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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)
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% for listing DSL
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% for inclusion of other PDF pages, in this case Frank's presentation
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%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/2005January/000780.html
%
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% % we are running LaTeX, not pdflatex
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% % we are running pdflatex, so convert .pdf files to .pdf
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% %*****************************************
%% ovo je za cirilicu
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% \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
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% or whatever
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% Or whatever. Note that the encoding and the font should match. If T1
% does not look nice, try deleting the line with the fontenc.
% Site Specific Dynamics of Structures:
%From Seismic Source to
%the Safety of Occupants and Content
\title[Uncertain Computational Mechanics]
{Forward and Backward \\
Uncertainty Propagation in \\
Computational Earthquake Engineering }
%\subtitle
%{Include Only If Paper Has a Subtitle}
%\author[Author, Another] % (optional, use only with lots of authors)
%{F.~Author\inst{1} \and S.~Another\inst{2}}
%  Give the names in the same order as the appear in the paper.
%  Use the \inst{?} command only if the authors have different
% affiliation.
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\pgfdeclareimage[height=0.7cm]{lbnllogo}{/home/jeremic/BG/amblemi/lbnllogo}
%\author[Jeremi{\'c} et al.] % (optional, use only with lots of authors)
\author[Jeremi{\'c} et al.] % (optional, use only with lots of authors)
%{Boris~Jeremi{\'c}}
{Boris Jeremi{\'c}
\\ \vspace*{1mm}
Han Yang, Hexiang Wang}
%\institute[Computational Geomechanics Group \hspace*{0.3truecm}
%\institute[\pgfuseimage{universitylogo}\hspace*{0.1truecm}\pgfuseimage{lbnllogo}] % (optional, but mostly needed)
\institute[\pgfuseimage{universitylogo}] % (optional, but mostly needed)
%{ Professor, University of California, Davis\\
{ University of California, Davis, CA}
% % and\\
% % Faculty Scientist, Lawrence Berkeley National Laboratory, Berkeley }
% Lawrence Berkeley National Laboratory, Berkeley, CA}
% %  Use the \inst command only if there are several affiliations.
%  Keep it simple, no one is interested in your street address.
\date[] % (optional, should be abbreviation of conference name)
{\small Tianjin University Lecture \\
24/25Nov2021}
\subject{}
% This is only inserted into the PDF information catalog. Can be left
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% If you have a file called "universitylogofilename.xxx", where xxx
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% Delete this, if you do not want the table of contents to pop up at
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\titlepage
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\begin{frame}
\frametitle{Outline}
\begin{scriptsize}
\tableofcontents
% You might wish to add the option [pausesections]
\end{scriptsize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Structuring a talk is a difficult task and the following structure
% may not be suitable. Here are some rules that apply for this
% solution:
%  Exactly two or three sections (other than the summary).
%  At *most* three subsections per section.
%  Talk about 30s to 2min per frame. So there should be between about
% 15 and 30 frames, all told.
%  A conference audience is likely to know very little of what you
% are going to talk about. So *simplify*!
%  In a 20min talk, getting the main ideas across is hard
% enough. Leave out details, even if it means being less precise than
% you think necessary.
%  If you omit details that are vital to the proof/implementation,
% just say so once. Everybody will be happy with that.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\subsection{Motivation}
\subsection{\ }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%dir
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}
\frametitle{Motivation}
\begin{itemize}
%\vspace*{0.3cm}
\item[] Improve modeling and simulation of infrastructure objects
% \vspace*{2mm}
% \item[] Expert numerical modeling and simulation tool
%
% \vspace*{1mm}
% \item[] Use of numerical models to
% analyze statics and dynamics of soil/rockstructure systems
%
\vspace*{4mm}
\item[] Modeling, epistemic uncertainty
\vspace*{4mm}
\item[] Parametric, aleatory uncertainty
\vspace*{4mm}
\item[] Goal is to Predict and Inform
% rather than (force) fit
\vspace*{4mm}
\item[] Engineer needs to know!
%
%
%
% \vspace*{1mm}
% \item[] Follow the flow, input and dissipation, of seismic energy,
% \vspace*{2mm}
% \item[]
% %System for
% {\bf Real}istic modeling and simulation of
% {\bf E}arthquakes and/or
% {\bf S}oils and/or
% {\bf S}tructures and their
% {\bf I}nteraction:\\
% RealESSI
% \hspace*{5mm}
% \url{http://realessi.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://msessi.info/}
% % %
%
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{Numerical Prediction under Uncertainty}
\begin{itemize}
%\vspace*{1mm}
\item[] {Modeling, Epistemic Uncertainty}
\begin{itemize}
\vspace*{1mm}
\item[] Modeling Simplifications
\vspace*{1mm}
\item[] Modeling sophistication for confidence in results
\vspace*{1mm}
\item[] Verification and Validation
%
%\vspace*{2mm}
% \item[] Choice of sophistication level for confidence in results
\end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace*{4mm}
\item[] {Parametric, Aleatory Uncertainty}
\begin{itemize}
\vspace*{1mm}
\item[] ${M} \ddot{u_i} + {C} \dot{u_i} + {K}^{ep} {u_i} = {F(t)}$,
\vspace*{1mm}
\item[] Uncertain: mass $M$, viscous damping $C$ and stiffness $K^{ep}$
\vspace*{1mm}
\item[] Uncertain loads, $F(t)$
\vspace*{1mm}
\item[] Results are PDFs and CDFs for $\sigma_{ij}$, $\epsilon_{ij}$, $u_i$, $\dot{u}_i$, $\ddot{u}_i$
\end{itemize}
\end{itemize}
%
%
% %Le doute n'est pas un {\'e}tat bien agr{\'e}able,\\
% mais l'assurance est un {\'e}tat ridicule. (Fran{\c c}oisMarie Arouet, Voltaire)
\end{frame}
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\begin{frame}
\frametitle{Modeling, Epistemic Uncertainty}
\begin{itemize}
\item[] Important (?!) features are simplified
\begin{itemize}
\vspace*{1mm}
\item[] 1C vs 3C seismic motions
\vspace*{1mm}
\item[] Elastic vs Inelastic behavior
\end{itemize}
%\vspace*{4mm}
% \item[] Unrealistic and unnecessary modeling simplifications
\vspace*{3mm}
\item[] Modeling simplifications are justifiable if one or two
level higher sophistication model demonstrates that features being
simplified out are less or not important
\end{itemize}
% local
% %\vspace*{2mm}
% \begin{center}
% \hspace*{7mm}
% %\movie[label=show3,width=8.8cm,poster,autostart,showcontrols]
% \movie[label=show3,width=5.5cm,poster,autostart, showcontrols]
% {\includegraphics[width=50mm]
% {/home/jeremic/tex/works/Conferences/2016/IAEA_TecDoc_February2016/My_Current_Work/movie_2_npps_mp4_icon.jpeg}}
% {/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Model01_ESSI_Response_May2015/movie_2_npps.mp4}
% %
% %\hfill
% \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}
% %
%
%
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\end{frame}
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\begin{frame}
\frametitle{1C vs 6C Free Field Motions}
\begin{itemize}
\item One component of motions, 1C from 6C
% or 3$\times$1D (it is done all the time!)
\item Excellent fit, wrong mechanics
% (goal is to predict and inform and not (force) fit)
\end{itemize}
% local
%\vspace*{2mm}
\begin{center}
\hspace*{16mm}
%\movie[label=show3,width=5.6cm,poster,autostart,showcontrols]
\movie[label=show3,width=61mm,poster, showcontrols]
{\includegraphics[width=60mm]{/home/jeremic/tex/works/Conferences/2016/IAEA_TecDoc_February2016/My_Current_Work/movie_ff_3d_mp4_icon.jpeg}}
{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Model01_ESSI_Response_May2015/movie_ff_3d.mp4}
%\hspace*{2mm}
%\hfill
%\movie[label=show3,width=5.6cm,poster,autostart,showcontrols]
\movie[label=show3,width=61mm,poster, showcontrols]
{\includegraphics[width=60mm]
{/home/jeremic/tex/works/Conferences/2016/IAEA_TecDoc_February2016/My_Current_Work/movie_ff_1d_mp4_icon.jpeg}}
{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Model01_ESSI_Response_May2015/movie_ff_1d.mp4}
\hspace*{16mm}
\end{center}
% local
% online
\begin{center}
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/ESSI_VisIt_movies_Jose_19May2015/movie_ff_3d.mp4}
{\tiny (MP4)}
%
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/ESSI_VisIt_movies_Jose_19May2015/movie_ff_1d.mp4}
{\tiny (MP4)}
\end{center}
% online
% out
% out % local
% out %\vspace*{2mm}
% out \begin{center}
% out \hspace*{16mm}
% out %\movie[label=show3,width=5.6cm,poster,autostart,showcontrols]
% out \movie[label=show3,width=61mm,poster, showcontrols]
% out {\includegraphics[width=60mm]{movie_ff_3d_mp4_icon.jpeg}}{movie_ff_3d.mp4}
% out % \hspace*{16mm}
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% out \movie[label=show3,width=61mm,poster, showcontrols]
% out {\includegraphics[width=60mm]{movie_ff_1d_mp4_icon.jpeg}}{movie_ff_1d.mp4}
% out \hspace*{16mm}
% out \end{center}
% out %
% out
% out \begin{flushleft}
% out %\vspace*{15mm}
% 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_50mmesh_45degree_Ormsby.mp4}
% out {\tiny (MP4)}
% out %
% out \hspace*{55mm}
% out \href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/ESSI_VisIt_movies_Jose_19May2015/movie_ff_1d.mp4}
% out % \href{./homo_50mmesh_45degree_Ormsby.mp4}
% out {\tiny (MP4)}
% out \end{flushleft}
% out %
% out
% out
% out
% out % online
% out % online \begin{center}
% 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_3d.mp4}
% 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}
% out % online {\includegraphics[width=50mm]{movie_ff_1d_mp4_icon.jpeg}}
% out % online \end{center}
% out % online
% out
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% out
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\end{frame}
%
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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]
% out {\includegraphics[width=69mm]{BJicon.png}}
% out {movie_2_npps.mp4}
% out \end{center}
% out
% out
% out \begin{flushleft}
% out \vspace*{15mm}
% out \href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/ESSI_VisIt_movies_Jose_19May2015/movie_2_npps.mp4}
% out {\tiny (MP4)}
% out \end{flushleft}
% out %
% out
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Elastic vs Inelastic NPP Response}
% Elastoplastic soil with contact elements
%% Both solid and contact elements dissipate energy
% \vspace*{5mm}
\begin{center}
% \hspace*{15mm}
\movie[label=show3,width=10cm,poster,autostart,showcontrols]
{\includegraphics[width=10cm]
{/home/jeremic/tex/works/Thesis/SumeetKumarSinha/Files_13Aug2017/NPP_Non_Linear_Effects_Sumeet.jpg}}
{/home/jeremic/tex/works/Thesis/SumeetKumarSinha/Files_13Aug2017/NPP_Non_Linear_Effects_Sumeet.mp4}
\end{center}
\begin{flushleft}
\vspace*{15mm}
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/NPP_animations_August2017/NPP_Non_Linear_Effects_Sumeet.mp4}
% \href{./homo_50mmesh_45degree_Ormsby.mp4}
{\tiny (MP4)}
\end{flushleft}
%
\end{frame}
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\begin{frame}
\frametitle{Energy Dissipation in a LargeScale Model}
% Elastoplastic soil with contact elements
%% Both solid and contact elements dissipate energy
% \vspace*{5mm}
\begin{center}
% \hspace*{15mm}
\movie[label=show3,width=10cm,poster,autostart,showcontrols]
{\includegraphics[width=10cm]
{/home/jeremic/tex/works/Conferences/2017/SMiRT_24/present/3D_Nonlinear_Modeling_and_it_Effects/NPP_Plastic_Dissipation_grab.jpg}}
{/home/jeremic/tex/works/Thesis/HanYang/Files_10Aug2017/NPP_Plastic_Dissipation.mp4}
\end{center}
\begin{flushleft}
\vspace*{15mm}
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Energy_Dissipation_Animations/NPP_Plastic_Dissipation.mp4}
% \href{./homo_50mmesh_45degree_Ormsby.mp4}
{\tiny (MP4)}
\end{flushleft}
%
\end{frame}
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% \begin{frame}
% \frametitle{Energy Dissipation for Design}
%
% \begin{figure}[!hbpt]
% \begin{center}
% \includegraphics[width=10.0truecm]{/home/jeremic/tex/works/Thesis/HanYang/Frame_animations_13Mar2019/2D_Frame_Model.pdf}
% \end{center}
% \end{figure}
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\begin{frame}
\frametitle{Design Alternatives}
% local
%\vspace*{2mm}
\begin{center}
\hspace*{16mm}
%\movie[label=show3,width=5.6cm,poster,autostart,showcontrols]
\movie[label=show3,width=61mm,poster, showcontrols]
{\includegraphics[width=60mm]{/home/jeremic/tex/works/Thesis/HanYang/Frame_animations_13Mar2019/Individual_Foundation_screen_grab.jpg}}
{/home/jeremic/tex/works/Thesis/HanYang/Frame_animations_13Mar2019/Individual_Foundation.mp4}
%\hspace*{2mm}
%\hfill
%\movie[label=show3,width=5.6cm,poster,autostart,showcontrols]
\movie[label=show3,width=61mm,poster, showcontrols]
{\includegraphics[width=61mm]{/home/jeremic/tex/works/Thesis/HanYang/Frame_animations_13Mar2019/Continuous_Foundation_screen_grab.jpg}}
{/home/jeremic/tex/works/Thesis/HanYang/Frame_animations_13Mar2019/Continuous_Foundation.mp4}
\hspace*{16mm}
\end{center}
% local
% online
\begin{center}
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/Energy_dissipation_frames/Individual_Foundation.mp4}
{\tiny (MP4)}
%
\hspace*{40mm}
%
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/Energy_dissipation_frames/Continuous_Foundation.mp4}
{\tiny (MP4)}
\end{center}
% online
\end{frame}
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\begin{frame}
\frametitle{Pine Flat Dam, Dynamic Response with Reservoir}
%\vspace*{5mm}
\begin{center}
% \hspace*{15mm}
\movie[label=show3,width=9cm,poster,autostart,showcontrols]
