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\title{Soil Uncertainty and Seismic Ground Motion}


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




\date
{\small GEESD IV \\
%Sacramento, California, U.S.A. \\ 
May 2008\\ ~\\    ~\\    
Support by NSF--CMMI--0600766}
% %\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{}
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% 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}}




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%  --   % $Header: /cvsroot/latex-beamer/latex-beamer/solutions/conference-talks/conference-ornate-20min.en.tex,v 1.6 2004/10/07 20:53:08 tantau Exp $
%  --   
%  --   \documentclass{beamer}
%  --   
%  --   % This file is a solution template for:
%  --   
%  --   % - Talk at a conference/colloquium.
%  --   % - Talk length is about 20min.
%  --   % - Style is ornate.
%  --   \definecolor{mycolor}{rgb}{0,0.08,0.45}%
%  --   %\graphicspath{{/home/kallol/publication/2007/Presentation/USNCCM9/}}
%  --   
%  --   
%  --   % Copyright 2004 by Till Tantau <tantau@users.sourceforge.net>.
%  --   %
%  --   % In principle, this file can be redistributed and/or modified under
%  --   % the terms of the GNU Public License, version 2.
%  --   %
%  --   % However, this file is supposed to be a template to be modified
%  --   % for your own needs. For this reason, if you use this file as a
%  --   % template and not specifically distribute it as part of a another
%  --   % package/program, I grant the extra permission to freely copy and
%  --   % modify this file as you see fit and even to delete this copyright
%  --   % notice. 
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%  --     % or ...
%  --   
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%  --     % or whatever (possibly just delete it)
%  --   }
%  --   
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%  --   % or whatever
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%  --   %\usepackage{sansserif}
%  --   %\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[Short Paper Title] % (optional, use only with long paper titles)
%  --   %{Title As It Is In the Proceedings}
%  --   \vspace*{0.4truecm}
%  --   \title{Uncertain Elasto--Plasticity}
%  --   
%  --   %\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=0.2cm]{university-logo}{ucdavis_logo_blue_sm}
%  --   
%  --   \author[Boris Jeremi{\'c}] % (optional, use only with lots of authors)
%  --   {
%  --   %\ \\
%  --   %\ \\
%  --   \Large{Boris~Jeremi{\'c}} \\
%  --   \large{Department of Civil and Environmental Engineering}\\
%  --   \large{University of California, Davis}
%  --   %\ \\
%  --   %\ \\
%  --   %\ \\
%  --   %\footnotesize
%  --   %{
%  --   %with guidance from \\
%  --   %\vspace*{1.0truecm} \\
%  --   %\ \\
%  --   %\hspace*{-1.9truecm} guidance from 
%  --   %Professors Boris Jeremi\'{c}}, M. Levent Kavvas, Ross Boulanger, and Niels Gr{\o}nbech Jensen 
%  --   %\hspace*{-1.9truecm}
%  --   }
%  --   % - Give the names in the same order as the appear in the paper.
%  --   % - Use the \inst{?} command only if the authors have different
%  --   %   affiliation.
%  --   
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%  --   {
%  --   %  \inst{1}%
%  --   %  Department of Civil and Environmental Engineering \\
%  --   %  University of California, Davis
%  --   %  \and
%  --   %  \inst{2}%
%  --   %  Department of Theoretical Philosophy\\
%  --   %  University of Elsewhere
%  --   }
%  --   % - Use the \inst command only if there are several affiliations.
%  --   % - Keep it simple, no one is interested in your street address.
%  --   %\begin{footnotesize}
%  --   %This work is supported by NSF (through Award \#) \\
%  --   %\end{footnotesize}
%  --   
%  --   %\vspace*{-1.4truecm}
%  --   \date
%  --   {
%  --   USC Seminar Series, December 2007 
%  --   %\vspace*{-1.1truecm}
%  --   \begin{footnotesize}
%  --   %\begin{tiny}
%  --   %\hspace*{-1.9truecm} 
%  --   %This work is supported by grant from NSF (CMMI-0600766) 
%  --   %\& EEC-9701568)
%  --   %\hspace*{-1.9truecm}
%  --   \end{footnotesize}
%  --   %\end{tiny}
%  --   }
%  --   % - 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{Theoretical Computer Science}
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%  --   % out. 
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%  --   % is a graphic format that can be processed by latex or pdflatex,
%  --   % resp., then you can add a logo as follows:
%  --   
%  --   %\pgfdeclareimage[height=0.5cm]{university-logo}{ucdavis_logo_gold_sm}
%  --   %\logo{\pgfuseimage{university-logo}}
%  --   
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%  --   % the beginning of each subsection:
%  --   \AtBeginSubsection[]
%  --   {
%  --     \begin{frame}<beamer>
%  --       \frametitle{Outline}
%  --       \tableofcontents[currentsection,currentsubsection]
%  --     \end{frame}
%  --   }
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%  --   % If you wish to uncover everything in a step-wise fashion, uncomment
%  --   % the following command: 
%  --   
%  --   %\beamerdefaultoverlayspecification{<+->}
%  --   

