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%  \usetheme{Hannover} % ima naslov i sadrzaj sa leve strane 
%  \usetheme{Singapore} % ima sadrzaj i tackice gore
%  \usetheme{Antibes}  % ima sadrzaj gore i kao graf ...
%  \usetheme{Berkeley} % ima sadrzaj desno
%  \usetheme{Berlin} % ima sadrzaj gore i tackice
%  \usetheme{Goettingen} % ima sadrzxaj za desne strane
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%%%%%%%
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%%%%  HYPERREF  HYPERREF  HYPERREF  HYPERREF  HYPERREF 
%%%%  HYPERREF  HYPERREF  HYPERREF  HYPERREF  HYPERREF 
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% does not look nice, try deleting the line with the fontenc.




%\title{NRC Staff Capacity Building: \\
%       Micromechanical Origins of Elasto-Plasticity }
\title{ 
Modeling and Simulation of \\
Earthquake Soil Structure Interaction for \\
Risk Informed Decision Making}


%\subtitle
%{Include Only If Paper Has a Subtitle}

%\author[Author, Another] % (optional, use only with lots of authors)
%{F.~Author\inst{1} \and S.~Another\inst{2}}
% - Give the names in the same order as the appear in the paper.
% - Use the \inst{?} command only if the authors have different
%   affiliation.
\pgfdeclareimage[height=0.2cm]{university-logo}{/home/jeremic/BG/amblemi/ucdavis_logo_blue_sm}
\pgfdeclareimage[height=0.7cm]{lbnl-logo}{/home/jeremic/BG/amblemi/lbnl-logo}

\author[Jeremi{\'c}] % (optional, use only with lots of authors)
{Boris~Jeremi{\'c} \\ 
{\small with
          Dr. Tafazzoli, 
          Mr. Abell, 
          Mr. Jeong, 
          Dr. Pisan{\`o}, 
          Prof. Sett (UA), 
          Prof. Taiebat (UBC), 
          Prof. Yang (UAA),
          Dr. Cheng (Itasca)} 
}
%{Boris~Jeremi{\'c},  Nima  Tafazzoli,  Babak Kamrani, Panagiota Tasiopoulou and
%Chang-Gyun Jeong}
% - 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}
\institute[\pgfuseimage{university-logo}\hspace*{0.1truecm}\pgfuseimage{lbnl-logo}] % (optional, but mostly needed)
{ Professor, University of California, Davis, CA\\
%  and\\ 
  Faculty Scientist, Lawrence Berkeley National Laboratory, Berkeley, CA }
% - Use the \inst command only if there are several affiliations.
% - Keep it simple, no one is interested in your street address.


\date[] % (optional, should be abbreviation of conference name)
{\small Idaho National Laboratory \\ July 2012}

\subject{}
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% If you have a file called "university-logo-filename.xxx", where xxx
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%\logo{\pgfuseimage{university-logo}}

% \pgfdeclareimage[height=0.5cm]{university-logo}{university-logo-filename}
% \logo{\pgfuseimage{university-logo}}




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\begin{frame}
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\section{Motivation}
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\subsection{Motivation}
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\begin{frame}
 \frametitle{The Problem}

\begin{itemize}
  

  \item Seismic response of Nuclear Power Plants


\vspace*{0.1cm}
  \item  3D, inclined seismic motions consisting of body and surface waves

\vspace*{0.1cm}
  \item   Inelastic (elastic, damage, plastic behavior of materials: soil, rock,
  concrete, steel, rubber, etc.)

\vspace*{0.1cm}
  \item  Full coupling of pore fluids (in soil and rock) with soil/rock skeleton

\vspace*{0.1cm}
  \item  Buoyant effects (foundations below water table)

\vspace*{0.1cm}
  \item    Uncertainty  in  seismic  sources, path, soil/rock response and structural
  response
  

\end{itemize}
\end{frame}
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\begin{frame}
 \frametitle{Potential Solution}

\begin{itemize}
  


%\vspace*{0.3cm}
  \item    {\bf Physics based  modeling  and  simulation} of seismic behavior of
  soil-structure systems (NPP structures, components and systems)

\vspace*{0.1cm}
  \item             Development  and  use  of  {\bf high  fidelity} time domain,
  nonlinear numerical  models,
  in {\bf deterministic} and  {\bf probabilistic} spaces


\vspace*{0.1cm}
  \item   Accurate following of  the  {\bf flow  of  seismic 
  energy}    (input  and  dissipation)   within soil-structure NPP system

\vspace*{0.1cm}
  \item  {\bf  Directing},      in   space   and   time,  with {\bf high (known)
  confidence}, seismic energy flow in the soil-foundation-structure system

%\vspace*{0.1cm}
%  \item  {\bf Education} for researchers, professional practice.
  

\end{itemize}
\end{frame}

% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%  \frametitle{Hypothesis}
% 
% \begin{itemize}
%   
% 
% 
% %\vspace*{0.5cm}
%   \item    Interplay   of  the  Earthquake  with the   
%   Soil, Foundation  and  Structure in time domain plays  major
%   role in  failures (and successes).
% 
% 
% \vspace*{0.2cm}
%   \item   Timing  and spatial location of energy dissipation determines location
%   and amount of damage.
% 
% \vspace*{0.2cm}
%   \item  If timing and spatial location of energy dissipation 
%          can be controlled (directed, designed),
%          we could  optimize soil-foundation-structure system for
%     \begin{itemize}
%       \item Safety and
%       \item Economy
%     \end{itemize}
% 
% \end{itemize}
% \end{frame}
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\section{Deterministic Modeling}


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\subsection{NRC ESSI Simulator System}

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% - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - \begin{frame} 
% - \frametitle{Project Goals}
% - 
% - \begin{itemize}
% - 
% - %\vspace*{0.5cm}
% - \item    Development    of    the    NRC ESSI Simulator System 
% - for Hi-Fi modeling and simulation of non-linear earthquake 
% - soil/rock structure interaction problems:
% -   \begin{itemize}
% - %    \item Time domain, nonlinear, parallel finite element program:
% -     \item  {\bf NRC-ESSI-Program}
% - %    \item  High  performance, parallel computer: 
% -     \item  {\bf NRC-ESSI-Computer}
% - %    \item  Educational endeavor, documentation:  
% -     \item  {\bf NRC-ESSI-Notes}
% -   \end{itemize}
% - 
% - \vspace*{0.1cm}
% - \item  Education:  NRC  Staff  Capacity  Building (seminars, short courses, 
% - NRC ESSI Notes, advising), targeting wider audience as well
% - 
% - \vspace*{0.1cm}
% - \item  Development  of ESSI case  studies: 
% - 3D, inclined seismic   motions,  soil/rock;  
% - foundation  interface  slip,  seismic  energy propagation dissipation
% - 
% - %\vspace*{0.2cm}
% - %\item 
% - %
% - %\vspace*{0.2cm}
% - %\item 
% - %
% - %\vspace*{0.2cm}
% - %\item 
% - 
% - \end{itemize}
% - 
% - \end{frame}
% - 
% - 
% - 
% - 
% - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}
\frametitle{NRC ESSI Simulator System}

\begin{itemize}

\item {\bf The NRC-ESSI-Program} is a 3D, nonlinear, time domain, 
parallel finite element program specifically developed for
Hi-Fi modeling  and  simulation  of  Earthquake  Soil/Rock  Structure
Interaction problems for NPPs on NRC-ESSI-Computer. \
%The NRC ESSI Program is based on
%a  number of public domain numerical libraries developed at UCD as well as those
%available  on  the  web,  that  are  compiled  and  linked  together to form the
%executable  program (NRC-ESSI-Program). Significant effort is devoted to development
%of  verification  and  validation  procedures,  as  well  as  on  development of
%extensive  documentation.  NRC-ESSI-Program  is  in  public  domain  and is licensed
%through the Lesser GPL.

%\vspace*{0.3cm}
\vspace*{0.1cm}
\item {\bf The NRC-ESSI-Computer} is a distributed memory 
parallel computer, a cluster of clusters with multiple performance  
processors and multiple performance networks.  
%Compute  nodes are Shared Memory Parallel
%(SMP)  computers,  that  are  connected,  using  high  speed  network(s), into a
%Distributed Memory Parallel (DMP) computer.

