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\title
[Simulations in Geomechanics: The Issue of Uncertainty]
{Simulations in Geomechanics: \\ The Issue of Uncertainty}
%\subtitle
%{Include Only If Paper Has a Subtitle}
%\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}}
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\institute[Computational Geomechanics Group \hspace*{0.3truecm}
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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}
\subject{}
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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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% \begin{itemize}
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% \item Diploma Engineering degree (Master of Engineering) Belgrade Univ.,
% Structural Engineering
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% \vspace*{0.3cm}
% \item Design of concrete and rock dams in Yugoslavia and Iraq
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% \vspace*{0.3cm}
% \item Design of residential and industrial buildings in
% Switzerland,
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% \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.
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% \item Great need for numerical simulation tools
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\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}
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% \begin{frame}
% \frametitle{Pile in Layered Soils}
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%
% \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}
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% \frametitle{Single Pile in Sand with Clay Layer: M, Q, p}
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% \begin{figure}[!htbp]
% \begin{center}
% \includegraphics[width=7.5cm]{/home/jeremic/tex/works/Thesis/ZhaohuiYang/SinglePiles/MQP_sand_wsfclay_UFmid2hdp0.jpg}
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\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}
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% \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}
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% \includegraphics[height=0.12\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_2_T010.jpg}
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% \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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% \\
% \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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\begin{frame}
\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}
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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_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_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_T080.jpg}
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\\
\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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\begin{frame}
\frametitle{EarthquakeSoilStructure 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/NorthridgeBent.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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% \vspace*{1cm}
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% \begin{center}
% \includegraphics[width=8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/MomentBent1Pile1.pdf}
% \\
% \includegraphics[width=8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/MomentBent1Pile2.pdf}
% %\caption{\label{BridgeSFSI01} FEM model for seismic response of a three bend
% %bridge.}
% \end{center}
% \end{figure}
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% \vspace*{2.9cm}
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% \includegraphics[width=1.5cm]{/home/jeremic/tex/works/Reports/2006/NEESDemoProject/PrototypeMesh.jpg}
% %\caption{\label{BridgeSFSI01} FEM model for seismic response of a three bend
% %bridge.}
% \end{flushright}
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\begin{frame}
\frametitle{PDD and Parallel Computer GeoWulf}
\begin{itemize}
%\vspace*{0.2cm}
\item {\bf P}lastic {\bf D}omain {\bf D}ecomposition \\
ElasticPlastic 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/NSFNuggets/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}
%
\end{frame}
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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 Pointwise \\
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 pointwise})
%
% \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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\begin{frame}
\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 tradeoff 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 $\rightarrow$ FokkerPlanckKolmogorov (FPK) for the
PDF (Kolmogorov 1941)
\vspace*{0.2cm}
\item Approach for random coefficient $\rightarrow$
Functional integration approach (Hopf 1952), Averaged equation approach (BharruchaReid 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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\begin{frame}
\frametitle{Recent StateoftheArt}
\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 ElastoPlasticity}
\end{itemize}
\end{frame}
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%  \begin{frame}
%  \frametitle{Objectives of Proposed Work}
% 
%  \begin{itemize}
% 
%  \item Constitutive Problem: Obtain secondorder (mean and variance) exact (analytical) PDF
%  of stress response $\rightarrow$ \textbf{FokkerPlanckKolmogorov (FPK) Equation}
% 
%  \item Boundary Value Problem: Model uncertain spatial variability of elasticplastic soil
%  $\rightarrow$ \textbf{Spectral Stochastic ElasticPlastic 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 ElastoPlasticity}
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\begin{frame}
\frametitle{Uncertainty Propagation through Constitutive Eq.}
%
\begin{itemize}
\item Incremental elpl 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 elasticplastic}}
%
\end{array} \right.