\begin{document}

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

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

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% 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}



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\subsection{Soils and Uncertainty}
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\begin{frame} 

\frametitle{Soils are Inherently Uncertain}



        \begin{itemize}

\vspace*{0.2cm}
        \item Spatial \\ 
              variability

\vspace*{0.2cm}
        \item Point-wise \\ 
              uncertainty, \\
%              testing \\
%              error, \\
%              transformation \\
%              error

\vspace*{0.2cm}
        \item 
%Point-wise \\ 
%              uncertainty, \\
              Testing \\
              error, \\
%              transformation \\
%              error
\vspace*{0.2cm}
        \item 
%        Point-wise \\ 
%              uncertainty, \\
%              testing \\
%              error, \\
              Transformation \\
              error

        \end{itemize}

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

\vspace*{-5cm}
\begin{figure}[!hbpt]
%\nonumber
%\begin{center}
\begin{flushright}
%\includegraphics[height=5.0cm]{/home/jeremic/tex/works/Conferences/2006/KragujevacSEECCM06/Presentation/MGMuzorak01.jpg}
%\includegraphics[height=5.5cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/FrictionAngleProfile.jpg}
\includegraphics[height=5.5cm]{/home/jeremic/tex/works/Conferences/2007/GeoDenver/SFEM/Presentation/FrictionAngleProfile.jpg}
\\
\mbox{(Mayne et al. (2000) }
\end{flushright}
%\end{center}
%\end{center}
\end{figure}


% - %\vspace*{2cm}
% - %
% - {%
% - \begin{beamercolorbox}{section in head/foot}
% - \usebeamerfont{framesubtitle}\tiny{
% - Stein Sture, Nicholas C. Costes, Susan N. Batiste, Mark R. Langton, Khalid A. Al-Shibli, Boris
% - Jeremi{\'c}, Roy A. Swanson and Melissa Frank. Mechanics of granular materials at low effective
% - stresses. \textit{ASCE Journal of Aerospace Engineering}, vol. 11, No. 3, pages 67-72, 1998.}
% - %\vskip2pt\insertnavigation{\paperwidth}\vskip2pt
% - \end{beamercolorbox}%
% - }

 

\end{frame}

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\begin{frame} 

\frametitle{On Uncertainties}

 

\begin{itemize}



\item Epistemic uncertainty - due to lack of knowledge

   \begin{itemize}

     \item Can be reduced by \\ collecting more data

     \item Mathematical tools not well \\
           developed, trade-off with \\
           aleatory uncertainty

   \end{itemize}

\vspace*{-2.9cm}
\begin{figure}[!hbpt]
\begin{flushright}
\includegraphics[height=4cm,angle=-90]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Present/uncertain03.pdf}
\end{flushright}
\end{figure}



\vspace*{0.15cm}
\item Aleatory uncertainty - inherent variation of physical system 

    \begin{itemize}

      \item Can not be reduced

      \item Has highly developed mathematical tools 

    \end{itemize}



\item Ergodicity -- exchange ensemble average
for time average? 

    \begin{itemize}

      \item Possibly  yes 

      \item Issues up for discussion (soil, concrete, rock, biomaterials...)

