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%\title{NRC Staff Capacity Building: \\
% Micromechanical Origins of ElastoPlasticity }
\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.
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\pgfdeclareimage[height=0.7cm]{lbnllogo}{/home/jeremic/BG/amblemi/lbnllogo}
\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
%ChangGyun Jeong}
%  Give the names in the same order as the appear in the paper.
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{ Professor, University of California, Davis, CA\\
% and\\
Faculty Scientist, Lawrence Berkeley National Laboratory, Berkeley, CA }
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{\small Idaho National Laboratory \\ July 2012}
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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
soilstructure 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 soilstructure NPP system
\vspace*{0.1cm}
\item {\bf Directing}, in space and time, with {\bf high (known)
confidence}, seismic energy flow in the soilfoundationstructure system
%\vspace*{0.1cm}
% \item {\bf Education} for researchers, professional practice.
\end{itemize}
\end{frame}
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% \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 soilfoundationstructure system for
% \begin{itemize}
% \item Safety and
% \item Economy
% \end{itemize}
%
% \end{itemize}
% \end{frame}
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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 HiFi modeling and simulation of nonlinear earthquake
%  soil/rock structure interaction problems:
%  \begin{itemize}
%  % \item Time domain, nonlinear, parallel finite element program:
%  \item {\bf NRCESSIProgram}
%  % \item High performance, parallel computer:
%  \item {\bf NRCESSIComputer}
%  % \item Educational endeavor, documentation:
%  \item {\bf NRCESSINotes}
%  \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 NRCESSIProgram} is a 3D, nonlinear, time domain,
parallel finite element program specifically developed for
HiFi modeling and simulation of Earthquake Soil/Rock Structure
Interaction problems for NPPs on NRCESSIComputer. \
%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 (NRCESSIProgram). Significant effort is devoted to development
%of verification and validation procedures, as well as on development of
%extensive documentation. NRCESSIProgram is in public domain and is licensed
%through the Lesser GPL.
%\vspace*{0.3cm}
\vspace*{0.1cm}
\item {\bf The NRCESSIComputer} 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 NRCESSINotes} 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
%NRCESSIProgram code API (application Programming Interface) and DSLs (Domain
%Specific Language).
%%NRCESSINotes, developed by Boris Jeremic and collaborators, are in public
%domain
%%and are licensed under a Creative Commons AttributionShareAlike 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://nrcessisimulator.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 costperformance \\
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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\begin{frame}
% \frametitle{Seismic Energy Dissipation for \underline{Soil}FoundationStructure Systems}
\frametitle{High Fidelity Modeling}
% \frametitle{Seismic Energy Dissipation for
% \underline{Soil}FoundationStructure 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}
% 
% 
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% 
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%  \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}
% 
% 
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\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 Elasticplastic 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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% \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}
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% \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}
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\begin{frame}
\frametitle{V \& V for ESSI Modeling and Simulations}
\begin{itemize}
\vspace*{0.3cm}
\item Material modeling and simulation (elastic, elasticplastic...)
\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}
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%
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\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: 1012 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, elasticplastic ?)
%
% \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}{ccccc}
\begin{tabular}{rm{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}cm{2.8cm}cm{2.8cm}cm{3.0cm}cm{4.0cm}c}
% \begin{tabularx}{\linewidth}{ccccc}
% \begin{tabular*}{0.75\textwidth}{@{\extracolsep{\fill}}ccccc}
\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_eK^{\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}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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 noslipping 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 noslipping 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/MeshIsolation.pdf}
\\
\includegraphics[width=4cm]{/home/jeremic/tex/works/Conferences/2009/CompDyn/Present/StackElementsCompare.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/timehistoryacc.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}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[!htbp]
\begin{center}
\hspace*{0.5cm}
\begin{tabular}{lllllll}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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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\end{frame}
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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 Pointwise \\
uncertainty, \\
testing \\
error, \\
transformation \\
error
\end{itemize}
% \vspace*{0.5cm}
% \item Failure mechanisms related to spatial variability (strain localization and
% bifurcation of response)
%
% \vspace*{0.5cm}
% \item Inverse problems
%
% \begin{itemize}
%
% \item New material design, ({\it pointwise})
%
% \item Solid and/or structure design (or retrofits), ({\it spatial})
%
% \end{itemize}
%\end{itemize}
\vspace*{5cm}
\begin{figure}[!hbpt]
%\nonumber
%\begin{center}
\begin{flushright}
%\includegraphics[height=5.0cm]{/home/jeremic/tex/works/Conferences/2006/KragujevacSEECCM06/Presentation/MGMuzorak01.jpg}
\includegraphics[height=5.5cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/FrictionAngleProfile.jpg}
\\
\mbox{(Mayne et al. (2000) }
\end{flushright}
%\end{center}
%\end{center}
\end{figure}
\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 tradeoff 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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\begin{frame}
\frametitle{Recent StateoftheArt}
\begin{itemize}
%\vspace*{0.5cm}
\item Governing equation
% \vspace*{0.5cm}
\begin{itemize}
\item Dynamic problems $\rightarrow$ $ M \ddot u + C \ddot u + K u = 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 ElastoPlasticity}
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{Uncertainty Propagation through Constitutive Eq.}
%
\begin{itemize}
\item Incremental elpl 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 elasticplastic}}
%
\end{array} \right.
