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% %% 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
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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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%             colorlinks=true,
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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[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.
\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)
\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}
%\institute[\pgfuseimage{university-logo}\hspace*{0.1truecm}\pgfuseimage{lbnl-logo}] % (optional, but mostly needed)
\institute[\pgfuseimage{university-logo}] % (optional, but mostly needed)
%{ 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.
% - Keep it simple, no one is interested in your street address.


\date[] % (optional, should be abbreviation of conference name)
{\small NHERI Lehigh Seminar\\
         11May2023}

\subject{}
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%\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}}




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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
% solution:

% - 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.
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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/rock-structure 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:\\
%     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}
\end{frame}
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\begin{frame}
\frametitle{ESSI Challenges}


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%
\begin{figure}[!htb]
\begin{center}
% %\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}
\includegraphics[width=3cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/_Chapter_Applications_ESSI_BOOK/NPP_Modeling_Issues_03.jpg}
\includegraphics[width=2.2cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/_Chapter_Applications_ESSI_BOOK/SMR_Modeling_Issues_02_pdf.pdf}
%\\
%\vspace*{-2mm}
\includegraphics[width=2.8cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/_Chapter_Applications_ESSI_BOOK/Building_modeling_Issues_03_pdf.pdf}
\includegraphics[width=2.5cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/_Chapter_Applications_ESSI_BOOK/Building_modeling_Issues_02_pdf.pdf}
\\
%\vspace*{2mm}
\includegraphics[width=1.0cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/_Chapter_Applications_ESSI_BOOK/Dam_modeling_Issues_01_pdf.pdf}
\includegraphics[width=3.5cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/_Chapter_Applications_ESSI_BOOK/Dam_modeling_Issues_02_pdf.pdf}
\includegraphics[width=2.0cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/_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 Building-foundation-soil system;
% Dam-Foundation-Fluid system;
% Bridge-soil  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 2-node/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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\begin{frame}
 \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}ois-Marie Arouet, Voltaire)



\end{frame}
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\begin{frame}
 \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}


% local
% %\vspace*{-2mm}
% \begin{center}
% \hspace*{-7mm}
% %\movie[label=show3,width=8.8cm,poster,autostart,showcontrols]
% \movie[label=show3,width=5.5cm,poster,autostart, showcontrols]
%   {\includegraphics[width=50mm]
%   {/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}
% %
% %\hfill
% \hspace*{-5mm}
% %
% \movie[label=show3,width=6.0cm,poster,autostart,showcontrols]
%   {\includegraphics[width=50mm]
%   {/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}
% \hspace*{-7mm}
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% %%
% \end{center}
% % local
% 
% % % \vspace*{-5mm}
% % \begin{center}
% % %\begin{flushleft}
% % % \hspace*{-15mm}
% % \movie[label=show3,width=5cm,poster,autostart,showcontrols]
% % {\includegraphics[width=5cm]
% % {/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{flushleft}
% % %%
% % \hfill
% % %%
% % %\begin{flushright}
% % % \hspace*{-15mm}
% % \movie[label=show3,width=5cm,poster,autostart,showcontrols]
% % {\includegraphics[width=5cm]
% % {/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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\begin{frame}
\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/JGGE-GoverGmax/figures/YoungModulus_RawData_and_MeanTrend_01-Ed.pdf}
\hspace*{-3mm}
% \hfill
\includegraphics[width=4.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGE-GoverGmax/figures/YoungModulus_Histogram_Normal_01-Ed.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}
%
\end{center}
\end{figure}


\vspace*{-5mm}
\begin{flushright}
{\tiny
(cf. Wang et al. (2019))\\
~}
\end{flushright}








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\section{ESSI Uncertainties}

      


















      









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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}ois-Marie Arouet, Voltaire)



\end{frame}
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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}
%
\hspace*{-10mm}
\includegraphics[width=5truecm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/Lotung_LSST07_FA25.jpeg}
\hspace*{5mm}
\includegraphics[width=5truecm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/Lotung_LSST12_FA25.jpeg}
\hspace*{-10mm}
%
\end{center}
\end{figure}
   



%\vspace*{3mm}
%\item[-] Effects of real earthquakes on soil-structure systems ?!




