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\usepackage{epstopdf}


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


\usepackage[absolute,overlay]{textpos}


\input{essi_listings_options.tex}



\usepackage{threeparttable}

%\usepackage{MnSymbol}%
% for smileys
\usepackage{wasysym}%
\usepackage{multirow}
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\usepackage{subfig}
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\usepackage[breakall]{truncate}
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\usepackage{ulem}
\usepackage{amsmath}
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\usepackage{IEEEtrantools}
\usepackage{setspace}
\usepackage{scalerel}



\usepackage{multimedia}

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

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


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% % for not showing eq numbers unless eq is references
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%  \usetheme{Marburg} % ima naslov i sadrzaj sa desne strane
%  \usetheme{Hannover} % ima naslov i sadrzaj sa leve strane
%  \usetheme{Singapore} % ima sadrzaj i tackice gore
%  \usetheme{Antibes}  % ima sadrzaj gore i kao graf ...
%  \usetheme{Berkeley} % ima sadrzaj desno
%  \usetheme{Berlin} % ima sadrzaj gore i tackice
%  \usetheme{Goettingen} % ima sadrzxaj za desne strane
%  \usetheme{Montpellier} % ima graf sadrzaj gore
%  \usetheme{Warsaw}
%  \usetheme{Warsaw}
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    \usecolortheme[RGB={20,0,128}]{structure}

% or ...

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

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


\usepackage[greek,english]{babel}



\usepackage{amsmath}
\usepackage{mathrsfs}
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\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
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\usepackage{hyperref}
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  pdftoolbar=true,
  pdfpagemode={None}
             colorlinks=true,
             linkcolor=webblue,
             citecolor=webblue,
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}
% \usepackage[pdfauthor={Boris Jeremic},
%             colorlinks=true,
%             linkcolor=webblue,
%             citecolor=webblue,
%             urlcolor=webblue,
%             linktocpage,
%             pdftex]{hyperref}


\usepackage{pause}
% or whatever
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%\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[Uncertain ESSI]
        {Uncertainties in Modeling and Simulation of
Earthquakes, Soils, Structures and their Interaction}


%\subtitle
%{Include Only If Paper Has a Subtitle}

%\author[Author, Another] % (optional, use only with lots of authors)
%{F.~Author\inst{1} \and S.~Another\inst{2}}
% - Give the names in the same order as the appear in the paper.
% - Use the \inst{?} command only if the authors have different
%   affiliation.
\pgfdeclareimage[height=0.2cm]{university-logo}{/home/jeremic/BG/amblemi/ucdavis_logo_blue_sm}
\pgfdeclareimage[height=0.7cm]{lbnl-logo}{/home/jeremic/BG/amblemi/lbnl-logo}


\author[Jeremi{\'c} et al.] % (optional, use only with lots of authors)
%{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)
\institute[\pgfuseimage{university-logo}] % (optional, but mostly needed)
%{ Professor, University of California, Davis\\
{ University of California, Davis, CA\\
%  and\\
%  Faculty Scientist, Lawrence Berkeley National Laboratory, Berkeley }
%   Lawrence Berkeley National Laboratory, Berkeley, CA
 }
% - Use the \inst command only if there are several affiliations.
% - Keep it simple, no one is interested in your street address.


\date[] % (optional, should be abbreviation of conference name)
{\small CU-Boulder, GEGM seminar series \\
        April 2021}

\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}}




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     \frametitle{Outline}
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 }


% 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{\ }



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







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

\begin{itemize}





\vspace*{3mm}
  \item[]  Improve  modeling  and  simulation  for  infrastructure objects


% \vspace*{2mm}
%  \item[]  Expert numerical modeling and simulation tool
%
% \vspace*{1mm}
%   \item[]      Use of numerical models to
%   analyze statics and dynamics of soil/rock-structure systems
%

\vspace*{1mm}
  \item[]  Control  of modeling, epistemic uncertainty

%\vspace*{3mm}
%  \item[]  Choice of analysis level of sophistication

\vspace*{1mm}
  \item[] Propagate parametric, aleatory  uncertainty



\vspace*{1mm}
  \item[] Goal: predict and inform


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


\vspace*{1mm}
  \item[] Design sustainable objects


\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*{-3mm}
 \hfill
\includegraphics[width=4.0truecm]{/home/jeremic/tex/works/Conferences/2021/CU-Boulder-GEGM-seminar-series-02Apr2021/present/IMG_0162.JPG}
%
\end{center}
\end{figure}




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

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




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










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


\begin{itemize}




%\vspace*{1mm}
  \item    \underline{Modeling, Epistemic 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, Aleatory Uncertainty},
  {\small ${M}  \ddot{u_i}  +  {C} \dot{u_i} + {K}^{ep} {u_i} = {F(t)}$},

  \begin{itemize}

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


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



  \item[-]  Important (?!)  features are simplified, 1C vs 3C, inelasticity

%\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 less or not important



\end{itemize}


% local
%\vspace*{-2mm}
\begin{center}
\hspace*{-7mm}
%\movie[label=show3,width=8.8cm,poster,autostart,showcontrols]
\movie[label=show3,width=5.5cm,poster,autostart, showcontrols]
  {\includegraphics[width=50mm]
  {/home/jeremic/tex/works/Conferences/2016/IAEA_TecDoc_February2016/My_Current_Work/movie_2_npps_mp4_icon.jpeg}}
  {/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Model01_ESSI_Response_May2015/movie_2_npps.mp4}
%
%\hfill
\hspace*{-5mm}
%
\movie[label=show3,width=6.0cm,poster,autostart,showcontrols]
  {\includegraphics[width=50mm]
  {/home/jeremic/tex/works/Conferences/2017/SMiRT_24/present/3D_Nonlinear_Modeling_and_it_Effects/NPP_Plastic_Dissipation_grab.jpg}}
  {/home/jeremic/tex/works/Thesis/HanYang/Files_10Aug2017/NPP_Plastic_Dissipation.mp4}
\hspace*{-7mm}
%\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}





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







\vspace*{-9mm}
\begin{figure}[!hbpt]
\begin{center}
%
%\hspace*{-7mm}
\includegraphics[width=5.00truecm]{/home/jeremic/tex/works/Thesis/HexiangWang/time_series_motionsn_06ug2019_SMIRT/Acc_realization_200.pdf}
%\hspace*{-3mm}
%\includegraphics[width=2cm]{/home/jeremic/tex/works/Papers/2019/Hexiang/1D_risk/version2/Figures/Acc_time_series_realiztion70.pdf}
%\includegraphics[width=2cm]{/home/jeremic/tex/works/Papers/2019/Hexiang/1D_risk/version2/Figures/Acc_time_series_realiztion100.pdf}
%% \includegraphics[width=0.31\textwidth]{Figures/Acc_time_series_realiztion350.pdf}
%\includegraphics[width=2cm]{/home/jeremic/tex/works/Papers/2019/Hexiang/1D_risk/version2/Figures/Acc_time_series_realiztion367.pdf}
\includegraphics[width=4cm]{/home/jeremic/tex/works/Papers/2019/Hexiang/1D_risk/version2/Figures/SA_GMPE_verification_std_08_no_smooth.pdf}
%
\end{center}
\end{figure}


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








\end{frame}







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



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

\frametitle{Real-ESSI Simulator System}

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


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




\end{frame}



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

\frametitle{Real-ESSI Simulator System}


\begin{itemize}


  \item Real-ESSI System Components
    \begin{itemize}
     \item[-] Real-ESSI Pre-processor (gmsh/gmESSI, X2ESSI)
     \item[-] Real-ESSI Program (local, remote, cloud)
     \item[-] Real-ESSI Post-Processor (Paraview/pvESSI, Python)

    \end{itemize}

\vspace*{3mm}
  \item Real-ESSI System availability: Windows/iOS/Linux docker, Linux Executables, AWS
%     \begin{itemize}
% %\vspace*{1mm}
%      \item[-] Educational  Institutions: AWS and Linux Executables, free
%      \item[-] Government Agencies, National Labs: AWS GovCloud, free
%      \item[-] Professional  Practice: AWS and Linux Executables, commercial
% %\vspace*{1mm}
% %%\vspace*{1mm}
% %      \item Sources available to collaborators
%     \end{itemize}



\vspace*{3mm}
  \item Real-ESSI education and training: theory and applications



\vspace*{3mm}
  \item Real-ESSI documentation and program available at
\url{http://real-essi.us/}
%\url{http://sokocalo.engr.ucdavis.edu/~jeremic/Real_ESSI_Simulator/}
%
%\url{http://real-essi.info/}
%


% \vspace*{2mm}
%   \item
%


\end{itemize}


\end{frame}




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

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

\begin{itemize}



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

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

%\vspace*{1mm}
  \item[-] Contact/interface/joint elements: bonded, shear/frictional (EPP, EPH,
  EPS); gap/normal; linear, nonlinear, dry, coupled/saturated,

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

%\vspace*{1mm}
  \item[-] Material models: soil, rock, concrete, steel...