%\vspace*{0.3cm}
\vspace*{0.1cm}
\item    {\bf        The    NRC-ESSI-Notes} represent a hypertext 
documentation system
%(Theory and Formulation, Software and Hardware, Verification and Validation, and
%Case  Studies  and  Practical  Examples)   
detailing modeling and simulation of NPP ESSI
problems.
%
%the
%NRC-ESSI-Program  code  API (application Programming Interface) and DSLs (Domain
%Specific Language).
%%NRC-ESSI-Notes,    developed  by Boris Jeremic and collaborators, are in public
%domain
%%and  are  licensed  under a Creative Commons Attribution-ShareAlike 3.0 Unported
%%License.
%
%\vspace*{0.3cm}
\end{itemize}

\end{frame}



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\begin{frame}
\frametitle{NRC ESSI Simulator Program}

\begin{itemize}

%\vspace*{0.2cm}
\item  Based  on  a  Collection  of Useful Libraries  (modular, portable)

\vspace*{0.1cm}
\item Library centric software design

\vspace*{0.1cm}
\item Various public domain licenses (GPL, LGPL, BSD, CC)

%\vspace*{0.3cm}
\vspace*{0.1cm}
\item Verification and Validation


\vspace*{0.1cm}
\item  Detailed program documentation  (part of NRC ESSI Notes)
    
\vspace*{0.1cm}
\item Target users: U.S.-NRC staff, UCD students, external users 
    
%\item Sources will be available through
%{\bf 
%\url{http://nrc-essi-simulator.info}}


\end{itemize}

\end{frame}



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\begin{frame}[fragile]
 \frametitle{NRC ESSI Simulator Computer}

  A distributed memory parallel (DMP) computer
   designed for high performance,  
   parallel finite element simulations

 \begin{itemize}
%\vspace*{0.1cm}
   \item Multiple performance CPUs \\
         and Networks
%\vspace*{0.1cm}
   \item Most cost-performance \\
         effective 
%\vspace*{0.1cm}
   \item Source compatibility with \\
         any DMP supercomputers
%\vspace*{0.1cm}
   \item Current system: 208 CPUs
%\vspace*{0.1cm}
   \item Near future: 784 CPUs

 \end{itemize}

  
\vspace*{-4.5cm}
\begin{flushright}
%\hspace*{-0.5cm}
\includegraphics[width=5.0cm]{/home/jeremic/public_html/NRC_ESSI_Simulator/NRC_ESSI_Computer/photos/IMG_2607.JPG}
%\includegraphics[width=6.0cm]{/home/jeremic/public_html/NRC_ESSI_Simulator/NRC_ESSI_Computer/photos/IMG_2609.JPG}
%\includegraphics[width=8.0cm]{/home/jeremic/public_html/NRC_ESSI_Simulator/NRC_ESSI_Computer/photos/IMG_2611.JPG}
\end{flushright}



\end{frame}

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\begin{frame}
\frametitle{NRC ESSI Simulator Notes}

\begin{itemize}

\item A hypertext documentation system  describing in detail 
modeling and simulations of NPP ESSI
problems 

\vspace*{0.1cm}
\item  Theoretical  and  Computational  Formulations  
\vspace*{0.1cm}
\item  Software and Hardware Platform Design 
\vspace*{0.1cm}
\item  Verification and Validation 
\vspace*{0.1cm}
\item Application Example


\end{itemize}

\end{frame}

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%

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\begin{frame}
% \frametitle{Seismic Energy Dissipation for \underline{Soil}-Foundation-Structure Systems}
 \frametitle{High Fidelity Modeling}
% \frametitle{Seismic Energy Dissipation for 
%             \underline{Soil}-Foundation-Structure Systems}


\begin{itemize}

  \item   Energy  influx,  \underline{body  and  surface waves, 3D, inclined} 
%  $E_{flux} = \rho A c \int_0^t \dot{u}_i^2 dt$ (Aki \& Richards)

\vspace*{0.1cm}
     \item Mechanical dissipation outside of SFS domain:
       \begin{itemize}
         \item \underline{Radiation} of reflected waves
         \item \underline{Radiation} of oscillating SFS system
       \end{itemize}
\vspace*{0.1cm}
     \item Mechanical dissipation inside SFS domain:
       \begin{itemize}
         \item \underline{Plasticity} of soil/rock subdomain
         \item \underline{Plasticity} of foundation -- soil/rock interface
         \item \underline{Viscous coupling} of porous solid with pore fluid (air,
         water)
         \item Plasticity/damage of the structure
         \item Viscous coupling of structure/foundation with fluids 
%         \item potential and kinetic energy
%         \item[] potential $\leftarrow \! \! \! \! \! \! \rightarrow$ kinetic energy
       \end{itemize}                           



\vspace*{0.1cm}
%   \item Numerical energy dissipation (numerical damping/production and period  errors)
%   \item Numerical energy dissipation (damping/production)
   \item Numerical energy dissipation/production


\end{itemize}

%
\end{frame}
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\begin{frame}
\frametitle{High Performance, Parallel Computing}

\begin{itemize}

\item The NRC ESSI Simulator can be used in both sequential and 
parallel modes

\vspace*{0.2cm}
\item For high fidelity models, parallel is really the only option

\vspace*{0.2cm}
\item  High  performance, parallel computing using 
Plastic Domain Decomposition
Method

\vspace*{0.2cm}
\item    Developed  for  multiple/variable capability CPUs and 
networks (DMP and
SMPs)

\end{itemize}

\end{frame}







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% - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - \begin{frame}
% - \frametitle{Earthquake Ground Motions}
% - 
% - 
% - 
% - \begin{itemize}
% - 
% - 
% - %\vspace*{0.5cm}
% - \item  Realistic earthquake ground motions 
% -    \begin{itemize}
% -      \item Body: P and S waves
% -      \item Surface: Rayleigh, Love waves, etc.
% -      \item Lack of correlation (incoherence)
% -      \item Inclined waves
% -      \item 3D waves
% -      \item Earthquake energy  dissipation
% - 
% -    \end{itemize}
% - 
% - 
% - 
% - 
% - 
% - \end{itemize}
% - 
% - 
% - 
% - 
% - \end{frame}
% - 
% - 
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% - 
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% - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - \begin{frame}
% - \frametitle{Body (P, S) and Surface (Rayleigh, Love) Waves}
% - 
% -  
% - \vspace*{-0.3cm}
% - \begin{figure}[!hbpt]
% - \begin{center}
% - \includegraphics[width=2.5cm, angle=45]{/home/jeremic/tex/works/consulting/2010/CanadianNuclearSafetyComission/Presentation/P_body_wave.jpeg}
% - \includegraphics[width=2.5cm, angle=45]{/home/jeremic/tex/works/consulting/2010/CanadianNuclearSafetyComission/Presentation/S_body_wave.jpeg}
% - \vspace*{-0.5cm}
% - \\
% - \includegraphics[width=3cm]{/home/jeremic/tex/works/consulting/2010/CanadianNuclearSafetyComission/Presentation/Rayleigh_surface_wave.jpeg}
% - \includegraphics[width=3cm]{/home/jeremic/tex/works/consulting/2010/CanadianNuclearSafetyComission/Presentation/Love_surface_wave.jpeg}
% - %\caption{\label{Love_surface_wave}   Visualization  of  propagation  of  a  Love
% - %surface seismic wave (illustrations are from MTU web site).}
% - \end{center}
% - \end{figure}
% - 
% - \end{frame}
% - 
% - 
% - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - 
% - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{frame}
\frametitle{Representative NPP Example Problem}

\vspace*{-0.50cm}

\begin{itemize}

\item Body and surface seismic waves

\item Seismic wave frequencies \\
      up to $50$Hz 

\item Elastic-plastic soil/rock and \\
      structural components,

\item Inelastic contact/gap 

\item Seismic isolator effects

\item Buoyant effects for deep foundation embedment 


\item High Fidelity Model: soil block: $230m \times 230m \times100m$, foundation
$90m  \times  90m$  Containment  Structure:  $40m \times 50m$, 2.1 Million DOFs,
700,000 elements,

\end{itemize}



\vspace*{-6.0cm}
\begin{flushright}
\hspace*{1cm}
\includegraphics[width=5.0cm]{/home/jeremic/tex/works/Conferences/2012/Seismic_research_issues_for_NPPs/Large_NPP_model_01A.jpg}
\end{flushright}








\end{frame}
































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\subsection{Verification and  Validation Suite}




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\begin{frame}
 \frametitle{Verification, Validation and Prediction}


\begin{itemize}

     \item Verification: the  process  of  determining  that    a  model
   implementation  accurately  represents the developer's conceptual description
   and specification. Mathematics issue. {\it Verification provides evidence that the
   model is solved correctly.}

     \item Validation:  The process of determining the degree to which a
   model  is  accurate  representation of the real world from the perspective of
   the  intended  uses  of  the  model.  Physics issue. {\it Validation provides
   evidence that the correct model is solved.}

   \item Prediction:  use  of  computational  model  to foretell the state of a
   physical   system   under  consideration  under  conditions  for  which  the
   computational model has not been validated
\end{itemize}

%
%\item Models available (some now, some later)
%\vspace*{-2.0cm}
\end{frame}
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% 
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%  \begin{frame}
%   \frametitle{Role of Verification and Validation}
%  
%  
%  \begin{figure}[!h]
%  \begin{center}
%  \hspace*{-2cm}
%  {\includegraphics[width=5.0cm]{/home/jeremic/tex/works/Conferences/2012/ASME_V_and_V_symposium/presentetation/RoleVV_NEW_knowledge.pdf}}
%  {\includegraphics[width=6.5cm]{/home/jeremic/tex/works/Conferences/2011/USNCCM11_Minneapolis/Coupled/Present/VandV_ODEN.jpg}}
%  \hspace*{-2cm}
%  \end{center}
%  \end{figure}
%  
%  {Oberkampf et al. \hspace*{4cm}  Oden et al.}
%  %
%  %\item Models available (some now, some later)
%  %\vspace*{-2.0cm}
%  \end{frame}
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%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  \begin{frame}
%  \frametitle{Importance of V \& V}
%  
%  
%  \begin{itemize}
%  
%  %   \vspace*{2.0truecm}
%     \item  V  \&  V  procedures  are  the  primary means of assessing accuracy in
%     modeling and computational simulations
%  
%     \vspace*{1.0truecm}
%     \item  V  \&  V  procedures  are the tools with which we build confidence and
%     credibility in modeling and computational simulations
%  
%  
%  \end{itemize}
%  
%  
%  
%  \end{frame}
%  
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{V \& V for ESSI Modeling and Simulations}


\begin{itemize}

   \vspace*{0.3cm}
   \item Material modeling and simulation (elastic, elastic-plastic...)


   \vspace*{0.3cm}
   \item Finite elements (solids, structural, special...)


   \vspace*{0.3cm}
   \item Solution advancement algorithms (static, dynamic...)


