      \begin{itemize}\scriptsize
        \item Kanalströmung mit komplexem Hindernis
        \item auftriebsgetriebene Nischenströmung (Adaption MIT - Benchmark)
      \end{itemize}
  \end{itemize}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Implementierung}
\begin{frame}\scriptsize
  \frametitle{Implementierung - Diskretisierung}
  \begin{itemize}
    \item Finite-Volumen Methode für stationäre, inkompressible Navier-Stokes Gleichungen
    \item standardmäßige Diskretisierung der konvektiven und viskosen Terme
    \item implizite Berücksichtigung des Druckgradienten
  \end{itemize}
  \begin{itemize}
    \item diskretisierte Massenerhaltungsgleichung
  \end{itemize}
\begin{align*}
  \label{eq:massbalance}
  \sum_{f \in \{w,s,b,t,n,e\}} u_{i,f} n_{f_i} S_f = 0
\end{align*}
  \begin{itemize}
    \item druckgewichtete Interpolation zur Massenflussberechnung
  \end{itemize}
\begin{align*}
  u_{i,f}^{(n)} 
  &=
  \left[\left(1 - \gamma_f\right) u_{i,P}^{(n)} + \gamma_f u_{i,Q}^{(n)} \right] \\
  & - \left(\left(1 - \gamma_f\right) \frac{V_P}{a_P^{u_i}} + \gamma_f \frac{ V_Q}{a_Q^{u_i}}\right)
  \left[ 
    \underline{ \left(\frac{\partial p}{\partial x_i}\right)_f^{(n)}}
  - \frac{1}{2} 
  \left( 
    \left( \frac{\partial p}{\partial x_i} \right)_P^{(n-1)} 
  + \left(\frac{\partial p}{\partial x_i}\right)_Q^{(n-1)} 
  \right)
  \right]
\end{align*}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}\scriptsize
  \frametitle{Implementierung - Temperaturkopplung}
  \begin{itemize}
    \item Diskretisierung der Boussinesq Approximation des Auftriebsterms \( \rho \beta (T - T_0 )\, g_i \)
    \item implizit oder explizit
  \end{itemize}
  \begin{align*}
    \frac{\partial}{\partial x_j} \left( \rho u_j T - \kappa \frac{\partial T}{\partial x_j} \right) =& q_T
  \end{align*}
  \begin{itemize}
    \item Newton-Raphson Linearisierung des konvektiven Terms der Temperaturgleichung \\(Galpin et al. 1986)
  \end{itemize}
\begin{align*}
  \left(u_{j,f} T_f \right)^{(n)} 
  &\approx 
  \underline{\vphantom{(u_j T)}{u_{j,f}}^{(n-1)} \vphantom{(u_{j,f})}{T_f}^{(n)}}  + \vphantom{(u_{j,f} T_f)}{u_{j,f}}^{(n)} \vphantom{(u_jT)}{T_f}^{(n-1)}  -  \vphantom{(u_j T)}{u_{j,f}}^{(n-1)} \vphantom{(u_j T)}{T_f}^{(n-1)}
\end{align*}
  \begin{itemize}
    \item Diskretisierung unter Verwendung der druckgewichteten Interpolation
  \end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}\scriptsize
\frametitle{Implementierung \\- System algebraischer Gleichungen}
\begin{align*}
  a_P^{u_i} u_{P,i}^{\vphantom{u_i}}
  + \sum_{F \in NB(P)} a_F^{u_i} u_{F,i}^{\vphantom{u_i}} 
  + \underbrace{\vphantom{\sum_F}{a_P^{u_i,p} p_P^{\vphantom{u_i,p}}
  + \sum_{F \in NB(P)} a_F^{u_i,p} p_F^{\vphantom{u_i,p}}}}_{\hbox{Pressure-velocity coupling}} 
  + \underbrace{\vphantom{\sum_F}{ a_P^{u_i,T} T_P^{\vphantom{u_i,T}}}}_{\hbox{Boussinesq approximation}} 
  &= b_{P,u_i}^{\vphantom{u_i,T}} \quad i=1,...,3  \\[2.0em]
  a_P^{\, p} p_{P}^{\vphantom{p}} 
  + \sum_{F \in NB(P)} a_F^{\,p} p_F^{\vphantom{\,p}} 
  + \underbrace{ \sum_{j=1}^3 \left(a_P^{\,p,u_j} u_{P,j}^{\vphantom{p,u_j}}
  + \sum_{F \in NB(P)} a_F^{\, p,u_j} u_{F,j}^{\vphantom{p,u_j}} \right)}_{\hbox{Pressure-velocity coupling}} 
  &= b_{P,p}^{\vphantom{p,u_j}}  \\[2.0em]
