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\title[Earthquake-Soil-Structure Systems]
% (optional, use only with long paper titles)
{Earthquake-Soil-Structure Systems}

%\subtitle
%{{\tiny full set of slides available at:}\\
%%\verb{http://sokocalo.engr.ucdavis.edu/~jeremic/}
%}

\pgfdeclareimage[height=0.2cm]{university-logo}{/home/jeremic/BG/amblemi/ucdavis_logo_blue_sm}

%\author[Boris Jeremi{\'c}, CompGeoMech \includegraphics[width=8cm]{/home/jeremic/BG/amblemi/ucdavis_logo_gold_lrg}] % (optional, use only with lots of authors)
\author[Boris Jeremi{\'c}] % (optional, use only with lots of authors)
{Boris~Jeremi{\'c} and 
Sashi~Kunnath \\ with\\ 
Kallol~Sett and 
Nima~Tafazzoli}
% - Give the names in the same order as the appear in the paper.
% - Use the \inst{?} command only if the authors have different
%   affiliation.

%\institute[Computational Geomechanics Group
\institute[Structural/Geotechnical Engineering
\pgfuseimage{university-logo} \hspace*{-0.3truecm}] % (optional, but mostly needed)
{ \vspace*{1cm} \\

  Department of Civil and Environmental Engineering\\
  University of California, Davis}
% - Use the \inst command only if there are several affiliations.
% - Keep it simple, no one is interested in your street address.

\date[] % (optional, should be abbreviation of conference name)
{ }
% - Either use conference name or its abbreviation.
% - Not really informative to the audience, more for people (including
%   yourself) who are reading the slides online

\subject{}
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% If you have a file called "university-logo-filename.xxx", where xxx
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%\logo{\pgfuseimage{university-logo}}

% \pgfdeclareimage[height=0.5cm]{university-logo}{university-logo-filename}
% \logo{\pgfuseimage{university-logo}}



% Delete this, if you do not want the table of contents to pop up at
% the beginning of each subsection:
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   \begin{frame}<beamer>
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  \frametitle{Outline}
  \tableofcontents
  % You might wish to add the option [pausesections]
\end{frame}


% Structuring a talk is a difficult task and the following structure
% may not be suitable. Here are some rules that apply for this
% solution: 

% - Exactly two or three sections (other than the summary).
% - At *most* three subsections per section.
% - Talk about 30s to 2min per frame. So there should be between about
%   15 and 30 frames, all told.

% - A conference audience is likely to know very little of what you
%   are going to talk about. So *simplify*!
% - In a 20min talk, getting the main ideas across is hard
%   enough. Leave out details, even if it means being less precise than
%   you think necessary.
% - If you omit details that are vital to the proof/implementation,
%   just say so once. Everybody will be happy with that.

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\section{Goals}

\subsection{}
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\begin{frame}
 \frametitle{Focus}

\begin{itemize}
  

  \item Steady progress in software and hardware allows 
        high  fidelity, detailed performance assessment (simulations) of
        critical  (and  other) infrastructure systems (bridges, dams, buildings,
        ports...)


  \item Interplay of Earthquake Soil Structure systems seems to play 
        a major role in catastrophic  failures (and successes)



  \item Quantify uncertainty and variablity in soil and structural behavior 

  \item  Provide  methodology  (formulation,  implementation)  for probabilistic
         performance based engineering (PEER type)

  \item Overcome   traditional  performance  assessment  approaches  used  in
        engineering practice (design using prescriptive code!)
  

\end{itemize}
\end{frame}
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\begin{frame}
 \frametitle{Historical Note (Full Circle?)}

\begin{itemize}
  

  \item  Soil--Structure  Interaction  phenomena first realized and described by
         Professor Kyoji Suyehiro
     \begin{itemize}
            \item Ship engineer (Professor of Naval Arch. at U. of Tokyo),
            \item  Earthquake  engineer  (First Director  of  the  Earthquake Research
                 Institute at U. of Tokyo),
            \item Was in Tokyo during Great Kant{\= o} earthquake
                  (11:58am (--12:08pm! slow?), 1st. Sept. 1923)
            \item Saw earthquake surface waves travel and buildings sway 
                  (ships in the ocean)
            \item  Presented  his new SSI work in the USA (Caltech,
                   UCB, Stanford, MIT) in 1931...
     \end{itemize}




  \item Slow and Fast earthquakes
        
  \item Uncertainty and variability (source, material...) 


\end{itemize}
\end{frame}
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%\subsection{Software Tools}
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%%%\begin{frame}
%%% \frametitle{Verification, Validation and Prediction !}
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%%%
%%%\begin{itemize}
%%%
%%%     \item Verification: the  process  of  determining  that    a  model
%%%   implementation  accurately  represents the developer's conceptual description
%%%   and specification. Mathematics issue. {\it Verification provides evidence that the
%%%   model is solved correctly.}
%%%
%%%     \item Validation:  The process of determining the degree to which a
%%%   model  is  accurate  representation of the real world from the perspective of
%%%   the  intended  uses  of  the  model.  Physics issue. {\it Validation provides
%%%   evidence that the correct model is solved.}
%%%
%%%   \item Prediction:  use  of  computational  model  to foretell the state of a
%%%   physical   system   under  consideration  under  conditions  for  which  the
%%%   computational model has not been validated
%%%\end{itemize}
%%%
%%%%
%%%%\item Models available (some now, some later) 
%%%%\vspace*{-2.0cm}
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\section{ESS Systems}


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\subsection{High Fidelity, 3D Models}

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\begin{frame}
 \frametitle{Detailed 3D, FEM model}

   \begin{itemize}

  \item Construction process

  \item Two types of soil: stiff soil (UT, UCD), soft soil (Bay Mud)

  \item Deconvolution of given surface ground motions

  \item Use of the DRM  (Prof. Bielak et al.) for seismic input

  \item Piles $\rightarrow$ beam-column elements in soil holes

  \item No artificial damping (only mat. dissipation, radiation)

  \item Structural model: collaboration UCD, UCB and UW

  \item Element size issues (filtering of frequencies) 

