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\title{The Case for Probabilistic ElastoPlasticity}
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\author[Jeremi{\'c}] % (optional, use only with lots of authors)
{Boris~Jeremi{\'c} \\ Kallol Sett (UA) and Lev Kavvas }
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\institute[Computational Geomechanics Group \hspace*{0.3truecm}
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{ Department of Civil and Environmental Engineering\\
University of California, Davis }
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{GheoMat \\
{\small Masseria Salamina \\
Italy, June 2009} }
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% just say so once. Everybody will be happy with that.
\section{Motivation}
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\subsection{Stochastic Systems: Historical Perspectives}
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\begin{frame}
\frametitle{Failure Mechanisms for Geomaterials}
\begin{center}
\vspace*{3.5cm}
\includegraphics[height=6.0cm]{/home/jeremic/tex/works/Conferences/2006/KragujevacSEECCM06/Presentation/MGMuzorak01.jpg}
\\
\small{
Soil: Inside Failure of "Uniform" MGM Specimen \\
(After Swanson et al. 1998)
}
\end{center}
\vspace*{6.0cm}
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\begin{frame}
\frametitle{Personal Motivation}
\begin{itemize}
\item Probabilistic fish counting
\vspace*{0.2cm}
\vspace*{0.2cm}
\item Williams' DEM simulations, differential displacement vortices
%\vspace*{0.2cm}
%\item Runesson's dilemma
\vspace*{0.2cm}
\item SFEM round table
\vspace*{0.2cm}
\item Kavvas' probabilistic hydrology
\end{itemize}
\end{frame}
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%
%
% \begin{frame}
% \frametitle{Geomaterials 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 \\
% \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (After 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 \\
% \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (After Lacasse and Nadim 1996)
% \end{flushleft}
%
% \end{frame}
%
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%
% \begin{frame}
% \frametitle{Brownian Motion}
%
% \begin{itemize}
%
% %\vspace*{1.5cm}
% \item Governing equation (Langevin equation):
%
%
% %\vspace*{0.5cm}
%
% \begin{equation}
% \nonumber
% m \frac{dv}{dt} = F(x)  \beta v + \eta(t)
% \end{equation}
%
%
% %\vspace*{0.5cm}
% \item Probability density function (PDF) of particle displacement obeys a
% simple diffusion equation (Einstein (1905)):
%
% %\vspace*{0.5cm}
%
% \begin{equation}
% \nonumber
% \frac{\partial f (x,t)}{\partial t} = D \frac{\partial^2 f(x,t)}{\partial x^2}
% \end{equation}
%
% %\vspace*{0.5cm}
% \item Addition of external forces (gravity, elastic or magnetic attraction) $\rightarrow$ FokkerPlanckKolmogorov (FPK)
% equation governs the PDF (Kolmogorov 1941)
%
% %\vspace*{0.5cm}
% \item Alternately, Monte Carlo method can be used for solution of Langevin equation $\rightarrow$ computationally very expensive
%
% \end{itemize}
%
%
% \end{frame}
%
%
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%
% \begin{frame}
% \frametitle{Classical SecondOrder Analysis}
%
%
% \begin{itemize}
% %\vspace*{1.5cm}
%
% \item Classical approach to secondorder analysis of random processes
%
% \begin{itemize}
%
% %\vspace*{0.5cm}
% \item Relationship between the autocorrelation function and spectral
% density function (Wiener 1930) $\rightarrow$ Paved the way to the solution of general
% stochastic differential equation (SDE)
%
% \end{itemize}
%
% %\vspace*{0.5cm}
% \item SDEs with random forcings $\rightarrow$ Highly developed mathematical theory for It{\^o} type equation:
%
% \vspace*{0.2cm}
% \begin{equation}
% \nonumber
% dx = a(x,t) dt + b(x,t) dW
% \end{equation}
%
% \begin{itemize}
%
% \item Solution is a Markov process
%
% \item PDF of solution process satisfies a FPK PDE
%
% \vspace*{0.3cm}
% \begin{equation}
% \nonumber
% \frac{\partial p\left(x,t\right)}{\partial t} = \frac{\partial}{\partial x} \left[a(x,t)p\left(x,t \right)\right]
% + \frac{1}{2}\frac{\partial^2}{\partial x^2}\left[b^2(x,t)p\left(x,t \right) \right]
% \end{equation}
%
% \end{itemize}
% \end{itemize}
%
% \end{frame}
%
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%
% \begin{frame}
% \frametitle{SDEs with Random Coefficients}
%
% \begin{itemize}
%
% \item Solution methods
%
% \begin{itemize}
%
% \item Functional integration approach (Hopf 1952)
%
% \item Averaged equation approach (BharruchaReid 1968)
%
% \item Numerical approaches (perturbation method, stochastic finite element etc.)
%
% \item Monte Carlo method.
%
% \end{itemize}
%
%
% % \vspace*{1.0cm}
% \item FPK equation for the characteristic functional of the solution for problem of
% wave propagation in random elastic media (Lee 1974)
%
% % \vspace*{1.0cm}
% % \item EulerianLagrangian form of FPK equation for probabilistic solution of flow through porous media (Kavvas 2003)
%
% \end{itemize}
%
%
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\begin{frame}
\frametitle{Types of Uncertainties}
\begin{itemize}
\item Epistemic uncertainty  due to lack of knowledge
\begin{itemize}
\item Can be reduced by collecting more data
\item Mathematical tools are not well developed
\item tradeoff with aleatory uncertainty
\end{itemize}
\vspace*{0.2cm}
\item Aleatory uncertainty  inherent variation of physical system
\begin{itemize}
\item Can not be reduced
\item Has highly developed mathematical tools
\end{itemize}
\end{itemize}
\vspace*{0.5cm}
\begin{figure}[!hbpt]
%\nonumber
\begin{center}
%\begin{center}
%\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
\includegraphics[height=4cm,angle=90]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Present/uncertain03.pdf}
%
%\mbox{\tiny{Lambe, T. W. and Whitman, R. V.,1969. Soil Mechanics. New York, John Wiley \& Sons}}
%
\end{center}
%\begin{flushright}
%Soil Variability in Relatively \\ Homogeneous Soil Deposit \\ (Clay Deposit of the Valley \\ of Mexico at a Typical \\ Spot in Mexico City)
%\includegraphics[width=14cm]{TypicalSoilCOV.jpg}
%
%\nonumber
%\caption{DruckerPrager Associative Linear Hardening Response with Random $G$}
%{\includegraphics[width=10.5cm]{QualSchematics4.jpg}}
%\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
%\end{flushright}
%\end{center}
\end{figure}
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\begin{frame}
\frametitle{Ergodicity}
\begin{itemize}
\item Exchange ensemble averages for time averages
\vspace*{0.2cm}
\item Is soil elastoplasticity ergodic?
\begin{itemize}
\item Can soil elasticplastic statistical properties be obtained by
temporal averaging?
\item Will soil elasticplastic statistical properties "renew" at each
occurrence?
\item Are soil elasticplastic statistical properties statistically
independent?
\end{itemize}
\item Claim in literature that structural nonlinear behavior is nonergodic
while earthquake characteristics are (?!)
\item However, earthquake characteristics is representing mechanics (fault slip)
on a different scale...
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{Historical Overview}
\begin{itemize}
%
\item Brownian motion, Langevin equation $\rightarrow$ PDF governed by simple diffusion Eq. (Einstein 1905)
%
%
% \begin{equation}
% \nonumber
% \frac{\partial f (x,t)}{\partial t} = D \frac{\partial^2 f(x,t)}{\partial x^2}
% \end{equation}
%
\vspace*{0.2cm}
\item With external forces $\rightarrow$ FokkerPlanckKolmogorov (FPK) for the PDF (Kolmogorov 1941)
\vspace*{0.2cm}
\item Approach for random forcing $\rightarrow$ relationship between the
autocorrelation function and spectral density function (Wiener 1930)
\vspace*{0.2cm}
\item Approach for random coefficient $\rightarrow$
Functional integration approach (Hopf 1952), Averaged equation approach (BharruchaReid 1968),
Numerical approaches, Monte Carlo method
%\item Alternately, Monte Carlo method can be used for solution of Langevin equation $\rightarrow$ computationally very expensive
\end{itemize}
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%tempout  \frametitle{Brownian Motions}
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%tempout 
%tempout 
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%tempout 
%tempout  %
%tempout  \item Governing (Langevin) equation:
%tempout 
%tempout 
%tempout 
%tempout  \begin{equation}
%tempout  \nonumber
%tempout  m \frac{dv}{dt} = F(x)  \beta v + \eta(t)
%tempout  \end{equation}
%tempout 
%tempout 
%tempout 
%tempout  %
%tempout  \item Probability density function (PDF) of particle displacement obeys a
%tempout  simple diffusion equation (Einstein (1905)):
%tempout 
%tempout  %
%tempout 
%tempout  \begin{equation}
%tempout  \nonumber
%tempout  \frac{\partial f (x,t)}{\partial t} = D \frac{\partial^2 f(x,t)}{\partial x^2}
%tempout  \end{equation}
%tempout 
%tempout 
%tempout  \item Addition of external forces (gravity, elastic or magnetic attraction) $\rightarrow$ FokkerPlanckKolmogorov (FPK)
%tempout  equation governs the PDF (Kolmogorov 1941)
%tempout 
%tempout 
%tempout  \item Alternately, Monte Carlo method can be used for solution of Langevin equation $\rightarrow$ computationally very expensive
%tempout 
%tempout  \end{itemize}
%tempout 
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%tempout  \frametitle{Stochastic Systems: Random Forcing}
%tempout 
%tempout  %
%tempout 
%tempout 
%tempout  \begin{itemize}
%tempout  %
%tempout 
%tempout  \item Classical approach: relationship between the autocorrelation function and
%tempout  spectral density function (Wiener 1930) $\rightarrow$ Paved the way to the
%tempout  solution of general stochastic differential equation (SDE)
%tempout 
%tempout 
%tempout  \item SDEs with random forcing $\rightarrow$ Highly developed mathematical theory for It{\^o} type equation:
%tempout  %
%tempout  %
%tempout  \begin{equation}
%tempout  \nonumber
%tempout  dx = a(x,t) dt + b(x,t) dW
%tempout  \end{equation}
%tempout 
%tempout  \begin{itemize}
%tempout 
%tempout  \item Solution is a Markov process
%tempout 
%tempout  \item PDF of solution process satisfies a FPK PDE
%tempout  %
%tempout  %
%tempout  \begin{equation}
%tempout  \nonumber
%tempout  \frac{\partial p\left(x,t\right)}{\partial t} = \frac{\partial}{\partial x} \left[a(x,t)p\left(x,t \right)\right]
%tempout  + \frac{1}{2}\frac{\partial^2}{\partial x^2}\left[b^2(x,t)p\left(x,t \right) \right]
%tempout  \end{equation}
%tempout 
%tempout  \end{itemize}
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%tempout 
%tempout  \frametitle{Stochastic Systems: Random Coefficient}
%tempout 
%tempout  %
%tempout 
%tempout 
%tempout 
%tempout  \begin{itemize}
%tempout 
%tempout  %\vspace*{0.5cm}
%tempout  \item Approximate Solution methods
%tempout 
%tempout  \begin{itemize}
%tempout 
%tempout  \item Functional integration approach (Hopf 1952)
%tempout 
%tempout  \item Averaged equation approach (BharruchaReid 1968)
%tempout 
%tempout  \item Numerical approaches
%tempout 
%tempout  \item Monte Carlo method
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%tempout 
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%tempout  \vspace*{0.5cm}
%tempout  \item FPK equation for the characteristic functional of the solution for problem of
%tempout  wave propagation in random media (Lee 1974)
%tempout 
%tempout 
%tempout  \vspace*{0.5cm}
%tempout  \item EulerianLagrangian form of FPK equation for probabilistic solution of flow through porous media (Kavvas 2003)
%tempout 
%tempout  \end{itemize}
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\subsection{Uncertainties in Material}
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\begin{frame}
\frametitle{Material Behavior Inherently Uncertain}
%\begin{itemize}
%\vspace*{0.5cm}
%\item
%Material behavior is inherently uncertain (concrete, metals, soil, rock,
%bone, foam, powder etc.)
