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%  \usetheme{Marburg} % ima naslov i sadrzaj sa desne strane
%  \usetheme{Hannover} % ima naslov i sadrzaj sa leve strane
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% \usetheme{Antibes}  % ima sadrzaj gore i kao graf ...
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%\title{NRC Staff Capacity Building: \\
%       Micromechanical Origins of Elasto-Plasticity }
\title[ESSI]
%      {Methods, Computational Platform, and          \\
%       Case Studies for Time-Domain                \\
%       Soil-Structures-Interaction \\
%       Modeling and Simulation \\
%       Incorporating Complex Seismic Loads\\
%       Introduction to the NRC ESSI Simulator System}
        {{\cyrsesnaest Numerichko modeliranje \\
           interakcije konstrukcije i tla u \\
           zemljotresnom inzhinjerstvu: \\
           KONSTRUKCIJA
}}


%\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}
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%\author[Boris Jeremi{\'c}, CompGeoMech \includegraphics[width=8cm]{/home/jeremic/BG/amblemi/ucdavis_logo_gold_lrg}] % (optional, use only with lots of authors)
\author[{\cyrosam Jeremi\cj{}}] % (optional, use only with lots of authors)
{\cyrsesnaest Boris~Jeremi\cj{}}
% - Give the names in the same order as the appear in the paper.
% - Use the \inst{?} command only if the authors have different
%   affiliation.

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

%\institute[Computational Geomechanics Group \hspace*{0.3truecm}
%\institute[\pgfuseimage{university-logo}\hspace*{0.1truecm}\pgfuseimage{lbnl-logo}\hspace*{0.1truecm}\pgfuseimage{ETH-logo}] % (optional, but mostly needed)
\institute[\pgfuseimage{university-logo}] % (optional, but mostly needed)
%{ Professor, University of California, Davis\\
{~\vspace*{-5mm} \\
 Professor, University of California, Davis, CA, USA \\
%  and\\
%  Faculty Scientist, Lawrence Berkeley National Laboratory, Berkeley }
  Faculty Scientist, Lawrence Berkeley National Laboratory, Berkeley, CA, USA \\
%  Visiting Professor, Swiss Federal Institute of Technology, Z{\" u}rich, CH 
  }
% - 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)
{~\vspace*{-5mm} \\ 
\cyrdvanaest Srpsko udruzhenje za zemljotresno inzhenjerstvo 
         \\
         SUZI 
        \\
        17 Septembar 2020 
%        \\ ~
%        \\ ~
%          \\ ~  \vspace*{-8mm}
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%\logo{\pgfuseimage{university-logo}}

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




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\begin{frame}
  \titlepage
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\begin{frame}
  \frametitle{Outline}

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  \tableofcontents
% You might wish to add the option [pausesections]
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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.
% - If you omit details that are vital to the proof/implementation,
%   just say so once. Everybody will be happy with that.



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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}


%\subsection*{Intro}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Motivation}

\begin{itemize}
  
           
%\vspace*{0.3cm}
  \item[-] Improve design and assessment of infrastructure objects

      
\vspace*{3mm}
  \item[-]  Use of high fidelity numerical models to  
  analyze behavior of 
  earthquake, soil, structure interacting (ESSI) systems


\vspace*{3mm}
  \item[-] Control modeling uncertainty




\vspace*{3mm}
  \item[-]  Propagate parametric uncertainty



\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




		 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Prediction under Uncertainty}




\vspace*{10mm}

\begin{itemize}




%\vspace*{1mm}
  \item    \underline{Modeling Uncertainty}, Simplifying assumptions
    \begin{itemize}

\vspace*{2mm}
      \item[] Low, medium, high  sophistication modeling and simulation

\vspace*{2mm}
      \item[] Choice of sophistication level for confidence in  results

    \end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace*{4mm}
  \item   \underline{Parametric    Uncertainty},  ${M}  \ddot{u_i}  +  {C}  \dot{u_i} + {K}^{ep} {u_i} = {F(t)}$,

  \begin{itemize}

\vspace*{2mm}
    \item[] Uncertain mass $M$, viscous damping  $C$ and stiffness $K^{ep}$


\vspace*{2mm}
    \item[] Propagation  of uncertainty in  loads, $F(t)$


\vspace*{2mm}
    \item[] Results are PDFs   and   CDFs  for  $\sigma_{ij}$,  $\epsilon_{ij}$,  $u_i$,  $\dot{u}_i$,  $\ddot{u}_i$

  \end{itemize}




\end{itemize}



%\vspace*{5mm}


\noindent
%
\rule[.1mm]{\textwidth}{.1mm}


{\scriptsize Le doute n'est pas un {\'e}tat bien agr{\'e}able,
\\ 
\vspace*{-2mm}
mais l'assurance est un {\'e}tat ridicule.  (Fran{\c c}ois-Marie Arouet, Voltaire)}