\vspace*{-0.3cm}
\begin{table}[!htbp]
%\caption{ }
\begin{center}
\begin{tabular}{|c|c|c|c|r|} 
\hline
model  size (el) &  el.  size  &  $f_{cutoff}$  & min. $G/Gmax$ & $\gamma$  \\
\hline
%
%
% 100cm element model, f_cuttof=10HZ, G/Gmax=1.0, 12K elements, Gmax model
12K         &  1.0~m          &  10~Hz      & 1.0      & <0.5~\% \\
%
%
%~90cm element model, f_cutoff ~= 3Hz, G/Gmax~=0.08, for epsilon ~= 1%, 15K
%elements, now running...
15K         &  0.9~m          &  >3~Hz     & 0.08            & 1.0~\% \\
%
%
%27.6cm element model, f_cutoff= 10HZ, G/Gmax=0.08, for epsilon = 1%, 150K
%elements, finest model
150K        &  0.3~m          &  10~Hz      & 0.08            & 1.0~\% \\
500K        &  0.15~m          &  10~Hz      & 0.02            & 5.0~\% \\
% 27.6cm element model, f_cutoff= 10HZ, G/Gmax=0.08, for epsilon = 1%, 150K
%elements, finest model
\hline
\end{tabular}
\end{center}
\end{table}

  \end{itemize}


\end{frame}
%- 
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\begin{frame}
 \frametitle{FEM Mesh (one of)}

%
\begin{figure}[!htbp] 
\begin{center}
\includegraphics[width=7cm]{/home/jeremic/tex/works/Reports/2006/NEESDemoProject/PrototypeMesh.jpg}
%\caption{\label{BridgeSFSI01}  FEM  model  for  seismic response of a three bend
%bridge.}
\end{center}
\end{figure}
%


%\vspace*{0.5cm}
{%
\begin{beamercolorbox}{section in head/foot}
\usebeamerfont{framesubtitle}\tiny{B.   Jeremi\'{c}   and   G.   Jie.}  "Parallel
Soil--Foundation--Structure Computations", 
Chapter  in  Book:  \textit{ Progress  in Computational Dynamics and
Earthquake Engineering}; {Taylor and Francis Publishers}, 2008.
%\vskip2pt\insertnavigation{\paperwidth}\vskip2pt
\end{beamercolorbox}%
}




\end{frame}
%- 

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\begin{frame}
  \frametitle{Parallel Computer GeoWulf}
    \begin{itemize}
%\vspace*{0.2cm}
      \item Distributed memory \\
            parallel computer
%\vspace*{0.2cm}

      \item Multiple generation \\
      compute nodes \\ 
      and networks
%\vspace*{0.2cm}

      \item Very cost effective!
%\vspace*{0.2cm}

      \item Same architecture as \\
            large parallel supercomputers \\
            (SDSC, TACC, EarthSimulator...)
%\vspace*{0.2cm}

      \item Local design, construction, \\
      available at all times!
%\vspace*{0.2cm}

%      \item 
%%\vspace*{0.2cm}

    \end{itemize}

%
\vspace*{-5.6cm}
\begin{figure}[!htbp] 
\hspace*{6cm}
%\begin{center}
\includegraphics[width=4truecm]{/home/jeremic/public_html/GeoWulf/Dec2006/IMG_0907.jpg}
\\
\hspace*{6cm}
\includegraphics[width=4truecm]{/home/jeremic/public_html/NSF-Nuggets/Students_develop_parallel_computer/StudentsConstructingGeoWulf.jpg}
%\end{center}
\end{figure}



\end{frame}


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\subsection{Behavior for Short and Long Period Motions}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
  \frametitle{Northridge and Kocaeli Input Motions}


\begin{figure}[!htbp] 
\begin{center}
\hspace*{-1cm}
\includegraphics[width=6.0truecm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/InputMotion_Northridge.pdf}
\includegraphics[width=6.0truecm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/LongMotion/InputMotion_Kocaeli.pdf}
\hspace*{-1cm}
\\
\hspace*{-1cm}
\includegraphics[width=6.0truecm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/InputMotion_Northridge_Spectrum.pdf}
\includegraphics[width=6.0truecm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/LongMotion/InputMotion_Kocaeli_Spectrum.pdf}
\hspace*{-1cm}
%\caption{\label{BridgeSFSI01}  FEM  model  for  seismic response of a three bend
%bridge.}
\end{center}
\end{figure}
%

\end{frame}
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\begin{frame}
  \frametitle{Parametric Simulation Results}


\vspace*{-0.5cm}
%\begin{landscape}
\begin{figure}[!htbp]
\begin{center}
\hspace*{-1cm}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/DispSoilBlock1.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/DispBent1.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/DispSoilBlock2.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/DispBent2.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/DispSoilBlock3.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/DispBent3.pdf}
\hspace*{-1cm}
\\
\hspace*{-1cm}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/AccelSoilBlock1.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/AccelBent1.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/AccelSoilBlock2.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/AccelBent2.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/AccelSoilBlock3.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/AccelBent3.pdf}
\hspace*{-1cm}
\\
\hspace*{-1cm}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/DispSoilBlock1_Spectrum.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/DispBent1_Spectrum.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/DispSoilBlock2_Spectrum.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/DispBent2_Spectrum.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/DispSoilBlock3_Spectrum.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/DispBent3_Spectrum.pdf}
\hspace*{-1cm}
\\
\hspace*{-1cm}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/AccelSoilBlock1_Spectrum.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/AccelBent1_Spectrum.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/AccelSoilBlock2_Spectrum.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/AccelBent2_Spectrum.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/AccelSoilBlock3_Spectrum.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/AccelBent3_Spectrum.pdf}
\hspace*{-1cm}
\\
\hspace*{-1cm}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/MomentBent1Pile1.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/MomentBent1Pile2.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/MomentBent2Pile1.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/MomentBent2Pile2.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/MomentBent3Pile1.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/MomentBent3Pile2.pdf}
\hspace*{-1cm}
\end{center}
\end{figure}

\vspace*{-0.8cm}
\begin{figure}[!htbp] 
\begin{center}
\includegraphics[width=3.2cm]{/home/jeremic/tex/works/Reports/2006/NEESDemoProject/PrototypeMesh.jpg}
%\caption{\label{BridgeSFSI01}  FEM  model  for  seismic response of a three bend
%bridge.}
\end{center}
\end{figure}
\vspace*{-2cm}


\clearpage




\end{frame}

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\begin{frame}
  \frametitle{Northridge Input Motions}