   \vspace*{0.3cm}
   \item Seismic input and radiation 


   \vspace*{0.3cm}
   \item Finite element model verification!


\end{itemize}



\end{frame}

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

%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Mesh Size Effects on Seismic Wave Propagation Modeling}

\begin{itemize}

  \item Finite element mesh "filters out" \\
       high frequencies

%\vspace*{0.2cm}
  \item Usual rule of thumb: 10-12 elements \\
        needed per wave length
%  (SASSI recommends only 5 ?!)

% 
%    \item  Maximum grid spacing should not exceed 
%    $\Delta h \;\le\; {\lambda}/{10}\;=\;{v}/({10\,f_{max}})$
%  where $v$ is the lowest wave velocity  (shear, elastic-plastic ?)
%  
%    \item  Tests  without and with numerical damping, for different element sizes 
%  

\item 1D wave propagation model
%\vspace*{0.2cm}
\item 3D finite elements (same in 3D)
%\vspace*{0.2cm}
\item Motions applied as displacements at the bottom


\end{itemize}

%\begin{figure}[H]
\vspace*{-4.0cm}
\begin{flushright}
\includegraphics[width=0.7cm]{/home/jeremic/tex/works/Conferences/2011/NRC_Staff_Capacity_Building_21Nov2011/model01.pdf}
\end{flushright}
%\end{figure}

\vspace*{-0.4cm}
\begin{small}
\begin{table}[!htbp]
\centering
% \begin{tabular}{|c|c|c|c|c|}
\begin{tabular}{|r|m{2.6cm}|m{1.5cm}|m{1.8cm}|m{2.3cm}|}
\hline
case  &  model  height  [m]  &  $V_s$ [m/s] & El.size [m] & $f_{max}$ (10el) [Hz]\\
\hline
%\hline
3 & 1000 & 1000 & 10 & 10\\
%\hline
4 & 1000 & 1000 & 20 & 5\\
%\hline
6 & 1000 & 1000 & 50 & 2\\
\hline
% \begin{tabular}{|m{1.5cm}c|m{2.8cm}c|m{2.8cm}c|m{3.0cm}c|m{4.0cm}c|}
% \begin{tabularx}{\linewidth}{|c|c|c|c|c|}
% \begin{tabular*}{0.75\textwidth}{@{\extracolsep{\fill}}|c|c|c|c|c|}
\end{tabular}
% \end{tabularx}
\end{table}
\end{small}




\end{frame}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Cases 3, 4, and 6, Ormsby Wavelet Input Motions}

 
\begin{figure}[H]
\begin{center}
\includegraphics[width=9cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/MeshSize/figs/3_4_6/Input_Displacement.pdf}
\end{center}
\end{figure}


 
\end{frame}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Cases 3, 4, and 6, Surface Motions}

 
\begin{figure}[H]
\begin{center}
\includegraphics[width=9cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/MeshSize/figs/3_4_6/displacement.pdf}
\end{center}
\end{figure}


 
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Cases 3, 4, and 6, Input and Surface Motions, FFT}

 
\begin{figure}[H]
\begin{center}
\includegraphics[width=9cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/MeshSize/figs/3_4_6/FFT.pdf}
\end{center}
\end{figure}


 
\end{frame}


 








%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{3D  Inclined  Body  and Surface Seismic Wave Fields}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Free Field, Inclined, 3D Body and Surface Waves}

  \begin{itemize}
    
    \item  Development  of  analytic  and  numerical  3D, inclined, uncorrelated
    seismic motions for verification

   \vspace*{0.2cm}
    \item Large scale models 

   \vspace*{0.2cm}
    \item Point shear source 

   \vspace*{0.2cm}
    \item Stress drop:
         \begin{itemize}

           \item Wavelet (Ricker, \\
           Ormsby, etc)
  
           \item Analytic 


         \end{itemize}

   \vspace*{0.2cm}
    \item Seismic input using DRM

  \end{itemize}



 
   \vspace*{-4.6cm}
\begin{flushright}
\includegraphics[width=6cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/wave_propagation/figs/FaultSlipModel2km.pdf}
\end{flushright}



\end{frame}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Domain Reduction Method}
%  \begin{itemize}
%    \item 
%  \end{itemize}

  

  The effective force  $P^{eff}$ \\
  is a dynamically consistent \\
  replacement the dynamic \\
  source forces  $P_{e}$

% \end{itemize}

   \vspace*{0.5cm}
  \begin{eqnarray}
P^{eff} = \left\{\begin{array}{c} P^{eff}_i \\ P^{eff}_b \\ P^{eff}_e \end{array}\right\}
= \left\{\begin{array}{c} 0 \\ -M^{\Omega+}_{be} \ddot{u}^0_e-K^{\Omega+}_{be}u^0_e
   \\ M^{\Omega+}_{eb}\ddot{u}^0_b+K^{\Omega+}_{eb}u^0_b\end{array}\right\}
\nonumber
\label{DRMeq09}
\end{eqnarray}
%

   \vspace*{-6.1cm}
\begin{flushright}
\includegraphics[width=6.5cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/wave_propagation/figs/DRMModel.pdf}
%\caption{Domain to be analyzed for the $2^{nd}$ analysis stage of DRM with smaller size 
%         comparing to the original model}
%\label{fig:DRMModel}
\end{flushright}



\end{frame}



% - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - \begin{frame}
% - \frametitle{Source Stress Drop (Ricker)}
% - %  \begin{itemize}
% - %    \item 
% - %  \end{itemize}
% - 
% - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - \begin{figure}[!htbp] 
% - \begin{center}
% - %\hline 
% - \includegraphics[width=5.5cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/wave_propagation/figs/input_ricker/displacement.pdf}
% - \includegraphics[width=5.5cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/wave_propagation/figs/input_ricker/FFT.pdf}
% - %\caption{Displacement time history and FFT of Ricker wave with dominant frequency of $1Hz$}
% - %\label{fig:input_ricker}
% - \end{center}
% - \end{figure}
% - 
% - 
% - 
% - 
% - \end{frame}
% - 
% - 
% - 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Verification: Displacements, Top Middle Point }
%  \begin{itemize}
%   \item 
%  \end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[!htbp] 
\begin{center}
\begin{tabular}{ccc} 
%\hline 
(X)
&
(Z)
\\
\includegraphics[width=5.0cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/wave_propagation/figs/ricker_2km/top_middle_comparison_disp_x.pdf}
&
\includegraphics[width=5.0cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/wave_propagation/figs/ricker_2km/top_middle_comparison_disp_z.pdf}
&
\end{tabular}
%\caption{Comparison of displacements for top middle point using Ricker wave $(f=1Hz)$ as an input motion}
%\label{fig:ricker_acc}
\end{center}
\end{figure}



\end{frame}


% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Verification: Accelerations, Top Middle Point }
% %  \begin{itemize}
% %    \item 
% %  \end{itemize}
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{figure}[!htbp] 
% \begin{center}
% \begin{tabular}{ccc} 
% %\hline 
% (X)
% &
% (Z)
% \\
% \includegraphics[width=5.0cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/wave_propagation/figs/ricker_2km/top_middle_comparison_accel_x.pdf}
% &
% \includegraphics[width=5.0cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/wave_propagation/figs/ricker_2km/top_middle_comparison_accel_z.pdf}
% &
% \end{tabular}
% %\caption{Comparison of accelerations for top middle point using Ricker wave $(f=1Hz)$ as an input motion}
% %\label{fig:ricker_acc}
% \end{center}
% \end{figure}
% 
% 
% 
% 
% \end{frame}
% 
% 
% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Verification: Disp. and Acc., Out of DRM }
%  \begin{itemize}
%    \item 
%  \end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[!htbp] 
\begin{center}
\begin{tabular}{ccc} 
%\hline 
Displacement
&
Acceleration
\\
\includegraphics[width=5.0cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/wave_propagation/figs/ricker_2km/10_40_disp_x.pdf}
&
\includegraphics[width=5.0cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/wave_propagation/figs/ricker_2km/10_40_accel_x.pdf}
&
\end{tabular}
%\caption{Displacement and acceleration time history for a point outside of DRM layer in (x) direction}
%\label{fig:out_ricker_disp}
\end{center}
\end{figure}



\end{frame}






























%
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\subsection{Examples}
%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Few Illustrative Examples}

\begin{itemize}
\item Slip between foundation slab and the soil/rock
underneath

\vspace*{0.2cm}
\item Passive seismic isolation by liquefaction

\vspace*{0.2cm}
\item Structural response in liquefied soil


\end{itemize}

\end{frame}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Nuclear Power Plant with Base Slip}

\begin{itemize}

  \item Low friction zone between \\
        concrete foundation and soil/rock

  \item Inclined, 3D, body and surface, \\
        seismic wave field (wavelets: \\
        Ricker, Ormsby; real seismic, etc.)


