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\title
[Simulations in Geomechanics: The Issue of Uncertainty]
{Simulations in Geomechanics: \\ The Issue of Uncertainty}


%\subtitle
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%\author[Author, Another] % (optional, use only with lots of authors)
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\author[{Jeremi{\'c}}] % (optional, use only with lots of authors)
{\large Boris~Jeremi{\'c}}
% - Give the names in the same order as the appear in the paper.
% - Use the \inst{?} command only if the authors have different
%   affiliation.

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{ 
%Department of Civil and Environmental Engineering\\
%University of California, Davis 
}
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{\small University of California, Davis}  
%{\small CompGeoMech}



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\section{Motivation}


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\subsection{The Need for Simulations in (Geo-) Mechanics}
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% \frametitle{My Engineering Background} 
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%   \item  Diploma  Engineering  degree  (Master  of  Engineering) Belgrade Univ.,
%   Structural Engineering 
% 
% \vspace*{0.3cm}
%   \item   Design of concrete and rock dams in Yugoslavia and Iraq
% 
% \vspace*{0.3cm}
%   \item   Design of residential  and  industrial  buildings  in
%   Switzerland, 
%   
% \vspace*{0.3cm}
%   \item M.S. and Ph.D. Univ. of Colorado, Computational (Geo)mechanics
% 
% \vspace*{0.3cm}
%   \item Prof. Univ. of California  and Clarkson Univ. 
% 
% \vspace*{0.3cm}
%   \item Great need for numerical simulation tools
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\begin{frame}
\frametitle{Numerical Simulation in Support of Design!} 

\begin{itemize}


\item Practical design experience 
\begin{itemize}
  
  \item[]   Design of concrete and rock dams, bridges (YU, IR, USA)

  \item[]   Design of residential  and  industrial  buildings (SUI, SA)
  
  \item[]  Design of buildings, tunnels, oil exploration equipment (USA) 


\end{itemize}

\vspace*{0.1cm}
  \item Verified, validated predictions

\vspace*{0.1cm}
  \item Proper  modeling of (multi-) physics

\vspace*{0.1cm}
  \item Flexible, usable, user friendly tools 

\vspace*{0.1cm}
  \item Detailed models that {\bf reduce} 
    
    \begin{itemize}
      \item[] Kolmogorov Complexity 
      \item[] Modeling uncertainty
    \end{itemize}

\end{itemize}


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% \begin{frame}
% \frametitle{Pile in Layered Soils} 
% 
%   
% \begin{figure}[!htbp]
% \begin{center}
% \includegraphics[width=4.0cm]{/home/jeremic/tex/works/Thesis/ZhaohuiYang/SinglePiles/SinglePile_CS_color.pdf}
% \hfill
% \includegraphics[width=4.0cm]{/home/jeremic/tex/works/Thesis/ZhaohuiYang/SinglePiles/spUFmesh.pdf}
% \end{center}
% %\vspace*{-0.3cm}
% \end{figure}
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% \begin{frame}
% \frametitle{Single Pile in Sand with Clay Layer: M, Q, p} 
% 
% \begin{figure}[!htbp]
% \begin{center}
% \includegraphics[width=7.5cm]{/home/jeremic/tex/works/Thesis/ZhaohuiYang/SinglePiles/MQP_sand_wsfclay_UFmid2hdp0.jpg}
% \end{center}
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\begin{frame}
 \frametitle{Pile in Layered Soil: Pressure Ratio Reduction}

\begin{figure}[!htbp]
\begin{flushleft}
\includegraphics[width=2.0cm]{/home/jeremic/tex/works/Thesis/ZhaohuiYang/SinglePiles/spUFmesh.pdf}
\hspace*{0.5cm}
\includegraphics[width=1.40cm]{/home/jeremic/tex/works/Thesis/ZhaohuiYang/SinglePiles/SinglePile_CS_color.pdf}
%\hspace*{-1cm}
\end{flushleft}
\end{figure}
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\vspace*{-2cm}
%
\begin{figure}[!htbp]
\begin{center}
\hspace*{-1cm}
\includegraphics[width=4.5cm]{/home/jeremic/tex/works/Thesis/ZhaohuiYang/SinglePiles/MQP_sand_wsfclay_UFmid2hdp0.jpg}
\includegraphics[width=6.5cm]{/home/jeremic/tex/works/Thesis/ZhaohuiYang/SinglePiles/P_ReductionRatio_sand3casesdp0_065.jpg}
\hspace*{-1cm}
\end{center}
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\begin{frame}
 \frametitle{Pile Group Interaction}

\begin{figure}[!hbpt]
\vspace*{-0.8cm}
\begin{flushleft}
%\hspace*{-1.0cm}
\includegraphics[width=3.5cm]{/home/jeremic/tex/works/Thesis/ZhaohuiYang/PileGroups/p4X3_Joey_iso.pdf}
\vspace*{-1.5cm}
\\
\includegraphics[width=3.5cm]{/home/jeremic/tex/works/Thesis/ZhaohuiYang/PileGroups/Plastified_3X3pg_skewCOLOR.pdf}
\end{flushleft}
\end{figure}


\begin{figure}[!h]
\vspace*{-11.0cm}
\hspace*{1.5cm}
\begin{flushright}
\includegraphics[width=9.5cm]{/home/jeremic/tex/works/Thesis/ZhaohuiYang/PileGroups/LoadRatio_per_pile_Dense_pg4x3.pdf}
\hspace*{-1.2cm}
\end{flushright}
\end{figure}
%\vspace*{-8.3cm}

\end{frame}

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% \begin{frame}
%  \frametitle{Pile in Liquefied Soil, Level Ground}
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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \includegraphics[width=0.04\textwidth,angle=0]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/Model_I.jpg}
% &
% \includegraphics[height=0.12\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_1_T002.jpg}
% &
% \includegraphics[height=0.12\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_1_T005.jpg}
% &
% \includegraphics[height=0.12\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_1_T010.jpg}
% &
% \includegraphics[height=0.12\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_1_T015.jpg}
% &
% \includegraphics[height=0.12\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_1_T020.jpg}
% &
% \includegraphics[height=0.12\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_1_T080.jpg}
% \\
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \includegraphics[width=0.04\textwidth,angle=0]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/Model_II.jpg}
% &
% \includegraphics[height=0.12\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_2_T002.jpg}
% &
% \includegraphics[height=0.12\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_2_T005.jpg}
% &
% \includegraphics[height=0.12\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_2_T010.jpg}
% &
% \includegraphics[height=0.12\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_2_T015.jpg}
% &
% \includegraphics[height=0.12\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_2_T020.jpg}
% &
% \includegraphics[height=0.12\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_2_T080.jpg}
% \\
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \includegraphics[width=0.045\textwidth,angle=0]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/Model_III.jpg}
% &
% \includegraphics[height=0.12\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_3_T002.jpg}
% &
% \includegraphics[height=0.12\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_3_T005.jpg}
% &
% \includegraphics[height=0.12\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_3_T010.jpg}
% &
% \includegraphics[height=0.12\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_3_T015.jpg}
% &
% \includegraphics[height=0.12\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_3_T020.jpg}
% &
% \includegraphics[height=0.12\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_3_T080.jpg}
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% \includegraphics[width=8cm,height=0.5cm]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/GMklot02.pdf}\hspace*{1cm}
% \\
% \vspace*{-0.5cm}
% \includegraphics[angle=-90,width=0.6\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_scale.pdf}
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 \frametitle{Pile in Liquefiable Sloping Ground}



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\begin{figure}[!htbp] 
\begin{center}
\hspace*{-0.5cm}
\begin{tabular}{lllllll} 
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\includegraphics[width=0.04\textwidth,angle=0]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/Model_IV.jpg}
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\includegraphics[height=0.12\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_4_T002.jpg}
&
\includegraphics[height=0.12\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_4_T005.jpg}
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\includegraphics[height=0.12\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_4_T010.jpg}
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\includegraphics[height=0.12\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_4_T015.jpg}
&
\includegraphics[height=0.12\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_4_T020.jpg}
&
\includegraphics[height=0.12\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_4_T080.jpg}
\\
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\includegraphics[width=0.04\textwidth,angle=0]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/Model_V.jpg}
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\includegraphics[height=0.12\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_5_T002.jpg}
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\includegraphics[height=0.12\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_5_T005.jpg}
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\includegraphics[height=0.12\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_5_T010.jpg}
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\includegraphics[height=0.12\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_5_T015.jpg}
&
\includegraphics[height=0.12\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_5_T020.jpg}
&
\includegraphics[height=0.12\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_5_T080.jpg}
\\
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\includegraphics[width=0.045\textwidth,angle=0]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/Model_VI.jpg}
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\includegraphics[height=0.12\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_6_T002.jpg}
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\includegraphics[height=0.12\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_6_T010.jpg}
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\includegraphics[height=0.12\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_6_T015.jpg}
&
\includegraphics[height=0.12\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_6_T020.jpg}
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\includegraphics[height=0.12\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_6_T080.jpg}
\\
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\includegraphics[width=8cm,height=0.5cm]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/GMklot02.pdf}\hspace*{1cm}
\\
\vspace*{-0.5cm}
\includegraphics[angle=-90,width=0.6\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_scale.pdf}
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\frametitle{Earthquake-Soil-Structure Interaction} 



\vspace*{-1.0cm}
\begin{figure}[!htbp] 
\begin{flushleft}
\includegraphics[width=3.7cm]{/home/jeremic/tex/works/Reports/2006/NEESDemoProject/PrototypeMesh.jpg}
%\caption{\label{BridgeSFSI01}  FEM  model  for  seismic response of a three bend
%bridge.}
\end{flushleft}
\end{figure}

 
\vspace*{-3.3cm}
\begin{figure}[!htbp] 
%\hspace*{5.0cm}
\begin{flushright}
\includegraphics[width=7cm,height=2.5cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/MomentBent1Pile1_25s_SC.pdf}
%\\
%\hspace*{1.2cm}
%\includegraphics[width=8cm,height=2cm]{/home/jeremic/tex/works/Conferences/2009/CompDyn/Present/Northridge-Bent.pdf}
%\caption{\label{BridgeSFSI01}  FEM  model  for  seismic response of a three bend
%bridge.}
\end{flushright}
\end{figure}

 
\vspace*{-0.9cm}
%\begin{landscape}
\begin{figure}[!htbp]
\begin{center}
\hspace*{-1.1cm}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/version_OKT/Images/DispSoilBlock1.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/version_OKT/Images/DispBent1.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/version_OKT/Images/DispSoilBlock2.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/version_OKT/Images/DispBent2.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/version_OKT/Images/DispSoilBlock3.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/version_OKT/Images/DispBent3.pdf}
\hspace*{-1.1cm}
\\
\hspace*{-1.1cm}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/version_OKT/Images/AccelSoilBlock1.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/version_OKT/Images/AccelBent1.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/version_OKT/Images/AccelSoilBlock2.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/version_OKT/Images/AccelBent2.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/version_OKT/Images/AccelSoilBlock3.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/version_OKT/Images/AccelBent3.pdf}
\hspace*{-1.1cm}
\\
\hspace*{-1.1cm}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/version_OKT/Images/DispSoilBlock1_Spectrum.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/version_OKT/Images/DispBent1_Spectrum.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/version_OKT/Images/DispSoilBlock2_Spectrum.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/version_OKT/Images/DispBent2_Spectrum.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/version_OKT/Images/DispSoilBlock3_Spectrum.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/version_OKT/Images/DispBent3_Spectrum.pdf}
\hspace*{-1.1cm}
\\
\hspace*{-1.1cm}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/version_OKT/Images/AccelSoilBlock1_Spectrum.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/version_OKT/Images/AccelBent1_Spectrum.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/version_OKT/Images/AccelSoilBlock2_Spectrum.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/version_OKT/Images/AccelBent2_Spectrum.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/version_OKT/Images/AccelSoilBlock3_Spectrum.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/version_OKT/Images/AccelBent3_Spectrum.pdf}
\hspace*{-1.1cm}
\\
\hspace*{-1.1cm}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/version_OKT/Images/MomentBent1Pile1.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/version_OKT/Images/MomentBent1Pile2.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/version_OKT/Images/MomentBent2Pile1.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/version_OKT/Images/MomentBent2Pile2.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/version_OKT/Images/MomentBent3Pile1.pdf}
\includegraphics[width=1.8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/version_OKT/Images/MomentBent3Pile2.pdf}
\hspace*{-1cm}
\end{center}
\end{figure}



\vspace*{-2cm}




\clearpage






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% %\caption{\label{BridgeSFSI01}  FEM  model  for  seismic response of a three bend
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    \begin{itemize}
%\vspace*{0.2cm}

      \item {\bf P}lastic {\bf D}omain {\bf D}ecomposition \\
      Elastic-Plastic Parallel \\
      Finite Element Method

      \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}

%\vspace*{0.2cm}

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

    \end{itemize}

%
\vspace*{-6.6cm}
\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}



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\begin{frame}
\frametitle{Collaboratory} 

\begin{itemize}

  \item  Prof. Zhaohui Yang (U. of Alaska) 
\vspace*{0.2cm}
  \item  Prof. Mahdi Taiebat (U. of British Columbia) 
\vspace*{0.2cm}
  \item  Dr. Zhao Cheng (EarthMechanics Inc.)
\vspace*{0.2cm}
  \item  Dr. Guanzhou  Jie  (Wells  Fargo Securities)
%  \item  Dr. Matthias Preisig (Ecole Polytechnique F{\'e}d{\'e}rale de  Lausanne)  
\vspace*{0.2cm}
  \item  Dr. Matthias Preisig (EPF de  Lausanne)  
\vspace*{0.2cm}
  \item  Prof. Kallol  Sett  (U. of  Akron)
%\vspace*{0.2cm}
%  \item  Prof. Lev Kavvas (UCD)


\end{itemize}


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\subsection{Uncertain Geomaterials}
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\begin{frame} 

\frametitle{Material Behavior Inherently Uncertain}


%\begin{itemize}

%\vspace*{0.5cm}
%\item  
%Material  behavior  is  inherently uncertain  (concrete,  metals,  soil,  rock, 
%bone, foam, powder etc.)

        \begin{itemize}

\vspace*{0.5cm}
        \item Spatial \\ 
              variability

\vspace*{0.5cm}
        \item Point-wise \\ 
              uncertainty, \\
              testing \\
              error, \\
              transformation \\
              error

        \end{itemize}

% \vspace*{0.5cm}
% \item  Failure mechanisms related to spatial variability (strain localization and
%        bifurcation of response)
% 
% \vspace*{0.5cm}
% \item Inverse problems
% 
%       \begin{itemize}
%       
%       \item New material design, ({\it point-wise})
% 
%       \item Solid and/or structure design (or retrofits),  ({\it spatial})
% 
%       \end{itemize}
                                  
%\end{itemize}

\vspace*{-5cm}
\begin{figure}[!hbpt]
%\nonumber
%\begin{center}
\begin{flushright}
%\includegraphics[height=5.0cm]{/home/jeremic/tex/works/Conferences/2006/KragujevacSEECCM06/Presentation/MGMuzorak01.jpg}
\includegraphics[height=5.5cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/FrictionAngleProfile.jpg}
\\
\mbox{(Mayne et al. (2000) }
\end{flushright}
%\end{center}
%\end{center}
\end{figure}

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\begin{frame} 
\frametitle{Soil Uncertainties and Quantification}

\begin{itemize}
%
%\vspace*{0.5cm}
\item Natural variability of soil deposit  (Fenton 1999) 

  \begin{itemize}

    \item Function of soil formation process 
 
  \end{itemize}


%
\vspace*{0.2cm}
\item Testing error (Stokoe et al. 2004)

  \begin{itemize}

    \item Imperfection of instruments 
 
    \item Error in methods to register quantities

  \end{itemize}

%
\vspace*{0.2cm}
\item Transformation error (Phoon and Kulhawy 1999)

  \begin{itemize}

    \item Correlation by empirical data fitting (e.g. CPT data $\rightarrow$ friction angle etc.)

