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\title{On Seismic Soil Structure \\
Interaction Simulations for \\
Nuclear Power Plants }
%\subtitle
%{Include Only If Paper Has a Subtitle}
%\author[Author, Another] % (optional, use only with lots of authors)
%{F.~Author\inst{1} \and S.~Another\inst{2}}
% - Give the names in the same order as the appear in the paper.
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% affiliation.
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\pgfdeclareimage[height=0.7cm]{lbnl-logo}{/home/jeremic/BG/amblemi/lbnl-logo}
\author[Jeremi{\'c} at al.] % (optional, use only with lots of authors)
{B.~Jeremi{\'c},
N.~Tafazzoli,
B.~Kamrani,
Y.C.~Chao,\\
C.G.~Jeong,
P.~Tasiopoulou,
K.~Sett,
A.~Kammerer,\\
N.~Orbovi{\'c}, and
A.~Blahoianu
}
%{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}] % (optional, but mostly needed)
{}
%{Univ. of California, Davis, Univ. of Akron, NTUA, U.S. NRC, CNSC }
% - Use the \inst command only if there are several affiliations.
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\date[] % (optional, should be abbreviation of conference name)
{\small OECD/NEA SSI workshop, Ottawa, October 2010}
\subject{}
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%\section{A Historical Note}
\section{A Hypothesis}
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% \subsection{Early Works}
%
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% \begin{frame}
%
% \frametitle{Historical Note on ESSI}
%
%
% \begin{itemize}
% \item Kyoji Suyehiro (Japanese Imperial
% Academy) Professor of Naval Engineering at Tokyo Imperial University, a
% director of Mitsubishi Research Institute for Shipbuilding
% \item Life changing event (was present in) Tokyo during Great
% Kant{\= o} earthquake, 01Sept1923, 11:58am(7.5), 12:01pm(7.3),
% 12.03pm(7.2)
% \item Saw earthquake surface waves travel and buildings sway
% \item Founding Director of the Earthquake Engineering Research
% Institute Univ. of Tokyo
% \item Observation of different damage to similar buildings on different
% soil/rock sites.
% % \item Records shows $4 \times$ (four) more damage to soft
% % wooden buildings on soft ground (nonlinear resonance ?)
% \item Toured USA in 1931 and gave a series of lectures at
% UCB, Stanford, CalTech, MIT, motivating
% Earthquake Engineering research in the U.S.A.
% \end{itemize}
%
%
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\subsection*{A Hypothesis}
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\begin{frame}
\frametitle{The ESSI Hypothesis}
\begin{itemize}
\item NPPSSS response is a function of a tightly coupled (in space and
time) triad of dynamic characteristic of
\begin{itemize}
\item Earthquake Ground Motions
\item Underlying Soil/Rock
\item NPP Structure, Systems and Components (NPPSSC)
\end{itemize}
%\item Interplay (in time and space) of dynamic characteristics of
% \begin{itemize}
% \item earthquake ground motions
% \item underlying soil/rock
% \item NPP structure
% \end{itemize}
%Determine the accumulation or prevention of damage
\item Energy balance: input (seismic) and dissipated (inelasticity, radiation,
coupling) will control fate of the NPPSSS
\item Better understanding of the timing and spatial location of energy
dissipation in Earthquake-Soil-Structure Interaction (ESSI) system
can add significant benefit to the safety and economy of NPPSSSs
\item High Fidelity Numerical Simulations of ESSI for NPPSSS
\end{itemize}
\end{frame}
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%\section{Seismic Energy Input, Dissipation, Mechanics, Modeling and Simulations}
\section{Seismic Energy}
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\subsection{Seismic Energy Input}
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%\section{Overview of the available computational tools for SSI analysis}
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Seismic Energy Source}
%
% \begin{itemize}
%
% %\vspace*{0.2cm}
% \item Large energy releases,
% \begin{itemize}
% \item Northridge, 1994, $M_{Richter} = 6.7$, $E_r = 6.8 \times 10^{16}J$
% % \item Loma Prieta, 1989, $M_{Richter} = 6.9$, $E_r =1.1 \times 10^{17}J$
% % \item LARGE JAPANESE EQ.
% % \item LARGE NEW ZEALAND EQ.
% \item Sumatra-Andaman, 2004, $M_{Richter} = 9.3$, $E_r =4.8 \times 10^{20}J$
% \item Valdivia, Chile, 1960, $M_{Richter} = 9.5$, $E_r =7.5 \times 10^{20}J$
% % \item Rhodes, 2008, $M_{Richter} = 6.5$, $E_r =2.4 \times 10^{16}J$
% \end{itemize}
% \vspace*{0.5cm}
% \item Part that energy is radiated as mechanical waves
% % ($\approx 1.6 \times 10^{-5}$)
% and makes it to the surface
% %\vspace*{0.2cm}
% % \item For comparison, specific energy of TNT is $4.2\times 10^6 J/kg$.