\end{eqnarray}
%\end{normalsize}
%\vspace{0.5cm}
% \item Nonlinear coupling in the ElPl modulus
% \item Focus on 1D $\rightarrow$ a nonlinear ODE with random coefficient and random forcing
%
%
%
% \begin{eqnarray}
% \nonumber
% \frac{d\sigma(x,t)}{dt} &=& \beta(\sigma(x,t),D^{el}(x),q(x),r(x);x,t) \frac{d\epsilon(x,t)}{dt} \\
% \nonumber
% &=& \eta(\sigma,D^{el},q,r,\epsilon; x,t) \mbox{\ \ \ \ with an I.C. $\sigma(0)=\sigma_0$}
% \end{eqnarray}
%
\end{itemize}
%
\end{frame}
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\begin{frame}
\frametitle{Probabilistic Stress Solution: \\ EulerianLagrangian 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 3D probabilistic stressstrain response
%
% \end{itemize}
%
%
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\begin{frame}
\frametitle{EulerianLagrangian FPK Equation and (SEP)FEM}
\begin{itemize}
\item Advectiondiffusion 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 Secondorder 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 elasticplasticdamage 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 ElasticPlastic Response}
\begin{figure}[!hbpt]
\begin{center}
%\includegraphics[width=8cm]{/home/jeremic/tex/works/Papers/2007/ProbabilisticYielding/figures/vonMises_G_and_cu_very_uncertain/Contour_PDFedited.pdf}
\includegraphics[width=8cm]{/home/jeremic/tex/works/Conferences/2012/DOELLNLworkshop2728Feb2012/ProbabilisticYielding_vonMises_G_and_cu_very_uncertain_Contour_PDFedited.pdf}
\end{center}
\end{figure}
\end{frame}
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\begin{frame}
\frametitle{Probabilistic ElasticPlastic Response}
\begin{figure}[!hbpt]
\begin{center}
%\includegraphics[height=6.0cm]{/home/jeremic/tex/works/Conferences/2011/ICASP11_Zurich/Present/PDF_PlotEd.pdf}
\includegraphics[width=9.5cm]{/home/jeremic/tex/works/Conferences/2012/DOELLNLworkshop2728Feb2012/ProbabilisticYielding_vonMises_G_and_cu_very_uncertain_PDFedited.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/JGGEGoverGmax/figures/ShearStrength_RawData_and_MeanTrendMod.pdf}
\hfill
\includegraphics[width=5.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGEGoverGmax/figures/ShearStrength_Histogram_PearsonIVFineTunedMod.pdf}
%
\end{center}
\end{figure}
\vspace*{0.3cm}
Transformation 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/JGGEGoverGmax/figures/YoungModulus_RawData_and_MeanTrend_01Ed.pdf}
\hfill
\includegraphics[width=5.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGEGoverGmax/figures/YoungModulus_Histogram_Normal_01Ed.pdf}
%
\end{center}
\end{figure}
\vspace*{0.3cm}
Transformation of SPT $N$value $\rightarrow$ 1D 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}
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\subsection{Probabilistic ElasticPlastic Response}
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\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 ($E$) $\rightarrow$
KarhunenLo{\`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}
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\begin{frame}
\frametitle{Spectral Stochastic ElasticPlastic FEM}
\begin{itemize}
\item Minimizing norm of error of finite representation using Galerkin
technique (Ghanem and Spanos 2003):
\vspace*{0.6truecm}
\begin{flushright}
\begin{equation}
\nonumber
\sum_{n = 1}^N K_{mn}^{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 FokkerPlanckKolmogorov approach based probabilistic constitutive integration
% at Gauss integration points
\end{itemize}
% \noindent Salient Features:
% \begin{itemize}
%
% \item Efficient representation of input random fields into finite number of random
% variables using KLexpansion
%
% \item Representation of (unknown) solution random variables using polynomial chaos of
% (known) input random variables
%
% \item FokkerPlanckKolmogorov approach based probabilistic constitutive integration
% at Gauss integration points
%
% \end{itemize}
%
%% \end{itemize}
%
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Inside SSEPFEM}
\begin{itemize}