\end{itemize}



\vspace*{-4.0cm}
\begin{figure}[!h]
\begin{flushright}
\includegraphics[width=2.50cm]{/home/jeremic/tex/works/Conferences/2011/NRC_LBNL_Review_Panel_Sept2011/2D_faul_slip_model_top_200m.pdf}
%{\includegraphics[width=4.0cm]{/home/jeremic/tex/works/Conferences/2011/NRC_LBNL_Review_Panel_Feb2011/Case_study_model/visit0002.jpeg}}
\end{flushright}
\end{figure}

\vspace*{-0.9cm}
\begin{figure}[!h]
\begin{flushright}
{\includegraphics[width=4.0cm]{/home/jeremic/tex/works/Conferences/2011/NRC_LBNL_Review_Panel_Feb2011/Case_study_model/visit0002.jpeg}}
\end{flushright}
\end{figure}




\vspace*{-3.6cm}
\begin{figure}[H]
\begin{center}
%\vspace*{-0.5cm}
%\includegraphics[width=2.0cm]{/home/jeremic/tex/works/Conferences/2011/NRC_LBNL_Review_Panel_Sept2011/2D_faul_slip_model_top_200m.pdf}
%\hspace*{0.5cm}
%\vspace*{0.2cm}
\includegraphics[width=6cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/FreeFieldInclinedMotionModels/FaultSlip_Ormsby/figs/surface_200m/middle_acceleration_x.pdf}
\hspace*{-0.5cm}
\includegraphics[width=6cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/FreeFieldInclinedMotionModels/FaultSlip_Ormsby/figs/surface_200m/middle_acceleration_z.pdf}
%
\vspace*{-0.5cm} 
\hspace*{0.8cm} 
\mbox{horizontal}
\hspace*{4cm} 
\mbox{vertical}
\hspace*{3cm} 
\end{center}
\end{figure}
\vspace*{-1.0cm} 
% 
%  %\vspace*{-3.5cm}
%  \begin{figure}[H]
%  \begin{center}
%  \includegraphics[width=6cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/FreeFieldInclinedMotionModels/FaultSlip_Ormsby/figs/surface_200m/middle_acceleration_x.pdf}
%  \hspace*{-0.5cm}
%  \includegraphics[width=6cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/FreeFieldInclinedMotionModels/FaultSlip_Ormsby/figs/surface_200m/middle_acceleration_z.pdf}
%  \end{center}
%  \end{figure}
%  
%  %\vspace*{-3.5cm}
%  \begin{figure}[H]
%  \begin{center}
%  \includegraphics[width=6cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/FaultModel_7seconds/xz_TimeHistory/5000_5000_x_acceleration.pdf}
%  \hspace*{-0.5cm}
%  \includegraphics[width=6cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/FaultModel_7seconds/xz_TimeHistory/5000_5000_z_acceleration.pdf}
%  \end{center}
%  \end{figure}
%  
%  \vspace*{-0.90cm}
%  {horizontal accelerations   \hfill vertical accelerations}
%  
%  
%  


\end{frame}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Acc. Response for a Full 3D (at $45^\circ$) Ricker Wavelet}

\begin{figure}[H]
\begin{center}
\begin{tabular}{rr} 
%\hline 
\mbox{\tiny top X}\includegraphics[width=4.0truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/91_97/top_structure_x_acceleration.pdf}
&
\mbox{\tiny top Z}\includegraphics[width=4.0truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/91_97/top_structure_z_acceleration.pdf}
\\
\mbox{\tiny bottom X}\includegraphics[width=4.0truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/91_97/bottom_structure_x_acceleration.pdf}
&
\mbox{\tiny bottom Z}\includegraphics[width=4.0truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/91_97/bottom_structure_z_acceleration.pdf}
\end{tabular}
%\caption{Comparison of acceleration time histories of the structure between 
%slipping and no-slipping models for Ricker wave}
\label{fig:3d_ricker_acc_1000}
\end{center}
\end{figure}



\end{frame}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{FFT Response for a Full 3D (at $45^\circ$) Ricker Wavelet}

\begin{figure}[H]
\begin{center}
\begin{tabular}{rr} 
\mbox{\tiny top X}\includegraphics[width=4.0truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/91_97/top_structure_x_acceleration_FFT.pdf}
&
\mbox{\tiny top Z}\includegraphics[width=4.0truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/91_97/top_structure_z_acceleration_FFT.pdf}
\\
\mbox{\tiny bottom X}\includegraphics[width=4.0truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/91_97/bottom_structure_x_acceleration_FFT.pdf}
&
\mbox{\tiny bottom Z}\includegraphics[width=4.0truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/91_97/bottom_structure_z_acceleration_FFT.pdf}
\end{tabular}
%\caption{Comparison of FFT of the acceleration of the structure between 
%slipping and no-slipping models for Ricker wave}
\label{fig:3d_ricker_fft_1000}
\end{center}
\end{figure}


\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Gaping Response ($45^\circ$ Ricker Wavelet)}

\vspace*{-0.1cm}
\begin{tiny}
\begin{figure}[H]
\begin{center}
\begin{tabular}{ccc} 
%\hline 
$4.5s$
&
$4.6s$
&
$4.7s$
\\
\includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_91_9pieces/gap450.pdf}
&
\includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_91_9pieces/gap460.pdf}
&
\includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_91_9pieces/gap470.pdf}
\\
$4.8s$
&
$4.9s$
&
$5.0s$
\\
\includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_91_9pieces/gap480.pdf}
&
\includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_91_9pieces/gap490.pdf}
&
\includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_91_9pieces/gap500.pdf}
\\
$5.1s$
&
$5.2s$
&
$5.3s$
\\
\includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_91_9pieces/gap510.pdf}
&
\includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_91_9pieces/gap520.pdf}
&
\includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_91_9pieces/gap530.pdf}
%\\
%
%\hline
\end{tabular}
%\caption{Distribution of gap openings along the contact interface for Ricker wave
%(gray scale given in meters)}
\label{fig:3d_ricker1000_gap_9}
\end{center}
\end{figure}
\end{tiny}



\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Slipping Response and Energy Dissipated ($45^\circ$ Ricker)}





\vspace*{-0.1cm}
\begin{tiny}
\begin{figure}[H]
\begin{flushleft}
\hspace*{-1cm}
\begin{tabular}{ccc} 
%\hline 
$4.5s$
&
$4.6s$
&
$4.7s$
\\
\includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_91_9pieces/slide450.pdf}
&
\includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_91_9pieces/slide460.pdf}
&
\includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_91_9pieces/slide470.pdf}
\\
$4.8s$
&
$4.9s$
&
$5.0s$
\\
\includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_91_9pieces/slide480.pdf}
&
\includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_91_9pieces/slide490.pdf}
&
\includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_91_9pieces/slide500.pdf}
\\
$5.1s$
&
$5.2s$
&
$5.3s$
\\
\includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_91_9pieces/slide510.pdf}
&
\includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_91_9pieces/slide520.pdf}
&
\includegraphics[width=2.3truecm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/Gap_Slide_Magnitude_91_9pieces/slide530.pdf}
%\\
%
%\hline
\end{tabular}
%\caption{Distribution of sliding along the contact interface for Ricker wave
%(gray scale given in meters)}
\label{fig:3d_ricker1000_slide_9}
\end{flushleft}
\end{figure}
\end{tiny}

\vspace*{-5cm}
\begin{figure}[!H]
\begin{flushright}
\includegraphics[width=5cm]{/home/jeremic/tex/works/Thesis/NimaTafazzoli/SSI_Contact_Element_01_13/figs/energy_sliding_92/energy_time.pdf}
\hspace*{-1cm}
\end{flushright}
\end{figure}



\end{frame}














%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
  \frametitle{Passive Base Isolation in Uniform and Layered Soils}
  



\vspace*{-0.5cm}
\begin{figure}[!htbp] 
\begin{flushleft}
\includegraphics[width=5cm,angle=90]{/home/jeremic/tex/works/Papers/2009/SeismicIsolationLiquefaction/Mesh-Isolation.pdf}
\\
\includegraphics[width=4cm]{/home/jeremic/tex/works/Conferences/2009/CompDyn/Present/StackElements-Compare.pdf} 
\end{flushleft}
\end{figure}
\vspace*{-1cm}


\vspace*{-8cm}
%\hspace*{-0.5cm}
\begin{figure}[!htbp] 
\begin{flushright}
\includegraphics[width=6cm]{/home/jeremic/tex/works/Papers/2009/SeismicIsolationLiquefaction/time-history-acc.jpg} \\
%\\
%\includegraphics[width=8cm]{/home/jeremic/tex/works/Thesis/GuanzhouJie/thesis/Verzija_Februar/Images/LongMotion/MomentBent1Pile2.pdf}
%\caption{\label{BridgeSFSI01}  FEM  model  for  seismic response of a three bend
%bridge.}
\end{flushright}
\end{figure}
\vspace*{-1cm}



\end{frame}
%



     
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\begin{frame}
 \frametitle{Pile in Liquefiable Sloping Ground}






 

\vspace*{-0.8cm}
\begin{figure}[!htbp]
\begin{flushright}
%  \includegraphics[width=6cm]{/home/jeremic/tex/works/Conferences/2007/PEERAnnualMeeting/Liquefaction/PileBridgeModel01.jpg} \\
  \includegraphics[width=3.3cm]{/home/jeremic/tex/works/Conferences/2007/PEERAnnualMeeting/Liquefaction/PileBridgeModel02.jpg} 
%  \includegraphics[width=4cm]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/GMklot02.pdf}
\end{flushright}
\end{figure}





