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% \usetheme{Antibes} % ima sadrzaj gore i kao graf ...
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% colorlinks=true,
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% does not look nice, try deleting the line with the fontenc.
% Site Specific Dynamics of Structures:
%From Seismic Source to
%the Safety of Occupants and Content
\title[Uncertain ESSI]
{Uncertainties in Modeling and Simulation of
Earthquakes, Soils, Structures and their Interaction}
%\subtitle
%{Include Only If Paper Has a Subtitle}
%\author[Author, Another] % (optional, use only with lots of authors)
%{F.~Author\inst{1} \and S.~Another\inst{2}}
%  Give the names in the same order as the appear in the paper.
%  Use the \inst{?} command only if the authors have different
% affiliation.
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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)
%{Boris~Jeremi{\'c}}
{Boris Jeremi{\'c}
}
%\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 CUBoulder, GEGM seminar series \\
April 2021}
\subject{}
% This is only inserted into the PDF information catalog. Can be left
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{
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\titlepage
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\begin{frame}
\frametitle{Outline}
\begin{scriptsize}
\tableofcontents
% You might wish to add the option [pausesections]
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% Structuring a talk is a difficult task and the following structure
% may not be suitable. Here are some rules that apply for this
% solution:
%  Exactly two or three sections (other than the summary).
%  At *most* three subsections per section.
%  Talk about 30s to 2min per frame. So there should be between about
% 15 and 30 frames, all told.
%  A conference audience is likely to know very little of what you
% are going to talk about. So *simplify*!
%  In a 20min talk, getting the main ideas across is hard
% enough. Leave out details, even if it means being less precise than
% you think necessary.
%  If you omit details that are vital to the proof/implementation,
% just say so once. Everybody will be happy with that.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Motivation}
%\subsection{\ }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}
\frametitle{Motivation}
\begin{itemize}
\vspace*{3mm}
\item[] Improve modeling and simulation for infrastructure objects
% \vspace*{2mm}
% \item[] Expert numerical modeling and simulation tool
%
% \vspace*{1mm}
% \item[] Use of numerical models to
% analyze statics and dynamics of soil/rockstructure systems
%
\vspace*{1mm}
\item[] Control of modeling, epistemic uncertainty
%\vspace*{3mm}
% \item[] Choice of analysis level of sophistication
\vspace*{1mm}
\item[] Propagate parametric, aleatory uncertainty
\vspace*{1mm}
\item[] Goal: predict and inform
\vspace*{1mm}
\item[] Engineer needs to know!
\vspace*{1mm}
\item[] Design sustainable objects
\begin{figure}[!hbpt]
\begin{center}
%
%\hspace*{7mm}
\includegraphics[width=5.0truecm]{/home/jeremic/tex/works/Conferences/2021/CUBoulderGEGMseminarseries02Apr2021/present/Saint_Sophia_Constantinopolis.jpg}
%\hspace*{3mm}
\hfill
\includegraphics[width=4.0truecm]{/home/jeremic/tex/works/Conferences/2021/CUBoulderGEGMseminarseries02Apr2021/present/IMG_0162.JPG}
%
\end{center}
\end{figure}
%
%
%
% \vspace*{1mm}
% \item[] Follow the flow, input and dissipation, of seismic energy,
% \vspace*{2mm}
% \item[]
% %System for
% {\bf Real}istic modeling and simulation of
% {\bf E}arthquakes and/or
% {\bf S}oils and/or
% {\bf S}tructures and their
% {\bf I}nteraction:\\
% 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 \underline{Modeling, Epistemic Uncertainty}, simplifying assumptions
\begin{itemize}
\vspace*{2mm}
\item[] Low, medium, high sophistication modeling and simulation
\vspace*{2mm}
\item[] Choice of sophistication level for confidence in results
\end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace*{4mm}
\item \underline{Parametric, Aleatory Uncertainty},
{\small ${M} \ddot{u_i} + {C} \dot{u_i} + {K}^{ep} {u_i} = {F(t)}$},
\begin{itemize}
\vspace*{2mm}
\item[] Uncertain mass $M$, viscous damping $C$ and stiffness $K^{ep}$
\vspace*{2mm}
\item[] Uncertain loads, $F(t)$
\vspace*{2mm}
\item[] Results are PDFs and CDFs
for $\sigma_{ij}$, $\epsilon_{ij}$, $u_i$, $\dot{u}_i$, $\ddot{u}_i$
\end{itemize}
\end{itemize}
%
%
% %Le doute n'est pas un {\'e}tat bien agr{\'e}able,\\
% mais l'assurance est un {\'e}tat ridicule. (Fran{\c c}oisMarie Arouet, Voltaire)
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}
\frametitle{Modeling, Epistemic Uncertainty}
\begin{itemize}
\item[] Important (?!) features are simplified, 1C vs 3C, inelasticity
%\vspace*{4mm}
% \item Unrealistic and unnecessary modeling simplifications
\vspace*{1mm}
\item[] Modeling simplifications are justifiable if one or two
level higher sophistication model demonstrates that features being
simplified out are less or not important
\end{itemize}
% local
%\vspace*{2mm}
\begin{center}
\hspace*{7mm}
%\movie[label=show3,width=8.8cm,poster,autostart,showcontrols]
\movie[label=show3,width=5.5cm,poster,autostart, showcontrols]
{\includegraphics[width=50mm]
{/home/jeremic/tex/works/Conferences/2016/IAEA_TecDoc_February2016/My_Current_Work/movie_2_npps_mp4_icon.jpeg}}
{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Model01_ESSI_Response_May2015/movie_2_npps.mp4}
%
%\hfill
\hspace*{5mm}
%
\movie[label=show3,width=6.0cm,poster,autostart,showcontrols]
{\includegraphics[width=50mm]
{/home/jeremic/tex/works/Conferences/2017/SMiRT_24/present/3D_Nonlinear_Modeling_and_it_Effects/NPP_Plastic_Dissipation_grab.jpg}}
{/home/jeremic/tex/works/Thesis/HanYang/Files_10Aug2017/NPP_Plastic_Dissipation.mp4}
\hspace*{7mm}
%\end{flushleft}
%%
\end{center}
% local
% % \vspace*{5mm}
% \begin{center}
% %\begin{flushleft}
% % \hspace*{15mm}
% \movie[label=show3,width=5cm,poster,autostart,showcontrols]
% {\includegraphics[width=5cm]
% {/home/jeremic/tex/works/Conferences/2017/SMiRT_24/present/3D_Nonlinear_Modeling_and_it_Effects/NPP_Plastic_Dissipation_grab.jpg}}
% {/home/jeremic/tex/works/Thesis/HanYang/Files_10Aug2017/NPP_Plastic_Dissipation.mp4}
% %\end{flushleft}
% %%
% \hfill
% %%
% %\begin{flushright}
% % \hspace*{15mm}
% \movie[label=show3,width=5cm,poster,autostart,showcontrols]
% {\includegraphics[width=5cm]
% {/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Energy_Dissipation_Animations/SMR_Energy_Dissipation_screen_grab.jpg}}
% {/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Energy_Dissipation_Animations/SMR_Energy_Dissipation.mp4}
% %\end{flushright}
% \end{center}
%
\end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Modeling Uncertainty, 6C vs 1C Motions}
%
%
% % local
% \vspace*{2mm}
% \begin{center}
% \hspace*{7mm}
% %\movie[label=show3,width=8.8cm,poster,autostart,showcontrols]
% \movie[label=show3,width=8.8cm,poster, showcontrols]
% {\includegraphics[width=92mm]
% {/home/jeremic/tex/works/Conferences/2016/IAEA_TecDoc_February2016/My_Current_Work/movie_2_npps_mp4_icon.jpeg}}
% {/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Model01_ESSI_Response_May2015/movie_2_npps.mp4}
% \end{center}
% % local
% % \vspace*{2mm}
% % \begin{center}
% % \hspace*{7mm}
% % \movie[label=show3,width=8.8cm,poster,autostart,showcontrols]
% % {\includegraphics[width=90mm]{movie_2_npps_mp4_icon.jpeg}}{movie_2_npps.mp4}
% % \end{center}
%
%
% % online
% \vspace*{12mm}
% \begin{flushleft}
% %\vspace*{15mm}
% \href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/ESSI_VisIt_movies_Jose_19May2015/movie_2_npps.mp4}
% {\tiny (MP4)}
% \end{flushleft}
% % online
%
%
%
%
%
% \end{frame}
%
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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/Hexiang/1D_risk/version2/Figures/SA_GMPE_verification_std_08_no_smooth.pdf}
%
\end{center}
\end{figure}
\vspace*{7mm}
\begin{flushright}
{\tiny
(cf. Wang et al. (2019))\\
~}
\end{flushright}
\end{frame}
%
%
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%\subsection*{RealESSI Simulator System}
\subsection{Real ESSI Simulator System}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{RealESSI Simulator System}
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,
%\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:
\vspace*{1mm}
\begin{itemize}
%\vspace*{1mm}
\item[] Design, linear elastic, load combinations, dimensioning
%\vspace*{1mm}
\item[] Assessment, nonlinear/inelastic, safety margins
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{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 RealESSI System availability: Windows/iOS/Linux docker, Linux Executables, AWS
% \begin{itemize}
% %\vspace*{1mm}
% \item[] Educational Institutions: AWS and Linux Executables, free
% \item[] Government Agencies, National Labs: AWS GovCloud, free
% \item[] Professional Practice: AWS and Linux Executables, commercial
% %\vspace*{1mm}
% %%\vspace*{1mm}
% % \item Sources available to collaborators
% \end{itemize}
\vspace*{3mm}
\item RealESSI education and training: theory and applications
\vspace*{3mm}
\item RealESSI documentation and program available at
\url{http://realessi.us/}
%\url{http://sokocalo.engr.ucdavis.edu/~jeremic/Real_ESSI_Simulator/}
%
%\url{http://realessi.info/}
%
% \vspace*{2mm}
% \item
%
\end{itemize}
\end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{RealESSI Modeling Features}
\begin{itemize}
%\vspace*{2mm}
\item[] Solid elements: dry, un/fullysaturated, elastic, inelastic
%\vspace*{1mm}
\item[] Structural elements: beams, shells, elastic, inelastic
%\vspace*{1mm}
\item[] Contact/interface/joint elements: bonded, shear/frictional (EPP, EPH,
EPS); gap/normal; linear, nonlinear, dry, coupled/saturated,
%\vspace*{1mm}
\item[] Super element: stiffness and mass matrices
%\vspace*{1mm}
\item[] Material models: soil, rock, concrete, steel...
