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

\usepackage{epstopdf}


\usepackage[absolute,overlay]{textpos}

\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
% \usepackage{graphicx}
% \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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{{\sf \uppercase{F}\kern-0.20em\uppercase{E}\kern-0.20em\uppercase{I}}}

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%  \usetheme{Marburg} % ima naslov i sadrzaj sa desne strane
%  \usetheme{Hannover} % ima naslov i sadrzaj sa leve strane
%  \usetheme{Singapore} % ima sadrzaj i tackice gore
%  \usetheme{Antibes}  % ima sadrzaj gore i kao graf ...
%  \usetheme{Berkeley} % ima sadrzaj desno
%  \usetheme{Berlin} % ima sadrzaj gore i tackice
%  \usetheme{Goettingen} % ima sadrzaj za desne strane
%  \usetheme{Montpellier} % ima graf sadrzaj gore
%  \usetheme{Warsaw}
%  \usetheme{Warsaw}
  \usetheme{Dresden}
    \usecolortheme[RGB={20,0,128}]{structure}

% or ...

  \setbeamercovered{transparent}


%  \setbeamercovered{transparent}
% or whatever (possibly just delete it)
%  \usecolortheme{albatross} % teget sa svetlim slovima
%  \usecolortheme{beetle} % siva pozadina (vrh plav)
  \usecolortheme{seagull} % sivo
%%%%%%%
%  \usecolortheme{BorisJeremic}
%%%%%%%
%  \usecolortheme{rose}
%   \usefonttheme[onlylarge]{structuresmallcapsserif}
%   \usefonttheme{structuresmallcapsserif}
}

\definecolor{mycolor}{rgb}{0,0.08,0.45}%


\usepackage[greek,english]{babel}



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

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


\usepackage{tikz}

\usetikzlibrary{calc}

\usepackage[export]{adjustbox}


\usepackage{array}



%%%%  HYPERREF  HYPERREF  HYPERREF  HYPERREF  HYPERREF
%%%%  HYPERREF  HYPERREF  HYPERREF  HYPERREF  HYPERREF
\definecolor{webgreen}{rgb}{0, 0.15, 0} % less intense green
\definecolor{webblue}{rgb}{0, 0, 0.45} % less intense blue
\definecolor{webred}{rgb}{0.15, 0, 0} % less intense red
%\usepackage[colorlinks=true,linkcolor=webblue,citecolor=webred,urlcolor=webgreen]{hyperref}

\usepackage{hyperref}
\hypersetup{
  pdfmenubar=true,
  pdftoolbar=true,
  pdfpagemode={None}
% Commenting this out fixes Windows acrobat problem (?!)
             colorlinks=true,
             linkcolor=webblue,
             citecolor=webblue,
             urlcolor=webblue,
}
% \usepackage[pdfauthor={Boris Jeremic},
%             colorlinks=true,
%             linkcolor=webblue,
%             citecolor=webblue,
%             urlcolor=webblue,
%             linktocpage,
%             pdftex]{hyperref}


\usepackage{pause}
% or whatever
%\usepackage{html}
%\usepackage{url}

\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[Real-ESSI, Uncertainty]
      {Aleatory Uncertainties in  \\
      Computational Earthquake Engineering}
%      {Real-ESSI Simulator System}
%      {Interface Between Earthquake Ground Motions and Structural
%Response: Numerical Modeling and Simulation of ESSI Behavior}


%\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}
% 
% 
% \pgfdeclareimage[height=0.20cm]{UCD-logo}{/home/jeremic/BG/amblemi/UCD/ucdavis_logo_blue_sm}
% %\pgfdeclareimage[height=0.18cm]{lbnl-logo}{/home/jeremic/BG/amblemi/LBNL/BerkeleyLab_Masterbrand_logo}
% \pgfdeclareimage[height=0.70cm]{LBNL-logo}{/home/jeremic/BG/amblemi/lbnl-logo}
% % % \pgfdeclareimage[height=0.15cm]{ESSI-C-logo}{/home/jeremic/BG/amblemi/ESSI/ESSI_Consultants_gray_02}
% %  \pgfdeclareimage[height=0.17cm]{ESSI-C-logo}{/home/jeremic/BG/amblemi/ESSI/ESSI_Consultants_white_01}
% % 
% %  \pgfdeclareimage[height=0.20cm]{UCD-logo}{/home/jeremic/BG/amblemi/UCD/UCD_white_01}
% %  \pgfdeclareimage[height=0.20cm]{LBNL-logo}{/home/jeremic/BG/amblemi/LBNL/LBNL_white_01}
% % % \pgfdeclareimage[height=0.15cm]{ESSI-C-logo}{/home/jeremic/BG/amblemi/ESSI/ESSI_Consultants_gray_02}
% %  \pgfdeclareimage[height=0.20cm]{ESSI-C-logo}{/home/jeremic/BG/amblemi/ESSI/ESSI-C_white_01}


 
% \pgfdeclareimage[height=0.40cm]{Best_PSHANI_2018_logo}{/home/jeremic/tex/works/Conferences/2018/BestPSHANI/Presentation/Best_PSHANI_2018_logo}