\end{frame}
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\section{Structure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Modeling}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Verification and Validation}

\begin{figure}[!ht]
\begin{center}
\vspace*{3mm}
%\hspace*{-2cm}
{\includegraphics[width=9.0cm]{/home/jeremic/tex/works/lecture_notes_LAKI/LectureNotes/Figure-files/_Chapter_Verification_and_Validation_Introduction/tex_works_Conferences_2011_USNCCM11_Minneapolis_Coupled_Present_VandV_ODEN.jpg}}
%\hspace*{-2cm}
%\caption{Role of Verification and Validation per \protect{\cite{Oden2010a}}.}  
\label{Role_VandV_Oden}  
\vspace*{-3mm}
\end{center}
\end{figure}

\begin{flushright}
%\footnotesize{(Oden, Moser and Ghattas 2010)}
\tiny{(Oden, Moser and Ghattas 2010)}
\end{flushright}

\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Component Verification: ANDES Shell}

 \begin{center}
% \includegraphics[width=0.6\textwidth]{./_Jose_Files/fig1_static_tests.pdf}
\hspace*{-10mm}
 \includegraphics[width=5cm]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Present06_figs/fig1_static_tests.pdf}
% \includegraphics[width=0.6\textwidth]{./_Jose_Files/fig2_static_tests.pdf}
 \includegraphics[width=5cm]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Present06_figs/fig2_static_tests.pdf}
\hspace*{-10mm}
\end{center}

\begin{flushleft}
    \begin{tabular}{|l|l|}
    \hline
 $N_{\texttt{subd}}$   &      $u_z$ \\ \hline
     2 &  96.2118 \\
     7 & 100.096 \\
   101 & 100.002 \\
       &            \\ \hline
    \end{tabular}
\end{flushleft}


\vspace*{-30mm}

 \begin{flushright}
% \includegraphics[width=0.9\textwidth]{./_Jose_Files/Test_shell_andes_1_free_vibration_EigenMode1.png}
%\hspace*{-10mm}
 \includegraphics[width=5cm]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Present06_figs/Test_shell_andes_1_free_vibration_EigenMode1.png}
 \\
\vspace*{-8mm}
 \includegraphics[width=5cm]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Present06_figs/Test_shell_andes_2_free_vibration_EigenMode1.png}
\hspace*{-10mm}
\end{flushright}

\end{frame}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


   
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Component Validation, Concrete Wall}
 
\hspace*{5mm}
   \begin{figure}[!h]
    \begin{center}
\hspace*{-15mm}
     \includegraphics[width=2truecm]{/home/jeremic/tex/works/Conferences/2018/DOE_LBNL_Advisory_Board_meet_08Jun2018/present/OECD_presentation/Figures/Rebar_Plan.pdf}
     \includegraphics[width=4truecm]{/home/jeremic/tex/works/Conferences/2018/DOE_LBNL_Advisory_Board_meet_08Jun2018/present/OECD_presentation/Figures/3D_mesh.pdf}
\includegraphics[width=2.8truecm]{/home/jeremic/tex/works/Conferences/2018/DOE_LBNL_Advisory_Board_meet_08Jun2018/present/OECD_presentation/Figures/Reg_A_Force_Displacement.pdf}
\includegraphics[width=2.8truecm]{/home/jeremic/tex/works/Conferences/2018/DOE_LBNL_Advisory_Board_meet_08Jun2018/present/OECD_presentation/Figures/ASR_A1_Force_Displacement.pdf}
\hspace*{-15mm}
    \end{center}
\end{figure}


\vspace{-2mm}
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=8.0truecm]{/home/jeremic/tex/works/Conferences/2018/DOE_Natural_Hazartd_Oct2018/Present/OECD_wall_damage_3_stages.jpg}
\end{center}
\end{figure}



% \vspace{-20mm}
% \begin{figure}[!htbp]
% \hspace*{-10mm}
% \begin{center}
% \includegraphics[width=4.0truecm]{/home/jeremic/tex/works/Conferences/2018/DOE_LBNL_Advisory_Board_meet_08Jun2018/present/OECD_presentation/Figures/Damage_3000.pdf}
% \hspace*{-5mm}
% \includegraphics[width=4.0truecm]{/home/jeremic/tex/works/Conferences/2018/DOE_LBNL_Advisory_Board_meet_08Jun2018/present/OECD_presentation/Figures/Damage_5000.pdf}
% \hspace*{-5mm}
% \includegraphics[width=4.0truecm]{/home/jeremic/tex/works/Conferences/2018/DOE_LBNL_Advisory_Board_meet_08Jun2018/present/OECD_presentation/Figures/Damage_10000.pdf}
% \end{center}
% \end{figure}
% 
% 
% \vspace*{-5mm}
% \hspace*{12mm}
% \begin{footnotesize}
% $u_y$ = 1.4 mm
% \hspace{12mm}
% $u_y$ = 1.8 mm
% \hspace{12mm}
% $u_y$ = 3.0 mm
% \end{footnotesize}
  

