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\mode<handout>{\setbeamercolor{background canvas}{bg=black!3}}

\usepackage{epstopdf}


%% Jose Antonio Abell Mena provided this for DSL descriptions
% (used in a file _Chapter_SoftwareHardware_Domain_Specific_Language_English.tex
% This is added for listing FEI DSL
%   since   he   customized   it,   it   needs   to   be   changed   (linked  to

 
\usepackage[absolute,overlay]{textpos}



%% Jose Antonio Abell Mena provided this for DSL descriptions
% (used in a file _Chapter_SoftwareHardware_Domain_Specific_Language_English.tex
% This is added for listing FEI DSL
%   since   he   customized   it,   it   needs   to   be   changed   (linked  to
% /usr/share/texmf/tex/latex/misc)
%\usepackage{myListings}
\input{essi_listings_options.tex}

 
\usepackage{threeparttable}

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\usetikzlibrary{tikzmark,calc}


      
\usepackage{multimedia}
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% from https://www.reddit.com/r/LaTeX/comments/x2j9jr/embed_video_to_latex_beamer_presentation/?rdt=48182

\usepackage{media9}

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\usepackage{pdfpages}

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%from http://mailman.mit.edu/pipermail/macpartners/2005-January/000780.html
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%% ovo je za cirilicu
\input vuk.def
\newfont{\cyr}{wncyr10 scaled 1000 }
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% FEI  LOGO definition
\newcommand{\FEI}
{{\sf \uppercase{F}\kern-0.20em\uppercase{E}\kern-0.20em\uppercase{I}}}

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%  \usetheme{Marburg} % ima naslov i sadrzaj sa desne strane
%  \usetheme{Hannover} % ima naslov i sadrzaj sa leve strane
%  \usetheme{Singapore} % ima sadrzaj i tackice gore
%  \usetheme{Antibes}  % ima sadrzaj gore i kao graf ...
%  \usetheme{Berkeley} % ima sadrzaj desno
%  \usetheme{Berlin} % ima sadrzaj gore i tackice
%  \usetheme{Goettingen} % ima sadrzxaj za desne strane
%  \usetheme{Montpellier} % ima graf sadrzaj gore
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}

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\usepackage[greek,english]{babel}



\usepackage{amsmath}
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%%%%  HYPERREF  HYPERREF  HYPERREF  HYPERREF  HYPERREF
%%%%  HYPERREF  HYPERREF  HYPERREF  HYPERREF  HYPERREF
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             colorlinks=true,
             linkcolor=webblue,
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             urlcolor=webblue,
}
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%             linktocpage,
%             pdftex]{hyperref}


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% does not look nice, try deleting the line with the fontenc.



%      Site Specific Dynamics of Structures: 
%From Seismic Source to 
%the Safety of Occupants and Content


\title[NPP Resilience]
        {Assessing and Increasing  Resilience of \\
         Nuclear Energy Facilities for \\
         Dynamic, Seismic Loads}
%        {Assessing and Increasing  Resilience of Nuclear Energy
%               Soil-Structure Systems for Dynamic/Seismic Loads}


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

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


\author[Jeremi{\'c} et al.] % (optional, use only with lots of authors)
%{Boris~Jeremi{\'c}}
{Boris Jeremi{\'c}
}

%\institute[Computational Geomechanics Group \hspace*{0.3truecm}
%\institute[] % (optional, but mostly needed)
\institute[\pgfuseimage{university-logo}\hspace*{0.1truecm}\pgfuseimage{lbnl-logo}] % (optional, but mostly needed)
%{ Professor, University of California, Davis\\
{ University of California, Davis 
%{ University of California, Davis, CA 
\\
Lawrence Berkeley National Lab}
% - Use the \inst command only if there are several affiliations.
% - Keep it simple, no one is interested in your street address.


\date[] % (optional, should be abbreviation of conference name)
{\small DOE-NRC NPH\\
        October 2024}

\subject{}
% This is only inserted into the PDF information catalog. Can be left
% out.



% If you have a file called "university-logo-filename.xxx", where xxx
% is a graphic format that can be processed by latex or pdflatex,
% resp., then you can add a logo as follows:

%\pgfdeclareimage[height=0.2cm]{university-logo}{/home/jeremic/BG/amblemi/ucdavis_logo_gold_lrg}
%\logo{\pgfuseimage{university-logo}}

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




% % Delete this, if you do not want the table of contents to pop up at
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\begin{frame}
 \frametitle{Fukushima NPP: Damage, Inelastic Behavior}


\begin{figure}[!hbpt]
\begin{center}
\includegraphics[width=7.0truecm]{/home/jeremic/tex/works/Conferences/2024/DOE_Natural_Hazard_Phenomena_at_NRC_29-30Oct2024/present/Fukushima_NPP_inside_photo_01.jpg}
\\
\includegraphics[width=7.0truecm]{/home/jeremic/tex/works/Conferences/2024/DOE_Natural_Hazard_Phenomena_at_NRC_29-30Oct2024/present/Fukushima_NPP_inside_photo_02.jpg}
\end{center}
\end{figure}
%\vspace*{-12mm}
%\begin{flushright}
%{\tiny[TEPCO]}
%\end{flushright}

%\vspace*{-12mm}
\begin{flushleft}
%{\tiny{Control Rod Drive System DAMAGE [TEPCO]}}
\hspace*{1mm}
Control Rod Drive System \underline{DAMAGE} [TEPCO, 2023]
\\
\vspace*{-1mm}
\hspace*{-2mm}
+
\vspace*{-1mm}
\\
\hspace*{1mm}
Loss of Coolant Accident, \underline{LOCA} [Japanese Diet NAIIC, 2012]
%Loss of Coolant Accident (LOCA) [Japanese Diet NAIIC]
\end{flushleft}








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\begin{frame}
 \frametitle{Fukushima NPP: Vertical Isolator Amplification  +80\% }

\vspace*{10mm}

\begin{figure}[!hbpt]
\begin{center}
\hspace*{-20mm}
\includegraphics[width=6.0truecm]{/home/jeremic/tex/works/Conferences/2024/DOE_Natural_Hazard_Phenomena_at_NRC_29-30Oct2024/present/Fukushima_NPP_vertical_amplification_01.jpg}
\hspace*{5mm}
\includegraphics[width=4.0truecm]{/home/jeremic/tex/works/Conferences/2024/DOE_Natural_Hazard_Phenomena_at_NRC_29-30Oct2024/present/Fukushima_NPP_vertical_amplification_02.jpg}
\hspace*{-20mm}
\end{center}
\end{figure}



\vspace*{5mm}
\begin{flushleft}
%{\tiny{Control Rod Drive System DAMAGE [TEPCO]}}
{{Base isolation, VERTICAL amplification (!)  [JAEE/LLNL, 2012]}}
\end{flushleft}
% 
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% {\tiny[LLNL]}
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% Structuring a talk is a difficult task and the following structure
% may not be suitable. Here are some rules that apply for this
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% - Exactly two or three sections (other than the summary).
% - At *most* three subsections per section.
% - Talk about 30s to 2min per frame. So there should be between about
%   15 and 30 frames, all told.