\vspace*{-0.7cm}
\begin{figure}[!htbp] 
\begin{center}
\hspace*{-1cm}
\includegraphics[width=8.0truecm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/InputMotion_Northridge.pdf}
\hspace*{-1cm}
\\
\hspace*{-1cm}
\includegraphics[width=8.0truecm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/InputMotion_Northridge_Spectrum.pdf}
\hspace*{-1cm}
%\caption{\label{BridgeSFSI01}  FEM  model  for  seismic response of a three bend
%bridge.}
\end{center}
\end{figure}
\vspace*{-1.0cm}
%

\end{frame}
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\begin{frame}
  \frametitle{Short Period E.: Left Bent, Structure and Soil, Disp.}
  



\vspace*{-1cm}
\begin{figure}[!htbp] 
\begin{center}
\includegraphics[width=8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/DispBent1.pdf}
%\includegraphics[width=8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/DispBent1.pdf}
\\
\includegraphics[width=8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/DispSoilBlock1.pdf}
%\includegraphics[width=8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/DispSoilBlock1.pdf}
%\caption{\label{BridgeSFSI01}  FEM  model  for  seismic response of a three bend
%bridge.}
\end{center}
\end{figure}


\vspace*{-2.9cm}
\hspace*{0.2cm}
\begin{figure}[!htbp] 
\begin{flushright}
\includegraphics[width=1.5cm]{/home/jeremic/tex/works/Reports/2006/NEESDemoProject/PrototypeMesh.jpg}
%\caption{\label{BridgeSFSI01}  FEM  model  for  seismic response of a three bend
%bridge.}
\end{flushright}
\end{figure}
\hspace*{-0.5cm}
\vspace*{-2.0cm}



\end{frame}
%-- 
%--  
%--  
%--  
%--  
%--  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%--  \begin{frame}
%--    \frametitle{Short Period E.:  Left Bent, Structure and Soil, Acc.}
%--    
%--  
%--  
%--  
%--  \vspace*{-1cm}
%--  \begin{figure}[!htbp] 
%--  \begin{center}
%--  \includegraphics[width=8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/AccelBent1.pdf}
%--  \\
%--  \includegraphics[width=8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/AccelSoilBlock1.pdf}
%--  %\caption{\label{BridgeSFSI01}  FEM  model  for  seismic response of a three bend
%--  %bridge.}
%--  \end{center}
%--  \end{figure}
%--  
%--  
%--  \vspace*{-2.9cm}
%--  \hspace*{0.2cm}
%--  \begin{figure}[!htbp] 
%--  \begin{flushright}
%--  \includegraphics[width=1.5cm]{/home/jeremic/tex/works/Reports/2006/NEESDemoProject/PrototypeMesh.jpg}
%--  %\caption{\label{BridgeSFSI01}  FEM  model  for  seismic response of a three bend
%--  %bridge.}
%--  \end{flushright}
%--  \end{figure}
%--  \hspace*{-0.5cm}
%--  \vspace*{-2.0cm}
%--  
%--  \end{frame}
%--  %-- 
%--  
%--  
%--  
%--  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%--  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
  \frametitle{Short Period E.: Left Bent, Structure and Soil, Acc.Sp.}
  



\vspace*{-1cm}
\begin{figure}[!htbp] 
\begin{center}
\includegraphics[width=8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/AccelBent1_Spectrum.pdf}
\\
\includegraphics[width=8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/AccelSoilBlock1_Spectrum.pdf}
%\caption{\label{BridgeSFSI01}  FEM  model  for  seismic response of a three bend
%bridge.}
\end{center}
\end{figure}


\vspace*{-2.9cm}
\hspace*{0.2cm}
\begin{figure}[!htbp] 
\begin{flushright}
\includegraphics[width=1.5cm]{/home/jeremic/tex/works/Reports/2006/NEESDemoProject/PrototypeMesh.jpg}
%\caption{\label{BridgeSFSI01}  FEM  model  for  seismic response of a three bend
%bridge.}
\end{flushright}
\end{figure}
\hspace*{-0.5cm}
\vspace*{-2.0cm}


\end{frame}
%-- 



%--  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%--  \begin{frame}
%--    \frametitle{Short Period E.: Left Bent, Structure and Soil, M.}
%--    
%--  
%--  
%--  
%--  \vspace*{-1cm}
%--  \begin{figure}[!htbp] 
%--  \begin{center}
%--  \includegraphics[width=8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/MomentBent1Pile1.pdf}
%--  \\
%--  \includegraphics[width=8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/MomentBent1Pile2.pdf}
%--  %\caption{\label{BridgeSFSI01}  FEM  model  for  seismic response of a three bend
%--  %bridge.}
%--  \end{center}
%--  \end{figure}
%--  
%--  
%--  
%--  
%--  \vspace*{-2.9cm}
%--  \hspace*{0.2cm}
%--  \begin{figure}[!htbp] 
%--  \begin{flushright}
%--  \includegraphics[width=1.5cm]{/home/jeremic/tex/works/Reports/2006/NEESDemoProject/PrototypeMesh.jpg}
%--  %\caption{\label{BridgeSFSI01}  FEM  model  for  seismic response of a three bend
%--  %bridge.}
%--  \end{flushright}
%--  \end{figure}
%--  \hspace*{-0.5cm}
%--  \vspace*{-2.0cm}
%--  
%--  
%--  \end{frame}
%--  %-- 
%--  
%--  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
  \frametitle{Short Period E.: Left Bent, Structure and Soil, M.}
  



\vspace*{-1cm}
\begin{figure}[!htbp] 
\begin{center}
\includegraphics[width=11cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/MomentBent1Pile1_25s_SC.pdf}
%\caption{\label{BridgeSFSI01}  FEM  model  for  seismic response of a three bend
%bridge.}
\end{center}
\end{figure}


\vspace*{-2.9cm}
\hspace*{0.2cm}
\begin{figure}[!htbp] 
\begin{flushright}
\includegraphics[width=1.5cm]{/home/jeremic/tex/works/Reports/2006/NEESDemoProject/PrototypeMesh.jpg}
%\caption{\label{BridgeSFSI01}  FEM  model  for  seismic response of a three bend
%bridge.}
\end{flushright}
\end{figure}
\hspace*{-0.5cm}
\vspace*{-2.0cm}