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




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
  \frametitle{Model Verification}



\begin{figure}[htbp]
\begin{center}
\vspace*{-6mm}
  \includegraphics[width = 3.0cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figure-files/nonlinear_analysis_steps/soil-structure/overview.png}
\vspace*{-1mm}
\\
  \includegraphics[width = 0.35cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figure-files/nonlinear_analysis_steps/free_field_1D/DRM_1D_motion_3D_just_column.jpg}
\hspace*{5mm}
%  \includegraphics[width = 0.1cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figure-files/nonlinear_analysis_steps/free_field_1D/DRM1D_Motion3D.png}
  \includegraphics[width = 2.5cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figure-files/nonlinear_analysis_steps/free_field_3D/motion3D_DRM3D_free_field.png}
\hspace*{5mm}
%  \includegraphics[width = 1cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figure-files/nonlinear_analysis_steps/soil-foundation/soil_foundation.png}
%  \includegraphics[width = 3cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figure-files/nonlinear_analysis_steps/soil-foundation/slice.png}
  \includegraphics[width = 2.5cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figure-files/nonlinear_analysis_steps/soil-foundation/foundation_results.png}
%  \includegraphics[width = 3cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figure-files/nonlinear_analysis_steps/soil-structure/overview.png}
\\
\vspace*{-3mm}
  \includegraphics[width = 1.0cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figure-files/nonlinear_analysis_steps/structure/eigen/structure-only.png}
\hfill
  \includegraphics[width = 1.2cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figure-files/nonlinear_analysis_steps/structure/eigen/eigen1.png}
\hfill
  \includegraphics[width = 1.2cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figure-files/nonlinear_analysis_steps/structure/eigen/eigen2.png}
\hfill
  \includegraphics[width = 1.2cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figure-files/nonlinear_analysis_steps/structure/eigen/eigen3.png}
\hfill
  \includegraphics[width = 1.2cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figure-files/nonlinear_analysis_steps/structure/eigen/eigen4.png}
\hfill
  \includegraphics[width = 1.2cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figure-files/nonlinear_analysis_steps/structure/eigen/eigen5.png}
\hfill
  \includegraphics[width = 1.2cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figure-files/nonlinear_analysis_steps/structure/eigen/eigen6.png}
\hfill
%  \includegraphics[width = 1.0cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figure-files/nonlinear_analysis_steps/structure/imposed_motion/structure-only.png}
%\hfill
  \includegraphics[width = 1.2cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figure-files/nonlinear_analysis_steps/structure/imposed_motion/imposed_motion_results.png}
%   \includegraphics[width = 0.1cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figure-files/nonlinear_analysis_steps/soil-structure/overview.png}
\\
\vspace*{-1mm}
  \includegraphics[width = 6cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figure-files/nonlinear_analysis_steps/soil-structure/DRM3D_motion3D_structure.png}
\end{center}
\end{figure}



\end{frame}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
%\frametitle{Ventura Hotel, Full Model}
\frametitle{Full ESSI Model}

%\vspace*{-5mm}
\begin{figure}[!htb]
\begin{center}
\includegraphics[width=12cm]{/home/jeremic/tex/works/Reports/2020/ATC-144-buildings/ATC_building_report_18Apr2020/ATC_building_report-master/images/Ventura_hotel_model_3D_View.jpg}
% \includegraphics[width=5cm]{/home/jeremic/tex/works/Reports/2020/ATC-144-buildings/ATC_building_report_18Apr2020/ATC_building_report-master/images/Ventura_hotel_model_3D_View_Closeup.jpg}
% \\
% %\includegraphics[width=5cm]{/home/jeremic/tex/works/Reports/2020/ATC-144-buildings/ATC_building_report_18Apr2020/ATC_building_report-master/images/Ventura_Hotel_View_3D.jpg}
% \includegraphics[width=5cm]{/home/jeremic/tex/works/Reports/2020/ATC-144-buildings/ATC_building_report_18Apr2020/ATC_building_report-master/images/Ventura_Hotel/Model/View_y.jpg}
% \includegraphics[width=5cm]{/home/jeremic/tex/works/Reports/2020/ATC-144-buildings/ATC_building_report_18Apr2020/ATC_building_report-master/images/Ventura_Hotel_View_3D_With_Annex.jpg}
\end{center}
\end{figure}

\end{frame}
%\vspace*{10mm}  




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
%\frametitle{Ventura Hotel, Structural Model}
\frametitle{Structure Only Model}

