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% Site Specific Dynamics of Structures:
%From Seismic Source to
%the Safety of Occupants and Content
\title[Uncertainties in ESSI Analysis]
{Epistemic and Aleatory Uncertainties in \\
Numerical Analysis of \\
Earthquake Soil Structure Interaction
}
%\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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%\author[Jeremi{\'c} et al.] % (optional, use only with lots of authors)
\author[Jeremi{\'c} et al.] % (optional, use only with lots of authors)
%{Boris~Jeremi{\'c}}
{Boris Jeremi{\'c} \\
Han Yang, Hexiang Wang, Sumeet Kumar Sinha }
%\institute[Computational Geomechanics Group \hspace*{0.3truecm}
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%{ Professor, University of California, Davis\\
{ University of California, Davis, CA}
% % and\\
% % Faculty Scientist, Lawrence Berkeley National Laboratory, Berkeley }
% Lawrence Berkeley National Laboratory, Berkeley, CA}
% %  Use the \inst command only if there are several affiliations.
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{\small NHERI Lehigh Seminar\\
11May2023}
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% may not be suitable. Here are some rules that apply for this
% solution:
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%  Talk about 30s to 2min per frame. So there should be between about
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% are going to talk about. So *simplify*!
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% you think necessary.
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% just say so once. Everybody will be happy with that.
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\section{Introduction}
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%\subsection{Motivation}
\subsection{\ }
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\begin{frame}
\frametitle{Motivation}
\begin{itemize}
\item[] Safety and economy of infrastructure objects
%\vspace*{0.3cm}
\vspace*{2mm}
\item[] Improve analysis of infrastructure objects
\vspace*{2mm}
\item[] Earthquakes, Soils, Structures and their Interaction
% \vspace*{2mm}
% \item[] Expert numerical modeling and simulation tool
%
% \vspace*{1mm}
% \item[] Use of numerical models to
% analyze statics and dynamics of soil/rockstructure systems
%
\vspace*{2mm}
\item[] Modeling, Epistemic uncertainty
\vspace*{2mm}
\item[] Parametric, Aleatory uncertainty
\vspace*{2mm}
\item[] Goal is to predict and inform
% rather than (force) fit
\vspace*{2mm}
\item[] Engineer needs to know!
%
%
%
% \vspace*{1mm}
% \item[] Follow the flow, input and dissipation, of seismic energy,
% \vspace*{2mm}
% \item[]
% %System for
% {\bf Real}istic modeling and simulation of
% {\bf E}arthquakes and/or
% {\bf S}oils and/or
% {\bf S}tructures and their
% {\bf I}nteraction:\\
% RealESSI
% \hspace*{5mm}
% \url{http://realessi.info/}
% % % % \hspace*{25mm}
% % \url{http://sokocalo.engr.ucdavis.edu/~jeremic/Real_ESSI_Simulator/}
% % % \href{http://sokocalo.engr.ucdavis.edu/~jeremic/Real_ESSI_Simulator/}{{http://sokocalo.engr.ucdavis.edu/~jeremic/Real_ESSI_Simulator/}
% % % % \url{http://msessi.info/}
% % %
%
\end{itemize}
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\begin{frame}
\frametitle{ESSI Challenges}
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%
\begin{figure}[!htb]
\begin{center}
% %\includegraphics[width=5cm]{Figurefiles/_Chapter_Applications_ESSI_BOOK/Building_modeling_Issues_01_pdf.pdf}
%\vspace*{3mm}
\includegraphics[width=2.5cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figurefiles/_Chapter_Applications_ESSI_BOOK/Seismic_Motions_modeling_Issues_01_pdf.pdf}
\includegraphics[width=5.5cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figurefiles/_Chapter_Applications_ESSI_BOOK/Seismic_Motions_modeling_Issues_02_pdf.pdf}
\\
%\vspace*{2mm}
%\vspace*{8mm}
\includegraphics[width=3cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figurefiles/_Chapter_Applications_ESSI_BOOK/NPP_Modeling_Issues_03.jpg}
\includegraphics[width=2.2cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figurefiles/_Chapter_Applications_ESSI_BOOK/SMR_Modeling_Issues_02_pdf.pdf}
%\\
%\vspace*{2mm}
\includegraphics[width=2.8cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figurefiles/_Chapter_Applications_ESSI_BOOK/Building_modeling_Issues_03_pdf.pdf}
\includegraphics[width=2.5cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figurefiles/_Chapter_Applications_ESSI_BOOK/Building_modeling_Issues_02_pdf.pdf}
\\
%\vspace*{2mm}
\includegraphics[width=1.0cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figurefiles/_Chapter_Applications_ESSI_BOOK/Dam_modeling_Issues_01_pdf.pdf}
\includegraphics[width=3.5cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figurefiles/_Chapter_Applications_ESSI_BOOK/Dam_modeling_Issues_02_pdf.pdf}
\includegraphics[width=2.0cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figurefiles/_Chapter_Applications_ESSI_BOOK/Bridge_modeling_Issues_01_pdf.pdf}
% %\vspace*{8mm}
% \caption{\label{ESSI_Models_and_Challenges}
% ESSI modeling and simulation challenges:
% Free field motions, 3C/6C vs 3$\times$1C;
% Nuclear Power Plant structure  soil/rock system, Small Modular Reactor structure  soil/rock system;
% Low and High Buildingfoundationsoil system;
% DamFoundationFluid system;
% Bridgesoil system;
% %
% Aspects of modeling:
% 1) Seismic motions,
% 2) Inelastic soil and rock,
% 3) Inelastic interface/contact/joints, foundation with soil/rock and
% interfaces/contacts/joints within structure,
% 4) Inelastic structure, systems and components,
% 5) Solid, Structure  Fluid interaction, external (reservoirs, fluid pools...) and internal
% (fully saturated and partially, (un)saturated soil, rock and concrete).}
\end{center}
\end{figure}
%
%
% \begin{itemize}
%
% \item[] Linear elastic, all elements
%
% %\vspace*{1mm}
% \item[] Nonlinear elastic, solids
%
% %\vspace*{1mm}
% \item[] Soil, solids/3D, dry, saturated and partially saturated)
%
% %\vspace*{1mm}
% \item[] Rock, solids/3D, dry, saturated and partially saturated)
%
% %\vspace*{1mm}
% \item[] Interface/Contact, soft, hard, gap, dry, saturated
%
% %\vspace*{1mm}
% \item[] Base isolator and dissipator 2node/3D
%
%
% %\vspace*{1mm}
% \item[] Concrete, solids/3D, wall/2D, beam/1D, ASR
%
% %\vspace*{1mm}
% \item[] Steel, beam/1D and solid/3D
% %
%
%
% %\vspace*{1mm}
% \item[] Seismic input, DRM
%
% %\vspace*{1mm}
% \item[] Energy Dissipation
%
%
% \end{itemize}
\end{frame}
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\frametitle{Numerical Prediction under Uncertainty}
\begin{itemize}
%\vspace*{1mm}
\item[] {Modeling, Epistemic Uncertainty}
\begin{itemize}
\vspace*{1mm}
\item[] Model simplifications
\vspace*{1mm}
\item[] Model sophistication level for confidence in results
%\vspace*{1mm}
% \item[] Verification and Validation
%
%
%\vspace*{2mm}
% \item[] Choice of sophistication level for confidence in results
\end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace*{4mm}
\item[] {Parametric, Aleatory Uncertainty}
\begin{itemize}
\vspace*{1mm}
\item[] ${M} \ddot{u_i} + {C} \dot{u_i} + {K}^{ep} {u_i} = {F(t)}$,
\vspace*{1mm}
\item[] Uncertain: mass $M$, viscous damping $C$ and stiffness $K^{ep}$
\vspace*{1mm}
\item[] Uncertain loads $F(t)$
\vspace*{1mm}
\item[] Results are PDFs and CDFs for $\sigma_{ij}$, $\epsilon_{ij}$, $u_i$, $\dot{u}_i$, $\ddot{u}_i$
\end{itemize}
\end{itemize}
%
%
% %Le doute n'est pas un {\'e}tat bien agr{\'e}able,\\
% mais l'assurance est un {\'e}tat ridicule. (Fran{\c c}oisMarie Arouet, Voltaire)
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\frametitle{Modeling, Epistemic Uncertainty}
\begin{itemize}
\item[] Important (?!) features are simplified
\begin{itemize}
\vspace*{1mm}
\item[] 3C/6C vs 1C seismic motions
\vspace*{1mm}
\item[] Elastic vs inelastic behavior
\end{itemize}
%\vspace*{4mm}
% \item[] Unrealistic and unnecessary modeling simplifications
\vspace*{4mm}
\item[] Modeling simplifications are justifiable if one or two
level higher sophistication model demonstrates that behavior being
simplified out is not important
\end{itemize}
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% % {/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Energy_Dissipation_Animations/SMR_Energy_Dissipation_screen_grab.jpg}}
% % {/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Energy_Dissipation_Animations/SMR_Energy_Dissipation.mp4}
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\frametitle{Parametric, Aleatory Uncertainty}
\vspace*{2mm}
%\vspace*{5mm}
\begin{figure}[!hbpt]
\begin{center}
%
\hspace*{7mm}
\includegraphics[width=5.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGEGoverGmax/figures/YoungModulus_RawData_and_MeanTrend_01Ed.pdf}
\hspace*{3mm}
% \hfill
\includegraphics[width=4.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGEGoverGmax/figures/YoungModulus_Histogram_Normal_01Ed.pdf}
%
\end{center}
\end{figure}
\vspace*{3mm}
%\vspace*{1.8cm}
%\hspace*{3.3cm}
\begin{flushright}
{\tiny
(cf. Phoon and Kulhawy (1999B))\\
~}
\end{flushright}
%
\vspace*{7mm}
\begin{figure}[!hbpt]
\begin{center}
%
%\hspace*{7mm}
\includegraphics[width=5.00truecm]{/home/jeremic/tex/works/Thesis/HexiangWang/time_series_motionsn_06ug2019_SMIRT/Acc_realization_200.pdf}
%\hspace*{3mm}
%\includegraphics[width=2cm]{/home/jeremic/tex/works/Papers/2019/Hexiang/1D_risk/version2/Figures/Acc_time_series_realiztion70.pdf}
%\includegraphics[width=2cm]{/home/jeremic/tex/works/Papers/2019/Hexiang/1D_risk/version2/Figures/Acc_time_series_realiztion100.pdf}
%% \includegraphics[width=0.31\textwidth]{Figures/Acc_time_series_realiztion350.pdf}
%\includegraphics[width=2cm]{/home/jeremic/tex/works/Papers/2019/Hexiang/1D_risk/version2/Figures/Acc_time_series_realiztion367.pdf}
\includegraphics[width=4cm]{/home/jeremic/tex/works/Papers/2019/1D_risk/version2/Figures/SA_GMPE_verification_std_08_no_smooth.pdf}
%
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\begin{flushright}
{\tiny
(cf. Wang et al. (2019))\\
~}
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%\subsection{Stochastic Modeling}
\subsection{Modeling, Epistemic Uncertainties}
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\begin{frame}
\frametitle{Modeling, Epistemic Uncertainties}
\begin{itemize}
\vspace*{1mm}
\item[] Modeling simplifications
%\vspace*{1mm}
% \item[] Modeling sophistication for confidence in results
%
% \vspace*{3mm}
% \item[] Important (?!) features are simplified
%
\begin{itemize}
\vspace*{3mm}
\item[] SSI vs nonSSI response
\vspace*{3mm}
\item[] Model geometry: 1D, 2D, 3D
% , solids vs structural elements
\vspace*{3mm}
\item[] 1C vs 3C/6C seismic motions
\vspace*{3mm}
\item[] Elastic vs Inelastic behavior
\vspace*{3mm}
\item[] Energy dissipation, inelastic vs viscous
\end{itemize}
%\vspace*{4mm}
% \item[] Unrealistic and unnecessary modeling simplifications
% \vspace*{3mm}
% \item[] Modeling simplifications are justifiable if one or two
% level higher sophistication model demonstrates that features being
% simplified out are less or not important
\end{itemize}
%
%
% %Le doute n'est pas un {\'e}tat bien agr{\'e}able,\\
% mais l'assurance est un {\'e}tat ridicule. (Fran{\c c}oisMarie Arouet, Voltaire)
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\begin{frame}
\frametitle{Real Earthquake Ground Motions}
\vspace*{1mm}
\begin{itemize}
\vspace*{1mm}
\item[] Inclined body waves: P and S waves
\vspace*{1mm}
\item[] Surface waves: Rayleigh, Love waves
\vspace*{1mm}
\item[] Near surface waves: Stoneley waves...
%\vspace*{1mm}
% \item[] Lack of correlation, incoherent motions
\vspace*{1mm}
\item[] All, most measured motions are full 3C/6C (3t, 3r)
\vspace*{1mm}
\item[] Example EQ: 2C LSST07(L); 3C/6C LSST12(R)
\vspace*{1mm}
%\vspace*{5mm}
\end{itemize}
\begin{figure}[!hbpt]
\begin{center}
%
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\includegraphics[width=5truecm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figurefiles/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/Lotung_LSST07_FA25.jpeg}
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\includegraphics[width=5truecm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figurefiles/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/Lotung_LSST12_FA25.jpeg}
\hspace*{10mm}
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\end{figure}
%\vspace*{3mm}
%\item[] Effects of real earthquakes on soilstructure systems ?!
\end{frame}
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% \frametitle{T{o}hoku Earthquake, 3D Motions}
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% {\includegraphics[width=3.0cm]{/home/jeremic/tex/works/Conferences/2011/NRC_Staff_Capacity_Building_25May2011/Tohoku_1Hz11locations.jpg}}
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% \frametitle{1C, 2C, 3$\times$1C, 3C/6C Seismic Motions}
%
% \vspace*{2mm}
%
% \begin{itemize}
%
% \item[] All, most measured motions are full 3C/6C (3t, 3r)
%
% \hspace*{5cm}
%
% \item[] What is the effect of neglecting, simplifying 3C/6C to 1C
%
% % \hspace*{5cm}
% % \item[] One example of an almost 2C motion (LSST07, LSST12)
%
% \vspace*{2mm}
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% %
% % \item[] 1D (?): M 6.9 San Pablo, Guatemala EQ, 14Jun2017
% %
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% \begin{frame}
% \frametitle{LSST Example, 2C, 3C/6C}
%
% %\vspace*{2mm}
%
% \begin{itemize}
%
% % \item[] All, most measured ground motions are full 3C (6C)
% %
% \item[] All, most measured motions are full 3C/6C (3t, 3r)
% \vspace*{1mm}
% \item[] Almost 2C motions, found one: LSST07(L)
% \vspace*{1mm}
% \item[] Same location, all other EQs, 3C motions: LSST12(R)
% \vspace*{3mm}
% %\vspace*{5mm}
%
%
% \begin{figure}[!hbpt]
% \begin{center}
% %
% \hspace*{10mm}
% \includegraphics[width=5truecm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figurefiles/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/Lotung_LSST07_FA25.jpeg}
% \hspace*{5mm}
% \includegraphics[width=5truecm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figurefiles/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/Lotung_LSST12_FA25.jpeg}
% \hspace*{10mm}
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% \frametitle{San Pablo Earthquake, 1C (?), 14Jun2017}
%
% Courtesy of \url{http://www.strongmotioncenter.org/}
%
% \vspace*{2mm}
%
% %\begin{itemize}
% %
% % \item[] San Pablo, Guatemala, 14June2017 (data from \url{http://www.strongmotioncenter.org/}
% %
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% % \end{itemize}
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% \includegraphics[width=5.5truecm]{/home/jeremic/oofep/LJUDI/PraveenMalhotra/EQ__01_Guatemala_14Jun2017.jpg}
% \\
% \includegraphics[width=1.7truecm]{/home/jeremic/oofep/LJUDI/PraveenMalhotra/EQ__02_Guatemala_14Jun2017.jpg}
% \includegraphics[width=1.6truecm]{/home/jeremic/oofep/LJUDI/PraveenMalhotra/EQ__03_Guatemala_14Jun2017.jpg}
% \includegraphics[width=1.7truecm]{/home/jeremic/oofep/LJUDI/PraveenMalhotra/EQ__04_Guatemala_14Jun2017.jpg}
% \includegraphics[width=1.7truecm]{/home/jeremic/oofep/LJUDI/PraveenMalhotra/EQ__05_Guatemala_14Jun2017.jpg}
% \includegraphics[width=1.7truecm]{/home/jeremic/oofep/LJUDI/PraveenMalhotra/EQ__06_Guatemala_14Jun2017.jpg}
% \includegraphics[width=1.7truecm]{/home/jeremic/oofep/LJUDI/PraveenMalhotra/EQ__07_Guatemala_14Jun2017.jpg}
% \hspace*{10mm}
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% \frametitle{Regional Geophysical Models}
%
% \begin{itemize}
%
% \item[] High fidelity free field seismic motions on regional scale
%
% \vspace*{3mm}
% \item[] Collaboration with LLNL: Dr.~Rodgers, Dr.~Pitarka and
% Dr.~Petersson, finite difference code SW4
%
%
% \vspace*{3mm}
% \item[] Knowledge of fault and geology needed
%
% \vspace*{3mm}
% \item[] Significant uncertainties (!!)