\end{frame}


% % \subsection*{Fundamentals}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{T{o}hoku Earthquake, Acc, Disp}
% 
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% \vspace*{-2cm}
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% {\includegraphics[width=6.0cm]{/home/jeremic/tex/works/Conferences/2011/NRC_Staff_Capacity_Building_25May2011/EW_acc4.jpg}}
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% \begin{frame}
% \frametitle{T{o}hoku  Earthquake (DSP)}
% 
% 
% \vspace*{-2cm}
% \begin{figure}[!h]
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% {\includegraphics[width=7.0cm]{/home/jeremic/tex/works/Conferences/2011/NRC_Staff_Capacity_Building_25May2011/EW_dsp4.jpg}}
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% \begin{frame}
% \frametitle{T{o}hoku Earthquake, 3D Motions}
% 
% 
% \vspace*{-2cm}
% \begin{figure}[!h]
% \begin{center}
% {\includegraphics[width=3.0cm]{/home/jeremic/tex/works/Conferences/2011/NRC_Staff_Capacity_Building_25May2011/Tohoku_1Hz-11-locations.jpg}}
% {\includegraphics[width=6.0cm]{/home/jeremic/tex/works/Conferences/2011/NRC_Staff_Capacity_Building_25May2011/Tohoku_1Hz-21.jpg}}
% \end{center}
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% \begin{frame}
%   \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}
% 
% 
% %
% %   \item[-]   1D (?): M 6.9 San Pablo, Guatemala EQ, 14Jun2017
% %
% \end{itemize}
% 
% 
% \end{frame}
% 


% 
%      
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  
%  \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/Figure-files/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/Lotung_LSST07_FA25.jpeg}
%  \hspace*{5mm}
%  \includegraphics[width=5truecm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/Lotung_LSST12_FA25.jpeg}
%  \hspace*{-10mm}
%  %
%  \end{center}
%  \end{figure}
%     
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%  \end{itemize}
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% 
% \begin{frame}
%   \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/}
% %
% %\vspace*{-5mm}
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% \hspace*{-10mm}
% \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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% \begin{frame}
% \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
% 
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% \vspace*{3mm}
%   \item[-] Knowledge of fault and geology needed
% 
% \vspace*{3mm}
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%     \begin{itemize}
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%        \item[-] fault rupture
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% \frametitle{Regional Geophysical  Models}
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%  \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 
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%   \includegraphics[width=10truecm]{/home/jeremic/tex/works/Conferences/2017/CompDyn2017/Present_PLENARY/USG-Bay_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}
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% USGS
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% %   \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}
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% % Rodgers and Pitarka 
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% % \end{frame}
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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 }
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% % -  \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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%   \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}
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% % \caption{\label{Fig:NPP_Model_In_Real_ESSI} Nuclear Power Plant Model with Shallow Foundation }
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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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% % online   
% \vspace*{-5mm}
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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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\begin{frame}
\frametitle{ESSI: 3C/6C vs 1C Seismic Motions}


\begin{itemize}


\item[-]  Assume: full 3C/6C motions, recorded only 1C 

\item[-]   From 1C motions, de-convolute/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}
\end{center}
\end{figure}



 \end{frame}



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\begin{frame}
 \frametitle{Realistic Ground Motions}

\vspace*{2mm}

\begin{itemize}

  \item[-]  Free field, regional scale models

\end{itemize}

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{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Free_Field_small_model_April2015/movie_input.mp4}
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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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% online  \href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Free_Field_small_model_April2015/movie_input.mp4}
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  {\includegraphics[width=80mm]{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Free_Field_small_model_April2015/movie_input_closeup.jpg}}
{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Free_Field_small_model_April2015/movie_input_closeup.mp4}
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\begin{frame}
\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)


\end{itemize}




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  {\includegraphics[width=60mm]{/home/jeremic/tex/works/Conferences/2016/IAEA_TecDoc_February2016/My_Current_Work/movie_ff_3d_mp4_icon.jpeg}}
  {/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Model01_ESSI_Response_May2015/movie_ff_3d.mp4}
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% local


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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_ff_3d.mp4}
{\tiny (MP4)}
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  {/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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% \begin{itemize}
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% \vspace*{2mm}
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\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: 
	             3D-deconvolution 

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%  \frametitle{Free Field, Variation in Input Wave Angle,  $f = 5$Hz}
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%  {/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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%  \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}
%  \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/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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%\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}
        
% online   
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\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/free_field_frequency.mp4}
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\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}
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% online  
   

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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}
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% online  
   

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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/SMR-ESSI_real_time_four_freq.mp4}
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%          
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%  \href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/SMR_animations_May2018/SMR-ESSI_real_time_four_freq.mp4}
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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}
% 
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% \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_50m-mesh_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})}
\end{flushleft}
% online
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%\vspace*{10mm}


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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, meta-materials
       \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 elastic-plastic element under cyclic shear loading
 
%\begin{itemize}
%\item[] Plastic work $\ne$ plastic dissipation
%\item[] 
Area of load-displacement 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_50m-mesh_45degree_Ormsby.mp4}
  {\tiny (MP4)}
\end{flushleft}
%


\end{frame}

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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_50m-mesh_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]
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\movie[label=show3,width=61mm,poster, showcontrols]
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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/Individual_Foundation.mp4}
{\tiny (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/Continuous_Foundation.mp4}
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\frametitle{ASCE-7-21, Low Building: BRB Energy Dissipation}