  a_P^{T} T_{P}^{\vphantom{T}} 
  + \sum_{F \in NB(P)} a_F^{T} T_F^{\vphantom{T}} 
  + \underbrace{\sum_{j=1}^3 \left(a_P^{T,u_j} u_{P,j}^{\vphantom{T,u_j}}
  + \sum_{F \in NB(P)} a_F^{T,u_j} u_{F,j}^{\vphantom{T,u_j}} \right) 
  + a_P^{T,p} p_P^{\vphantom{T,p}} 
+ \sum_{F \in NB(P) } a_F^{T,p} p_F^{\vphantom{T,p}}}_{\hbox{Newton-Raphson linearization}} 
  &= b_{P,T}^{\vphantom{T,p}}
\end{align*}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
  \frametitle{Implementierung \\- Blockränder und Parallelisierung}
  \vspace{0.5cm}
  \begin{itemize}\scriptsize
    \item implizite Behandlung der Blockränder nach Lilek et al. (1997)
    \item Portable Extensible Toolkit for Scientific Computation (PETSc)
    \item PETSc Datenstrukturen zur MPI-Parallelisierung des Lösers
    \item spezielle Vektorobjekte zur Verwaltung von Ghost-Values
  \end{itemize}
  \vspace{-0.5cm}
  \begin{columns}
  \begin{column}{4cm}
  \begin{figure}
    \centering
    \label{fig:nonmatching}
    \resizebox{1.0\linewidth}{!}{\input{./img/nonmatching.tikz.tex}}
  %\caption{Non-matching grid cells with hanging nodes at a two-dimensional block boundary. Indexing is based on the face segments $S_l$}
  \end{figure}
  \end{column}
  \begin{column}{8cm}
  \begin{figure}
    \centering
    \label{fig:segassemble}
    \resizebox{\linewidth}{!}{\input{./img/ghosting.tikz.tex}}
  %\caption{Storage and update of ghost values in vectors related to variables on multi block domains. The blocks have been assigned to two different Processes, \emph{Proc} $1$ and \emph{Proc} $2$. The control volumes of the two-dimensional problem domain are indexed with respect to the process local indexing.}
    \label{fig:ghosting}
  \end{figure}
\end{column}
\end{columns}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Implementierung - Assemblierung}
\begin{columns}
  \begin{column}{0.5cm}
  \end{column}
  \begin{column}{3.5cm}
  \begin{figure}
    \centering
    \resizebox{1.2\linewidth}{!}{\input{./img/blockstruc.tikz.tex}}
    \caption{\tiny{Blockstrukturiertes Gitter und resultierende Matrixbelegung für eine Variable ohne Kopplung}}
  \end{figure}
  \end{column}
  \begin{column}{10cm}
    \vspace{-0.3cm}
  \begin{figure}
    \centering
    \label{fig:segassemble}
    \resizebox{.7\linewidth}{!}{\input{./img/matrix.tikz.tex}}
%\caption{Non-zero structure of the linear systems used in the SIMPLE algorithm for a block-structured grid consisting of one  $2\times2\times2$ cell and one $3\times3\times3$ cell block}
  \end{figure}
  \end{column}
\end{columns}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}\scriptsize
\frametitle{Implementierung \\- Assemblierung mit Kopplung}
%\vspace{-0.5cm}
\begin{columns}
\begin{column}{7cm}
\begin{figure}
  \centering
  \label{fig:cpldassemble}
  \resizebox{\linewidth}{!}{\input{./img/coupledmat.tikz.tex}}
  %\caption{Matrixstruktur für Blockmatrizen}
\end{figure}
\end{column}
  \vspace{1cm}
\begin{column}{5cm}
  unterschiedliche Kopplungsterme:\\
  \quad \sqa \, Druck-Geschwindigkeit \\
  \quad \sqb \, Geschwindigkeit-Temperatur \\
  \quad \sqc \, Temperatur-Geschwindigkeit/Druck \\
  \vspace{3cm}
  \hspace*{-3cm} \scriptsize{$\color{fnbblue}\blacksquare$} an Stelle eines Eintrags steht eine dicht besetzte \(5x5\) Matrix
