\documentclass{beamer}
%\documentclass[handout]{beamer}
%\usepackage{pgfpages}
%\pgfpagelayout{2 on 1}[letterpaper, border shrink=5mm]

\mode<handout>{\setbeamercolor{background canvas}{bg=black!3}}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% ovo je za cirilicu
\input vuk.def
%\newfont{\cyr}{wncyr10 scaled 1200}
%\newfont{\cyrimena}{wncyb10 scaled 1800}
%\newfont{\cyrnaslov}{wncyb10 scaled 2600}
%\newfont{\cyrpodnaslov}{wncyr10 scaled 1400}

\newfont{\cyr}{wncyss10 scaled 1200}
\newfont{\cyrimena}{wncyss10 scaled 2000}
\newfont{\cyrnaslov}{wncyss10 scaled 2800}
\newfont{\cyrpodnaslov}{wncyss10 scaled 1400}

\newfont{\cyn}{wncyss10 scaled 2200}
%% JB sig
\newfont{\cyrjb}{wncyr10 scaled 600}
\newcommand{\JB}
{{\cyrjb \lower0.50ex\hbox{\uppercase{J}}\kern-.58em\hbox{\uppercase{B}}}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\setbeamertemplate{navigation symbols}{}
\mode<presentation>
{
%  \usetheme{Marburg} % ima naslov i sadrzaj sa desne strane
%  \usetheme{Hannover} % ima naslov i sadrzaj sa leve strane 
%  \usetheme{Singapore} % ima sadrzaj i tackice gore
%  \usetheme{Antibes}  % ima sadrzaj gore i kao graf ...
%  \usetheme{Berkeley} % ima sadrzaj desno
%  \usetheme{Berlin} % ima sadrzaj gore i tackice
%  \usetheme{Goettingen} % ima sadrzxaj za desne strane
%  \usetheme{Montpellier} % ima graf sadrzaj gore
%  \usetheme{Warsaw}
%  \usetheme{Warsaw}
  \usetheme{Dresden}
    \usecolortheme[RGB={20,0,128}]{structure}

  % or ...

  \setbeamercovered{transparent}


%  \setbeamercovered{transparent}
% or whatever (possibly just delete it)
%  \usecolortheme{albatross} % teget sa svetlim slovima
%  \usecolortheme{beetle} % siva pozadina (vrh plav)
%  \usecolortheme{seagull} % sivo
%%%%%%%
%  \usecolortheme{BorisJeremic}
%%%%%%%
%  \usecolortheme{rose}
%   \usefonttheme[onlylarge]{structuresmallcapsserif}
%   \usefonttheme{structuresmallcapsserif}
}


\usepackage[english]{babel}

%%%%  HYPERREF  HYPERREF  HYPERREF  HYPERREF  HYPERREF 
%%%%  HYPERREF  HYPERREF  HYPERREF  HYPERREF  HYPERREF 
\definecolor{webgreen}{rgb}{0, 0.15, 0} % less intense green
\definecolor{webblue}{rgb}{0, 0, 0.15} % less intense blue
\definecolor{webred}{rgb}{0.15, 0, 0} % less intense red
%\usepackage[colorlinks=true,linkcolor=webblue,citecolor=webred,urlcolor=webgreen]{hyperref}

\usepackage{hyperref}
\hypersetup{
  pdfmenubar=true,
  pdftoolbar=true,
  pdfpagemode={None}
}
\usepackage{pause}
% or whatever
%\usepackage{html}
%\usepackage{url}

\usepackage[latin1]{inputenc}
% or whatever

\usepackage{times}
\usepackage[T1]{fontenc}
% Or whatever. Note that the encoding and the font should match. If T1
% does not look nice, try deleting the line with the fontenc.



\title{Seismic Wave Propagation in Stochastic Soils}

%\subtitle
%{Include Only If Paper Has a Subtitle}

%\author[Author, Another] % (optional, use only with lots of authors)
%{F.~Author\inst{1} \and S.~Another\inst{2}}
% - Give the names in the same order as the appear in the paper.
% - Use the \inst{?} command only if the authors have different
%   affiliation.
\pgfdeclareimage[height=0.15cm]{university-logo}{/home/jeremic/BG/amblemi/ucdavis_logo_blue_sm}

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


\institute[Computational                   Geomechanics                   Group
\pgfuseimage{university-logo} \hspace*{-0.3truecm}] % (optional, but mostly needed)
{
 Department of Civil and Environmental Engineering\\
  University of California, \\ Davis, California, U.S.A. 
  }
% - Use the \inst command only if there are several affiliations.
% - Keep it simple, no one is interested in your street address.