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%\movie[label=show3,width=10cm,poster,autostart,showcontrols]
\movie[label=show3,width=10cm,poster,showcontrols]
{\includegraphics[width=10cm]
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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_plasticity.pdf}
%      \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 Contact/interface/joint, inelastic, soil/rock -- foundation
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%  The nuclear power plant structure comprise of
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%   \item Containment/Shield building, $f^{cont}_{1}= 4Hz$
%   \item Concrete raft foundation: $3.5m$ thick
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%   \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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%   \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 (concrete-soil), gaping and nonlinear shear
%    \end{itemize}
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%   \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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%    \includegraphics[width=7.0cm]{/home/jeremic/tex/works/Thesis/SumeetKumarSinha/Files_10Aug2017/Npp_Non_Linear_Effects/images/Elastic_Without_Contact_SMIRT_2017.pdf}
%   \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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% %  \movie[label=show3,width=10cm,poster,autostart,showcontrols]
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% %  {/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}
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% \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}
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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}
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% %        \begin{center}
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% %       \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}
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% %  %   \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}
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% %  % %    \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}
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% %  % %    \vspace{-0.3cm}
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% %         \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}
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% %       \includegraphics[width=5cm]{/home/jeremic/tex/works/Thesis/HexiangWang/Files_SMiRT_11Aug2017/pic/bouyant_displacement.pdf}
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%\subsection{Stochastic Modeling}
\subsection{Aleatory, Parametric Uncertainties}





\begin{frame}
 \frametitle{Aleatory, Parametric Uncertainties}


  \begin{itemize}


\vspace*{3mm}
    \item[]  ${M}  \ddot{u_i}  +  {C}  \dot{u_i} + {K}^{ep} {u_i} = {F(t)}$, 

\vspace*{5mm}
    \item[] Uncertain: mass $M$, viscous damping  $C$ and stiffness $K^{ep}$


\vspace*{5mm}
    \item[] Uncertain  loads $F(t)$


\vspace*{5mm}
    \item[] Results are PDFs   and   CDFs  for  $\sigma_{ij}$,  $\epsilon_{ij}$,  $u_i$,  $\dot{u}_i$,  $\ddot{u}_i$

  \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}ois-Marie Arouet, Voltaire)
% 


\end{frame}
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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}




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\begin{frame} 

\frametitle{Forward Uncertain Inelasticity}

% 

\begin{itemize}



\item[-] Incremental el--pl 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}








  

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\begin{frame} 

\frametitle{Probabilistic Elastic-Plastic Response}


\begin{figure}[!hbpt]
\begin{center}
%\includegraphics[width=8cm]{/home/jeremic/tex/works/Papers/2007/ProbabilisticYielding/figures/vonMises_G_and_cu_very_uncertain/Contour_PDF-edited.pdf}
\includegraphics[width=8cm]{/home/jeremic/tex/works/Conferences/2012/DOE-LLNL-workshop-27-28-Feb-2012/ProbabilisticYielding_vonMises_G_and_cu_very_uncertain_Contour_PDF-edited.pdf}
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\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_Randomp0-m.pdf}
%\hspace*{-2mm}
\includegraphics[width=6.0cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourHighOCR_RandomG_RandomM-m.pdf}
\hspace*{-15mm}
\end{center}
\end{figure}


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\end{frame}
% --  %%%%%%









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\begin{frame}{Stochastic Elastic-Plastic 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{(non-Gaussian, heterogeneous/ non-stationary)}}:
          Multi-dimensional Hermite Polynomial Chaos (PC) with {known coefficients}
%\vspace{0.05in}
\vspace*{2mm}
\item[-] Output response process: Multi-dimensional 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 elastic-plastic 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 zero-mean unit-variance 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}
% 
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% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \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 elasto-plasticity with stochastic Armstrong-Frederick
% 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 Elastic-Plastic 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}
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% % \hspace*{-15mm}
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% {\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
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% {/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Probabilistic_Elasto_Plasticity_and_Stochastic_Elastic_Plastic_Finite_Element_Method/PEP_Animation.mp4}
% \end{center}
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% \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_50m-mesh_45degree_Ormsby.mp4}
%   {\tiny (MP4)}
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%\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 Elastic-Plastic 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}}
% 
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% 
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\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_50m-mesh_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_50m-mesh_45degree_Ormsby.mp4}
%   {\tiny (MP4)}
% \end{flushleft}
% %
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% 





















% 
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  \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 
%  
%  % \[\lambda(EDP>z) = \int_{IM} \underbrace{|\frac{d\lambda(IM)}{dIM}|}_\text{PSHA} \underbrace{G(EDP|IM)}_{\text{fragility}} dIM \]
%  
%  \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}
%  
%  % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  % 
%  % \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}
%  % 
%  % \end{itemize}
%  % \end{frame}
%  % 
%  
%  
%  
%  

















