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\title[Probabilistic Elasto--Plasticity] % (optional, use only with long paper titles)
{The Role of Material Variability and Uncertainty in 
Elastic-Plastic Finite Element Simulations
}

\subtitle
{}

\author[Boris Jeremi{\'c} and Kallol 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.
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\institute[UC Davis] % (optional, but mostly needed)
{
  Department of Civil and Environmental Engineering\\
  University of California, Davis}
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\date[SEECCM-06, June 28-30, 2006, Kragujevac, Serbia and Montenegro] % (optional, should be abbreviation of conference name)
{1st South-East European \\ Conference on Computational Mechanics,\\
June 28-30, 2006, Kragujevac, Serbia and Montenegro}
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\section{Motivation}

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\subsection{Historical Overview}
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%   \begin{frame} \frametitle{Presentation Overview}
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%   \begin{itemize}
%   
%   %
%   \item Motivation
%   
%   %
%   %\item Soil Uncertainties and their Quantifications
%   
%   %
%   \item Stochastic Systems: Historical Perspectives
%   
%   \item Material (Soil) Uncertainties and Quantifications
%   
%   \item Propagation of Uncertainties in Mechanics (Geomechanics)
%   
%   
%   %
%   \item Probabilistic Elasto-Plasticity
%   
%           \begin{itemize}
%   
%   %             
%                \item Stochastic Differential Equation Approach
%   
%   %            
%                \item Examples and Verifications
%   
%            \end{itemize}
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%   %\item Global, SFEM Level Simulation
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%   \item Conclusions
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\begin{frame} \frametitle{\Large{Types of Uncertainties}}

 

\begin{itemize}



\item Epistemic uncertainty - uncertainty due to lack of knowledge

\begin{itemize}


%
\item Can be reduced by collecting more data

%
\item Mathematical tools (neural network, fuzzy logic etc.) are not well developed $\rightarrow$ trade-off with aleatory uncertainty

\end{itemize}


\item Aleatory uncertainty - inherent variation of physical system 


\begin{itemize}


\item Can not be reduced

%
\item Has highly developed mathematical tools (classical second-order analysis) to deal with

\end{itemize}



\end{itemize}

 



\begin{figure}[!hbpt]
%\nonumber
\begin{center}
%\begin{center}
%\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
\includegraphics[height=2cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/uncertain02.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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\begin{frame} \frametitle{\Large{Brownian Motions}}


 

\begin{itemize}

%
\item Governing (Langevin) equation:



\begin{equation}
\nonumber
m \frac{dv}{dt} = F(x) - \beta v + \eta(t)
\end{equation} 



%
\item Probability density function (PDF) of particle displacement obeys a 
simple diffusion equation (Einstein (1905)):

%

\begin{equation}
\nonumber
\frac{\partial f (x,t)}{\partial t} = D \frac{\partial^2 f(x,t)}{\partial x^2}
\end{equation}


\item Addition of external forces (gravity, elastic or magnetic attraction) $\rightarrow$ Fokker-Planck-Kolmogorov (FPK)
equation governs the PDF (Kolmogorov 1941)


\item Alternately, Monte Carlo method can be used for solution of Langevin equation  $\rightarrow$ computationally very expensive

\end{itemize}

 


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\begin{frame} \frametitle{\Large{Stochastic Systems: Random Forcing}}

%
 

\begin{itemize}
%

\item  Classical approach: relationship between the autocorrelation function and
spectral  density  function  (Wiener  1930)  $\rightarrow$  Paved the way to the
solution of general stochastic differential equation (SDE)


\item SDEs with random forcing $\rightarrow$ Highly developed mathematical theory for It{\^o} type equation:
%
%
\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
%
%
\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}

 


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\begin{frame} \frametitle{\Large{Stochastic Systems: Random Coefficient}}

%
 


\begin{itemize}

%\vspace*{0.5cm}
\item Approximate Solution methods
              
      \begin{itemize}

      \item Functional integration approach (Hopf 1952)

      \item Averaged equation approach (Bharrucha-Reid 1968) 