% %%\vspace*{0.2cm}
% %% \item Rhodes earthquake was $\approx 0.1 kt$.
%
% \end{itemize}
%
%
% \end{frame}
% %-
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\begin{frame}
\frametitle{Seismic Energy Input Into the NPPSSS}
\begin{itemize}
\vspace*{-1.0cm}
\item Seismic energy propagates to the NPPSSS
\item Kinetic energy flux through closed surface $\Gamma$ includes both incoming
and outgoing waves (using DRM)
\begin{eqnarray*}
E_{flux} =
\left[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 \right]_i
%
% \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]^{T}
\times u_i
%\left[
%\begin{array}{c}
%0 \\
%{u}_b\\
%{u}_e
%\end{array}
%\right]
\end{eqnarray*}
\item Alternatively, $E_{flux} = \rho A c \int_0^t \dot{u}_i^2 dt$
\item Outgoing kinetic energy \\
is obtained from outgoing \\
wave field ($w_i$, in DRM)
\item Incoming kinetic energy \\
is then the difference.
\end{itemize}
\vspace*{-3.0cm}
% \begin{figure}[!hbpt]
\begin{flushright}
\hfill \includegraphics[width=5.5cm]{/home/jeremic/tex/works/Conferences/2010/NRC-LBL-ProjectReviewMeeting_21_22_Sept_2010/DRMideaNPP03.pdf}
\end{flushright}
%\end{figure}
\vspace*{-2cm}
\end{frame}
%-
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\subsection{Seismic Energy Dissipation}
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\begin{frame}
% \frametitle{Seismic Energy Dissipation for \underline{Soil}-Foundation-Structure Systems}
\frametitle{Seismic Energy Dissipation within NPPSSS}
% \frametitle{Seismic Energy Dissipation for
% \underline{Soil}-Foundation-Structure Systems}
\begin{itemize}
\vspace*{0.2cm}
\item Mechanical dissipation outside of NPPSSS domain:
\begin{itemize}
\item reflected wave radiation
\item NPP system oscillation radiation
\end{itemize}
\vspace*{0.2cm}
\item Mechanical dissipation/conversion inside NPPSSS:
\begin{itemize}
\item plasticity of the soil/rock subdomain
\item viscous coupling of porous solid with pore fluid (air, water)
\item plasticity/damage of parts of the structure/foundation
\item viscous coupling of structure/foundation with fluids
% \item potential and kinetic energy
% \item potential $\leftarrow \! \! \! \! \! \! \rightarrow$ kinetic energy
\end{itemize}
\vspace*{0.2cm}
% \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}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Energy Dissipation by Soil Plasticity}
%
%
% \begin{itemize}
%
% % \item Plastic work ($W = \int_{0}^{t} \sigma_{ij} d \epsilon_{ij}^{pl}$)
% \item Plastic work ($W = \int \sigma_{ij} d \epsilon_{ij}^{pl}$)
%
% \item Energy dissipation capacity for different soils
%
%
% \end{itemize}
%
% %\vspace*{-1.0cm}
% \begin{center}
% \vspace*{-0.9cm}
% \includegraphics[width=8.5cm]{/home/jeremic/tex/works/Conferences/2009/CompDyn/Present/Energy-Capacity2.pdf}
% \hspace*{-0.2cm}
% %\includegraphics[width=6.0cm]{/home/jeremic/tex/works/Conferences/2009/CompDyn/Present/Energy-Capacity10.pdf}
% \vspace*{-1.0cm}
% \end{center}
%
%
% \end{frame}
% %-
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Energy Dissipation by Soil Viscous Coupling}
%
%
% \begin{itemize}
%
% \item Viscous coupling of porous solid and fluid
% \item Energy loss per unit volume is $E_{vc}= n^2 k^{-1} (\dot{U}_i - \dot{u}_i)^2$
% % \item Natural in $u-p-U$ formulation:
%
% \end{itemize}
% %\vspace*{-0.4cm}
%
%
%
%
% \begin{footnotesize}
% %\begin{tiny}
% \begin{eqnarray*}
% %& &\left[ \begin{array}{ccc}
% \left[ \begin{array}{ccc}
% (M_s)_{KijL} & 0 & 0 \\
% 0 & 0 & 0 \\
% 0 & 0 & (M_f)_{KijL}
% \end{array} \right]
% \left[ \begin{array}{c}
% \ddot{\overline{u}}_{Lj} \\
% \ddot{\overline{p}}_N \\