%\vspace*{1mm}
  \item[-] Seismic input: 1C and 3C/6C, deterministic or probabilistic

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

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

%\vspace*{1mm}
  \item[-] Intrusive probabilistic inelastic modeling

%
%\vspace*{1mm}
%  \item Modeling features listed at
%  \hspace*{5mm}
%  \href{http://real-essi.info/}{http://real-essi.info/}
%%  \hspace*{5mm}
%%  and
%%  \hspace*{5mm}
%%  \href{http://real-essi.info/}{http://real-essi.info/}


\end{itemize}

\end{frame}

%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Real-ESSI Simulation Features}
%
%
%
% %\vspace*{-10mm}
%
% \begin{itemize}
%
%   \item[-] Static loading stages
%
% \vspace*{2mm}
%   \item[-] Dynamic loading stages
%
% \vspace*{2mm}
%   \item[-] Restart, simulation tree
%
% \vspace*{2mm}
%   \item[-] Solution advancement methods/algorithms, \\
%         on global and constitutive levels, \\
%         with and without enforcing equilibrium
%
%
% %\vspace*{1mm}
% %  \item Load combinations, elastic, for design
%
% \vspace*{2mm}
%   \item[-] High Performance Computing
% %  clusters, cloud, supercomputers
%     \begin{itemize}
% \vspace*{1mm}
%       \item[.] Fine grained, template mataprograms, small matrix library
% \vspace*{1mm}
%       \item[.] Coarse grained, distributed memory parallel
%     \end{itemize}
%
%
% % \vspace*{1mm}
% %   \item All  Simulation Features are listed at
% %   \hspace*{5mm}
% %   \href{http://real-essi.info/}{http://real-essi.info/}
% % %  \hspace*{5mm}
% % %  and
% % %  \hspace*{5mm}
% % %  \href{http://real-essi.info/}{http://real-essi.info/}
%
%
%
% \end{itemize}
%
%
%
% \vspace*{-60mm}
% %\begin{figure}[!hbpt]
% \begin{flushright}
% \includegraphics[width=2.5cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figure-files/_Chapter_Theory_Introduction/tex_works_psfigures_loading_stage-increments-iterations.pdf}
% \end{flushright}
% %\vspace*{-0.5cm}
% %\end{figure}
% %
%
%
%
%
%
% \end{frame}
%
%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Real ESSI Simulator: Domain Specific Language, DSL}

 \begin{itemize}
   \item[] Domain Specific Language (DSL), Yacc \& Lex
\vspace*{3mm}
   \item[] English like modeling and simulation language
\vspace*{3mm}
   \item[] Parser and compiler, can define functions, models, etc.
\vspace*{3mm}
   \item[] Can extend models and methods
\vspace*{3mm}
   \item[] Requires units!
 \end{itemize}
%
\end{frame}

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


























%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Modeling and Simulation}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Seismic Motions}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Earthquake Ground Motions}



\begin{itemize}


\vspace*{0.5cm}
\item  Real earthquake ground motions
   \begin{itemize}
     \item[] Body, P and S waves
     \item[] Surface, Rayleigh and Love waves
     \item[] Lack of correlation, incoherent motions
     \item[] Inclined seismic waves
     \item[] 3D, 3C/6C waves
%     \item Earthquake energy  dissipation

   \end{itemize}



\vspace*{0.5cm}
\item What are the  effects of real earthquake ground motions on soil-structure systems ?!



\end{itemize}




\end{frame}


% \subsection*{Fundamentals}

%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{ESSI: 6C or 1C Seismic Motions}


\begin{itemize}


\item Assume that a full 6C (3C) motions at the surface are only recorded in one
horizontal direction


\item  From such recorded motions one can develop a vertically propagating shear
wave (1C) in 1D

\item  Apply  such  vertically propagating shear wave to same soil-structure
system

\end{itemize}

\vspace*{-3mm}
\begin{figure}[!H]
\begin{center}
 \includegraphics[width=6.5cm]{/home/jeremic/tex/works/Conferences/2015/CompDyn/Present/6D_to_1D_01.jpg}
\end{center}
\end{figure}



 \end{frame}



% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%  \frametitle{Realistic Ground Motions}
%
% \begin{itemize}
%
%   \item Free field seismic motion models
%
% \end{itemize}
%
% % local
% % local
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% \vspace*{-2mm}
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% % \movie[label=show3,width=6.0cm,poster]
% %   {\includegraphics[width=60mm]{movie_input_mp4_icon.jpeg}}{movie_input.mp4}
% % \hspace*{-2mm}
% % %\hfill
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% \movie[label=show3,width=100mm,poster,autostart,showcontrols]
% %\movie[label=show3,width=90mm,poster]
%   {\includegraphics[width=90mm]{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Free_Field_small_model_April2015/movie_input_mp4_icon.jpeg}}
% {/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Free_Field_small_model_April2015/movie_input.mp4}
% \hspace*{-12mm}
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% % local
% % local
% % local
% %
%
%
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%
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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%  \frametitle{Development of Realistic Motions}
%
% \begin{itemize}
%
%   \item Sources will send both P and S waves
%
% \end{itemize}
%
% % online
% % online  \begin{center}
% % online  \href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Free_Field_small_model_April2015/movie_input.mp4}
% % online    {\includegraphics[width=55mm]{movie_input_mp4_icon.jpeg}}
% % online  %
% % online  \href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Free_Field_small_model_April2015/movie_input_closeup.mp4}
% % online    {\includegraphics[width=50mm]{movie_input_closeup_mp4_icon.jpeg}}
% % online  \end{center}
% % online
%
% % local
% % local
% % local
% \vspace*{-2mm}
% \begin{center}
% %\movie[label=show3,width=8.5cm,poster,autostart,showcontrols,loop]
% \hspace*{-12mm}
% % %\movie[label=show3,width=6.0cm,autostart,showcontrols]
% % \movie[label=show3,width=6.0cm,poster]
% %   {\includegraphics[width=60mm]{movie_input_mp4_icon.jpeg}}{movie_input.mp4}
% % \hspace*{-2mm}
% % %\hfill
% % %\hspace*{-7mm}
% % %\movie[label=show3,width=5.3cm,poster,autostart,showcontrols]
% \movie[label=show3,width=80mm,poster, showcontrols]
%   {\includegraphics[width=80mm]{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Free_Field_small_model_April2015/movie_input_closeup.jpg}}
% {/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Free_Field_small_model_April2015/movie_input_closeup.mp4}
% \hspace*{-12mm}
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% % local
% % local
% % local
% %
%
%
% \end{frame}
%
%


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



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{1C vs 6C Free Field Motions}




\begin{itemize}

  \item One component of motions, 1C from  6C
%  or 3$\times$1D (it is done all the time!)

  \item  Excellent fit
%  (goal is to predict and inform and not (force) fit)


\end{itemize}




% local
%\vspace*{-2mm}
\begin{center}
\hspace*{-16mm}
%\movie[label=show3,width=5.6cm,poster,autostart,showcontrols]
\movie[label=show3,width=61mm,poster, showcontrols]
  {\includegraphics[width=60mm]{/home/jeremic/tex/works/Conferences/2016/IAEA_TecDoc_February2016/My_Current_Work/movie_ff_3d_mp4_icon.jpeg}}
  {/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Model01_ESSI_Response_May2015/movie_ff_3d.mp4}
%\hspace*{-2mm}
%\hfill
%\movie[label=show3,width=5.6cm,poster,autostart,showcontrols]
\movie[label=show3,width=61mm,poster, showcontrols]
  {\includegraphics[width=60mm]
  {/home/jeremic/tex/works/Conferences/2016/IAEA_TecDoc_February2016/My_Current_Work/movie_ff_1d_mp4_icon.jpeg}}
  {/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Model01_ESSI_Response_May2015/movie_ff_1d.mp4}
\hspace*{-16mm}
\end{center}
% local


% online
\begin{center}
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/ESSI_VisIt_movies_Jose_19May2015/movie_ff_3d.mp4}
{\tiny (MP4)}
%
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/ESSI_VisIt_movies_Jose_19May2015/movie_ff_1d.mp4}
 {\tiny (MP4)}
\end{center}
% online






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% out    {\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_3d.mp4}
% out  %  \href{./homo_50m-mesh_45degree_Ormsby.mp4}
% out    {\tiny (MP4)}
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% out  % online    {\includegraphics[width=50mm]{movie_ff_3d_mp4_icon.jpeg}}
% out  % online  %
% out  % online  \href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/ESSI_VisIt_movies_Jose_19May2015/movie_ff_1d.mp4}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{6C vs 1C NPP ESSI Response Comparison}


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



% out
% out  \vspace*{-2mm}
% out  \begin{center}
% out  \hspace*{-7mm}
% out  \movie[label=show3,width=8.8cm,poster,autostart,showcontrols]
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% out    {\tiny (MP4)}
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% out  %
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 \end{frame}





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Ventura Hotel, Northridge Earthquake, nonSSI vs SSI}


% local

\vspace*{-5mm}
\begin{center}
\hspace*{-16mm}
%\movie[label=show3,width=5.6cm,poster,autostart,showcontrols]
\movie[label=show3,width=120mm,showcontrols]
  {\includegraphics[width=120mm]{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_Buildings/Ventura_Hotel_SSI_vs_nonSSI_screen_grab.jpg}}
  {/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_Buildings/Ventura_Hotel_SSI_vs_nonSSI.mp4}
\hspace*{-16mm}
\end{center}
% local


% online
\vspace*{-12mm}
\hspace*{-12mm}
\begin{flushleft}
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_ESSI_for_Buildings/Ventura_Hotel_SSI_vs_nonSSI.mp4}
{\tiny ({\bf MP4})}
\end{flushleft}
% online
\vspace*{-12mm}


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







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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Energy Dissipation}