      \item Numerical approaches  

      \item Monte Carlo method

      \end{itemize}



\vspace*{0.5cm}
\item FPK equation for the characteristic functional of the solution for problem of 
      wave propagation in random media (Lee 1974)


\vspace*{0.5cm}
\item Eulerian-Lagrangian form of FPK equation for probabilistic solution of flow through porous media (Kavvas 2003)

\end{itemize}


 

           

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\subsection{Uncertainties in Material}
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\begin{frame} \frametitle{\Large{Material Uncertainties}}


\begin{itemize}

%\vspace*{0.5cm}
\item  Material's  (concrete,  metals,  soil,  rock,  bone,  foam,  powder etc.)
       behavior is inherently uncertain

        \begin{itemize}

        \item Spatial variability

        \item Point-wise uncertainty -  testing error, transformation error

        \end{itemize}

\vspace*{0.5cm}
\item  Failure mechanisms related to spatial variability (strain localization and
       bifurcation of response)

\vspace*{0.5cm}
\item Inverse problems

      \begin{itemize}
      
      \item New material design, ({\it point-wise})

      \item Solid and/or structure design (or retrofits),  ({\it spatial})

      \end{itemize}
                                  
\end{itemize}

 


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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=6.5cm]{/home/jeremic/tex/works/Conferences/2006/KragujevacSEECCM06/Presentation/MGMuzorak01.jpg}
\end{center}
%\end{center}
\end{figure}



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\begin{frame} \frametitle{{Soil: Spatial Variation (Mayne et al. (2000))}}


\begin{figure}[!hbpt]
%\nonumber
\begin{center}
%\includegraphics[height=4cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/FrictionAngleProfile.jpg}
\includegraphics[height=6.5cm]{/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}



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%  \begin{frame} \frametitle{\Large{Motivation}}
%  
%  
%  \center{Typical Coefficients of Variation of Different Soil Properties} 
%  
%  \center{(After Lacasse and Nadim 1996)}
%  \begin{figure}[!hbpt]
%  %\nonumber
%  %\begin{flushleft}
%  \begin{center}
%  %\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
%  %\includegraphics[height=12cm]{TypicalSoilProfile.jpg}
%  %\caption{Soil Variability in Relatively Homogeneous Soil Deposit}
%  %
%  %\end{flushleft}
%  %\begin{flushright}
%  \includegraphics[width=25cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/TableTypicalCOV.jpg}
%  %\tiny{Lacasse and Nadim (1996), "Uncertainties in Characterizing Soil Properties", Proceedings of Uncertainty '96, July 31-August 3, 
%  %1996, Madison, Wisconsin } 
%  %
%  %\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{\Large{Soil Uncertainties and Quantifications}}


 

\begin{itemize}

%
%\vspace*{0.5cm}
\item Natural variability of soil deposit  (Fenton 1999) $\rightarrow$ function of soil formation process 

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

\begin{itemize}

\item COVs and covariance structures of inherent variabilities

\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}


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

\end{itemize}

 




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\section{Boundary Value Problem}

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\subsection{Propagation of Uncertainties in Mechanics (Geomechanics) }
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\begin{frame} \frametitle{\Large{Propagation of Uncertainties in Mechanics}}

%


\noindent
Governing equation

       
       \begin{itemize}

       \item Dynamic problems  $\rightarrow$ $ M \ddot u + C \ddot u + K u = \phi $

       \item Static problems  $\rightarrow$   $ K u = \phi $

       \end{itemize}


\noindent
Existing solution methods

       
       \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 

             \end{itemize}

       
       \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 (Perturbation method,
               fails  if  COVs  of soil $>$ 20\% ; Spectral method $\rightarrow$
               elastic material)