% \begin{itemize}
%
% \item[] fault rupture
% \item[] deep and shallow geology
%
% \end{itemize}
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% \begin{frame}
% \frametitle{Regional Geophysical Models}
%
% \begin{figure}[!htb]
% \begin{center}
% \includegraphics[width=5truecm]{/home/jeremic/tex/works/Conferences/2017/CompDyn2017/Present_PLENARY/San_Francisco__Regional_Model_BIG.jpg}
% \includegraphics[width=5.2truecm]{/home/jeremic/tex/works/Conferences/2017/CompDyn2017/Present_PLENARY/San_Francisco__Regional_Model.jpg}
% \end{center}
% % \caption{\label{Fig:NPP_Model_In_Real_ESSI} Nuclear Power Plant Model with Shallow Foundation }
% \end{figure}
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% Rodgers and Pitarka
%
% \end{frame}
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% \begin{frame}
% \frametitle{Regional Geophysical Models}
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% \begin{figure}[!htb]
% \begin{center}
% \includegraphics[width=10truecm]{/home/jeremic/tex/works/Conferences/2017/CompDyn2017/Present_PLENARY/USGBay_Area_Model_CC_det2_sm.jpg}
% \end{center}
% % \caption{\label{Fig:NPP_Model_In_Real_ESSI} Nuclear Power Plant Model with Shallow Foundation }
% \end{figure}
%
% USGS
%
% \end{frame}
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% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \begin{frame}
% % \frametitle{Regional Geophysical Models}
% %
% % \begin{figure}[!htb]
% % \begin{center}
% % \includegraphics[width=8truecm]{/home/jeremic/tex/works/Conferences/2017/CompDyn2017/Present_PLENARY/San_Francisco_Model_Geology_Base_and_Stoch.jpg}
% % \end{center}
% % % \caption{\label{Fig:NPP_Model_In_Real_ESSI} Nuclear Power Plant Model with Shallow Foundation }
% % \end{figure}
% %
% % Rodgers and Pitarka
% %
% % \end{frame}
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% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% %  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %  \begin{frame}
% %  \frametitle{Regional Geophysical Simulations using SW4}
% % 
% %  \begin{figure}[!htb]
% %  \begin{center}
% %  \includegraphics[width=8truecm]{/home/jeremic/tex/works/Conferences/2017/CompDyn2017/Present_PLENARY/CPU_scaling_CORI.png}
% %  \end{center}
% %  % \caption{\label{Fig:NPP_Model_In_Real_ESSI} Nuclear Power Plant Model with Shallow Foundation }
% %  \end{figure}
% %  \vspace*{2mm}
% % 
% %  Rodgers and Petersson
% % 
% % 
% %  \end{frame}
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% %  \begin{frame}
% %  \frametitle{HPC on CORI (LBNL/NERSC)}
% % 
% %  \begin{figure}[!htb]
% %  \begin{center}
% %  \includegraphics[width=8truecm]{/home/jeremic/tex/works/Conferences/2017/CompDyn2017/Present_PLENARY/CORI_picture.jpg}
% %  \end{center}
% %  % \caption{\label{Fig:NPP_Model_In_Real_ESSI} Nuclear Power Plant Model with Shallow Foundation }
% %  \end{figure}
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% \frametitle{Example Regional Model}
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% \begin{figure}[!htb]
% \begin{center}
% \includegraphics[width=5.5truecm]{/home/jeremic/tex/works/Conferences/2017/CompDyn2017/Present_PLENARY/Vs_at_top.jpg}
% % \includegraphics[width=5truecm]{/home/jeremic/tex/works/Conferences/2017/CompDyn2017/Present_PLENARY/Horizontal_Velocity_at_12s.jpg}
% \includegraphics[width=5truecm]{/home/jeremic/tex/works/Conferences/2017/CompDyn2017/Present_PLENARY/Peak_Velocity.jpg}
% \end{center}
% % \caption{\label{Fig:NPP_Model_In_Real_ESSI} Nuclear Power Plant Model with Shallow Foundation }
% \end{figure}
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% \frametitle{Example Regional Model (Rodgers)}
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% %\vspace*{5mm}
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% %
% %\movie[label=show3,width=10.0cm,poster,autostart,showcontrols]
% \movie[label=show3,width=8.0cm,poster]
% %\movie[label=show3,width=6.0cm,poster,showcontrols]
% {\includegraphics[width=80mm]{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/NPP_animations_August2017/M6_5_s500_BASIN_STOCHASTIC_mag_SLOW.jpg}}
% {/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/NPP_animations_August2017/M6_5_s500_BASIN_STOCHASTIC_mag_SLOW.mpg}
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% %
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% % online
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% \href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/NPP_animations_August2017/M6_5_s500_BASIN_STOCHASTIC_mag_SLOW.mpg}
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% \includegraphics[width=11truecm]{/home/jeremic/tex/works/Thesis/HexiangWang/Files_08June2017/pic/SW42DRM.pdf}
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\frametitle{ESSI: 3C/6C vs 1C Seismic Motions}
\begin{itemize}
\item[] Assume: full 3C/6C motions, recorded only 1C
\item[] From 1C motions, deconvolute/convolute 1C
\item[] Apply 6C and 1C to ESSI system
\end{itemize}
%\vspace*{3mm}
\begin{figure}[!H]
\begin{center}
\includegraphics[width=6.5cm]{/home/jeremic/tex/works/Conferences/2015/CompDyn/Present/6D_to_1D_01.jpg}
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\frametitle{Realistic Ground Motions}
\vspace*{2mm}
\begin{itemize}
\item[] Free field, regional scale models
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\frametitle{Development of Realistic Motions}
\begin{itemize}
\item[] Sources will send both P and S waves
\end{itemize}
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\frametitle{1C vs 6C Free Field Motions}
\begin{itemize}
\item[] One component of motions, 1C from 6C
% or 3$\times$1D (it is done all the time!)
\item[] Excellent fit, wrong mechanics
% (goal is to predict and inform and not (force) fit)
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%\movie[label=show3,width=8.8cm,poster,autostart,showcontrols]
\movie[label=show3,width=8.8cm,poster, showcontrols]
{\includegraphics[width=92mm]
{/home/jeremic/tex/works/Conferences/2016/IAEA_TecDoc_February2016/My_Current_Work/movie_2_npps_mp4_icon.jpeg}}
{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Model01_ESSI_Response_May2015/movie_2_npps.mp4}
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\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/ESSI_VisIt_movies_Jose_19May2015/movie_2_npps.mp4}
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% \frametitle{Real Wave Field from Surface Measurements}
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% \begin{itemize}
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% \item[] Use surface and shallow measurements to develop full 6C wave field
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\begin{frame}
\frametitle{Realistic Seismic Wave Fields}
\begin{itemize}
\vspace*{2mm}
\item[] Stress test motions:
\begin{itemize}
\vspace*{1mm}
\item[] Variable wave length, frequency
\vspace*{1mm}
\item[] Variable wave inclination
\end{itemize}
\vspace*{4mm}
\item[] Use surface and shallow motion measurements to develop full 6C wave field:
3Ddeconvolution
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% \frametitle{Free Field, Variation in Input Wave Angle, $f = 5$Hz}
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% % Elastoplastic soil with contact elements
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% {\includegraphics[width=10cm]
% {/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Free_Field_animations_angle_or_frequency_variation/Free_Field_variation_in_wave_angle.jpg}}
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% % Elastoplastic soil with contact elements
% %% Both solid and contact elements dissipate energy
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% {\includegraphics[width=10cm]
% {/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/SMR_animations_May2018/SMR_ESSI_4_inclinations.jpg}}
% {/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/SMR_animations_May2018/SMR_ESSI_4_inclindations.mp4}
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\begin{frame}
%\frametitle{Free Field, Variation in Input Frequency, $\theta = 60^{o}$}
\frametitle{Free Field, Variable Wave Length, $\theta = 60^{o}$}
% Elastoplastic soil with contact elements
%% Both solid and contact elements dissipate energy
% \vspace*{5mm}
\begin{center}
% \hspace*{15mm}
%\movie[label=show3,width=10cm,poster,autostart,showcontrols]
\movie[label=show3,width=10cm,poster,showcontrols]
{\includegraphics[width=10cm]
{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Free_Field_animations_angle_or_frequency_variation/Free_Field_variation_in_wave_frequency.jpg}}
{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Free_Field_animations_angle_or_frequency_variation/free_field_frequency.mp4}
\end{center}
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\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Free_Field_animations_angle_or_frequency_variation/free_field_frequency.mp4}
{\tiny (MP4)}
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\begin{frame}
\frametitle{SMR ESSI, Variable Wave Length, $\theta = 60^{o}$}
% Elastoplastic soil with contact elements
%% Both solid and contact elements dissipate energy
% \vspace*{5mm}
\begin{center}
% \hspace*{15mm}
%\movie[label=show3,width=10cm,poster,autostart,showcontrols]
\movie[label=show3,width=10cm,poster,showcontrols]
{\includegraphics[width=10cm]
{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Free_Field_animations_angle_or_frequency_variation/ESSI_SMR_variation_in_wave_frequency.jpg}}
{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Free_Field_animations_angle_or_frequency_variation/SMR_frequency.mp4}
\end{center}
% online
\vspace*{12mm}
\begin{flushleft}
\hspace*{4mm}
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Free_Field_animations_angle_or_frequency_variation/SMR_frequency.mp4}
{\tiny (MP4)}
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\begin{frame}
%\frametitle{SMR ESSI, Variation in Input Frequency, $\theta = 60^{o}$}
\frametitle{SMR ESSI, 3C vs 3$\times$1C}
% Elastoplastic soil with contact elements
%% Both solid and contact elements dissipate energy
% \vspace*{5mm}
\begin{center}
% \hspace*{15mm}
%\movie[label=show3,width=10cm,poster,autostart,showcontrols]
\movie[label=show3,width=10cm,poster,showcontrols]
{\includegraphics[width=10cm]
{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/SMR_animations_May2018/3Dvs1D_deconvolution.jpg}}
{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/SMR_animations_May2018/3Dvs1D_deconvolution.ogv}
\end{center}
% online
\vspace*{12mm}
\begin{flushleft}
\hspace*{4mm}
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/SMR_animations_May2018/3Dvs1D_deconvolution.ogv}
{\tiny (OGV)}
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% \begin{frame}
%
% %\frametitle{SMR ESSI, Variation in Input Frequency, $\theta = 60^{o}$}
% \frametitle{SMR ESSI, Variation in Input Frequency, REAL TIME}
%
% % Elastoplastic soil with contact elements
% %% Both solid and contact elements dissipate energy
%
%
% % \vspace*{5mm}
% \begin{center}
% % \hspace*{15mm}
% %\movie[label=show3,width=10cm,poster,autostart,showcontrols]
% \movie[label=show3,width=10cm,poster,showcontrols]
% {\includegraphics[width=10cm]
% {/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Free_Field_animations_angle_or_frequency_variation/ESSI_SMR_variation_in_wave_frequency.jpg}}
% {/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/SMR_animations_May2018/SMRESSI_real_time_four_freq.mp4}
% \end{center}
%
% % online
% \vspace*{12mm}
% \begin{flushleft}
% \hspace*{4mm}
% \href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/SMR_animations_May2018/SMRESSI_real_time_four_freq.mp4}
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% \frametitle{Pine Flat Dam, Inclined Plane Waves}
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% \begin{center}
% % \hspace*{15mm}
% %\movie[label=show3,width=11cm,poster,autostart,showcontrols]
% \movie[label=show3,width=11cm,autostart,showcontrols]
% {\includegraphics[width=11.0cm]
% {/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_Concrete_Dams/Pine_Flat_Dam/dynamic_response_inclination.jpg}}
% {/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_Concrete_Dams/Pine_Flat_Dam/dynamic_response_inclination.mp4}
% \end{center}
%
% \begin{flushleft}
% \vspace*{15mm}
% \href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Concrete_Dams/Pine_Flat_Dam/dynamic_response_inclination.mp4}
% % \href{./homo_50mmesh_45degree_Ormsby.mp4}
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\frametitle{Ventura Hotel, Northridge Earthquake, nonSSI vs SSI}
% local
\vspace*{5mm}
\begin{center}
\hspace*{16mm}
%\movie[label=show3,width=5.6cm,poster,autostart,showcontrols]
\movie[label=show3,width=120mm,showcontrols]
{\includegraphics[width=120mm]{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_Buildings/Ventura_Hotel_SSI_vs_nonSSI_screen_grab.jpg}}
{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_Buildings/Ventura_Hotel_SSI_vs_nonSSI.mp4}
\hspace*{16mm}
\end{center}
% local
% online
\vspace*{12mm}
\hspace*{12mm}
\begin{flushleft}
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_ESSI_for_Buildings/Ventura_Hotel_SSI_vs_nonSSI.mp4}
{\tiny ({\bf MP4})}
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\begin{frame}
\frametitle{Energy Input and Dissipation}
\begin{itemize}
\vspace*{1mm}
\item[] Energy input, dynamic forcing
\vspace*{4mm}
\item[] Energy dissipation outside SSI domain:
\begin{itemize}
\item[] SSI system oscillation radiation
\item[] Reflected wave radiation
\end{itemize}
\vspace*{2mm}
\item[] Energy dissipation/conversion inside SSI domain:
\begin{itemize}
\vspace*{1mm}
\item[] Inelasticity of soil, interfaces, structure, dissipators
\vspace*{1mm}
\item[] Viscous coupling with internal/pore and external fluids
% % \item[] potential and kinetic energy
% \item[] potential $\leftarrow \! \! \! \! \! \! \rightarrow$ kinetic energy
\vspace*{1mm}
\item[] Energy deflectors, metamaterials
\end{itemize}
\vspace*{2mm}
