\documentclass[fleqn]{beamer}
%\documentclass[handout]{beamer}
%\usepackage{pgfpages}
%\pgfpagelayout{2 on 1}[letterpaper, border shrink=5mm]

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

\usepackage{epstopdf}

\usepackage[absolute,overlay]{textpos}

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




 
\usepackage{threeparttable}

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


      
\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
\input vuk.def
\newfont{\cyr}{wncyr10 scaled 1000 }
\newfont{\cyb}{wncyb10 scaled 1000 }
\newfont{\cyi}{wncyi10 scaled 1000 }
\newfont{\cysc}{wncysc10 scaled 1000 }
\newfont{\cyss}{wncyss10 scaled 1200 }
\newfont{\romjba}{cmss10 scaled 1000}
\newfont{\cyrjbnaslov}{wncyss10 scaled 1400 }



% % for not showing eq numbers unless eq is references
% \usepackage[fleqn,tbtags]{mathtools}
% \mathtoolsset{
%   showonlyrefs
%   }

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% FEI  LOGO definition
\newcommand{\FEI}
{{\sf \uppercase{F}\kern-0.20em\uppercase{E}\kern-0.20em\uppercase{I}}}

% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % % ovo je za cirilicu
% % \input vuk.def
% % %\newfont{\cyr}{wncyr10 scaled 1200}
% % %\newfont{\cyrimena}{wncyb10 scaled 1800}
% % %\newfont{\cyrnaslov}{wncyb10 scaled 2600}
% % %\newfont{\cyrpodnaslov}{wncyr10 scaled 1400}
%
% \newfont{\cyr}{wncyss10 scaled 1200}
% \newfont{\cyrimena}{wncyss10 scaled 2000}
% \newfont{\cyrnaslov}{wncyss10 scaled 2800}
% \newfont{\cyrpodnaslov}{wncyss10 scaled 1400}
%
% \newfont{\cyn}{wncyss10 scaled 2200}
% %% JB sig
% \newfont{\cyrjb}{wncyr10 scaled 600}
% \newcommand{\JB}
% {{\cyrjb \lower0.50ex\hbox{\uppercase{J}}\kern-.58em\hbox{\uppercase{B}}}}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\setbeamertemplate{navigation symbols}{}
\mode<presentation>
{
%  \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}
  \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{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
\definecolor{webred}{rgb}{0.35, 0, 0} % less intense red
%\usepackage[colorlinks=true,linkcolor=webblue,citecolor=webred,urlcolor=webgreen]{hyperref}


\usepackage{hyperref}
\hypersetup{
  pdfmenubar=true,
  pdftoolbar=true,
  pdfpagemode={None}
             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[Risk Analysis - SEPFEM]
      {Seismic Risk Analysis: \\
       Time Domain, Intrusive, \\
       Stochastic Elastic Plastic \\
       Finite Element Method}
%      {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}


%\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}}
{Jeremi{\'c}, Wang, Wang, Yang}
%Fangbo Wang, David McCallen}
%Floriana Petrone, Francis McKenna, David McCallen}
 
%
%\author[Jeremi{\'c} et al.] % (optional, use only with lots of authors)
%%{Boris~Jeremi{\'c}}
%{Boris Jeremi{\'c}
%}

%\institute[Computational Geomechanics Group \hspace*{0.3truecm}
\institute[\pgfuseimage{university-logo}\hspace*{0.1truecm}\pgfuseimage{lbnl-logo}] % (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 CompDyn\\ Crete, June 2019}

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


%
%\date[] % (optional, should be abbreviation of conference name)
%{\small United States Society on Dams\\
%         Chicago, April 2019}
%
%\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}{3}
 \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:





% -  \documentclass[fleqn]{beamer}
% -  %\documentclass[handout]{beamer}
% -  %\usepackage{pgfpages}
% -  %\pgfpagelayout{2 on 1}[letterpaper, border shrink=5mm]
% -  
% -  \mode<handout>{\setbeamercolor{background canvas}{bg=black!3}}
% -  
% -  % for extra, backup slides, see  https://www.yesterdayscoffee.de/2013/03/20/backup-slides-in-latex-beamer/
% -  % then in front of extra slides put 
% -  % \appendix
% -  % start backup slides here
% -  \usepackage{appendixnumberbeamer}
% -  
% -  \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
% -  \input vuk.def
% -  \newfont{\cyr}{wncyr10 scaled 1000 }
% -  \newfont{\cyb}{wncyb10 scaled 1000 }
% -  \newfont{\cyi}{wncyi10 scaled 1000 }
% -  \newfont{\cysc}{wncysc10 scaled 1000 }
% -  \newfont{\cyss}{wncyss10 scaled 1200 }
% -  \newfont{\romjba}{cmss10 scaled 1000}
% -  \newfont{\cyrjbnaslov}{wncyss10 scaled 1400 }
% -  
% -  
% -  
% -  % % for not showing eq numbers unless eq is references
% -  % \usepackage[fleqn,tbtags]{mathtools}
% -  % \mathtoolsset{
% -  %   showonlyrefs
% -  %   }
% -  
% -  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  % FEI  LOGO definition
% -  \newcommand{\FEI}
% -  {{\sf \uppercase{F}\kern-0.20em\uppercase{E}\kern-0.20em\uppercase{I}}}
% -  
% -  % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  % % % ovo je za cirilicu
% -  % % \input vuk.def
% -  % % %\newfont{\cyr}{wncyr10 scaled 1200}
% -  % % %\newfont{\cyrimena}{wncyb10 scaled 1800}
% -  % % %\newfont{\cyrnaslov}{wncyb10 scaled 2600}
% -  % % %\newfont{\cyrpodnaslov}{wncyr10 scaled 1400}
% -  %
% -  % \newfont{\cyr}{wncyss10 scaled 1200}
% -  % \newfont{\cyrimena}{wncyss10 scaled 2000}
% -  % \newfont{\cyrnaslov}{wncyss10 scaled 2800}
% -  % \newfont{\cyrpodnaslov}{wncyss10 scaled 1400}
% -  %
% -  % \newfont{\cyn}{wncyss10 scaled 2200}
% -  % %% JB sig
% -  % \newfont{\cyrjb}{wncyr10 scaled 600}
% -  % \newcommand{\JB}
% -  % {{\cyrjb \lower0.50ex\hbox{\uppercase{J}}\kern-.58em\hbox{\uppercase{B}}}}
% -  % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% -  
% -  
% -  \setbeamertemplate{navigation symbols}{}
% -  \mode<presentation>
% -  {
% -  %  \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}
% -    \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
% -  \definecolor{gray}{rgb}{0.6, 0.6, 0.6} % gray
% -  %\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[Risk Analysis - SEPFEM]
% -        {Seismic Risk Analysis: \\
% -         Time Domain, Intrusive, \\
% -         Stochastic Elastic Plastic \\
% -         Finite Element Method}
% -  %      {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=0.15cm]{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=0.15cm]{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}}
% -  {Jeremi{\'c}, Wang, Wang, Yang}
% -  %Fangbo Wang, David McCallen}
% -  %Floriana Petrone, Francis McKenna, David McCallen}
% -   
% -  %\institute[Computational Geomechanics Group \hspace*{0.3truecm}
% -  \institute[\pgfuseimage{UCD-logo}\hspace*{0.1truecm}\pgfuseimage{lbnl-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.
% -  
% -  
% -  %\institute[Computational Geomechanics Group \hspace*{0.3truecm}
% -  %\institute[] % (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}
% -  %{ 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)
% -  {\small CompDyn 2019 \\ Crete, June 2019}
% -  
% -  \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}{3}
% -  \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}

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

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

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

\begin{frame}
  \frametitle{Outline}

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

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

% 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{\ }
%\subsection{Motivation}



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\begin{frame}
 \frametitle{Motivation}

\begin{itemize}



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

\vspace*{5mm}
  \item[]  Use of numerical modeling for design and assessment
 


% \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*{5mm}
  \item[]  Analysis uncertainties

    \begin{itemize}


\vspace*{2mm}
           \item[]  Modeling uncertainty
% 
\vspace*{2mm}
           \item[]  Parametric uncertainty
% 


       \end{itemize}

%  %
%  %\vspace*{2mm}
%  %   \item[]  Choice of analysis level of sophistication
%  
%  
%  \vspace*{3mm}
%    \item[] Goal: Predict and Inform rather than fit
%  
%  
%  
%  %\vspace*{2mm}
%  %  \item[]  Follow  the  flow, input and dissipation,  of seismic  energy, 
%  %
%  
%  \vspace*{3wmm}
%    \item[]  System  for modeling and simulation of
%    Earthquakes and/or    Soils and/or    Structures    and    their    Interaction 
%  %  Real-ESSI/MS-ESSI:
%   \\
%   \vspace*{2mm}
%    \hspace*{25mm} 
%      \href{http://real-essi.info/}{http://real-essi.info/}
%  % \url{http://ms-essi.info/}





\end{itemize}
\end{frame}
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\begin{frame}
 \frametitle{Personal Risk}

\begin{itemize}



%\vspace*{0.3cm}
  \item[] Personal exposure to natural and man made hazard 
 

\vspace*{5mm}
  \item[]  Decision making under uncertainty 
 

\vspace*{5mm}
  \item[] Importance of education: engineering with uncertainty 

\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{Motivation: Seismic Hazard}



\begin{figure}[!hbpt]
\begin{center}
% %
%\hspace*{-3mm}
\includegraphics[width=11.0truecm]{/home/jeremic/tex/works/Conferences/2018/GeoMEast2018/Short_Course/present/Global_Seismic_Hazard_Map_1999.jpg}
\end{center}
\end{figure}

\end{frame}


% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%  \frametitle{Global Seismic Hazard Map}
% 
% 
% 10\% probability of exceedance in 50 years (Giardini et al.)
% %, 475 year return-period
% 
% 
% 
% %\vspace*{5mm}
% \begin{figure}[!hbpt]
% \begin{center}
% % %
% %\hspace*{-3mm}
% \includegraphics[width=11.5truecm]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/World_earthquake_hazard.jpg}
% \end{center}
% \end{figure}
% 
% 
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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\begin{frame}
 \frametitle{Mediterranean Seismic Hazard}