  \end{itemize}

\end{itemize}


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\frametitle{Types of Uncertainties}

 

\begin{itemize}



\item Aleatory uncertainty - inherent variation of physical system 
%
    \begin{itemize}
%
      \item Can not be reduced
%
      \item Has highly developed mathematical tools 
%
    \end{itemize}
%
\vspace*{0.2cm}
\item Epistemic uncertainty - due to lack of knowledge

   \begin{itemize}

     \item Can be reduced by \\
           collecting more data

     \item Mathematical tools \\
           are not well developed 

     \item trade-off with \\
           aleatory uncertainty

   \end{itemize}


%
\vspace*{-3.2cm}
\begin{figure}[!hbpt]
\begin{flushright}
\includegraphics[height=5cm,angle=-90]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Present/uncertain03.pdf}
\end{flushright}
\end{figure}

%\vspace*{1.0cm}

\item Ergodicity (exchanging ensemble averages for time average) assumed to hold


\end{itemize}

 




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\frametitle{Historical Overview}


 

\begin{itemize}

%
\item Brownian motion, Langevin equation $\rightarrow$  PDF governed by simple diffusion Eq. (Einstein 1905)


%
%  
%  \begin{equation}
%  \nonumber
%  \frac{\partial f (x,t)}{\partial t} = D \frac{\partial^2 f(x,t)}{\partial x^2}
%  \end{equation}
%  


\vspace*{0.2cm}
\item  Approach  for  random  forcing  $\rightarrow$  relationship  between  the
autocorrelation function and spectral density function (Wiener 1930)

\vspace*{0.2cm}
\item  With external forces $\rightarrow$ Fokker-Planck-Kolmogorov (FPK) for the
PDF (Kolmogorov 1941)


\vspace*{0.2cm}
\item Approach for random coefficient $\rightarrow$ 
Functional integration approach (Hopf 1952), Averaged equation approach (Bharrucha-Reid 1968), 
Numerical approaches, Monte Carlo method


%\item Alternately, Monte Carlo method can be used for solution of Langevin equation  $\rightarrow$ computationally very expensive

\end{itemize}

 


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\frametitle{Recent State-of-the-Art}

\begin{itemize}

%\vspace*{-0.5cm}
\item Governing equation

%       \vspace*{-0.5cm}
       \begin{itemize}

       \item Dynamic problems  $\rightarrow$ $ M \ddot u + C \ddot u + K u = F $

       \item Static problems  $\rightarrow$   $ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \   K u = F $

       \end{itemize}

%\vspace{-0.4cm}
\item Existing solution methods

%       \vspace*{-0.5cm}
       \begin{itemize}

       \item \textbf{Random r.h.s} (external force random)

             \begin{itemize}

      \item FPK equation approach 

      \item Use of fragility curves with deterministic FEM (DFEM)

      \end{itemize}

%       \vspace*{0.2cm}
       \item \textbf{Random l.h.s} (material properties random)

             \begin{itemize}

      \item Monte Carlo approach with DFEM $\rightarrow$ CPU expensive

%      \item Stochastic finite element method (e.g. Perturbation method  
%             $\rightarrow$ a linearized expansion! Error increases as a function
%      of COV; Spectral method 
%             $\rightarrow$ developed for elastic materials so far)

      \item Perturbation method 
             $\rightarrow$ a linearized expansion! Error increases as a function
      of COV 
      
      \item Spectral method 
             $\rightarrow$ developed for elastic materials so far

%             \begin{itemize}
%
%             \item Perturbation method  $\rightarrow$ fails if COVs of soil $>$ 20\%
%
%      \item Spectral method $\rightarrow$ only for elastic material
%
%             \end{itemize}

      \end{itemize}
             
       \end{itemize}      

\item  New developments for {\bf Probabilistic Elasto--Plasticity}


\end{itemize}

\end{frame}


% -  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  \begin{frame}
% -  \frametitle{Objectives of Proposed Work}
% -  
% -  \begin{itemize}
% -  
% -  \item Constitutive Problem: Obtain second-order (mean and variance) exact (analytical) PDF 
% -  of stress response $\rightarrow$ \textbf{Fokker--Planck--Kolmogorov (FPK) Equation}
% -  
% -  \item Boundary Value Problem: Model uncertain spatial variability of elastic--plastic soil
% -  $\rightarrow$ \textbf{Spectral Stochastic Elastic--Plastic Finite Element Method (SSEPFEM)}
% -  
% -  
% -  \vspace*{0.5truecm}
% -          
% -  \item Overcome the drawbacks of \textit{Monte Carlo Technique} and \textit{Perturbation Method}
% -  
% -  \item Obtain complete probabilistic description (PDF): Materials often fail at low 
% -  probability (tails of PDF)
% -  
% -  %         \begin{itemize}
% -  
% -  %          \item Materials often fail at low probability (tails of PDF)
% -  
% -  %         \end{itemize} 
% -  
% -  \item Carry out sensitivity analysis: Advanced constitutive models are sometimes highly sensitive to 
% -  fluctuations in material parameters
% -  
% -  %          \begin{itemize}
% -  
% -  %        \item Advanced models are highly sensitive to fluctuations in soil parameters
% -  
% -  \end{itemize}
% -  
% -  %\end{itemize}
% -  
% -  \end{frame}
% -  %
% -  
% -  
% -  
% -  
% -  





\section{Probabilistic Elasto--Plasticity}

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\subsection{PEP Formulation}
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\begin{frame} 

\frametitle{Uncertainty Propagation through Constitutive Eq.}

% 

\begin{itemize}

 


\item Incremental el--pl constitutive equation  
     $\displaystyle \frac{d\sigma_{ij}}{dt} = D_{ijkl} \displaystyle \frac{d\epsilon_{kl}}{dt}$

 
  
%\begin{normalsize}

%    
%    \begin{equation}
%    \nonumber
%    \frac{d\sigma_{ij}}{dt} = D_{ijkl} \frac{d\epsilon_{kl}}{dt}
%    \end{equation}
    
    
    \begin{eqnarray}
    \nonumber
     D_{ijkl} = \left\{\begin{array}{ll}
%
     D^{el}_{ijkl}
%
%
        \;\;\; & \mbox{\large{~for elastic}} \\
%
           \\
%
           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_*}

           \;\;\; & \mbox{\large{~for elastic--plastic}}
%
           \end{array} \right.
           \end{eqnarray}

%\end{normalsize}

 



%\vspace{0.5cm}
% \item Non--linear coupling in the El--Pl modulus


%-  \item Focus on 1-D $\rightarrow$ a nonlinear ODE with random coefficient and random forcing
%-  
%-  
%- 
%-  \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)  \mbox{\ \ \ \ with an I.C. $\sigma(0)=\sigma_0$}
%-  \end{eqnarray}
%- 
 

\end{itemize}

% 


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\end{frame}






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\begin{frame} 

\frametitle{Previous Work}

 

\begin{itemize}

\item 
Linear algebraic or differential equations  $\rightarrow$ Analytical solution:

         \begin{itemize}
        \item Variable Transf. Method (Montgomery and Runger 2003)
        \item Cumulant Expansion Method (Gardiner 2004)
        \end{itemize}

\item 
Nonlinear differential equations (elasto-plastic/viscoelastic-viscoplastic):
     
      \begin{itemize}

      \item  Monte  Carlo Simulation (Schueller 1997, De Lima et al 2001, Mellah
      et al. 2000, Griffiths et al. 2005...) \\ $\rightarrow$ accurate, very costly

      \item  Perturbation  Method  (Anders and Hori 2000, Kleiber and Hien 1992,
      Matthies  et  al. 1997) \\ $\rightarrow$ first and second order Taylor series
      expansion  about mean - limited to problems with small C.O.V. and inherits
      "closure problem"

      \end{itemize}

% 
%  \item 
%  Monte Carlo method: accurate, very costly
%  
%  \item 
%  Perturbation method: 

\end{itemize}


 



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\end{frame}






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\begin{frame} 

\frametitle{Problem Statement and Solution}
 


 
 
\begin{itemize}

\item Incremental 3D elastic-plastic stress--strain:
% 
%    
$d\sigma_{ij} = \left[ 
    D^{el}_{ijkl}
           -
           {( D^{el}_{ijmn} m_{mn} n_{pq} D^{el}_{pqkl}})/
           ({n_{rs} D^{el}_{rstu} m_{tu} - \xi_* r_*})
           \right] d\epsilon_{kl} $
   

\item Define stress density $\rho(\sigma,t)$  evolves in probabilistic space according
to the constitutive equation
 



\item Stress density $\rho(\sigma,t)$ varies in pseudo-time according to a continuity 
Liouville equation (Kubo 1963) ${\partial \rho (\sigma(x,t),t)}/{\partial t}
= 
-{\partial  \eta (\sigma(x,t), D^{el}(x), q(x), r(x), \epsilon(x,t))  }
\
{\partial \sigma} \rho[\sigma(x,t),t]$


\item 
Continuity  equation can be written  in  ensemble  average  form  
%(eg. cumulant expansion method 
(Kavvas and Karakas 1996)
%)

\item 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


% 
% \item
% %\noindent
% Using 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 we can finally obtain
% 





 \end{itemize}

\end{frame}



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\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}


\end{frame}







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\begin{frame} 

\frametitle{Eulerian--Lagrangian 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  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}

% 
% \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, 2007, In press (published
% online in the \textit{Online First} section)}
% %\vskip2pt\insertnavigation{\paperwidth}\vskip2pt
% \end{beamercolorbox}%
% }



\end{frame}



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\subsection{Probabilistic Elastic--Plastic Response}
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\begin{frame} 

%\frametitle{Elastic Response with Random $G$}
\frametitle{Elastic Material with Uncertain Shear Modulus $G$}

 

\begin{itemize}

\item General form of elastic constitutive rate equation
%
\begin{eqnarray}
\nonumber
\frac{d \sigma_{12}}{dt} = 2G \frac{d \epsilon_{12}}{dt} = \eta(G, \epsilon_{12};t)
\end{eqnarray}

\item Advection and diffusion coefficients of FPK equation
%
\begin{equation}
\nonumber
N_{(1)}=2\frac{d \epsilon_{12}}{dt} <G>
\;\;\;\;\; 
\mbox{;} 
\;\;\;\;\;
N_{(2)}=4t\left(\displaystyle \frac{d \epsilon_{12}}{dt} \right)^2 Var[G]
\end{equation}

\item {Example:}
$<G>$ = 2.5 MPa;
Std. Deviation$[G]$ = 0.5 MPa



\end{itemize}
 








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\end{frame}






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\begin{frame} 

%\frametitle{Elastic Response with Random $G$}
\frametitle{Probabilistic Elastic Response}


\begin{figure}[!hbpt]
\vspace*{-1.5cm}
\begin{center}
\includegraphics[height=6.7cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/PDFElastic_RandomG-m.pdf}
\end{center}
\vspace*{-1.5cm}
\end{figure}


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\begin{frame} 

\frametitle{Verification -- Variable Transformation Method}


\begin{figure}[!hbpt]
\begin{center}
\includegraphics[height=6cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/MonteCarlo_Elastic-m.pdf}
%  %
%  \hspace*{-3cm}
%  \vspace*{4cm}
%  %
%  \includegraphics[height=4cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ReasonOfNotMatchingMonteCarlo-m.pdf}
\end{center}
\end{figure}
%  
%  \vspace*{-4cm}
%  
%  \begin{flushleft}
%  Effect of Approximation of I.C on the \\ PDF of Stress at 0.0426 \% Strain
%  \end{flushleft}

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\end{frame}






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% \begin{frame} 
% 
% \frametitle{Drucker-Prager Linear Hardening with Random $G$}
% 
%  
% 
% %
% %  
% %  
% %  General form of Drucker-Prager elastic-plastic linear hardening constitutive rate equation
% %  
% %  
% 
% \begin{eqnarray}
% \nonumber
% \frac{d \sigma_{12}}{dt} = 
% G^{ep} \frac{d \epsilon_{12}}{dt} = 
% \eta(\sigma_{12}, G, K, \alpha, \alpha', \epsilon_{12};t)
% \end{eqnarray}
% 
% 
% 
% \noindent
% Advection and diffusion coefficients of FPK equation
% 
% 
% 
% \begin{equation}
% \nonumber
% N_{(1)}=\displaystyle \frac{d \epsilon_{12}}{dt} \left< 2G - \displaystyle \frac{G^2}{G+9K\alpha^2+\displaystyle \frac{1}{\sqrt{3}}I_1
% \alpha'} \right>
% \end{equation}
% 
% 
% 
% \begin{equation}
% \nonumber
% N_{(2)}=t\left(\displaystyle \frac{d \epsilon_{12}}{dt} \right)^2 Var\left[2G - \displaystyle \frac{G^2}{G+9K\alpha^2+\displaystyle 
% \frac{1}{\sqrt{3}}I_1 \alpha'} \right]
% \end{equation}
% 
% %  \end{itemize}
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% \begin{frame} 
% 
% \frametitle{{Drucker-Prager Linear Hardening, Random $G$}}
% 
% \vspace*{-1.0cm}
% 
% \begin{figure}[!hbpt]
% \begin{center}
% \includegraphics[height=6.5cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/PDFDruckerPrager_RandomG-m.pdf}
% %\hfill
% %\includegraphics[height=6.0cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourDruckerPrager_RandomG-m.pdf}
% \end{center}
% \end{figure}
% 
% \vspace*{-1.5cm}
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% \begin{frame} 
% 
% \frametitle{{Drucker-Prager Linear Hardening, Random $G$}}
% 
% \vspace*{-1.0cm}
% 
% \begin{figure}[!hbpt]
% \begin{center}
% %\includegraphics[height=4.5cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/PDFDruckerPrager_RandomG-m.pdf}
% %\hfill
% \includegraphics[height=6.50cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourDruckerPrager_RandomG-m.pdf}
% \end{center}
% \end{figure}
% 
% \vspace*{-1.5cm}
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% \begin{frame} 
% 
% \frametitle{Verification of D--P E--P Response - Monte Carlo}
% 
% 
% \begin{figure}[!hbpt]
% %\nonumber
% %\begin{flushleft}
% \begin{center}
% %\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
% \includegraphics[height=6cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/MonteCarlo_DruckerPrager-m.pdf}
% %
% %\end{flushleft}
% %\begin{flushright}
% %\includegraphics[height=11.5cm]{ContourDruckerPrager_RandomG-m.pdf}
% %
% %\nonumber
% %\caption{Drucker-Prager Associative Linear Hardening Response with Random $G$}
% %{\includegraphics[width=10.5cm]{QualSchematics4.jpg}}
% %\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
% %\end{flushleft}
% %\end{flushright}
% \end{center}
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\begin{frame} 

\frametitle{Modified Cam Clay Constitutive Model}

 

\begin{small}
\begin{equation}
\nonumber
\frac{d \sigma_{12}}{dt} = G^{ep} \frac{d \epsilon_{12}}{dt} = \eta(\sigma_{12}, G, M, e_0, p_0, \lambda, \kappa, \epsilon_{12};t)
\end{equation}
%
%
%
%
%
\begin{equation}
\nonumber
\eta = \left[2G - \displaystyle \frac{\left(36 \displaystyle \frac{G^2}{M^4} \right) \sigma_{12}^2}
{\displaystyle \frac{(1+e_0)p(2p-p_0)^2}{\kappa} + \left(18 \displaystyle \frac{G}{M^4}\right) \sigma_{12}^2
+ \displaystyle \frac{1+e_0}{\lambda-\kappa} p p_0 (2p-p_0)} \right] 
\end{equation}
\end{small}



\noindent
Advection and diffusion coefficients of FPK equation



\begin{equation}
\nonumber
N_{(1)}^{(i)}=\left<\eta^{(i)}(t)\right> + \int_0^t d\tau cov\left[\displaystyle \frac{\partial \eta^{(i)}(t)}{\partial t};
\eta^{(i)} (t-\tau)\right]
\end{equation}



\begin{equation}
\nonumber
N_{(2)}^{(i)} = \int_0^t d\tau cov\left[\eta^{(i)}(t); \eta^{(i)} (t-\tau)\right]
\end{equation}

 

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% -- \begin{frame} 
% -- 
% -- \frametitle{Low OCR Cam Clay with \\ Random $G$}
% -- 
% -- \vspace*{-1.5cm}
% -- 
% -- \begin{figure}[!hbpt]
% -- %\nonumber
% -- \begin{center}
% -- \includegraphics[height=3.5cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/PDFLowOCR_RandomG-m.pdf}
% -- \hfill
% -- \includegraphics[height=5.4cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourLowOCR_RandomG-m.pdf}
% -- \end{center}
% -- \end{figure}
% -- 
% -- \begin{itemize}
% -- 
% -- \item Approximation of I.C.
% -- 
% -- 
% -- %\item Wide transition between el. \& el.-pl.
% -- 
% -- 
% -- \item Non-symmetry in probability distribution!
% -- 
% -- 
% -- \item Response at critical state fairly certain but different than deterministic
% -- 
% -- \end{itemize}
% -- 
% -- 
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% -- 
% -- 
% -- \end{frame}
% -- 
% -- 
% -- 



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\begin{frame} 

\frametitle{{Low OCR Cam Clay  with  Random $G$, $M$ and $p_0$}}





\begin{itemize}

%\item Narrow transition between el. \& el.-pl.