%\vspace*{1mm}
% \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{Energy Dissipation Control}
\vspace*{3mm}
\begin{figure}[!H]
%\hspace*{10mm}
% \includegraphics[width=3cm]{/home/jeremic/tex/works/Thesis/HanYang/Files_Energy_dissipation_01Dec2017/case_a.pdf}
% \includegraphics[width=3cm]{/home/jeremic/tex/works/Thesis/HanYang/Files_Energy_dissipation_01Dec2017/case_b.pdf}
\includegraphics[width=8cm]{/home/jeremic/tex/works/Thesis/HanYang/Files_Energy_dissipation_01Dec2017/case_g.pdf}
% \includegraphics[width=3cm]{/home/jeremic/tex/works/Thesis/HanYang/Files_Energy_dissipation_01Dec2017/case_e.pdf}
\end{figure}
\end{frame}
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\begin{frame}
\frametitle{Plastic Energy Dissipation $\ne$ Plastic Work }
\vspace*{3mm}
% Single elasticplastic element under cyclic shear loading
%\begin{itemize}
%\item[] Plastic work $\ne$ plastic dissipation
%\item[]
Area of loaddisplacement loop is NOT plastic dissipation
%\end{itemize}
\vspace*{4mm}
\begin{figure}[!hbpt]
\begin{center}
\hspace*{5mm}
\includegraphics[width=11.0truecm]{/home/jeremic/tex/works/Thesis/HanYang/Files_06June2017/DOE_Annual_2017/Figures/Dissipation_Material.png}
\end{center}
\end{figure}
\end{frame}
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\begin{frame}
\frametitle{Elastic vs Inelastic NPP Response}
% Elastoplastic soil with contact elements
%% Both solid and contact elements dissipate energy
% \vspace*{5mm}
\begin{center}
% \hspace*{15mm}
\movie[label=show3,width=10cm,poster,autostart,showcontrols]
{\includegraphics[width=10cm]
{/home/jeremic/tex/works/Thesis/SumeetKumarSinha/Files_13Aug2017/NPP_Non_Linear_Effects_Sumeet.jpg}}
{/home/jeremic/tex/works/Thesis/SumeetKumarSinha/Files_13Aug2017/NPP_Non_Linear_Effects_Sumeet.mp4}
\end{center}
\begin{flushleft}
\vspace*{15mm}
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/NPP_animations_August2017/NPP_Non_Linear_Effects_Sumeet.mp4}
% \href{./homo_50mmesh_45degree_Ormsby.mp4}
{\tiny (MP4)}
\end{flushleft}
%
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\begin{frame}
\frametitle{NPP Seismic Reponse, Energy Dissipation}
% Elastoplastic soil with contact elements
%% Both solid and contact elements dissipate energy
% \vspace*{5mm}
\begin{center}
% \hspace*{15mm}
\movie[label=show3,width=10cm,poster,autostart,showcontrols]
{\includegraphics[width=10cm]
{/home/jeremic/tex/works/Conferences/2017/SMiRT_24/present/3D_Nonlinear_Modeling_and_it_Effects/NPP_Plastic_Dissipation_grab.jpg}}
{/home/jeremic/tex/works/Thesis/HanYang/Files_10Aug2017/NPP_Plastic_Dissipation.mp4}
\end{center}
\begin{flushleft}
\vspace*{15mm}
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Energy_Dissipation_Animations/NPP_Plastic_Dissipation.mp4}
% \href{./homo_50mmesh_45degree_Ormsby.mp4}
{\tiny (MP4)}
\end{flushleft}
%
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}
\frametitle{SMR Seismic Reponse, Energy Dissipation}
% Elastoplastic soil with contact elements
%% Both solid and contact elements dissipate energy
% \vspace*{5mm}
\begin{center}
% \hspace*{15mm}
\movie[label=show3,width=10cm,poster,autostart,showcontrols]
{\includegraphics[width=10cm]
{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Energy_Dissipation_Animations/SMR_Energy_Dissipation_screen_grab.jpg}}
{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Energy_Dissipation_Animations/SMR_Energy_Dissipation.mp4}
\end{center}
\begin{flushleft}
\vspace*{15mm}
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Energy_Dissipation_Animations/SMR_Energy_Dissipation.mp4}
% \href{./homo_50mmesh_45degree_Ormsby.mp4}
{\tiny (MP4)}
\end{flushleft}
%
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% \begin{frame}
% \frametitle{Energy Dissipation for Design}
%
% \begin{figure}[!hbpt]
% \begin{center}
% \includegraphics[width=10.0truecm]{/home/jeremic/tex/works/Thesis/HanYang/Frame_animations_13Mar2019/2D_Frame_Model.pdf}
% \end{center}
% \end{figure}
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\begin{frame}
\frametitle{Design Alternatives}
% local
%\vspace*{2mm}
\begin{center}
\hspace*{16mm}
%\movie[label=show3,width=5.6cm,poster,autostart,showcontrols]
\movie[label=show3,width=61mm,poster, showcontrols]
{\includegraphics[width=60mm]{/home/jeremic/tex/works/Thesis/HanYang/Frame_animations_13Mar2019/Individual_Foundation_screen_grab.jpg}}
{/home/jeremic/tex/works/Thesis/HanYang/Frame_animations_13Mar2019/Individual_Foundation.mp4}
%\hspace*{2mm}
%\hfill
%\movie[label=show3,width=5.6cm,poster,autostart,showcontrols]
\movie[label=show3,width=61mm,poster, showcontrols]
{\includegraphics[width=61mm]{/home/jeremic/tex/works/Thesis/HanYang/Frame_animations_13Mar2019/Continuous_Foundation_screen_grab.jpg}}
{/home/jeremic/tex/works/Thesis/HanYang/Frame_animations_13Mar2019/Continuous_Foundation.mp4}
\hspace*{16mm}
\end{center}
% local
% online
\begin{center}
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/Energy_dissipation_frames/Individual_Foundation.mp4}
{\tiny (MP4)}
%
\hspace*{40mm}
%
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/Energy_dissipation_frames/Continuous_Foundation.mp4}
{\tiny (MP4)}
\end{center}
% online
\end{frame}
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\begin{frame}
\frametitle{ASCE721, Low Building: BRB Energy Dissipation}
% \vspace*{5mm}
\begin{center}
% \hspace*{15mm}
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{\includegraphics[width=10cm]
{/home/jeremic/tex/works/Thesis/HanYang/ASCE721_low_building_energy_dissipation/ATC_Short_Building_PD.jpg}}
{/home/jeremic/tex/works/Thesis/HanYang/ASCE721_low_building_energy_dissipation/ATC_Short_Building_PD.mp4}
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\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/Energy_dissipation_frames/ATC_Short_Building_PD.mp4}
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% \includegraphics[width=3.4cm]{/home/jeremic/tex/works/Thesis/HanYang/Files_Energy_dissipation_01Dec2017/case_Rayleigh.pdf}
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% \hspace*{10mm} Numerical \hspace*{20mm} Viscous \hspace*{20mm} Plasticity
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% \item[] Dry, single phase
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% \item Contact/interface/joint, inelastic, soil/rock  foundation
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% \item[] Normal, hard and soft, gap open/close
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% \item Structure, inelastic, damage, cracks
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% \includegraphics[width=8.5cm]{/home/jeremic/tex/works/Thesis/SumeetKumarSinha/Files_10Aug2017/Npp_Non_Linear_Effects/images/NPP_With_Shallow_Foundation.pdf}
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% % \caption{\label{Fig:NPP_Model_In_Real_ESSI} Nuclear Power Plant Model with Shallow Foundation }
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% \frametitle{Structure Model}
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% The nuclear power plant structure comprise of
% \begin{itemize}
% \item Auxiliary building, $f^{aux}_{1}= 8Hz$
% \item Containment/Shield building, $f^{cont}_{1}= 4Hz$
% \item Concrete raft foundation: $3.5m$ thick
% \end{itemize}
% \begin{figure}[!h]
% \begin{center}
% \includegraphics[width=0.8\textwidth]{/home/jeremic/tex/works/Thesis/SumeetKumarSinha/Files_10Aug2017/Npp_Non_Linear_Effects/images/NPP_Model_Auxiliary_And_Containment_Building.pdf}
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% \frametitle{Inelastic Soil and Inelastic Contact/Interface/Joint}
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% \begin{itemize}
% \item Shear velocity of soil $V_s=500m/s$
% \item Undrained shear strength (Dickenson 1994) $V_s [m/s] = 23 (S_u [kPa])^{0.475}$
% \item For $V_s=500m/s$ Undrained Strength $S_u=650kPa$ and Young's Modulus of $E=1.3GPa$
% \item von Mises, Armstrong Frederick kinematic hardening
% ($S_u=650kPa$ at $\gamma=0.01\%$; $h_a = 30MPa$, $c_r = 25$)
% \item Soft contact (concretesoil), gaping and nonlinear shear
% \end{itemize}
%
% \begin{figure}[!h]
% \begin{center}
% \includegraphics[width=4cm]{/home/jeremic/tex/works/Thesis/SumeetKumarSinha/Files_10Aug2017/Npp_Non_Linear_Effects/images/Von_Mies_non_Linear_Hardening.pdf}
% \includegraphics[width=3.5cm]{/home/jeremic/tex/works/Thesis/SumeetKumarSinha/Files_10Aug2017/Npp_Non_Linear_Effects/images/SoftContact.pdf}
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% \includegraphics[width=3.54cm]{/home/jeremic/tex/works/Thesis/SumeetKumarSinha/Files_10Aug2017/Npp_Non_Linear_Effects/images/D_Acceleration_X.pdf}
% \includegraphics[width=3.54cm]{/home/jeremic/tex/works/Thesis/SumeetKumarSinha/Files_10Aug2017/Npp_Non_Linear_Effects/images/D_Acceleration_Y.pdf}
% \includegraphics[width=3.54cm]{/home/jeremic/tex/works/Thesis/SumeetKumarSinha/Files_10Aug2017/Npp_Non_Linear_Effects/images/D_Acceleration_Z.pdf}
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% \includegraphics[width=3.54cm]{/home/jeremic/tex/works/Thesis/SumeetKumarSinha/Files_10Aug2017/Npp_Non_Linear_Effects/images/FFT_D_Acceleration_X.pdf}
% \includegraphics[width=3.54cm]{/home/jeremic/tex/works/Thesis/SumeetKumarSinha/Files_10Aug2017/Npp_Non_Linear_Effects/images/FFT_D_Acceleration_Y.pdf}
% \includegraphics[width=3.54cm]{/home/jeremic/tex/works/Thesis/SumeetKumarSinha/Files_10Aug2017/Npp_Non_Linear_Effects/images/FFT_D_Acceleration_Z.pdf}
% % \caption{\label{Fig:Response_of_Top_of_Containment_Building} Seismic Response at Top of Containment Building}
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% \includegraphics[width=7.0cm]{/home/jeremic/tex/works/Thesis/SumeetKumarSinha/Files_10Aug2017/Npp_Non_Linear_Effects/images/Acceleration_Inelastic_With_Contact_SMIRT_2017.pdf}
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% \hspace*{20mm}
% \includegraphics[width=7.0cm]{/home/jeremic/tex/works/Thesis/SumeetKumarSinha/Files_10Aug2017/Npp_Non_Linear_Effects/images/Inelastic_With_Contact_SMIRT_2017.pdf}
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% % \frametitle{Elastic and Inelastic Response: Differences}
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% % % Elastoplastic soil with contact elements
% % %% Both solid and contact elements dissipate energy
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% % % \vspace*{5mm}
% % \begin{center}
% % % \hspace*{15mm}
% % \movie[label=show3,width=10cm,poster,autostart,showcontrols]
% % {\includegraphics[width=10cm]
% % {/home/jeremic/tex/works/Thesis/SumeetKumarSinha/Files_13Aug2017/NPP_Non_Linear_Effects_Sumeet.jpg}}
% % {/home/jeremic/tex/works/Thesis/SumeetKumarSinha/Files_13Aug2017/NPP_Non_Linear_Effects_Sumeet.mp4}
% % \end{center}
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% % \begin{flushleft}
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% % \href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/NPP_animations_August2017/NPP_Non_Linear_Effects_Sumeet.mp4}
% % % \href{./homo_50mmesh_45degree_Ormsby.mp4}
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% \frametitle{NPP: Plastic Energy Dissipation}
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% % Elastoplastic soil with contact elements
% %% Both solid and contact elements dissipate energy
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% % \vspace*{5mm}
% \begin{center}
% % \hspace*{15mm}
% \movie[label=show3,width=10cm,poster,autostart,showcontrols]
% {\includegraphics[width=10cm]
% {/home/jeremic/tex/works/Conferences/2017/SMiRT_24/present/3D_Nonlinear_Modeling_and_it_Effects/NPP_Plastic_Dissipation_grab.jpg}}
% {/home/jeremic/tex/works/Thesis/HanYang/Files_10Aug2017/NPP_Plastic_Dissipation.mp4}
% \end{center}
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% \begin{flushleft}
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% \href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Energy_Dissipation_Animations/NPP_Plastic_Dissipation.mp4}
% % \href{./homo_50mmesh_45degree_Ormsby.mp4}
% {\tiny (MP4)}
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% % \begin{frame}
% % \frametitle{Deeply Embedded Structures}
% % \begin{columns}[T]
% % \begin{column}{.5\textwidth}
% % \vspace{0.5cm}
% % \begin{figure}[!H]
% % \begin{center}
% % \includegraphics[width=\textwidth]{/home/jeremic/tex/works/Thesis/HexiangWang/Files_SMiRT_11Aug2017/pic/Points_configuration.pdf}
% % \end{center}
% % \end{figure}
% % \end{column}
% % \begin{column}{.5\textwidth}
% % \vspace{0.6cm}
% % \centerline{\scriptsize Location of points}
% % \vspace{0.6cm}
% % \begin{figure}[!H]
% % \begin{center}
% % \includegraphics[width=\textwidth]{/home/jeremic/tex/works/Thesis/HexiangWang/Files_SMiRT_11Aug2017/pic/representative_points.pdf}
% % \end{center}
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% % \begin{frame}
% % \frametitle{SMR: Inelastic ESSI Effects, Top Center}
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% %
% % \vspace*{30mm}
% %
% % \begin{figure}[!H]
% % \begin{flushleft}
% % \includegraphics[width=3.2cm]{/home/jeremic/tex/works/Thesis/HexiangWang/Files_SMiRT_11Aug2017/pic/AF_yuan_point1_ax_time_series.pdf}
% % \includegraphics[width=3.2cm]{/home/jeremic/tex/works/Thesis/HexiangWang/Files_SMiRT_11Aug2017/pic/AF_yuan_point1_ay_time_series.pdf}
% % \includegraphics[width=3.2cm]{/home/jeremic/tex/works/Thesis/HexiangWang/Files_SMiRT_11Aug2017/pic/AF_yuan_point1_az_time_series.pdf}
% % \\
% % \includegraphics[width=3.2cm]{/home/jeremic/tex/works/Thesis/HexiangWang/Files_SMiRT_11Aug2017/pic/AF_yuan_Point1_ax_fft.pdf}
% % \includegraphics[width=3.2cm]{/home/jeremic/tex/works/Thesis/HexiangWang/Files_SMiRT_11Aug2017/pic/AF_yuan_Point1_ay_fft.pdf}
% % \includegraphics[width=3.2cm]{/home/jeremic/tex/works/Thesis/HexiangWang/Files_SMiRT_11Aug2017/pic/AF_yuan_Point1_az_fft.pdf}
% % \end{flushleft}
% % \end{figure}
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% % \begin{figure}[!H]
% % \begin{flushright}
% % \hspace*{30mm}