{\includegraphics[width=8cm]
{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_Concrete_Dams/Pine_Flat_Dam/soil_fluid_interaction_dam.jpg}}
{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_Concrete_Dams/Pine_Flat_Dam/soil_fluid_interaction_dam.mp4}
\end{center}
\begin{flushleft}
\vspace*{15mm}
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Concrete_Dams/Pine_Flat_Dam/soil_fluid_interaction_dam.mp4}
% \href{./homo_50mmesh_45degree_Ormsby.mp4}
{\tiny (MP4)}
\end{flushleft}
%
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Pine Flat Dam, Hydrodynamic Pressure}
%\vspace*{5mm}
\begin{center}
% \hspace*{15mm}
\movie[label=show3,width=10cm,poster,autostart,showcontrols]
{\includegraphics[width=10.0cm]
{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_Concrete_Dams/Pine_Flat_Dam/dynamic_pressure.jpg}}
{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_Concrete_Dams/Pine_Flat_Dam/dynamic_pressure.mp4}
\end{center}
\begin{flushleft}
\vspace*{15mm}
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Concrete_Dams/Pine_Flat_Dam/dynamic_pressure.mp4}
% \href{./homo_50mmesh_45degree_Ormsby.mp4}
{\tiny (MP4)}
\end{flushleft}
%
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
% \frametitle{Pine Flat Dam, Inelastic Interface, Hydrostatic}
\frametitle{Pine Flat Dam, Inelastic Interface}
%\vspace*{5mm}
\begin{center}
% \hspace*{15mm}
\movie[label=show3,width=10cm,poster,autostart,showcontrols]
{\includegraphics[width=10.0cm]
{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_Concrete_Dams/Pine_Flat_Dam/Case_D3_Gap_Open_more.jpg}}
{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_Concrete_Dams/Pine_Flat_Dam/Case_D3_Gap_Open_more.mp4}
\end{center}
\begin{flushleft}
\vspace*{15mm}
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Concrete_Dams/Pine_Flat_Dam/Case_D3_Gap_Open_more.mp4}
% \href{./homo_50mmesh_45degree_Ormsby.mp4}
{\tiny (MP4)}
\end{flushleft}
%
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}
\frametitle{Pine Flat Dam, Inclined Plane Waves}
%\vspace*{5mm}
\begin{center}
% \hspace*{15mm}
\movie[label=show3,width=10cm,poster,autostart,showcontrols]
{\includegraphics[width=10.0cm]
{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_Concrete_Dams/Pine_Flat_Dam/dynamic_response_inclination.jpg}}
{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_Concrete_Dams/Pine_Flat_Dam/dynamic_response_inclination.mp4}
\end{center}
\begin{flushleft}
\vspace*{15mm}
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Concrete_Dams/Pine_Flat_Dam/dynamic_response_inclination.mp4}
% \href{./homo_50mmesh_45degree_Ormsby.mp4}
{\tiny (MP4)}
\end{flushleft}
%
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\begin{frame}
\frametitle{Parametric, Aleatory Uncertainty}
\vspace*{2mm}
%\vspace*{5mm}
\begin{figure}[!hbpt]
\begin{center}
%
\hspace*{7mm}
\includegraphics[width=5.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGEGoverGmax/figures/YoungModulus_RawData_and_MeanTrend_01Ed.pdf}
\hspace*{3mm}
% \hfill
\includegraphics[width=4.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGEGoverGmax/figures/YoungModulus_Histogram_Normal_01Ed.pdf}
%
\end{center}
\end{figure}
\vspace*{5mm}
%\vspace*{1.8cm}
%\hspace*{3.3cm}
\begin{flushright}
{\tiny
(cf. Phoon and Kulhawy (1999B))\\
~}
\end{flushright}
%
\vspace*{9mm}
\begin{figure}[!hbpt]
\begin{center}
%
%\hspace*{7mm}
\includegraphics[width=5.00truecm]{/home/jeremic/tex/works/Thesis/HexiangWang/time_series_motionsn_06ug2019_SMIRT/Acc_realization_200.pdf}
%\hspace*{3mm}
%\includegraphics[width=2cm]{/home/jeremic/tex/works/Papers/2019/Hexiang/1D_risk/version2/Figures/Acc_time_series_realiztion70.pdf}
%\includegraphics[width=2cm]{/home/jeremic/tex/works/Papers/2019/Hexiang/1D_risk/version2/Figures/Acc_time_series_realiztion100.pdf}
%% \includegraphics[width=0.31\textwidth]{Figures/Acc_time_series_realiztion350.pdf}
%\includegraphics[width=2cm]{/home/jeremic/tex/works/Papers/2019/Hexiang/1D_risk/version2/Figures/Acc_time_series_realiztion367.pdf}
\includegraphics[width=4cm]{/home/jeremic/tex/works/Papers/2019/1D_risk/version2/Figures/SA_GMPE_verification_std_08_no_smooth.pdf}
%
\end{center}
\end{figure}
\vspace*{7mm}
\begin{flushright}
{\tiny
(cf. Wang et al. (2019))\\
~}
\end{flushright}
\end{frame}
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% % \frametitle{Sensitivity Analysis}
% \frametitle{Engineer Needs to Know!}
%
%
% \begin{itemize}
%
%
%
% \item[] Forward propagation of uncertainty, full probabilistic,
% nonlinear/inelastic EarthquakeSoilStructureInteraction, ESSI response in
% time domain (Jeremic et al 2011, Wang et al 2019)
% % %\vspace*{1mm}
% % \item[] \underline{Sensitivity Analysis} quantifies the relative importance
% % of input uncertain parameters and their contributions to the probabilistic
% % system response
% %
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \vspace*{4mm}
% % \item[] \underline{Local sensitivity analysis} focuses on the local impact of
% % input uncertain parameters on model response, quantified by the gradient of
% % system response with respect to the variation of input parameters
% %
%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \vspace*{4mm}
% % \item[] \underline{Global sensitivity analysis} studies the
% % respective
% \vspace*{4mm}
% \item[] Backward propagation, sensitivity analysis, quantifies the relative importance
% of input uncertain parameters on the variance of the probabilistic system
% response
% (Sobol 2001, Sudret 2008, Jeremic et al 2021)
% %Sobol, {\cyss Sobol{p1}} indices (Sobol 2001), Sudret (2008)
% % encoding for soft b is {p1}, see
%
%
%
% \end{itemize}
%
%
%
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Parametric Uncertainty: Soil Stiffness and Strength}
%
%
% \vspace*{2mm}
% %\vspace*{3mm}
% \begin{figure}[!hbpt]
% \begin{center}
% %
% \hspace*{7mm}
% \includegraphics[width=5.5truecm]{/home/jeremic/tex/works/Papers/2008/JGGEGoverGmax/figures/YoungModulus_RawData_and_MeanTrend_01Ed.pdf}
% \hspace*{3mm}
% % \hfill
% \includegraphics[width=3.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGEGoverGmax/figures/YoungModulus_Histogram_Normal_01Ed.pdf}
% %
% \end{center}
% \end{figure}
%
% \vspace*{5mm}
% \begin{figure}[!hbpt]
% \begin{center}
% %
% \hspace*{7mm}
% \includegraphics[width=5.00truecm]{/home/jeremic/tex/works/Papers/2008/JGGEGoverGmax/figures/ShearStrength_RawData_and_MeanTrendMod.pdf}
% \hspace*{3mm}
% % \hfill
% \includegraphics[width=3.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGEGoverGmax/figures/ShearStrength_Histogram_PearsonIVFineTunedMod.pdf}
% %
% \end{center}
% \end{figure}
%
% %\vspace*{5mm}
% %\vspace*{1.8cm}
% %\hspace*{3.3cm}
% \begin{flushright}
% {\tiny
% (cf. Phoon and Kulhawy (1999B))\\
% ~}
% \end{flushright}
% %
%
%
% \end{frame}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Parametric Uncertainty: Material Properties}
\vspace*{5mm}
\begin{figure}[!hbpt]
\begin{center}
% %
\hspace*{3mm}
\includegraphics[width=2.5truecm]{/home/jeremic/tex/works/Thesis/KonstantinosKarapiperis/Soil_Uncertainty_Report_Pdf_Cdf_Figures/FieldPhiPdf.pdf}
\hspace*{3mm}
\includegraphics[width=2.5truecm]{/home/jeremic/tex/works/Thesis/KonstantinosKarapiperis/Soil_Uncertainty_Report_Pdf_Cdf_Figures/FieldPhiCdf.pdf}
\hspace*{3mm}
\includegraphics[width=2.5truecm]{/home/jeremic/tex/works/Thesis/KonstantinosKarapiperis/Soil_Uncertainty_Report_Pdf_Cdf_Figures/FieldSuPdf.pdf}
\hspace*{3mm}
\includegraphics[width=2.5truecm]{/home/jeremic/tex/works/Thesis/KonstantinosKarapiperis/Soil_Uncertainty_Report_Pdf_Cdf_Figures/FieldSuCdf.pdf}
\\
%\vspace*{2mm}
\hspace*{2.5cm} \mbox{\tiny Field $\phi$} \hspace*{3.5cm} \mbox{\tiny Field $c_u$}
\\
\vspace*{10mm}
\hspace*{3mm}
\includegraphics[width=2.5truecm]{/home/jeremic/tex/works/Thesis/KonstantinosKarapiperis/Soil_Uncertainty_Report_Pdf_Cdf_Figures/LabPhiPdf.pdf}
\hspace*{3mm}
\includegraphics[width=2.5truecm]{/home/jeremic/tex/works/Thesis/KonstantinosKarapiperis/Soil_Uncertainty_Report_Pdf_Cdf_Figures/LabPhiCdf.pdf}
\hspace*{3mm}
\includegraphics[width=2.5truecm]{/home/jeremic/tex/works/Thesis/KonstantinosKarapiperis/Soil_Uncertainty_Report_Pdf_Cdf_Figures/LabSuPdf.pdf}
\hspace*{3mm}
\includegraphics[width=2.5truecm]{/home/jeremic/tex/works/Thesis/KonstantinosKarapiperis/Soil_Uncertainty_Report_Pdf_Cdf_Figures/LabSuCdf.pdf}
\\
%\vspace*{8mm}
\hspace*{2.5cm} \mbox{\tiny Lab $\phi$} \hspace*{3.5cm} \mbox{\tiny Lab $c_u$}
\end{center}
\end{figure}
\end{frame}
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% \begin{frame}
%
% \frametitle{RealESSI Simulator System}
%
% %
%
% \vspace*{2mm}
% The RealESSI,
% {\underline {\bf Real}}istic
% %{\underline {\bf M}}odeling and
% %{\underline {\bf S}}imulation of
% {M}odeling and
% {S}imulation of
% {\underline {\bf E}}arthquakes,
% {\underline {\bf S}}oils,
% {\underline {\bf S}}tructures and their
% {\underline {\bf I}}nteraction. Simulator is a software, hardware and
% documentation system for time domain,
% linear and nonlinear, inelastic, deterministic or probabilistic, 3D,
% modeling and simulation of:
%
% \vspace*{1mm}
% \begin{itemize}
% %\vspace*{1mm}
% \item[] statics and dynamics of soil,
% % %\vspace*{1mm}
% % \item[] statics and dynamics of rock,
% %\vspace*{1mm}
% \item[] statics and dynamics of structures,
% %\vspace*{1mm}
% \item[] statics of soilstructure systems, and
% %\vspace*{1mm}
% \item[] dynamics of earthquakesoilstructure system interaction
% \end{itemize}
%
%
%
% Used for:
% \begin{itemize}
% %\vspace*{1mm}
% \item[] Design: linear elastic, load combinations, dimensioning
%
%
% %\vspace*{1mm}
% \item[] Assessment: nonlinear/inelastic, risk, safety margins
% \end{itemize}
%
%
%
%
% \end{frame}
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%
% \frametitle{RealESSI Simulator System}
%
%
% \begin{itemize}
%
%
% \item RealESSI System Components
% \begin{itemize}
% \item[] RealESSI Preprocessor (gmsh/gmESSI, X2ESSI)
% \item[] RealESSI Program (local, remote, cloud)
% \item[] RealESSI PostProcessor (Paraview/pvESSI, Python)
%
% \end{itemize}
%
% \vspace*{1mm}
% \item RealESSI System availability:
% \begin{itemize}
% %\vspace*{1mm}
% \item[] Educational Institutions: AWS, Linux Image, free
% \item[] Government Agencies, National Labs: AWS GovCloud
% \item[] Professional Practice: AWS, commercial
% %\vspace*{1mm}
% %%\vspace*{1mm}
% % \item Sources available to collaborators
% \end{itemize}
%
%
%
% \vspace*{1mm}
% \item RealESSI education and training: theory and applications
%
%
%
% \vspace*{1mm}
% \item RealESSI documentation and program available at
% \url{http://realessi.info/}
% %\url{http://sokocalo.engr.ucdavis.edu/~jeremic/Real_ESSI_Simulator/}
% %
% %\url{http://realessi.info/}
% %
%
%
% % \vspace*{2mm}
% % \item
% %
%
%
% \end{itemize}
%
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% %\subsection*{RealESSI Simulator System}
% %\subsection{RealESSI}
% \subsection{\ }
%
%
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\begin{frame}
\frametitle{RealESSI Simulator System}
\vspace*{1mm}
%
The RealESSI,
{\underline {\bf Real}}istic
%{\underline {\bf M}}odeling and
%{\underline {\bf S}}imulation of
{M}odeling and
{S}imulation of
{\underline {\bf E}}arthquakes,
{\underline {\bf S}}oils,
{\underline {\bf S}}tructures and their
{\underline {\bf I}}nteraction Simulator is a software, hardware and
documentation system for time domain,
linear and nonlinear,
elastic and inelastic,
deterministic or probabilistic,
3D,
modeling and simulation of:
\vspace*{1mm}
\begin{itemize}
\vspace*{1mm}
\item[] statics and dynamics of soil and rock,
\vspace*{1mm}
\item[] statics and dynamics of rock,
\vspace*{1mm}
\item[] statics and dynamics of structures,
\vspace*{1mm}
\item[] statics of soilstructure systems, and
\vspace*{1mm}
\item[] dynamics of earthquakesoilstructure system interaction
\end{itemize}
\vspace*{1mm}
Used for:
\begin{itemize}
\vspace*{1mm}
\item[] Design, linear elastic, load combinations, dimensioning
\vspace*{1mm}
\item[] Assessment, nonlinear/inelastic, safety margins
\end{itemize}
\end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%
% \frametitle{RealESSI Simulator System}
%
%
% \begin{itemize}
%
%
% \item RealESSI System Components
% \begin{itemize}
% \item[] RealESSI Preprocessor (gmsh/gmESSI, X2ESSI)
% \item[] RealESSI Program (local, remote, cloud)
% \item[] RealESSI PostProcessor (Paraview/pvESSI, Python)
%
% \end{itemize}
%
% \vspace*{3mm}
% \item Availability: Windows/MacOS/Linux Docker, Linux execs, AWS
% % \begin{itemize}
% % %\vspace*{1mm}
% % \item[] Educational Institutions: AWS and Linux Executables, free
% % \item[] Government Agencies, National Labs: AWS GovCloud, free
% % \item[] Professional Practice: AWS and Linux Executables, commercial
% % %\vspace*{1mm}
% % %%\vspace*{1mm}
% % % \item Sources available to collaborators
% % \end{itemize}
%
%
%
% \vspace*{3mm}
% \item RealESSI education and training: theory and applications
%
%
%
% \vspace*{3mm}
% \item RealESSI documentation and program available at
% \url{http://realessi.info/}
% %\url{http://sokocalo.engr.ucdavis.edu/~jeremic/Real_ESSI_Simulator/}
% %
% %\url{http://realessi.info/}
% %
%
%
% % \vspace*{2mm}
% % \item
% %
%
%
% \end{itemize}
%
%
% \end{frame}
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Quality Assurance}
%
% \begin{itemize}
%
% \item Full verification suit for each element, model, algorithm
%
% \vspace*{5mm}
% \item Certification process
%
% \begin{itemize}
%
% \vspace*{2mm}
% \item[] ASME NQA1
%
% \vspace*{2mm}
% \item[] ISO900032014
%
% \end{itemize}
%
% %\vspace*{3mm}
% %\item[] Verification examples given below
%
% \end{itemize}
%
%
%
%
% \end{frame}
%
% % %
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{RealESSI}
%
% \begin{itemize}
%
%
%
% %\vspace*{2mm}
% \item A system for time domain, nonlinear/inelastic, deterministic or
% probabilistic, modeling and simulation of
%
% \begin{itemize}
% \item statics and dynamics of soil,
% \item statics and dynamics of rock,
% \item statics and dynamics of structures,
% \item statics of soilstructure systems, and
% \item dynamics of earthquakesoilstructure system interaction.