%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \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}
% 
% 
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsection{Issues in State-of-the-art SRA}

%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  \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}
%  
%  \end{frame}
%  
%  % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  
%  
%  
%  
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  \begin{frame}
%  
%  \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  $\sigma_{ij}$, 
%  $\epsilon_{ij}$, $u_i$... 
%  
%  
%  
%  
%  
%  
%  
%  
%  \end{itemize}
%  
%  \end{frame}
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  
%  
%  
%  
%  
%  
%  
%  
%  
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  \begin{frame}
%  \frametitle{Stochastic Elastic-Plastic 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}
%  
%  
%  
%  
%  
%  \end{itemize}
%  
%  
%  
%  \end{frame}
%  
%  
%  
%  
%  % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  % 
%  % \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}}
%  % 
%  % 
%  % \end{frame}
%  % 
%  % 
%  % 
%  % 
%  
%  
%  
%  
%   
%  %  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %  \begin{frame}
%  %  
%  %  \frametitle{Monte Carlo (MC)}
%  %  
%  %  \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 
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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}


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\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}

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\begin{textblock}{15}(4.6, 13.2)
%$\Leftarrow$
{\Large $\leftarrow$}
\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=-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 $\lambda(EDP>z)=$

$\quad \sum N_i(M_i, R_i) P(EDP>z|M_i, R_i)$
\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}
% 
% % \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}
% 
% \end{frame}
% 
% \endgroup
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \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}(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}
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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}
% 
% \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}(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}
% 
% \endgroup
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 


% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% 
% \begin{frame}
% 
% \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 Inter-frequency 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 Inter-frequency 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}
% 
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%





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% %\subsection{Illustrative Example}
% %
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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 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}
%



\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 \\
%                   El-Pl 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}
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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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%  
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%  
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%  \item[-] Incremental el--pl 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} \ddot{ u_i} +
%  { C} \dot{ u_i} +
%  { K}^{ep} { u_i} =
%  { F(t)}
%      \nonumber
%  \end{equation}
%  
%  
%  \vspace*{2mm}
%  \item[-] Material and loads are uncertain
%  
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%  \end{itemize}
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%  \frametitle{{Cam Clay  with  Random $G$, $M$ and $p_0$}}
%  
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%  \begin{figure}[!hbpt]
%  \begin{center}
%  \hspace*{-15mm}
%  \includegraphics[width=6.0cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourLowOCR_RandomG_RandomM_Randomp0-m.pdf}
%  %\hspace*{-2mm}
%  \includegraphics[width=6.0cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourHighOCR_RandomG_RandomM-m.pdf}
%  \hspace*{-15mm}
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%  \begin{frame}{Time Domain Stochastic Galerkin Method}
%  
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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{(non-Gaussian, heterogeneous/ non-stationary)}}:
%            Multi-dimensional Hermite Polynomial Chaos (PC) with {known coefficients}
%  %\vspace{0.05in}
%  \vspace*{2mm}
%  \item[-] Output response process: Multi-dimensional 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 elastic-plastic constitutive law at each time step
%   
%  \end{itemize}
%  
%  %\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 zero-mean unit-variance 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}
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%  % \subsection{Direct Solution for Probabilistic Stiffness and Stress in 1D}
%  % 
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%  %%%%%%%%%%%%%%%%%%%%%%%%%%    BEGGINING PEP    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%  \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 elasto-plasticity with stochastic Armstrong-Frederick
%  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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%  \end{itemize}
%  
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%  % \vspace{0.5cm}
%  
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%  
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%  
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%  % \end{itemize}
%  % 
%  \end{frame}
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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}
%    
%    
%  
%  \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}
%  % 
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%  \item[-] Incremental stress: each  Polynomial  Chaos  component  is updated
%  incrementally 
%  % via stochastic Galerkin projection
%  
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%   
%    {$\Delta\sigma_1^{n+1}  = \frac{1}{<\Phi_1\Phi_1> }\{\sum_{i=1}^{P_h} \sum_{k=1}^{P_e}\ h_i  \Delta \epsilon_k^n <\Phi_i \Phi_k \Phi_1>- \sum_{j=1}^{P_g} \sum_{k=1}^{P_e}\sum_{l=1}^{P_{\sigma}} \ c_j \Delta \epsilon_k^n \sigma_l^n <\Phi_j \Phi_k \Phi_l \Phi_1>\}$}
%  \\
%  . . . 
%  \\
%  % ${\vdots}$ 
%  {$\Delta\sigma_P^{n+1}  = \frac{1}{<\Phi_P\Phi_P> }\{\sum_{i=1}^{P_h} \sum_{k=1}^{P_e}\ h_i  \Delta \epsilon_k^n <\Phi_i \Phi_k \Phi_P>- \sum_{j=1}^{P_g} \sum_{k=1}^{P_e}\sum_{l=1}^{P_{\sigma}} \ c_j \Delta \epsilon_k^n \sigma_l^n <\Phi_j \Phi_k \Phi_l \Phi_P>\}$}
%  
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%  \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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%  %\section[Formulation]{Stochastic Dynamic Finite Element Formulation}
%  %\subsection[Time domain stochastic Galerkin method]{Time domain stochastic Galerkin method}
%  %\frame{\tableofcontents[currentsubsection,sectionstyle=show/shaded]}
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%  \frametitle{Stochastic Elastic-Plastic Finite Element Method}
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%  \begin{itemize}
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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 
%  %Stochastic system of equation resulting from Galerkin approach (static example):
%  %
%  %\item  Time domain integration  using Newmark and/or HHT, in probabilistic spaces
%  
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%  \item[-] Jeremi{\'c} et al. 2011
%  
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%  \end{itemize}
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%  \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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%  \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 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{Backward  Uncertainty Propagation, Sensitivities}
%%\subsection{Sobol Sensitivity Analysis}


