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% % (used in a file _Chapter_SoftwareHardware_Domain_Specific_Language_English.tex
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% \usetheme{Antibes} % ima sadrzaj gore i kao graf ...
% \usetheme{Berkeley} % ima sadrzaj desno
% \usetheme{Berlin} % ima sadrzaj gore i tackice
% \usetheme{Goettingen} % ima sadrzxaj za desne strane
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% colorlinks=true,
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% linktocpage,
% pdftex]{hyperref}
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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)
%{ 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 Winter School, Ascona CH \\
January 2020}
\subject{}
% This is only inserted into the PDF information catalog. Can be left
% out.
% If you have a file called "universitylogofilename.xxx", where xxx
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%\logo{\pgfuseimage{universitylogo}}
% \pgfdeclareimage[height=0.5cm]{universitylogo}{universitylogofilename}
% \logo{\pgfuseimage{universitylogo}}
% Delete this, if you do not want the table of contents to pop up at
% the beginning of each subsection:
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{
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\begin{document}
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\begin{frame}
\titlepage
\end{frame}
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\begin{frame}
\frametitle{Outline}
\begin{scriptsize}
\tableofcontents
% You might wish to add the option [pausesections]
\end{scriptsize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 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.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Motivation}
\subsection{\ }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Motivation}
\begin{itemize}
%\vspace*{0.3cm}
\item[] Improve modeling and simulation for infrastructure objects
% \vspace*{2mm}
% \item[] Expert numerical modeling and simulation tool
%
% \vspace*{1mm}
% \item[] Use of numerical models to
% analyze statics and dynamics of soil/rockstructure systems
%
\vspace*{3mm}
\item[] Reduction of modeling uncertainty
\vspace*{3mm}
\item[] Choice of analysis level of sophistication
\vspace*{3mm}
\item[] Account for parametric uncertainty
\vspace*{3mm}
\item[] Goal: Predict and Inform rather than (force) fit
\vspace*{3mm}
\item[] Engineer needs to know!
%
%
%
% \vspace*{1mm}
% \item[] Follow the flow, input and dissipation, of seismic energy,
% \vspace*{2mm}
% \item[]
% %System for
% {\bf Real}istic modeling and simulation of
% {\bf E}arthquakes and/or
% {\bf S}oils and/or
% {\bf S}tructures and their
% {\bf I}nteraction:\\
% RealESSI
% \hspace*{5mm}
% \url{http://realessi.info/}
% % % % \hspace*{25mm}
% % \url{http://sokocalo.engr.ucdavis.edu/~jeremic/Real_ESSI_Simulator/}
% % % \href{http://sokocalo.engr.ucdavis.edu/~jeremic/Real_ESSI_Simulator/}{{http://sokocalo.engr.ucdavis.edu/~jeremic/Real_ESSI_Simulator/}
% % % % \url{http://msessi.info/}
% % %
%
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Numerical Prediction under Uncertainty}
\begin{itemize}
%\vspace*{1mm}
\item \underline{Modeling Uncertainty}, Simplifying assumptions
\begin{itemize}
\vspace*{2mm}
\item[] Low, medium, high sophistication modeling and simulation
\vspace*{2mm}
\item[] Choice of sophistication level for confidence in results
\end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace*{4mm}
\item \underline{Parametric Uncertainty}, ${M} \ddot{u_i} + {C} \dot{u_i} + {K}^{ep} {u_i} = {F(t)}$,
\begin{itemize}
\vspace*{2mm}
\item[] Uncertain 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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Modeling 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}
%OVDE
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Modeling Uncertainty, 6C vs 1C Motions}
%
%
% % local
% \vspace*{2mm}
% \begin{center}
% \hspace*{7mm}
% %\movie[label=show3,width=8.8cm,poster,autostart,showcontrols]
% \movie[label=show3,width=8.8cm,poster, showcontrols]
% {\includegraphics[width=92mm]
% {/home/jeremic/tex/works/Conferences/2016/IAEA_TecDoc_February2016/My_Current_Work/movie_2_npps_mp4_icon.jpeg}}
% {/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Applications_ESSI_for_NPPs/Model01_ESSI_Response_May2015/movie_2_npps.mp4}
% \end{center}
% % local
% % \vspace*{2mm}
% % \begin{center}
% % \hspace*{7mm}
% % \movie[label=show3,width=8.8cm,poster,autostart,showcontrols]
% % {\includegraphics[width=90mm]{movie_2_npps_mp4_icon.jpeg}}{movie_2_npps.mp4}
% % \end{center}