%\author[Jeremi{\'c} et al. Best PSHANI] % (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}}
%\\

%\institute[Computational Geomechanics Group \hspace*{0.3truecm}
%\institute[] % (optional, but mostly needed)
\institute[\pgfuseimage{university-logo}] % (optional, but mostly needed)
%\institute[\pgfuseimage{university-logo}\hspace*{0.1truecm}\pgfuseimage{lbnl-logo}] % (optional, but mostly needed)
%\institute[\pgfuseimage{UCD-logo}\hspace*{0.15truecm}\pgfuseimage{LBNL-logo}] % (optional, but mostly needed)
%\institute[\pgfuseimage{UCD-logo}\hspace*{0.15truecm}\pgfuseimage{LBNL-logo}\hspace*{0.15truecm}\pgfuseimage{ESSI-C-logo}] % (optional, but mostly needed)
%\institute[\pgfuseimage{UCD-logo}\mbox{;}\pgfuseimage{LBNL-logo}\mbox{;}\pgfuseimage{ESSI-C-logo}] % (optional, but mostly needed)
%{ Professor, University of California, Davis\\
%     { University of California, Davis, CA, USA}
%{UCD and LBNL}
{University of California, Davis}
%{ University of California, Davis, CA\\
%  and\\
%  Faculty Scientist, Lawrence Berkeley National Laboratory, Berkeley }
%  Lawrence Berkeley National Laboratory, Berkeley, CA}
%  ESSI Consultants, Davis, 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)
{\footnotesize The Tenth Kwang-Hua Forum\\ 
         Tongji University, Shanghai, China\\
         December 2023}

\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
% - % the beginning of each subsection:
% - % \AtBeginSubsection[]
% - \setcounter{tocdepth}{1}
% - %\AtBeginSubsection[]
% - \AtBeginSection[]
% -  {
% -    \begin{scriptsize}
% -    \begin{frame}<beamer>
% -      \frametitle{Outline}
% -      \tableofcontents[currentsection,currentsubsection]
% - %    \tableofcontents[currentsection]
% -    \end{frame}
% -    \end{scriptsize}
% -  }
% - 

% If you wish to uncover everything in a step-wise fashion, uncomment
% the following command:





\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{\ }



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\begin{frame}
 \frametitle{Motivation}

\begin{itemize}



 
\vspace*{2mm}
  \item[-] Safety and economy of infrastructure

%\vspace*{0.3cm}
 
%\vspace*{2mm}
%  \item[]  Improve  analysis of infrastructure objects
 
 
%  %\vspace*{1mm}
%  %  \item[-]  Analysis of Earthquakes and/or Soils and/or Structures and their Interaction
%  %  \item[-]  Analysis of Earthquakes, Soils, Structures and their Interaction
%  %  \item[-]  Soils, Structures and their Interaction, statics and dynamics
%    \item[-]  Soils, Structures, statics and dynamics
 

% \vspace*{1mm}
%  \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*{1mm}
%    \item[-] Analysis to  predict and inform 
%  % rather than (force) fit


%\vspace*{1mm}
%  \item[-] Design, build and maintain sustainable objects
\vspace*{1mm}
  \item[-] Design, build and maintain sustainable infrastructure


    
\vspace*{1mm}
  \item[-] Responsible Engineer, with Executive Powers 

%\vspace*{1mm}
% \item[] Engineer with executive powers


%\vspace*{2mm}
\vspace*{1mm}
  \item[-] Engineer with versatile, quality assured analysis tool to

  \begin{itemize}
\vspace*{1mm}
%  \item[-] Explore various design concepts
  \item[-] Explore design concepts

\vspace*{1mm}
  \item[-] Assess infrastructure performance 

  \end{itemize}


% \vspace*{2mm}
%   \item[-]  Choice of analysis/modeling level of sophistication
% 
% \vspace*{1mm}
%   \item[] Modeling simplifications, Epistemic uncertainty
% 
% \vspace*{1mm}
%   \item[] Variable, random material and loads, Aleatory  uncertainty
% 

% % 
% \vspace*{1mm}
%   \item[-] Infrastructure digital twins

% 
% \vspace*{2mm}
%   \item[-] Engineering Analysis to Predict and Inform
% 



% \vspace*{2mm}
%   \item[] Modeling sophistication level, epistemic uncertainty
% 
% \vspace*{2mm}
%   \item[] Parametric, aleatory  uncertainty
% 

% 
% \vspace*{2mm}
%   \item[-] Goal: Predict and Inform 
% % rather than (force) fit
% 
\vspace*{1mm}
  \item[-] Engineer needs to know!