\date
{4th ICEGE, \\ 
Thessaloniki, Greece, \\ 
June, 2007}
% %\date[] % (optional, should be abbreviation of conference name)
% %{
% %\small Graduate Student: Guanzhou Jie 
% %\\ ~\\
% %Funding: NSF--CMS--0324661, NSF--TeraGrid, NSF--EEC--9701568 \\
% %%\texttt{http://sokocalo.engr.ucdavis.edu/$\tilde{~}$jeremic/}
% }
% % - Either use conference name or its abbreviation.
% % - Not really informative to the audience, more for people (including
% %   yourself) who are reading the slides online

\subject{}
% This is only inserted into the PDF information catalog. Can be left
% out. 



% If you have a file called "university-logo-filename.xxx", where xxx
% is a graphic format that can be processed by latex or pdflatex,
% resp., then you can add a logo as follows:

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

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



%% Delete this, if you do not want the table of contents to pop up at
%% the beginning of each subsection:
% \AtBeginSubsection[]
% {
%   \begin{frame}<beamer>
%     \frametitle{Outline}
%     \tableofcontents[currentsection,currentsubsection]
%%    \tableofcontents[currentsection]
%   \end{frame}
% }


% If you wish to uncover everything in a step-wise fashion, uncomment
% the following command: 

%\beamerdefaultoverlayspecification{<+->}


\begin{document}

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

\begin{frame}
  \titlepage
\end{frame}

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

\begin{frame}
  \frametitle{Outline}
  \tableofcontents
  % You might wish to add the option [pausesections]
\end{frame}

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

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

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

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Motivation}

% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Failure Mechanisms of Geomaterials} 
% \begin{flushleft}
% \vspace*{-0.5cm}
% \includegraphics[height=5.0cm]{/home/jeremic/tex/works/Conferences/2007/GeoDenver/SFEM/Presentation/MGMuzorak01.jpg}
% \\
% \small{
% Soil: Inside Failure of "Uniform" MGM Specimen \\
% (After Swanson et al. 1998)
% }
% \end{flushleft}
% \vspace*{-6.0cm}
% %
% \begin{flushleft}
% \begin{small}
% \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Failure \\
% \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $\downarrow$ \\
% \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Strain localization \\
% \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \  and \\
% \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ bifurcation of response \\
% \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $\downarrow$ \\
% \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Stems from \\
% \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ material inhomogeneity
% \end{small}
% \end{flushleft}
% 
% \end{frame}
% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Soils Behavior is Uncertain}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Soils are Inherently Uncertain}
\begin{flushleft}
%\begin{center}
%\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
\vspace*{-0.1cm}
\includegraphics[height=4.0cm]{/home/jeremic/tex/works/Conferences/2007/GeoDenver/SFEM/Presentation/FrictionAngleProfile.jpg}
%\vspace*{-5.0cm}
%\mbox{\tiny{Lambe, T. W. and Whitman, R. V.,1969. Soil Mechanics. New York, John Wiley \& Sons}}
%\vspace*{-6.5cm}
\end{flushleft}
\vspace*{-4.0cm}
%
\begin{flushleft}
\begin{small}
\ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Spatial Variation of Friction Angle\\
%\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Undrained Shear Strength \\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (Mayne et al. (2000))
\end{small}
\end{flushleft}
\vspace*{0.2cm}
\begin{flushright}
\includegraphics[width=7.1cm]{/home/jeremic/tex/works/Conferences/2007/GeoDenver/SFEM/Presentation/TableTypicalCOV.jpg}
\end{flushright}

\begin{flushleft}
\ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ Typical COVs of Different Soil Properties \\
\ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (Lacasse and Nadim 1996)
\end{flushleft}