%
%
% % online
% \vspace*{12mm}
% \begin{flushleft}
% %\vspace*{15mm}
% \href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Applications_Earthquake_Soil_Structure_Interaction_General_Aspects/ESSI_VisIt_movies_Jose_19May2015/movie_2_npps.mp4}
% {\tiny (MP4)}
% \end{flushleft}
% % online
%
%
%
%
%
% \end{frame}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Parametric Uncertainty: Material and Motions}
\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}
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Parametric Uncertainty: Soil Stiffness and Strength}
%
%
% \vspace*{2mm}
% %\vspace*{3mm}
% \begin{figure}[!hbpt]
% \begin{center}
% %
% \hspace*{7mm}
% \includegraphics[width=5.5truecm]{/home/jeremic/tex/works/Papers/2008/JGGEGoverGmax/figures/YoungModulus_RawData_and_MeanTrend_01Ed.pdf}
% \hspace*{3mm}
% % \hfill
% \includegraphics[width=3.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGEGoverGmax/figures/YoungModulus_Histogram_Normal_01Ed.pdf}
% %
% \end{center}
% \end{figure}
%
% \vspace*{5mm}
% \begin{figure}[!hbpt]
% \begin{center}
% %
% \hspace*{7mm}
% \includegraphics[width=5.00truecm]{/home/jeremic/tex/works/Papers/2008/JGGEGoverGmax/figures/ShearStrength_RawData_and_MeanTrendMod.pdf}
% \hspace*{3mm}
% % \hfill
% \includegraphics[width=3.0truecm]{/home/jeremic/tex/works/Papers/2008/JGGEGoverGmax/figures/ShearStrength_Histogram_PearsonIVFineTunedMod.pdf}
% %
% \end{center}
% \end{figure}
%
% %\vspace*{5mm}
% %\vspace*{1.8cm}
% %\hspace*{3.3cm}
% \begin{flushright}
% {\tiny
% (cf. Phoon and Kulhawy (1999B))\\
% ~}
% \end{flushright}
% %
%
%
% \end{frame}
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Parametric Uncertainty: Material Properties}
%
%
%
% \vspace*{5mm}
% \begin{figure}[!hbpt]
% \begin{center}
% % %
% \hspace*{3mm}
% \includegraphics[width=2.5truecm]{/home/jeremic/tex/works/Thesis/KonstantinosKarapiperis/Soil_Uncertainty_Report_Pdf_Cdf_Figures/FieldPhiPdf.pdf}
% \hspace*{3mm}
% \includegraphics[width=2.5truecm]{/home/jeremic/tex/works/Thesis/KonstantinosKarapiperis/Soil_Uncertainty_Report_Pdf_Cdf_Figures/FieldPhiCdf.pdf}
% \hspace*{3mm}
% \includegraphics[width=2.5truecm]{/home/jeremic/tex/works/Thesis/KonstantinosKarapiperis/Soil_Uncertainty_Report_Pdf_Cdf_Figures/FieldSuPdf.pdf}
% \hspace*{3mm}
% \includegraphics[width=2.5truecm]{/home/jeremic/tex/works/Thesis/KonstantinosKarapiperis/Soil_Uncertainty_Report_Pdf_Cdf_Figures/FieldSuCdf.pdf}
% \\
% %\vspace*{2mm}
% \hspace*{2.5cm} \mbox{\tiny Field $\phi$} \hspace*{3.5cm} \mbox{\tiny Field $c_u$}
% \\
% \vspace*{10mm}
% \hspace*{3mm}
% \includegraphics[width=2.5truecm]{/home/jeremic/tex/works/Thesis/KonstantinosKarapiperis/Soil_Uncertainty_Report_Pdf_Cdf_Figures/LabPhiPdf.pdf}
% \hspace*{3mm}
% \includegraphics[width=2.5truecm]{/home/jeremic/tex/works/Thesis/KonstantinosKarapiperis/Soil_Uncertainty_Report_Pdf_Cdf_Figures/LabPhiCdf.pdf}
% \hspace*{3mm}
% \includegraphics[width=2.5truecm]{/home/jeremic/tex/works/Thesis/KonstantinosKarapiperis/Soil_Uncertainty_Report_Pdf_Cdf_Figures/LabSuPdf.pdf}
% \hspace*{3mm}
% \includegraphics[width=2.5truecm]{/home/jeremic/tex/works/Thesis/KonstantinosKarapiperis/Soil_Uncertainty_Report_Pdf_Cdf_Figures/LabSuCdf.pdf}
% \\
% %\vspace*{8mm}
% \hspace*{2.5cm} \mbox{\tiny Lab $\phi$} \hspace*{3.5cm} \mbox{\tiny Lab $c_u$}
% \end{center}
% \end{figure}
%
%
%
% \end{frame}
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{RealESSI Simulator System}
%
\vspace*{2mm}
The RealESSI,
{\underline {\bf Real}}istic
%{\underline {\bf M}}odeling and
%{\underline {\bf S}}imulation of
{M}odeling and
{S}imulation of
{\underline {\bf E}}arthquakes,
{\underline {\bf S}}oils,
{\underline {\bf S}}tructures and their
{\underline {\bf I}}nteraction. Simulator is a software, hardware and
documentation system for time domain,
linear and nonlinear, inelastic, deterministic or probabilistic, 3D,
modeling and simulation of:
\vspace*{1mm}
\begin{itemize}
%\vspace*{1mm}
\item[] statics and dynamics of soil,
% %\vspace*{1mm}
% \item[] statics and dynamics of rock,
%\vspace*{1mm}
\item[] statics and dynamics of structures,
%\vspace*{1mm}
\item[] statics of soilstructure systems, and
%\vspace*{1mm}
\item[] dynamics of earthquakesoilstructure system interaction
\end{itemize}
Used for:
\begin{itemize}
%\vspace*{1mm}
\item[] Design: linear elastic, load combinations, dimensioning
%\vspace*{1mm}
\item[] Assessment: nonlinear/inelastic, risk, safety margins
\end{itemize}
\end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%
% \frametitle{RealESSI Simulator System}
%
%
% \begin{itemize}
%
%
% \item RealESSI System Components