\end{column}
\end{columns}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{frame}
%  \frametitle{Implementierung - Lösungsalgorithmus}
%\alglanguage{pseudocode}
%\begin{algorithm}[H]
%\label{al:simple}
%\caption{SIMPLE Algorithm}
%\begin{algorithmic}\scriptsize
%\State{\textit{INITIALIZE} variables}
%\While{(convergence criterion not accomplished)}
%\State{\textit{SOLVE} linearized momentum balances}
%\State{\textit{CALCULATE} mass fluxes}
%\State{\textit{SOLVE} pressure correction equation to assure continuity}
%\State{\textit{UPDATE} pressure}
%\State{\textit{UPDATE} velocities and mass fluxes}
%\If{(coupled scalar equation)}
%  \State{\textit{SOLVE} scalar equation}
%\EndIf
%\EndWhile
%\If{(decoupled scalar equation)}
%  \State{\textit{SOLVE} scalar equation}
%\EndIf
%\end{algorithmic}
%\end{algorithm}
%\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
  \frametitle{Implementierung - Lösungsalgorithmus}
\alglanguage{pseudocode}
\begin{algorithm}[H]
\caption{Fully Coupled Solution Algorithm}
\begin{algorithmic}\scriptsize
\State{\textit{INITIALIZE} variables}
\While{(convergence criterion not accomplished)}
\If{(temperature coupling)}
  \State{\textit{SOLVE} the linear system for velocities, pressure and temperature}
\Else
  \State{\textit{SOLVE} the linear system for velocities and pressure}
\EndIf
\State{\textit{CALCULATE} mass fluxes }
\If{(coupled scalar equation)}
  \State{\textit{SOLVE} scalar equation}
\EndIf
\EndWhile
\end{algorithmic}
\end{algorithm}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Performanceuntersuchung}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
  \frametitle{McCalpin STREAM Benchmark \\- Sustainable Memory Bandwidth}

\vspace{-0.2cm}
\begin{columns}
  \begin{column}{8.5cm}
  \begin{figure} \centering
  \pgfplotsset{every axis/.append style={
      font=\tiny,
      line width=1pt,
      tick style={line width=0.8pt}}}
      \begin{tikzpicture}
        \pgfplotsset{every axis legend/.append style={
        at={(0.5,1.03)},
        anchor=south}}
      \begin{axis}[
        ylabel={Sustainable memory bandwidth MB/s},
        xlabel={Number of bound cores},
        xtick={1,4,8,12,16},
        %ytick={1.7e-003,1.75e-3,1.8e-003,1.85e-3},
        %yticklabels={1.7E-3,1.75E-3,1.8E-3,1.85E-3},
        %ymin=1.65e-003,ymax=1.9e-003,
        xmin=0,xmax=17,
        %ymin=0.5e4,ymax=1.3e5,
        ymin=0,ymax=8e4,
        %legend pos=outer north east,
        %height=20cm,width=10cm
        width=9.0cm,
        height=6.5cm,
        grid=major,
        ]
    %    \addplot[color=black,mark=o] file {./files/mpi1.def};
    %    \addplot[color=black,mark=square] file {./files/mpi1.op1};
    %    \addplot[color=tud2a,mark=*] file {./files/mpi1.core};
    \addplot[color=tud4a,mark=*] file {./files/mpi1.def};
    \addplot[color=tud9a,mark=square*] file {./files/mpi1.op1};
    \addplot[color=tud2a,mark=triangle*] file {./files/mpi1.core};
        \addlegendentry{Default binding};
        \addlegendentry{map-by ppr:8:node map-by ppr:4:socket}
        \addlegendentry{map-by core}
      \end{axis}
      \end{tikzpicture}
    \end{figure}
    \end{column}
    \begin{column}{4cm}
      \begin{itemize}\scriptsize
        \item Nutzung eines Knotens der MPI1-Sektion des HHLR 