% - A conference audience is likely to know very little of what you
%   are going to talk about. So *simplify*!
% - In a 20min talk, getting the main ideas across is hard
%   enough. Leave out details, even if it means being less precise than
%   you think necessary.
% - If you omit details that are vital to the proof/implementation,
%   just say so once. Everybody will be happy with that.



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\section{Motivation}




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%\subsection{Motivation}
\subsection{\ }



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\begin{frame}
 \frametitle{Motivation}

\begin{itemize}



 

%\vspace*{0.3cm}
  \item[]  Improve static and dynamic analysis of 
           Soil-Structure Interaction (SSI) for             
           Nuclear Energy Facilities (NEF)
 
\vspace*{2mm}
  \item[] Inelastic behavior of NEF SSI systems is always present

% \vspace*{2mm}
%  \item[]  Expert numerical modeling and simulation tool
% 
% \vspace*{2mm}
%   \item[]      Use of numerical models to
%   analyze statics and dynamics of soil/rock-structure systems
% 

\vspace*{2mm}
  \item[]  Reduce  modeling uncertainty

\vspace*{2mm}
  \item[]  Improve analysis sophistication level 

\vspace*{2mm}
  \item[]  Analyze, propagate parametric uncertainty

\vspace*{2mm}
  \item[] Goal: Predict and Inform 

\vspace*{2mm}
  \item[] Engineer needs to know all \underline{what ifs!}



%
%  
%  
%  \vspace*{1mm}
%    \item[]  Follow  the  flow, input and dissipation,  of seismic  energy, 

% \vspace*{1mm}
%   \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:\\
%    Real-ESSI 
%  \hspace*{5mm} 
%   \url{http://real-essi.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://ms-essi.info/}
% % %  
% % 




\end{itemize}
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\begin{frame}
 \frametitle{Dedication}


% Thick fbox lines 
\setlength{\fboxrule}{1.8pt} 
 
    
\vspace*{5mm}

\noindent
\framebox{Robert P. Kennedy, 1939-2018}
%\hspace*{5mm}
\vspace*{-18mm}
\begin{flushright}
\includegraphics[height=15mm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/_Chapter_Dedication_Photos/Robert_P_Kennedy/Robert_P_Kennedy_07Dec2018.jpg}
\end{flushright}

\vspace*{-1mm}
\noindent
"Response of a soil structure system 
 is nonlinear, and I would really 
 like to know what that response is!"

%\vspace*{1mm}
%\noindent
%"There are engineers and then there are Engineers!"



\vspace*{12mm}
\noindent
\fbox{Neboj{\v s}a Orbovi{\' c},  1962-2021}
%\hspace*{7mm}
\vspace*{-15mm}
\begin{flushright}
\includegraphics[width=18mm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/_Chapter_Dedication_Photos/Nebojsa_Orbovic/Nebojsa_Orbovic_26Feb2016.jpg}
\end{flushright}

%\noindent
%"As an engineer, I have to know, \\
%with good accuracy, what will happen \\
%to the structure during loading, \\
%hence numerical analysis and \\
%verification and validation for \\
%numerical analysis is really important"

\vspace*{-1mm}
\noindent
"As an engineer, I have to know  what are response sensitivities 
 to modeling choices and model parameters."


% back to standard fbox lines 
 \setlength{\fboxrule}{0.2pt} 



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\begin{frame} 

\frametitle{Analysis of NEF Soil-Structure Systems}

\vspace*{-1mm}
    \begin{itemize}
\vspace*{2mm}
      \item[-]  Nonlinear, inelastic    material  behavior  of  SSI systems
        \begin{itemize}
          \item[-] Soil: always inelastic
          \item[-] Interfaces: always inelastic 
          \item[-] Concrete: usually inelastic 
          \item[-] Steel: sometimes  inelastic 
        \end{itemize}
\vspace*{1mm}
%      \item[-] Numerical analysis \underline{needs to} follow physics
      \item[-] Numerical analysis \underline{has to} follow physics
\vspace*{1mm}
      \item[-] Numerical analysis programs to provide all necessary 
               modeling (models, elements) and simulation (algorithms, methods) features
\vspace*{1mm}
      \item[-] Verification and Validation (QA) is really important
        \begin{itemize}
          \item[-] Quality of numerical discretization, algorithms (math.)
%          \item[-] Assess quality of numerical algorithms, methods
          \item[-] Quality of models (phys.)
        \end{itemize}
%\vspace*{2mm}
%      \item[-] 
    \end{itemize}



\end{frame}


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% \begin{frame}
% % \frametitle{Nonlinear Material Behavior}
% % \frametitle{Soil, Interface, Concrete Wall Behavior}
% % \frametitle{Nonlinear: Soil, Interface, Concrete, Steel}
%  \frametitle{All Materials are Inelastic}
% % \frametitle{All Materials are Inelastic: Soil, Interface, Concrete, Steel}
% 
% 
%  
% %\vspace*{-3mm}
% \begin{figure}[!hbpt]
% \begin{center}
% \includegraphics[width=3.7truecm]{/home/jeremic/tex/works/Conferences/2024/SMiRT27_Yokohama_03-07Mar2024/present/Workshop_Nonlinear_ESSI/Soil-cyclic-reponse.jpg}
% \includegraphics[width=4.2truecm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/_Chapter_Theory_Small_Deformation_Elasto_Plasticity/contact_Cyclic_test_Shahrour_1991.pdf}
% \\
% \includegraphics[width=3.7cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/_Chapter_Applications_Concrete_Structures/Shear_Wall_Membrane/Reg_B_Force_Displacement.pdf}
% \includegraphics[width=3.7cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/_Chapter_Applications_Concrete_Structures/Shear_Wall_Membrane/ASR_A1_Force_Displacement.pdf}
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% 
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%  \begin{frame}
%   \frametitle{Nonlinear Response of Soil}
%  
%  
%   
%  \begin{figure}[!hbpt]
%  \begin{center}
%  \includegraphics[width=6.0truecm]{/home/jeremic/tex/works/Conferences/2024/SMiRT27_Yokohama_03-07Mar2024/present/Workshop_Nonlinear_ESSI/Soil-cyclic-reponse.jpg}
%  % \caption{\label{fig:Cruas-element-tests} Solid element tests: a)~triaxial compression,
%  % b)~simple shear, c)~pure shear
%  % (isochoric with no rotation of the principle stress and strain directions),
%  % d)~pure shear (in soil mechanics, commonly understood as isochoric shear).}
%  \end{center}
%  \end{figure}
%  
%  
%  
%   
%  
%  %
%  \end{frame}
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%  \begin{frame}
%   \frametitle{Nonlinear Response of Interfaces}
%  
%  
%   
%  \vspace*{4mm}
%  \begin{figure}[!hbpt]
%  \begin{center}
%  \includegraphics[width=7.5truecm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/_Chapter_Theory_Small_Deformation_Elasto_Plasticity/contact_Cyclic_test_Shahrour_1991.pdf}
%  % \caption{\label{fig:Cruas-element-tests} Solid element tests: a)~triaxial compression,
%  % b)~simple shear, c)~pure shear
%  % (isochoric with no rotation of the principle stress and strain directions),
%  % d)~pure shear (in soil mechanics, commonly understood as isochoric shear).}
%  \end{center}
%  \end{figure}
%  
%  
%  
%   
%  
%  %
%  \end{frame}
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%  
%  
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%  \begin{frame}
%   \frametitle{Nonlinear Response of Concrete Shear Walls}
%  
%  
%   
%  \vspace*{5mm}
%  \begin{figure}[!hbpt]
%  \begin{center}
%  %{\includegraphics[width=0.49\columnwidth]{Figure-files/_Chapter_Applications_Concrete_Structures/Shear_Wall_Membrane/Reg_A_Force_Displacement.pdf}} 
%  \hspace*{-10mm}
%  \includegraphics[width=5.5cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/_Chapter_Applications_Concrete_Structures/Shear_Wall_Membrane/Reg_B_Force_Displacement.pdf}
%  \includegraphics[width=5.5cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/_Chapter_Applications_Concrete_Structures/Shear_Wall_Membrane/ASR_A1_Force_Displacement.pdf}
%  \hspace*{-10mm}
%  \end{center}
%  \end{figure}
%  %
%  
%  \noindent
%  \hspace*{15mm}{\small Regular Concrete} \hspace*{25mm} {\small ASR Concrete}
%  
%   
%  
%  %
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%  
%  

