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\usepackage{epstopdf}


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


                                                               

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\usepackage[absolute,overlay]{textpos}



 
\usepackage{threeparttable}

%\usepackage{MnSymbol}%
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\usepackage{subfig}
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\usepackage{ulem}
\usepackage{amsmath}
\usepackage{bm}
\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}
% \usepackage{epstopdf}
% \fi
% %*****************************************


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

% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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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}
  \usetheme{Dresden}
    \usecolortheme[RGB={20,0,128}]{structure}

% or ...

  \setbeamercovered{transparent}


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

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


\usepackage[greek,english]{babel}



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

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


\usepackage{array}



%%%%  HYPERREF  HYPERREF  HYPERREF  HYPERREF  HYPERREF
%%%%  HYPERREF  HYPERREF  HYPERREF  HYPERREF  HYPERREF
\definecolor{webgreen}{rgb}{0, 0.35, 0} % less intense green
\definecolor{webblue}{rgb}{0, 0, 0.50} % less intense blue
\definecolor{webred}{rgb}{0.35, 0, 0} % less intense red
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\usepackage{hyperref}
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  pdfmenubar=true,
  pdftoolbar=true,
  pdfpagemode={None}
             colorlinks=true,
             linkcolor=webblue,
             citecolor=webblue,
             urlcolor=webblue,
}
% \usepackage[pdfauthor={Boris Jeremic},
%             colorlinks=true,
%             linkcolor=webblue,
%             citecolor=webblue,
%             urlcolor=webblue,
%             linktocpage,
%             pdftex]{hyperref}


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

\usepackage[latin1]{inputenc}
% or whatever
\usepackage{times}
\usepackage[T1]{fontenc}
% Or whatever. Note that the encoding and the font should match. If T1
% does not look nice, try deleting the line with the fontenc.



%      Site Specific Dynamics of Structures: 
%From Seismic Source to 
%the Safety of Occupants and Content


\title[Real-ESSI]
        { Stochastic Site Response Analysis \\
          Through Uncertain Elastoplastic Soil
}


%\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}}
{Fangbo Wang, Hexiang Wang, Maxime Lacoure, \\ 
Han Yang, Yuan Feng,  \\
\vspace*{2mm}
Boris Jeremi{\'c}
}

%\institute[Computational Geomechanics Group \hspace*{0.3truecm}
\institute[\pgfuseimage{university-logo}\hspace*{0.1truecm}\pgfuseimage{lbnl-logo}] % (optional, but mostly needed)
%{ Professor, University of California, Davis\\
{ University of California, Davis, CA\\
%  and\\
%  Faculty Scientist, Lawrence Berkeley National Laboratory, Berkeley }
  Lawrence Berkeley National Laboratory, Berkeley, CA}
% - Use the \inst command only if there are several affiliations.
% - Keep it simple, no one is interested in your street address.


\date[] % (optional, should be abbreviation of conference name)
{\small SMiRT25\\
         Charlotte, NC, USA, August 2019}

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



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

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

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




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

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

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\begin{frame}
  \frametitle{Outline}

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  \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{\ }



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\begin{frame}
 \frametitle{Motivation}

\begin{itemize}



 

%\vspace*{0.3cm}
  \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*{2mm}
  \item[]  Reduction  of modeling uncertainty

\vspace*{2mm}
  \item[]  Choice of analysis level of sophistication


\vspace*{2mm}
  \item[] Goal: Predict and Inform rather than fit

\vspace*{2mm}
  \item[] Engineer needs to know!