% \end{itemize}
%
%
% \vspace*{1mm}
% \item Design, linear elastic, load combinations, dimensioning
%
%
% \vspace*{1mm}
% \item Assessment, nonlinear/inelastic, safety margins
%
%
%
% %\vspace*{1mm}
% % \item Develops methods and models that inform and predict rather than (force) fit.
%
% \vspace*{1mm}
% \item Collaboration and financial support from the USDOE, USNRC,
% USNSF, Caltrans, CNSCCCSN, UNIAEA, Shimizu, Basler\&Hofmann, etc.
%
%
%
% \end{itemize}
%
% \end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{RealESSI Modeling Features}
\begin{itemize}
%\vspace*{2mm}
\item[] Solid elements: dry, (un)saturated, elastic, inelastic
%\vspace*{1mm}
\item[] Structural elements: beams, shells, elastic, inelastic
%\vspace*{1mm}
\item[] Contact/interface/joint elements: Bonded, Shear/Frictional (EPP, EPH,
EPS); Gap/Normal; linear, nonlinear, dry, coupled/saturated,
%\vspace*{1mm}
\item[] Super element: stiffness and mass matrices
%\vspace*{1mm}
\item[] Material models: soil, rock, concrete, steel...
%\vspace*{1mm}
\item[] Seismic input: 1C and 3C, deterministic or probabilistic
%\vspace*{1mm}
\item[] Energy dissipation: elasticplastic, viscous, algorithmic
%\vspace*{1mm}
\item[] Solid/StructureFluid interaction, full coupling, OpenFOAM
%\vspace*{1mm}
\item[] Intrusive, forw. and backw. probabilistic inelastic modeling
%\vspace*{1mm}
\item[] Detailed Verification and partial Validation
%\vspace*{1mm}
\item[] RealESSI system:
\hspace*{2mm}
\underline{\href{http://realessi.info/}{http://realessi.info/}}
% \hspace*{5mm}
% and
% \hspace*{5mm}
% \href{http://realessi.info/}{http://realessi.info/}
\end{itemize}
\end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{RealESSI 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://realessi.info/}{http://realessi.info/}
% % % \hspace*{5mm}
% % % and
% % % \hspace*{5mm}
% % % \href{http://realessi.info/}{http://realessi.info/}
%
%
%
% \end{itemize}
%
%
%
% \vspace*{60mm}
% %\begin{figure}[!hbpt]
% \begin{flushright}
% \includegraphics[width=2.5cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figurefiles/_Chapter_Theory_Introduction/tex_works_psfigures_loading_stageincrementsiterations.pdf}
% \end{flushright}
% %\vspace*{0.5cm}
% %\end{figure}
% %
%
%
%
%
%
% \end{frame}
%
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Real ESSI Simulator: Domain Specific Language, DSL}
%
% \begin{itemize}
% \item Domain Specific Language (DSL), Yacc \& Lex
% \vspace*{3mm}
% \item English like modeling and simulation language
% \vspace*{3mm}
% \item Parser and compiler, can define functions, models, etc.
% \vspace*{3mm}
% \item Can extend models and methods
% \vspace*{3mm}
% \item Requires units!
% \end{itemize}
% %
% \end{frame}
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%
%
% %
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \begin{frame}[fragile]
% % \frametitle{DSL: English Language Binding Modeling Parser}
% % {\footnotesize
% % \begin{lstlisting}
% % // Defining variables
% % x = 7; // x is doublevalued variable with adimenisional units.
% % y = 3.972e+2; // Decimal and scientific notation is supported.
% % // Operations: All standard arithmetic operations (Unites!)
% % a = x + y; // Addition
% % b = x  y; // Subtraction
% % c = x*y; // Product
% % d = x/y; // Quotient
% % e = y%x; // Modulus (how many times x fits in y)
% % // Predefined variables. For example, the variable 'm' defines 'meter'.
% % L1 = 1*m;
% % L2 = 40*mm; // Defines L2 to be 40 millimiters.
% % L3 = 3.14*cm;
% % L4 = 3.14;
% % A5 = 3.14*cm^2;
% % \end{lstlisting}
% % }
% % \end{frame}
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \begin{frame}[fragile]
% % \frametitle{DSL: English Language Modeling Parser}
% % {\footnotesize
% % \begin{lstlisting}
% % F1 = 10*kN; // Define few forces.
% % F2 = 300*N;
% % F3 = 4*kg*g; // Here g is the predefined acceleration
% % // due to gravity.
% % // Operations are sensitive to units. For example,
% % foo = L1 + F1; // Produces an error because units are
% % // not compatible. However,
% % L4 = L1 + L2 + L3; // is OK.
% % // Multiplication (division and modulus) always work
% % // because the result produces a quantity with new units
% % // (except when the adimensional quantity is involved).
% % A = L1*L2;
% % pressure = F1 / A;
% % // All numbers are converted to SI units (kg  m  s)
% % // and internally stored in that system.
% % \end{lstlisting}
% % }
% % \end{frame}
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \begin{frame}
% % \frametitle{DSL: ESSI Input Language, Basics}
% %
% %
% % \begin{itemize}
% %
% % \item Angle brackets \lstinline<> denotes user input
% %
% % \item Expected unit (dimension) is given (example:
% % \lstinline, for length unit)
% %
% % \item Symbol \lstinline<.> represents the adimensional quantity.
% %
% % \item Vertical bar \lstinline++ (``OR'' sign)) is used to separate two or more keyword
% % options, i.e. \lstinline+[abc]+ is used indicate keyword options
% % \lstinline+a+ or \lstinline+b+ or \lstinline+c+.
% %
% % \item The symbol \lstinline+...+ is used to denote where several long options
% % exist and are explained elsewhere (an example of this is available below in a
% % material model definitions).
% %
% % \end{itemize}
% %
% % \end{frame}
% %
%
%
%
% %
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \begin{frame}
% % \frametitle{DSL: ESSI Input Language, Units}
% %
% %
% % All commands require unit consistency. Base units, SI or other (British
% % Imperial) can be used
% % \begin{itemize}
% % \item length, symbol $L$, units [m, in, ft]
% % \item mass, symbol $M$, units [kg, lb],
% % \item time, symbol $T$, units [s]
% % \end{itemize}
% %
% % Derived units can also be used:
% %
% % \begin{itemize}
% % \item angle, symbol rad (radian), unit [$dimensionless, L/L$]
% % \item force, symbol N (Newton), units [$N, kN, MN, M*L/T^2$],
% % \item stress, symbol Pa (Pascal), units [$Pa, kPa, MPa, N/L^2, M/L/T^2$]
% % \item strain, symbol (no symbol), units [$L/L$]
% % \item mass density, symbol (no symbol), units [$M/L^3$]
% % \item force density, symbol (no symbol), units [$M/L^2/T^2$]
% % \end{itemize}
% %
% % \end{frame}
% %
% %
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \begin{frame}
% % \frametitle{DSL: ESSI Input Language, Loading Stages}
% %
% %
% % Start a new loading stage with
% %
% % \lstinlinenew loading stage "loading_stage_name";
% %
% % \vspace*{0.5cm}
% % Example, starting a new loading stage called {\it "self weight load"}
% %
% % \lstinlinenew loading stage "self weight load";
% %
% %
% %
% %
% %
% %
% % \end{frame}
% %
% %
% %
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%
% % %% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % %\begin{frame}[fragile]
% % % \frametitle{DSL: Beam Example, Model}
% % %
% % %
% % %
% % %\begin{figure}[!h]
% % %\begin{center}
% % %\includegraphics[width=10cm]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Cantilever_Beam.pdf}
% % %%\caption{8 node brick element}
% % %%\label{fig:8node_command}
% % %\end{center}
% % %\end{figure}
% % %
% % %
% % %%}
% % %\end{frame}
% % %
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}[fragile]
% \frametitle{Real ESSI DSL Example}
%
%
% \vspace*{10mm}
% \begin{flushright}
% \includegraphics[width=3cm]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Cantilever_Beam.pdf}
% %\caption{8 node brick element}
% %\label{fig:8node_command}
% \end{flushright}
%
% \vspace*{5mm}
% \begin{lstlisting}
% model name "SmallTestModel";
% new loading stage "First_static";
% // Nodal Coordinates
% add node # 1 at (0*m, 0*m, 0*m) with 6 dofs;
% add node # 2 at (0*m, 0*m, 1*m) with 6 dofs;
% add element # 1 type beam_elastic with
% nodes (1, 2) cross_section=1.0*m^2
% elastic_modulus=1.0e5*KN/m^2
% shear_modulus=2.0e4*KN/m^2
% torsion_Jx=2*0.083*m^4
% bending_Iy=0.083*m^4 bending_Iz=0.083*m^4
% mass_density=2500.0*kg/m^3
% xz_plane_vector = (0, 1, 0)
% joint_1_offset = (0.0*m, 0.0*m, 0.0*m)
% joint_2_offset = (0.0*m, 0.0*m, 0.0*m);
% \end{lstlisting}
%
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}[fragile]
% \frametitle{Real ESSI DSL Example}
%
%
% \begin{lstlisting}
% fix node No 1 dofs all;
% add load # 1 to node # 2 type
% linear Fx=9*kN;
% define load factor increment 0.01;
%
% define solver UMFPack;
% define convergence test
% Norm_Displacement_Increment
% tolerance = 1e5
% maximum_iterations = 20
% verbose_level = 4;
% define algorithm Newton;
% simulate 100 steps using static algorithm;
%
% //bye;
% zaijian;
% \end{lstlisting}
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \begin{frame}
% % \frametitle{RealESSI Model Development}
% %
% % \begin{itemize}
% %
% % \item[] PreProcessing, model development gmsh/gmESSI
% %
% % \vspace*{1mm}
% % \item[] Existing model translation, SASSI$\rightarrow$RealESSI
% %
% %
% % \vspace*{1mm}
% % \item[] Self documenting input language
% %
% % \vspace*{1mm}
% % \item[] Units required for all input variables
% %
% % \vspace*{1mm}
% % \item[] All variables and constants need to be defined by user
% %
% %
% %
% % \vspace*{1mm}
% % \item[] Sophistication level of choice
% %
% % %\vspace*{1mm}
% % % \item[] Reduce modeling uncertainty
% %
% % \vspace*{1mm}
% % \item[] Model developed in phases
% %
% % \vspace*{1mm}
% % \item[] Verify model components
% %
% %
% % \vspace*{1mm}
% % \item[] Build confidence in inelastic modeling
% %
% % \end{itemize}
% %
% % \end{frame}
% %
% %
% %
% %
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \begin{frame}
% % \frametitle{RealESSI Modeling Phases}
% %
% %
% %
% % \begin{figure}[htbp]
% % \begin{center}
% % \includegraphics[width = 2.3cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figurefiles/nonlinear_analysis_steps/soilstructure/overview.png}
% % \vspace*{1mm}
% % \\
% % \includegraphics[width = 0.35cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figurefiles/nonlinear_analysis_steps/free_field_1D/DRM_1D_motion_3D_just_column.jpg}
% % \hspace*{5mm}
% % % \includegraphics[width = 0.1cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figurefiles/nonlinear_analysis_steps/free_field_1D/DRM1D_Motion3D.png}
% % \includegraphics[width = 2.5cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figurefiles/nonlinear_analysis_steps/free_field_3D/motion3D_DRM3D_free_field.png}
% % \hspace*{5mm}
% % % \includegraphics[width = 1cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figurefiles/nonlinear_analysis_steps/soilfoundation/soil_foundation.png}
% % % \includegraphics[width = 3cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figurefiles/nonlinear_analysis_steps/soilfoundation/slice.png}
% % \includegraphics[width = 2.5cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figurefiles/nonlinear_analysis_steps/soilfoundation/foundation_results.png}
% % % \includegraphics[width = 3cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figurefiles/nonlinear_analysis_steps/soilstructure/overview.png}
% % \\
% % \vspace*{3mm}
% % \includegraphics[width = 1.0cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figurefiles/nonlinear_analysis_steps/structure/eigen/structureonly.png}
% % \hfill
% % \includegraphics[width = 1.2cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figurefiles/nonlinear_analysis_steps/structure/eigen/eigen1.png}
% % \hfill
% % \includegraphics[width = 1.2cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figurefiles/nonlinear_analysis_steps/structure/eigen/eigen2.png}
% % \hfill
% % \includegraphics[width = 1.2cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figurefiles/nonlinear_analysis_steps/structure/eigen/eigen3.png}
% % \hfill
% % \includegraphics[width = 1.2cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figurefiles/nonlinear_analysis_steps/structure/eigen/eigen4.png}
% % \hfill
% % \includegraphics[width = 1.2cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figurefiles/nonlinear_analysis_steps/structure/eigen/eigen5.png}
% % \hfill
% % \includegraphics[width = 1.2cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figurefiles/nonlinear_analysis_steps/structure/eigen/eigen6.png}
% % \hfill
% % % \includegraphics[width = 1.0cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figurefiles/nonlinear_analysis_steps/structure/imposed_motion/structureonly.png}
% % %\hfill
% % \includegraphics[width = 1.2cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figurefiles/nonlinear_analysis_steps/structure/imposed_motion/imposed_motion_results.png}
% % % \includegraphics[width = 0.1cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figurefiles/nonlinear_analysis_steps/soilstructure/overview.png}
% % \\
% % \vspace*{1mm}
% % \includegraphics[width = 6cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Real_ESSI_short_course_examples_day_123/short_course_document/Figurefiles/nonlinear_analysis_steps/soilstructure/DRM3D_motion3D_structure.png}
% % \end{center}
% % \end{figure}
% %
% %
% %
% % \end{frame}
% %
% %
% %
% %
% %
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \begin{frame}
% % \frametitle{RealESSI Results Post Processing}
% %
% % \begin{itemize}
% %
% %
% % \item All output is saved (stress, strain, displacements, energy...)