\end{eqnarray}
%\end{normalsize}
%\vspace{0.5cm}
% \item Nonlinear coupling in the ElPl modulus
% \item Focus on 1D $\rightarrow$ 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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\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 (elastoplastic/viscoelasticviscoplastic):
\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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\begin{frame}
\frametitle{Problem Statement and Solution}
\begin{itemize}
\item Incremental 3D elasticplastic stressstrain:
%
%
$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 pseudotime 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{EulerianLagrangian FPK Equation}
%
%\begin{itemize}
\begin{footnotesize}
%
% %\noindent
% van Kampen's Lemma (van Kampen 1976) $\rightarrow$ $ <\rho(\sigma,t)>=P(\sigma,t) $,
% ensemble average of phase density
% %(in stress space here)
% is the probability density;
\begin{eqnarray}
\nonumber
&&\displaystyle \frac{\partial P(\sigma(x_t,t), t)}{\partial t}=
 \displaystyle \frac{\partial}{\partial \sigma} \left[ \left\{\left< \vphantom{\int_{0}^{t} d\tau} \eta(\sigma(x_t,t), D^{el}(x_t),
q(x_t), r(x_t), \epsilon(x_t,t)) \right> \right. \right. \\
\nonumber
&+& \left. \left. \int_{0}^{t} d\tau Cov_0 \left[ \displaystyle \frac{\partial \eta(\sigma(x_t,t), D^{el}(x_t), q(x_t), r(x_t),
\epsilon(x_t,t))}{\partial \sigma}; \right. \right. \right. \\
\nonumber
& & \left. \left. \left. \eta(\sigma(x_{t\tau},t\tau), D^{el}(x_{t\tau}), q(x_{t\tau}), r(x_{t\tau}),
\epsilon(x_{t\tau},t\tau) \vphantom{\int_{0}^{t} d\tau} \right] \right \} P(\sigma(x_t,t),t) \right] \\
\nonumber
&+& \displaystyle \frac{\partial^2}{\partial \sigma^2} \left[ \left\{ \int_{0}^{t} d\tau Cov_0 \left[ \vphantom{\int_{0}^{t}}
\eta(\sigma(x_t,t), D^{el}(x_t), q(x_t), r(x_t), \epsilon(x_t,t)); \right. \right. \right. \\
\nonumber
& & \left. \left. \left. \eta (\sigma(x_{t\tau},t\tau), D^{el}(x_{t\tau}), q(x_{t\tau}), r(x_{t\tau}),
\epsilon(x_{t\tau},t\tau)) \vphantom{\int_{0}^{t}} \right] \vphantom{\int_{0}^{t}} \right\} P(\sigma (x_t,t),t) \right] \\
\nonumber
\end{eqnarray}
\end{footnotesize}
\end{frame}
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\begin{frame}
\frametitle{EulerianLagrangian FPK Equation}
\begin{itemize}
\item Advectiondiffusion equation
%
\begin{equation}
\nonumber
\frac{\partial P(\sigma,t)}{\partial t} = \frac{\partial}{\partial \sigma}\left[N_{(1)}P(\sigma,t)\frac{\partial}{\partial \sigma}
\left\{N_{(2)} P(\sigma,t)\right\} \right]
\end{equation}
%
\item Complete probabilistic description of response
\item Solution PDF is secondorder exact to covariance of time (exact mean and variance)
\item It is deterministic equation in probability density space
\item It is linear PDE in probability density space
%$\rightarrow$ simplifies the numerical solution process
\item FPK diffusionadvection 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
% ElastoPlasticity: Formulation in 1D", \textit{Acta Geotechnica}, Vol. 2, No. 3, 2007, In press (published
% online in the \textit{Online First} section)}
% %\vskip2pt\insertnavigation{\paperwidth}\vskip2pt
% \end{beamercolorbox}%
% }
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\subsection{Probabilistic ElasticPlastic 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}
\;\;\;\;\;
\mbox{;}
\;\;\;\;\;
N_{(2)}=4t\left(\displaystyle \frac{d \epsilon_{12}}{dt} \right)^2 Var[G]
\end{equation}
\item {Example:}
$$ = 2.5 MPa;
Std. Deviation$[G]$ = 0.5 MPa
\end{itemize}
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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_RandomGm.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_Elasticm.pdf}
% %
% \hspace*{3cm}
% \vspace*{4cm}
% %
% \includegraphics[height=4cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ReasonOfNotMatchingMonteCarlom.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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%
% \frametitle{DruckerPrager Linear Hardening with Random $G$}
%
%
%
% %
% %
% %
% % General form of DruckerPrager elasticplastic 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{{DruckerPrager Linear Hardening, Random $G$}}
%
% \vspace*{1.0cm}
%
% \begin{figure}[!hbpt]
% \begin{center}
% \includegraphics[height=6.5cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/PDFDruckerPrager_RandomGm.pdf}
% %\hfill
% %\includegraphics[height=6.0cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourDruckerPrager_RandomGm.pdf}
% \end{center}
% \end{figure}
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% \vspace*{1.5cm}
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% \begin{frame}
%
% \frametitle{{DruckerPrager Linear Hardening, Random $G$}}
%
% \vspace*{1.0cm}
%
% \begin{figure}[!hbpt]
% \begin{center}
% %\includegraphics[height=4.5cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/PDFDruckerPrager_RandomGm.pdf}
% %\hfill
% \includegraphics[height=6.50cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourDruckerPrager_RandomGm.pdf}