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


\vspace*{2mm}
  \item[]  System  for {\bf Real}istic modeling and simulation of
  {\bf E}arthquakes,  {\bf S}oils,    {\bf S}tructures and  
  their {\bf I}nteraction: \\
\vspace*{3mm}
   {\bf Real-ESSI Simulator}  \hspace*{2mm}  
  {\bf \url{http://real-essi.info/}} 





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







                       

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


\begin{itemize}




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

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


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

    \end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace*{4mm}
  \item   \underline{Parametric    Uncertainty},  \hspace*{2mm} ${M}  \ddot{u_i}  +  {C}  \dot{u_i} + {K}^{ep} {u_i} = {F(t)}$

  \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}
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\begin{frame}
 \frametitle{Modeling Uncertainty}

\begin{itemize}



  \item[-]  Important (?!)  features are simplified, 1C vs 6C, 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 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=60mm]
  {/home/jeremic/tex/works/Conferences/2017/SMiRT_24/present/3D_Nonlinear_Modeling_and_it_Effects/NPP_Plastic_Dissipation_grab.jpg}}
  {/home/jeremic/tex/works/Thesis/HanYang/Files_10Aug2017/NPP_Plastic_Dissipation.mp4}
\hspace*{-7mm}
%\end{flushleft}
%%
\end{center}
% local

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



\end{frame}




      
%OVDE

% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%  \frametitle{Modeling Uncertainty, 6C vs 1C Motions}
% 
%      
% % local
% \vspace*{-2mm}
% \begin{center}
% \hspace*{-7mm}
% %\movie[label=show3,width=8.8cm,poster,autostart,showcontrols]
% \movie[label=show3,width=8.8cm,poster, showcontrols]
%   {\includegraphics[width=92mm]
%   {/home/jeremic/tex/works/Conferences/2016/IAEA_TecDoc_February2016/My_Current_Work/movie_2_npps_mp4_icon.jpeg}}
%   {/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Model01_ESSI_Response_May2015/movie_2_npps.mp4}
% \end{center}
% % local
% %  \vspace*{-2mm}
% %  \begin{center}
% %  \hspace*{-7mm}
% %  \movie[label=show3,width=8.8cm,poster,autostart,showcontrols]
% %    {\includegraphics[width=90mm]{movie_2_npps_mp4_icon.jpeg}}{movie_2_npps.mp4}
% %  \end{center}
% 
% 
% % online
% \vspace*{-12mm}
% \begin{flushleft}
% %\vspace*{-15mm}
% \href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/ESSI_VisIt_movies_Jose_19May2015/movie_2_npps.mp4}
%   {\tiny (MP4)}
% \end{flushleft}
% % online
% 
% 
% 
% 
% 
% \end{frame}
% 





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Parametric Uncertainty: Soil Stiffness and Strength}


\vspace*{2mm}
%\vspace*{-3mm}
\begin{figure}[!hbpt]
\begin{center}
%
\hspace*{-7mm}
\includegraphics[width=5.5truecm]{/home/jeremic/tex/works/Papers/2008/JGGE-GoverGmax/figures/YoungModulus_RawData_and_MeanTrend_01-Ed.pdf}
\hspace*{-3mm}
% \hfill
\includegraphics[width=3.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGE-GoverGmax/figures/YoungModulus_Histogram_Normal_01-Ed.pdf}
%
\end{center}
\end{figure}

\vspace*{-5mm}
\begin{figure}[!hbpt]
\begin{center}
%
\hspace*{-7mm}
\includegraphics[width=5.00truecm]{/home/jeremic/tex/works/Papers/2008/JGGE-GoverGmax/figures/ShearStrength_RawData_and_MeanTrend-Mod.pdf}
\hspace*{-3mm}
% \hfill
\includegraphics[width=3.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGE-GoverGmax/figures/ShearStrength_Histogram_PearsonIV-FineTuned-Mod.pdf}
%
\end{center}
\end{figure}

