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% does not look nice, try deleting the line with the fontenc.
\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]{universitylogo}{/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{universitylogo} \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 NSFCMMI0600766}
% %\date[] % (optional, should be abbreviation of conference name)
% %{
% %\small Graduate Student: Guanzhou Jie
% %\\ ~\\
% %Funding: NSFCMS0324661, NSFTeraGrid, NSFEEC9701568 \\
% %%\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 "universitylogofilename.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]{universitylogo}{/home/jeremic/BG/amblemi/ucdavis_logo_gold_lrg}
%\logo{\pgfuseimage{universitylogo}}
% \pgfdeclareimage[height=0.5cm]{universitylogo}{universitylogofilename}
% \logo{\pgfuseimage{universitylogo}}
% Delete this, if you do not want the table of contents to pop up at
% the beginning of each subsection:
\AtBeginSubsection[]
{
\begin{frame}
\frametitle{Outline}
\tableofcontents[currentsection,currentsubsection]
% \tableofcontents[currentsection]
\end{frame}
}
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% the following command:
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% 
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% 
%  % $Header: /cvsroot/latexbeamer/latexbeamer/solutions/conferencetalks/conferenceornate20min.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 .
%  %
%  % 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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%  }
% 
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%  % or whatever
% 
%  \usepackage[latin1]{inputenc}
%  % or whatever
% 
%  %%\usepackage{times}
%  %\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 ElastoPlasticity}
% 
%  %\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]{universitylogo}{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.
% 
%  \institute[Computational Geomechanics Group \pgfuseimage{universitylogo} \hspace*{0.3truecm}] % (optional, but mostly needed)
%  {
%  % \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 (CMMI0600766)
%  %\& EEC9701568)
%  %\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}
%  % This is only inserted into the PDF information catalog. Can be left
%  % out.
% 
%  % If you have a file called "universitylogofilename.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=0.5cm]{universitylogo}{ucdavis_logo_gold_sm}
%  %\logo{\pgfuseimage{universitylogo}}
% 
%  % Delete this, if you do not want the table of contents to pop up at
%  % the beginning of each subsection:
%  \AtBeginSubsection[]
%  {
%  \begin{frame}
%  \frametitle{Outline}
%  \tableofcontents[currentsection,currentsubsection]
%  \end{frame}
%  }
% 
% 
%  % If you wish to uncover everything in a stepwise 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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 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 Pointwise \\
uncertainty, \\
% testing \\
% error, \\
% transformation \\
% error
\vspace*{0.2cm}
\item
%Pointwise \\
% uncertainty, \\
Testing \\
error, \\
% transformation \\
% error
\vspace*{0.2cm}
\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}
\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. AlShibli, 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 6772, 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, tradeoff 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 ElastoPlasticity}
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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 elpl:
%
%
% \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}
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%
%
% \frametitle{Stochastic Continuity (Liouville) Equation}
%
%
%
% \begin{itemize}
%
%
% \item phase density $\rho$ of $\sigma(x,t)$ varies in time according to a continuity
% Liouville equation (Kubo 1963):
% %
% \begin{eqnarray}
% \frac{\partial \rho (\sigma(x,t),t)}{\partial t}
% =
% \nonumber
% \\
% \frac{\partial \eta (\sigma(x,t), D^{el}(x), q(x), r(x), \epsilon(x,t)) }{\partial \sigma}
% \;\;
% \rho[\sigma(x,t),t]
% \nonumber
% \end{eqnarray}
%
% \vspace{0.5cm}
% \item with initial conditions $\rho(\sigma,0) = \delta(\sigma\sigma_0)$
%
%
% \end{itemize}
%
% \end{frame}
%
%
%
%
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%
%
% \frametitle{{Ensemble Average form of Liouville Equation}}
%
%
%
%
%
%
%
% \noindent
% Continuity equation written in ensemble average form (eg. cumulant
% expansion method (Kavvas and Karakas 1996)):
%
%
% %\vspace*{0.5cm}
%
% \begin{footnotesize}
%
% \begin{eqnarray}
% \nonumber
% &&\displaystyle \frac{\partial \left < \rho(\sigma(x_t,t), t) \right >}{\partial t}=
%  \displaystyle \frac{\partial}{\partial \sigma} \left[ \left\{\left< \vphantom{\int_{0}^{t} d\tau} \eta(\sigma(x_t,t), D^{el}(x_t),
% q(x_t), r(x_t), \epsilon(x_t,t)) \right> \right. \right. \\
% \nonumber
% &+& \left. \left. \int_{0}^{t} d\tau Cov_0 \left[ \displaystyle \frac{\partial \eta(\sigma(x_t,t), D^{el}(x_t), q(x_t), r(x_t),
% \epsilon(x_t,t))}{\partial \sigma}; \right. \right. \right. \\
% \nonumber
% & & \left. \left. \left. \eta(\sigma(x_{t\tau},t\tau), D^{el}(x_{t\tau}), q(x_{t\tau}), r(x_{t\tau}),
% \epsilon(x_{t\tau},t\tau) \vphantom{\int_{0}^{t} d\tau} \right] \right \} \left < \rho (\sigma(x_t,t),t) \right > \right] \\
% \nonumber
% &+& \displaystyle \frac{\partial^2}{\partial \sigma^2} \left[ \left\{ \int_{0}^{t} d\tau Cov_0 \left[ \vphantom{\int_{0}^{t}}