\vspace*{0.3cm}
\item Non-symmetry in \\
      probability \\
      distribution


\vspace*{0.3cm}
\item Difference \\
      between \\
      mean, mode and \\
      deterministic 



\vspace*{0.3cm}
\item Divergence at \\
      critical state \\
      because $M$ might be \\
      (is) uncertain?

\end{itemize}


\vspace*{-6.3cm}
%\hspace*{0.5cm}
\begin{figure}[!hbpt]
\begin{flushright}
%\includegraphics[height=3.5cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/PDFLowOCR_RandomG_RandomM_Randomp0-m.pdf}
%\hfill
\includegraphics[height=6.5cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourLowOCR_RandomG_RandomM_Randomp0-m.pdf}
\end{flushright}
\end{figure}


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% --  
% --  \begin{frame} 
% --  
% --  \frametitle{{Low OCR Cam Clay  with \\ Random $G$, $M$ and $p_0$}}
% --  
% --  \vspace*{-1.5cm}
% --  
% --  \begin{figure}[!hbpt]
% --  \begin{center}
% --  \includegraphics[height=3.5cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/PDFLowOCR_RandomG_RandomM_Randomp0-m.pdf}
% --  \hfill
% --  \includegraphics[height=5.5cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourLowOCR_RandomG_RandomM_Randomp0-m.pdf}
% --  \end{center}
% --  \end{figure}
% --  
% --  
% --  
% --  \vspace*{-0.5cm}
% --  
% --  \begin{itemize}
% --  
% --  %\item Narrow transition between el. \& el.-pl.
% --  
% --  
% --  \item Non-symmetry in probability distribution
% --  
% --  
% --  \item Difference between mean, mode and deterministic 
% --  
% --  
% --  
% --  \item Divergence at critical state because $M$ is uncertain
% --  
% --  \end{itemize}
% --  
% --  
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% --  
% --  \end{frame}
% --  





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\begin{frame} 

\frametitle{Comparison of Low OCR Cam Clay at $\epsilon$ = 1.62 \%}



%\vspace*{-4.50cm}

\begin{figure}[!hbpt]
\begin{center}
%\includegraphics[height=14cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/CamClayPDFComparison-m.pdf}
\includegraphics[height=5.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Present/LowOCR_ComparisonPDF-m_ps.pdf}
\end{center}
\end{figure}                                                          

\vspace*{-0.5cm}

\begin{itemize}

\item None coincides with deterministic

\item Some  very uncertain, some very certain

\item Either on safe or unsafe side

\end{itemize}

 


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\end{frame}






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% \begin{frame} 
% 
% \frametitle{{High OCR Cam Clay with Random $G$ and $M$}}
% 
% 
% \begin{itemize}
% 
% %\item Approximation of I.C.
% 
% %
% %\item Very uncertain transition between el. \& el.-pl.
% 
% 
% \vspace*{0.5cm}
% \item Large non-symmetry \\
%      in probability \\
%      distribution
% 
% 
% \vspace*{0.5cm}
% \item Significant \\
%       differences in \\
%       mean, mode, \\
%       and deterministic 
% 
% 
% \vspace*{0.5cm}
% \item Divergence at \\
%       critical state, \\
%       $M$ is uncertain
% 
% \end{itemize}
% 
% 
% \vspace*{-6.5cm}
% %
% \begin{figure}[!hbpt]
% \begin{flushright}
% %\includegraphics[height=3.4cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/PDFHighOCR_RandomG_RandomM-m.pdf}
% %\hfill
% \includegraphics[height=6.5cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourHighOCR_RandomG_RandomM-m.pdf}
% \end{flushright}
% \end{figure}                                                          
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\begin{frame} 


\frametitle{Probabilistic Yielding}

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\begin{itemize}

%
\item Weighted elastic and elastic--plastic solution
${\partial P(\sigma,t)}/{\partial t} 
= 
-{\partial \left(N^w_{(1)}P(\sigma,t)
-{\partial \left(N^w_{(2)} P(\sigma,t)\right) }/{\partial \sigma} \right)}/
{\partial \sigma}$






%\vspace*{0.5cm}
\item Weighted advection and diffusion coefficients are then 
$N_{(1,2)}^{w} (\sigma) 
= 
(1 - P[\Sigma_y \leq \sigma]) N_{(1)}^{el} + P[\Sigma_y \leq \sigma] N_{(1)}^{el-pl} $

%\vspace*{0.5cm}
\item Cumulative Density  Function \\
 (CDF) of the yield function 
% \vspace*{-0.5cm}
% \begin{figure}[!h]
% %\hspace*{7cm}
% \includegraphics[width=5.5cm]{/home/jeremic/tex/works/Papers/2007/ProbabilisticYielding/vonMises_YieldCDF_Combined-edited.pdf} 
% \end{figure}
% \vspace*{-0.5cm}

 
\item {Similar to European Pricing \\
       Option in financial simulations \\
  (Black--Scholes options \\
   pricing model '73, Nobel prize for \\
   Economics '97)}


\end{itemize}

\vspace*{-4.5cm}
\begin{figure}[!h]
\hspace*{6.2cm}
\includegraphics[width=5.5cm]{/home/jeremic/tex/works/Papers/2007/ProbabilisticYielding/vonMises_YieldCDF_Combined-edited.pdf} 
\end{figure}
\vspace*{-0.5cm}


\end{frame}



%--  
  

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% \begin{frame}
% 
% 
% 
% \frametitle{Bi--Linear von Mises Response}
% 
% 
% 
% \vspace*{0.50cm}
% \begin{figure}[!htbp]
% \begin{center}
% \includegraphics[width=9.5cm]{/home/jeremic/tex/works/Papers/2007/ProbabilisticYielding/Final/vonMises_G_and_cu_very_uncertain_PDF-edited.pdf}
% % %\vspace*{-0.5cm}
% % \caption{von  Mises  associative  plasticity  model  with uncertain shear modulus and
% % shear strength (yield parameter): (a) Evolution of PDF of stress with 
% % strain (PDF=10000 was used as a cutoff for surface plot) and 
% % (b) Contours of evolution of stress PDF with strain.}
% % \label{vonMises_G_and_cu_very_uncertain}
% \end{center}
% \end{figure}
% %
% 
% 
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% 
% 
% 
% %--  
% 
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\begin{frame}


\frametitle{Bi--Linear von Mises Response}


\begin{figure}[!hbpt]
\begin{center}
\includegraphics[width=8cm]{/home/jeremic/tex/works/Papers/2007/ProbabilisticYielding/figures/vonMises_G_and_cu_very_uncertain/Contour_PDF-edited.pdf}
\end{center}
\end{figure}                                                          



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% \begin{frame}
% 
% 
% \frametitle{Cyclic Bi--Linear von Mises, Yield Stress PDF}
% 
% 
% \begin{figure}[!hbpt]
% \begin{center}
% \includegraphics[width=7cm]{/home/jeremic/tex/works/Papers/2008/CyclicProbabilisticBehaviorGeomaterials/figures/IJNAMG_Review02_Plot02-Edited_ps.jpg}
% \end{center}
% \end{figure}                                                          
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% \begin{frame}
% 
% \frametitle{Cyclic Bi--Linear von Mises Response}
% 
% \begin{figure}[!hbpt]
% \begin{center}
% \includegraphics[width=11cm]{/home/jeremic/tex/works/Papers/2008/CyclicProbabilisticBehaviorGeomaterials/figures/PDF07_of_ShearStress_Cyclic_PerfectlyPlastic_IJNAMG-Ed.pdf}
% \end{center}
% \end{figure}                                                          
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% % \begin{frame}
% % 
% % \frametitle{Cyclic Bi--Linear von Mises Response}
% % 
% % \begin{figure}[!hbpt]
% % \begin{center}
% % \includegraphics[width=7cm]{/home/jeremic/tex/works/Papers/2008/CyclicProbabilisticBehaviorGeomaterials/figures/PDF06_of_ShearStress_Cyclic_PerfectlyPlastic_IJNAMG-Ed.pdf}
% % \end{center}
% % \end{figure}                                                          
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% \begin{frame}
% 
% \frametitle{Cyclic Bi--Linear von Mises Response}
% 
% \begin{figure}[!hbpt]
% \begin{center}
% \includegraphics[width=11cm]{/home/jeremic/tex/works/Papers/2008/CyclicProbabilisticBehaviorGeomaterials/figures/PDF06_of_ShearStress_Cyclic_PerfectlyPlastic_IJNAMG-Ed.pdf}
% \end{center}
% \end{figure}                                                          
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% 
% 
% \frametitle{Drucker Prager: Uncertain $G$, Certain $\phi$ }
% 
% %
% \begin{figure}[!htbp]
% \begin{center}
% \includegraphics[width=8.0cm]{/home/jeremic/tex/works/Papers/2007/ProbabilisticYielding/Final/DruckerPrager_G_very_uncertain_but_alpha_fairly_certain_Contour_PDF-edited.pdf}
% \end{center}
% \end{figure}
% 
% \vspace*{-0.1cm}
% 
% 
% 
% \end{frame}
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% \begin{frame}
% 
% 
% 
% \frametitle{Drucker Prager: Certain $G$, Uncertain $\phi$ }
% 
% %
% \begin{figure}[!htbp]
% \begin{center}
% \includegraphics[width=8.0cm]{/home/jeremic/tex/works/Papers/2007/ProbabilisticYielding/Final/DruckerPrager_G_fairly_certain_but_alpha_very_uncertain_Contour_PDF-edited.pdf}
% \end{center}
% \end{figure}
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% \vspace*{-0.1cm}
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\begin{frame}



\frametitle{SPT Based Determination of Shear Strength}




\begin{figure}[!hbpt]
\begin{center}
% 
\includegraphics[width=5.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGE-GoverGmax/figures/ShearStrength_RawData_and_MeanTrend-Mod.pdf}
\hfill
\includegraphics[width=5.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGE-GoverGmax/figures/ShearStrength_Histogram_PearsonIV-FineTuned-Mod.pdf}
%
\end{center}
\end{figure}                                                          

\vspace*{-0.3cm}
Transformation of SPT $N$-value $\rightarrow$ undrained shear
strength, $s_u$ (cf. Phoon and Kulhawy (1999B)

Histogram  of  the  residual  
(w.r.t  the  deterministic transformation
equation) undrained strength, 
along with fitted probability density function 
(Pearson IV)
\end{frame}


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



\frametitle{SPT Based Determination of Young's Modulus}


\begin{figure}[!hbpt]
\begin{center}
%
\includegraphics[width=5.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGE-GoverGmax/figures/YoungModulus_RawData_and_MeanTrend_01-Ed.pdf} 
\hfill
\includegraphics[width=5.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGE-GoverGmax/figures/YoungModulus_Histogram_Normal_01-Ed.pdf}
%
\end{center}
\end{figure}                                                          

\vspace*{-0.3cm}
 Transformation of SPT $N$-value $\rightarrow$ 1-D Young's modulus, $E$ (cf. Phoon and Kulhawy (1999B))

 Histogram of the residual (w.r.t the deterministic transformation equation) Young's modulus, along with fitted probability density function

\end{frame}



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\begin{frame}

%\frametitle{Cyclic Response of Such Uncertain Material}
\frametitle{{Probabilistic Material Response} (von--Mises)}


\begin{figure}[!hbpt]
\begin{center}
%
\includegraphics[width=8.15truecm]{/home/jeremic/tex/works/Papers/2008/JGGE-GoverGmax/figures/meanStressStrainPlot_1Point026-Mod.pdf}
%\hfill
%\includegraphics[width=4.35truecm]{/home/jeremic/tex/works/Papers/2008/JGGE-GoverGmax/figures/sdStressStrainPlot_1Point026-Mod.pdf}
%
\end{center}
\end{figure}                                                          

\vspace*{-0.3cm}
\end{frame}



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\begin{frame}



%\frametitle{Cyclic Response of Such Uncertain Material}
\frametitle{Probabilistic Material Response, Standard Deviation}


\begin{figure}[!hbpt]
\begin{center}
%
\includegraphics[width=5.15truecm]{/home/jeremic/tex/works/Papers/2008/JGGE-GoverGmax/figures/meanStressStrainPlot_1Point026-Mod.pdf}
\hfill
\includegraphics[width=4.35truecm]{/home/jeremic/tex/works/Papers/2008/JGGE-GoverGmax/figures/sdStressStrainPlot_1Point026-Mod.pdf}
%
\end{center}
\end{figure}                                                          

\vspace*{-0.3cm}
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%--  
  


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\begin{frame}



\frametitle{$G/G_{max}$ Response}


\begin{figure}[!hbpt]
\begin{center}
%
\includegraphics[width=10.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGE-GoverGmax/figures/GoverGmax-Mod.pdf}
%\hfill
%\includegraphics[width=6.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGE-GoverGmax/figures/Damping-Mod.pdf}
%
\end{center}
\end{figure}                                                          

\vspace*{-0.3cm}

\end{frame}



%--  
  





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\begin{frame}



\frametitle{Damping Response}


\begin{figure}[!hbpt]
\begin{center}
%
%\includegraphics[width=6.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGE-GoverGmax/figures/GoverGmax-Mod.pdf}
%\hfill
\includegraphics[width=10.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGE-GoverGmax/figures/Damping-Mod.pdf}
%
\end{center}
\end{figure}                                                          

\vspace*{-0.3cm}

\end{frame}



%--  
  





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%\subsection{Boundary Value Problem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section[SSEPFEM]{Stochastic Elastic--Plastic Finite Element Method}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{SSEPFEM Formulation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

















