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

\usepackage{epstopdf}


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

 
\usepackage[absolute,overlay]{textpos}


  
% for tikzpicture
\usepackage{tikz}
\usetikzlibrary{tikzmark,calc}




 
\usepackage{threeparttable}

%\usepackage{MnSymbol}%
% for smileys
\usepackage{wasysym}%
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% for subfloat figures:
\usepackage{subfig}
\usepackage{tabularx}
%used for truncated section title in headers
\usepackage[breakall]{truncate}
%\usepackage{caption}


\usepackage{ulem}
\usepackage{amsmath}
\usepackage{amsmath}
\usepackage{amssymb}
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% solution from https://texfaq.org/FAQ-manymathalph
\newcommand{\bmmax}{3}
% \newcommand\bmmax{0} % default 4
\usepackage{bm}

\usepackage{IEEEtrantools}
\usepackage{setspace}
\usepackage{scalerel}



% for listing DSL
\input{essi_listings_options.tex}



      
\usepackage{multimedia}

% for inclusion of other PDF pages, in this case Frank's presentation
\usepackage{pdfpages}

%This is a macro to  convert eps to pdf files on the fly.
% make sure figure syntax uses graphicx syntax NOT epsfig syntax
%from http://mailman.mit.edu/pipermail/macpartners/2005-January/000780.html
%
% \ifx\pdfoutput\undefined
% % we are running LaTeX, not pdflatex
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% \else
% % we are running pdflatex, so convert .pdf files to .pdf
% \usepackage[pdftex]{graphicx}
% \usepackage{epstopdf}
% \fi
% %*****************************************


%% ovo je za cirilicu
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% % for not showing eq numbers unless eq is references
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% FEI  LOGO definition
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{{\sf \uppercase{F}\kern-0.20em\uppercase{E}\kern-0.20em\uppercase{I}}}

% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % % ovo je za cirilicu
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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
%  \usetheme{Warsaw}
%  \usetheme{Warsaw}
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    \usecolortheme[RGB={20,0,128}]{structure}

% or ...

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%  \setbeamercovered{transparent}
% or whatever (possibly just delete it)
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%  \usecolortheme{beetle} % siva pozadina (vrh plav)
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%%%%%%%
%  \usecolortheme{BorisJeremic}
%%%%%%%
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%   \usefonttheme[onlylarge]{structuresmallcapsserif}
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\usepackage[greek,english]{babel}



\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}

\newcommand{\ud}{{\rm d}}


\usepackage{array}



%%%%  HYPERREF  HYPERREF  HYPERREF  HYPERREF  HYPERREF
%%%%  HYPERREF  HYPERREF  HYPERREF  HYPERREF  HYPERREF
\definecolor{webgreen}{rgb}{0, 0.35, 0} % less intense green
\definecolor{webblue}{rgb}{0, 0, 0.50} % less intense blue
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}
% \usepackage[pdfauthor={Boris Jeremic},
%             colorlinks=true,
%             linkcolor=webblue,
%             citecolor=webblue,
%             urlcolor=webblue,
%             linktocpage,
%             pdftex]{hyperref}


\usepackage{pause}
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\usepackage[latin1]{inputenc}
% or whatever
\usepackage{times}
\usepackage[T1]{fontenc}
% Or whatever. Note that the encoding and the font should match. If T1
% 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[Uncertain Computational Mechanics]
        {Forward and Backward \\
Uncertainty Propagation in \\
Computational Earthquake Engineering  }


%\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} 
\\ \vspace*{1mm}
Han Yang, Hexiang Wang}


%\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 Tianjin University Lecture \\
         24/25Nov2021}

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



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

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

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




% Delete this, if you do not want the table of contents to pop up at
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\setcounter{tocdepth}{3}
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 }


% If you wish to uncover everything in a step-wise fashion, uncomment
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\begin{document}

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

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\begin{frame}
  \frametitle{Outline}

   \begin{scriptsize}
  \tableofcontents
% You might wish to add the option [pausesections]
   \end{scriptsize}
\end{frame}

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




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\subsection{Motivation}
\subsection{\ }



%%%%%%%%%%%%%%%%%%%%%%%%%%%%dir
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%





 

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

\begin{itemize}



 

%\vspace*{0.3cm}
  \item[]  Improve  modeling  and  simulation  of infrastructure objects
 

% \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*{4mm}
  \item[] Modeling, epistemic uncertainty

\vspace*{4mm}
  \item[] Parametric, aleatory  uncertainty



\vspace*{4mm}
  \item[] Goal is to  Predict and Inform 
% rather than (force) fit

\vspace*{4mm}
  \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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  











                       

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


\begin{itemize}




%\vspace*{1mm}
  \item[-]    {Modeling, Epistemic Uncertainty}
    \begin{itemize}

\vspace*{1mm}
      \item[] Modeling Simplifications

\vspace*{1mm}
      \item[] Modeling sophistication 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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Modeling, Epistemic Uncertainty}

\begin{itemize}



  \item[-]  Important (?!)  features are simplified
    \begin{itemize}
\vspace*{1mm}
        \item[] 1C vs 3C seismic motions
\vspace*{1mm}
        \item[] Elastic vs Inelastic behavior
    \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}


% 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
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% %
% \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}
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% % {\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}
% % %%
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% % {/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Energy_Dissipation_Animations/SMR_Energy_Dissipation_screen_grab.jpg}}
% % {/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Energy_Dissipation_Animations/SMR_Energy_Dissipation.mp4}
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% %  
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\end{frame}



        
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}




% local
%\vspace*{-2mm}
\begin{center}
\hspace*{-16mm}
%\movie[label=show3,width=5.6cm,poster,autostart,showcontrols]
\movie[label=show3,width=61mm,poster, showcontrols]
  {\includegraphics[width=60mm]{/home/jeremic/tex/works/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}
%\hspace*{-2mm}
%\hfill
%\movie[label=show3,width=5.6cm,poster,autostart,showcontrols]
\movie[label=show3,width=61mm,poster, showcontrols]
  {\includegraphics[width=60mm]
  {/home/jeremic/tex/works/Conferences/2016/IAEA_TecDoc_February2016/My_Current_Work/movie_ff_1d_mp4_icon.jpeg}}
  {/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Model01_ESSI_Response_May2015/movie_ff_1d.mp4}
\hspace*{-16mm}
\end{center}
% local


% online
\begin{center}
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/ESSI_VisIt_movies_Jose_19May2015/movie_ff_3d.mp4}
{\tiny (MP4)}
%
\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_1d.mp4}
 {\tiny (MP4)}
\end{center}
% online






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% out  %  \href{./homo_50m-mesh_45degree_Ormsby.mp4}
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% 
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\begin{frame}
\frametitle{6C vs 1C NPP ESSI Response Comparison}

 
% local
\vspace*{-2mm}
\begin{center}
\hspace*{-7mm}
%\movie[label=show3,width=8.8cm,poster,autostart,showcontrols]
\movie[label=show3,width=8.8cm,poster, showcontrols]
  {\includegraphics[width=92mm]
  {/home/jeremic/tex/works/Conferences/2016/IAEA_TecDoc_February2016/My_Current_Work/movie_2_npps_mp4_icon.jpeg}}
  {/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Model01_ESSI_Response_May2015/movie_2_npps.mp4}
\end{center}
% local
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%  \movie[label=show3,width=8.8cm,poster,autostart,showcontrols]
%    {\includegraphics[width=90mm]{movie_2_npps_mp4_icon.jpeg}}{movie_2_npps.mp4}
%  \end{center}


% online
\vspace*{-12mm}
\begin{flushleft}
%\vspace*{-15mm}
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/ESSI_VisIt_movies_Jose_19May2015/movie_2_npps.mp4}
  {\tiny (MP4)}
\end{flushleft}
% online



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% out  \href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/ESSI_VisIt_movies_Jose_19May2015/movie_2_npps.mp4}
% out    {\tiny (MP4)}
% out  \end{flushleft}
% out  %
% out  


 \end{frame}




        

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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{Energy Dissipation in a Large-Scale Model}

% 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{Energy Dissipation for Design}
% 
% \begin{figure}[!hbpt]
% \begin{center}
% \includegraphics[width=10.0truecm]{/home/jeremic/tex/works/Thesis/HanYang/Frame_animations_13Mar2019/2D_Frame_Model.pdf}
% \end{center}
% \end{figure}
% 
% 
% \end{frame}
% 
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\begin{frame}
\frametitle{Design Alternatives}


% local
%\vspace*{-2mm}
\begin{center}
\hspace*{-16mm}
%\movie[label=show3,width=5.6cm,poster,autostart,showcontrols]
\movie[label=show3,width=61mm,poster, showcontrols]
  {\includegraphics[width=60mm]{/home/jeremic/tex/works/Thesis/HanYang/Frame_animations_13Mar2019/Individual_Foundation_screen_grab.jpg}}
  {/home/jeremic/tex/works/Thesis/HanYang/Frame_animations_13Mar2019/Individual_Foundation.mp4}
%\hspace*{-2mm}
%\hfill
%\movie[label=show3,width=5.6cm,poster,autostart,showcontrols]
\movie[label=show3,width=61mm,poster, showcontrols]
  {\includegraphics[width=61mm]{/home/jeremic/tex/works/Thesis/HanYang/Frame_animations_13Mar2019/Continuous_Foundation_screen_grab.jpg}}
  {/home/jeremic/tex/works/Thesis/HanYang/Frame_animations_13Mar2019/Continuous_Foundation.mp4}
\hspace*{-16mm}
\end{center}
% local


% online
\begin{center}
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/Energy_dissipation_frames/Individual_Foundation.mp4}
{\tiny (MP4)}
%
\hspace*{40mm}
%
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/Energy_dissipation_frames/Continuous_Foundation.mp4}
 {\tiny (MP4)}
\end{center}
% online





\end{frame}
 
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\begin{frame}
 \frametitle{Pine Flat Dam, Dynamic Response with Reservoir}

%\vspace*{-5mm}
\begin{center}
% \hspace*{-15mm}
\movie[label=show3,width=9cm,poster,autostart,showcontrols]
{\includegraphics[width=8cm]
{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_Concrete_Dams/Pine_Flat_Dam/soil_fluid_interaction_dam.jpg}}
{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_Concrete_Dams/Pine_Flat_Dam/soil_fluid_interaction_dam.mp4}
\end{center}

\begin{flushleft}
\vspace*{-15mm}
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Concrete_Dams/Pine_Flat_Dam/soil_fluid_interaction_dam.mp4}
%  \href{./homo_50m-mesh_45degree_Ormsby.mp4}
  {\tiny (MP4)}
\end{flushleft}
%

\end{frame}

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


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

\begin{frame}
 \frametitle{Pine Flat Dam, Hydrodynamic Pressure}

%\vspace*{-5mm}
\begin{center}
% \hspace*{-15mm}
\movie[label=show3,width=10cm,poster,autostart,showcontrols]
{\includegraphics[width=10.0cm]
{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_Concrete_Dams/Pine_Flat_Dam/dynamic_pressure.jpg}}
{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_Concrete_Dams/Pine_Flat_Dam/dynamic_pressure.mp4}
\end{center}

\begin{flushleft}
\vspace*{-15mm}
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Concrete_Dams/Pine_Flat_Dam/dynamic_pressure.mp4}
%  \href{./homo_50m-mesh_45degree_Ormsby.mp4}
  {\tiny (MP4)}
\end{flushleft}
%

\end{frame}

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

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

\begin{frame}
% \frametitle{Pine Flat Dam, Inelastic Interface, Hydrostatic}
 \frametitle{Pine Flat Dam, Inelastic Interface}

%\vspace*{-5mm}
\begin{center}
% \hspace*{-15mm}
\movie[label=show3,width=10cm,poster,autostart,showcontrols]
{\includegraphics[width=10.0cm]
{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_Concrete_Dams/Pine_Flat_Dam/Case_D3_Gap_Open_more.jpg}}
{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_Concrete_Dams/Pine_Flat_Dam/Case_D3_Gap_Open_more.mp4}