\begin{itemize}
\vspace*{0.5cm}
\item Spatial \\
variability
\vspace*{0.5cm}
\item Pointwise \\
uncertainty, \\
testing \\
error, \\
transformation \\
error
\end{itemize}
% \vspace*{0.5cm}
% \item Failure mechanisms related to spatial variability (strain localization and
% bifurcation of response)
%
% \vspace*{0.5cm}
% \item Inverse problems
%
% \begin{itemize}
%
% \item New material design, ({\it pointwise})
%
% \item Solid and/or structure design (or retrofits), ({\it spatial})
%
% \end{itemize}
%\end{itemize}
\vspace*{5cm}
\begin{figure}[!hbpt]
%\nonumber
%\begin{center}
\begin{flushright}
%\includegraphics[height=5.0cm]{/home/jeremic/tex/works/Conferences/2006/KragujevacSEECCM06/Presentation/MGMuzorak01.jpg}
\includegraphics[height=5.5cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/FrictionAngleProfile.jpg}
\\
\mbox{(Mayne et al. (2000) }
\end{flushright}
%\end{center}
%\end{center}
\end{figure}
%  %\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}%
%  }
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%\slide{\Large{Motivation}}
%
%\begin{figure}[!hbpt]
%\begin{flushleft}
%\includegraphics[height=17cm]{TypicalSoilProfile.jpg}
%
%\mbox{\tiny{Lambe, T. W. and Whitman, R. V.,1969. Soil Mechanics. New York, John Wiley \& Sons}}
%
%\end{flushleft}
%\begin{flushright}
%Soil Variability in Relatively \\ Homogeneous Soil Deposit \\ (Clay Deposit of the Valley \\ of Mexico at a Typical \\ Spot in Mexico City)
%\end{flushright}
%\end{figure}
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%  \begin{frame} \frametitle{{Soil: Inside Failure (MGM)}}
% 
% 
%  \begin{figure}[!hbpt]
%  %\nonumber
%  \begin{center}
%  \includegraphics[height=6.0cm]{/home/jeremic/tex/works/Conferences/2006/KragujevacSEECCM06/Presentation/MGMuzorak01.jpg}
%  \end{center}
%  %\end{center}
%  \end{figure}
% 
%  {%
%  \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}%
%  }
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% 
%  \frametitle{{Soil: Spatial and Point Variation}}
% 
% 
%  \begin{figure}[!hbpt]
%  %\nonumber
%  \begin{center}
%  %\includegraphics[height=4cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/FrictionAngleProfile.jpg}
%  \includegraphics[height=5cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/FrictionAngleProfile.jpg}
%  %\includegraphics[height=4cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ShearStrengthProfile.jpg}
%  %\includegraphics[height=4cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ShearStrengthProfile.jpg}
%  \end{center}
%  %\end{center}
%  \end{figure}
% 
%  (Mayne et al. (2000)
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\begin{frame} \frametitle{\Large{Motivation}}
\center{Typical Coefficients of Variation of Different Soil Properties}
\begin{figure}[!hbpt]
\begin{center}
\includegraphics[width=10cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/TableTypicalCOV.jpg}
\end{center}
\end{figure}
\flushright{(After Lacasse and Nadim 1996)}
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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}
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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}
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\begin{frame}
\frametitle{Recent StateoftheArt}
\begin{itemize}
%\vspace*{0.5cm}
\item Governing equation
% \vspace*{0.5cm}
\begin{itemize}
\item Dynamic problems $\rightarrow$ $ M \ddot u + C \ddot u + K u = \phi $
\item Static problems $\rightarrow$ $ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ K u = \phi $
\end{itemize}
%\vspace{0.4cm}
\item Existing solution methods
% \vspace*{0.5cm}
\begin{itemize}
\item \textbf{Random r.h.s} (external force random)
\begin{itemize}
\item FPK equation approach
\item Use of fragility curves with deterministic FEM (DFEM)
\end{itemize}
% \vspace*{0.2cm}
\item \textbf{Random l.h.s} (material properties random)
\begin{itemize}
\item Monte Carlo approach with DFEM $\rightarrow$ CPU expensive
% \item Stochastic finite element method (e.g. Perturbation method
% $\rightarrow$ a linearized expansion! Error increases as a function
% of COV; Spectral method
% $\rightarrow$ developed for elastic materials so far)
\item Perturbation method
$\rightarrow$ a linearized expansion! Error increases as a function
of COV
\item Spectral method
$\rightarrow$ developed for elastic materials so far
% \begin{itemize}
%
% \item Perturbation method $\rightarrow$ fails if COVs of soil $>$ 20\%
%
% \item Spectral method $\rightarrow$ only for elastic material
%
% \end{itemize}
\end{itemize}
\end{itemize}
\item New developments for elastoplastic applications
\end{itemize}
\end{frame}
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%  \begin{frame}
%  \frametitle{Objectives of Proposed Work}
% 
%  \begin{itemize}
% 
%  \item Constitutive Problem: Obtain secondorder (mean and variance) exact (analytical) PDF
%  of stress response $\rightarrow$ \textbf{FokkerPlanckKolmogorov (FPK) Equation}
% 
%  \item Boundary Value Problem: Model uncertain spatial variability of elasticplastic soil
%  $\rightarrow$ \textbf{Spectral Stochastic ElasticPlastic Finite Element Method (SSEPFEM)}
% 
% 
%  \vspace*{0.5truecm}
% 
%  \item Overcome the drawbacks of \textit{Monte Carlo Technique} and \textit{Perturbation Method}
% 
%  \item Obtain complete probabilistic description (PDF): Materials often fail at low
%  probability (tails of PDF)
% 
%  % \begin{itemize}
% 
%  % \item Materials often fail at low probability (tails of PDF)
% 
%  % \end{itemize}
% 
%  \item Carry out sensitivity analysis: Advanced constitutive models are sometimes highly sensitive to
%  fluctuations in material parameters
% 
%  % \begin{itemize}
% 
%  % \item Advanced models are highly sensitive to fluctuations in soil parameters
% 
%  \end{itemize}
% 
%  %\end{itemize}
% 
%  \end{frame}
%  %
% 
% 
% 
% 
% 
\section{Probabilistic ElastoPlasticity}
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\subsection{PEP Formulations}
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\begin{frame}
\frametitle{Uncertainty Propagation through Constitutive Eq.}
%
\begin{itemize}
\item Incremental elpl constitutive equation
$\displaystyle \frac{d\sigma_{ij}}{dt} = D_{ijkl} \displaystyle \frac{d\epsilon_{kl}}{dt}$
%\begin{normalsize}
%
% \begin{equation}
% \nonumber
% \frac{d\sigma_{ij}}{dt} = D_{ijkl} \frac{d\epsilon_{kl}}{dt}
% \end{equation}
\begin{eqnarray}
\nonumber
D_{ijkl} = \left\{\begin{array}{ll}
%
D^{el}_{ijkl}
%
%
\;\;\; & \mbox{\large{~for elastic}} \\
%
\\
%
D^{el}_{ijkl}

\frac{\displaystyle D^{el}_{ijmn} m_{mn} n_{pq} D^{el}_{pqkl}}
{\displaystyle n_{rs} D^{el}_{rstu} m_{tu}  \xi_* r_*}
\;\;\; & \mbox{\large{~for elasticplastic}}
%
\end{array} \right.