             \end{itemize}
             


\end{itemize}

%
%\noindent
%Probabilistic inelastic (elastic--plastic, damage...) missing
%


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%--
%-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%-- \begin{frame} \frametitle{{It would be nice to be able to...}}
%-- 
%--  
%-- 
%-- \begin{itemize}
%-- 
%-- \item Constitutive label: Able to deal with probabilistic non--linear material (soil) properties  $\rightarrow$ Probabilistic Elasto--Plasticity
%-- 
%-- \begin{itemize}
%--                
%-- \item Overcome the drawbacks of \textit{Monte Carlo Technique} and \textit{Perturbation Method}
%-- 
%-- \item Obtain complete probabilistic description (PDF)
%-- 
%--          \begin{itemize}
%-- 
%--          \item Materials often fail at low probability (tails of PDF)
%-- 
%--          \end{itemize} 
%-- 
%-- \item Carry out sensitivity analysis 
%-- 
%--           \begin{itemize}
%-- 
%--           \item Advanced models are highly sensitive to fluctuations in soil parameters
%-- 
%--           \end{itemize}
%-- 
%-- \end{itemize}
%-- 
%-- 
%-- \item Finite element label: Overcome the drawbacks of \textit{Monte Carlo Technique} (computationally very expensive) and \textit{Perturbation Method} (fails
%-- if materials (soils) exhibit large COV) \\
%-- $\rightarrow$ Spectral Stochastic Elastic--Plastic Finite Element Method (SSEPFEM)
%-- 
%-- 
%-- \end{itemize}
%-- 
%--  
%-- 
%-- 
%-- 
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%-- \end{frame}
%-- 
%-- 




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\subsection{Stochastic Finite Element Method}
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\begin{frame} \frametitle{{Truncated Karhunen--Loeve Expansion \\ 
for Input Field }}


 

              \begin{itemize}

%              \vspace*{1cm}
              \item Input random fields represented in eigen-modes of covariance kernel
              
%--              $ S_u(x,\theta) = \bar S_u(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 [S_u(x,\theta) - \bar S_u (x)] f_i (x) dx $
%--  %             
%--  %             \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*{1cm}
               \item Error minimizing property

%              \vspace*{1cm}
               \item Minimizes  number of stochastic dimensions


                \end{itemize}
             


\vspace*{-0.5cm}

%
\begin{figure}[!hbpt]
\begin{center}
\includegraphics[height=1.7cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ActualExponentialCovarianveSurface.jpg}
\hfill
%\hspace*{0.3cm}
\includegraphics[height=1.7cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/KL_ApproxWith_1Term_CovarianveSurface.jpg}
\hfill
%\hspace*{0.3cm}
\includegraphics[height=1.7cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/KL_ApproxWith_2Terms_CovarianveSurface.jpg}
\hfill
%\hspace*{0.3cm}
\includegraphics[height=1.7cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/KL_ApproxWith_3Terms_CovarianveSurface.jpg}
\end{center}
\end{figure}
%
\vspace*{-0.5cm}
\small{ Exact cov. surface \hspace*{0.5cm}  1-term \hspace*{1.00cm}  2-terms  \hspace*{1.35cm} 3-terms }

\vspace*{0.3cm}


% %\begin{flushright}
% 
% %
% \begin{figure}[!hbpt]
% \begin{center}
% \includegraphics[height=1.0cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ShearStrengthProfile.jpg}
% \end{center}
% \end{figure}
% 
% 


%\vspace*{1cm} 
Model covariance kernel: $ C(x_1, x_2) = e^{-| x_1 - x_2 | /b} $  
\\
Truncated K-L approximation: $ C(x_1, x_2) = \sum_{k =1}^M \lambda_k f_k(x_1) f_k(x_2) $

%
%\small{Two-terms approximation \ \ \ \ \ \ \ \ \ \ \ \  Three-terms approximation}


%\begin{center}
%{K-L Expansion of Covariance Kernel}
%\end{center}

%\end{itemize}

 


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\begin{frame} \frametitle{\Large{Polynomial Chaos (PC) Expansion for DOFs}}


 

%\begin{itemize}


%
%\item Spectral stochastic finite element, Contd.....

