% vim: set foldmethod=marker foldmarker=<<<,>>>:

  \section{Introduction} %<<<

  \begin{FIframe}{Slender Body Theory}{} %<<<

    \vspace{-1.5em}
    \begin{columns}[t]
      \column{0.5\textwidth}

        \vspace{1em}
        Stokes simulations with fibers are key to modeling complex fluids
        (suspensions, rheology, industrial, biomedical, cellular biophysics).

        %\only<2->
        {
        \vspace{2em}
        {\bf Slender Body Theory (SBT):}
        \begin{itemize}
          \item Asymptotic expansion in radius ($\varepsilon$) \\
                as $\varepsilon \to\ 0$ (Keller-Rubinow '76).

                \vspace{1em}
          \item Doublet correction to make velocity theta-independent (Johnson '80).

        \end{itemize}

        %\vspace{1em}
        %The force rep w/ plain Stokeslets doesn't make velocity theta-independent on the surface, so the doublet is added to do that better.
        %With doublet correction , error $\sim r^2. \log(r)$.
        }

        %\only<3->{
        %\vspace{1em}
        %SBT has only very recently been placed on rigorous footing.
        %(Koens-Lauga '18, Mori-Ohm-Spirn '19). %(error $\sim r \log^k(r)$)
        %}

      \column{0.5\textwidth}

        \begin{columns}
          \column{0.5\textwidth}
            \only<1>{\embedvideo{\includegraphics[width=0.99\textwidth]{videos/starfish}}{videos/starfish.mov}}%
            %\starttext
            %    \setupinteraction[state=start]
            %    \enabletrackers[graphics.locating]
            %    \externalfigure[sample.mov][width=10cm, height=10cm]
            %\stoptext
            %\only<2->{\includegraphics[width=0.99\textwidth]{videos/starfish1}}
            \\Starfish larvae \\
            (Gilpin et al. 2016)

          \column{0.5\textwidth}
            \vspace{1em}
            \includegraphics[width=0.99\textwidth]{figs/oocyte} \\
            Drosophila oocyte (Stein et al. 2021)
        \end{columns}

        \centering
        \includegraphics[width=0.6\textwidth]{figs/mitosis} \\
        Mitotic spindle (Nazockdast et al. 2015)

    \end{columns}

  \end{FIframe} %>>>

  \begin{FIframe}{Slender Body Theory Error Estimates}{} %<<<

    {\bf Error estimates:} Rigorous analysis difficult (few very recent studies)
    \begin{itemize}
      \item classical asymptotics claims: $\varepsilon^2 \log(\varepsilon)$
      \item rigorous analysis: $\varepsilon \log^{3/2}(\varepsilon)$ \qquad (Mori-Ohm-Spirn '19)
      \item numerical tests: $\varepsilon^{1.7}$ \qquad (Mitchell et al. '21 -- verify close-touching breakdown)\\
        \quad close-to-touching with gap of 10$\varepsilon$,~~ only 2.5-digits in the infty-norm.\\
        %\quad $\varepsilon$=1e-2 ~~only 1-2 digits achievable by SBT.\\
    \end{itemize}

    \only<1>{
      \centering
      \includegraphics[width=0.30\textwidth]{figs/cilia.jpg}

      \vspace{-2ex}
      {\tiny Source: http://remf.dartmouth.edu/imagesindex.html}
    }
    \only<2>{
      \vspace{1em}
      \begin{columns}
        \column{0.5\textwidth}

            \begin{tabular}{| r r r|}
              \hline
              $\varepsilon$ &     $\vct{u}_{exact}$ & Rel-Error \\
              \hline
              1e-1      &   6.1492138359856e-2 &   0.5e-2 \\
              1e-2      &   9.0984522324584e-2 &   0.1e-3 \\
              1e-3      &   1.2015655889904e-1 &   0.2e-5 \\
              1e-4      &   1.4931932907587e-1 &   0.2e-7 \\
              1e-5      &   1.7848191313097e-1 &   0.3e-9 \\
              \hline
            \end{tabular}