% \end{center}
% \end{figure}
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%
% \frametitle{Verification of DP EP 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_DruckerPragerm.pdf}
% %
% %\end{flushleft}
% %\begin{flushright}
% %\includegraphics[height=11.5cm]{ContourDruckerPrager_RandomGm.pdf}
% %
% %\nonumber
% %\caption{DruckerPrager Associative Linear Hardening Response with Random $G$}
% %{\includegraphics[width=10.5cm]{QualSchematics4.jpg}}
% %\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
% %\end{flushleft}
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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(2pp_0)^2}{\kappa} + \left(18 \displaystyle \frac{G}{M^4}\right) \sigma_{12}^2
+ \displaystyle \frac{1+e_0}{\lambda\kappa} p p_0 (2pp_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_RandomGm.pdf}
%  \hfill
%  \includegraphics[height=5.4cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourLowOCR_RandomGm.pdf}
%  \end{center}
%  \end{figure}
% 
%  \begin{itemize}
% 
%  \item Approximation of I.C.
% 
% 
%  %\item Wide transition between el. \& el.pl.
% 
% 
%  \item Nonsymmetry 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 Nonsymmetry 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_Randomp0m.pdf}
%\hfill
\includegraphics[height=6.5cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourLowOCR_RandomG_RandomM_Randomp0m.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_Randomp0m.pdf}
%  \hfill
%  \includegraphics[height=5.5cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourLowOCR_RandomG_RandomM_Randomp0m.pdf}
%  \end{center}
%  \end{figure}
% 
% 
% 
%  \vspace*{0.5cm}
% 
%  \begin{itemize}
% 
%  %\item Narrow transition between el. \& el.pl.
% 
% 
%  \item Nonsymmetry 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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\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/CamClayPDFComparisonm.pdf}
\includegraphics[height=5.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Present/LowOCR_ComparisonPDFm_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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% \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 nonsymmetry \\
% 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_RandomMm.pdf}
% %\hfill
% \includegraphics[height=6.5cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourHighOCR_RandomG_RandomMm.pdf}
% \end{flushright}
% \end{figure}
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\begin{frame}
\frametitle{Probabilistic Yielding}
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\begin{itemize}
%
\item Weighted elastic and elasticplastic 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)}^{elpl} $
%\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_Combinededited.pdf}
% \end{figure}
% \vspace*{0.5cm}
\item {Similar to European Pricing \\
Option in financial simulations \\
(BlackScholes 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_Combinededited.pdf}
\end{figure}
\vspace*{0.5cm}
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% \begin{frame}
%
%
%
% \frametitle{BiLinear 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_PDFedited.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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\begin{frame}
\frametitle{BiLinear 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_PDFedited.pdf}
\end{center}
\end{figure}
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% \begin{frame}
%
%
% \frametitle{Cyclic BiLinear von Mises, Yield Stress PDF}
%
%
% \begin{figure}[!hbpt]
% \begin{center}
% \includegraphics[width=7cm]{/home/jeremic/tex/works/Papers/2008/CyclicProbabilisticBehaviorGeomaterials/figures/IJNAMG_Review02_Plot02Edited_ps.jpg}
% \end{center}
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% \begin{frame}
%
% \frametitle{Cyclic BiLinear von Mises Response}
%
% \begin{figure}[!hbpt]
% \begin{center}
% \includegraphics[width=11cm]{/home/jeremic/tex/works/Papers/2008/CyclicProbabilisticBehaviorGeomaterials/figures/PDF07_of_ShearStress_Cyclic_PerfectlyPlastic_IJNAMGEd.pdf}
% \end{center}
% \end{figure}
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% % \begin{frame}
% %
% % \frametitle{Cyclic BiLinear von Mises Response}
% %
% % \begin{figure}[!hbpt]
% % \begin{center}
% % \includegraphics[width=7cm]{/home/jeremic/tex/works/Papers/2008/CyclicProbabilisticBehaviorGeomaterials/figures/PDF06_of_ShearStress_Cyclic_PerfectlyPlastic_IJNAMGEd.pdf}
% % \end{center}
% % \end{figure}
% %
% % \end{frame}
% %
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% \begin{frame}
%
% \frametitle{Cyclic BiLinear von Mises Response}
%
% \begin{figure}[!hbpt]
% \begin{center}
% \includegraphics[width=11cm]{/home/jeremic/tex/works/Papers/2008/CyclicProbabilisticBehaviorGeomaterials/figures/PDF06_of_ShearStress_Cyclic_PerfectlyPlastic_IJNAMGEd.pdf}
% \end{center}
% \end{figure}
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% \end{frame}
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% \begin{frame}