%  
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%  \begin{frame} 
%  
%  \frametitle{Expert Engineer, Analyst}
%  
%  \vspace*{-1mm}
%      \begin{itemize}
%  \vspace*{4mm}
%        \item[-] Nonlinear analysis requires expert engineers
%  %\vspace*{2mm}
%  %      \item[-] Nonlinear analysis requires sophisticated programs
%  \vspace*{4mm}
%        \item[-] Experienced engineers
%  \vspace*{4mm}
%        \item[-] Training and education
%  \vspace*{4mm}
%        \item[-] Training, education and experience takes time!
%  %\vspace*{2mm}
%  %      \item[-] 
%      \end{itemize}
%  
%  
%  
%  \end{frame}
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%  
%  

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\section{Asses and Increase Resilience of NEF SSI}
% \section{Resilient of NEF SSI}


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%\subsection{Motivation}
%\subsection{Benefits and Detriments}
%\subsection{\ }
\subsection*{Resilient NEF SSI Systems}



 


 
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\frametitle{Stress Test Motions, Inclined Wave $\theta = 60^{o}$}
%\frametitle{Stress Test MotionsFree Field, Variation in Input Frequency, $\theta = 60^{o}$}

% Elastoplastic soil with contact elements
%% Both solid and contact elements dissipate energy

     
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% 
% \frametitle{SMR ESSI, Variation in Input Wave Angle,  $f = 5$Hz}
% 
% % Elastoplastic soil with contact elements
% %% Both solid and contact elements dissipate energy
% 
%      
% % \vspace*{-5mm}
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\frametitle{SMR ESSI, Variation in Input Frequency, $\theta = 60^{o}$}

% Elastoplastic soil with contact elements
%% Both solid and contact elements dissipate energy

     
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% %\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}
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\begin{frame}
% \frametitle{Seismic Energy Input and Dissipation in ESSI System}
 \frametitle{Seismic Energy Propagation in ESSI System}

\begin{itemize}

\vspace*{1mm}
     \item[] Energy input, forces/loads, static/dynamic


\vspace*{4mm}
     \item[] Energy dissipation outside SSI domain:
       \begin{itemize}
         \item[] SSI system oscillation radiation
         \item[] Reflected waves radiation
       \end{itemize}
\vspace*{2mm}
     \item[] Energy dissipation/conversion inside SSI domain:
       \begin{itemize}
\vspace*{1mm}
         \item[] Inelasticity: soil/rock, interfaces, structure, dissipators
%         \item[] Inelasticity/damage: soil/rock, interfaces, structure, dissipators
\vspace*{1mm}
         \item[] Viscous coupling: internal, pore and external fluids
% %         \item[] potential and kinetic energy
%          \item[] potential $\leftarrow \! \! \! \! \! \! \rightarrow$ kinetic energy
\vspace*{1mm}
         \item[] Energy deflectors, meta-materials/meta-devices
       \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}