%
\vspace*{-0.4cm}
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\begin{figure}[!htbp] 
\begin{center}
\hspace*{-0.5cm}
\begin{tabular}{lllllll} 
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\includegraphics[width=0.03\textwidth,angle=0]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/Model_IV.jpg}
&
\includegraphics[height=0.09\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_4_T002.jpg}
&
\includegraphics[height=0.09\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_4_T005.jpg}
&
\includegraphics[height=0.09\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_4_T010.jpg}
&
\includegraphics[height=0.09\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_4_T015.jpg}
&
\includegraphics[height=0.09\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_4_T020.jpg}
&
\includegraphics[height=0.09\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_4_T080.jpg}
\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\includegraphics[width=0.03\textwidth,angle=0]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/Model_V.jpg}
&
\includegraphics[height=0.09\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_5_T002.jpg}
&
\includegraphics[height=0.09\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_5_T005.jpg}
&
\includegraphics[height=0.09\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_5_T010.jpg}
&
\includegraphics[height=0.09\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_5_T015.jpg}
&
\includegraphics[height=0.09\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_5_T020.jpg}
&
\includegraphics[height=0.09\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_5_T080.jpg}
\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\includegraphics[width=0.035\textwidth,angle=0]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/Model_VI.jpg}
&
\includegraphics[height=0.09\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_6_T002.jpg}
&
\includegraphics[height=0.09\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_6_T005.jpg}
&
\includegraphics[height=0.09\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_6_T010.jpg}
&
\includegraphics[height=0.09\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_6_T015.jpg}
&
\includegraphics[height=0.09\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_6_T020.jpg}
&
\includegraphics[height=0.09\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_6_T080.jpg}
\\
t=
&
2~sec
&
5~sec 
&
10~sec
&
15~sec
&
20~sec
&
80~sec
\end{tabular}
\includegraphics[width=7cm,height=0.45cm]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/GMklot02.pdf}\hspace*{1cm}
\\
\vspace*{-0.5cm}
\includegraphics[angle=-90,width=0.4\textwidth]{/home/jeremic/tex/works/Papers/2008/Pile_in_liquefied_soil_upU/NewFiga/Snap_scale.pdf}
\end{center}
\end{figure}
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\section{Probabilistic Modeling}



 

\subsection{Uncertain Geomaterials}
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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, \\
              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}

\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}

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% - \begin{frame} 
% - \frametitle{Soil Uncertainties and Quantification}
% - 
% - \begin{itemize}
% - %
% - %\vspace*{0.5cm}
% - \item Natural variability of soil deposit  (Fenton 1999) 
% - 
% -   \begin{itemize}
% - 
% -     \item Function of soil formation process 
% -  
% -   \end{itemize}
% - 
% - 
% - %
% - \vspace*{0.2cm}
% - \item Testing error (Stokoe et al. 2004)
% - 
% -   \begin{itemize}
% - 
% -     \item Imperfection of instruments 
% -  
% -     \item Error in methods to register quantities
% - 
% -   \end{itemize}
% - 
% - %
% - \vspace*{0.2cm}
% - \item Transformation error (Phoon and Kulhawy 1999)
% - 
% -   \begin{itemize}
% - 
% -     \item Correlation by empirical data fitting (e.g. CPT data $\rightarrow$ friction angle etc.)
% - 
% -   \end{itemize}
% - 
% - \end{itemize}
% - 
% - 
% - \end{frame}
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\begin{frame} 

\frametitle{Types of Uncertainties}

 

\begin{itemize}



\item Aleatory uncertainty - inherent variation of physical system 
%
    \begin{itemize}
%
      \item Can not be reduced
%
      \item Has highly developed mathematical tools 
%
    \end{itemize}
%
\vspace*{0.2cm}
\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*{-3.2cm}
\begin{figure}[!hbpt]
\begin{flushright}
\includegraphics[height=5cm,angle=-90]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Present/uncertain03.pdf}
\end{flushright}
\end{figure}

%\vspace*{1.0cm}

\item Ergodicity (exchanging ensemble averages for time average) assumed to hold


\end{itemize}

 




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\end{frame}







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\begin{frame}


\frametitle{Recent State-of-the-Art}

\begin{itemize}

%\vspace*{-0.5cm}
\item Governing equation

%       \vspace*{-0.5cm}
       \begin{itemize}

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

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

       \end{itemize}

%\vspace{-0.4cm}
\item Existing solution methods

%       \vspace*{-0.5cm}
       \begin{itemize}

       \item \textbf{Random r.h.s} (external force random)

             \begin{itemize}

      \item FPK equation approach 

      \item Use of fragility curves with deterministic FEM (DFEM)

      \end{itemize}

%       \vspace*{0.2cm}
       \item \textbf{Random l.h.s} (material properties random)

             \begin{itemize}

      \item Monte Carlo approach with DFEM $\rightarrow$ CPU expensive

%      \item Stochastic finite element method (e.g. Perturbation method  
%             $\rightarrow$ a linearized expansion! Error increases as a function
%      of COV; Spectral method 
%             $\rightarrow$ developed for elastic materials so far)

      \item Perturbation method 
             $\rightarrow$ a linearized expansion! Error increases as a function
      of COV 
      
      \item Spectral method 
             $\rightarrow$ developed for elastic materials so far

%             \begin{itemize}
%
%             \item Perturbation method  $\rightarrow$ fails if COVs of soil $>$ 20\%
%
%      \item Spectral method $\rightarrow$ only for elastic material
%
%             \end{itemize}

      \end{itemize}
             
       \end{itemize}      

\item  Original development of  {\bf Probabilistic Elasto--Plasticity}


\end{itemize}

\end{frame}



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\begin{frame} 

\frametitle{Uncertainty Propagation through Constitutive Eq.}

% 

\begin{itemize}

 


\item Incremental el--pl constitutive equation  
     $\Delta \sigma_{ij} = E_{ijkl} \Delta \epsilon_{kl}$

 
  
%\begin{normalsize}

%    
%    \begin{equation}
%    \nonumber
%    \frac{d\sigma_{ij}}{dt} = E_{ijkl} \frac{d\epsilon_{kl}}{dt}
%    \end{equation}
    
    
    \begin{eqnarray}
    \nonumber
     E_{ijkl} = \left\{\begin{array}{ll}
%
     E^{el}_{ijkl}
%
%
        \;\;\; & \mbox{\large{~for elastic}} \\
%
           \\
%
           E^{el}_{ijkl}
           -
           \frac{\displaystyle  E^{el}_{ijmn} m_{mn} \; n_{pq} E^{el}_{pqkl}}
           {\displaystyle n_{rs} E^{el}_{rstu} m_{tu} - \xi_* h_*}

           \;\;\; & \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 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}

% 

\end{frame}

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\begin{frame} 

\frametitle{Probabilistic Stress Solution: \\ Eulerian--Lagrangian form of 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), E_{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), E_{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), E_{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), E_{mnrs}(x_t), \epsilon_{rs}(x_t,t)); \right. \right. \right. \\
\nonumber
& & \left. \left. \left. \eta_{ab} (\sigma_{ab}(x_{t-\tau}, t-\tau), E_{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 and (SEP)FEM}

 

\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}

%

\vspace*{0.1cm}
\item {\bf  Complete} probabilistic description of response 


\vspace*{0.1cm}
\item {\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*{0.1cm}
\item  Any uncertain FEM problem
(${\bf M} \ddot{\bf u}
+
{\bf C} \dot{\bf u}
+
{\bf K} {\bf u}
=
{\bf F}
$)  
with
  \begin{itemize}
    \item uncertain material parameters (stiffness matrix ${\bf K}$),
    \item uncertain loading (load vector ${\bf F}$)
  \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


%\vspace*{0.2truecm}

\end{itemize}


\end{frame}






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\begin{frame} 

\frametitle{Probabilistic Elastic-Plastic Response}


\begin{figure}[!hbpt]
\begin{center}
%\includegraphics[width=8cm]{/home/jeremic/tex/works/Papers/2007/ProbabilisticYielding/figures/vonMises_G_and_cu_very_uncertain/Contour_PDF-edited.pdf}
\includegraphics[width=8cm]{/home/jeremic/tex/works/Conferences/2012/DOE-LLNL-workshop-27-28-Feb-2012/ProbabilisticYielding_vonMises_G_and_cu_very_uncertain_Contour_PDF-edited.pdf}
\end{center}
\end{figure}

\end{frame}

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\begin{frame} 

\frametitle{Probabilistic Elastic-Plastic Response}


\begin{figure}[!hbpt]
\begin{center}
%\includegraphics[height=6.0cm]{/home/jeremic/tex/works/Conferences/2011/ICASP11_Zurich/Present/PDF_Plot-Ed.pdf}
\includegraphics[width=9.5cm]{/home/jeremic/tex/works/Conferences/2012/DOE-LLNL-workshop-27-28-Feb-2012/ProbabilisticYielding_vonMises_G_and_cu_very_uncertain_PDF-edited.pdf}
\end{center}
\end{figure}

\end{frame}




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\begin{frame}



\frametitle{SPT Based Determination of Shear Strength}




\begin{figure}[!hbpt]
\begin{center}
% 
\includegraphics[width=5.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGE-GoverGmax/figures/ShearStrength_RawData_and_MeanTrend-Mod.pdf}
\hfill
\includegraphics[width=5.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGE-GoverGmax/figures/ShearStrength_Histogram_PearsonIV-FineTuned-Mod.pdf}
%
\end{center}
\end{figure}                                                          

\vspace*{-0.3cm}
Transformation of SPT $N$-value $\rightarrow$ undrained shear
strength, $s_u$ (cf. Phoon and Kulhawy (1999B)

Histogram  of  the  residual  
(w.r.t  the  deterministic transformation
equation) undrained strength, 
along with fitted probability density function 
(Pearson IV)
\end{frame}


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\begin{frame}



\frametitle{SPT Based Determination of Young's Modulus}


\begin{figure}[!hbpt]
\begin{center}
%
\includegraphics[width=5.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGE-GoverGmax/figures/YoungModulus_RawData_and_MeanTrend_01-Ed.pdf} 
\hfill
\includegraphics[width=5.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGE-GoverGmax/figures/YoungModulus_Histogram_Normal_01-Ed.pdf}
%
\end{center}
\end{figure}                                                          