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


\end{frame}

% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Parametric Uncertainty: Material Properties}
% 
%           
% 
% \vspace*{5mm}
% \begin{figure}[!hbpt]
% \begin{center}
% % %
% \hspace*{-3mm}
% \includegraphics[width=2.5truecm]{/home/jeremic/tex/works/Thesis/KonstantinosKarapiperis/Soil_Uncertainty_Report_Pdf_Cdf_Figures/FieldPhiPdf.pdf}
% \hspace*{-3mm}
% \includegraphics[width=2.5truecm]{/home/jeremic/tex/works/Thesis/KonstantinosKarapiperis/Soil_Uncertainty_Report_Pdf_Cdf_Figures/FieldPhiCdf.pdf}
% \hspace*{3mm}
% \includegraphics[width=2.5truecm]{/home/jeremic/tex/works/Thesis/KonstantinosKarapiperis/Soil_Uncertainty_Report_Pdf_Cdf_Figures/FieldSuPdf.pdf}
% \hspace*{-3mm}
% \includegraphics[width=2.5truecm]{/home/jeremic/tex/works/Thesis/KonstantinosKarapiperis/Soil_Uncertainty_Report_Pdf_Cdf_Figures/FieldSuCdf.pdf}
% \\
% %\vspace*{-2mm}
% \hspace*{-2.5cm} \mbox{\tiny Field $\phi$} \hspace*{3.5cm} \mbox{\tiny Field $c_u$}
% \\
% \vspace*{10mm}
% \hspace*{-3mm}
% \includegraphics[width=2.5truecm]{/home/jeremic/tex/works/Thesis/KonstantinosKarapiperis/Soil_Uncertainty_Report_Pdf_Cdf_Figures/LabPhiPdf.pdf}
% \hspace*{-3mm}
% \includegraphics[width=2.5truecm]{/home/jeremic/tex/works/Thesis/KonstantinosKarapiperis/Soil_Uncertainty_Report_Pdf_Cdf_Figures/LabPhiCdf.pdf}
% \hspace*{3mm}
% \includegraphics[width=2.5truecm]{/home/jeremic/tex/works/Thesis/KonstantinosKarapiperis/Soil_Uncertainty_Report_Pdf_Cdf_Figures/LabSuPdf.pdf}
% \hspace*{-3mm}
% \includegraphics[width=2.5truecm]{/home/jeremic/tex/works/Thesis/KonstantinosKarapiperis/Soil_Uncertainty_Report_Pdf_Cdf_Figures/LabSuCdf.pdf}
% \\
% %\vspace*{-8mm}
% \hspace*{-2.5cm} \mbox{\tiny Lab $\phi$} \hspace*{3.5cm} \mbox{\tiny Lab $c_u$}
% \end{center}
% \end{figure}
% 
% 
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% 
% 
% 
% 
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% \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,  inelastic,  deterministic  or probabilistic, 3D, finite
% element modeling and simulation of:
% 
%     \begin{itemize}
% %\vspace*{1mm}
%       \item statics and dynamics of soil,
% %\vspace*{1mm}
%       \item    statics and dynamics of rock,
% %\vspace*{1mm}
%       \item    statics and dynamics of structures,
% %\vspace*{1mm}
%       \item    statics of soil-structure systems, and
% %\vspace*{1mm}
%       \item    dynamics of earthquake-soil-structure system interaction
%     \end{itemize}
% 
% 
% 
% Used for:
% \begin{itemize}
% %\vspace*{1mm}
%   \item Design, linear elastic, load combinations, dimensioning
% 
% 
% %\vspace*{1mm}
%   \item Assessment, nonlinear/inelastic, 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, Python, Matlab)
% 
%     \end{itemize} 
% 
% \vspace*{1mm}
%   \item Real-ESSI System availability:                   
%     \begin{itemize}
% %\vspace*{1mm}
%      \item Educational  Institutions: Amazon Web Services (AWS), free 
%      \item Government Agencies, National Labs: AWS GovCloud
%      \item Professional  Practice: AWS, commercial
% %\vspace*{1mm}
% %%\vspace*{1mm}
% %      \item Sources available to collaborators
%     \end{itemize}
% 
% 
% 
% \vspace*{1mm}
%   \item Real-ESSI Short Courses, online, worldwide
% 
% 
% 
% \vspace*{1mm}
%   \item \url{http://real-essi.info/}
% %
% 
% 
% % \vspace*{2mm}
% %   \item
% % 
% 
% 
% \end{itemize}
% 
% 
% \end{frame}
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\section{Stochastic ESSI}