\end{frame}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Soil Uncertainties}

\begin{frame}
\frametitle{Characterization and Quantification}

\begin{itemize}

%\vspace*{1.5cm}
\item Natural variability of soil deposit  (Phoon and Kulhawy 1999, Fenton 1999) 
$\rightarrow$ function of soil formation process 

\begin{itemize}

\item Ergodic assumption might not strictly apply (Der Kiureghian 2005)

\end{itemize}

%\vspace*{1.5cm}
\item Testing error (Phoon and Kulhawy 1999, Marosi and Hiltunen 2004, Stokoe et al. 2004)

\begin{itemize}

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

\end{itemize}

%\vspace*{1.5cm}
\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}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %
% %\begin{frame}
% %\frametitle{Probabilistic Site Characterization}
% %%\textbf{For complete probabilistic site characterization a very large amount of data is needed...... Need alternate strategies!!!}
% %
% %\begin{itemize}
% %
% %%\vspace*{-0.6cm}
% %
% %\item Large (physically large but statistically (?)) amount of data 
% %
% %\begin{itemize}
% %
% %%\vspace*{-0.5cm}
% %\item Mean and COV from site-specific data
% %
% %\item Covariance structure from detailed analysis from similar sites (e.g. Fenton 1999)
% %
% %\end{itemize}
% %
% %%\vspace*{-0.4cm}
% %
% %\item Minimal data 
% %
% %\begin{itemize}
% %
% %%\vspace*{-0.6cm}
% %\item Phoon and Kulhawy (1999) developed general guidelines for typical sites and test methods
% %
% %\begin{itemize}
% %
% %\item COVs and covariance structures of inherent variabilities
% %
% %\item COVs of testing errors and transformation uncertainties.
% %
% %\end{itemize}
% %
% %\item Marosi and Hiltunen (2004) and Stokoe et al. (2004) extended the general guidelines for SASW method and $G/G_{max}$ curve
% %
% %%\vspace*{-0.8cm}
% %\end{itemize}
% %
% %%\vspace*{-0.6cm}
% %\item Moderate amount of data $\rightarrow$ Bayesian updating (e.g. Phoon and Kulhawy 1999, Baecher and 
% %Christian 2003)
% %
% %\end{itemize}
% %
% %\end{frame}
% %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Uncertain Soil Simulations}

\subsection{Probabilistic Soil Elasto--Plasticity}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Our Original Developments}

\begin{description}

\item[\underline{Constitutive}]: Second-order accurate (exact mean and variance) PDF 
of stress-strain response (Fokker -- Planck -- Kolmogorov Equation)

\item[\underline{Spatial}]: Spectral Stochastic Elastic -- Plastic Finite Element Method,
to simulate uncertain spatial variability of elastic--plastic soils


\end{description}

%\vspace*{0.5truecm}
        
%\item Overcome the drawbacks of \textit{Monte Carlo Technique} and \textit{Perturbation Method}

\begin{itemize}
\item Obtain complete probabilistic description (PDF) for:
  \begin{itemize}
    \item Stresses--Strain  response
    \item Displacements (velocities, accelerations)
  \end{itemize}

\item Use for:
  \begin{itemize}
    \item Sensitivity analysis
    \item Probability of failure (tails of PDF)
    \item Probabilistic site characterization design
  \end{itemize}

\end{itemize}

\end{frame}

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

%\subsubsection{Spectral Stochastic Elastic--Plastic Finite Element Method (SSEPFEM)}

\begin{frame}
\frametitle{Input Soil Parameters Random Fields}
              
       \begin{itemize}                                                                                                                         
                                                                                                                                                      
       \item Truncated Karhunen--Loev\'{e} (KL) expansion                                                                
       \item Representation of input random fields in eigen-modes of covariance kernel
                                                                                                                                                      