%\vspace*{1mm}
\item[] Seismic input: 1C and 3C/6C, deterministic or probabilistic
%\vspace*{1mm}
\item[] Energy dissipation calculation: elasticplastic, viscous, algorithmic
%\vspace*{1mm}
\item[] Solid/StructureFluid interaction, full coupling, OpenFOAM
%\vspace*{1mm}
\item[] Intrusive probabilistic inelastic modeling
%
%\vspace*{1mm}
% \item Modeling features 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}
\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}
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\section{Modeling and Simulation}
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\subsection{Seismic Motions}
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\begin{frame}
\frametitle{Earthquake Ground Motions}
\begin{itemize}
\vspace*{0.5cm}
\item Real earthquake ground motions
\begin{itemize}
\item[] Body, P and S waves
\item[] Surface, Rayleigh and Love waves
\item[] Lack of correlation, incoherent motions
\item[] Inclined seismic waves
\item[] 3D, 3C/6C waves
% \item Earthquake energy dissipation
\end{itemize}
\vspace*{0.5cm}
\item What are the effects of real earthquake ground motions on soilstructure systems ?!
\end{itemize}
\end{frame}
% \subsection*{Fundamentals}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}
\frametitle{ESSI: 6C or 1C Seismic Motions}
\begin{itemize}
\item Assume that a full 6C (3C) motions at the surface are only recorded in one
horizontal direction
\item From such recorded motions one can develop a vertically propagating shear
wave (1C) in 1D
\item Apply such vertically propagating shear wave to same soilstructure
system
\end{itemize}
\vspace*{3mm}
\begin{figure}[!H]
\begin{center}
\includegraphics[width=6.5cm]{/home/jeremic/tex/works/Conferences/2015/CompDyn/Present/6D_to_1D_01.jpg}
\end{center}
\end{figure}
\end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Realistic Ground Motions}
%
% \begin{itemize}
%
% \item Free field seismic motion models
%
% \end{itemize}
%
% % local
% % local
% % local
% \vspace*{2mm}
% \begin{center}
% %\movie[label=show3,width=8.5cm,poster,autostart,showcontrols,loop]
% \hspace*{12mm}
% % %\movie[label=show3,width=6.0cm,autostart,showcontrols]
% % \movie[label=show3,width=6.0cm,poster]
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% % \hspace*{2mm}
% % %\hfill
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% %\movie[label=show3,width=90mm,poster]
% {\includegraphics[width=90mm]{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Free_Field_small_model_April2015/movie_input_mp4_icon.jpeg}}
% {/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Free_Field_small_model_April2015/movie_input.mp4}
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% \begin{frame}
% \frametitle{Development of Realistic Motions}
%
% \begin{itemize}
%
% \item Sources will send both P and S waves
%
% \end{itemize}
%
% % online
% % online \begin{center}
% % online \href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Free_Field_small_model_April2015/movie_input.mp4}
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% % online %
% % online \href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Free_Field_small_model_April2015/movie_input_closeup.mp4}
% % online {\includegraphics[width=50mm]{movie_input_closeup_mp4_icon.jpeg}}
% % online \end{center}
% % online
%
% % local
% % local
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% \hspace*{12mm}
% % %\movie[label=show3,width=6.0cm,autostart,showcontrols]
% % \movie[label=show3,width=6.0cm,poster]
% % {\includegraphics[width=60mm]{movie_input_mp4_icon.jpeg}}{movie_input.mp4}
% % \hspace*{2mm}
% % %\hfill
% % %\hspace*{7mm}
% % %\movie[label=show3,width=5.3cm,poster,autostart,showcontrols]
% \movie[label=show3,width=80mm,poster, showcontrols]
% {\includegraphics[width=80mm]{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Free_Field_small_model_April2015/movie_input_closeup.jpg}}
% {/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Free_Field_small_model_April2015/movie_input_closeup.mp4}
% \hspace*{12mm}
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\begin{frame}
\frametitle{1C vs 6C Free Field Motions}
\begin{itemize}
\item One component of motions, 1C from 6C
% or 3$\times$1D (it is done all the time!)
\item Excellent fit
% (goal is to predict and inform and not (force) fit)
\end{itemize}
% local
%\vspace*{2mm}
\begin{center}
\hspace*{16mm}
%\movie[label=show3,width=5.6cm,poster,autostart,showcontrols]
\movie[label=show3,width=61mm,poster, showcontrols]
{\includegraphics[width=60mm]{/home/jeremic/tex/works/Conferences/2016/IAEA_TecDoc_February2016/My_Current_Work/movie_ff_3d_mp4_icon.jpeg}}
{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Model01_ESSI_Response_May2015/movie_ff_3d.mp4}
%\hspace*{2mm}
%\hfill
%\movie[label=show3,width=5.6cm,poster,autostart,showcontrols]
\movie[label=show3,width=61mm,poster, showcontrols]
{\includegraphics[width=60mm]
{/home/jeremic/tex/works/Conferences/2016/IAEA_TecDoc_February2016/My_Current_Work/movie_ff_1d_mp4_icon.jpeg}}
{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Model01_ESSI_Response_May2015/movie_ff_1d.mp4}
\hspace*{16mm}
\end{center}
% local
% online
\begin{center}
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/ESSI_VisIt_movies_Jose_19May2015/movie_ff_3d.mp4}
{\tiny (MP4)}
%
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/ESSI_VisIt_movies_Jose_19May2015/movie_ff_1d.mp4}
{\tiny (MP4)}
\end{center}
% online
% out
% out % local
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% 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 \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 %
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% out \href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/ESSI_VisIt_movies_Jose_19May2015/movie_ff_1d.mp4}
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% 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}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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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
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% out {movie_2_npps.mp4}
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% out \href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/ESSI_VisIt_movies_Jose_19May2015/movie_2_npps.mp4}
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\begin{frame}
\frametitle{Ventura Hotel, Northridge Earthquake, nonSSI vs SSI}
% local
\vspace*{5mm}
\begin{center}
\hspace*{16mm}
%\movie[label=show3,width=5.6cm,poster,autostart,showcontrols]
\movie[label=show3,width=120mm,showcontrols]
{\includegraphics[width=120mm]{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_Buildings/Ventura_Hotel_SSI_vs_nonSSI_screen_grab.jpg}}
{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_Buildings/Ventura_Hotel_SSI_vs_nonSSI.mp4}
\hspace*{16mm}
\end{center}
% local
% online
\vspace*{12mm}
\hspace*{12mm}
\begin{flushleft}
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_ESSI_for_Buildings/Ventura_Hotel_SSI_vs_nonSSI.mp4}
{\tiny ({\bf MP4})}
\end{flushleft}
% online
\vspace*{12mm}
\end{frame}
%\vspace*{10mm}
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\subsection{Energy Dissipation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Energy Input and Dissipation}
\begin{itemize}
\vspace*{1mm}
\item[] Energy input, dynamic forcing
\vspace*{4mm}
\item[] Energy dissipation outside SSI domain:
\begin{itemize}
\item[] SSI system oscillation radiation
\item[] Reflected wave radiation
\end{itemize}
%\vspace*{1mm}
\item[] Energy dissipation/conversion inside SSI domain:
\begin{itemize}
\item[] Inelasticity of soil, interfaces, structure, dissipators
\item[] Viscous coupling with internal/pore, and external fluids
% % \item[] potential and kinetic energy
% \item[] potential $\leftarrow \! \! \! \! \! \! \rightarrow$ kinetic energy
\end{itemize}
%\vspace*{1mm}
% \item[] Numerical energy dissipation (numerical damping/production and period errors)
% \item[] Numerical energy dissipation (damping/production)
\item[] Numerical energy dissipation/production
\end{itemize}
%
\end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Plastic Energy Dissipation}
\vspace*{2mm}
Single elasticplastic element under cyclic shear loading
\begin{itemize}
\item[] Difference between plastic work and plastic dissipation
\item[] Plastic work can decrease
\item[] Plastic dissipation always increases
\end{itemize}
%\vspace*{7mm}
\begin{figure}[!hbpt]
\begin{center}
\hspace*{5mm}
\includegraphics[width=11.0truecm]{/home/jeremic/tex/works/Thesis/HanYang/Files_06June2017/DOE_Annual_2017/Figures/Dissipation_Material.png}
\end{center}
\end{figure}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%
% \frametitle{Energy Dissipation Control Mechanisms}
%
% \begin{figure}[!H]
% %\hspace*{10mm}
% \includegraphics[width=3.4cm]{/home/jeremic/tex/works/Thesis/HanYang/Files_Energy_dissipation_01Dec2017/case_plasticity.pdf}
% \includegraphics[width=3.4cm]{/home/jeremic/tex/works/Thesis/HanYang/Files_Energy_dissipation_01Dec2017/case_Rayleigh.pdf}
% \includegraphics[width=3.4cm]{/home/jeremic/tex/works/Thesis/HanYang/Files_Energy_dissipation_01Dec2017/case_Newmark.pdf}
% \end{figure}
%
%
% % \hspace*{10mm} Numerical \hspace*{20mm} Viscous \hspace*{20mm} Plasticity
% \hspace*{10mm} Plasticity \hspace*{20mm} Viscous \hspace*{20mm} Numerical
%
%
%
% \end{frame}
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}
\frametitle{Energy Dissipation Control}
\begin{figure}[!H]
%\hspace*{10mm}
% \includegraphics[width=3cm]{/home/jeremic/tex/works/Thesis/HanYang/Files_Energy_dissipation_01Dec2017/case_a.pdf}
% \includegraphics[width=3cm]{/home/jeremic/tex/works/Thesis/HanYang/Files_Energy_dissipation_01Dec2017/case_b.pdf}
\includegraphics[width=9cm]{/home/jeremic/tex/works/Thesis/HanYang/Files_Energy_dissipation_01Dec2017/case_g.pdf}
% \includegraphics[width=3cm]{/home/jeremic/tex/works/Thesis/HanYang/Files_Energy_dissipation_01Dec2017/case_e.pdf}
\end{figure}
\end{frame}
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%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Inelastic Modeling of Soil Structure System}
%
% \begin{itemize}
%
% \vspace*{1mm}
% \item[] Soil, inelastic, elasticplastic
% \begin{itemize}
% \item[] Dry, single phase
% \item[] Unsaturated, partially saturated, and fully saturated
% \end{itemize}
%
%
% \vspace*{1mm}
% \item[] Interface/Contact/Joint, inelastic, gap open/close, slip
% \begin{itemize}
% \item[] Dry, single phase,
% % \begin{itemize}
% % \item[] Normal, hard and soft, gap open/close
% % \item[] Friction, slip, nonlinear
% % \end{itemize}
% \item[] Fully saturated, suction, excess pressure, buoyant force
% \end{itemize}
%
%
% \vspace*{1mm}
% \item[] Structure, inelastic, damage, cracks, ASR...