%\vspace*{-5mm}
\begin{figure}[!htb]
\begin{center}
% \includegraphics[width=5cm]{/home/jeremic/tex/works/Reports/2020/ATC-144-buildings/ATC_building_report_18Apr2020/ATC_building_report-master/images/Ventura_hotel_model_3D_View.jpg}
% \includegraphics[width=5cm]{/home/jeremic/tex/works/Reports/2020/ATC-144-buildings/ATC_building_report_18Apr2020/ATC_building_report-master/images/Ventura_hotel_model_3D_View_Closeup.jpg}
% \\
% %\includegraphics[width=5cm]{/home/jeremic/tex/works/Reports/2020/ATC-144-buildings/ATC_building_report_18Apr2020/ATC_building_report-master/images/Ventura_Hotel_View_3D.jpg}
\includegraphics[width=10cm]{/home/jeremic/tex/works/Reports/2020/ATC-144-buildings/ATC_building_report_18Apr2020/ATC_building_report-master/images/Ventura_Hotel/Model/View_y.jpg}
%\includegraphics[width=19cm]{/home/jeremic/tex/works/Reports/2020/ATC-144-buildings/ATC_building_report_18Apr2020/ATC_building_report-master/images/Ventura_Hotel_View_3D_With_Annex.jpg}
\end{center}
\end{figure}

\end{frame}
%\vspace*{10mm}  







% - 
% -  
% -  
% -  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  \begin{frame} 
% -  \frametitle{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,
% -  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}
% -  
% -  \vspace{4mm}
% -  \item[-] Miscommunication between seismologists and structural engineers,
% -  $Sa(T_0)$ not compatible with nonlinear FEM
% -  
% -  \vspace{4mm}
% -  \item[-] IMs difficult to choose, Spectral Acc, PGA, PGV...
% -  
% -  
% -  \vspace{4mm}
% -  \item[-] Single IM does not contain all/most uncertainty
% -  
% -  \vspace{4mm}
% -  \item[-] Monte Carlo, not accurate enough for tails 
% -  
% -  \vspace{4mm}
% -  \item[-] Monte Carlo, computationally expensive, CyberShake for LA, 20,000
% -  cases, 100y runtime, (Maechling et al. 2007)
% -  
% -  
% -  
% -  
% -  \end{itemize}
% -  
% -  \end{frame}
% -  
% -  % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}

\frametitle{Time Domain Intrusive Seismic Risk Analysis}

%  

\begin{itemize}

\vspace*{2mm}
\item[-] Stochastic Elastic-Plastic Finite Element Method, SEPFEM, 
 ${M}  \ddot{u_i}  +  {C}  \dot{u_i} + {K}^{ep} {u_i} = {F(t)}$, 


\vspace*{2mm}
\item[-] Uncertain seismic loads, from  uncertain seismic motions, using Domain
Reduction Method


\vspace*{2mm}
\item[-] Uncertain elastic-plastic material, stress and stiffness solution using
Forward Kolmogorov, Fokker-Planck equation

\vspace*{2mm}
\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 uncertainty expanded along stochastic shape functions: 
%-  $D(x,t,\theta) = \sum_{i=0}^{P_d} r_i(x,t) * \Phi_i[\{\xi_1, ..., \xi_m\}]$
%-  
%-  \vspace*{4mm}
%-  \item[] Loading uncertainty expanded along stochastic shape functions: 
%-  $f(x,t,\theta) = \sum_{i=0}^{P_f} f_i(x,t) * \zeta_i[\{\xi_{m+1}, ..., \xi_f]$
%-  
%-  \vspace*{4mm} 
%-  \item[] Displacement expanded along stochastic shape functions: 
%-  $u(x,t,\theta) = \sum_{i=0}^{P_u} u_i(x,t) * \Psi_i[\{\xi_1, ..., \xi_m, \xi_{m+1}, ..., \xi_f\}]$
%-  
%-  \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{Stochastic Seismic Motion Development}
% - 
% - \begin{itemize}
% -  \item UCERF3 (Field et al. 2014)
% -  \item Stochastic motions (Boore 2003) 
% -  \item Polynomial Chaos Karhunen-Lo{\`e}ve expansion
% -  \item Domain Reduction Method for $P_{eff}$ (Bielak et al. 2003)
% - \end{itemize}
% - 
% - \begin{figure}[!hbpt]
% - \begin{center}
% - \includegraphics[width=4cm]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/UCERF3.pdf}
% - \includegraphics[width=3.0cm]{/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}
% - 
% - % 
% - %  \begin{figure}[!hbpt]
% - %  \begin{center}
% - %  \includegraphics[width=10cm]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/structural_uncertainty.pdf}
% - %  \end{center}
% - %  \end{figure}
% - %  
% - %  
% - %  \begin{figure}[!hbpt]
% - %  \begin{center}
% - %  \includegraphics[width=10cm]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/probabilsitc_evolution.png}
% - %  \end{center}
% - %  \end{figure}
% - %  
% - %  
% - %  
% - %  \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}
% - %  
% - %  
% - \end{frame}
% - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - 