%       \vspace*{-0.1cm}                                                                                                                               
%              \begin{figure}[!hbpt]                                                                                                                   
              \begin{flushleft}                                                                                                                       
              \includegraphics[height=3.0cm]{/home/jeremic/tex/works/Conferences/2007/GeoDenver/SFEM/Presentation/ShearStrengthProfile.jpg}                                                                                
              \end{flushleft}                                                                                                                         
              \vspace*{-3.15cm}                                                                                                                        
       \begin{flushright}                                                                                                                             
%       \begin{equation}                                                                                                                              
%       \nonumber                                                                                                                                     
%       \begin{normalsize}                                                                                                                            
       $ S_u(x,\theta) = \bar S_u(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 [S_u(x,\theta) - \bar S_u (x)] 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{Stochastic Elastic--Plastic Finite Elements}


\begin{normalsize}
\begin{equation}
\nonumber
\sum_{n = 1}^N K_{mn}^{e,ep} d_{ni} 
+ 
\sum_{n = 1}^N \sum_{j = 0}^P d_{nj} \sum_{k = 1}^M C_{ijk} K_{mnk}^{'e,ep}  
= 
\left< F_m \psi_i[\{\zeta_r\}] \right >
\end{equation}
\end{normalsize}

\vspace*{-0.5cm}
\begin{normalsize}
\begin{equation}
\nonumber
K_{mn} = \int_D B_n D  B_m dV
\;\;\;\; 
\mbox{;} 
\;\;\;\; 
K'_{mnk} = \int_D B_n \sqrt \lambda_k h_k B_m dV
\end{equation}
\end{normalsize}

\vspace*{-1.0cm}
\begin{equation}
\nonumber
C_{ijk} = \left < \zeta_k(\theta) \psi_i[\{\zeta_r\}] \psi_j[\{\zeta_r\}] \right >  
\;\;\;\; 
\mbox{;} 
\;\;\;\; 
F_m = \int_D \phi N_m dV
\end{equation}

\begin{itemize}

      \item Based on SSFEM (Ghanem and Spanos (2003))

      \item Random {\bf material} and random {\bf forcing}
      
      \item Efficient representation of input random fields into finite number of random
            variables using KL-expansion

      \item Representation of (unknown) solution random variables using polynomial chaos of
            (known) input random variables

\end{itemize}

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

%\subsubsection{Fokker--Planck--Kolmogorov (FPK) Equation Approach}

\begin{frame}
\frametitle{Probabilistic Elasto--Plasticity}

%\vspace*{-2.99truecm}

\begin{itemize}
%\item Probabilistic soil nonlinearity at constitutive level

\item Probabilistic elastic--plastic constitutive incremental equation 
      $\Delta \sigma_{ij} = D_{ijkl}^{e,ep} \Delta \epsilon_{kl}$
    \begin{itemize}
      \item Random stiffness $D_{ijkl}^{e,ep}$
      \item Random strain increment $\Delta \epsilon_{kl}$
    \end{itemize}
    

\vspace*{0.2truecm}
\item Use of Euler Lagrange form of Fokker--Planck--Kolmogorov (FPK) equation 
      (Kavvas 2003) to obtain

\vspace*{0.2truecm}
\item Second-order accurate (exact mean and variance) stress--strain solution (PDF) 

\vspace*{0.2truecm}
\item Complete probabilistic description of response $\rightarrow$ PDF


%\vspace*{0.2truecm}

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

%\vspace*{0.2truecm}

\vspace*{0.2truecm}
\item FPK equation is applicable to any elastic--plastic material model 
%$\rightarrow$ 
%Only the coefficients $N_{(1)}$ and $N_{(2)}$ are different for different material models

\end{itemize}

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

\begin{frame}
\frametitle{1--D Low and High OCR Cam Clay}

\vspace*{-0.1cm}
%\begin{figure}[!hbpt]
%\nonumber
\begin{center}
%\begin{center}
%\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
\hspace*{-0.7cm}
\includegraphics[height=5.5cm]{/home/jeremic/tex/works/Conferences/2007/GeoDenver/SFEM/Presentation/ContourLowOCR_RandomG_RandomM_Randomp0-m.pdf}
\hfill
\includegraphics[height=5.5cm]{/home/jeremic/tex/works/Conferences/2007/GeoDenver/SFEM/Presentation/ContourHighOCR_RandomG_RandomM-m.pdf}
\hspace*{-0.7cm}
\end{center}
\vspace*{-0.1cm}


random $G$, $M$ and $p_0$

\end{frame}

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










%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Stochastic Soil Dynamics}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{1--D Shear Column Example}

