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\title{On Probabilistic Yielding of (Geo)Materials}
%\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
{WCCM8 / ECCOMAS2008 \\
1st. July 2008, \\
Venezia, Italia}
%\\
%June/July 2008}
%giugno/luglio 2008}
% %\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/}
% }
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\subject{}
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% Structuring a talk is a difficult task and the following structure
% may not be suitable. Here are some rules that apply for this
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\section{Motivation}
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\subsection{}
%\subsection{Stochastic Systems: Historical Perspectives}
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% \frametitle{History}
%
%
%
% \begin{itemize}
%
%
% \item Probabilistic fish counting
%
% \vspace*{0.2cm}
%
% \vspace*{0.2cm}
% \item Williams' DEM simulations, differential displacement vortexes
%
% %\vspace*{0.2cm}
% %\item Runesson's dilemma
%
% \vspace*{0.2cm}
% \item SFEM round table
%
% \vspace*{0.2cm}
% \item Kavvas' probabilistic hydrology
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\frametitle{Uncertain (Geo)materials}
\begin{center}
%\vspace*{0.5cm}
\includegraphics[height=4.3cm]{/home/jeremic/tex/works/Conferences/2007/GeoDenver/SFEM/Presentation/FrictionAngleProfile.jpg}
\hfill
\includegraphics[height=3.9cm]{/home/jeremic/tex/works/Conferences/2006/KragujevacSEECCM06/Presentation/MGMuzorak01.jpg}
\\
\small{
Mayne et al. (2000)
\hspace*{4.3cm}
Sture et al. (1998)
}
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% 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.}
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\frametitle{Background}
\begin{itemize}
%
\item Brownian motion, Langevin equation (Einstein 1905)
\item Random forcing (FokkerPlanck, 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 Random coefficient (Hopf 1952),
(BharruchaReid 1968), Monte Carlo method
%\item Alternately, Monte Carlo method can be used for solution of Langevin equation $\rightarrow$ computationally very expensive
\item Epistemic uncertainty, 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*{0.15cm}
\item Aleatory uncertainty, inherent variation
%\vspace*{1.9cm}
%\begin{figure}[!hbpt]
%\begin{flushright}
%\includegraphics[height=3.5cm,angle=90]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Present/uncertain03.pdf}
%\end{flushright}
%\end{figure}
% \begin{itemize}
%
% \item Can not be reduced
%
% \item Has highly developed mathematical tools
%
% \end{itemize}
\item Methods for treating
epistemic uncertainty
not well developed,
tradeoff with
aleatory uncertainty
\item Ergodicity of geomaterials !(?)
% \begin{itemize}
%
% \item Are geomaterials ergodic? Possibly yes
%
%% \item Issues up for discussion (soil, concrete, rock, biomaterials...)
%
% \end{itemize}
\item Prof. Einav question
\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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\section{Uncertain Material Nonlinear Modeling}
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\subsection{Probabilistic ElastoPlasticity}
%\subsection{PEP Formulations}
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\frametitle{Uncertainty Propagation through Constitutive Eq.}
%
\begin{itemize}
\item Incremental elpl constitutive equation
$\Delta \sigma_{ij} = D_{ijkl} \Delta \epsilon_{kl}$
%\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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% 
%  \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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% 
%  \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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% 
% 
%  \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}
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% 
% 
%  \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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%  \end{frame}
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\begin{frame}
\frametitle{EulerianLagrangian FPK Equation}
%
%\begin{itemize}
\begin{footnotesize}
%\noindent
Liouville continuity equation (Kubo 1963);
ensemble average form
(Kavvas and Karakas 1996);
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}
% %
% %
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% \end{frame}
%
%
%
%
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\begin{frame}
\frametitle{Compact Form of EulerianLagrangian FPK Equation}
\begin{itemize}
\item Advectiondiffusion equation
%
\begin{equation}
\nonumber