\end{eqnarray}
%\end{normalsize}
%\vspace{0.5cm}
% \item Nonlinear coupling in the ElPl modulus
% \item Focus on 1D $\rightarrow$ a nonlinear ODE with random coefficient and random forcing
%
%
%
% \begin{eqnarray}
% \nonumber
% \frac{d\sigma(x,t)}{dt} &=& \beta(\sigma(x,t),D^{el}(x),q(x),r(x);x,t) \frac{d\epsilon(x,t)}{dt} \\
% \nonumber
% &=& \eta(\sigma,D^{el},q,r,\epsilon; x,t) \mbox{\ \ \ \ with an I.C. $\sigma(0)=\sigma_0$}
% \end{eqnarray}
%
\end{itemize}
%
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\end{frame}
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\begin{frame}
\frametitle{Previous Work}
\begin{itemize}
\item
Linear algebraic or differential equations $\rightarrow$ Analytical solution:
\begin{itemize}
\item Variable Transf. Method (Montgomery and Runger 2003)
\item Cumulant Expansion Method (Gardiner 2004)
\end{itemize}
\item
Nonlinear differential equations (elastoplastic/viscoelasticviscoplastic):
\begin{itemize}
\item Monte Carlo Simulation (Schueller 1997, De Lima et al 2001, Mellah
et al. 2000, Griffiths et al. 2005...) \\ $\rightarrow$ accurate, very costly
\item Perturbation Method (Anders and Hori 2000, Kleiber and Hien 1992,
Matthies et al. 1997) \\ $\rightarrow$ first and second order Taylor series
expansion about mean  limited to problems with small C.O.V. and inherits
"closure problem"
\end{itemize}
%
% \item
% Monte Carlo method: accurate, very costly
%
% \item
% Perturbation method:
\end{itemize}
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\end{frame}
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\begin{frame}
\frametitle{Problem Statement}
\begin{itemize}
\item Incremental 3D elasticplastic stressstrain:
%
%
\begin{equation}
\nonumber
\frac{ 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 \}
\frac{ d\epsilon_{kl}}{d t}
\end{equation}
\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}
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\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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\begin{frame}
\frametitle{EulerianLagrangian FPK Equation}
%
%\begin{itemize}
\begin{footnotesize}
%\noindent
van Kampen's Lemma (van Kampen 1976) $\rightarrow$ $ <\rho(\sigma,t)>=P(\sigma,t) $,
ensemble average of phase density
%(in stress space here)
is the probability density;
\begin{eqnarray}
\nonumber
&&\displaystyle \frac{\partial P(\sigma(x_t,t), t)}{\partial t}=
 \displaystyle \frac{\partial}{\partial \sigma} \left[ \left\{\left< \vphantom{\int_{0}^{t} d\tau} \eta(\sigma(x_t,t), D^{el}(x_t),
q(x_t), r(x_t), \epsilon(x_t,t)) \right> \right. \right. \\
\nonumber
&+& \left. \left. \int_{0}^{t} d\tau Cov_0 \left[ \displaystyle \frac{\partial \eta(\sigma(x_t,t), D^{el}(x_t), q(x_t), r(x_t),
\epsilon(x_t,t))}{\partial \sigma}; \right. \right. \right. \\
\nonumber
& & \left. \left. \left. \eta(\sigma(x_{t\tau},t\tau), D^{el}(x_{t\tau}), q(x_{t\tau}), r(x_{t\tau}),
\epsilon(x_{t\tau},t\tau) \vphantom{\int_{0}^{t} d\tau} \right] \right \} P(\sigma(x_t,t),t) \right] \\
\nonumber
&+& \displaystyle \frac{\partial^2}{\partial \sigma^2} \left[ \left\{ \int_{0}^{t} d\tau Cov_0 \left[ \vphantom{\int_{0}^{t}}
\eta(\sigma(x_t,t), D^{el}(x_t), q(x_t), r(x_t), \epsilon(x_t,t)); \right. \right. \right. \\
\nonumber
& & \left. \left. \left. \eta (\sigma(x_{t\tau},t\tau), D^{el}(x_{t\tau}), q(x_{t\tau}), r(x_{t\tau}),
\epsilon(x_{t\tau},t\tau)) \vphantom{\int_{0}^{t}} \right] \vphantom{\int_{0}^{t}} \right\} P(\sigma (x_t,t),t) \right] \\
\nonumber
\end{eqnarray}
\end{footnotesize}
\end{frame}
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%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  \begin{frame}
% 
%  \frametitle{EulerianLagrangian Format}
% 
% 
%  \vspace*{0.5cm}
% 
% 
%  \begin{itemize}
% 
%  % \item $Cov_0[\cdot]$ $\rightarrow$ Timeordered covariance function
%  % %
%  % %
%  % \begin{eqnarray}
%  % \nonumber
%  % &&Cov_0[\eta(x,t_1);\eta(x,t_2)] \\
%  % && \ \ \ \ \ \ \ \ \ \ \ \ \ \ = \left < \eta(x,t_1)\eta(x,t_2) \right >  \left < \eta(x,t_1) \right > \cdot \left < \eta(x,t_2) \right >
%  % \nonumber
%  % \end{eqnarray}
%  %
%  \vspace*{0.5cm}
%  \item Realspace location (Lagrangian) $x_t$ is known but pullback to Eulerian location $x_{t\tau}$ is unknown
% 
%  \vspace*{0.5cm}
%  \item Can be related using strain rate $\dot \epsilon \ (=d\epsilon/dt)$
% 
% 
%  \begin{equation}
%  \nonumber
%  d\epsilon = \dot \epsilon \tau =\frac{x_tx_{t\tau}}{x_t} \mbox{; \ \ \ \ or, \ \ \ } x_{t\tau}=(1\dot \epsilon \tau)x_t
%  \end{equation}
% 
%  \end{itemize}
% 
% 
%  \end{frame}
% 
% 
% 
% 
% 
% 
%  %
%  % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame} \frametitle{Equivalent It{\^o} Stochastic Differential Equation}
%
% \begin{itemize}
%
%
%
% \item Equivalency between It{\^o} stochastic differential equation and FPK equation (Gardiner 2004):
%
%
%
% \end{itemize}
%
%
%
%
%
%
% \begin{normalsize}
%
% \begin{eqnarray}
% \nonumber
% \lefteqn{d\sigma (x,t)=
% \left\{ \left< \vphantom{\displaystyle \frac{\partial \eta_1 (\sigma(x_t,t), A(x_t,t), \epsilon(x_t,t))}{\partial
% \sigma}} \eta (\sigma(x_t,t), D(x_t), q(x_t), r(x_t), \epsilon(x_t,t)) \right> \right.} \\
% \nonumber
% &+& \left. \int_0^t d \tau Cov_0 \left[\displaystyle \frac{\partial \eta (\sigma(x_t,t), D(x_t), q(x_t), r(x_t), \epsilon(x_t,t))}
% {\partial \sigma}; \right. \right. \\
% \nonumber
% & & \left. \left. \eta (\sigma (x_{t\tau},t\tau), D(x_{t\tau}), q(x_{t\tau}), r(x_{t\tau}), \epsilon (x_{t\tau},
% t\tau)) \vphantom{\displaystyle \frac{\partial \eta_1 (\sigma(x_t,t), A(x_t,t), \epsilon(x_t,t))}{\partial \sigma}}\right] \right\} dt
% + b(\sigma,t)dW(t)
% \end{eqnarray}
% %
% \noindent \flushleft{where},
% %
% \begin{eqnarray}
% \nonumber
% b^2(\sigma,t) &=& 2 \int_0^t d \tau Cov_0 \left[\vphantom{\int_0^t d \tau} \eta(\sigma(x_t,t), D(x_t), q(x_t), r(x_t),
% \epsilon(x_t,t)); \right. \\
% \nonumber
% & & \left. \eta(\sigma (x_{t\tau},t\tau), D (x_{t\tau}), q(x_{t\tau}), r(x_{t\tau}), \epsilon
% (x_{t\tau},t\tau))
% \vphantom{\int_0^t d \tau} \right]
% \end{eqnarray}
%
% \end{normalsize}
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% \end{frame}
%
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame} \frametitle{{3D FPK Equation}}
%
% \begin{footnotesize}
%
% \begin{eqnarray}
% \nonumber
% \lefteqn{\displaystyle \frac{\partial P(\sigma_{ij}(x_t,t), t)}{\partial t} = \displaystyle \frac{\partial}{\partial \sigma_{mn}}
% \left[ \left\{\left< \vphantom{\int_{0}^{t}} \eta_{mn}(\sigma_{mn}(x_t,t), D_{mnrs}(x_t), \epsilon_{rs}(x_t,t))\right> \right. \right.} \\
% \nonumber
% &+& \left. \left. \int_{0}^{t} d\tau Cov_0 \left[\displaystyle \frac{\partial \eta_{mn}(\sigma_{mn}(x_t,t), D_{mnrs}(x_t),
% \epsilon_{rs}(x_t,t))} {\partial \sigma_{ab}}; \right. \right. \right. \\
% \nonumber
% & & \left. \left. \left. \eta_{ab} (\sigma_{ab}(x_{t\tau}, t\tau), D_{abcd}(x_{t\tau}), \epsilon_{cd}(x_{t\tau}, t\tau)
% \vphantom{\int_{0}^{t}} \right] \right \} P(\sigma_{ij}(x_t,t),t) \right] \\
% \nonumber
% &+& \displaystyle \frac{\partial^2}{\partial \sigma_{mn} \partial \sigma_{ab}} \left[ \left\{ \int_{0}^{t} d\tau Cov_0 \left[
% \vphantom{\int_{0}^{t}} \eta_{mn}(\sigma_{mn}(x_t,t), D_{mnrs}(x_t), \epsilon_{rs}(x_t,t)); \right. \right. \right. \\
% \nonumber
% & & \left. \left. \left. \eta_{ab} (\sigma_{ab}(x_{t\tau}, t\tau), D_{abcd}(x_{t\tau}), \epsilon_{cd}(x_{t\tau}, t\tau))
% \vphantom{\int_{0}^{t}} \right] \vphantom{\int_{0}^{t}} \right\} P(\sigma_{ij}(x_t,t),t) \right]
% \end{eqnarray}
%
%
% \end{footnotesize}
%
%
%
% % \begin{itemize}
% %
% %
% %
% % \item 6 equations
% %
% % \item Complete description of 3D probabilistic stressstrain response
% %
% % \end{itemize}
% %
% %
%
%
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%
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\begin{frame}
\frametitle{EL 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
%\vspace*{0.2truecm}
\end{itemize}
%
% \vspace*{0.5cm}
% {%
% \begin{beamercolorbox}{section in head/foot}
% \usebeamerfont{framesubtitle}\tiny{B. Jeremi\'{c}, K. Sett, and M. L. Kavvas, "Probabilistic
% ElastoPlasticity: Formulation in 1D", \textit{Acta Geotechnica}, Vol. 2, No. 3, 2007, In press (published
% online in the \textit{Online First} section)}
% %\vskip2pt\insertnavigation{\paperwidth}\vskip2pt
% \end{beamercolorbox}%
% }
\end{frame}
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\begin{frame}
\frametitle{Template Solution of FPK Equation}
\begin{itemize}
\item FPK diffusionadvection equation is applicable to any material model $\rightarrow$
only the coefficients $N_{(1)}$ and $N_{(2)}$ are different for different material models
%
%
%
%\begin{normalsize}
\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]
%\nonumber
= \frac{\partial \zeta}{\partial \sigma}
\end{equation}
%\end{normalsize}
%
\item Initial condition
\begin{itemize}
\item Deterministic $\rightarrow$ Dirac delta function $\rightarrow$ $ P(\sigma,0)=\delta(\sigma) $
\item Random $\rightarrow$ Any given distribution
\end{itemize}
\item Boundary condition: Reflecting BC $\rightarrow$ conserves probability mass
$\zeta(\sigma,t)_{At \ Boundaries}=0$
\item Finite Differences used for solution (among many others)
\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}
% %\vskip2pt\insertnavigation{\paperwidth}\vskip2pt
% \end{beamercolorbox}%
% }
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%==\begin{frame} \frametitle{Solution of FPK Equation: Numerical Scheme}
%==
%==
%==
%==\begin{itemize}
%==
%==
%== \item \textit{Method of Lines} $\rightarrow$ semidiscretization of stress domain by \textit{Finite Difference Technique}
%==
%== \begin{itemize}
%==
%== \item Has inherent drawbacks $\rightarrow$ nor very efficient
%==
%==
%==\begin{center}
%== \begin{figure}[!hbpt]
%== \nonumber
%==% \begin{center}
%==% \begin{flushleft}
%==% \begin{center}
%== \includegraphics[width=10cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/AnticipatedInfluence.jpg}
%== \includegraphics[width=16.0cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/NumericalScheme01.jpg}
%==
%==% \end{flushleft}
%==% \begin{flushright}
%==\includegraphics[height=13cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourLowOCR_RandomGm.pdf}
%==
%==\nonumber
%==\caption{Low OCR Cam Clay Response with Random $G$}
%=={\includegraphics[width=10.5cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/QualSchematics4.jpg}}
%==\includegraphics[width=10cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/AnticipatedInfluence.jpg}
%==% \end{flushleft}
%==
%==% \end{flushright}
%==%
%==%\begin{flushright}
%==
%==\begin{itemize}
%==
%==\* Adaptive technique \\
%==
%==\ \\
%==
%==\* Krylov subspace \ \ \ \ \ \\
%==
%==\ \\
%==
%==\* Meshfree technique
%==
%==\end{itemize}
%==
%==\end{flushright}
%==
%==\end{center}
%==\end{figure}
%==
%==\item Possible improvements through:
%==
%==\begin{itemize}
%==
%==\item Adaptive techniques (FEM, meshfree etc.)