%      \item Solution (displacement) random field $\rightarrow$ Can not use K-L expansion directly

              \begin{itemize}

              \item  DOF  covariance  kernel  is  not  known  a  priori (unknown
              eigenvalues $e_j$ and eigenvectors $b_j(x)$) 
              $u(x,\theta)=\sum_{j=1}^L e_j \;\; \chi_j(\theta) \;\; b_j(x)$
              
              \item DOFs expressed  as  functionals  of  known  input random
              variables and unknown deterministic function 
              $u(x,\theta)=\zeta[\xi_i(\theta),x]$
              


              \item Need a basis of known random variables $\rightarrow$ PC expansion
              $\chi_j(\theta)=\sum_{i=0}^P\gamma_i^{(j)}\psi_i\left[\left\{\xi_r\right\}\right]$; 
              \\
              $u(x,\theta)=\sum_{j=1}^L \sum_{i=0}^P \gamma_i^{(j)} \psi_i[\{\xi_r\}]e_j b_j(x) = 
              \sum_{i=0}^P \psi_i[\{\xi_r\}] d_i(x)$
              
              \item Deterministic coefficients can be found by minimizing norm of error of finite 
                   representation (e.g. using Galerkin scheme) 
              
               


              \end{itemize}
 



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\begin{frame} \frametitle{\Large{SSEPFEM Formulation}}

%
 

%\begin{itemize}


%
%\item Spectral stochastic finite element, Contd.....

\vspace*{-0.8cm}

      
      \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*{-0.3cm}

      \begin{normalsize}
      \begin{equation}
      \nonumber
      K_{mn} = \int_D B_n D  B_m dV   \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ K'_{mnk} = \int_D B_n \sqrt \lambda_k h_k B_m dV
      \end{equation}

%      \begin{equation}
%      \nonumber
%      K'_{mnk} = \int_D B_n(x) \sqrt \lambda_k h_k(x) B_m (x) dV
%      \end{equation}

\vspace*{-0.3cm}
      
      \begin{equation}
      \nonumber
      C_{ijk} = \left < \zeta_k(\theta) \psi_i[\{\zeta_r\}] \psi_j[\{\zeta_r\}] \right >  \ \ \ \ \ \ \ \ \ \ \ \  F_m = \int_D \phi N_m dV  \ \ \ \ \ \ \ \ \ \ \ \
      \end{equation}

      \end{normalsize}

      \begin{itemize}

\item Generalized DOF

%-- 
%--                 \begin{figure}[!hbpt]
%--   %              \begin{flushleft}
%--                 \begin{flushright}
%--                 \includegraphics[height=6cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/DamMesh.jpg}
%--                  \end{flushright}
%--   %              \end{flushleft}
%--   %              
%--   %             \begin{flushright}
%--   %             \begin{equation}
%--   %             \nonumber
%--   %             \begin{normalsize}
%--   %             Material (Soil) nonlinearity \ \ \ \ \\
%--   %             \ \\
%--   %             $\rightarrow$ Constitutive integration \\
%--   %             at the Gauss points \\
%--   %             (Probabilistic Elasto--Plasticity)
%--   %             \end{equation}
%--   %              \end{normalsize}
%--   %             \end{flushright}
%--                 \end{figure}
%--   
\item Material (soil) nonlinearity $\rightarrow$ Constitutive integration at Gauss point 
$\rightarrow$ Probabilistic Elasto--Plasticity
      
\end{itemize}
      
%\end{itemize}

 




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\section{Probabilistic Elasto--Plasticity}

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\subsection{Probabilistic Elastic--Plastic:  Differential Equation}
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\begin{frame} \frametitle{\Large{Uncertainty Propagation through Constitutive Eq.}}

% 

\begin{itemize}

 


\item General 3-D elastic-plastic constitutive law $\rightarrow$  $ \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}
           -
           \displaystyle \displaystyle \frac{D^{el}_{ijmn}
           \displaystyle \frac{\partial U}{\partial \sigma_{mn}}
           \displaystyle \frac{\partial f}{\partial \sigma_{pq}}
           D^{el}_{pqkl}}
           {\displaystyle \frac{\partial f}{\partial \sigma_{rs}}
           D^{el}_{rstu}
           \displaystyle \frac{\partial U}{\partial \sigma_{tu}}
           -
           \displaystyle \frac{\partial f}{\partial q_*}r_*}