        \item low-level Benchmark
        \item für bandbreitenlimitierte Programmperformance
        \item TRIAD KERNEL: \( a[i] = b[i] + scalar * c[i] \) 
      \end{itemize}
    \end{column}
    \end{columns}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}\scriptsize
  \frametitle{Manufactured Solution}
  \begin{itemize}
    \item divergenzfreie Lösung für Geschwindigkeit
    \item \(\vec{u} = \nabla \times \vecg{\Psi}\)
  \end{itemize}
  \begin{align*}
    u_1(\vec{x}) &= 2\,\cos \left( {x_1}^{2}+{x_2}^{2}+{x_3}^{2} \right) x_2+2\,\sin \left( {x_1}^{2}+{x_2}^{2}+{x_3}^{2} \right) x_3  \\
    u_2(\vec{x}) &= 2\,\cos \left( {x_1}^{2}+{x_2}^{2}+{x_3}^{2} \right) x_3-2\,\cos \left( {x_1}^{2 }+{x_2}^{2}+{x_3}^{2} \right) x_1 \\
    u_3(\vec{x}) &= -2\,\sin \left( {x_1}^{2}+{x_2}^{2}+{x_3}^{2} \right) x_1-2\,\cos \left( {x_1}^{ 2}+{x_2}^{2}+{x_3}^{2} \right) x_2
  \end{align*}
  \begin{itemize}
    \item Problemgebiet so wählen, dass Kontinuität im diskreten Sinne global erfüllt wird
  \end{itemize}
  \begin{itemize} 
    \item Lösungen für Druck (und Temperatur)
  \end{itemize}
  \begin{align*}
    p(\vec{x}) &= \sin \left( {x_1}^{2}+{x_2}^{2}+{x_3}^{2} \right) \cos \left( {x_1}^{2}+ {x_2}^{2 }+{x_3}^{2} \right) \\
    \Big(T(\vec{x}) &= \sin \left( {x_1}^{2} \right) \cos \left( {x_2}^{2} \right) \sin \left( {x_3 }^{2} \right) \Big)
  \end{align*}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Manufactured Solution - Strong-Scaling}
\begin{columns}
\begin{column}{6.0cm}
\begin{figure}
  %\setlength{\abovecaptionskip}{5pt plus 3pt minus 2pt}
  \begin{center}
  \begin{tikzpicture}
      \begin{loglogaxis}[
        legend columns=-1,
        legend entries={SIMPLE algorithm,Fully coupled algorithm},
        legend to name=named,
        xlabel=Number of processes,
        ylabel=Wall-clock time s,
        xtick={1,2,4,8,16,32,64,128,256,512,1024},
        xticklabels={1,2,4,8,16,32,64,128,256,512,1024},
        xmin=1,xmax=1200,
        width=6.5cm,
        height=6cm,
        font=\tiny,
      ]
      \addplot[color=black,mark=*] file {./files/seg.128};
      \addplot[color=black,mark=square*] file {./files/cpld.gamg.128};
      \ref{named}
    \end{loglogaxis}
  % \caption{Wall-clock time comparison for segregated and fully coupled solution algorithm solving for an analytical solution on a grid with $128^3$ cells}
  % \label{fig:wall}
  \end{tikzpicture}
  \end{center}
\end{figure}
\end{column}
\begin{column}{6.5cm}
  \begin{figure}
  \begin{center}
  \begin{tikzpicture}
      \begin{loglogaxis}[
        xlabel=Number of processes,
        ylabel=Speed-Up,
        xtick={1,2,4,8,16,32,64,128,256,512,1024},
        xticklabels={1,2,4,8,16,32,64,128,256,512,1024,},
        ytick={1,2,4,8,16,32,64,128,256,512},
        yticklabels={1,2,4,8,16,32,64,128,256,512},
        grid=major,
        xmin=8,xmax=1200,
        width=5.5cm,
        height=6cm,
        ymax=75,
        font=\tiny,
        legend style={ cells={anchor=east}, legend pos=outer north east, }
      ]
      \addplot[color=black,mark=*] file {./files/speedup.seg.128};
      \addlegendentry {SEG}
      \addplot[color=black,mark=square*] file {./files/speedup.cpld.128};
      \addlegendentry {CPLD}
      \addplot[mark=none,black] file {./files/speedup.ideal};