% \ddot{\overline{U}}_{Lj}
% \end{array} \right]
% +
% \left[ \begin{array}{ccc}
% (C_1)_{KijL} & 0 & -(C_2)_{KijL} \\
% 0 & 0 & 0 \\
% -(C_2)_{LjiK} & 0 & (C_3)_{KijL} \\
% \end{array} \right]
% \left[ \begin{array}{c}
% \dot{\overline{u}}_{Lj} \\
% \dot{\overline{p}}_N \\
% \dot{\overline{U}}_{Lj}
% \end{array} \right]
% \nonumber\\
% +
% %& &\left[ \begin{array}{ccc}
% \left[ \begin{array}{ccc}
% (K^{EP})_{KijL} & -(G_1)_{KiM} & 0 \\
% -(G_1)_{LjM} & -P_{MN} & -(G_2)_{LjM} \\
% 0 & -(G_2)_{KiL} & 0
% \end{array} \right]
% \left[ \begin{array}{c}
% \overline{u}_{Lj} \\
% \overline{p}_M \\
% \overline{U}_{Lj}
% \end{array} \right]
% =
% \left[ \begin{array}{c}
% \overline{f}_{Ki}^{solid} \\
% 0 \\
% \overline{f}_{Ki}^{fluid}
% \end{array} \right] \nonumber\\
% \label{68}
% \end{eqnarray*}
% %\end{tiny}
% \end{footnotesize}
%
% \vspace*{-1cm}
%
% \begin{footnotesize}
% \begin{eqnarray*}
% %%%%%%%%
% (C_{(1,2,3)})_{KijL}
% =
% \int_{\Omega} N_K^{(u,u,U)}
% n^2 k_{ij}^{-1}
% N_L^{(u,U,U)} d\Omega
% % (C_1)_{KijL} =\int_{\Omega} N_K^u n^2 k_{ij}^{-1} N_L^u d\Omega
% % \;\; \mbox{;} \;\;
% % (C_2)_{KijL} =\int_{\Omega} N_K^u n^2 k_{ij}^{-1} N_L^U d\Omega
% % %%%%%%%%
% % %\;\; \mbox{;}\;\;
% % \\
% % %%%%%%%%
% % (C_3)_{KijL} =\int_{\Omega} N_K^U n^2 k_{ij}^{-1} N_L^U d\Omega
% \end{eqnarray*}
% \end{footnotesize}
% %
% %
% %
% %
% %\newpage
%
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%
%
% \end{frame}
% %-
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% \begin{frame}
% \frametitle{Numerical Energy Dissipation}
%
% \begin{itemize}
%
% \item Newmark and Hilber-Hughes-Taylor can be made non-dissipative for
% elastic system $\alpha=0.0, \beta = 0.25 ; \gamma = 0.5,$
%
% \item Or dissipative (for elastic) for higher frequency modes:
% %Newmark ($\gamma \ge 0.5, \;\;\; \beta = 0.25(\gamma+0.5)^2$ ),
% %Hilber-Hughes-Taylor ($-0.3\dot{3}\le\alpha \le0, \;\;\;\gamma =
% %0.5(1-2\alpha), \;\;\; \beta = 0.25(1-\alpha)^2$)
% \begin{itemize}
% \item N: $\gamma \ge 0.5, \;\;\; \beta = 0.25(\gamma+0.5)^2$,
% \item HHT: $-0.3\dot{3}\le\alpha \le0, \;\;\;\gamma =
% 0.5(1-2\alpha), \;\;\; \beta = 0.25(1-\alpha)^2$
% \end{itemize}
%
% \item For nonlinear problems,
% energy cannot be maintained
% \begin{itemize}
% \item Energy dissipation for steps with reduction of stiffness
% \item Energy production for steps with increase of stiffness
%
% \hspace{1cm}
% \includegraphics[width=1.80cm, angle=-90]{/home/jeremic/tex/works/Conferences/2009/CompDyn/Present/EnergyDissipationReducationStiffness.pdf}
% \hspace{1cm}
% \includegraphics[width=1.80cm, angle=-90]{/home/jeremic/tex/works/Conferences/2009/CompDyn/Present/EnergyProductionIncreaseStiffness.pdf}
% \hspace{1cm}
%
%
%
% \end{itemize}
%
%
% \end{itemize}
%
%
% \end{frame}
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\section{ESSI Modeling}
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\subsection{Frequency and Time Domain Techniques}
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\begin{frame}
\frametitle{ESSI Modeling Approaches}
\begin{itemize}
\item Analytical (closed form) solutions
\begin{itemize}
\item Limited application for realistic NPPSSCs
\item Excellent for verification studies
\item Good for initial insight
\item Potentially large modeling uncertainty!
\end{itemize}
\item Numerical solutions
\begin{itemize}
\item Integral Equations (Boundary Element Method, CLASSI)
\item Finite Element Methods
\begin{itemize}
\item Frequency domain (SASSI, etc.), widely used, linear elastic, etc.
\item Time domain (LS-DYNA, NRC ESSI Simulator, etc.), gaining
popularity, full non-linear, etc.