           \;\;\; & \mbox{\large{~for elastic--plastic}}
%
           \end{array} \right.
           \end{eqnarray}

%\end{normalsize}

 


\item Non--linear coupling in the coefficient (elastic--plastic modulus)


\item Focusing on 1-D constitutive Behavior $\rightarrow$ 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{\Large{Previous Works}}

 

\begin{itemize}

\item Linear algebraic or differential equations  $\rightarrow$ Analytical solution:

         \begin{itemize}
        \item Variable Transformation Method (Montgomery and Runger 2003)
        \item Cumulant Expansion Method (Gardiner 2004)
        \end{itemize}

\item Nonlinear differential equations (elasto-plastic/viscoelastic-viscoplastic):
     
      \begin{itemize}
      \item Monte Carlo Simulation (Schueller 1997, De Lima et al 2001, Mellah et al. 2000, Griffiths et al. 2005...)
      \item Perturbation Method (Anders and Hori 2000, Kleiber and Hien 1992, Matthies et al. 1997)
      \end{itemize}


\item Monte Carlo method: accurate, very costly

\item Perturbation method: first and second order Taylor series expansion about mean - 
limited to problems having small C.O.V. and inherits 'closure problem' 

\end{itemize}

 



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\begin{frame} \frametitle{\Large{Problem Statement}}
 


 
 
\begin{itemize}

\item General 3-D elastic-plastic constitutive law:
% 
%    
    \begin{equation}
    \nonumber
    d\sigma_{ij} = \left \{ D_{ijkl}^{el} - \displaystyle \frac{D_{ijmn}^{el} \displaystyle \frac{\partial U}{\partial \sigma_{mn}}
    \displaystyle \frac{\partial f}{\partial \sigma_{pq}} D_{pqkl}^{el}} 
     {\displaystyle \frac{\partial f}{\partial \sigma_{rs}}D_{rstu}^{el}\displaystyle \frac{\partial U}{\partial \sigma_{tu}}
     - \displaystyle \frac{\partial f}{\partial q_*} r_*} \right \}  d\epsilon_{kl}
    \end{equation}
   


 

 \item Focusing on 1-D constitutive Behavior $\rightarrow$ 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)
 \end{eqnarray}
%
with an initial condition $\sigma(0)=\sigma_0$

 \end{itemize}

 
 

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\end{frame}






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\begin{frame} \frametitle{\Large{Stochastic Continuity (Liouville) Equation}}

 

%\begin{itemize}

%\item 
              
              \begin{figure}[!hbpt]
              \begin{flushleft}
              \includegraphics[height=5.5cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/CloudOfPoints01.jpg}
\hspace*{0.5cm}
              \end{flushleft}
              
              \begin{flushright}
%             \begin{equation}
%             \nonumber
%             \begin{normalsize}
\vspace*{-6cm}
              $ \displaystyle \frac{\partial \rho (\sigma(x,t),t)}{\partial t} = \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ \\
              $ - \displaystyle \frac{\partial}{\partial \sigma} \left [ \vphantom{\displaystyle \frac{\partial}{\partial \sigma}} \eta (\sigma(x,t),  D^{el}(x), q(x), \right. \ \ $ \\
              $  \left.  r(x), \epsilon(x,t)) \vphantom{\displaystyle \frac{\partial}{\partial \sigma}} \right ] \rho[\sigma(x,t),t]    $ \\
              \ \\
              Initial condition: \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
              \ \\
              $\rho(\sigma,0)=\delta(\sigma-\sigma_0) \ \ \ \ \ \ \ \ \ \ \ \ $
              \ \\
              \ \\
              $\rho(\sigma,0) \rightarrow$ density  of \ \ \ \ \ \ \ \ \ \ \ \\
              probabilistic solutions in \ \ \ \ \ \ \ \ \ \ \ \\  
              $\sigma$ space  \ \ \ \ \ \ \ \ \ \ \  
%             \end{equation}
%              \end{normalsize}
              \end{flushright}
              \end{figure}