    \end{loglogaxis}
  \end{tikzpicture}
  \end{center}
  \end{figure}
\end{column}
\end{columns}

%\vspace{-0.5cm}
\begin{itemize}\scriptsize
  \item \(128 \times 128 \times 128\) Unbekannte auf Prozesse verteilt
  \item ab 256 Prozessen zu wenig Unbekannte pro Prozess
\end{itemize}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
  \frametitle{Manufactured Solution - Weak-Scaling}
\begin{columns}
\begin{column}{5cm}
  \vspace{-0.5cm}
\begin{figure}
  %\setlength{\abovecaptionskip}{5pt plus 3pt minus 2pt}
  \begin{center}
  \begin{tikzpicture}
      \begin{loglogaxis}[
        legend columns=-1,
        legend entries={SIMPLE algorithm,Fully coupled algorithm},
        legend to name=named,
        xlabel=Number of processes,
        ylabel=Number of outer iterations,
        xtick={1,2,4,8,16,32,64,128,256,512,1024},
        xticklabels={1,2,4,8,16,32,64,128,256,512,1024,},
        %ytick={1,2,4,8,16,32,64,128,256,512},
        %yticklabels={1,2,4,8,16,32,64,128,256,512},
        %grid=major,
        xmin=8,xmax=1200,
        height=5.0cm,
        font=\tiny,
      ]
      \addplot[color=black,mark=*] file {./files/weak.seg.iter};
      \addplot[color=black,mark=square*] file {./files/weak.iter};
      \ref{named}
    \end{loglogaxis}
  % \caption{Wall-clock time comparison for segregated and fully coupled solution algorithm solving for an analytical solution on a grid with $128^3$ cells}
  % \label{fig:wall}
  \end{tikzpicture}
  \end{center}
\end{figure}
\end{column}
\begin{column}{7cm}
  \begin{figure}
  \begin{center}
  \begin{tikzpicture}
      \begin{loglogaxis}[
        xlabel=Number of processes,
        ylabel=Wall-clock time s,
        xtick={1,2,4,8,16,32,64,128,256,512,1024},
        xticklabels={1,2,4,8,16,32,64,128,256,512,1024,},
        xmin=8,xmax=1200,
        height=5.0cm,
        font=\tiny,
        legend style={ cells={anchor=east}, legend pos=outer north east, }
      ]
      \addplot[color=black,mark=*] file {./files/weak.seg.time};
      \addlegendentry {SEG}
      \addplot[color=black,mark=square*] file {./files/weak.time};
      \addlegendentry {CPLD}
    \end{loglogaxis}
  \end{tikzpicture}
  \end{center}
  \end{figure}
\end{column}
\end{columns}

\vspace{0.5cm}
\begin{itemize}\scriptsize
  \item \(32 \times 32 \times 32\) Unbekannte pro Prozess
  \item gekoppelter Algorithmus skaliert 
  \item Implementierung skaliert nicht (Präkonditionierer)
\end{itemize}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
  \frametitle{Kanalströmung - Problemstellung}
  \begin{columns}
    \begin{column}{5cm}
\begin{figure}
  \centering
  \resizebox{1.0\linewidth}{!}{\includegraphics{./img/channel3d.pdf}}
  \setlength{\abovecaptionskip}{-15pt plus 3pt minus 2pt}
  \caption{Skizze der Kanalströmung}
  \label{fig:sketch}
\end{figure}
\end{column}
\begin{column}{5cm}
  \begin{itemize}\scriptsize
    \item Würfel als Hindernisse
    \item parabolisches Einstromprofil
    \item \(Re \approx 20 \)
    \item laminare Strömung
    \item nichttriviale Blockübergänge
  \end{itemize}
\end{column}
\end{columns}
\vspace{-0.3cm}
\begin{figure}
   \centering
    \subfigure{
      \resizebox{0.3\linewidth}{!}{\input{./img/blocking2.tikz.tex}}
      \raggedleft{}
    }
    \hfil