% % \includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/HexiangWang/Files_SMiRT_11Aug2017/pic/Points_configuration.pdf}
% % \hspace*{10mm}
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% % % \begin{frame}
% % % \frametitle{SMR: ESSI Effects, Location}
% % %
% % %
% % %
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% % %
% % % \vspace*{30mm}
% % %
% % % \scriptsize \hspace*{20mm} Point 3 \hspace*{25mm} Point 4 \hspace*{20mm} Point 5
% % % \begin{figure}[!H]
% % % \begin{flushleft}
% % % \includegraphics[width=3.6cm]{/home/jeremic/tex/works/Thesis/HexiangWang/Files_SMiRT_11Aug2017/pic/AF_yuan_point3_ax_time_series.pdf}
% % % \includegraphics[width=3.6cm]{/home/jeremic/tex/works/Thesis/HexiangWang/Files_SMiRT_11Aug2017/pic/AF_yuan_point4_ax_time_series.pdf}
% % % \includegraphics[width=3.6cm]{/home/jeremic/tex/works/Thesis/HexiangWang/Files_SMiRT_11Aug2017/pic/AF_yuan_point5_ax_time_series.pdf}
% % % \\
% % % \includegraphics[width=3.6cm]{/home/jeremic/tex/works/Thesis/HexiangWang/Files_SMiRT_11Aug2017/pic/AF_yuan_Point3_ax_fft.pdf}
% % % \includegraphics[width=3.6cm]{/home/jeremic/tex/works/Thesis/HexiangWang/Files_SMiRT_11Aug2017/pic/AF_yuan_Point4_ax_fft.pdf}
% % % \includegraphics[width=3.6cm]{/home/jeremic/tex/works/Thesis/HexiangWang/Files_SMiRT_11Aug2017/pic/AF_yuan_Point5_ax_fft.pdf}
% % % \end{flushleft}
% % % \end{figure}
% % %
% % %
% % % \vspace*{75mm}
% % %
% % % \begin{figure}[!H]
% % % \begin{flushright}
% % % \hspace*{30mm}
% % % \includegraphics[width=2cm]{/home/jeremic/tex/works/Thesis/HexiangWang/Files_SMiRT_11Aug2017/pic/Points_configuration.pdf}
% % % \hspace*{10mm}
% % % \end{flushright}
% % % \end{figure}
% % %
% % %
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% % % % \begin{columns}[T]
% % % % \begin{column}{0.8\textwidth}
% % % % \vspace{0.6cm}
% % % % \scriptsize \quad \quad \quad \quad Point 3\quad \quad \quad \quad \quad \quad \quad Point 4\quad \quad \quad \quad \quad \quad Point 5
% % % % \vspace{0.3cm}
% % % % \begin{figure}[!H]
% % % % \begin{center}
% % % % \includegraphics[width=0.3\textwidth]{/home/jeremic/tex/works/Thesis/HexiangWang/Files_SMiRT_11Aug2017/pic/AF_yuan_point3_ax_time_series.pdf}
% % % % \includegraphics[width=0.3\textwidth]{/home/jeremic/tex/works/Thesis/HexiangWang/Files_SMiRT_11Aug2017/pic/AF_yuan_point4_ax_time_series.pdf}
% % % % \includegraphics[width=0.3\textwidth]{/home/jeremic/tex/works/Thesis/HexiangWang/Files_SMiRT_11Aug2017/pic/AF_yuan_point5_ax_time_series.pdf} \\
% % % % \includegraphics[width=0.3\textwidth]{/home/jeremic/tex/works/Thesis/HexiangWang/Files_SMiRT_11Aug2017/pic/AF_yuan_Point3_ax_fft.pdf}
% % % % \includegraphics[width=0.3\textwidth]{/home/jeremic/tex/works/Thesis/HexiangWang/Files_SMiRT_11Aug2017/pic/AF_yuan_Point4_ax_fft.pdf}
% % % % \includegraphics[width=0.3\textwidth]{/home/jeremic/tex/works/Thesis/HexiangWang/Files_SMiRT_11Aug2017/pic/AF_yuan_Point5_ax_fft.pdf}
% % % % \end{center}
% % % % \end{figure}
% % % % \vspace{0.3cm}
% % % % \begin{itemize}
% % % % \item Nonlinear effects attenuate along the depth
% % % % \end{itemize}
% % % % \end{column}
% % % % \begin{column}{0.3\textwidth}
% % % % \begin{figure}[!H]
% % % % \begin{center}
% % % % \includegraphics[width=\textwidth]{/home/jeremic/tex/works/Thesis/HexiangWang/Files_SMiRT_11Aug2017/pic/Points_configuration.pdf}
% % % % \end{center}
% % % % \end{figure}
% % % % \end{column}
% % % % \end{columns}
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% % % \end{frame}
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% % \begin{frame}
% % \frametitle{SMR: ESSI Effects, Material Modeling}
% % \begin{columns}[T]
% % \begin{column}{0.6\textwidth}
% % \scriptsize \quad \quad \quad \quad \quad Material A\quad \quad \quad \quad \quad \quad \quad Material B
% % \vspace{3mm}
% % \begin{figure}[!H]
% % \begin{center}
% % \includegraphics[width=3cm]{/home/jeremic/tex/works/Thesis/HexiangWang/Files_SMiRT_11Aug2017/pic/AF_yuan_point1_ax_time_series.pdf}
% % \includegraphics[width=3cm]{/home/jeremic/tex/works/Thesis/HexiangWang/Files_SMiRT_11Aug2017/pic/bilinear_point1_ax_time_series.pdf} \\
% % \includegraphics[width=3cm]{/home/jeremic/tex/works/Thesis/HexiangWang/Files_SMiRT_11Aug2017/pic/AF_yuan_Point1_ax_fft.pdf}
% % \includegraphics[width=3cm]{/home/jeremic/tex/works/Thesis/HexiangWang/Files_SMiRT_11Aug2017/pic/bilinear_Point1_ax_fft.pdf}
% % \end{center}
% % \end{figure}
% % \vspace{0.3cm}
% % % \begin{itemize}
% % % \item
% % % \end{itemize}
% % \end{column}
% % \hspace{0.3cm}
% % \begin{column}{0.3\textwidth}
% % \vspace{0.35cm}
% % \begin{figure}[!H]
% % \begin{center}
% % \includegraphics[width=3cm]{/home/jeremic/tex/works/Thesis/HexiangWang/Files_SMiRT_11Aug2017/pic/ha=30Mpa.png} \\
% % \tiny \quad Material A: nonlinear, vM  AF\\
% % \vspace{0.2cm}
% % \includegraphics[width=3cm]{/home/jeremic/tex/works/Thesis/HexiangWang/Files_SMiRT_11Aug2017/pic/material_stress_strain_behavior.pdf} \\
% % \tiny \quad Material B: Bilinear\\
% % \end{center}
% % \end{figure}
% % \end{column}
% % \end{columns}
% % \end{frame}
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% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \begin{frame}
% % \frametitle{SMR: Accelerations Along Depth}
% %
% % \vspace{0.6cm}
% %
% % \begin{columns}[T]
% %
% % \hspace{0.6cm}
% %
% % \begin{column}{1.1\textwidth}
% %
% % \begin{figure}[!H]
% % \begin{center}
% % \includegraphics[width=5cm]{/home/jeremic/tex/works/Thesis/HexiangWang/Files_SMiRT_11Aug2017/pic/free_field_acceleration_depth_variation.pdf}\hspace{0.7cm}
% % \includegraphics[width=5cm]{/home/jeremic/tex/works/Thesis/HexiangWang/Files_SMiRT_11Aug2017/pic/free_field_a_depth_variation_AF.pdf}
% % \end{center}
% % \end{figure}
% %
% % \vspace{1.1cm}
% %
% % \begin{figure}[!H]
% % \begin{center}
% % \includegraphics[width=5cm]{/home/jeremic/tex/works/Thesis/HexiangWang/Files_SMiRT_11Aug2017/pic/SMR_acceleration_depth_variation_elastic.pdf}\hspace{0.7cm}
% % \includegraphics[width=5cm]{/home/jeremic/tex/works/Thesis/HexiangWang/Files_SMiRT_11Aug2017/pic/SMR_acceleration_depth_variation.pdf}
% % \end{center}
% % \end{figure}
% % \end{column}
% %
% %
% % \hspace{0.7cm}
% % \begin{column}{0.15\textwidth}
% % \vspace{1.5cm}
% % Nonlinear site effects\\
% % \vspace{2.5cm}
% % SSI effects
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\begin{frame}
\frametitle{Forward Uncertainty Propagation}
\begin{itemize}
\vspace*{3mm}
\item[] Given uncertain material and uncertain loads
\vspace*{5mm}
\item[] Determine uncertain response, $u_i, \dot{u}_i, \ddot{u}_i,
\epsilon_{ij}, \sigma_{ij}$, PDFs/CDFs
\vspace*{5mm}
\item[] Intrusive, analytic development, SEPFEM
\vspace*{5mm}
\item[] Avoid Monte Carlo inefficiencies
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Forward Uncertain Inelasticity}
%
\begin{itemize}
\item[] Incremental elpl constitutive equation
%
\begin{eqnarray}
\nonumber
\Delta \sigma_{ij}
=
% E^{EP}_{ijkl}
E^{EP}_{ijkl} \; \Delta \epsilon_{kl}
=
\left[
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_*}
\right]
\Delta \epsilon_{kl}
\end{eqnarray}
\vspace*{2mm}
\item[] Dynamic Finite Elements
%
\begin{equation}
{ M} \Delta \ddot{ u_i}
+
{ C} \Delta \dot{ u_i}
+
{ K}^{ep} \Delta { u_i}
=
\Delta { F(t)}
\nonumber
\end{equation}
\vspace*{2mm}
\item[] Material and loads are uncertain
\end{itemize}
\end{frame}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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{{Cam Clay with Random $G$, $M$ and $p_0$}}
\begin{figure}[!hbpt]
\begin{center}
\hspace*{15mm}
\includegraphics[width=6.0cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourLowOCR_RandomG_RandomM_Randomp0m.pdf}
%\hspace*{2mm}
\includegraphics[width=6.0cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourHighOCR_RandomG_RandomMm.pdf}
\hspace*{15mm}
\end{center}
\end{figure}
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\end{frame}
%  %%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Stochastic ElasticPlastic Finite Element Method}
%\vspace*{2mm}
Dynamic Finite Elements
$
{ M} \ddot{ u_i} +
{ C} \dot{ u_i} +
{ K}^{ep} { u_i} =
{ F(t)}$
\begin{itemize}
\vspace*{2mm}
\item[] Input random field/process{\normalsize{(nonGaussian, heterogeneous/ nonstationary)}}:
Multidimensional Hermite Polynomial Chaos (PC) with {known coefficients}
%\vspace{0.05in}
\vspace*{2mm}
\item[] Output response process: Multidimensional Hermite PC with {unknown coefficients}
% % \vspace{0.05in}
% %\vspace*{2mm}
% \item[] Galerkin projection: minimize the error to compute unknown coefficients of response process
% % %\vspace{0.05in}
% % \vspace*{2mm}
% % \item[] Time integration using Newmark's method
% % : Update coefficients following
% % an elasticplastic constitutive law at each time step
\vspace*{2mm}
\item[] Complete probabilistic response
\vspace*{2mm}
\item[] NO need to decide/define Intensity Measures (IMs) !
\end{itemize}
%
% \vspace*{1mm}
% \tiny{Jeremi{\'c} et al. 2011}
%
%\scriptsize
%Note: PC = Polynomial Chaos
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% \begin{frame}{Discretization of Input Random Process/Field $\beta(x,\theta)$}
% \begin{center}
% \includegraphics[scale=0.35]{/home/jeremic/tex/works/Thesis/FangboWang/slides_13Mar2019/Fangbo_slides/figs/PC_KL_explanation.PNG} \\
% \end{center}
%
%
% \footnotesize{Note: $\beta(x,\theta)$ is an input random process with any
% marginal distribution, \\ \hspace{21mm} with any covariance structure;} \\
% \footnotesize{\hspace{8mm} $\gamma(x,\theta)$ is a zeromean unitvariance Gaussian random process.} \\
%
% \end{frame}
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}{Polynomial Chaos Representation}
%
% %\scriptsize{
% Material random field: \\
% %\vspace{0.3cm}
% %\begin{equation*}
% $D(x, \theta)= \sum_{i=1}^{P1} a_i(x) \Psi_i(\left\{\xi_r(\theta)\right\})$
% %\end{equation*}
%
%
% \vspace{4mm}
%
% Seismic loads/motions random process: \\
% %\vspace{0.3cm}
% %\begin{equation*}
% $f_m(t, \theta)=\sum_{j=1}^{P_2} f_{mj}(t) \Psi_j(\{\xi_k(\theta)\})$
% %\end{equation*}
%
% \vspace{4mm}
%
% Displacement response: \\
% %\vspace{0.3cm}
% %\begin{equation*}
% $u_n(t, \theta)=\sum_{k=1}^{P_3} d_{nk}(t) \Psi_k(\{\xi_l(\theta)\})$
% %\end{equation*}
%
% \vspace{3mm}
%
% %Acceleration response:
% %%\vspace{0.3cm}
% %%\begin{equation*}
% %$\ddot u_n(t, \theta)=\sum_{k=1}^{P_3} \ddot d_{nk}(t) \Psi_k(\{\xi_l(\theta)\})$
% %%\end{equation*}
%
% %\vspace{3mm}
% \vspace{5mm}
%
% where $a_i(x), f_{mj}(t)$ are {known PC coefficients}, while $d_{nk}(t)$
% are {unknown PC coefficients}.
% %}
%
% \end{frame}
%
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \subsection{Direct Solution for Probabilistic Stiffness and Stress in 1D}
% %
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %%%%%%%%%%%%%%%%%%%%%%%%%% BEGGINING PEP %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
% \begin{frame}{Direct Probabilistic Constitutive Solution in 1D}
%
%
% % \begin{itemize}
% %
% % \vspace{0.5cm}
% %
% % \item<1> Probabilistic constitutive modeling : \vspace{0.5cm}
%
% \begin{itemize}
%
%
% \vspace*{4mm}
% \item[] Zero elastic region elastoplasticity with stochastic ArmstrongFrederick
% kinematic hardening
%
% $ \Delta\sigma =\ H_a \Delta \epsilon  c_r \sigma \Delta \epsilon ;
% \hspace{0.5cm}
% E_t = {d\sigma}/{d\epsilon} = H_a \pm c_r \sigma $
%
% \vspace*{4mm}
% \item[] Uncertain:
% init. stiff. $H_a$,
% shear strength $H_a/c_r$,
% strain $\Delta \epsilon$:
%
% $ H_a = \Sigma h_i \Phi_i; \;\;\;
% C_r = \Sigma c_i \Phi_i; \;\;\;
% \Delta\epsilon = \Sigma \Delta\epsilon_i \Phi_i $
%
%
%
% \vspace*{4mm}
% \item[] Resulting stress and stiffness are also uncertain
%
% % 
% %  $ \sum_{l=1}^{P_{\sigma}} \Delta\sigma_i \Phi_i = \sum_{i=1}^{P_h} \sum_{k=1}^{P_e}\ h_i \Delta \epsilon_k \Phi_i \Phi_k  \sum_{j=1}^{P_g} \sum_{k=1}^{P_e}\sum_{l=1}^{P_{\sigma}} \ c_i \Delta \epsilon_k \sigma_l \Phi_j \Phi_k \Phi_l$
% % 
% %  $ \sum_{l=1}^{P_{E_t}} \Delta E_{t_i} \Phi_i = \sum_{i=1}^{P_h} h_i \Phi_i \pm \sum_{i=1}^{P_c} \sum_{l=1}^{P_{\sigma}} \ c_i \sigma_l \Phi_i \Phi_l$
% % 
%
%
% \end{itemize}
%
%
% % \vspace{0.5cm}
%
%
%
% % \vspace{1cm}
%
% %\item<1> Time integration is done via Newmark algorithm
%
% %
% % \end{itemize}
% %
% \end{frame}
%
%
% % % % % % % % % % % % % % % % %
%
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}{Direct Probabilistic Stiffness Solution}
%
% \begin{itemize}
%
%
% \item[] Analytic product for all the components,
%
% $ E^{EP}_{ijkl}
% =
% \left[
% 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_*}
% \right]
% $
%
%
%
%
% \vspace*{2mm}
% \item[] Stiffness: each Polynomial Chaos component is updated incrementally
% % at each Gauss Point via stochastic Galerkin projection
%
%
%
% \small{$E_{t_1}^{n+1} = \frac{1}{<\Phi_1\Phi_1> }\{\sum_{i=1}^{P_h} \ h_i <\Phi_i \Phi_1> \pm \sum_{j=1}^{P_c} \sum_{l=1}^{P_{\sigma}} \ c_j \sigma_l^{n+1} <\Phi_j \Phi_l \Phi_1>\}$}
% \\
% . . .
% %
% %
% % $\large{\vdots}$
% \\
% \small{$E_{t_P}^{n+1} = \frac{1}{<\Phi_1\Phi_P> }\{\sum_{i=1}^{P_h} \ h_i <\Phi_i \Phi_P> \pm \sum_{j=1}^{P_c} \sum_{l=1}^{P_{\sigma}} \ c_j \sigma_l^{n+1} <\Phi_j \Phi_l \Phi_P>\}$}
%
%
% \vspace*{2mm}
% \item[] Total stiffness is :
%
% $ E_{t}^{n+1} = \sum_{l=1}^{P_{E}} E_{t_i}^{n+1} \Phi_i $
%
%
%
%
% \end{itemize}
%
%
% \end{frame}
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}{Direct Probabilistic Stress Solution}
%
% \begin{itemize}
%
%
%
% \item[] Analytic product, for each stress component,
%
% $ \Delta \sigma_{ij} = E^{EP}_{ijkl} \; \Delta \epsilon_{kl} $
% % =
% % \left[
% % 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_*}
% % \right]
% % \Delta \epsilon_{kl}
% %
%
%
% \vspace*{2mm}
% \item[] Incremental stress: each Polynomial Chaos component is updated
% incrementally
% % via stochastic Galerkin projection
%
%
%
%
% {$\Delta\sigma_1^{n+1} = \frac{1}{<\Phi_1\Phi_1> }\{\sum_{i=1}^{P_h} \sum_{k=1}^{P_e}\ h_i \Delta \epsilon_k^n <\Phi_i \Phi_k \Phi_1> \sum_{j=1}^{P_g} \sum_{k=1}^{P_e}\sum_{l=1}^{P_{\sigma}} \ c_j \Delta \epsilon_k^n \sigma_l^n <\Phi_j \Phi_k \Phi_l \Phi_1>\}$}
% \\
% . . .