% %
% % \vspace*{5mm}
% % \item Time histories, scripts to plot or extract in preferred format
% %
% % \vspace*{5mm}
% % \item 3D visualization, Paraview with pvESSI plugin
% %
% %
% %
% % \end{itemize}
% %
% % \end{frame}
% %
% %
% %
% %
% %
% %
% %
% %
% %
% %
% %
% %
% %
% %
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \begin{frame}
% % \frametitle{RealESSI Core Functionality}
% %
% % \begin{itemize}
% %
% %
% %
% % %\vspace*{2mm}
% % \item [] Introduction to inelastic, nonlinear analysis for practicing engineers
% %
% % %
% % % % \vspace*{3mm}
% % % \item Usable models for professional practice
% % %
% % % %\vspace*{2mm}
% % % \item Core functionality needed for nonlinear modeling in professional
% % % practice
% % % %
% % %
% %
% %
% % % %\vspace*{0.3cm}
% % % %\vspace*{2mm}
% % % \item Hierarchy of modeling capabilities,
% % %
% % % \begin{itemize}
% % %
% % % %\vspace*{1mm}
% % % \item Linear elastic models, elastic constants, viscous damping
% % %
% % % %\vspace*{1mm}
% % % \item Nonlinear models, core functionality, does not require much
% % % material data however, sensitivity study is advised
% % %
% % % %\vspace*{1mm}
% % % \item High sophistication nonlinear models, require material data
% % %
% % %
% % % \end{itemize}
% % %
% %
% %
% % \vspace*{2mm}
% % \item[] Use of prescribed, required (low, medium, high) fidelity numerical
% % models to analyze ESSI behavior
% %
% %
% % \vspace*{2mm}
% % \item[] Set of suggested modeling and simulation parameters
% %
% % \vspace*{2mm}
% % \item[] Investigate sensitivity of response to model sophistication
% %
% % \vspace*{2mm}
% % \item[] Investigate sensitivity of response to model parameters
% %
% % %
% % % \vspace*{1mm}
% % % \item[] Accurately follow the flow of seismic energy in a
% % % soil structure system
% % %
% % % \vspace*{1mm}
% % % \item[] The goal is to create methodology and numerical tool that is used to
% % % predict and inform and not to fit
% % %
% % %
% % %
% % % %\vspace*{1mm}
% % % % \item[] Directing, in space and time, seismic energy flow in the
% % % % soil structure system
% % %
% %
% %
% % \end{itemize}
% % \end{frame}
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %
% %
% %
% %
% %
% %
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \begin{frame}
% % \frametitle{RealESSI Core Functionality Components}
% %
% % \begin{itemize}
% %
% %
% % \item[] Structural elements: Truss, Beam, Shell, SuperElement
% %
% % \vspace*{2mm}
% % \item[] Soil, solids: elastic, $G/G_{max}$
% %
% % \vspace*{2mm}
% % \item[] Contacts/interfaces/joints: Bonded, Frictional (EPP, EPH, EPS), Gap
% % open/close
% %
% %
% % \vspace*{2mm}
% % \item[] Loads: Static, Dynamic (earthquake, 1C or 3$\times$1C), Restart
% %
% % \vspace*{2mm}
% % \item[] Simulation: Explicit noequilibrium, Implicit equilibrium
% %
% % \vspace*{2mm}
% % \item[] Core Functionality Application programs: APPs
% % %
% %
% %
% % \end{itemize}
% %
% %
% % \end{frame}
% %
% %
% %
% %
% %
% %
% %
% %
% %
% %
% %
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \begin{frame}
% % \frametitle{RealESSI Education and Training}
% %
% % \begin{itemize}
% %
% %
% % \item RealESSI Education
% % \begin{itemize}
% % \vspace*{1mm}
% % \item[] Online courses
% % \vspace*{1mm}
% % \item[] Educational short videos
% % \vspace*{1mm}
% % \item[] Professional practice
% % % \vspace*{1mm}
% % % \item Developers
% % \vspace*{1mm}
% % \item[] Practical examples available in lecture notes, and documentation
% % \vspace*{1mm}
% % \item[] Documentation, Lecture Notes:
% % (I) Theory and Computational Formulation,
% % (II) Software and Hardware System,
% % (III) Verification and Validation,
% % (IV) Modeling and Simulation Examples,
% % (V) Application to Practical Engineering Problems.
% %
% % \end{itemize}
% %
% % %\vspace{1mm}
% % % \item Documentation, extensive
% %
% % \vspace{2mm}
% % \item Lecture notes available online through
% % \url{http://realessi.info/}
% %
% %
% % \end{itemize}
% %
% % \end{frame}
% %
% %
% %
% %
% %
% %
% %
% %
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
% % \subsection{Verification and Validation}
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Verification and Validation is Essential!}
%
% \begin{itemize}
%
% \item V\&V most important for providing confidence in results
%
% \vspace*{3mm}
% \item Numerical modeling program(s) should not be used without extensive/full V\&V
%
% \vspace*{3mm}
% \item V\&V of FEM models is also essential
%
%
% \vspace*{3mm}
% \item Real ESSI Simulator has an extensive Verification database, and a
% smaller Validation database
%
% %
% % \item Education is the key to successful use of realistic nonlinear
% % Earthquake Soil Structure Interaction modeling and simulation
% %
% % \item Development of the Real ESSI Simulator system
% % \href{http://realessi.info}{http://realessi.info}
%
% % \item Importance of using proper models correctly (verification, validation, level of sophistication)
% %
%
%
%
%
% \end{itemize}
%
%
%
% \end{frame}
% %
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Verification and Validation}
% \begin{itemize}
%
%
%
%
%
% \item {{ Verification:} provides evidence that the model is solved
% correctly.} Mathematics issue. Well developed
% for the Real ESSI Simulator.
%
% \vspace*{4mm}
% \item {{ Validation:} provides evidence that the correct model is
% solved.} Physics issue. Work in progress, UCSD, TJU labs
%
% \vspace*{4mm}
% \item { Prediction:} use of computational model
% to foretell the state of a physical system under consideration under
% conditions for which the computational model has not been validated.
%
% % \item { Prediction under Uncertainty}: use of computational model
% % to predict the state of SSI system under
% % conditions for which the computational model has not been validated.
% %
% %
% %
% % \vspace*{1mm}
% % \item Modeling and parametric uncertainties are always present, need to be
% % addressed
% %
% % %\vspace*{1mm}
% % % \item Predictive capabilities with {low Kolmogorov Complexity}
% % %
% %
% % \vspace*{1mm}
% % \item Goal: Predict and Inform and rather than (force) Fit
% %
% %
%
%
%
% \end{itemize}
% \end{frame}
%
%
%
%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \begin{frame}
% % \frametitle{V \& V Motivation}
% %
% %
% %
% % \begin{itemize}
% %
% %
% % %\vspace*{0.5cm}
% % \item How much can (should) we trust model implementations (verification)?
% %
% % %\vspace*{0.5cm}
% % \item How much can (should) we trust numerical simulations (validation)?
% %
% % %\vspace*{0.5cm}
% % \item How good are our numerical predictions?
% %
% % %\vspace*{0.5cm}
% % \item Can simulation tools be used for improving safety and economy?
% %
% % % \vspace*{2.0truecm}
% % \item V \& V procedures are the primary means of assessing accuracy in
% % modeling and computational simulations
% %
% % % \vspace*{1.0truecm}
% % \item V \& V procedures are the tools with which we build confidence and
% % credibility in modeling and computational simulations
% %
% %
% %
% %
% % %\vspace*{2.5cm}
% % %\item How do we use experimental simulations to improve models
% %
% %
% %
% %
% % \end{itemize}
% %
% %
% %
% %
% % \end{frame}
% %
% %
% %
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \begin{frame}
% % \frametitle{Role of Verification and Validation}
% %
% %
% %
% % \begin{figure}[!h]
% % \begin{center}
% % \hspace*{2cm}
% % %{\includegraphics[width=5.0cm]{/home/jeremic/tex/works/Conferences/2012/ASME_V_and_V_symposium/presentetation/RoleVV_NEW_knowledge.pdf}}
% % {\includegraphics[width=5.0cm]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Present06_figs/RoleVV_NEW_knowledge.pdf}}
% % %{\includegraphics[width=6.5cm]{/home/jeremic/tex/works/Conferences/2011/USNCCM11_Minneapolis/Coupled/Present/VandV_ODEN.jpg}}
% % {\includegraphics[width=6.5cm]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Present06_figs/VandV_ODEN.jpg}}
% % \hspace*{2cm}
% % \end{center}
% % \end{figure}
% %
% % {Oberkampf et al. \hspace*{4cm} Oden et al.}
% % %
% % %\item Models available (some now, some later)
% % %\vspace*{2.0cm}
% %
% % %
% % %\item Models available (some now, some later)
% % %\vspace*{2.0cm}
% % \end{frame}
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%
% \begin{frame}
% \frametitle{Verification and Validation}
%
%
%
% %
%
%
% \begin{figure}[!h]
% % \vspace*{0.5cm}
% \hspace*{8mm}
% %\begin{center}
% %{\includegraphics[width=11cm]{/home/jeremic/tex/works/Conferences/2005/OpenSeesWorkshopAugust/DeveloperSymposium/VerifValidFund01.pdf}}
% {\includegraphics[width=12cm]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Present06_figs/VerifValidFund01.pdf}}
% %\end{center}
% \end{figure}
%
%
% \end{frame}
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \begin{frame}
% % \frametitle{Important Sources}
% %
% %
% %
% % \begin{itemize}
% %
% %
% %
% %
% % %\item {\it Short Course on Verification and Validation in
% % %Computational Mechanics}, by {\sc Dr. William Oberkampf}, Sandia National Laboratories
% % %July 27th, 2003, Albuquerque, New Mexico.
% %
% % \item
% % {\sc W.~L. Oberkampf, T.~G. Trucano, and C. Hirsch.}
% % Verification, validation and predictive capability in computational
% % engineering and physics.
% % In {\em Proceedings of the Foundations for Verification and
% % Validation on the 21st Century Workshop}, pages 174, Laurel, Maryland,
% % October 2223 2002. Johns Hopkins University / Applied Physics Laboratory.
% % %{\href{http://sokocalo.engr.ucdavis.edu/~jeremic/UsefulReadings/OberkampfTrucanoHirsch.pdf}
% % %{PDF available here}}
% %
% %
% % \item
% % {\sc P.~J. Roache.}
% % {\em Verification and Validation in Computational Science and
% % Engineering}.
% % Hermosa publishers, 1998.
% % ISBN 0913478083.
% % %
% %
% %
% % % \item Material from {\it Verification and Validation in Computational Mechanics}
% % % web site \texttt{http://www.usacm.org/vnvcsm/} at the USACM.
% %
% % \item William L. Oberkampf and Christopher J. Roy. Verification and Validation
% % in Scientific Computing. Cambridge University Press, 2010.
% %
% %
% %
% % \end{itemize}
% %
% %
% %
% %
% % \end{frame}
% %
% %
% % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % %
% % %
% % %
% %
% %
% %
% %
% %
% %
% %
% %
% %
% %
% %
% %
% %
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \begin{frame}
% % \frametitle{Verification}
% %
% % \begin{itemize}
% %
% % \item Source code management
% % \item Source code verification
% %
% % \item Constitutive integration
% % \item Static and dynamic behavior of single phase solids
% % \item Static and dynamic behavior of fully and partially saturated, fully
% % coupled, porous solidpore fluid problems
% % \item Static and dynamic behavior of structural elements
% % \item Static and dynamic behavior of special elements
% % (contactsinterface/gapfrictional/drysaturated, isolators/dissipators)
% % \item Static and dynamic FEM solution advancement
% % \item Seismic wave propagation problems
% % \item FEM Model verification, hierarchy of models
% % %\item Static and Dynamic Behavior of SoilStructureInteraction
% %
% % \end{itemize}
% %
% % \end{frame}
% %
% %
% %
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \begin{frame}
% % \frametitle{Constitutive Integration Verification}
% %
% %
% % \begin{itemize}
% %
% % \item Asymptotic regime of convergence
% %
% % \item Richardson extrapolation
% %
% % \item Grid convergence index
% %
% % \end{itemize}
% %
% %
% % \begin{center}
% % \begin{figure}[!htbp]
% % \vspace*{3mm}
% % \includegraphics[width=5cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figurefiles/asymptotic_regime.pdf}
% % \includegraphics[width=3.5truecm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figurefiles/asymptotic_nonassociate.pdf}
% % \\
% % \includegraphics[width=3.0truecm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figurefiles/errormap/DPAF_implicitErrMap.pdf}
% % \includegraphics[width=3cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figurefiles/GCI/NonAssociate.pdf}
% % \vspace*{3mm}
% % \end{figure}
% % \end{center}
% %
% %
% %
% %
% %
% %
% %
% %
% % \end{frame}
% %
% %
% %
% %
% %
% %
% %
% % %
% % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % % \begin{frame}
% % % \frametitle{Seismic Energy Input and Dissipation}
% % %
% % % \begin{itemize}
% % %
% % % \vspace*{1mm}
% % % \item[] Seismic energy input, through a closed boundary (DRM)
% % %
% % %
% % % \vspace*{4mm}
% % % \item[] Mechanical dissipation outside SSI domain:
% % % \begin{itemize}
% % % \item[] Reflected wave radiation
% % % \item[] SSI system oscillation radiation
% % % \end{itemize}
% % % %\vspace*{1mm}
% % % \item[] Mechanical dissipation/conversion inside SSI domain:
% % % \begin{itemize}
% % % \item[] Inelasticity of soil and contact zone
% % % \item[] Inelasticity/damage of structure and foundation
% % % \item[] Viscous coupling of fluids and soils and structure
% % % % % \item[] potential and kinetic energy
% % % % \item[] potential $\leftarrow \! \! \! \! \! \! \rightarrow$ kinetic energy
% % % \end{itemize}
% % %
% % %
% % %
% % % %\vspace*{1mm}
% % % % \item[] Numerical energy dissipation (numerical damping/production and period errors)
% % % % \item[] Numerical energy dissipation (damping/production)
% % % \item[] Numerical energy dissipation/production
% % %
% % %
% % % \end{itemize}
% % %
% % % %
% % % \end{frame}
% % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % %
% %
% %
% %
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \begin{frame}
% %
% % \frametitle{Energy Dissipation Verification: \\ Plastic Work $\neq$ Plastic Dissipation}
% %
% % %\begin{itemize}
% % %\item Negative incremental energy dissipation
% % %\item Plastic work is NOT plastic dissipation
% % %\end{itemize}
% %
% % \begin{figure}[!H]
% % \begin{center}
% % \includegraphics[height=4.5cm]{/home/jeremic/tex/works/Thesis/HanYang/Files_06June2017/DOE_Annual_2017/Figures/Negative_Dissipation_Problem.png}
% % \end{center}
% % \end{figure}
% %
% % \vspace*{2mm}
% % \begin{itemize}
% %
% % \item[] Direct violation of the second law of thermodynamics
% %
% % %\vspace*{3mm}
% % \item[] 600 papers since 1990 (!?!) repeat this error
% %
% % % \vspace*{3mm}
% % % \item[] Important form of energy missing: {Plastic Free Energy}
% % %
% % % \vspace*{3mm}
% % % \item[] First described by Taylor and Quinney in 1934!
% % %
% % % \vspace*{3mm}
% % % \item[] Plastic Work vs. {Plastic Energy Dissipation}
% %
% %
% % %\vspace*{4mm}
% % %\item[] However it seems to be forgotten
% %
% % \end{itemize}
% %
% %
% % \end{frame}
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %
% %
% %
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \begin{frame}
% % \frametitle{Dynamic Time Stepping Verification}
% %
% % %\footnotesize
% % Based on the amplification matrix ${\bf A}$, to calculate the analytical solution of
% % damping ratios and period shift.