\end{center}

\begin{flushleft}
\vspace*{-15mm}
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Concrete_Dams/Pine_Flat_Dam/Case_D3_Gap_Open_more.mp4}
%  \href{./homo_50m-mesh_45degree_Ormsby.mp4}
  {\tiny (MP4)}
\end{flushleft}
%

\end{frame}

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

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

\begin{frame}
 \frametitle{Pine Flat Dam, Inclined Plane Waves}

%\vspace*{-5mm}
\begin{center}
% \hspace*{-15mm}
\movie[label=show3,width=10cm,poster,autostart,showcontrols]
{\includegraphics[width=10.0cm]
{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_Concrete_Dams/Pine_Flat_Dam/dynamic_response_inclination.jpg}}
{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_Concrete_Dams/Pine_Flat_Dam/dynamic_response_inclination.mp4}
\end{center}

\begin{flushleft}
\vspace*{-15mm}
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Concrete_Dams/Pine_Flat_Dam/dynamic_response_inclination.mp4}
%  \href{./homo_50m-mesh_45degree_Ormsby.mp4}
  {\tiny (MP4)}
\end{flushleft}
%

\end{frame}

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%       
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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*{-5mm}
%\vspace*{-1.8cm}
%\hspace*{-3.3cm}
\begin{flushright}
{\tiny
(cf. Phoon and Kulhawy (1999B))\\
~}
\end{flushright}
%





   

\vspace*{-9mm}
\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*{-7mm}
\begin{flushright}
{\tiny
(cf. Wang et al. (2019))\\
~}
\end{flushright}








\end{frame}













%                        
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% % \frametitle{Sensitivity Analysis}
%  \frametitle{Engineer Needs to Know!}
% 
% 
% \begin{itemize}
% 
% 
% 
%   \item[-]    Forward    propagation    of   uncertainty,   full   probabilistic,
%   nonlinear/inelastic  Earthquake-Soil-Structure-Interaction, ESSI response in
%   time domain (Jeremic et al 2011, Wang et al 2019)
% % %\vspace*{1mm}
% %   \item[]      \underline{Sensitivity Analysis} quantifies the relative importance
% %   of  input  uncertain  parameters  and their contributions to the probabilistic
% %   system response
% % 
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \vspace*{4mm}
% %   \item[]    \underline{Local sensitivity analysis} focuses on the local impact of
% %   input  uncertain  parameters  on model response, quantified by the gradient of
% %   system response with respect to the variation of input parameters
% % 
% 
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \vspace*{4mm}
% %   \item[]    \underline{Global sensitivity analysis} studies the
% % respective
% \vspace*{4mm}
%   \item[-] Backward propagation, sensitivity analysis, quantifies the relative importance
%  of  input  uncertain  parameters on the variance of the probabilistic system
%   response  
% (Sobol 2001, Sudret 2008, Jeremic et al 2021)
% %Sobol, {\cyss Sobol{p1}} indices (Sobol 2001), Sudret (2008)
% % encoding for soft b is {p1}, see
% 
% 
% 
% \end{itemize}
% 
% 
% 
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 


% 
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Parametric Uncertainty: Soil Stiffness and Strength}
% 
% 
% \vspace*{2mm}
% %\vspace*{-3mm}
% \begin{figure}[!hbpt]
% \begin{center}
% %
% \hspace*{-7mm}
% \includegraphics[width=5.5truecm]{/home/jeremic/tex/works/Papers/2008/JGGE-GoverGmax/figures/YoungModulus_RawData_and_MeanTrend_01-Ed.pdf}
% \hspace*{-3mm}
% % \hfill
% \includegraphics[width=3.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGE-GoverGmax/figures/YoungModulus_Histogram_Normal_01-Ed.pdf}
% %
% \end{center}
% \end{figure}
% 
% \vspace*{-5mm}
% \begin{figure}[!hbpt]
% \begin{center}
% %
% \hspace*{-7mm}
% \includegraphics[width=5.00truecm]{/home/jeremic/tex/works/Papers/2008/JGGE-GoverGmax/figures/ShearStrength_RawData_and_MeanTrend-Mod.pdf}
% \hspace*{-3mm}
% % \hfill
% \includegraphics[width=3.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGE-GoverGmax/figures/ShearStrength_Histogram_PearsonIV-FineTuned-Mod.pdf}
% %
% \end{center}
% \end{figure}
% 
% %\vspace*{-5mm}
% %\vspace*{-1.8cm}
% %\hspace*{-3.3cm}
% \begin{flushright}
% {\tiny
% (cf. Phoon and Kulhawy (1999B))\\
% ~}
% \end{flushright}
% %
% 
% 
% \end{frame}
% 
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\begin{frame}
\frametitle{Parametric Uncertainty: Material Properties}

          

\vspace*{5mm}
\begin{figure}[!hbpt]
\begin{center}
% %
\hspace*{-3mm}
\includegraphics[width=2.5truecm]{/home/jeremic/tex/works/Thesis/KonstantinosKarapiperis/Soil_Uncertainty_Report_Pdf_Cdf_Figures/FieldPhiPdf.pdf}
\hspace*{-3mm}
\includegraphics[width=2.5truecm]{/home/jeremic/tex/works/Thesis/KonstantinosKarapiperis/Soil_Uncertainty_Report_Pdf_Cdf_Figures/FieldPhiCdf.pdf}
\hspace*{3mm}
\includegraphics[width=2.5truecm]{/home/jeremic/tex/works/Thesis/KonstantinosKarapiperis/Soil_Uncertainty_Report_Pdf_Cdf_Figures/FieldSuPdf.pdf}
\hspace*{-3mm}
\includegraphics[width=2.5truecm]{/home/jeremic/tex/works/Thesis/KonstantinosKarapiperis/Soil_Uncertainty_Report_Pdf_Cdf_Figures/FieldSuCdf.pdf}
\\
%\vspace*{-2mm}
\hspace*{-2.5cm} \mbox{\tiny Field $\phi$} \hspace*{3.5cm} \mbox{\tiny Field $c_u$}
\\
\vspace*{10mm}
\hspace*{-3mm}
\includegraphics[width=2.5truecm]{/home/jeremic/tex/works/Thesis/KonstantinosKarapiperis/Soil_Uncertainty_Report_Pdf_Cdf_Figures/LabPhiPdf.pdf}
\hspace*{-3mm}
\includegraphics[width=2.5truecm]{/home/jeremic/tex/works/Thesis/KonstantinosKarapiperis/Soil_Uncertainty_Report_Pdf_Cdf_Figures/LabPhiCdf.pdf}
\hspace*{3mm}
\includegraphics[width=2.5truecm]{/home/jeremic/tex/works/Thesis/KonstantinosKarapiperis/Soil_Uncertainty_Report_Pdf_Cdf_Figures/LabSuPdf.pdf}
\hspace*{-3mm}
\includegraphics[width=2.5truecm]{/home/jeremic/tex/works/Thesis/KonstantinosKarapiperis/Soil_Uncertainty_Report_Pdf_Cdf_Figures/LabSuCdf.pdf}
\\
%\vspace*{-8mm}
\hspace*{-2.5cm} \mbox{\tiny Lab $\phi$} \hspace*{3.5cm} \mbox{\tiny Lab $c_u$}
\end{center}
\end{figure}



\end{frame}



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% 
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  \begin{frame} 
%  
%  \frametitle{Real-ESSI Simulator System}
%  
%  % 
%  
%  \vspace*{2mm}
%  The  Real-ESSI,
%  {\underline {\bf Real}}istic  
%  %{\underline {\bf M}}odeling  and  
%  %{\underline {\bf S}}imulation  of  
%  {M}odeling  and  
%  {S}imulation  of  
%  {\underline {\bf E}}arthquakes,  
%  {\underline {\bf S}}oils,
%  {\underline {\bf S}}tructures  and  their  
%  {\underline {\bf I}}nteraction.  Simulator  is  a  software,  hardware  and
%  documentation   system   for   time  domain,
%  linear  and  nonlinear,  inelastic,  deterministic  or probabilistic, 3D, 
%  modeling and simulation of:
%  
%  \vspace*{-1mm}
%      \begin{itemize}
%  %\vspace*{1mm}
%        \item[-] statics and dynamics of soil,
%  % %\vspace*{1mm}
%  %       \item[-]    statics and dynamics of rock,
%  %\vspace*{1mm}
%        \item[-]    statics and dynamics of structures,
%  %\vspace*{1mm}
%        \item[-]    statics of soil-structure systems, and
%  %\vspace*{1mm}
%        \item[-]    dynamics of earthquake-soil-structure system interaction
%      \end{itemize}
%  
%  
%  
%  Used for:
%  \begin{itemize}
%  %\vspace*{1mm}
%    \item[-] Design: linear elastic, load combinations, dimensioning
%  
%  
%  %\vspace*{1mm}
%    \item[-] Assessment: nonlinear/inelastic, risk, safety margins
%  \end{itemize}
%  
%  
%  
%  
%  \end{frame}
%  
%  
%  
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame} 
% 
% \frametitle{Real-ESSI Simulator System}
% 
%  
% \begin{itemize}
% 
% 
%   \item Real-ESSI System Components
%     \begin{itemize} 
%      \item[-] Real-ESSI Pre-processor (gmsh/gmESSI, X2ESSI) 
%      \item[-] Real-ESSI Program (local, remote, cloud)
%      \item[-] Real-ESSI Post-Processor (Paraview/pvESSI, Python)
% 
%     \end{itemize} 
% 
% \vspace*{1mm}
%   \item Real-ESSI System availability:                   
%     \begin{itemize}
% %\vspace*{1mm}
%      \item[-] Educational  Institutions: AWS, Linux Image, free 
%      \item[-] Government Agencies, National Labs: AWS GovCloud
%      \item[-] Professional  Practice: AWS, commercial
% %\vspace*{1mm}
% %%\vspace*{1mm}
% %      \item Sources available to collaborators
%     \end{itemize}
% 
% 
% 
% \vspace*{1mm}
%   \item Real-ESSI education and training: theory and applications
% 
% 
% 
% \vspace*{1mm}
%   \item Real-ESSI documentation and program available at  
% \url{http://real-essi.info/}    
% %\url{http://sokocalo.engr.ucdavis.edu/~jeremic/Real_ESSI_Simulator/}
% %
% %\url{http://real-essi.info/}
% %
% 
% 
% % \vspace*{2mm}
% %   \item
% % 
% 
% 
% \end{itemize}
% 
% 
% \end{frame}
% 
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%\section{Real-ESSI Simulator}

      
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %\subsection*{Real-ESSI Simulator System}
% %\subsection{Real-ESSI}
% \subsection{\ }
% 
% 

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

\frametitle{Real-ESSI Simulator System}

\vspace*{1mm}
%
The  Real-ESSI, 
{\underline {\bf Real}}istic  
%{\underline {\bf M}}odeling  and  
%{\underline {\bf S}}imulation  of  
{M}odeling  and  
{S}imulation  of  
{\underline {\bf E}}arthquakes,  
{\underline {\bf S}}oils,
{\underline {\bf S}}tructures  and  their  
{\underline {\bf I}}nteraction  Simulator  is  a  software,  hardware  and
documentation   system   for   time  domain,
linear and  nonlinear,  
elastic and inelastic,  
deterministic  or probabilistic,
3D, 
modeling and simulation of:

\vspace*{-1mm}
    \begin{itemize}
\vspace*{-1mm}
     \item[-] statics and dynamics of soil and rock,
\vspace*{-1mm}
      \item[-]    statics and dynamics of rock,
\vspace*{-1mm}
     \item[-]    statics and dynamics of structures,
\vspace*{-1mm}
     \item[-]    statics of soil-structure systems, and
\vspace*{-1mm}
      \item[-]    dynamics of earthquake-soil-structure system interaction
    \end{itemize}