 
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\subsection{\ }





% 
% 
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Uncertainties are inevitable in predicting behaviors of structures during future earthquakes}
%\vspace{0.5cm}
% 
% \begin{tabular}{c  c}
% \begin{minipage}{0.45\textwidth}
% %
% \hspace{-1cm} {\color{blue} Soils are spatially non-uniform}
% \begin{itemize}
% 
% \footnotesize
% \item Spatial non-uniformity is a function of soil formation process.
% \vspace*{0.1cm}
% \item Spatial non-uniformity is usually 'uncertain', due to limited data! 
% \vspace*{0.1cm}
% \item Soil parameters are ideally described as non-Gaussian random fields 
% \end{itemize}
% 
% \end{minipage} &
% %
% \hspace*{0.75truecm} 
% \begin{minipage}{0.50\textwidth}
% 
% \begin{figure}
% \includegraphics[height=5.45cm]{Fangbo_figs/TypicalSoilVariability.jpg} 
% \end{figure}
% \vspace*{-0.5truecm}
% \centering
% \scriptsize Schematic profile of a soil deposit \\
% \hspace*{0.1cm} {\tiny(Terzaghi, Peck and Mesri, 1996)}
% \end{minipage}
% \end{tabular}
% 
% \end{frame}
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Uncertainties are inevitable in predicting behaviors of structures during future earthquakes}
% 
% \begin{columns}
% 
% \column{0.5\textwidth}
% 
% {\color{blue} Bedrock motion is also uncertain}
% \begin{itemize}
% \small
% \item Factors lead to uncertainty: 
% \begin{itemize}
% \item Mechanism of source
% \item Source parameters
% \item Earthquake propagation path
% \item ...
% \end{itemize}
% 
% \item Bedrock motion is ideally described as a non-stationary random process
% \end{itemize}
% \vspace{0.5cm}
% 
% 
% \column{0.5\textwidth}
% \includegraphics[scale=0.2]{Fangbo_figs/Three_stages_of_ground_mtion_Jie_Li.pdf} \\
% \vspace{-0.6cm}
% \begin{flushright}
% \scriptsize{Three stages explicitly indicate the non-stationarity of seismic wave}
% \end{flushright}
% \centering
% \vspace{-0.3cm}
% \hspace{0.5cm} \tiny(Li and Chen, 2009)
% 
% 
% \end{columns}
% 
% \end{frame}
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% \begin{frame}
% \frametitle{Uncertainties are inevitable in predicting behaviors of structures during future earthquakes}
% \vspace{0.2cm}
% {\color{blue}{Design frameworks can account for uncertainties:}}
% \footnotesize
% \begin{itemize}
% \item ASD $\rightarrow$ a single factor of safety
% \item LRFD $\rightarrow$ load and resistance factors
% \item Performance-based design (PBD):
% \end{itemize}
% 
% \vspace{-0.4cm}
% \begin{figure}
% \includegraphics[scale=0.3]{Fangbo_figs/Performance_design_framework.jpg} 
% \vspace{-0.3cm}
% \caption{ \scriptsize{Schematic of PBD framework} {\tiny(Moehle and Deierlein, 2004)} }
% \end{figure}
% 
% \end{frame}
% 
% 
% \begin{frame}[t]
% \frametitle{Uncertainties are inevitable in predicting behaviors of structures during future earthquakes}
% 
% {\footnotesize \color{blue}{However, numerical simulations -- which are used to feed design framework -- still remain exclusively deterministic:}}