\frac{\partial P(\sigma,t)}{\partial t} = \frac{\partial}{\partial \sigma}\left[N_{(1)}P(\sigma,t)\frac{\partial}{\partial \sigma}
\left\{N_{(2)} P(\sigma,t)\right\} \right]
\end{equation}
%
\item Complete probabilistic description of response
\item Solution PDF is secondorder exact to covariance of time (exact mean and variance)
\item It is deterministic equation in probability density space
\item It is linear PDE in probability density space
$\rightarrow$ simplifies the numerical solution process
%\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}
%\vskip2pt\insertnavigation{\paperwidth}\vskip2pt
\end{beamercolorbox}%
}
\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Template Solution of EL 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 random or deterministic
%
% \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)
%
\item Solution for both
\begin{itemize}
\item probabilistic elastic (PEL) and
\item probabilistic elasticplastic (PELPL) problems
\end{itemize}
\end{itemize}
\vspace*{1.0cm}
{%
\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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\end{frame}
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%==%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Probabilistic Yielding}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Probabilistic Yielding: Weighted Coefficients}
\begin{itemize}
%
%\vspace*{0.5cm}
\item Probabilistic yielding: take both PEL and PELPL into account at the same
time with some (un)certainty
\vspace*{0.3cm}
\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*{1.0cm}
\vspace*{0.3cm}
\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}
\vspace*{0.3cm}
\item Similar to European Pricing Option in financial simulations (BlackScholes options
pricing model '73, Nobel prize for Economics '97)
\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 print in \textit{Communications in Numerical Methods in
Engineering}, 2008.}
%\vskip2pt\insertnavigation{\paperwidth}\vskip2pt
\end{beamercolorbox}%
}
\end{frame}
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\begin{frame}
\frametitle{Probability of Yielding}
\vspace*{1.00cm}
\begin{itemize}
\item Probabilistic von Mises
\item Probability of yielding at \\
$\sigma=0.0012~{\rm MPa}$\\
$P[\Sigma_y \leq (\sigma=0.0012~{\rm MPa})]$
\\
$ = 0.8$
\item Equivalent advection and \\
diffusion coefficients are
%
\begin{equation}
N_{(1)}^{eq}_{\sigma=0.0012~MPa}
=
(10.8)N_{(1)}^{el}+0.8N_{(1)}^{ep}
\nonumber
\end{equation}
%
\begin{equation}
N_{(2)}^{eq}_{\sigma=0.0012~MPa}
=
(10.8)N_{(2)}^{el}+0.8N_{(2)}^{ep}
\nonumber
\end{equation}
%
\end{itemize}
\vspace*{7.0cm}
\begin{figure}[!htbp]
\begin{flushright}
\includegraphics[height=4.2cm]{/home/jeremic/tex/works/Papers/2007/ProbabilisticYielding/vonMises_YieldCDF_Combinededited.pdf}
%\vspace*{0.8cm}
%\caption{CDF of shear strength for von Mises model: (a) very uncertain case,
%(b) fairly certain case.}
\label{vonMises_Yield_CDF}
\end{flushright}
\end{figure}
%
%
%\begin{figure}[!hbpt]
%\begin{center}
%\includegraphics[width=8cm]{/home/jeremic/tex/works/Papers/2007/ProbabilisticYielding/figures/vonMises_G_and_cu_very_uncertain/Contour_PDFedited.pdf}
%\end{center}
%\end{figure}
%
%\vspace*{0.3cm}
%linear elastic  linear hardening plastic von Mises
%
\end{frame}
%
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\begin{frame}
\frametitle{BiLinear von Mises Response}
\vspace*{0.20cm}
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=8.0cm]{/home/jeremic/tex/works/Papers/2007/ProbabilisticYielding/Final/vonMises_G_and_cu_very_uncertain_Contour_PDFedited.pdf}
% %\vspace*{0.5cm}
% \caption{von Mises associative plasticity model with uncertain shear modulus and
% shear strength (yield parameter): (a) Evolution of PDF of stress with
% strain (PDF=10000 was used as a cutoff for surface plot) and
% (b) Contours of evolution of stress PDF with strain.}
% \label{vonMises_G_and_cu_very_uncertain}
\end{center}
\end{figure}
%
\end{frame}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{BiLinear von Mises Response}
\vspace*{0.50cm}
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=9.5cm]{/home/jeremic/tex/works/Papers/2007/ProbabilisticYielding/Final/vonMises_G_and_cu_very_uncertain_PDFedited.pdf}
% %\vspace*{0.5cm}
% \caption{von Mises associative plasticity model with uncertain shear modulus and
% shear strength (yield parameter): (a) Evolution of PDF of stress with
% strain (PDF=10000 was used as a cutoff for surface plot) and
% (b) Contours of evolution of stress PDF with strain.}
% \label{vonMises_G_and_cu_very_uncertain}
\end{center}
\end{figure}
%
\end{frame}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{von Mises, Certain $c_u$, Uncertain $G$}