10\% probability of exceedance in 50 years (Giardini et al.)
%, 475 year return-period



%\vspace*{5mm}
\begin{figure}[!hbpt]
\begin{center}
% %
%\hspace*{-3mm}
\includegraphics[width=9.5truecm]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/Mediteranean_Earthquake_Hazard.jpg}
\end{center}
\end{figure}


\end{frame}
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\begin{frame}
 \frametitle{Crete Seismic Hazard}


10\% probability of exceedance in 50 years (Giardini et al.)
%, 475 year return-period



%\vspace*{5mm}
\begin{figure}[!hbpt]
\begin{center}
% %
%\hspace*{-3mm}
\includegraphics[width=9.5truecm]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/Crete_Seismic_Hazard.jpg}
\end{center}
\end{figure}


\end{frame}
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% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%  \frametitle{Predictive Capabilities}
% % \frametitle{High Fidelity Modeling of SFS System:
% % Verification, Validation and Prediction}
% 
% 
% \end{frame}
% 















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\section{Modeling}





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\subsection{Prediction of Behavior}




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\begin{frame}
 \frametitle{Predictive Capabilities}
% \frametitle{High Fidelity Modeling of SFS System:
% Verification, Validation and Prediction}

\begin{itemize}

   \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*{4mm}
     \item[]  {{  Verification:}  provides  evidence  that  the  model is solved
     correctly.} Mathematics issue.

\vspace*{4mm}
     \item[]  {{  Validation:}  provides  evidence  that  the  correct  model is
     solved.} Physics issue.
% 
\end{itemize}


\end{frame}



% 





% 










% \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 rather than (force) Fit
% 



  


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


\begin{itemize}




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

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


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

    \end{itemize}

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

  \begin{itemize}

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


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


\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)



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\subsection{Uncertainties}



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\begin{frame}
 \frametitle{Modeling Uncertainty}

\begin{itemize}



  \item[]  Important (?!)  features are simplified

     \begin{itemize}

\vspace*{2mm}
       \item[] Seismic motions, 1C vs 3C, 

\vspace*{2mm}
       \item[] Inelastic response
  
   \end{itemize}




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

\vspace*{6mm}
     \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
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%   {\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}
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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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% % local
% % 
% % % \vspace*{-5mm}
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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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% % %\begin{flushright}
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% % {\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}


 

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\begin{frame}
\frametitle{Modeling Uncertainty: Free Field Motions}



 
\begin{itemize}

  \item One component of motions, 1C, from  3C
%  or 3$\times$1D (it is done all the time!)

  \item  Excellent fit
%  (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)}
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% out    {\includegraphics[width=60mm]{movie_ff_3d_mp4_icon.jpeg}}{movie_ff_3d.mp4}
% out  % \hspace*{-16mm}
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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_ff_1d.mp4}
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\begin{frame}
 \frametitle{Modeling Uncertainty:  6C vs 1C}

     
% 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}




  
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\begin{frame}

\frametitle{Modeling Uncertainty: Input Frequency}

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

     
% \vspace*{-5mm}
\begin{center}
% \hspace*{-15mm}
%\movie[label=show3,width=10cm,poster,autostart,showcontrols]
\movie[label=show3,width=10cm,poster,showcontrols]
{\includegraphics[width=10cm]
{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Free_Field_animations_angle_or_frequency_variation/Free_Field_variation_in_wave_frequency.jpg}}
{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Free_Field_animations_angle_or_frequency_variation/free_field_frequency.mp4}
\end{center}
        
% online   
\vspace*{-12mm}
\begin{flushleft}
\hspace*{-4mm}
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Free_Field_animations_angle_or_frequency_variation/free_field_frequency.mp4}
  {\tiny (MP4)}
\end{flushleft}
% online  
   

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\begin{frame}

\frametitle{Modeling Uncertainty: SMR ESSI}

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

     
% \vspace*{-5mm}
\begin{center}
% \hspace*{-15mm}
%\movie[label=show3,width=10cm,poster,autostart,showcontrols]
\movie[label=show3,width=10cm,poster,showcontrols]
{\includegraphics[width=10cm]
{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Free_Field_animations_angle_or_frequency_variation/ESSI_SMR_variation_in_wave_frequency.jpg}}
{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Free_Field_animations_angle_or_frequency_variation/SMR_frequency.mp4}
\end{center}
        
% online   
\vspace*{-12mm}
\begin{flushleft}
\hspace*{-4mm}
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Free_Field_animations_angle_or_frequency_variation/SMR_frequency.mp4}
  {\tiny (MP4)}
\end{flushleft}
% online  
   

\end{frame}

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\begin{frame}

%\frametitle{SMR ESSI, Variation in Input Frequency, $\theta = 60^{o}$}
\frametitle{Modeling Uncertainty: 3C vs 3$\times$1C}

