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\title{Stochastic Elastic-Plastic Finite Element Method for 
Performance Risk Simulations}


%\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.
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%   affiliation.
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\pgfdeclareimage[height=0.35cm]{UA-logo}{/home/jeremic/BG/amblemi/uakron_logo_03}

\author[Jeremi{\'c} and Sett] % (optional, use only with lots of authors)
{Boris~Jeremi{\'c}\inst{1} \and  Kallol Sett\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.





\institute[Computational Geomechanics Group \hspace*{0.3truecm}
\pgfuseimage{university-logo}\hspace*{0.1truecm}\pgfuseimage{UA-logo} \hspace*{-0.3truecm}] % (optional, but mostly needed)
{  \inst{1}University of California, Davis 
  \and
  \inst{2}University of Akron, Ohio}
% - Use the \inst command only if there are several affiliations.
% - Keep it simple, no one is interested in your street address.


\date[ICASP11] % (optional, should be abbreviation of conference name)
{ICASP \\
 {\small Z{\"u}rich, Switzerland  \\
 August 2011} }


\subject{}
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  \frametitle{Outline}
  \tableofcontents
  % You might wish to add the option [pausesections]
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% Structuring a talk is a difficult task and the following structure
% may not be suitable. Here are some rules that apply for this
% solution: 

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

% - A conference audience is likely to know very little of what you
%   are going to talk about. So *simplify*!
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%   you think necessary.
% - If you omit details that are vital to the proof/implementation,
%   just say so once. Everybody will be happy with that.





\section{Motivation}


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%\subsection{Risk and Civil Engineering}
\subsection{}
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\begin{frame} 

\frametitle{Determining Risk for Civil Engineering Object Behavior}

 

\begin{itemize}



%\vspace*{0.4cm}
\item Risk: inherent, intrinsic, constitutive part of civil engineering


\vspace*{0.4cm}
\item Uncertain loads (!)


\vspace*{0.4cm}
\item Uncertain materials (!!)


\vspace*{0.4cm}
\item Uncertain human factor (!)


\end{itemize}

\begin{equation}
{\bf M} \ddot{\bf u}
+
{\bf C} \dot{\bf u}
+
{\bf K} {\bf u}
=
{\bf F}
\nonumber
\end{equation}

 
\end{frame}


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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 trade-off 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{Drucker-Prager 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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% 
% \end{frame}
% 
% 
% 
% 
% 
% 
% 
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% \begin{frame} 
% 
% \frametitle{Ergodicity}
% 
%  
% 
% \begin{itemize}
% 
% 
% 
% \item Exchange ensemble averages for time averages
% 
% \vspace*{0.2cm}
% \item Is soil elasto-plasticity ergodic? 
% 
%     \begin{itemize}
% 
%       \item  Can  soil  elastic--plastic  statistical  properties be obtained by
%       temporal averaging?
% 
% 
%       \item Will soil  elastic--plastic  statistical  properties "renew" at each
%       occurrence?
% 
%       \item  Are  soil   elastic--plastic  statistical  properties statistically
%       independent?
% 
%     \end{itemize}
% 
% 
% \item  Claim  in  literature  that structural nonlinear behavior is non--ergodic
% 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{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 Point-wise \\ 
              uncertainty \\
              \begin{itemize}
                \item testing \\
                error \\
                \item transformation \\
                error
              \end{itemize}

        \end{itemize}

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

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



\end{frame}






% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \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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% 
% \end{frame}
% 
% 
% 
% 
% 
% 
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\section{Probabilistic Elasto--Plasticity}

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\subsection{PEP Formulations}
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\begin{frame} 

\frametitle{Uncertainty Propagation through Constitutive Eq.}

% 

\begin{itemize}

 


\item Incremental el--pl 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 elastic--plastic}}
%
           \end{array} \right.
           \end{eqnarray}

%\end{normalsize}

 