% 
%  
%  
%  \vspace*{1mm}
%  \begin{figure}[!hbpt]
%  \begin{center}
%  %
%  %\hspace*{-7mm}
%  %\includegraphics[width=5.0truecm]{/home/jeremic/tex/works/Conferences/2021/CU-Boulder-GEGM-seminar-series-02Apr2021/present/Saint_Sophia_Constantinopolis.jpg}
%  %\hspace*{1mm}
%  \includegraphics[height=1.2truecm]{/home/jeremic/tex/works/Conferences/2023/5_SEECCM_Vrnjacka_Banja_5-6Jul2023/present/Parthenon_on_Acropolis_June2023.jpg}
%  %\hspace*{2mm}
%  \hspace*{2mm}
%  \includegraphics[height=1.2truecm]{/home/jeremic/tex/works/Conferences/2021/ASCE-4_Kennedy_Lecture/present/Aya-Sofia_03_1990.jpg}
%  %\hspace*{2mm}
%  \hspace*{2mm}
%  \includegraphics[height=1.2truecm]{/home/jeremic/tex/works/Conferences/2021/ASCE-4_Kennedy_Lecture/present/ZhaozhouBridge.jpg}
%  \hspace*{2mm}
%  %\hspace*{2mm}
%  \includegraphics[height=1.2truecm]{/home/jeremic/tex/works/Conferences/2021/ASCE-4_Kennedy_Lecture/present/Four_Water_Wheels_Hama_Syria.jpg}
%  \hspace*{2mm}
%  %\hspace*{2mm}
%  \includegraphics[height=1.2truecm]{/home/jeremic/tex/works/Conferences/2023/5_SEECCM_Vrnjacka_Banja_5-6Jul2023/present/Eiffel_tower_design_drawings/Eiffel_tower_02.jpg}
%  %
%  %\vspace*{3mm}
%  \end{center}
%  \end{figure}
%  %\vspace*{3mm}
%  
%  %\vspace*{1mm}
%  \begin{figure}[!hbpt]
%  \begin{center}
%  %
%  %\hspace*{-7mm}
%  %\includegraphics[width=5.0truecm]{/home/jeremic/tex/works/Conferences/2021/CU-Boulder-GEGM-seminar-series-02Apr2021/present/Saint_Sophia_Constantinopolis.jpg}
%  %\includegraphics[height=1.8truecm]{/home/jeremic/tex/works/Conferences/2021/ASCE-4_Kennedy_Lecture/present/Aya-Sofia_03_1990.jpg}
%  %\hspace*{2mm}
%  \includegraphics[height=3.5truecm]{/home/jeremic/tex/works/Conferences/2023/5_SEECCM_Vrnjacka_Banja_5-6Jul2023/present/Eiffel_tower_design_drawings/Eiffel_tower_force_diagram_shape.jpg}
%  \hspace*{4mm}
%  \includegraphics[height=3.5truecm]{/home/jeremic/tex/works/Conferences/2023/5_SEECCM_Vrnjacka_Banja_5-6Jul2023/present/Eiffel_tower_design_drawings/Eiffel_tower_foundation.jpg}
%  %\hspace*{2mm}
%  %
%  %\vspace*{3mm}
%  \vspace*{-2mm}
%  \end{center}
%  \end{figure}
%  %\vspace*{3mm}



%
%
%
%  \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.us/}
%  % % %  \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.us/}
%  % %
%


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

%  \vspace*{2mm}
%    \item[]  
%  %System  for 
%   {\bf Real}istic modeling and simulation of
%    {\bf E}arthquakes and/or    
%    {\bf S}oils and/or    
%    {\bf S}tructures    and    their    
%    {\bf I}nteraction:\\
%     Real-ESSI 
%   \hspace*{5mm} 
%    \url{http://real-essi.info/} 
%  % % %  \hspace*{25mm} 
%  %     \url{http://sokocalo.engr.ucdavis.edu/~jeremic/Real_ESSI_Simulator/}
%  % %    \href{http://sokocalo.engr.ucdavis.edu/~jeremic/Real_ESSI_Simulator/}{{http://sokocalo.engr.ucdavis.edu/~jeremic/Real_ESSI_Simulator/}
%  % %  % \url{http://ms-essi.info/}
%  % %  
% 




\end{itemize}
\end{frame}
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%  




  


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\begin{frame}
%\frametitle{ESSI Challenges}
\frametitle{Civil Engineering Analysis Challenges}
           