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

     
% \vspace*{-5mm}
\begin{center}
% \hspace*{-15mm}
%\movie[label=show3,width=10cm,poster,autostart,showcontrols]
\movie[label=show3,width=10cm,poster,showcontrols]
{\includegraphics[width=10cm]
{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/SMR_animations_May2018/3Dvs1D_deconvolution.jpg}}
{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/SMR_animations_May2018/3Dvs1D_deconvolution.ogv}
\end{center}
        
% online   
\vspace*{-12mm}
\begin{flushleft}
\hspace*{-4mm}
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/SMR_animations_May2018/3Dvs1D_deconvolution.ogv}
  {\tiny (OGV)}
\end{flushleft}
% online  
   

\end{frame}

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

 \frametitle{Modeling Uncertainty: Elastic vs Inelastic}


\vspace*{5mm}
Accelerations

\hspace*{-35mm}
  \begin{figure}[!h]
  \vspace*{-2mm}
  \begin{center}
  \hspace*{-15mm}
   \includegraphics[width=7.0cm]{/home/jeremic/tex/works/Thesis/SumeetKumarSinha/Files_10Aug2017/Npp_Non_Linear_Effects/images/Acceleration_Elastic_Without_Contact_SMIRT_2017.pdf}
  \hspace*{-20mm}
   \includegraphics[width=7.0cm]{/home/jeremic/tex/works/Thesis/SumeetKumarSinha/Files_10Aug2017/Npp_Non_Linear_Effects/images/Acceleration_Inelastic_With_Contact_SMIRT_2017.pdf}
  \hspace*{-25mm}
  \end{center}
 \end{figure}
\hspace*{-45mm}

\vspace*{-10mm}

\hspace*{10mm} Elastic \hspace*{40mm} Inelastic \hspace*{40mm} 

\end{frame}

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\begin{frame}

 
 \frametitle{Modeling Uncertainty: Elastic vs Inelastic}

\vspace*{5mm}
Displacements
 
\hspace*{-35mm}
  \begin{figure}[!h]
  \vspace*{-2mm}
  \begin{center}
  \hspace*{-15mm}
   \includegraphics[width=7.0cm]{/home/jeremic/tex/works/Thesis/SumeetKumarSinha/Files_10Aug2017/Npp_Non_Linear_Effects/images/Elastic_Without_Contact_SMIRT_2017.pdf}
  \hspace*{-20mm}
   \includegraphics[width=7.0cm]{/home/jeremic/tex/works/Thesis/SumeetKumarSinha/Files_10Aug2017/Npp_Non_Linear_Effects/images/Inelastic_With_Contact_SMIRT_2017.pdf}
  \hspace*{-25mm}
  \end{center}
 \end{figure}
\hspace*{-35mm}

\vspace*{-10mm}

\hspace*{10mm} Elastic  \hspace*{40mm} Inelastic \hspace*{40mm} 

\end{frame}
 
 

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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% 
% \frametitle{Modeling Uncertainty: Elastic and Inelastic 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{Modeling Uncertainty: Energy Dissipation}

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

% \vspace*{-5mm}
\begin{center}
% \hspace*{-15mm}
\movie[label=show3,width=10cm,poster,autostart,showcontrols]
{\includegraphics[width=10cm]
{/home/jeremic/tex/works/Conferences/2017/SMiRT_24/present/3D_Nonlinear_Modeling_and_it_Effects/NPP_Plastic_Dissipation_grab.jpg}}
{/home/jeremic/tex/works/Thesis/HanYang/Files_10Aug2017/NPP_Plastic_Dissipation.mp4}
\end{center}




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



\end{frame}

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\begin{frame}
\frametitle{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}







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

\frametitle{Uncertain Results: Energy Dissipation}

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

% \vspace*{-5mm}
\begin{center}
% \hspace*{-15mm}
\movie[label=show3,width=10cm,poster,autostart,showcontrols]
{\includegraphics[width=10cm]
{/home/jeremic/tex/works/Thesis/HanYang/Stocahstic_animations_14Jun2019/Blur_Videos/NPP_Blured_more_screeen_grab.jpg}}
{/home/jeremic/tex/works/Thesis/HanYang/Stocahstic_animations_14Jun2019/Blur_Videos/NPP_Blurred_More.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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\section{Time Domain Intrusive Seismic Risk Analysis}

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\subsection{Modeling Framework}









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\begin{frame} 
\frametitle{Current State of Art Seismic Risk Analysis (SRA)}


\begin{itemize}
%\vspace{2mm}

\item[-] Intensity measure (IM) selected as a proxy for ground motions, 
usually Spectral acceleration $Sa(T_0)$

\vspace{4mm}
\item[-] Ground Motion Prediction Equations (GMPEs) need development, ergodic or site specific 

\vspace{4mm}
\item[-] Probabilistic seismic hazard analysis (PSHA)
% for ground motion $\lambda(Sa>z)$
% \begin{equation*}
% \resizebox{0.85\hsize}{!}{%
% $\lambda(Sa>z) = \sum_{i=1}^{NFL} \underbrace{N_i \int\int f_{mi}(M) f_{ri}(R|M)}_\text{seismic source characterization (SSC)} \underbrace{P(Sa>z|M, R)}_\text{GMPE} dM dR$}
% \end{equation*}

\vspace{4mm}
\item[-] Fragility analysis $P(EDP>x|IM=z)$, deterministic time domain FEM,
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}
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% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsection{Issues in State-of-the-art SRA}

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

\frametitle{Seismic  Risk Analysis Challenges}

\begin{itemize}

\vspace{4mm}
\item[-] Miscommunication between seismologists and structural engineers,
$Sa(T_0)$ not compatible with nonlinear FEM

\vspace{4mm}
\item[-] IMs difficult to choose, Spectral Acc, PGA, PGV...