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=8.5cm]{/home/jeremic/tex/works/Papers/2007/ProbabilisticYielding/Final/vonMises_G_very_uncertain_but_cu_fairly_certain_Contour_PDFedited.pdf}
\end{center}
\end{figure}
%
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{von Mises, Uncertain $c_u$, Certain $G$}
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=9.5cm]{/home/jeremic/tex/works/Papers/2007/ProbabilisticYielding/Final/vonMises_G_fairly_certain_but_cu_very_uncertain_Mean_SD_Mode_DeterSoledited.pdf}
\end{center}
\end{figure}
%
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  \begin{frame}
% 
% 
% 
%  \frametitle{Drucker Prager}
% 
% 
%  \begin{figure}[!htbp]
%  \begin{center}
%  \includegraphics[height=4cm]{/home/jeremic/tex/works/Papers/2007/ProbabilisticYielding/Final/DruckerPrager_YieldCDF_Combinededited.pdf}
%  \vspace*{0.8cm}
%  \caption{CDF of yield stresses for DruckerPrager model:
%  (a) very uncertain and (b) fairly certain}
%  \label{DruckerPrager_Yield_CDF}
%  \end{center}
%  \end{figure}
% 
% 
% 
%  \end{frame}
% 
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  \begin{frame}
% 
% 
% 
%  \frametitle{Drucker Prager}
% 
%  %
%  \begin{figure}[!htbp]
%  \begin{center}
%  \mbox{a)}\includegraphics[width=4.5cm]{/home/jeremic/tex/works/Papers/2007/ProbabilisticYielding/Final/DruckerPrager_G_and_alpha_very_uncertain_PDFedited.pdf}
%  \hfill
%  \mbox{b)}\includegraphics[width=4.5cm]{/home/jeremic/tex/works/Papers/2007/ProbabilisticYielding/Final/DruckerPrager_G_and_alpha_very_uncertain_Contour_PDFedited.pdf}
%  %\vspace*{0.5cm}
%  \caption{DruckerPrager associative elasticplastic model with uncertain shear
%  modulus and frictional coefficient: (a) Evolution of probability density function (PDF) of stress with
%  strain (PDF=10000 was used as a cutoff for surface plot) and (b) Contours of evolution of PDF with strain}
%  \label{DruckerPrager_G_and_cu_very_uncertain}
%  \end{center}
%  \end{figure}
% 
% 
% 
% 
%  \end{frame}
% 
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Drucker Prager: Uncertain $G$, Certain $\phi$ }
%
\begin{figure}[!htbp]
\begin{center}
\mbox{a)}\includegraphics[width=8.5cm]{/home/jeremic/tex/works/Papers/2007/ProbabilisticYielding/Final/DruckerPrager_G_very_uncertain_but_alpha_fairly_certain_Contour_PDFedited.pdf}
\hfill
\end{center}
\end{figure}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Drucker Prager: Certain $G$, Uncertain $\phi$ }
%
\begin{figure}[!htbp]
\begin{center}
\mbox{b)}\includegraphics[width=8.5cm]{/home/jeremic/tex/works/Papers/2007/ProbabilisticYielding/Final/DruckerPrager_G_fairly_certain_but_alpha_very_uncertain_Contour_PDFedited.pdf}
\end{center}
\end{figure}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Summary}
\begin{frame}
\frametitle{Summary}
\begin{itemize}
\item Secondorder (mean and variance) exact, method
to account for probabilistic elasticplastic material simulation
\vspace*{0.3cm}
\item Probabilities of material yielding, probably govern underlying mechanics
of elastoplasticity
%\vspace*{0.3cm}
\vspace*{0.3cm}
\item Probabilities of material yielding (spatial distributions) also probably
govern underlying mechanics of failure, localization of deformation...
%\vspace*{0.3cm}
\vspace*{0.3cm}
\item Probably numerous applications
\end{itemize}
\end{frame}
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% \begin{frame}
%
%\begin{itemize}
%\item Language used by Beamer: L\uncover<2>{A}TEX
%\item Language used by Beamer: L\only<2>{A}TEX
%\end{itemize}
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%\end{frame}
%
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\begin{frame}
\begin{center}
%\only<1>{\includegraphics[width=4.0cm]{/home/jeremic/tex/works/ThankYou.jpg}}
%\only<2>{\includegraphics[width=4.0cm]{/home/jeremic/tex/works/ThankYou01.jpg}}
%\only<3>
{\includegraphics[width=6.0cm]{/home/jeremic/tex/works/ThankYou02.jpg}}
\end{center}
\end{frame}
% All of the following is optional and typically not needed.
%\appendix
%\section*{\appendixname}
%\subsection*{For Further Reading}
%
%\begin{frame}[allowframebreaks]
% \frametitle{For Further Reading}
%
% \begin{thebibliography}{10}
%
% \beamertemplatebookbibitems
% % Start with overview books.
%
% \bibitem{Author1990}
% A.~Author.
% \newblock {\em Handbook of Everything}.
% \newblock Some Press, 1990.
%
%
% \beamertemplatearticlebibitems
% % Followed by interesting articles. Keep the list short.
%
% \bibitem{Someone2000}
% S.~Someone.
% \newblock On this and that.
% \newblock {\em Journal of This and That}, 2(1):50100,
% 2000.
% \end{thebibliography}
%\end{frame}
\end{document}