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%
\begin{figure}[!htb]
\begin{center}
% %\includegraphics[width=5cm]{Figure-files/_Chapter_Applications_ESSI_BOOK/Building_modeling_Issues_01_pdf.pdf}
%\vspace*{-3mm}
\includegraphics[width=2.5cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/_Chapter_Applications_ESSI_BOOK/Seismic_Motions_modeling_Issues_01_pdf.pdf}
\includegraphics[width=5.5cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/_Chapter_Applications_ESSI_BOOK/Seismic_Motions_modeling_Issues_02_pdf.pdf}
\\
%\vspace*{-2mm}
%\vspace*{-8mm}
\vspace*{2mm}
\includegraphics[width=3.0cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/_Chapter_Applications_ESSI_BOOK/Building_modeling_Issues_03_pdf.pdf}
\includegraphics[width=2.5cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/_Chapter_Applications_ESSI_BOOK/Building_modeling_Issues_02_pdf.pdf}
\includegraphics[width=2.5cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/_Chapter_Applications_ESSI_BOOK/Bridge_modeling_Issues_01_pdf.pdf}
\includegraphics[width=2.5cm]{/home/jeremic/tex/Classes/2020/Spring_semester_ETH/ESSI/Term_Projects_from_Students/Tunnel02.pdf}
\\
\vspace*{2mm}
%\includegraphics[width=1.2cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/_Chapter_Applications_ESSI_BOOK/Dam_modeling_Issues_01_pdf.pdf}
\includegraphics[width=4.0cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/_Chapter_Applications_ESSI_BOOK/Dam_modeling_Issues_02_pdf.pdf}
\includegraphics[width=2.8cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/_Chapter_Applications_ESSI_BOOK/NPP_Modeling_Issues_03.jpg}
\includegraphics[width=2.0cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/_Chapter_Applications_ESSI_BOOK/SMR_Modeling_Issues_02_pdf.pdf}
% %\vspace*{-8mm}
% \caption{\label{ESSI_Models_and_Challenges}
% ESSI modeling and simulation challenges:
% Free field motions, 3C/6C vs 3$\times$1C;
% Nuclear Power  Plant structure -- soil/rock system, Small Modular Reactor structure -- soil/rock system;
% Low and High Building-foundation-soil system;
% Dam-Foundation-Fluid system;
% Bridge-soil  system;
% %
% Aspects of modeling:
% 1) Seismic motions,
% 2) Inelastic soil and rock,
% 3) Inelastic interface/contact/joints, foundation with soil/rock and
%    interfaces/contacts/joints within structure,
% 4) Inelastic structure, systems and components,
% 5) Solid, Structure -- Fluid interaction, external (reservoirs, fluid pools...) and internal
%    (fully saturated and partially, (un-)saturated soil, rock and concrete).}
\end{center}
\end{figure}

%
%  
%  \begin{itemize}
%  
%      \item[-] Linear elastic, all elements
%  
%  %\vspace*{1mm}
%      \item[-] Nonlinear elastic, solids
%  
%  %\vspace*{1mm}
%      \item[-] Soil, solids/3D, dry, saturated and partially saturated)
%  
%  %\vspace*{1mm}
%      \item[-] Rock, solids/3D, dry, saturated and partially saturated) 
%  
%  %\vspace*{1mm}
%      \item[-] Interface/Contact, soft, hard, gap, dry, saturated
%  
%  %\vspace*{1mm}
%      \item[-] Base isolator and dissipator 2-node/3D
%  
%  
%  %\vspace*{1mm}
%      \item[-] Concrete, solids/3D, wall/2D, beam/1D, ASR
%  
%  %\vspace*{1mm}
%      \item[-] Steel, beam/1D and solid/3D
%  %
%  
%  
%  %\vspace*{1mm}
%      \item[-] Seismic input, DRM
%  
%  %\vspace*{1mm}
%      \item[-]  Energy Dissipation
%  
%  
%  \end{itemize}


\end{frame}
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\begin{frame}
 \frametitle{Progress, Infrastructure}
                 

\begin{center}
{\tiny Nov1990}
\includegraphics[width=48mm]{/home/jeremic/public_html/NaseSlike/1990/Kina-PutSvile/7022.jpg}
\includegraphics[width=48mm]{/home/jeremic/public_html/NaseSlike/1990/Kina-PutSvile/7021.jpg} 
\\
{\tiny Jul2019}
\includegraphics[width=48mm]{/home/jeremic/public_html/NaseSlike/2019/Jul/put_Peking_Shijiazhuang_13-20Jul2019_06/IMG_0348.JPG}
\includegraphics[width=48mm]{/home/jeremic/public_html/NaseSlike/2019/Jul/put_Peking_Shijiazhuang_13-20Jul2019_06/IMG_0350.JPG}
\end{center}


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








 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Infrastructure Digital Twin}

\begin{itemize}




 

%\vspace*{0.3cm}
\vspace*{1mm}
  \item[-] Infrastructure exists in  three dimensions 3D

\vspace*{2mm}
  \item[-] Material behavior is nonlinear, inelastic

\vspace*{2mm}
  \item[-] Soil and Structure work together





%\vspace*{4mm}
\vspace*{2mm}
  \item[-] Modeling, epistemic uncertainty, analysis sophistication






%\vspace*{4mm}
\vspace*{2mm}
  \item[-] Parametric, aleatory  uncertainty


\begin{itemize}
\vspace*{1mm}
  \item[-] Uncertain material parameters

\vspace*{1mm}
  \item[-] Uncertain loads
% 

\end{itemize}


%  \item[] Analysis model is everything

 

% \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[] Goal: 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{Motivation}
% 
%  
% \begin{itemize}
% 
% % 
% %  \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[]  Fran{\c c}ois-Marie Arouet,  Voltaire: 
% "Le doute n'est pas une condition agr{\'e}able, mais la certitude est absurde."
% 
%  \vspace*{9mm}
% 
% \item[]  Niklaus  Wirth:   
% "Software is getting slower more rapidly than hardware becomes faster."
% 
% \end{itemize}
% 
% 
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%   
% 
% 
% 
% 
% 
% 
%                        