    \end{itemize}





\end{itemize}

 


\end{frame}



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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*{0.5cm}
\item Testing error (Stokoe et al. 2004)

\begin{itemize}

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

\end{itemize}

%
\vspace*{0.5cm}
\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}






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\begin{frame} 

\frametitle{Probabilistic material (Soil Site) Characterization}

 


\begin{itemize}

\item Ideal: complete probabilistic site characterization
%  a very large amount of data is needed...... Need alternate strategies!!!}




\item Large (physically large but not statistically) amount of data 

  \begin{itemize}
  
  
    \item Site specific mean and coefficient of variation (COV)
    
    \item  Covariance  structure from similar sites (e.g. Fenton 1999)
  
  \end{itemize}


\item Moderate amount of data $\rightarrow$ Bayesian updating (e.g. Phoon and Kulhawy 1999, Baecher and Christian 2003)

\item Minimal data: general guidelines for typical sites and test methods (Phoon and Kulhawy (1999))

  \begin{itemize}
  
    \item COVs and covariance structures of inherent variability
    
    \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

%
%\end{itemize}



\end{itemize}

 
\end{frame}

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%  

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\section{Stochastic Elasto--Plasticity}
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\subsection{Constitutive and Finite Element Levels}
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\begin{frame} 

\frametitle{Problem Setup}
 


 
 
\begin{itemize}

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

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

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


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

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

 \end{itemize}

\end{frame}






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





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\begin{frame} 

\frametitle{Eulerian--Lagrangian FPK Equation}

%

%\begin{itemize}

\begin{footnotesize}

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

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

\end{footnotesize}


\vspace*{0.5cm}
{%
\begin{beamercolorbox}{section in head/foot}
\usebeamerfont{framesubtitle}\tiny{B. Jeremi\'{c}, K. Sett, and M. L. Kavvas, "Probabilistic
Elasto--Plasticity: Formulation in 1--D", \textit{Acta Geotechnica}, Vol. 2, No. 3, 2007, In press (published
online in the \textit{Online First} section)}
%\vskip2pt\insertnavigation{\paperwidth}\vskip2pt
\end{beamercolorbox}%
}


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\begin{frame} 

\frametitle{Euler--Lagrange FPK Equation}

 

\begin{itemize}

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

%

\item Complete probabilistic description of response 


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


\item It is deterministic equation in probability density space

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

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

%\vspace*{0.2truecm}

\end{itemize}



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



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%GEESD out 
%GEESD out 
%GEESD out \begin{itemize}
%GEESD out 
%GEESD out %
%GEESD out \vspace*{0.2cm}
%GEESD out \item Weighted elastic and elastic--plastic Solution
%GEESD out ${\partial P(\sigma,t)}/{\partial t} 
%GEESD out = 
%GEESD out -{\partial \left(N^w_{(1)}P(\sigma,t)
%GEESD out -{\partial \left(N^w_{(2)} P(\sigma,t)\right) }/{\partial \sigma} \right)}/
%GEESD out {\partial \sigma}$
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%GEESD out 
%GEESD out 
%GEESD out 
%GEESD out 
%GEESD out 
%GEESD out \vspace*{0.1cm}
%GEESD out \item Weighted advection and diffusion coefficients are then 
%GEESD out $N_{(1,2)}^{w} (\sigma) 
%GEESD out = 
%GEESD out (1 - P[\Sigma_y \leq \sigma]) N_{(1)}^{el} + P[\Sigma_y \leq \sigma] N_{(1)}^{el-pl} $
%GEESD out \begin{footnotesize}
%GEESD out (Black--Scholes options \\
%GEESD out  pricing model '73, \\
%GEESD out Nobel Economics Prize '97)
%GEESD out \end{footnotesize}
%GEESD out 
%GEESD out 
%GEESD out \vspace*{0.1cm}
%GEESD out %\vspace*{0.5cm}
%GEESD out \item   
%GEESD out Cumulative Probability \\
%GEESD out Density  function  (CDF) \\
%GEESD out of the yield function 
%GEESD out 
%GEESD out \end{itemize}
%GEESD out 
%GEESD out \vspace*{-3.5cm}
%GEESD out \begin{flushright}
%GEESD out %\begin{figure}[!h]
%GEESD out %\hspace*{7cm}
%GEESD out \includegraphics[width=5cm]{/home/jeremic/tex/works/Papers/2007/ProbabilisticYielding/vonMises_YieldCDF_Combined-edited.pdf} 
%GEESD out %\end{figure}
%GEESD out \end{flushright}
%GEESD out \vspace*{-0.5cm}
%GEESD out 
%GEESD out 
%GEESD out %\vspace*{0.5cm}
%GEESD out {%
%GEESD out \begin{beamercolorbox}{section in head/foot}
%GEESD out \usebeamerfont{framesubtitle}\tiny{B.  Jeremi{\'c} and K. Sett. On Probabilistic
%GEESD out Yielding  of  Materials.  in print in  \textit{Communications in Numerical Methods in
%GEESD out Engineering}, 2007.}
%GEESD out %\vskip2pt\insertnavigation{\paperwidth}\vskip2pt
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%GEESD out }
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%GEESD out \begin{frame}
%GEESD out 
%GEESD out 
%GEESD out 
%GEESD out \frametitle{Transformation of a Bi--Linear (von Mises) Response}
%GEESD out 
%GEESD out 
%GEESD out \begin{figure}[!hbpt]
%GEESD out \begin{center}
%GEESD out \includegraphics[width=8cm]{/home/jeremic/tex/works/Papers/2007/ProbabilisticYielding/figures/vonMises_G_and_cu_very_uncertain/Contour_PDF-edited.pdf}
%GEESD out \end{center}
%GEESD out \end{figure}                                                          
%GEESD out 
%GEESD out \vspace*{-0.3cm}
%GEESD out linear elastic -- linear hardening plastic von Mises
%GEESD out 
%GEESD out 
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%GEESD out 
%GEESD out 
%GEESD out 
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\begin{frame}
\frametitle{Spectral Stochastic Elastic--Plastic FEM}