\end{itemize}
% \item Can be very powerful but can also be misused
\item Educated developers/modelers/analysts are a must
\end{itemize}
\end{itemize}
\end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%
% \frametitle{Frequency and Time Domain Approaches}
%
%
% \begin{itemize}
% \item Frequency Domain Approach
% \begin{itemize}
% \item Widely used in seismic analysis of NPPSSCs
% \item Rely on superposition (linear elasticity assumed)
% \item Energy dissipation mimicked through viscous damping
% \item Can be very powerful, however, can be unintentionally misused and
% needs very educated modelers/analysts
% \end{itemize}
%
% \item Time Domain Approach
%
% \begin{itemize}
% \item Not as widely used for seismic analysis of NPPSSCs
% \item Offer accurate, nonlinear modeling of mechanics
% of Earthquake-Soil-Structure Interaction for NPPSSSs
% \item Can be very powerful, however, can be
% misused and needs very educated modelers/analysts
% \item Require use of new (already available and/or in
% development) software and hardware tools
% \end{itemize}
%
% \end{itemize}
%
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\begin{frame}
\frametitle{NRC ESSI Simulator Program: Library Centric Design}
%Levels of abstraction
\begin{itemize}
\item A full 3D, non-linear Earthquake-Soil-Structure Interaction program,
computer and documentation system
\item MOSS library (UCD Modified OpenSees Services: trimmed, debugged,
verified, documented),
\item Plastic Domain Decomposition for Parallel Computing
\item Finite element and material libraries (FEMtools, Template3DEP)
\item Numerical utility libraries (BLAS, lapack, nDarray, matrix...)
\item Solver libraries (UMFPACK, PETSc, SuperLU...)
\item Graph libraries (ParMETIS)
\item Domain Specific Language (DSL) library
\item Verification, Validation, Educational, and Real NPPSSS Examples library
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{NRC ESSI Simulator Program: Management}
\begin{itemize}
\vspace*{0.2cm}
\item Application Programming Interface (API): well documented, for all
libraries and examples
\vspace*{0.2cm}
\item Detailed theory background
\vspace*{0.2cm}
\item Verification examples, extensive
\vspace*{0.2cm}
\item Validation examples, as available
\vspace*{0.2cm}
\item Educational examples, extensive
\vspace*{0.2cm}
\item NRC ESSI Simulation in public domain, an open source
license (LPGL)
\vspace*{0.2cm}
\item Source files management by subversion for a large number of
developers and users
%, currently at code.google.com,
% probable move to LBNL
% \item Editors appointed for source libraries, example libraries, documentation
% libraries...
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{NRC ESSI Simulator Computer}
\begin{itemize}
%\vspace*{0.2cm}
\item Distributed memory parallel \\
computer
%\vspace*{0.2cm}
% \vspace*{0.3cm}
\item Very cost effective, \\
affordable, high availability,\\
design exportable to Companies, \\
Regulatory Agencies, Universities
%\vspace*{0.2cm}
% \vspace*{0.3cm}
\item Same architecture as large \\
parallel supercomputers\\
(SDSC, TACC, EarthSimulator...)
%\vspace*{0.2cm}
% \vspace*{0.3cm}
\item Current version at UCD, \\
new version to be acquired soon
%
\vspace*{-5.6cm}
\begin{figure}[!htbp]
%\hspace*{6cm}
\begin{flushright}
\includegraphics[width=4.2cm]{/home/jeremic/tex/works/Conferences/2010/Ottawa_SSI_conference/Present/Geowulf.jpg}
%\includegraphics[width=5cm]{/home/jeremic/public_html/GeoWulf/Dec2006/IMG_0907.jpg}
\end{flushright}
\hspace*{-2cm}
\end{figure}
% \item Local design, construction,
% available at all times!