% \\
% % ${\vdots}$
% {$\Delta\sigma_P^{n+1} = \frac{1}{<\Phi_P\Phi_P> }\{\sum_{i=1}^{P_h} \sum_{k=1}^{P_e}\ h_i \Delta \epsilon_k^n <\Phi_i \Phi_k \Phi_P> \sum_{j=1}^{P_g} \sum_{k=1}^{P_e}\sum_{l=1}^{P_{\sigma}} \ c_j \Delta \epsilon_k^n \sigma_l^n <\Phi_j \Phi_k \Phi_l \Phi_P>\}$}
%
%
% \vspace*{2mm}
% \item[] Stress update:
%
% $ \sum_{l=1}^{P_{\sigma}} \sigma_i^{n+1} \Phi_i = \sum_{l=1}^{P_{\sigma}} \sigma_i^{n} \Phi_i + \sum_{l=1}^{P_{\sigma}} \Delta\sigma_i^{n+1} \Phi_i$
%
%
%
% \end{itemize}
%
% \end{frame}
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% \begin{frame}
% \frametitle{Probabilistic ElasticPlastic Response}
%
%
% % % \vspace*{5mm}
% % \begin{center}
% % % \hspace*{15mm}
% % \movie[label=show3,width=7cm,poster,autostart,showcontrols]
% % {\includegraphics[width=7cm]
% % {/home/jeremic/tex/works/Thesis/HanYang/Files_06June2017/DOE_Annual_2017/Figures/NPP_Plastic_Dissipation_Density.png}}
% % %{/home/jeremic/tex/works/Thesis/HanYang/Files_06June2017/DOE_Annual_2017/Figures/NPP_without_Contact_vonMises.mp4}
% % {NPP_without_Contact_vonMises.mp4}
% % \end{center}
%
% %\vspace*{5mm}
% \begin{center}
% % \hspace*{15mm}
% \movie[label=show3,width=9cm,poster,autostart,showcontrols]
% {\includegraphics[width=9cm]
% {/home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Latex/img/figure_PEP_25.png}}
% % /home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Latex/img/figure_PEP_25.pdf
% %{/home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Animations/PEP_Animation.mp4}
% {/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Probabilistic_Elasto_Plasticity_and_Stochastic_Elastic_Plastic_Finite_Element_Method/PEP_Animation.mp4}
% \end{center}
%
%
%
%
% \begin{flushleft}
% \vspace*{15mm}
% \href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Probabilistic_Elasto_Plasticity_and_Stochastic_Elastic_Plastic_Finite_Element_Method/PEP_Animation.mp4}
% % \href{./homo_50mmesh_45degree_Ormsby.mp4}
% {\tiny (MP4)}
% \end{flushleft}
% %
%
%
%
%
% %
% % \includegraphics[width = 12cm]{./img/figure_PEP_25.pdf}
%
%
% \end{frame}
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section[Formulation]{Stochastic Dynamic Finite Element Formulation}
%\subsection[Time domain stochastic Galerkin method]{Time domain stochastic Galerkin method}
%\frame{\tableofcontents[currentsubsection,sectionstyle=show/shaded]}
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Stochastic ElasticPlastic Finite Element Method}
%
%
%
% \begin{itemize}
%
% \vspace*{4mm}
% \item[] Material uncertainty expanded into stochastic shape funcs.
% %$E(x,t,\theta) = \sum_{i=0}^{P_d} r_i(x,t) * \Phi_i[\{\xi_1, ..., \xi_m\}]$
%
% \vspace*{4mm}
% \item[] Loading uncertainty expanded into stochastic shape funcs.
% %$f(x,t,\theta) = \sum_{i=0}^{P_f} f_i(x,t) * \zeta_i[\{\xi_{m+1}, ..., \xi_f]$
%
% \vspace*{4mm}
% \item[] Displacement expanded into stochastic shape funcs.
% %$u(x,t,\theta) = \sum_{i=0}^{P_u} u_i(x,t) * \Psi_i[\{\xi_1, ..., \xi_m, \xi_{m+1}, ..., \xi_f\}]$
%
% \vspace*{4mm}
% \item[] Complete probabilistic response
%
%
% \vspace*{4mm}
% \item[] NO need to decide, define Intensity Measures (IMs) !
%
% %$u(x,t,\theta) = \sum_{i=0}^{P_u} u_i(x,t) * \Psi_i[\{\xi_1, ..., \xi_m, \xi_{m+1}, ..., \xi_f\}]$
% %\item
% %Stochastic system of equation resulting from Galerkin approach (static example):
% %
% %\item Time domain integration using Newmark and/or HHT, in probabilistic spaces
%
%
% % %
% % \vspace*{1mm}
% % \item[] Jeremi{\'c} et al. 2011
%
%
% \end{itemize}
%
% %
% % \begin{tiny}
% % \[
% % %$
% % \begin{bmatrix}
% % \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_0> K^{(k)} & \dots & \sum_{k=0}^{P_d} <\Phi_k \Psi_P \Psi_0> K^{(k)}\\
% % \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_1> K^{(k)} & \dots & \sum_{k=0}^{P_d} <\Phi_k \Psi_P \Psi_1> K^{(k)}\\ \\
% % \vdots & \vdots & \vdots & \vdots\\
% % \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_P> K^{(k)} & \dots & \sum_{k=0}^{M} <\Phi_k \Psi_P \Psi_P> K^{(k)}
% % \end{bmatrix}
% % \begin{bmatrix}
% % \Delta u_{10} \\
% % \vdots \\
% % \Delta u_{N0}\\
% % \vdots \\
% % \Delta u_{1P_u}\\
% % \vdots \\
% % \Delta u_{NP_u}
% % \end{bmatrix}
% % =
% % %\]
% % %\[
% % \begin{bmatrix}
% % \sum_{i=0}^{P_f} f_i <\Psi_0\zeta_i> \\
% % \sum_{i=0}^{P_f} f_i <\Psi_1\zeta_i> \\
% % \sum_{i=0}^{P_f} f_i <\Psi_2\zeta_i> \\
% % \vdots \\
% % \sum_{i=0}^{P_f} f_i <\Psi_{P_u}\zeta_i>\\
% % \end{bmatrix}
% % %$
% % \]
% % \end{tiny}
% %
% %
% %
% %
%
% \end{frame}
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% \begin{frame}
% \frametitle{SEPFEM: System Size}
%
% \begin{itemize}
%
% \item[] SEPFEM offers a complete probabilistic solution
%
% \item[] It is NOT based on Monte Carlo approach
%
% \item[] System of equations does grow (!)
%
% \end{itemize}
%
%
% % \normalsize{Typical number of terms required for a SEPFEM problem} \vspace{1cm}\\
% \scalebox{0.7}{
% \begin{tabular}{ c c c c}
% \# KL terms material & \# KL terms load & PC order displacement& Total \# terms per DoF\\ \hline
% 4 & 4 & 10 & 43758 \\
% 4 & 4 & 20 & 3 108 105 \\
% 4 & 4 & 30 & 48 903 492 \\
% 6 & 6 & 10 & 646 646 \\
% 6 & 6 & 20 & 225 792 840 \\
% 6 & 6 & 30 & 1.1058 $10^{10}$ \\
% 8 & 8 & 10 & 5 311 735 \\
% 8 & 8 & 20 & 7.3079 $10^{9}$ \\
% 8 & 8 & 30 & 9.9149 $10^{11}$\\
%
% ... & ... & ... & ...\\
% % \hline
% \end{tabular}}
%
%
% \end{frame}
%
%
%
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{SEPFEM: Example in 1D}
\vspace*{2mm}
\begin{center}
% \hspace*{15mm}
\movie[label=show3,width=9cm,showcontrols]
{\includegraphics[width=9cm]{/home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Latex/img/figure_elastic_900.png}}
% /home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Latex/img/figure_PEP_25.pdf
{/home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Animations/SEPFEM_Animation_Elastic.mp4}
% {SEPFEM_Animation_Elastic.mp4}
\end{center}
%
% \vspace*{2mm}
% \begin{center}
% % \hspace*{15mm}
% \movie[label=show3,width=9cm,poster,autostart,showcontrols]
% {\includegraphics[width=9cm]{/home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Latex/img/figure_elastic_900.png}}
% % /home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Latex/img/figure_PEP_25.pdf
% {/home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Animations/SEPFEM_Animation_Elastic.mp4}
% \end{center}
%
% \includegraphics[width = 12cm]{./img/figure_elastic_900.pdf}
\begin{flushleft}
\vspace*{15mm}
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Probabilistic_Elasto_Plasticity_and_Stochastic_Elastic_Plastic_Finite_Element_Method/SEPFEM_Animation_Elastic.mp4}
% \href{./homo_50mmesh_45degree_Ormsby.mp4}
{\tiny (MP4)}
\end{flushleft}
%
\end{frame}
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% \begin{frame}
% \frametitle{SEPFEM: Example in 3D}
%
%
%
% %\vspace*{5mm}
% \begin{center}
% % \hspace*{15mm}
% \movie[label=show3,width=10cm,poster,autostart,showcontrols]
% {\includegraphics[width=10cm]
% {/home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Latex/img/SFEM_3D.png}}
% % /home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Latex/img/figure_PEP_25.pdf
% {/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Probabilistic_Elasto_Plasticity_and_Stochastic_Elastic_Plastic_Finite_Element_Method/SFEM_Animation_3D.mp4}
% %{/home/jeremic/tex/works/Thesis/MaximeLacour/Files_27Jun2017/Summer_Slides/Animations/SFEM_Animation_3D.mp4}
% \end{center}
%
% % \includegraphics[width = 12cm]{./img/SFEM_3D.pdf}
%
% \begin{flushleft}
% %\hspace*{15mm}
% \vspace*{15mm}
% \href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Probabilistic_Elasto_Plasticity_and_Stochastic_Elastic_Plastic_Finite_Element_Method/SFEM_Animation_3D.mp4}
% % \href{./homo_50mmesh_45degree_Ormsby.mp4}
% {\tiny (MP4)}
% \end{flushleft}
% %
%
%
% \end{frame}
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsection{Formulation}
% \subsection[TDNIPSRA Formulation]{Formulation}
% %\subsection{TDNIPSRA Formulation}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%
% \frametitle{Probabilistic Seismic Risk Analysis (PSRA)}
%
% \begin{textblock}{15}(0.8, 3.9)
% \scriptsize
% Uncertain source, path, \\
% \quad site and structure
% \end{textblock}
%
% \begin{textblock}{15}(4.8, 4.2)
% $\Longrightarrow$
% \end{textblock}
%
% \begin{textblock}{15}(5.8, 3.7)
% \scriptsize
% %Acceptably small probability\\
% Probabilities of \\
% engineering demand parameters (EDP) \\
% damage measures (DM), loss, etc
% \end{textblock}
%
% \begin{textblock}{15}(0.4, 11.8)
% \tiny
% \textbf{Uncertain source rupture}
% \end{textblock}
%
% \begin{textblock}{15}(3.55, 4.8)
% \begin{figure}[H]
% \flushleft
% \includegraphics[width=0.7\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/problem_statement.pdf}
% \end{figure}
% \end{textblock}
%
% \begin{textblock}{15}(12.5, 5.5)
% \tiny
% adapted from Taga(1982)
% \end{textblock}
%
% \begin{textblock}{15}(11.7, 10.6)
% \tiny
% \textbf{Uncertain material properties}
% \end{textblock}
%
%
% \begin{textblock}{15}(0.4, 8.0)
% \begin{figure}[H]
% \flushleft
% \includegraphics[width=0.2\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/4231447_Northridge_eq.pdf}
% \end{figure}
% \end{textblock}
%
%
% \begin{textblock}{15}(10.7, 10.2)
% \begin{figure}[H]
% \flushleft
% \includegraphics[width=0.3\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/uncertain_SPT.pdf}
% \end{figure}
% \end{textblock}
%
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%
% %\frametitle{State of the Art Probabilistic Seismic Risk Analysis}
% \frametitle{Probabilistic Seismic Risk Analysis}
%
% \begin{itemize}
%
% % \item Target: safe design, acceptably small failure probability $\lambda (EDP)$
%
% \item[]
% %Powerful tool
% %Allows for
% Objective, quantitative decision making based on exceedance rate
% $\lambda (EDP>z)$
%
% \item[] PSRA: convolution of PSHA and fragility
%
% % \[\lambda(EDP>z) = \int_{IM} \underbrace{\frac{d\lambda(IM)}{dIM}}_\text{PSHA} \underbrace{G(EDPIM)}_{\text{fragility}} dIM \]
%
% \vspace{0.1cm}
%
% \[\lambda(EDP>z) = \int \underbrace{\frac{d\lambda(IM>x)}{dx}}_\text{\textbf{PSHA}} \underbrace{G(EDP>zIM=x)}_{\text{\textbf{fragility analysis}}} dx\]
%
% \small{$\lambda(\cdot)$ : rate of exceedance\\
% \vspace{0.07cm}
% $EDP$: engineering demand parameter\\
% \vspace{0.07cm}
% $PSHA$: probabilistic seismic hazard analysis\\
% \vspace{0.07cm}
% $IM$: intensity measure}
%
% \end{itemize}
%
% \end{frame}
%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %
% % \begin{frame}
% % \frametitle{Intensity Measure (IM)}
% % IM serves as the proxy of damaging ground motions
% %
% % \vspace{0.3cm}
% %
% % \begin{itemize}
% % \item[] Does a single IM, e.g., $Sa(T_0)$, represent all uncertainty?
% % %% influencing EDP?
% % %\begin{itemize}
% % %\item[] \small Structure nonlinearity
% % %% \item[] \small Liquefaction: PGA and duration
% % %\item[] \small Higher mode response
% % %\end{itemize}
% %
% % \vspace{3mm}
% %
% % \item[] Practically difficult/contentious to choose
% %
% % % \begin{itemize}
% % % % \item[] \small Geohazard: Liquefaction, slope deformation
% % % % \item[] \small PGA v.s. AI v.s. RMS for liquefaction
% % % \item[] \small AI v.s. PGV v.s. CAV for dam embankment
% % % \end{itemize}
% %
% % \vspace{3mm}
% %
% % % \item Additional effort for new GMPEs
% %
% % % \begin{itemize}
% % % \item[] \small vector hazard: GMPE with covariance of IMs, fragility as function of IMs, rarely used
% % % \end{itemize}
% %
% % % \item[] Miscommunication: seismologists and engineers
% % %
% % % \begin{itemize}
% % % \item[] \small $Sa(T_0)$ not compatible with time domain nonlinear analysis
% % % \end{itemize}
% %
% % \end{itemize}
% % \end{frame}
% %
%
%
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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Current State of Art Seismic Risk Analysis (SRA)}
%
%
% \begin{itemize}
% %\vspace{2mm}
%
% \item[] Intensity measure (IM) selected as a proxy for ground motions,
% usually Spectral acceleration $Sa(T_0)$
%
% \vspace{4mm}
% \item[] Ground Motion Prediction Equations (GMPEs) need development, ergodic or site specific
%
% \vspace{4mm}
% \item[] Probabilistic seismic hazard analysis (PSHA)
% % for ground motion $\lambda(Sa>z)$
% % \begin{equation*}
% % \resizebox{0.85\hsize}{!}{%
% % $\lambda(Sa>z) = \sum_{i=1}^{NFL} \underbrace{N_i \int\int f_{mi}(M) f_{ri}(RM)}_\text{seismic source characterization (SSC)} \underbrace{P(Sa>zM, R)}_\text{GMPE} dM dR$}
% % \end{equation*}
%
% \vspace{4mm}
% \item[] Fragility analysis $P(EDP>xIM=z)$, deterministic time domain FEM,
% perhaps using Monte Carlo (MC)
%
% % \begin{itemize}
% %
% % \item[] Records selection: Spectrummatching technique UHS, etc
% %
% % \item[] Incremental dynamic analysis: Monte Carlo
% %
% % \end{itemize}
%
%
%
% \end{itemize}
%
% % \begin{textblock}{15}(2.2, 9.2)
% % \begin{figure}[H]
% % \flushleft
% % % \includegraphics[width=0.38\linewidth]{pic/hazard_curve.png}
% % \includegraphics[width=0.38\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/hazard_curve.pdf}
% % \enspace
% % \includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/design_spectra.png}
% % \end{figure}
% % \end{textblock}
%
%
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsection{Issues in Stateoftheart SRA}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%
% \frametitle{Seismic Risk Analysis Challenges}
%
%
% \begin{itemize}
%
%
%
% \item[] IM serves as the proxy of damaging ground motions
% \vspace{2mm}
% \item[] Does a single IM, e.g., $Sa(T_0)$, represent all uncertainty?
% %\item[] Practically difficult/contentious to choose
%
%
% %\vspace{3mm}
% \vspace{2mm}
% \item[] IMs difficult to choose, Spectral Acc, PGA, PGV...
%
%
% %%\vspace{3mm}
% %\item[] Single IM does not contain all/most uncertainty
%
%
%
%
% \vspace{2mm}
% \item[] Fragility analysis: incremental dynamic analysis (IDA)
% % using Monte Carlo method
%
% \vspace{2mm}
% \item[] Use of Monte Carlo method, accuracy, efficiency...
%
% %\vspace{3mm}
% \vspace{2mm}
% \item[] Monte Carlo, computationally expensive, CyberShake for LA, 20,000
% cases, 100Y runtime, (Maechling et al. 2007)
%
% %
% %
% % \vspace{3mm}
% % \item[] Miscommunication between seismologists and struct/geotech engineers,
% % $Sa(T_0)$ not compatible with nonlinear FEM
%
%
%
%
% \end{itemize}
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% \end{frame}
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% \begin{frame}
%
% \frametitle{Time Domain Intrusive PSRA Framework}
%
% %
%
% \begin{itemize}
%
% %\vspace*{2mm}
% \item[] Stochastic ElasticPlastic Finite Element Method, SEPFEM,
% ${M} \ddot{u_i} + {C} \dot{u_i} + {K}^{ep} {u_i} = {F(t)}$,
% (Sett et al. 2011)
%
%
% \vspace*{4mm}
% \item[] Uncertain elasticplastic material
% %stress and stiffness solution using
% %Forward Kolmogorov, FokkerPlanck equation
%
%
% \vspace*{4mm}
% \item[] Uncertain seismic loads/motions
% % using Domain Reduction Method
%
%
% \vspace*{4mm}
% \item[] Results, probability distribution functions for $\sigma_{ij}$,
% $\epsilon_{ij}$, $u_i$...