\vspace*{2.0cm}
{%
\begin{beamercolorbox}{section in head/foot}
\usebeamerfont{framesubtitle}\tiny{B.   Jeremi\'{c},  G.   Jie, 
M. Preisig and N. Tafazzoli.}  "Soil--Foundation--Structure  Interaction
in non--Uniform Soils", 
in review in \textit{Earthquake Engineering and Structural Dynamics}, 2008.
%\vskip2pt\insertnavigation{\paperwidth}\vskip2pt
\end{beamercolorbox}%
}



\end{frame}
%-- 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%--  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%--  \begin{frame}
%--    \frametitle{Short Period E.: Left Bent, Free Field vs Real Disp.}
%--    
%--  
%--  
%--  
%--  \vspace*{-1cm}
%--  \begin{figure}[!htbp] 
%--  \begin{center}
%--  \includegraphics[width=11cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/DispSoilBlock1_15cm_25s_SC.pdf}
%--  %\caption{\label{BridgeSFSI01}  FEM  model  for  seismic response of a three bend
%--  %bridge.}
%--  \end{center}
%--  \end{figure}
%--  
%--  
%--  \vspace*{-2.9cm}
%--  \hspace*{0.2cm}
%--  \begin{figure}[!htbp] 
%--  \begin{flushright}
%--  \includegraphics[width=1.5cm]{/home/jeremic/tex/works/Reports/2006/NEESDemoProject/PrototypeMesh.jpg}
%--  %\caption{\label{BridgeSFSI01}  FEM  model  for  seismic response of a three bend
%--  %bridge.}
%--  \end{flushright}
%--  \end{figure}
%--  \hspace*{-0.5cm}
%--  \vspace*{-2.0cm}
%--  
%--  
%--  \end{frame}
%--  %-- 
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%--  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%--  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
  \frametitle{Kocaeli Input Motions}


\vspace*{-0.7cm}
\begin{figure}[!htbp] 
\begin{center}
\hspace*{-1cm}
\includegraphics[width=8.0truecm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/LongMotion/InputMotion_Kocaeli.pdf}
\hspace*{-1cm}
\\
\hspace*{-1cm}
\includegraphics[width=8.0truecm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/LongMotion/InputMotion_Kocaeli_Spectrum.pdf}
\hspace*{-1cm}
%\caption{\label{BridgeSFSI01}  FEM  model  for  seismic response of a three bend
%bridge.}
\end{center}
\end{figure}
%
\vspace*{-1cm}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%--  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%--  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%--  \begin{frame}
%--    \frametitle{Long Period E.: Left Bent, Structure and Soil, Disp.}
%--    
%--  
%--  
%--  
%--  \vspace*{-1cm}
%--  \begin{figure}[!htbp] 
%--  \begin{center}
%--  \includegraphics[width=8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/LongMotion/DispBent1.pdf}
%--  \\
%--  \includegraphics[width=8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/LongMotion/DispSoilBlock1.pdf}
%--  %\caption{\label{BridgeSFSI01}  FEM  model  for  seismic response of a three bend
%--  %bridge.}
%--  \end{center}
%--  \end{figure}
%--  
%--  
%--  \vspace*{-2.9cm}
%--  \hspace*{0.2cm}
%--  \begin{figure}[!htbp] 
%--  \begin{flushright}
%--  \includegraphics[width=1.5cm]{/home/jeremic/tex/works/Reports/2006/NEESDemoProject/PrototypeMesh.jpg}
%--  %\caption{\label{BridgeSFSI01}  FEM  model  for  seismic response of a three bend
%--  %bridge.}
%--  \end{flushright}
%--  \end{figure}
%--  \hspace*{-0.5cm}
%--  \vspace*{-2.0cm}
%--  
%--  
%--  
%--  \end{frame}
%--  %-- 
%--  
%--  
%--  
%--  
%--  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%--  \begin{frame}
%--    \frametitle{Long Period E.: Left Bent, Structure and Soil, Acc.}
%--    
%--  
%--  
%--  
%--  \vspace*{-1cm}
%--  \begin{figure}[!htbp] 
%--  \begin{center}
%--  \includegraphics[width=8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/LongMotion/AccelBent1.pdf}
%--  \\
%--  \includegraphics[width=8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/LongMotion/AccelSoilBlock1.pdf}
%--  %\caption{\label{BridgeSFSI01}  FEM  model  for  seismic response of a three bend
%--  %bridge.}
%--  \end{center}
%--  \end{figure}
%--  
%--  
%--  
%--  \vspace*{-2.9cm}
%--  \hspace*{0.2cm}
%--  \begin{figure}[!htbp] 
%--  \begin{flushright}
%--  \includegraphics[width=1.5cm]{/home/jeremic/tex/works/Reports/2006/NEESDemoProject/PrototypeMesh.jpg}
%--  %\caption{\label{BridgeSFSI01}  FEM  model  for  seismic response of a three bend
%--  %bridge.}
%--  \end{flushright}
%--  \end{figure}
%--  \hspace*{-0.5cm}
%--  \vspace*{-2.0cm}
%--  
%--  
%--  \end{frame}
%--  %-- 
%--  
%--  
%--  
%--  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%--  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%--  
%--  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%--  \begin{frame}
%--    \frametitle{Long Period E.: Left Bent, Structure and Soil, Acc.Sp.}
%--    
%--  
%--  
%--  
%--  \vspace*{-1cm}
%--  \begin{figure}[!htbp] 
%--  \begin{center}
%--  \includegraphics[width=8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/LongMotion/AccelBent1_Spectrum.pdf}
%--  \\
%--  \includegraphics[width=8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/LongMotion/AccelSoilBlock1_Spectrum.pdf}
%--  %\caption{\label{BridgeSFSI01}  FEM  model  for  seismic response of a three bend
%--  %bridge.}
%--  \end{center}
%--  \end{figure}
%--  
%--  
%--  \vspace*{-2.9cm}
%--  \hspace*{0.2cm}
%--  \begin{figure}[!htbp] 
%--  \begin{flushright}
%--  \includegraphics[width=1.5cm]{/home/jeremic/tex/works/Reports/2006/NEESDemoProject/PrototypeMesh.jpg}
%--  %\caption{\label{BridgeSFSI01}  FEM  model  for  seismic response of a three bend
%--  %bridge.}
%--  \end{flushright}
%--  \end{figure}
%--  \hspace*{-0.5cm}
%--  \vspace*{-2.0cm}
%--  
%--  
%--  \end{frame}
%--  %-- 
%--  
%--  
%--  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
  \frametitle{Long Period E.: Left Bent, Structure and Soil, M.}
  