%\vspace*{0.2cm}
% \item
%%\vspace*{0.2cm}
\end{itemize}
\end{frame}
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% \begin{frame}
% \frametitle{NRC ESSI Simulator Computer: Architecture}
%
%
% %
% %\vspace*{-0.6cm}
% \begin{figure}[!htbp]
% \begin{center}
% \includegraphics[width=11truecm]{/home/jeremic/tex/works/psfigures/ParallelGeoWulf07.pdf}
% %\hspace*{0.1cm}
% %\includegraphics[width=3truecm]{/home/jeremic/public_html/NSF-Nuggets/Students_develop_parallel_computer/StudentsConstructingGeoWulf.jpg}
% \end{center}
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% \begin{frame}
% \frametitle{NRC ESSI Simulator Computer: Space and Cooling}
%
% %
% %\vspace*{-5.6cm}
% \begin{figure}[!htbp]
% %\hspace*{6cm}
% %\begin{center}
% \includegraphics[width=8truecm]{/home/jeremic/public_html/GeoWulf/Dec2006/IMG_0907.jpg}
% %\hfill
% %%\hspace*{6cm}
% %\includegraphics[width=6truecm]{/home/jeremic/public_html/NSF-Nuggets/Students_develop_parallel_computer/StudentsConstructingGeoWulf.jpg}
% %\end{center}
% \end{figure}
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\begin{frame}
\frametitle{Illustrative Example: Free Field}
%
%\vspace*{-5.6cm}
\begin{figure}[!htbp]
%\hspace*{6cm}
%\begin{center}
\includegraphics[width=10truecm]{/home/jeremic/tex/works/Conferences/2010/Ottawa_SSI_conference/Paper/model_FF.jpg}
%
%/home/jeremic/tex/works/Conferences/2010/NRC-LBL-ProjectReviewMeeting_21_22_Sept_2010/VisIt_Full_Model_DRM_1.jpeg}
%\hfill
%%\hspace*{6cm}
%\includegraphics[width=6truecm]{/home/jeremic/public_html/NSF-Nuggets/Students_develop_parallel_computer/StudentsConstructingGeoWulf.jpg}
%\end{center}
\end{figure}
%
\end{frame}
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\begin{frame}
\frametitle{Illustrative Example: ESSI for NPPs}
%
%\vspace*{-5.6cm}
\begin{figure}[!htbp]
%\hspace*{6cm}
%\begin{center}
\includegraphics[width=10truecm]{/home/jeremic/tex/works/Conferences/2010/Ottawa_SSI_conference/Paper/model_SSI.jpg}
%\includegraphics[width=10truecm]{/home/jeremic/tex/works/Conferences/2010/Ottawa_SSI_conference/Paper/model_FF.jpg}
%/home/jeremic/tex/works/Conferences/2010/NRC-LBL-ProjectReviewMeeting_21_22_Sept_2010/VisIt_Excavated_Material_DRM_3.jpeg}
%\hfill
%%\hspace*{6cm}
%\includegraphics[width=6truecm]{/home/jeremic/public_html/NSF-Nuggets/Students_develop_parallel_computer/StudentsConstructingGeoWulf.jpg}
%\end{center}
\end{figure}
%
\end{frame}
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\begin{frame}
\frametitle{Seismic Input: Green's Function and DRM}
%
%\vspace*{-5.6cm}
\begin{figure}[!htbp]
%\hspace*{6cm}
%\begin{center}
\includegraphics[width=9truecm]{/home/jeremic/tex/works/Conferences/2010/NRC-LBL-ProjectReviewMeeting_21_22_Sept_2010/Map_of_Points.pdf}
%\hfill
%%\hspace*{6cm}
%\includegraphics[width=6truecm]{/home/jeremic/public_html/NSF-Nuggets/Students_develop_parallel_computer/StudentsConstructingGeoWulf.jpg}
%\end{center}
\end{figure}
%
\end{frame}
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\begin{frame}
\frametitle{Free field Motions: Lack of Correlation}
%
\vspace*{-0.5cm}
\begin{figure}[!htbp]
%\hspace*{6cm}
%\begin{center}
\includegraphics[width=10truecm]{/home/jeremic/tex/works/Conferences/2010/NRC-LBL-ProjectReviewMeeting_21_22_Sept_2010/Time_History_Plots.pdf}
%\hfill
%%\hspace*{6cm}
%\includegraphics[width=6truecm]{/home/jeremic/public_html/NSF-Nuggets/Students_develop_parallel_computer/StudentsConstructingGeoWulf.jpg}
%\end{center}
\end{figure}
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\end{frame}
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\begin{frame}
\frametitle{Free field Motions: Lack of Correlation}
%
\vspace*{-0.5cm}
%\vspace*{-5.6cm}
\begin{figure}[!htbp]
%\hspace*{6cm}
%\begin{center}
\includegraphics[width=10truecm]{/home/jeremic/tex/works/Conferences/2010/NRC-LBL-ProjectReviewMeeting_21_22_Sept_2010/Time_History_Plots_Zoomed.pdf}
%\hfill
%%\hspace*{6cm}
%\includegraphics[width=6truecm]{/home/jeremic/public_html/NSF-Nuggets/Students_develop_parallel_computer/StudentsConstructingGeoWulf.jpg}
%\end{center}
\end{figure}
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\subsection{Verification and Validation}
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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
an NPPSSS under conditions for which the
computational model has not been validated
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{Role of Verification and Validation}
\vspace*{0.5truecm}
\begin{figure}[!h]
\begin{center}
{\includegraphics[width=8.0cm]{/home/jeremic/tex/works/Presentation/2003/Verif_and_Valid/RoleVV.pdf}}
\end{center}
\end{figure}
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\end{frame}
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\begin{frame}
\frametitle{Verification and Validation for Prediction}
\begin{itemize}
%\vspace*{0.5cm}
\item How much can (should) we trust model implementations (verification)?