%\vspace{0.5cm}
% \item Non--linear coupling in the El--Pl modulus


\item What if all (any) material parameters are uncertain

\item Since material {\bf is} inherently spatially variable and uncertain at the
point, PEP and SEPFEM methods were developed


%-  \item Focus on 1-D $\rightarrow$ 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{Problem Statement}
%  
% 
% 
%  
%  
% \begin{itemize}
% 
% \item Incremental 3D elastic-plastic stress--strain:
% % 
% %    
% 
%     \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 (3D is also available) $\rightarrow$  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{Solution to Probabilistic Elastic-Plastic Problem}

 

\begin{itemize}


\item Use of stochastic continuity (Liouiville) equation (Kubo 1963)

%\vspace{0.1cm}
\item With cumulant expansion method  (Kavvas and Karakas 1996)

%\vspace{0.1cm}
\item To obtain ensemble average form of Liouville Equation

%\vspace{0.1cm}
\item  Which,  with  van  Kampen's  Lemma (van Kampen 1976): ensemble average of
phase density is the probability density


%\vspace{0.1cm}
\item           Yields      Eulerian-Lagrangian  form  of the Forward Kolmogorov
(Fokker-Planck-Kolmogorov) equation

\end{itemize}
 
\end{frame}





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\begin{frame} 

\frametitle{Eulerian--Lagrangian FPK Equation}

%

%\begin{itemize}
% 3D
\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}




% 1D % 1D
% 1D \begin{footnotesize}
% 1D \begin{eqnarray}
% 1D \nonumber
% 1D &&\displaystyle \frac{\partial P(\sigma(x_t,t), t)}{\partial t}=
% 1D - \displaystyle \frac{\partial}{\partial \sigma} \left[ \left\{\left< \vphantom{\int_{0}^{t} d\tau} \eta(\sigma(x_t,t), D^{el}(x_t),
% 1D q(x_t), r(x_t), \epsilon(x_t,t)) \right> \right. \right. \\
% 1D \nonumber
% 1D &+& \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),
% 1D \epsilon(x_t,t))}{\partial \sigma}; \right. \right. \right. \\
% 1D \nonumber
% 1D & & \left. \left. \left. \eta(\sigma(x_{t-\tau},t-\tau), D^{el}(x_{t-\tau}), q(x_{t-\tau}), r(x_{t-\tau}),
% 1D \epsilon(x_{t-\tau},t-\tau) \vphantom{\int_{0}^{t} d\tau} \right] \right \} P(\sigma(x_t,t),t) \right] \\
% 1D \nonumber
% 1D &+& \displaystyle \frac{\partial^2}{\partial \sigma^2} \left[ \left\{ \int_{0}^{t} d\tau Cov_0 \left[ \vphantom{\int_{0}^{t}}
% 1D \eta(\sigma(x_t,t), D^{el}(x_t), q(x_t), r(x_t), \epsilon(x_t,t)); \right. \right. \right. \\
% 1D \nonumber
% 1D & & \left. \left. \left. \eta (\sigma(x_{t-\tau},t-\tau), D^{el}(x_{t-\tau}), q(x_{t-\tau}), r(x_{t-\tau}),
% 1D \epsilon(x_{t-\tau},t-\tau)) \vphantom{\int_{0}^{t}} \right] \vphantom{\int_{0}^{t}} \right\} P(\sigma (x_t,t),t) \right] \\
% 1D \nonumber
% 1D \end{eqnarray}
% 1D 
% 1D \end{footnotesize}





\end{frame}







%-- 
%-- \item 6 equations
%-- 
%-- \item Complete description of 3-D probabilistic stress-strain response
%-- 
%-- \end{itemize}
%-- 
%--  


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\begin{frame} 

\frametitle{Eulerian--Lagrangian FPK Equation}

 

\begin{itemize}

\item Advection-diffusion equation 
%
\begin{equation}                
\nonumber
\frac{\partial P(\sigma_{ij},t)}{\partial t} 
= 
-\frac{\partial}{\partial\sigma_{ab}}
\left[N_{ab}^{(1)}P(\sigma_{ij},t)
-
\frac{\partial}{\partial \sigma_{cd}}
\left\{N_{abcd}^{(2)} P(\sigma_{ij},t)\right\} \right]
\end{equation}