\vspace{4mm}
\item[-] Single IM does not contain all/most uncertainty

\vspace{4mm}
\item[-] Monte Carlo, not accurate enough for tails 

\vspace{4mm}
\item[-] Monte Carlo, computationally expensive, CyberShake for LA, 20,000
cases, 100y runtime, (Maechling et al. 2007)




\end{itemize}

\end{frame}

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




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

\frametitle{Time Domain Intrusive SRA Framework}

%  

\begin{itemize}

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


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


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

\vspace*{2mm}
\item[-] Results, probability distribution functions for  $\sigma_{ij}$, 
$\epsilon_{ij}$, $u_i$... 








\end{itemize}

\end{frame}
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% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Stochastic Elastic-Plastic Finite Element Method (SEPFEM)}
% 
% 
% 
% \begin{itemize}
% 
% \item[] Material uncertainty expanded along stochastic shape functions: 
% $D(x,t,\theta) = \sum_{i=0}^{P_d} r_i(x,t) * \Phi_i[\{\xi_1, ..., \xi_m\}]$
% 
% \vspace*{4mm}
% \item[] Loading uncertainty expanded along stochastic shape functions: 
% $f(x,t,\theta) = \sum_{i=0}^{P_f} f_i(x,t) * \zeta_i[\{\xi_{m+1}, ..., \xi_f]$
% 
% \vspace*{4mm} 
% \item[] Displacement expanded along stochastic shape functions: 
% $u(x,t,\theta) = \sum_{i=0}^{P_u} u_i(x,t) * \Psi_i[\{\xi_1, ..., \xi_m, \xi_{m+1}, ..., \xi_f\}]$
% 
% \end{itemize}
% 
% 
% 
% \end{frame}
% 
% 
% 

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

\begin{frame}
\frametitle{Stochastic Elastic-Plastic Finite Element Method (SEPFEM)}
%\frametitle{SEPFEM : Formulation}

Stochastic system of equations 

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



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


\end{frame}


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

\frametitle{Stochastic Seismic Motion Development}

\begin{itemize}
 \item UCERF3 (Field et al. 2014)
 \item Stochastic motions (Boore 2003) 
 \item Polynomial Chaos Karhunen-Lo{\`e}ve expansion
 \item Domain Reduction Method for $P_{eff}$ (Bielak et al. 2003)
\end{itemize}

\begin{figure}[!hbpt]
\begin{center}
\includegraphics[width=4cm]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/UCERF3.pdf}
\includegraphics[width=3.0cm]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/SMSIM.pdf}
\includegraphics[width=3.5cm]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/KL_exact_dis_correlation_from_dis.pdf}
\end{center}
\end{figure}

% 
%  \begin{figure}[!hbpt]
%  \begin{center}
%  \includegraphics[width=10cm]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/structural_uncertainty.pdf}
%  \end{center}
%  \end{figure}
%  
%  
%  \begin{figure}[!hbpt]
%  \begin{center}
%  \includegraphics[width=10cm]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/probabilsitc_evolution.png}
%  \end{center}
%  \end{figure}
%  
%  
%  
%  \begin{figure}[!hbpt]
%  \begin{center}
%  \includegraphics[width=10cm]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/seismic_risk_result_framework.png}
%  \end{center}
%  \end{figure}
%  
%  
\end{frame}
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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 Ground Motion Modeling}
% 
% Boore (2003) approach for median of log-normal random FAS field: 
% 
% \[FAS(f) = \underbrace{S(M_0, f)}_\text{source}\underbrace{P(R, f)}_\text{path} \underbrace{A(f)D(f)}_\text{site}\]
% 
% \vspace{-3mm}
% 
% where:
% \begin{itemize}
% \small
% 
% % \item $S(M_0,f) = \frac{CM_0(2\pi f)^2}{1+(f/f_0)^2}$ 
% 
% \item $S(M_0,f) = CM_0(2\pi f)^2/[1+f^2/f_0^2]$ 
% 
% \vspace{1mm}
% 
% \item $P(R, f) = Z(R)exp(-\pi fR/Q\beta)$
% 
% \vspace{1mm}
% 
% \item $A(f)$: quarter wavelength approximation. 
% 
% \vspace{1mm}
% 
% \item $D(f)= exp(-\pi \kappa_0f)$
% 
% \end{itemize}
% 
% Improvements: 
% 
% \begin{itemize}
% \small
% 
% \item Inter-frequency correlation by \textit{Bayless \& Abrahamson (2018)}
% 
% \vspace{1mm}
% 
% \item Phase derivative model by \textit{Baglio \& Abrahamson (2017)}
% 
% \end{itemize}
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 

% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% 
% \frametitle{Stochastic Ground Motion Modeling}
% 
% \begin{textblock}{15}(0.5, 4.2)
% Verification with GMPE for $M=7$, $R=15km$ scenario
% \end{textblock}
% 
% \begin{textblock}{15}(0.0, 5.0)
% \includegraphics[width=0.4\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/SA_GMPE_verification_std_08_no_smooth.pdf}
% \end{textblock}
% 
% \begin{textblock}{15}(5.3, 5.0)
% \includegraphics[width=0.4\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/Goodness_fit_std_08_no_smooth.pdf}
% \end{textblock}
% 
% \begin{textblock}{15}(10.5, 5.0)
% \includegraphics[width=0.4\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/Standard_deviation_std_08_no_smooth.pdf}
% \end{textblock}
% 
% \begin{textblock}{15}(0.5, 11.0)
% \begin{itemize}
% \item $\Delta \sigma= 84bar$, $\kappa=0.03s$ with total $\sigma=0.8ln$. 
% \item Simulated median is not biased. 
% \item Consistent total uncertainties with GMPE.  
% \end{itemize}
% \end{textblock}
% 
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Uncertainty Representation \& Propagation}
% \begin{itemize}
% \vspace{-2mm}
% \item \small Unified uncertainty representation for forcing and media $D(\boldsymbol{x}, \theta)$. 
% 
% \begin{itemize}
% \item [--] Hermite polynomial chaos (PC) for marginal distribution  
% \item [--] Karhunen-Lo$\grave{e}$ve (KL) expansion for correlation structure 
% \end{itemize}
% 
% \item \small Hermite PCs: Non-Gaussian random field with underlying Gaussian random field 
% 
% \[D(\boldsymbol{x}, \theta) = \sum_{i=0}^{P} D_i(\boldsymbol{x}) \Omega_i(\gamma(\boldsymbol{x},\theta)) \]
% 
% where: \[\Omega_i = 1, \gamma, \gamma^2-1, \gamma^3-3\gamma, ... \]
% 
% \[<\Omega_i> =0; \ \ <\Omega_i\Omega_j> = 0 \ \ for \ \ i \neq j \]
% 
% \end{itemize}
% 
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Uncertainty Representation \& Propagation}
% 
% \begin{textblock}{15}(0.5, 5.0)
% \includegraphics[width=0.4\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/Hermite_PC_order.png}
% \end{textblock}
% 
% \begin{textblock}{15}(7.5, 4.0)
% \includegraphics[width=0.5\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/Gamma_distribution_hermite.png}
% \end{textblock}
% 
% \begin{textblock}{15}(5.5, 12.0)
% \includegraphics[width=0.7\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/Gamma_distribution.png}
% \end{textblock}
% 
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Uncertainty Representation \& Propagation}
% 
% \vspace{0.1cm}
% 
% \small Karhunen-Lo$\grave{e}$ve (KL) expansion: capture correlation structure 
% 
% \vspace{-0.3cm}
% 
% \[Cov_D(x_1, x_2) = \sum_{i=1}^{P} D_i(x_1)D_i(x_2)i!Cov_{\gamma}(x_1, x_2)\]
% 
% \vspace{-0.2cm}
% 
% \small Eigen-decomposed Gaussian covariance kernel: $ \lambda_{i}$ eigenvalue, $f_i(\boldsymbol{x})$ eigen-function
% 
% \vspace{-0.7cm}
% 
% \[\gamma(\boldsymbol{x}, \theta)=\sum_{i=1}^{M}\sqrt{\lambda_i}f_i(\boldsymbol{x})\xi_i \]
% 
% \vspace{-0.2cm}
% 
% \small Fredholm's integral equation of the second kind: 
% 
% \[\int_V Cov_{\gamma}(x_1, x_2) f_i(x_1) \, dx_1 = \lambda_{i} f_i(x_2)\]
% 
% \vspace{-0.1cm}
% 
% \small \{$\xi_i$\} are zero mean, unit variance, mutually orthogonal Gaussian random variables
% 
% \vspace{-0.2cm}
% 
% \[<\xi_i> =0; \ \ <\xi_i\xi_j> = \delta_{ij}\]
% 
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Uncertainty Representation \& Propagation}
% \begin{textblock}{15}(0.5, 6.0)
% \includegraphics[width=0.4\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/1D_KL_exp_illustration.png}
% \end{textblock}
% \begin{textblock}{15}(7.5, 4.0)
% \includegraphics[width=0.5\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/KL_eigen_modes.png}
% \end{textblock}
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% 
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Uncertainty Representation \& Propagation}
% 
% \small Random field $D(\boldsymbol{x}, \theta)$ of arbitrary marginal distribution \& correlation structure can be represented as: 
% 
% \begin{textblock}{14}(-2.4, 6.0)
% \[D(\boldsymbol{x}, \theta) = \sum_{i=0}^{K} d_i(\boldsymbol{x}) \Psi_i(\{\xi_j(\theta)\})\]
% \[d_i(\boldsymbol{x}) = \frac{P!}{(\Psi_i^2)} D_P(\boldsymbol{x}) \prod_{j=1}^{P} \frac{\sqrt{\lambda_{k(j)}}f_{k(j)}(\boldsymbol{x})}{\sqrt{\sum_{m=1}^{M}(\sqrt{\lambda_m}f_m(\boldsymbol{x}))^2}}\]
% \end{textblock}
% 
% \vspace{3.5cm}
% where: \\
% $\{\Psi_i\}$  are multi-dimensional Hermite \\
% PC bases of order $P$, dimension $M$. 
% 
% \begin{textblock}{15}(8.9, 5.5)
% \includegraphics[width=0.45\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/Multidimensional_Hermite_PC.png}
% \end{textblock}
% 
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Uncertainty Representation \& Propagation}
% 
% \small 
% 
% Galerkin stochastic finite element method (SFEM)
% 
% \[M_{IacJ} \ddot{U}_{Jc} + \boldsymbol{K^{ep}_{IacJ}} U_{Jc} = \boldsymbol{F_{Ia}}\]
% 
% Probabilistic discretization:
% 
% \vspace{-0.2cm}
% 
% \[K^{ep}_{IacJ}(\theta)= K^{ep}_{IacJi} \Psi_i (\theta)\]
% 
% \vspace{-0.3cm}
% 
% \[F_{Ia}(\theta)= F_{Iaj} \psi_j (\theta)\]
% 
% \vspace{-0.3cm}
% 
% \[U_{Jc}(\theta) = U_{Jck} \phi_k(\theta)\]
% 
% \vspace{-0.3cm}
% 
% Galerkin projection:
% 
% \[M_{IacJ} (\phi_k \phi_m) \ddot{U}_{Jck} + K^{ep}_{IacJi} (\Psi_i\phi_k\phi_m) U_{Jck} = F_{Iaj} (\psi_j \phi_m) \]
% 
% \vspace{0.2cm}
% \scriptsize where $(\Psi_i\phi_k\phi_m)$ is the triple product of multi-dimensional Hermite PCs 
% 
% \end{frame}
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\subsection{Illustrative Example}