%
%
%
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% \end{itemize}
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% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% \begin{frame}
% \frametitle{Stochastic ElasticPlastic Finite Element Method}
%
%
%
% \begin{itemize}
%
% %\item[] Material uncertainties: expanded along stochastic shape functions:
% \item[] Material uncertainties: stochastic shape functions:
% $E^{ep}(x,t,\theta) = \sum_{i=0}^{P_d} E_i(x,t) * \Phi_i[\{\xi_1, ..., \xi_m\}]$
%
% \vspace*{1mm}
% \item[] Loading uncertainties: stochastic shape functions
% $F(x,t,\theta) = \sum_{i=0}^{P_f} F_i(x,t) * \zeta_i[\{\xi_{m+1}, ..., \xi_f]$
%
% \vspace*{1mm}
% \item[] Displacement expanded: stochastic shape functions:
% $u(x,t,\theta) = \sum_{i=0}^{P_u} u_i(x,t) * \Psi_i[\{\xi_1, ..., \xi_m, \xi_{m+1}, ..., \xi_f\}]$
%
%
% \vspace*{1mm}
% \item[]
% Stochastic system of equations
% \vspace*{2mm}
% \begin{tiny}
% \[
% \begin{bmatrix}
% \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_0> K^{(k)} & \dots & \sum_{k=0}^{P_d} <\Phi_k \Psi_P \Psi_0> K^{(k)}\\
% \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_1> K^{(k)} & \dots & \sum_{k=0}^{P_d} <\Phi_k \Psi_P \Psi_1> K^{(k)}\\ \\
% \vdots & \vdots & \vdots & \vdots\\
% \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_P> K^{(k)} & \dots & \sum_{k=0}^{M} <\Phi_k \Psi_P \Psi_P> K^{(k)}
% \end{bmatrix}
% \begin{bmatrix}
% u_{10} \\
% \vdots \\
% u_{N0}\\
% \vdots \\
% u_{1P_u}\\
% \vdots \\
% u_{NP_u}
% \end{bmatrix}
% =
% %\]
% %\[
% \begin{bmatrix}
% \sum_{i=0}^{P_f} f_i <\Psi_0\zeta_i> \\
% \sum_{i=0}^{P_f} f_i <\Psi_1\zeta_i> \\
% \sum_{i=0}^{P_f} f_i <\Psi_2\zeta_i> \\
% \vdots \\
% \sum_{i=0}^{P_f} f_i <\Psi_{P_u}\zeta_i>\\
% \end{bmatrix}
% \]
% \end{tiny}
%
%
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% \end{itemize}
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% \end{frame}
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% %
% % \begin{frame}
% % \frametitle{Stochastic ElasticPlastic Finite Element Method}
% % %\frametitle{SEPFEM : Formulation}
% %
% % Stochastic system of equations
% %
% % \begin{tiny}
% % \[
% % \begin{bmatrix}
% % \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_0> K^{(k)} & \dots & \sum_{k=0}^{P_d} <\Phi_k \Psi_P \Psi_0> K^{(k)}\\
% % \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_1> K^{(k)} & \dots & \sum_{k=0}^{P_d} <\Phi_k \Psi_P \Psi_1> K^{(k)}\\ \\
% % \vdots & \vdots & \vdots & \vdots\\
% % \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_P> K^{(k)} & \dots & \sum_{k=0}^{M} <\Phi_k \Psi_P \Psi_P> K^{(k)}
% % \end{bmatrix}
% % \begin{bmatrix}
% % u_{10} \\
% % \vdots \\
% % u_{N0}\\
% % \vdots \\
% % u_{1P_u}\\
% % \vdots \\
% % u_{NP_u}
% % \end{bmatrix}
% % =
% % %\]
% % %\[
% % \begin{bmatrix}
% % \sum_{i=0}^{P_f} f_i <\Psi_0\zeta_i> \\
% % \sum_{i=0}^{P_f} f_i <\Psi_1\zeta_i> \\
% % \sum_{i=0}^{P_f} f_i <\Psi_2\zeta_i> \\
% % \vdots \\
% % \sum_{i=0}^{P_f} f_i <\Psi_{P_u}\zeta_i>\\
% % \end{bmatrix}
% % \]
% % \end{tiny}
% %
% %
% %
% % % \normalsize{Typical number of terms required for a SEPFEM problem} \vspace{1cm}\\
% % \scalebox{0.7}{
% % \begin{tabular}{ c c c c}
% % \# KL terms material & \# KL terms load & PC order displacement& Total \# terms per DoF\\ \hline
% % 4 & 4 & 10 & 43758 \\
% % 4 & 4 & 20 & 3 108 105 \\
% % % 4 & 4 & 30 & 48 903 492 \\
% % 6 & 6 & 10 & 646 646 \\
% % % 6 & 6 & 20 & 225 792 840 \\
% % % 6 & 6 & 30 & 1.1058 $10^{10}$ \\
% % % 8 & 8 & 10 & 5 311 735 \\
% % % 8 & 8 & 20 & 7.3079 $10^{9}$ \\
% % % 8 & 8 & 30 & 9.9149 $10^{11}$\\
% %
% % ... & ... & ... & ...\\ \hline
% % \end{tabular}}
% %
% %
% % \end{frame}
% %
% %
% %
% %
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% % \begin{frame}
% %
% % \frametitle{Monte Carlo (MC)}
% %
% % \begin{itemize}
% %
% % \item Monte Carlo simulations: nonintrusive approach
% %
% % \begin{itemize}
% % \item [] \small Slow convergence rate $1/\sqrt{N}$
% % \item [] \small Hard for stable tail distribution toward lowrisk level
% % \end{itemize}
% %
% % \item Fragility curve: incremental dynamic analysis (IDA)
% %
% % \begin{itemize}
% % \item [] \small Impractical for large $3D$ nonlinear ESSI system
% % \end{itemize}
% %
% % \item Uncertain seismic wave propagation over regional geology
% %
% % \begin{itemize}
% % \item [] \small CyberShake from SCEC
% % \item [] \small Los Angeles, over 20,000 scenarios within 200 km, \textbf{300 million CPUhours and over 100 years} (Maechling et al. 2007)
% % \end{itemize}
% %
% % \end{itemize}
% % \end{frame}
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%
% %\subsection{TDNIPSRA Example}
% \subsection[TDNIPSRA Example]{Example}
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begingroup
\setbeamertemplate{footline}{}
\begin{frame}
%\frametitle{TDNIPSRA Framework}
\frametitle{Application: Seismic Hazard}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{textblock}{15}(0, 4.0)
\includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/UCERF3.pdf}
\end{textblock}
\begin{textblock}{15}(0.3, 3.5)
\scriptsize{Seismic source characterization}
\end{textblock}
\begin{textblock}{15}(2.9, 5.2)
\tiny{UCERF3 (2014)}
\end{textblock}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{textblock}{15}(5.1, 6.5)
%$\Rightarrow$
{\Large $\rightarrow$}
\end{textblock}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{textblock}{15}(5.8, 3.9)
\vspace*{1mm}
\includegraphics[width=0.27\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/SMSIM.pdf}
\end{textblock}
\begin{textblock}{15}(7.1, 6.2)
\scalebox{.9}{\tiny{Fourier spectra}}
\\
\vspace*{0.2cm}
\scalebox{.9}{\tiny{\hspace{0.14cm} Boore(2003)}}
\end{textblock}
\begin{textblock}{15}(6.1, 3.5)
\scriptsize{Stochastic ground motion}
\end{textblock}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{textblock}{15}(9.9, 6.5)
%{\bf $\Rightarrow$}
{\Large $\rightarrow$}
\end{textblock}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{textblock}{15}(10.5, 4.2)
% \includegraphics[width=0.35\linewidth]{pic/KL_exact_dis_correlation_from_dis.pdf}
\includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Acc_realization_200.pdf}
\end{textblock}
\begin{textblock}{15}(11.1, 9.6)
\scriptsize{Uncertainty characterization \\
\hspace{0.1cm} Hermite polynomial chaos}
\end{textblock}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{textblock}{15}(10.2, 13.2)
{\Large $\leftarrow$}
\end{textblock}
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\begin{textblock}{15}(11, 11.2)
\includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/structural_uncertainty.pdf}
\end{textblock}
\begin{textblock}{15}(5.3, 10.75)
\includegraphics[width=0.33\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/probabilsitc_evolution.png}
\end{textblock}
\begin{textblock}{15}(5.4, 9.6)
\scriptsize{\quad \quad Uncertainty propagation \\
\quad \quad \quad \quad SEPFEM}
\end{textblock}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{textblock}{15}(4.6, 13.2)
%$\Leftarrow$
{\Large $\leftarrow$}
\end{textblock}
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\begin{textblock}{15}(0.3, 11.0)
\includegraphics[width=0.29\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/seismic_risk_result_framework.png}
\end{textblock}
\begin{tikzpicture}[remember picture, overlay]
\draw[line width=1pt, draw=black, rounded corners=4pt, fill=gray!20, fill opacity=1]
([xshift=25pt,yshift=55pt]$(pic cs:a) + (0pt,8pt)$) rectangle ([xshift=95pt,yshift=18pt]$(pic cs:b)+(0pt,2pt)$);
\end{tikzpicture}
\begin{textblock}{15}(0.1, 9.3)
\scriptsize
\quad \quad \quad \quad $\lambda(EDP>z)=$
$\quad \sum N_i(M_i, R_i) P(EDP>zM_i, R_i)$
\end{textblock}
\begin{textblock}{15}(1.6, 10.7)
\scriptsize{EDP hazard/risk}
\end{textblock}
\end{frame}
\endgroup
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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begingroup
%
% \setbeamertemplate{footline}{}
%
% \begin{frame}
%
% \frametitle{Time Domain Intrusive PSRA Framework}
%
%
% \begin{textblock}{15}(0, 4.0)
% \includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/UCERF3.pdf}
% \end{textblock}
%
% \begin{textblock}{15}(0.3, 3.5)
% \scriptsize{Seismic source characterization}
% \end{textblock}
%
% \begin{textblock}{15}(2.9, 5.2)
% \tiny{UCERF3 (2014)}
% \end{textblock}
%
% \begin{textblock}{15}(5.1, 6.5)
% $\Rightarrow$
% \end{textblock}
%
%
% \begin{textblock}{15}(5.8, 3.9)
% \vspace*{1mm}
% \includegraphics[width=0.27\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/SMSIM.pdf}
% \end{textblock}
%
%
% \begin{textblock}{15}(7.1, 6.2)
% \scalebox{.9}{\tiny{Fourier spectra}}
% \\
% \vspace*{0.2cm}
% \scalebox{.9}{\tiny{\hspace{0.14cm} Boore(2003)}}
% \end{textblock}
%
% \begin{textblock}{15}(6.1, 3.5)
% \scriptsize{Stochastic ground motion}
% \end{textblock}
%
% \begin{textblock}{15}(9.9, 6.5)
% $\Rightarrow$
% \end{textblock}
%
% \begin{textblock}{15}(10.5, 4.2)
% % \includegraphics[width=0.35\linewidth]{pic/KL_exact_dis_correlation_from_dis.pdf}
% \includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Acc_realization_200.pdf}
% \end{textblock}
%
% \begin{textblock}{15}(11.1, 9.6)
% \scriptsize{Uncertainty characterization \\
% \hspace{0.1cm} Hermite polynomial chaos}
% \end{textblock}
%
% \begin{textblock}{15}(11, 11.2)
% \includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/structural_uncertainty.pdf}
% \end{textblock}
%
% % \begin{textblock}{15}(10.2, 13.2)
% % $\Leftarrow$
% % \end{textblock}
%
% % \begin{textblock}{15}(5.3, 10.75)
% % \includegraphics[width=0.33\linewidth]{pic/probabilsitc_evolution.png}
% % \end{textblock}
%
% % \begin{textblock}{15}(5.4, 9.6)
% % \scriptsize{\quad \quad Uncertainty propagation \\
% % \quad \quad \quad \quad stochastic FEM}
% % \end{textblock}
%
% % \begin{textblock}{15}(4.6, 13.2)
% % $\Leftarrow$
% % \end{textblock}
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% % \begin{textblock}{15}(0.3, 11.0)
% % \includegraphics[width=0.29\linewidth]{pic/seismic_risk_result_framework.png}
% % \end{textblock}
%
% % \begin{tikzpicture}[remember picture, overlay]
% % \draw[line width=1pt, draw=black, rounded corners=4pt, fill=gray!20, fill opacity=1]
% % ([xshift=25pt,yshift=52pt]$(pic cs:a) + (0pt,8pt)$) rectangle ([xshift=95pt,yshift=18pt]$(pic cs:b)+(0pt,2pt)$);
% % \end{tikzpicture}
%
%
% % \begin{textblock}{15}(0.1, 9.3)
% % \scriptsize
% % \quad \quad \quad \quad $\lambda(EDP>z)=$
%
% % $\quad \sum N_i(M_i, R_i) P(EDP>zM_i, R_i)$
% % \end{textblock}
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% % \begin{textblock}{15}(1.6, 10.7)
% % \scriptsize{EDP hazard/risk}
% % \end{textblock}
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% \end{frame}
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% \begingroup
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% \begin{frame}
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% \frametitle{Time Domain Intrusive PSRA Framework}
%
%
% \begin{textblock}{15}(0, 4.0)
% \includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/UCERF3.pdf}
% \end{textblock}
%
% \begin{textblock}{15}(0.3, 3.5)
% \scriptsize{Seismic source characterization}
% \end{textblock}
%
% \begin{textblock}{15}(2.9, 5.2)
% \tiny{UCERF3 (2014)}
% \end{textblock}
%
% \begin{textblock}{15}(5.1, 6.5)
% $\Rightarrow$
% \end{textblock}
%
%
% \begin{textblock}{15}(5.8, 3.9)
% \vspace*{1mm}
% \includegraphics[width=0.27\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/SMSIM.pdf}
% \end{textblock}
%
%
% \begin{textblock}{15}(7.1, 6.2)
% \scalebox{.9}{\tiny{Fourier spectra}}
% \\
% \vspace*{0.2cm}
% \scalebox{.9}{\tiny{\hspace{0.14cm} Boore(2003)}}
% \end{textblock}
%
% \begin{textblock}{15}(6.1, 3.5)
% \scriptsize{Stochastic ground motion}
% \end{textblock}
%
% \begin{textblock}{15}(9.9, 6.5)
% $\Rightarrow$
% \end{textblock}
%
% \begin{textblock}{15}(10.5, 4.2)
% % \includegraphics[width=0.35\linewidth]{pic/KL_exact_dis_correlation_from_dis.pdf}
% \includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Acc_realization_200.pdf}
% \end{textblock}
%
% \begin{textblock}{15}(11.1, 9.6)
% \scriptsize{Uncertainty characterization \\
% \hspace{0.1cm} Hermite polynomial chaos}
% \end{textblock}
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%
% \begin{textblock}{15}(10.2, 13.2)
% $\Leftarrow$
% \end{textblock}
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% \begin{textblock}{15}(11, 11.2)
% \includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/structural_uncertainty.pdf}
% \end{textblock}
%
% \begin{textblock}{15}(5.3, 10.75)
% \includegraphics[width=0.33\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/probabilsitc_evolution.png}
% \end{textblock}
%
% \begin{textblock}{15}(5.4, 9.6)
% \scriptsize{\quad \quad Uncertainty propagation \\
% \quad \quad \quad \quad stochastic FEM}
% \end{textblock}
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% % \begin{textblock}{15}(4.6, 13.2)
% % $\Leftarrow$
% % \end{textblock}
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% % \begin{textblock}{15}(0.3, 11.0)
% % \includegraphics[width=0.29\linewidth]{pic/seismic_risk_result_framework.png}
% % \end{textblock}
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% % \begin{tikzpicture}[remember picture, overlay]
% % \draw[line width=1pt, draw=black, rounded corners=4pt, fill=gray!20, fill opacity=1]
% % ([xshift=25pt,yshift=52pt]$(pic cs:a) + (0pt,8pt)$) rectangle ([xshift=95pt,yshift=18pt]$(pic cs:b)+(0pt,2pt)$);
% % \end{tikzpicture}
%
%
% % \begin{textblock}{15}(0.1, 9.3)
% % \scriptsize
% % \quad \quad \quad \quad $\lambda(EDP>z)=$
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% % $\quad \sum N_i(M_i, R_i) P(EDP>zM_i, R_i)$
% % \end{textblock}
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% % \begin{textblock}{15}(1.6, 10.7)
% % \scriptsize{EDP hazard/risk}
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% \begingroup
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% \setbeamertemplate{footline}{}
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% \begin{frame}
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% \frametitle{Time Domain Intrusive PSRA Framework}
%
%
% \begin{textblock}{15}(0, 4.0)
% \includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/UCERF3.pdf}
% \end{textblock}
%
% \begin{textblock}{15}(0.3, 3.5)
% \scriptsize{Seismic source characterization}
% \end{textblock}
%
% \begin{textblock}{15}(2.9, 5.2)
% \tiny{UCERF3 (2014)}
% \end{textblock}
%
% \begin{textblock}{15}(5.1, 6.5)
% $\Rightarrow$
% \end{textblock}
%
%
% \begin{textblock}{15}(5.8, 3.9)
% \vspace*{1mm}
% \includegraphics[width=0.27\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/SMSIM.pdf}
% \end{textblock}
%
%
% \begin{textblock}{15}(7.1, 6.2)
% \scalebox{.9}{\tiny{Fourier spectra}}
% \\
% \vspace*{0.2cm}
% \scalebox{.9}{\tiny{\hspace{0.14cm} Boore(2003)}}
% \end{textblock}
%
% \begin{textblock}{15}(6.1, 3.5)
% \scriptsize{Stochastic ground motion}
% \end{textblock}
%
% \begin{textblock}{15}(9.9, 6.5)
% $\Rightarrow$
% \end{textblock}
%
% \begin{textblock}{15}(10.5, 4.2)
% % \includegraphics[width=0.35\linewidth]{pic/KL_exact_dis_correlation_from_dis.pdf}
% \includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Acc_realization_200.pdf}
% \end{textblock}
%
% \begin{textblock}{15}(11.1, 9.6)
% \scriptsize{Uncertainty characterization \\
% \hspace{0.1cm} Hermite polynomial chaos}
% \end{textblock}
%
%
% \begin{textblock}{15}(10.2, 13.2)
% $\Leftarrow$
% \end{textblock}
%
% \begin{textblock}{15}(11, 11.2)
% \includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/structural_uncertainty.pdf}
% \end{textblock}
%
% \begin{textblock}{15}(5.3, 10.75)
% \includegraphics[width=0.33\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/probabilsitc_evolution.png}
% \end{textblock}
%
% \begin{textblock}{15}(5.4, 9.6)
% \scriptsize{\quad \quad Uncertainty propagation \\
% \quad \quad \quad \quad stochastic FEM}
% \end{textblock}
%
% \begin{textblock}{15}(4.6, 13.2)
% $\Leftarrow$
% \end{textblock}
%
% \begin{textblock}{15}(0.3, 11.0)
% \includegraphics[width=0.29\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/seismic_risk_result_framework.png}
% \end{textblock}
%
% \begin{tikzpicture}[remember picture, overlay]
% \draw[line width=1pt, draw=black, rounded corners=4pt, fill=gray!20, fill opacity=1]
% ([xshift=25pt,yshift=52pt]$(pic cs:a) + (0pt,8pt)$) rectangle ([xshift=95pt,yshift=18pt]$(pic cs:b)+(0pt,2pt)$);
% \end{tikzpicture}
%
%
% \begin{textblock}{15}(0.1, 9.25)
% \scriptsize
% \quad \quad \quad \quad $\lambda(EDP>z)=$
%
% $\quad \sum N_i(M_i, R_i) P(EDP>zM_i, R_i)$
% \end{textblock}
%
% \begin{textblock}{15}(1.6, 10.7)
% \scriptsize{EDP hazard/risk}
% \end{textblock}
%
% \end{frame}
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% \endgroup
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% \begin{frame}
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% \frametitle{Stochastic Ground Motion Modeling}
%
% % \vspace{0.4cm}
%
% \begin{itemize}
%
% %\item[] \normalsize Shift from modeling specific IM to fundamental characteristics of ground motions
% \item[] \normalsize Shift from modeling specific IM to fundamental characteristics of ground motions
%
% \begin{itemize}
%
% \item[] \normalsize Uncertain Fourier amplitude spectra (FAS)
%
% \item[] \normalsize Uncertain Fourier phase spectra (FPS)
%
% \end{itemize}
%
% % \item \scriptsize Mean behavior of stochastic FAS
%
% % \begin{itemize}
%
% % \item[] \scriptsize $w^2$ source radiation spectrum by \textit{Brune(1970)}
%
% % \item[] \scriptsize Systematic studies by \textit{ \textbf{Boore}(1983, \textbf{2003}, 2015)}.