%==
%==\item Krylov subspace technique (reducedorder modeling)
%==
%==\end{itemize}
%==
%==\end{itemize}
%==
%==\end{itemize}
%==
%==
%==
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\begin{frame}
\frametitle{Application of FPK equation to Material Models}
\begin{itemize}
\vspace*{0.5cm}
\item FPK equation is applicable to any incremental elasticplastic material
model
%\item Only the coefficients $N_{(1)}$ and $N_{(2)}$ differ
\vspace*{0.5cm}
\item Solution in terms of PDF, not a single value of stress
\vspace*{0.5cm}
\item Influence of initial condition on the PDF of stress
\vspace*{0.5cm}
\item Mean stress yielding or
\vspace*{0.5cm}
\item Probabilistic yielding
% \item Differences in mean, mode and deterministic solution of stress
\end{itemize}
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\end{frame}
\subsection{Probabilistic ElasticPlastic Response}
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\begin{frame}
\frametitle{Elastic Response with Random $G$}
\begin{itemize}
\item General form of elastic constitutive rate equation
\begin{eqnarray}
\nonumber
\frac{d \sigma_{12}}{dt} &=& 2G \frac{d \epsilon_{12}}{dt} \\
\nonumber
&=& \eta(G, \epsilon_{12};t)
\end{eqnarray}
\item Advection and diffusion coefficients of FPK equation
\begin{equation}
\nonumber
N_{(1)}=2\frac{d \epsilon_{12}}{dt}
\end{equation}
\begin{equation}
\nonumber
N_{(2)}=4t\left(\displaystyle \frac{d \epsilon_{12}}{dt} \right)^2 Var[G]
\end{equation}
\end{itemize}
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\begin{frame}
\frametitle{Elastic Response with Random $G$}
\begin{figure}[!hbpt]
%\nonumber
%\begin{flushleft}
\begin{center}
%\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
\includegraphics[height=5.5cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/PDFElastic_RandomGm.pdf}
%
%\end{flushleft}
%\begin{flushright}
%\includegraphics[height=9cm]{ReasonOfNotMatchingMonteCarlom.pdf}
%
%\nonumber
%\caption{DruckerPrager Associative Linear Hardening Response with Random $G$}
%{\includegraphics[width=10.5cm]{QualSchematics4.jpg}}
%\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
%\end{flushright}
\end{center}
\end{figure}
$$ = 2.5 MPa;
Std. Deviation$[G]$ = 0.5 MPa
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\begin{frame}
\frametitle{Verification  Variable Transformation Method}
\begin{figure}[!hbpt]
\begin{center}
\includegraphics[height=6cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/MonteCarlo_Elasticm.pdf}
% %
% \hspace*{3cm}
% \vspace*{4cm}
% %
% \includegraphics[height=4cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ReasonOfNotMatchingMonteCarlom.pdf}
\end{center}
\end{figure}
%
% \vspace*{4cm}
%
% \begin{flushleft}
% Effect of Approximation of I.C on the \\ PDF of Stress at 0.0426 \% Strain
% \end{flushleft}
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%  \begin{frame}
% 
%  \frametitle{DruckerPrager Linear Hardening with Random $G$}
% 
% 
% 
%  %
%  %
%  %
%  % General form of DruckerPrager elasticplastic linear hardening constitutive rate equation
%  %
%  %
% 
%  \begin{eqnarray}
%  \nonumber
%  \frac{d \sigma_{12}}{dt} =
%  G^{ep} \frac{d \epsilon_{12}}{dt} =
%  \eta(\sigma_{12}, G, K, \alpha, \alpha', \epsilon_{12};t)
%  \end{eqnarray}
% 
% 
% 
%  \noindent
%  Advection and diffusion coefficients of FPK equation
% 
% 
% 
%  \begin{equation}
%  \nonumber
%  N_{(1)}=\displaystyle \frac{d \epsilon_{12}}{dt} \left< 2G  \displaystyle \frac{G^2}{G+9K\alpha^2+\displaystyle \frac{1}{\sqrt{3}}I_1
%  \alpha'} \right>
%  \end{equation}
% 
% 
% 
%  \begin{equation}
%  \nonumber
%  N_{(2)}=t\left(\displaystyle \frac{d \epsilon_{12}}{dt} \right)^2 Var\left[2G  \displaystyle \frac{G^2}{G+9K\alpha^2+\displaystyle
%  \frac{1}{\sqrt{3}}I_1 \alpha'} \right]
%  \end{equation}
% 
%  % \end{itemize}
% 
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% 
% 
%  \end{frame}
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% 
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%  %  \begin{frame}
%  % 
%  %  \frametitle{{DruckerPrager Linear Hardening \\ with Random $G$}}
%  % 
%  %  \vspace*{1.5cm}
%  % 
%  %  \begin{figure}[!hbpt]
%  %  \begin{center}
%  %  \includegraphics[height=3.5cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/PDFDruckerPrager_RandomGm.pdf}
%  %  \hfill
%  %  \includegraphics[height=5.0cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourDruckerPrager_RandomGm.pdf}
%  %  \end{center}
%  %  \end{figure}
%  % 
%  % 
%  % 
%  % 
%  %  \begin{itemize}
%  % 
%  %  \item Mean stress yielding
%  % 
%  %  \item Approximation of I.C.
%  % 
%  % 
%  %  \item Smooth transition between el. \& el.pl.
%  % 
%  % 
%  %  \item Symmetry in probability distribution
%  % 
%  %  \end{itemize}
%  % 
%  %  %
%  %  % \begin{flushright}
%  %  % \includegraphics[height=13cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourDruckerPrager_RandomGm.pdf}
%  %  % %
%  %  % \nonumber
%  %  % %\caption{DruckerPrager Associative Linear Hardening Response with Random $G$}
%  %  % %{\includegraphics[width=10.5cm]{QualSchematics4.jpg}}
%  %  % %\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
%  %  % %\end{flushleft}
%  %  % \end{flushright}
%  %  % \end{figure}
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% 
% 
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%  \begin{frame}
% 
%  \frametitle{Verification of DP EP Response  Monte Carlo}
% 
% 
%  \begin{figure}[!hbpt]
%  %\nonumber
%  %\begin{flushleft}
%  \begin{center}
%  %\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
%  \includegraphics[height=6cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/MonteCarlo_DruckerPragerm.pdf}
%  %
%  %\end{flushleft}
%  %\begin{flushright}
%  %\includegraphics[height=11.5cm]{ContourDruckerPrager_RandomGm.pdf}
%  %
%  %\nonumber
%  %\caption{DruckerPrager Associative Linear Hardening Response with Random $G$}
%  %{\includegraphics[width=10.5cm]{QualSchematics4.jpg}}
%  %\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
%  %\end{flushleft}
%  %\end{flushright}
%  \end{center}
%  \end{figure}
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\begin{frame}
\frametitle{Modified Cam Clay Constitutive Model}
\begin{small}
\begin{equation}
\nonumber
\frac{d \sigma_{12}}{dt} = G^{ep} \frac{d \epsilon_{12}}{dt} = \eta(\sigma_{12}, G, M, e_0, p_0, \lambda, \kappa, \epsilon_{12};t)
\end{equation}
%
%
%
%
%
\begin{equation}
\nonumber
\eta = \left[2G  \displaystyle \frac{\left(36 \displaystyle \frac{G^2}{M^4} \right) \sigma_{12}^2}
{\displaystyle \frac{(1+e_0)p(2pp_0)^2}{\kappa} + \left(18 \displaystyle \frac{G}{M^4}\right) \sigma_{12}^2
+ \displaystyle \frac{1+e_0}{\lambda\kappa} p p_0 (2pp_0)} \right]
\end{equation}
\end{small}
\noindent
Advection and diffusion coefficients of FPK equation
\begin{equation}
\nonumber
N_{(1)}^{(i)}=\left<\eta^{(i)}(t)\right> + \int_0^t d\tau cov\left[\displaystyle \frac{\partial \eta^{(i)}(t)}{\partial t};
\eta^{(i)} (t\tau)\right]
\end{equation}
\begin{equation}
\nonumber
N_{(2)}^{(i)} = \int_0^t d\tau cov\left[\eta^{(i)}(t); \eta^{(i)} (t\tau)\right]
\end{equation}
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%  \begin{frame}
% 
%  \frametitle{Low OCR Cam Clay with \\ Random $G$}
% 
%  \vspace*{1.5cm}
% 
%  \begin{figure}[!hbpt]
%  %\nonumber
%  \begin{center}
%  \includegraphics[height=3.5cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/PDFLowOCR_RandomGm.pdf}
%  \hfill
%  \includegraphics[height=5.4cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourLowOCR_RandomGm.pdf}
%  \end{center}
%  \end{figure}
% 
%  \begin{itemize}
% 
%  \item Approximation of I.C.
% 
% 
%  %\item Wide transition between el. \& el.pl.
% 
% 
%  \item Nonsymmetry in probability distribution!