\vspace*{-1mm}
Used for:
\begin{itemize}
\vspace*{-1mm}
  \item[-] Design, linear elastic, load combinations, dimensioning


\vspace*{-1mm}
  \item[-] Assessment, nonlinear/inelastic, safety margins
\end{itemize}




\end{frame}



% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame} 
% 
% \frametitle{Real-ESSI Simulator System}
% 
%  
% \begin{itemize}
% 
% 
%   \item Real-ESSI System Components
%     \begin{itemize} 
%      \item[-] Real-ESSI Pre-processor (gmsh/gmESSI, X2ESSI) 
%      \item[-] Real-ESSI Program (local, remote, cloud)
%      \item[-] Real-ESSI Post-Processor (Paraview/pvESSI, Python)
% 
%     \end{itemize} 
% 
% \vspace*{3mm}
%   \item Availability: Windows/MacOS/Linux Docker, Linux execs, AWS
% %     \begin{itemize}
% % %\vspace*{1mm}
% %      \item[-] Educational  Institutions: AWS and Linux Executables, free 
% %      \item[-] Government Agencies, National Labs: AWS GovCloud, free
% %      \item[-] Professional  Practice: AWS and Linux Executables, commercial
% % %\vspace*{1mm}
% % %%\vspace*{1mm}
% % %      \item Sources available to collaborators
% %     \end{itemize}
% 
% 
% 
% \vspace*{3mm}
%   \item Real-ESSI education and training: theory and applications
% 
% 
% 
% \vspace*{3mm}
%   \item Real-ESSI documentation and program available at  
% \url{http://real-essi.info/}    
% %\url{http://sokocalo.engr.ucdavis.edu/~jeremic/Real_ESSI_Simulator/}
% %
% %\url{http://real-essi.info/}
% %
% 
% 
% % \vspace*{2mm}
% %   \item
% % 
% 
% 
% \end{itemize}
% 
% 
% \end{frame}
% 
% 


% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Quality Assurance}
% 
% \begin{itemize}
% 
% \item Full verification suit for each element, model, algorithm
% 
% \vspace*{5mm}
% \item Certification process
% 
%  \begin{itemize}
% 
% \vspace*{2mm}
%     \item[-] ASME  NQA-1
%  
% \vspace*{2mm}
%     \item[-] ISO-90003-2014
% 
%  \end{itemize}
% 
% %\vspace*{3mm}
% %\item[] Verification examples given below
% 
% \end{itemize}
% 
% 
% 
% 
% \end{frame}
% 
% %  % 
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  \begin{frame}
%  \frametitle{Real-ESSI}
%  
%  \begin{itemize}
%  
%  
%  
%  %\vspace*{2mm}
%    \item    A  system  for  time domain, nonlinear/inelastic,  deterministic  or 
%    probabilistic, modeling and simulation of
%  
%      \begin{itemize}
%        \item  statics and dynamics of soil,
%        \item  statics and dynamics of rock,
%        \item  statics and dynamics of structures,
%        \item  statics of soil-structure systems, and
%        \item  dynamics of earthquake-soil-structure system interaction.
%        \end{itemize}
%  
%  
%  \vspace*{1mm}
%    \item Design, linear elastic, load combinations, dimensioning
%  
%  
%  \vspace*{1mm}
%    \item Assessment, nonlinear/inelastic, safety margins
%  
%  
%  
%  %\vspace*{1mm}
%  %  \item  Develops methods and models that inform and predict rather than (force) fit.
%  
%  \vspace*{1mm}
%          \item Collaboration and financial support from the US-DOE, US-NRC,
%          US-NSF, Caltrans, CNSC-CCSN, UN-IAEA, Shimizu, Basler\&Hofmann, etc.
%  
%  
%  
%  \end{itemize}
%   
%  \end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Real-ESSI Modeling Features}

\begin{itemize}



%\vspace*{2mm}
  \item[-] Solid elements: dry, (un-)saturated, elastic, inelastic

%\vspace*{1mm}
  \item[-] Structural elements: beams, shells, elastic, inelastic 

%\vspace*{1mm}
  \item[-] Contact/interface/joint elements: Bonded, Shear/Frictional (EPP, EPH,
  EPS); Gap/Normal; linear, nonlinear, dry, coupled/saturated,

%\vspace*{1mm}
  \item[-] Super element: stiffness and mass matrices

%\vspace*{1mm}
  \item[-] Material models: soil, rock, concrete, steel... 
  
%\vspace*{1mm}
  \item[-] Seismic input: 1C and 3C, deterministic or probabilistic

%\vspace*{1mm}
  \item[-] Energy dissipation: elastic-plastic, viscous, algorithmic

%\vspace*{1mm}
  \item[-] Solid/Structure-Fluid interaction, full coupling, OpenFOAM

%\vspace*{1mm}
  \item[-] Intrusive, forw. and backw.  probabilistic inelastic modeling


%\vspace*{1mm}
  \item[-] Detailed Verification and partial Validation


%\vspace*{1mm}
  \item[-] Real-ESSI system: 
\hspace*{2mm} 
  \underline{\href{http://real-essi.info/}{http://real-essi.info/}}
%  \hspace*{5mm} 
%  and 
%  \hspace*{5mm} 
%  \href{http://real-essi.info/}{http://real-essi.info/}