% tempout
% tempout 
% tempout %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% tempout \begin{frame}
% tempout 
% tempout \frametitle{Governing Equations}
% tempout 
% tempout 
% tempout combining equilibrium, strain compatibility, and constitutive equations
% tempout 
% tempout \begin{equation}
% tempout \label{eqno_Dis01}
% tempout \Xi(x) u(x) = \phi(x)
% tempout \end{equation}
% tempout 
% tempout \noindent where $\Xi(x)$ is a linear/non-linear differential operator, $\phi(x)$ is the
% tempout external force and $u(x)$ is the response.
% tempout %
% tempout If the material properties are random (uncertain), $\Xi$ in Eq.~(\ref{eqno_Dis01}) becomes a stochastic
% tempout linear/non-linear differential operator and as a result, the response, $u$ becomes a random field.
% tempout %
% tempout 
% tempout 
% tempout \end{frame}
% tempout 
% tempout %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% tempout \begin{frame}
% tempout 
% tempout \frametitle{Separate Deterministic and Random Components}
% tempout 
% tempout 
% tempout stochastic operator $\Xi$ $\rightarrow$: deterministic  $L$ $+$  random $\Pi$,
% tempout % whose coefficients are zero-mean random fields, one can write Eq.~(\ref{eqno_Dis01}) as:
% tempout 
% tempout \begin{equation}
% tempout \label{eqno_Dis02}
% tempout \left[L (x) + \Pi(x,\theta) \right] u(x, \theta) = \phi (x)
% tempout \end{equation}
% tempout  
% tempout  Further,  if one split the input material properties
% tempout random field into a deterministic trend and zero-mean random (uncertain)
% tempout residual             about             trend,  $\mathbb{D}  (x, \theta)  =
% tempout \hat{\mathbb{D}}(x)  +  \mathbb{R}  (x, \theta)$, one can write
% tempout Eq.~(\ref{eqno_Dis02}) as:
% tempout 
% tempout \begin{equation}
% tempout \label{eqno_Dis03}
% tempout \left[L_1 (x) \hat{\mathbb{D}}(x) + L_2 (x) \mathbb{R} (x,\theta) \right] u(x, \theta) = \phi (x)
% tempout \end{equation}
% tempout 
% tempout \noindent  where  $L_1  (x)$  and  $L_2  (x)$ are deterministic
% tempout differential   operators. 
% tempout 
% tempout \end{frame}
% tempout 
% tempout %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% tempout \begin{frame}
% tempout 
% tempout \frametitle{Spatial and Stochastic Discretization}
% tempout 
% tempout Spectral approach to stochastic
% tempout finite element formulation necessities discretization of 
% tempout Eq.~(\ref{eqno_Dis02}) in both stochastic and spatial dimensions, as follows:
% tempout 
% tempout \begin{enumerate}
% tempout 
% tempout \item Karhunen--Lo\`{e}ve (KL) expansion (Karhunen \cite{Karhunen:1947};
% tempout Lo\`{e}ve \cite{Loeve:1948};  Ghanem and Spanos  \cite{book:Ghanem}): 
% tempout It discretizes the zero-mean fluctuating part
% tempout of the input material properties random field ($\mathbb{R} (x, \theta)$) 
% tempout into finite number of independent basic random variables, 
% tempout 
% tempout \begin{equation}
% tempout \label{eqno_c1}
% tempout \mathbb{R} (x, \theta) = \sum_{n=1}^L \sqrt \lambda_n \xi_n (\theta) f_n(x)
% tempout \end{equation}
% tempout 
% tempout %
% tempout where,   $\lambda_n$   and   $f_n(x)$  are  the  eigenvalues  and  eigenvector,
% tempout respectively of the covariance kernel of the zero-mean random field ($\mathbb{R}
% tempout (x,  \theta)$).  The  zero-mean  random  variables, $\xi_n(\theta)$ are mutually
% tempout independent and have unit variances.
% tempout 
% tempout \item    Polynomial    chaos    (PC)    expansion   (Wiener \cite{Wiener:1938};      
% tempout Ghanem and Spanos \cite{book:Ghanem}):  It expands any unknown random variable
% tempout in terms of known random variables. For example, any unknown random variable,
% tempout $\chi(\theta)$ can be expanded, truncating after $P$ terms, in functional (polynomial chaos, $\psi_i\left[\left\{\xi_r\right\}\right]$) of
% tempout known random variables, $\xi_r$ and unknown deterministic coefficient,
% tempout 
% tempout \begin{equation}
% tempout \label{eqno_c2}
% tempout \chi_j(\theta)=\sum_{i=0}^P\gamma_i^{(j)}\psi_i\left[\left\{\xi_r\right\}\right]
% tempout \end{equation}
% tempout 
% tempout Using PC expansion, the unknown partially-discretized -- in spatial dimension, using KL expansion -- displacement 
% tempout random field ($u(x, \theta)$) in Eq.~(\ref{eqno_Dis03}) can be further discretized in the stochastic dimension,
% tempout  
% tempout \begin{eqnarray}
% tempout \label{eqno_c3}
% tempout u(x,\theta)&=&\sum_{j=1}^L e_j \chi_j(\theta)b_j(x) = \sum_{j=1}^L \sum_{i=0}^P \gamma_i^{(j)} \psi_i[\{\xi_r\}]c_j(x) \\
% tempout \label{eqno_c6}
% tempout &=&\sum_{i=0}^P \psi_i[\{\xi_r\}] \sum_{j=1}^L \gamma_i^{(j)}c_j(x) = \sum_{i=0}^P \psi_i[\{\xi_r\}] d_i(x)
% tempout \end{eqnarray}
% tempout 
% tempout where, $c_j(x) = e_j b_j(x)$ and $d_i(x)=\sum_{j=1}^L\gamma_i^{(j)} c_j(x)$
% tempout 
% tempout \item  Regular shape function expansion (Zienkiewicz  and  Taylor  \cite{book:Zienkiewicz},  Bathe
% tempout \cite{book:Bathe}, Ghanem and Spanos \cite{book:Ghanem}): The spatial component
% tempout ($d_i  (x)$,  in  the  above polynomial chaos expansion, Eq.~(\ref{eqno_c6})) of the
% tempout unknown random field can be expanded as follows:
% tempout 
% tempout \begin{equation}
% tempout \label{eqno_c7}
% tempout d_i(x)=\sum_{n=1}^N d_{ni} l_n(x)
% tempout \end{equation}
% tempout 
% tempout \noindent where $l_m$ are the shape functions.
% tempout 
% tempout \end{enumerate}
% tempout 
% tempout 
% tempout 
% tempout \end{frame}
% tempout 
% tempout %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% tempout \begin{frame}
% tempout 
% tempout \frametitle{Stochastic Finite Elements}
% tempout 
% tempout Discretizing Eq.~(\ref{eqno_Dis03}) in both spatial and stochastic dimensions using KL, PC, and shape function expansions, one can write
% tempout Eq.~(\ref{eqno_Dis03}) as, 
% tempout 
% tempout \begin{equation}
% tempout \label{eqno_c8}
% tempout \sum_{i=0}^P \sum_{n=1}^N \left[\psi_i[\{\xi_r\}] L_1 (x) \hat {\mathbb{D}} (x) l_n(x) + 
% tempout \sum_{m=1}^{M} \xi_m (\theta) \psi_i[\{\xi_r\}] ~L_2 (x) \sqrt \lambda_m f_m(x) l_n(x) \right]  d_{ni} 
% tempout = \phi (x)
% tempout \end{equation}
% tempout 
% tempout 
% tempout \noindent Galerkin     type     procedure     may     be    applied    to
% tempout Eq.~(\ref{eqno_c8})  to solve for the unknown coefficients ($d_{mi}$) of
% tempout the PC-expansion of the displacement random field and after some algebra, one may write 
% tempout Eq.~(\ref{eqno_c8}) as (cf. Ghanem and Spanos \cite{book:Ghanem}),
% tempout 
% tempout \begin{equation}
% tempout \label{eqno_c9}
% tempout \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} 
% tempout = \Phi_m \left< \psi_i[\{\xi_r\}]\right>
% tempout \end{equation}
% tempout 
% tempout 
% tempout \noindent where, $\Phi_{k} = \int_{D} \phi (x) l_k(x) dx$ and $c_{ijk} = \left <\xi_k(\theta) \psi_i[\{\xi_r\}] \psi_j[\{\xi_r\}]  \right >$. 
% tempout In the above equation (Eq.~(\ref{eqno_c9})), one may note that the expected values the polynomial
% tempout chaos ($\psi_i[\{\xi_r\}]$) and the product of orthonormal random variables and polynomial chaos ($\xi_k(\theta) \psi_i[\{\xi_r\}] \psi_j[\{\xi_r\}]$) can easily be
% tempout pre-calculated symbolically using Mathematica \cite{software:Mathematica}. 
% tempout %
% tempout Eq.~(\ref{eqno_c9}) can be written in more familiar matrix form as,
% tempout 
% tempout \begin{equation}
% tempout \label{eqno_c10}
% tempout \bar K \bar{u} = \bar F
% tempout \end{equation}
% tempout 
% tempout \noindent where, $\bar{u}$ is the generalized displacement vector, $\bar F$ is the generalized force vector, 
% tempout $K'$ is the deterministic component of the generalized stiffness matrix ($\bar K$):
% tempout 
% tempout \begin{equation}
% tempout \label{eqno_c11}
% tempout K'_{nk} = \int_{D} L_1 (x) l_n(x)~\hat {\mathbb{D}}~l_k(x) dx
% tempout \end{equation}
% tempout 
% tempout \noindent and, $K''$ is the stochastic component of the generalized stiffness matrix ($\bar K$):
% tempout 
% tempout \begin{equation}
% tempout \label{eqno_c12}
% tempout K''_{mnk} = \int_{D} L_2 (x) l_n(x) \left\{\sqrt \lambda_m f_m(x) \right\} l_k(x) dx
% tempout \end{equation}
% tempout 
% tempout 
% tempout \end{frame}
% tempout 
% tempout %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% tempout \begin{frame}
% tempout 
% tempout \frametitle{Force-Residual Form}
% tempout 
% tempout For non-linear problems (as attempted in this paper), Eq.~(\ref{eqno_c10})
% tempout can be written in force-residual form and solved incrementally. 
% tempout One  can  write Eq.~(\ref{eqno_c10}) in force-residual
% tempout form as:
% tempout 
% tempout \begin{equation}
% tempout \label{eqno_1EP}
% tempout \bar r(\bar u, \lambda) = 0
% tempout \end{equation}
% tempout 
% tempout \noindent where, $\bar u$ are the generalized degrees of freedom and $\lambda$ is the control parameter.
% tempout Differentiating Eq.~(\ref{eqno_1EP}), one can write the rate form of force-residual equation 
% tempout (Eq.~(\ref{eqno_1EP})) as,
% tempout 
% tempout \begin{equation}
% tempout \label{eqno_2EP}
% tempout \dot {\bar r} = \bar K \dot {\bar u} - \bar q \dot \lambda = 0
% tempout \end{equation}
% tempout 
% tempout \noindent where, $\bar K$ (=$\partial r_i/\partial u_j$) is the tangent stiffness matrix and $\bar q$ (= $-\partial r_i/\partial \lambda$) is
% tempout the load vector. At regular points of $\bar u-\lambda$ space, the tangent stiffness matrix is non--singular and 
% tempout hence one can solve the force-residual rate equation for $\dot {\bar u}$ as,
% tempout 
% tempout \begin{equation}
% tempout \label{eqno_5EP}
% tempout \dot {\bar u} = \left({\bar K}^{-1} \bar q \right) \dot \lambda
% tempout \end{equation} 
% tempout 
% tempout \noindent or,
% tempout 
% tempout \begin{equation}
% tempout \label{eqno_6EP}
% tempout {\bar u}^{'} = \frac{d \bar u}{d \lambda} =  {\bar K}^{-1} \bar q 
% tempout \end{equation}
% tempout 
% tempout \noindent  Eq.~(\ref{eqno_6EP}), which is a
% tempout set  of  non--linear ODE can be solved numerically for $\bar u$
% tempout with  a  set of initial condition (at $\lambda$ = 0, $\bar u$ =
% tempout $\bar  u_0$)  using  either pure  incremental  method (forward Euler
% tempout method) or incremental--iterative method (Newton method). This
% tempout paper only considers the forward Euler method by which knowing the 
% tempout solution of $\bar u$ at the $n^{th}$ step ($\bar u_n$), the solution 
% tempout at $(n+1)^{th}$ step ($\bar u_{n+1}$) can be obtained as,
% tempout 
% tempout \begin{equation}
% tempout \label{eqno_7EP}
% tempout \bar u_{n+1} = \bar u_n + \Delta \lambda_n {\bar u_n}^{'}
% tempout \end{equation}
% tempout 
% tempout \noindent or, the incremental solution ($\Delta \bar u_n$) as,
% tempout 
% tempout \begin{equation}
% tempout \label{eqno_8EP}
% tempout \Delta \bar u_n = \bar u_{n+1} - \bar u_n = \left( {\bar K_n}^{-1} \bar q_n \right) \Delta \lambda_n 
% tempout \end{equation}
% tempout 
% tempout 
% tempout \end{frame}
% tempout 
% tempout %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% tempout \begin{frame}
% tempout 
% tempout \frametitle{Generalized Tangent Stiffness Matrix}
% tempout 
% tempout In Eq.~(\ref{eqno_8EP}), the  generalized tangent stiffness matrix