% \begin{itemize}
% \item[] RealESSI Preprocessor (gmsh/gmESSI, X2ESSI)
% \item[] RealESSI Program (local, remote, cloud)
% \item[] RealESSI PostProcessor (Paraview/pvESSI, Python)
%
% \end{itemize}
%
% \vspace*{1mm}
% \item RealESSI System availability:
% \begin{itemize}
% %\vspace*{1mm}
% \item[] Educational Institutions: AWS, Linux Image, free
% \item[] Government Agencies, National Labs: AWS GovCloud
% \item[] Professional Practice: AWS, commercial
% %\vspace*{1mm}
% %%\vspace*{1mm}
% % \item Sources available to collaborators
% \end{itemize}
%
%
%
% \vspace*{1mm}
% \item RealESSI education and training: theory and applications
%
%
%
% \vspace*{1mm}
% \item RealESSI documentation and program available at
% \url{http://realessi.info/}
% %\url{http://sokocalo.engr.ucdavis.edu/~jeremic/Real_ESSI_Simulator/}
% %
% %\url{http://realessi.info/}
% %
%
%
% % \vspace*{2mm}
% % \item
% %
%
%
% \end{itemize}
%
%
% \end{frame}
%
%
%
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\section{Uncertain Inelastic Computational Mechanics}
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%\subsection{Stochastic Modeling}
\subsection{\ }
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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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%\frame{\tableofcontents[currentsubsection,sectionstyle=show/shaded]}
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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsection{Direct Solution for Probabilistic Stiffness and Stress in 1D}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%% BEGGINING PEP %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Direct Probabilistic Constitutive Solution in 1D}
% \begin{itemize}
%
% \vspace{0.5cm}
%
% \item<1> Probabilistic constitutive modeling : \vspace{0.5cm}
\begin{itemize}
\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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
%
%
% \includegraphics[width = 12cm]{./img/figure_PEP_25.pdf}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section[Formulation]{Stochastic Dynamic Finite Element Formulation}
%\subsection[Time domain stochastic Galerkin method]{Time domain stochastic Galerkin method}
%\frame{\tableofcontents[currentsubsection,sectionstyle=show/shaded]}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Stochastic ElasticPlastic Finite Element Method}
\begin{itemize}
\item Material uncertainty expanded into stochastic shape funcs.
%$E(x,t,\theta) = \sum_{i=0}^{P_d} r_i(x,t) * \Phi_i[\{\xi_1, ..., \xi_m\}]$
\vspace*{1mm}
\item Loading uncertainty expanded into stochastic shape funcs.
%$f(x,t,\theta) = \sum_{i=0}^{P_f} f_i(x,t) * \zeta_i[\{\xi_{m+1}, ..., \xi_f]$
\vspace*{1mm}
\item Displacement expanded into stochastic shape funcs.
%$u(x,t,\theta) = \sum_{i=0}^{P_u} u_i(x,t) * \Psi_i[\{\xi_1, ..., \xi_m, \xi_{m+1}, ..., \xi_f\}]$
%\item
%Stochastic system of equation resulting from Galerkin approach (static example):
%
%\item Time domain integration using Newmark and/or HHT, in probabilistic spaces
\vspace*{1mm}
\item Jeremi{\'c} et al. 2011
\end{itemize}
\begin{tiny}
\[
%$
\begin{bmatrix}
\sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_0> K^{(k)} & \dots & \sum_{k=0}^{P_d} <\Phi_k \Psi_P \Psi_0> K^{(k)}\\
\sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_1> K^{(k)} & \dots & \sum_{k=0}^{P_d} <\Phi_k \Psi_P \Psi_1> K^{(k)}\\ \\
\vdots & \vdots & \vdots & \vdots\\
\sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_P> K^{(k)} & \dots & \sum_{k=0}^{M} <\Phi_k \Psi_P \Psi_P> K^{(k)}
\end{bmatrix}
\begin{bmatrix}
\Delta u_{10} \\
\vdots \\
\Delta u_{N0}\\
\vdots \\
\Delta u_{1P_u}\\
\vdots \\
\Delta u_{NP_u}
\end{bmatrix}
=
%\]
%\[
\begin{bmatrix}
\sum_{i=0}^{P_f} f_i <\Psi_0\zeta_i> \\
\sum_{i=0}^{P_f} f_i <\Psi_1\zeta_i> \\
\sum_{i=0}^{P_f} f_i <\Psi_2\zeta_i> \\
\vdots \\
\sum_{i=0}^{P_f} f_i <\Psi_{P_u}\zeta_i>\\
\end{bmatrix}
%$
\]
\end{tiny}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{SEPFEM: System Size}
\begin{itemize}
\item SEPFEM offers a complete 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}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
{SEPFEM_Animation_Elastic.mp4}
\end{center}
%
% \vspace*{2mm}
% \begin{center}
% % \hspace*{15mm}
% \movie[label=show3,width=9cm,poster,autostart,showcontrols]
% {\includegraphics[width=9cm]{/home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Latex/img/figure_elastic_900.png}}
% % /home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Latex/img/figure_PEP_25.pdf
% {/home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Animations/SEPFEM_Animation_Elastic.mp4}