\end{itemize}
 
\end{frame}


% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Real-ESSI Simulation Features}
% 
% 
% 
% %\vspace*{-10mm}
% 
% \begin{itemize}
% 
%   \item[-] Static loading stages
% 
% \vspace*{2mm}
%   \item[-] Dynamic loading stages
% 
% \vspace*{2mm}
%   \item[-] Restart, simulation tree
% 
% \vspace*{2mm}
%   \item[-] Solution advancement methods/algorithms, \\
%         on global and constitutive levels, \\
%         with and without enforcing equilibrium
% 
% 
% %\vspace*{1mm}
% %  \item Load combinations, elastic, for design
% 
% \vspace*{2mm}
%   \item[-] High Performance Computing
% %  clusters, cloud, supercomputers 
%     \begin{itemize}
% \vspace*{1mm}
%       \item[.] Fine grained, template mataprograms, small matrix library
% \vspace*{1mm}
%       \item[.] Coarse grained, distributed memory parallel 
%     \end{itemize}
% 
% 
% % \vspace*{1mm}
% %   \item All  Simulation Features are listed at  
% %   \hspace*{5mm}
% %   \href{http://real-essi.info/}{http://real-essi.info/} 
% % %  \hspace*{5mm} 
% % %  and 
% % %  \hspace*{5mm} 
% % %  \href{http://real-essi.info/}{http://real-essi.info/}
% 
% 
% 
% \end{itemize}
% 
% 
%  
% \vspace*{-60mm}
% %\begin{figure}[!hbpt]
% \begin{flushright}
% \includegraphics[width=2.5cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/_Chapter_Theory_Introduction/tex_works_psfigures_loading_stage-increments-iterations.pdf}
% \end{flushright}
% %\vspace*{-0.5cm}
% %\end{figure}
% %
% 
% 
% 
% 
%  
% \end{frame}
% 
% 
% 
%   
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Real ESSI Simulator: Domain Specific Language, DSL}
% 
%  \begin{itemize}
%    \item Domain Specific Language (DSL), Yacc \& Lex
% \vspace*{3mm}
%    \item English like modeling and simulation language
% \vspace*{3mm}
%    \item Parser and compiler, can define functions, models, etc.
% \vspace*{3mm}
%    \item Can extend models and methods
% \vspace*{3mm}
%    \item Requires units!
%  \end{itemize}
% %
% \end{frame}
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% 
% 
% 
% % 
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \begin{frame}[fragile]
% %  \frametitle{DSL: English Language Binding Modeling Parser}
% % {\footnotesize
% % \begin{lstlisting}
% % // Defining variables
% % x = 7; // x is double-valued variable with adimenisional units.
% % y = 3.972e+2; // Decimal and scientific notation is supported.
% % // Operations: All standard arithmetic operations (Unites!)
% % a = x + y;  // Addition
% % b = x - y;  // Subtraction
% % c = x*y;  // Product
% % d = x/y;  // Quotient
% % e = y%x;  // Modulus (how many times x fits in y)
% % // Predefined variables. For example, the variable 'm' defines 'meter'.
% % L1 = 1*m;
% % L2 = 40*mm;    // Defines L2 to be 40 millimiters.
% % L3 = 3.14*cm;
% % L4 = 3.14;
% % A5 = 3.14*cm^2;
% % \end{lstlisting}
% % }
% % \end{frame}
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \begin{frame}[fragile]
% %  \frametitle{DSL: English Language Modeling Parser}
% % {\footnotesize
% % \begin{lstlisting}
% % F1 = 10*kN; // Define few forces.
% % F2 = 300*N;
% % F3 = 4*kg*g; // Here g is the predefined acceleration
% %              // due to gravity.
% % // Operations are sensitive to units. For example,
% % foo = L1 + F1; // Produces an error because units are
% %                // not compatible. However,
% % L4 = L1 + L2 + L3;  // is OK.
% % // Multiplication (division and modulus) always work
% % // because the result produces a quantity with new units
% % // (except when the adimensional quantity is involved).
% % A = L1*L2;
% % pressure = F1 / A;
% % // All numbers are converted to SI units (kg - m - s)
% % // and internally stored in that system.
% % \end{lstlisting}
% % }
% % \end{frame}
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \begin{frame}
% % \frametitle{DSL: ESSI Input Language, Basics}
% % 
% % 
% % \begin{itemize}
% % 
% % \item Angle  brackets  \lstinline|<>| denotes  user input
% % 
% % \item Expected   unit (dimension)  is  given (example: 
% % \lstinline|<L>|, for length unit)
% % 
% % \item Symbol \lstinline|<.>| represents the adimensional quantity.
% % 
% % \item Vertical  bar  \lstinline+|+ (``OR'' sign)) is used to separate two or more keyword
% % options,   i.e.   \lstinline+[a|b|c]+   is   used   indicate   keyword   options
% % \lstinline+a+ or \lstinline+b+ or \lstinline+c+.
% % 
% % \item The  symbol  \lstinline+|...|+  is  used  to  denote  where several long options
% % exist  and  are  explained elsewhere (an example of this is available below in a
% % material model definitions).
% % 
% % \end{itemize}
% % 
% % \end{frame}
% % 
% 
% 
% 
% % 
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \begin{frame}
% % \frametitle{DSL: ESSI Input Language, Units}
% % 
% % 
% % All    commands  require  unit  consistency.  Base units, SI  or  other (British
% % Imperial)  can be used
% % \begin{itemize}
% %   \item  length, symbol  $L$, units [m, in, ft]
% %   \item  mass, symbol    $M$, units [kg, lb],
% %   \item  time, symbol    $T$, units [s]
% % \end{itemize}
% % 
% % Derived units can also be used:
% % 
% % \begin{itemize}
% %   \item angle, symbol rad (radian), unit [$dimensionless, L/L$]
% %   \item force, symbol N (Newton), units [$N, kN, MN, M*L/T^2$],
% %   \item stress, symbol Pa (Pascal), units [$Pa, kPa, MPa, N/L^2, M/L/T^2$]
% %   \item strain, symbol (no symbol), units [$L/L$]
% %   \item mass density, symbol (no symbol), units [$M/L^3$]
% %   \item force density, symbol (no symbol), units [$M/L^2/T^2$]
% % \end{itemize}
% % 
% % \end{frame}
% % 
% % 
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \begin{frame}
% % \frametitle{DSL: ESSI Input Language, Loading Stages}
% % 
% % 
% % Start         a    new    loading    stage  with
% % 
% % \lstinline|new loading stage "loading_stage_name";|
% % 
% % \vspace*{0.5cm}
% % Example,  starting a new loading stage called {\it "self weight load"}
% % 
% % \lstinline|new loading stage "self weight load";|
% % 
% % 
% % 
% % 
% % 
% % 
% % \end{frame}
% % 
% % 
% % 
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% 
% 
% % %% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % %\begin{frame}[fragile]
% % % \frametitle{DSL: Beam Example, Model}
% % %
% % %
% % %
% % %\begin{figure}[!h]
% % %\begin{center}
% % %\includegraphics[width=10cm]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Cantilever_Beam.pdf}
% % %%\caption{8 node brick element}
% % %%\label{fig:8node_command}
% % %\end{center}
% % %\end{figure}
% % %
% % %
% % %%}
% % %\end{frame}
% % %
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}[fragile]
%  \frametitle{Real ESSI DSL Example}
% 
% 
% \vspace*{-10mm}
% \begin{flushright}
% \includegraphics[width=3cm]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Cantilever_Beam.pdf}
% %\caption{8 node brick element}
% %\label{fig:8node_command}
% \end{flushright}
% 
% \vspace*{-5mm}
% \begin{lstlisting}
% model name "SmallTestModel";
% new loading stage "First_static";
% // Nodal  Coordinates
% add node # 1 at (0*m, 0*m, 0*m) with 6 dofs;
% add node # 2 at (0*m, 0*m, 1*m) with 6 dofs;
% add element # 1 type beam_elastic with
%   nodes (1, 2) cross_section=1.0*m^2
%   elastic_modulus=1.0e5*KN/m^2
%   shear_modulus=2.0e4*KN/m^2
%   torsion_Jx=2*0.083*m^4
%   bending_Iy=0.083*m^4 bending_Iz=0.083*m^4
%   mass_density=2500.0*kg/m^3
%   xz_plane_vector = (0, -1, 0)
%   joint_1_offset = (0.0*m, 0.0*m, 0.0*m)
%   joint_2_offset = (0.0*m, 0.0*m, 0.0*m);
% \end{lstlisting}
% 
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% 
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}[fragile]
%  \frametitle{Real ESSI DSL Example}
% 
% 
% \begin{lstlisting}
% fix node No 1 dofs all;
% add load # 1 to node # 2 type
%   linear Fx=-9*kN;
% define load factor increment 0.01;
% 
% define solver UMFPack;
% define convergence test
%   Norm_Displacement_Increment
%   tolerance = 1e-5
%   maximum_iterations = 20
%   verbose_level = 4;
% define algorithm Newton;
% simulate 100 steps using static algorithm;
% 
% //bye;
% zaijian;
% \end{lstlisting}
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% 
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \begin{frame}
% % \frametitle{Real-ESSI Model Development}
% % 
% % \begin{itemize}
% % 
% %     \item[-] Pre-Processing, model development gmsh/gmESSI
% % 
% % \vspace*{1mm}
% %     \item[-] Existing model translation, SASSI$\rightarrow$Real-ESSI
% % 
% % 
% % \vspace*{1mm}
% %     \item[-] Self documenting input language
% % 
% % \vspace*{1mm}
% %     \item[-] Units required for all input variables
% % 
% % \vspace*{1mm}
% %     \item[-] All variables and constants need to be defined by user
% % 
% % 
% % 
% % \vspace*{1mm}
% %     \item[-] Sophistication level of choice
% % 
% % %\vspace*{1mm}
% % %     \item[-] Reduce modeling uncertainty
% % 
% % \vspace*{1mm}
% %     \item[-] Model developed in phases
% % 
% % \vspace*{1mm}
% %     \item[-] Verify model components
% % 
% % 
% % \vspace*{1mm}
% %     \item[-]  Build confidence in inelastic modeling
% % 
% % \end{itemize}
% %  
% % \end{frame}
% % 
% % 
% % 
% % 
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \begin{frame}
% %   \frametitle{Real-ESSI  Modeling Phases}
% % 
% % 
% % 
% % \begin{figure}[htbp]
% % \begin{center}
% %   \includegraphics[width = 2.3cm]{/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*{-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{Real-ESSI Results Post Processing}
% % 
% % \begin{itemize}
% % 
% % 
% %       \item All output is saved (stress, strain, displacements, energy...)
% % 
% % \vspace*{5mm}
% %       \item Time histories, scripts to plot or extract in preferred format
% % 
% % \vspace*{5mm}
% %       \item 3D visualization, Paraview with pvESSI plugin
% % 
% % 
% % 
% % \end{itemize}
% %  
% % \end{frame}
% % 
% % 
% % 
% % 
% % 
% % 
% % 
% % 
% % 
% % 
% % 
% % 
% % 
% % 
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \begin{frame}
% %  \frametitle{Real-ESSI Core Functionality}
% % 
% % \begin{itemize}
% % 
% % 
% % 
% % %\vspace*{2mm}
% %   \item [-] Introduction to inelastic, nonlinear analysis for practicing engineers
% % 
% % % 
% % %  % \vspace*{3mm}
% % %    \item Usable models for professional practice
% % %  
% % %  %\vspace*{2mm}
% % %   \item  Core  functionality  needed  for  nonlinear  modeling  in professional
% % %   practice
% % %  % 
% % %  
% % 
% % 
% % %  %\vspace*{0.3cm}
% % %  %\vspace*{2mm}
% % %    \item Hierarchy of modeling capabilities, 
% % %    
% % %       \begin{itemize}
% % %  
% % %  %\vspace*{1mm}
% % %          \item Linear elastic models, elastic constants, viscous damping
% % %  
% % %  %\vspace*{1mm}
% % %          \item    Nonlinear  models,  core  functionality,  does not require much
% % %          material data however, sensitivity study is advised
% % %    
% % %  %\vspace*{1mm}
% % %          \item High sophistication  nonlinear models, require material data
% % %       
% % %       
% % %       \end{itemize}
% % %  
% % 
% % 
% % \vspace*{2mm}
% %   \item[-]   Use  of prescribed, required (low, medium, high) fidelity numerical
% %   models to analyze ESSI behavior
% % 
% % 
% % \vspace*{2mm}
% %   \item[-]  Set of suggested modeling and simulation parameters
% % 
% % \vspace*{2mm}
% %   \item[-]  Investigate  sensitivity  of  response  to model sophistication
% % 
% % \vspace*{2mm}
% %   \item[-]  Investigate  sensitivity  of response  to model parameters
% % 
% % % 
% % % \vspace*{1mm}
% % %   \item[]  Accurately   follow  the  flow   of  seismic  energy  in  a
% % %   soil structure system
% % % 
% % % \vspace*{1mm}
% % %   \item[]   The  goal is to create methodology and numerical tool that is used to 
% % %   predict and inform and not to fit
% % % 
% % % 
% % % 
% % % %\vspace*{1mm}
% % % %  \item[]   Directing,   in   space   and   time,   seismic  energy  flow  in  the
% % % %  soil structure system
% % % 
% % 
% % 
% % \end{itemize}
% % \end{frame}
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % 
% % 
% % 
% % 
% % 
% % 
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % 
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \begin{frame}
% %  \frametitle{Real-ESSI Core Functionality Components}
% % 
% % \begin{itemize}
% % 
% % 
% %    \item[-] Structural elements: Truss, Beam, Shell, Super-Element
% % 
% % \vspace*{2mm}
% %    \item[-] Soil, solids:  elastic, $G/G_{max}$
% %         
% % \vspace*{2mm}
% %    \item[-] Contacts/interfaces/joints: Bonded, Frictional (EPP, EPH, EPS), Gap
% %    open/close
% % 
% % 
% % \vspace*{2mm}
% %    \item[-] Loads: Static, Dynamic (earthquake, 1C or 3$\times$1C), Restart   
% % 
% % \vspace*{2mm}
% %    \item[-] Simulation: Explicit no-equilibrium, Implicit equilibrium
% % 
% % \vspace*{2mm}
% %    \item[-] Core Functionality Application programs: APPs
% % % 
% % 
% % 
% % \end{itemize}
% % 
% % 
% % \end{frame}
% % 
% % 
% % 
% % 
% % 
% % 
% % 
% % 
% % 
% % 
% % 
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \begin{frame}
% % \frametitle{Real-ESSI Education and Training}
% % 
% % \begin{itemize}
% % 
% % 
% %   \item Real-ESSI Education 
% %     \begin{itemize}
% % \vspace*{1mm}
% %       \item[-] Online courses
% % \vspace*{1mm}
% %       \item[-] Educational short videos
% % \vspace*{1mm}
% %      \item[-] Professional practice 
% % %   \vspace*{1mm}
% % %        \item Developers 
% % \vspace*{1mm}
% %      \item[-] Practical examples available in lecture notes, and documentation
% % \vspace*{1mm}
% %      \item[-] Documentation, Lecture Notes:
% %              (I) Theory and Computational Formulation,
% %              (II) Software and Hardware System,
% %              (III) Verification and Validation,
% %              (IV) Modeling and Simulation Examples,
% %              (V) Application to Practical Engineering Problems.
% % 
% %     \end{itemize}
% % 
% % %\vspace{1mm}
% % %  \item Documentation, extensive
% % 
% % \vspace{2mm}
% %   \item Lecture notes available online through 
% % \url{http://real-essi.info/}    
% %   
% % 
% % \end{itemize}
% %  
% % \end{frame}
% % 
% % 
% % 
% % 
% % 
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% 
% % \subsection{Verification  and Validation}
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%   \frametitle{Verification and Validation is Essential!}
% 
%     \begin{itemize}
%        
%       \item  V\&V most important for providing confidence in results
% 
% \vspace*{3mm}
%       \item Numerical modeling program(s) should not be used without extensive/full V\&V
% 
% \vspace*{3mm}
%       \item V\&V of FEM models is also essential
% 
% 
% \vspace*{3mm}
%       \item  Real  ESSI  Simulator has an extensive Verification database, and a
%       smaller Validation database
% 
% % 
% %       \item    Education  is  the  key  to successful use of realistic nonlinear
% %       Earthquake Soil Structure Interaction modeling and simulation
% % 
% %     \item     Development     of    the    Real    ESSI    Simulator    system
% %     \href{http://real-essi.info}{http://real-essi.info}
% 
% %     \item Importance of using proper models correctly (verification, validation, level of sophistication)
% % 
% 
% 
%       
% 
% \end{itemize}
% 
% 
% 
% \end{frame}
% %
% 
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%  \frametitle{Verification and Validation}