% \eta(\sigma(x_t,t), D^{el}(x_t), q(x_t), r(x_t), \epsilon(x_t,t)); \right. \right. \right. \\
% \nonumber
% & & \left. \left. \left. \eta (\sigma(x_{t\tau},t\tau), D^{el}(x_{t\tau}), q(x_{t\tau}), r(x_{t\tau}),
% \epsilon(x_{t\tau},t\tau)) \vphantom{\int_{0}^{t}} \right] \vphantom{\int_{0}^{t}} \right\} \left < \rho (\sigma (x_t,t),t) \right > \right] \\
% \nonumber
% \end{eqnarray}
%
% \end{footnotesize}
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
% \end{frame}
%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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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}
\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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\begin{frame}
\frametitle{EulerLagrange 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 Template 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}
{%
\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 ElastoPlasticity", \textit{International Journal for Numerical
and Analytical Methods in Geomechanics}, Vol. 31, No. 7, pp. 953975, 2007}
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%GEESD out
%GEESD out \frametitle{Probabilistic Yielding}
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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 elasticplastic 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)}^{elpl} $
%GEESD out \begin{footnotesize}
%GEESD out (BlackScholes 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_Combinededited.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
%GEESD out \end{beamercolorbox}%
%GEESD out }
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%GEESD out \begin{frame}
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%GEESD out
%GEESD out
%GEESD out \frametitle{Transformation of a BiLinear (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_PDFedited.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
%GEESD out \end{frame}
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%
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\begin{frame}
\frametitle{Spectral Stochastic ElasticPlastic 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 KLexpansion
\item Unknown solution random variables represented using polynomial chaos of
(known) input random variables
\item FokkerPlanckKolmogorov approach based probabilistic constitutive
integration at Gauss integration points
\end{itemize}
%% \end{itemize}
%
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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/EastWestProfileEdited.pdf}
\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 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/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{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}
\end{itemize}
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% BJtempout\setbeamercovered{invisible}
\begin{frame}
\frametitle{Surface Displacement Time History}
\vspace*{12.0truecm}
%\hspace*{0.7cm} \uncover<1>{\includegraphics[width=12.0cm]{Chapter9_ElasticPlasticResponse_MeanNew.pdf}}
%%
%\hspace*{6.0cm} \uncover<2>{\includegraphics[width=12.0cm]{Chapter9_ElasticPlasticResponse_SDNew.pdf}} \hspace*{2.0cm} \\
%%
%\uncover<1>{
\hspace*{0.7cm}
\includegraphics[width=12.0cm]{/home/jeremic/tex/works/Conferences/2008/EM/Present/Chapter9_ElasticPlasticResponse_MeanNew.pdf}
%}
% BJtempout\uncover<3>{\hspace*{6.0cm} \includegraphics[width=12.0cm]{Chapter9_ElasticPlasticResponse_SDNew.pdf} \hspace*{6.0truecm}}
{\hspace*{6.0cm} \includegraphics[width=12.0cm]{/home/jeremic/tex/works/Conferences/2008/EM/Present/Chapter9_ElasticPlasticResponse_SDNew.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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\frametitle{Mean $\pm$ Standard Deviation}
\vspace*{19.0truecm}
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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_MeanNew.pdf}
%\hspace*{4.0cm} \includegraphics[width=9.0cm]{Chapter9_ElasticPlasticResponse_SDNew.pdf}
%
%%\vspace*{0.5truecm}
%%\hspace*{1.0cm}
%\tiny{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Mean~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Standard Deviation}
%
%\vspace*{8.35truecm}
%
%
%\hspace*{0.05cm} \includegraphics[width=9.0cm]{Chapter9_ElasticPlasticResponseNew.pdf}
%\hspace*{3.55cm} \includegraphics[width=9.0cm]{Chapter9_ElasticPlasticResponse_COVNew.pdf}
%
%%\vspace*{0.1truecm}
%%\hspace*{0.5cm}
%\tiny{~~~~~~~~~~~~~~~~~~~~~~~~~~Mean$\pm$ Standard Deviation~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~COV}
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%\end{frame}
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\begin{frame}
\frametitle{PDF of Surface Displacement Time History}
\begin{itemize}
\item PDF at the finite \\
element nodes can be \\
obtained using, e.g., \\
Edgeworth expansion \\
(Ghanem and Spanos \\
2003)
\vspace*{0.1truecm}
\item Numerous applications, \\
especially where extreme \\
statistics are critical
\end{itemize}
\vspace*{11.0cm}
\hspace*{5.5truecm} \includegraphics[height=12.0cm]{/home/jeremic/tex/works/Conferences/2008/EM/Present/Chapter9_ElasticPlasticResponse_PDFNewEdited.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}
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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 elasticplastic 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}