\vspace*{-0.3cm}
 Transformation of SPT $N$-value $\rightarrow$ 1-D Young's modulus, $E$ (cf. Phoon and Kulhawy (1999B))

 Histogram of the residual (w.r.t the deterministic transformation equation) Young's modulus, along with fitted probability density function

\end{frame}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Probabilistic Elastic--Plastic Response}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




 \begin{frame}



\frametitle{Stochastic Finite Element Formulation}

\begin{itemize}

\item Governing equations:


\begin{equation}
\nonumber
A\sigma = \phi(t);~~~ Bu = \epsilon; ~~~\sigma = E \epsilon
\end{equation}

\vspace*{0.2cm}
\item {\bf Spatial} and 
{\bf stochastic} discretization

\begin{itemize}


\vspace*{0.2cm}
\item  Deterministic  spatial  differential operators ($A$ \& $B$) $\rightarrow$
Regular shape function method with Galerkin scheme

\vspace*{0.2cm}
\item    Input    random   field   material   properties   ($E$)   $\rightarrow$
Karhunen--Lo{\`e}ve (KL) expansion, optimal expansion, error minimizing property

\vspace*{0.2cm}
\item  Unknown  solution  random field ($u$) $\rightarrow$ Polynomial Chaos (PC)
expansion



\end{itemize}



\end{itemize}

\end{frame}

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

\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}^{ep} d_{ni} + \sum_{n = 1}^N \sum_{j = 0}^P d_{nj} \sum_{k = 1}^M C_{ijk} K_{mnk}^{'ep}  = \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}^{ep} = \int_D B_n \textcolor{mycolor}{E}^{ep}  B_m dV    
      \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
      K_{mnk}^{'ep} = \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}

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

\begin{frame}
\frametitle{Inside SSEPFEM}

\begin{itemize}

\item Explicit stochastic elastic--plastic 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  Excellent for parallelization, both at the element and global levels


\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}















% - 
% -  \begin{frame}
% -  
% -  
% -  
% -  \frametitle{Governing Equations \& Discretization Scheme}
% -  
% -  \begin{itemize}
% -  
% -  \item Governing equations in mechanics:
% -  
% -  
% -  \begin{equation}
% -  \nonumber
% -  A\sigma = \phi(t);~~~ Bu = \epsilon; ~~~\sigma = D \epsilon
% -  \end{equation}
% -  
% -  \item  Discretization (spatial and stochastic) schemes
% -  
% -  \begin{itemize}
% -  
% -  \item    Input    random   field   material   properties   ($D$)   $\rightarrow$
% -  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}
% -  
% -  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  % - % - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  % - 
% -  % - \begin{frame}
% -  % - 
% -  % - 
% -  % - \frametitle{Truncated Karhunen--Lo{\`e}ve (KL) expansion}
% -  % -               
% -  % -        \begin{itemize}                                                                                                                         
% -  % -                                                                                                                                                       
% -  % -        \item Representation of input random fields in eigen-modes of covariance kernel                                                                
% -  % -                                                                                                                                                       
% -  % - %       \vspace*{-0.1cm}                                                                                                                               
% -  % - %              \begin{figure}[!hbpt]                                                                                                                   
% -  % -               \begin{flushleft}                                                                                                                       
% -  % -               \includegraphics[height=3.0cm]{/home/jeremic/tex/works/Conferences/2008/GeoCongress/Probabilistic/Paper/TypicalDataPlotBH1-Edited.jpg}
% -  % - %       ShearStrengthProfile.jpg}                                                                                
% -  % -               \end{flushleft}                                                                                                                         
% -  % -               \vspace*{-3.15cm}                                                                                                                        
% -  % -        \begin{flushright}                                                                                                                             
% -  % - %       \begin{equation}                                                                                                                              
% -  % - %       \nonumber                                                                                                                                     
% -  % - %       \begin{normalsize}                                                                                                                            
% -  % -        $ q_T(x,\theta) = \bar q_T(x) + \sum_{n=1}^M \sqrt{\lambda_n} \xi_n(\theta) f_n(x) $ \\                                                        
% -  % -        \ \\                                                                                                                                           
% -  % -        $ \int_D C(x_1, x_2) f (x_2) dx_2 = \lambda f (x_1) \ \ \ \ \ \ \ \ \ \  \ \ $ \\                                                              
% -  % -        \ \\                                                                                                                                           
% -  % -        $ \xi_i(\theta) = \displaystyle \frac{1}{\sqrt \lambda_i} \int_D \left [q_T(x,\theta) - \bar q_T (x) \right] f_i (x) dx $                                   
% -  % - %       \end{equation}                                                                                                                                
% -  % - %              \end{normalsize}                                                                                                                       
% -  % -        \end{flushright}                                                                                                                               
% -  % - %       \end{figure}                                                                                                                                   
% -  % -                                                                                                                                                       
% -  % - %       \vspace{-6.0cm}                                                                                                                               
% -  % - %       \begin{flushright}                                                                                                                            
% -  % - %       \begin{equation}                                                                                                                              
% -  % - %       \nonumber                                                                                                                                     
% -  % - %        w(x,\theta) = \bar w(x) + \sum_{n=0}^M \sqrt{\lambda_n} \zeta_n(\theta) f_n(x)                                                               
% -  % - %       \end{equation}                                                                                                                                
% -  % - %        \end{flushright}                                                                                                                             
% -  % -                                                                                                                                                       
% -  % - %        \vspace*{0.8cm}                                                                                                                               
% -  % -         \item Error minimizing property                                                                                                               
% -  % -                                                                                                                                                       
% -  % -         \item Optimal expansion $\rightarrow$ minimization of number of stochastic dimensions                                                         
% -  % -                                                                                                                                                       
% -  % -                                                                                                                                                       
% -  % -          \end{itemize}                                                                                                                                                                                                                                                                   
% -  % -                                                                                                                                                                                                                                                                                          
% -  % - 
% -  % - 
% -  % - \end{frame}
% -  % - 
% -  % - 
% -  % - % - % - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  % - % - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  % - % - \begin{frame}
% -  % - % - \frametitle{KL Expansion (of Covariance Kernel)}
% -  % - % - 
% -  % - % - \begin{flushleft}
% -  % - % - %\begin{center}
% -  % - % - %\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
% -  % - % - \includegraphics[height=2.2cm]{/home/jeremic/tex/works/Conferences/2007/GeoDenver/SFEM/Presentation/ActualExponentialCovarianveSurface.jpg}
% -  % - % - \hspace*{0.3cm}
% -  % - % - \includegraphics[height=2.2cm]{/home/jeremic/tex/works/Conferences/2007/GeoDenver/SFEM/Presentation/KL_ApproxWith_1Term_CovarianveSurface.jpg}
% -  % - % - %\vspace*{1.0cm}
% -  % - % - %\mbox{Exact covariance surface}
% -  % - % - %\vspace*{-4.0cm}
% -  % - % - \end{flushleft}
% -  % - % - \vspace*{-0.4cm}
% -  % - % - \small{\ \ \ \ \ \ \ \ \ \ \ \ \ Exact \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1 term approx.} \\
% -  % - % - \small{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (8.49\% error)}
% -  % - % - %\begin{flushright}
% -  % - % - %\end{center}
% -  % - % - %\end{figure}
% -  % - % - 
% -  % - % - %\vspace*{-0.5cm}
% -  % - % - %\small{Exact covariance surface \ \ \ \ \ \ \ \ \ \ \ \ \ \ One-term approximation}
% -  % - % - %
% -  % - % - %\vspace*{0.5cm}
% -  % - % - %
% -  % - % - %\begin{figure}[!hbpt]
% -  % - % - %\begin{center}
% -  % - % - \begin{flushleft}
% -  % - % - \includegraphics[height=2.2cm]{/home/jeremic/tex/works/Conferences/2007/GeoDenver/SFEM/Presentation/KL_ApproxWith_2Terms_CovarianveSurface.jpg}
% -  % - % - \hspace*{-0.2cm}
% -  % - % - \includegraphics[height=2.2cm]{/home/jeremic/tex/works/Conferences/2007/GeoDenver/SFEM/Presentation/KL_ApproxWith_3Terms_CovarianveSurface.jpg}
% -  % - % - \end{flushleft}
% -  % - % - %\vspace*{-0.5cm}
% -  % - % - \small{\ \ \ \ \ 2 terms approx. \ \ \ \ \ \ \ \ \  3 terms approx.} \\
% -  % - % - \small{\ \ \ \ \ \ (1.15\% error) \ \ \ \ \ \ \ \ \ \ \ \ \ (1.13\% error)} 
% -  % - % - %
% -  % - % - \vspace{-6.5cm}
% -  % - % - \begin{flushright}
% -  % - % - 
% -  % - % - \includegraphics[height=2.0cm]{/home/jeremic/tex/works/Reports/2006/SEPFEM/figures/CPT_DataAnalysis_Plots/TypicalAutoCovariancePlotBH1_FiniteScale-Edited.jpg} \hspace*{-0.7cm}
% -  % - % - %ShearStrengthProfile.jpg}
% -  % - % - 
% -  % - % - covariance function \\
% -  % - % - (exponential):  \ \ \ \ \  \\
% -  % - % - %\ \\
% -  % - % - $ C(x_1, x_2) = \sigma^2 e^{-| x_1 - x_2 | /b} $  \hspace*{-0.5cm} \\
% -  % - % - \ \\
% -  % - % - \ \\
% -  % - % - KL approximation: \ \ \ \ \ \ \ \  \\
% -  % - % - \ \\
% -  % - % - $ C(x_1, x_2) \ \ \ \ \ \ \ \ \ \ \ \ \  $ \\
% -  % - % -     $ = \sum_{k =1}^M \lambda_k f_k(x_1) f_k(x_2) $
% -  % - % - 
% -  % - % - \end{flushright}
% -  % - % - %\end{figure}
% -  % - % - 
% -  % - % - %\vspace*{2.5cm}
% -  % - % - %\small{Two-terms approximation \ \ \ \ \ \ \ \ \ \ \ \  Three-terms approximation}
% -  % - % - 
% -  % - % - \vspace*{4.0cm}
% -  % - % - \begin{center}
% -  % - % - \large{KL Expansion of Covariance Kernel}
% -  % - % - \end{center}
% -  % - % - 
% -  % - % - \end{frame}
% -  % - % - 
% -  % - % - 
% -  % - % - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  % - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  % - 
% -  % - \begin{frame}
% -  % - 
% -  % - \frametitle{Polynomial Chaos (PC) Expansion}
% -  % - 
% -  % - %\begin{itemize}
% -  % - 
% -  % - %\vspace*{-0.35cm}
% -  % - %\item Solution (displacement) random field $\rightarrow$ Can not use K-L expansion directly
% -  % - 
% -  % -        \begin{itemize}
% -  % - 
% -  % -        \item Covariance kernel is not known a priori
% -  % - 
% -  % -        \vspace*{-0.5cm}
% -  % -        \begin{normalsize}
% -  % -        \begin{equation}
% -  % -        \nonumber
% -  % -        u(x,\theta)=\sum_{j=1}^L e_j \chi_j(\theta)b_j(x)
% -  % -        \end{equation}
% -  % -        \end{normalsize}
% -  % - 
% -  % -        \vspace*{-0.3cm}
% -  % -        \item Can be expressed as functional of known random variables and unknown deterministic function
% -  % - 
% -  % -        \vspace*{-0.5cm}
% -  % -        \begin{normalsize}
% -  % -        \begin{equation}
% -  % -        \nonumber
% -  % -        u(x,\theta)=\zeta[\xi_i(\theta),x]
% -  % -        \end{equation}
% -  % -        \end{normalsize}
% -  % - 
% -  % -        \vspace*{-0.5cm}
% -  % -        \item Need a basis of known random variables $\rightarrow$ PC expansion
% -  % - 
% -  % -        \vspace*{-0.2cm}
% -  % -        \begin{normalsize}
% -  % -        \begin{equation}
% -  % -        \nonumber
% -  % -        \chi_j(\theta)=\sum_{i=0}^P\gamma_i^{(j)}\psi_i\left[\left\{\xi_r\right\}\right]
% -  % -        \end{equation}
% -  % - 
% -  % -        \vspace*{-0.5cm}
% -  % -        \begin{equation}
% -  % -        \nonumber
% -  % -        u(x,\theta)=\sum_{j=1}^L \sum_{i=0}^P \gamma_i^{(j)} \psi_i[\{\xi_r\}]e_j b_j(x) = \sum_{i=0}^P \psi_i[\{\xi_r\}] d_i(x)
% -  % -        \end{equation}
% -  % -        \end{normalsize}
% -  % - 
% -  % - %       \vspace*{-0.6cm}
% -  % - %       \item Deterministic coefficients can be found by minimizing norm of error of finite 
% -  % - %            representation (e.g. using Galerkin scheme) 
% -  % -        
% -  % -         
% -  % - 
% -  % - 
% -  % - %       \end{itemize}
% -  % - \end{itemize}
% -  % - 
% -  % - 
% -  % - %\end{itemize}
% -  % - 
% -  % - \end{frame}
% -  % - 
% -  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  
% -  \begin{frame}
% -  \frametitle{Spectral Stochastic 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}{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}
% -  
% -  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  
% -  \begin{frame}
% -  \frametitle{Inside SSEPFEM}
% -  
% -  \begin{itemize}
% -  
% -  \item Explicit stochastic elastic--plastic 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}
% -  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Seismic Wave Propagation Through Uncertain Soils}


% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% 
% 
% \frametitle{Applications}
% 
% 
% 
% 
% \begin{itemize}
% 
% \vspace*{0.3cm}
% \item Stochastic elastic--plastic simulations of soils and structures
% 
% \vspace*{0.3cm}
% \item Probabilistic inverse problems
% 
% \vspace*{0.3cm}
% \item Geotechnical site characterization design
% 
% \vspace*{0.3cm}
% \item Optimal material design
% 
% 
% \end{itemize}
% 
% \end{frame}
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% - +
% - + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - + 
% - + \begin{frame}
% - + 
% - + 
% - + \frametitle{Random Field Modeling of Uncertain Soil Properties}
% - + 
% - + \begin{itemize}
% - + 
% - + \item Finite scale model 
% - + 
% - + \begin{itemize}
% - + 
% - + \item Short memory, finite correlation length
% - + 
% - + \item Common autocovariance model $\rightarrow$ exponential, spherical, triangular, linear--exponential
% - + 
% - + \end{itemize}
% - + 
% - + \item Fractal model 
% - + 
% - + \begin{itemize}
% - + 
% - + \item long memory, infinite correlation length $\rightarrow$ more realistic for modeling horizontal 
% - + spatial uncertainty
% - + 
% - + \item 1/f-type noise process with power spectral density, $P(\omega)~=~P_0~\omega^{-\gamma}$, with
% - + upper and/or lower frequency cut-off.
% - + 
% - + \end{itemize}
% - + 
% - + \end{itemize}
% - + 
% - + \end{frame}
% - + 
% - + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}


\frametitle{Seismic Wave Propagation through Stochastic Soil}


\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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\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{Random Field Parameters from Site Data}


%\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{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{Decision About Site (Material) Characterization}




\begin{itemize}

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


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


\vspace*{0.3cm}
\item Improve site (material) characterization if  probabilities of exceedance are unacceptable!


\end{itemize}

\end{frame}

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\begin{frame}


\frametitle{Full PDFs of all DOFs (and $\sigma_{ij}$, $\epsilon_{ij}$, etc.)}
%\frametitle{Full PDFs for Real Data Case}


\begin{itemize}
\vspace*{0.7cm}
  \item Stochastic Elastic-Plastic\\
        Finite Element Method \\
        (SEPFEM)  \\    
\vspace*{0.5cm}
  \item Dynamic case
\vspace*{0.5cm}
  \item Full PDF at \\
        each time step $\Delta t$
\end{itemize}


\vspace*{-4.60cm}
\begin{flushright}
\includegraphics[width=6.0cm]{/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{flushright}

% 
\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{PDF at each $\Delta t$ (say 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}

% 
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\begin{frame}


\frametitle{PDF $\rightarrow$ CDF (Fragility) 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=9.0cm]{/home/jeremic/tex/works/Conferences/2009/UNION-Univ-BGD/Present/Plots_with_Labels/ProbabilityOfExceedance20cm_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}
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% \begin{frame}
% 
% 
% \frametitle{Probability of Exceedance of $50$cm}
% 
% 
% 
% \begin{figure}
% \begin{center}
% %\hspace*{-0.75cm}                                                                                               
% \includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2009/UNION-Univ-BGD/Present/Plots_with_Labels/ProbabilityOfExceedance50cm_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}
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% \begin{frame}
% 
% 
% \frametitle{Probabilities of Exceedance vs. Displacements}
% 
% 
% 
% \begin{figure}
% \begin{center}
% %\hspace*{-0.75cm}                                                                                               
% \includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2009/UNION-Univ-BGD/Present/Plots_with_Labels/ProbabilityOfExceedance_vs_Displacement_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}
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% % 
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% 
% \begin{frame}
% 
% \frametitle{Probabilities of Unacceptable Deformation}
% 
% \begin{figure}
% \begin{center}
% \vspace*{-0.3cm}                                                                                               
% \includegraphics[width=10.5cm]{/home/jeremic/tex/works/Conferences/2009/UNION-Univ-BGD/Present/NewPlots/with_legends_and_labels/Exceedance20cm_LomaPrieta-Edited_ps.pdf}
% \vspace*{-0.5cm}
% %\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 Unacceptable Deformation ($50$cm)}

\begin{figure}
\begin{center}
\vspace*{-0.3cm}
%\hspace*{-0.75cm}                                                                                               
\includegraphics[width=10.50cm]{/home/jeremic/tex/works/Conferences/2009/UNION-Univ-BGD/Present/NewPlots/with_legends_and_labels/Exceedance50cm_LomaPrieta-Edited_ps.pdf}
\vspace*{-0.5cm}
%\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 Informed Decision Process}

\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}



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\begin{frame}
  \frametitle{Summary}
  
  \begin{itemize}


%  \item  High fidelity
%  modeling and simulations for performance assessment of infrastructure systems


%\vspace*{0.5cm}
  \item              {\bf    Interplay}        of     {\bf Uncertain} {\bf 
  Earthquake}, {\bf Uncertain}  {\bf Soil/Rock},
%  {Foundation} 
  and  {\bf  Uncertain}   {\bf Structure} in  time domain {\bf probably} plays a
  decisive role in seismic performance of NPPs

\vspace*{0.1cm}
  \item           Improve {\bf risk informed decision making}  through  high 
  fidelity  {\bf  Deterministic}  and  {\bf  Stochastic  Elastic-Plastic  Finite
  Element} modeling and simulation

 
\vspace*{0.1cm}
  \item  {\bf Education and training} of users will prove essential
  

\vspace*{0.1cm}
  \item    {\bf Acknowledgement:} funding and collaboration with the US-NRC, and
  funding from NSF, DOE.