% \end{center}
%
% \includegraphics[width = 12cm]{./img/figure_elastic_900.pdf}
\begin{flushleft}
\vspace*{15mm}
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Probabilistic_Elasto_Plasticity_and_Stochastic_Elastic_Plastic_Finite_Element_Method/SEPFEM_Animation_Elastic.mp4}
% \href{./homo_50mmesh_45degree_Ormsby.mp4}
{\tiny (MP4)}
\end{flushleft}
%
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{SEPFEM: Example in 3D}
%\vspace*{5mm}
\begin{center}
% \hspace*{15mm}
\movie[label=show3,width=10cm,poster,autostart,showcontrols]
{\includegraphics[width=10cm]
{/home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Latex/img/SFEM_3D.png}}
% /home/jeremic/tex/works/Thesis/MaximeLacour/Files_06Jun2017/Panel_Review_Slides_ML/Latex/img/figure_PEP_25.pdf
{/home/jeremic/public_html/lecture_notes_online_material/_Chapter_Probabilistic_Elasto_Plasticity_and_Stochastic_Elastic_Plastic_Finite_Element_Method/SFEM_Animation_3D.mp4}
%{/home/jeremic/tex/works/Thesis/MaximeLacour/Files_27Jun2017/Summer_Slides/Animations/SFEM_Animation_3D.mp4}
\end{center}
% \includegraphics[width = 12cm]{./img/SFEM_3D.pdf}
\begin{flushleft}
%\hspace*{15mm}
\vspace*{15mm}
\href{http://sokocalo.engr.ucdavis.edu/~jeremic/lecture_notes_online_material/_Chapter_Probabilistic_Elasto_Plasticity_and_Stochastic_Elastic_Plastic_Finite_Element_Method/SFEM_Animation_3D.mp4}
% \href{./homo_50mmesh_45degree_Ormsby.mp4}
{\tiny (MP4)}
\end{flushleft}
%
\end{frame}
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\section{Applications}
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%\subsection{Uncertain Inelasticity}
\subsection{\ }
% 
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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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,
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}
\vspace{4mm}
\item[] Miscommunication between seismologists and structural engineers,
$Sa(T_0)$ not compatible with nonlinear FEM
\vspace{4mm}
\item[] IMs difficult to choose, Spectral Acc, PGA, PGV...
\vspace{4mm}
\item[] Single IM does not contain all/most uncertainty
\vspace{4mm}
\item[] Monte Carlo, not accurate enough for tails
\vspace{4mm}
\item[] Monte Carlo, computationally expensive, CyberShake for LA, 20,000
cases, 100y runtime, (Maechling et al. 2007)
\end{itemize}
\end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Time Domain Intrusive SRA Framework}
%
\begin{itemize}
\vspace*{2mm}
\item[] Stochastic ElasticPlastic Finite Element Method, SEPFEM,
${M} \ddot{u_i} + {C} \dot{u_i} + {K}^{ep} {u_i} = {F(t)}$,
\vspace*{2mm}
\item[] Uncertain seismic loads, from uncertain seismic motions, using Domain
Reduction Method
\vspace*{2mm}
\item[] Uncertain elasticplastic material, stress and stiffness solution using
Forward Kolmogorov, FokkerPlanck equation
\vspace*{2mm}
\item[] Results, probability distribution functions for $\sigma_{ij}$,
$\epsilon_{ij}$, $u_i$...
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Stochastic ElasticPlastic Finite Element Method (SEPFEM)}
%
%
%
% \begin{itemize}
%
% \item[] Material uncertainty expanded along stochastic shape functions:
% $D(x,t,\theta) = \sum_{i=0}^{P_d} r_i(x,t) * \Phi_i[\{\xi_1, ..., \xi_m\}]$
%
% \vspace*{4mm}
% \item[] Loading uncertainty expanded along stochastic shape functions:
% $f(x,t,\theta) = \sum_{i=0}^{P_f} f_i(x,t) * \zeta_i[\{\xi_{m+1}, ..., \xi_f]$
%
% \vspace*{4mm}
% \item[] Displacement expanded along stochastic shape functions:
% $u(x,t,\theta) = \sum_{i=0}^{P_u} u_i(x,t) * \Psi_i[\{\xi_1, ..., \xi_m, \xi_{m+1}, ..., \xi_f\}]$
%
% \end{itemize}
%
%
%
% \end{frame}
%
%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% \begin{frame}
% \frametitle{Stochastic ElasticPlastic Finite Element Method (SEPFEM)}
% %\frametitle{SEPFEM : Formulation}
%
% Stochastic system of equations
%
% \begin{tiny}
% \[
% \begin{bmatrix}
% \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_0> K^{(k)} & \dots & \sum_{k=0}^{P_d} <\Phi_k \Psi_P \Psi_0> K^{(k)}\\
% \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_1> K^{(k)} & \dots & \sum_{k=0}^{P_d} <\Phi_k \Psi_P \Psi_1> K^{(k)}\\ \\
% \vdots & \vdots & \vdots & \vdots\\
% \sum_{k=0}^{P_d} <\Phi_k \Psi_0 \Psi_P> K^{(k)} & \dots & \sum_{k=0}^{M} <\Phi_k \Psi_P \Psi_P> K^{(k)}
% \end{bmatrix}
% \begin{bmatrix}
% u_{10} \\
% \vdots \\
% u_{N0}\\
% \vdots \\
% u_{1P_u}\\
% \vdots \\
% u_{NP_u}
% \end{bmatrix}
% =
% %\]
% %\[
% \begin{bmatrix}
% \sum_{i=0}^{P_f} f_i <\Psi_0\zeta_i> \\
% \sum_{i=0}^{P_f} f_i <\Psi_1\zeta_i> \\
% \sum_{i=0}^{P_f} f_i <\Psi_2\zeta_i> \\
% \vdots \\