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%  
%  \begin{frame}
%  \frametitle{Energy Dissipation within ESSI System}
%  
%  \begin{itemize}
%  
%  
%  
%  \vspace*{4mm}
%  \item[-] 
%  %Increment of 
%  Plastic energy dissipation:
%  %$\Phi =  \sigma_{ij} \dot{\epsilon}_{ij} - \sigma_{ij} \dot{\epsilon}_{ij}^{el} - \rho \dot{\psi}_{pl} \ge 0$
%  $\Delta  \Phi =  \sigma_{ij} \Delta {\epsilon}_{ij} - 
%            \sigma_{ij} \Delta {\epsilon}_{ij}^{el} - 
%  %          \rho 
%            \Delta {\psi}_{pl} \ge 0$
%   
%  
%  % %Following  the first and second laws of thermodynamics, the equation for plastic
%  % %energy  dissipation  in  decoupled  material models was presented by 
%  % %%\citeN7{Yang2017a}:
%  % %
%  % \begin{equation}
%  % \Phi =  \sigma_{ij} \dot{\epsilon}_{ij} - \sigma_{ij} \dot{\epsilon}_{ij}^{el} - \rho \dot{\psi}_{pl} \ge 0
%  % \label{equation_plastic_dissipation_final}
%  % \end{equation}
%  % %
%  % where  $\Phi$  is  the  rate  of  plastic  energy  dissipation  per unit volume,
%  % $\sigma_{ij}$  is  the stress  tensor,  $\epsilon_{ij}$  is  the strain tensor,
%  % $\epsilon_{ij}^{el}$  is  the  elastic  part of the strain tensor, $\rho$ is the
%  % mass  density of the material, ${\psi}_{pl}$ is the plastic free energy per unit
%  % volume.  
%  %
%  
%  
%  
%  
%  
%  
%  \vspace*{6mm}
%  \item[-] 
%  %Increment of 
%  Viscous energy dissipation/damping: \\
%  %dissipation $\Delta {D}_V$,  
%  $\Delta {D}_V =  C_{ij} \dot{u}_j \Delta {u}_i $
%  
%  % 
%  % To  compute  the  energy  dissipation  due to viscous damping, we start with the
%  % general form of the equation of motion:
%  % %
%  % \begin{equation}
%  % M_{ij} \ddot{u}_j(t) + C_{ij} \dot{u}_j(t) + K_{ij}^{elpl}(t) u_j(t) = f_i(t)
%  % \label{equation_of_motion}
%  % \end{equation}
%  % %
%  % where  $u_j(t)$ is the vector of generalized displacements, $M_{ij}$ is the mass
%  % matrix,   $C_{ij}$   is   the   damping   matrix,   $K_{ij}^{elpl}(t)$   is  the
%  % inelastic  stiffness  matrix that generally evolves with time, $f_i(t)$ is
%  % the external load vector.
%  % %
%  % For linear viscous damping of the Rayleigh type, the damping matrix is
%  % expressed as
%  % %
%  % \begin{equation}
%  % C_{ij} = a_M M_{ij} + a_{K} K_{ij}^{el}
%  % \label{equation_damping_matrix}
%  % \end{equation}
%  % %
%  % where  $a_M$  and  $a_{K}$  are  damping constants with units of s$^{-1}$ and s,
%  % respectively.
%  % %
%  % 
%  %     
%  % %
%  % The incremental form of energy balance for a dynamic system with viscous damping can be expressed as
%  % %
%  % \begin{equation}
%  % \Delta {W}_{Input} = \Delta {E}_K + \Delta {D}_V + \Delta {W}_M
%  % \label{equation_energy_balance}
%  % \end{equation}
%  % %
%  % The left hand side of Equation~\ref{equation_energy_balance}  is the
%  % increment of external input work
%  % %
%  % \begin{equation}
%  % \Delta {W}_{Input}=  f_i \Delta {u}_i
%  % \label{equation_energy_balance_W_Input}
%  % \end{equation}
%  %
%  %
%  % %%
%  % The three terms on the right hand side of Equation~\ref{equation_energy_balance} are
%  % the  increment of kinetic energy $\Delta {E}_K$, the increment of viscous energy 
%  % dissipation $\Delta {D}_V$, and the increment of material work of the system $\Delta {W}_M$
%  % %
%  % \begin{equation}
%  % \begin{aligned}
%  % \Delta {E}_K &= M_{ij} \ddot{u}_j \Delta u_i \\
%  % \Delta {D}_V &=  C_{ij} \dot{u}_j \Delta {u}_i \\
%  % \Delta {W}_M &= K_{ij}^{elpl} {u}_j \Delta {u}_i = \Delta {E}_S + \Delta {E}_P + \Delta {D}_P
%  % \end{aligned}
%  % \label{equation_energy_balance_components}
%  % \end{equation}
%  % %
%  % Note  that  the term of material work $W_M$ can be separated into an elastic part
%  % and  a  plastic  part.
%  % These two components are known as  the elastic strain energy $E_S$ and the plastic
%  % work of the system, respectively.
%  % %
%  % Then,  as  mentioned  in  the  previous  section,  plastic  work can be further
%  % decomposed into plastic free energy $E_P$ and plastic energy dissipation $D_P$.
%  
%  
%  \vspace*{6mm}
%  \item[-] 
%  Algorithmic, numerical energy dissipation/production:
%  \\
%   Newmark, Hilber-Hughes-Taylor, Houbolt, Bathe, Wilson...
%  
%  
%  %     
%  
%  
%  \end{itemize}
%  
%  
%  
%  
%  
%  
%  \end{frame}
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%  
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\frametitle{Energy Dissipation Control, Analysis}

\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}




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% \begin{frame}
%  
% \frametitle{Plastic Energy Dissipation}
% 
% 
% \vspace*{2mm}
%  
% % Single elastic-plastic element under cyclic shear loading
%  
% \begin{itemize}
% \item[] Plastic work is NOT plastic dissipation !
% \item[] Surface area of $F-\Delta$ or $\sigma-\epsilon$ is NOT plastic dissipation !
% % \item[] Plastic dissipation always increases
% \end{itemize}
%  
% %\vspace*{-7mm}
% \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}
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%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%  
%  \begin{frame}
%  \frametitle{Plastic Energy Dissipation}
%  
%  
%  \vspace*{6mm}
%  
%  Increment of  plastic energy dissipation:
%  %$\Phi =  \sigma_{ij} \dot{\epsilon}_{ij} - \sigma_{ij} \dot{\epsilon}_{ij}^{el} - \rho \dot{\psi}_{pl} \ge 0$
%  $\Delta \Phi =  \sigma_{ij} \Delta {\epsilon}_{ij}^{pl} - 
%  %          \sigma_{ij} \Delta {\epsilon}_{ij}^{el} - 
%  %           \rho 
%             \Delta {\psi}_{pl} \ge 0$
%   
%  
%  
%  \vspace*{6mm}
%  
%  
%   Plastic Free Energy $  \Delta {\psi}_{pl} $ 
%  
%  \begin{itemize}
%  
%  \vspace*{4mm}
%  \item[-]  3D Drucker-Prager El=Pl, Armstrong Frederick (soil) \\
%          $\Delta {\psi}_{pl}  
%          = 
%          \left(( 3/(2 h_a)) \alpha_{ij} \Delta  \alpha_{ij}  
%          - 
%          m_{ii}^{vol} \Delta \lambda \right) (-\sigma_{kk}/3 ) $
%  %        m_{ii}^{vol} \Delta \lambda \right) p $
%  
%  
%  \vspace*{4mm}
%  \item[-]  1D steel fiber (BRB) \\
%          $\Delta {\psi}_{pl}  = \frac{1}{2} 
%           \left( 
%              (\sigma + \sigma^{r})\Delta \epsilon) 
%              +
%              (\epsilon^{pl} - \epsilon^{r}) \Delta \sigma \right) $
%  
%  \vspace*{4mm}
%  
%  
%  
%     \end{itemize}
%  
%  
%  
%  
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% 
% \frametitle{ASCE-7, Steel Building with BRBs}
% 
% \vspace*{3mm}
% 
% 
% \begin{figure}[!H]
% \hspace*{-5mm}
% %\hspace*{-10mm}
%      \includegraphics[width=12cm]{/home/jeremic/tex/works/Conferences/2023/CIVIL-COMP-28-31Aug2023/present/ASCE-7-Steel-building-BRB_model.jpg}
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% \begin{frame}
% 
% \frametitle{ASCE-7, Steel Building with BRBs, Soil}
% 
% 
% Parabolic Drucker-Prager with Armstong-Frederick rotational kinematic hardening
% 
% %\vspace*{3mm}
% \begin{figure}[!H]
% %\hspace*{-10mm}
%      \includegraphics[width=6cm]{/home/jeremic/tex/works/Conferences/2023/CIVIL-COMP-28-31Aug2023/present/ASCE-7-Steel-building-BRB_soil_profile.jpg}
% \\
%      \includegraphics[width=3.5cm]{/home/jeremic/tex/works/Conferences/2023/CIVIL-COMP-28-31Aug2023/present/ASCE-7-Steel-building-BRB_upper_soil_response.jpg}
%      \includegraphics[width=3.5cm]{/home/jeremic/tex/works/Conferences/2023/CIVIL-COMP-28-31Aug2023/present/ASCE-7-Steel-building-BRB_lower_soil_response.jpg}
% %     \includegraphics[width=3cm]{/home/jeremic/tex/works/Conferences/2023/CIVIL-COMP-28-31Aug2023/present/from_Han_24Aug2023/}
% %     \includegraphics[width=3cm]{/home/jeremic/tex/works/Conferences/2023/CIVIL-COMP-28-31Aug2023/present/from_Han_24Aug2023/}
% %     \includegraphics[width=3cm]{/home/jeremic/tex/works/Conferences/2023/CIVIL-COMP-28-31Aug2023/present/from_Han_24Aug2023/}
% \end{figure}
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% 
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% \begin{frame}
% 
% \frametitle{ASCE-7, Steel Building with BRBs, Steel}
% 
% 
% Giuffr{\'e}-Menegotto-Pinto 1D Fiber steel model
% 
% \vspace*{3mm}
% \begin{figure}[!H]
% %\hspace*{-10mm}
%      \includegraphics[width=8cm]{/home/jeremic/tex/works/Conferences/2023/CIVIL-COMP-28-31Aug2023/present/ASCE-7-Steel-building-BRB_BRB_calibration.jpg}
% \end{figure}
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% \begin{frame}
% 
% \frametitle{Buckling Restrained Braces (BRB) Energy Dissipation}
% 
% 
% %\vspace*{3mm}
% 
% \begin{flushleft}
% \begin{figure}[!H]
% \hspace*{-20mm}
% %\hspace*{-10mm}
%      \includegraphics[width=5cm]{/home/jeremic/tex/works/Conferences/2023/CIVIL-COMP-28-31Aug2023/present/ASCE-7-Steel-building-BRB_BRB_elements.jpg}
% \hspace*{50mm}
% \end{figure}
% \end{flushleft}
% 
% \vspace*{-12mm}
% \begin{flushright}
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\noindent
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{\small Energy dissipation in structure}
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{\small  Energy dissipation in soil} 
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\vspace*{-4mm}