%
% % \end{itemize}
%
% \vspace{0.05cm}
%
% %\item[] \normalsize Recent GMPE study of FAS,
% %(FAS marginal median \& variability GMPEs by \textit{{Bora et al.
% %(2018)}} and {\textit{Bayless \& Abrahamson (2019)}} ;
% %FAS Interfrequency correlation GMPE by \textit{Stafford(2017)} and
% %{\textit{Bayless \& Abrahamson (2018)}})
%
%
% \vspace*{1mm}
% \item[] \normalsize GMPE studies of FAS,
% (
% \textit{{Bora et al. (2018)}},
% \textit{Bayless \& Abrahamson (2018,2019)},
% \textit{Stafford(2017)},
% %{\textit{Bayless \& Abrahamson (2018)}
% )
%
% % \begin{itemize}
% %
% % \item[] \scriptsize FAS marginal median \& variability GMPEs by \textit{\textbf{Bora et al. (2018)}} and \textbf{\textit{Bayless \& Abrahamson (2019)}}
% %
% % %\vspace{0.1cm}
% %
% % \item[] \scriptsize FAS Interfrequency correlation GMPE by \textit{Stafford(2017)} and \textbf{\textit{Bayless \& Abrahamson (2018)}}.
% %
% % \end{itemize}
%
%
% %\vspace{0.05cm}
%
% %\item[] \normalsize Stochastic FPS by phase derivative (Boore,2005)
% %(Logistic phase derivative model by {\textit{Baglio \& Abrahamson (2017)}})
% \vspace*{1mm}
% \item[] \normalsize Stochastic FPS by phase derivative (Boore,2005)
% (Logistic phase derivative model by {\textit{Baglio \& Abrahamson (2017)}})
%
% \vspace*{1mm}
% \item[] \normalsize Near future change from \textbf{ $\boldsymbol{Sa(T_0)}$} to \textbf{FAS} and \textbf{FPS}
% %next five years
% % as envisioned by Abrahamson (2018)
%
% \end{itemize}
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% %\subsection{Illustrative Example}
% %
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\begin{frame}
%\frametitle{TDNIPSRA Example Object}
\frametitle{Example Object}
\begin{textblock}{15}(0.5, 4.0)
\includegraphics[width=0.47\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/faults_configuration_new.pdf}
\end{textblock}
\begin{textblock}{15}(7.5, 3.4)
\includegraphics[width=0.55\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/SSC_legend.pdf}
\end{textblock}
\begin{textblock}{15}(0.8, 11.6)
\scriptsize
\begin{itemize}
\item Fault 1: San Gregorio fault
\item Fault 2: Calaveras fault
\item Uncertainty: Segmentation, \\ slip rate, rupture geometry, etc.
\end{itemize}
\end{textblock}
\begin{textblock}{15}(8.5, 11.6)
\scriptsize
\begin{itemize}
\item 371 total seismic scenarios
\item $M \ 5 \sim 5.5$ and $6.5 \sim 7.0$
\item $R_{jb} \ 20km \sim 40km$
\end{itemize}
\end{textblock}
\end{frame}
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\begin{frame}
\frametitle{Stochastic Ground Motion Modeling}
\begin{textblock}{15}(1.0, 3.7)
\small Realizations of simulated uncertain motions for scenario $M=7$, $R=15km$:
\end{textblock}
\begin{textblock}{15}(0.5, 4.0)
\includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Acc_time_series100.pdf}
\end{textblock}
\begin{textblock}{15}(5.5, 4.0)
\includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Acc_time_series343.pdf} \enspace
\end{textblock}
\begin{textblock}{15}(10.5, 4.0)
\includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Acc_time_series439.pdf}
\end{textblock}
\begin{textblock}{15}(1.0, 9.2)
\small Verification with GMPE:
\end{textblock}
\begin{textblock}{15}(0.3, 9.5)
\includegraphics[width=0.36\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/SA_GMPE_verification_std_08_no_smooth.pdf}
\end{textblock}
\begin{textblock}{15}(5.6, 9.5)
\includegraphics[width=0.36\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Goodness_fit_std_08_no_smooth.pdf}
\end{textblock}
\begin{textblock}{15}(10.8, 9.5)
\includegraphics[width=0.36\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/Standard_deviation_std_08_no_smooth_new.pdf}
\end{textblock}
% \begin{textblock}{15}(0.5, 11.0)
% \begin{itemize}
% \item $\Delta \sigma= 84bar$, $\kappa=0.03s$ with total $\sigma=0.8ln$.
% \item Simulated median is not biased.
% \item Consistent total uncertainties with GMPE.
% \end{itemize}
% \end{textblock}
\end{frame}
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\begin{frame}
\frametitle{Stochastic Ground Motion Characterization}
{\begin{textblock}{15}(0.1, 3.62)
\scriptsize
\includegraphics[width=0.3\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/KL_mean_acc_from_acc.pdf}
\quad \quad \quad Acc. marginal mean
\end{textblock}
\begin{textblock}{15}(3.7, 3.62)
\scriptsize
\includegraphics[width=0.3\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/KL_var_acc_from_acc.pdf}
\quad \quad \quad Acc. marginal S.D.
\end{textblock}
\begin{textblock}{15}(7.6, 3.8)
\scriptsize
\includegraphics[width=0.3\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/KL_exact_acc_correlation_from_acc.pdf}
\quad \quad \quad Acc. realization Cov.
\end{textblock}
\begin{textblock}{15}(11.8, 3.9)
\scriptsize
\includegraphics[width=0.3\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/KL_simulated_acc_correlation_from_acc.pdf}
\quad \quad Acc. synthesized Cov.
\end{textblock}}
\begin{textblock}{15}(0.1, 9.3)
\scriptsize
\includegraphics[width=0.31\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/KL_mean_dis_from_dis.pdf}
\end{textblock}
\begin{textblock}{15}(0.9, 13.75)
\scriptsize
Dis. marginal mean
\end{textblock}
\begin{textblock}{15}(4.2, 9.4)
\scriptsize
\includegraphics[width=0.3\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/KL_var_dis_from_dis.pdf}
\end{textblock}
\begin{textblock}{15}(5.1, 13.75)
\scriptsize
Dis. marginal S.D.
\end{textblock}
\begin{textblock}{15}(8.2, 9.5)
\scriptsize
\includegraphics[width=0.27\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/KL_exact_dis_correlation_from_dis.pdf}
\quad \quad Dis. realization Cov.
\end{textblock}
\begin{textblock}{15}(12.2, 9.6)
\scriptsize
\includegraphics[width=0.27\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/KL_simulated_dis_correlation_from_dis.pdf}
\quad Dis. synthesized Cov.
\end{textblock}
\end{frame}
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\begin{frame}
\frametitle{Stochastic Soil, Structure Modeling}
%Uncertain 1D shear response
\begin{figure}[!htbp]
\centering
%\subfloat[Uncertain $H_a$]{
%\hspace{0.8cm}
%\includegraphics[width=0.53\textwidth]{/home/jeremic/tex/works/Papers/2019/Hexiang/1D_risk/version6/Figures/constitutive_relation_uncertainHa_certainCr_MC_verification.pdf}}
%\subfloat[Uncertain $H_a$ and $C_r$]{
%\hspace{0.2cm}
%\includegraphics[width=0.53\textwidth]{/home/jeremic/tex/works/Papers/2019/Hexiang/1D_risk/version6/Figures/constitutive_relation_uncertainHa_uncertainCr_MC_verification.pdf}}
%\vspace{2mm}
%\caption{\label{figure_probabilisitc_constitutive_relation} Intrusive probabilistic modeling of ArmstrongFrederick hysteretic behavior and verification with Monte Carlo simulation: (a) Gaussian distributed $Ha$ with mean 1.76 $\times 10^{7} \ N/m$ and 15\% coefficient of variation (COV), $C_r = 17.6$. (b) Gaussian distributed $Ha$ with mean 1.76 $\times 10^{7} \ N/m$ and 15\% coefficient of variation (COV), Gaussian distributed $C_r$ with mean 17.6 and 15\% COV.}
\subfloat[Frame]{
\hspace{0.8cm}
\includegraphics[width=2.5cm]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/ShearFrame8levels.jpg}}
\subfloat[Interstory response]{
\hspace{10mm}
\includegraphics[width=6cm]{/home/jeremic/tex/works/Papers/2019/1D_risk/version6/Figures/constitutive_relation_uncertainHa_uncertainCr_MC_verification.pdf}}
%\vspace{2mm}
%\caption{\label{figure_probabilisitc_constitutive_relation} Intrusive probabilistic modeling of ArmstrongFrederick hysteretic behavior and verification with Monte Carlo simulation: (a) Gaussian distributed $Ha$ with mean 1.76 $\times 10^{7} \ N/m$ and 15\% coefficient of variation (COV), $C_r = 17.6$. (b) Gaussian distributed $Ha$ with mean 1.76 $\times 10^{7} \ N/m$ and 15\% coefficient of variation (COV), Gaussian distributed $C_r$ with mean 17.6 and 15\% COV.}
\end{figure}
%
\end{frame}
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% \begin{frame}
%
% \frametitle{Probabilistic Dynamic Structural Response}
%
% \begin{textblock}{15}(0.7, 4.5)
% \scriptsize
% \includegraphics[width=0.46\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/shear_frame_illustration_update.pdf}
% \end{textblock}
%
% \begin{textblock}{15}(0.2, 10.8)
% \begin{itemize}
% \scriptsize \item Coefficient of variation 15$\%$ for $H_a$ and $C_r$
% %\scriptsize \item Exponential correlation with correlation \\
% %length $l_c = 10$ floors
% \scriptsize \item Time domain stochastic \\
% ElPl FEM analysis (SEPFEM)
% % : uncertain \\ structure with uncertain excitations
% \end{itemize}
% \end{textblock}
%
% \begin{textblock}{15}(7.7, 5)
% \scriptsize
% \includegraphics[width=0.52\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Probabilistic_Response_Node_1_new.pdf}
% \end{textblock}
%
% \begin{textblock}{15}(8.3, 4.2)
% \scriptsize Probabilistic response of top floor from SFEM
% \end{textblock}
% \end{frame}
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\begingroup
\setbeamertemplate{footline}{}
\begin{frame}
\frametitle{Seismic Risk Analysis}
\begin{textblock}{15}(1.9,3.8)
\scriptsize
\includegraphics[width=0.42\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/MIDR_PDF_evolution.pdf}
\includegraphics[width=0.42\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/PDF_MIDR_combine.pdf}
\end{textblock}
\begin{textblock}{15}(1.9,9.5)
\scriptsize
\includegraphics[width=0.42\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/MIDR_distribution_different_floors.pdf}
\includegraphics[width=0.42\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/Risk_MIDR.pdf}
\end{textblock}
\begin{textblock}{15}(0.8, 3.8)
\scriptsize Engineering demand parameter (EDP): Maximum interstory drift ratio (MIDR)
\end{textblock}
\end{frame}
\endgroup
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\begingroup
\setbeamertemplate{footline}{}
\begin{frame}
\frametitle{Seismic Risk Analysis}
\begin{textblock}{15}(1.9,3.8)
\scriptsize
\includegraphics[width=0.42\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/PFA_distribution.pdf}
\includegraphics[width=0.42\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/Risk_PFA.pdf}
\end{textblock}
\begin{textblock}{15}(1.9,9.3)
\scriptsize
\includegraphics[width=0.41\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/2D_EDP_PDF_1e5.pdf}
\end{textblock}
\begin{textblock}{15}(8.7,9.4)
\scriptsize
\includegraphics[width=0.40\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/2D_EDP_PDF_downview_1e5.pdf}
\end{textblock}
\begin{textblock}{15}(0.8, 3.8)
\scriptsize Engineering demand parameter (EDP): Peak floor acceleration (PFA)
\end{textblock}
\end{frame}
\endgroup
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\begin{frame}
\frametitle{Seismic Risk Analysis}
%\vspace{0.5cm}
\vspace*{2mm}
\begin{itemize}
% \item[] \small Damage measure (DM) defined on multiple EDPs:
% \item[] \small Damage measure (DM) defined on single EDP:
\item[] Damage measure defined on single EDP:
\vspace*{3mm}
%\begin{textblock}{15}(0.7,8.9)
\begin{table}[!htbp]
\small
\resizebox{0.98\hsize}{!}{
\begin{tabular}{ccccccc}
%\hline
\textbf{DM} & MIDR\textgreater{}0.5\% & \textbf{MIDR\textgreater{}1\%} & MIDR\textgreater{}2\% & PFA\textgreater{}0.5${\rm m/s^2}$ & \textbf{PFA\textgreater{}1\boldsymbol{${\rm m/s^2}$}} & PFA\textgreater{}1.5${\rm m/s^2}$ \\
\hline
\textbf{Risk [/yr]} & 6.66$\times 10^{3}$ & \textbf{3.83\boldsymbol{$\times 10^{3}$}} & 9.97$\times 10^{5}$ & 6.65$\times 10^{3}$ & \textbf{1.92 \boldsymbol{$\times 10^{3}$}} & 9.45$\times 10^{5}$ \\
%\hline
\end{tabular}}
\end{table}
%\end{textblock}
\vspace{4mm}
\item[] Damage measure (DM) defined on multiple EDPs:
% \vspace{2mm}
{\scriptsize $DM: \{\text{MIDR}>1\%\, \cup \,\text{PFA}>1{\rm m/s^2} \}$, seismic risk is \boldsymbol{$4.2 \times 10^{3}/yr$} }