% 
% 
%  \item Response at critical state fairly certain but different than deterministic
% 
%  \end{itemize}
% 
% 
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% 
% 
% 
%  \end{frame}
% 
% 
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\begin{frame}
\frametitle{{Low OCR Cam Clay with Random $G$, $M$ and $p_0$}}
\begin{itemize}
%\item Narrow transition between el. \& el.pl.
\vspace*{0.3cm}
\item Nonsymmetry in \\
probability \\
distribution
\vspace*{0.3cm}
\item Difference \\
between \\
mean, mode and \\
deterministic
\vspace*{0.3cm}
\item Divergence at \\
critical state \\
because $M$ is \\
uncertain
\end{itemize}
\vspace*{6.3cm}
%\hspace*{0.5cm}
\begin{figure}[!hbpt]
\begin{flushright}
%\includegraphics[height=3.5cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/PDFLowOCR_RandomG_RandomM_Randomp0m.pdf}
%\hfill
\includegraphics[height=6.5cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourLowOCR_RandomG_RandomM_Randomp0m.pdf}
\end{flushright}
\end{figure}
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% 
%  \begin{frame}
% 
%  \frametitle{{Low OCR Cam Clay with \\ Random $G$, $M$ and $p_0$}}
% 
%  \vspace*{1.5cm}
% 
%  \begin{figure}[!hbpt]
%  \begin{center}
%  \includegraphics[height=3.5cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/PDFLowOCR_RandomG_RandomM_Randomp0m.pdf}
%  \hfill
%  \includegraphics[height=5.5cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourLowOCR_RandomG_RandomM_Randomp0m.pdf}
%  \end{center}
%  \end{figure}
% 
% 
% 
%  \vspace*{0.5cm}
% 
%  \begin{itemize}
% 
%  %\item Narrow transition between el. \& el.pl.
% 
% 
%  \item Nonsymmetry in probability distribution
% 
% 
%  \item Difference between mean, mode and deterministic
% 
% 
% 
%  \item Divergence at critical state because $M$ is uncertain
% 
%  \end{itemize}
% 
% 
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% 
%  \end{frame}
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\begin{frame}
\frametitle{Comparison of Low OCR Cam Clay at $\epsilon$ = 1.62 \%}
%\vspace*{4.50cm}
\begin{figure}[!hbpt]
\begin{center}
%\includegraphics[height=14cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/CamClayPDFComparisonm.pdf}
\includegraphics[height=5.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Present/LowOCR_ComparisonPDFm_ps.pdf}
\end{center}
\end{figure}
\vspace*{0.5cm}
\begin{itemize}
\item None coincides with deterministic
\item Some very uncertain, some very certain
\item Either on safe or unsafe side
\end{itemize}
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\begin{frame}
\frametitle{{High OCR Cam Clay with Random $G$ and $M$}}
\begin{itemize}
%\item Approximation of I.C.
%
%\item Very uncertain transition between el. \& el.pl.
\vspace*{0.5cm}
\item Large nonsymmetry \\
in probability \\
distribution
\vspace*{0.5cm}
\item Significant \\
differences in \\
mean, mode, \\
and deterministic
\vspace*{0.5cm}
\item Divergence at \\
critical state, \\
$M$ is uncertain
\end{itemize}
\vspace*{6.5cm}
%
\begin{figure}[!hbpt]
\begin{flushright}
%\includegraphics[height=3.4cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/PDFHighOCR_RandomG_RandomMm.pdf}
%\hfill
\includegraphics[height=6.5cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourHighOCR_RandomG_RandomMm.pdf}
\end{flushright}
\end{figure}
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\begin{frame}
\frametitle{Probabilistic Yielding}
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\begin{itemize}
%
\item Weighted elastic and elasticplastic Solution
${\partial P(\sigma,t)}/{\partial t}
=
{\partial \left(N^w_{(1)}P(\sigma,t)
{\partial \left(N^w_{(2)} P(\sigma,t)\right) }/{\partial \sigma} \right)}/
{\partial \sigma}$
%\vspace*{0.5cm}
\item Weighted advection and diffusion coefficients are then
$N_{(1,2)}^{w} (\sigma)
=
(1  P[\Sigma_y \leq \sigma]) N_{(1)}^{el} + P[\Sigma_y \leq \sigma] N_{(1)}^{elpl} $
%\vspace*{0.5cm}
\item
%\vspace*{10cm}
Cumulative Probability Density function (CDF) of the yield function
\vspace*{0.5cm}
\begin{figure}[!h]
%\hspace*{7cm}
\includegraphics[width=4cm]{/home/jeremic/tex/works/Papers/2007/ProbabilisticYielding/vonMises_YieldCDF_Combinededited.pdf}
\end{figure}
\vspace*{0.5cm}
\end{itemize}
% %\vspace*{0.5cm}
% {%
% \begin{beamercolorbox}{section in head/foot}
% \usebeamerfont{framesubtitle}\tiny{B. Jeremi{\'c} and K. Sett. On Probabilistic
% Yielding of Materials. in review in \textit{Communications in Numerical Methods in
% Engineering}, 2007.}
% %\vskip2pt\insertnavigation{\paperwidth}\vskip2pt
% \end{beamercolorbox}%
% }
\end{frame}
%
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\begin{frame}
\frametitle{Transformation of a BiLinear (von Mises) Response}
\begin{figure}[!hbpt]
\begin{center}
\includegraphics[width=7cm]{/home/jeremic/tex/works/Papers/2007/ProbabilisticYielding/figures/vonMises_G_and_cu_very_uncertain/Contour_PDFedited.pdf}
\end{center}
\end{figure}
\vspace*{0.3cm}
linear elastic  linear hardening plastic von Mises
\end{frame}
%
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\begin{frame}
\frametitle{SPT Based Determination of Shear Strength}
\begin{figure}[!hbpt]
\begin{center}
%
\includegraphics[width=5.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGEGoverGmax/figures/ShearStrength_RawData_and_MeanTrendMod.pdf}
\hfill
\includegraphics[width=5.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGEGoverGmax/figures/ShearStrength_Histogram_PearsonIVFineTunedMod.pdf}
%
\end{center}
\end{figure}
\vspace*{0.3cm}
Transformation relationship between SPT $N$value and undrained shear
strength, $s_u$ (cf. Phoon and Kulhawy (1999B)
Histogram of the residual (w.r.t the deterministic transformation
equation) undrained strength, along with fitted probability density function
\end{frame}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{SPT Based Determination of Young's Modulus}
\begin{figure}[!hbpt]
\begin{center}
%
\includegraphics[width=5.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGEGoverGmax/figures/YoungModulus_RawData_and_MeanTrend_01Ed.pdf}
\hfill
\includegraphics[width=5.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGEGoverGmax/figures/YoungModulus_Histogram_Normal_01Ed.pdf}
%
\end{center}
\end{figure}
\vspace*{0.3cm}
Transformation relationship between SPT $N$value and pressuremeter Young's modulus, $E$ (cf. Phoon and Kulhawy (1999B))
Histogram of the residual (w.r.t the deterministic transformation equation) Young's modulus, along with fitted probability density function
\end{frame}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Cyclic Response of Such Uncertain Material}
\begin{figure}[!hbpt]
\begin{center}
%
\includegraphics[width=5.15truecm]{/home/jeremic/tex/works/Papers/2008/JGGEGoverGmax/figures/meanStressStrainPlot_1Point026Mod.pdf}
\hfill
\includegraphics[width=4.35truecm]{/home/jeremic/tex/works/Papers/2008/JGGEGoverGmax/figures/sdStressStrainPlot_1Point026Mod.pdf}
%
\end{center}
\end{figure}
\vspace*{0.3cm}
\end{frame}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{$G/G_{max}$ Response}
\begin{figure}[!hbpt]
\begin{center}
%
\includegraphics[width=10.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGEGoverGmax/figures/GoverGmaxMod.pdf}
%\hfill
%\includegraphics[width=6.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGEGoverGmax/figures/DampingMod.pdf}
%
\end{center}
\end{figure}
\vspace*{0.3cm}
\end{frame}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Damping Response}
\begin{figure}[!hbpt]
\begin{center}
%
%\includegraphics[width=6.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGEGoverGmax/figures/GoverGmaxMod.pdf}
%\hfill
\includegraphics[width=10.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGEGoverGmax/figures/DampingMod.pdf}
%
\end{center}
\end{figure}
\vspace*{0.3cm}
\end{frame}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Boundary Value Problem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section[SEPFEM]{Stochastic ElasticPlastic Finite Element Method}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{SEPFEM Formulations}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Governing Equations \& Discretization Scheme}
\begin{itemize}
\item Governing equations in geomechanics:
\begin{equation}
\nonumber
A\sigma = \phi(t);~~~ Bu = \epsilon; ~~~\sigma = D \epsilon
\end{equation}
\item Discretization (spatial and stochastic) schemes
\begin{itemize}
\item Input random field material properties ($D$) $\rightarrow$
KarhunenLo{\`e}ve (KL) expansion, optimal expansion, error minimizing property
\item Unknown solution random field ($u$) $\rightarrow$ Polynomial Chaos (PC)
expansion
\item Deterministic spatial differential operators ($A$ \& $B$) $\rightarrow$
Regular shape function method with Galerkin scheme
\end{itemize}
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