% \begin{itemize}
%     
%     
% 
% 
% 
%      \item  {{  Verification:}  provides  evidence  that  the  model is solved
%      correctly.} Mathematics issue.  Well developed 
%      for the Real ESSI Simulator.
% 
% \vspace*{4mm}
%      \item  {{  Validation:}  provides  evidence  that  the  correct  model is
%      solved.} Physics issue. Work in progress, UCSD, TJU labs 
%         
% \vspace*{4mm}
%    \item {  Prediction:}  use  of  computational  model
%    to  foretell  the  state  of  a  physical  system  under  consideration under
%    conditions for which the computational model has not been validated.
%   
% %   \item {  Prediction under Uncertainty}:  use  of  computational  model
% %   to  predict  the  state  of SSI system under
% %   conditions for which the computational model has not been validated.
% % 
% % 
% %   
% % \vspace*{1mm}
% %    \item  Modeling  and  parametric uncertainties are always present, need to be
% %    addressed 
% % 
% % %\vspace*{1mm}
% % %   \item Predictive capabilities with {low Kolmogorov Complexity}
% % %
% % 
% % \vspace*{1mm}
% %      \item Goal: Predict and Inform and rather than (force) Fit
% % 
% % 
% 
% 
% 
% \end{itemize}
% \end{frame}
% 
% 
% 
% 
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \begin{frame}
% % \frametitle{V \& V Motivation}
% % 
% % 
% % 
% % \begin{itemize}
% % 
% % 
% % %\vspace*{0.5cm}
% % \item How much can (should) we trust  model implementations (verification)?
% % 
% % %\vspace*{0.5cm}
% % \item How much can (should) we trust numerical simulations (validation)?
% % 
% % %\vspace*{0.5cm}
% % \item How good are our numerical predictions?
% % 
% % %\vspace*{0.5cm}
% % \item    Can      simulation  tools  be used for improving safety and economy?
% % 
% % %   \vspace*{2.0truecm}
% %    \item  V  \&  V  procedures  are  the  primary means of assessing accuracy in
% %    modeling and computational simulations
% % 
% % %   \vspace*{1.0truecm}
% %    \item  V  \&  V  procedures  are the tools with which we build confidence and
% %    credibility in modeling and computational simulations
% % 
% % 
% % 
% % 
% % %\vspace*{2.5cm}
% % %\item How do we  use experimental simulations   to improve models
% % 
% % 
% % 
% % 
% % \end{itemize}
% % 
% % 
% % 
% % 
% % \end{frame}
% % 
% % 
% % 
%    
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% 
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \begin{frame}
% %  \frametitle{Role of Verification and Validation}
% % 
% % 
% % 
% % \begin{figure}[!h]
% % \begin{center}
% % \hspace*{-2cm}
% % %{\includegraphics[width=5.0cm]{/home/jeremic/tex/works/Conferences/2012/ASME_V_and_V_symposium/presentetation/RoleVV_NEW_knowledge.pdf}}
% % {\includegraphics[width=5.0cm]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Present06_figs/RoleVV_NEW_knowledge.pdf}}
% % %{\includegraphics[width=6.5cm]{/home/jeremic/tex/works/Conferences/2011/USNCCM11_Minneapolis/Coupled/Present/VandV_ODEN.jpg}}
% % {\includegraphics[width=6.5cm]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Present06_figs/VandV_ODEN.jpg}}
% % \hspace*{-2cm}
% % \end{center}
% % \end{figure}
% % 
% % {Oberkampf et al. \hspace*{4cm}  Oden et al.}
% % %
% % %\item Models available (some now, some later)
% % %\vspace*{-2.0cm}
% % 
% % %
% % %\item Models available (some now, some later)
% % %\vspace*{-2.0cm}
% % \end{frame}
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% 
% 
% \begin{frame}
%   \frametitle{Verification and Validation}
% 
% 
% 
% %--
% 
% 
% \begin{figure}[!h]
% % \vspace*{-0.5cm}
% \hspace*{-8mm}
% %\begin{center}
% %{\includegraphics[width=11cm]{/home/jeremic/tex/works/Conferences/2005/OpenSeesWorkshopAugust/DeveloperSymposium/VerifValidFund01.pdf}}
% {\includegraphics[width=12cm]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Present06_figs/VerifValidFund01.pdf}}
% %\end{center}
% \end{figure}
% 
% 
% \end{frame}
% 
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% 
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \begin{frame}
% % \frametitle{Important Sources}
% % 
% % 
% % 
% % \begin{itemize}
% % 
% % 
% % 
% % 
% % %\item {\it  Short Course on Verification and Validation in
% % %Computational Mechanics}, by {\sc Dr. William Oberkampf}, Sandia National Laboratories
% % %July 27th, 2003, Albuquerque, New Mexico.
% % 
% % \item
% % {\sc W.~L. Oberkampf, T.~G. Trucano, and C. Hirsch.}
% % Verification, validation and predictive capability in computational
% %   engineering and physics.
% % In {\em Proceedings of the Foundations for Verification and
% %   Validation on the 21st Century Workshop}, pages 1--74, Laurel, Maryland,
% %   October 22-23 2002. Johns Hopkins University / Applied Physics Laboratory.
% % %{\href{http://sokocalo.engr.ucdavis.edu/~jeremic/UsefulReadings/Oberkampf-Trucano-Hirsch.pdf}
% % %{PDF available here}}
% % 
% % 
% % \item
% % {\sc P.~J. Roache.}
% %  {\em Verification and Validation in Computational Science and
% %   Engineering}.
% % Hermosa publishers, 1998.
% % ISBN 0-913478-08-3.
% % %
% % 
% % 
% % % \item Material from {\it Verification and Validation in Computational Mechanics}
% % % web site \texttt{http://www.usacm.org/vnvcsm/} at the USACM.
% % 
% %  \item  William L. Oberkampf and Christopher J. Roy. Verification and Validation
% %  in Scientific Computing. Cambridge University Press, 2010.
% % 
% % 
% % 
% % \end{itemize}
% % 
% % 
% % 
% % 
% % \end{frame}
% % 
% % 
% % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % % 
% % % 
% % % 
% % 
% % 
% % 
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% % 
% % 
% % 
% % 
% % 
% % 
% % 
% % 
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \begin{frame}
% %  \frametitle{Verification}
% % 
% % \begin{itemize}
% % 
% % \item   Source code management
% % \item   Source code verification
% % 
% % \item  Constitutive integration
% % \item   Static and dynamic behavior of single phase solids
% % \item       Static  and dynamic behavior of fully and partially saturated, fully
% % coupled, porous solid-pore fluid problems
% % \item   Static and dynamic behavior of structural elements
% % \item            Static    and    dynamic    behavior    of   special   elements
% % (contacts-interface/gap-frictional/dry-saturated, isolators/dissipators)
% % \item   Static and dynamic FEM solution advancement
% % \item   Seismic wave propagation problems
% % \item FEM Model verification, hierarchy of models
% % %\item   Static and Dynamic Behavior of Soil-Structure-Interaction
% % 
% % \end{itemize}
% % 
% % \end{frame}
% % 
% % 
% % 
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% % 
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \begin{frame}
% %  \frametitle{Constitutive Integration Verification}
% % 
% % 
% % \begin{itemize}
% % 
% %   \item Asymptotic regime of convergence
% % 
% %   \item Richardson extrapolation
% % 
% %   \item Grid convergence index
% % 
% % \end{itemize}
% % 
% % 
% %  \begin{center}
% %  \begin{figure}[!htbp]
% % \vspace*{-3mm}
% %  \includegraphics[width=5cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figure-files/asymptotic_regime.pdf}
% %  \includegraphics[width=3.5truecm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figure-files/asymptotic_nonassociate.pdf}
% % \\
% %  \includegraphics[width=3.0truecm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figure-files/error-map/DPAF_implicitErrMap.pdf}
% %  \includegraphics[width=3cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figure-files/GCI/NonAssociate.pdf}
% % \vspace*{-3mm}
% %  \end{figure}
% %  \end{center}
% % 
% % 
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% % 
% % 
% % 
% % 
% % 
% % \end{frame}
% % 
% % 
% % 
% % 
% % 
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% % 
% % % 
% % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % % \begin{frame}
% % %  \frametitle{Seismic Energy Input and Dissipation}
% % % 
% % % \begin{itemize}
% % % 
% % % \vspace*{1mm}
% % %      \item[] Seismic energy input, through a closed boundary (DRM)
% % % 
% % % 
% % % \vspace*{4mm}
% % %      \item[] Mechanical dissipation outside SSI domain:
% % %        \begin{itemize}
% % %          \item[] Reflected wave radiation
% % %          \item[] SSI system oscillation radiation
% % %        \end{itemize}
% % % %\vspace*{1mm}
% % %      \item[] Mechanical dissipation/conversion inside SSI domain:
% % %        \begin{itemize}
% % %          \item[] Inelasticity of soil and contact zone
% % %          \item[] Inelasticity/damage of structure and foundation
% % %          \item[] Viscous coupling of fluids and  soils and structure
% % % % %         \item[] potential and kinetic energy
% % % %          \item[] potential $\leftarrow \! \! \! \! \! \! \rightarrow$ kinetic energy
% % %        \end{itemize}                           
% % % 
% % % 
% % % 
% % % %\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}
% % % %  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % % 
% % 
% % 
% %  
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \begin{frame}
% %  
% % \frametitle{Energy Dissipation Verification: \\ Plastic Work $\neq$ Plastic Dissipation}
% %  
% % %\begin{itemize}
% % %\item Negative incremental energy dissipation
% % %\item Plastic work is NOT plastic dissipation
% % %\end{itemize}
% %  
% % \begin{figure}[!H]
% % \begin{center}
% % \includegraphics[height=4.5cm]{/home/jeremic/tex/works/Thesis/HanYang/Files_06June2017/DOE_Annual_2017/Figures/Negative_Dissipation_Problem.png}
% % \end{center}
% % \end{figure}
% %  
% %  \vspace*{-2mm}
% % \begin{itemize}
% % 
% % \item[] Direct violation of the second law of thermodynamics
% % 
% % %\vspace*{3mm}
% % \item[] 600 papers since 1990 (!?!) repeat this error
% % 
% % % \vspace*{3mm}
% % % \item[] Important form of energy missing: {Plastic Free Energy}
% % % 
% % % \vspace*{3mm}
% % % \item[] First described by Taylor and Quinney in 1934! 
% % % 
% % % \vspace*{3mm}
% % % \item[] Plastic Work vs. {Plastic Energy Dissipation}
% % 
% % 
% % %\vspace*{4mm}
% % %\item[] However it seems to be forgotten
% % 
% % \end{itemize}
% % 
% % 
% % \end{frame}
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % 
% % 
% % 
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \begin{frame}
% %  \frametitle{Dynamic Time Stepping Verification}
% % 
% % %\footnotesize
% % Based  on  the  amplification  matrix ${\bf A}$,  to  calculate the analytical solution of
% % damping ratios and period shift.
% % 
% % Example: Hilber-Hughes-Taylor $\alpha=-0.1$
% % 
% % \begin{figure}
% % \hspace*{-5mm}
% % \includegraphics[width=6cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figure-files/dynamic/hht/HHT-alpha-01-xi.jpg}
% % \includegraphics[width=6cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figure-files/dynamic/hht/HHT-alpha-01-period-shift.jpg}
% % \end{figure}
% % % HHT Integration $\alpha=-0.2$
% % % \begin{figure}
% % % \includegraphics[width=0.36\textwidth]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figure-files/dynamic/hht/HHT-alpha-02-xi.jpg}
% % % \includegraphics[width=0.36\textwidth]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figure-files/dynamic/hht/HHT-alpha-02-period-shift.jpg}
% % % \end{figure}
% % 
% % 
% % \end{frame}
% % 
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% % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% % \begin{frame}
% % \frametitle{Seismic Input Verification, DRM}
% % % \begin{figure}
% % % % \includegraphics[scale=0.35]{Present06_figs/3DModel.eps}
% % %  \includegraphics[width=3cm]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Present06_figs/3DModel.eps}
% % % \end{figure}
% % 
% % \begin{figure}
% % %\hspace*{-5mm}
% %  \includegraphics[width=5cm]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Present06_figs/90m_Northridge_EW.eps}
% % \\
% %  \includegraphics[width=5cm]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Present06_figs/90m_Northridge_NS.eps}
% % \\
% %  \includegraphics[width=5cm]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Present06_figs/90m_Northridge_UD.eps}
% % \end{figure}
% % 
% % 
% % 
% % 
% % % local
% % % local% local
% % % local%\vspace*{-2mm}
% % % local\begin{center}
% % % local\hspace*{-16mm}
% % % local%\movie[label=show3,width=5.6cm,poster,autostart,showcontrols]
% % % local\movie[label=show3,width=61mm,poster,showcontrols]
% % % local  {\includegraphics[width=60mm]
% % % local  {movie_ff_3d_mp4_icon.jpeg}}
% % % local  {/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Model01_ESSI_Response_May2015/movie_ff_3d.mp4}
% % % local%\hspace*{-2mm}
% % % local%\hfill
% % % local%\movie[label=show3,width=5.6cm,poster,autostart,showcontrols]
% % % local\movie[label=show3,width=61mm,poster,showcontrols]
% % % local  {\includegraphics[width=60mm]{movie_ff_1d_mp4_icon.jpeg}}
% % % local  {/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Model01_ESSI_Response_May2015/movie_ff_1d.mp4}
% % % local\hspace*{-16mm}
% % % local\end{center}
% % % local% local  
% % 
% % 
% % % online % online
% % % online % online
% % % online % online
% % % online \begin{center}
% % % online \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}
% % % online   {\includegraphics[width=50mm]{movie_ff_3d_mp4_icon.jpeg}}
% % % online %
% % % online \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_1d.mp4}
% % % online   {\includegraphics[width=50mm]{movie_ff_1d_mp4_icon.jpeg}}
% % % online \end{center}
% % % online % online
% % % online % online
% % % online % online
% % % online 
% % 
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% % 
% % \end{frame}
% % 
% % 
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% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \begin{frame}
% % \frametitle{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{Verification: Irregular Solids and Poisson's Ratio}
% % 
% % 
% % % \begin{itemize}
% % %  \item Irregular Mesh
% % %  \item Mesh Refinement
% % % \end{itemize}
% % 
% %  \begin{figure}
% % %  \hspace*{-25mm}
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% %  \includegraphics[width=6.5cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figure-files/beam/beam_brick27_shape2_vertical.pdf}
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% % \includegraphics[width=4.0cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figure-files/beam/error27brick_beam_different_poisson_ratio_disp_div100.jpeg}
% % \hspace*{3mm}
% % \includegraphics[width=4.0cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figure-files/beam/high_poisson.png}
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% % \includegraphics[width=3cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figure-files/beam/circular_plate4.pdf}
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% % \hspace*{3mm}
% % \includegraphics[width=3.5cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figure-files/beam/plate_error_plot_clamped.jpeg}
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% % \hspace*{3mm}
% % \includegraphics[width=2.5cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figure-files/boussinesq/model3Dview.JPG}
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% % \hspace*{5mm}
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\section{Uncertain Inelastic Dynamics}