% - 
% - 
% - 
% - 
% - 
% - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% - \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 Ground Motion Modeling}
% 
% Boore (2003) approach for median of log-normal random FAS field: 
% 
% \[FAS(f) = \underbrace{S(M_0, f)}_\text{source}\underbrace{P(R, f)}_\text{path} \underbrace{A(f)D(f)}_\text{site}\]
% 
% \vspace{-3mm}
% 
% where:
% \begin{itemize}
% \small
% 
% % \item $S(M_0,f) = \frac{CM_0(2\pi f)^2}{1+(f/f_0)^2}$ 
% 
% \item $S(M_0,f) = CM_0(2\pi f)^2/[1+f^2/f_0^2]$ 
% 
% \vspace{1mm}
% 
% \item $P(R, f) = Z(R)exp(-\pi fR/Q\beta)$
% 
% \vspace{1mm}
% 
% \item $A(f)$: quarter wavelength approximation. 
% 
% \vspace{1mm}
% 
% \item $D(f)= exp(-\pi \kappa_0f)$
% 
% \end{itemize}
% 
% Improvements: 
% 
% \begin{itemize}
% \small
% 
% \item Inter-frequency correlation by \textit{Bayless \& Abrahamson (2018)}
% 
% \vspace{1mm}
% 
% \item Phase derivative model by \textit{Baglio \& Abrahamson (2017)}
% 
% \end{itemize}
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 

% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% 
% \frametitle{Stochastic Ground Motion Modeling}
% 
% \begin{textblock}{15}(0.5, 4.2)
% Verification with GMPE for $M=7$, $R=15km$ scenario
% \end{textblock}
% 
% \begin{textblock}{15}(0.0, 5.0)
% \includegraphics[width=0.4\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/SA_GMPE_verification_std_08_no_smooth.pdf}
% \end{textblock}
% 
% \begin{textblock}{15}(5.3, 5.0)
% \includegraphics[width=0.4\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/Goodness_fit_std_08_no_smooth.pdf}
% \end{textblock}
% 
% \begin{textblock}{15}(10.5, 5.0)
% \includegraphics[width=0.4\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/Standard_deviation_std_08_no_smooth.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}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Uncertainty Representation \& Propagation}
\begin{itemize}
\vspace{-2mm}
\item \small Unified uncertainty representation for forces and material 
%$D(\boldsymbol{x}, \theta)$. 

\begin{itemize}
\item [--] Hermite polynomial chaos (PC) for marginal distribution  
\item [--] Karhunen-Lo$\grave{e}$ve (KL) expansion for correlation structure 
\end{itemize}

\item \small Hermite PCs: Non-Gaussian random field with underlying Gaussian
random field $D(\boldsymbol{x}, \theta) = \sum_{i=0}^{P} D_i(\boldsymbol{x}) \Omega_i(\gamma(\boldsymbol{x},\theta))$
%
\\
with: 
\\
$\Omega_i = 1, \gamma, \gamma^2-1, \gamma^3-3\gamma, ... $
%
\\
%
$<\Omega_i> =0; \ \ <\Omega_i\Omega_j> = 0 \ \ for \ \ i \neq j $

\end{itemize}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% 
%- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%- \begin{frame}
%- \frametitle{Uncertainty Representation \& Propagation}
%- 
%- \begin{textblock}{15}(0.5, 5.0)
%- \includegraphics[width=0.4\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/Hermite_PC_order.png}
%- \end{textblock}
%- 
%- \begin{textblock}{15}(7.5, 4.0)
%- \includegraphics[width=0.5\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/Gamma_distribution_hermite.png}
%- \end{textblock}
%- 
%- \begin{textblock}{15}(5.5, 12.0)
%- \includegraphics[width=0.7\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/Gamma_distribution.png}
%- \end{textblock}
%- 
%- \end{frame}
%- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%- 

%- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%- \begin{frame}
%- \frametitle{Uncertainty Representation \& Propagation}
%- 
%- \vspace{0.1cm}
%- 
%- \small Karhunen-Lo$\grave{e}$ve (KL) expansion: capture correlation structure 
%- 
%- \vspace{-0.3cm}
%- 
%- \[Cov_D(x_1, x_2) = \sum_{i=1}^{P} D_i(x_1)D_i(x_2)i!Cov_{\gamma}(x_1, x_2)\]
%- 
%- \vspace{-0.2cm}
%- 
%- \small Eigen-decomposed Gaussian covariance kernel: $ \lambda_{i}$ eigenvalue, $f_i(\boldsymbol{x})$ eigen-function
%- 
%- \vspace{-0.7cm}
%- 
%- \[\gamma(\boldsymbol{x}, \theta)=\sum_{i=1}^{M}\sqrt{\lambda_i}f_i(\boldsymbol{x})\xi_i \]
%- 
%- \vspace{-0.2cm}
%- 
%- \small Fredholm's integral equation of the second kind: 
%- 
%- \[\int_V Cov_{\gamma}(x_1, x_2) f_i(x_1) \, dx_1 = \lambda_{i} f_i(x_2)\]
%- 
%- \vspace{-0.1cm}
%- 
%- \small \{$\xi_i$\} are zero mean, unit variance, mutually orthogonal Gaussian random variables
%- 
%- \vspace{-0.2cm}
%- 
%- \[<\xi_i> =0; \ \ <\xi_i\xi_j> = \delta_{ij}\]
%- 
%- \end{frame}
%- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%- 
%- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%- \begin{frame}
%- \frametitle{Uncertainty Representation \& Propagation}
%- \begin{textblock}{15}(0.5, 6.0)
%- \includegraphics[width=0.4\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/1D_KL_exp_illustration.png}
%- \end{textblock}
%- \begin{textblock}{15}(7.5, 4.0)
%- \includegraphics[width=0.5\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/KL_eigen_modes.png}
%- \end{textblock}
%- \end{frame}
%- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%- 
%- 
%- 
%- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%- \begin{frame}
%- \frametitle{Uncertainty Representation \& Propagation}
%- 
%- \small Random field $D(\boldsymbol{x}, \theta)$ of arbitrary marginal distribution \& correlation structure can be represented as: 
%- 
%- \begin{textblock}{14}(-2.4, 6.0)
%- \[D(\boldsymbol{x}, \theta) = \sum_{i=0}^{K} d_i(\boldsymbol{x}) \Psi_i(\{\xi_j(\theta)\})\]
%- \[d_i(\boldsymbol{x}) = \frac{P!}{(\Psi_i^2)} D_P(\boldsymbol{x}) \prod_{j=1}^{P} \frac{\sqrt{\lambda_{k(j)}}f_{k(j)}(\boldsymbol{x})}{\sqrt{\sum_{m=1}^{M}(\sqrt{\lambda_m}f_m(\boldsymbol{x}))^2}}\]
%- \end{textblock}
%- 
%- \vspace{3.5cm}
%- where: \\
%- $\{\Psi_i\}$  are multi-dimensional Hermite \\
%- PC bases of order $P$, dimension $M$. 
%- 
%- \begin{textblock}{15}(8.9, 5.5)
%- \includegraphics[width=0.45\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/Multidimensional_Hermite_PC.png}
%- \end{textblock}
%- 
%- \end{frame}
%- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%- \begin{frame}
%- \frametitle{Uncertainty Representation \& Propagation}
%- 
%- \small 
%- 
%- Galerkin stochastic finite element method (SFEM)
%- 
%- \[M_{IacJ} \ddot{U}_{Jc} + \boldsymbol{K^{ep}_{IacJ}} U_{Jc} = \boldsymbol{F_{Ia}}\]
%- 
%- Probabilistic discretization:
%- 
%- \vspace{-0.2cm}
%- 
%- \[K^{ep}_{IacJ}(\theta)= K^{ep}_{IacJi} \Psi_i (\theta)\]
%- 
%- \vspace{-0.3cm}
%- 
%- \[F_{Ia}(\theta)= F_{Iaj} \psi_j (\theta)\]
%- 
%- \vspace{-0.3cm}
%- 
%- \[U_{Jc}(\theta) = U_{Jck} \phi_k(\theta)\]
%- 
%- \vspace{-0.3cm}
%- 
%- Galerkin projection:
%- 
%- \[M_{IacJ} (\phi_k \phi_m) \ddot{U}_{Jck} + K^{ep}_{IacJi} (\Psi_i\phi_k\phi_m) U_{Jck} = F_{Iaj} (\psi_j \phi_m) \]
%- 
%- \vspace{0.2cm}
%- \scriptsize where $(\Psi_i\phi_k\phi_m)$ is the triple product of multi-dimensional Hermite PCs 
%- 
%- \end{frame}
%- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%- 