% %
% % Example: HilberHughesTaylor $\alpha=0.1$
% %
% % \begin{figure}
% % \hspace*{5mm}
% % \includegraphics[width=6cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figurefiles/dynamic/hht/HHTalpha01xi.jpg}
% % \includegraphics[width=6cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figurefiles/dynamic/hht/HHTalpha01periodshift.jpg}
% % \end{figure}
% % % HHT Integration $\alpha=0.2$
% % % \begin{figure}
% % % \includegraphics[width=0.36\textwidth]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figurefiles/dynamic/hht/HHTalpha02xi.jpg}
% % % \includegraphics[width=0.36\textwidth]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figurefiles/dynamic/hht/HHTalpha02periodshift.jpg}
% % % \end{figure}
% %
% %
% % \end{frame}
% %
% %
% %
% %
% %
% %
% %
% %
% %
% % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \begin{frame}
% % \frametitle{Seismic Input Verification, DRM}
% % % \begin{figure}
% % % % \includegraphics[scale=0.35]{Present06_figs/3DModel.eps}
% % % \includegraphics[width=3cm]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Present06_figs/3DModel.eps}
% % % \end{figure}
% %
% % \begin{figure}
% % %\hspace*{5mm}
% % \includegraphics[width=5cm]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Present06_figs/90m_Northridge_EW.eps}
% % \\
% % \includegraphics[width=5cm]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Present06_figs/90m_Northridge_NS.eps}
% % \\
% % \includegraphics[width=5cm]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Present06_figs/90m_Northridge_UD.eps}
% % \end{figure}
% %
% %
% %
% %
% % % local
% % % local% local
% % % local%\vspace*{2mm}
% % % local\begin{center}
% % % local\hspace*{16mm}
% % % local%\movie[label=show3,width=5.6cm,poster,autostart,showcontrols]
% % % local\movie[label=show3,width=61mm,poster,showcontrols]
% % % local {\includegraphics[width=60mm]
% % % local {movie_ff_3d_mp4_icon.jpeg}}
% % % local {/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Model01_ESSI_Response_May2015/movie_ff_3d.mp4}
% % % local%\hspace*{2mm}
% % % local%\hfill
% % % local%\movie[label=show3,width=5.6cm,poster,autostart,showcontrols]
% % % local\movie[label=show3,width=61mm,poster,showcontrols]
% % % local {\includegraphics[width=60mm]{movie_ff_1d_mp4_icon.jpeg}}
% % % local {/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Model01_ESSI_Response_May2015/movie_ff_1d.mp4}
% % % local\hspace*{16mm}
% % % local\end{center}
% % % local% local
% %
% %
% % % online % online
% % % online % online
% % % online % online
% % % online \begin{center}
% % % online \href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/ESSI_VisIt_movies_Jose_19May2015/movie_ff_3d.mp4}
% % % online {\includegraphics[width=50mm]{movie_ff_3d_mp4_icon.jpeg}}
% % % online %
% % % online \href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/ESSI_VisIt_movies_Jose_19May2015/movie_ff_1d.mp4}
% % % online {\includegraphics[width=50mm]{movie_ff_1d_mp4_icon.jpeg}}
% % % online \end{center}
% % % online % online
% % % online % online
% % % online % online
% % % online
% %
% %
% %
% % \end{frame}
% %
% %
% %
% %
% %
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \begin{frame}
% % \frametitle{Verification: ANDES Shell}
% %
% % \begin{center}
% % % \includegraphics[width=0.6\textwidth]{./_Jose_Files/fig1_static_tests.pdf}
% % \hspace*{10mm}
% % \includegraphics[width=5cm]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Present06_figs/fig1_static_tests.pdf}
% % % \includegraphics[width=0.6\textwidth]{./_Jose_Files/fig2_static_tests.pdf}
% % \includegraphics[width=5cm]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Present06_figs/fig2_static_tests.pdf}
% % \hspace*{10mm}
% % \end{center}
% %
% % \begin{flushleft}
% % \begin{tabular}{ll}
% % \hline
% % $N_{\texttt{subd}}$ & $u_z$ \\ \hline
% % 2 & 96.2118 \\
% % 7 & 100.096 \\
% % 101 & 100.002 \\
% % & \\ \hline
% % \end{tabular}
% % \end{flushleft}
% %
% %
% % \vspace*{30mm}
% %
% % \begin{flushright}
% % % \includegraphics[width=0.9\textwidth]{./_Jose_Files/Test_shell_andes_1_free_vibration_EigenMode1.png}
% % %\hspace*{10mm}
% % \includegraphics[width=5cm]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Present06_figs/Test_shell_andes_1_free_vibration_EigenMode1.png}
% % \\
% % \vspace*{8mm}
% % \includegraphics[width=5cm]{/home/jeremic/tex/works/Conferences/2013/NRC_Short_Course_May2013/Present/Present06_figs/Test_shell_andes_2_free_vibration_EigenMode1.png}
% % \hspace*{10mm}
% % \end{flushright}
% %
% % \end{frame}
% %
% %
% %
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %
% % \begin{frame}
% % \frametitle{Verification: Irregular Solids and Poisson's Ratio}
% %
% %
% % % \begin{itemize}
% % % \item Irregular Mesh
% % % \item Mesh Refinement
% % % \end{itemize}
% %
% % \begin{figure}
% % % \hspace*{25mm}
% % % \includegraphics[width=5.5cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figurefiles/beam/beam_8brick.pdf}
% % % % \hspace*{25mm}
% % % % \includegraphics[width=5.5cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figurefiles/beam/beam_8brick_more_2.pdf}
% % \hspace*{25mm}
% % \includegraphics[width=6.5cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figurefiles/beam/beam_brick27_shape1_vertical.pdf}
% % \hspace*{35mm}
% % \includegraphics[width=6.5cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figurefiles/beam/beam_brick27_shape2_vertical.pdf}
% % \hspace*{35mm}
% % \includegraphics[width=6.5cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figurefiles/beam/beam_brick27_shape3_vertical.pdf}
% % \hspace*{25mm}
% % \end{figure}
% %
% %
% % \vspace*{6mm}
% %
% % \begin{figure}
% % \includegraphics[width=8cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figurefiles/beam/irregular_shape.png}
% % \end{figure}
% %
% % % \begin{itemize}
% % % \item Verification Results of High Poisson's ratios
% % % \end{itemize}
% %
% % \vspace*{5mm}
% %
% % \begin{figure}
% % \hspace*{9mm}
% % \includegraphics[width=4.0cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figurefiles/beam/error27brick_beam_different_poisson_ratio_disp_div100.jpeg}
% % \hspace*{3mm}
% % \includegraphics[width=4.0cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figurefiles/beam/high_poisson.png}
% % \end{figure}
% %
% %
% % \end{frame}
% %
% %
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \begin{frame}
% % \frametitle{Verification of Solid Shell/Plate}
% %
% %
% % \begin{figure}
% % \hspace*{9mm}
% % \includegraphics[width=3cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figurefiles/beam/circular_plate1.pdf}
% % \hspace*{5mm}
% % \includegraphics[width=3cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figurefiles/beam/circular_plate3.pdf}
% % \hspace*{5mm}
% % \includegraphics[width=3cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figurefiles/beam/circular_plate4.pdf}
% % \hspace*{5mm}
% % \includegraphics[width=3cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figurefiles/beam/circular_plate6.pdf}
% % \hspace*{9mm}
% % \end{figure}
% %
% % \vspace*{2mm}
% %
% % \begin{itemize}
% % \item Simply supported and clamped ends
% % \item Timoshenko's analytic solutions
% % \end{itemize}
% %
% %
% %
% % \vspace*{3mm}
% %
% % \begin{figure}
% % \hspace*{2mm}
% % \includegraphics[width=3.5cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figurefiles/beam/plate_error_plot.jpeg}
% % \hspace*{3mm}
% % \includegraphics[width=3.5cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figurefiles/beam/plate_error_plot_clamped.jpeg}
% % \end{figure}
% %
% %
% % \end{frame}
% %
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \begin{frame}
% % \frametitle{Verification of Boussinesq Problem}
% %
% %
% % \begin{figure}
% % \hspace*{9mm}
% % \includegraphics[width=5cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figurefiles/boussinesq/point_load_on_half_space.png}
% % \hspace*{3mm}
% % \includegraphics[width=2.5cm]{/home/jeremic/tex/works/Thesis/YuanFeng/Files_18Aug2017/V_V_slides_SMiRT/Figurefiles/boussinesq/model3Dview.JPG}
% % \end{figure}
% %
% %
% % \vspace*{3mm}
% %
% %
% % \begin{figure}
% % \hspace*{9mm}
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% % {/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Energy_Dissipation_Animations/NPP_Plastic_Dissipation.mp4}
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% \includegraphics[width=7cm]{/home/jeremic/tex/works/lecture_notes_SOKOCALO/Figurefiles/_Chapter_Theory_Parallel_Computing_in_Coputational_Geomechanics/tex_works_Thesis_GuanzhouJie_thesis_Verzija_Februar_Images_Speedup01.pdf}
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\section{Uncertain Inelastic Dynamics}
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%\subsection{Stochastic Modeling}
\subsection{Forward Uncertainty Propagation}
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\begin{frame}
\frametitle{Forward Uncertainty Propagation}
\begin{itemize}
\vspace*{3mm}
\item[] Given uncertain material and uncertain loads
\vspace*{3mm}
\item[] Determine uncertain response, $u_i, \dot{u}_i, \ddot{u}_i,
\epsilon_{ij}, \sigma_{ij}$, PDFs/CDFs
\vspace*{3mm}
\item[] Intrusive, analytic development, to circumvent Monte Carlo
inefficiencies
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}
\frametitle{Forward Uncertain Inelasticity}
%
\begin{itemize}
\item[] Incremental elpl constitutive equation
%
\begin{eqnarray}
\nonumber
\Delta \sigma_{ij}
=
% E^{EP}_{ijkl}
E^{EP}_{ijkl} \; \Delta \epsilon_{kl}
=
\left[
E^{el}_{ijkl}

\frac{\displaystyle E^{el}_{ijmn} m_{mn} n_{pq} E^{el}_{pqkl}}
{\displaystyle n_{rs} E^{el}_{rstu} m_{tu}  \xi_* h_*}
\right]
\Delta \epsilon_{kl}
\end{eqnarray}
\vspace*{2mm}
\item[] Dynamic Finite Elements
%
\begin{equation}
{ M} \ddot{ u_i} +
{ C} \dot{ u_i} +
{ K}^{ep} { u_i} =
{ F(t)}
\nonumber
\end{equation}
\vspace*{2mm}
\item[] Material and loads are uncertain
\end{itemize}
\end{frame}
%
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\begin{frame}
\frametitle{Probabilistic ElasticPlastic Response}
\begin{figure}[!hbpt]
\begin{center}
%\includegraphics[width=8cm]{/home/jeremic/tex/works/Papers/2007/ProbabilisticYielding/figures/vonMises_G_and_cu_very_uncertain/Contour_PDFedited.pdf}
\includegraphics[width=8cm]{/home/jeremic/tex/works/Conferences/2012/DOELLNLworkshop2728Feb2012/ProbabilisticYielding_vonMises_G_and_cu_very_uncertain_Contour_PDFedited.pdf}
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\frametitle{{Cam Clay with Random $G$, $M$ and $p_0$}}
\begin{figure}[!hbpt]
\begin{center}
\hspace*{15mm}
\includegraphics[width=6.0cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourLowOCR_RandomG_RandomM_Randomp0m.pdf}
%\hspace*{2mm}
\includegraphics[width=6.0cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourHighOCR_RandomG_RandomMm.pdf}
\hspace*{15mm}
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%  %%%%%%
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\begin{frame}{Time Domain Stochastic Galerkin Method}
\vspace*{2mm}
Dynamic Finite Elements
$
{ M} \ddot{ u_i} +
{ C} \dot{ u_i} +
{ K}^{ep} { u_i} =
{ F(t)}$
\begin{itemize}
\vspace*{2mm}
\item[] Input random field/process{\normalsize{(nonGaussian, heterogeneous/ nonstationary)}}:
Multidimensional Hermite Polynomial Chaos (PC) with {known coefficients}
%\vspace{0.05in}
\vspace*{2mm}
\item[] Output response process: Multidimensional Hermite PC with {unknown coefficients}
% \vspace{0.05in}
\vspace*{2mm}
\item[] Galerkin projection: minimize the error to compute unknown coefficients of response process
% %\vspace{0.05in}
% \vspace*{2mm}
% \item[] Time integration using Newmark's method
% % : Update coefficients following
% % an elasticplastic constitutive law at each time step
\end{itemize}
%\scriptsize
%Note: PC = Polynomial Chaos
\end{frame}
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% \begin{frame}{Discretization of Input Random Process/Field $\beta(x,\theta)$}
% \begin{center}
% \includegraphics[scale=0.35]{/home/jeremic/tex/works/Thesis/FangboWang/slides_13Mar2019/Fangbo_slides/figs/PC_KL_explanation.PNG} \\
% \end{center}
%
%
% \footnotesize{Note: $\beta(x,\theta)$ is an input random process with any
% marginal distribution, \\ \hspace{21mm} with any covariance structure;} \\
% \footnotesize{\hspace{8mm} $\gamma(x,\theta)$ is a zeromean unitvariance Gaussian random process.} \\
%
% \end{frame}
%
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\begin{frame}{Polynomial Chaos Representation}
%\scriptsize{
Material random field: \\
%\vspace{0.3cm}
%\begin{equation*}
$D(x, \theta)= \sum_{i=1}^{P1} a_i(x) \Psi_i(\left\{\xi_r(\theta)\right\})$
%\end{equation*}
\vspace{4mm}
Seismic loads/motions random process: \\
%\vspace{0.3cm}
%\begin{equation*}
$f_m(t, \theta)=\sum_{j=1}^{P_2} f_{mj}(t) \Psi_j(\{\xi_k(\theta)\})$
%\end{equation*}
\vspace{4mm}
Displacement response: \\
%\vspace{0.3cm}
%\begin{equation*}
$u_n(t, \theta)=\sum_{k=1}^{P_3} d_{nk}(t) \Psi_k(\{\xi_l(\theta)\})$
%\end{equation*}
\vspace{3mm}
%Acceleration response:
%%\vspace{0.3cm}
%%\begin{equation*}
%$\ddot u_n(t, \theta)=\sum_{k=1}^{P_3} \ddot d_{nk}(t) \Psi_k(\{\xi_l(\theta)\})$
%%\end{equation*}
%\vspace{3mm}
\vspace{5mm}
where $a_i(x), f_{mj}(t)$ are {known PC coefficients}, while $d_{nk}(t)$
are {unknown PC coefficients}.
%}
\end{frame}
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% \subsection{Direct Solution for Probabilistic Stiffness and Stress in 1D}
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\begin{frame}{Direct Probabilistic Constitutive Solution in 1D}
% \begin{itemize}
%
% \vspace{0.5cm}
%
% \item<1> Probabilistic constitutive modeling : \vspace{0.5cm}
\begin{itemize}
\vspace*{4mm}
\item[] Zero elastic region elastoplasticity with stochastic ArmstrongFrederick
kinematic hardening
$ \Delta\sigma =\ H_a \Delta \epsilon  c_r \sigma \Delta \epsilon ;
\hspace{0.5cm}
E_t = {d\sigma}/{d\epsilon} = H_a \pm c_r \sigma $
\vspace*{4mm}
\item[] Uncertain:
init. stiff. $H_a$,
shear strength $H_a/c_r$,
strain $\Delta \epsilon$:
$ H_a = \Sigma h_i \Phi_i; \;\;\;
C_r = \Sigma c_i \Phi_i; \;\;\;
\Delta\epsilon = \Sigma \Delta\epsilon_i \Phi_i $
\vspace*{4mm}
\item[] Resulting stress and stiffness are also uncertain
% 
%  $ \sum_{l=1}^{P_{\sigma}} \Delta\sigma_i \Phi_i = \sum_{i=1}^{P_h} \sum_{k=1}^{P_e}\ h_i \Delta \epsilon_k \Phi_i \Phi_k  \sum_{j=1}^{P_g} \sum_{k=1}^{P_e}\sum_{l=1}^{P_{\sigma}} \ c_i \Delta \epsilon_k \sigma_l \Phi_j \Phi_k \Phi_l$
% 
%  $ \sum_{l=1}^{P_{E_t}} \Delta E_{t_i} \Phi_i = \sum_{i=1}^{P_h} h_i \Phi_i \pm \sum_{i=1}^{P_c} \sum_{l=1}^{P_{\sigma}} \ c_i \sigma_l \Phi_i \Phi_l$
% 
\end{itemize}
% \vspace{0.5cm}
% \vspace{1cm}
%\item<1> Time integration is done via Newmark algorithm
%
% \end{itemize}
%
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% % % % % % % % % % % % % % % %
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\begin{frame}{Direct Probabilistic Stiffness Solution}
\begin{itemize}
\item[] Analytic product for all the components,
$ E^{EP}_{ijkl}
=
\left[
E^{el}_{ijkl}

\frac{\displaystyle E^{el}_{ijmn} m_{mn} n_{pq} E^{el}_{pqkl}}
{\displaystyle n_{rs} E^{el}_{rstu} m_{tu}  \xi_* h_*}
\right]
$
\vspace*{2mm}
\item[] Stiffness: each Polynomial Chaos component is updated incrementally
% at each Gauss Point via stochastic Galerkin projection
\small{$E_{t_1}^{n+1} = \frac{1}{<\Phi_1\Phi_1> }\{\sum_{i=1}^{P_h} \ h_i <\Phi_i \Phi_1> \pm \sum_{j=1}^{P_c} \sum_{l=1}^{P_{\sigma}} \ c_j \sigma_l^{n+1} <\Phi_j \Phi_l \Phi_1>\}$}
\\
. . .