% \vspace{-0.2cm}
% \begin{columns}
% \column{0.5\textwidth}
% \minipage[c][0.6\textheight][s]{\columnwidth}
% \begin{figure}
%   \includegraphics[width=0.8\textwidth, angle=90]{Fangbo_figs/SSI_model_deterministic}
%   \tiny{\linespread{1.0} Schematic of a Soil-Foundation-Structure system subjected to earthquake}
% \end{figure}
% 
% \endminipage
% 
% \column{0.5\textwidth}
% \footnotesize
% 
% \begin{itemize}
% \item Although high-fidelity models that reduce modeling uncertainty are increasingly being used
% \item Uncertainties in model parameters are ignored:
%  \begin{itemize}
%      \scriptsize
%      \item Material parameters
%      \item Forcing function
%  \end{itemize}
%  \item Resulting in negation of the effects of nonlinear stochastic dynamics
% \end{itemize}
% 
% \end{columns}
% 
% \end{frame}
% 
% 
% 
% 
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% 
% \begin{frame}{Effects of nonlinear stochastic dynamics must be captured for accurate prediction}
% 
% \begin{columns}
% 
% \column{0.4\textwidth}
% \scriptsize
% 
% %\begin{center}
% %traditional SSI model \\
% %+ \\
% %uncertain soil deposit \\
% %+ \\
% %uncertain bedrock motion \\
% %$\Downarrow$ \\
% %stochastic SSI model
% %\end{center}
% Dynamic interaction of:
% \vspace{0.3cm}
% \begin{itemize}
% \item Non-Gaussian characteristics and spatial correlation of material parameters
% \item Temporal correlation of the forcing function
% \item Probabilistic evolution of material parameters as materials plastify
% \end{itemize}
% 
% \column{0.60\textwidth}
% \vspace{-1cm}
% %  \hspace{-0.9cm}
% \begin{figure}
%   \begin{flushleft}
%   \includegraphics[width=0.7\textwidth, angle=90]{Fangbo_figs/SSI_model_stochastic}
%   \end{flushleft}
%   \caption{\scriptsize Schematic illustration of a stochastic soil-foundation-structure system.}
% \end{figure}
% 
% \end{columns}
% 
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% %\begin{frame}{Governing equations of solid mechanics}
% %\footnotesize
% %{
% %\begin{itemize}
% % \item Equilibrium equation: 
% %   \begin{flushleft}
% %     \begin{equation*}
% %        \frac{\partial \sigma_{ij}}{\partial x_j}+ {\color{blue}{b_j}} = \rho u_i
% %     \end{equation*}
% %   \end{flushleft}
% % \item Strain compatibility equation: 
% %     \begin{equation*}
% %        \epsilon_{ij} = \frac{1}{2} \left( \frac{\partial u_{i}}{\partial x_j}+ \frac{\partial u_{j}}{\partial x_i} \right)
% %     \end{equation*}
% % \item Constitutive equation:   
% %           \begin{equation*}
% %       \dot{\sigma}_{ij} = {\color{blue}{D_{ijkl}}} \dot{\epsilon}_{kl}
% %     \end{equation*}
% %\end{itemize}
% %}
% %\vspace{0.5cm}
% %\centering Input uncertainties (${\color{blue}{D_{ijkl}}}$, ${\color{blue}{b_j}}$) $\rightarrow$ stochastic PDE
% %\end{frame}
% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Simulation Methods for Stochastic PDEs}