% tempout ($\bar K$)  needs  to  be  re-evaluated  at  each step because the
% tempout constitutive  properties ($\hat
% tempout {\mathbb{D}}$  in  Eq.~(\ref{eqno_c11}) and $\sqrt \lambda_m$,  where  the  subscript  $m$  denotes  the  KL-space, in
% tempout Eq.~(\ref{eqno_c12})) of the material change as the material
% tempout plastifies.
% tempout %
% tempout The evolutions of these material properties, as the material plastifies, are
% tempout governed by the constitutive rate equation (Eq.~(\ref{eqno_1})). 
% tempout %
% tempout In this paper, at each  integration   point   (Gauss   point), 
% tempout the constitutive equation, written in probability density space (the FPKE, refer Eq.~(\ref{eqno_3})),
% tempout is solved once for each KL-space\footnote{Note that KL-spaces are orthogonal to each other} to obtain 
% tempout updated $\hat{\mathbb{D}}$ (refer Eq.~(\ref{eqno_c11})) 
% tempout and $\sqrt \lambda_m$ (refer Eq.~(\ref{eqno_c12})), and thereby the generalized tangent stiffness matrix
% tempout ($\bar K$; refer Eq.~(\ref{eqno_8EP})) at each load step. For example, for 1--D problem, 
% tempout knowing the (random) strain increment at the ${k-1}^{th}$ global load step as,
% tempout 
% tempout \begin{equation}
% tempout \label{eqno_13EP}
% tempout \Delta \epsilon^{k-1} (x, \theta) 
% tempout = \sum_{i=0}^P \psi_i[\{\xi_r\}] (\theta) \sum_{n=1}^N \Delta d^{k-1}_{ni} B_n(x)
% tempout \end{equation}
% tempout 
% tempout \noindent the advection and diffusion coefficients for the FPKE, assuming 1--D von Mises elastic--perfectly plastic 
% tempout material behavior, at the $k^{th}$ global load step can be calculated as follows:
% tempout 
% tempout \begin{itemize}
% tempout 
% tempout \item[] For the zeroth KL-space (mean space):
% tempout %
% tempout \vspace*{-0.1truecm}
% tempout %
% tempout \begin{eqnarray}
% tempout \begin{array}{l}
% tempout {N_{(1)}^{{eq}^{vM}}}^k (\sigma) = (1/\Delta t)~\hat{\mathbb{D}}~\left(1 - P[\Sigma_y \leq \sigma]\right) \sum_{i=0}^P \left <\vphantom{\sum_{i=0}^P} \psi_i[\{\xi_r\}] \right > \sum_{n=1}^N \Delta d^{k-1}_{ni} B_n  \\
% tempout {N_{(2)}^{{eq}^{vM}}}^k (\sigma) = t~(1/\Delta t)^2~\hat{\mathbb{D}}^2~\left(1 - P[\Sigma_y \leq \sigma]\right) \sum_{i=0}^P Var \left[\vphantom{\sum_{i=0}^P} \psi_i[\{\xi_r\}] \right] \left(\sum_{n=1}^N \Delta d^{k-1}_{ni} B_n \right)^2 
% tempout \end{array}
% tempout \label{ADcoeffs01}
% tempout \end{eqnarray}
% tempout 
% tempout \item[] For any other KL-spaces:
% tempout %
% tempout \vspace*{-0.1truecm}
% tempout %
% tempout \begin{eqnarray}
% tempout \begin{array}{l}
% tempout {N_{(1)}^{{eq}^{vM}}}^k (\sigma) = (1/\Delta t)~\sqrt \lambda~f ~\left(1 - P[\Sigma_y \leq \sigma]\right) \sum_{i=0}^P \left <\vphantom{\sum_{i=0}^P} \xi ~\psi_i[\{\xi_r\}] \right > \sum_{n=1}^N \Delta d^{k-1}_{ni} B_n \\
% tempout {N_{(2)}^{{eq}^{vM}}}^k (\sigma) = t~(1/\Delta t)^2~\lambda~f^2~\left(1 - P[\Sigma_y \leq \sigma]\right) \sum_{i=0}^P Var \left[\vphantom{\sum_{i=0}^P} \xi ~\psi_i[\{\xi_r\}] \right] \left(\sum_{n=1}^N \Delta d^{k-1}_{ni} B_n \right)^2
% tempout \end{array}
% tempout \label{ADcoeffs02}
% tempout \end{eqnarray}
% tempout 
% tempout \end{itemize}
% tempout 
% tempout \noindent In Eqs.~(\ref{eqno_13EP})-(\ref{ADcoeffs02}), $B$ is the derivative of shape function. 
% tempout %
% tempout Also, one may note that the mean and variances of polynomial chaos ($\psi_i[\{\xi_r\}]$) and the mean and variances of the product of orthonormal
% tempout Gaussian random variables ($\xi_m$, where the subscript $m$ denotes the KL-space) and polynomial chaos ($\psi_i[\{\xi_r\}]$) could be 
% tempout easily pre-calculated symbolically using Mathematica \cite{software:Mathematica}. 
% tempout % 
% tempout Assuming any value of $\Delta t$, the FPKEs
% tempout corresponding to each KL-space (including the mean space) can be solved at $t = \Delta t$ to obtain the corresponding probability density 
% tempout functions of stress for the strain increment given by Eq.~(\ref{eqno_13EP}).
% tempout %
% tempout For  advanced global level solution schemes, such as Newton-type
% tempout incremental-iterative   scheme,  these  probability  densities  of
% tempout stress at  each KL-space (including the mean space) could be used to compute the internal
% tempout forces.  However,  in  this paper, only pure incremental
% tempout solution  scheme  is  considered. The pure incremental solution
% tempout scheme  only  needs  information  on  the  new tangent material
% tempout properties  at  each KL-space (for example, $\hat {\mathbb{D}}$
% tempout in  Eq.~(\ref{eqno_c11}) and $\sqrt \lambda_m$, where
% tempout the     subscript     $m$     denotes     the    KL-mode,    in
% tempout Eq.~(\ref{eqno_c12}))  to increment forward. 
% tempout %
% tempout In this paper, the new tangent
% tempout material properties at each KL-space has been 
% tempout calculated from the corresponding mean\footnote{obtained from probability density (solution of FPKE) of evolutionary stress
% tempout by standard integration technique} of evolutionary stress at each KL-space. 
% tempout %
% tempout To this end, one     can    write    the    constitutive    rate    equation
% tempout (Eq.~(\ref{eqno_1})),  after taking expectation on both sides:
% tempout 
% tempout \begin{equation}
% tempout \label{eqno_14EPa}
% tempout \frac{d \left< \sigma_{ij}(t)\right>}{d t}
% tempout =
% tempout \left< D_{ijkl}(\sigma_{ij}, t) \frac{d \epsilon_{kl}}{d t} \right > +  \int_{t-s}^t ds 
% tempout       Cov\left[D_{ijkl}({\sigma_{ij}}_{(s)}, s) \frac{d \epsilon_{kl}}{d t}; D_{ijkl}({\sigma_{ij}}_{(t-s)}, t-s) \frac{d \epsilon_{kl}}{d t} \right]
% tempout \end{equation}
% tempout 
% tempout \noindent Assuming $\delta$-correlation, the covariance term on the r.h.s of  Eq.~(\ref{eqno_14EPa}) 
% tempout becomes variance and hence, one can write Eq.~(\ref{eqno_14EPa}) in incremental form as:
% tempout 
% tempout \begin{equation}
% tempout \label{eqno_14EPb}
% tempout \frac{\Delta \left< \sigma_{ij}(t)\right>}{\Delta t}
% tempout =
% tempout \left< D_{ijkl}(\sigma_{ij}, t) \frac{\Delta \epsilon_{kl}}{\Delta t} \right > +
% tempout Var\left[D_{ijkl}(\sigma_{ij}, t) \frac{\Delta \epsilon_{kl}}{\Delta t} \right]
% tempout \end{equation}
% tempout 
% tempout \noindent  In  Eq.~(\ref{eqno_14EPb}),  the rate  of  change of mean stress can be easily computed from the
% tempout probability density function of stress by standard integration.
% tempout Having   known   the   l.h.s  of  Eq.~(\ref{eqno_14EPb}) over few sub-steps within a global load step increment and noting that incremental 
% tempout strain is given by Eq.~(\ref{eqno_13EP}), the evolved tangent material
% tempout properties of each space can be can be
% tempout obtained  using  a least square type procedure (e.g. Levenberg-Marquardt technique \cite{Levenberg:1944, Marquardt:1963}). For example, 
% tempout for 1-D problem, for zeroth KL-space (mean space), Eq.~(\ref{eqno_14EPb}) simplifies to:
% tempout 
% tempout \begin{equation}
% tempout \label{eqno_15EP}
% tempout {\left . \frac{\Delta \left< \sigma (t)\right>}{\Delta t}\right |}_{zero^{th} KL-space}
% tempout =
% tempout \hat{\mathbb{D}} (t) \left< \frac{\Delta \epsilon }{\Delta t} \right > +
% tempout {\hat{\mathbb{D}} (t)}^2 Var\left[\frac{\Delta \epsilon }{\Delta t} \right]
% tempout \end{equation}
% tempout 
% tempout \noindent If one assumes that the mean tangent material property ($\hat{\mathbb{D}} (t)$) evolves within the load step as 
% tempout $ \hat{\mathbb{D}} (t)~=~a_1~+~a_2~t^2 $, where $a_1$ and $a_1$ are unknown deterministic coefficients, then, one can write  
% tempout Eq.~(\ref{eqno_15EP}) as,
% tempout 
% tempout \begin{eqnarray}
% tempout \nonumber
% tempout {\left. \frac{\Delta \left< \sigma (t)\right>}{\Delta t} \right |}_{zero^{th} KL-space}
% tempout &=&  \left ( a_1 \left< \frac{\Delta \epsilon }{\Delta t} \right > + a_1^2 Var\left[\frac{\Delta \epsilon }{\Delta t} \right]\right) + \\
% tempout \nonumber
% tempout &&
% tempout \left ( a_2 \left< \frac{\Delta \epsilon }{\Delta t} \right > + 2 a_1 a_2 Var\left[\frac{\Delta \epsilon }{\Delta t} \right] \right) t^2 + \\
% tempout \label{eqno_16EP}
% tempout && a_2^2 Var\left[\frac{\Delta \epsilon }{\Delta t} \right] t^4
% tempout \end{eqnarray}
% tempout 
% tempout \noindent Using Eq.~(\ref{eqno_13EP}) and the same $\Delta t$ as used for solving the FPKE, the mean and variance terms of the r.h.s of 
% tempout Eq.~(\ref{eqno_16EP}) can be easily evaluated and hence, knowing the l.h.s of  Eq.~(\ref{eqno_16EP}) over few sub-steps, the unknown 
% tempout deterministic coefficients ($a_1$ and $a_2$) can be estimated using Levenberg-Marquardt technique \cite{Levenberg:1944, Marquardt:1963}.  
% tempout Then, the evolved mean tangent material property ($\hat{\mathbb{D}} (t)$) at the end of global load step can be evaluated as
% tempout $ \hat{\mathbb{D}} (t)~=~a_1~+~a_2~t^2 $ by substituting $t = \Delta t$. 
% tempout 
% tempout Similarly, for any other KL-spaces, for 1-D problem, Eq.~(\ref{eqno_14EPb}) simplifies to:
% tempout 
% tempout \begin{equation}
% tempout \label{eqno_17EP}
% tempout {\left. \frac{\Delta \left< \sigma (t)\right>}{\Delta t} \right |}_{non-zero KL-space}
% tempout =
% tempout \sqrt \lambda (t) f \left< \xi \frac{\Delta \epsilon }{\Delta t} \right > +
% tempout \lambda (t) f^2 Var\left[\xi \frac{\Delta \epsilon }{\Delta t} \right]
% tempout \end{equation}
% tempout 
% tempout \noindent Assuming the tangent material property at the $m^{th}$ KL-space ($\sqrt \lambda_m (t)$) evolves within the load step as,
% tempout $ \sqrt \lambda (t)~=~a_3~+~a_4~t^2 $, where $a_3$ and $a_4$ are unknown deterministic coefficients, one can write  
% tempout Eq.~(\ref{eqno_17EP}) as,
% tempout 
% tempout \begin{eqnarray}
% tempout \nonumber
% tempout {\left. \frac{\Delta \left< \sigma (t)\right>}{\Delta t} \right |}_{non-zero KL-space}
% tempout &=&  \left ( f a_3 \left< \xi \frac{\Delta \epsilon }{\Delta t} \right > + f^2 a_3^2 Var\left[\xi \frac{\Delta \epsilon }{\Delta t} \right]\right)  + \\
% tempout \nonumber
% tempout &&
% tempout \left ( f a_4 \left< \xi \frac{\Delta \epsilon }{\Delta t} \right > + 2 f^2 a_3 a_4 Var\left[\xi \frac{\Delta \epsilon }{\Delta t} \right] \right) t^2 + \\
% tempout \label{eqno_18EP}
% tempout &&
% tempout f^2 a_4^2 Var\left[\xi \frac{\Delta \epsilon }{\Delta t} \right] t^4
% tempout \end{eqnarray}
% tempout 
% tempout \noindent Following the same procedure, as described for the zeroth KL-space, from Eq.~(\ref{eqno_18EP}), the evolved tangent material 
% tempout property at the $m^{th}$ KL-space ($\sqrt \lambda_m (t))$ at the end of global load step can be estimated using Levenberg-Marquardt 
% tempout technique \cite{Levenberg:1944, Marquardt:1963}. 
% tempout 
% tempout 
% tempout 
% tempout 
% tempout 
% tempout 
% tempout \end{frame}
% tempout 
% tempout 
% tempout 
% tempout 
% tempout 
% tempout 
% tempout 
% tempout 
% tempout 
% tempout 
% tempout 
% tempout 
% tempout 
% tempout 
% tempout 
% tempout 
% tempout 
% tempout 
% tempout 
% tempout 
% tempout 
% tempout 
% tempout 
% tempout 
% tempout 
% tempout 
% tempout 
% tempout 
% tempout 
% tempout 
% tempout 
% tempout 
% tempout 
% tempout 
% tempout 



