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{\small More structural damage}
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%\frametitle{Important Infrastructure, Energy Dissipation}
\frametitle{LWR NEF, Energy Dissipation}

% Elastoplastic soil with contact elements
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\frametitle{SMR NEF, Energy Dissipation}
%\frametitle{SMR Seismic Response, Energy Dissipation}

% Elastoplastic soil with contact elements
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\begin{frame}{Forward and Bakward Uncertainty Propagation}


\vspace*{2mm}
Time Domain Stochastic Elastic-Plastic FEM
%Dynamic Finite Elements: 
$
{ M} \ddot{ u_i} +
{ C} \dot{ u_i} +
{ K}^{ep} { u_i} =
{ F(t)}$


\begin{itemize}
\vspace*{1mm}
\item[-] Input random field and random process, non-Gaussian,
         heterogeneous/non-stationary: Multi-dimensional Hermite Polynomial
         Chaos (PC) with {known coefficients}
%\vspace{0.05in}
\vspace*{1mm}
\item[-] Output response process: Multi-dimensional Hermite PC with {unknown
         coefficients}
% \vspace{0.05in}


 
\end{itemize}

\vspace*{3mm}
\underline{Results} $\rightarrow$ Probability Distribution Functions (PDF)  ($u_i, \sigma_{ij}, \epsilon_{ij}$...)

\vspace*{3mm}
\underline{Sensitivities} $\leftarrow$ Sobol sensitivities to input uncertainties

%\scriptsize 
%Note: PC = Polynomial Chaos

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%  \subsection[TDNIPSRA Formulation]{Formulation}
%  %\subsection{TDNIPSRA Formulation}

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%  \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}
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%  \begin{textblock}{15}(4.8, 4.2)
%  $\Longrightarrow$
%  \end{textblock}
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%  \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}
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%  \tiny
%  adapted from Taga(1982)
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%  \begin{textblock}{15}(11.7, 10.6)
%  \tiny
%  \textbf{Uncertain material properties}
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%  \begin{figure}[H]
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%  \includegraphics[width=0.2\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/4231447_Northridge_eq.pdf}
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%  \includegraphics[width=0.3\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/uncertain_SPT.pdf}
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%  \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 
%  
%  % \[\lambda(EDP>z) = \int_{IM} \underbrace{|\frac{d\lambda(IM)}{dIM}|}_\text{PSHA} \underbrace{G(EDP|IM)}_{\text{fragility}} dIM \]
%  
%  \vspace{-0.1cm}
%  
%  \[\lambda(EDP>z) = \int \underbrace{|\frac{d\lambda(IM>x)}{dx}|}_\text{\textbf{PSHA}} \underbrace{G(EDP>z|IM=x)}_{\text{\textbf{fragility analysis}}} dx\]
%  
%  \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}
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%  % 
%  % \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 Geo-hazard: 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}
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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}{!}{%
% % $\lambda(Sa>z) = \sum_{i=1}^{NFL} \underbrace{N_i \int\int f_{mi}(M) f_{ri}(R|M)}_\text{seismic source characterization (SSC)} \underbrace{P(Sa>z|M, R)}_\text{GMPE} dM dR$}
% % \end{equation*}
% 
% \vspace{4mm}
% \item[-] Fragility analysis $P(EDP>x|IM=z)$, deterministic time domain FEM,
% perhaps using Monte Carlo (MC)
% 
% %  \begin{itemize}
% % 
% %     \item[-] Records selection: Spectrum-matching 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}
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% \subsection{Issues in State-of-the-art SRA}

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%  \begin{frame} 
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%  \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
%  
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%  \end{itemize}
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%  \frametitle{Time Domain Intrusive PSRA Framework}
%  
%  %  
%  
%  \begin{itemize}
%  
%  %\vspace*{2mm}
%  \item[-] Stochastic Elastic-Plastic 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 elastic-plastic material
%  %stress and stiffness solution using
%  %Forward Kolmogorov, Fokker-Planck 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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%  
%  
%  
%  \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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%  % \begin{frame}
%  % \frametitle{Stochastic Elastic-Plastic 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}}
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%  %  
%  %  \begin{itemize}
%  %  
%  %  \item Monte Carlo simulations: non-intrusive approach 
%  %  
%  %  \begin{itemize}
%  %  \item [--] \small Slow convergence rate $1/\sqrt{N}$
%  %  \item [--] \small Hard for stable tail distribution toward low-risk 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 CPU-hours and over 100 years} (Maechling et al. 2007)
%  %  \end{itemize}
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% 
% %\subsection{TDNIPSRA Example}
% \subsection[TDNIPSRA Example]{Example}