% \sum_{i=0}^{P_f} f_i <\Psi_{P_u}\zeta_i>\\
% \end{bmatrix}
% \]
% \end{tiny}
%
%
%
% % \normalsize{Typical number of terms required for a SEPFEM problem} \vspace{1cm}\\
% \scalebox{0.7}{
% \begin{tabular}{ c c c c}
% \# KL terms material & \# KL terms load & PC order displacement& Total \# terms per DoF\\ \hline
% 4 & 4 & 10 & 43758 \\
% 4 & 4 & 20 & 3 108 105 \\
% 4 & 4 & 30 & 48 903 492 \\
% 6 & 6 & 10 & 646 646 \\
% 6 & 6 & 20 & 225 792 840 \\
% 6 & 6 & 30 & 1.1058 $10^{10}$ \\
% % 8 & 8 & 10 & 5 311 735 \\
% % 8 & 8 & 20 & 7.3079 $10^{9}$ \\
% % 8 & 8 & 30 & 9.9149 $10^{11}$\\
%
% ... & ... & ... & ...\\ \hline
% \end{tabular}}
%
%
% \end{frame}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Stochastic Seismic Motion Development}
\begin{itemize}
\item UCERF3 (Field et al. 2014)
\item Stochastic motions (Boore 2003)
\item Polynomial Chaos KarhunenLo{\`e}ve expansion
\item Domain Reduction Method for $P_{eff}$ (Bielak et al. 2003)
\end{itemize}
\begin{figure}[!hbpt]
\begin{center}
\includegraphics[width=4cm]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/UCERF3.pdf}
\includegraphics[width=3.0cm]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/SMSIM.pdf}
\includegraphics[width=3.5cm]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/KL_exact_dis_correlation_from_dis.pdf}
\end{center}
\end{figure}
%
% \begin{figure}[!hbpt]
% \begin{center}
% \includegraphics[width=10cm]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/structural_uncertainty.pdf}
% \end{center}
% \end{figure}
%
%
% \begin{figure}[!hbpt]
% \begin{center}
% \includegraphics[width=10cm]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/probabilsitc_evolution.png}
% \end{center}
% \end{figure}
%
%
%
% \begin{figure}[!hbpt]
% \begin{center}
% \includegraphics[width=10cm]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/seismic_risk_result_framework.png}
% \end{center}
% \end{figure}
%
%
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Stochastic Ground Motion Modeling}
\begin{itemize}
%\vspace*{1mm}
\item[] Modeling fundamental characteristics of uncertain ground motions,
Stochastic Fourier amplitude spectra (FAS). and Stochastic Fourier phase spectra
(FPS) and not specific IM
\vspace*{1mm}
\item[] Mean behavior of stochastic FAS, $w^2$ source radiation spectrum by
Brune(1970), and Boore(1983, 2003, 2015).
\vspace*{1mm}
\item[] Variability models for stochastic FAS,
FAS GMPEs by Bora et al. (2015, 2018), Bayless \&
Abrahamson (2019),
% for marginal median \& variability,
%Interfrequency correlation structure by
Stafford(2017) and Bayless \& Abrahamson (2018).
\vspace*{1mm}
\item[] Stochastic FPS by phase derivative (Boore,2005), Logistic phase derivative
model by Baglio \& Abrahamson (2017)
%\item Upcoming: a major change from \textbf{ $\boldsymbol{Sa(T_0)}$ to FAS} in next five years as envisioned by Abrahamson (2018)
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%
% \frametitle{Stochastic Ground Motion Modeling}
%
% Boore (2003) approach for median of lognormal random FAS field:
%
% \[FAS(f) = \underbrace{S(M_0, f)}_\text{source}\underbrace{P(R, f)}_\text{path} \underbrace{A(f)D(f)}_\text{site}\]
%
% \vspace{3mm}
%
% where:
% \begin{itemize}
% \small
%
% % \item $S(M_0,f) = \frac{CM_0(2\pi f)^2}{1+(f/f_0)^2}$
%
% \item $S(M_0,f) = CM_0(2\pi f)^2/[1+f^2/f_0^2]$
%
% \vspace{1mm}
%
% \item $P(R, f) = Z(R)exp(\pi fR/Q\beta)$
%
% \vspace{1mm}
%
% \item $A(f)$: quarter wavelength approximation.
%
% \vspace{1mm}
%
% \item $D(f)= exp(\pi \kappa_0f)$
%
% \end{itemize}
%
% Improvements:
%
% \begin{itemize}
% \small
%
% \item Interfrequency correlation by \textit{Bayless \& Abrahamson (2018)}
%
% \vspace{1mm}
%
% \item Phase derivative model by \textit{Baglio \& Abrahamson (2017)}
%
% \end{itemize}
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
%
% \frametitle{Stochastic Ground Motion Modeling}
%
% \begin{textblock}{15}(0.5, 4.2)
% Verification with GMPE for $M=7$, $R=15km$ scenario
% \end{textblock}
%
% \begin{textblock}{15}(0.0, 5.0)
% \includegraphics[width=0.4\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/SA_GMPE_verification_std_08_no_smooth.pdf}
% \end{textblock}
%
% \begin{textblock}{15}(5.3, 5.0)
% \includegraphics[width=0.4\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/Goodness_fit_std_08_no_smooth.pdf}
% \end{textblock}
%
% \begin{textblock}{15}(10.5, 5.0)
% \includegraphics[width=0.4\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/Standard_deviation_std_08_no_smooth.pdf}
% \end{textblock}
%
% \begin{textblock}{15}(0.5, 11.0)
% \begin{itemize}
% \item $\Delta \sigma= 84bar$, $\kappa=0.03s$ with total $\sigma=0.8ln$.