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\begin{frame} 
\frametitle{Uncertain Model Description}

\begin{textblock}{15}(0.3, 3.4)
\includegraphics[width=0.70\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/example_illustration.png}
\end{textblock}


\begin{textblock}{15}(1.2, 11.6)
\scriptsize
\begin{itemize}
\item Fault 1: San Gregorio fault 
\item Fault 2: Calaveras fault
\item Uncertainty: Segmentation, \\ slip rate, rupture geometry, etc.
\end{itemize} 
\end{textblock}


\begin{textblock}{15}(7.2, 11.7)
\scriptsize
\begin{itemize}
\item $Vs_{30}=620m/s$
\item $m=100kips/g$
\item $\overline{k} = 168kip/in$
\end{itemize}
\end{textblock}

\begin{textblock}{15}(9.7, 6.5)
\scriptsize
\[Cov_k  = 
\begin{bmatrix}
1.0 & 0.6 & 0.3 & 0.2\\
0.6 & 1.0 & 0.5 & 0.2\\
0.3 & 0.5 & 1.0 & 0.7\\
0.2 & 0.2 & 0.7 & 1.0
\end{bmatrix}
\]
\end{textblock}

\end{frame}
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\begin{frame}
\frametitle{Seismic Source Characterization}

\begin{textblock}{15}(0.3, 3.4)
\includegraphics[width=0.65\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/SSC_legend.pdf}
\end{textblock}

\begin{textblock}{15}(6.5, 12.1)
\scriptsize
\begin{itemize}
\item 371 total seismic scenarios 
\item $M \ 5 \sim 5.5$ and $6.5 \sim 7.0$
\item $R_{jb} \ 20km \sim 40km$
\end{itemize}
\end{textblock}

\begin{textblock}{15}(11.7, 3.3)
\includegraphics[width=0.22\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/Acc_time_series100.pdf}\\
\includegraphics[width=0.22\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/Acc_time_series343.pdf}\\
\includegraphics[width=0.22\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/Acc_time_series439.pdf}
\end{textblock}
\end{frame}
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\begin{frame} 
\frametitle{Stochastic Ground  Representation}

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

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

\begin{textblock}{15}(3.7, 3.62)
\scriptsize
\includegraphics[width=0.3\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/KL_var_acc_from_acc.pdf}

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

\begin{textblock}{15}(7.6, 3.8)
\scriptsize
\includegraphics[width=0.3\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/KL_exact_acc_correlation_from_acc.pdf}

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

\begin{textblock}{15}(11.8, 3.9)
\scriptsize
\includegraphics[width=0.3\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/KL_simulated_acc_correlation_from_acc.pdf}

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

\begin{textblock}{15}(-0.1, 9.3)
\scriptsize
\includegraphics[width=0.31\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/KL_mean_dis_from_dis.pdf}
\end{textblock}

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

\begin{textblock}{15}(4.2, 9.4)
\scriptsize
\includegraphics[width=0.3\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/KL_var_dis_from_dis.pdf}
\end{textblock}

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

\begin{textblock}{15}(8.2, 9.5)
\scriptsize
\includegraphics[width=0.27\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/KL_exact_dis_correlation_from_dis.pdf}