% \begin{itemize}
% \item[] Nonlinear/inelastic 1D concrete, steel, 3D fiber beams
% \item[] Nonlinear/inelastic 3D concrete, steel, 3D solids, 3D shells
% % \item[] Nonlinear/inelastic 3D concrete, steel shell element
% % \item[] Alcali Silica Reaction concrete modeling
% \end{itemize}
%
%
% \vspace*{1mm}
% \item[] Solid/StructureFluid interaction, open surface
%
%
%
% \end{itemize}
%
% %
% \end{frame}
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%
% \frametitle{NPP Model }
%
% \begin{figure}[!h]
% \begin{center}
% \includegraphics[width=8.5cm]{/home/jeremic/tex/works/Thesis/SumeetKumarSinha/Files_10Aug2017/Npp_Non_Linear_Effects/images/NPP_With_Shallow_Foundation.pdf}
% \end{center}
% % \caption{\label{Fig:NPP_Model_In_Real_ESSI} Nuclear Power Plant Model with Shallow Foundation }
% \end{figure}
%
%
% % \begin{tikzpicture}[remember picture,overlay]
% % \node[xshift=3.5cm,yshift=0.6cm] at (current page.center) {\includegraphics[width=0.5\textwidth]{images/Contact_In_Industry}};
% % \end{tikzpicture}
%
% \end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%
% \frametitle{Structure Model}
%
% The nuclear power plant structure comprise of
% \begin{itemize}
% \item[]Auxiliary building, $f^{aux}_{1}= 8Hz$
% \item[]Containment/Shield building, $f^{cont}_{1}= 4Hz$
% \item[]Concrete raft foundation: $3.5m$ thick
% \end{itemize}
% \begin{figure}[!h]
% \begin{center}
% \includegraphics[width=0.8\textwidth]{/home/jeremic/tex/works/Thesis/SumeetKumarSinha/Files_10Aug2017/Npp_Non_Linear_Effects/images/NPP_Model_Auxiliary_And_Containment_Building.pdf}
% \end{center}
% \caption{\label{Fig:NPP_Structure_Model_In_Real_ESSI} Auxiliary and Containment Building }
% \end{figure}
%
% \end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%
% \frametitle{Inelastic Soil and Inelastic Contact/Interface/Joint}
%
% \begin{itemize}
% \item[] Shear velocity of soil $V_s=500m/s$
% \item[] Undrained shear strength (Dickenson 1994) $V_s [m/s] = 23 (S_u [kPa])^{0.475}$
% \item[] For $V_s=500m/s$ Undrained Strength $S_u=650kPa$ and Young's Modulus of $E=1.3GPa$
% \item[] von Mises, Armstrong Frederick kinematic hardening
% ($S_u=650kPa$ at $\gamma=0.01\%$; $h_a = 30MPa$, $c_r = 25$)
% \item[] Soft contact (concretesoil), gaping and nonlinear shear
% \end{itemize}
%
% \begin{figure}[!h]
% \begin{center}
% \includegraphics[width=4cm]{/home/jeremic/tex/works/Thesis/SumeetKumarSinha/Files_10Aug2017/Npp_Non_Linear_Effects/images/Von_Mies_non_Linear_Hardening.pdf}
% \includegraphics[width=3.5cm]{/home/jeremic/tex/works/Thesis/SumeetKumarSinha/Files_10Aug2017/Npp_Non_Linear_Effects/images/SoftContact.pdf}
% \end{center}
% \end{figure}
%
%
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%
% \frametitle{Acceleration Traces, Elastic vs Inelastic }
%
%
% \hspace*{35mm}
% \begin{figure}[!h]
% \vspace*{2mm}
% \begin{center}
% \hspace*{15mm}
% \includegraphics[width=7.0cm]{/home/jeremic/tex/works/Thesis/SumeetKumarSinha/Files_10Aug2017/Npp_Non_Linear_Effects/images/Acceleration_Elastic_Without_Contact_SMIRT_2017.pdf}
% \hspace*{20mm}
% \includegraphics[width=7.0cm]{/home/jeremic/tex/works/Thesis/SumeetKumarSinha/Files_10Aug2017/Npp_Non_Linear_Effects/images/Acceleration_Inelastic_With_Contact_SMIRT_2017.pdf}
% \hspace*{25mm}
% \end{center}
% \end{figure}
% \hspace*{35mm}
%
% \hspace*{10mm} Elastic \hspace*{40mm} Inelastic \hspace*{40mm}
%
% \end{frame}
%
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\begin{frame}
\frametitle{Energy Dissipation for Design}
\begin{figure}[!hbpt]
\begin{center}
\includegraphics[width=10.0truecm]{/home/jeremic/tex/works/Thesis/HanYang/Frame_animations_13Mar2019/2D_Frame_Model.pdf}
\end{center}
\end{figure}
\end{frame}
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\begin{frame}
\frametitle{Design Alternatives}
% local
%\vspace*{2mm}
\begin{center}
\hspace*{16mm}
%\movie[label=show3,width=5.6cm,poster,autostart,showcontrols]
\movie[label=show3,width=61mm,poster, showcontrols]
{\includegraphics[width=60mm]{/home/jeremic/tex/works/Thesis/HanYang/Frame_animations_13Mar2019/Individual_Foundation_screen_grab.jpg}}
{/home/jeremic/tex/works/Thesis/HanYang/Frame_animations_13Mar2019/Individual_Foundation.mp4}
%\hspace*{2mm}
%\hfill
%\movie[label=show3,width=5.6cm,poster,autostart,showcontrols]
\movie[label=show3,width=61mm,poster, showcontrols]
{\includegraphics[width=61mm]{/home/jeremic/tex/works/Thesis/HanYang/Frame_animations_13Mar2019/Continuous_Foundation_screen_grab.jpg}}
{/home/jeremic/tex/works/Thesis/HanYang/Frame_animations_13Mar2019/Continuous_Foundation.mp4}
\hspace*{16mm}
\end{center}
% local
% online
\begin{center}
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/Energy_dissipation_frames/Individual_Foundation.mp4}
{\tiny (MP4)}
%
\hspace*{40mm}
%
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/Energy_dissipation_frames/Continuous_Foundation.mp4}
{\tiny (MP4)}
\end{center}
% online
\end{frame}
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\begin{frame}
\frametitle{Energy Dissipation for an SMR Model}
% Elastoplastic soil with contact elements
%% Both solid and contact elements dissipate energy
% \vspace*{5mm}
\begin{center}
% \hspace*{15mm}
\movie[label=show3,width=10cm,poster,autostart,showcontrols]
{\includegraphics[width=10cm]
{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Energy_Dissipation_Animations/SMR_Energy_Dissipation_screen_grab.jpg}}
{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Energy_Dissipation_Animations/SMR_Energy_Dissipation.mp4}
\end{center}
\begin{flushleft}
\vspace*{15mm}
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Energy_Dissipation_Animations/SMR_Energy_Dissipation.mp4}
% \href{./homo_50mmesh_45degree_Ormsby.mp4}
{\tiny (MP4)}
\end{flushleft}
%
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\end{frame}
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% \begin{frame}
% \frametitle{ASCE721: Buildings and Models, Low Building}
%
% %\vspace*{5mm}
% \begin{figure}[!htb]
% \begin{center}
% \includegraphics[width=10cm]{/home/jeremic/tex/works/Thesis/HexiangWang/ATC144_buildings_results_plots_15Sep2019/steel_frame2.pdf}
% \end{center}
% \end{figure}
%
% \end{frame}
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% \begin{frame}
% \frametitle{ASCE721: Low Building Energy Dissipation}
%
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% % \vspace*{5mm}
% \begin{center}
% % \hspace*{15mm}
% %\movie[label=show3,width=10cm,poster,autostart,showcontrols]
% \movie[label=show3,width=10cm,poster,showcontrols]
% {\includegraphics[width=10cm]
% {/home/jeremic/tex/works/Thesis/HanYang/ASCE721_low_building_energy_dissipation/ATC_Short_Building_PD.jpg}}
% {/home/jeremic/tex/works/Thesis/HanYang/ASCE721_low_building_energy_dissipation/ATC_Short_Building_PD.mp4}
% \end{center}
%
% % online
% \vspace*{12mm}
% \begin{flushleft}
% \hspace*{4mm}
% \href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/Energy_dissipation_frames/ATC_Short_Building_PD.mp4}
% {\tiny (MP4)}
% \end{flushleft}
% % online
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\section{Uncertain Inelastic Computational Mechanics}
%OVDE
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%\subsection{Stochastic Modeling}
\subsection{Probabilistic Computational Mechanics}
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\begin{frame}
\frametitle{Uncertainty Propagation through
Inelastic System}
%
\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{Previous Work}
\begin{itemize}
\item
Linear algebraic or differential equations:
\begin{itemize}
\item Variable Transf. Method (Montgomery and Runger 2003)
\item Cumulant Expansion Method (Gardiner 2004)
\end{itemize}
\item
Nonlinear differential equations:
\begin{itemize}
\item Monte Carlo Simulation (Schueller 1997, De Lima et al 2001, Mellah
et al. 2000, Griffiths et al. 2005...) \\ $\rightarrow$ can be accurate, very costly
\item Perturbation Method (Anders and Hori 2000, Kleiber and Hien 1992,
Matthies et al. 1997) \\ $\rightarrow$ first and second order Taylor series
expansion about mean  limited to problems with small C.O.V. and inherits
"closure problem"
\item SFEM (Ghanem and Spanos 1989, Matthies et al, 2004, 2005, 2014...)