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


      
%\vspace*{-0.5truecm}
%\hspace*{-3.0truecm}
\begin{itemize}

  \item Static pushover of a \\
        Stochastic \\
        shear column \\
        10m high (deep)
 
 
  \item Small correlation \\
        length results in \\
        mean  that tends to \\
        deterministic solution

  \item High correlation \\
        length increases \\
        mean and \\
        standard deviation

\end{itemize}
      

\vspace*{-5.5truecm}
\hspace*{5.0truecm}
%
%\begin{flushright}
%\hspace*{-1.0cm}
%\includegraphics[height=4.0cm]{/home/jeremic/tex/works/Conferences/2007/GeoDenver/SFEM/Presentation/PEER2007_3.jpg}
%\hspace*{0.0cm}
%\includegraphics[height=3.0cm]{/home/jeremic/tex/works/Conferences/2007/GeoDenver/SFEM/Presentation/PEER2007.jpg}
%\hspace*{0.3cm}
%\vspace*{-0.7truecm}
\includegraphics[height=6.5cm]{/home/jeremic/tex/works/Conferences/2007/GeoDenver/SFEM/Presentation/PEER2007_2.jpg}
%\end{flushright}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Stochastic Seismic Ground Motions}

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

%\vspace*{-0.5truecm}

%\begin{flushleft}
\vspace*{-5.2cm}
\includegraphics[height=9.0cm]{/home/jeremic/tex/works/Conferences/2007/4ICEGE-Solun/StochasticGroundMotions/Present/KallolsPlots/SinusoidalBaseShaking/evolution_of_PDF_at_top_node_view01-Edited.pdf}
%\end{flushleft}


\vspace*{-3.5cm}
\hspace*{5.0cm}
\includegraphics[height=2.8cm]{/home/jeremic/tex/works/Conferences/2007/4ICEGE-Solun/StochasticGroundMotions/Present/KallolsPlots/SinusoidalBaseShaking/evolution_of_mean_and_SD_at_top_node-Edited_01_ps.pdf}
\\
%\vspace*{-3.0cm}
\hspace*{6.0cm}
\includegraphics[height=3.0cm]{/home/jeremic/tex/works/Conferences/2007/4ICEGE-Solun/StochasticGroundMotions/Present/KallolsPlots/SinusoidalBaseShaking/evolution_of_SD_at_top_node-Edited_01_ps.pdf}

\vspace*{-2.0truecm}


\begin{itemize}

\item Sinusoidal motions example

\item Complete PDF of motions \\
      for each time step.

\item In general, increase in system uncertainty

\end{itemize}

\end{frame}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Stochastic Seismic Ground Motions}

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

\vspace*{-0.2truecm}

\begin{center}
\includegraphics[height=4.50cm]{/home/jeremic/tex/works/Conferences/2007/4ICEGE-Solun/StochasticGroundMotions/Present/KallolsPlots/ImperialValley1938BaseShaking/ImperialValley1938_mean_and_SD_at_top_node_Zoomed_m_ps.pdf}
\end{center}

\vspace*{-0.4truecm}


\begin{itemize}

\item Example motions: Imperial Valley
 
\item Mean and SD of ground motions

\item Large uncertainty at ground motion peaks

\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Summary}

\begin{frame}
\frametitle{Summary}

\begin{itemize}

\item A new, second-order accurate (exact mean and variance) formulation 
for probabilistic elastic--plastic soil simulation

%\vspace*{-0.3cm}
%\item Methodology, which combines SSFEM and FPK equation approach, 
%overcomes the drawbacks of \textit{Monte Carlo Method} and \textit{Perturbation Technique}
\vspace*{0.2cm}

\item Analytic modeling and simulations of 
  \begin{itemize}
    \item spatial variability and 
    \item point-wise uncertainty
  \end{itemize}
   of soil elastic of elastic--plastic properties for static and dynamic problems

%\item Results in second order accurate probabilistic stress-strain and displacement predictions

\vspace*{0.2cm}
\item Application to:
  \begin{itemize}
    \item Sensitivity analysis
    \item Probability of failure (tails of PDF)
    \item Site characterization design (probabilistic)
    \item General stochastic modeling in elastic--plastic 
          solid and structural mechanics
  \end{itemize}


\end{itemize}


\end{frame}


\end{document}