% \item Simulated median is not biased.
% \item Consistent total uncertainties with GMPE.
% \end{itemize}
% \end{textblock}
%
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Uncertainty Representation \& Propagation}
% \begin{itemize}
% \vspace{2mm}
% \item \small Unified uncertainty representation for forcing and media $D(\boldsymbol{x}, \theta)$.
%
% \begin{itemize}
% \item [] Hermite polynomial chaos (PC) for marginal distribution
% \item [] KarhunenLo$\grave{e}$ve (KL) expansion for correlation structure
% \end{itemize}
%
% \item \small Hermite PCs: NonGaussian random field with underlying Gaussian random field
%
% \[D(\boldsymbol{x}, \theta) = \sum_{i=0}^{P} D_i(\boldsymbol{x}) \Omega_i(\gamma(\boldsymbol{x},\theta)) \]
%
% where: \[\Omega_i = 1, \gamma, \gamma^21, \gamma^33\gamma, ... \]
%
% \[<\Omega_i> =0; \ \ <\Omega_i\Omega_j> = 0 \ \ for \ \ i \neq j \]
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% \end{itemize}
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% \begin{frame}
% \frametitle{Uncertainty Representation \& Propagation}
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% \begin{textblock}{15}(0.5, 5.0)
% \includegraphics[width=0.4\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/Hermite_PC_order.png}
% \end{textblock}
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% \begin{textblock}{15}(7.5, 4.0)
% \includegraphics[width=0.5\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/Gamma_distribution_hermite.png}
% \end{textblock}
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% \begin{textblock}{15}(5.5, 12.0)
% \includegraphics[width=0.7\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/Gamma_distribution.png}
% \end{textblock}
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% \begin{frame}
% \frametitle{Uncertainty Representation \& Propagation}
%
% \vspace{0.1cm}
%
% \small KarhunenLo$\grave{e}$ve (KL) expansion: capture correlation structure
%
% \vspace{0.3cm}
%
% \[Cov_D(x_1, x_2) = \sum_{i=1}^{P} D_i(x_1)D_i(x_2)i!Cov_{\gamma}(x_1, x_2)\]
%
% \vspace{0.2cm}
%
% \small Eigendecomposed Gaussian covariance kernel: $ \lambda_{i}$ eigenvalue, $f_i(\boldsymbol{x})$ eigenfunction
%
% \vspace{0.7cm}
%
% \[\gamma(\boldsymbol{x}, \theta)=\sum_{i=1}^{M}\sqrt{\lambda_i}f_i(\boldsymbol{x})\xi_i \]
%
% \vspace{0.2cm}
%
% \small Fredholm's integral equation of the second kind:
%
% \[\int_V Cov_{\gamma}(x_1, x_2) f_i(x_1) \, dx_1 = \lambda_{i} f_i(x_2)\]
%
% \vspace{0.1cm}
%
% \small \{$\xi_i$\} are zero mean, unit variance, mutually orthogonal Gaussian random variables
%
% \vspace{0.2cm}
%
% \[<\xi_i> =0; \ \ <\xi_i\xi_j> = \delta_{ij}\]
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% \end{frame}
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% \begin{frame}
% \frametitle{Uncertainty Representation \& Propagation}
% \begin{textblock}{15}(0.5, 6.0)
% \includegraphics[width=0.4\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/1D_KL_exp_illustration.png}
% \end{textblock}
% \begin{textblock}{15}(7.5, 4.0)
% \includegraphics[width=0.5\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/KL_eigen_modes.png}
% \end{textblock}
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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{frame}
% \frametitle{Uncertainty Representation \& Propagation}
%
% \small Random field $D(\boldsymbol{x}, \theta)$ of arbitrary marginal distribution \& correlation structure can be represented as:
%
% \begin{textblock}{14}(2.4, 6.0)
% \[D(\boldsymbol{x}, \theta) = \sum_{i=0}^{K} d_i(\boldsymbol{x}) \Psi_i(\{\xi_j(\theta)\})\]
% \[d_i(\boldsymbol{x}) = \frac{P!}{(\Psi_i^2)} D_P(\boldsymbol{x}) \prod_{j=1}^{P} \frac{\sqrt{\lambda_{k(j)}}f_{k(j)}(\boldsymbol{x})}{\sqrt{\sum_{m=1}^{M}(\sqrt{\lambda_m}f_m(\boldsymbol{x}))^2}}\]
% \end{textblock}
%
% \vspace{3.5cm}
% where: \\
% $\{\Psi_i\}$ are multidimensional Hermite \\
% PC bases of order $P$, dimension $M$.