\vspace*{-1cm}
\begin{figure}[!htbp] 
\begin{center}
\includegraphics[width=11cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/LongMotion/MomentBent1Pile1.pdf}
%\\
%\includegraphics[width=8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/LongMotion/MomentBent1Pile2.pdf}
%\caption{\label{BridgeSFSI01}  FEM  model  for  seismic response of a three bend
%bridge.}
\end{center}
\end{figure}


\vspace*{-2.9cm}
\hspace*{0.2cm}
\begin{figure}[!htbp] 
\begin{flushright}
\includegraphics[width=1.5cm]{/home/jeremic/tex/works/Reports/2006/NEESDemoProject/PrototypeMesh.jpg}
%\caption{\label{BridgeSFSI01}  FEM  model  for  seismic response of a three bend
%bridge.}
\end{flushright}
\end{figure}
\hspace*{-0.5cm}
\vspace*{-2.0cm}


\end{frame}
%-- 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%   \frametitle{Soft Soils: Left Bent, Structure and Soil, Moments}
%   
% 
% 
% 
% \vspace*{-1cm}
% \begin{figure}[!htbp] 
% \begin{center}
% \includegraphics[width=8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/LongMotion/MomentBent1Pile1_25s_SC.pdf}
% %\caption{\label{BridgeSFSI01}  FEM  model  for  seismic response of a three bend
% %bridge.}
% \end{center}
% \end{figure}
% \vspace*{-2cm}
% 
% 
% \end{frame}
% %-- 
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%   \frametitle{Soft Soils: Left Bent, Free Field vs Real Displacements}
%   
% 
% 
% 
% \vspace*{-1cm}
% \begin{figure}[!htbp] 
% \begin{center}
% \includegraphics[width=8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/LongMotion/DispSoilBlock1_15cm_25s_SC.pdf}
% %\caption{\label{BridgeSFSI01}  FEM  model  for  seismic response of a three bend
% %bridge.}
% \end{center}
% \end{figure}
% \vspace*{-2cm}
% 
% 
% \end{frame}
% %-- 
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%





\section{Uncertain Seismic Motions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\subsection{Constitutive and Spatial Uncertainties}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame} 

\frametitle{Problem Setup}
 


 
 
\begin{itemize}

\vspace*{0.3cm}
\item Incr. 3D el--pl:
%
%
%    \begin{equation}
$
    \nonumber
      d\sigma_{ij} = \left \{ 
    D^{el}_{ijkl}
           -
           \frac{\displaystyle  D^{el}_{ijmn} m_{mn} n_{pq} D^{el}_{pqkl}}
           {\displaystyle n_{rs} D^{el}_{rstu} m_{tu} - \xi_* r_*}
           \right \}  
          {d\epsilon_{kl}}
$    
%   \end{equation}
   

\vspace*{0.3cm}
\item phase density $\rho$ of $\sigma(x,t)$ varies in time according to a continuity 
Liouville equation (Kubo 1963) 

\vspace*{0.3cm}
\item 
Continuity  equation  written  in  ensemble  average  form  (eg. cumulant
expansion method (Kavvas and Karakas 1996))


\vspace*{0.3cm}
\item 
van  Kampen's  Lemma (van Kampen 1976) $\rightarrow$  $  <\rho(\sigma,t)>=P(\sigma,t) $, 
ensemble average of phase density is the probability density

%  
%  
%   \item  Focus  on  1D  $\rightarrow$  a  nonlinear  ODE  with random coefficient
%   (material) and random forcing ($\epsilon$)
%  %
%  % 
%  %
%   \begin{eqnarray}
%   \nonumber
%   \frac{d\sigma(x,t)}{dt} &=& \beta(\sigma(x,t),D^{el}(x),q(x),r(x);x,t) \frac{d\epsilon(x,t)}{dt} \\
%   \nonumber
%   &=& \eta(\sigma,D^{el},q,r,\epsilon; x,t)
%   \end{eqnarray}
%  %
%  with initial condition $\sigma(0)=\sigma_0$