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


\begin{itemize}




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

\vspace*{2mm}
      \item[] Modeling Simplifications

\vspace*{2mm}
      \item[] Modeling sophistication for confidence in  results 

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

    \end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace*{4mm}
  \item[-]   {Parametric, Aleatory    Uncertainty}

  \begin{itemize}


\vspace*{2mm}
    \item[]  ${M}  \ddot{u_i}  +  {C}  \dot{u_i} + {K}^{ep} {u_i} = {F(t)}$, 

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


\vspace*{2mm}
    \item[] Uncertain  loads, $F(t)$


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

  \end{itemize}




\end{itemize}



% 
% 
% %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}



\vspace*{1mm}
  \item[-] Simplified modeling, 3D/2D/1D, 1C/2C/3C/6C. damping, viscous,
   elastic/el-pl, algorithmic

%\vspace*{4mm}
%     \item   Unrealistic   and   unnecessary  modeling simplifications

% \vspace*{1mm}
     \item[-]  Modeling  simplifications   are  justifiable  if  one  or  two 
     level higher sophistication model demonstrates that features being
     simplified out are 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]
\movie[label=show3,width=5.5cm,showcontrols]
  {\includegraphics[width=50mm]
  {/home/jeremic/tex/works/Conferences/2016/IAEA_TecDoc_February2016/My_Current_Work/movie_2_npps_mp4_icon.jpeg}}
  {/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Model01_ESSI_Response_May2015/movie_2_npps.mp4}
%
%\hfill
\hspace*{-5mm}
%
%\movie[label=show3,width=6.0cm,poster,autostart,showcontrols]
\movie[label=show3,width=6.0cm,showcontrols]
  {\includegraphics[width=50mm]
  {/home/jeremic/tex/works/Conferences/2017/SMiRT_24/present/3D_Nonlinear_Modeling_and_it_Effects/NPP_Plastic_Dissipation_grab.jpg}}
  {/home/jeremic/tex/works/Thesis/HanYang/Files_10Aug2017/NPP_Plastic_Dissipation.mp4}
\hspace*{-7mm}
%\end{flushleft}
%%
\end{center}
% local

% % \vspace*{-5mm}
% \begin{center}
% %\begin{flushleft}
% % \hspace*{-15mm}
% \movie[label=show3,width=5cm,poster,autostart,showcontrols]
% {\includegraphics[width=5cm]
% {/home/jeremic/tex/works/Conferences/2017/SMiRT_24/present/3D_Nonlinear_Modeling_and_it_Effects/NPP_Plastic_Dissipation_grab.jpg}}
% {/home/jeremic/tex/works/Thesis/HanYang/Files_10Aug2017/NPP_Plastic_Dissipation.mp4}
% %\end{flushleft}
% %%
% \hfill
% %%
% %\begin{flushright}
% % \hspace*{-15mm}
% \movie[label=show3,width=5cm,poster,autostart,showcontrols]
% {\includegraphics[width=5cm]
% {/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Energy_Dissipation_Animations/SMR_Energy_Dissipation_screen_grab.jpg}}
% {/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Energy_Dissipation_Animations/SMR_Energy_Dissipation.mp4}
% %\end{flushright}
% \end{center}
%  



\end{frame}







      
%OVDE

% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%  \frametitle{Modeling Uncertainty, 6C vs 1C Motions}
% 
%      
% % 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
% %  \vspace*{-2mm}
% %  \begin{center}
% %  \hspace*{-7mm}
% %  \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
% 
% 
% 
% 
% 
% \end{frame}
% 

    
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Parametric, Aleatory Uncertainty}



\vspace*{2mm}
%\vspace*{-5mm}
\begin{figure}[!hbpt]
\begin{center}
%
\hspace*{-7mm}
\includegraphics[width=5.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGE-GoverGmax/figures/YoungModulus_RawData_and_MeanTrend_01-Ed.pdf}
\hspace*{-3mm}
% \hfill
\includegraphics[width=4.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGE-GoverGmax/figures/YoungModulus_Histogram_Normal_01-Ed.pdf}
%
\end{center}
\end{figure}


\vspace*{-3mm}
%\vspace*{-1.8cm}
%\hspace*{-3.3cm}
\begin{flushright}
{\tiny
(cf. Phoon and Kulhawy (1999B))\\
~}
\end{flushright}
%





   

\vspace*{-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/Hexiang/1D_risk/version2/Figures/SA_GMPE_verification_std_08_no_smooth.pdf}
%
\end{center}
\end{figure}


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








\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}
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \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{Uncertain Inelastic Dynamics}

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





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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 behavior (LHS) is uncertain

\vspace*{2mm}
\item[-] Loads (RHS) are uncertain



\end{itemize}



\end{frame}














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\begin{frame} 

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


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




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


%\begin{frame}{Time Domain Stochastic Galerkin Method}
\begin{frame}{Stochastic Elastic-Plastic FEM}