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


 \begin{frame}



\frametitle{Stochastic Finite Element Formulation}

\begin{itemize}

\item Governing equations:


\begin{equation}
\nonumber
A\sigma = \phi(t);~~~ Bu = \epsilon; ~~~\sigma = D \epsilon
\end{equation}

\vspace*{0.2cm}
\item {\bf Spatial} and 
{\bf stochastic} discretization

\begin{itemize}


\vspace*{0.2cm}
\item  Deterministic  spatial  differential operators ($A$ \& $B$) $\rightarrow$
Regular shape function method with Galerkin scheme

\vspace*{0.2cm}
\item    Input    random   field   material   properties   ($D$)   $\rightarrow$
Karhunen--Lo{\`e}ve (KL) expansion, optimal expansion, error minimizing property

\vspace*{0.2cm}
\item  Unknown  solution  random field ($u$) $\rightarrow$ Polynomial Chaos (PC)
expansion



\end{itemize}



\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - 
% - \begin{frame}
% - 
% - 
% - \frametitle{Truncated Karhunen--Lo{\`e}ve (KL) expansion}
% -               
% -        \begin{itemize}                                                                                                                         
% -                                                                                                                                                       
% -        \item Representation of input random fields in eigen-modes of covariance kernel                                                                
% -                                                                                                                                                       
% - %       \vspace*{-0.1cm}                                                                                                                               
% - %              \begin{figure}[!hbpt]                                                                                                                   
% -               \begin{flushleft}                                                                                                                       
% -               \includegraphics[height=3.0cm]{/home/jeremic/tex/works/Conferences/2008/GeoCongress/Probabilistic/Paper/TypicalDataPlotBH1-Edited.jpg}
% - %       ShearStrengthProfile.jpg}                                                                                
% -               \end{flushleft}                                                                                                                         
% -               \vspace*{-3.15cm}                                                                                                                        
% -        \begin{flushright}                                                                                                                             
% - %       \begin{equation}                                                                                                                              
% - %       \nonumber                                                                                                                                     
% - %       \begin{normalsize}                                                                                                                            
% -        $ q_T(x,\theta) = \bar q_T(x) + \sum_{n=1}^M \sqrt{\lambda_n} \xi_n(\theta) f_n(x) $ \\                                                        
% -        \ \\                                                                                                                                           
% -        $ \int_D C(x_1, x_2) f (x_2) dx_2 = \lambda f (x_1) \ \ \ \ \ \ \ \ \ \  \ \ $ \\                                                              
% -        \ \\                                                                                                                                           
% -        $ \xi_i(\theta) = \displaystyle \frac{1}{\sqrt \lambda_i} \int_D \left [q_T(x,\theta) - \bar q_T (x) \right] f_i (x) dx $                                   
% - %       \end{equation}                                                                                                                                
% - %              \end{normalsize}                                                                                                                       
% -        \end{flushright}                                                                                                                               
% - %       \end{figure}                                                                                                                                   
% -                                                                                                                                                       
% - %       \vspace{-6.0cm}                                                                                                                               
% - %       \begin{flushright}                                                                                                                            
% - %       \begin{equation}                                                                                                                              
% - %       \nonumber                                                                                                                                     
% - %        w(x,\theta) = \bar w(x) + \sum_{n=0}^M \sqrt{\lambda_n} \zeta_n(\theta) f_n(x)                                                               
% - %       \end{equation}                                                                                                                                
% - %        \end{flushright}                                                                                                                             
% -                                                                                                                                                       
% - %        \vspace*{0.8cm}                                                                                                                               
% -         \item Error minimizing property                                                                                                               
% -                                                                                                                                                       
% -         \item Optimal expansion $\rightarrow$ minimization of number of stochastic dimensions                                                         
% -                                                                                                                                                       
% -                                                                                                                                                       
% -          \end{itemize}                                                                                                                                                                                                                                                                   
% -                                                                                                                                                                                                                                                                                          
% - 
% - 
% - \end{frame}
% - 
% - 
% - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - \begin{frame}
% - \frametitle{KL Expansion (of Covariance Kernel)}
% - 
% - \begin{flushleft}
% - %\begin{center}
% - %\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
% - \includegraphics[height=2.2cm]{/home/jeremic/tex/works/Conferences/2007/GeoDenver/SFEM/Presentation/ActualExponentialCovarianveSurface.jpg}
% - \hspace*{0.3cm}
% - \includegraphics[height=2.2cm]{/home/jeremic/tex/works/Conferences/2007/GeoDenver/SFEM/Presentation/KL_ApproxWith_1Term_CovarianveSurface.jpg}
% - %\vspace*{1.0cm}
% - %\mbox{Exact covariance surface}
% - %\vspace*{-4.0cm}
% - \end{flushleft}
% - \vspace*{-0.4cm}
% - \small{\ \ \ \ \ \ \ \ \ \ \ \ \ Exact \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1 term approx.} \\
% - \small{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (8.49\% error)}
% - %\begin{flushright}
% - %\end{center}
% - %\end{figure}
% - 
% - %\vspace*{-0.5cm}
% - %\small{Exact covariance surface \ \ \ \ \ \ \ \ \ \ \ \ \ \ One-term approximation}
% - %
% - %\vspace*{0.5cm}
% - %
% - %\begin{figure}[!hbpt]
% - %\begin{center}
% - \begin{flushleft}
% - \includegraphics[height=2.2cm]{/home/jeremic/tex/works/Conferences/2007/GeoDenver/SFEM/Presentation/KL_ApproxWith_2Terms_CovarianveSurface.jpg}
% - \hspace*{-0.2cm}
% - \includegraphics[height=2.2cm]{/home/jeremic/tex/works/Conferences/2007/GeoDenver/SFEM/Presentation/KL_ApproxWith_3Terms_CovarianveSurface.jpg}
% - \end{flushleft}
% - %\vspace*{-0.5cm}
% - \small{\ \ \ \ \ 2 terms approx. \ \ \ \ \ \ \ \ \  3 terms approx.} \\
% - \small{\ \ \ \ \ \ (1.15\% error) \ \ \ \ \ \ \ \ \ \ \ \ \ (1.13\% error)} 
% - %
% - \vspace{-6.5cm}
% - \begin{flushright}
% - 
% - \includegraphics[height=2.0cm]{/home/jeremic/tex/works/Reports/2006/SEPFEM/figures/CPT_DataAnalysis_Plots/TypicalAutoCovariancePlotBH1_FiniteScale-Edited.jpg} \hspace*{-0.7cm}
% - %ShearStrengthProfile.jpg}
% - 
% - covariance function \\
% - (exponential):  \ \ \ \ \  \\
% - %\ \\
% - $ C(x_1, x_2) = \sigma^2 e^{-| x_1 - x_2 | /b} $  \hspace*{-0.5cm} \\
% - \ \\
% - \ \\
% - KL approximation: \ \ \ \ \ \ \ \  \\
% - \ \\
% - $ C(x_1, x_2) \ \ \ \ \ \ \ \ \ \ \ \ \  $ \\
% -     $ = \sum_{k =1}^M \lambda_k f_k(x_1) f_k(x_2) $
% - 
% - \end{flushright}
% - %\end{figure}
% - 
% - %\vspace*{2.5cm}
% - %\small{Two-terms approximation \ \ \ \ \ \ \ \ \ \ \ \  Three-terms approximation}
% - 
% - \vspace*{4.0cm}
% - \begin{center}
% - \large{KL Expansion of Covariance Kernel}
% - \end{center}
% - 
% - \end{frame}
% - 
% - 
% - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - 
% - \begin{frame}
% - 
% - \frametitle{Polynomial Chaos (PC) Expansion}
% - 
% - %\begin{itemize}
% - 
% - %\vspace*{-0.35cm}
% - %\item Solution (displacement) random field $\rightarrow$ Can not use K-L expansion directly
% - 
% -        \begin{itemize}
% - 
% -        \item Covariance kernel is not known a priori
% - 
% -        \vspace*{-0.5cm}
% -        \begin{normalsize}
% -        \begin{equation}
% -        \nonumber
% -        u(x,\theta)=\sum_{j=1}^L e_j \chi_j(\theta)b_j(x)
% -        \end{equation}
% -        \end{normalsize}
% - 
% -        \vspace*{-0.3cm}
% -        \item Can be expressed as functional of known random variables and unknown deterministic function
% - 
% -        \vspace*{-0.5cm}
% -        \begin{normalsize}
% -        \begin{equation}
% -        \nonumber
% -        u(x,\theta)=\zeta[\xi_i(\theta),x]
% -        \end{equation}
% -        \end{normalsize}
% - 
% -        \vspace*{-0.5cm}
% -        \item Need a basis of known random variables $\rightarrow$ PC expansion
% - 
% -        \vspace*{-0.2cm}
% -        \begin{normalsize}
% -        \begin{equation}
% -        \nonumber
% -        \chi_j(\theta)=\sum_{i=0}^P\gamma_i^{(j)}\psi_i\left[\left\{\xi_r\right\}\right]
% -        \end{equation}
% - 
% -        \vspace*{-0.5cm}
% -        \begin{equation}
% -        \nonumber
% -        u(x,\theta)=\sum_{j=1}^L \sum_{i=0}^P \gamma_i^{(j)} \psi_i[\{\xi_r\}]e_j b_j(x) = \sum_{i=0}^P \psi_i[\{\xi_r\}] d_i(x)
% -        \end{equation}
% -        \end{normalsize}
% - 
% - %       \vspace*{-0.6cm}
% - %       \item Deterministic coefficients can be found by minimizing norm of error of finite 
% - %            representation (e.g. using Galerkin scheme) 
% -        
% -         
% - 
% - 
% - %       \end{itemize}
% - \end{itemize}
% - 
% - 
% - %\end{itemize}
% - 
% - \end{frame}
% - 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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

\begin{itemize}

\item  Minimizing    norm  of  error  of  finite  representation  using Galerkin
technique (Ghanem and Spanos 2003):

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

%      \begin{itemize}
%
%      \vspace*{0.5cm}
%      \item Final eqn.:
%      
%      \vspace*{-0.4cm}
%      \begin{flushright}
%      \begin{normalsize}
%      \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[\{\zeta_r\}] \right >
%      \end{equation}
%      \end{normalsize}
%      \end{flushright}

      \vspace*{-0.5cm}
      \begin{equation}
      \nonumber
      K_{mn}^{ep} = \int_D B_n \textcolor{mycolor}{D}^{ep}  B_m dV    
      \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
      K_{mnk}^{'ep} = \int_D B_n {\sqrt \lambda_k h_k} B_m dV
      \end{equation}
      
      \vspace*{-1.0cm}
      \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}

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


\end{itemize}

%  \noindent Salient Features:
       
%      \begin{itemize}
%
%      \item Efficient representation of input random fields into finite number of random
%            variables using KL-expansion
%      
%      \item Representation of (unknown) solution random variables 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}
%
\end{frame}

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

\begin{frame}
\frametitle{Inside SSEPFEM}

\begin{itemize}

\item Explicit stochastic elastic--plastic finite element computations

\vspace*{0.2cm}
\item FPK probabilistic constitutive integration at Gauss integration points

\vspace*{0.2cm}
\item   Increase   in  (stochastic)  dimensions  (KL  and  PC)  of  the  problem



\vspace*{0.2cm}
\item  Excellent for parallelization, both at the element and global levels


\vspace*{0.2cm}
\item Development of the probabilistic elastic--plastic stiffness tensor


\end{itemize}

%  \noindent Salient Features:
       
%      \begin{itemize}
%
%      \item Efficient representation of input random fields into finite number of random
%            variables using KL-expansion
%      
%      \item Representation of (unknown) solution random variables 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}
%
\end{frame}















% - 
% -  \begin{frame}
% -  
% -  
% -  
% -  \frametitle{Governing Equations \& Discretization Scheme}
% -  
% -  \begin{itemize}
% -  
% -  \item Governing equations in mechanics:
% -  
% -  
% -  \begin{equation}
% -  \nonumber
% -  A\sigma = \phi(t);~~~ Bu = \epsilon; ~~~\sigma = D \epsilon
% -  \end{equation}
% -  
% -  \item  Discretization (spatial and stochastic) schemes
% -  
% -  \begin{itemize}
% -  
% -  \item    Input    random   field   material   properties   ($D$)   $\rightarrow$
% -  Karhunen--Lo{\`e}ve (KL) expansion, optimal expansion, error minimizing property
% -  
% -  \item  Unknown  solution  random field ($u$) $\rightarrow$ Polynomial Chaos (PC)
% -  expansion
% -  
% -  \item  Deterministic  spatial  differential operators ($A$ \& $B$) $\rightarrow$
% -  Regular shape function method with Galerkin scheme
% -  
% -  
% -  \end{itemize}
% -  
% -  
% -  
% -  \end{itemize}
% -  
% -  \end{frame}
% -  
% -  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  % - % - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  % - 
% -  % - \begin{frame}
% -  % - 
% -  % - 
% -  % - \frametitle{Truncated Karhunen--Lo{\`e}ve (KL) expansion}
% -  % -               
% -  % -        \begin{itemize}                                                                                                                         
% -  % -                                                                                                                                                       
% -  % -        \item Representation of input random fields in eigen-modes of covariance kernel                                                                
% -  % -                                                                                                                                                       
% -  % - %       \vspace*{-0.1cm}                                                                                                                               
% -  % - %              \begin{figure}[!hbpt]                                                                                                                   
% -  % -               \begin{flushleft}                                                                                                                       
% -  % -               \includegraphics[height=3.0cm]{/home/jeremic/tex/works/Conferences/2008/GeoCongress/Probabilistic/Paper/TypicalDataPlotBH1-Edited.jpg}
% -  % - %       ShearStrengthProfile.jpg}                                                                                
% -  % -               \end{flushleft}                                                                                                                         
% -  % -               \vspace*{-3.15cm}                                                                                                                        
% -  % -        \begin{flushright}                                                                                                                             
% -  % - %       \begin{equation}                                                                                                                              
% -  % - %       \nonumber                                                                                                                                     
% -  % - %       \begin{normalsize}                                                                                                                            
% -  % -        $ q_T(x,\theta) = \bar q_T(x) + \sum_{n=1}^M \sqrt{\lambda_n} \xi_n(\theta) f_n(x) $ \\                                                        
% -  % -        \ \\                                                                                                                                           
% -  % -        $ \int_D C(x_1, x_2) f (x_2) dx_2 = \lambda f (x_1) \ \ \ \ \ \ \ \ \ \  \ \ $ \\                                                              
% -  % -        \ \\                                                                                                                                           
% -  % -        $ \xi_i(\theta) = \displaystyle \frac{1}{\sqrt \lambda_i} \int_D \left [q_T(x,\theta) - \bar q_T (x) \right] f_i (x) dx $                                   
% -  % - %       \end{equation}                                                                                                                                
% -  % - %              \end{normalsize}                                                                                                                       
% -  % -        \end{flushright}                                                                                                                               
% -  % - %       \end{figure}                                                                                                                                   
% -  % -                                                                                                                                                       
% -  % - %       \vspace{-6.0cm}                                                                                                                               
% -  % - %       \begin{flushright}                                                                                                                            
% -  % - %       \begin{equation}                                                                                                                              
% -  % - %       \nonumber                                                                                                                                     
% -  % - %        w(x,\theta) = \bar w(x) + \sum_{n=0}^M \sqrt{\lambda_n} \zeta_n(\theta) f_n(x)                                                               
% -  % - %       \end{equation}                                                                                                                                
% -  % - %        \end{flushright}                                                                                                                             
% -  % -                                                                                                                                                       
% -  % - %        \vspace*{0.8cm}                                                                                                                               
% -  % -         \item Error minimizing property                                                                                                               
% -  % -                                                                                                                                                       
% -  % -         \item Optimal expansion $\rightarrow$ minimization of number of stochastic dimensions                                                         
% -  % -                                                                                                                                                       
% -  % -                                                                                                                                                       
% -  % -          \end{itemize}                                                                                                                                                                                                                                                                   
% -  % -                                                                                                                                                                                                                                                                                          
% -  % - 
% -  % - 
% -  % - \end{frame}
% -  % - 
% -  % - 
% -  % - % - % - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  % - % - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  % - % - \begin{frame}
% -  % - % - \frametitle{KL Expansion (of Covariance Kernel)}
% -  % - % - 
% -  % - % - \begin{flushleft}
% -  % - % - %\begin{center}
% -  % - % - %\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
% -  % - % - \includegraphics[height=2.2cm]{/home/jeremic/tex/works/Conferences/2007/GeoDenver/SFEM/Presentation/ActualExponentialCovarianveSurface.jpg}
% -  % - % - \hspace*{0.3cm}
% -  % - % - \includegraphics[height=2.2cm]{/home/jeremic/tex/works/Conferences/2007/GeoDenver/SFEM/Presentation/KL_ApproxWith_1Term_CovarianveSurface.jpg}
% -  % - % - %\vspace*{1.0cm}
% -  % - % - %\mbox{Exact covariance surface}
% -  % - % - %\vspace*{-4.0cm}
% -  % - % - \end{flushleft}
% -  % - % - \vspace*{-0.4cm}
% -  % - % - \small{\ \ \ \ \ \ \ \ \ \ \ \ \ Exact \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1 term approx.} \\
% -  % - % - \small{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (8.49\% error)}
% -  % - % - %\begin{flushright}
% -  % - % - %\end{center}
% -  % - % - %\end{figure}
% -  % - % - 
% -  % - % - %\vspace*{-0.5cm}
% -  % - % - %\small{Exact covariance surface \ \ \ \ \ \ \ \ \ \ \ \ \ \ One-term approximation}
% -  % - % - %
% -  % - % - %\vspace*{0.5cm}
% -  % - % - %
% -  % - % - %\begin{figure}[!hbpt]
% -  % - % - %\begin{center}
% -  % - % - \begin{flushleft}
% -  % - % - \includegraphics[height=2.2cm]{/home/jeremic/tex/works/Conferences/2007/GeoDenver/SFEM/Presentation/KL_ApproxWith_2Terms_CovarianveSurface.jpg}
% -  % - % - \hspace*{-0.2cm}
% -  % - % - \includegraphics[height=2.2cm]{/home/jeremic/tex/works/Conferences/2007/GeoDenver/SFEM/Presentation/KL_ApproxWith_3Terms_CovarianveSurface.jpg}
% -  % - % - \end{flushleft}
% -  % - % - %\vspace*{-0.5cm}
% -  % - % - \small{\ \ \ \ \ 2 terms approx. \ \ \ \ \ \ \ \ \  3 terms approx.} \\
% -  % - % - \small{\ \ \ \ \ \ (1.15\% error) \ \ \ \ \ \ \ \ \ \ \ \ \ (1.13\% error)} 
% -  % - % - %
% -  % - % - \vspace{-6.5cm}
% -  % - % - \begin{flushright}
% -  % - % - 
% -  % - % - \includegraphics[height=2.0cm]{/home/jeremic/tex/works/Reports/2006/SEPFEM/figures/CPT_DataAnalysis_Plots/TypicalAutoCovariancePlotBH1_FiniteScale-Edited.jpg} \hspace*{-0.7cm}
% -  % - % - %ShearStrengthProfile.jpg}
% -  % - % - 
% -  % - % - covariance function \\
% -  % - % - (exponential):  \ \ \ \ \  \\
% -  % - % - %\ \\
% -  % - % - $ C(x_1, x_2) = \sigma^2 e^{-| x_1 - x_2 | /b} $  \hspace*{-0.5cm} \\
% -  % - % - \ \\
% -  % - % - \ \\
% -  % - % - KL approximation: \ \ \ \ \ \ \ \  \\
% -  % - % - \ \\
% -  % - % - $ C(x_1, x_2) \ \ \ \ \ \ \ \ \ \ \ \ \  $ \\
% -  % - % -     $ = \sum_{k =1}^M \lambda_k f_k(x_1) f_k(x_2) $
% -  % - % - 
% -  % - % - \end{flushright}
% -  % - % - %\end{figure}
% -  % - % - 
% -  % - % - %\vspace*{2.5cm}
% -  % - % - %\small{Two-terms approximation \ \ \ \ \ \ \ \ \ \ \ \  Three-terms approximation}
% -  % - % - 
% -  % - % - \vspace*{4.0cm}
% -  % - % - \begin{center}
% -  % - % - \large{KL Expansion of Covariance Kernel}
% -  % - % - \end{center}
% -  % - % - 
% -  % - % - \end{frame}
% -  % - % - 
% -  % - % - 
% -  % - % - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  % - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  % - 
% -  % - \begin{frame}
% -  % - 
% -  % - \frametitle{Polynomial Chaos (PC) Expansion}
% -  % - 
% -  % - %\begin{itemize}
% -  % - 
% -  % - %\vspace*{-0.35cm}
% -  % - %\item Solution (displacement) random field $\rightarrow$ Can not use K-L expansion directly
% -  % - 
% -  % -        \begin{itemize}
% -  % - 
% -  % -        \item Covariance kernel is not known a priori
% -  % - 
% -  % -        \vspace*{-0.5cm}
% -  % -        \begin{normalsize}
% -  % -        \begin{equation}
% -  % -        \nonumber
% -  % -        u(x,\theta)=\sum_{j=1}^L e_j \chi_j(\theta)b_j(x)
% -  % -        \end{equation}
% -  % -        \end{normalsize}
% -  % - 
% -  % -        \vspace*{-0.3cm}
% -  % -        \item Can be expressed as functional of known random variables and unknown deterministic function
% -  % - 
% -  % -        \vspace*{-0.5cm}
% -  % -        \begin{normalsize}
% -  % -        \begin{equation}
% -  % -        \nonumber
% -  % -        u(x,\theta)=\zeta[\xi_i(\theta),x]
% -  % -        \end{equation}
% -  % -        \end{normalsize}
% -  % - 
% -  % -        \vspace*{-0.5cm}
% -  % -        \item Need a basis of known random variables $\rightarrow$ PC expansion
% -  % - 
% -  % -        \vspace*{-0.2cm}
% -  % -        \begin{normalsize}
% -  % -        \begin{equation}
% -  % -        \nonumber
% -  % -        \chi_j(\theta)=\sum_{i=0}^P\gamma_i^{(j)}\psi_i\left[\left\{\xi_r\right\}\right]
% -  % -        \end{equation}
% -  % - 
% -  % -        \vspace*{-0.5cm}
% -  % -        \begin{equation}
% -  % -        \nonumber
% -  % -        u(x,\theta)=\sum_{j=1}^L \sum_{i=0}^P \gamma_i^{(j)} \psi_i[\{\xi_r\}]e_j b_j(x) = \sum_{i=0}^P \psi_i[\{\xi_r\}] d_i(x)
% -  % -        \end{equation}
% -  % -        \end{normalsize}
% -  % - 
% -  % - %       \vspace*{-0.6cm}
% -  % - %       \item Deterministic coefficients can be found by minimizing norm of error of finite 
% -  % - %            representation (e.g. using Galerkin scheme) 
% -  % -        
% -  % -         
% -  % - 
% -  % - 
% -  % - %       \end{itemize}
% -  % - \end{itemize}
% -  % - 
% -  % - 
% -  % - %\end{itemize}
% -  % - 
% -  % - \end{frame}
% -  % - 
% -  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  
% -  \begin{frame}
% -  \frametitle{Spectral Stochastic Elastic--Plastic FEM}
% -  
% -  \begin{itemize}
% -  
% -  \item  Minimizing    norm  of  error  of  finite  representation  using Galerkin
% -  technique (Ghanem and Spanos 2003):
% -  
% -        \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}
% -  
% -  %      \begin{itemize}
% -  %
% -  %      \vspace*{0.5cm}
% -  %      \item Final eqn.:
% -  %      
% -  %      \vspace*{-0.4cm}
% -  %      \begin{flushright}
% -  %      \begin{normalsize}
% -  %      \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[\{\zeta_r\}] \right >
% -  %      \end{equation}
% -  %      \end{normalsize}
% -  %      \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*{-1.0cm}
% -        \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}
% -  
% -  %\item Fokker--Planck--Kolmogorov approach based probabilistic constitutive integration
% -  %      at Gauss integration points
% -  
% -  
% -  \end{itemize}
% -  
% -  %  \noindent Salient Features:
% -         
% -  %      \begin{itemize}
% -  %
% -  %      \item Efficient representation of input random fields into finite number of random
% -  %            variables using KL-expansion
% -  %      
% -  %      \item Representation of (unknown) solution random variables 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}
% -  %
% -  \end{frame}
% -  
% -  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  
% -  \begin{frame}
% -  \frametitle{Inside SSEPFEM}
% -  
% -  \begin{itemize}
% -  
% -  \item Explicit stochastic elastic--plastic finite element computations
% -  
% -  \vspace*{0.2cm}
% -  \item FPK probabilistic constitutive integration at Gauss integration points
% -  
% -  \vspace*{0.2cm}
% -  \item   Increase   in  (stochastic)  dimensions  (KL  and  PC)  of  the  problem
% -  
% -  
% -  \vspace*{0.2cm}
% -  \item Development of the probabilistic elastic--plastic stiffness tensor
% -  
% -  
% -  \end{itemize}
% -  
% -  %  \noindent Salient Features:
% -         
% -  %      \begin{itemize}
% -  %
% -  %      \item Efficient representation of input random fields into finite number of random
% -  %            variables using KL-expansion
% -  %      
% -  %      \item Representation of (unknown) solution random variables 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}
% -  %
% -  \end{frame}
% -  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{SSEPFEM Example}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{frame}