    \subfigure{
      \resizebox{0.3\linewidth}{!}{\input{./img/blocking.tikz.tex}}
      \raggedleft{}
    }
    \caption{Blockaufteilung um die Hindernisse im Kanal}
    \label{fig:blocking}
\end{figure}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
  \frametitle{Kanalströmung \\- Druck-Geschwindigkeits-Kopplung}
  \vspace{0.3cm}
\begin{figure}
  \centering
  \resizebox{0.8\linewidth}{!}{\input{./img/channel.tikz.tex}}
  \caption{Numerisches Gitter am West- und Ostrand der Kanalströmung}
  \label{fig:channel1}
\end{figure}
\vspace{-0.7cm}
\begin{table}[h!]\centering
  \caption{Performanceuntersuchung der Kanalströmung zum Vergleich des SIMPLE (SEG) und vollständig gekoppelten (CPLD) Lösungsalgorithmus}
\ra{1.3}
\resizebox{7cm}{!}{
  \begin{tabular}{lcccc}\toprule
    No. of unknowns & SEG - time s & CPLD - time s & SEG - its. & CPLD - its. \\
    \midrule
    \rowcolor{tud0b} 75768    & 0.2226E+02 & 0.2674E+02 & 151  & 67 \\
    \rowcolor{black!00} 408040   & 0.4053E+03 & 0.1499E+03 & 355  & 42 \\
    \rowcolor{tud0b} 2611080  & 1.1352E+05 & 0.3105E+04 & 1592 & 39 \\
  \end{tabular}
}
  \label{tab:channelcompare}
\end{table}
%
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Temperaturgetriebene Nischenströmung \\- Problemstellung}
\begin{columns}
  \begin{column}{6cm}
    \vspace{-0.1cm}
    \begin{figure}[h!]
    \centering
    \resizebox{0.75\linewidth}{!}{\includegraphics[trim=3.5cm 5cm 0cm 5cm, scale=0.5,clip=true]{./img/cavity.pdf}}
    \end{figure}
    \vspace{-0.9cm}
    \begin{figure}[h!]
    \setlength{\abovecaptionskip}{-5pt plus 3pt minus 2pt}
    \centering
    \hspace*{0.45cm}\resizebox{0.75\linewidth}{!}{\includegraphics[trim=3.5cm 5cm 0cm 5cm, scale=0.5,clip=true]{./img/cavityw.pdf}}
    \caption{Temperatur-und $w$-Geschwindigkeitsfeld}
    \end{figure}
  \end{column}
 \begin{column}{6cm}
   \begin{itemize}\scriptsize
     \item Testfall mit natürlicher Konvektion
     \item starke Kopplung, da  Strömung getrieben durch Temperaturdifferenzen 
     \item 3d Kavität mit isothermer Ost- (kalt) und Westwand (heiß)
     \item alle übrigen Wände adiabat
     \item \(Ra \approx 10^4 \), \(Pr = 0.71 \)
     \item stationäre, laminare Strömung
   \end{itemize}
 \end{column}
\end{columns}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
  \frametitle{Temperaturgetriebene Nischenströmung \\- Temperaturkopplung}

\begin{table}[h!]\centering
\ra{1.3}
\caption{Performance des SIMPLE Algorithmus (SEG) und des gekoppelten Algorithmus (CPLD) mit impliziter Boussinesq Approximation (TCPLD) und semi-impliziter Temperatur-Geschwindigkeits/Druck-Kopplung (NRCPLD).}
\begin{columns}
\begin{column}{7cm}
\resizebox{7cm}{!}{
\begin{tabular}{cccc}\toprule
    Resolution & Solver configuration & Time s & No. of nonlinear its. \\
    \midrule
    \rowcolor{tud0b}\multirow{4}{*}{}            & SEG    & 0.3719E+02 & 203 \\
    \rowcolor{tud0b}                             & CPLD   & 0.6861E+02 & 62  \\
    \rowcolor{tud0b}                             & TCPLD  & 0.1012E+03 & 31  \\
    \rowcolor{tud0b} \multirow{-4}{*}{32x32x32}  & NRCPLD & 0.2153E+02 & 22  \\ %\hline
    %
    \rowcolor{black!00}\multirow{4}{*}{}            & SEG    & 0.1997E+04 &  804 \\
    \rowcolor{black!00}                             & CPLD   & 0.7687E+03 &  63  \\