\vspace*{2mm}
Dynamic Finite Elements 
$
{ M} \ddot{ u_i} +
{ C} \dot{ u_i} +
{ K}^{ep} { u_i} =
{ F(t)}$


\begin{itemize}
\vspace*{2mm}
\item[-] Input random field/process{\normalsize{(non-Gaussian, heterogeneous/ non-stationary)}}:
          Multi-dimensional Hermite Polynomial Chaos (PC) with {known coefficients}
%\vspace{0.05in}
\vspace*{2mm}
\item[-] Output response process: Multi-dimensional Hermite PC with {unknown coefficients}
% \vspace{0.05in}
\vspace*{2mm}
 \item[-] Galerkin projection: minimize the error to compute unknown coefficients of response process
% %\vspace{0.05in}
% \vspace*{2mm}
%  \item[-] Time integration using Newmark's method
% % : Update coefficients following
% % an elastic-plastic constitutive law at each time step
 


\vspace*{2mm}
 \item[-] SEPFEM eliminates Monte-Carlo inefficiency and inaccuracy 


\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$
% - 
% - 
% - 
% - \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}
% 
% 







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


\vspace*{-5mm} 
\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 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}}
% - 
% - 
% - \end{frame}
% - 
% - 














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%\subsection{Backward Propagation, Sensitivities}
%\subsection{Sobol Sensitivity Analysis}





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Sobol Sensitivity Analysis}




\begin{itemize}


\item[-] 
The ANalysis Of VAriance representation
% of $f(x)$ 
(Sobol 2001)

% 
% \item[-] ANOVA representation is unique!
% 
\item[-]
Total variance of the probabilistic model response
$y=f(\boldsymbol{X})$ 
%
\begin{eqnarray*}
D = \mathrm{Var}[f(\boldsymbol{X})] = \int_{I^{n}} f^2(\boldsymbol{x}) \ d\boldsymbol{x} - f_0^2
\end{eqnarray*}

% \item[-]
% %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*}
% %
% 
% \item[-]
% 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{itemize}
% 
% \end{frame}
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \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} 
\end{equation*}
%
\end{itemize}


\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Sensitivity Analysis}
% 
%        
% The Sobol indices for global sensitivity analysis can be used for
% 
% Ranking of input uncertain parameters $\boldsymbol{X} = [X_1, ..., X_n]$. 
% 
% Neglect of inessential uncertain parameters and high order terms in Equation~\ref{eq_ANOVA_representation}.   
% %
% 
% 
% 
% The total sensitivity indices $S^{\mathrm{total}}_i$ is defined to evaluate the total influence of a certain input parameter $X_i$ as: 
% %
% \begin{equation*}
% S^{\mathrm{total}}_i = \sum_{\mathscr{S}_i}D_{i_1...i_s} 
% \end{equation*}
% %
% 
% set $\mathscr{S}_i$ contains all the indexes involving index $i$ 
% and Sobol' indices set $\{D_{i_1...i_s}\}$ is the
% collection of all partial sensitivity indices that are related to parameter
% $X_i$.
% %
% \begin{equation*}
% \mathscr{S}_i = \{(i_1, ..., i_s): \exists k, 1\leq k \leq s, i_k = i\}
% \label{eq_index_set}
% \end{equation*}
% 
% 
% \end{frame}
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
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\begin{frame}
\frametitle{Sobol-Sudret Sensitivity Analysis}


\begin{itemize}

\item[-]
PC expansion of response in ANOVA form (Sudret 2008)

\item[-]
Multi-dimensional PC bases $\{\Psi_j(\boldsymbol{\xi})\}$ 
decomposition
%  into products of single dimension PC chaos bases of different orders
%
\vspace*{-3mm}
\begin{equation*}
\Psi_j(\boldsymbol{\xi}) = \prod_{i=1}^{n} \phi_{\alpha_i}(\xi_i) 
\end{equation*}
%
% $\phi_{\alpha_i}(\xi_i)$ is the single dimensional, order $\alpha_i$, polynomial function of underlying basic random variable $\xi_i$. 
%

%


\vspace*{-1mm}
\item[-] 
ANOVA representation $\rightarrow$
% of probabilistic model response, 
PC-based Sobol'
indices $S_{i_1...i_s}^{PC}$ 
%
\vspace*{-3mm}
\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}
\end{equation*}
%



    
\vspace*{-1mm}
\item[-]
Total Sobol' indices $S_{j_1...j_t}^{PC, \mathrm{total}}$
% for any group of
%parameters $\{\xi_{j_1}, ..., \xi_{j_t}\}$ is
%
\vspace*{-2mm}
\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}   
\end{equation*}
%
% where $\mathscr{S}_{j_1,...,j_t} = \{(i_1, ..., i_s): (j_1,...,j_t) \subset (i_1, ..., i_s)\}$

% $\mathscr{S}_{j_1,...,j_t}$ 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}
% %



\item[-]
%Using PC representation of probabilistic model response,
Sobol-Sudret sensitivity indices within SEPFEM are analytic and inexpensive 
%evaluated with very small computational expense. 
%

\end{itemize}


\end{frame}








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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% \frametitle{Sobol Sensitivity Analysis}
% 
% 
% 
% \end{frame}
% 
% 
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Sobol Sensitivity Analysis}
% 
% 
% 
% \end{frame}
% 
% 
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Sobol Sensitivity Analysis}
% 
% 
% 
% \end{frame}
% 
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% \section{Sensitivity Analysis Example}
\subsection{Example}