      
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%\subsection{Stochastic Modeling}
\subsection{Forward Uncertainty Propagation}



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\begin{frame}
\frametitle{Forward Uncertainty Propagation}

\begin{itemize}

\vspace*{3mm}
  \item[-] Given uncertain material and uncertain loads

\vspace*{3mm}
  \item[-] Determine uncertain response, $u_i, \dot{u}_i, \ddot{u}_i,
  \epsilon_{ij}, \sigma_{ij}$, PDFs/CDFs

\vspace*{3mm}
  \item[-] Intrusive, analytic development, to circumvent 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} \ddot{ u_i} +
{ C} \dot{ u_i} +
{ K}^{ep} { u_i} =
{ 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}
\end{center}
\end{figure}

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\begin{frame} 

\frametitle{{Cam Clay  with  Random $G$, $M$ and $p_0$}}


\begin{figure}[!hbpt]
\begin{center}
\hspace*{-15mm}
\includegraphics[width=6.0cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourLowOCR_RandomG_RandomM_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}{Time Domain Stochastic Galerkin 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
 
\end{itemize}

%\scriptsize 
%Note: PC = Polynomial Chaos

\end{frame}

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% 
% \begin{frame}{Discretization of Input Random Process/Field $\beta(x,\theta)$}
% \begin{center}
% \includegraphics[scale=0.35]{/home/jeremic/tex/works/Thesis/FangboWang/slides_13Mar2019/Fangbo_slides/figs/PC_KL_explanation.PNG} \\
% \end{center}
% 
% 
% \footnotesize{Note: $\beta(x,\theta)$ is an input random process with any
% marginal distribution,  \\ \hspace{21mm} with any covariance structure;} \\
% \footnotesize{\hspace{8mm} $\gamma(x,\theta)$ is a 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    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\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}
\begin{center}
% \hspace*{-15mm}
\movie[label=show3,width=9cm,poster,autostart,showcontrols]
{\includegraphics[width=9cm]
{/home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Latex/img/figure_PEP_25.png}}
% /home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Latex/img/figure_PEP_25.pdf
%{/home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Animations/PEP_Animation.mp4}
{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Probabilistic_Elasto_Plasticity_and_Stochastic_Elastic_Plastic_Finite_Element_Method/PEP_Animation.mp4}
\end{center}


 

\begin{flushleft}
\vspace*{-15mm}
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Probabilistic_Elasto_Plasticity_and_Stochastic_Elastic_Plastic_Finite_Element_Method/PEP_Animation.mp4}
%  \href{./homo_50m-mesh_45degree_Ormsby.mp4}
  {\tiny (MP4)}
\end{flushleft}
%




%
% \includegraphics[width = 12cm]{./img/figure_PEP_25.pdf}


\end{frame}









%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section[Formulation]{Stochastic Dynamic Finite Element Formulation}
%\subsection[Time domain stochastic Galerkin method]{Time domain stochastic Galerkin method}
%\frame{\tableofcontents[currentsubsection,sectionstyle=show/shaded]}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Stochastic Elastic-Plastic Finite Element Method}



\begin{itemize}

\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


%
\vspace*{1mm} 
\item[-] Jeremi{\'c} et al. 2011


\end{itemize}


\begin{tiny}
\[
%$
\begin{bmatrix}
    \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_0> K^{(k)}    & \dots & \sum_{k=0}^{P_d} <\Phi_k \Psi_P \Psi_0> K^{(k)}\\
    \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_1> K^{(k)}    & \dots & \sum_{k=0}^{P_d} <\Phi_k \Psi_P \Psi_1> K^{(k)}\\  \\
    \vdots & \vdots & \vdots & \vdots\\
    \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_P> K^{(k)}     & \dots & \sum_{k=0}^{M} <\Phi_k \Psi_P \Psi_P> K^{(k)}
\end{bmatrix}
\begin{bmatrix}
    \Delta u_{10} \\
    \vdots \\
    \Delta u_{N0}\\
    \vdots \\
    \Delta u_{1P_u}\\
    \vdots \\
    \Delta u_{NP_u}
\end{bmatrix}
=
%\]
%\[
\begin{bmatrix}
     \sum_{i=0}^{P_f} f_i  <\Psi_0\zeta_i> \\
    \sum_{i=0}^{P_f} f_i   <\Psi_1\zeta_i> \\
   \sum_{i=0}^{P_f} f_i    <\Psi_2\zeta_i> \\
    \vdots \\
     \sum_{i=0}^{P_f} f_i  <\Psi_{P_u}\zeta_i>\\
\end{bmatrix}
%$
\]
\end{tiny}





\end{frame}








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

\begin{frame}
\frametitle{SEPFEM: System Size}

\begin{itemize}

  \item[-] SEPFEM offers a complete probabilistic solution 

  \item[-]  It is NOT based on Monte Carlo approach

  \item[-]  System of equations does grow (!)

\end{itemize}


%   \normalsize{Typical number of terms required for a SEPFEM problem} \vspace{1cm}\\
  \scalebox{0.7}{
  \begin{tabular}{ c c c c}
   \# KL terms material & \# KL terms load & PC order displacement& Total \# terms per DoF\\ \hline
   4 & 4 & 10 & 43758 \\  
   4 & 4 & 20 & 3 108 105 \\    
   4 & 4 & 30 & 48 903 492 \\  
   6 & 6 & 10 & 646 646 \\    
   6 & 6 & 20 & 225 792 840 \\  
   6 & 6 & 30 & 1.1058 $10^{10}$ \\    
   8 & 8 & 10 & 5 311 735 \\  
   8 & 8 & 20 & 7.3079 $10^{9}$ \\    
   8 & 8 & 30 & 9.9149 $10^{11}$\\  
   
   ...  & ... & ... & ...\\   
%    \hline
  \end{tabular}}


\end{frame}







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

\begin{frame}
\frametitle{SEPFEM: Example in 1D}




\vspace*{-2mm}
\begin{center}
% \hspace*{-15mm}
\movie[label=show3,width=9cm,showcontrols]
{\includegraphics[width=9cm]{/home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Latex/img/figure_elastic_900.png}}
% /home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Latex/img/figure_PEP_25.pdf
{/home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Animations/SEPFEM_Animation_Elastic.mp4}
% {SEPFEM_Animation_Elastic.mp4}
\end{center}


% 
%  \vspace*{-2mm}
%  \begin{center}
%  % \hspace*{-15mm}
%  \movie[label=show3,width=9cm,poster,autostart,showcontrols]
%  {\includegraphics[width=9cm]{/home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Latex/img/figure_elastic_900.png}}
%  % /home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Latex/img/figure_PEP_25.pdf
%  {/home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Animations/SEPFEM_Animation_Elastic.mp4}
%  \end{center}
%  
% \includegraphics[width = 12cm]{./img/figure_elastic_900.pdf}

 

\begin{flushleft}
\vspace*{-15mm}
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Probabilistic_Elasto_Plasticity_and_Stochastic_Elastic_Plastic_Finite_Element_Method/SEPFEM_Animation_Elastic.mp4}
%  \href{./homo_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}
%


\end{frame}

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






















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

%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  \begin{frame} 
%  
%  \frametitle{Probabilistic Seismic Risk Analysis (PSRA)}
%  
%  \begin{textblock}{15}(0.8, 3.9)
%  \scriptsize
%  Uncertain source, path, \\
%  \quad site and structure
%  \end{textblock}
%  
%  \begin{textblock}{15}(4.8, 4.2)
%  $\Longrightarrow$
%  \end{textblock}
%  
%  \begin{textblock}{15}(5.8, 3.7)
%  \scriptsize
%  %Acceptably small probability\\
%  Probabilities of \\
%  engineering demand parameters (EDP) \\
%   damage measures (DM), loss, etc   
%  \end{textblock}
%  
%  \begin{textblock}{15}(0.4, 11.8)
%  \tiny
%  \textbf{Uncertain source rupture}
%  \end{textblock}
%  
%  \begin{textblock}{15}(3.55, 4.8)
%  \begin{figure}[H]
%  \flushleft
%  \includegraphics[width=0.7\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/problem_statement.pdf}
%  \end{figure}
%  \end{textblock}
%  
%  \begin{textblock}{15}(12.5, 5.5)
%  \tiny
%  adapted from Taga(1982)
%  \end{textblock}
%  
%  \begin{textblock}{15}(11.7, 10.6)
%  \tiny
%  \textbf{Uncertain material properties}
%  \end{textblock}
%  
%  
%  \begin{textblock}{15}(0.4, 8.0)
%  \begin{figure}[H]
%  \flushleft
%  \includegraphics[width=0.2\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/4231447_Northridge_eq.pdf}
%  \end{figure}
%  \end{textblock}
%  
%  
%  \begin{textblock}{15}(10.7, 10.2)
%  \begin{figure}[H]
%  \flushleft
%  \includegraphics[width=0.3\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/uncertain_SPT.pdf}
%  \end{figure}
%  \end{textblock}
%  
%  \end{frame}
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  \begin{frame} 
%  
%  %\frametitle{State of the Art Probabilistic Seismic Risk Analysis}
%  \frametitle{Probabilistic Seismic Risk Analysis}
%  
%  \begin{itemize}
%  
%  % \item Target: safe design, acceptably small failure probability $\lambda (EDP)$
%  
%  \item[-] 
%  %Powerful tool 
%  %Allows for 
%  Objective, quantitative decision making based on exceedance rate
%  $\lambda (EDP>z)$
%  
%  \item[-] PSRA: convolution of PSHA and fragility 
%  
%  % \[\lambda(EDP>z) = \int_{IM} \underbrace{|\frac{d\lambda(IM)}{dIM}|}_\text{PSHA} \underbrace{G(EDP|IM)}_{\text{fragility}} dIM \]
%  
%  \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  $\sigma_{ij}$, 
%  $\epsilon_{ij}$, $u_i$... 
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%  \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}
%  
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%  % 
%  % \begin{frame}
%  % \frametitle{Stochastic Elastic-Plastic Finite Element Method}
%  % %\frametitle{SEPFEM : Formulation}
%  % 
%  % Stochastic system of equations 
%  % 
%  % \begin{tiny}
%  % \[
%  % \begin{bmatrix}
%  %     \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_0> K^{(k)}    & \dots & \sum_{k=0}^{P_d} <\Phi_k \Psi_P \Psi_0> K^{(k)}\\
%  %     \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_1> K^{(k)}    & \dots & \sum_{k=0}^{P_d} <\Phi_k \Psi_P \Psi_1> K^{(k)}\\  \\
%  %     \vdots & \vdots & \vdots & \vdots\\
%  %     \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_P> K^{(k)}     & \dots & \sum_{k=0}^{M} <\Phi_k \Psi_P \Psi_P> K^{(k)}
%  % \end{bmatrix}
%  % \begin{bmatrix}
%  %     u_{10} \\
%  %     \vdots \\
%  %     u_{N0}\\
%  %     \vdots \\
%  %     u_{1P_u}\\
%  %     \vdots \\
%  %     u_{NP_u}
%  % \end{bmatrix}
%  % =
%  % %\]
%  % %\[
%  % \begin{bmatrix}
%  %      \sum_{i=0}^{P_f} f_i  <\Psi_0\zeta_i> \\
%  %     \sum_{i=0}^{P_f} f_i   <\Psi_1\zeta_i> \\
%  %    \sum_{i=0}^{P_f} f_i    <\Psi_2\zeta_i> \\
%  %     \vdots \\
%  %      \sum_{i=0}^{P_f} f_i  <\Psi_{P_u}\zeta_i>\\
%  % \end{bmatrix}
%  % \]
%  % \end{tiny}
%  % 
%  % 
%  % 
%  % %   \normalsize{Typical number of terms required for a SEPFEM problem} \vspace{1cm}\\
%  %   \scalebox{0.7}{
%  %   \begin{tabular}{ c c c c}
%  %    \# KL terms material & \# KL terms load & PC order displacement& Total \# terms per DoF\\ \hline
%  %    4 & 4 & 10 & 43758 \\  
%  %    4 & 4 & 20 & 3 108 105 \\    
%  % %   4 & 4 & 30 & 48 903 492 \\  
%  %    6 & 6 & 10 & 646 646 \\    
%  % %   6 & 6 & 20 & 225 792 840 \\  
%  % %   6 & 6 & 30 & 1.1058 $10^{10}$ \\    
%  % %   8 & 8 & 10 & 5 311 735 \\  
%  % %   8 & 8 & 20 & 7.3079 $10^{9}$ \\    
%  % %   8 & 8 & 30 & 9.9149 $10^{11}$\\  
%  %    
%  %    ...  & ... & ... & ...\\    \hline
%  %   \end{tabular}}
%  % 
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%  % \end{frame}
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%  %  \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 
%  %  \item [--] \small Los Angeles, over 20,000 scenarios within 200 km, \textbf{300 million CPU-hours and over 100 years} (Maechling et al. 2007)
%  %  \end{itemize}
%  %  
%  %  \end{itemize}
%  %  \end{frame}
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% 
% %\subsection{TDNIPSRA Example}
% \subsection[TDNIPSRA Example]{Example}