%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \frametitle{Energy Input and Dissipation}

\begin{itemize}

\vspace*{1mm}
     \item[] Energy input, dynamic forcing


\vspace*{4mm}
     \item[] Energy dissipation outside SSI domain:
       \begin{itemize}
         \item[] SSI system oscillation radiation
         \item[] Reflected wave radiation
       \end{itemize}
%\vspace*{1mm}
     \item[] Energy dissipation/conversion inside SSI domain:
       \begin{itemize}
         \item[] Inelasticity of soil, interfaces, structure, dissipators
         \item[] Viscous coupling with internal/pore, and external fluids
% %         \item[] potential and kinetic energy
%          \item[] potential $\leftarrow \! \! \! \! \! \! \rightarrow$ kinetic energy
       \end{itemize}



%\vspace*{1mm}
%   \item[] Numerical energy dissipation (numerical damping/production and period  errors)
%   \item[] Numerical energy dissipation (damping/production)
   \item[] Numerical energy dissipation/production


\end{itemize}

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



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

\frametitle{Plastic Energy Dissipation}


\vspace*{2mm}

 Single elastic-plastic element under cyclic shear loading

\begin{itemize}
\item[] Difference between plastic work and plastic dissipation
\item[] Plastic work can decrease
\item[] Plastic dissipation always increases
\end{itemize}

%\vspace*{-7mm}
\begin{figure}[!hbpt]
\begin{center}
\hspace*{-5mm}
\includegraphics[width=11.0truecm]{/home/jeremic/tex/works/Thesis/HanYang/Files_06June2017/DOE_Annual_2017/Figures/Dissipation_Material.png}
\end{center}
\end{figure}

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




% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%
% \frametitle{Energy Dissipation Control Mechanisms}
%
% \begin{figure}[!H]
% %\hspace*{-10mm}
%      \includegraphics[width=3.4cm]{/home/jeremic/tex/works/Thesis/HanYang/Files_Energy_dissipation_01Dec2017/case_plasticity.pdf}
%      \includegraphics[width=3.4cm]{/home/jeremic/tex/works/Thesis/HanYang/Files_Energy_dissipation_01Dec2017/case_Rayleigh.pdf}
%      \includegraphics[width=3.4cm]{/home/jeremic/tex/works/Thesis/HanYang/Files_Energy_dissipation_01Dec2017/case_Newmark.pdf}
% \end{figure}
%
%
% % \hspace*{10mm} Numerical  \hspace*{20mm}  Viscous  \hspace*{20mm} Plasticity
%  \hspace*{10mm} Plasticity \hspace*{20mm}  Viscous  \hspace*{20mm} Numerical
%
%
%
% \end{frame}
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


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

\frametitle{Energy Dissipation Control}

\begin{figure}[!H]
%\hspace*{-10mm}
%     \includegraphics[width=3cm]{/home/jeremic/tex/works/Thesis/HanYang/Files_Energy_dissipation_01Dec2017/case_a.pdf}
%     \includegraphics[width=3cm]{/home/jeremic/tex/works/Thesis/HanYang/Files_Energy_dissipation_01Dec2017/case_b.pdf}
     \includegraphics[width=9cm]{/home/jeremic/tex/works/Thesis/HanYang/Files_Energy_dissipation_01Dec2017/case_g.pdf}
%     \includegraphics[width=3cm]{/home/jeremic/tex/works/Thesis/HanYang/Files_Energy_dissipation_01Dec2017/case_e.pdf}
\end{figure}




\end{frame}

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




% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%  \frametitle{Inelastic Modeling of Soil Structure System}
%
% \begin{itemize}
%
% \vspace*{1mm}
%      \item[-] Soil, inelastic, elastic-plastic
%        \begin{itemize}
%          \item[] Dry, single phase
%          \item[] Unsaturated, partially saturated, and fully saturated
%        \end{itemize}
%
%
% \vspace*{1mm}
%      \item[-] Interface/Contact/Joint, inelastic, gap open/close, slip
%        \begin{itemize}
%          \item[]  Dry,  single  phase,
% %           \begin{itemize}
% %             \item[]  Normal, hard  and  soft, gap open/close
% %             \item[]  Friction, slip, nonlinear
% %           \end{itemize}
%          \item[] Fully saturated, suction, excess pressure, buoyant force
%        \end{itemize}
%
%
% \vspace*{1mm}
%      \item[-] Structure, inelastic, damage, cracks, ASR...
%        \begin{itemize}
%          \item[] Nonlinear/inelastic 1D  concrete, steel, 3D fiber beams
%          \item[] Nonlinear/inelastic 3D  concrete, steel,  3D solids, 3D shells
% %         \item[] Nonlinear/inelastic 3D  concrete, steel  shell element
% %         \item[] Alcali Silica Reaction concrete modeling
%        \end{itemize}
%
%
% \vspace*{1mm}
%      \item[-] Solid/Structure-Fluid interaction, open surface
%
%
%
% \end{itemize}
%
% %
% \end{frame}
%

% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%
%  \frametitle{NPP Model }
%
%   \begin{figure}[!h]
%   \begin{center}
%    \includegraphics[width=8.5cm]{/home/jeremic/tex/works/Thesis/SumeetKumarSinha/Files_10Aug2017/Npp_Non_Linear_Effects/images/NPP_With_Shallow_Foundation.pdf}
%   \end{center}
% %  \caption{\label{Fig:NPP_Model_In_Real_ESSI} Nuclear Power Plant Model with Shallow Foundation }
%   \end{figure}
%
%
% % \begin{tikzpicture}[remember picture,overlay]
% %     \node[xshift=3.5cm,yshift=-0.6cm] at (current page.center) {\includegraphics[width=0.5\textwidth]{images/Contact_In_Industry}};
% % \end{tikzpicture}
%
% \end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%
%  \frametitle{Structure Model}
%
%  The nuclear power plant structure comprise of
%  \begin{itemize}
%   \item[-]Auxiliary building, $f^{aux}_{1}= 8Hz$
%   \item[-]Containment/Shield building, $f^{cont}_{1}= 4Hz$
%   \item[-]Concrete raft foundation: $3.5m$ thick
%  \end{itemize}
%   \begin{figure}[!h]
%   \begin{center}
%    \includegraphics[width=0.8\textwidth]{/home/jeremic/tex/works/Thesis/SumeetKumarSinha/Files_10Aug2017/Npp_Non_Linear_Effects/images/NPP_Model_Auxiliary_And_Containment_Building.pdf}
%   \end{center}
%   \caption{\label{Fig:NPP_Structure_Model_In_Real_ESSI} Auxiliary and Containment Building }
%   \end{figure}
%
% \end{frame}
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% \begin{frame}
%
%  \frametitle{Inelastic Soil and Inelastic Contact/Interface/Joint}
%
%  \begin{itemize}
%   \item[-] Shear velocity of soil $V_s=500m/s$
%   \item[-] Undrained shear strength (Dickenson 1994) $V_s [m/s] = 23 (S_u [kPa])^{0.475}$
%   \item[-] For $V_s=500m/s$ Undrained Strength $S_u=650kPa$ and Young's Modulus of $E=1.3GPa$
%   \item[-] von Mises, Armstrong Frederick kinematic hardening
%   ($S_u=650kPa$ at $\gamma=0.01\%$; $h_a = 30MPa$, $c_r = 25$)
%   \item[-] Soft contact (concrete-soil), gaping and nonlinear shear
%   \end{itemize}
%
%  \begin{figure}[!h]
%  \begin{center}
%   \includegraphics[width=4cm]{/home/jeremic/tex/works/Thesis/SumeetKumarSinha/Files_10Aug2017/Npp_Non_Linear_Effects/images/Von_Mies_non_Linear_Hardening.pdf}
%   \includegraphics[width=3.5cm]{/home/jeremic/tex/works/Thesis/SumeetKumarSinha/Files_10Aug2017/Npp_Non_Linear_Effects/images/SoftContact.pdf}
%  \end{center}
%   \end{figure}
%
%
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%





%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  \begin{frame}
%
%   \frametitle{Acceleration Traces, Elastic vs Inelastic }
%
%
%  \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*{-35mm}
%
%  \hspace*{10mm} Elastic  \hspace*{40mm} Inelastic \hspace*{40mm}
%
%  \end{frame}
%
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\begin{frame}
\frametitle{Energy Dissipation for Design}

\begin{figure}[!hbpt]
\begin{center}
\includegraphics[width=10.0truecm]{/home/jeremic/tex/works/Thesis/HanYang/Frame_animations_13Mar2019/2D_Frame_Model.pdf}
\end{center}
\end{figure}


\end{frame}






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\begin{frame}
\frametitle{Design Alternatives}


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


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





\end{frame}

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\begin{frame}

\frametitle{Energy Dissipation for an SMR Model}

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


% \vspace*{-5mm}
\begin{center}
% \hspace*{-15mm}
\movie[label=show3,width=10cm,poster,autostart,showcontrols]
{\includegraphics[width=10cm]
{/home/jeremic/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{center}