%\vspace*{0.5cm}
\item How much can (should) we trust numerical simulations (validation)?
%\vspace*{0.5cm}
\item How good are our numerical predictions?
%\vspace*{0.5cm}
\item Can a simulation tool (NRC ESSI Simulator) be used for assessing {\bf
public safety}?
% \vspace*{2.0truecm}
\item V\&V procedures are the primary means of assessing accuracy,
building confidence and credibility in
modeling and computational simulations
% \vspace*{0.2truecm}
\item Ever present uncertainties need to be modeled and propagated through
the simulation process
% %\vspace*{1.0truecm}
% \item V \& V procedures are the tools with which we build confidence and
% credibility in modeling and computational simulations
%
%\vspace*{2.5cm}
%\item How do we use experimental simulations to improve models
\end{itemize}
\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}
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%
%
%
% \begin{frame}
% \frametitle{Fundamentals of Verification and Validation}
%
%
%
% %--
%
%
% \begin{figure}[!h]
% %\vspace*{-0.5cm}
% %\hspace*{-0.5cm}
% %\begin{center}
% {\includegraphics[width=11cm]{/home/jeremic/tex/works/Conferences/2005/OpenSeesWorkshopAugust/DeveloperSymposium/VerifValidFund01.pdf}}
% %\end{center}
% \end{figure}
%
%
% \end{frame}
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% \begin{frame}
% \frametitle{Prediction}
%
%
% \begin{itemize}
%
% % \vspace*{0.0truecm}
% \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
%
% % \vspace*{0.5truecm}
% % \item Validation does not directly make a claim about the accuracy of a prediction
% \item Validation does not claim accuracy of a prediction
% \begin{itemize}
% % \item Computational models are easily misused (unintentionally or intentionally)
% \item How closely related are the conditions of the prediction and
% specific cases in validation database
% \item How well is physics of the problem understood
%
% \end{itemize}
%
%
% % \vspace*{0.2truecm}
% \item Ever present uncertainty needs to be estimated for predictions
%
% % \vspace*{0.2truecm}
% \item Identify all relevant sources of uncertainty
%
% % \vspace*{0.2truecm}
% \item Create mathematical representation of individual sources
%
% % \vspace*{0.2truecm}
% \item Propagate representation of sources through modeling and simulation process
%
%
%
%
% \end{itemize}
%
%
%
% \end{frame}
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\section{Uncertainty Aspects}
\subsection{Uncertain Engineering Materials}
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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}
\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{Eulerian--Lagrangian FPK Equation}
%
% %
%
% %\begin{itemize}
%
% \begin{footnotesize}
% %
% % %\noindent
% % van Kampen's Lemma (van Kampen 1976) $\rightarrow$ $ <\rho(\sigma,t)>=P(\sigma,t) $,
% % ensemble average of phase density
% % %(in stress space here)
% % is the probability density;
%
%
% \begin{eqnarray}
% \nonumber
% &&\displaystyle \frac{\partial P(\sigma(x_t,t), t)}{\partial t}=
% - \displaystyle \frac{\partial}{\partial \sigma} \left[ \left\{\left< \vphantom{\int_{0}^{t} d\tau} \eta(\sigma(x_t,t), D^{el}(x_t),
% q(x_t), r(x_t), \epsilon(x_t,t)) \right> \right. \right. \\
% \nonumber
% &+& \left. \left. \int_{0}^{t} d\tau Cov_0 \left[ \displaystyle \frac{\partial \eta(\sigma(x_t,t), D^{el}(x_t), q(x_t), r(x_t),
% \epsilon(x_t,t))}{\partial \sigma}; \right. \right. \right. \\
% \nonumber
% & & \left. \left. \left. \eta(\sigma(x_{t-\tau},t-\tau), D^{el}(x_{t-\tau}), q(x_{t-\tau}), r(x_{t-\tau}),
% \epsilon(x_{t-\tau},t-\tau) \vphantom{\int_{0}^{t} d\tau} \right] \right \} P(\sigma(x_t,t),t) \right] \\
% \nonumber
% &+& \displaystyle \frac{\partial^2}{\partial \sigma^2} \left[ \left\{ \int_{0}^{t} d\tau Cov_0 \left[ \vphantom{\int_{0}^{t}}
% \eta(\sigma(x_t,t), D^{el}(x_t), q(x_t), r(x_t), \epsilon(x_t,t)); \right. \right. \right. \\
% \nonumber
% & & \left. \left. \left. \eta (\sigma(x_{t-\tau},t-\tau), D^{el}(x_{t-\tau}), q(x_{t-\tau}), r(x_{t-\tau}),
% \epsilon(x_{t-\tau},t-\tau)) \vphantom{\int_{0}^{t}} \right] \vphantom{\int_{0}^{t}} \right\} P(\sigma (x_t,t),t) \right] \\
% \nonumber
% \end{eqnarray}
%
% \end{footnotesize}
%
%
% \end{frame}
%
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\begin{frame}
\frametitle{Probabilistic Elasto-Plasticity (PEP) and