%
%
%
% \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_PDFedited.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_PDFedited.pdf}
% \end{center}
% \end{figure}
%
%
% \vspace*{0.1cm}
%
%
% \end{frame}
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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/JGGEGoverGmax/figures/ShearStrength_RawData_and_MeanTrendMod.pdf}
\hfill
\includegraphics[width=5.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGEGoverGmax/figures/ShearStrength_Histogram_PearsonIVFineTunedMod.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/JGGEGoverGmax/figures/YoungModulus_RawData_and_MeanTrend_01Ed.pdf}
\hfill
\includegraphics[width=5.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGEGoverGmax/figures/YoungModulus_Histogram_Normal_01Ed.pdf}
%
\end{center}
\end{figure}
\vspace*{0.3cm}
Transformation of SPT $N$value $\rightarrow$ 1D 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} (vonMises)}
\begin{figure}[!hbpt]
\begin{center}
%
\includegraphics[width=8.15truecm]{/home/jeremic/tex/works/Papers/2008/JGGEGoverGmax/figures/meanStressStrainPlot_1Point026Mod.pdf}
%\hfill
%\includegraphics[width=4.35truecm]{/home/jeremic/tex/works/Papers/2008/JGGEGoverGmax/figures/sdStressStrainPlot_1Point026Mod.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/JGGEGoverGmax/figures/meanStressStrainPlot_1Point026Mod.pdf}
\hfill
\includegraphics[width=4.35truecm]{/home/jeremic/tex/works/Papers/2008/JGGEGoverGmax/figures/sdStressStrainPlot_1Point026Mod.pdf}
%
\end{center}
\end{figure}
\vspace*{0.3cm}
\end{frame}
%
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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/JGGEGoverGmax/figures/GoverGmaxMod.pdf}
%\hfill
%\includegraphics[width=6.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGEGoverGmax/figures/DampingMod.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/JGGEGoverGmax/figures/GoverGmaxMod.pdf}
%\hfill
\includegraphics[width=10.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGEGoverGmax/figures/DampingMod.pdf}
%
\end{center}
\end{figure}
\vspace*{0.3cm}
\end{frame}
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%\subsection{Boundary Value Problem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section[SSEPFEM]{Stochastic ElasticPlastic 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/nonlinear 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/nonlinear 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 zeromean 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 zeromean 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 KarhunenLo\`{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 zeromean 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 zeromean random field ($\mathbb{R}
% tempout (x, \theta)$). The zeromean 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 partiallydiscretized  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 PCexpansion 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 precalculated 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{ForceResidual Form}
% tempout
% tempout For nonlinear problems (as attempted in this paper), Eq.~(\ref{eqno_c10})
% tempout can be written in forceresidual form and solved incrementally.
% tempout One can write Eq.~(\ref{eqno_c10}) in forceresidual
% 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 forceresidual 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 nonsingular and
% tempout hence one can solve the forceresidual 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 nonlinear 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 incrementaliterative 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 reevaluated 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 KLspace, 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 KLspace\footnote{Note that KLspaces 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 1D problem,
% tempout knowing the (random) strain increment at the ${k1}^{th}$ global load step as,
% tempout
% tempout \begin{equation}
% tempout \label{eqno_13EP}
% tempout \Delta \epsilon^{k1} (x, \theta)
% tempout = \sum_{i=0}^P \psi_i[\{\xi_r\}] (\theta) \sum_{n=1}^N \Delta d^{k1}_{ni} B_n(x)
% tempout \end{equation}
% tempout
% tempout \noindent the advection and diffusion coefficients for the FPKE, assuming 1D von Mises elasticperfectly 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 KLspace (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^{k1}_{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^{k1}_{ni} B_n \right)^2
% tempout \end{array}
% tempout \label{ADcoeffs01}
% tempout \end{eqnarray}
% tempout
% tempout \item[] For any other KLspaces:
% 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^{k1}_{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^{k1}_{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 KLspace) and polynomial chaos ($\psi_i[\{\xi_r\}]$) could be
% tempout easily precalculated symbolically using Mathematica \cite{software:Mathematica}.