%
%\item The 1-D constitutive equation can be visualized that from each initial point in $\sigma$-space a trajectory starts out which 
%describes the corresponding solution of the stochastic process

%
%\item When we consider a cloud of initial points described by density $ \rho (\sigma,0) $ in $\sigma$-space, and movement of all these points is 
%dictated by the constitutive equation, the phase density $ \rho $ of $\sigma(x,t)$ varies in time according to a continuity equation 
%(Liouville equation): 

%
%\begin{equation}
%\nonumber
%\frac{\partial \rho (\sigma(x,t),t)}{\partial t}=-\frac{\partial}{\partial \sigma} \eta (\sigma(x,t), D^{el}(x), q(x), r(x), \epsilon(x,t)) \ \ \rho[\sigma(x,t),t]
%\end{equation}

%

%with initial condition

%

%\begin{equation}
%\nonumber
%\rho(\sigma,0)=\delta(\sigma-\sigma_0)
%\end{equation}

%\end{itemize}

 

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\end{frame}






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\begin{frame} \frametitle{{Ensemble Average form of Liouville Equation}}



 

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


%\noindent
$\rightarrow$ 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{\Large{Eulerian--Lagrangian FPK Equation}}

%

%\begin{itemize}

\begin{footnotesize}



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

\end{footnotesize}


 
\vspace*{-0.5cm}

\begin{itemize}


\item Complete probabilistic description of response 


\item Second-order exact to covariance of time


\item Deterministic equation (in probability density space)

\end{itemize}

 


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\end{frame}






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%tempout %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%tempout %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%tempout \begin{frame} \frametitle{\Large{Eulerian-Lagrangian FPK Equation}}
%tempout 
%tempout 
%tempout \vspace*{-0.5cm}
%tempout  
%tempout 
%tempout \begin{itemize}
%tempout 
%tempout \item $Cov_0[\cdot]$ $\rightarrow$ Time-ordered covariance function
%tempout %
%tempout %
%tempout \begin{eqnarray}
%tempout \nonumber
%tempout &&Cov_0[\eta(x,t_1);\eta(x,t_2)] \\
%tempout && \ \ \ \ \ \ \ \ \ \ \ \ \ \ = \left < \eta(x,t_1)\eta(x,t_2) \right > - \left < \eta(x,t_1) \right > \cdot \left < \eta(x,t_2) \right >
%tempout \nonumber
%tempout \end{eqnarray}
%tempout 
%tempout \vspace*{0.5cm}
%tempout \item Real-space location (Lagrangian) $x_t$ is known but pull-back to Eulerian location $x_{t-\tau}$ is unknown 
%tempout 
%tempout \item Can be related using strain rate $\dot \epsilon \ (=d\epsilon/dt)$
%tempout 
%tempout 
%tempout \begin{equation}
%tempout \nonumber
%tempout d\epsilon = \dot \epsilon \tau =\frac{x_t-x_{t-\tau}}{x_t}    \mbox{; \ \ \ \  or, \ \ \ }  x_{t-\tau}=(1-\dot \epsilon \tau)x_t 
%tempout \end{equation}
%tempout 
%tempout \end{itemize}
%tempout 
%tempout  
%tempout 
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%tempout 
%tempout \end{frame}
%tempout 
%tempout 
%tempout 
%tempout 
%tempout 
%tempout %-- 
%--  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%--  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%--  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%--  \begin{frame} \frametitle{\Large{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}
%--  
%--  
%--  



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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \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 3-D probabilistic stress-strain response
% %-- 
% %-- \end{itemize}
% %-- 
% %--  
% 
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% \end{frame}
% 
% 
% 
% 
% 

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\begin{frame} \frametitle{\Large{Solution of FPK Equation}}

 

\begin{itemize}


\item FPK equation $\rightarrow$ advection-diffusion equation or continuity equation
%
%
%\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 Numerical scheme $\rightarrow$ \textit{Finite Difference Technique}

\end{itemize}

 