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

\begin{textblock}{15}(12.2, 9.6)
\scriptsize
\includegraphics[width=0.27\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/KL_simulated_dis_correlation_from_dis.pdf}

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

\frametitle{Probabilistic Dynamic Response}

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

\begin{textblock}{15}(4.8, 3.8)
\scriptsize Probabilistic response of top floor from SFEM
\end{textblock}
\end{frame}
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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame} 
% 
% \frametitle{Probabilistic Dynamic Response}
% 
% \begin{textblock}{15}(0.9, 4.4)
% \scriptsize
% \includegraphics[width=0.55\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/probability_evolution.jpg}
% \includegraphics[width=0.43\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/Contour_PDF_evolution_top_floor_deformation.pdf}
% \end{textblock}
% 
% \begin{textblock}{15}(4.8, 4.2)
% \scriptsize Probabilistic density of displacements evolution of top floor
% \end{textblock}
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
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\begin{frame} 

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

% \begin{textblock}{15}(0.7, 4.8)
% \scriptsize
% \includegraphics[width=0.55\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/PDF_evolution.png}
% \end{textblock}
% 
\vspace*{-5mm}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=10cm]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/MIDR_distribution.pdf}
\end{center}
\end{figure}

\end{frame}
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\begin{frame} 

\frametitle{Seismic Risk Analysis}

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

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

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

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

\end{frame}
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% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame} 
% 
% \frametitle{Sensitivity Study}
% 
% \begin{textblock}{15}(1.7, 3.0)
% \scriptsize
% \includegraphics[width=0.42\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/risk_curve_different_stressdrop.pdf}
% \end{textblock}
% 
% \begin{textblock}{15}(9.1, 3.0)
% \scriptsize
% \includegraphics[width=0.42\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/risk_curve_different_kappa.pdf}
% \end{textblock}
% 
% 
% \begin{textblock}{15}(1.7, 8.7)
% \scriptsize
% \includegraphics[width=0.42\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/median_FAS_stressdrop_comparison.pdf}
% \end{textblock}
% 
% \begin{textblock}{15}(9.1, 8.7)
% \scriptsize
% \includegraphics[width=0.42\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/median_FAS_kappa_comparison1.pdf}
% \end{textblock}
% 
% \begin{textblock}{15}(7.9, 8.7)
% \scriptsize Site $\kappa_0$:
% \end{textblock}
% 
% \begin{textblock}{15}(0.3, 8.7)
% \scriptsize Source $\Delta \sigma$:
% \end{textblock}
% 
% \end{frame}
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% 
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame} 
% \frametitle{Sensitivity Study}
% 
% \begin{textblock}{15}(1.3, 4.2)
% Fundamental frequency $f$ increases from 1.6Hz to 8Hz:
% \end{textblock}
% 
% \begin{textblock}{15}(4.7, 5)
% \scriptsize
% \includegraphics[width=0.55\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/risk_curve_different_kappa_highfrequency.pdf}
% \end{textblock}
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% \end{frame}
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\begin{frame} 
\frametitle{Seismic Risk, Uncertain Material}


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


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\section{Summary}

%\subsection*{Summary}
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%\subsection{Summary}

\subsection{\ }

% 
  


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\begin{frame}
  \frametitle{Method Summary}

  \begin{itemize}

\vspace*{1mm}
    \item[-] Full probabilistic modeling and simulation of earthquake soil structure
    interacting (ESSI) systems


\vspace*{2mm}
  \item[-] Intrusive modeling using SEPFEM 


\vspace*{2mm}
    \item[-] Input: uncertain seismic ground motions

\vspace*{2mm}
   \item[-]  Input:  uncertain material
   
\vspace*{2mm}
  \item[-] Output: accurate probability density functions for 
        $\sigma_{ij}$,  
        $\epsilon_{ij}$,  
        $u_i$,  
        $\dot{u}_i$,  
        $\ddot{u}_i$, 
        and any of their combinations


\vspace*{2mm}
  \item[-]  Objective risk analysis, removing subjective choices

\vspace*{2mm}
  \item[-] Requires data and is computationally intensive, however,
   less  demanding and more accurate than MC approach



\end{itemize}

\end{frame}
%


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\begin{frame}
  \frametitle{Conclusion}

  \begin{itemize}

%\vspace*{1mm}
    \item[-] Numerical modeling to predict and inform, rather than fit

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

\vspace*{2mm}
   \item[-]     Collaborators:   Wang, Wang, Feng,  Yang,  Behbehani,  Sinha, 
   Pisan{\'o}, Abell, Cheng, Jie, Luo, Tafazzoli, Preisig, Tasiopoulou, Watanabe
   
\vspace*{2mm}
  \item[-]  Funding  from  and  collaboration  with  the US-DOE, US-NRC, US-NSF, 
  CNSC-CCSN, UN-IAEA, Shimizu Corp. and ILEE is greatly appreciated,

\vspace*{2mm}
  \item[-] Real-ESSI Simulator System: \\
  \hspace*{20mm} \textbf{\url{http://real-essi.info/}} 

\end{itemize}

\end{frame}
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\end{document}

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