%
%
% $\large{\vdots}$
\\
\small{$E_{t_P}^{n+1} = \frac{1}{<\Phi_1\Phi_P> }\{\sum_{i=1}^{P_h} \ h_i <\Phi_i \Phi_P> \pm \sum_{j=1}^{P_c} \sum_{l=1}^{P_{\sigma}} \ c_j \sigma_l^{n+1} <\Phi_j \Phi_l \Phi_P>\}$}
\vspace*{2mm}
\item[] Total stiffness is :
$ E_{t}^{n+1} = \sum_{l=1}^{P_{E}} E_{t_i}^{n+1} \Phi_i $
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Direct Probabilistic Stress Solution}
\begin{itemize}
\item[] Analytic product, for each stress component,
$ \Delta \sigma_{ij} = E^{EP}_{ijkl} \; \Delta \epsilon_{kl} $
% =
% \left[
% E^{el}_{ijkl}
% 
% \frac{\displaystyle E^{el}_{ijmn} m_{mn} n_{pq} E^{el}_{pqkl}}
% {\displaystyle n_{rs} E^{el}_{rstu} m_{tu}  \xi_* h_*}
% \right]
% \Delta \epsilon_{kl}
%
\vspace*{2mm}
\item[] Incremental stress: each Polynomial Chaos component is updated
incrementally
% via stochastic Galerkin projection
{$\Delta\sigma_1^{n+1} = \frac{1}{<\Phi_1\Phi_1> }\{\sum_{i=1}^{P_h} \sum_{k=1}^{P_e}\ h_i \Delta \epsilon_k^n <\Phi_i \Phi_k \Phi_1> \sum_{j=1}^{P_g} \sum_{k=1}^{P_e}\sum_{l=1}^{P_{\sigma}} \ c_j \Delta \epsilon_k^n \sigma_l^n <\Phi_j \Phi_k \Phi_l \Phi_1>\}$}
\\
. . .
\\
% ${\vdots}$
{$\Delta\sigma_P^{n+1} = \frac{1}{<\Phi_P\Phi_P> }\{\sum_{i=1}^{P_h} \sum_{k=1}^{P_e}\ h_i \Delta \epsilon_k^n <\Phi_i \Phi_k \Phi_P> \sum_{j=1}^{P_g} \sum_{k=1}^{P_e}\sum_{l=1}^{P_{\sigma}} \ c_j \Delta \epsilon_k^n \sigma_l^n <\Phi_j \Phi_k \Phi_l \Phi_P>\}$}
\vspace*{2mm}
\item[] Stress update:
$ \sum_{l=1}^{P_{\sigma}} \sigma_i^{n+1} \Phi_i = \sum_{l=1}^{P_{\sigma}} \sigma_i^{n} \Phi_i + \sum_{l=1}^{P_{\sigma}} \Delta\sigma_i^{n+1} \Phi_i$
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{Probabilistic ElasticPlastic Response}
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% % \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_50mmesh_45degree_Ormsby.mp4}
{\tiny (MP4)}
\end{flushleft}
%
%
% \includegraphics[width = 12cm]{./img/figure_PEP_25.pdf}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section[Formulation]{Stochastic Dynamic Finite Element Formulation}
%\subsection[Time domain stochastic Galerkin method]{Time domain stochastic Galerkin method}
%\frame{\tableofcontents[currentsubsection,sectionstyle=show/shaded]}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Stochastic ElasticPlastic Finite Element Method}
\begin{itemize}
\item[] Material uncertainty expanded into stochastic shape funcs.
%$E(x,t,\theta) = \sum_{i=0}^{P_d} r_i(x,t) * \Phi_i[\{\xi_1, ..., \xi_m\}]$
\vspace*{1mm}
\item[] Loading uncertainty expanded into stochastic shape funcs.
%$f(x,t,\theta) = \sum_{i=0}^{P_f} f_i(x,t) * \zeta_i[\{\xi_{m+1}, ..., \xi_f]$
\vspace*{1mm}
\item[] Displacement expanded into stochastic shape funcs.
%$u(x,t,\theta) = \sum_{i=0}^{P_u} u_i(x,t) * \Psi_i[\{\xi_1, ..., \xi_m, \xi_{m+1}, ..., \xi_f\}]$
%\item
%Stochastic system of equation resulting from Galerkin approach (static example):
%
%\item Time domain integration using Newmark and/or HHT, in probabilistic spaces
%
\vspace*{1mm}
\item[] Jeremi{\'c} et al. 2011
\end{itemize}
\begin{tiny}
\[
%$
\begin{bmatrix}
\sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_0> K^{(k)} & \dots & \sum_{k=0}^{P_d} <\Phi_k \Psi_P \Psi_0> K^{(k)}\\
\sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_1> K^{(k)} & \dots & \sum_{k=0}^{P_d} <\Phi_k \Psi_P \Psi_1> K^{(k)}\\ \\
\vdots & \vdots & \vdots & \vdots\\
\sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_P> K^{(k)} & \dots & \sum_{k=0}^{M} <\Phi_k \Psi_P \Psi_P> K^{(k)}
\end{bmatrix}
\begin{bmatrix}
\Delta u_{10} \\
\vdots \\
\Delta u_{N0}\\
\vdots \\
\Delta u_{1P_u}\\
\vdots \\
\Delta u_{NP_u}
\end{bmatrix}
=
%\]
%\[
\begin{bmatrix}
\sum_{i=0}^{P_f} f_i <\Psi_0\zeta_i> \\
\sum_{i=0}^{P_f} f_i <\Psi_1\zeta_i> \\
\sum_{i=0}^{P_f} f_i <\Psi_2\zeta_i> \\
\vdots \\
\sum_{i=0}^{P_f} f_i <\Psi_{P_u}\zeta_i>\\
\end{bmatrix}
%$
\]
\end{tiny}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{SEPFEM: System Size}
\begin{itemize}
\item[] SEPFEM offers a complete probabilistic solution
\item[] It is NOT based on Monte Carlo approach
\item[] System of equations does grow (!)
\end{itemize}
% \normalsize{Typical number of terms required for a SEPFEM problem} \vspace{1cm}\\
\scalebox{0.7}{
\begin{tabular}{ c c c c}
\# KL terms material & \# KL terms load & PC order displacement& Total \# terms per DoF\\ \hline
4 & 4 & 10 & 43758 \\
4 & 4 & 20 & 3 108 105 \\
4 & 4 & 30 & 48 903 492 \\
6 & 6 & 10 & 646 646 \\
6 & 6 & 20 & 225 792 840 \\
6 & 6 & 30 & 1.1058 $10^{10}$ \\
8 & 8 & 10 & 5 311 735 \\
8 & 8 & 20 & 7.3079 $10^{9}$ \\
8 & 8 & 30 & 9.9149 $10^{11}$\\
... & ... & ... & ...\\
% \hline
\end{tabular}}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{SEPFEM: Example in 1D}
\vspace*{2mm}
\begin{center}
% \hspace*{15mm}
\movie[label=show3,width=9cm,showcontrols]
{\includegraphics[width=9cm]{/home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Latex/img/figure_elastic_900.png}}
% /home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Latex/img/figure_PEP_25.pdf
{/home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Animations/SEPFEM_Animation_Elastic.mp4}
% {SEPFEM_Animation_Elastic.mp4}
\end{center}
%
% \vspace*{2mm}
% \begin{center}
% % \hspace*{15mm}
% \movie[label=show3,width=9cm,poster,autostart,showcontrols]
% {\includegraphics[width=9cm]{/home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Latex/img/figure_elastic_900.png}}
% % /home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Latex/img/figure_PEP_25.pdf
% {/home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Animations/SEPFEM_Animation_Elastic.mp4}
% \end{center}
%
% \includegraphics[width = 12cm]{./img/figure_elastic_900.pdf}
\begin{flushleft}
\vspace*{15mm}
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Probabilistic_Elasto_Plasticity_and_Stochastic_Elastic_Plastic_Finite_Element_Method/SEPFEM_Animation_Elastic.mp4}
% \href{./homo_50mmesh_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_50mmesh_45degree_Ormsby.mp4}
{\tiny (MP4)}
\end{flushleft}
%
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsection{Formulation}
% \subsection[TDNIPSRA Formulation]{Formulation}
% %\subsection{TDNIPSRA Formulation}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%
% \frametitle{Probabilistic Seismic Risk Analysis (PSRA)}
%
% \begin{textblock}{15}(0.8, 3.9)
% \scriptsize
% Uncertain source, path, \\
% \quad site and structure
% \end{textblock}
%
% \begin{textblock}{15}(4.8, 4.2)
% $\Longrightarrow$
% \end{textblock}
%
% \begin{textblock}{15}(5.8, 3.7)
% \scriptsize
% %Acceptably small probability\\
% Probabilities of \\
% engineering demand parameters (EDP) \\
% damage measures (DM), loss, etc
% \end{textblock}
%
% \begin{textblock}{15}(0.4, 11.8)
% \tiny
% \textbf{Uncertain source rupture}
% \end{textblock}
%
% \begin{textblock}{15}(3.55, 4.8)
% \begin{figure}[H]
% \flushleft
% \includegraphics[width=0.7\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/problem_statement.pdf}
% \end{figure}
% \end{textblock}
%
% \begin{textblock}{15}(12.5, 5.5)
% \tiny
% adapted from Taga(1982)
% \end{textblock}
%
% \begin{textblock}{15}(11.7, 10.6)
% \tiny
% \textbf{Uncertain material properties}
% \end{textblock}
%
%
% \begin{textblock}{15}(0.4, 8.0)
% \begin{figure}[H]
% \flushleft
% \includegraphics[width=0.2\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/4231447_Northridge_eq.pdf}
% \end{figure}
% \end{textblock}
%
%
% \begin{textblock}{15}(10.7, 10.2)
% \begin{figure}[H]
% \flushleft
% \includegraphics[width=0.3\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/uncertain_SPT.pdf}
% \end{figure}
% \end{textblock}
%
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%
% %\frametitle{State of the Art Probabilistic Seismic Risk Analysis}
% \frametitle{Probabilistic Seismic Risk Analysis}
%
% \begin{itemize}
%
% % \item Target: safe design, acceptably small failure probability $\lambda (EDP)$
%
% \item[]
% %Powerful tool
% %Allows for
% Objective, quantitative decision making based on exceedance rate
% $\lambda (EDP>z)$
%
% \item[] PSRA: convolution of PSHA and fragility
%
% % \[\lambda(EDP>z) = \int_{IM} \underbrace{\frac{d\lambda(IM)}{dIM}}_\text{PSHA} \underbrace{G(EDPIM)}_{\text{fragility}} dIM \]
%
% \vspace{0.1cm}
%
% \[\lambda(EDP>z) = \int \underbrace{\frac{d\lambda(IM>x)}{dx}}_\text{\textbf{PSHA}} \underbrace{G(EDP>zIM=x)}_{\text{\textbf{fragility analysis}}} dx\]
%
% \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 Geohazard: Liquefaction, slope deformation
% % % % \item[] \small PGA v.s. AI v.s. RMS for liquefaction
% % % \item[] \small AI v.s. PGV v.s. CAV for dam embankment
% % % \end{itemize}
% %
% % \vspace{3mm}
% %
% % % \item Additional effort for new GMPEs
% %
% % % \begin{itemize}
% % % \item[] \small vector hazard: GMPE with covariance of IMs, fragility as function of IMs, rarely used
% % % \end{itemize}
% %
% % % \item[] Miscommunication: seismologists and engineers
% % %
% % % \begin{itemize}
% % % \item[] \small $Sa(T_0)$ not compatible with time domain nonlinear analysis
% % % \end{itemize}
% %
% % \end{itemize}
% % \end{frame}
% %
%
%
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Current State of Art Seismic Risk Analysis (SRA)}
%
%
% \begin{itemize}
% %\vspace{2mm}
%
% \item[] Intensity measure (IM) selected as a proxy for ground motions,
% usually Spectral acceleration $Sa(T_0)$
%
% \vspace{4mm}
% \item[] Ground Motion Prediction Equations (GMPEs) need development, ergodic or site specific
%
% \vspace{4mm}
% \item[] Probabilistic seismic hazard analysis (PSHA)
% % for ground motion $\lambda(Sa>z)$
% % \begin{equation*}
% % \resizebox{0.85\hsize}{!}{%
% % $\lambda(Sa>z) = \sum_{i=1}^{NFL} \underbrace{N_i \int\int f_{mi}(M) f_{ri}(RM)}_\text{seismic source characterization (SSC)} \underbrace{P(Sa>zM, R)}_\text{GMPE} dM dR$}
% % \end{equation*}
%
% \vspace{4mm}
% \item[] Fragility analysis $P(EDP>xIM=z)$, deterministic time domain FEM,
% perhaps using Monte Carlo (MC)
%
% % \begin{itemize}
% %
% % \item[] Records selection: Spectrummatching technique UHS, etc
% %
% % \item[] Incremental dynamic analysis: Monte Carlo
% %
% % \end{itemize}
%
%
%
% \end{itemize}
%
% % \begin{textblock}{15}(2.2, 9.2)
% % \begin{figure}[H]
% % \flushleft
% % % \includegraphics[width=0.38\linewidth]{pic/hazard_curve.png}
% % \includegraphics[width=0.38\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/hazard_curve.pdf}
% % \enspace
% % \includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/design_spectra.png}
% % \end{figure}
% % \end{textblock}
%
%
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsection{Issues in Stateoftheart SRA}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%
% \frametitle{Seismic Risk Analysis Challenges}
%
%
% \begin{itemize}
%
%
%
% \item[] IM serves as the proxy of damaging ground motions
% \vspace{2mm}
% \item[] Does a single IM, e.g., $Sa(T_0)$, represent all uncertainty?
% %\item[] Practically difficult/contentious to choose
%
%
% %\vspace{3mm}
% \vspace{2mm}
% \item[] IMs difficult to choose, Spectral Acc, PGA, PGV...
%
%
% %%\vspace{3mm}
% %\item[] Single IM does not contain all/most uncertainty
%
%
%
%
% \vspace{2mm}
% \item[] Fragility analysis: incremental dynamic analysis (IDA)
% % using Monte Carlo method
%
% \vspace{2mm}
% \item[] Use of Monte Carlo method, accuracy, efficiency...
%
% %\vspace{3mm}
% \vspace{2mm}
% \item[] Monte Carlo, computationally expensive, CyberShake for LA, 20,000
% cases, 100Y runtime, (Maechling et al. 2007)
%
% %
% %
% % \vspace{3mm}
% % \item[] Miscommunication between seismologists and struct/geotech engineers,
% % $Sa(T_0)$ not compatible with nonlinear FEM
%
%
%
%
% \end{itemize}
%
% \end{frame}
%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%
% \frametitle{Time Domain Intrusive PSRA Framework}
%
% %
%
% \begin{itemize}
%
% %\vspace*{2mm}
% \item[] Stochastic ElasticPlastic 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 elasticplastic material
% %stress and stiffness solution using
% %Forward Kolmogorov, FokkerPlanck equation
%
%
% \vspace*{4mm}
% \item[] Uncertain seismic loads/motions
% % using Domain Reduction Method
%
%
% \vspace*{4mm}
% \item[] Results, probability distribution functions for $\sigma_{ij}$,
% $\epsilon_{ij}$, $u_i$...