 \end{itemize}

\end{frame}






%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  \begin{frame} 
%  
%  
%  
%  \frametitle{Evolution of the Density $\rho(\sigma,t)$ }
%  
%   
%  
%  \begin{itemize}
%  
%  \vspace*{1cm}
%    \item  From each initial point in \\
%           $\sigma$-space a trajectory \\
%           starts out describing \\
%           the corresponding solution \\
%           of the stochastic process
%    
%  \vspace*{0.3cm}
%    \item  Movement of a cloud of initial\\
%           points  described  by density \\
%           $\rho(\sigma,0)$ in $\sigma$-space, \\
%           is governed by the \\
%           constitutive equation,
%  
%  \end{itemize}
%  
%  
%  %\begin{figure}[!hbpt]
%  %\begin{center}
%  \vspace*{-5cm}
%  \hspace*{6cm}
%  \includegraphics[height=4.5cm,angle=-90]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Present/Cloud_of_Points.pdf}
%  %\end{center}
%  %\end{figure}
%                
%  
%  
%  \end{frame}
%  
%  
%  
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  \begin{frame} 
%  
%  
%  \frametitle{Stochastic Continuity (Liouville) Equation}
%  
%   
%  
%  \begin{itemize}
%  
%  
%  \item phase density $\rho$ of $\sigma(x,t)$ varies in time according to a continuity 
%  Liouville equation (Kubo 1963): 
%  %
%  \begin{eqnarray}
%  \frac{\partial \rho (\sigma(x,t),t)}{\partial t}
%  = 
%  \nonumber
%  \\
%  -\frac{\partial  \eta (\sigma(x,t), D^{el}(x), q(x), r(x), \epsilon(x,t))  }{\partial \sigma}
%  \;\; 
%  \rho[\sigma(x,t),t]
%  \nonumber
%  \end{eqnarray}
%  
%  \vspace{0.5cm}
%  \item with initial conditions $\rho(\sigma,0) = \delta(\sigma-\sigma_0)$
%  
%  
%  \end{itemize}
%   
%  \end{frame}
%  
%  
%  
%  
%  
%  
%  
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  \begin{frame} 
%  
%  
%  \frametitle{{Ensemble Average form of Liouville Equation}}
%  
%  
%  
%   
%  
%  
%  
%  \noindent
%  Continuity  equation  written  in  ensemble  average  form  (eg. cumulant
%  expansion method (Kavvas and Karakas 1996)):
%  
%   
%  %\vspace*{-0.5cm}
%  
%  \begin{footnotesize}
%  
%  \begin{eqnarray}
%  \nonumber
%  &&\displaystyle \frac{\partial \left < \rho(\sigma(x_t,t), t) \right >}{\partial t}=
%  - \displaystyle \frac{\partial}{\partial \sigma} \left[ \left\{\left< \vphantom{\int_{0}^{t} d\tau} \eta(\sigma(x_t,t), D^{el}(x_t),
%  q(x_t), r(x_t), \epsilon(x_t,t)) \right> \right. \right. \\
%  \nonumber
%  &+& \left. \left. \int_{0}^{t} d\tau Cov_0 \left[ \displaystyle \frac{\partial \eta(\sigma(x_t,t), D^{el}(x_t), q(x_t), r(x_t),
%  \epsilon(x_t,t))}{\partial \sigma}; \right. \right. \right. \\
%  \nonumber
%  & & \left. \left. \left. \eta(\sigma(x_{t-\tau},t-\tau), D^{el}(x_{t-\tau}), q(x_{t-\tau}), r(x_{t-\tau}),
%  \epsilon(x_{t-\tau},t-\tau) \vphantom{\int_{0}^{t} d\tau} \right] \right \} \left < \rho (\sigma(x_t,t),t) \right > \right] \\
%  \nonumber
%  &+& \displaystyle \frac{\partial^2}{\partial \sigma^2} \left[ \left\{ \int_{0}^{t} d\tau Cov_0 \left[ \vphantom{\int_{0}^{t}}
%  \eta(\sigma(x_t,t), D^{el}(x_t), q(x_t), r(x_t), \epsilon(x_t,t)); \right. \right. \right. \\
%  \nonumber
%  & & \left. \left. \left. \eta (\sigma(x_{t-\tau},t-\tau), D^{el}(x_{t-\tau}), q(x_{t-\tau}), r(x_{t-\tau}),
%  \epsilon(x_{t-\tau},t-\tau)) \vphantom{\int_{0}^{t}} \right] \vphantom{\int_{0}^{t}} \right\} \left < \rho (\sigma (x_t,t),t) \right > \right] \\
%  \nonumber
%  \end{eqnarray}
%  
%  \end{footnotesize}
%  
%  
%  
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  
%  
%  \end{frame}
%  





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

\frametitle{Eulerian--Lagrangian FPK Equation}

%

%\begin{itemize}

\begin{footnotesize}

% %\noindent
% van  Kampen's  Lemma (van Kampen 1976) $\rightarrow$  $  <\rho(\sigma,t)>=P(\sigma,t) $, 
% ensemble average of phase density 
% %(in stress space here) 
% is the probability density;
% 

\begin{eqnarray}
\nonumber
&&\displaystyle \frac{\partial P(\sigma(x_t,t), t)}{\partial t}=
- \displaystyle \frac{\partial}{\partial \sigma} \left[ \left\{\left< \vphantom{\int_{0}^{t} d\tau} \eta(\sigma(x_t,t), D^{el}(x_t),
q(x_t), r(x_t), \epsilon(x_t,t)) \right> \right. \right. \\
\nonumber
&+& \left. \left. \int_{0}^{t} d\tau Cov_0 \left[ \displaystyle \frac{\partial \eta(\sigma(x_t,t), D^{el}(x_t), q(x_t), r(x_t),
\epsilon(x_t,t))}{\partial \sigma}; \right. \right. \right. \\
\nonumber
& & \left. \left. \left. \eta(\sigma(x_{t-\tau},t-\tau), D^{el}(x_{t-\tau}), q(x_{t-\tau}), r(x_{t-\tau}),
\epsilon(x_{t-\tau},t-\tau) \vphantom{\int_{0}^{t} d\tau} \right] \right \} P(\sigma(x_t,t),t) \right] \\
\nonumber
&+& \displaystyle \frac{\partial^2}{\partial \sigma^2} \left[ \left\{ \int_{0}^{t} d\tau Cov_0 \left[ \vphantom{\int_{0}^{t}}
\eta(\sigma(x_t,t), D^{el}(x_t), q(x_t), r(x_t), \epsilon(x_t,t)); \right. \right. \right. \\
\nonumber
& & \left. \left. \left. \eta (\sigma(x_{t-\tau},t-\tau), D^{el}(x_{t-\tau}), q(x_{t-\tau}), r(x_{t-\tau}),
\epsilon(x_{t-\tau},t-\tau)) \vphantom{\int_{0}^{t}} \right] \vphantom{\int_{0}^{t}} \right\} P(\sigma (x_t,t),t) \right] \\
\nonumber
\end{eqnarray}

\end{footnotesize}


\vspace*{0.5cm}
{%
\begin{beamercolorbox}{section in head/foot}
\usebeamerfont{framesubtitle}\tiny{B. Jeremi\'{c}, K. Sett, and M. L. Kavvas, "Probabilistic
Elasto--Plasticity:  Formulation  in  1--D",  \textit{Acta Geotechnica}, Vol.~2,
No.~3, pp~197-210,  2007.}
%\vskip2pt\insertnavigation{\paperwidth}\vskip2pt
\end{beamercolorbox}%
}


\end{frame}






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

\frametitle{Euler--Lagrange FPK Equation}

 

\begin{itemize}

\item Advection-diffusion equation 
%
\begin{equation}                
\nonumber
\frac{\partial P(\sigma,t)}{\partial t} = -\frac{\partial}{\partial \sigma}\left[N_{(1)}P(\sigma,t)-\frac{\partial}{\partial \sigma}
\left\{N_{(2)} P(\sigma,t)\right\} \right]
\end{equation}

%

\item Complete probabilistic description of response 


\item Solution PDF is second-order exact to covariance of time (exact mean and variance)


\item It is deterministic equation in probability density space

\item It is linear PDE in probability density space 
$\rightarrow$ Simplifies the numerical solution process

\item Template FPK diffusion--advection equation is applicable to any material model $\rightarrow$ 
only the coefficients $N_{(1)}$ and $N_{(2)}$ are different for different material models