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%   \begingroup
%   
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%   
%   \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 $\rightarrow$}
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%   
%   \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 $\rightarrow$}
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%   \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)
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%   \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}
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%   {\Large $\leftarrow$}
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%   \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}
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%   \end{tikzpicture}
%   
%   
%   \begin{textblock}{15}(-0.1, 9.3)
%   \scriptsize
%   \quad \quad \quad \quad $\lambda(EDP>z)=$
%   
%   $\quad \sum N_i(M_i, R_i) P(EDP>z|M_i, R_i)$
%   \end{textblock}
%   
%   \begin{textblock}{15}(1.6, 10.7)
%   \scriptsize{EDP hazard/risk}
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% \begingroup
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% \begin{frame}
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% \frametitle{Time Domain Intrusive PSRA Framework}
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% \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}
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% \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}
% 
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% \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}
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% % \begin{textblock}{15}(10.2, 13.2)
% % $\Leftarrow$
% % \end{textblock}
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% % \begin{textblock}{15}(5.3, 10.75)
% % \includegraphics[width=0.33\linewidth]{pic/probabilsitc_evolution.png}
% % \end{textblock}
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% % \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}
% 
% % \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}
% 
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% % \begin{textblock}{15}(-0.1, 9.3)
% % \scriptsize
% % \quad \quad \quad \quad $\lambda(EDP>z)=$
% 
% % $\quad \sum N_i(M_i, R_i) P(EDP>z|M_i, R_i)$
% % \end{textblock}
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% % \begin{textblock}{15}(1.6, 10.7)
% % \scriptsize{EDP hazard/risk}
% % \end{textblock}
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% \begingroup
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% \setbeamertemplate{footline}{}
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% \begin{frame}
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% \frametitle{Time Domain Intrusive PSRA Framework}
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% 
% \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}
% 
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% \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}
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% \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}
% 
% % \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 $\lambda(EDP>z)=$
% 
% % $\quad \sum N_i(M_i, R_i) P(EDP>z|M_i, R_i)$
% % \end{textblock}
% 
% % \begin{textblock}{15}(1.6, 10.7)
% % \scriptsize{EDP hazard/risk}
% % \end{textblock}
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% \end{frame}
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% \endgroup
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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}
% 
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% \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 $\lambda(EDP>z)=$
% 
% $\quad \sum N_i(M_i, R_i) P(EDP>z|M_i, R_i)$
% \end{textblock}
% 
% \begin{textblock}{15}(1.6, 10.7)
% \scriptsize{EDP hazard/risk}
% \end{textblock}
% 
% \end{frame}
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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}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \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}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% 
% 
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% 
% \frametitle{Stochastic Material 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 Armstrong-Frederick 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/Shear-Frame-8-levels.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 Armstrong-Frederick 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}
% %
% 
% 
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% \end{frame}
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%   % \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
%   % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%   
%   
%   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%   \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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% 
% %\frametitle{Sensitivity Analysis, Backward Propagation}
% \frametitle{Sensitivity Analysis}
% 
% %  
% 
% \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
% 
% %\begin{itemize}
% %
% %
% % \vspace*{2mm}
% \vspace*{1mm}
% \item[-] Total variance in PGA, \underline{in this case}, dominated by uncertain ground motions
%     \begin{itemize}
% \vspace*{1mm}
%       \item[] $49$\% from uncertain rock motions at depth
% \vspace*{1mm}
%       \item[] $2$\% from uncertain soil
% \vspace*{1mm}
%       \item[] $49$\% from interaction of uncertain rock motions and  uncertain soil 
% 
% 
%     \end{itemize}
% 
% \end{itemize}
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% %  \item 
% %\end{itemize}
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\begin{frame}

\frametitle{Sensitivity Example: Probabilistic SSI}

\vspace*{5mm}
\begin{itemize}
% \item[-] Three-layered ground with \\
%       uncertain material properties and \\
%       uncertain seismic rock motions

 \item[-] Uncertain material: \\
       uncertain random field, \\
       marginally lognormal \\
       distribution, \\
       exponential correlation \\
       length 10m

\vspace*{2mm}
\item[-]  Uncertain  seismic \\
          rock motions:  \\
          seismic  scenario \\
           M=7,  R=50km \\
%       Stochastic Fourier amplitude spectra, \\
%       Stochastic Fourier phase derivative

\end{itemize}


\vspace*{-50mm}
\begin{figure}[!hbpt]
\begin{flushright}
\includegraphics[width=5.5cm]{/home/jeremic/tex/works/Conferences/2021/CompDyn_8th_Athens_21-23Jun2020/present/Three-Layer-Model.jpg}
\\
\includegraphics[width=5.5cm]{/home/jeremic/tex/works/Conferences/2021/CompDyn_8th_Athens_21-23Jun2020/present/Input-motions.jpg}
\end{flushright}
\end{figure}

% 
%  \begin{figure}[!hbpt]
%  \begin{center}
%  \includegraphics[width=10cm]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/structural_uncertainty.pdf}
%  \end{center}
%  \end{figure}
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%  
%  \begin{figure}[!hbpt]
%  \begin{center}
%  \includegraphics[width=10cm]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/probabilsitc_evolution.png}
%  \end{center}
%  \end{figure}
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%  \begin{figure}[!hbpt]
%  \begin{center}
%  \includegraphics[width=10cm]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/seismic_risk_result_framework.png}
%  \end{center}
%  \end{figure}
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%  
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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 inter-story drift ratio (MIDR) 
\end{textblock}
\end{frame}
\endgroup
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% 
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% 
% \frametitle{Stochastic Material Parameters}
% 
% %  
% 
% \vspace*{5mm}
% Lognormal distributed random field with PC Dim. 3 Order 2
% 
% 
% \vspace*{-5mm}
% \begin{figure}[!hbpt]
% \begin{center}
% \hspace*{-10mm}
% \includegraphics[width=13cm]{/home/jeremic/tex/works/Conferences/2021/CompDyn_8th_Athens_21-23Jun2020/present/Stochastic_material_parameters.jpg}
% \end{center}
% \end{figure}
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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% 
% 
% 
% 
%  
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% 
% \frametitle{Stochastic Seismic Motion Development}
% 
% \vspace*{5mm}
% \begin{itemize}
%  \item[-] UCERF3 (Field et al. 2014)
%  \item[-] Stochastic motions (Boore 2003) 
%  \item[-] Polynomial Chaos Karhunen-Lo{\`e}ve expansion
%  \item[-] Probabilistic DRM (Bielak et al. 2003, Wang et al. 2021)
% \end{itemize}
% 
% \vspace*{-3mm}
% \begin{figure}[!hbpt]
% \begin{center}
% \includegraphics[width=5cm]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/UCERF3.pdf}
% \includegraphics[width=4.0cm]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/SMSIM.pdf}
% %\includegraphics[width=3.5cm]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/KL_exact_dis_correlation_from_dis.pdf}
% \end{center}
% \end{figure}
% 
% % 
% %  
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% 
% 
% 
% 
% 
% 
% 
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% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% 
% \frametitle{Stochastic Ground Motion Modeling}
% 
% \begin{itemize}
% 
% %\vspace*{1mm}
% \item[-] Modeling fundamental characteristics of uncertain ground motions,
% Stochastic Fourier amplitude spectra (FAS). and Stochastic Fourier phase spectra
% (FPS) and not specific IM
% 
% \vspace*{1mm}
% \item[-]  Mean behavior of stochastic FAS,  $w^2$ source radiation spectrum by
% Brune(1970), and Boore(1983, 2003, 2015).
% 
% \vspace*{1mm}
% \item[-] Variability models for stochastic FAS, 
%  FAS GMPEs by Bora et al. (2015, 2018), Bayless \&
%  Abrahamson (2019), 
% % for marginal median \& variability, 
% %Inter-frequency correlation structure by 
% Stafford(2017) and Bayless \& Abrahamson (2018).  
% 
% \vspace*{1mm}
% \item[-] Stochastic FPS by phase derivative (Boore,2005), Logistic phase derivative
% model by Baglio \& Abrahamson (2017)
% 
% %\item Upcoming: a major change from \textbf{ $\boldsymbol{Sa(T_0)}$ to FAS} in next five years as envisioned by Abrahamson (2018)
% 
% \end{itemize}
% 
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% 
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \begin{frame}
% % 
% % \frametitle{Stochastic Seismic Motions}
% % 
% % %  
% % 
% % %Lognormal distributed random field with PC Dim. 3 Order 2
% % 
% % 
% % 
% % %\vspace*{-50mm}
% % \begin{figure}[!hbpt]
% % \begin{center}
% % \hspace*{-10mm}
% % \includegraphics[width=12.5cm]{/home/jeremic/tex/works/Conferences/2021/CompDyn_8th_Athens_21-23Jun2020/present/Stochastic_seismic_motions.jpg}
% % \end{center}
% % \end{figure}
% % 
% % \end{frame}
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
%          
% 
% % 
% % 
% % 
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% % \begin{frame}
% % 
% % \frametitle{Stochastic Seismic Motions, Displacement}
% % 
% % %  
% % 
% % %\vspace*{-50mm}
% % \begin{figure}[!hbpt]
% % \begin{center}
% % \hspace*{-10mm}
% % \includegraphics[width=12.5cm]{/home/jeremic/tex/works/Conferences/2021/CompDyn_8th_Athens_21-23Jun2020/present/Stochastic-motions-DISP.jpg}
% % \end{center}
% % \end{figure}
% % 
% % \end{frame}
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % 
%          
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% 
% \frametitle{Stochastic Seismic Motions, Accelerations}
% 
% %  
% 
% \vspace*{-1mm}
% \begin{figure}[!hbpt]
% \begin{center}
% \hspace*{-10mm}
% \includegraphics[width=12.5cm]{/home/jeremic/tex/works/Conferences/2021/CompDyn_8th_Athens_21-23Jun2020/present/Stochastic-motions-ACC.jpg}
% \end{center}
% \end{figure}
% 
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% 
% 
% % 
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \begin{frame}
% % 
% % \frametitle{Stochastic Site Response}
% % 
% % %  
% % 
% % 
% % \vspace*{-3mm}
% % \begin{figure}[!hbpt]
% % \begin{center}
% % \hspace*{-10mm}
% % \includegraphics[width=12.5cm]{/home/jeremic/tex/works/Conferences/2021/CompDyn_8th_Athens_21-23Jun2020/present/Stochastic-site-response.jpg}
% % \end{center}
% % \end{figure}
% % 
% % 
% % \vspace*{-2mm}
% % PC dimensions $\xi_1$, - $\xi_3$ are from uncertain soil
% % 
% % PC dimensions $\xi_4$, - $\xi_{153}$ are from uncertain rock motions
% % 
% % \end{frame}
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % 
% 
% 
% 
% 

% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% 
% \frametitle{Sensitivity of PGA from Uncertain Soil}
% 
% %  
% 
% 
% \vspace*{10mm}
% \begin{itemize}
%   \item[-] First 10 terms \\
%         from soil uncertainty
% 
% 
% 
%   \item[-] Total Sobol     \\
%         sensitivity index  \\
%         $S_{1-3}^{PC,total} = 0.51$
% 
% 
% \end{itemize}
% 
% 
% 
% \vspace*{-40mm}
% \begin{small}
% %\begin{center}
% \begin{table}[!htb]
% \hspace*{50mm}
% \begin{tabular}{l l}
% Sobol Index           &  Value  \\ \hline 
% $S_{1,123}^{PC}$      &   0,04389  \\
% $S_{1,118}^{PC}$      &   0,02605  \\
% $S_{1,127}^{PC}$      &   0,02370  \\
% $S_{1,100}^{PC}$      &   0,01759  \\
% $S_{1,103}^{PC}$      &   0,01700  \\
% $S_{1,134}^{PC}$      &   0,01680  \\
% $S_{1,141}^{PC}$      &   0,01611  \\
% $S_{1,110}^{PC}$      &   0,01358  \\
% $S_{1,130}^{PC}$      &   0,01303  \\
% $S_{1,132}^{PC}$      &   0,01068 \\
% . . .                &   . . .     \\ \hline  
%     \end{tabular}
% \end{table}
% %\end{center}
% \end{small}
% 
% 
% 
% 
% %
% %
% %PC dimensions $\xi_1$, - $\xi_3$ are from uncertain soil
% %
% %PC dimensions $\xi_4$, - $\xi_{153}$ are from uncertain motions
% 
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% 


% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% 
% \frametitle{Sensitivity of  PGA from Uncertain Rock Motions}
% 
% %  
% 
% 
% \vspace*{10mm}
% \begin{itemize}
%   \item[-] First 10 terms \\
%         from motions uncertainty
% 
% 
%   \item[-] Total Sobol     \\
%         sensitivity index \\
%         $S_{1-153}^{PC,total} = 0.98$
% 
% 
% \end{itemize}
% 
% 
% 
% \vspace*{-40mm}
% \begin{small}
% %\begin{center}
% \begin{table}[!htb]
% \hspace*{50mm}
% \begin{tabular}{l l}
% Sobol Index           &  Value  \\ \hline 
% $S_{1,123}^{PC}$      &   0,04389  \\
% $S_{137}^{PC}$      &   0,03459  \\
% $S_{110}^{PC}$      &   0,03061  \\
% $S_{118}^{PC}$      &   0,02698  \\
% $S_{1,118}^{PC}$      &   0,02605  \\
% $S_{108}^{PC}$      &   0,02482   \\
% $S_{141}^{PC}$      &   0,02373  \\
% $S_{1,127}^{PC}$      &   0,02370  \\
% $S_{1,100}^{PC}$      &   0,01759  \\
% $S_{1,103}^{PC}$      &   0,01700  \\
% . . .                &   . . .     \\ \hline  
%     \end{tabular}
% \end{table}
% %\end{center}
% \end{small}
% 
% 
% 
% 
% %
% %
% %PC dimensions $\xi_1$, - $\xi_3$ are from uncertain soil
% %
% %PC dimensions $\xi_4$, - $\xi_{153}$ are from uncertain motions
% 
% \end{frame}
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\begin{frame}

\frametitle{Sensitivity Analysis}

%  


%\begin{itemize}
%
%
\vspace*{4mm}
%  \item[-] 
Total variance in PGA, in this particular case (!), 
dominated by uncertain ground motions
    \begin{itemize}
\vspace*{4mm}
      \item[] $49$\% from uncertain rock motions at depth
\vspace*{4mm}
      \item[] $2$\% from uncertain soil
\vspace*{4mm}
      \item[] $49$\% from interaction of uncertain rock motions and  uncertain soil 


    \end{itemize}


%  \item 
%\end{itemize}



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\begin{frame}
%\frametitle{ESSI Challenges}
\frametitle{Increase Resilience of NEF SSI Systems}
           