%  \begin{frame}
% 
% 
%  \frametitle{Truncated KarhunenLo{\`e}ve (KL) expansion}
% 
%  \begin{itemize}
% 
%  \item Representation of input random fields in eigenmodes of covariance kernel
% 
%  % \vspace*{0.1cm}
%  % \begin{figure}[!hbpt]
%  \begin{flushleft}
%  \includegraphics[height=3.0cm]{/home/jeremic/tex/works/Conferences/2008/GeoCongress/Probabilistic/Paper/TypicalDataPlotBH1Edited.jpg}
%  % ShearStrengthProfile.jpg}
%  \end{flushleft}
%  \vspace*{3.15cm}
%  \begin{flushright}
%  % \begin{equation}
%  % \nonumber
%  % \begin{normalsize}
%  $ q_T(x,\theta) = \bar q_T(x) + \sum_{n=1}^M \sqrt{\lambda_n} \xi_n(\theta) f_n(x) $ \\
%  \ \\
%  $ \int_D C(x_1, x_2) f (x_2) dx_2 = \lambda f (x_1) \ \ \ \ \ \ \ \ \ \ \ \ $ \\
%  \ \\
%  $ \xi_i(\theta) = \displaystyle \frac{1}{\sqrt \lambda_i} \int_D \left [q_T(x,\theta)  \bar q_T (x) \right] f_i (x) dx $
%  % \end{equation}
%  % \end{normalsize}
%  \end{flushright}
%  % \end{figure}
% 
%  % \vspace{6.0cm}
%  % \begin{flushright}
%  % \begin{equation}
%  % \nonumber
%  % w(x,\theta) = \bar w(x) + \sum_{n=0}^M \sqrt{\lambda_n} \zeta_n(\theta) f_n(x)
%  % \end{equation}
%  % \end{flushright}
% 
%  % \vspace*{0.8cm}
%  \item Error minimizing property
% 
%  \item Optimal expansion $\rightarrow$ minimization of number of stochastic dimensions
% 
% 
%  \end{itemize}
% 
% 
% 
%  \end{frame}
% 
% 
%  %  %  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %  \begin{frame}
%  %  \frametitle{KL Expansion (of Covariance Kernel)}
%  % 
%  %  \begin{flushleft}
%  %  %\begin{center}
%  %  %\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
%  %  \includegraphics[height=2.2cm]{/home/jeremic/tex/works/Conferences/2007/GeoDenver/SFEM/Presentation/ActualExponentialCovarianveSurface.jpg}
%  %  \hspace*{0.3cm}
%  %  \includegraphics[height=2.2cm]{/home/jeremic/tex/works/Conferences/2007/GeoDenver/SFEM/Presentation/KL_ApproxWith_1Term_CovarianveSurface.jpg}
%  %  %\vspace*{1.0cm}
%  %  %\mbox{Exact covariance surface}
%  %  %\vspace*{4.0cm}
%  %  \end{flushleft}
%  %  \vspace*{0.4cm}
%  %  \small{\ \ \ \ \ \ \ \ \ \ \ \ \ Exact \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1 term approx.} \\
%  %  \small{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (8.49\% error)}
%  %  %\begin{flushright}
%  %  %\end{center}
%  %  %\end{figure}
%  % 
%  %  %\vspace*{0.5cm}
%  %  %\small{Exact covariance surface \ \ \ \ \ \ \ \ \ \ \ \ \ \ Oneterm approximation}
%  %  %
%  %  %\vspace*{0.5cm}
%  %  %
%  %  %\begin{figure}[!hbpt]
%  %  %\begin{center}
%  %  \begin{flushleft}
%  %  \includegraphics[height=2.2cm]{/home/jeremic/tex/works/Conferences/2007/GeoDenver/SFEM/Presentation/KL_ApproxWith_2Terms_CovarianveSurface.jpg}
%  %  \hspace*{0.2cm}
%  %  \includegraphics[height=2.2cm]{/home/jeremic/tex/works/Conferences/2007/GeoDenver/SFEM/Presentation/KL_ApproxWith_3Terms_CovarianveSurface.jpg}
%  %  \end{flushleft}
%  %  %\vspace*{0.5cm}
%  %  \small{\ \ \ \ \ 2 terms approx. \ \ \ \ \ \ \ \ \ 3 terms approx.} \\
%  %  \small{\ \ \ \ \ \ (1.15\% error) \ \ \ \ \ \ \ \ \ \ \ \ \ (1.13\% error)}
%  %  %
%  %  \vspace{6.5cm}
%  %  \begin{flushright}
%  % 
%  %  \includegraphics[height=2.0cm]{/home/jeremic/tex/works/Reports/2006/SEPFEM/figures/CPT_DataAnalysis_Plots/TypicalAutoCovariancePlotBH1_FiniteScaleEdited.jpg} \hspace*{0.7cm}
%  %  %ShearStrengthProfile.jpg}
%  % 
%  %  covariance function \\
%  %  (exponential): \ \ \ \ \ \\
%  %  %\ \\
%  %  $ C(x_1, x_2) = \sigma^2 e^{ x_1  x_2  /b} $ \hspace*{0.5cm} \\
%  %  \ \\
%  %  \ \\
%  %  KL approximation: \ \ \ \ \ \ \ \ \\
%  %  \ \\
%  %  $ C(x_1, x_2) \ \ \ \ \ \ \ \ \ \ \ \ \ $ \\
%  %  $ = \sum_{k =1}^M \lambda_k f_k(x_1) f_k(x_2) $
%  % 
%  %  \end{flushright}
%  %  %\end{figure}
%  % 
%  %  %\vspace*{2.5cm}
%  %  %\small{Twoterms approximation \ \ \ \ \ \ \ \ \ \ \ \ Threeterms approximation}
%  % 
%  %  \vspace*{4.0cm}
%  %  \begin{center}
%  %  \large{KL Expansion of Covariance Kernel}
%  %  \end{center}
%  % 
%  %  \end{frame}
%  % 
%  % 
%  %  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
%  \begin{frame}
% 
%  \frametitle{Polynomial Chaos (PC) Expansion}
% 
%  %\begin{itemize}
% 
%  %\vspace*{0.35cm}
%  %\item Solution (displacement) random field $\rightarrow$ Can not use KL expansion directly
% 
%  \begin{itemize}
% 
%  \item Covariance kernel is not known a priori
% 
%  \vspace*{0.5cm}
%  \begin{normalsize}
%  \begin{equation}
%  \nonumber
%  u(x,\theta)=\sum_{j=1}^L e_j \chi_j(\theta)b_j(x)
%  \end{equation}
%  \end{normalsize}
% 
%  \vspace*{0.3cm}
%  \item Can be expressed as functional of known random variables and unknown deterministic function
% 
%  \vspace*{0.5cm}
%  \begin{normalsize}
%  \begin{equation}
%  \nonumber
%  u(x,\theta)=\zeta[\xi_i(\theta),x]
%  \end{equation}
%  \end{normalsize}
% 
%  \vspace*{0.5cm}
%  \item Need a basis of known random variables $\rightarrow$ PC expansion
% 
%  \vspace*{0.2cm}
%  \begin{normalsize}
%  \begin{equation}
%  \nonumber
%  \chi_j(\theta)=\sum_{i=0}^P\gamma_i^{(j)}\psi_i\left[\left\{\xi_r\right\}\right]
%  \end{equation}
% 
%  \vspace*{0.5cm}
%  \begin{equation}
%  \nonumber
%  u(x,\theta)=\sum_{j=1}^L \sum_{i=0}^P \gamma_i^{(j)} \psi_i[\{\xi_r\}]e_j b_j(x) = \sum_{i=0}^P \psi_i[\{\xi_r\}] d_i(x)
%  \end{equation}
%  \end{normalsize}
% 
%  % \vspace*{0.6cm}
%  % \item Deterministic coefficients can be found by minimizing norm of error of finite
%  % representation (e.g. using Galerkin scheme)
% 
% 
% 
% 
%  % \end{itemize}
%  \end{itemize}
% 
% 
%  %\end{itemize}
% 
%  \end{frame}
% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Spectral Stochastic ElasticPlastic FEM}
\begin{itemize}
\item Minimizing norm of error of finite representation using Galerkin
technique (Ghanem and Spanos 2003):
\vspace*{0.6truecm}
\begin{flushright}
\begin{equation}
\nonumber
\sum_{n = 1}^N K_{mn} d_{ni} + \sum_{n = 1}^N \sum_{j = 0}^P d_{nj} \sum_{k = 1}^M C_{ijk} K'_{mnk} = \left< F_m \psi_i[\{\xi_r\}] \right >
\end{equation}
\end{flushright}
% \begin{itemize}
%
% \vspace*{0.5cm}
% \item Final eqn.:
%
% \vspace*{0.4cm}
% \begin{flushright}
% \begin{normalsize}
% \begin{equation}
% \nonumber
% \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \sum_{n = 1}^N K_{mn} d_{ni} + \sum_{n = 1}^N \sum_{j = 0}^P d_{nj} \sum_{k = 1}^M C_{ijk} K'_{mnk} = \left< F_m \psi_i[\{\zeta_r\}] \right >
% \end{equation}
% \end{normalsize}
% \end{flushright}
\vspace*{0.5cm}
\begin{equation}
\nonumber
K_{mn} = \int_D B_n \textcolor{mycolor}{D} B_m dV \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ K'_{mnk} = \int_D B_n {\sqrt \lambda_k h_k} B_m dV
\end{equation}
\vspace*{1.0cm}
\begin{equation}
\nonumber
C_{ijk} = \left < \xi_k(\theta) \psi_i[\{\xi_r\}] \psi_j[\{\xi_r\}] \right > \ \ \ \ \ \ \ \ \ \ \ \ F_m = \int_D \phi N_m dV \ \ \ \ \ \ \ \ \ \ \ \
\end{equation}
%\item FokkerPlanckKolmogorov approach based probabilistic constitutive integration
% at Gauss integration points
\end{itemize}
% \noindent Salient Features:
% \begin{itemize}
%
% \item Efficient representation of input random fields into finite number of random
% variables using KLexpansion
%
% \item Representation of (unknown) solution random variables using polynomial chaos of
% (known) input random variables
%
% \item FokkerPlanckKolmogorov approach based probabilistic constitutive integration
% at Gauss integration points
%
% \end{itemize}
%
%% \end{itemize}
%
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Inside SEPFEM}
\begin{itemize}
\item Explicit stochastic elasticplastic finite element computations
\vspace*{0.2cm}
\item FPK probabilistic constitutive integration at Gauss integration points
\vspace*{0.2cm}
\item Increase in (stochastic) dimensions (KL and PC) of the problem
\vspace*{0.2cm}
\item Development of the probabilistic elasticplastic stiffness tensor
\end{itemize}
% \noindent Salient Features:
% \begin{itemize}
%
% \item Efficient representation of input random fields into finite number of random
% variables using KLexpansion
%
% \item Representation of (unknown) solution random variables using polynomial chaos of
% (known) input random variables
%
% \item FokkerPlanckKolmogorov approach based probabilistic constitutive integration
% at Gauss integration points
%
% \end{itemize}
%
%% \end{itemize}
%
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{SEPFEM Verification Example}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{1D Static Pushover Test Example}
\begin{itemize}
\vspace*{0.3cm}
\item Linear elastic model: \\
$ = 2.5$~kPa, \\
$Var[G]=0.15~{\rm kPa}^2$,\\
correlation length for $G=0.3$~m.
\vspace*{0.3cm}
\item Elasticplastic material model,\\
von Mises, linear hardening,\\
$ = 2.5$~kPa, \\
$Var[G]=0.15~{\rm kPa}^2$,\\
correlation length for $G=0.3$~m, \\
$C_u = 5$~kPa, \\
$C^{'}_u = 2$~kPa.