            %\begin{tabular}{r r r r | c r r r r} // these are for elipse
            %  \hline
            %  $\varepsilon$ & $\bm u_0$ & Error \\
            %  \hline
            %  $0.1$     & $0.0518$ & $0.7e-2$ \\
            %  $0.01$    & $0.0736$ & $0.2e-3$ \\
            %  $0.001$   & $0.0950$ & $0.3e-5$ \\
            %  $0.0001$  & $0.1163$ & $0.4e-7$ \\
            %  %$0.00001$ & $0.1377$ & $0.6e-9$ \\
            %  \hline
            %\end{tabular}
            % ellipse (semiaxes 2,0.5) radius eps=0.1...
            % N=480: L=8.578421775156826 drag force
            % F.     = 19.17234313264176
            % Fexact = 19.31188135187
            %
            % ellipse (semiaxes 2,0.5) radius eps=0.01...
            % N=480: L=8.578421775156826 drag force
            % F      = 13.58844162453679
            % Fexact = 13.59082284902
            %
            % ellipse (semiaxes 2,0.5) radius eps=0.001...
            % N=480: L=8.578421775156826 drag force
            % F      = 10.52899298797188
            % Fexact = 10.52902479066
            %
            % ellipse (semiaxes 2,0.5) radius eps=0.0001...
            % N=480: L=8.578421775156826 drag force
            % F.     = 8.594914613917958
            % Fexact = 8.594914990618
            %
            % ellipse (semiaxes 2,0.5) radius eps=1e-05...
            % N=480: L=8.578421775156826 drag force
            % F      = 7.261368067858561
            % Fexact = 7.2613680720
        \column{0.5\textwidth}
          \includegraphics[width=0.95\textwidth]{figs/ring-sed}
      \end{columns}
    }
    \only<3>{
      \centering
      \includegraphics[align=c,width=0.50\textwidth]{figs/sbt-close-error2}
      \includegraphics[align=c,width=0.40\textwidth]{figs/sbt-close-error1}
    }

  \end{FIframe} %>>>

  \begin{FIframe}{Convergent Slender Body Theory}{} %<<<

    %Goals: Develop a boundary integral framework
    %\begin{itemize}
    %  \item to actually solve the slender body BVP \\
    %        (in convergent way, not just asymptotically)
    %  \item with efficient quadratures \\
    %        (effort independent of radius)
    %  \item 

    %\end{itemize}
    \vspace{1.5em}
    {\bf Goals:} Develop boundary integral methods to solve the slender body BVP
    \begin{itemize}
      \item in a convergent way.
      \item adaptively when fibers get close.
      \item efficiently with effort independent of radius.
    \end{itemize}

    %\begin{itemize}
    %  \item No convergence analysis for fibers of given nonzero radius. %, you do not know errors in simulation .
    %  \item Uncontrolled errors when fibers close $O(\varepsilon)$. %, SBT assumptions break down.
    %\end{itemize}

    %%Efficient convergent BIE method needed, allowing adaptivity for close interactions.

    %\only<2->{
    %  \vspace{1.5em}
    %  {\bf Goals:} Develop boundary integral methods to solve the slender body BVP
    %  \begin{itemize}
    %    \item in a convergent way.
    %    \item adaptively when fibers get close.
    %    \item efficiently with effort independent of radius.
    %  \end{itemize}
    %  Validate current SBT simulations.
    %}

    %%\vspace{0.5em}
    %%Most existing qudaratures cannot resolve high aspect ratio geometries.

    \only<2->{
      \vspace{3.5em}
      Focus on rigid fibers in this talk ~~--~~ flexible fibers for future.

      \vspace{0.5em}
      {\em Related work:} ~~ Mitchell et al, '21 (mixed-BVP corresponding to flexible fiber loop)
    }


    %Only loops for now, to avoids complications with endpoint singularities.

    %\textcolor{blue}{\bf Quadratures for slender bodies}
    %\begin{itemize}
    %  \item compute interactions of filaments (eg. microtubules) in viscous fluids without asymptotic approximations.
    %  \item fully resolved boundary-integral formulation; have to deal with highly anisotropic elements.
    %\end{itemize}
  \end{FIframe} %>>>

  %\begin{FIframe}{Challenges for Boundary Integral Methods}{} %<<<

  %  Slender body aspect ratio $\sim$ $\mathcal{O}(10)$ to $\mathcal{O}(10^5)$

  %  \vspace{1em}
  %  {\bf Layer-potential quadrature}
  %  \begin{itemize}
  %    \item efficient with cost independent of aspect ratio.
  %  \end{itemize}

  %  \vspace{1em}
  %  {\bf Boundary integral equation formulations}
  %  \begin{itemize}
  %    \item remain well-conditioned as $\epsilon \rightarrow 0$
  %  \end{itemize}

  %\end{FIframe}

  %%\begin{FIframe}{Motivation}{} %<<<
  %%  \begin{itemize}
  %%    \item aspect ratios of $10^4$ or greater
  %%    \item existing quadrature schemes are not efficient in this regime
  %%  \end{itemize}
  %%\end{FIframe} %>>>
  %%\begin{FIframe}{Outline}{} %<<<
  %%{\large

  %%  \begin{itemize}
  %%    \item Slender Body Quadrature

  %%    \vspace{1em}
  %%    \item Stokes Mobility Problem

  %%  \end{itemize}

  %%}
  %%\end{FIframe} %>>>

  %%>>>