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


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




\begin{itemize}

  \item SFEM: Ghanem and Spanos 2003

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

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

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

\end{itemize}

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

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\section{An Application}


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\subsection{Seismic Wave Propagation Through Uncertain Soils}


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\begin{frame}




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


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


\end{frame}


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\begin{frame}


\frametitle{Random Field Parameters from Site Data}

\begin{itemize}

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

\item Maximum likelihood estimates of correlation length

\vspace*{0.3truecm}

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

\vspace*{-4.9truecm}

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

\vspace*{0.02truecm}

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

\end{itemize}

\end{frame}


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\begin{frame}


\frametitle{Seismic Wave Propagation through Stochastic Soil}

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

\begin{itemize}

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

\begin{itemize}

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

\end{itemize}

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

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

\end{itemize}


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

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

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

\end{itemize}

\end{frame}

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



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

\vspace*{-12.0truecm}

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

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

\vspace*{-19.0truecm}

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

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

\begin{itemize}

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

\vspace*{0.1truecm}

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

\end{itemize}

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


\end{frame}

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

\vspace*{-10.25cm}

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

\end{frame}

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

\vspace*{-10.25cm}

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

\end{frame}

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

\vspace*{-10.25cm}

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

\end{frame}

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\begin{frame}
\frametitle{Derivative Applications}

\vspace*{-0.25truecm}

\begin{itemize}

\item Performance based engineering

\begin{itemize}

\item Reliability index ($\beta$)

\item Probability of failure ($p_f$)

\end{itemize}

\vspace*{-12.0truecm}

\hspace*{-1.0truecm} \includegraphics[height=15.0cm]{/home/jeremic/tex/works/Conferences/2008/EM/Present/SchematicReliability.pdf} \hspace*{-1.5truecm}

\vspace*{0.5truecm}

\item Sensitivity analysis

\item Financial risk analysis


%\vspace*{-8.0truecm}

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

\end{itemize}

\end{frame}


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\section{Summary}

\begin{frame}


\frametitle{Summary}

\begin{itemize}

\item    Development  of  probabilistic elastic--plastic modeling and simulation
methodology

\vspace*{0.5cm}
\item Takes into account 

\begin{itemize}

\item uncertainty of (measured) soil parameters and 
\item soil spatial variability

\end{itemize}

\vspace*{0.5cm}
\item Possibly useful for applications where mean, mode and extreme statistics is important




\end{itemize}


\end{frame}

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\end{document}