%\vspace*{0.2truecm}

\end{itemize}



  
{%
\begin{beamercolorbox}{section in head/foot}
\usebeamerfont{framesubtitle}\tiny{K. Sett, B. Jeremi{\'c} and M.L.  Kavvas, "The Role of Nonlinear 
Hardening/Softening in Probabilistic Elasto--Plasticity", \textit{International Journal for Numerical 
and Analytical Methods in Geomechanics}, Vol.~31, No.~7, pp~953-975, 2007}
%\vskip2pt\insertnavigation{\paperwidth}\vskip2pt
\end{beamercolorbox}%
}



\end{frame}




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\begin{frame}
\frametitle{Spectral Stochastic Elastic--Plastic FEM}

      \vspace*{-0.6truecm}
      \begin{flushright}
      \begin{equation}
      \nonumber
      \sum_{n = 1}^N K_{mn} d_{ni} + \sum_{n = 1}^N \sum_{j = 0}^P d_{nj} \sum_{k = 1}^M C_{ijk} K'_{mnk}  = \left< F_m \psi_i[\{\xi_r\}] \right >
      \end{equation}
      \end{flushright}


      \vspace*{-0.5cm}
      \begin{equation}
      \nonumber
      K_{mn} = \int_D B_n \textcolor{mycolor}{D}  B_m dV    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ K'_{mnk} = \int_D B_n {\sqrt \lambda_k h_k} B_m dV
      \end{equation}
      
      \vspace*{-0.5cm}
      \begin{equation}
      \nonumber
      C_{ijk} = \left < \xi_k(\theta) \psi_i[\{\xi_r\}] \psi_j[\{\xi_r\}] \right >  \ \ \ \ \ \ \ \ \ \ \ \  F_m = \int_D \phi N_m dV  \ \ \ \ \ \ \ \ \ \ \ \
      \end{equation}




\begin{itemize}

  \item SFEM: Ghanem and Spanos 2003

  \item  Material  variables random field represented through a finite number of
  random variables using KL-expansion

  \item  Unknown solution random variables represented using polynomial chaos of
  (known) input random variables

  \item  Fokker--Planck--Kolmogorov  approach  based  probabilistic constitutive
  integration at Gauss integration points

\end{itemize}

%%   \end{itemize}
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\end{frame}

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%\section{An Application}


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\subsection{Seismic Wave Propagation Through Uncertain Soils}


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\frametitle{``Uniform'' CPT  Site Data}


\vspace*{-0.7cm}
%\begin{figure}
\begin{center}
\includegraphics[height=6.7cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/CPT_DataAnalysis_Plots/EastWestProfile-Edited.pdf}
\end{center}
%\end{figure}


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\begin{frame}


\frametitle{Random Field Parameters from Site Data}

\begin{itemize}

%\item maximizing the log--likelihood of observing the spatial data under assumed joined distribution (for finite 
%scale model) or maximizing the log--likelihood of observing the periodogram estimates (for fractal model)

\item Maximum likelihood estimates of correlation length

\vspace*{0.3truecm}

%\begin{figure}
\begin{flushleft}
\hspace*{-1.7cm}
\includegraphics[height=4.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/CPT_DataAnalysis_Plots/SamplingPlan-Edited.jpg}
\hspace*{0.0cm}
\includegraphics[height=4.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/CPT_DataAnalysis_Plots/TypicalDataPlotBH1-Edited.jpg} \\
\small{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Typical CPT $q_T$}
\end{flushleft}
%\end{figure}

\vspace*{-4.9truecm}

%\begin{figure}
\begin{flushright}
\includegraphics[width=4.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/CPT_DataAnalysis_Plots/TypicalAutoCovariancePlotBH1_FiniteScale-Edited.jpg} \\
\vspace*{0.01truecm}
\small{Finite Scale}
\end{flushright}
%\end{figure}

\vspace*{0.02truecm}

%\begin{figure}
\begin{flushright}
\includegraphics[width=4.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/CPT_DataAnalysis_Plots/TypicalAutoCovariancePlotBH1_Fractal-Edited.jpg} \\
\small{Fractal}
\end{flushright}
%\end{figure}

\end{itemize}

\end{frame}


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\begin{frame}


\frametitle{Seismic Wave Propagation through Stochastic Soil}

%\begin{flushleft}
%\includegraphics[height=5.0cm]{PEER2007_3.jpg}
%\end{flushleft}
%\vspace*{-0.5truecm}

\begin{itemize}

\item Soil as 12.5 m deep 1--D soil column (von Mises Material) 

\begin{itemize}

\item Properties (including testing uncertainty) obtained through random field modeling of CPT $q_T$
%
$\left<q_T\right> = 4.99 ~MPa;~~Var[q_T] = 25.67 ~MPa^2; $\\
Cor. ~Length $[q_T] = 0.61 ~m; $ Testing~Error $= 2.78 ~MPa^2$

\end{itemize}

\vspace*{0.2cm}
\item $q_T$ was transformed to obtain $G$: ~~$G/(1-\nu)~=~2.9q_T$ 

\begin{itemize}
\item Assumed transformation uncertainty = 5\%
%
$\left<G\right> = 11.57MPa; Var[G] = 142.32 MPa^2$ \\
Cor.~Length $[G] = 0.61 m$

\end{itemize}


%\begin{center}
%\hspace*{-1.7cm}
%\includegraphics[height=3.5cm]{Chapter9_Schematic.jpg}
%\hspace*{0.0cm}
%\includegraphics[height=3.5cm]{Chapter9_BaseDisplacement.jpg} \\
%\small{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Base Displacement}
%\end{center}

\vspace*{0.2cm}
\item Input motions: modified 1938 Imperial Valley

% \vspace*{-0.2cm}
% \begin{center}
% \includegraphics[height=2.0cm]{Chapter9_BaseDisplacement.jpg} 
% \end{center}

\end{itemize}

\end{frame}

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%--  \subsection{Seismic Wave Propagation Through Uncertain Soils}



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% BJtempout\setbeamercovered{invisible}

\begin{frame}
\frametitle{Surface Displacement Time History}

\vspace*{-12.0truecm}

%\hspace*{-0.7cm} \uncover<1->{\includegraphics[width=12.0cm]{Chapter9_ElasticPlasticResponse_Mean-New.pdf}} 
%%
%\hspace*{-6.0cm} \uncover<2->{\includegraphics[width=12.0cm]{Chapter9_ElasticPlasticResponse_SD-New.pdf}} \hspace*{-2.0cm} \\
%%
%\uncover<1->{
\hspace*{-0.7cm}
\includegraphics[width=12.0cm]{/home/jeremic/tex/works/Conferences/2008/EM/Present/Chapter9_ElasticPlasticResponse_Mean-New.pdf}
%}
% BJtempout\uncover<3->{\hspace*{-6.0cm} \includegraphics[width=12.0cm]{Chapter9_ElasticPlasticResponse_SD-New.pdf} \hspace*{-6.0truecm}}
{\hspace*{-6.0cm} \includegraphics[width=12.0cm]{/home/jeremic/tex/works/Conferences/2008/EM/Present/Chapter9_ElasticPlasticResponse_SD-New.pdf} \hspace*{-6.0truecm}}