% tempout %
% tempout Assuming any value of $\Delta t$, the FPKEs
% tempout corresponding to each KLspace (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 Newtontype
% tempout incrementaliterative scheme, these probability densities of
% tempout stress at each KLspace (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 KLspace (for example, $\hat {\mathbb{D}}$
% tempout in Eq.~(\ref{eqno_c11}) and $\sqrt \lambda_m$, where
% tempout the subscript $m$ denotes the KLmode, in
% tempout Eq.~(\ref{eqno_c12})) to increment forward.
% tempout %
% tempout In this paper, the new tangent
% tempout material properties at each KLspace 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 KLspace.
% 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_{ts}^t ds
% tempout Cov\left[D_{ijkl}({\sigma_{ij}}_{(s)}, s) \frac{d \epsilon_{kl}}{d t}; D_{ijkl}({\sigma_{ij}}_{(ts)}, ts) \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 substeps 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. LevenbergMarquardt technique \cite{Levenberg:1944, Marquardt:1963}). For example,
% tempout for 1D problem, for zeroth KLspace (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} KLspace}
% 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} KLspace}
% 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 substeps, the unknown
% tempout deterministic coefficients ($a_1$ and $a_2$) can be estimated using LevenbergMarquardt 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 KLspaces, for 1D 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 }_{nonzero KLspace}
% 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 $m^{th}$ KLspace ($\sqrt \lambda_m (t)$) 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 }_{nonzero KLspace}
% 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 KLspace, from Eq.~(\ref{eqno_18EP}), the evolved tangent material
% tempout property at the $m^{th}$ KLspace ($\sqrt \lambda_m (t))$ at the end of global load step can be estimated using LevenbergMarquardt
% 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 ($D$) $\rightarrow$
KarhunenLo{\`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 KarhunenLo{\`e}ve (KL) expansion}
% 
%  \begin{itemize}
% 
%  \item Representation of input random fields in eigenmodes of covariance kernel
% 
%  % \vspace*{0.1cm}
%  % \begin{figure}[!hbpt]
%  \begin{flushleft}
%  \includegraphics[height=3.0cm]{/home/jeremic/tex/works/Conferences/2008/GeoCongress/Probabilistic/Paper/TypicalDataPlotBH1Edited.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 \ \ \ \ \ \ \ \ \ \ \ \ \ \ Oneterm 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_FiniteScaleEdited.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{Twoterms approximation \ \ \ \ \ \ \ \ \ \ \ \ Threeterms 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 KL 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 ElasticPlastic 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 FokkerPlanckKolmogorov 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 KLexpansion
%
% \item Representation of (unknown) solution random variables using polynomial chaos of
% (known) input random variables
%
% \item FokkerPlanckKolmogorov 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 elasticplastic 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 elasticplastic stiffness tensor
\end{itemize}
% \noindent Salient Features:
% \begin{itemize}
%
% \item Efficient representation of input random fields into finite number of random
% variables using KLexpansion
%
% \item Representation of (unknown) solution random variables using polynomial chaos of
% (known) input random variables
%
% \item FokkerPlanckKolmogorov 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 ($D$) $\rightarrow$
%  KarhunenLo{\`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 KarhunenLo{\`e}ve (KL) expansion}
%  % 
%  %  \begin{itemize}
%  % 
%  %  \item Representation of input random fields in eigenmodes of covariance kernel
%  % 
%  %  % \vspace*{0.1cm}
%  %  % \begin{figure}[!hbpt]
%  %  \begin{flushleft}
%  %  \includegraphics[height=3.0cm]{/home/jeremic/tex/works/Conferences/2008/GeoCongress/Probabilistic/Paper/TypicalDataPlotBH1Edited.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 \ \ \ \ \ \ \ \ \ \ \ \ \ \ Oneterm 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_FiniteScaleEdited.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{Twoterms approximation \ \ \ \ \ \ \ \ \ \ \ \ Threeterms 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 KL 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 ElasticPlastic 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 FokkerPlanckKolmogorov 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 KLexpansion
%  %
%  % \item Representation of (unknown) solution random variables using polynomial chaos of
%  % (known) input random variables
%  %
%  % \item FokkerPlanckKolmogorov 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 elasticplastic 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 elasticplastic stiffness tensor
% 
% 
%  \end{itemize}
% 
%  % \noindent Salient Features:
% 
%  % \begin{itemize}
%  %
%  % \item Efficient representation of input random fields into finite number of random
%  % variables using KLexpansion
%  %
%  % \item Representation of (unknown) solution random variables using polynomial chaos of
%  % (known) input random variables
%  %
%  % \item FokkerPlanckKolmogorov approach based probabilistic constitutive integration
%  % at Gauss integration points
%  %
%  % \end{itemize}
%  %
%  %% \end{itemize}
%  %
%  \end{frame}
% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{SSEPFEM Example}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{1D Static Pushover Test Example}
\begin{itemize}
\vspace*{0.3cm}
\item Linear elastic model: \\
$ = 2.5$~kPa, \\
$Var[G]=0.15~{\rm kPa}^2$,\\
correlation length for $G=0.3$~m.