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%\subsection{Uncertain Inelasticity}
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\begin{frame}

\frametitle{Example: 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 log-normal \\
       distribution, \\
       exponential correlation \\
       length 10m

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

\end{itemize}


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

% 
%  \begin{figure}[!hbpt]
%  \begin{center}
%  \includegraphics[width=10cm]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/structural_uncertainty.pdf}
%  \end{center}
%  \end{figure}
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%  
%  \begin{figure}[!hbpt]
%  \begin{center}
%  \includegraphics[width=10cm]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/probabilsitc_evolution.png}
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%  \begin{figure}[!hbpt]
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%  \includegraphics[width=10cm]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/seismic_risk_result_framework.png}
%  \end{center}
%  \end{figure}
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\end{frame}
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\begin{frame}

\frametitle{Stochastic Material Parameters}

%  

\vspace*{5mm}
Log-normal 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}

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\end{frame}
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% -  
% -  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  \begin{frame}
% -  
% -  \frametitle{Stochastic Ground Motion Modeling}
% -  
% -  \begin{itemize}
% -  
% -  %\vspace*{1mm}
% -  \item[-] Modeling fundamental characteristics of uncertain ground motions,
% -  Stochastic Fourier amplitude spectra (FAS). and Stochastic Fourier phase spectra
% -  (FPS) and not specific IM
% -  
% -  \vspace*{1mm}
% -  \item[-]  Mean behavior of stochastic FAS,  $w^2$ source radiation spectrum by
% -  Brune(1970), and Boore(1983, 2003, 2015).
% -  
% -  \vspace*{1mm}
% -  \item[-] Variability models for stochastic FAS, 
% -   FAS GMPEs by Bora et al. (2015, 2018), Bayless \&
% -   Abrahamson (2019), 
% -  % for marginal median \& variability, 
% -  %Inter-frequency correlation structure by 
% -  Stafford(2017) and Bayless \& Abrahamson (2018).  
% -  
% -  \vspace*{1mm}
% -  \item[-] Stochastic FPS by phase derivative (Boore,2005), Logistic phase derivative
% -  model by Baglio \& Abrahamson (2017)
% -  
% -  %\item Upcoming: a major change from \textbf{ $\boldsymbol{Sa(T_0)}$ to FAS} in next five years as envisioned by Abrahamson (2018)
% -  
% -  \end{itemize}
% -  
% -  \end{frame}
% -  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  
% -  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}

\frametitle{Stochastic Seismic Motions}

%  

%Log-normal distributed random field with PC Dim. 3 Order 2



%\vspace*{-50mm}
\begin{figure}[!hbpt]
\begin{center}
\hspace*{-10mm}
\includegraphics[width=12.5cm]{/home/jeremic/tex/works/Conferences/2021/CompDyn_8th_Athens_21-23Jun2020/present/Stochastic_seismic_motions.jpg}
\end{center}
\end{figure}

\end{frame}
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% -  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  \begin{frame}
% -  
% -  \frametitle{Stochastic Seismic Motions, Displacement}
% -  
% -  %  
% -  
% -  %\vspace*{-50mm}
% -  \begin{figure}[!hbpt]
% -  \begin{center}
% -  \hspace*{-10mm}
% -  \includegraphics[width=12.5cm]{/home/jeremic/tex/works/Conferences/2021/CompDyn_8th_Athens_21-23Jun2020/present/Stochastic-motions-DISP.jpg}
% -  \end{center}
% -  \end{figure}
% -  
% -  \end{frame}
% -  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  
% -           
% -  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  \begin{frame}
% -  
% -  \frametitle{Stochastic Seismic Motions, Accelerations}
% -  
% -  %  
% -  
% -  \vspace*{-1mm}
% -  \begin{figure}[!hbpt]
% -  \begin{center}
% -  \hspace*{-10mm}
% -  \includegraphics[width=12.5cm]{/home/jeremic/tex/works/Conferences/2021/CompDyn_8th_Athens_21-23Jun2020/present/Stochastic-motions-ACC.jpg}
% -  \end{center}
% -  \end{figure}
% -  
% -  \end{frame}
% -  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  
% -  