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





% \vspace*{-5mm}



\end{frame}




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% \begin{frame}
% \frametitle{ASCE-7-21: Buildings and Models, Low Building}
%
% %\vspace*{-5mm}
% \begin{figure}[!htb]
% \begin{center}
% \includegraphics[width=10cm]{/home/jeremic/tex/works/Thesis/HexiangWang/ATC-144_buildings_results_plots_15Sep2019/steel_frame2.pdf}
% \end{center}
% \end{figure}
%
% \end{frame}
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{ASCE-7-21: Low Building Energy Dissipation}
%
%
% % \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/tex/works/Thesis/HanYang/ASCE-7-21_low_building_energy_dissipation/ATC_Short_Building_PD.jpg}}
% {/home/jeremic/tex/works/Thesis/HanYang/ASCE-7-21_low_building_energy_dissipation/ATC_Short_Building_PD.mp4}
% \end{center}
%
% % online
% \vspace*{-12mm}
% \begin{flushleft}
% \hspace*{-4mm}
% \href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/Energy_dissipation_frames/ATC_Short_Building_PD.mp4}
%   {\tiny (MP4)}
% \end{flushleft}
% % online
%
%
%
%
% \end{frame}
% %\vspace*{10mm}
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\section{Uncertain Inelastic Computational Mechanics}

%OVDE


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%\subsection{Stochastic Modeling}
\subsection{Probabilistic Computational Mechanics}





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

\frametitle{Uncertainty Propagation through
Inelastic System}

%

\begin{itemize}



\item Incremental el--pl constitutive equation
%
    \begin{eqnarray}
    \nonumber
  \Delta \sigma_{ij}
  =
%  E^{EP}_{ijkl}
     E^{EP}_{ijkl} \; \Delta \epsilon_{kl}
     =
     \left[
     E^{el}_{ijkl}
       -
       \frac{\displaystyle  E^{el}_{ijmn} m_{mn} n_{pq} E^{el}_{pqkl}}
           {\displaystyle n_{rs} E^{el}_{rstu} m_{tu} - \xi_* h_*}
           \right]
    \Delta \epsilon_{kl}
    \end{eqnarray}




\vspace*{2mm}
\item Dynamic Finite Elements
%
\begin{equation}
{ M} \ddot{ u_i} +
{ C} \dot{ u_i} +
{ K}^{ep} { u_i} =
{ F(t)}
    \nonumber
\end{equation}


\vspace*{2mm}
\item Material and loads are uncertain



\end{itemize}



\end{frame}














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\begin{frame}

\frametitle{Previous Work}



\begin{itemize}

\item
Linear algebraic or differential equations:

     \begin{itemize}
        \item Variable Transf. Method (Montgomery and Runger 2003)
        \item Cumulant Expansion Method (Gardiner 2004)
     \end{itemize}

\item
Nonlinear differential equations:

      \begin{itemize}

      \item  Monte  Carlo Simulation (Schueller 1997, De Lima et al 2001, Mellah
      et al. 2000, Griffiths et al. 2005...) \\ $\rightarrow$ can be accurate, very costly

      \item  Perturbation  Method  (Anders and Hori 2000, Kleiber and Hien 1992,
      Matthies  et  al. 1997) \\ $\rightarrow$ first and second order Taylor series
      expansion  about mean - limited to problems with small C.O.V. and inherits
      "closure problem"

      \item SFEM (Ghanem and Spanos 1989, Matthies et al,  2004, 2005, 2014...)


      \end{itemize}

%
%  \item
%  Monte Carlo method: accurate, very costly
%
%  \item
%  Perturbation method:

\end{itemize}






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\end{frame}





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\begin{frame} \frametitle{{3D Fokker-Planck-Kolmogorov Equation}}

\begin{footnotesize}

\begin{eqnarray}
\nonumber
\lefteqn{\displaystyle \frac{\partial P(\sigma_{ij}(x_t,t), t)}{\partial t} = \displaystyle \frac{\partial}{\partial \sigma_{mn}}
\left[ \left\{\left< \vphantom{\int_{0}^{t}} \eta_{mn}(\sigma_{mn}(x_t,t), E_{mnrs}(x_t), \epsilon_{rs}(x_t,t))\right> \right. \right.} \\
\nonumber
&+& \left. \left. \int_{0}^{t} d\tau Cov_0 \left[\displaystyle \frac{\partial \eta_{mn}(\sigma_{mn}(x_t,t), E_{mnrs}(x_t),
\epsilon_{rs}(x_t,t))} {\partial \sigma_{ab}}; \right.  \right. \right. \\
\nonumber
& & \left. \left. \left. \eta_{ab} (\sigma_{ab}(x_{t-\tau}, t-\tau), E_{abcd}(x_{t-\tau}), \epsilon_{cd}(x_{t-\tau}, t-\tau)
\vphantom{\int_{0}^{t}} \right] \right \} P(\sigma_{ij}(x_t,t),t) \right] \\
\nonumber
&+& \displaystyle \frac{\partial^2}{\partial \sigma_{mn} \partial \sigma_{ab}} \left[ \left\{ \int_{0}^{t} d\tau Cov_0 \left[
\vphantom{\int_{0}^{t}} \eta_{mn}(\sigma_{mn}(x_t,t), E_{mnrs}(x_t), \epsilon_{rs}(x_t,t)); \right. \right. \right. \\
\nonumber
& & \left. \left. \left. \eta_{ab} (\sigma_{ab}(x_{t-\tau}, t-\tau), E_{abcd}(x_{t-\tau}), \epsilon_{cd}(x_{t-\tau}, t-\tau))
\vphantom{\int_{0}^{t}} \right] \vphantom{\int_{0}^{t}} \right\} P(\sigma_{ij}(x_t,t),t) \right]
\end{eqnarray}


\end{footnotesize}

\begin{flushright}
(Jeremi{\'c} et al. 2007)
\end{flushright}


%-- \begin{itemize}
%--
%--
%--
%-- \item 6 equations
%--
%-- \item Complete description of 3-D probabilistic stress-strain response
%--
%-- \end{itemize}
%--
%--


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\end{frame}






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\begin{frame}

\frametitle{FPK Equation}



\begin{itemize}

\item Advection-diffusion equation
%
\begin{equation}
\nonumber
\frac{\partial P(\sigma,t)}{\partial t} = -\frac{\partial}{\partial \sigma}\left[N_{(1)}P(\sigma,t)-\frac{\partial}{\partial \sigma}
\left\{N_{(2)} P(\sigma,t)\right\} \right]
\end{equation}

%

\item Complete probabilistic description of response


\item Solution PDF is second-order exact to covariance of time (exact mean and variance)


\item It is deterministic equation in probability density space

\item It is linear PDE in probability density space
$\rightarrow$ simplifies the numerical solution process

%\vspace*{0.2truecm}

\end{itemize}

%
% \vspace*{0.5cm}
% {%
% \begin{beamercolorbox}{section in head/foot}
% \usebeamerfont{framesubtitle}\tiny{B. Jeremi\'{c}, K. Sett, and M. L. Kavvas, "Probabilistic
% Elasto--Plasticity: Formulation in 1--D", \textit{Acta Geotechnica}, Vol. 2, No. 3, 2007, In press (published
% online in the \textit{Online First} section)}
% %\vskip2pt\insertnavigation{\paperwidth}\vskip2pt
% \end{beamercolorbox}%
% }



\end{frame}



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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%
%
%
% \frametitle{Template Solution of FPK Equation}
%
%
%
% \begin{itemize}
%
%
%
%
% \item FPK diffusion--advection equation is applicable to any material model $\rightarrow$
% only the coefficients $N_{(1)}$ and $N_{(2)}$ are different for different material models
% % %
% % %
%
%
%
% % %
% \item Initial condition
%
%   \begin{itemize}
%
%     \item Deterministic $\rightarrow$ Dirac delta function $\rightarrow$ $ P(\sigma,0)=\delta(\sigma) $
%
%     \item Random $\rightarrow$ Any given distribution
%
%   \end{itemize}
%
% \item Boundary condition: Reflecting BC $\rightarrow$ conserves probability mass
% $\zeta(\sigma,t)|_{At \ Boundaries}=0$
%
% \item Solve using finite differences and/or finite elements
%
%
% \item  However  (!!)  it  is a stress solution and probabilistic stiffness is an
% approximation!
%
% \end{itemize}
%
%
% \end{frame}
%
%
%
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%
% \frametitle{Probabilistic Elastic-Plastic Response}
%
%
% \begin{figure}[!hbpt]
% \begin{center}
% %\includegraphics[width=8cm]{/home/jeremic/tex/works/Papers/2007/ProbabilisticYielding/figures/vonMises_G_and_cu_very_uncertain/Contour_PDF-edited.pdf}
% \includegraphics[width=8cm]{/home/jeremic/tex/works/Conferences/2012/DOE-LLNL-workshop-27-28-Feb-2012/ProbabilisticYielding_vonMises_G_and_cu_very_uncertain_Contour_PDF-edited.pdf}
% \end{center}
% \end{figure}
%
% \end{frame}
%
%
%
%
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\begin{frame}

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


\begin{figure}[!hbpt]
\begin{center}
\hspace*{-10mm}
\includegraphics[width=5.5cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourLowOCR_RandomG_RandomM_Randomp0-m.pdf}
\includegraphics[width=5.5cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourHighOCR_RandomG_RandomM-m.pdf}
\hspace*{-10mm}
\end{center}
\end{figure}


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




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

\begin{frame}{Time Domain Stochastic Galerkin Method}


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


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

\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{3mm}

Seismic 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{3mm}

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{3mm}

where $a_i(x), f_{mj}(t)$ are {known PC coefficients}, while $d_{nk}(t)$
are {unknown PC coefficients}.
%}