    \rowcolor{black!00}                             & TCPLD  & 0.1278E+04 &  59  \\
    \rowcolor{black!00} \multirow{-4}{*}{64x64x64}  & NRCPLD & 0.4240E+03 &  17  \\ %\hline
    %
    \rowcolor{tud0b}\multirow{4}{*}{}               & SEG    & 0.5197E+05 &  3060 \\
    \rowcolor{tud0b}                                & CPLD   & 0.1860E+05 &  74   \\
    \rowcolor{tud0b}                                & TCPLD  & 0.1950E+05 &  50   \\
    \rowcolor{tud0b} \multirow{-4}{*}{128x128x128}  & NRCPLD & 0.6155E+04 &  18   \\ %\hline
    %
  \end{tabular}
}
  \label{tab:cavitycompare}
\end{column}
\begin{column}{2cm}
\begin{figure}
  \centering
  \label{fig:cpldassemble}
  \resizebox{1.3\linewidth}{!}{\input{./img/coupledmat.tikz.beamer.tex}}
  %\caption{Matrixstruktur für Blockmatrizen}
\end{figure}
\end{column}
  \vspace{1cm}
\end{columns}
\end{table}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Fazit und Ausblick}
\begin{frame}\scriptsize
  \frametitle{Fazit und Ausblick}
  \begin{itemize}
    \item MPI parallelisierte Implementierung eines vollständig gekoppelten Lösungsverfahrens
    \item 3d blockstrukturierte Gitter mit hängenden Knoten
    \item unterschiedliche Methoden zur Temperaturkopplung
    \item Untersuchung der Skalierbarkeit auf HHLR
    \item Performancevergleich mit SIMPLE-Löser 
  \end{itemize}
  \vspace{1cm}
  weiterer Forschungsbedarf:
  \begin{itemize}
    \item instationäre laminare Strömungen bei hohen Rayleighzahlen
    \item implizite Kopplung im Kontext der Turbulenz, Mehrphasenströmungen, etc.
    \item matrixfreie Präkonditionierer
    \item physikbasierte Präkonditionierer (SIMPLE(R) oder andere Schur-Komplement Präkonditionierung)
    \item spezielle algebraische Mehrgitter-Präkonditionierer
  \end{itemize}
\end{frame}
% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[allowframebreaks]
  \frametitle<presentation>{Literatur}    
  \begin{thebibliography}{10} 
  \beamertemplatearticlebibitems
  \setbeamertemplate{bibliography item}[online]

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  S., Adams,
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  Kaushik D.,
  Knepley, M.~G.,
  Curfman McInnes, L.,
  Rupp, K.,
  Smith, B.~F. 
  and Zhang H.,}
   \newblock {PETS}c {W}eb Page.
   \newblock \url{http://www.mcs.anl.gov/petsc} (zuletzt besucht am \today).

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  \beamertemplatearticlebibitems
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    Galpin, P. F. und Raithby, G. D.,
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  \bibitem{klaij13}
    Klaij, C. M. und Vuik, C.
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  \bibitem{lilek97}
    Lilek, \u{Z}., Muzaferija, S., Peri\'c, M. und Seidl, V.,
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    \newblock {\em Numerical Heat Transfer, Part B: Fundamentals}, 32(4):385--401, 1997.

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  \bibitem{vakilipour12}
    Vakilipour, S. und Ormiston, S. J.,
    \newblock A Coupled Pressure-Based Co-Located Finite-Volume Solution Method for Natural-Convection Flows.
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  \end{thebibliography}

\end{frame}

\end{document}