%

\item Complete probabilistic description of response 


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


\item Deterministic equation in probability density space

\item Linear PDE in probability density space 
$\rightarrow$ simplifies the numerical solution process


\item  Applicable  to any elastic-plastic-damage material model (only coefficients $N_{ab}^{(1)}$
and $N_{abcd}^{(2)}$ differ)


%\vspace*{0.2truecm}

\end{itemize}


\end{frame}






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\begin{frame} 

\frametitle{Probabilistic Elastic-Plastic Response}


\begin{figure}[!hbpt]
%\nonumber
%\begin{flushleft}
\begin{center}
%\includegraphics[height=3.0cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/PDFElastic_RandomG-m.pdf}
%\hspace*{0.3cm}
%\includegraphics[height=6.0cm]{/home/jeremic/tex/works/Conferences/2011/ICASP11_Zurich/Present/Probabilistic_El_Pl_example.jpg}
\includegraphics[height=6.0cm]{/home/jeremic/tex/works/Conferences/2011/ICASP11_Zurich/Present/PDF_Plot-Ed.pdf}
%
%\end{flushleft}
%\begin{flushright}
%\includegraphics[height=9cm]{ReasonOfNotMatchingMonteCarlo-m.pdf}
%
%\nonumber
%\caption{Drucker-Prager Associative Linear Hardening Response with Random $G$}
%{\includegraphics[width=10.5cm]{QualSchematics4.jpg}}
%\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
%\end{flushright}
\end{center}
\end{figure}

%Probabilistic Elastic   \hspace*{2cm}  Probabilistic Elastic-Plastic
%%$<G>$ = 2.5 MPa;
%%Std. Deviation$[G]$ = 0.5 MPa

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\end{frame}







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%\subsection{Boundary Value Problem}
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\subsection[SEPFEM]{Stochastic Elastic--Plastic Finite Element Method}
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\begin{frame}



\frametitle{Governing Equations \& Discretization Scheme}

\begin{itemize}

\item Governing equations:
%
%
\begin{equation}
\nonumber
A\sigma = \phi(t) ~~~~~~~ Bu = \epsilon ~~~~~~~ \sigma = \textcolor{mycolor}{\bf D} \epsilon
\end{equation}

\item  Discretization (spatial and stochastic) schemes

\begin{itemize}

\item    Input    random   field   material   properties   ($\textcolor{mycolor}{\bf D}$)   $\rightarrow$
Karhunen--Lo{\`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}

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\begin{frame}
\frametitle{Spectral Stochastic Elastic--Plastic 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}{\bf 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 Fokker--Planck--Kolmogorov 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 KL-expansion
%      
%      \item Representation of (unknown) solution random variables using polynomial chaos of
%            (known) input random variables
%
%      \item Fokker--Planck--Kolmogorov approach based probabilistic constitutive integration
%      at Gauss integration points
%
%      \end{itemize}
%      
%%   \end{itemize}
%
\end{frame}

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\begin{frame}
\frametitle{Inside SEPFEM}

\begin{itemize}

\item Stochastic elastic--plastic (explicit) 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 elastic--plastic stiffness tensor


\end{itemize}

%  \noindent Salient Features:
       
%      \begin{itemize}
%
%      \item Efficient representation of input random fields into finite number of random
%            variables using KL-expansion
%      
%      \item Representation of (unknown) solution random variables using polynomial chaos of
%            (known) input random variables
%
%      \item Fokker--Planck--Kolmogorov approach based probabilistic constitutive integration
%      at Gauss integration points
%
%      \end{itemize}
%      
%%   \end{itemize}
%
\end{frame}

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\section{Applications to Risk Analysis}