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\vspace*{5mm}

\begin{itemize}

%\vspace*{-2mm}
  \item[1.]  
%  MCDEEP   
Plasticity of soil, \\
structure, and the \\
  soil-foundation interface
%\citep{Jeremic2009b,Jeremic2020},

%\vspace*{-2mm}
  \item[2.]   
%  MCDEEP 
  Energy  dissipators, \\
  energy sinks
%, within structure, 
%  for example buckling restrained braces (BRBs), frictional pendulum,
%  lead core elastomers,  
%  etc.
%\citep{Symans2008,Nucera2007,Jeremic2020}, 
%  and/or
%  other nonlinear energy sinks

%\vspace*{-2mm}
  \item[3.] 
%  MCDEEP 
Viscous dampers and \\
viscous coupling
% between   fluid and structure
%\cite{Symans1998,Symans2008,Roh2010},

%\vspace*{-2mm}
  \item[4.]  
%   MCDEEP 
External   trenches  
%\\ surrounding the
%  structure
%%\citep{Woods1968,Bolt1981},

%\vspace*{-2mm}
   \item[5.]  
%   MCDEEP 
Meta-materials, \\
meta-devices
%, for example resonant unit cells, 
%   negative stiffness meta-materials, etc., adjacent to or within the
%   structure
%\citep{Kanellopoulos2020, Chondrogiannis2021},

\end{itemize}
%


%
\vspace*{-50mm}
\begin{figure}[!htb]
\begin{flushright}
% % %\includegraphics[width=5cm]{Figure-files/_Chapter_Applications_ESSI_BOOK/Building_modeling_Issues_01_pdf.pdf}
% %\vspace*{-3mm}
% \includegraphics[width=2.5cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/_Chapter_Applications_ESSI_BOOK/Seismic_Motions_modeling_Issues_01_pdf.pdf}
% \includegraphics[width=5.5cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/_Chapter_Applications_ESSI_BOOK/Seismic_Motions_modeling_Issues_02_pdf.pdf}
% \\
%\vspace*{-2mm}
%\vspace*{-8mm}
%\vspace*{-2mm}
%\hspace*{-15mm}
\includegraphics[width=7cm]{/home/jeremic/tex/works/Proposals/2022/DOE_NSRD_program_FY22/Energy_Balance/proposal/figures/seismic_shielding_NPP_01.pdf}
%\hspace*{-5mm}
%\vspace*{-5mm}
\hspace*{-10mm}
\end{flushright}
%\vspace*{-5mm}
\end{figure}


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%    \frametitle{Meta-Devices for Resilient NEF SSI}
%    
%    
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%    %\vspace*{-2mm}
%    \begin{center}
%    \hspace*{-16mm}
%    %\movie[label=show3,width=5.6cm,poster,autostart,showcontrols]
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%      {\includegraphics[width=60mm,trim={0 0 0 0},clip]{
%       /home/jeremic/tex/works/Conferences/2024/DOE_Natural_Hazard_Phenomena_at_NRC_29-30Oct2024/present/moment_frame_building-without_metamaterials-inelastic_soil-tied_interfaces-inelastic_fiber_beam_elements-ormsby_05-10Hz_view1.jpg}}
%    %  {/home/jeremic/tex/works/Conferences/2024/DOE_Natural_Hazard_Phenomena_at_NRC_29-30Oct2024/present/moment_frame_building-without_metamaterials-inelastic_soil-tied_interfaces-inelastic_fiber_beam_elements-ormsby_05-10Hz_view1.mp4}
%      {/home/jeremic/tex/works/Conferences/2024/DOE_Natural_Hazard_Phenomena_at_NRC_29-30Oct2024/present/moment_frame_building-without_metamaterials-inelastic_soil-tied_interfaces-inelastic_fiber_beam_elements-ormsby_05-10Hz_view1.ogv}
%    %
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%      {\includegraphics[width=60mm]{/home/jeremic/tex/works/Conferences/2024/DOE_Natural_Hazard_Phenomena_at_NRC_29-30Oct2024/present/moment_frame_building-with_metamaterials-inelastic_soil-tied_interfaces-inelastic_fiber_beam_elements-ormsby_05-10Hz_view1.jpg}}
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\section{Summary}

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\subsection{\ }


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\frametitle{Summary}

\begin{itemize}
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\item[-] Sophisticated, realistic analysis: improve design, safety
% of NEF SSI systems
%\item[-] Sophisticated analysis of NEF SSI systems

% \item[-] Invest in analysis tools, computers, Education and Training

% %\vspace*{1mm}
% \vspace*{1mm}
%       \item[-] Realistic analysis: better design, improved safety

%\vspace*{1mm}
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      \item[-] Improved infrastructure economy, and (!) $CO_2$ $\downarrow$ 
% %\vspace*{1mm}
% %      \item[-] Engineering \$ $\uparrow$
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%       \item[-] Construction material savings, $CO_2$ $\downarrow$
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%       \item[-] Sustainable infrastracture
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%     \item[-] Engineering analysis to \underline{predict} and \underline{inform}

%\vspace*{1mm}
\vspace*{1mm}
  \item[-] Education and Training is the Key !
%\vspace*{1mm}
%   \item[-] Analysis tool: \underline{\url{http://real-essi.us}}
%    All available within \url{http://real-essi.us}
      




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%     \item[-] Engineering analysis to \underline{predict} and \underline{inform}
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% %    \item[-] Engineers  need to know and improve Safety and Economy
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%   \item[-] Stress test, shake-out infrastructure systems
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%   \item[-] Direct, control static and dynamic energy propagation

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% %  \item[-]  
%   \item[-] Mechanical energy dissipation within infrastructure
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%   \item[-] Education and Training is the Key !
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   \item[-] Analysis tool: Real-ESSI Simulator
%   \item[-] Analysis tool: Real-ESSI Simulator (\underline{\url{http://real-essi.us}})
% \underline{\url{http://real-essi.us}}
%    All available within \url{http://real-essi.us}
      




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   \item[-]     
%Collaborators:   
            Students/PostDocs:
            Yang, Cheng, 
            Tafazzoli, Feng,  Yang,  
            Sinha,  Wang, 
            Pisan{\'o}, Abell, Tafazzoli, Sett, 
            Vilhar, Jeong, Jie, Preisig, Liu, Jain, Liao, Wu, Li, 
            Tasiopoulou, Watanabe, Luo, 
            Cheng, Yang, Kanellopoulos, Staszewska ...
   

      

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  \item[-]  Collaboration/Funding, much appreciated:
%  \item[-]  Collaboration and Funding is much appreciated:
% Appreciate funds and  collab from
  US-DOE, 
  US-NRC, 
  US-NSF, 
  US-DOD,
  Caltrans,
  US-FEMA/ATC, 
  CNSC-CCSN, 
  UN-IAEA, 
  CERN,
  ETH,
%  CH-ENSI and 
%  CH-ENSI/Basler\&Hofmann AG,
  CH-ENSI,
  Basler\&Hofmann AG,
  Shimizu Corp.



\end{itemize}




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