\frametitle{1--D Static Pushover Test Example}




\begin{itemize}

\vspace*{-0.3cm}
  \item  Linear elastic model: \\
         $<G> = 2.5$~kPa, \\
         $Var[G]=0.15~{\rm kPa}^2$,\\ 
         correlation length for $G=0.3$~m.
  
\vspace*{0.3cm}
  \item  Elastic--plastic material model,\\
         von Mises, linear hardening,\\
         $<G> = 2.5$~kPa, \\
         $Var[G]=0.15~{\rm kPa}^2$,\\ 
         correlation length for $G=0.3$~m, \\
         $C_u = 5$~kPa, \\
         $C^{'}_u = 2$~kPa.

\end{itemize}


%\begin{figure}[!hbpt]
%\begin{center}
\vspace*{-6cm}
\hspace*{7.5cm}
\includegraphics[height=2.5cm,angle=-90]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Present/ShearBeamModel.pdf}
%\end{center}
%\end{figure}

% % 
% \begin{center}
% %\hspace*{-1.7cm}
% \includegraphics[height=5.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter8Plots/Chapter8_SchematicStaticProblem.jpg}
% %\hspace*{0.7cm}
% %\includegraphics[height=3.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter8Plots/Chapter8_StaticExample_Elastic-Edited.jpg} \\
% %\vspace*{0.1truecm}
% %%\small{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Base Displacement}
% %\includegraphics[height=3.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter8Plots/Chapter8_StaticExample_ElasticPlastic-Edited.pdf}
% %\hspace*{0.0cm}
% %\includegraphics[height=3.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter8Plots/PushOverTestPDF.jpg}
% \end{center}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% 
% 
% \frametitle{Linear Elastic FEM Verification}
% 
% 
% \vspace*{-0.5cm}
% \begin{figure}[!htbp]
% \begin{center}
% \includegraphics[width=0.8\textwidth]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter8Plots/Chapter8_StaticExample_Elastic-Edited.pdf}
% \label{figure:StaticElasticSimulation}
% \end{center}
% \end{figure}
% \vspace*{-2.5cm}
% 
% Mean and standard deviations of displacement at the top node,\\
% linear elastic material model, \\
% KL-dimension=2, order of PC=2. 
% %Monte Carlo simulation
% %is also shown.
% 
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}


\frametitle{SSEPFEM Response}



%\vspace*{-0.2cm}
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.65\textwidth]{/home/jeremic/tex/works/Papers/2010/SSEPFEM_static/meanPMsd_ps.pdf}
\label{figure:StaticElasticPlasticSimulation}
\end{center}
\end{figure}
\vspace*{-0.2cm}

Mean and standard deviations of displacement at the top node,\\
von Mises elastic-perfectly plastic material model,\\ 
KL-dimension=2, order of PC=2.
% Monte Carlo
%simulation is also shown}


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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Evolution of Probabilistic Stiffness at $-6.645$m }
  
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.48\textwidth]{/home/jeremic/tex/works/Papers/2010/SSEPFEM_static/figures/tangentModulus_ps.pdf}
\hspace*{0.1truecm}
\includegraphics[width=0.48\textwidth]{/home/jeremic/tex/works/Papers/2010/SSEPFEM_static/figures/lambda_ps.pdf}
\end{center}
\end{figure}

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


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Probability for Softening!}
  
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.485\textwidth]{/home/jeremic/tex/works/Papers/2010/SSEPFEM_static/figures/tangentModulusEvolution_ps.pdf}
%\hspace*{0.1truecm}
\includegraphics[width=0.50\textwidth]{/home/jeremic/tex/works/Papers/2010/SSEPFEM_static/figures/SDtangentModulusEvolution_ps.pdf}
\end{center}
\end{figure}

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

% -- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -- \begin{frame}
% -- 
% -- 
% -- \frametitle{Stochastic Response at the Top}
% -- 
% -- 
% -- 
% -- 
% -- 
% -- \begin{figure}[!htbp]
% -- \begin{center}
% -- \includegraphics[width=0.65\textwidth]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter8Plots/Chapter8_StaticExample_PDFComparison-Edited.pdf}
% -- \label{figure:StaticProblem_PDFComparison}
% -- \end{center}
% -- \end{figure}
% -- 
% -- 
% -- Comparison of PDF of top node displacement.
% -- 
% -- 
% -- 
% -- \end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%-- \begin{frame}
%-- \frametitle{1--D Static Pushover Test Example}
%-- 
%-- 
%-- 
%-- \begin{figure}[!htbp]
%-- \begin{center}
%-- \includegraphics[width=0.35\textwidth]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter8Plots/Chapter8_StaticExample_CorLength_andKL_LargeCorLength_Mean-Edited.pdf}
%-- \includegraphics[width=0.35\textwidth]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter8Plots/Chapter8_StaticExample_CorLength_andKL_LargeCorLength_SD-Edited.pdf}
%-- 
%-- \label{figure:CorLength_andKL_LargeCorLength_Mean}
%-- \end{center}
%-- \end{figure}
%-- 
%-- 
%-- 
%-- Correlation length and KL dimension: Mean and standard deviation of displacement along depth of the 1--D soil column with
%-- linear elastic material model, having very small variance (COV = 1\%) of shear modulus and very large 
%-- ratio of correlation length of shear modulus to domain length (= 100)}
%-- 
%-- Correlation length and KL dimension: Standard deviation of displacement along depth of the 1--D 
%-- soil column with linear elastic material model, having very small variance (COV = 1\%) of shear modulus 
%-- and very large ratio of correlation length of shear modulus to domain length (= 100)}
%-- 
%-- \end{frame}
%-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%-- \begin{frame}
%-- \frametitle{1--D Static Pushover Test Example}
%-- 
%-- 
%-- 
%-- \begin{figure}[!htbp]
%-- \begin{center}
%-- \includegraphics[width=0.7\textwidth]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter8Plots/Chapter8_StaticExample_CorLength_andKL_SmallCorLength_Mean-Edited.pdf}
%-- \caption{Correlation length and KL dimension: Mean displacement along depth of the 1--D soil column with
%-- linear elastic material model, having very small variance (COV = 1\%) of shear modulus and very small 
%-- ratio of correlation length of shear modulus to domain length (= 0.0001)}
%-- \label{figure:CorLength_andKL_SmallCorLength_Mean}
%-- \end{center}
%-- \end{figure}
%-- 
%-- 
%-- 
%-- \end{frame}
%-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%-- \begin{frame}
%-- \frametitle{1--D Static Pushover Test Example}
%-- 
%-- 
%-- 
%-- \begin{figure}[!htbp]
%-- \begin{center}
%-- \includegraphics[width=0.7\textwidth]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter8Plots/Chapter8_StaticExample_CorLength_andKL_SmallCorLength_SD-Edited.pdf}
%-- \caption{Correlation length and KL dimension: Standard deviation of displacement along depth of the 1--D 
%-- soil column with linear elastic material model, having very small variance (COV = 1\%) of shear modulus 
%-- and very small ratio of correlation length of shear modulus to domain length (= 0.0001)}
%-- \label{figure:CorLength_andKL_SmallCorLength_SD}
%-- \end{center}
%-- \end{figure}
%-- 
%-- 
%-- 
%-- 
%-- \end{frame}
%-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%-- \begin{frame}
%-- \frametitle{1--D Static Pushover Test Example}
%-- 
%-- 
%-- 
%-- \begin{figure}[!htbp]
%-- \begin{center}
%-- \includegraphics[width=0.7\textwidth]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter8Plots/Chapter8_StaticExample_Variance_andPC_LargeVariance_Mean-Edited.pdf}
%-- \caption{Variance and order of PC: Mean displacement along depth of the 1--D soil column with
%-- linear elastic material model, having large variance (COV = 20\%) of shear modulus and  
%-- ratio of correlation length of shear modulus to domain length = 0.1}
%-- \label{figure:Variance_andPC_LargeVariance_Mean}
%-- \end{center}
%-- \end{figure}
%-- 
%-- 
%-- \end{frame}
%-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%-- \begin{frame}
%-- \frametitle{1--D Static Pushover Test Example}
%-- 
%-- 
%-- 
%-- \begin{figure}[!htbp]
%-- \begin{center}
%-- \includegraphics[width=0.7\textwidth]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter8Plots/Chapter8_StaticExample_Variance_andPC_LargVariance_SD-Edited.pdf}
%-- \caption{Variance and order of PC: Standard deviation of displacement along depth of the 1--D 
%-- soil column with linear elastic material model, having large variance (COV = 20\%) of shear modulus 
%-- and ratio of correlation length of shear modulus to domain length = 0.1}
%-- \label{figure:Variance_andPC_LargeVariance_SD}
%-- \end{center}
%-- \end{figure}
%-- 
%-- 
%-- \end{frame}
%-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%-- \begin{frame}
%-- \frametitle{1--D Static Pushover Test Example}
%-- 
%-- 
%-- 
%-- \begin{figure}[!htbp]
%-- \begin{center}
%-- \includegraphics[width=0.7\textwidth]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter8Plots/Chapter8_StaticExample_Variance_andPC_SmallVariance_Mean-Edited.pdf}
%-- \caption{Variance and order of PC: Mean displacement along depth of the 1--D soil column with
%-- linear elastic material model, having very small variance (COV = 1\%) of shear modulus and  
%-- ratio of correlation length of shear modulus to domain length = 0.1}
%-- \label{figure:Variance_andPC_SmallVariance_Mean}
%-- \end{center}
%-- \end{figure}
%-- 
%-- 
%-- 
%-- \end{frame}
%-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%-- \begin{frame}
%-- \frametitle{1--D Static Pushover Test Example}
%-- 
%-- 
%-- 
%-- \begin{figure}[!htbp]
%-- \begin{center}
%-- \includegraphics[width=0.7\textwidth]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter8Plots/Chapter8_StaticExample_Variance_andPC_SmallVariance_SD-Edited.pdf}
%-- \caption{Variance and order of PC: Standard deviation of displacement along depth of the 1--D 
%-- soil column with linear elastic material model, having very small variance (COV = 1\%) of shear modulus 
%-- and ratio of correlation length of shear modulus to domain length = 0.1}
%-- \label{figure:Variance_andPC_SmallVariance_SD}
%-- \end{center}
%-- \end{figure}
%-- 
%-- % \begin{table}
%-- % \begin{center}
%-- % \caption{Comparison of results (at top node) of FPKE-based spectral stochastic finite element with
%-- % direct spectral stochastic finite element, for 1--D soil column example, with linear elastic material}
%-- % \label{Table:Comparison}
%-- % \begin{tabular}{|c|ccc|ccc|}
%-- %  \hline
%-- % Load Step & \multicolumn{3}{c|}{Mean of solution} & \multicolumn{3}{c|}{Standard deviation of solution} \\
%-- %  \cline{2-7}
%-- % & Direct (mm) & FPKE (mm) & Error (\%) & Direct (mm) & FPKE (mm) & Error (\%) \\
%-- %  \hline \hline
%-- % 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
%-- %  \hline
%-- % 1 & 2.015 & 2.015 & 0 & 0.2005 & 0.2005 & 0 \\
%-- %  \hline
%-- % 2 & 4.03 & 4.00 & 0.7 & 0.40 & 0.39 & 0.7 \\
%-- %  \hline
%-- % 3 & 6.04 & 5.95 & 1.4 & 0.60 & 0.59 & 1.4 \\
%-- %  \hline
%-- % 4 & 8.06 & 7.89 & 2.1 & 0.80 & 0.78 & 2.1 \\
%-- %  \hline
%-- % 5 & 10.00 & 9.84 & 1.7 & 1.00 & 0.98 & 1.7 \\
%-- %  \hline
%-- % 6 & 12.09 & 11.88 & 1.7 & 1.20 & 1.21 & 0.6 \\
%-- %  \hline
%-- % \end{tabular}        
%-- % \end{center}
%-- % \end{table}
%--  
%-- 
%-- %\begin{figure}[htbp]
%-- %\begin{center}
%-- %\includegraphics[width=\textwidth]{EffectOfCorLength_6_3_01}
%-- %\caption{}
%-- %\label{figure:EffectOfKL_on_CorLength_Mean}
%-- %\end{center}
%-- %\end{figure}
%-- %
%-- %
%-- %\begin{figure}[htbp]
%-- %\begin{center}
%-- %\includegraphics[width=\textwidth]{EffectOfCorLength_6_3_02}
%-- %\caption{}
%-- %\label{figure:EffectOfKL_on_CorLength_Mean}
%-- %\end{center}
%-- %\end{figure}
%-- %
%-- %\begin{figure}[htbp]
%-- %\begin{center}
%-- %\includegraphics[width=\textwidth]{EffectOfCorLength_6_3_03}
%-- %\caption{}
%-- %\label{figure:EffectOfKL_on_CorLength_Mean}
%-- %\end{center}
%-- %\end{figure}
%-- %
%-- %\begin{figure}[htbp]
%-- %\begin{center}
%-- %\includegraphics[width=\textwidth]{EffectOfCorLength_6_3_04}
%-- %\caption{}
%-- %\label{figure:EffectOfKL_on_CorLength_Mean}
%-- %\end{center}
%-- %\end{figure}
%-- %
%-- %
%-- %\begin{figure}[htbp]
%-- %\begin{center}
%-- %\includegraphics[width=\textwidth]{EffectOfCorLength_4_4_01}
%-- %\caption{}
%-- %\label{figure:EffectOfKL_on_CorLength_Mean}
%-- %\end{center}
%-- %\end{figure}
%-- %
%-- %\begin{figure}[htbp]
%-- %\begin{center}
%-- %\includegraphics[width=\textwidth]{EffectOfCorLength_4_4_02}
%-- %\caption{}
%-- %\label{figure:EffectOfKL_on_CorLength_Mean}
%-- %\end{center}
%-- %\end{figure}
%-- %
%-- %
%-- %\begin{figure}[htbp]
%-- %\begin{center}
%-- %\includegraphics[width=\textwidth]{EffectOfCorLength_4_4_03}
%-- %\caption{}
%-- %\label{figure:EffectOfKL_on_CorLength_Mean}
%-- %\end{center}
%-- %\end{figure}
%-- %
%-- %
%-- %\begin{figure}[htbp]
%-- %\begin{center}
%-- %\includegraphics[width=\textwidth]{EffectOfKL_on_CorLength_Mean}
%-- %\caption{}
%-- %\label{figure:EffectOfKL_on_CorLength_Mean}
%-- %\end{center}
%-- %\end{figure}
%-- %
%-- %
%-- %\begin{figure}[htbp]
%-- %\begin{center}
%-- %\includegraphics[width=\textwidth]{EffectOfKL_on_CorLength_SD}
%-- %\caption{}
%-- %\label{figure:EffectOfKL_on_CorLength_SD}
%-- %\end{center}
%-- %\end{figure}
%-- %
%-- %
%-- %\begin{figure}[htbp]
%-- %\begin{center}
%-- %\includegraphics[width=\textwidth]{EffectOfPC_on_COV_Mean}
%-- %\caption{}
%-- %\label{figure:EffectOfKL_on_CorLength_SD}
%-- %\end{center}
%-- %\end{figure}
%-- %
%-- %\begin{figure}[htbp]
%-- %\begin{center}
%-- %\includegraphics[width=\textwidth]{EffectOfPC_on_COV_SD}
%-- %\caption{}
%-- %\label{figure:EffectOfKL_on_CorLength_SD}
%-- %\end{center}
%-- %\end{figure}
%-- %
%-- 
%-- 
%-- \end{frame}
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\section{Applications}