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\begin{frame}
\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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% 
%  
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%  \begin{frame}
%  \frametitle{ANOVA Representation}
%  
%  
%  
%  
%   
%  
%  %\vspace*{10mm}
%  
%  Model with $n$ uncertain inputs ($\boldsymbol{x}$) and scalar output $y$:
%  
%  
%  \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 $I^{n}$
%  %
%  
%  \vspace*{5mm}
%   
%  The ANalysis Of VAriance representation
%  % of $f(x)$ 
%  (Sobol 2001):
%  
%  \vspace*{-6mm}
%  \begin{eqnarray*}
%  f(x_1, ... x_n) = f_0 + \sum_{i=1}^{n} f_i(x_i) + 
%  \sum_{1\leq i<j\leq n} 
%  f_{ij}(x_i, x_j) 
%  +...
%  %\\
%  %+ 
%  f_{1,...n}(x_1,...x_n) 
%  \nonumber
%  \label{eq_ANOVA_representation}
%  \end{eqnarray*}
%  
%  
%  \end{frame}
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%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%  \begin{frame}
%  \frametitle{Sensitivity Analysis, ANOVA Representation}
%  
%  
%  
%  
%  Total of $2^n$ summands
%  
%  \vspace*{4mm}
%  %
%  Mean value $f_0 = \int_{I^{n}} f(\boldsymbol{x}) d \boldsymbol{x}$ 
%  %
%  
%  \vspace*{4mm}
%  Integral of each summand 
%  $\int_{0}^{1} f_{i_1,... i_s}({x_{i_1},... x_{i_s}}) dx_{i_k} = 0 \mbox{;~}
%   1 \leq k \leq s$ 
%  %in Equation \ref{eq_ANOVA_representation} over any of its independent variable is zero: 
%  
%  %\begin{equation}
%  %\int_{0}^{1} f_{i_1, ..., i_s}({x_{i_1}, x_{i_2}, ..., x_{i_s}}) \ dx_{i_k} = 0 \ \ \ \mathrm{for} \ \ \ 1 \leq k \leq s 
%  %\label{eq_summand_integral_zero}
%  %\end{equation}
%  % 
%  % 
%  % 
%  
%  
%  \vspace*{4mm}
%  Summands are orthogonal to each other
%  
%  \vspace*{-6mm}
%  %
%  \begin{eqnarray*}
%  \int_{I^{n}} f_{i_1,...i_s}({x_{i_1}, x_{i_2},...x_{i_s}}) 
%  f_{j_1,...j_t}({x_{j_1}, x_{j_2},...x_{j_t}}) d \boldsymbol{x} = 0 
%  \mbox{;}
%  %\mathrm{for} \ \ \
%  \\
%  \{i_1,...i_s\} \neq \{j_1,...j_t\}  
%  \nonumber
%  \label{eq_orthogonal}
%  \end{eqnarray*}
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%  \begin{frame}
%  \frametitle{Sensitivity Analysis, ANOVA Representation}
%  
%  ANOVA representation is unique!
%  % and derived analytically
%  
%  \vspace*{5mm}
%   
%  Univariate terms: 
%  %
%  \begin{eqnarray*}
%  f_i(x_i) = \int_{I^{n-1}} f(\boldsymbol{x}) \ d\boldsymbol{x}_{\sim i} - f_0 
%  \end{eqnarray*}
%  
%  %$\int_{I^{n-1}} (\cdot) \ d\boldsymbol{x}_{\sim i}$ is integral
%  %over all dimensions except $x_i$.
%  %
%  
%  \vspace*{5mm}
%  
%  
%  Bivariate terms:
%  % 
%  \begin{eqnarray*}
%  f_{ij}(x_i, x_j) = \int_{I^{n-2}} f(\boldsymbol{x}) \ d\boldsymbol{x}_{\sim [ij]} - f_i(x_i) - f_j(x_j) - f_0 
%  \end{eqnarray*}
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%  \begin{frame}
%  \frametitle{Sensitivity Analysis, Variance}
%  
%  \vspace*{3mm}
%  
%  Total variance of the probabilistic model response
%  $y=f(\boldsymbol{X})$ is
%  %
%  \begin{eqnarray*}