\item Explicit stochastic elasticplastic finite element computations
\vspace*{0.2cm}
\item FPK probabilistic constitutive integration at Gauss integration points
\vspace*{0.2cm}
\item Increase in (stochastic) dimensions (KL and PC) of the problem
\vspace*{0.2cm}
\item Excellent for parallelization, both at the element and global levels
\vspace*{0.2cm}
\item Development of the probabilistic elasticplastic stiffness tensor
\end{itemize}
% \noindent Salient Features:
% \begin{itemize}
%
% \item Efficient representation of input random fields into finite number of random
% variables using KLexpansion
%
% \item Representation of (unknown) solution random variables using polynomial chaos of
% (known) input random variables
%
% \item FokkerPlanckKolmogorov approach based probabilistic constitutive integration
% at Gauss integration points
%
% \end{itemize}
%
%% \end{itemize}
%
\end{frame}
% 
%  \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 ($D$) $\rightarrow$
%  KarhunenLo{\`e}ve (KL) expansion, optimal expansion, error minimizing property
% 
%  \item Unknown solution random field ($u$) $\rightarrow$ Polynomial Chaos (PC)
%  expansion
% 
%  \item Deterministic spatial differential operators ($A$ \& $B$) $\rightarrow$
%  Regular shape function method with Galerkin scheme
% 
% 
%  \end{itemize}
% 
% 
% 
%  \end{itemize}
% 
%  \end{frame}
% 
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %  %  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  % 
%  %  \begin{frame}
%  % 
%  % 
%  %  \frametitle{Truncated KarhunenLo{\`e}ve (KL) expansion}
%  % 
%  %  \begin{itemize}
%  % 
%  %  \item Representation of input random fields in eigenmodes of covariance kernel
%  % 
%  %  % \vspace*{0.1cm}
%  %  % \begin{figure}[!hbpt]
%  %  \begin{flushleft}
%  %  \includegraphics[height=3.0cm]{/home/jeremic/tex/works/Conferences/2008/GeoCongress/Probabilistic/Paper/TypicalDataPlotBH1Edited.jpg}
%  %  % ShearStrengthProfile.jpg}
%  %  \end{flushleft}
%  %  \vspace*{3.15cm}
%  %  \begin{flushright}
%  %  % \begin{equation}
%  %  % \nonumber
%  %  % \begin{normalsize}
%  %  $ q_T(x,\theta) = \bar q_T(x) + \sum_{n=1}^M \sqrt{\lambda_n} \xi_n(\theta) f_n(x) $ \\
%  %  \ \\
%  %  $ \int_D C(x_1, x_2) f (x_2) dx_2 = \lambda f (x_1) \ \ \ \ \ \ \ \ \ \ \ \ $ \\
%  %  \ \\
%  %  $ \xi_i(\theta) = \displaystyle \frac{1}{\sqrt \lambda_i} \int_D \left [q_T(x,\theta)  \bar q_T (x) \right] f_i (x) dx $
%  %  % \end{equation}
%  %  % \end{normalsize}
%  %  \end{flushright}
%  %  % \end{figure}
%  % 
%  %  % \vspace{6.0cm}
%  %  % \begin{flushright}
%  %  % \begin{equation}
%  %  % \nonumber
%  %  % w(x,\theta) = \bar w(x) + \sum_{n=0}^M \sqrt{\lambda_n} \zeta_n(\theta) f_n(x)
%  %  % \end{equation}
%  %  % \end{flushright}
%  % 
%  %  % \vspace*{0.8cm}
%  %  \item Error minimizing property
%  % 
%  %  \item Optimal expansion $\rightarrow$ minimization of number of stochastic dimensions
%  % 
%  % 
%  %  \end{itemize}
%  % 
%  % 
%  % 
%  %  \end{frame}
%  % 
%  % 
%  %  %  %  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %  %  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %  %  \begin{frame}
%  %  %  \frametitle{KL Expansion (of Covariance Kernel)}
%  %  % 
%  %  %  \begin{flushleft}
%  %  %  %\begin{center}
%  %  %  %\includegraphics[width=10cm]{AnticipatedInfluence.jpg}
%  %  %  \includegraphics[height=2.2cm]{/home/jeremic/tex/works/Conferences/2007/GeoDenver/SFEM/Presentation/ActualExponentialCovarianveSurface.jpg}
%  %  %  \hspace*{0.3cm}
%  %  %  \includegraphics[height=2.2cm]{/home/jeremic/tex/works/Conferences/2007/GeoDenver/SFEM/Presentation/KL_ApproxWith_1Term_CovarianveSurface.jpg}
%  %  %  %\vspace*{1.0cm}
%  %  %  %\mbox{Exact covariance surface}
%  %  %  %\vspace*{4.0cm}
%  %  %  \end{flushleft}
%  %  %  \vspace*{0.4cm}
%  %  %  \small{\ \ \ \ \ \ \ \ \ \ \ \ \ Exact \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1 term approx.} \\
%  %  %  \small{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (8.49\% error)}
%  %  %  %\begin{flushright}
%  %  %  %\end{center}