\end{itemize}
%
% \item
% Monte Carlo method: accurate, very costly
%
% \item
% Perturbation method:
\end{itemize}
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\end{frame}
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\begin{frame} \frametitle{{3D FokkerPlanckKolmogorov Equation}}
\begin{footnotesize}
\begin{eqnarray}
\nonumber
\lefteqn{\displaystyle \frac{\partial P(\sigma_{ij}(x_t,t), t)}{\partial t} = \displaystyle \frac{\partial}{\partial \sigma_{mn}}
\left[ \left\{\left< \vphantom{\int_{0}^{t}} \eta_{mn}(\sigma_{mn}(x_t,t), E_{mnrs}(x_t), \epsilon_{rs}(x_t,t))\right> \right. \right.} \\
\nonumber
&+& \left. \left. \int_{0}^{t} d\tau Cov_0 \left[\displaystyle \frac{\partial \eta_{mn}(\sigma_{mn}(x_t,t), E_{mnrs}(x_t),
\epsilon_{rs}(x_t,t))} {\partial \sigma_{ab}}; \right. \right. \right. \\
\nonumber
& & \left. \left. \left. \eta_{ab} (\sigma_{ab}(x_{t\tau}, t\tau), E_{abcd}(x_{t\tau}), \epsilon_{cd}(x_{t\tau}, t\tau)
\vphantom{\int_{0}^{t}} \right] \right \} P(\sigma_{ij}(x_t,t),t) \right] \\
\nonumber
&+& \displaystyle \frac{\partial^2}{\partial \sigma_{mn} \partial \sigma_{ab}} \left[ \left\{ \int_{0}^{t} d\tau Cov_0 \left[
\vphantom{\int_{0}^{t}} \eta_{mn}(\sigma_{mn}(x_t,t), E_{mnrs}(x_t), \epsilon_{rs}(x_t,t)); \right. \right. \right. \\
\nonumber
& & \left. \left. \left. \eta_{ab} (\sigma_{ab}(x_{t\tau}, t\tau), E_{abcd}(x_{t\tau}), \epsilon_{cd}(x_{t\tau}, t\tau))
\vphantom{\int_{0}^{t}} \right] \vphantom{\int_{0}^{t}} \right\} P(\sigma_{ij}(x_t,t),t) \right]
\end{eqnarray}
\end{footnotesize}
\begin{flushright}
(Jeremi{\'c} et al. 2007)
\end{flushright}
% \begin{itemize}
%
%
%
% \item 6 equations
%
% \item Complete description of 3D probabilistic stressstrain response
%
% \end{itemize}
%
%
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\end{frame}
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\begin{frame}
\frametitle{FPK Equation}
\begin{itemize}
\item Advectiondiffusion equation
%
\begin{equation}
\nonumber
\frac{\partial P(\sigma,t)}{\partial t} = \frac{\partial}{\partial \sigma}\left[N_{(1)}P(\sigma,t)\frac{\partial}{\partial \sigma}
\left\{N_{(2)} P(\sigma,t)\right\} \right]
\end{equation}
%
\item Complete probabilistic description of response
\item Solution PDF is secondorder exact to covariance of time (exact mean and variance)
\item It is deterministic equation in probability density space
\item It is linear PDE in probability density space
$\rightarrow$ simplifies the numerical solution process
%\vspace*{0.2truecm}
\end{itemize}
%
% \vspace*{0.5cm}
% {%
% \begin{beamercolorbox}{section in head/foot}
% \usebeamerfont{framesubtitle}\tiny{B. Jeremi\'{c}, K. Sett, and M. L. Kavvas, "Probabilistic
% ElastoPlasticity: Formulation in 1D", \textit{Acta Geotechnica}, Vol. 2, No. 3, 2007, In press (published
% online in the \textit{Online First} section)}
% %\vskip2pt\insertnavigation{\paperwidth}\vskip2pt
% \end{beamercolorbox}%
% }
\end{frame}
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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%
%
%
% \frametitle{Template Solution of FPK Equation}
%
%
%
% \begin{itemize}
%
%
%
%
% \item FPK diffusionadvection equation is applicable to any material model $\rightarrow$
% only the coefficients $N_{(1)}$ and $N_{(2)}$ are different for different material models
% % %
% % %
%
%
%
% % %
% \item Initial condition
%
% \begin{itemize}
%
% \item Deterministic $\rightarrow$ Dirac delta function $\rightarrow$ $ P(\sigma,0)=\delta(\sigma) $
%
% \item Random $\rightarrow$ Any given distribution
%
% \end{itemize}
%
% \item Boundary condition: Reflecting BC $\rightarrow$ conserves probability mass
% $\zeta(\sigma,t)_{At \ Boundaries}=0$
%
% \item Solve using finite differences and/or finite elements
%
%
% \item However (!!) it is a stress solution and probabilistic stiffness is an
% approximation!
%
% \end{itemize}
%
%
% \end{frame}
%
%
%
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%
% \frametitle{Probabilistic 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}
% \end{center}
% \end{figure}
%
% \end{frame}
%
%
%
%
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\begin{frame}
\frametitle{{Cam Clay with Random $G$, $M$ and $p_0$}}
\begin{figure}[!hbpt]
\begin{center}
\hspace*{10mm}
\includegraphics[width=5.5cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourLowOCR_RandomG_RandomM_Randomp0m.pdf}
\includegraphics[width=5.5cm]{/home/jeremic/tex/works/Conferences/2006/KallolsPresentationGaTech/ContourHighOCR_RandomG_RandomMm.pdf}
\hspace*{10mm}
\end{center}
\end{figure}
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\end{frame}
%  %%%%%%
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\begin{frame}{Time Domain Stochastic Galerkin Method}
Dynamic Finite Elements $
{ M} \ddot{ u_i} +
{ C} \dot{ u_i} +
{ K}^{ep} { u_i} =
{ F(t)}$
\begin{itemize}
\item Input random field/process{\normalsize{(nonGaussian, heterogeneous/ nonstationary)}}
\begin{itemize}
\item[] Multidimensional Hermite Polynomial Chaos (PC) with {known coefficients}
\end{itemize}
%\vspace{0.05in}
\item Output response process
\begin{itemize}
\item[] Multidimensional Hermite PC with {unknown coefficients}
\end{itemize}
% \vspace{0.05in}
\item Galerkin projection: minimize the error to compute unknown coefficients of response process
%\vspace{0.05in}
\item Time integration using Newmark's method
\begin{itemize}
\item[] Update coefficients following an elasticplastic constitutive law at each time step
\end{itemize}
\end{itemize}
%\scriptsize
%Note: PC = Polynomial Chaos
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% \begin{frame}{Discretization of Input Random Process/Field $\beta(x,\theta)$}
% \begin{center}
% \includegraphics[scale=0.35]{/home/jeremic/tex/works/Thesis/FangboWang/slides_13Mar2019/Fangbo_slides/figs/PC_KL_explanation.PNG} \\
% \end{center}
%
%
% \footnotesize{Note: $\beta(x,\theta)$ is an input random process with any
% marginal distribution, \\ \hspace{21mm} with any covariance structure;} \\
% \footnotesize{\hspace{8mm} $\gamma(x,\theta)$ is a 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{3mm}
Seismic motions random process: \\
%\vspace{0.3cm}
%\begin{equation*}
$f_m(t, \theta)=\sum_{j=1}^{P_2} f_{mj}(t) \Psi_j(\{\xi_k(\theta)\})$
%\end{equation*}
\vspace{3mm}
Displacement response: \\
%\vspace{0.3cm}
%\begin{equation*}
$u_n(t, \theta)=\sum_{k=1}^{P_3} d_{nk}(t) \Psi_k(\{\xi_l(\theta)\})$
%\end{equation*}
\vspace{3mm}
%Acceleration response:
%%\vspace{0.3cm}
%%\begin{equation*}
%$\ddot u_n(t, \theta)=\sum_{k=1}^{P_3} \ddot d_{nk}(t) \Psi_k(\{\xi_l(\theta)\})$
%%\end{equation*}
\vspace{3mm}
\vspace{3mm}
where $a_i(x), f_{mj}(t)$ are {known PC coefficients}, while $d_{nk}(t)$
are {unknown PC coefficients}.