  
\end{itemize}


\end{frame}
%








%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
  \frametitle{References}
  


\begin{tiny}


\begin{itemize}{\leftmargin=1em}
%\begin{list}{\labelitemi}{\leftmargin=1em}

%\begin{enumerate}
%
\setlength{\itemindent}{12pt}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item[][]
\vspace*{-0.1cm}
\item[]
%\item[][CV-J30]
K. Sett, 
B. Jeremi{\'c}.
and M. L. Kavvas.
  Stochastic Elastic--Plastic Finite Elements.
  {\em  Computer  Methods  in  Applied  Mechanics  and
Engineering}, Vol 200, No. 9-12, pp 997-1007,  
2011
{.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item[][]
\vspace*{-0.1cm}
\item[]
%\item[][CV-J29]
K. Sett, B. Unutmaz, K. {\"O}. {\c C}etin, S. Koprivica and B. Jeremi{\'c}.
   Soil  Uncertainty  and  its  Influence  on  Simulated $G/G_{max}$ and
Damping Behavior.
  {\em  {ASCE} Journal of Geotechnical and Geoenvironmental Engineering}, Volume
  137, Issue 3, pp 218-226, March, 2011{.}

% 2009{\href{http://sokocalo.engr.ucdavis.edu/~jeremic/wwwpublications/Seismic_isolation_by_liquefaction.pdf}
% {.}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item[][]
\vspace*{-0.1cm}
\item[]
%\item[][CV-J28]
M. Taiebat, 
B. Jeremi{\'c}.
Y. F. Dafalias, 
A. M. Kaynia, and
Z. Cheng.
    Propagation of Seismic Waves through Liquefied Soils.
   {\em Soil Dynamics and Earthquake Engineering}, No.~30, pp~236-257,
2010
{.}
%     In   review,  {\em Soil Dynamics and Earthquake Engineering}, 
% 2009{\href{http://sokocalo.engr.ucdavis.edu/~jeremic/wwwpublications/Seismic_isolation_by_liquefaction.pdf}
% {.}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item[][]
\vspace*{-0.1cm}
\item[]
%\item[][CV-J27]
K. Sett and 
B. Jeremi{\'c}.
  Probabilistic Yielding and Cyclic Behavior of Geomaterials. 
  {\em International Journal for Numerical and Analytical
Methods in Geomechanics},  Vol.~34, No.~15, pp~1541-1559, 
2010
{.}


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


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item[][]
\vspace*{-0.1cm}
\item[]
%\item[][CV-J26]
Z. Cheng and  B. Jeremi{\'c}.
   Numerical Simulations of Piles in Liquefied Soils.
  {\em Soil Dynamics and Earthquake Engineering}, No.~29, pp~1405-1416,
2009
{.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item[][]
\vspace*{-0.1cm}
\item[]
%\item[][CV-J24]
  B. Jeremi{\'c},
G. Jie, M. Preisig and N. Tafazzoli.
     Time  domain simulation of soil-foundation-structure interaction in
non-uniform soils.
  {\em Earthquake Engineering and Structural Dynamics}, 
Volume 38, Issue 5, pp 699-718,
2009
{.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item[][]
\vspace*{-0.1cm}
\item[]
%\item[][CV-J22]
B. Jeremi{\'c} and K. Sett.
  On Probabilistic Yielding of Materials. 
    {\em Communications in Numerical Methods
in Engineering}, Volume 25, No. 3, pp 291-300, 
2009
{.}

%{\tiny (UCD--pub53) }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item[][]
\vspace*{-0.1cm}
\item[]
%\item[][CV-J20]
B. Jeremi{\'c}, Z. Cheng, M. Taiebat and Yannis Dafalias.
  Numerical Simulation of Fully Saturated Porous Materials.
   {\em International Journal for Numerical and Analytical
Methods in Geomechanics}, Volume 32, No. 13, pp 1635-1660, 
2008
{.}



%{\tiny (UCD--pub52) }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item[][]
\vspace*{-0.1cm}
\item[]
%\item[][CV-J18]
K. Sett, B. Jeremi{\' c}, and 
M. L. Kavvas.
  Probabilistic Elasto-Plasticity: 
      Solution and Verification in 1D. 
  {\em Acta Geotechnica}, Volume 2., No. 3. pp 211-220,   October
2007
{.}

%{\tiny (UCD--pub47) }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item[][]
\vspace*{-0.1cm}
\item[]
%\item[][CV-J17]
B. Jeremi{\' c}, K. Sett and M. L. Kavvas.
  Probabilistic  Elasto-Plasticity:  Formulation in 1D.
  {\em Acta Geotechnica}, Volume 2., No. 3. pp 197-210, October 
2007
{.}

%{\tiny (UCD--pub46) }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item[][]
\vspace*{-0.1cm}
\item[]
%\item[][CV-J15]
Z. Yang and  B. Jeremi{\' c}.
  Study of Soil Layering Effects on Lateral Loading Behavior of Piles
   {\em  {ASCE}  Journal  of  Geotechnical  and
Geoenvironmental                      Engineering}, Volume 131, No. 6, June 
2005, pp. 762-770
{.}

%{\tiny (UCD--pub35) }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item[][]
\vspace*{-0.1cm}
\item[]
%\item[][CV-J14]
 B. Jeremi{\' c}, S. Kunnath and F. Xiong.
   Influence  of    Soil--Structure  interaction on Seismic Response of
Bridges.
    {\em  International Journal for Engineering
Structures},      Volume      26,      Issue     3,     February     2004,     pp.
391-402
{.}



% - 
% -  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  %\item[][]
% -  \item[]
% -  %\item[][CV-J20]
% -  B. Jeremi{\'c}, Z. Cheng, M. Taiebat and Yannis Dafalias.
% -    Numerical Simulation of Fully Saturated Porous Materials.
% -     {\em International Journal for Numerical and Analytical
% -  Methods in Geomechanics}, Volume 32, No. 13, pp 1635-1660, 
% -  2008
% -  {.}
% -  
% -  
% -  
% -  %{\tiny (UCD--pub52) }
% -  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  
% -  
% -  
% -  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  %\item[][]
% -  \item[]
% -  %\item[][CV-J18]
% -  K. Sett, B. Jeremi{\' c}, and 
% -  M. L. Kavvas.
% -    Probabilistic Elasto-Plasticity: 
% -        Solution and Verification in 1D. 
% -    {\em Acta Geotechnica}, Volume 2., No. 3. pp 211-220,   October
% -  2007
% -  {.}
% -  
% -  %{\tiny (UCD--pub47) }
% -  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  
% -  
% -  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  %\item[][]
% -  \item[]
% -  %\item[][CV-J17]
% -  B. Jeremi{\' c}, K. Sett and M. L. Kavvas.
% -    Probabilistic  Elasto-Plasticity:  Formulation in 1D.
% -    {\em Acta Geotechnica}, Volume 2., No. 3. pp 197-210, October 
% -  2007
% -  {.}
% -  
% -  %{\tiny (UCD--pub46) }
% -  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  
% -  
% -  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  %\item[][]
% -  \item[]
% -  %\item[][CV-J16]
% -  B. Jeremi{\' c} and Z. Cheng.
% -    Significance of Equal Principal Stretches in Computational Hyperelasticity.
% -       {\em  Communications in Numerical Methods in Engineering}, Volume 21,
% -  Issue 9, pp 477-486, September 
% -  2005
% -  {.}
% -  
% -  %{\tiny (UCD--pub37) }
% -  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  
% -  
% -  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  %\item[][]
% -  \item[]
% -  %\item[][CV-J15]
% -  Z. Yang and  B. Jeremi{\' c}.
% -    Study of Soil Layering Effects on Lateral Loading Behavior of Piles
% -     {\em  {ASCE}  Journal  of  Geotechnical  and
% -  Geoenvironmental                      Engineering}, Volume 131, No. 6, June 
% -  2005, pp. 762-770
% -  {.}
% -  
% -  %{\tiny (UCD--pub35) }
% -  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  
% -  
% -  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  
% -  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  %\item[][]
% -  \item[]
% -  %\item[][CV-J14]
% -   B. Jeremi{\' c}, S. Kunnath and F. Xiong.
% -     Influence  of    Soil--Structure  interaction on Seismic Response of
% -  Bridges.
% -      {\em  International Journal for Engineering
% -  Structures},      Volume      26,      Issue     3,     February     2004,     pp.
% -  391-402
% -  {.}
% -  
% -  %{\tiny (UCD--pub27) }
% -  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  
% -  
% -  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  %\item[][]
% -  \item[]
% -  %\item[][CV-J12]
% -  Z. Yang and    B. Jeremi{\' c}.
% -     Numerical Study of the Effective Stiffness for  Pile
% -  Groups.
% -   {\em  International  Journal for
% -  Numerical  and  Analytical  Methods  in  Geomechanics},  Volume  27,  Issue 15, pp
% -  1255-1276,                                                                  Dec.
% -  2003
% -  {.}
% -  
% -  %{\tiny (UCD--pub24)}
% -  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  
% -  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  %\item[][]
% -  \item[]
% -  %\item[][CV-J11 ]
% -  Z. Yang and    B. Jeremi{\' c}.
% -     Numerical    analysis    of    pile  behavior under lateral loads in
% -             layered elastic-plastic soils.
% -     {\em  International  Journal for
% -  Numerical  and  Analytical  Methods  in  Geomechanics},  Volume  26,  Issue 14, pp
% -  1385-1406,                                                                  Dec.
% -  2002{.}
% -  
 
\end{itemize}        
%\end{list}

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