\end{itemize}
%\begin{figure}[!hbpt]
%\begin{center}
\vspace*{6cm}
\hspace*{7.5cm}
\includegraphics[height=2.5cm,angle=90]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Present/ShearBeamModel.pdf}
%\end{center}
%\end{figure}
% %
% \begin{center}
% %\hspace*{1.7cm}
% \includegraphics[height=5.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter8Plots/Chapter8_SchematicStaticProblem.jpg}
% %\hspace*{0.7cm}
% %\includegraphics[height=3.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter8Plots/Chapter8_StaticExample_ElasticEdited.jpg} \\
% %\vspace*{0.1truecm}
% %%\small{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Base Displacement}
% %\includegraphics[height=3.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter8Plots/Chapter8_StaticExample_ElasticPlasticEdited.pdf}
% %\hspace*{0.0cm}
% %\includegraphics[height=3.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter8Plots/PushOverTestPDF.jpg}
% \end{center}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Linear Elastic FEM Verification}
\vspace*{0.5cm}
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.8\textwidth]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter8Plots/Chapter8_StaticExample_ElasticEdited.pdf}
\label{figure:StaticElasticSimulation}
\end{center}
\end{figure}
\vspace*{2.5cm}
Mean and standard deviations of displacement at the top node,\\
linear elastic material model, \\
KLdimension=2, order of PC=2.
%Monte Carlo simulation
%is also shown.
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{SEPFEM verification}
%\vspace*{0.2cm}
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.75\textwidth]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter8Plots/Chapter8_StaticExample_ElasticPlasticEdited.pdf}
\label{figure:StaticElasticPlasticSimulation}
\end{center}
\end{figure}
\vspace*{0.2cm}
Mean and standard deviations of displacement at the top node,\\
von Mises elasticplastic linear hardening material model,\\
KLdimension=2, order of PC=2.
% Monte Carlo
%simulation is also shown}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  \begin{frame}
% 
% 
%  \frametitle{Stochastic Response at the Top}
% 
% 
% 
% 
% 
%  \begin{figure}[!htbp]
%  \begin{center}
%  \includegraphics[width=0.65\textwidth]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter8Plots/Chapter8_StaticExample_PDFComparisonEdited.pdf}
%  \label{figure:StaticProblem_PDFComparison}
%  \end{center}
%  \end{figure}
% 
% 
%  Comparison of PDF of top node displacement.
% 
% 
% 
%  \end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{1D Static Pushover Test Example}
%
%
%
% \begin{figure}[!htbp]
% \begin{center}
% \includegraphics[width=0.35\textwidth]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter8Plots/Chapter8_StaticExample_CorLength_andKL_LargeCorLength_MeanEdited.pdf}
% \includegraphics[width=0.35\textwidth]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter8Plots/Chapter8_StaticExample_CorLength_andKL_LargeCorLength_SDEdited.pdf}
%
% \label{figure:CorLength_andKL_LargeCorLength_Mean}
% \end{center}
% \end{figure}
%
%
%
% Correlation length and KL dimension: Mean and stadard deviation of displacement along depth of the 1D soil column with
% linear elastic material model, having very small variance (COV = 1\%) of shear modulus and very large
% ratio of correlation length of shear modulus to domain length (= 100)}
%
% Correlation length and KL dimension: Standard deviation of displacement along depth of the 1D
% soil column with linear elastic material model, having very small variance (COV = 1\%) of shear modulus
% and very large ratio of correlation length of shear modulus to domain length (= 100)}
%
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{1D Static Pushover Test Example}
%
%
%
% \begin{figure}[!htbp]
% \begin{center}
% \includegraphics[width=0.7\textwidth]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter8Plots/Chapter8_StaticExample_CorLength_andKL_SmallCorLength_MeanEdited.pdf}
% \caption{Correlation length and KL dimension: Mean displacement along depth of the 1D soil column with
% linear elastic material model, having very small variance (COV = 1\%) of shear modulus and very small
% ratio of correlation length of shear modulus to domain length (= 0.0001)}
% \label{figure:CorLength_andKL_SmallCorLength_Mean}
% \end{center}
% \end{figure}
%
%
%
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{1D Static Pushover Test Example}
%
%
%
% \begin{figure}[!htbp]
% \begin{center}
% \includegraphics[width=0.7\textwidth]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter8Plots/Chapter8_StaticExample_CorLength_andKL_SmallCorLength_SDEdited.pdf}
% \caption{Correlation length and KL dimension: Standard deviation of displacement along depth of the 1D
% soil column with linear elastic material model, having very small variance (COV = 1\%) of shear modulus
% and very small ratio of correlation length of shear modulus to domain length (= 0.0001)}
% \label{figure:CorLength_andKL_SmallCorLength_SD}
% \end{center}
% \end{figure}
%
%
%
%
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{1D Static Pushover Test Example}
%
%
%
% \begin{figure}[!htbp]
% \begin{center}
% \includegraphics[width=0.7\textwidth]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter8Plots/Chapter8_StaticExample_Variance_andPC_LargeVariance_MeanEdited.pdf}
% \caption{Variance and order of PC: Mean displacement along depth of the 1D soil column with
% linear elastic material model, having large variance (COV = 20\%) of shear modulus and
% ratio of correlation length of shear modulus to domain length = 0.1}
% \label{figure:Variance_andPC_LargeVariance_Mean}
% \end{center}
% \end{figure}
%
%
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{1D Static Pushover Test Example}
%
%
%
% \begin{figure}[!htbp]
% \begin{center}
% \includegraphics[width=0.7\textwidth]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter8Plots/Chapter8_StaticExample_Variance_andPC_LargVariance_SDEdited.pdf}
% \caption{Variance and order of PC: Standard deviation of displacement along depth of the 1D
% soil column with linear elastic material model, having large variance (COV = 20\%) of shear modulus
% and ratio of correlation length of shear modulus to domain length = 0.1}
% \label{figure:Variance_andPC_LargeVariance_SD}
% \end{center}
% \end{figure}
%
%
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{1D Static Pushover Test Example}
%
%
%
% \begin{figure}[!htbp]
% \begin{center}
% \includegraphics[width=0.7\textwidth]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter8Plots/Chapter8_StaticExample_Variance_andPC_SmallVariance_MeanEdited.pdf}
% \caption{Variance and order of PC: Mean displacement along depth of the 1D soil column with
% linear elastic material model, having very small variance (COV = 1\%) of shear modulus and
% ratio of correlation length of shear modulus to domain length = 0.1}
% \label{figure:Variance_andPC_SmallVariance_Mean}
% \end{center}
% \end{figure}
%
%
%
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{1D Static Pushover Test Example}
%
%
%
% \begin{figure}[!htbp]
% \begin{center}
% \includegraphics[width=0.7\textwidth]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter8Plots/Chapter8_StaticExample_Variance_andPC_SmallVariance_SDEdited.pdf}
% \caption{Variance and order of PC: Standard deviation of displacement along depth of the 1D
% soil column with linear elastic material model, having very small variance (COV = 1\%) of shear modulus
% and ratio of correlation length of shear modulus to domain length = 0.1}
% \label{figure:Variance_andPC_SmallVariance_SD}
% \end{center}
% \end{figure}
%
% % \begin{table}
% % \begin{center}
% % \caption{Comparison of results (at top node) of FPKEbased spectral stochastic finite element with
% % direct spectral stochastic finite element, for 1D soil column example, with linear elastic material}
% % \label{Table:Comparison}
% % \begin{tabular}{ccccccc}
% % \hline
% % Load Step & \multicolumn{3}{c}{Mean of solution} & \multicolumn{3}{c}{Standard deviation of solution} \\
% % \cline{27}
% % & Direct (mm) & FPKE (mm) & Error (\%) & Direct (mm) & FPKE (mm) & Error (\%) \\
% % \hline \hline
% % 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
% % \hline
% % 1 & 2.015 & 2.015 & 0 & 0.2005 & 0.2005 & 0 \\
% % \hline
% % 2 & 4.03 & 4.00 & 0.7 & 0.40 & 0.39 & 0.7 \\
% % \hline
% % 3 & 6.04 & 5.95 & 1.4 & 0.60 & 0.59 & 1.4 \\
% % \hline
% % 4 & 8.06 & 7.89 & 2.1 & 0.80 & 0.78 & 2.1 \\
% % \hline
% % 5 & 10.00 & 9.84 & 1.7 & 1.00 & 0.98 & 1.7 \\
% % \hline
% % 6 & 12.09 & 11.88 & 1.7 & 1.20 & 1.21 & 0.6 \\
% % \hline
% % \end{tabular}
% % \end{center}
% % \end{table}
%
%
% %\begin{figure}[htbp]
% %\begin{center}
% %\includegraphics[width=\textwidth]{EffectOfCorLength_6_3_01}
% %\caption{}
% %\label{figure:EffectOfKL_on_CorLength_Mean}
% %\end{center}
% %\end{figure}
% %
% %
% %\begin{figure}[htbp]
% %\begin{center}
% %\includegraphics[width=\textwidth]{EffectOfCorLength_6_3_02}
% %\caption{}
% %\label{figure:EffectOfKL_on_CorLength_Mean}
% %\end{center}
% %\end{figure}
% %
% %\begin{figure}[htbp]
% %\begin{center}
% %\includegraphics[width=\textwidth]{EffectOfCorLength_6_3_03}
% %\caption{}
% %\label{figure:EffectOfKL_on_CorLength_Mean}
% %\end{center}
% %\end{figure}
% %
% %\begin{figure}[htbp]
% %\begin{center}
% %\includegraphics[width=\textwidth]{EffectOfCorLength_6_3_04}
% %\caption{}
% %\label{figure:EffectOfKL_on_CorLength_Mean}
% %\end{center}
% %\end{figure}
% %
% %
% %\begin{figure}[htbp]
% %\begin{center}
% %\includegraphics[width=\textwidth]{EffectOfCorLength_4_4_01}
% %\caption{}
% %\label{figure:EffectOfKL_on_CorLength_Mean}
% %\end{center}
% %\end{figure}
% %
% %\begin{figure}[htbp]
% %\begin{center}
% %\includegraphics[width=\textwidth]{EffectOfCorLength_4_4_02}
% %\caption{}
% %\label{figure:EffectOfKL_on_CorLength_Mean}
% %\end{center}
% %\end{figure}
% %
% %
% %\begin{figure}[htbp]
% %\begin{center}
% %\includegraphics[width=\textwidth]{EffectOfCorLength_4_4_03}
% %\caption{}
% %\label{figure:EffectOfKL_on_CorLength_Mean}
% %\end{center}
% %\end{figure}
% %
% %
% %\begin{figure}[htbp]
% %\begin{center}
% %\includegraphics[width=\textwidth]{EffectOfKL_on_CorLength_Mean}
% %\caption{}
% %\label{figure:EffectOfKL_on_CorLength_Mean}
% %\end{center}
% %\end{figure}
% %
% %
% %\begin{figure}[htbp]
% %\begin{center}
% %\includegraphics[width=\textwidth]{EffectOfKL_on_CorLength_SD}
% %\caption{}
% %\label{figure:EffectOfKL_on_CorLength_SD}
% %\end{center}
% %\end{figure}
% %
% %
% %\begin{figure}[htbp]
% %\begin{center}
% %\includegraphics[width=\textwidth]{EffectOfPC_on_COV_Mean}
% %\caption{}
% %\label{figure:EffectOfKL_on_CorLength_SD}
% %\end{center}
% %\end{figure}
% %
% %\begin{figure}[htbp]
% %\begin{center}
% %\includegraphics[width=\textwidth]{EffectOfPC_on_COV_SD}
% %\caption{}
% %\label{figure:EffectOfKL_on_CorLength_SD}
% %\end{center}
% %\end{figure}
% %
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\begin{frame}
\frametitle{Applications}
\begin{itemize}
\vspace*{0.3cm}
\item Stochastic elasticplastic simulations of soils and structures
\vspace*{0.3cm}
\item Probabilistic inverse problems
\vspace*{0.3cm}
\item Geotechnical site characterization design
\vspace*{0.3cm}
\item Optimal material design
\end{itemize}
\end{frame}
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%  +
%  + \begin{frame}
%  +
%  +
%  + \frametitle{Random Field Modeling of Uncertain Soil Properties}
%  +
%  + \begin{itemize}
%  +
%  + \item Finite scale model
%  +
%  + \begin{itemize}
%  +
%  + \item Short memory, finite correlation length
%  +
%  + \item Common autocovariance model $\rightarrow$ exponential, spherical, triangular, linearexponential
%  +
%  + \end{itemize}
%  +
%  + \item Fractal model
%  +
%  + \begin{itemize}
%  +
%  + \item long memory, infinite correlation length $\rightarrow$ more realistic for modeling horizontal
%  + spatial uncertainty
%  +
%  + \item 1/ftype noise process with power spectral density, $P(\omega)~=~P_0~\omega^{\gamma}$, with
%  + upper and/or lower frequency cutoff.