% 
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begingroup

\setbeamertemplate{footline}{}

\begin{frame}

%\frametitle{TDNIPSRA Framework}
\frametitle{Application: Seismic Hazard}

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

\begin{textblock}{15}(0, 4.0)
\includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/UCERF3.pdf}
\end{textblock}

\begin{textblock}{15}(0.3, 3.5)
\scriptsize{Seismic source characterization}
\end{textblock}

\begin{textblock}{15}(2.9, 5.2)
\tiny{UCERF3 (2014)}
\end{textblock}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{textblock}{15}(5.1, 6.5)
%$\Rightarrow$
{\Large $\rightarrow$}
\end{textblock}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{textblock}{15}(5.8, 3.9)
\vspace*{1mm}
\includegraphics[width=0.27\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/SMSIM.pdf}
\end{textblock}


\begin{textblock}{15}(7.1, 6.2)
\scalebox{.9}{\tiny{Fourier spectra}}
\\
\vspace*{-0.2cm}
\scalebox{.9}{\tiny{\hspace{0.14cm} Boore(2003)}}
\end{textblock}

\begin{textblock}{15}(6.1, 3.5)
\scriptsize{Stochastic ground motion}
\end{textblock}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{textblock}{15}(9.9, 6.5)
%{\bf $\Rightarrow$}
{\Large $\rightarrow$}
\end{textblock}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{textblock}{15}(10.5, 4.2)
% \includegraphics[width=0.35\linewidth]{pic/KL_exact_dis_correlation_from_dis.pdf}
\includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Acc_realization_200.pdf}
\end{textblock}

\begin{textblock}{15}(11.1, 9.6)
\scriptsize{Uncertainty characterization \\
\hspace{0.1cm} Hermite polynomial chaos}
\end{textblock}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{textblock}{15}(10.2, 13.2)
{\Large $\leftarrow$}
\end{textblock}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{textblock}{15}(11, 11.2)
\includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/structural_uncertainty.pdf}
\end{textblock}

\begin{textblock}{15}(5.3, 10.75)
\includegraphics[width=0.33\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/probabilsitc_evolution.png}
\end{textblock}

\begin{textblock}{15}(5.4, 9.6)
\scriptsize{\quad \quad Uncertainty propagation \\ 
\quad \quad \quad \quad SEPFEM}
\end{textblock}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{textblock}{15}(4.6, 13.2)
%$\Leftarrow$
{\Large $\leftarrow$}
\end{textblock}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\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
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% 
% 
% 
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \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
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% \begingroup
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% \setbeamertemplate{footline}{}
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% \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]{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}
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% \begingroup
% 
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% 
% \frametitle{Time Domain Intrusive PSRA Framework}
% 
% 
% \begin{textblock}{15}(0, 4.0)
% \includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/UCERF3.pdf}
% \end{textblock}
% 
% \begin{textblock}{15}(0.3, 3.5)
% \scriptsize{Seismic source characterization}
% \end{textblock}
% 
% \begin{textblock}{15}(2.9, 5.2)
% \tiny{UCERF3 (2014)}
% \end{textblock}
% 
% \begin{textblock}{15}(5.1, 6.5)
% $\Rightarrow$
% \end{textblock}
% 
% 
% \begin{textblock}{15}(5.8, 3.9)
% \vspace*{1mm}
% \includegraphics[width=0.27\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/SMSIM.pdf}
% \end{textblock}
% 
% 
% \begin{textblock}{15}(7.1, 6.2)
% \scalebox{.9}{\tiny{Fourier spectra}}
% \\
% \vspace*{-0.2cm}
% \scalebox{.9}{\tiny{\hspace{0.14cm} Boore(2003)}}
% \end{textblock}
% 
% \begin{textblock}{15}(6.1, 3.5)
% \scriptsize{Stochastic ground motion}
% \end{textblock}
% 
% \begin{textblock}{15}(9.9, 6.5)
% $\Rightarrow$
% \end{textblock}
% 
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% \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}
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% ([xshift=-25pt,yshift=-52pt]$(pic cs:a) + (0pt,8pt)$) rectangle ([xshift=95pt,yshift=-18pt]$(pic cs:b)+(0pt,-2pt)$);
% \end{tikzpicture}
% 
% 
% \begin{textblock}{15}(-0.1, 9.25)
% \scriptsize
% \quad \quad \quad \quad $\lambda(EDP>z)=$
% 
% $\quad \sum N_i(M_i, R_i) P(EDP>z|M_i, R_i)$
% \end{textblock}
% 
% \begin{textblock}{15}(1.6, 10.7)
% \scriptsize{EDP hazard/risk}
% \end{textblock}
% 
% \end{frame}
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% \endgroup
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% 


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\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}

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\begin{frame} 
%\frametitle{TDNIPSRA Example Object}
\frametitle{Example Object}

\begin{textblock}{15}(0.5, 4.0)
\includegraphics[width=0.47\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/faults_configuration_new.pdf}
\end{textblock}

\begin{textblock}{15}(7.5, 3.4)
\includegraphics[width=0.55\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/SSC_legend.pdf}
\end{textblock}

\begin{textblock}{15}(0.8, 11.6)
\scriptsize
\begin{itemize}
\item Fault 1: San Gregorio fault 
\item Fault 2: Calaveras fault
\item Uncertainty: Segmentation, \\ slip rate, rupture geometry, etc.
\end{itemize} 
\end{textblock}

\begin{textblock}{15}(8.5, 11.6)
\scriptsize
\begin{itemize}
\item 371 total seismic scenarios 
\item $M \ 5 \sim 5.5$ and $6.5 \sim 7.0$
\item $R_{jb} \ 20km \sim 40km$
\end{itemize}
\end{textblock}

\end{frame}
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\begin{frame}

\frametitle{Stochastic Ground Motion Modeling}


\begin{textblock}{15}(1.0, 3.7)
\small Realizations of simulated uncertain motions for scenario $M=7$, $R=15km$: 
\end{textblock}

\begin{textblock}{15}(0.5, 4.0)
\includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Acc_time_series100.pdf}
\end{textblock}

\begin{textblock}{15}(5.5, 4.0)
\includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Acc_time_series343.pdf} \enspace
\end{textblock}

\begin{textblock}{15}(10.5, 4.0)
\includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Acc_time_series439.pdf}
\end{textblock}

\begin{textblock}{15}(1.0, 9.2)
\small Verification with GMPE: 
\end{textblock}

\begin{textblock}{15}(0.3, 9.5)
\includegraphics[width=0.36\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/SA_GMPE_verification_std_08_no_smooth.pdf}
\end{textblock}

\begin{textblock}{15}(5.6, 9.5)
\includegraphics[width=0.36\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Goodness_fit_std_08_no_smooth.pdf}
\end{textblock}

\begin{textblock}{15}(10.8, 9.5)
\includegraphics[width=0.36\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/Standard_deviation_std_08_no_smooth_new.pdf}
\end{textblock}

% \begin{textblock}{15}(0.5, 11.0)
% \begin{itemize}
% \item $\Delta \sigma= 84bar$, $\kappa=0.03s$ with total $\sigma=0.8ln$. 
% \item Simulated median is not biased. 
% \item Consistent total uncertainties with GMPE.  
% \end{itemize}
% \end{textblock}

\end{frame}
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\begin{frame} 
\frametitle{Stochastic Ground Motion Characterization}

{\begin{textblock}{15}(-0.1, 3.62)
\scriptsize
\includegraphics[width=0.3\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/KL_mean_acc_from_acc.pdf}

\quad \quad \quad Acc. marginal mean
\end{textblock}

\begin{textblock}{15}(3.7, 3.62)
\scriptsize
\includegraphics[width=0.3\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/KL_var_acc_from_acc.pdf}

\quad \quad \quad Acc. marginal S.D. 
\end{textblock}

\begin{textblock}{15}(7.6, 3.8)
\scriptsize
\includegraphics[width=0.3\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/KL_exact_acc_correlation_from_acc.pdf}

\quad \quad \quad Acc. realization Cov.
\end{textblock}

\begin{textblock}{15}(11.8, 3.9)
\scriptsize
\includegraphics[width=0.3\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/KL_simulated_acc_correlation_from_acc.pdf}

\quad \quad Acc. synthesized Cov.
\end{textblock}}

\begin{textblock}{15}(-0.1, 9.3)
\scriptsize
\includegraphics[width=0.31\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/KL_mean_dis_from_dis.pdf}
\end{textblock}

\begin{textblock}{15}(0.9, 13.75)
\scriptsize
Dis. marginal mean
\end{textblock}

\begin{textblock}{15}(4.2, 9.4)
\scriptsize
\includegraphics[width=0.3\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/KL_var_dis_from_dis.pdf}
\end{textblock}

\begin{textblock}{15}(5.1, 13.75)
\scriptsize
Dis. marginal S.D. 
\end{textblock}

\begin{textblock}{15}(8.2, 9.5)
\scriptsize
\includegraphics[width=0.27\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/KL_exact_dis_correlation_from_dis.pdf}

\quad \quad Dis. realization Cov.
\end{textblock}

\begin{textblock}{15}(12.2, 9.6)
\scriptsize
\includegraphics[width=0.27\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/KL_simulated_dis_correlation_from_dis.pdf}

\quad Dis. synthesized Cov.
\end{textblock}
\end{frame}
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\begin{frame}