% - 
% -  
% -    
% -  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  \begin{frame} 
% -  \frametitle{Uncertain Model Description}
% -  
% -  \begin{textblock}{15}(0.3, 3.4)
% -  \includegraphics[width=0.70\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/example_illustration.png}
% -  \end{textblock}
% -  
% -  
% -  \begin{textblock}{15}(1.2, 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}(7.2, 11.7)
% -  \scriptsize
% -  \begin{itemize}
% -  \item $Vs_{30}=620m/s$
% -  \item $m=100kips/g$
% -  \item $\overline{k} = 168kip/in$
% -  \end{itemize}
% -  \end{textblock}
% -  
% -  \begin{textblock}{15}(9.7, 6.5)
% -  \scriptsize
% -  \[Cov_k  = 
% -  \begin{bmatrix}
% -  1.0 & 0.6 & 0.3 & 0.2\\
% -  0.6 & 1.0 & 0.5 & 0.2\\
% -  0.3 & 0.5 & 1.0 & 0.7\\
% -  0.2 & 0.2 & 0.7 & 1.0
% -  \end{bmatrix}
% -  \]
% -  \end{textblock}
% -  
% -  \end{frame}
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% -  
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% -  \begin{frame}
% -  \frametitle{Seismic Source Characterization}
% -  
% -  \begin{textblock}{15}(0.3, 3.4)
% -  \includegraphics[width=0.65\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/SSC_legend.pdf}
% -  \end{textblock}
% -  
% -  \begin{textblock}{15}(6.5, 12.1)
% -  \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}
% -  
% -  \begin{textblock}{15}(11.7, 3.3)
% -  \includegraphics[width=0.22\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/Acc_time_series100.pdf}\\
% -  \includegraphics[width=0.22\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/Acc_time_series343.pdf}\\
% -  \includegraphics[width=0.22\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/Acc_time_series439.pdf}
% -  \end{textblock}
% -  \end{frame}
% -  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  
% -  
% -  
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% -  \begin{frame} 
% -  \frametitle{Stochastic Ground Motion Realizations}
% -  
% -  
% -  \begin{figure}[!htbp]
% -  \centering
% -  \includegraphics[width=9cm]{/home/jeremic/tex/works/Papers/2019/Hexiang/1D_risk/version6/Figures/Acc_realization_200.pdf}
% -  %
% -  %
% -  % \includegraphics[width=0.31\textwidth]{/home/jeremic/tex/works/Papers/2019/Hexiang/1D_risk/version6/Figures/Acc_time_series_realiztion70.pdf}
% -  % \includegraphics[width=0.29\textwidth]{/home/jeremic/tex/works/Papers/2019/Hexiang/1D_risk/version6/Figures/Acc_time_series_realiztion100.pdf}
% -  % \includegraphics[width=0.32\textwidth]{/home/jeremic/tex/works/Papers/2019/Hexiang/1D_risk/version6/Figures/Acc_time_series_realiztion367.pdf}
% -  % %\caption{\label{figure_acc_realizations}  Realizations of uncertain acceleration
% -  % %time series population.}
% -  \end{figure}
% -  
% -  
% -  
% -  
% -  
% -  
% -  \end{frame}
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% -  










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\subsection{Uncertain ESSI}









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\begin{frame} 
\frametitle{Stochastic Ground Motion Representation}

{\begin{textblock}{15}(-0.1, 3.62)
\scriptsize
\includegraphics[width=0.3\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/KL_mean_acc_from_acc.pdf}

\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/2019/CompDyn/present/pic/KL_var_acc_from_acc.pdf}

\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/2019/CompDyn/present/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/2019/CompDyn/present/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/2019/CompDyn/present/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/2019/CompDyn/present/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/2019/CompDyn/present/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/2019/CompDyn/present/pic/KL_simulated_dis_correlation_from_dis.pdf}

\quad Dis. synthesized Cov.
\end{textblock}
\end{frame}
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\begin{frame} 
\frametitle{Stochastic Structural Response}


\begin{figure}[!htbp]
\centering
\includegraphics[width=8cm]{/home/jeremic/tex/works/Papers/2019/Hexiang/1D_risk/version6/Figures/constitutive_relation_uncertainHa_uncertainCr_MC_verification.pdf}
%
%
% \includegraphics[width=0.31\textwidth]{/home/jeremic/tex/works/Papers/2019/Hexiang/1D_risk/version6/Figures/Acc_time_series_realiztion70.pdf}
% \includegraphics[width=0.29\textwidth]{/home/jeremic/tex/works/Papers/2019/Hexiang/1D_risk/version6/Figures/Acc_time_series_realiztion100.pdf}
% \includegraphics[width=0.32\textwidth]{/home/jeremic/tex/works/Papers/2019/Hexiang/1D_risk/version6/Figures/Acc_time_series_realiztion367.pdf}
% %\caption{\label{figure_acc_realizations}  Realizations of uncertain acceleration
% %time series population.}
\end{figure}

%  
% \begin{figure}[!htbp]
% \centering
% \subfloat[Uncertain $H_a$]{
% \hspace{-0.8cm}
% \includegraphics[width=0.53\textwidth]{Figures/constitutive_relation_uncertainHa_certainCr_MC_verification.pdf}}
% \subfloat[Uncertain $H_a$ and $C_r$]{
% \hspace{-0.2cm}
% \includegraphics[width=0.53\textwidth]{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 Response}

\begin{textblock}{15}(2.9, 4)
\scriptsize
\includegraphics[width=0.65\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/Probabilistic_Response_Node_1_new.pdf}
\end{textblock}

\begin{textblock}{15}(4.8, 3.8)
\scriptsize Probabilistic response of top floor from SFEM
\end{textblock}
\end{frame}
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\begin{frame} 