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\end{frame}






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%==%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%==\begin{frame} \frametitle{\Large{Solution of FPK Equation: Numerical Scheme}}
%==
%== 
%==
%==\begin{itemize}
%==
%==  
%==  \item \textit{Method of Lines} $\rightarrow$ semi-discretization 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_RandomG-m.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 (reduced-order modeling)
%==
%==\end{itemize}
%==
%==\end{itemize}
%==
%==\end{itemize}
%==
%== 
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%==
%==\end{frame}






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\begin{frame} \frametitle{\Large{Application of FPK equation to Material Models}}

 

\begin{itemize}

\item  FPK  equation  is applicable to any incremental elastic--plastic material
model (only the coefficients $N_{(1)}$ and $N_{(2)}$ differ)


\item Unique attributes of probabilistic solution


      \begin{itemize}

      \item Solution in terms of PDF, not a single value of stress

      \item Influence of initial condition on the PDF of stress



       \item Transition between elastic and elastic--plastic

       \item Symmetry and non--symmetry in PDF of stress

       \item Differences in mean, mode and deterministic solution of stress

       \item Interaction of random soil properties on the PDF of stress


\end{itemize}

\end{itemize}

 


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\end{frame}







\subsection{Probabilistic Elastic--Plastic Response}
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\begin{frame} \frametitle{\Large{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} <G>
\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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\end{frame}






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\begin{frame} \frametitle{\Large{Elastic Response with Random $G$}}


\begin{figure}[!hbpt]
%\nonumber
%\begin{flushleft}
\begin{center}
%\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
\includegraphics[height=6cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/PDFElastic_RandomG-m.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}
$<G>$ = 2.5 MPa;
Std. Deviation$[G]$ = 0.5 MPa

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\end{frame}






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\begin{frame} \frametitle{\Large{Verification -- Variable Transformation Method}}


\begin{figure}[!hbpt]
\begin{center}
\includegraphics[height=6cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/MonteCarlo_Elastic-m.pdf}
%  %
%  \hspace*{-3cm}
%  \vspace*{4cm}
%  %
%  \includegraphics[height=4cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ReasonOfNotMatchingMonteCarlo-m.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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\end{frame}






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\begin{frame} \frametitle{\Large{Drucker-Prager Linear Hardening with Random $G$}}

 

%
%  
%  
%  General form of Drucker-Prager elastic-plastic 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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\begin{frame} \frametitle{\large{Drucker-Prager 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_RandomG-m.pdf}
\hfill
\includegraphics[height=5.0cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourDruckerPrager_RandomG-m.pdf}
\end{center}
\end{figure}




\begin{itemize}

\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_RandomG-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{flushleft}
% \end{flushright}
% \end{figure}
% 
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\end{frame}






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\begin{frame} \frametitle{\Large{Verification of D--P E--P 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_DruckerPrager-m.pdf}
%
%\end{flushleft}
%\begin{flushright}
%\includegraphics[height=11.5cm]{ContourDruckerPrager_RandomG-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{flushleft}
%\end{flushright}
\end{center}
\end{figure}

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\end{frame}






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\begin{frame} \frametitle{\Large{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(2p-p_0)^2}{\kappa} + \left(18 \displaystyle \frac{G}{M^4}\right) \sigma_{12}^2
+ \displaystyle \frac{1+e_0}{\lambda-\kappa} p p_0 (2p-p_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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\end{frame}






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\begin{frame} \frametitle{\Large{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_RandomG-m.pdf}
\hfill
\includegraphics[height=5.4cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourLowOCR_RandomG-m.pdf}
\end{center}
\end{figure}

\begin{itemize}

\item Approximation of I.C.


%\item Wide transition between el. \& el.-pl.


\item Non-symmetry 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{\large{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_Randomp0-m.pdf}
\hfill
\includegraphics[height=5.5cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourLowOCR_RandomG_RandomM_Randomp0-m.pdf}
\end{center}
\end{figure}





\begin{itemize}

%\item Narrow transition between el. \& el.-pl.