%  +
%  + \end{itemize}
%  +
%  + \end{itemize}
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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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\frametitle{Random Field Parameters from Site Data}
\begin{itemize}
%\item maximizing the loglikelihood of observing the spatial data under assumed joined distribution (for finite
%scale model) or maximizing the loglikelihood of observing the periodogram estimates (for fractal model)
\item Maximum likelihood estimates
\vspace*{0.3truecm}
%\begin{figure}
\begin{flushleft}
\hspace*{1.7cm}
\includegraphics[height=4.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/CPT_DataAnalysis_Plots/SamplingPlanEdited.jpg}
\hspace*{0.0cm}
\includegraphics[height=4.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/CPT_DataAnalysis_Plots/TypicalDataPlotBH1Edited.jpg} \\
\small{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Typical CPT $q_T$}
\end{flushleft}
%\end{figure}
\vspace*{4.9truecm}
%\begin{figure}
\begin{flushright}
\includegraphics[width=4.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/CPT_DataAnalysis_Plots/TypicalAutoCovariancePlotBH1_FiniteScaleEdited.jpg} \\
\vspace*{0.01truecm}
\small{Finite Scale}
\end{flushright}
%\end{figure}
\vspace*{0.02truecm}
%\begin{figure}
\begin{flushright}
\includegraphics[width=4.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/CPT_DataAnalysis_Plots/TypicalAutoCovariancePlotBH1_FractalEdited.jpg} \\
\small{Fractal}
\end{flushright}
%\end{figure}
\end{itemize}
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\begin{frame}
\frametitle{"Uniform" CPT Site Data}
\vspace*{0.7cm}
%\begin{figure}
\begin{center}
\includegraphics[height=6.7cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/CPT_DataAnalysis_Plots/EastWestProfileEdited.pdf}
\end{center}
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\begin{frame}
\frametitle{Seismic Wave Propagation through Stochastic Soil}
\begin{figure}
\begin{center}
\hspace*{0.75cm}
\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter9Plots/Chapter9_ElasticPlasticResponseNew.pdf}
%\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
\end{center}
\end{figure}
Mean$\pm$ Standard Deviation
%\begin{flushleft}
%\includegraphics[height=5.0cm]{PEER2007_3.jpg}
%\end{flushleft}
% \hspace*{1.0cm} \noindent Statistics of Top Node Displacement:
%
% \vspace*{0.5truecm}
%
% \begin{figure}
% \begin{flushleft}
% \hspace*{1.0cm}
% \includegraphics[width=4.0cm]{/home/kallol/publication/2007/Presentation/PhDExitSeminar/Chapter9_ElasticPlasticResponse_MeanNew.jpg}
% \hspace*{0.1cm}
% \includegraphics[width=4.0cm]{/home/kallol/publication/2007/Presentation/PhDExitSeminar/Chapter9_ElasticPlasticResponse_SDNew.jpg}
% \end{flushleft}
% \end{figure}
% \vspace*{0.5truecm}
% \hspace*{1.0cm} \tiny{~~~~~~~~~~~~~~~~~~~~~~~~~~~~Mean~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Standard Deviation}
%
% \vspace*{0.3truecm}
%
% \begin{figure}
% \begin{flushleft}
% \hspace*{0.75cm}
% \includegraphics[width=4.0cm]{/home/kallol/publication/2007/Presentation/PhDExitSeminar/Chapter9_ElasticPlasticResponseNew.jpg}
% \hspace*{0.4cm}
% \includegraphics[width=4.0cm]{/home/kallol/publication/2007/Presentation/PhDExitSeminar/Chapter9_ElasticPlasticResponse_COVNew.jpg}
% \end{flushleft}
% \end{figure}
% \vspace*{0.3truecm}
% \hspace*{0.5cm} \tiny{~~~~~~~Mean$\pm$ Standard Deviation~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~COV}
%
%
% \vspace*{6.0cm}
% \begin{flushright}
% \includegraphics[height=4.5cm]{/home/kallol/publication/2007/Presentation/PhDExitSeminar/Chapter9_ElasticPlasticResponse_PDFNewEdited.jpg} \hspace*{1.0cm}
% \end{flushright}
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\begin{frame}
\frametitle{Three Approaches to Modeling}
\begin{itemize}
\vspace*{0.3cm}
\item Do nothing about site characterization (rely on experience): conservative
{\bf guess} of soil data, $COV = 225$\%, correlation length $= 12$m.
\vspace*{0.3cm}
\item Do better than standard site characterization: $COV = 103$\%, correlation length $= 0.61$m)
\vspace*{0.3cm}
\item Improve site characterization if probabilities of exceedance are unacceptable!
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{Evolution of Mean $\pm$ SD for Guess Case}
\begin{figure}
\begin{center}
\hspace*{0.75cm}
\includegraphics[width=10.0cm]{/home/jeremic/tex/works/Conferences/2009/UNIONUnivBGD/Present/Plots_with_Labels/Evolutionary_Mean_pm_SD_NoDataEdited.pdf}
\hspace*{0.75cm}
%\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
\end{center}
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\begin{frame}
\frametitle{Evolution of Mean $\pm$ SD for Real Data Case}
\begin{figure}
\begin{center}
\hspace*{0.75cm}
\includegraphics[width=10.0cm]{/home/jeremic/tex/works/Conferences/2009/UNIONUnivBGD/Present/Plots_with_Labels/Evolutionary_Mean_pm_SD_ActualEdited.pdf}
\hspace*{0.75cm}
%\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
\end{center}
\end{figure}
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\begin{frame}
\frametitle{Full PDFs for Real Data Case}
\begin{figure}
\begin{center}
\vspace*{0.75cm}
\includegraphics[width=7.0cm]{/home/jeremic/tex/works/Conferences/2009/UNIONUnivBGD/Present/Plots_with_Labels/EvolutionaryPDF_ActualEdited.pdf}
\vspace*{0.75cm}
%\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
\end{center}
\end{figure}
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\begin{frame}
\frametitle{Example: PDF at $6$ s}
\begin{figure}
\begin{center}
\hspace*{1.75cm}
\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2009/UNIONUnivBGD/Present/Plots_with_Labels/PDFs_at6sec_Actual_vs_NoDataEdited.pdf}
\vspace*{0.75cm}
%\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
\end{center}
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\begin{frame}
\frametitle{Example: CDF at $6$ s}
\begin{figure}
\begin{center}
%\hspace*{0.75cm}
\includegraphics[width=8.0cm]{/home/jeremic/tex/works/Conferences/2009/UNIONUnivBGD/Present/Plots_with_Labels/CDFs_at6sec_Actual_vs_NoDataEdited.pdf}
\vspace*{0.75cm}
%\hspace*{0.75cm}
%\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
\end{center}
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\begin{frame}
\frametitle{Probability of Exceedance of $20$cm}
\begin{figure}
\begin{center}
%\hspace*{0.75cm}
\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2009/UNIONUnivBGD/Present/Plots_with_Labels/ProbabilityOfExceedance20cm_Actual_vs_NoDataEdited.pdf}
\vspace*{0.75cm}
%\hspace*{0.75cm}
%\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
\end{center}
\end{figure}
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\begin{frame}
\frametitle{Probability of Exceedance of $50$cm}
\begin{figure}
\begin{center}
%\hspace*{0.75cm}
\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2009/UNIONUnivBGD/Present/Plots_with_Labels/ProbabilityOfExceedance50cm_Actual_vs_NoDataEdited.pdf}
\vspace*{0.75cm}
%\hspace*{0.75cm}
%\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
\end{center}
\end{figure}
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\begin{frame}
\frametitle{Probabilities of Exceedance vs. Displacements}
\begin{figure}
\begin{center}
%\hspace*{0.75cm}
\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2009/UNIONUnivBGD/Present/Plots_with_Labels/ProbabilityOfExceedance_vs_Displacement_Actual_vs_NoDataEdited.pdf}
\vspace*{0.75cm}
%\hspace*{0.75cm}
%\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
\end{center}
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\section{Summary}
\begin{frame}
\frametitle{Summary}
\begin{itemize}
\item Behavior of materials is probably probabilistic!
\vspace*{0.6cm}
\item Technical developments are available and are being refined
\vspace*{0.6cm}
\item Human nature: how much do you want to know about potential problem?
\end{itemize}
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%\item Language used by Beamer: L\uncover<2>{A}TEX
%\item Language used by Beamer: L\only<2>{A}TEX
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%
% % All of the following is optional and typically not needed.
%\appendix
%\section*{\appendixname}
%\subsection*{For Further Reading}
%
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