\vspace{1mm}
{\scriptsize $DM: \{\text{MIDR}>1\%\, \cap \,\text{PFA}>1{\rm m/s^2} \}$, seismic risk is \boldsymbol{$1.71 \times 10^{3}/yr$}}
\vspace{3mm}
%\vspace{20mm}
\vspace{4mm}
\item[] \small Seismic risk for DM defined on multiple EDPs can be quite
different from that defined on single EDP
\end{itemize}
\end{frame}
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% \item[] Incremental elpl constitutive equation
% %
% \begin{eqnarray}
% \nonumber
% \Delta \sigma_{ij}
% =
% % E^{EP}_{ijkl}
% E^{EP}_{ijkl} \; \Delta \epsilon_{kl}
% =
% \left[
% 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_*}
% \right]
% \Delta \epsilon_{kl}
% \end{eqnarray}
%
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% \vspace*{2mm}
% \item[] Dynamic Finite Elements
% %
% \begin{equation}
% { M} \ddot{ u_i} +
% { C} \dot{ u_i} +
% { K}^{ep} { u_i} =
% { F(t)}
% \nonumber
% \end{equation}
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% \vspace*{2mm}
% \item[] Material and loads are uncertain
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% \includegraphics[width=6.0cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourLowOCR_RandomG_RandomM_Randomp0m.pdf}
% %\hspace*{2mm}
% \includegraphics[width=6.0cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourHighOCR_RandomG_RandomMm.pdf}
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% Dynamic Finite Elements
% $
% { M} \ddot{ u_i} +
% { C} \dot{ u_i} +
% { K}^{ep} { u_i} =
% { F(t)}$
%
%
% \begin{itemize}
% \vspace*{2mm}
% \item[] Input random field/process{\normalsize{(nonGaussian, heterogeneous/ nonstationary)}}:
% Multidimensional Hermite Polynomial Chaos (PC) with {known coefficients}
% %\vspace{0.05in}
% \vspace*{2mm}
% \item[] Output response process: Multidimensional Hermite PC with {unknown coefficients}
% % \vspace{0.05in}
% \vspace*{2mm}
% \item[] Galerkin projection: minimize the error to compute unknown coefficients of response process
% % %\vspace{0.05in}
% % \vspace*{2mm}
% % \item[] Time integration using Newmark's method
% % % : Update coefficients following
% % % an elasticplastic constitutive law at each time step
%
% \end{itemize}
%
% %\scriptsize
% %Note: PC = Polynomial Chaos
%
% \end{frame}
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% %
% % \begin{frame}{Discretization of Input Random Process/Field $\beta(x,\theta)$}
% % \begin{center}
% % \includegraphics[scale=0.35]{/home/jeremic/tex/works/Thesis/FangboWang/slides_13Mar2019/Fangbo_slides/figs/PC_KL_explanation.PNG} \\
% % \end{center}
% %
% %
% % \footnotesize{Note: $\beta(x,\theta)$ is an input random process with any
% % marginal distribution, \\ \hspace{21mm} with any covariance structure;} \\
% % \footnotesize{\hspace{8mm} $\gamma(x,\theta)$ is a zeromean unitvariance Gaussian random process.} \\
% %
% % \end{frame}
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% \begin{frame}{Polynomial Chaos Representation}
%
% %\scriptsize{
% Material random field: \\
% %\vspace{0.3cm}
% %\begin{equation*}
% $D(x, \theta)= \sum_{i=1}^{P1} a_i(x) \Psi_i(\left\{\xi_r(\theta)\right\})$
% %\end{equation*}
%
%
% \vspace{4mm}
%
% Seismic loads/motions random process: \\
% %\vspace{0.3cm}
% %\begin{equation*}
% $f_m(t, \theta)=\sum_{j=1}^{P_2} f_{mj}(t) \Psi_j(\{\xi_k(\theta)\})$
% %\end{equation*}
%
% \vspace{4mm}
%
% Displacement response: \\
% %\vspace{0.3cm}
% %\begin{equation*}
% $u_n(t, \theta)=\sum_{k=1}^{P_3} d_{nk}(t) \Psi_k(\{\xi_l(\theta)\})$
% %\end{equation*}
%
% \vspace{3mm}
%
% %Acceleration response:
% %%\vspace{0.3cm}
% %%\begin{equation*}
% %$\ddot u_n(t, \theta)=\sum_{k=1}^{P_3} \ddot d_{nk}(t) \Psi_k(\{\xi_l(\theta)\})$
% %%\end{equation*}
%
% %\vspace{3mm}
% \vspace{5mm}
%
% where $a_i(x), f_{mj}(t)$ are {known PC coefficients}, while $d_{nk}(t)$
% are {unknown PC coefficients}.
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% % \subsection{Direct Solution for Probabilistic Stiffness and Stress in 1D}
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% \begin{frame}{Direct Probabilistic Constitutive Solution in 1D}
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% % \begin{itemize}
% %
% % \vspace{0.5cm}
% %
% % \item<1> Probabilistic constitutive modeling : \vspace{0.5cm}
%
% \begin{itemize}
%
%
% \vspace*{4mm}
% \item[] Zero elastic region elastoplasticity with stochastic ArmstrongFrederick
% kinematic hardening
%
% $ \Delta\sigma =\ H_a \Delta \epsilon  c_r \sigma \Delta \epsilon ;
% \hspace{0.5cm}
% E_t = {d\sigma}/{d\epsilon} = H_a \pm c_r \sigma $
%
% \vspace*{4mm}
% \item[] Uncertain:
% init. stiff. $H_a$,
% shear strength $H_a/c_r$,
% strain $\Delta \epsilon$:
%
% $ H_a = \Sigma h_i \Phi_i; \;\;\;
% C_r = \Sigma c_i \Phi_i; \;\;\;
% \Delta\epsilon = \Sigma \Delta\epsilon_i \Phi_i $
%
%
%
% \vspace*{4mm}
% \item[] Resulting stress and stiffness are also uncertain
%
% % 
% %  $ \sum_{l=1}^{P_{\sigma}} \Delta\sigma_i \Phi_i = \sum_{i=1}^{P_h} \sum_{k=1}^{P_e}\ h_i \Delta \epsilon_k \Phi_i \Phi_k  \sum_{j=1}^{P_g} \sum_{k=1}^{P_e}\sum_{l=1}^{P_{\sigma}} \ c_i \Delta \epsilon_k \sigma_l \Phi_j \Phi_k \Phi_l$
% % 
% %  $ \sum_{l=1}^{P_{E_t}} \Delta E_{t_i} \Phi_i = \sum_{i=1}^{P_h} h_i \Phi_i \pm \sum_{i=1}^{P_c} \sum_{l=1}^{P_{\sigma}} \ c_i \sigma_l \Phi_i \Phi_l$
% % 
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% %\item<1> Time integration is done via Newmark algorithm
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% \begin{frame}{Direct Probabilistic Stiffness Solution}
%
% \begin{itemize}
%
%
% \item[] Analytic product for all the components,
%
% $ E^{EP}_{ijkl}
% =
% \left[
% 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_*}
% \right]
% $
%
%
%
%
% \vspace*{2mm}
% \item[] Stiffness: each Polynomial Chaos component is updated incrementally
% % at each Gauss Point via stochastic Galerkin projection
%
%
%
% \small{$E_{t_1}^{n+1} = \frac{1}{<\Phi_1\Phi_1> }\{\sum_{i=1}^{P_h} \ h_i <\Phi_i \Phi_1> \pm \sum_{j=1}^{P_c} \sum_{l=1}^{P_{\sigma}} \ c_j \sigma_l^{n+1} <\Phi_j \Phi_l \Phi_1>\}$}
% \\
% . . .
% %
% %
% % $\large{\vdots}$
% \\
% \small{$E_{t_P}^{n+1} = \frac{1}{<\Phi_1\Phi_P> }\{\sum_{i=1}^{P_h} \ h_i <\Phi_i \Phi_P> \pm \sum_{j=1}^{P_c} \sum_{l=1}^{P_{\sigma}} \ c_j \sigma_l^{n+1} <\Phi_j \Phi_l \Phi_P>\}$}
%
%
% \vspace*{2mm}
% \item[] Total stiffness is :
%
% $ E_{t}^{n+1} = \sum_{l=1}^{P_{E}} E_{t_i}^{n+1} \Phi_i $
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% \begin{frame}{Direct Probabilistic Stress Solution}
%
% \begin{itemize}
%
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%
% \item[] Analytic product, for each stress component,
%
% $ \Delta \sigma_{ij} = E^{EP}_{ijkl} \; \Delta \epsilon_{kl} $
% % =
% % \left[
% % 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_*}
% % \right]
% % \Delta \epsilon_{kl}
% %
%
%
% \vspace*{2mm}
% \item[] Incremental stress: each Polynomial Chaos component is updated
% incrementally
% % via stochastic Galerkin projection
%
%
%
%
% {$\Delta\sigma_1^{n+1} = \frac{1}{<\Phi_1\Phi_1> }\{\sum_{i=1}^{P_h} \sum_{k=1}^{P_e}\ h_i \Delta \epsilon_k^n <\Phi_i \Phi_k \Phi_1> \sum_{j=1}^{P_g} \sum_{k=1}^{P_e}\sum_{l=1}^{P_{\sigma}} \ c_j \Delta \epsilon_k^n \sigma_l^n <\Phi_j \Phi_k \Phi_l \Phi_1>\}$}
% \\
% . . .
% \\
% % ${\vdots}$
% {$\Delta\sigma_P^{n+1} = \frac{1}{<\Phi_P\Phi_P> }\{\sum_{i=1}^{P_h} \sum_{k=1}^{P_e}\ h_i \Delta \epsilon_k^n <\Phi_i \Phi_k \Phi_P> \sum_{j=1}^{P_g} \sum_{k=1}^{P_e}\sum_{l=1}^{P_{\sigma}} \ c_j \Delta \epsilon_k^n \sigma_l^n <\Phi_j \Phi_k \Phi_l \Phi_P>\}$}
%
%
% \vspace*{2mm}
% \item[] Stress update:
%
% $ \sum_{l=1}^{P_{\sigma}} \sigma_i^{n+1} \Phi_i = \sum_{l=1}^{P_{\sigma}} \sigma_i^{n} \Phi_i + \sum_{l=1}^{P_{\sigma}} \Delta\sigma_i^{n+1} \Phi_i$
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% \item[] Material uncertainty expanded into stochastic shape funcs.
% %$E(x,t,\theta) = \sum_{i=0}^{P_d} r_i(x,t) * \Phi_i[\{\xi_1, ..., \xi_m\}]$
%
% \vspace*{1mm}
% \item[] Loading uncertainty expanded into stochastic shape funcs.
% %$f(x,t,\theta) = \sum_{i=0}^{P_f} f_i(x,t) * \zeta_i[\{\xi_{m+1}, ..., \xi_f]$
%
% \vspace*{1mm}
% \item[] Displacement expanded into stochastic shape funcs.
% %$u(x,t,\theta) = \sum_{i=0}^{P_u} u_i(x,t) * \Psi_i[\{\xi_1, ..., \xi_m, \xi_{m+1}, ..., \xi_f\}]$
%
% %\item
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% %
% %\item Time domain integration using Newmark and/or HHT, in probabilistic spaces
%
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% \vspace*{1mm}
% \item[] Jeremi{\'c} et al. 2011
%
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% \end{itemize}
%
%
% \begin{tiny}
% \[
% %$
% \begin{bmatrix}
% \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_0> K^{(k)} & \dots & \sum_{k=0}^{P_d} <\Phi_k \Psi_P \Psi_0> K^{(k)}\\
% \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_1> K^{(k)} & \dots & \sum_{k=0}^{P_d} <\Phi_k \Psi_P \Psi_1> K^{(k)}\\ \\
% \vdots & \vdots & \vdots & \vdots\\
% \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_P> K^{(k)} & \dots & \sum_{k=0}^{M} <\Phi_k \Psi_P \Psi_P> K^{(k)}
% \end{bmatrix}
% \begin{bmatrix}
% \Delta u_{10} \\
% \vdots \\
% \Delta u_{N0}\\
% \vdots \\
% \Delta u_{1P_u}\\
% \vdots \\
% \Delta u_{NP_u}
% \end{bmatrix}
% =
% %\]
% %\[
% \begin{bmatrix}
% \sum_{i=0}^{P_f} f_i <\Psi_0\zeta_i> \\
% \sum_{i=0}^{P_f} f_i <\Psi_1\zeta_i> \\
% \sum_{i=0}^{P_f} f_i <\Psi_2\zeta_i> \\
% \vdots \\
% \sum_{i=0}^{P_f} f_i <\Psi_{P_u}\zeta_i>\\
% \end{bmatrix}
% %$
% \]
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% \item[] SEPFEM offers a complete probabilistic solution
%
% \item[] It is NOT based on Monte Carlo approach
%
% \item[] System of equations grows (!)
%
% \end{itemize}
%
%
% % \normalsize{Typical number of terms required for a SEPFEM problem} \vspace{1cm}\\
% \scalebox{0.7}{
% \begin{tabular}{ c c c c}
% \# KL terms material & \# KL terms load & PC order displacement& Total \# terms per DoF\\ \hline
% 4 & 4 & 10 & 43758 \\
% 4 & 4 & 20 & 3 108 105 \\
% 4 & 4 & 30 & 48 903 492 \\
% 6 & 6 & 10 & 646 646 \\
% 6 & 6 & 20 & 225 792 840 \\
% 6 & 6 & 30 & 1.1058 $10^{10}$ \\
% 8 & 8 & 10 & 5 311 735 \\
% 8 & 8 & 20 & 7.3079 $10^{9}$ \\
% 8 & 8 & 30 & 9.9149 $10^{11}$\\
%
% ... & ... & ... & ...\\
% % \hline
% \end{tabular}}
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%%\subsection{Sobol Sensitivity Analysis}
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\frametitle{Backward Uncertainty Propagation, Sensitivities}
\begin{itemize}
\vspace*{2mm}
\item[] Given forward uncertain response, PDFs, CDFs...
\vspace*{4mm}
\item[] Contributions of uncertain input to forward uncertainties
\vspace*{4mm}
\item[] Sensitivity of uncertain response to input uncertainties
\vspace*{4mm}
\item[] Sobol indices
\end{itemize}
\end{frame}
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% \begin{frame}
% \frametitle{ANOVA Representation}
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% %\vspace*{10mm}
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% Model with $n$ uncertain inputs ($\boldsymbol{x}$) and scalar output $y$:
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% \vspace*{6mm}
% \begin{equation}
% y = f(\boldsymbol{x}) \mbox{;} \ \ \boldsymbol{x} \in I^{n}
% \nonumber
% \end{equation}
%
% % input parameters $\boldsymbol{x}$ are defined in $n$ dimensional unit
% % cube $I^{n}$
% %
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% \vspace*{5mm}
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% The ANalysis Of VAriance representation
% % of $f(x)$
% (Sobol 2001):
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% \vspace*{6mm}
% \begin{eqnarray*}
% f(x_1, ... x_n) = f_0 + \sum_{i=1}^{n} f_i(x_i) +
% \sum_{1\leq i