\item Non-symmetry in probability distribution


\item Difference between mean, mode and deterministic responses



\item Divergence at critical state because $M$ is uncertain

\end{itemize}


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\end{frame}






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\begin{frame} \frametitle{\Large{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/CamClayPDFComparison-m.pdf}
\end{center}
\end{figure}                                                          

\vspace*{-5.5cm}

\begin{itemize}

\item None coincides with deterministic

\item Some cases are very uncertain while some are fairly certain

\item Either on safe or unsafe side

\end{itemize}

 


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\end{frame}






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\begin{frame} \frametitle{\large{High OCR Cam Clay with \\ Random $G$ and $M$}}

\vspace*{-1.5cm}

\begin{figure}[!hbpt]
\begin{center}
\includegraphics[height=3.4cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/PDFHighOCR_RandomG_RandomM-m.pdf}
\hfill
\includegraphics[height=5.4cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourHighOCR_RandomG_RandomM-m.pdf}
\end{center}
\end{figure}                                                          


\begin{itemize}

%\item Approximation of I.C.

%
\item Very uncertain transition between el. \& el.-pl.


%\item Large non-symmetry in probability distribution


\item {\bf Differences} between mean, mode, and deterministic responses


\item Divergence at critical state, $M$ is uncertain

\end{itemize}


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\end{frame}






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\begin{frame} \frametitle{\Large{Conclusions}}
%
 

\begin{itemize}

\item A new approach to account for uncertainties in elastic--plastic material simulation


\item Methodology, which results in a FPK equation, overcomes the drawbacks of \textit{Monte Carlo Method} and
\textit{Perturbation Technique}


\item Advantage of FPK equation is evident as it transforms the original non--linear stochastic ODE to a linear
deterministic PDE


\item Developed methodology is capable of providing complete probabilistic description (PDF) of the solution


\item Development is general in nature and applicable to any incremental elastic--plastic material model

\end{itemize}

 


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\end{frame}






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% \begin{frame} \frametitle{\Large{Inverse problem: Design of Geotechnical Site Characterization}}
% %
%  
% 
% \begin{itemize}
% 
% \item Client specified probability of failure $\rightarrow$ How detailed site characterization is necessary?
% (how many boreholes to perform, at which locations etc.)
% 
% 
% \begin{figure}[!hbpt]
% %\nonumber
% %\begin{flushleft}
% \begin{center}
% \includegraphics[width=17cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/InverseProblemSketch.jpg}
% \end{center}
% \end{figure}
% 
% \end{itemize}
% 
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% \begin{frame} \frametitle{\Large{Random Wave Propagation Through Stochastic Soil}}
% 
% 
%  
% 
% \begin{itemize}
% 
% \item Probabilistic seismic hazard analysis $\rightarrow$ How much of total uncertainty is due  to the 
% uncertainty in earthquake forcing and how much is due to uncertainty in soil?
% 
% \begin{figure}[!hbpt]
% %\nonumber
% %\begin{flushleft}
% \begin{center}
% \includegraphics[width=20cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/WavePropagationProblemSketch.jpg}
% \end{center}
% \end{figure}
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% \end{itemize}
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% \begin{frame} \frametitle{\Large{Probabilistic Simulation of Liquefaction and Lateral Spreading}}
% %
%  
% 
% \begin{itemize}
% 
% \item What's the probability that certain soil will liquefy ?
% 
% \item Use of u--p--U formulation (solid--fluid coupled formulation) for SSEPFEM and large deformation formulation of
% probabilistic elasto--plasticity
% 
% 
% \begin{figure}[!hbpt]
% %\nonumber
% %\begin{flushleft}
% \begin{center}
% \includegraphics[width=20cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/LiquefactionProblemSketch01.jpg}
% \end{center}
% \end{figure}
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% 
% \end{itemize}
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% \begin{frame} \frametitle{~}
% 
% 
% \begin{Huge}
% 
% \begin{center}
% 
% \textbf{Thank You}
% 
% \end{center}
% \end{Huge}
% 
% \end{frame}

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\end{document}
\bye


        