\vspace*{0.3cm}
\item Elasticplastic material model,\\
von Mises, linear hardening,\\
$ = 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_ElasticEdited.jpg} \\
% %\vspace*{0.1truecm}
% %%\small{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Base Displacement}
% %\includegraphics[height=3.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter8Plots/Chapter8_StaticExample_ElasticPlasticEdited.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_ElasticEdited.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, \\
% KLdimension=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 elasticperfectly plastic material model,\\
KLdimension=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_PDFComparisonEdited.pdf}
%  \label{figure:StaticProblem_PDFComparison}
%  \end{center}
%  \end{figure}
% 
% 
%  Comparison of PDF of top node displacement.
% 
% 
% 
%  \end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{1D 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_MeanEdited.pdf}
% \includegraphics[width=0.35\textwidth]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter8Plots/Chapter8_StaticExample_CorLength_andKL_LargeCorLength_SDEdited.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 1D 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 1D
% 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{1D 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_MeanEdited.pdf}
% \caption{Correlation length and KL dimension: Mean displacement along depth of the 1D 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{1D 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_SDEdited.pdf}
% \caption{Correlation length and KL dimension: Standard deviation of displacement along depth of the 1D
% 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{1D 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_MeanEdited.pdf}
% \caption{Variance and order of PC: Mean displacement along depth of the 1D 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{1D 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_SDEdited.pdf}
% \caption{Variance and order of PC: Standard deviation of displacement along depth of the 1D
% 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{1D 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_MeanEdited.pdf}
% \caption{Variance and order of PC: Mean displacement along depth of the 1D 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{1D 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_SDEdited.pdf}
% \caption{Variance and order of PC: Standard deviation of displacement along depth of the 1D
% 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 FPKEbased spectral stochastic finite element with
% % direct spectral stochastic finite element, for 1D soil column example, with linear elastic material}
% % \label{Table:Comparison}
% % \begin{tabular}{ccccccc}
% % \hline
% % Load Step & \multicolumn{3}{c}{Mean of solution} & \multicolumn{3}{c}{Standard deviation of solution} \\
% % \cline{27}
% % & 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}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%
%
% \frametitle{Applications}
%
%
%
%
% \begin{itemize}
%
% \vspace*{0.3cm}
% \item Stochastic elasticplastic 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}
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  +
%  + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  +
%  + \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, linearexponential
%  +
%  + \end{itemize}
%  +
%  + \item Fractal model
%  +
%  + \begin{itemize}
%  +
%  + \item long memory, infinite correlation length $\rightarrow$ more realistic for modeling horizontal
%  + spatial uncertainty
%  +
%  + \item 1/ftype noise process with power spectral density, $P(\omega)~=~P_0~\omega^{\gamma}$, with
%  + upper and/or lower frequency cutoff.