\end{frame}




% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}{FEM and Stochastic Elastic-Plastic FEM, SEPFEM}
%
%
% %\vspace{4mm}
% \vspace{-2mm}
%
% \small
% {
% %FEM:
% \begin{eqnarray*}
%   \sum_{e} [ \int_{D_e} N_m(x)\rho(x)N_n(x)d\Omega \; {\color{blue}{\ddot{u}_n(t)}} +
%   \\
%   \int_{D_e}\nabla N_m(x) {\color{blue}{E(x)}} \nabla N_n(x)d\Omega \; {\color{blue}{u_n(t)}} - {\color{blue}{f_m(t)}} ]=0
%  \end{eqnarray*}
%
% %SEPFEM:
%  \begin{eqnarray*}
%      &&\lefteqn{\sum_{n=1}^N \sum_{k=1}^{P_3} \langle \Psi_k \Psi_l \rangle \int_{D_e}N_m(x)\rho(x)N_n(x)d\Omega \; \; \ddot{d}_{nk}(t)  \; \; +}
%      \\
%      &&\sum_{n=1}^N \sum_{k=1}^{P_3} \sum_{i=1}^{P_1} \langle \Psi_i \Psi_k \Psi_l \rangle
%      \int_{D_e}B_m(x) {\color{blue}{a_i(x,t)}} B_n(x)d\Omega \; \; d_{nk}(t)
%      =
%      \\
%      &&\sum_{j=1}^{P_2} \langle \Psi_j \Psi_l \rangle f_{mj}(t) \\
% \end{eqnarray*}
% }
% \vspace{-0.3cm}
%
% %\scriptsize{Note: update \textcolor{blue}{$a_i(x,t)$} for elastic-plastic material}
%
% \vspace{0.4cm}
%
% %\begin{beamercolorbox}{section in head/foot}
% %\usebeamerfont{framesubtitle}\tiny{Wang, F. and Sett, K., "Time-Domain Stochastic Finite Element Simulation of Uncertain Seismic Wave Propagation through Uncertain Heterogeneous Solids", \textit{Soil Dynamics and Earthquake Engineering}, 88:369-385, 2016.}
% %\end{beamercolorbox}
%
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}{SEPFEM}
% Matrix form:
% \begin{equation*}
% \bm{M} \ddot{\bm{d}} + \bm{K} \bm{d} = \bm{f}
% \end{equation*}
% For damped systems:
% \begin{equation*}
% \label{eqno_19}
% \bm{M} \ddot{\bm{d}} + \bm{C} \dot{\bm{d}} + \bm{K} \bm{d} = \bm{f}
% \end{equation*}
% \hspace{5cm}    {\huge $\Downarrow$  }   \\
% \hspace{2.2cm}   Newmark's method to solve in time domain \\
%
% \vspace{1cm}
% \scriptsize{
% \noindent where $\bm{M}$, $\bm{C}$ and $\bm{K}$ are generalized mass, damping and stiffness matrices,\\
%  \hspace{0.9cm} $\bm{f}$, $\bm{d}$, and $\ddot {\bm{d}}$ are generalized force, displacement, and acceleration vectors.
% }
%
% \end{frame}
%

%\begin{frame}{Physical significance of stochastic DOFs} \label{matrix_form}
%
%{\scriptsize Finite element system of equation --- \only<1> {deterministic} \only<2> {stochastic}}
%\vspace{-0.5cm}
%
%\begin{equation}
%\nonumber
%\tiny
%\left[ \begin{array}{ccccc}
%\uncover<2-> {\displaystyle \sum_{i=1}^{P_1} C_{i11}} \only<1> {K_1} \only<2> {K_i} & \uncover<2-> {\displaystyle \sum_{i=1}^{P_1} C_{i12} K_i} & \uncover<2-> \dots & \uncover<2-> \dots & \uncover<2-> {\displaystyle \sum_{i=1}^{P_1} C_{i 1 P_{\scaleto{3}{3pt}}} K_i} \\
%\uncover<2-> {\displaystyle \sum_{i=1}^{P_1} C_{i21} K_i} & \uncover<2-> {\displaystyle \sum_{i=1}^{P_1} C_{i22} K_i} & \uncover<2-> \dots & \uncover<2-> \dots & \uncover<2-> \vdots \\
%\uncover<2-> \vdots & \uncover<2-> \vdots & \uncover<2-> \ddots &  & \uncover<2-> \vdots \\
%\uncover<2-> \vdots & \uncover<2-> \vdots &  & \uncover<2-> \ddots & \uncover<2-> \vdots \\
%\uncover<2-> {\displaystyle \sum_{i=1}^{P_1} C_{i P_{\scaleto{3}{3pt}} 1} K_i} & \uncover<2-> {\dots} & \uncover<2->{\dots} & \uncover<2-> {\dots}  & \uncover<2-> {\displaystyle \sum_{i=1}^{P_1} C_{i P_{\scaleto{3}{3pt}} P_{\scaleto{3}{3pt}}} K_i}
%\end{array} \right]
%%
%\left[ \begin{array}{c}
%\vec{d}_1 \\
%\\
%\uncover<2-> {\vec{d}_2} \\
%\\
%\uncover<2-> \vdots \\
%\\
%\uncover<2-> \vdots \\
%\\
%\uncover<2-> {\vec{d}_{P_{\scaleto{3}{3pt}}}}
%\end{array} \right]
%%
%=
%%
%\left[ \begin{array}{c}
%\uncover<2-> {\displaystyle \sum_{j=1}^{P_{\scaleto{2}{3pt}}} C_{j1}} \only<1> {\vec{f}_1} \only<2> {\vec{f}_j}\\
%\uncover<2-> {\displaystyle \sum_{j=1}^{P_{\scaleto{2}{3pt}}} C_{j2} \vec{f}_j} \\
%\uncover<2-> \vdots \\
%\uncover<2-> \vdots \\
%\uncover<2-> {\displaystyle \sum_{j=1}^{P_{\scaleto{2}{3pt}}} C_{j P_{\scaleto{3}{3pt}}} \vec{f}_j}
%\end{array} \right]
%\end{equation}
%
%\tiny
%{
%
%\only<1>
%{
%Note: $K_1$ is the deterministic stiffness matrix; \\
%\vspace{0.1cm}
%\hspace{0.55cm} $\vec{d}_1$ is the displacement vector for all the nodes; \\
%\vspace{0.1cm}
%\hspace{0.55cm} $\vec{f}_1$ is the forcing vector for all the nodes; \\
%\vspace{0.1cm}
%\hspace{0.55cm} Size of the stiffness matrix is $N \times N$, $N$ is the number of deterministic DOFs. \\
%\vspace{0.1cm}
%\hspace{0.55cm} \color{white}{deterministic DOFs;} \\
%}
%\only<2>
%{
%Note: $C_{ijk}= \langle \Psi_i \Psi_j \Psi_k \rangle$, $C_{ij}= \langle \Psi_i \Psi_j \rangle$;  \\
%\vspace{0.1cm}
%\hspace{0.55cm} $K_i$ is the block stiffness matrix with $i$th PC coefficients of modulus, {\tiny for example, $K_1$ is the deterministic matrix};  \\
%\vspace{0.1cm}
%\hspace{0.55cm} $\vec{d}_i$ is the block vector for $i$th PC coefficients of displacement for all the nodes; \\
%\vspace{0.1cm}
%\hspace{0.55cm} $\vec{f}_i$ is the block vector for $i$th PC coefficients of forcing for all the nodes; \\
%\vspace{0.1cm}
%\hspace{0.55cm} Size of the global matrix is $(N \times P_3) \times (N \times P_3)$, $N$ is the number of deterministic DOFs; \\
%
%}
%}
%
%\begin{flushright}
%\tiny
%\hyperlink{3D_matrix}{\beamergotobutton{}}
%\end{flushright}
%
%
%\end{frame}


%\begin{frame}{Size of the stochastic stiffness matrix}
%\begin{itemize}
%
%\item Governed by:
%\begin{itemize}
%\item PC dimension $\rightarrow$ function of correlation length
%\item PC order $\rightarrow$ function of COV
%\end{itemize}
%
%\begin{small}
%
%\begin{table}
%\begin{center}
%\begin{tabular}{|c|l|r|}
%\hline
%PC dimension           & Order of PC         & Size of Stiffness Matrix \\
% \hline \hline
%2                   & 1         & Real DOFs x 3 \\
%      & 2  & x 6 \\
%      & 4         & x 15 \\
% \hline
%4      & 1         & Real DOFs x 5 \\
%      & 2  & x 15 \\
%                    & 4         & x 70 \\
% \hline
%6      & 1         & Real DOFs x 7 \\
%      & 2  & x 28 \\
%                    & 4         & x 210 \\
% \hline
%\end{tabular}
%\end{center}
%\end{table}
%
%\end{small}
%
%\end{itemize}
%\end{frame}


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%\subsection[Constitutive update]{Intrusive constitutive update}
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%\frame{\tableofcontents[currentsubsection,sectionstyle=show/shaded]}
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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsection{Direct Solution for Probabilistic Stiffness and Stress in 1D}
%


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%%%%%%%%%%%%%%%%%%%%%%%%%%    BEGGINING PEP    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}{Direct Probabilistic Constitutive Solution in 1D}


% \begin{itemize}
%
%  \vspace{0.5cm}
%
% \item<1-> Probabilistic constitutive modeling : \vspace{0.5cm}

\begin{itemize}


\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*{2mm}
\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*{2mm}
\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]
$