%
% \begin{textblock}{15}(8.9, 5.5)
% \includegraphics[width=0.45\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/Multidimensional_Hermite_PC.png}
% \end{textblock}
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% \end{frame}
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% \begin{frame}
% \frametitle{Uncertainty Representation \& Propagation}
%
% \small
%
% Galerkin stochastic finite element method (SFEM)
%
% \[M_{IacJ} \ddot{U}_{Jc} + \boldsymbol{K^{ep}_{IacJ}} U_{Jc} = \boldsymbol{F_{Ia}}\]
%
% Probabilistic discretization:
%
% \vspace{0.2cm}
%
% \[K^{ep}_{IacJ}(\theta)= K^{ep}_{IacJi} \Psi_i (\theta)\]
%
% \vspace{0.3cm}
%
% \[F_{Ia}(\theta)= F_{Iaj} \psi_j (\theta)\]
%
% \vspace{0.3cm}
%
% \[U_{Jc}(\theta) = U_{Jck} \phi_k(\theta)\]
%
% \vspace{0.3cm}
%
% Galerkin projection:
%
% \[M_{IacJ} (\phi_k \phi_m) \ddot{U}_{Jck} + K^{ep}_{IacJi} (\Psi_i\phi_k\phi_m) U_{Jck} = F_{Iaj} (\psi_j \phi_m) \]
%
% \vspace{0.2cm}
% \scriptsize where $(\Psi_i\phi_k\phi_m)$ is the triple product of multidimensional Hermite PCs
%
% \end{frame}
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{frame}
\frametitle{Uncertain Model Description}
\begin{textblock}{15}(0.3, 3.4)
\includegraphics[width=0.70\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/example_illustration.png}
\end{textblock}
\begin{textblock}{15}(1.2, 11.6)
\scriptsize
\begin{itemize}
\item Fault 1: San Gregorio fault
\item Fault 2: Calaveras fault
\item Uncertainty: Segmentation, \\ slip rate, rupture geometry, etc.
\end{itemize}
\end{textblock}
\begin{textblock}{15}(7.2, 11.7)
\scriptsize
\begin{itemize}
\item $Vs_{30}=620m/s$
\item $m=100kips/g$
\item $\overline{k} = 168kip/in$
\end{itemize}
\end{textblock}
\begin{textblock}{15}(9.7, 6.5)
\scriptsize
\[Cov_k =
\begin{bmatrix}
1.0 & 0.6 & 0.3 & 0.2\\
0.6 & 1.0 & 0.5 & 0.2\\
0.3 & 0.5 & 1.0 & 0.7\\
0.2 & 0.2 & 0.7 & 1.0
\end{bmatrix}
\]
\end{textblock}
\end{frame}
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\begin{frame}
\frametitle{Seismic Source Characterization}
\begin{textblock}{15}(0.3, 3.4)
\includegraphics[width=0.65\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/SSC_legend.pdf}
\end{textblock}
\begin{textblock}{15}(6.5, 12.1)
\scriptsize
\begin{itemize}
\item 371 total seismic scenarios
\item $M \ 5 \sim 5.5$ and $6.5 \sim 7.0$
\item $R_{jb} \ 20km \sim 40km$
\end{itemize}
\end{textblock}
\begin{textblock}{15}(11.7, 3.3)
\includegraphics[width=0.22\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/Acc_time_series100.pdf}\\
\includegraphics[width=0.22\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/Acc_time_series343.pdf}\\
\includegraphics[width=0.22\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/Acc_time_series439.pdf}
\end{textblock}
\end{frame}
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\begin{frame}
\frametitle{Stochastic Ground Representation}
{\begin{textblock}{15}(0.1, 3.62)
\scriptsize
\includegraphics[width=0.3\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/KL_mean_acc_from_acc.pdf}
\quad \quad Acc. marginal mean
\end{textblock}
\begin{textblock}{15}(3.7, 3.62)
\scriptsize
\includegraphics[width=0.3\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/KL_var_acc_from_acc.pdf}
\quad \quad Acc. marginal S.D.
\end{textblock}
\begin{textblock}{15}(7.6, 3.8)
\scriptsize
\includegraphics[width=0.3\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/KL_exact_acc_correlation_from_acc.pdf}
\quad \quad \quad Acc. realization Cov.
\end{textblock}
\begin{textblock}{15}(11.8, 3.9)
\scriptsize
\includegraphics[width=0.3\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/KL_simulated_acc_correlation_from_acc.pdf}
\quad \quad Acc. synthesized Cov.
\end{textblock}}
\begin{textblock}{15}(0.1, 9.3)
\scriptsize
\includegraphics[width=0.31\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/KL_mean_dis_from_dis.pdf}
\end{textblock}
\begin{textblock}{15}(0.9, 13.75)
\scriptsize
Dis. marginal mean
\end{textblock}
\begin{textblock}{15}(4.2, 9.4)
\scriptsize
\includegraphics[width=0.3\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/KL_var_dis_from_dis.pdf}
\end{textblock}
\begin{textblock}{15}(5.1, 13.75)
\scriptsize
Dis. marginal S.D.
\end{textblock}
\begin{textblock}{15}(8.2, 9.5)
\scriptsize
\includegraphics[width=0.27\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/KL_exact_dis_correlation_from_dis.pdf}
\quad \quad Dis. realization Cov.
\end{textblock}
\begin{textblock}{15}(12.2, 9.6)
\scriptsize
\includegraphics[width=0.27\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/KL_simulated_dis_correlation_from_dis.pdf}
\quad Dis. synthesized Cov.