\frametitle{Stochastic Material Modeling}

%Uncertain 1D shear response 



\begin{figure}[!htbp]
\centering
%\subfloat[Uncertain $H_a$]{
%\hspace{-0.8cm}
%\includegraphics[width=0.53\textwidth]{/home/jeremic/tex/works/Papers/2019/Hexiang/1D_risk/version6/Figures/constitutive_relation_uncertainHa_certainCr_MC_verification.pdf}}
%\subfloat[Uncertain $H_a$ and $C_r$]{
%\hspace{-0.2cm}
%\includegraphics[width=0.53\textwidth]{/home/jeremic/tex/works/Papers/2019/Hexiang/1D_risk/version6/Figures/constitutive_relation_uncertainHa_uncertainCr_MC_verification.pdf}}
%\vspace{-2mm}
%\caption{\label{figure_probabilisitc_constitutive_relation} Intrusive probabilistic modeling of Armstrong-Frederick hysteretic behavior and  verification  with Monte Carlo simulation: (a) Gaussian distributed  $Ha$  with  mean 1.76 $\times 10^{7} \ N/m$ and 15\% coefficient of variation (COV), $C_r = 17.6$. (b) Gaussian distributed  $Ha$  with  mean 1.76 $\times 10^{7} \ N/m$ and 15\% coefficient of variation (COV), Gaussian distributed $C_r$ with mean 17.6 and 15\% COV.}
\subfloat[Frame]{
\hspace{-0.8cm}
\includegraphics[width=2.5cm]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/Shear-Frame-8-levels.jpg}}
\subfloat[Interstory response]{
\hspace{10mm}
\includegraphics[width=6cm]{/home/jeremic/tex/works/Papers/2019/1D_risk/version6/Figures/constitutive_relation_uncertainHa_uncertainCr_MC_verification.pdf}}
%\vspace{-2mm}
%\caption{\label{figure_probabilisitc_constitutive_relation} Intrusive probabilistic modeling of Armstrong-Frederick hysteretic behavior and  verification  with Monte Carlo simulation: (a) Gaussian distributed  $Ha$  with  mean 1.76 $\times 10^{7} \ N/m$ and 15\% coefficient of variation (COV), $C_r = 17.6$. (b) Gaussian distributed  $Ha$  with  mean 1.76 $\times 10^{7} \ N/m$ and 15\% coefficient of variation (COV), Gaussian distributed $C_r$ with mean 17.6 and 15\% COV.}
\end{figure}
%



\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}

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\begingroup

\setbeamertemplate{footline}{}

\begin{frame} 

\frametitle{Seismic Risk Analysis}

\begin{textblock}{15}(1.9,3.8)
\scriptsize
\includegraphics[width=0.42\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/MIDR_PDF_evolution.pdf}
\includegraphics[width=0.42\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/PDF_MIDR_combine.pdf}
\end{textblock}

\begin{textblock}{15}(1.9,9.5)
\scriptsize
\includegraphics[width=0.42\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/MIDR_distribution_different_floors.pdf}
\includegraphics[width=0.42\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/Risk_MIDR.pdf}
\end{textblock}

\begin{textblock}{15}(0.8, 3.8)
\scriptsize Engineering demand parameter (EDP): Maximum inter-story drift ratio (MIDR) 
\end{textblock}
\end{frame}
\endgroup
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\begingroup
\setbeamertemplate{footline}{}
\begin{frame} 

\frametitle{Seismic Risk Analysis}

\begin{textblock}{15}(1.9,3.8)
\scriptsize
\includegraphics[width=0.42\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/PFA_distribution.pdf}
\includegraphics[width=0.42\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/Risk_PFA.pdf}
\end{textblock}

\begin{textblock}{15}(1.9,9.3)
\scriptsize
\includegraphics[width=0.41\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/2D_EDP_PDF_1e5.pdf}
\end{textblock}

\begin{textblock}{15}(8.7,9.4)
\scriptsize
\includegraphics[width=0.40\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/2D_EDP_PDF_downview_1e5.pdf}
\end{textblock}

\begin{textblock}{15}(0.8, 3.8)
\scriptsize Engineering demand parameter (EDP): Peak floor acceleration (PFA) 
\end{textblock}
\end{frame}
\endgroup
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\begin{frame} 

\frametitle{Seismic Risk Analysis}

%\vspace{-0.5cm}
\vspace*{2mm}

\begin{itemize}

%  \item[-] \small Damage measure (DM) defined on multiple EDPs: 
%  \item[-] \small Damage measure (DM) defined on single EDP:
  \item[-] Damage measure defined on single EDP:

\vspace*{-3mm}

%\begin{textblock}{15}(0.7,8.9)
\begin{table}[!htbp]
\small
\resizebox{0.98\hsize}{!}{
\begin{tabular}{ccccccc}
%\hline
\textbf{DM} & MIDR\textgreater{}0.5\% & \textbf{MIDR\textgreater{}1\%} & MIDR\textgreater{}2\% & PFA\textgreater{}0.5${\rm m/s^2}$ & \textbf{PFA\textgreater{}1\boldsymbol{${\rm m/s^2}$}} & PFA\textgreater{}1.5${\rm m/s^2}$ \\ 
\hline
\textbf{Risk [/yr]} & 6.66$\times 10^{-3}$                    & \textbf{3.83\boldsymbol{$\times 10^{-3}$}}                  & 9.97$\times 10^{-5}$                   & 6.65$\times 10^{-3}$                 & \textbf{1.92 \boldsymbol{$\times 10^{-3}$}}               & 9.45$\times 10^{-5}$                 \\ 
%\hline
\end{tabular}}
\end{table}
%\end{textblock}


\vspace{4mm}
  \item[-] Damage measure (DM) defined on multiple EDPs: 

%    \vspace{2mm}
    {\scriptsize $DM: \{\text{MIDR}>1\%\,  \cup \,\text{PFA}>1{\rm m/s^2} \}$, seismic risk is \boldsymbol{$4.2 \times 10^{-3}/yr$} }

    \vspace{1mm}
    {\scriptsize $DM: \{\text{MIDR}>1\%\,  \cap \,\text{PFA}>1{\rm m/s^2} \}$, seismic risk is \boldsymbol{$1.71 \times 10^{-3}/yr$}}

    \vspace{3mm}


%\vspace{20mm}

\vspace{4mm}
\item[-] \small Seismic risk for DM defined on multiple EDPs can be quite
different from that defined on single EDP

\end{itemize}
\end{frame}
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%  \begin{equation}
%  { M} \ddot{ u_i} +
%  { C} \dot{ u_i} +
%  { K}^{ep} { u_i} =
%  { F(t)}
%      \nonumber
%  \end{equation}
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%  \vspace*{2mm}
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%  $
%  { 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}
%  
%  
%  
%  
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  % \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$
%  
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%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%  
%  \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}
%  \begin{center}
%  % \hspace*{-15mm}
%  \movie[label=show3,width=9cm,poster,autostart,showcontrols]
%  {\includegraphics[width=9cm]
%  {/home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Latex/img/figure_PEP_25.png}}
%  % /home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Latex/img/figure_PEP_25.pdf
%  %{/home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Animations/PEP_Animation.mp4}
%  {/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Probabilistic_Elasto_Plasticity_and_Stochastic_Elastic_Plastic_Finite_Element_Method/PEP_Animation.mp4}
%  \end{center}
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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)}
%  \end{flushleft}
%  %
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%  % \includegraphics[width = 12cm]{./img/figure_PEP_25.pdf}
%  
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%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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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}
%  
%  \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
%  
%  
%  
%  \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}
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%  \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 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*{6mm}
  \item[-] Contributions of uncertain input to forward uncertainties

\vspace*{6mm}
  \item[-] Sensitivity of forward uncertain response to input uncertainties


\end{itemize}


\end{frame}




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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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\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*}


%



\end{frame}

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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*}







\end{frame}

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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*}
%    


\end{frame}

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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}


\end{frame}

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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*}
% 
% 
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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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% \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*}
%



\end{frame}

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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. 
%


\end{frame}


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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}
%  \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}
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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}

\end{frame}
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\begin{frame}

\frametitle{Stochastic Seismic Motion Development}

\vspace*{5mm}
\begin{itemize}
 \item[-] UCERF3 (Field et al. 2014)
 \item[-] Stochastic motions (Boore 2003) 
 \item[-] Polynomial Chaos Karhunen-Lo{\`e}ve expansion
 \item[-] Probabilistic DRM (Bielak et al. 2003, Wang et al. 2021)
\end{itemize}

\vspace*{-3mm}
\begin{figure}[!hbpt]
\begin{center}
\includegraphics[width=5cm]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/UCERF3.pdf}
\includegraphics[width=4.0cm]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/SMSIM.pdf}
%\includegraphics[width=3.5cm]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/KL_exact_dis_correlation_from_dis.pdf}
\end{center}
\end{figure}

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

\frametitle{Stochastic Ground Motion Modeling}

\begin{itemize}

%\vspace*{1mm}
\item[-] Modeling fundamental characteristics of uncertain ground motions,
Stochastic Fourier amplitude spectra (FAS). and Stochastic Fourier phase spectra
(FPS) and not specific IM

\vspace*{1mm}
\item[-]  Mean behavior of stochastic FAS,  $w^2$ source radiation spectrum by
Brune(1970), and Boore(1983, 2003, 2015).

\vspace*{1mm}
\item[-] Variability models for stochastic FAS, 
 FAS GMPEs by Bora et al. (2015, 2018), Bayless \&
 Abrahamson (2019), 
% for marginal median \& variability, 
%Inter-frequency correlation structure by 
Stafford(2017) and Bayless \& Abrahamson (2018).  

\vspace*{1mm}
\item[-] Stochastic FPS by phase derivative (Boore,2005), Logistic phase derivative
model by Baglio \& Abrahamson (2017)

%\item Upcoming: a major change from \textbf{ $\boldsymbol{Sa(T_0)}$ to FAS} in next five years as envisioned by Abrahamson (2018)

\end{itemize}

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





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

\frametitle{Sensitivity of  PGA from Uncertain Rock Motions}

%  


\vspace*{10mm}
\begin{itemize}
  \item[-] First 10 terms \\
        from motions uncertainty


  \item[-] Total Sobol     \\
        sensitivity index \\
        $S_{1-153}^{PC,total} = 0.98$


\end{itemize}



\vspace*{-40mm}
\begin{small}
%\begin{center}
\begin{table}[!htb]
\hspace*{50mm}
\begin{tabular}{l l}
Sobol Index           &  Value  \\ \hline 
$S_{1,123}^{PC}$      &   0,04389  \\
$S_{137}^{PC}$      &   0,03459  \\
$S_{110}^{PC}$      &   0,03061  \\
$S_{118}^{PC}$      &   0,02698  \\
$S_{1,118}^{PC}$      &   0,02605  \\
$S_{108}^{PC}$      &   0,02482   \\
$S_{141}^{PC}$      &   0,02373  \\
$S_{1,127}^{PC}$      &   0,02370  \\
$S_{1,100}^{PC}$      &   0,01759  \\
$S_{1,103}^{PC}$      &   0,01700  \\
. . .                &   . . .     \\ \hline  
    \end{tabular}
\end{table}
%\end{center}
\end{small}




%
%
%PC dimensions $\xi_1$, - $\xi_3$ are from uncertain soil
%
%PC dimensions $\xi_4$, - $\xi_{153}$ are from uncertain motions

\end{frame}
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\begin{frame}

\frametitle{Sensitivity Analysis}

%  


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


    \end{itemize}


%  \item 
%\end{itemize}



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\section{Summary}



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    \subsection{\ }






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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}


\end{frame}
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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*{2mm}
    \item[-] Analysis of uncertainties and sensitivities

\vspace*{4mm}
    \item[-] Predict and Inform


\vspace*{4mm}
%  \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*{4mm}
  \item[-]  
%    \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}



\end{frame}
%


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%  RIP:  Neboj{\v s}a Orbovi{\' c}  (1962-2021) \\
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