%  %  %  %\end{figure}
%  %  % 
%  %  %  %\vspace*{0.5cm}
%  %  %  %\small{Exact covariance surface \ \ \ \ \ \ \ \ \ \ \ \ \ \ Oneterm approximation}
%  %  %  %
%  %  %  %\vspace*{0.5cm}
%  %  %  %
%  %  %  %\begin{figure}[!hbpt]
%  %  %  %\begin{center}
%  %  %  \begin{flushleft}
%  %  %  \includegraphics[height=2.2cm]{/home/jeremic/tex/works/Conferences/2007/GeoDenver/SFEM/Presentation/KL_ApproxWith_2Terms_CovarianveSurface.jpg}
%  %  %  \hspace*{0.2cm}
%  %  %  \includegraphics[height=2.2cm]{/home/jeremic/tex/works/Conferences/2007/GeoDenver/SFEM/Presentation/KL_ApproxWith_3Terms_CovarianveSurface.jpg}
%  %  %  \end{flushleft}
%  %  %  %\vspace*{0.5cm}
%  %  %  \small{\ \ \ \ \ 2 terms approx. \ \ \ \ \ \ \ \ \ 3 terms approx.} \\
%  %  %  \small{\ \ \ \ \ \ (1.15\% error) \ \ \ \ \ \ \ \ \ \ \ \ \ (1.13\% error)}
%  %  %  %
%  %  %  \vspace{6.5cm}
%  %  %  \begin{flushright}
%  %  % 
%  %  %  \includegraphics[height=2.0cm]{/home/jeremic/tex/works/Reports/2006/SEPFEM/figures/CPT_DataAnalysis_Plots/TypicalAutoCovariancePlotBH1_FiniteScaleEdited.jpg} \hspace*{0.7cm}
%  %  %  %ShearStrengthProfile.jpg}
%  %  % 
%  %  %  covariance function \\
%  %  %  (exponential): \ \ \ \ \ \\
%  %  %  %\ \\
%  %  %  $ C(x_1, x_2) = \sigma^2 e^{ x_1  x_2  /b} $ \hspace*{0.5cm} \\
%  %  %  \ \\
%  %  %  \ \\
%  %  %  KL approximation: \ \ \ \ \ \ \ \ \\
%  %  %  \ \\
%  %  %  $ C(x_1, x_2) \ \ \ \ \ \ \ \ \ \ \ \ \ $ \\
%  %  %  $ = \sum_{k =1}^M \lambda_k f_k(x_1) f_k(x_2) $
%  %  % 
%  %  %  \end{flushright}
%  %  %  %\end{figure}
%  %  % 
%  %  %  %\vspace*{2.5cm}
%  %  %  %\small{Twoterms approximation \ \ \ \ \ \ \ \ \ \ \ \ Threeterms approximation}
%  %  % 
%  %  %  \vspace*{4.0cm}
%  %  %  \begin{center}
%  %  %  \large{KL Expansion of Covariance Kernel}
%  %  %  \end{center}
%  %  % 
%  %  %  \end{frame}
%  %  % 
%  %  % 
%  %  %  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  % 
%  %  \begin{frame}
%  % 
%  %  \frametitle{Polynomial Chaos (PC) Expansion}
%  % 
%  %  %\begin{itemize}
%  % 
%  %  %\vspace*{0.35cm}
%  %  %\item Solution (displacement) random field $\rightarrow$ Can not use KL expansion directly
%  % 
%  %  \begin{itemize}
%  % 
%  %  \item Covariance kernel is not known a priori
%  % 
%  %  \vspace*{0.5cm}
%  %  \begin{normalsize}
%  %  \begin{equation}
%  %  \nonumber
%  %  u(x,\theta)=\sum_{j=1}^L e_j \chi_j(\theta)b_j(x)
%  %  \end{equation}
%  %  \end{normalsize}
%  % 
%  %  \vspace*{0.3cm}
%  %  \item Can be expressed as functional of known random variables and unknown deterministic function
%  % 
%  %  \vspace*{0.5cm}
%  %  \begin{normalsize}
%  %  \begin{equation}
%  %  \nonumber
%  %  u(x,\theta)=\zeta[\xi_i(\theta),x]
%  %  \end{equation}
%  %  \end{normalsize}
%  % 
%  %  \vspace*{0.5cm}
%  %  \item Need a basis of known random variables $\rightarrow$ PC expansion
%  % 
%  %  \vspace*{0.2cm}
%  %  \begin{normalsize}
%  %  \begin{equation}
%  %  \nonumber
%  %  \chi_j(\theta)=\sum_{i=0}^P\gamma_i^{(j)}\psi_i\left[\left\{\xi_r\right\}\right]
%  %  \end{equation}
%  % 
%  %  \vspace*{0.5cm}
%  %  \begin{equation}
%  %  \nonumber
%  %  u(x,\theta)=\sum_{j=1}^L \sum_{i=0}^P \gamma_i^{(j)} \psi_i[\{\xi_r\}]e_j b_j(x) = \sum_{i=0}^P \psi_i[\{\xi_r\}] d_i(x)
%  %  \end{equation}
%  %  \end{normalsize}
%  % 
%  %  % \vspace*{0.6cm}
%  %  % \item Deterministic coefficients can be found by minimizing norm of error of finite
%  %  % representation (e.g. using Galerkin scheme)
%  % 
%  % 
%  % 
%  % 
%  %  % \end{itemize}
%  %  \end{itemize}
%  % 
%  % 
%  %  %\end{itemize}
%  % 
%  %  \end{frame}
%  % 
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
%  \begin{frame}
%  \frametitle{Spectral Stochastic ElasticPlastic FEM}
% 
%  \begin{itemize}
% 
%  \item Minimizing norm of error of finite representation using Galerkin
%  technique (Ghanem and Spanos 2003):
% 
%  \vspace*{0.6truecm}
%  \begin{flushright}
%  \begin{equation}
%  \nonumber
%  \sum_{n = 1}^N K_{mn} d_{ni} + \sum_{n = 1}^N \sum_{j = 0}^P d_{nj} \sum_{k = 1}^M C_{ijk} K'_{mnk} = \left< F_m \psi_i[\{\xi_r\}] \right >