\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{$E_{t_P}^{n+1}  =  \frac{1}{<\Phi_1\Phi_P> }\{\sum_{i=1}^{P_h} \ h_i   <\Phi_i \Phi_P> \pm \sum_{j=1}^{P_c} \sum_{l=1}^{P_{\sigma}} \ c_j  \sigma_l^{n+1} <\Phi_j  \Phi_l \Phi_P>\}$}


\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*{1mm}
\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*{1mm}
\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}


\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 solution (single step)

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

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

\end{itemize}


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

   ...  & ... & ... & ...\\
%    \hline
  \end{tabular}}


\end{frame}












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




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


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



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



\end{frame}








%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% \begin{frame}
% \frametitle{SEPFEM: Example in 3D}
%
%
%
% %\vspace*{-5mm}
% \begin{center}
% % \hspace*{-15mm}
% \movie[label=show3,width=10cm,poster,autostart,showcontrols]
% {\includegraphics[width=10cm]
% {/home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Latex/img/SFEM_3D.png}}
% % /home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Latex/img/figure_PEP_25.pdf
% {/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Probabilistic_Elasto_Plasticity_and_Stochastic_Elastic_Plastic_Finite_Element_Method/SFEM_Animation_3D.mp4}
% %{/home/jeremic/tex/works/Thesis/MaximeLacour/Files_27Jun2017/Summer_Slides/Animations/SFEM_Animation_3D.mp4}
% \end{center}
%
% %  \includegraphics[width = 12cm]{./img/SFEM_3D.pdf}
%
% \begin{flushleft}
% %\hspace*{-15mm}
% \vspace*{-15mm}
% \href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Probabilistic_Elasto_Plasticity_and_Stochastic_Elastic_Plastic_Finite_Element_Method/SFEM_Animation_3D.mp4}
% %  \href{./homo_50m-mesh_45degree_Ormsby.mp4}
%   {\tiny (MP4)}
% \end{flushleft}
% %
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% \end{frame}
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%\subsection{Uncertain Inelasticity}
\subsection{Risk Analysis}









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\begin{frame}

\frametitle{Probabilistic Seismic Risk Analysis (PSRA)}

\begin{textblock}{15}(0.8, 3.9)
\scriptsize
Uncertain source, path, \\
\quad site and structure
\end{textblock}

\begin{textblock}{15}(4.8, 4.2)
$\Longrightarrow$
\end{textblock}

\begin{textblock}{15}(5.8, 3.7)
\scriptsize
%Acceptably small probability\\
Probabilities of \\
engineering demand parameters (EDP) \\
 damage measures (DM), loss, etc
\end{textblock}

\begin{textblock}{15}(0.4, 11.8)
\tiny
\textbf{Uncertain source rupture}
\end{textblock}

\begin{textblock}{15}(3.55, 4.8)
\begin{figure}[H]
\flushleft
\includegraphics[width=0.7\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/problem_statement.pdf}
\end{figure}
\end{textblock}

\begin{textblock}{15}(12.5, 5.5)
\tiny
adapted from Taga(1982)
\end{textblock}

\begin{textblock}{15}(11.7, 10.6)
\tiny
\textbf{Uncertain material properties}
\end{textblock}


\begin{textblock}{15}(0.4, 8.0)
\begin{figure}[H]
\flushleft
\includegraphics[width=0.2\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/4231447_Northridge_eq.pdf}
\end{figure}
\end{textblock}


\begin{textblock}{15}(10.7, 10.2)
\begin{figure}[H]
\flushleft
\includegraphics[width=0.3\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/uncertain_SPT.pdf}
\end{figure}
\end{textblock}

\end{frame}
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\begin{frame}

%\frametitle{State of the Art Probabilistic Seismic Risk Analysis}
\frametitle{Probabilistic Seismic Risk Analysis}

\begin{itemize}

% \item Target: safe design, acceptably small failure probability $\lambda (EDP)$

\item[-]
%Powerful tool
%Allows for
Objective, quantitative decision making based on exceedance rate
$\lambda (EDP>z)$

\item[-] PSRA: convolution of PSHA and fragility

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

\vspace{-0.1cm}

\[\lambda(EDP>z) = \int \underbrace{|\frac{d\lambda(IM>x)}{dx}|}_\text{\textbf{PSHA}} \underbrace{G(EDP>z|IM=x)}_{\text{\textbf{fragility analysis}}} dx\]

\small{$\lambda(\cdot)$ : rate of exceedance\\
\vspace{0.07cm}
$EDP$: engineering demand parameter\\
\vspace{0.07cm}
$PSHA$: probabilistic seismic hazard analysis\\
\vspace{0.07cm}
$IM$: intensity measure}

\end{itemize}

\end{frame}

% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% \begin{frame}
% \frametitle{Intensity Measure (IM)}
% IM serves as the proxy of damaging ground motions
%
% \vspace{0.3cm}
%
% \begin{itemize}
% \item[-] Does a single IM, e.g., $Sa(T_0)$, represent all uncertainty?
% %%  influencing EDP?
% %\begin{itemize}
% %\item[--] \small Structure nonlinearity
% %% \item[--] \small Liquefaction: PGA and duration
% %\item[--] \small Higher mode response
% %\end{itemize}
%
% \vspace{3mm}
%
% \item[-] Practically difficult/contentious to choose
%
% % \begin{itemize}
% % % \item[--] \small Geo-hazard: Liquefaction, slope deformation
% % % \item[--] \small PGA v.s. AI v.s. RMS for liquefaction
% % \item[--] \small AI v.s. PGV v.s. CAV for dam embankment
% % \end{itemize}
%
% \vspace{3mm}
%
% % \item Additional effort for new GMPEs
%
% % \begin{itemize}
% % \item[--] \small vector hazard: GMPE with covariance of IMs, fragility as function of IMs, rarely used
% % \end{itemize}
%
% % \item[-] Miscommunication: seismologists and engineers
% %
% % \begin{itemize}
% % \item[] \small $Sa(T_0)$ not compatible with time domain nonlinear analysis
% % \end{itemize}
%
% \end{itemize}
% \end{frame}
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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,
% perhaps using Monte Carlo (MC)
%
% %  \begin{itemize}
% %
% %     \item[-] Records selection: Spectrum-matching technique UHS, etc
% %
% %     \item[-] Incremental dynamic analysis: Monte Carlo
% %
% %   \end{itemize}
%
%
%
% \end{itemize}
%
% % \begin{textblock}{15}(2.2, 9.2)
% % \begin{figure}[H]
% % \flushleft
% % % \includegraphics[width=0.38\linewidth]{pic/hazard_curve.png}
% % \includegraphics[width=0.38\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/hazard_curve.pdf}
% % \enspace
% % \includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/design_spectra.png}
% % \end{figure}
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\frametitle{Seismic  Risk Analysis Challenges}


\begin{itemize}



\item[-] Intensity Measure(s) (IM) serves as the proxy of damaging ground motions
\vspace{2mm}
\item[-] Does a single IM, e.g., $Sa(T_0)$, represent all uncertainty?
%\item[-] Practically difficult/contentious to choose


%\vspace{3mm}
\vspace{2mm}
\item[-] IMs difficult to choose, Spectral Acc, PGA, PGV...


%%\vspace{3mm}
%\item[-] Single IM does not contain all/most uncertainty




\vspace{2mm}
\item[-] Fragility analysis: incremental dynamic analysis (IDA)
% using Monte Carlo method

\vspace{2mm}
\item[-] Use of Monte Carlo method, accuracy, efficiency...

%\vspace{3mm}
\vspace{2mm}
\item[-] Monte Carlo, computationally expensive, CyberShake for LA, 20,000
cases, 100Y runtime, (Maechling et al. 2007)

%
%
% \vspace{3mm}
% \item[-] Miscommunication between seismologists and struct/geotech engineers,
% $Sa(T_0)$ not compatible with nonlinear FEM