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\subsection{Seismic Wave Propagation Through Uncertain Soils}


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\begin{frame}


\frametitle{Risk Assessment Applications}



\begin{itemize}

%\vspace*{0.2cm}
\item  Any  problem 
(${\bf M} \ddot{\bf u}
+
{\bf C} \dot{\bf u}
+
{\bf K} {\bf u}
=
{\bf F}
$)  
with known
  \begin{itemize}
    \item PDFs of material parameters,
    \item PDFs of loading 
  \end{itemize}
 can  be  analyzed using PEP and SEPFEM to obtain PDFs of DOFs,
 stress, strain...

%\vspace*{0.2cm}
\item PEP solution is second order accurate (exact mean and standard deviation)

%\vspace*{0.2cm}
\item SEPFEM solution (PDFs) can be made as accurate as need be


\item Tails of PDFs can than be used to develop accurate risk


\item Application to a realistic case of seismic wave propagation


\end{itemize}

\end{frame}

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\begin{frame}




\frametitle{"Uniform" CPT  Site Data}


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


\end{frame}


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\begin{frame}


\frametitle{Seismic Wave Propagation through Stochastic Soil}

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

\begin{itemize}

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

\begin{itemize}

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

\end{itemize}

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

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

\end{itemize}


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

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

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

\end{itemize}

\end{frame}

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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %\begin{frame}
% %
% %
% %\frametitle{Random Field Parameters from Site Data}
% %
% %\begin{itemize}
% %
% %%\item maximizing the log--likelihood of observing the spatial data under assumed joined distribution (for finite 
% %%scale model) or maximizing the log--likelihood of observing the periodogram estimates (for fractal model)
% %
% %\item Maximum likelihood estimates
% %
% %\vspace*{0.3truecm}
% %
% %%\begin{figure}
% %\begin{flushleft}
% %\hspace*{-1.7cm}
% %\includegraphics[height=4.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/CPT_DataAnalysis_Plots/SamplingPlan-Edited.jpg}
% %\hspace*{0.0cm}
% %\includegraphics[height=4.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/CPT_DataAnalysis_Plots/TypicalDataPlotBH1-Edited.jpg} \\
% %\small{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Typical CPT $q_T$}
% %\end{flushleft}
% %%\end{figure}
% %
% %\vspace*{-4.9truecm}
% %
% %%\begin{figure}
% %\begin{flushright}
% %\includegraphics[width=4.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/CPT_DataAnalysis_Plots/TypicalAutoCovariancePlotBH1_FiniteScale-Edited.jpg} \\
% %\vspace*{0.01truecm}
% %\small{Finite Scale}
% %\end{flushright}
% %%\end{figure}
% %
% %\vspace*{0.02truecm}
% %
% %%\begin{figure}
% %\begin{flushright}
% %\includegraphics[width=4.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/CPT_DataAnalysis_Plots/TypicalAutoCovariancePlotBH1_Fractal-Edited.jpg} \\
% %\small{Fractal}
% %\end{flushright}
% %%\end{figure}
% %
% %\end{itemize}
% %
% %\end{frame}
% %
% %
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%-- 
%--  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%--  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%--  \subsection{Seismic Wave Propagation Through Uncertain Soils}



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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_ElasticPlasticResponse-New.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_Mean-New.jpg}
% \hspace*{-0.1cm}
% \includegraphics[width=4.0cm]{/home/kallol/publication/2007/Presentation/PhDExitSeminar/Chapter9_ElasticPlasticResponse_SD-New.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_ElasticPlasticResponse-New.jpg}
% \hspace*{0.4cm}
% \includegraphics[width=4.0cm]{/home/kallol/publication/2007/Presentation/PhDExitSeminar/Chapter9_ElasticPlasticResponse_COV-New.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_PDF-New-Edited.jpg} \hspace*{-1.0cm}
% \end{flushright}
% 
\end{frame}