%
%
%
%
%
%
%
%
% \end{itemize}
%
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
%
%
%
%
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Stochastic ElasticPlastic 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}
%
%
%
%
%
% \end{itemize}
%
%
%
% \end{frame}
%
%
%
%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %
% % \begin{frame}
% % \frametitle{Stochastic ElasticPlastic 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: nonintrusive approach
% %
% % \begin{itemize}
% % \item [] \small Slow convergence rate $1/\sqrt{N}$
% % \item [] \small Hard for stable tail distribution toward lowrisk 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 CPUhours and over 100 years} (Maechling et al. 2007)
% % \end{itemize}
% %
% % \end{itemize}
% % \end{frame}
% %
% %
% %
% %
% %
% %
% %
%
%
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% %\subsection{TDNIPSRA Example}
% \subsection[TDNIPSRA Example]{Example}
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begingroup
\setbeamertemplate{footline}{}
\begin{frame}
%\frametitle{TDNIPSRA Framework}
\frametitle{Application: Seismic Hazard}
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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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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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\begin{textblock}{15}(10.2, 13.2)
{\Large $\leftarrow$}
\end{textblock}
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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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\begin{textblock}{15}(4.6, 13.2)
%$\Leftarrow$
{\Large $\leftarrow$}
\end{textblock}
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\begin{textblock}{15}(0.3, 11.0)
\includegraphics[width=0.29\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/seismic_risk_result_framework.png}
\end{textblock}
\begin{tikzpicture}[remember picture, overlay]
\draw[line width=1pt, draw=black, rounded corners=4pt, fill=gray!20, fill opacity=1]
([xshift=25pt,yshift=55pt]$(pic cs:a) + (0pt,8pt)$) rectangle ([xshift=95pt,yshift=18pt]$(pic cs:b)+(0pt,2pt)$);
\end{tikzpicture}
\begin{textblock}{15}(0.1, 9.3)
\scriptsize
\quad \quad \quad \quad $\lambda(EDP>z)=$
$\quad \sum N_i(M_i, R_i) P(EDP>zM_i, R_i)$
\end{textblock}
\begin{textblock}{15}(1.6, 10.7)
\scriptsize{EDP hazard/risk}
\end{textblock}
\end{frame}
\endgroup
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% \begingroup
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% \setbeamertemplate{footline}{}
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% \begin{frame}
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% \frametitle{Time Domain Intrusive PSRA 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}
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% \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}
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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}
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% \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}
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% \begin{textblock}{15}(6.1, 3.5)
% \scriptsize{Stochastic ground motion}
% \end{textblock}
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% \begin{textblock}{15}(9.9, 6.5)
% $\Rightarrow$
% \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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% \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}
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% % \begin{textblock}{15}(10.2, 13.2)
% % $\Leftarrow$
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% % \begin{textblock}{15}(5.3, 10.75)
% % \includegraphics[width=0.33\linewidth]{pic/probabilsitc_evolution.png}
% % \end{textblock}
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% % \begin{textblock}{15}(5.4, 9.6)
% % \scriptsize{\quad \quad Uncertainty propagation \\
% % \quad \quad \quad \quad stochastic FEM}
% % \end{textblock}
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% % \begin{textblock}{15}(4.6, 13.2)
% % $\Leftarrow$
% % \end{textblock}
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% % \begin{textblock}{15}(0.3, 11.0)
% % \includegraphics[width=0.29\linewidth]{pic/seismic_risk_result_framework.png}
% % \end{textblock}
%
% % \begin{tikzpicture}[remember picture, overlay]
% % \draw[line width=1pt, draw=black, rounded corners=4pt, fill=gray!20, fill opacity=1]
% % ([xshift=25pt,yshift=52pt]$(pic cs:a) + (0pt,8pt)$) rectangle ([xshift=95pt,yshift=18pt]$(pic cs:b)+(0pt,2pt)$);
% % \end{tikzpicture}
%
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% % \begin{textblock}{15}(0.1, 9.3)
% % \scriptsize
% % \quad \quad \quad \quad $\lambda(EDP>z)=$
%
% % $\quad \sum N_i(M_i, R_i) P(EDP>zM_i, R_i)$
% % \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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% \endgroup
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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begingroup
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% \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}
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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}
%
%
% \begin{textblock}{15}(10.2, 13.2)
% $\Leftarrow$
% \end{textblock}
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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 stochastic FEM}
% \end{textblock}
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% % \begin{textblock}{15}(4.6, 13.2)
% % $\Leftarrow$
% % \end{textblock}
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% % \begin{textblock}{15}(0.3, 11.0)
% % \includegraphics[width=0.29\linewidth]{pic/seismic_risk_result_framework.png}
% % \end{textblock}
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% % \begin{tikzpicture}[remember picture, overlay]
% % \draw[line width=1pt, draw=black, rounded corners=4pt, fill=gray!20, fill opacity=1]
% % ([xshift=25pt,yshift=52pt]$(pic cs:a) + (0pt,8pt)$) rectangle ([xshift=95pt,yshift=18pt]$(pic cs:b)+(0pt,2pt)$);
% % \end{tikzpicture}
%
%
% % \begin{textblock}{15}(0.1, 9.3)
% % \scriptsize
% % \quad \quad \quad \quad $\lambda(EDP>z)=$
%
% % $\quad \sum N_i(M_i, R_i) P(EDP>zM_i, R_i)$
% % \end{textblock}
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% % \begin{textblock}{15}(1.6, 10.7)
% % \scriptsize{EDP hazard/risk}
% % \end{textblock}
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% \end{frame}
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% \endgroup
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% \begingroup
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% \setbeamertemplate{footline}{}
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% \begin{frame}
%
% \frametitle{Time Domain Intrusive PSRA Framework}
%
%
% \begin{textblock}{15}(0, 4.0)
% \includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/UCERF3.pdf}
% \end{textblock}
%
% \begin{textblock}{15}(0.3, 3.5)
% \scriptsize{Seismic source characterization}
% \end{textblock}
%
% \begin{textblock}{15}(2.9, 5.2)
% \tiny{UCERF3 (2014)}
% \end{textblock}
%
% \begin{textblock}{15}(5.1, 6.5)
% $\Rightarrow$
% \end{textblock}
%
%
% \begin{textblock}{15}(5.8, 3.9)
% \vspace*{1mm}
% \includegraphics[width=0.27\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/SMSIM.pdf}
% \end{textblock}
%
%
% \begin{textblock}{15}(7.1, 6.2)
% \scalebox{.9}{\tiny{Fourier spectra}}
% \\
% \vspace*{0.2cm}
% \scalebox{.9}{\tiny{\hspace{0.14cm} Boore(2003)}}
% \end{textblock}
%
% \begin{textblock}{15}(6.1, 3.5)
% \scriptsize{Stochastic ground motion}
% \end{textblock}
%
% \begin{textblock}{15}(9.9, 6.5)
% $\Rightarrow$
% \end{textblock}
%
% \begin{textblock}{15}(10.5, 4.2)
% % \includegraphics[width=0.35\linewidth]{pic/KL_exact_dis_correlation_from_dis.pdf}
% \includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Acc_realization_200.pdf}
% \end{textblock}
%
% \begin{textblock}{15}(11.1, 9.6)
% \scriptsize{Uncertainty characterization \\
% \hspace{0.1cm} Hermite polynomial chaos}
% \end{textblock}
%
%
% \begin{textblock}{15}(10.2, 13.2)
% $\Leftarrow$
% \end{textblock}
%
% \begin{textblock}{15}(11, 11.2)
% \includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/structural_uncertainty.pdf}
% \end{textblock}
%
% \begin{textblock}{15}(5.3, 10.75)
% \includegraphics[width=0.33\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/probabilsitc_evolution.png}
% \end{textblock}
%
% \begin{textblock}{15}(5.4, 9.6)
% \scriptsize{\quad \quad Uncertainty propagation \\
% \quad \quad \quad \quad stochastic FEM}
% \end{textblock}
%
% \begin{textblock}{15}(4.6, 13.2)
% $\Leftarrow$
% \end{textblock}
%
% \begin{textblock}{15}(0.3, 11.0)
% \includegraphics[width=0.29\linewidth]{/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 $\lambda(EDP>z)=$
%
% $\quad \sum N_i(M_i, R_i) P(EDP>zM_i, R_i)$
% \end{textblock}
%
% \begin{textblock}{15}(1.6, 10.7)
% \scriptsize{EDP hazard/risk}
% \end{textblock}
%
% \end{frame}
%
% \endgroup
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
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\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 Interfrequency 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 Interfrequency correlation GMPE by \textit{Stafford(2017)} and \textbf{\textit{Bayless \& Abrahamson (2018)}}.
%
% \end{itemize}
%\vspace{0.05cm}
%\item[] \normalsize Stochastic FPS by phase derivative (Boore,2005)
%(Logistic phase derivative model by {\textit{Baglio \& Abrahamson (2017)}})
\vspace*{1mm}
\item[] \normalsize Stochastic FPS by phase derivative (Boore,2005)
(Logistic phase derivative model by {\textit{Baglio \& Abrahamson (2017)}})
\vspace*{1mm}
\item[] \normalsize Near future change from \textbf{ $\boldsymbol{Sa(T_0)}$} to \textbf{FAS} and \textbf{FPS}
%next five years
% as envisioned by Abrahamson (2018)
\end{itemize}
\end{frame}
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% %\subsection{Illustrative Example}
% %
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
%\frametitle{TDNIPSRA Example Object}
\frametitle{Example Object}
\begin{textblock}{15}(0.5, 4.0)
\includegraphics[width=0.47\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/faults_configuration_new.pdf}
\end{textblock}
\begin{textblock}{15}(7.5, 3.4)
\includegraphics[width=0.55\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/SSC_legend.pdf}
\end{textblock}
\begin{textblock}{15}(0.8, 11.6)
\scriptsize
\begin{itemize}
\item Fault 1: San Gregorio fault
\item Fault 2: Calaveras fault
\item Uncertainty: Segmentation, \\ slip rate, rupture geometry, etc.
\end{itemize}
\end{textblock}
\begin{textblock}{15}(8.5, 11.6)
\scriptsize
\begin{itemize}
\item 371 total seismic scenarios
\item $M \ 5 \sim 5.5$ and $6.5 \sim 7.0$
\item $R_{jb} \ 20km \sim 40km$
\end{itemize}
\end{textblock}
\end{frame}
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\begin{frame}
\frametitle{Stochastic Ground Motion Modeling}
\begin{textblock}{15}(1.0, 3.7)
\small Realizations of simulated uncertain motions for scenario $M=7$, $R=15km$:
\end{textblock}
\begin{textblock}{15}(0.5, 4.0)
\includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Acc_time_series100.pdf}
\end{textblock}
\begin{textblock}{15}(5.5, 4.0)
\includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Acc_time_series343.pdf} \enspace
\end{textblock}
\begin{textblock}{15}(10.5, 4.0)
\includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Acc_time_series439.pdf}
\end{textblock}
\begin{textblock}{15}(1.0, 9.2)
\small Verification with GMPE:
\end{textblock}
\begin{textblock}{15}(0.3, 9.5)
\includegraphics[width=0.36\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/SA_GMPE_verification_std_08_no_smooth.pdf}
\end{textblock}
\begin{textblock}{15}(5.6, 9.5)
\includegraphics[width=0.36\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Goodness_fit_std_08_no_smooth.pdf}
\end{textblock}
\begin{textblock}{15}(10.8, 9.5)
\includegraphics[width=0.36\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/Standard_deviation_std_08_no_smooth_new.pdf}
\end{textblock}
% \begin{textblock}{15}(0.5, 11.0)
% \begin{itemize}
% \item $\Delta \sigma= 84bar$, $\kappa=0.03s$ with total $\sigma=0.8ln$.
% \item Simulated median is not biased.
% \item Consistent total uncertainties with GMPE.
% \end{itemize}
% \end{textblock}
\end{frame}
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\begin{frame}
\frametitle{Stochastic Ground Motion Characterization}
{\begin{textblock}{15}(0.1, 3.62)
\scriptsize
\includegraphics[width=0.3\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/KL_mean_acc_from_acc.pdf}
\quad \quad \quad Acc. marginal mean
\end{textblock}
\begin{textblock}{15}(3.7, 3.62)
\scriptsize
\includegraphics[width=0.3\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/KL_var_acc_from_acc.pdf}
\quad \quad \quad Acc. marginal S.D.
\end{textblock}
\begin{textblock}{15}(7.6, 3.8)
\scriptsize
\includegraphics[width=0.3\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/KL_exact_acc_correlation_from_acc.pdf}
\quad \quad \quad Acc. realization Cov.
\end{textblock}
\begin{textblock}{15}(11.8, 3.9)
\scriptsize
\includegraphics[width=0.3\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/KL_simulated_acc_correlation_from_acc.pdf}
\quad \quad Acc. synthesized Cov.
\end{textblock}}
\begin{textblock}{15}(0.1, 9.3)
\scriptsize
\includegraphics[width=0.31\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/KL_mean_dis_from_dis.pdf}
\end{textblock}
\begin{textblock}{15}(0.9, 13.75)
\scriptsize
Dis. marginal mean
\end{textblock}
\begin{textblock}{15}(4.2, 9.4)
\scriptsize
\includegraphics[width=0.3\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/KL_var_dis_from_dis.pdf}
\end{textblock}
\begin{textblock}{15}(5.1, 13.75)
\scriptsize
Dis. marginal S.D.
\end{textblock}
\begin{textblock}{15}(8.2, 9.5)
\scriptsize
\includegraphics[width=0.27\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/KL_exact_dis_correlation_from_dis.pdf}
\quad \quad Dis. realization Cov.
\end{textblock}
\begin{textblock}{15}(12.2, 9.6)
\scriptsize
\includegraphics[width=0.27\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/KL_simulated_dis_correlation_from_dis.pdf}
\quad Dis. synthesized Cov.