\end{itemize}

\end{frame}

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% \frametitle{Time Domain Intrusive PSRA Framework}
%
% %
%
% \begin{itemize}
%
% %\vspace*{2mm}
% \item[-] Stochastic Elastic-Plastic Finite Element Method, SEPFEM,
%  ${M}  \ddot{u_i}  +  {C}  \dot{u_i} + {K}^{ep} {u_i} = {F(t)}$,
%  (Sett et al. 2011)
%
%
% \vspace*{4mm}
% \item[-] Uncertain elastic-plastic material
% %stress and stiffness solution using
% %Forward Kolmogorov, Fokker-Planck equation
%
%
% \vspace*{4mm}
% \item[-] Uncertain seismic loads/motions
% % using Domain Reduction Method
%
%
% \vspace*{4mm}
% \item[-] Results, probability distribution functions for  $\sigma_{ij}$,
% $\epsilon_{ij}$, $u_i$...
%
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% \end{itemize}
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% \begin{frame}
% \frametitle{Stochastic Elastic-Plastic Finite Element Method}
%
%
%
% \begin{itemize}
%
% %\item[-] Material uncertainties:  expanded along stochastic shape functions:
% \item[-] Material uncertainties: stochastic shape functions:
% $E^{ep}(x,t,\theta) = \sum_{i=0}^{P_d} E_i(x,t) * \Phi_i[\{\xi_1, ..., \xi_m\}]$
%
% \vspace*{1mm}
% \item[-] Loading uncertainties: stochastic shape functions
% $F(x,t,\theta) = \sum_{i=0}^{P_f} F_i(x,t) * \zeta_i[\{\xi_{m+1}, ..., \xi_f]$
%
% \vspace*{1mm}
% \item[-] Displacement expanded: stochastic shape functions:
% $u(x,t,\theta) = \sum_{i=0}^{P_u} u_i(x,t) * \Psi_i[\{\xi_1, ..., \xi_m, \xi_{m+1}, ..., \xi_f\}]$
%
%
% \vspace*{1mm}
% \item[-]
% Stochastic system of equations
% \vspace*{-2mm}
% \begin{tiny}
% \[
% \begin{bmatrix}
%     \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_0> K^{(k)}    & \dots & \sum_{k=0}^{P_d} <\Phi_k \Psi_P \Psi_0> K^{(k)}\\
%     \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_1> K^{(k)}    & \dots & \sum_{k=0}^{P_d} <\Phi_k \Psi_P \Psi_1> K^{(k)}\\  \\
%     \vdots & \vdots & \vdots & \vdots\\
%     \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_P> K^{(k)}     & \dots & \sum_{k=0}^{M} <\Phi_k \Psi_P \Psi_P> K^{(k)}
% \end{bmatrix}
% \begin{bmatrix}
%     u_{10} \\
%     \vdots \\
%     u_{N0}\\
%     \vdots \\
%     u_{1P_u}\\
%     \vdots \\
%     u_{NP_u}
% \end{bmatrix}
% =
% %\]
% %\[
% \begin{bmatrix}
%      \sum_{i=0}^{P_f} f_i  <\Psi_0\zeta_i> \\
%     \sum_{i=0}^{P_f} f_i   <\Psi_1\zeta_i> \\
%    \sum_{i=0}^{P_f} f_i    <\Psi_2\zeta_i> \\
%     \vdots \\
%      \sum_{i=0}^{P_f} f_i  <\Psi_{P_u}\zeta_i>\\
% \end{bmatrix}
% \]
% \end{tiny}
%
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% \end{itemize}
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% \end{frame}
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%
% \begin{frame}
% \frametitle{Stochastic Elastic-Plastic Finite Element Method}
% %\frametitle{SEPFEM : Formulation}
%
% Stochastic system of equations
%
% \begin{tiny}
% \[
% \begin{bmatrix}
%     \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_0> K^{(k)}    & \dots & \sum_{k=0}^{P_d} <\Phi_k \Psi_P \Psi_0> K^{(k)}\\
%     \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_1> K^{(k)}    & \dots & \sum_{k=0}^{P_d} <\Phi_k \Psi_P \Psi_1> K^{(k)}\\  \\
%     \vdots & \vdots & \vdots & \vdots\\
%     \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_P> K^{(k)}     & \dots & \sum_{k=0}^{M} <\Phi_k \Psi_P \Psi_P> K^{(k)}
% \end{bmatrix}
% \begin{bmatrix}
%     u_{10} \\
%     \vdots \\
%     u_{N0}\\
%     \vdots \\
%     u_{1P_u}\\
%     \vdots \\
%     u_{NP_u}
% \end{bmatrix}
% =
% %\]
% %\[
% \begin{bmatrix}
%      \sum_{i=0}^{P_f} f_i  <\Psi_0\zeta_i> \\
%     \sum_{i=0}^{P_f} f_i   <\Psi_1\zeta_i> \\
%    \sum_{i=0}^{P_f} f_i    <\Psi_2\zeta_i> \\
%     \vdots \\
%      \sum_{i=0}^{P_f} f_i  <\Psi_{P_u}\zeta_i>\\
% \end{bmatrix}
% \]
% \end{tiny}
%
%
%
% %   \normalsize{Typical number of terms required for a SEPFEM problem} \vspace{1cm}\\
%   \scalebox{0.7}{
%   \begin{tabular}{ c c c c}
%    \# KL terms material & \# KL terms load & PC order displacement& Total \# terms per DoF\\ \hline
%    4 & 4 & 10 & 43758 \\
%    4 & 4 & 20 & 3 108 105 \\
% %   4 & 4 & 30 & 48 903 492 \\
%    6 & 6 & 10 & 646 646 \\
% %   6 & 6 & 20 & 225 792 840 \\
% %   6 & 6 & 30 & 1.1058 $10^{10}$ \\
% %   8 & 8 & 10 & 5 311 735 \\
% %   8 & 8 & 20 & 7.3079 $10^{9}$ \\
% %   8 & 8 & 30 & 9.9149 $10^{11}$\\
%
%    ...  & ... & ... & ...\\    \hline
%   \end{tabular}}
%
%
% \end{frame}
%
%
%
%





%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  \begin{frame}
%
%  \frametitle{Monte Carlo (MC)}
%
%  \begin{itemize}
%
%  \item Monte Carlo simulations: non-intrusive approach
%
%  \begin{itemize}
%  \item [--] \small Slow convergence rate $1/\sqrt{N}$
%  \item [--] \small Hard for stable tail distribution toward low-risk level
%  \end{itemize}
%
%  \item Fragility curve: incremental dynamic analysis (IDA)
%
%  \begin{itemize}
%  \item [--] \small Impractical for large $3D$ nonlinear ESSI system
%  \end{itemize}
%
%  \item Uncertain seismic wave propagation over regional geology
%
%  \begin{itemize}
%  \item [--] \small CyberShake from SCEC
%  \item [--] \small Los Angeles, over 20,000 scenarios within 200 km, \textbf{300 million CPU-hours and over 100 years} (Maechling et al. 2007)
%  \end{itemize}
%
%  \end{itemize}
%  \end{frame}
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\begin{frame}

%\frametitle{TDNIPSRA Framework}
\frametitle{Risk Framework}

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

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

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

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\begin{textblock}{15}(5.1, 6.5)
%$\Rightarrow$
{\Large $\rightarrow$}
\end{textblock}
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\begin{textblock}{15}(5.8, 3.9)
\vspace*{1mm}
\includegraphics[width=0.27\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/SMSIM.pdf}
\end{textblock}


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

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

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\begin{textblock}{15}(9.9, 6.5)
%{\bf $\Rightarrow$}
{\Large $\rightarrow$}
\end{textblock}
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\begin{textblock}{15}(10.5, 4.2)
% \includegraphics[width=0.35\linewidth]{pic/KL_exact_dis_correlation_from_dis.pdf}
\includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Acc_realization_200.pdf}
\end{textblock}

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


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

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

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

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%$\Leftarrow$
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\includegraphics[width=0.29\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/seismic_risk_result_framework.png}
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\end{tikzpicture}


\begin{textblock}{15}(-0.1, 9.3)
\scriptsize
\quad \quad \quad \quad $\lambda(EDP>z)=$