Stochastic Elastic-Plastic Finite Element Method (SEPFEM)}
\begin{itemize}
%\vspace*{0.3cm}
\item PEP: Eulerian--Lagrangian form of the Fokker-Planck-Kolmogorov (FPK)
equation
%
% \begin{equation}
% \nonumber
% \frac{\partial P(\sigma,t)}{\partial t} = -\frac{\partial}{\partial \sigma}\left[N_{(1)}P(\sigma,t)-\frac{\partial}{\partial \sigma}
% \left\{N_{(2)} P(\sigma,t)\right\} \right]
% \end{equation}
%
%
\begin{itemize}
\item Input, probability distribution of material properties
%\vspace*{0.3cm}
\item Output: Complete probabilistic description of
response, solution is a Probability Density Function (PDF) of stress
%\vspace*{0.3cm}
\item Solution PDF is second-order {\bf exact} to covariance of time (exact mean and variance)
\end{itemize}
% \item It is deterministic equation in probability density space
%
% \item It is linear PDE in probability density space
% %$\rightarrow$ simplifies the numerical solution process
%
% %\vspace*{0.3cm}
% \item PEP: Probabilistic Elastic-Plastic solution applicable to any incremental
% elasto-plasticity material model
%
\item PEP $+$ Spectral Stochastic Finite Element Method
%
\begin{itemize}
\item Input: PDF for material properties (LHS), probabilistic
seismic loading (RHS)
\item Output: accurate, full PDF of displacements (and
$\dot{u}_i$, $\ddot{u}_i$), stress, strain, etc.
\end{itemize}
%\vspace*{0.2truecm}
\end{itemize}
%
% \vspace*{0.5cm}
% {%
% \begin{beamercolorbox}{section in head/foot}
% \usebeamerfont{framesubtitle}\tiny{B. Jeremi\'{c}, K. Sett, and M. L. Kavvas, "Probabilistic
% Elasto--Plasticity: Formulation in 1--D", \textit{Acta Geotechnica}, Vol. 2, No. 3, 2007, In press (published
% online in the \textit{Online First} section)}
% %\vskip2pt\insertnavigation{\paperwidth}\vskip2pt
% \end{beamercolorbox}%
% }
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% \begin{frame}
%
% \frametitle{{Low OCR Cam Clay with Random $G$, $M$ and $p_0$}}
%
%
%
%
%
% \begin{itemize}
%
% %\item Narrow transition between el. \& el.-pl.
%
%
% \vspace*{0.3cm}
% \item Non-symmetry in \\
% probability \\
% distribution
%
%
% \vspace*{0.3cm}
% \item Difference \\
% between \\
% mean, mode and \\
% deterministic
%
%
%
% \vspace*{0.3cm}
% \item Divergence at \\
% critical state \\
% because $M$ might be \\
% (is) uncertain?
%
% \end{itemize}
%
%
% \vspace*{-6.3cm}
% %\hspace*{0.5cm}
% \begin{figure}[!hbpt]
% \begin{flushright}
% %\includegraphics[height=3.5cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/PDFLowOCR_RandomG_RandomM_Randomp0-m.pdf}
% %\hfill
% \includegraphics[height=6.5cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourLowOCR_RandomG_RandomM_Randomp0-m.pdf}
% \end{flushright}
% \end{figure}
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% \begin{frame}
%
% \frametitle{Comparison of Low OCR Cam Clay at $\epsilon$ = 1.62 \%}
%
%
%
% %\vspace*{-4.50cm}
%
% \begin{figure}[!hbpt]
% \begin{center}
% %\includegraphics[height=14cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/CamClayPDFComparison-m.pdf}
% \includegraphics[height=5.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Present/LowOCR_ComparisonPDF-m_ps.pdf}
% \end{center}
% \end{figure}
%
% \vspace*{-0.5cm}
%
% \begin{itemize}
%
% \item None coincides with deterministic
%
% \item Some very uncertain, some very certain
%
% \item Either on safe or unsafe side
%
% \end{itemize}
%
%
%
%
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%
%
%
%
% \end{frame}
%
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%
% \begin{frame}
% \frametitle{Spectral Stochastic Elastic--Plastic FEM}
%
% \begin{itemize}
%
% \item Minimizing norm of error of finite representation using Galerkin
% technique (Ghanem and Spanos 2003):
%
% \vspace*{-0.6truecm}
% \begin{flushright}
% \begin{equation}
% \nonumber
% \sum_{n = 1}^N K_{mn}^{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}{D}^{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}
%
% %
% \end{frame}
%
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\subsection{Uncertain Seismic Motions}
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\begin{frame}
\frametitle{Decision About Site (Material) Characterization}
\begin{itemize}
\vspace*{0.3cm}
\item Do an inadequate site characterization (rely on experience): conservative
{\bf guess} for soil data, $COV = 225$\%, large correlation length (length of
a model).