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

\begin{frame}


\frametitle{Full PDFs All Results}



\begin{figure}
\begin{center}
\vspace*{-0.75cm}
\includegraphics[width=7.0cm]{/home/jeremic/tex/works/Conferences/2009/UNION-Univ-BGD/Present/Plots_with_Labels/EvolutionaryPDF_Actual-Edited.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}

% 
\end{frame}

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\subsection{Probabilistic Analysis for Decision Making}


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\begin{frame}


\frametitle{Example: Three Approaches to Modeling}




\begin{itemize}

\vspace*{0.3cm}
\item    {\bf  Do  nothing}  about  site  (material)  characterization  (rely on
experience):  conservative  {guess}  for  soil  data, $COV = 225$\%, correlation
length $= 12$m.


\vspace*{0.3cm}
\item    {\bf Do better} than standard site  (material) characterization: $COV =
103$\%, correlation length $= 0.61$m)


\vspace*{0.3cm}
\item        {\bf     Do   the best} site  (material) characterization to reduce
probabilities of exceedance


\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/UNION-Univ-BGD/Present/Plots_with_Labels/Evolutionary_Mean_pm_SD_NoData-Edited.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}
% %
% %% 
% %\end{frame}
% %
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %
% %



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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/UNION-Univ-BGD/Present/Plots_with_Labels/Evolutionary_Mean_pm_SD_Actual-Edited.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}
% %
% %% 
% %\end{frame}
% %
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %

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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/UNION-Univ-BGD/Present/Plots_with_Labels/PDFs_at6sec_Actual_vs_NoData-Edited.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}

% 
\end{frame}

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\begin{frame}


\frametitle{Example: CDF (Non-Exceedance) at $6$ s}



\begin{figure}
\begin{center}
%\hspace*{-0.75cm}                                                                                               
\includegraphics[width=8.0cm]{/home/jeremic/tex/works/Conferences/2009/UNION-Univ-BGD/Present/Plots_with_Labels/CDFs_at6sec_Actual_vs_NoData-Edited.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}

% 
\end{frame}

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


\begin{frame}


\frametitle{Probability of Exceedance of $20$cm}



\begin{figure}
\begin{center}
%\hspace*{-0.75cm}                                                                                               
\includegraphics[width=11.0cm]{/home/jeremic/tex/works/Conferences/2009/UNION-Univ-BGD/Present/NewPlots/with_legends_and_labels/Exceedance20cm_LomaPrieta-Edited.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}
 
\end{frame}

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\begin{frame}

\frametitle{Probability of Exceedance of $50$cm}

\begin{figure}
\begin{center}
%\hspace*{-0.75cm}                                                                                               
\includegraphics[width=11.0cm]{/home/jeremic/tex/works/Conferences/2009/UNION-Univ-BGD/Present/NewPlots/with_legends_and_labels/Exceedance50cm_LomaPrieta-Edited.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}
 
\end{frame}

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\begin{frame}

\frametitle{Risk of Unacceptable Deformation}

\begin{figure}
\begin{center}
%\hspace*{-0.75cm}                                                                                               
\includegraphics[width=8.0cm]{/home/jeremic/tex/works/Conferences/2009/UNION-Univ-BGD/Present/NewPlots/with_legends_and_labels/Summary_LomaPrieta-Edited.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}
 
\end{frame}

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\section{Summary}

\begin{frame}


\frametitle{Summary}

\begin{itemize}

\item  Behavior  of  all  civil  engineering  objects  (structures, soils...) is
probably probabilistic!

\vspace*{0.3cm}
\item    Presented  methodology  (PEP  and  SEPFEM)  allows  for (very) accurate
numerical  simulation  of  PDFs of DOFs (and stress, strain) from
known (given) PDFs of material properties and PDFs of loads.


\vspace*{0.3cm}
\item Human nature: how much do you want to know about potential problem?

\end{itemize}


\end{frame}

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%    2000.
%  \end{thebibliography}
%\end{frame}

\end{document}