%  %D = \mathrm{Var}[f(\boldsymbol{X})] = \int_{I^{n}} f^2(\boldsymbol{x}) \ d\boldsymbol{x} - f_0^2
%  D = {Var}[f(\boldsymbol{X})] = \int_{I^{n}} f^2(\boldsymbol{x}) \ d\boldsymbol{x} - f_0^2
%  \end{eqnarray*}
%  
%  %Using Equations~\ref{eq_ANOVA_representation} and \ref{eq_orthogonal}, 
%  Total variance $D$, decomposed 
%  %
%  \begin{eqnarray*}
%  D = \sum_{i=1}^{n} D_i + \sum_{1\leq i <j \leq n} D_{ij} + ... + D_{1,2,...,n} = \sum_{s=1}^{n}\sum_{i_1<...<i_s}D_{i_1...i_s}
%  \label{eq_variance_decomposition}
%  \end{eqnarray*}
%  %
%  
%  Variance contribution from individual summand
%  %
%  \begin{eqnarray*}
%  D_{i_1..i_s} = \int_{I^s} f^2_{i_1...i_s}(x_{i_1}, ..., x_{i_s}) \ dx_{i_1}, ...dx_{i_s}, 
%  \\
%  1\leq i_1 < ... < i_s \leq n \mbox{;~} s= 1,...n
%  \end{eqnarray*}
%  %    
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%  \begin{frame}
%  \frametitle{Sobol Indices}
%  
%  
%  \begin{itemize}
%  
%  \item[-] Sobol' indices $S_{i_1...i_s}$,
%  fractional contributions from 
%  random inputs $\{X_{i_1}, ..., X_{i_s}\}$ to the total variance $D$:
%  $S_{i_1...i_s} = D_{i_1...i_s} / D$
%  % %The Sobol indices $S_{i_1...i_s}$: 
%  % %
%  % \begin{equation*}
%  % S_{i_1...i_s} = D_{i_1...i_s} / D
%  % \end{equation*}
%  % %
%  
%  
%  \vspace*{2mm}
%  
%  \item[-] First order indices $S_i$ $\rightarrow$ individual influence of each uncertain input
%  parameter 
%  %$X_i$
%  %
%  
%  \vspace*{2mm}
%  \item[-] Higher order indices $S_{i_1...i_s}$ $\rightarrow$ mixed influence from groups
%  of uncertain input parameters
%  %
%  
%  
%  
%  \vspace*{2mm}
%  \item[-] Total sensitivity indices,
%  % $S^{\mathrm{total}}_i$
%  %$\rightarrow$ 
%  influence of input parameter $X_i$
%  %
%  \begin{equation*}
%  %S^{\mathrm{total}}_i = \sum_{\mathscr{S}_i}D_{i_1...i_s} 
%  S^{{total}}_i = \sum_{\mathscr{S}_i}D_{i_1...i_s} 
%  \end{equation*}
%  %
%  \end{itemize}
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%  % \begin{frame}
%  % \frametitle{Sensitivity Analysis}
%  % 
%  %        
%  % The Sobol indices for global sensitivity analysis can be used for
%  % 
%  % Ranking of input uncertain parameters $\boldsymbol{X} = [X_1, ..., X_n]$. 
%  % 
%  % Neglect of inessential uncertain parameters and high order terms in Equation~\ref{eq_ANOVA_representation}.   
%  % %
%  % 
%  % 
%  % 
%  % The total sensitivity indices $S^{\mathrm{total}}_i$ is defined to evaluate the total influence of a certain input parameter $X_i$ as: 
%  % %
%  % \begin{equation*}
%  % S^{\mathrm{total}}_i = \sum_{\mathscr{S}_i}D_{i_1...i_s} 
%  % \end{equation*}
%  % %
%  % 
%  % set $\mathscr{S}_i$ contains all the indexes involving index $i$ 
%  % and Sobol' indices set $\{D_{i_1...i_s}\}$ is the
%  % collection of all partial sensitivity indices that are related to parameter
%  % $X_i$.
%  % %
%  % \begin{equation*}
%  % \mathscr{S}_i = \{(i_1, ..., i_s): \exists k, 1\leq k \leq s, i_k = i\}
%  % \label{eq_index_set}
%  % \end{equation*}
%  % 
%  % 
%  % \end{frame}
%  % 
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%  % 