%  \end{equation}
%  \end{flushright}
% 
%  % \begin{itemize}
%  %
%  % \vspace*{0.5cm}
%  % \item Final eqn.:
%  %
%  % \vspace*{0.4cm}
%  % \begin{flushright}
%  % \begin{normalsize}
%  % \begin{equation}
%  % \nonumber
%  % \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \sum_{n = 1}^N K_{mn} d_{ni} + \sum_{n = 1}^N \sum_{j = 0}^P d_{nj} \sum_{k = 1}^M C_{ijk} K'_{mnk} = \left< F_m \psi_i[\{\zeta_r\}] \right >
%  % \end{equation}
%  % \end{normalsize}
%  % \end{flushright}
% 
%  \vspace*{0.5cm}
%  \begin{equation}
%  \nonumber
%  K_{mn} = \int_D B_n \textcolor{mycolor}{D} B_m dV \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ K'_{mnk} = \int_D B_n {\sqrt \lambda_k h_k} B_m dV
%  \end{equation}
% 
%  \vspace*{1.0cm}
%  \begin{equation}
%  \nonumber
%  C_{ijk} = \left < \xi_k(\theta) \psi_i[\{\xi_r\}] \psi_j[\{\xi_r\}] \right > \ \ \ \ \ \ \ \ \ \ \ \ F_m = \int_D \phi N_m dV \ \ \ \ \ \ \ \ \ \ \ \
%  \end{equation}
% 
%  %\item FokkerPlanckKolmogorov approach based probabilistic constitutive integration
%  % at Gauss integration points
% 
% 
%  \end{itemize}
% 
%  % \noindent Salient Features:
% 
%  % \begin{itemize}
%  %
%  % \item Efficient representation of input random fields into finite number of random
%  % variables using KLexpansion
%  %
%  % \item Representation of (unknown) solution random variables using polynomial chaos of
%  % (known) input random variables
%  %
%  % \item FokkerPlanckKolmogorov approach based probabilistic constitutive integration
%  % at Gauss integration points
%  %
%  % \end{itemize}
%  %
%  %% \end{itemize}
%  %
%  \end{frame}
% 
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
%  \begin{frame}
%  \frametitle{Inside SSEPFEM}
% 
%  \begin{itemize}
% 
%  \item Explicit stochastic elasticplastic finite element computations
% 
%  \vspace*{0.2cm}
%  \item FPK probabilistic constitutive integration at Gauss integration points
% 
%  \vspace*{0.2cm}
%  \item Increase in (stochastic) dimensions (KL and PC) of the problem
% 
% 
%  \vspace*{0.2cm}
%  \item Development of the probabilistic elasticplastic stiffness tensor
% 
% 
%  \end{itemize}
% 
%  % \noindent Salient Features:
% 
%  % \begin{itemize}
%  %
%  % \item Efficient representation of input random fields into finite number of random
%  % variables using KLexpansion
%  %
%  % \item Representation of (unknown) solution random variables using polynomial chaos of
%  % (known) input random variables
%  %
%  % \item FokkerPlanckKolmogorov approach based probabilistic constitutive integration
%  % at Gauss integration points
%  %
%  % \end{itemize}
%  %
%  %% \end{itemize}
%  %
%  \end{frame}
% 
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\subsection{Seismic Wave Propagation Through Uncertain Soils}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%
%
% \frametitle{Applications}
%
%
%
%
% \begin{itemize}
%
% \vspace*{0.3cm}
% \item Stochastic elasticplastic simulations of soils and structures
%
% \vspace*{0.3cm}
% \item Probabilistic inverse problems
%
% \vspace*{0.3cm}
% \item Geotechnical site characterization design
%
% \vspace*{0.3cm}
% \item Optimal material design
%
%
% \end{itemize}
%
% \end{frame}
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  +
%  + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  +
%  + \begin{frame}
%  +
%  +
%  + \frametitle{Random Field Modeling of Uncertain Soil Properties}
%  +
%  + \begin{itemize}
%  +
%  + \item Finite scale model
%  +
%  + \begin{itemize}
%  +
%  + \item Short memory, finite correlation length
%  +
%  + \item Common autocovariance model $\rightarrow$ exponential, spherical, triangular, linearexponential
%  +
%  + \end{itemize}
%  +
%  + \item Fractal model
%  +
%  + \begin{itemize}
%  +
%  + \item long memory, infinite correlation length $\rightarrow$ more realistic for modeling horizontal
%  + spatial uncertainty
%  +
%  + \item 1/ftype noise process with power spectral density, $P(\omega)~=~P_0~\omega^{\gamma}$, with
%  + upper and/or lower frequency cutoff.