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

\frametitle{Stochastic Site Response, Mean and St.Dev.}

%  


\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}
% -  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  
% -  
% -  
% -  
% -  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  \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 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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%  
%  
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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*{1mm}
%      \item[-] Predict and inform
%  
%  
%  \vspace*{1mm}
%    \item[-] Engineer needs to know!
%  
%  
%   
%  \vspace*{1mm}
%    \item[-]  
%      \url{http://real-essi.us/}
%        
%  %
%  %\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}
%  %      
%  
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%  
%  % 
%  % \vspace*{2mm}
%  % \begin{center}
%  % \includegraphics[width=20mm]{/home/jeremic/tex/works/Conferences/2021/CompDyn_8th_Athens_21-23Jun2020/present/Efcharisto-thanks_you_in_Greek.jpg}
%  % \\ ~ \\
%  % %\includegraphics[width=1cm]{/home/jeremic/tex/works/Conferences/2021/CompDyn_8th_Athens_21-23Jun2020/present/Slike_iz_Atine/IMG_0552.JPG}
%  % %\includegraphics[width=1cm]{/home/jeremic/tex/works/Conferences/2021/CompDyn_8th_Athens_21-23Jun2020/present/Slike_iz_Atine/IMG_0549.JPG}
%  % %\includegraphics[width=1cm]{/home/jeremic/tex/works/Conferences/2021/CompDyn_8th_Athens_21-23Jun2020/present/Slike_iz_Atine/IMG_0505.JPG}
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%  % %\includegraphics[width=1cm]{/home/jeremic/tex/works/Conferences/2021/CompDyn_8th_Athens_21-23Jun2020/present/Slike_iz_Atine/IMG_0473.JPG}
%  % \includegraphics[width=2cm]{/home/jeremic/tex/works/Conferences/2021/CompDyn_8th_Athens_21-23Jun2020/present/Slike_iz_Atine/IMG_0464.JPG}
%  % %\includegraphics[width=2cm]{/home/jeremic/tex/works/Conferences/2021/CompDyn_8th_Athens_21-23Jun2020/present/Slike_iz_Atine/IMG_0418.JPG}
%  % %\includegraphics[width=2cm]{/home/jeremic/tex/works/Conferences/2021/CompDyn_8th_Athens_21-23Jun2020/present/Slike_iz_Atine/IMG_0381.JPG}
%  % %\includegraphics[width=2cm]{/home/jeremic/tex/works/Conferences/2021/CompDyn_8th_Athens_21-23Jun2020/present/Slike_iz_Atine/IMG_0374.JPG}
%  % %\includegraphics[width=2cm]{/home/jeremic/tex/works/Conferences/2021/CompDyn_8th_Athens_21-23Jun2020/present/Slike_iz_Atine/IMG_0371.JPG}
%  % \includegraphics[width=2cm]{/home/jeremic/tex/works/Conferences/2021/CompDyn_8th_Athens_21-23Jun2020/present/Slike_iz_Atine/IMG_0356.JPG}
%  % %\includegraphics[width=2cm]{/home/jeremic/tex/works/Conferences/2021/CompDyn_8th_Athens_21-23Jun2020/present/Slike_iz_Atine/IMG_0342.JPG}
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%  % \includegraphics[width=2cm]{/home/jeremic/tex/works/Conferences/2021/CompDyn_8th_Athens_21-23Jun2020/present/Slike_iz_Atine/IMG_0252.JPG}
%  % \includegraphics[width=2cm]{/home/jeremic/tex/works/Conferences/2021/CompDyn_8th_Athens_21-23Jun2020/present/Slike_iz_Atine/IMG_0248.JPG}
%  % \includegraphics[width=2cm]{/home/jeremic/tex/works/Conferences/2021/CompDyn_8th_Athens_21-23Jun2020/present/Slike_iz_Atine/IMG_0393.JPG}
%  % \includegraphics[width=2cm]{/home/jeremic/tex/works/Conferences/2021/CompDyn_8th_Athens_21-23Jun2020/present/Slike_iz_Atine/IMG_0763.JPG}
%  % \includegraphics[width=2cm]{/home/jeremic/tex/works/Conferences/2021/CompDyn_8th_Athens_21-23Jun2020/present/Slike_iz_Atine/IMG_0372.JPG}
%  % \end{center}
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\begin{frame}
  \frametitle{Summary}

  \begin{itemize}

%\vspace*{1mm}
    \item[-] Numerical analysis to predict and inform


% %\vspace*{1mm}
%     \item[-] Analysis of uncertainties and sensitivities
%
%\vspace*{1mm}
%    \item[-] Predict and inform
%

\vspace*{1mm}
  \item[-] Engineer needs to know!


% 
% \vspace*{2mm}
%     \item[-] Education and Training is the key!





\vspace*{1mm}
  \item[-] Real-ESSI Simulator System: 
   \url{http://real-essi.info}
% %\\                       



% \vspace*{1mm}
%         \item[-]  Collaborators at UCDavis:  Yang,  Behbehani,  Sinha, 
%         Wang, Lacour, Wang, Pisan{\'o}, Abell, Tafazzoli, Jie, Preisig,
%         Tasiopoulou, Watanabe, Luo, Cheng, Yang, Kanellopoulos, Staszewska
   

\vspace*{1mm}
        \item[-]  Collaborators at UCDavis:  Wang, Yang, Lacour, Staszewska


\vspace*{1mm}
        \item[-]  Collaboration with and  financial  support  from  the 
        US-NRC,
        US-NSF,  
        US-DOE, 
%         PEER,
%  %       Caltrans,  
%          CNSC-CCSN,  
        UN-IAEA, 
%  %        ILEE,
%          Shimizu, 
%          ENSI-CH-Basler\&Hofmann
is much appreciated



\vspace*{1mm}
\item[-] "Le doute n'est pas une condition agr{\'e}able, mais la certitude est
         absurde"
         (Voltaire)
%         (Fran{\c c}ois-Marie Arouet, Voltaire)
   


\end{itemize}



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