%}
\end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}{FEM and Stochastic ElasticPlastic FEM, SEPFEM}
%
%
% %\vspace{4mm}
% \vspace{2mm}
%
% \small
% {
% %FEM:
% \begin{eqnarray*}
% \sum_{e} [ \int_{D_e} N_m(x)\rho(x)N_n(x)d\Omega \; {\color{blue}{\ddot{u}_n(t)}} +
% \\
% \int_{D_e}\nabla N_m(x) {\color{blue}{E(x)}} \nabla N_n(x)d\Omega \; {\color{blue}{u_n(t)}}  {\color{blue}{f_m(t)}} ]=0
% \end{eqnarray*}
%
% %SEPFEM:
% \begin{eqnarray*}
% &&\lefteqn{\sum_{n=1}^N \sum_{k=1}^{P_3} \langle \Psi_k \Psi_l \rangle \int_{D_e}N_m(x)\rho(x)N_n(x)d\Omega \; \; \ddot{d}_{nk}(t) \; \; +}
% \\
% &&\sum_{n=1}^N \sum_{k=1}^{P_3} \sum_{i=1}^{P_1} \langle \Psi_i \Psi_k \Psi_l \rangle
% \int_{D_e}B_m(x) {\color{blue}{a_i(x,t)}} B_n(x)d\Omega \; \; d_{nk}(t)
% =
% \\
% &&\sum_{j=1}^{P_2} \langle \Psi_j \Psi_l \rangle f_{mj}(t) \\
% \end{eqnarray*}
% }
% \vspace{0.3cm}
%
% %\scriptsize{Note: update \textcolor{blue}{$a_i(x,t)$} for elasticplastic material}
%
% \vspace{0.4cm}
%
% %\begin{beamercolorbox}{section in head/foot}
% %\usebeamerfont{framesubtitle}\tiny{Wang, F. and Sett, K., "TimeDomain Stochastic Finite Element Simulation of Uncertain Seismic Wave Propagation through Uncertain Heterogeneous Solids", \textit{Soil Dynamics and Earthquake Engineering}, 88:369385, 2016.}
% %\end{beamercolorbox}
%
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}{SEPFEM}
% Matrix form:
% \begin{equation*}
% \bm{M} \ddot{\bm{d}} + \bm{K} \bm{d} = \bm{f}
% \end{equation*}
% For damped systems:
% \begin{equation*}
% \label{eqno_19}
% \bm{M} \ddot{\bm{d}} + \bm{C} \dot{\bm{d}} + \bm{K} \bm{d} = \bm{f}
% \end{equation*}
% \hspace{5cm} {\huge $\Downarrow$ } \\
% \hspace{2.2cm} Newmark's method to solve in time domain \\
%
% \vspace{1cm}
% \scriptsize{
% \noindent where $\bm{M}$, $\bm{C}$ and $\bm{K}$ are generalized mass, damping and stiffness matrices,\\
% \hspace{0.9cm} $\bm{f}$, $\bm{d}$, and $\ddot {\bm{d}}$ are generalized force, displacement, and acceleration vectors.
% }
%
% \end{frame}
%
%\begin{frame}{Physical significance of stochastic DOFs} \label{matrix_form}
%
%{\scriptsize Finite element system of equation  \only<1> {deterministic} \only<2> {stochastic}}
%\vspace{0.5cm}
%
%\begin{equation}
%\nonumber
%\tiny
%\left[ \begin{array}{ccccc}
%\uncover<2> {\displaystyle \sum_{i=1}^{P_1} C_{i11}} \only<1> {K_1} \only<2> {K_i} & \uncover<2> {\displaystyle \sum_{i=1}^{P_1} C_{i12} K_i} & \uncover<2> \dots & \uncover<2> \dots & \uncover<2> {\displaystyle \sum_{i=1}^{P_1} C_{i 1 P_{\scaleto{3}{3pt}}} K_i} \\
%\uncover<2> {\displaystyle \sum_{i=1}^{P_1} C_{i21} K_i} & \uncover<2> {\displaystyle \sum_{i=1}^{P_1} C_{i22} K_i} & \uncover<2> \dots & \uncover<2> \dots & \uncover<2> \vdots \\
%\uncover<2> \vdots & \uncover<2> \vdots & \uncover<2> \ddots & & \uncover<2> \vdots \\
%\uncover<2> \vdots & \uncover<2> \vdots & & \uncover<2> \ddots & \uncover<2> \vdots \\
%\uncover<2> {\displaystyle \sum_{i=1}^{P_1} C_{i P_{\scaleto{3}{3pt}} 1} K_i} & \uncover<2> {\dots} & \uncover<2>{\dots} & \uncover<2> {\dots} & \uncover<2> {\displaystyle \sum_{i=1}^{P_1} C_{i P_{\scaleto{3}{3pt}} P_{\scaleto{3}{3pt}}} K_i}
%\end{array} \right]
%%
%\left[ \begin{array}{c}
%\vec{d}_1 \\
%\\
%\uncover<2> {\vec{d}_2} \\
%\\
%\uncover<2> \vdots \\
%\\
%\uncover<2> \vdots \\
%\\
%\uncover<2> {\vec{d}_{P_{\scaleto{3}{3pt}}}}
%\end{array} \right]
%%
%=
%%
%\left[ \begin{array}{c}
%\uncover<2> {\displaystyle \sum_{j=1}^{P_{\scaleto{2}{3pt}}} C_{j1}} \only<1> {\vec{f}_1} \only<2> {\vec{f}_j}\\
%\uncover<2> {\displaystyle \sum_{j=1}^{P_{\scaleto{2}{3pt}}} C_{j2} \vec{f}_j} \\
%\uncover<2> \vdots \\
%\uncover<2> \vdots \\
%\uncover<2> {\displaystyle \sum_{j=1}^{P_{\scaleto{2}{3pt}}} C_{j P_{\scaleto{3}{3pt}}} \vec{f}_j}
%\end{array} \right]
%\end{equation}
%
%\tiny
%{
%
%\only<1>
%{
%Note: $K_1$ is the deterministic stiffness matrix; \\
%\vspace{0.1cm}
%\hspace{0.55cm} $\vec{d}_1$ is the displacement vector for all the nodes; \\
%\vspace{0.1cm}
%\hspace{0.55cm} $\vec{f}_1$ is the forcing vector for all the nodes; \\
%\vspace{0.1cm}
%\hspace{0.55cm} Size of the stiffness matrix is $N \times N$, $N$ is the number of deterministic DOFs. \\
%\vspace{0.1cm}
%\hspace{0.55cm} \color{white}{deterministic DOFs;} \\
%}
%\only<2>
%{
%Note: $C_{ijk}= \langle \Psi_i \Psi_j \Psi_k \rangle$, $C_{ij}= \langle \Psi_i \Psi_j \rangle$; \\
%\vspace{0.1cm}
%\hspace{0.55cm} $K_i$ is the block stiffness matrix with $i$th PC coefficients of modulus, {\tiny for example, $K_1$ is the deterministic matrix}; \\
%\vspace{0.1cm}
%\hspace{0.55cm} $\vec{d}_i$ is the block vector for $i$th PC coefficients of displacement for all the nodes; \\
%\vspace{0.1cm}
%\hspace{0.55cm} $\vec{f}_i$ is the block vector for $i$th PC coefficients of forcing for all the nodes; \\
%\vspace{0.1cm}
%\hspace{0.55cm} Size of the global matrix is $(N \times P_3) \times (N \times P_3)$, $N$ is the number of deterministic DOFs; \\
%
%}
%}
%
%\begin{flushright}
%\tiny
%\hyperlink{3D_matrix}{\beamergotobutton{}}
%\end{flushright}
%
%
%\end{frame}
%\begin{frame}{Size of the stochastic stiffness matrix}
%\begin{itemize}
%
%\item Governed by:
%\begin{itemize}
%\item PC dimension $\rightarrow$ function of correlation length
%\item PC order $\rightarrow$ function of COV
%\end{itemize}
%
%\begin{small}
%
%\begin{table}
%\begin{center}
%\begin{tabular}{clr}
%\hline
%PC dimension & Order of PC & Size of Stiffness Matrix \\
% \hline \hline
%2 & 1 & Real DOFs x 3 \\
% & 2 & x 6 \\
% & 4 & x 15 \\
% \hline
%4 & 1 & Real DOFs x 5 \\
% & 2 & x 15 \\
% & 4 & x 70 \\
% \hline
%6 & 1 & Real DOFs x 7 \\
% & 2 & x 28 \\
% & 4 & x 210 \\
% \hline
%\end{tabular}
%\end{center}
%\end{table}
%
%\end{small}
%
%\end{itemize}
%\end{frame}