%  +
%  + \end{itemize}
%  +
%  + \end{itemize}
%  +
%  + \end{frame}
%  +
%  + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Seismic Wave Propagation through Stochastic Soil}
\begin{itemize}
%\item maximizing the loglikelihood of observing the spatial data under assumed joined distribution (for finite
%scale model) or maximizing the loglikelihood of observing the periodogram estimates (for fractal model)
\item Maximum likelihood estimates
\vspace*{0.3truecm}
%\begin{figure}
\begin{flushleft}
\hspace*{1.7cm}
\includegraphics[height=4.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/CPT_DataAnalysis_Plots/SamplingPlanEdited.jpg}
\hspace*{0.0cm}
\includegraphics[height=4.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/CPT_DataAnalysis_Plots/TypicalDataPlotBH1Edited.jpg} \\
\small{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Typical CPT $q_T$}
\end{flushleft}
%\end{figure}
\vspace*{4.9truecm}
%\begin{figure}
\begin{flushright}
\includegraphics[width=4.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/CPT_DataAnalysis_Plots/TypicalAutoCovariancePlotBH1_FiniteScaleEdited.jpg} \\
\vspace*{0.01truecm}
\small{Finite Scale}
\end{flushright}
%\end{figure}
\vspace*{0.02truecm}
%\begin{figure}
\begin{flushright}
\includegraphics[width=4.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/CPT_DataAnalysis_Plots/TypicalAutoCovariancePlotBH1_FractalEdited.jpg} \\
\small{Fractal}
\end{flushright}
%\end{figure}
\end{itemize}
\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/EastWestProfileEdited.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 1D soil column (von Mises Material)
\begin{itemize}
\item Properties (including testing uncertainty) obtained through random field modeling of CPT $q_T$
%
$\left = 4.99 ~MPa;~~Var[q_T] = 25.67 ~MPa^2; $\\
Cor. ~Length $[q_T] = 0.61 ~m; $ Testing~Error $= 2.78 ~MPa^2$
\end{itemize}
\vspace*{0.2cm}
\item $q_T$ was transformed to obtain $G$: ~~$G/(1\nu)~=~2.9q_T$
\begin{itemize}
\item Assumed transformation uncertainty = 5\%
%
$\left = 11.57MPa; Var[G] = 142.32 MPa^2$ \\
Cor.~Length $[G] = 0.61 m$
\end{itemize}
%\begin{center}
%\hspace*{1.7cm}
%\includegraphics[height=3.5cm]{Chapter9_Schematic.jpg}
%\hspace*{0.0cm}
%\includegraphics[height=3.5cm]{Chapter9_BaseDisplacement.jpg} \\
%\small{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Base Displacement}
%\end{center}
\vspace*{0.2cm}
\item Input motions: modified 1938 Imperial Valley
% \vspace*{0.2cm}
% \begin{center}
% \includegraphics[height=2.0cm]{Chapter9_BaseDisplacement.jpg}
% \end{center}
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% \begin{frame}
%
%
% \frametitle{Seismic Wave Propagation through Stochastic Soil}
%
%
%
% \begin{figure}
% \begin{center}
% \hspace*{0.75cm}
% \includegraphics[width=9.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/Chapter9Plots/Chapter9_ElasticPlasticResponseNew.pdf}
% %\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
% \end{center}
% \end{figure}
%
% Mean$\pm$ Standard Deviation
%
%
%
% %\begin{flushleft}
% %\includegraphics[height=5.0cm]{PEER2007_3.jpg}
% %\end{flushleft}
%
% % \hspace*{1.0cm} \noindent Statistics of Top Node Displacement:
% %
% % \vspace*{0.5truecm}
% %
% % \begin{figure}
% % \begin{flushleft}
% % \hspace*{1.0cm}
% % \includegraphics[width=4.0cm]{/home/kallol/publication/2007/Presentation/PhDExitSeminar/Chapter9_ElasticPlasticResponse_MeanNew.jpg}
% % \hspace*{0.1cm}
% % \includegraphics[width=4.0cm]{/home/kallol/publication/2007/Presentation/PhDExitSeminar/Chapter9_ElasticPlasticResponse_SDNew.jpg}
% % \end{flushleft}
% % \end{figure}
% % \vspace*{0.5truecm}
% % \hspace*{1.0cm} \tiny{~~~~~~~~~~~~~~~~~~~~~~~~~~~~Mean~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Standard Deviation}
% %
% % \vspace*{0.3truecm}
% %
% % \begin{figure}
% % \begin{flushleft}
% % \hspace*{0.75cm}
% % \includegraphics[width=4.0cm]{/home/kallol/publication/2007/Presentation/PhDExitSeminar/Chapter9_ElasticPlasticResponseNew.jpg}
% % \hspace*{0.4cm}
% % \includegraphics[width=4.0cm]{/home/kallol/publication/2007/Presentation/PhDExitSeminar/Chapter9_ElasticPlasticResponse_COVNew.jpg}
% % \end{flushleft}
% % \end{figure}
% % \vspace*{0.3truecm}
% % \hspace*{0.5cm} \tiny{~~~~~~~Mean$\pm$ Standard Deviation~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~COV}
% %
% %
% % \vspace*{6.0cm}
% % \begin{flushright}
% % \includegraphics[height=4.5cm]{/home/kallol/publication/2007/Presentation/PhDExitSeminar/Chapter9_ElasticPlasticResponse_PDFNewEdited.jpg} \hspace*{1.0cm}
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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}
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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 ElasticPlastic\\