\begin{itemize}

\item Analytical, stochastic differential equation approach: difficult to solve with complex random coefficients

\vspace*{2mm}
\item Monte Carlo method : Computationally expensive, \\ 
       will show some results at SMiRT 43026

\vspace*{2mm}
\item Perturbation approach: Small variation with respect to mean, closure problem

\vspace*{2mm}
\item Stochastic collocation method: Global error minimization

\vspace*{2mm}
\item Stochastic Galerkin method: Local error minimization, 
					 Stochastic Elastic-Plastic Finite Element  Method (SEPFEM)
%    \begin{itemize}
%         \item \color{blue}{ \footnotesize Previous studies$\rightarrow$ Uncertain static loading, linear elastic material.}
%         \item \color{blue}{ \footnotesize My research$\rightarrow$ Uncertain dynamic loading, nonlinear material.}
%    \end{itemize}
\end{itemize}

\end{frame}




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\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*{3mm}
\item Dynamic Finite Elements
%
\begin{equation}
{ M} \ddot{ u_i} +
{ C} \dot{ u_i} +
{ K}^{ep} { u_i} =
{ F(t)}
    \nonumber
\end{equation}


\vspace*{3mm}
\item Material and load parameters are uncertain



\end{itemize}



\end{frame}






 
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\begin{frame} 

\frametitle{Probabilistic Elastic-Plastic Response}


\begin{figure}[!hbpt]
\begin{center}
%\includegraphics[width=8cm]{/home/jeremic/tex/works/Papers/2007/ProbabilisticYielding/figures/vonMises_G_and_cu_very_uncertain/Contour_PDF-edited.pdf}
\includegraphics[width=8cm]{/home/jeremic/tex/works/Conferences/2012/DOE-LLNL-workshop-27-28-Feb-2012/ProbabilisticYielding_vonMises_G_and_cu_very_uncertain_Contour_PDF-edited.pdf}
\end{center}
\end{figure}

\end{frame}



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\begin{frame}
\frametitle{Probabilistic Elastic-Plastic Modeling}


% % \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}

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


\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 (Matthies et al,  2004, 2005, 2014...)
% 
% 
%       \end{itemize}
% 
% % 
% %  \item 
% %  Monte Carlo method: accurate, very costly
% %  
% %  \item 
% %  Perturbation method: 
% 
% \end{itemize}
% 
% 
%  
% 
% 
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% \end{frame}
% 
% 


% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame} \frametitle{{3D FPK 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), D_{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), D_{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), D_{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), D_{mnrs}(x_t), \epsilon_{rs}(x_t,t)); \right. \right. \right. \\
% \nonumber
% & & \left. \left. \left. \eta_{ab} (\sigma_{ab}(x_{t-\tau}, t-\tau), D_{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{itemize}
% %-- 
% %-- 
% %-- 
% %-- \item 6 equations
% %-- 
% %-- \item Complete description of 3-D probabilistic stress-strain response
% %-- 
% %-- \end{itemize}
% %-- 
% %--  
% 
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% \end{frame}
% 
% 
% 



% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \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}
% 
% 
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \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
% % %
% % %
% % %
% % %\begin{normalsize}
% % \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]
% % %\nonumber
% %  = -\frac{\partial \zeta}{\partial \sigma}
% % \end{equation}
% % %\end{normalsize}
% 
% %
% 
% \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}
% 
% 
% 
% 
% 
% 
% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsection{Direct Solution for Probabilistic Stiffness and Stress in 1D}
% 
% 
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%    BEGGINING PEP    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% 
% \begin{frame}{Direct Probabilistic Constitutive Modeling 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}{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}{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}
% 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\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}{Time Domain SEPFEM}

\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{5mm}

Motion 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{5mm}

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}
\hspace*{4mm}
$a_i(x), f_{mj}(t)$ are {known PC coefficients}, 

\vspace{3mm}
\hspace*{4mm}
$d_{nk}(t)$ are {unknown PC coefficients, results}.
%}

\end{frame}












 



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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


\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) 

\vspace*{2mm}
  \item  It is NOT based on Monte Carlo approach

\vspace*{2mm}
  \item  System of equations does grow (!)

\end{itemize}


\vspace*{4mm}
%   \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 & 43,758 \\  
   4 & 4 & 20 & 3,108,105 \\    
   4 & 4 & 30 & 48,903,492 \\  
   6 & 6 & 10 & 646,646 \\    
   6 & 6 & 20 & 225,792,840 \\  
%   6 & 6 & 30 & 1.1058 $10^{10}$ \\    
%   8 & 8 & 10 & 5 311 735 \\  
%   8 & 8 & 20 & 7.3079 $10^{9}$ \\    
%   8 & 8 & 30 & 9.9149 $10^{11}$\\  
    ...  & ... & ... & ...\\   
%    \hline
  \end{tabular}}