%  +
%  + \end{itemize}
%  +
%  + \end{itemize}
%  +
%  + \end{frame}
%  +
%  + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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 1D soil column (von Mises Material)
\begin{itemize}
\item Properties (including testing uncertainty) obtained through random field modeling of CPT $q_T$
%
$\left = 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\%
%
$\left = 11.57MPa; Var[G] = 142.32 MPa^2$ \\
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}
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\begin{frame}
\frametitle{Random Field Parameters from Site Data}
\begin{itemize}
%\item maximizing the loglikelihood of observing the spatial data under assumed joined distribution (for finite
%scale model) or maximizing the loglikelihood 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/SamplingPlanEdited.jpg}
\hspace*{0.0cm}
\includegraphics[height=4.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/CPT_DataAnalysis_Plots/TypicalDataPlotBH1Edited.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_FiniteScaleEdited.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_FractalEdited.jpg} \\
\small{Fractal}
\end{flushright}
%\end{figure}
\end{itemize}
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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/EastWestProfileEdited.pdf}
\end{center}
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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_ElasticPlasticResponseNew.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_MeanNew.jpg}
% \hspace*{0.1cm}
% \includegraphics[width=4.0cm]{/home/kallol/publication/2007/Presentation/PhDExitSeminar/Chapter9_ElasticPlasticResponse_SDNew.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_ElasticPlasticResponseNew.jpg}
% \hspace*{0.4cm}
% \includegraphics[width=4.0cm]{/home/kallol/publication/2007/Presentation/PhDExitSeminar/Chapter9_ElasticPlasticResponse_COVNew.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_PDFNewEdited.jpg} \hspace*{1.0cm}
% \end{flushright}
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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}
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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/UNIONUnivBGD/Present/Plots_with_Labels/Evolutionary_Mean_pm_SD_NoDataEdited.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}
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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/UNIONUnivBGD/Present/Plots_with_Labels/Evolutionary_Mean_pm_SD_ActualEdited.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{Full PDFs for Real Data Case}
\begin{figure}
\begin{center}
\vspace*{0.75cm}
\includegraphics[width=7.0cm]{/home/jeremic/tex/works/Conferences/2009/UNIONUnivBGD/Present/Plots_with_Labels/EvolutionaryPDF_ActualEdited.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}
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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/UNIONUnivBGD/Present/Plots_with_Labels/PDFs_at6sec_Actual_vs_NoDataEdited.pdf}
\vspace*{0.75cm}
%\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
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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/UNIONUnivBGD/Present/Plots_with_Labels/CDFs_at6sec_Actual_vs_NoDataEdited.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}
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% \begin{frame}
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%
% \frametitle{Probability of Exceedance of $20$cm}
%
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%
% \begin{figure}
% \begin{center}
% %\hspace*{0.75cm}
% \includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2009/UNIONUnivBGD/Present/Plots_with_Labels/ProbabilityOfExceedance20cm_Actual_vs_NoDataEdited.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}
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% \frametitle{Probability of Exceedance of $50$cm}
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% \begin{figure}
% \begin{center}
% %\hspace*{0.75cm}
% \includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2009/UNIONUnivBGD/Present/Plots_with_Labels/ProbabilityOfExceedance50cm_Actual_vs_NoDataEdited.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}
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% \frametitle{Probabilities of Exceedance vs. Displacements}
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%
% \begin{figure}
% \begin{center}
% %\hspace*{0.75cm}
% \includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2009/UNIONUnivBGD/Present/Plots_with_Labels/ProbabilityOfExceedance_vs_Displacement_Actual_vs_NoDataEdited.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}
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% \frametitle{Probabilities of Unacceptable Deformation}
%
% \begin{figure}
% \begin{center}
% \vspace*{0.3cm}
% \includegraphics[width=10.5cm]{/home/jeremic/tex/works/Conferences/2009/UNIONUnivBGD/Present/NewPlots/with_legends_and_labels/Exceedance20cm_LomaPrietaEdited_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}
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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/UNIONUnivBGD/Present/NewPlots/with_legends_and_labels/Exceedance50cm_LomaPrietaEdited_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}
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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/UNIONUnivBGD/Present/NewPlots/with_legends_and_labels/Summary_LomaPrietaEdited.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}
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\section{Summary and Future}
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\begin{frame}
\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}
\end{frame}
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\begin{frame}
\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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%\item Language used by Beamer: L\uncover<2>{A}TEX
%\item Language used by Beamer: L\only<2>{A}TEX
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% % All of the following is optional and typically not needed.
%\appendix
%\section*{\appendixname}
%\subsection*{For Further Reading}
%
%\begin{frame}[allowframebreaks]
% \frametitle{For Further Reading}
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% A.~Author.
% \newblock {\em Handbook of Everything}.
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% \bibitem{Someone2000}
% S.~Someone.
% \newblock On this and that.
% \newblock {\em Journal of This and That}, 2(1):50100,
% 2000.
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\end{document}