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/UNIONUnivBGD/Present/Plots_with_Labels/EvolutionaryPDF_ActualEdited.pdf}
%\vspace*{0.75cm}
%\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
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\frametitle{Evolution of Mean $\pm$ SD for Guess Case}
\begin{figure}
\begin{center}
\hspace*{0.75cm}
\includegraphics[width=10.0cm]{/home/jeremic/tex/works/Conferences/2009/UNIONUnivBGD/Present/Plots_with_Labels/Evolutionary_Mean_pm_SD_NoDataEdited.pdf}
\hspace*{0.75cm}
%\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
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\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/UNIONUnivBGD/Present/Plots_with_Labels/PDFs_at6sec_Actual_vs_NoDataEdited.pdf}
\vspace*{0.75cm}
%\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
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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/UNIONUnivBGD/Present/Plots_with_Labels/CDFs_at6sec_Actual_vs_NoDataEdited.pdf}
\vspace*{0.75cm}
%\hspace*{0.75cm}
%\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
\end{center}
\end{figure}
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%
%
% \frametitle{Probability of Exceedance of $20$cm}
%
%
%
% \begin{figure}
% \begin{center}
% %\hspace*{0.75cm}
% \includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2009/UNIONUnivBGD/Present/Plots_with_Labels/ProbabilityOfExceedance20cm_Actual_vs_NoDataEdited.pdf}
% \vspace*{0.75cm}
% %\hspace*{0.75cm}
% %\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
% \end{center}
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%
%
% \frametitle{Probability of Exceedance of $50$cm}
%
%
%
% \begin{figure}
% \begin{center}
% %\hspace*{0.75cm}
% \includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2009/UNIONUnivBGD/Present/Plots_with_Labels/ProbabilityOfExceedance50cm_Actual_vs_NoDataEdited.pdf}
% \vspace*{0.75cm}
% %\hspace*{0.75cm}
% %\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
% \end{center}
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%
%
% \frametitle{Probabilities of Exceedance vs. Displacements}
%
%
%
% \begin{figure}
% \begin{center}
% %\hspace*{0.75cm}
% \includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2009/UNIONUnivBGD/Present/Plots_with_Labels/ProbabilityOfExceedance_vs_Displacement_Actual_vs_NoDataEdited.pdf}
% \vspace*{0.75cm}
% %\hspace*{0.75cm}
% %\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
% \end{center}
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%
% \frametitle{Probabilities of Unacceptable Deformation}
%
% \begin{figure}
% \begin{center}
% \vspace*{0.3cm}
% \includegraphics[width=10.5cm]{/home/jeremic/tex/works/Conferences/2009/UNIONUnivBGD/Present/NewPlots/with_legends_and_labels/Exceedance20cm_LomaPrietaEdited_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}
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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/UNIONUnivBGD/Present/NewPlots/with_legends_and_labels/Exceedance50cm_LomaPrietaEdited_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/UNIONUnivBGD/Present/NewPlots/with_legends_and_labels/Summary_LomaPrietaEdited.pdf}
\vspace*{0.75cm}
%\hspace*{0.75cm}
%\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
\end{center}
\end{figure}
\end{frame}
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\section{Summary}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Summary}
\begin{itemize}
% \item 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 ElasticPlastic 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 USNRC, and
funding from NSF, DOE.
\end{itemize}
\end{frame}
%
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\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[][CVJ30]
K. Sett,
B. Jeremi{\'c}.
and M. L. Kavvas.
Stochastic ElasticPlastic Finite Elements.
{\em Computer Methods in Applied Mechanics and
Engineering}, Vol 200, No. 912, pp 9971007,
2011
{.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item[][]
\vspace*{0.1cm}
\item[]
%\item[][CVJ29]
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 218226, March, 2011{.}
% 2009{\href{http://sokocalo.engr.ucdavis.edu/~jeremic/wwwpublications/Seismic_isolation_by_liquefaction.pdf}
% {.}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item[][]
\vspace*{0.1cm}
\item[]
%\item[][CVJ28]
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~236257,
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[][CVJ27]
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~15411559,
2010
{.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item[][]
\vspace*{0.1cm}
\item[]
%\item[][CVJ26]
Z. Cheng and B. Jeremi{\'c}.
Numerical Simulations of Piles in Liquefied Soils.