\end{frame}












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


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

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


\vspace*{-2mm}
\begin{center}
% \hspace*{-15mm}
\movie[label=show3,width=9cm,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/public_html/lecture_notes_online_material/_Chapter_Probabilistic_Elasto_Plasticity_and_Stochastic_Elastic_Plastic_Finite_Element_Method/SEPFEM_Animation_Elastic.mp4}
\end{center}

% \includegraphics[width = 12cm]{./img/figure_elastic_900.pdf}




\end{frame}




































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\section{Conclusion}



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    \subsection{\ }





 
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame} 
% 
% \frametitle{Real-ESSI Simulator System}
% 
% The  Real-ESSI, Realistic  
% {\underline {\bf M}}odeling  and  
% {\underline {\bf 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  high  fidelity,  high  performance,  time  domain,
% nonlinear/inelastic, deterministic or probabilistic, 3D, finite element modeling
% and simulation of:
% 
%     \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}
% 
% 
% \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, Python, Matlab)
% 
%     \end{itemize} 
% 
% \vspace*{1mm}
%   \item Real-ESSI System availability:                   
%     \begin{itemize}
% %\vspace*{1mm}
%      \item Educational  Institutions: Amazon Web Services (AWS), free 
%      \item Government Agencies, National Labs: AWS GovCloud
%      \item Professional  Practice: AWS, commercial
% %\vspace*{1mm}
% %%\vspace*{1mm}
% %      \item Sources available to collaborators
%     \end{itemize}
% 
% 
% 
% \vspace*{1mm}
% \item Quality Management System, ASME-NQA-1, ISO9003-2018, Certification in progress
% 
% 
% \vspace*{1mm}
%   \item Real-ESSI Short Courses (online, this Fall)
% 
% 
% 
% \vspace*{1mm}
%   \item System description and documentation at  
%     \url{http://sokocalo.engr.ucdavis.edu/~jeremic/Real_ESSI_Simulator/}
% %
% %\url{http://real-essi.info/}
% %
% 
% 
% % \vspace*{2mm}
% %   \item
% % 
% 
% 
% \end{itemize}
% 
% 
% \end{frame}
% 
% 
% 
% 
% 
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%







%\subsection*{Summary}


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




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% 
% 
% \begin{frame} 
% 
% \frametitle{Science Quotes}
% 
% \begin{itemize}
% 
% 
% % \item Max  Planck: 
% % "A    new    scientific  truth does not triumph by convincing  its opponents and
% % making  them see the light, but rather because its opponents eventually die, and
% % a  new  generation  grows  up  that  is familiar with it." (Science advances one
% % funeral at a time)
% % 
% % 
%  \vspace*{3mm}
% 
% \item  Fran{\c c}ois-Marie Arouet,  Voltaire: 
% "Le doute n'est pas une condition agr{\'e}able, mais la certitude est absurde."
% 
%  \vspace*{3mm}
% 
% \item  Niklaus  Wirth:   
% "Software is getting slower more rapidly than hardware becomes faster."
% \end{itemize}
% 
% 
% \end{frame}
% 
% 
% 
% 
% 



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
  \frametitle{Summary}

  \begin{itemize}

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

\vspace*{1mm}
    \item Analytic modeling and simulation of uncertain ESSI

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


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


\vspace*{1mm}
    \item Education and Training is the key!

\vspace*{1mm}
    \item Real-ESSI short course this Fall! 


\vspace*{1mm}
    \item  Funding  from  and  collaboration  with  the 
           US-DOE, US-NRC, US-NSF, Caltrans,  US-BR, US-FEMA, 
           CNSC-CCSN, UN-IAEA, Shimizu C. and ENSI/Basler\&Hofmann 
           is greatly appreciated,

\vspace*{1mm}
    \item {\bf \url{http://real-essi.info/}}
      
\end{itemize}

\end{frame}
%







\end{document}

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