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

\begin{textblock}{15}(1.6, 10.7)
\scriptsize{EDP hazard/risk}
\end{textblock}

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% \begin{textblock}{15}(0, 4.0)
% \includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/UCERF3.pdf}
% \end{textblock}
%
% \begin{textblock}{15}(0.3, 3.5)
% \scriptsize{Seismic source characterization}
% \end{textblock}
%
% \begin{textblock}{15}(2.9, 5.2)
% \tiny{UCERF3 (2014)}
% \end{textblock}
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% \begin{textblock}{15}(5.1, 6.5)
% $\Rightarrow$
% \end{textblock}
%
%
% \begin{textblock}{15}(5.8, 3.9)
% \vspace*{1mm}
% \includegraphics[width=0.27\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/SMSIM.pdf}
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%
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% \vspace*{-0.2cm}
% \scalebox{.9}{\tiny{\hspace{0.14cm} Boore(2003)}}
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%
% \begin{textblock}{15}(6.1, 3.5)
% \scriptsize{Stochastic ground motion}
% \end{textblock}
%
% \begin{textblock}{15}(9.9, 6.5)
% $\Rightarrow$
% \end{textblock}
%
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% % \includegraphics[width=0.35\linewidth]{pic/KL_exact_dis_correlation_from_dis.pdf}
% \includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Acc_realization_200.pdf}
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% \scriptsize{Uncertainty characterization \\
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% \includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/structural_uncertainty.pdf}
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% % $\Leftarrow$
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% % \end{textblock}
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% \end{textblock}
%
% \begin{textblock}{15}(0.3, 3.5)
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% \end{textblock}
%
% \begin{textblock}{15}(2.9, 5.2)
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% \end{textblock}
%
% \begin{textblock}{15}(5.1, 6.5)
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% \end{textblock}
%
%
% \begin{textblock}{15}(5.8, 3.9)
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% \includegraphics[width=0.27\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/SMSIM.pdf}
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%
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%
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%
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% \includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Acc_realization_200.pdf}
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%
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% \scriptsize{\quad \quad Uncertainty propagation \\
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% \end{textblock}
%
% % \begin{textblock}{15}(4.6, 13.2)
% % $\Leftarrow$
% % \end{textblock}
%
% % \begin{textblock}{15}(0.3, 11.0)
% % \includegraphics[width=0.29\linewidth]{pic/seismic_risk_result_framework.png}
% % \end{textblock}
%
% % \begin{tikzpicture}[remember picture, overlay]
% % \draw[line width=1pt, draw=black, rounded corners=4pt, fill=gray!20, fill opacity=1]
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% % \end{tikzpicture}
%
%
% % \begin{textblock}{15}(-0.1, 9.3)
% % \scriptsize
% % \quad \quad \quad \quad $\lambda(EDP>z)=$
%
% % $\quad \sum N_i(M_i, R_i) P(EDP>z|M_i, R_i)$
% % \end{textblock}
%
% % \begin{textblock}{15}(1.6, 10.7)
% % \scriptsize{EDP hazard/risk}
% % \end{textblock}
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% \end{frame}
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%
%
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% \begingroup
%
% \setbeamertemplate{footline}{}
%
% \begin{frame}
%
% \frametitle{Time Domain Intrusive PSRA Framework}
%
%
% \begin{textblock}{15}(0, 4.0)
% \includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/UCERF3.pdf}
% \end{textblock}
%
% \begin{textblock}{15}(0.3, 3.5)
% \scriptsize{Seismic source characterization}
% \end{textblock}
%
% \begin{textblock}{15}(2.9, 5.2)
% \tiny{UCERF3 (2014)}
% \end{textblock}
%
% \begin{textblock}{15}(5.1, 6.5)
% $\Rightarrow$
% \end{textblock}
%
%
% \begin{textblock}{15}(5.8, 3.9)
% \vspace*{1mm}
% \includegraphics[width=0.27\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/SMSIM.pdf}
% \end{textblock}
%
%
% \begin{textblock}{15}(7.1, 6.2)
% \scalebox{.9}{\tiny{Fourier spectra}}
% \\
% \vspace*{-0.2cm}
% \scalebox{.9}{\tiny{\hspace{0.14cm} Boore(2003)}}
% \end{textblock}
%
% \begin{textblock}{15}(6.1, 3.5)
% \scriptsize{Stochastic ground motion}
% \end{textblock}
%
% \begin{textblock}{15}(9.9, 6.5)
% $\Rightarrow$
% \end{textblock}
%
% \begin{textblock}{15}(10.5, 4.2)
% % \includegraphics[width=0.35\linewidth]{pic/KL_exact_dis_correlation_from_dis.pdf}
% \includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Acc_realization_200.pdf}
% \end{textblock}
%
% \begin{textblock}{15}(11.1, 9.6)
% \scriptsize{Uncertainty characterization \\
% \hspace{0.1cm} Hermite polynomial chaos}
% \end{textblock}
%
%
% \begin{textblock}{15}(10.2, 13.2)
% $\Leftarrow$
% \end{textblock}
%
% \begin{textblock}{15}(11, 11.2)
% \includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/structural_uncertainty.pdf}
% \end{textblock}
%
% \begin{textblock}{15}(5.3, 10.75)
% \includegraphics[width=0.33\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/probabilsitc_evolution.png}
% \end{textblock}
%
% \begin{textblock}{15}(5.4, 9.6)
% \scriptsize{\quad \quad Uncertainty propagation \\
% \quad \quad \quad \quad stochastic FEM}
% \end{textblock}
%
% \begin{textblock}{15}(4.6, 13.2)
% $\Leftarrow$
% \end{textblock}
%
% \begin{textblock}{15}(0.3, 11.0)
% \includegraphics[width=0.29\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/seismic_risk_result_framework.png}
% \end{textblock}
%
% \begin{tikzpicture}[remember picture, overlay]
% \draw[line width=1pt, draw=black, rounded corners=4pt, fill=gray!20, fill opacity=1]
% ([xshift=-25pt,yshift=-52pt]$(pic cs:a) + (0pt,8pt)$) rectangle ([xshift=95pt,yshift=-18pt]$(pic cs:b)+(0pt,-2pt)$);
% \end{tikzpicture}
%
%
% \begin{textblock}{15}(-0.1, 9.25)
% \scriptsize
% \quad \quad \quad \quad $\lambda(EDP>z)=$
%
% $\quad \sum N_i(M_i, R_i) P(EDP>z|M_i, R_i)$
% \end{textblock}
%
% \begin{textblock}{15}(1.6, 10.7)
% \scriptsize{EDP hazard/risk}
% \end{textblock}
%
% \end{frame}
%
% \endgroup
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%


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\begin{frame}

\frametitle{Stochastic Ground Motion Modeling}

% \vspace{-0.4cm}

\begin{itemize}

%\item[-] \normalsize Shift from modeling specific IM to fundamental characteristics of ground motions
\item[-] \normalsize Shift from modeling specific IM to fundamental characteristics of ground motions

\begin{itemize}

\item[-] \normalsize Uncertain Fourier amplitude spectra (FAS)

\item[-] \normalsize Uncertain Fourier phase spectra (FPS)

\end{itemize}

% \item \scriptsize Mean behavior of stochastic FAS

% \begin{itemize}

% \item[--] \scriptsize $w^2$ source radiation spectrum by \textit{Brune(1970)}

% \item[--] \scriptsize Systematic studies by \textit{ \textbf{Boore}(1983, \textbf{2003}, 2015)}.

% \end{itemize}

\vspace{0.05cm}

%\item[-] \normalsize Recent GMPE study of FAS,
%(FAS marginal median \& variability GMPEs by \textit{{Bora et al.
%(2018)}} and {\textit{Bayless \& Abrahamson (2019)}} ;
%FAS Inter-frequency correlation GMPE by \textit{Stafford(2017)} and
%{\textit{Bayless \& Abrahamson (2018)}})


\vspace*{1mm}
\item[-] \normalsize GMPE studies of FAS,
(
\textit{{Bora et al. (2018)}},
\textit{Bayless \& Abrahamson (2018,2019)},
\textit{Stafford(2017)},
%{\textit{Bayless \& Abrahamson (2018)}
)

% \begin{itemize}
%
% \item[-] \scriptsize FAS marginal median \& variability GMPEs by \textit{\textbf{Bora et al. (2018)}} and \textbf{\textit{Bayless \& Abrahamson (2019)}}
%
% %\vspace{0.1cm}
%
% \item[-] \scriptsize FAS Inter-frequency correlation GMPE by \textit{Stafford(2017)} and \textbf{\textit{Bayless \& Abrahamson (2018)}}.
%
% \end{itemize}


%\vspace{0.05cm}

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

\vspace*{1mm}
\item[-] \normalsize Near future change from \textbf{ $\boldsymbol{Sa(T_0)}$} to \textbf{FAS}
%next five years
% as envisioned by Abrahamson (2018)

\end{itemize}

\end{frame}
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% %\subsection{Illustrative Example}
% %
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
%\frametitle{TDNIPSRA Example Object}
\frametitle{Risk Example}

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

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

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

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

\end{frame}
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\begin{frame}

\frametitle{Stochastic Ground Motion Modeling}


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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\frametitle{Stochastic Material Modeling}

%Uncertain 1D shear response



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



\end{frame}
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\begin{frame}

\frametitle{Probabilistic Dynamic Structural Response}

\begin{textblock}{15}(0.7, 4.5)
\scriptsize
\includegraphics[width=0.46\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/shear_frame_illustration_update.pdf}
\end{textblock}

\begin{textblock}{15}(0.2, 10.8)
\begin{itemize}
\scriptsize \item Coefficient of variation 15$\%$ for $H_a$ and $C_r$
%\scriptsize \item Exponential correlation with correlation  \\
%length $l_c = 10$ floors
\scriptsize \item Time domain stochastic \\
                  El-Pl FEM analysis (SEPFEM)
% : uncertain \\ structure with uncertain excitations
\end{itemize}
\end{textblock}

\begin{textblock}{15}(7.7, 5)
\scriptsize
\includegraphics[width=0.52\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Probabilistic_Response_Node_1_new.pdf}
\end{textblock}

\begin{textblock}{15}(8.3, 4.2)
\scriptsize Probabilistic response of top floor from SFEM
\end{textblock}
\end{frame}

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\begingroup

\setbeamertemplate{footline}{}

\begin{frame}

\frametitle{Seismic Risk Analysis}

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

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

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

\frametitle{Seismic Risk Analysis}

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

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

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

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

\frametitle{Seismic Risk Analysis}

\vspace{-0.5cm}

\begin{itemize}

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

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

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

    \vspace*{3mm}
%  \item[-] \small Damage measure (DM) defined on single EDP:
  \item[-] Damage measure defined on single EDP:

\vspace*{-15mm}

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




\vspace*{30mm}

\item[-] \small Seismic risk for DM defined on multiple EDPs can be quite
different from that defined on single EDP.

\end{itemize}
\end{frame}
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\section{Summary}



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\frametitle{Appropriate Science Quotes}

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%  \item Max  Planck:
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\item[]  Fran{\c c}ois-Marie Arouet,  Voltaire:
"Le doute n'est pas une condition agr{\'e}able, mais la certitude est absurde."

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\item[]  Niklaus  Wirth:
"Software is getting slower more rapidly than hardware becomes faster."

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  \frametitle{Summary}

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%    \item Importance   of   using  proper  models  correctly  (verification,
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%    \item Reduction of modeling uncertainty
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    \item[-] Numerical modeling to predict and inform
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    \item[-] Engineer needs to know!
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%    \item[-]  Sophisticated inelastic determinstic/probabilistic modeling
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\vspace*{1mm}
    \item[-] Education and Training is the key!


\vspace*{1mm}
   \item[-]     Collaborators:    Feng,  Yang,  Behbehani,  Sinha,  Wang,
   Karapiperis, Wang, Lacoure, Pisan{\'o}, Abell, Tafazzoli, Jie, Preisig,
   Tasiopoulou, Watanabe, Cheng, Yang.



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


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


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