{\em Soil Dynamics and Earthquake Engineering}, No.~29, pp~14051416,
2009
{.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item[][]
\vspace*{0.1cm}
\item[]
%\item[][CVJ24]
B. Jeremi{\'c},
G. Jie, M. Preisig and N. Tafazzoli.
Time domain simulation of soilfoundationstructure interaction in
nonuniform soils.
{\em Earthquake Engineering and Structural Dynamics},
Volume 38, Issue 5, pp 699718,
2009
{.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item[][]
\vspace*{0.1cm}
\item[]
%\item[][CVJ22]
B. Jeremi{\'c} and K. Sett.
On Probabilistic Yielding of Materials.
{\em Communications in Numerical Methods
in Engineering}, Volume 25, No. 3, pp 291300,
2009
{.}
%{\tiny (UCDpub53) }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item[][]
\vspace*{0.1cm}
\item[]
%\item[][CVJ20]
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 16351660,
2008
{.}
%{\tiny (UCDpub52) }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item[][]
\vspace*{0.1cm}
\item[]
%\item[][CVJ18]
K. Sett, B. Jeremi{\' c}, and
M. L. Kavvas.
Probabilistic ElastoPlasticity:
Solution and Verification in 1D.
{\em Acta Geotechnica}, Volume 2., No. 3. pp 211220, October
2007
{.}
%{\tiny (UCDpub47) }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item[][]
\vspace*{0.1cm}
\item[]
%\item[][CVJ17]
B. Jeremi{\' c}, K. Sett and M. L. Kavvas.
Probabilistic ElastoPlasticity: Formulation in 1D.
{\em Acta Geotechnica}, Volume 2., No. 3. pp 197210, October
2007
{.}
%{\tiny (UCDpub46) }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item[][]
\vspace*{0.1cm}
\item[]
%\item[][CVJ15]
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. 762770
{.}
%{\tiny (UCDpub35) }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\item[][]
\vspace*{0.1cm}
\item[]
%\item[][CVJ14]
B. Jeremi{\' c}, S. Kunnath and F. Xiong.
Influence of SoilStructure interaction on Seismic Response of
Bridges.
{\em International Journal for Engineering
Structures}, Volume 26, Issue 3, February 2004, pp.
391402
{.}
% 
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %\item[][]
%  \item[]
%  %\item[][CVJ20]
%  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 16351660,
%  2008
%  {.}
% 
% 
% 
%  %{\tiny (UCDpub52) }
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% 
% 
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %\item[][]
%  \item[]
%  %\item[][CVJ18]
%  K. Sett, B. Jeremi{\' c}, and
%  M. L. Kavvas.
%  Probabilistic ElastoPlasticity:
%  Solution and Verification in 1D.
%  {\em Acta Geotechnica}, Volume 2., No. 3. pp 211220, October
%  2007
%  {.}
% 
%  %{\tiny (UCDpub47) }
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% 
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %\item[][]
%  \item[]
%  %\item[][CVJ17]
%  B. Jeremi{\' c}, K. Sett and M. L. Kavvas.
%  Probabilistic ElastoPlasticity: Formulation in 1D.
%  {\em Acta Geotechnica}, Volume 2., No. 3. pp 197210, October
%  2007
%  {.}
% 
%  %{\tiny (UCDpub46) }
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% 
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %\item[][]
%  \item[]
%  %\item[][CVJ16]
%  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 477486, September
%  2005
%  {.}
% 
%  %{\tiny (UCDpub37) }
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% 
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %\item[][]
%  \item[]
%  %\item[][CVJ15]
%  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. 762770
%  {.}
% 
%  %{\tiny (UCDpub35) }
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% 
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %\item[][]
%  \item[]
%  %\item[][CVJ14]
%  B. Jeremi{\' c}, S. Kunnath and F. Xiong.
%  Influence of SoilStructure interaction on Seismic Response of
%  Bridges.
%  {\em International Journal for Engineering
%  Structures}, Volume 26, Issue 3, February 2004, pp.
%  391402
%  {.}
% 
%  %{\tiny (UCDpub27) }
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% 
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %\item[][]
%  \item[]
%  %\item[][CVJ12]
%  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
%  12551276, Dec.
%  2003
%  {.}
% 
%  %{\tiny (UCDpub24)}
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %\item[][]
%  \item[]
%  %\item[][CVJ11 ]
%  Z. Yang and B. Jeremi{\' c}.
%  Numerical analysis of pile behavior under lateral loads in
%  layered elasticplastic soils.
%  {\em International Journal for
%  Numerical and Analytical Methods in Geomechanics}, Volume 26, Issue 14, pp
%  13851406, Dec.
%  2002{.}
% 
\end{itemize}
%\end{list}
\end{tiny}
\end{frame}
%
\end{document}