\end{textblock}
\end{frame}
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\begin{frame}
\frametitle{Stochastic Material Modeling}
%Uncertain 1D shear response
\begin{figure}[!htbp]
\centering
%\subfloat[Uncertain $H_a$]{
%\hspace{0.8cm}
%\includegraphics[width=0.53\textwidth]{/home/jeremic/tex/works/Papers/2019/Hexiang/1D_risk/version6/Figures/constitutive_relation_uncertainHa_certainCr_MC_verification.pdf}}
%\subfloat[Uncertain $H_a$ and $C_r$]{
%\hspace{0.2cm}
%\includegraphics[width=0.53\textwidth]{/home/jeremic/tex/works/Papers/2019/Hexiang/1D_risk/version6/Figures/constitutive_relation_uncertainHa_uncertainCr_MC_verification.pdf}}
%\vspace{2mm}
%\caption{\label{figure_probabilisitc_constitutive_relation} Intrusive probabilistic modeling of ArmstrongFrederick 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/ShearFrame8levels.jpg}}
\subfloat[Interstory response]{
\hspace{10mm}
\includegraphics[width=6cm]{/home/jeremic/tex/works/Papers/2019/1D_risk/version6/Figures/constitutive_relation_uncertainHa_uncertainCr_MC_verification.pdf}}
%\vspace{2mm}
%\caption{\label{figure_probabilisitc_constitutive_relation} Intrusive probabilistic modeling of ArmstrongFrederick 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 \\
ElPl 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 interstory drift ratio (MIDR)
\end{textblock}
\end{frame}
\endgroup
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\begingroup
\setbeamertemplate{footline}{}
\begin{frame}
\frametitle{Seismic Risk Analysis}
\begin{textblock}{15}(1.9,3.8)
\scriptsize
\includegraphics[width=0.42\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/PFA_distribution.pdf}
\includegraphics[width=0.42\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/Risk_PFA.pdf}
\end{textblock}
\begin{textblock}{15}(1.9,9.3)
\scriptsize
\includegraphics[width=0.41\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/2D_EDP_PDF_1e5.pdf}
\end{textblock}
\begin{textblock}{15}(8.7,9.4)
\scriptsize
\includegraphics[width=0.40\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/2D_EDP_PDF_downview_1e5.pdf}
\end{textblock}
\begin{textblock}{15}(0.8, 3.8)
\scriptsize Engineering demand parameter (EDP): Peak floor acceleration (PFA)
\end{textblock}
\end{frame}
\endgroup
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\begin{frame}
\frametitle{Seismic Risk Analysis}
%\vspace{0.5cm}
\vspace*{2mm}
\begin{itemize}
% \item[] \small Damage measure (DM) defined on multiple EDPs:
% \item[] \small Damage measure (DM) defined on single EDP:
\item[] Damage measure defined on single EDP:
\vspace*{3mm}
%\begin{textblock}{15}(0.7,8.9)
\begin{table}[!htbp]
\small
\resizebox{0.98\hsize}{!}{
\begin{tabular}{ccccccc}
%\hline
\textbf{DM} & MIDR\textgreater{}0.5\% & \textbf{MIDR\textgreater{}1\%} & MIDR\textgreater{}2\% & PFA\textgreater{}0.5${\rm m/s^2}$ & \textbf{PFA\textgreater{}1\boldsymbol{${\rm m/s^2}$}} & PFA\textgreater{}1.5${\rm m/s^2}$ \\
\hline
\textbf{Risk [/yr]} & 6.66$\times 10^{3}$ & \textbf{3.83\boldsymbol{$\times 10^{3}$}} & 9.97$\times 10^{5}$ & 6.65$\times 10^{3}$ & \textbf{1.92 \boldsymbol{$\times 10^{3}$}} & 9.45$\times 10^{5}$ \\
%\hline
\end{tabular}}
\end{table}
%\end{textblock}
\vspace{4mm}
\item[] Damage measure (DM) defined on multiple EDPs:
% \vspace{2mm}
{\scriptsize $DM: \{\text{MIDR}>1\%\, \cup \,\text{PFA}>1{\rm m/s^2} \}$, seismic risk is \boldsymbol{$4.2 \times 10^{3}/yr$} }
\vspace{1mm}
{\scriptsize $DM: \{\text{MIDR}>1\%\, \cap \,\text{PFA}>1{\rm m/s^2} \}$, seismic risk is \boldsymbol{$1.71 \times 10^{3}/yr$}}
\vspace{3mm}
%\vspace{20mm}
\vspace{4mm}
\item[] \small Seismic risk for DM defined on multiple EDPs can be quite
different from that defined on single EDP
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
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% \begin{frame}
%
% \frametitle{Forward Uncertain Inelasticity}
%
%
%
% \begin{itemize}
%
%
%
% \item[] Incremental elpl 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{{Cam Clay with Random $G$, $M$ and $p_0$}}
%
%
% \begin{figure}[!hbpt]
% \begin{center}
% \hspace*{15mm}
% \includegraphics[width=6.0cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourLowOCR_RandomG_RandomM_Randomp0m.pdf}
% %\hspace*{2mm}
% \includegraphics[width=6.0cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourHighOCR_RandomG_RandomMm.pdf}
% \hspace*{15mm}
% \end{center}
% \end{figure}
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% \end{frame}
% %  %%%%%%
%
%
%
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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% \begin{frame}{Time Domain Stochastic Galerkin Method}
%
%
% \vspace*{2mm}
% Dynamic Finite Elements
% $
% { M} \ddot{ u_i} +
% { C} \dot{ u_i} +
% { K}^{ep} { u_i} =
% { F(t)}$
%
%
% \begin{itemize}
% \vspace*{2mm}
% \item[] Input random field/process{\normalsize{(nonGaussian, heterogeneous/ nonstationary)}}:
% Multidimensional Hermite Polynomial Chaos (PC) with {known coefficients}
% %\vspace{0.05in}
% \vspace*{2mm}
% \item[] Output response process: Multidimensional Hermite PC with {unknown coefficients}
% % \vspace{0.05in}
% \vspace*{2mm}
% \item[] Galerkin projection: minimize the error to compute unknown coefficients of response process
% % %\vspace{0.05in}
% % \vspace*{2mm}
% % \item[] Time integration using Newmark's method
% % % : Update coefficients following
% % % an elasticplastic constitutive law at each time step
%
% \end{itemize}
%
% %\scriptsize
% %Note: PC = Polynomial Chaos
%
% \end{frame}
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %
% % \begin{frame}{Discretization of Input Random Process/Field $\beta(x,\theta)$}
% % \begin{center}
% % \includegraphics[scale=0.35]{/home/jeremic/tex/works/Thesis/FangboWang/slides_13Mar2019/Fangbo_slides/figs/PC_KL_explanation.PNG} \\
% % \end{center}
% %
% %
% % \footnotesize{Note: $\beta(x,\theta)$ is an input random process with any
% % marginal distribution, \\ \hspace{21mm} with any covariance structure;} \\
% % \footnotesize{\hspace{8mm} $\gamma(x,\theta)$ is a zeromean unitvariance Gaussian random process.} \\
% %
% % \end{frame}
% %
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}{Polynomial Chaos Representation}
%
% %\scriptsize{
% Material random field: \\
% %\vspace{0.3cm}
% %\begin{equation*}
% $D(x, \theta)= \sum_{i=1}^{P1} a_i(x) \Psi_i(\left\{\xi_r(\theta)\right\})$
% %\end{equation*}
%
%
% \vspace{4mm}
%
% Seismic loads/motions random process: \\
% %\vspace{0.3cm}
% %\begin{equation*}
% $f_m(t, \theta)=\sum_{j=1}^{P_2} f_{mj}(t) \Psi_j(\{\xi_k(\theta)\})$
% %\end{equation*}
%
% \vspace{4mm}
%
% Displacement response: \\
% %\vspace{0.3cm}
% %\begin{equation*}
% $u_n(t, \theta)=\sum_{k=1}^{P_3} d_{nk}(t) \Psi_k(\{\xi_l(\theta)\})$
% %\end{equation*}
%
% \vspace{3mm}
%
% %Acceleration response:
% %%\vspace{0.3cm}
% %%\begin{equation*}
% %$\ddot u_n(t, \theta)=\sum_{k=1}^{P_3} \ddot d_{nk}(t) \Psi_k(\{\xi_l(\theta)\})$
% %%\end{equation*}
%
% %\vspace{3mm}
% \vspace{5mm}
%
% where $a_i(x), f_{mj}(t)$ are {known PC coefficients}, while $d_{nk}(t)$
% are {unknown PC coefficients}.
% %}
%
% \end{frame}
%
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% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% % \subsection{Direct Solution for Probabilistic Stiffness and Stress in 1D}
% %
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%% BEGGINING PEP %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
% \begin{frame}{Direct Probabilistic Constitutive Solution in 1D}
%
%
% % \begin{itemize}
% %
% % \vspace{0.5cm}
% %
% % \item<1> Probabilistic constitutive modeling : \vspace{0.5cm}
%
% \begin{itemize}
%
%
% \vspace*{4mm}
% \item[] Zero elastic region elastoplasticity with stochastic ArmstrongFrederick
% kinematic hardening
%
% $ \Delta\sigma =\ H_a \Delta \epsilon  c_r \sigma \Delta \epsilon ;
% \hspace{0.5cm}
% E_t = {d\sigma}/{d\epsilon} = H_a \pm c_r \sigma $
%
% \vspace*{4mm}
% \item[] Uncertain:
% init. stiff. $H_a$,
% shear strength $H_a/c_r$,
% strain $\Delta \epsilon$:
%
% $ H_a = \Sigma h_i \Phi_i; \;\;\;
% C_r = \Sigma c_i \Phi_i; \;\;\;
% \Delta\epsilon = \Sigma \Delta\epsilon_i \Phi_i $
%
%
%
% \vspace*{4mm}
% \item[] Resulting stress and stiffness are also uncertain
%
% % 
% %  $ \sum_{l=1}^{P_{\sigma}} \Delta\sigma_i \Phi_i = \sum_{i=1}^{P_h} \sum_{k=1}^{P_e}\ h_i \Delta \epsilon_k \Phi_i \Phi_k  \sum_{j=1}^{P_g} \sum_{k=1}^{P_e}\sum_{l=1}^{P_{\sigma}} \ c_i \Delta \epsilon_k \sigma_l \Phi_j \Phi_k \Phi_l$
% % 
% %  $ \sum_{l=1}^{P_{E_t}} \Delta E_{t_i} \Phi_i = \sum_{i=1}^{P_h} h_i \Phi_i \pm \sum_{i=1}^{P_c} \sum_{l=1}^{P_{\sigma}} \ c_i \sigma_l \Phi_i \Phi_l$
% % 
%
%
% \end{itemize}
%
%
% % \vspace{0.5cm}
%
%
%
% % \vspace{1cm}
%
% %\item<1> Time integration is done via Newmark algorithm
%
% %
% % \end{itemize}
% %
% \end{frame}
%
%
% % % % % % % % % % % % % % % % %
%
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}{Direct Probabilistic Stiffness Solution}
%
% \begin{itemize}
%
%
% \item[] Analytic product for all the components,
%
% $ E^{EP}_{ijkl}
% =
% \left[
% E^{el}_{ijkl}
% 
% \frac{\displaystyle E^{el}_{ijmn} m_{mn} n_{pq} E^{el}_{pqkl}}
% {\displaystyle n_{rs} E^{el}_{rstu} m_{tu}  \xi_* h_*}
% \right]
% $
%
%
%
%
% \vspace*{2mm}
% \item[] Stiffness: each Polynomial Chaos component is updated incrementally
% % at each Gauss Point via stochastic Galerkin projection
%
%
%
% \small{$E_{t_1}^{n+1} = \frac{1}{<\Phi_1\Phi_1> }\{\sum_{i=1}^{P_h} \ h_i <\Phi_i \Phi_1> \pm \sum_{j=1}^{P_c} \sum_{l=1}^{P_{\sigma}} \ c_j \sigma_l^{n+1} <\Phi_j \Phi_l \Phi_1>\}$}
% \\
% . . .
% %
% %
% % $\large{\vdots}$
% \\
% \small{$E_{t_P}^{n+1} = \frac{1}{<\Phi_1\Phi_P> }\{\sum_{i=1}^{P_h} \ h_i <\Phi_i \Phi_P> \pm \sum_{j=1}^{P_c} \sum_{l=1}^{P_{\sigma}} \ c_j \sigma_l^{n+1} <\Phi_j \Phi_l \Phi_P>\}$}
%
%
% \vspace*{2mm}
% \item[] Total stiffness is :
%
% $ E_{t}^{n+1} = \sum_{l=1}^{P_{E}} E_{t_i}^{n+1} \Phi_i $
%
%
%
%
% \end{itemize}
%
%
% \end{frame}
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}{Direct Probabilistic Stress Solution}
%
% \begin{itemize}
%
%
%
% \item[] Analytic product, for each stress component,
%
% $ \Delta \sigma_{ij} = E^{EP}_{ijkl} \; \Delta \epsilon_{kl} $
% % =
% % \left[
% % E^{el}_{ijkl}
% % 
% % \frac{\displaystyle E^{el}_{ijmn} m_{mn} n_{pq} E^{el}_{pqkl}}
% % {\displaystyle n_{rs} E^{el}_{rstu} m_{tu}  \xi_* h_*}
% % \right]
% % \Delta \epsilon_{kl}
% %
%
%
% \vspace*{2mm}
% \item[] Incremental stress: each Polynomial Chaos component is updated
% incrementally
% % via stochastic Galerkin projection
%
%
%
%
% {$\Delta\sigma_1^{n+1} = \frac{1}{<\Phi_1\Phi_1> }\{\sum_{i=1}^{P_h} \sum_{k=1}^{P_e}\ h_i \Delta \epsilon_k^n <\Phi_i \Phi_k \Phi_1> \sum_{j=1}^{P_g} \sum_{k=1}^{P_e}\sum_{l=1}^{P_{\sigma}} \ c_j \Delta \epsilon_k^n \sigma_l^n <\Phi_j \Phi_k \Phi_l \Phi_1>\}$}
% \\
% . . .
% \\
% % ${\vdots}$
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% \item[] Material uncertainty expanded into stochastic shape funcs.
% %$E(x,t,\theta) = \sum_{i=0}^{P_d} r_i(x,t) * \Phi_i[\{\xi_1, ..., \xi_m\}]$
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% \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.
% %$u(x,t,\theta) = \sum_{i=0}^{P_u} u_i(x,t) * \Psi_i[\{\xi_1, ..., \xi_m, \xi_{m+1}, ..., \xi_f\}]$
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% %Stochastic system of equation resulting from Galerkin approach (static example):
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% \item[] Jeremi{\'c} et al. 2011
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% \[
% %$
% \begin{bmatrix}
% \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_0> K^{(k)} & \dots & \sum_{k=0}^{P_d} <\Phi_k \Psi_P \Psi_0> K^{(k)}\\
% \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_1> K^{(k)} & \dots & \sum_{k=0}^{P_d} <\Phi_k \Psi_P \Psi_1> K^{(k)}\\ \\
% \vdots & \vdots & \vdots & \vdots\\
% \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_P> K^{(k)} & \dots & \sum_{k=0}^{M} <\Phi_k \Psi_P \Psi_P> K^{(k)}
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% \begin{bmatrix}
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% \vdots \\
% \Delta u_{N0}\\
% \vdots \\
% \Delta u_{1P_u}\\
% \vdots \\
% \Delta u_{NP_u}
% \end{bmatrix}
% =
% %\]
% %\[
% \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>\\
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% \item[] SEPFEM offers a complete probabilistic solution
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% % \normalsize{Typical number of terms required for a SEPFEM problem} \vspace{1cm}\\
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% \# 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 $10^{11}$\\
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\subsection{Backward Uncertainty Propagation, Sensitivities}
%\subsection{Sobol Sensitivity Analysis}
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\begin{frame}
\frametitle{Backward Uncertainty Propagation, Sensitivities}
\begin{itemize}
\vspace*{2mm}
\item[] Given forward uncertain response, PDFs, CDFs...
\vspace*{6mm}
\item[] Contributions of uncertain input to forward uncertainties
\vspace*{6mm}
\item[] Sensitivity of forward uncertain response to input uncertainties
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\begin{frame}
\frametitle{ANOVA Representation}
%\vspace*{10mm}
Model with $n$ uncertain inputs ($\boldsymbol{x}$) and scalar output $y$:
\vspace*{6mm}
\begin{equation}
y = f(\boldsymbol{x}) \mbox{;} \ \ \boldsymbol{x} \in I^{n}
\nonumber
\end{equation}
% input parameters $\boldsymbol{x}$ are defined in $n$ dimensional unit
% cube $I^{n}$
%
\vspace*{5mm}
The ANalysis Of VAriance representation
% of $f(x)$
(Sobol 2001):
\vspace*{6mm}
\begin{eqnarray*}
f(x_1, ... x_n) = f_0 + \sum_{i=1}^{n} f_i(x_i) +
\sum_{1\leq i