\frametitle{Probabilistic Dynamic Response}

%\vspace*{10mm}
\begin{textblock}{15}(0.9, 6.4)
\scriptsize
\includegraphics[width=0.55\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/probability_evolution.jpg}
\includegraphics[width=0.43\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/Contour_PDF_evolution_top_floor_deformation.pdf}
\end{textblock}

\begin{textblock}{15}(4.8, 6.2)
\scriptsize Probabilistic density of displacements evolution of top floor
\end{textblock}
\end{frame}
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\begin{frame} 

\frametitle{Maximum Inter-Story Drift Ratio (MIDR)}

% \begin{textblock}{15}(0.7, 4.8)
% \scriptsize
% \includegraphics[width=0.55\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/PDF_evolution.png}
% \end{textblock}
% 
%\vspace*{-5mm}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=9cm]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/MIDR_distribution.pdf}
\end{center}
\end{figure}

\end{frame}
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\begin{frame} 

\frametitle{Seismic Risk Analysis}

\begin{textblock}{15}(0.7, 4.8)
\scriptsize
\includegraphics[width=0.45\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/combined_risk_curve.pdf}
\end{textblock}

\begin{textblock}{15}(7.9, 4.8)
\scriptsize
\includegraphics[width=0.48\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/risk_deaggregation_MIDR1.pdf}
\end{textblock}

\begin{textblock}{15}(0.9, 11.9)
\begin{itemize}
\item[] \scriptsize $\lambda(MIDR >1 \%) = 9.7 \times 10^{-3}$
\item[] \scriptsize $\lambda(MIDR >2 \%) = 1.7 \times 10^{-3}$
\item[] \scriptsize $\lambda(MIDR >4 \%) = 5.9 \times 10^{-5}$
\end{itemize}
\end{textblock}

\begin{textblock}{15}(9.3, 12.5)
\scriptsize Risk de-aggregation for $\lambda(MIDR >1 \%)$
\end{textblock}

\end{frame}
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\begin{frame} 

\frametitle{Sensitivity Study}

\begin{textblock}{15}(1.7, 6.2)
\scriptsize
\includegraphics[width=0.40\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/risk_curve_different_stressdrop.pdf}
\end{textblock}

\begin{textblock}{15}(9.1, 6.2)
\scriptsize
\includegraphics[width=0.40\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/risk_curve_different_kappa.pdf}
\end{textblock}

%-
%- \begin{textblock}{15}(1.7, 8.9)
%- \scriptsize
%- \includegraphics[width=0.40\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/median_FAS_stressdrop_comparison.pdf}
%- \end{textblock}
%- 
%- \begin{textblock}{15}(9.1, 8.9)
%- \scriptsize
%- \includegraphics[width=0.40\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/median_FAS_kappa_comparison1.pdf}
%- \end{textblock}
%- 
\begin{textblock}{15}(7.9, 5.7)
\scriptsize Site $\kappa_0$:
\end{textblock}

 \begin{textblock}{15}(0.3, 5.7)
 \scriptsize Source $\Delta \sigma$:
 \end{textblock}

\end{frame}
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\begin{frame} 
\frametitle{Sensitivity Study}

\begin{textblock}{15}(1.3, 4.2)
If first mode structural frequency increases from 1.6Hz to 8Hz, \\
significant effect of site $\kappa_0$: 
\end{textblock}

%\begin{textblock}{15}(4.7, 5)
\begin{figure}[h]
\vspace*{7mm}
\begin{center}
%\scriptsize
\includegraphics[width=8cm]{/home/jeremic/tex/works/Papers/2019/Hexiang/GMPE_FAS_risk/Figures/risk_curve_different_kappa_highfrequency.pdf}
\end{center}
\end{figure}
%\end{textblock}



%\begin{textblock}{15}(4.7, 5)
%\scriptsize
%\includegraphics[width=0.55\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/risk_curve_different_kappa_highfrequency.pdf}
%\end{textblock}

\end{frame}
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\begin{frame} 
\frametitle{Seismic Risk, Uncertain Material}


\begin{figure}[htb]
\begin{center}
\includegraphics[width=8cm]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/seismic_risk_compare.pdf}
\end{center}
\end{figure}


\end{frame}
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\section{Summary}



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\begin{frame}
  \frametitle{Summary}


\begin{itemize}


\vspace*{2mm}
     \item[-] Importance of using  realistic models

\vspace*{3mm}
     \item[-] Reduce modeling uncertainty

\vspace*{3mm}
     \item[-] Propagation of parametric, aleatory uncertainty

\vspace*{3mm}
     \item[-] Probabilistic response for decisions making

%\vspace*{0.2cm}
%     \item[-] Help in checking hidden (most if not all) failure (problem) modes



%
% \vspace*{0.2cm}
%      \item[-]  NRC  ESSI  Simulator  will  (can)  take  into consideration (use for
%      modeling/simulation) all of the above motions

\end{itemize}

\end{frame}
%--


\end{document}


