2024-06-13-talk-CSBQ/conclusions.tex

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  \section{Conclusions} %<<<

  \begin{FIframe}{Conclusions}{} %<<<

    %\vspace{1.2em}
    \begin{itemize}
      \item Convergent boundary integral formulation for slender bodies,
        \begin{itemize}
          \item unlike SBT, boundary conditions enforced to high accuracy.
        \end{itemize}
        \vspace{0.5ex}
      \item Special quadrature - efficient for aspect ratios as large as $10^5$.
        \begin{itemize}
          %\item fast computation of modal/toroidal Green's function.
          %\item special (Chebyshev) quadratures for singular integrals along length of fibers.
          \item quadrature setup rates $\sim 20,\!000$ unknowns/s/core (at 7-digits).
        \end{itemize}
        \vspace{0.5ex}
      \item Combined field BIE formulations,
        \begin{itemize}
          \item well-conditioned for slender-body geometries.
          %\item high-order time stepping (SDC), Krylov subspace preconditioner.
        \end{itemize}
    \end{itemize}

    \only<2>{
    \vspace{1.5em}
    {\bf Limitations and ongoing work:}
    \begin{itemize}
      \item Flexible fibers -- applications in biological fluids.
      %\item Open problems: collision handling.

      %\item Open fibers (singularities at ends). %Special elements (and quadratures) for fiber endpoints (non-loop geometries).
      %%\item Replace Chebyshev quadratures with generalized Gaussian quadratures% of Bremer, Gimbutas and Rokhlin - SISC 2010.
      %%\item Parallelisation with proper load balancing.
      %%\item FMM acceleration of far-field computation.
      %%\item Apply to problems in biological fluids.
      %\item Mobility problem and flexible fibers.
      %\item Comparison w/ SBT efficiency when SBT is sufficiently accurate.
    \end{itemize}
    }



    % - end-caps so that we can have non-loop geometries
    % - replace Chebyshev quadratures with Generalized Gaussian Quadratures
    %%%%%%%% - develop preconditioners for close to touching geometries?
    % direct comparison with slender-body theory
    % develop applications ...
    % parallelization



    %\vspace{1em}
    %\textcolor{blue}{\bf Future directions}

    %\vspace{0.5em}
    %\begin{columns}
    %  \column{0.9\textwidth}
    %    \begin{itemize}
    %      \item apply quadratures to numerical simulations of biological processes (collaboration with CCB).
    %    \end{itemize}
    %  \column{0.1\textwidth}
    %\end{columns}

    %\vspace{0.5em}
    %\begin{columns}
    %  \column{0.65\textwidth}
    %    \begin{itemize}
    %      \item study convergence in close-to-touching setups; ~~ require adaptivity in length as well as $\theta$-dimensions.
    %    \end{itemize}
    %  \column{0.35\textwidth}
    %    \includegraphics[width=0.9\textwidth]{figs/close-touching}

    %    {\small (fig from Morse et al.)}
    %\end{columns}

    % - end-caps so that we can have non-loop geometries
    % - replace Chebyshev quadratures with Generalized Gaussian Quadratures
    %%%%%%%% - develop preconditioners for close to touching geometries?
    % direct comparison with slender-body theory
    % develop applications ...
    % parallelization

    %\vspace{0.75em}
    %\begin{columns}
    %  \column{0.9\textwidth}
    %    \begin{itemize}
    %      \item develop similar ideas for other special cases and more generally for high aspect ratio panels

    %      \begin{center}
    %        \includegraphics[width=0.6\textwidth]{slender-body/high-aspect-panels.png}
    %      \end{center}
    %    \end{itemize}
    %  \column{0.1\textwidth}
    %\end{columns}

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  \begin{FIframe}{Numerical Results - Sedimentation Flow}{} %<<<

    \vspace{-1.9em}
    \centering
    \only<1>{ \embedvideo{\includegraphics[width=0.94\textwidth]{videos/bacteria2_.png}}{videos/bacteria2_.mov} }%
    \only<2>{ \embedvideo{\includegraphics[width=0.47\textwidth]{videos/bacteria64-density.png}}{videos/bacteria64-density.mov} }

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