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%  \begin{frame}
%  \frametitle{Sobol Indices and Polynomial Chaos}
%  
%  
%  \vspace*{3mm}
%  
%  PC expansion of response, ANOVA form (Sudret 2008)
%  
%  %
%  %\vspace*{1mm}
%  
%  Multi-dimensional PC bases $\{\Psi_j(\boldsymbol{\xi})\}$ 
%  decomposed into products of single dimension PC chaos bases of different orders
%  %
%  \vspace*{-1mm}
%  \begin{equation*}
%  \Psi_j(\boldsymbol{\xi}) = \prod_{i=1}^{n} \phi_{\alpha_i}(\xi_i) 
%  \nonumber
%  \end{equation*}
%  %
%  $\phi_{\alpha_i}(\xi_i)$ is the single dimensional, order $\alpha_i$, polynomial function of underlying basic random variable $\xi_i$. 
%  %
%  
%  %
%  
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%  % \end{frame}
%  % 
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%  % \begin{frame}
%  % \frametitle{Sobol Sensitivity Analysis}
%  % 
%              
%  From ANOVA representation of probabilistic model response, the PC-based Sobol'
%  indices $S_{i_1...i_s}^{PC}$ are
%  %
%  
%  \vspace*{-1mm}
%  \begin{equation*}
%  %S_{i_1...i_s}^{PC} = \sum_{\boldsymbol{\alpha} \in \mathscr{S}_{{i_1},...,i_s}} y^2_{\boldsymbol{\alpha}} \boldsymbol{E} \left[\Psi_{\boldsymbol{\alpha}}^2\right] / D^{PC}
%  S_{i_1...i_s}^{PC} = \sum_{\boldsymbol{\alpha} \in {S}_{{i_1},...,i_s}} y^2_{\boldsymbol{\alpha}} \boldsymbol{E} \left[\Psi_{\boldsymbol{\alpha}}^2\right] / D^{PC}
%  \end{equation*}
%  %
%  
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%  
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%  
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%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  \begin{frame}
%  \frametitle{Sobol Sensitivity Analysis}
%  
%      
%  %Total Sobol' indices $S_{j_1...j_t}^{PC, \mathrm{total}}$
%  Total Sobol' indices $S_{j_1...j_t}^{PC, {total}}$
%  % for any group of
%  %parameters $\{\xi_{j_1}, ..., \xi_{j_t}\}$ is
%  %
%  \begin{equation*}
%  %S_{j_1...j_t}^{PC, \mathrm{total}} = \sum_{(i_1, ..., i_s) \in \mathscr{S}_{j_1,...,j_t}} S_{i_1...i_s}^{PC}   
%  S_{j_1...j_t}^{PC, {total}} = \sum_{(i_1, ..., i_s) \in {S}_{j_1,...,j_t}} S_{i_1...i_s}^{PC}   
%  \end{equation*}
%  %
%  %where $\mathscr{S}_{j_1,...,j_t} = \{(i_1, ..., i_s): (j_1,...,j_t) \subset (i_1, ..., i_s)\}$
%  where ${S}_{j_1,...,j_t} = \{(i_1, ..., i_s): (j_1,...,j_t) \subset (i_1, ..., i_s)\}$
%  
%  % $\mathscr{S}_{j_1,...,j_t}$ is defined as: 
%  % %
%  % \begin{equation}
%  % \mathscr{S}_{j_1,...,j_t} = \{(i_1, ..., i_s): (j_1,...,j_t) \subset (i_1, ..., i_s)\}
%  % \end{equation}
%  % %
%  
%  
%  \vspace*{10mm}
%  Using PC representation of probabilistic model response,
%  Sobol' sensitivity indices are \underline{analytic} and \underline{inexpensive}
%  %evaluated with very small computational expense. 
%  %
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% {/home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Latex/img/SFEM_3D.png}}
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%\subsection{Sensitivity Analysis Example}