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%\subsection[Constitutive update]{Intrusive constitutive update}
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% \subsection{Direct Solution for Probabilistic Stiffness and Stress in 1D}
%
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%%%%%%%%%%%%%%%%%%%%%%%%%% BEGGINING PEP %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}{Direct Probabilistic Constitutive Solution in 1D}
% \begin{itemize}
%
% \vspace{0.5cm}
%
% \item<1> Probabilistic constitutive modeling : \vspace{0.5cm}
\begin{itemize}
\item Zero elastic region 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*{2mm}
\item Uncertain:
init. stiff. $H_a$,
shear strength $H_a/c_r$,
strain $\Delta \epsilon$:
$ H_a = \Sigma h_i \Phi_i; \;\;\;
C_r = \Sigma c_i \Phi_i; \;\;\;
\Delta\epsilon = \Sigma \Delta\epsilon_i \Phi_i $
\vspace*{2mm}
\item Resulting stress and stiffness are also uncertain
% 
%  $ \sum_{l=1}^{P_{\sigma}} \Delta\sigma_i \Phi_i = \sum_{i=1}^{P_h} \sum_{k=1}^{P_e}\ h_i \Delta \epsilon_k \Phi_i \Phi_k  \sum_{j=1}^{P_g} \sum_{k=1}^{P_e}\sum_{l=1}^{P_{\sigma}} \ c_i \Delta \epsilon_k \sigma_l \Phi_j \Phi_k \Phi_l$
% 
%  $ \sum_{l=1}^{P_{E_t}} \Delta E_{t_i} \Phi_i = \sum_{i=1}^{P_h} h_i \Phi_i \pm \sum_{i=1}^{P_c} \sum_{l=1}^{P_{\sigma}} \ c_i \sigma_l \Phi_i \Phi_l$
% 
\end{itemize}
% \vspace{0.5cm}
% \vspace{1cm}
%\item<1> Time integration is done via Newmark algorithm
%
% \end{itemize}
%
\end{frame}
% % % % % % % % % % % % % % % %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Direct Probabilistic Stiffness Solution}
\begin{itemize}
\item Analytic product for all the components,
$ E^{EP}_{ijkl}
=
\left[
E^{el}_{ijkl}

\frac{\displaystyle E^{el}_{ijmn} m_{mn} n_{pq} E^{el}_{pqkl}}
{\displaystyle n_{rs} E^{el}_{rstu} m_{tu}  \xi_* h_*}
\right]
$
\item Stiffness: each Polynomial Chaos component is updated incrementally
% at each Gauss Point via stochastic Galerkin projection
\small{$E_{t_1}^{n+1} = \frac{1}{<\Phi_1\Phi_1> }\{\sum_{i=1}^{P_h} \ h_i <\Phi_i \Phi_1> \pm \sum_{j=1}^{P_c} \sum_{l=1}^{P_{\sigma}} \ c_j \sigma_l^{n+1} <\Phi_j \Phi_l \Phi_1>\}$}
$\large{\vdots}$
\small{$E_{t_P}^{n+1} = \frac{1}{<\Phi_1\Phi_P> }\{\sum_{i=1}^{P_h} \ h_i <\Phi_i \Phi_P> \pm \sum_{j=1}^{P_c} \sum_{l=1}^{P_{\sigma}} \ c_j \sigma_l^{n+1} <\Phi_j \Phi_l \Phi_P>\}$}
\item Total stiffness is :
$ E_{t}^{n+1} = \sum_{l=1}^{P_{E}} E_{t_i}^{n+1} \Phi_i $
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Direct Probabilistic Stress Solution}
\begin{itemize}
\item Analytic product, for each stress component,
$ \Delta \sigma_{ij} = E^{EP}_{ijkl} \; \Delta \epsilon_{kl} $
% =
% \left[
% E^{el}_{ijkl}
% 
% \frac{\displaystyle E^{el}_{ijmn} m_{mn} n_{pq} E^{el}_{pqkl}}
% {\displaystyle n_{rs} E^{el}_{rstu} m_{tu}  \xi_* h_*}
% \right]
% \Delta \epsilon_{kl}
%
\vspace*{1mm}
\item Incremental stress: each Polynomial Chaos component is updated
incrementally
% via stochastic Galerkin projection
{$\Delta\sigma_1^{n+1} = \frac{1}{<\Phi_1\Phi_1> }\{\sum_{i=1}^{P_h} \sum_{k=1}^{P_e}\ h_i \Delta \epsilon_k^n <\Phi_i \Phi_k \Phi_1> \sum_{j=1}^{P_g} \sum_{k=1}^{P_e}\sum_{l=1}^{P_{\sigma}} \ c_j \Delta \epsilon_k^n \sigma_l^n <\Phi_j \Phi_k \Phi_l \Phi_1>\}$}
${\vdots}$
{$\Delta\sigma_P^{n+1} = \frac{1}{<\Phi_P\Phi_P> }\{\sum_{i=1}^{P_h} \sum_{k=1}^{P_e}\ h_i \Delta \epsilon_k^n <\Phi_i \Phi_k \Phi_P> \sum_{j=1}^{P_g} \sum_{k=1}^{P_e}\sum_{l=1}^{P_{\sigma}} \ c_j \Delta \epsilon_k^n \sigma_l^n <\Phi_j \Phi_k \Phi_l \Phi_P>\}$}
\vspace*{1mm}
\item Stress update:
$ \sum_{l=1}^{P_{\sigma}} \sigma_i^{n+1} \Phi_i = \sum_{l=1}^{P_{\sigma}} \sigma_i^{n} \Phi_i + \sum_{l=1}^{P_{\sigma}} \Delta\sigma_i^{n+1} \Phi_i$
\end{itemize}
\end{frame}
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%
% \begin{frame}
% \frametitle{Probabilistic ElasticPlastic Response}
%
%
% % % \vspace*{5mm}
% % \begin{center}
% % % \hspace*{15mm}
% % \movie[label=show3,width=7cm,poster,autostart,showcontrols]
% % {\includegraphics[width=7cm]
% % {/home/jeremic/tex/works/Thesis/HanYang/Files_06June2017/DOE_Annual_2017/Figures/NPP_Plastic_Dissipation_Density.png}}
% % %{/home/jeremic/tex/works/Thesis/HanYang/Files_06June2017/DOE_Annual_2017/Figures/NPP_without_Contact_vonMises.mp4}
% % {NPP_without_Contact_vonMises.mp4}
% % \end{center}
%
% %\vspace*{5mm}
% \begin{center}
% % \hspace*{15mm}
% \movie[label=show3,width=9cm,poster,autostart,showcontrols]
% {\includegraphics[width=9cm]
% {/home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Latex/img/figure_PEP_25.png}}
% % /home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Latex/img/figure_PEP_25.pdf
% %{/home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Animations/PEP_Animation.mp4}
% {/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Probabilistic_Elasto_Plasticity_and_Stochastic_Elastic_Plastic_Finite_Element_Method/PEP_Animation.mp4}
% \end{center}
%
%
%
%
% \begin{flushleft}
% \vspace*{15mm}
% \href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Probabilistic_Elasto_Plasticity_and_Stochastic_Elastic_Plastic_Finite_Element_Method/PEP_Animation.mp4}
% % \href{./homo_50mmesh_45degree_Ormsby.mp4}
% {\tiny (MP4)}
% \end{flushleft}
% %
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% % \includegraphics[width = 12cm]{./img/figure_PEP_25.pdf}
%
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%\section[Formulation]{Stochastic Dynamic Finite Element Formulation}
%\subsection[Time domain stochastic Galerkin method]{Time domain stochastic Galerkin method}
%\frame{\tableofcontents[currentsubsection,sectionstyle=show/shaded]}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Stochastic 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 solution (single step)
\item[] It is NOT based on Monte Carlo approach
\item[] System of equations does grow (!)