\vspace*{0.3cm}
\item Do a good site characterization: $COV = 103$\%, correlation
length calculated ($= 0.61$m)
\vspace*{0.3cm}
\item Do an excellent (much improved) site characterization if probabilities of exceedance
are unacceptable!
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{Random Field Parameters from Site Data}
\begin{itemize}
%\item maximizing the log--likelihood of observing the spatial data under assumed joined distribution (for finite
%scale model) or maximizing the log--likelihood of observing the periodogram estimates (for fractal model)
\item Maximum likelihood estimates
\vspace*{0.3truecm}
%\begin{figure}
\begin{flushleft}
\hspace*{-1.7cm}
\includegraphics[height=4.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/CPT_DataAnalysis_Plots/SamplingPlan-Edited.jpg}
\hspace*{0.0cm}
\includegraphics[height=4.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/CPT_DataAnalysis_Plots/TypicalDataPlotBH1-Edited.jpg} \\
\small{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Typical CPT $q_T$}
\end{flushleft}
%\end{figure}
\vspace*{-4.9truecm}
%\begin{figure}
\begin{flushright}
\includegraphics[width=4.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/CPT_DataAnalysis_Plots/TypicalAutoCovariancePlotBH1_FiniteScale-Edited.jpg} \\
\vspace*{0.01truecm}
\small{Finite Scale}
\end{flushright}
%\end{figure}
\vspace*{0.02truecm}
%\begin{figure}
\begin{flushright}
\includegraphics[width=4.0cm]{/home/jeremic/tex/works/Thesis/KallolSett/Dissertation/figures/CPT_DataAnalysis_Plots/TypicalAutoCovariancePlotBH1_Fractal-Edited.jpg} \\
\small{Fractal}
\end{flushright}
%\end{figure}
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{"Uniform" CPT Site Data (Courtesy of USGS)}
\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}
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\begin{frame}
\frametitle{Statistics of Stochastic Soil Profile(s)}
%\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 = 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}
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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}
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\begin{frame}
\frametitle{Evolution of Mean $\pm$ SD for Real Data Case}
\begin{figure}
\begin{center}
\hspace*{-0.75cm}
\includegraphics[width=10.0cm]{/home/jeremic/tex/works/Conferences/2009/UNION-Univ-BGD/Present/Plots_with_Labels/Evolutionary_Mean_pm_SD_Actual-Edited.pdf}
\hspace*{-0.75cm}
%\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
\end{center}
\end{figure}
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\begin{frame}
\frametitle{Full PDFs for Real Data Case}
\begin{figure}
\begin{center}
\vspace*{-0.75cm}
\includegraphics[width=7.0cm]{/home/jeremic/tex/works/Conferences/2009/UNION-Univ-BGD/Present/Plots_with_Labels/EvolutionaryPDF_Actual-Edited.pdf}
\vspace*{-0.75cm}
%\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
\end{center}
\end{figure}
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\begin{frame}
\frametitle{Example: PDF at $6$ s}
\begin{figure}
\begin{center}
\hspace*{1.75cm}
\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2009/UNION-Univ-BGD/Present/Plots_with_Labels/PDFs_at6sec_Actual_vs_NoData-Edited.pdf}
\vspace*{-0.75cm}
%\includegraphics[width=9.0cm]{/home/jeremic/tex/works/Conferences/2007/USC_seminar/Application_figs/Mean_and_SDElasticPlastic_ps.pdf}
\end{center}
\end{figure}
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\begin{frame}
\frametitle{Example: CDF 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}
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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}
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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}
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\section{Summary}
\subsection{}
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\begin{frame}
\frametitle{Summary}
\begin{itemize}
\item There is a need for high fidelity modeling and simulations
(verified and validated, deterministic and probabilistic) for NPPSSSs
%\vspace*{0.2cm}
\item Such high fidelity modeling and simulations will improve safety and
economy
%\vspace*{0.2cm}
\item Education for Developers, Modelers/Analysts, Researchers, Consultants,
Regulators is very important
%\vspace*{0.2cm}
\item Presented research was/is funded in part and performed in collaboration
with the Caltrans, NSF, U.S. NRC and CNSC
%\vspace*{0.2cm}
\item CompDyn2011 Corfu, Greece, 26-28 May, Soil-Structure Interaction
Mini-Symposium
\end{itemize}
\end{frame}
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\end{document}