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\subsection{Seismic Wave Propagation Through Uncertain Soils}


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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% 
% 
% \frametitle{Applications}
% 
% 
% 
% 
% \begin{itemize}
% 
% \vspace*{0.3cm}
% \item Stochastic elastic--plastic simulations of soils and structures
% 
% \vspace*{0.3cm}
% \item Probabilistic inverse problems
% 
% \vspace*{0.3cm}
% \item Geotechnical site characterization design
% 
% \vspace*{0.3cm}
% \item Optimal material design
% 
% 
% \end{itemize}
% 
% \end{frame}
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% - +
% - + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - + 
% - + \begin{frame}
% - + 
% - + 
% - + \frametitle{Random Field Modeling of Uncertain Soil Properties}
% - + 
% - + \begin{itemize}
% - + 
% - + \item Finite scale model 
% - + 
% - + \begin{itemize}
% - + 
% - + \item Short memory, finite correlation length
% - + 
% - + \item Common autocovariance model $\rightarrow$ exponential, spherical, triangular, linear--exponential
% - + 
% - + \end{itemize}
% - + 
% - + \item Fractal model 
% - + 
% - + \begin{itemize}
% - + 
% - + \item long memory, infinite correlation length $\rightarrow$ more realistic for modeling horizontal 
% - + spatial uncertainty
% - + 
% - + \item 1/f-type noise process with power spectral density, $P(\omega)~=~P_0~\omega^{-\gamma}$, with
% - + upper and/or lower frequency cut-off.
% - + 
% - + \end{itemize}
% - + 
% - + \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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\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

\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{"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}


\end{frame}


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



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\begin{frame}


\frametitle{Seismic Wave Propagation through Stochastic Soil}



\begin{figure}
\begin{center}
\hspace*{-0.75cm}
\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter9Plots/Chapter9_ElasticPlasticResponse-New.pdf}
%\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
\end{center}
\end{figure}

Mean$\pm$ Standard Deviation



%\begin{flushleft}
%\includegraphics[height=5.0cm]{PEER2007_3.jpg}
%\end{flushleft}

% \hspace*{-1.0cm} \noindent Statistics of Top Node Displacement:
% 
% \vspace*{-0.5truecm}
% 
% \begin{figure}
% \begin{flushleft}
% \hspace*{-1.0cm}
% \includegraphics[width=4.0cm]{/home/kallol/publication/2007/Presentation/PhDExitSeminar/Chapter9_ElasticPlasticResponse_Mean-New.jpg}
% \hspace*{-0.1cm}
% \includegraphics[width=4.0cm]{/home/kallol/publication/2007/Presentation/PhDExitSeminar/Chapter9_ElasticPlasticResponse_SD-New.jpg}
% \end{flushleft}
% \end{figure}
% \vspace*{-0.5truecm}
% \hspace*{-1.0cm} \tiny{~~~~~~~~~~~~~~~~~~~~~~~~~~~~Mean~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Standard Deviation}
% 
% \vspace*{-0.3truecm}
% 
% \begin{figure}
% \begin{flushleft}
% \hspace*{-0.75cm}
% \includegraphics[width=4.0cm]{/home/kallol/publication/2007/Presentation/PhDExitSeminar/Chapter9_ElasticPlasticResponse-New.jpg}
% \hspace*{0.4cm}
% \includegraphics[width=4.0cm]{/home/kallol/publication/2007/Presentation/PhDExitSeminar/Chapter9_ElasticPlasticResponse_COV-New.jpg}
% \end{flushleft}
% \end{figure}
% \vspace*{-0.3truecm}
% \hspace*{-0.5cm} \tiny{~~~~~~~Mean$\pm$ Standard Deviation~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~COV}
% 
% 
% \vspace*{-6.0cm}
% \begin{flushright}
% \includegraphics[height=4.5cm]{/home/kallol/publication/2007/Presentation/PhDExitSeminar/Chapter9_ElasticPlasticResponse_PDF-New-Edited.jpg} \hspace*{-1.0cm}
% \end{flushright}
% 
\end{frame}

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\subsection{Probabilistic Analysis for Decision Making}


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\begin{frame}


\frametitle{Decision About Site (Material) Characterization}




\begin{itemize}

\vspace*{0.3cm}
\item  Do nothing about site characterization (rely on experience): conservative
{\bf guess}  of  soil  data, $COV  = 225$\%, correlation length $= 12$m.


\vspace*{0.3cm}
\item  Do better than standard site characterization: $COV = 103$\%, correlation
length $= 0.61$m)


\vspace*{0.3cm}
\item Improve site (material) characterization if  probabilities of exceedance are unacceptable!


\end{itemize}

\end{frame}

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\begin{frame}


\frametitle{Evolution of Mean $\pm$ SD for Guess Case}



\begin{figure}
\begin{center}
\hspace*{-0.75cm}
\includegraphics[width=10.0cm]{/home/jeremic/tex/works/Conferences/2009/UNION-Univ-BGD/Present/Plots_with_Labels/Evolutionary_Mean_pm_SD_NoData-Edited.pdf}
\hspace*{-0.75cm}
%\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
\end{center}
\end{figure}

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\begin{frame}


\frametitle{Evolution of Mean $\pm$ SD for Real Data Case}



\begin{figure}
\begin{center}
\hspace*{-0.75cm}
\includegraphics[width=10.0cm]{/home/jeremic/tex/works/Conferences/2009/UNION-Univ-BGD/Present/Plots_with_Labels/Evolutionary_Mean_pm_SD_Actual-Edited.pdf}
\hspace*{-0.75cm}
%\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
\end{center}
\end{figure}

% 
\end{frame}

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\begin{frame}


\frametitle{Full PDFs for Real Data Case}



\begin{figure}
\begin{center}
\vspace*{-0.75cm}
\includegraphics[width=7.0cm]{/home/jeremic/tex/works/Conferences/2009/UNION-Univ-BGD/Present/Plots_with_Labels/EvolutionaryPDF_Actual-Edited.pdf}
\vspace*{-0.75cm}
%\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
\end{center}
\end{figure}

% 
\end{frame}

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\begin{frame}


\frametitle{Example: PDF at $6$ s}



\begin{figure}
\begin{center}
\hspace*{1.75cm}
\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2009/UNION-Univ-BGD/Present/Plots_with_Labels/PDFs_at6sec_Actual_vs_NoData-Edited.pdf}
\vspace*{-0.75cm}
%\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
\end{center}
\end{figure}

% 
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\begin{frame}


\frametitle{Example: CDF at $6$ s}



\begin{figure}
\begin{center}
%\hspace*{-0.75cm}                                                                                               
\includegraphics[width=8.0cm]{/home/jeremic/tex/works/Conferences/2009/UNION-Univ-BGD/Present/Plots_with_Labels/CDFs_at6sec_Actual_vs_NoData-Edited.pdf}
\vspace*{-0.75cm}
%\hspace*{-0.75cm}
%\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
\end{center}
\end{figure}

% 
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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% 
% \begin{frame}
% 
% 
% \frametitle{Probability of Exceedance of $20$cm}
% 
% 
% 
% \begin{figure}
% \begin{center}
% %\hspace*{-0.75cm}                                                                                               
% \includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2009/UNION-Univ-BGD/Present/Plots_with_Labels/ProbabilityOfExceedance20cm_Actual_vs_NoData-Edited.pdf}
% \vspace*{-0.75cm}
% %\hspace*{-0.75cm}
% %\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
% \end{center}
% \end{figure}
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% % 
% \end{frame}
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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% \begin{frame}
% 
% 
% \frametitle{Probability of Exceedance of $50$cm}
% 
% 
% 
% \begin{figure}
% \begin{center}
% %\hspace*{-0.75cm}                                                                                               
% \includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2009/UNION-Univ-BGD/Present/Plots_with_Labels/ProbabilityOfExceedance50cm_Actual_vs_NoData-Edited.pdf}
% \vspace*{-0.75cm}
% %\hspace*{-0.75cm}
% %\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
% \end{center}
% \end{figure}
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% % 
% \end{frame}
% 
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% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% \begin{frame}
% 
% 
% \frametitle{Probabilities of Exceedance vs. Displacements}
% 
% 
% 
% \begin{figure}
% \begin{center}
% %\hspace*{-0.75cm}                                                                                               
% \includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2009/UNION-Univ-BGD/Present/Plots_with_Labels/ProbabilityOfExceedance_vs_Displacement_Actual_vs_NoData-Edited.pdf}
% \vspace*{-0.75cm}
% %\hspace*{-0.75cm}
% %\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
% \end{center}
% \end{figure}
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% % 
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% 
% \begin{frame}
% 
% \frametitle{Probabilities of Unacceptable Deformation}
% 
% \begin{figure}
% \begin{center}
% \vspace*{-0.3cm}                                                                                               
% \includegraphics[width=10.5cm]{/home/jeremic/tex/works/Conferences/2009/UNION-Univ-BGD/Present/NewPlots/with_legends_and_labels/Exceedance20cm_LomaPrieta-Edited_ps.pdf}
% \vspace*{-0.5cm}
% %\hspace*{-0.75cm}
% %\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
% \end{center}
% \end{figure}
%  
% \end{frame}
% 
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\begin{frame}

\frametitle{Probability of Unacceptable Deformation ($50$cm)}

\begin{figure}
\begin{center}
\vspace*{-0.3cm}
%\hspace*{-0.75cm}                                                                                               
\includegraphics[width=10.50cm]{/home/jeremic/tex/works/Conferences/2009/UNION-Univ-BGD/Present/NewPlots/with_legends_and_labels/Exceedance50cm_LomaPrieta-Edited_ps.pdf}
\vspace*{-0.5cm}
%\hspace*{-0.75cm}
%\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
\end{center}
\end{figure}
 
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\begin{frame}

\frametitle{Risk Informed Decision Process}

\begin{figure}
\begin{center}
%\hspace*{-0.75cm}                                                                                               
\includegraphics[width=8.0cm]{/home/jeremic/tex/works/Conferences/2009/UNION-Univ-BGD/Present/NewPlots/with_legends_and_labels/Summary_LomaPrieta-Edited.pdf}
\vspace*{-0.75cm}
%\hspace*{-0.75cm}
%\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
\end{center}
\end{figure}
 
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\section{Summary and Future}



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\frametitle{Summary}

\begin{itemize}

\item  Material (solids and structures) behavior is uncertain (probably!)

\vspace*{0.2cm}
\item Simulation of behavior for Geotechnical/Structural system needs to be done
probabilistically 
%(Banks and Insurance industry will force us into it)


\vspace*{0.2cm}
\item Methods for such simulations do exist (shown today)
%  and are being refined 




\vspace*{0.2cm}
\item Problem might be with the Human Nature! (how much do you want or do not want
to know about potential problem?!)

\end{itemize}


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\frametitle{Risk Information}

\begin{itemize}

\item  Risk informed decisions, very valuable and sought after in

  \begin{itemize}
    \item[] Nuclear Engineering
    \item[] Aerospace Engineering
    \item[] Mechanical Engineering
    \item[] Biomechanics
    \item[] Civil Engineering (Geotech/Struct)
  \end{itemize}

\vspace*{0.2cm}
\item Owners, Banks and Insurance agencies (will) require it


\vspace*{0.2cm}
\item Improve infrastructure economy and safety through rational probabilistic mechanics



\end{itemize}


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