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%\subsection{Uncertain Inelasticity}
%\subsection{\ }









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\begin{frame}

\frametitle{Application: Stochastic Site Response}

\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}
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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}
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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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\end{frame}
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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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% 
% 


 
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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}

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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}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

         

% 
% 
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \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 rock motions at depth
    \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}



\end{frame}
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\section{Summary}



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% \begin{frame}
%  \frametitle{Appropriate Science and Engineering Quotes}
% 
%  
% \begin{itemize}
% 
% 
% 
% % \vspace*{2mm}
% 
% \begin{small}
% 
% \vspace*{2mm}
% \item[]  Fran{\c c}ois-Marie Arouet,  Voltaire: \\
% "Le doute n'est pas une condition agr{\'e}able, 
% mais la certitude est absurde"
% 
% 
% % \vspace*{2mm}
% 
% 
% 
% % 
% %  \vspace*{1mm}
% % %1895   
% % \item[-] William Thomson, Lord Kelvin:   
% % "Heavier-than-air flying machines are impossible."
% % 
% % 
% 
% % 
% % 
% %  \vspace*{1mm}
% % %1943   
% % \item[-] Thomas Watson, IBM Chairman:   
% % "I think there is a world market for maybe five computers."
% % 
% 
%  \vspace*{3mm}
% \item[] Max  Planck:  \\
% "A    new    scientific  truth does not triumph by convincing  its opponents and
% making  them see the light, but rather because its opponents eventually die, and
% a  new  generation  grows  up  that  is familiar with it" 
% %(Science advances one funeral at a time)
% 
% 
% % 
% %  \vspace*{3mm}
% % \item[-]  Niklaus  Wirth:   
% % "Software is getting slower more rapidly than hardware becomes faster."
% 
% 
% 
%  \vspace*{3mm}
% \item[] Theodore Von K{\'a}rm{\'a}n: \\
% "The Scientist studies what is,
%  the engineer creates what has never been"
% 
% 
% \end{small}
% 
% 
% \end{itemize}
% 
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\begin{frame}
  \frametitle{Summary}

  \begin{itemize}

%    \item Importance   of   using  proper  models  correctly  (verification,
%    validation, level of sophistication)
%     
 
%    \item Reduction of modeling uncertainty 
%

\vspace*{4mm}
    \item[-] Engineering analysis uncertainties
      \begin{itemize}
\vspace*{1mm}
           \item[] Modeling, Epistemic 
\vspace*{1mm}
           \item[] Parametric, Aleatory
      \end{itemize}

\vspace*{2mm}
    \item[-] Engineering analysis to \underline{predict} and \underline{inform}


\vspace*{2mm}
%  \item[-]  
%    \url{http://real-essi.us/}
%     \url{http://real-essi.info/}
  \item[-] Engineer needs to know!


 
%\vspace*{4mm}
%  \item[-]  Real-ESSI Simulator Systems
%% {\includegraphics[width=12mm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/Real_ESSI_in_different_langauges/Real_ESSI_Chinese.jpeg}}
%%    \url{http://real-essi.us/}
%
%
\vspace*{2mm}
  \item[-] Real-ESSI Simulator $\rightarrow$ \hspace*{3mm}  
    \url{http://real-essi.us/}
%    \url{http://real-essi.info/}
      
%
%\vspace*{1mm}
%    \item[-] Sensitivity analysis


%\vspace*{1mm}
%   \item[-]     Collaborators:    Feng,  Yang, Wang,  Yang.
   
   

% \vspace*{1mm}
%   \item[-]  Funding  from  and  collaboration  with the ATC/US-FEMA, US-DOE, US-NRC, US-NSF, 
%   CNSC-CCSN, UN-IAEA, and Shimizu Corp. is greatly appreciated,

% 
%\vspace*{1mm}
%  \item[-]  
%    \url{http://sokocalo.engr.ucdavis.edu/~jeremic}
%      

\end{itemize}







\vspace*{4mm}
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\begin{center}
%
\includegraphics[height=1.1truecm]{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Free_Field_small_model_April2015/movie_input_mp4_icon.jpeg}
\includegraphics[height=1.1truecm]{/home/jeremic/tex/works/Conferences/2016/IAEA_TecDoc_February2016/My_Current_Work/movie_2_npps_mp4_icon.jpeg}
\includegraphics[height=1.1truecm]{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_Buildings/Ventura_Hotel_SSI_vs_nonSSI_screen_grab.jpg}
% \includegraphics[height=1.1truecm]{/home/jeremic/tex/works/Reports/2019/Pine_Flats_Dam_USSD/USSD_Dam_Report_2019/Figures/Model_Mesh_No_Reservior.pdf}
\includegraphics[height=1.1truecm]{/home/jeremic/tex/works/Thesis/HanYang/ASCE-7-21_low_building_energy_dissipation/ATC_Short_Building_PD.jpg}
%\includegraphics[height=1.0truecm]{/home/jeremic/tex/works/Conferences/2018/Oersted-DONG-Energy/present/Pile_in_liquefied_soil.jpg}
\\
%
%\hspace*{-4mm}
%\includegraphics[width=1.3truecm]{/home/jeremic/tex/works/Conferences/2021/CU-Boulder-GEGM-seminar-series-02Apr2021/present/Saint_Sophia_Constantinopolis.jpg}
%\includegraphics[height=1.3truecm]{/home/jeremic/tex/works/Conferences/2022/USBR_V_and_V_Mar-Apr-May_2022/present/Lecture01_Modeling/Kuca_Bajina_Basta-Drina.jpg}
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