\end{itemize}
% \normalsize{Typical number of terms required for a SEPFEM problem} \vspace{1cm}\\
\scalebox{0.7}{
\begin{tabular}{ c c c c}
\# KL terms material & \# KL terms load & PC order displacement& Total \# terms per DoF\\ \hline \\
4 & 4 & 10 & 43758 \\
4 & 4 & 20 & 3 108 105 \\
4 & 4 & 30 & 48 903 492 \\
6 & 6 & 10 & 646 646 \\
6 & 6 & 20 & 225 792 840 \\
6 & 6 & 30 & 1.1058 $10^{10}$ \\
8 & 8 & 10 & 5 311 735 \\
8 & 8 & 20 & 7.3079 $10^{9}$ \\
8 & 8 & 30 & 9.9149 $10^{11}$\\
... & ... & ... & ...\\
% \hline
\end{tabular}}
\end{frame}
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\begin{frame}
\frametitle{SEPFEM: Example in 1D}
\vspace*{2mm}
\begin{center}
% \hspace*{15mm}
\movie[label=show3,width=9cm,poster,autostart,showcontrols]
{\includegraphics[width=9cm]{/home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Latex/img/figure_elastic_900.png}}
% /home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Latex/img/figure_PEP_25.pdf
%{/home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Animations/SEPFEM_Animation_Elastic.mp4}
{/home/jeremic/lecture_notes_online_material/_Chapter_Probabilistic_Elasto_Plasticity_and_Stochastic_Elastic_Plastic_Finite_Element_Method/SEPFEM_Animation_Elastic.mp4}
%{SEPFEM_Animation_Elastic.mp4}
\end{center}
%
% \vspace*{2mm}
% \begin{center}
% % \hspace*{15mm}
% \movie[label=show3,width=9cm,poster,autostart,showcontrols]
% {\includegraphics[width=9cm]{/home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Latex/img/figure_elastic_900.png}}
% % /home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Latex/img/figure_PEP_25.pdf
% {/home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Animations/SEPFEM_Animation_Elastic.mp4}
% \end{center}
%
% \includegraphics[width = 12cm]{./img/figure_elastic_900.pdf}
\begin{flushleft}
\vspace*{15mm}
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Probabilistic_Elasto_Plasticity_and_Stochastic_Elastic_Plastic_Finite_Element_Method/SEPFEM_Animation_Elastic.mp4}
% \href{./homo_50mmesh_45degree_Ormsby.mp4}
{\tiny (MP4)}
\end{flushleft}
%
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%
% \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}
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%\subsection{Uncertain Inelasticity}
\subsection{Risk Analysis}
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\begin{frame}
\frametitle{Probabilistic Seismic Risk Analysis (PSRA)}
\begin{textblock}{15}(0.8, 3.9)
\scriptsize
Uncertain source, path, \\
\quad site and structure
\end{textblock}
\begin{textblock}{15}(4.8, 4.2)
$\Longrightarrow$
\end{textblock}
\begin{textblock}{15}(5.8, 3.7)
\scriptsize
%Acceptably small probability\\
Probabilities of \\
engineering demand parameters (EDP) \\
damage measures (DM), loss, etc
\end{textblock}
\begin{textblock}{15}(0.4, 11.8)
\tiny
\textbf{Uncertain source rupture}
\end{textblock}
\begin{textblock}{15}(3.55, 4.8)
\begin{figure}[H]
\flushleft
\includegraphics[width=0.7\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/problem_statement.pdf}
\end{figure}
\end{textblock}
\begin{textblock}{15}(12.5, 5.5)
\tiny
adapted from Taga(1982)
\end{textblock}
\begin{textblock}{15}(11.7, 10.6)
\tiny
\textbf{Uncertain material properties}
\end{textblock}
\begin{textblock}{15}(0.4, 8.0)
\begin{figure}[H]
\flushleft
\includegraphics[width=0.2\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/4231447_Northridge_eq.pdf}
\end{figure}
\end{textblock}
\begin{textblock}{15}(10.7, 10.2)
\begin{figure}[H]
\flushleft
\includegraphics[width=0.3\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/uncertain_SPT.pdf}
\end{figure}
\end{textblock}
\end{frame}
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\begin{frame}
%\frametitle{State of the Art Probabilistic Seismic Risk Analysis}
\frametitle{Probabilistic Seismic Risk Analysis}
\begin{itemize}
% \item Target: safe design, acceptably small failure probability $\lambda (EDP)$
\item[]
%Powerful tool
%Allows for
Objective, quantitative decision making based on exceedance rate
$\lambda (EDP>z)$
\item[] PSRA: convolution of PSHA and fragility
% \[\lambda(EDP>z) = \int_{IM} \underbrace{\frac{d\lambda(IM)}{dIM}}_\text{PSHA} \underbrace{G(EDPIM)}_{\text{fragility}} dIM \]
\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}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Current State of Art Seismic Risk Analysis (SRA)}
%
%
% \begin{itemize}
% %\vspace{2mm}
%
% \item[] Intensity measure (IM) selected as a proxy for ground motions,
% usually Spectral acceleration $Sa(T_0)$
%
% \vspace{4mm}
% \item[] Ground Motion Prediction Equations (GMPEs) need development, ergodic or site specific
%
% \vspace{4mm}
% \item[] Probabilistic seismic hazard analysis (PSHA)
% % for ground motion $\lambda(Sa>z)$
% % \begin{equation*}
% % \resizebox{0.85\hsize}{!}{%
% % $\lambda(Sa>z) = \sum_{i=1}^{NFL} \underbrace{N_i \int\int f_{mi}(M) f_{ri}(RM)}_\text{seismic source characterization (SSC)} \underbrace{P(Sa>zM, R)}_\text{GMPE} dM dR$}
% % \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[] Intensity Measure(s) (IM) serves as the proxy of damaging ground motions
\vspace{2mm}
\item[] Does a single IM, e.g., $Sa(T_0)$, represent all uncertainty?
%\item[] Practically difficult/contentious to choose
%\vspace{3mm}
\vspace{2mm}
\item[] IMs difficult to choose, Spectral Acc, PGA, PGV...
%%\vspace{3mm}
%\item[] Single IM does not contain all/most uncertainty
\vspace{2mm}
\item[] Fragility analysis: incremental dynamic analysis (IDA)
% using Monte Carlo method
\vspace{2mm}
\item[] Use of Monte Carlo method, accuracy, efficiency...
%\vspace{3mm}
\vspace{2mm}
\item[] Monte Carlo, computationally expensive, CyberShake for LA, 20,000
cases, 100Y runtime, (Maechling et al. 2007)
%
%
% \vspace{3mm}
% \item[] Miscommunication between seismologists and struct/geotech engineers,
% $Sa(T_0)$ not compatible with nonlinear FEM
\end{itemize}
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% %
%
% \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 $\sigma_{ij}$,
% $\epsilon_{ij}$, $u_i$...
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% \end{itemize}
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%
%
% \begin{itemize}
%
% %\item[] Material uncertainties: expanded along stochastic shape functions:
% \item[] Material uncertainties: stochastic shape functions:
% $E^{ep}(x,t,\theta) = \sum_{i=0}^{P_d} E_i(x,t) * \Phi_i[\{\xi_1, ..., \xi_m\}]$
%
% \vspace*{1mm}
% \item[] Loading uncertainties: stochastic shape functions
% $F(x,t,\theta) = \sum_{i=0}^{P_f} F_i(x,t) * \zeta_i[\{\xi_{m+1}, ..., \xi_f]$
%
% \vspace*{1mm}
% \item[] Displacement expanded: stochastic shape functions:
% $u(x,t,\theta) = \sum_{i=0}^{P_u} u_i(x,t) * \Psi_i[\{\xi_1, ..., \xi_m, \xi_{m+1}, ..., \xi_f\}]$
%
%
% \vspace*{1mm}
% \item[]
% Stochastic system of equations
% \vspace*{2mm}
% \begin{tiny}
% \[
% \begin{bmatrix}
% \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_0> K^{(k)} & \dots & \sum_{k=0}^{P_d} <\Phi_k \Psi_P \Psi_0> K^{(k)}\\
% \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_1> K^{(k)} & \dots & \sum_{k=0}^{P_d} <\Phi_k \Psi_P \Psi_1> K^{(k)}\\ \\
% \vdots & \vdots & \vdots & \vdots\\
% \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_P> K^{(k)} & \dots & \sum_{k=0}^{M} <\Phi_k \Psi_P \Psi_P> K^{(k)}
% \end{bmatrix}
% \begin{bmatrix}
% u_{10} \\
% \vdots \\
% u_{N0}\\
% \vdots \\
% u_{1P_u}\\
% \vdots \\
% u_{NP_u}
% \end{bmatrix}
% =
% %\]
% %\[
% \begin{bmatrix}
% \sum_{i=0}^{P_f} f_i <\Psi_0\zeta_i> \\
% \sum_{i=0}^{P_f} f_i <\Psi_1\zeta_i> \\
% \sum_{i=0}^{P_f} f_i <\Psi_2\zeta_i> \\
% \vdots \\
% \sum_{i=0}^{P_f} f_i <\Psi_{P_u}\zeta_i>\\
% \end{bmatrix}
% \]
% \end{tiny}
%
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% \end{itemize}
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% \end{frame}
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%
% \begin{frame}
% \frametitle{Stochastic 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}
%
%
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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \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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\begin{frame}
%\frametitle{TDNIPSRA Framework}
\frametitle{Risk Framework}
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\begin{textblock}{15}(0, 4.0)
\includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/UCERF3.pdf}
\end{textblock}
\begin{textblock}{15}(0.3, 3.5)
\scriptsize{Seismic source characterization}
\end{textblock}
\begin{textblock}{15}(2.9, 5.2)
\tiny{UCERF3 (2014)}
\end{textblock}
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\begin{textblock}{15}(5.1, 6.5)
%$\Rightarrow$
{\Large $\rightarrow$}
\end{textblock}
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\begin{textblock}{15}(5.8, 3.9)
\vspace*{1mm}
\includegraphics[width=0.27\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/SMSIM.pdf}
\end{textblock}
\begin{textblock}{15}(7.1, 6.2)
\scalebox{.9}{\tiny{Fourier spectra}}
\\
\vspace*{0.2cm}
\scalebox{.9}{\tiny{\hspace{0.14cm} Boore(2003)}}
\end{textblock}
\begin{textblock}{15}(6.1, 3.5)
\scriptsize{Stochastic ground motion}
\end{textblock}
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\begin{textblock}{15}(9.9, 6.5)
%{\bf $\Rightarrow$}
{\Large $\rightarrow$}
\end{textblock}
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\begin{textblock}{15}(10.5, 4.2)
% \includegraphics[width=0.35\linewidth]{pic/KL_exact_dis_correlation_from_dis.pdf}
\includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Acc_realization_200.pdf}
\end{textblock}
\begin{textblock}{15}(11.1, 9.6)
\scriptsize{Uncertainty characterization \\
\hspace{0.1cm} Hermite polynomial chaos}
\end{textblock}
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{\Large $\leftarrow$}