\end{textblock}
\end{frame}
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\begin{frame}
\frametitle{Probabilistic Dynamic Response}
\begin{textblock}{15}(2.9, 4)
\scriptsize
\includegraphics[width=0.65\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/Probabilistic_Response_Node_1_new.pdf}
\end{textblock}
\begin{textblock}{15}(4.8, 3.8)
\scriptsize Probabilistic response of top floor from SFEM
\end{textblock}
\end{frame}
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\begin{frame}
\frametitle{Probabilistic Dynamic Response}
\begin{textblock}{15}(0.9, 4.4)
\scriptsize
\includegraphics[width=0.55\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/probability_evolution.jpg}
\includegraphics[width=0.43\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/Contour_PDF_evolution_top_floor_deformation.pdf}
\end{textblock}
\begin{textblock}{15}(4.8, 4.2)
\scriptsize Probabilistic density of displacements evolution of top floor
\end{textblock}
\end{frame}
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\begin{frame}
\frametitle{Maximum InterStory Drift Ratio (MIDR)}
% \begin{textblock}{15}(0.7, 4.8)
% \scriptsize
% \includegraphics[width=0.55\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/PDF_evolution.png}
% \end{textblock}
%
\vspace*{5mm}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=10cm]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/MIDR_distribution.pdf}
\end{center}
\end{figure}
\end{frame}
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\begin{frame}
\frametitle{Seismic Risk Analysis}
\begin{textblock}{15}(0.7, 4.8)
\scriptsize
\includegraphics[width=0.45\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/combined_risk_curve.pdf}
\end{textblock}
\begin{textblock}{15}(7.9, 4.8)
\scriptsize
\includegraphics[width=0.48\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/risk_deaggregation_MIDR1.pdf}
\end{textblock}
\begin{textblock}{15}(0.9, 11.9)
\begin{itemize}
\item[] \scriptsize $\lambda(MIDR >1 \%) = 9.7 \times 10^{3}$
\item[] \scriptsize $\lambda(MIDR >2 \%) = 1.7 \times 10^{3}$
\item[] \scriptsize $\lambda(MIDR >4 \%) = 5.9 \times 10^{5}$
\end{itemize}
\end{textblock}
\begin{textblock}{15}(9.3, 12.5)
\scriptsize Risk deaggregation for $\lambda(MIDR >1 \%)$
\end{textblock}
\end{frame}
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\begin{frame}
\frametitle{Sensitivity Study}
\begin{textblock}{15}(1.7, 3.0)
\scriptsize
\includegraphics[width=0.42\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/risk_curve_different_stressdrop.pdf}
\end{textblock}
\begin{textblock}{15}(9.1, 3.0)
\scriptsize
\includegraphics[width=0.42\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/risk_curve_different_kappa.pdf}
\end{textblock}
\begin{textblock}{15}(1.7, 8.7)
\scriptsize
\includegraphics[width=0.42\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/median_FAS_stressdrop_comparison.pdf}
\end{textblock}
\begin{textblock}{15}(9.1, 8.7)
\scriptsize
\includegraphics[width=0.42\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/median_FAS_kappa_comparison1.pdf}
\end{textblock}
\begin{textblock}{15}(7.9, 8.7)
\scriptsize Site $\kappa_0$:
\end{textblock}
\begin{textblock}{15}(0.3, 8.7)
\scriptsize Source $\Delta \sigma$:
\end{textblock}
\end{frame}
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\begin{frame}
\frametitle{Sensitivity Study}
\begin{textblock}{15}(1.3, 4.2)
Fundamental frequency $f$ increases from 1.6Hz to 8Hz:
\end{textblock}
\begin{textblock}{15}(4.7, 5)
\scriptsize
\includegraphics[width=0.55\linewidth]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/risk_curve_different_kappa_highfrequency.pdf}
\end{textblock}
\end{frame}
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\begin{frame}
\frametitle{Seismic Risk, Uncertain Material}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=8cm]{/home/jeremic/tex/works/Conferences/2019/CompDyn/present/pic/seismic_risk_compare.pdf}
\end{center}
\end{figure}
\end{frame}
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\section{Summary}
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%\subsection*{Summary}
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\begin{frame}
\frametitle{Appropriate Science Quotes}
\begin{itemize}
%
% \item Max Planck:
% "A new scientific truth does not triumph by convincing its opponents and
% making them see the light, but rather because its opponents eventually die, and
% a new generation grows up that is familiar with it." (Science advances one
% funeral at a time)
%
\vspace*{3mm}
\item Fran{\c c}oisMarie Arouet, Voltaire:
"Le doute n'est pas une condition agr{\'e}able, mais la certitude est absurde."
\vspace*{9mm}
\item Niklaus Wirth:
"Software is getting slower more rapidly than hardware becomes faster."
\end{itemize}
\end{frame}
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\begin{frame}
\frametitle{Summary}
\begin{itemize}
% \item Importance of using proper models correctly (verification,
% validation, level of sophistication)
%
% \item Reduction of modeling uncertainty
%
\vspace*{1mm}
\item[] Numerical modeling to predict and inform, rather than fit
%\vspace*{1mm}
% \item Change of demand due to inelastic effects
%\vspace*{1mm}
% \begin{itemize}
% \item Reduction of dynamic motions
% \item Increase in deformations
% \end{itemize}
\vspace*{1mm}
\item[] Sophisticated inelastic/nonlinear determinstic/probabilistic modeling
and simulations needs to be done carefully and in phases
\vspace*{1mm}
\item[] Education and Training is the key!
\vspace*{1mm}
\item[] Collaborators: Feng, Yang, Behbehani, Sinha, Wang,
Karapiperis, Wang, Lacoure, Pisan{\'o}, Abell, Tafazzoli, Jie, Preisig,
Tasiopoulou, Watanabe, Cheng, Yang.
\vspace*{1mm}
\item[] Funding from and collaboration with the ATC/USFEMA, USDOE, USNRC, USNSF,
CNSCCCSN, UNIAEA, and Shimizu Corp. is greatly appreciated,
\vspace*{1mm}
\item[]
\url{http://sokocalo.engr.ucdavis.edu/~jeremic}
\end{itemize}
\end{frame}
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\end{document}
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