\vspace*{0.25truecm}
% BJtempout\hspace*{0.70cm} Deterministic\uncover<2->{/Mean} \uncover<3->{\hspace*{2.35cm} Standard Deviation \hspace*{-1.0cm}}
\hspace*{0.70cm} Deterministic/Mean {\hspace*{2.35cm} Standard Deviation \hspace*{-1.0cm}}
\end{frame}

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\begin{frame}
\frametitle{Mean $\pm$ Standard Deviation}

\vspace*{-19.0truecm}

\hspace*{0.15truecm} \includegraphics[width=19.0cm]{/home/jeremic/tex/works/Conferences/2008/EM/Present/Chapter9_ElasticPlasticResponse-New.pdf}

\end{frame}

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%
%\begin{frame}
%\frametitle{Evolution of Statistics of Top Node Displacement}
%
%\vspace*{-9.0truecm}
%
%\hspace*{-0.5cm} \includegraphics[width=9.0cm]{Chapter9_ElasticPlasticResponse_Mean-New.pdf}
%\hspace*{-4.0cm} \includegraphics[width=9.0cm]{Chapter9_ElasticPlasticResponse_SD-New.pdf}
%
%%\vspace*{-0.5truecm}
%%\hspace*{-1.0cm} 
%\tiny{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Mean~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Standard Deviation}
%
%\vspace*{-8.35truecm}
%
%
%\hspace*{-0.05cm} \includegraphics[width=9.0cm]{Chapter9_ElasticPlasticResponse-New.pdf}
%\hspace*{-3.55cm} \includegraphics[width=9.0cm]{Chapter9_ElasticPlasticResponse_COV-New.pdf}
%
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%%\hspace*{-0.5cm} 
%\tiny{~~~~~~~~~~~~~~~~~~~~~~~~~~Mean$\pm$ Standard Deviation~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~COV}
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%\end{frame}
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\begin{frame}
\frametitle{PDF of Surface Displacement Time History}

\begin{itemize}

\item PDF at the finite \\
element nodes can be \\
obtained using, e.g., \\
Edgeworth expansion \\
(Ghanem and Spanos \\ 
2003)

\vspace*{0.1truecm}

\item Numerous applications,  \\
especially where extreme \\
statistics are critical

\end{itemize}

\vspace*{-11.0cm}
\hspace*{5.5truecm} \includegraphics[height=12.0cm]{/home/jeremic/tex/works/Conferences/2008/EM/Present/Chapter9_ElasticPlasticResponse_PDF-New-Edited.pdf} \hspace*{-2.0cm}


\end{frame}

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\begin{frame}
\frametitle{Most Probable Solution}

\vspace*{-10.25cm}

\includegraphics[height=16.0cm]{/home/jeremic/tex/works/Conferences/2008/EM/Present/SamplePDF_DynamicProblem_Mode.pdf}  \hspace*{-5.0truecm} At $t~=~1.1$ sec \hspace*{-2.0truecm}

\end{frame}

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\begin{frame}
\frametitle{Tails of PDF}

\vspace*{-10.25cm}

\includegraphics[height=16.0cm]{/home/jeremic/tex/works/Conferences/2008/EM/Present/SamplePDF_DynamicProblem_TailsOfPDF.pdf}  \hspace*{-5.0truecm} At $t~=~1.1$ sec \hspace*{-2.0truecm}

\end{frame}

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\begin{frame}
\frametitle{Probability of Exceedance}

\vspace*{-10.25cm}

\includegraphics[height=16.0cm]{/home/jeremic/tex/works/Conferences/2008/EM/Present/SampleCDF_DynamicProblem.pdf}  \hspace*{-5.0truecm} At $t~=~1.1$ sec \hspace*{-2.0truecm}

\end{frame}

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\begin{frame}
\frametitle{Derivative Applications}

%\vspace*{-0.25truecm}

\begin{itemize}

\item Performance based engineering

\begin{itemize}

\item Reliability index ($\beta$)

\item Probability of damage and/or failure ($p_f$)

\end{itemize}

\vspace*{-10.5truecm}

\hspace*{-1.0truecm} \includegraphics[height=13.0cm]{/home/jeremic/tex/works/Conferences/2008/EM/Present/SchematicReliability.pdf} \hspace*{-1.5truecm}

%\vspace*{0.5truecm}

\item Sensitivity analysis

\item Financial risk analysis

\item  In  general,  useful  for  applications  where  mean,  mode  and  extreme
       statistics are important

%\vspace*{-8.0truecm}

%\hspace*{4.0truecm} \includegraphics[angle=90,height=10.0cm]{SchematicFinancialRisk.pdf} \hspace*{-1.5truecm}

\end{itemize}

\end{frame}


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\section{Summary}



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\begin{frame}
  \frametitle{Summary}
  
  \begin{itemize}


  \item  {\bf Steady progress} in software and hardware allows for high fidelity Model
  Based Simulations for performance assessment of infrastructure systems


\vspace*{0.2cm}
  \item {\bf Interplay} of  {\bf Earthquake(s)} with {\bf Soil} and {\bf Structure} 
        \underline{\bf Systems} plays a major role in catastrophic  failures (and successes)


\vspace*{0.2cm}
  \item  {\bf Probabilistic}    performance based engineering (uncertainty in soil and
  structural behavior, earthquake motions...)

\vspace*{0.2cm}
  \item {\bf Overcome} traditional  performance  assessment  approaches  used  in
        engineering practice (design using prescriptive code!)
  

%
%    \item {\bf Matching Triad}: 
%    \underline{Earthquake}, 
%    \underline{Soil} and 
%    \underline{Structure}  
%    ({\bf ESS}) interaction determines 
%          possible benefits or detriments of SFSI



\end{itemize}



\end{frame}
%-- 

\end{document}