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\begin{textblock}{15}(11, 11.2)
\includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/structural_uncertainty.pdf}
\end{textblock}
\begin{textblock}{15}(5.3, 10.75)
\includegraphics[width=0.33\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/probabilsitc_evolution.png}
\end{textblock}
\begin{textblock}{15}(5.4, 9.6)
\scriptsize{\quad \quad Uncertainty propagation \\
\quad \quad \quad \quad SEPFEM}
\end{textblock}
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%$\Leftarrow$
{\Large $\leftarrow$}
\end{textblock}
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\includegraphics[width=0.29\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/seismic_risk_result_framework.png}
\end{textblock}
\begin{tikzpicture}[remember picture, overlay]
\draw[line width=1pt, draw=black, rounded corners=4pt, fill=gray!20, fill opacity=1]
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\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}
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% \begin{textblock}{15}(0, 4.0)
% \includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/UCERF3.pdf}
% \end{textblock}
%
% \begin{textblock}{15}(0.3, 3.5)
% \scriptsize{Seismic source characterization}
% \end{textblock}
%
% \begin{textblock}{15}(2.9, 5.2)
% \tiny{UCERF3 (2014)}
% \end{textblock}
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% \begin{textblock}{15}(5.1, 6.5)
% $\Rightarrow$
% \end{textblock}
%
%
% \begin{textblock}{15}(5.8, 3.9)
% \vspace*{1mm}
% \includegraphics[width=0.27\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/SMSIM.pdf}
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%
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% \begin{textblock}{15}(7.1, 6.2)
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% \\
% \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}
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% \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}
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% % \quad \quad \quad \quad stochastic FEM}
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% \begin{textblock}{15}(2.9, 5.2)
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% \end{textblock}
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% \begin{textblock}{15}(5.1, 6.5)
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% \includegraphics[width=0.27\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/SMSIM.pdf}
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% \scriptsize{\quad \quad Uncertainty propagation \\
% \quad \quad \quad \quad stochastic FEM}
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% \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}
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% \begin{textblock}{15}(1.6, 10.7)
% \scriptsize{EDP hazard/risk}
% \end{textblock}
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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}
%next five years
% as envisioned by Abrahamson (2018)
\end{itemize}
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\begin{frame}
%\frametitle{TDNIPSRA Example Object}
\frametitle{Risk Example}
\begin{textblock}{15}(0.5, 4.0)
\includegraphics[width=0.47\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/faults_configuration_new.pdf}
\end{textblock}
\begin{textblock}{15}(7.5, 3.4)
\includegraphics[width=0.55\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/SSC_legend.pdf}
\end{textblock}
\begin{textblock}{15}(0.8, 11.6)
\scriptsize
\begin{itemize}
\item Fault 1: San Gregorio fault
\item Fault 2: Calaveras fault
\item Uncertainty: Segmentation, \\ slip rate, rupture geometry, etc.
\end{itemize}
\end{textblock}
\begin{textblock}{15}(8.5, 11.6)
\scriptsize
\begin{itemize}
\item 371 total seismic scenarios
\item $M \ 5 \sim 5.5$ and $6.5 \sim 7.0$
\item $R_{jb} \ 20km \sim 40km$
\end{itemize}
\end{textblock}
\end{frame}
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\begin{frame}
\frametitle{Stochastic Ground Motion Modeling}
\begin{textblock}{15}(1.0, 3.7)
\small Realizations of simulated uncertain motions for scenario $M=7$, $R=15km$:
\end{textblock}
\begin{textblock}{15}(0.5, 4.0)
\includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Acc_time_series100.pdf}
\end{textblock}
\begin{textblock}{15}(5.5, 4.0)
\includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Acc_time_series343.pdf} \enspace
\end{textblock}
\begin{textblock}{15}(10.5, 4.0)
\includegraphics[width=0.35\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Acc_time_series439.pdf}
\end{textblock}
\begin{textblock}{15}(1.0, 9.2)
\small Verification with GMPE:
\end{textblock}
\begin{textblock}{15}(0.3, 9.5)
\includegraphics[width=0.36\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/SA_GMPE_verification_std_08_no_smooth.pdf}
\end{textblock}
\begin{textblock}{15}(5.6, 9.5)
\includegraphics[width=0.36\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Goodness_fit_std_08_no_smooth.pdf}
\end{textblock}
\begin{textblock}{15}(10.8, 9.5)
\includegraphics[width=0.36\linewidth]{/home/jeremic/tex/works/Conferences/2020/Natural_Phenomena_Hazard_Oct2020/present/from_Hexiang_17Oct2020/pic/Lec4/Standard_deviation_std_08_no_smooth_new.pdf}
\end{textblock}
% \begin{textblock}{15}(0.5, 11.0)
% \begin{itemize}
% \item $\Delta \sigma= 84bar$, $\kappa=0.03s$ with total $\sigma=0.8ln$.
% \item Simulated median is not biased.
% \item Consistent total uncertainties with GMPE.
% \end{itemize}
% \end{textblock}
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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}
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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/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.}
\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}
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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}
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\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)
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\frametitle{Seismic Risk Analysis}
\vspace{0.5cm}
\begin{itemize}
% \item[] \small Damage measure (DM) defined on multiple EDPs:
\item[] Damage measure (DM) defined on multiple EDPs:
\vspace{2mm}
{\scriptsize $DM: \{\text{MIDR}>1\%\, \cup \,\text{PFA}>1{\rm m/s^2} \}$, seismic risk is \boldsymbol{$4.2 \times 10^{3}/yr$} }
\vspace{1mm}
{\scriptsize $DM: \{\text{MIDR}>1\%\, \cap \,\text{PFA}>1{\rm m/s^2} \}$, seismic risk is \boldsymbol{$1.71 \times 10^{3}/yr$}}
\vspace*{3mm}
% \item[] \small Damage measure (DM) defined on single EDP:
\item[] Damage measure defined on single EDP:
\vspace*{15mm}
\begin{textblock}{15}(0.7,8.9)
\begin{table}[!htbp]
\small
\resizebox{0.98\hsize}{!}{
\begin{tabular}{ccccccc}
%\hline
\textbf{DM} & MIDR\textgreater{}0.5\% & \textbf{MIDR\textgreater{}1\%} & MIDR\textgreater{}2\% & PFA\textgreater{}0.5${\rm m/s^2}$ & \textbf{PFA\textgreater{}1\boldsymbol{${\rm m/s^2}$}} & PFA\textgreater{}1.5${\rm m/s^2}$ \\
\hline
\textbf{Risk [/yr]} & 6.66$\times 10^{3}$ & \textbf{3.83\boldsymbol{$\times 10^{3}$}} & 9.97$\times 10^{5}$ & 6.65$\times 10^{3}$ & \textbf{1.92 \boldsymbol{$\times 10^{3}$}} & 9.45$\times 10^{5}$ \\
%\hline
\end{tabular}}
\end{table}
\end{textblock}
\vspace*{30mm}
\item[] \small Seismic risk for DM defined on multiple EDPs can be quite
different from that defined on single EDP.
\end{itemize}
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% % \includegraphics[width=0.55\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/PDF_evolution.png}
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% \includegraphics[width=0.45\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/combined_risk_curve.pdf}
% \end{textblock}
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% \includegraphics[width=0.48\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/risk_deaggregation_MIDR1.pdf}
% \end{textblock}
%
% \begin{textblock}{15}(0.9, 11.9)
% \begin{itemize}
% \item[] \scriptsize $\lambda(MIDR >1 \%) = 9.7 \times 10^{3}$
% \item[] \scriptsize $\lambda(MIDR >2 \%) = 1.7 \times 10^{3}$
% \item[] \scriptsize $\lambda(MIDR >4 \%) = 5.9 \times 10^{5}$
% \end{itemize}
% \end{textblock}
%
% \begin{textblock}{15}(9.3, 12.5)
% \scriptsize Risk deaggregation for $\lambda(MIDR >1 \%)$
% \end{textblock}
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\section{Summary}
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\frametitle{Appropriate Science Quotes}
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\item[] Fran{\c c}oisMarie Arouet, Voltaire:
"Le doute n'est pas une condition agr{\'e}able, mais la certitude est absurde."
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\item[] Niklaus Wirth:
"Software is getting slower more rapidly than hardware becomes faster."
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\item[] Numerical modeling to predict and inform
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\item[] Engineer needs to know!
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\item[] Education and Training is the key!
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\item[] Collaborators: Feng, Yang, Behbehani, Sinha, Wang,
Karapiperis, Wang, Lacoure, Pisan{\'o}, Abell, Tafazzoli, Jie, Preisig,
Tasiopoulou, Watanabe, Cheng, Yang.
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\item[] Funding from and collaboration with the
ATC/USFEMA, USDOE, USNRC, USNSF,
CNSCCCSN, UNIAEA, ENSICHB\&H and Shimizu Corp. is greatly appreciated,
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\item[]
\url{http://sokocalo.engr.ucdavis.edu/~jeremic}
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